Comparison theory for Lipschitz spacetimes

COMP ARISON THEOR Y F OR LIPSCHITZ SP A CETIMES MA THIAS BRAUN AND MAR T A SÁLAMO CANDAL Abstract. W e pro v e a globally h yperb olic spacetime with locally Lipsc hitz contin uous metric and timelike distributional Ricci curv ature bounded from below ob eys the timelike measure con traction property . The remarkable class of examples of spacetimes that are cov ered by this result includes impulsive gravit y w av es, thin shells, and matc hed spacetimes. As applications, we get new comparison theorems for Lipsc hitz spacetimes in sharp form: d’Alembert, timelike Brunn–Minko wski, and timelike Bishop– Gromov. Under appropriate non branc hing assumptions (conjectured to hold in even low er regularit y), our results also yield the timelik e curv ature-dimension condition, a v olume incompleteness theorem, as well as exact representation formulas and sharp comparison estimates for d’Alembertians of Lorentz distance functions from general spacelike submanifolds. Moreov er, we establish the sharp timelik e Bonnet–Myers inequality ad ho c using the lo calization technique from con vex geometry . Alongside, we prov e a timelike diameter estimate for spacetimes whose timelik e Ricci curv ature is positive up to a “small” deviation (in an L p -sense). This adapts prior theo- rems for Riemannian manifolds by Petersen–Sprouse and A ubry to Lorentzian geometry , a transition the former t wo anticipated almost 30 years ago. Contents 1. In tro duction 2 1.1. Bac kground 2 1.2. Sharp timelike Bonnet–Myers inequality 3 1.3. Analytic and syn thetic spacetime geometry 4 1.4. Applications to comparison theory 6 1.5. Organization 8 2. Analytic timelike Ricci curv ature low er b ounds 8 2.1. Lipsc hitz spacetime geometry 8 2.2. Regularization 9 2.3. Regularit y of time separation function 15 3. Syn thetic timelik e Ricci curv ature low er b ounds 16 3.1. One-dimensional considerations 16 3.2. Loren tzian optimal transport 19 3.3. V ariable timelike curv ature-dimension condition 22 3.4. V ariable timelike measure contraction prop ert y 24 4. Smo oth considerations 25 4.1. Timelik e cut locus 25 4.2. Lo calization 27 2020 Mathematics Subje ct Classiﬁc ation. Primary 49Q22, 53C50; Secondary 46F10, 53C23, 83C75. K ey wor ds and phr ases. Lipsc hitz spacetime; Regularization; Timelike Ricci curv ature; Com- parison theory; Lorentzian optimal transp ort. MB is supported b y the EPFL through a Bernoulli Instructorship. MSC w as funded in part by the Austrian Science F und (FWF) [Grant DOI 10.55776/EFP6 ] and the Vienna School of Mathematics. F or open access purp oses, the authors hav e applied a CC BY public copyrigh t license to any author accepted man uscript v ersion arising from this submission. 2 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL 4.3. Displacemen t con vexit y with defect 30 5. Sharp timelike Bonnet–Myers inequality 35 6. Main results and applications 36 6.1. F rom analytic to synthetic 36 6.2. Applications of Theorem 6.1 38 App endix A. F urther sharp timelike diameter estimates 41 App endix B. Details ab out Theorems 6.5 and 6.6 43 B.1. Go od geo desics 43 B.2. A Brenier–McCann theorem 45 B.3. Pro ofs of Theorems 6.5 and 6.6 45 References 49 1. Introduction The goal of this work is t wofold. First, we establish several sharp comparison theorems for globally hyperb olic spacetimes with a locally Lipsc hitz con tin uous metric, which we refer to as Lipschitz spacetimes. Second, we prov e that analytic timelik e Ricci curv ature b ounds for such spacetimes, form ulated through the theory of tensor distributions, imply synthetic timelike Ricci curv ature b ounds in terms of optimal transp ort and metric geometry . The former results are obtained as applications of the latter. A key ingredient of our approach is a new link b etw een lo w regularity spacetime geometry and the well-dev eloped theory of Riemannian manifolds with integral curv ature b ounds. 1.1. Bac kground. There is a certain agreement to call a spacetime ( M , g ) “non- smo oth” if its metric tensor g fails to hav e regularity C 2 (in which key concepts like geo desics and curv ature would still b e classically deﬁned). As p oin ted out already b y Lichnero wicz [ 55 ] in the 1950s, nonsmo oth spacetimes arise in man y natural w a ys: one source of motiv ation comes from the initial v alue problem for the Einstein equations, cf. Rendall [ 72 ]. Moreov er, Hawking–Ellis [ 43 ] stated in the 1970s that the p ersistence of the classical incompleteness theorems b elow regularity C 2 w ould rule out the undesired possibility that observ ers cease to exist merely due to a loss of regularit y , thereb y underlining the physical signiﬁcance of “singularities” . Progress on the incompleteness asp ect, ho w ev er, has only b een made rather recently , cf. Calisti et al. [ 17 ] and the references therein. The framework of our contribution are Lipschitz spacetimes. There, the geo- desic equation was studied by Sämann–Steinbauer [ 75 ], Graf–Ling [ 41 ], and Lange– Lytc hak–Sämann [ 54 ]. On the other hand, only recen tly Calisti et al. [ 17 ] prov ed the ﬁrst result in volving curv ature hypotheses on Lipschitz spacetimes, namely Ha wking’s incompleteness theorem. Curved Lipschitz spacetimes arise in numerous prominen t examples, suc h as impulsive gravit y w av es (cf. e.g. Cho quet-Bruhat [ 27 ] and Penrose [ 68 ]) and thin shells and matched spacetimes (cf. e.g. Israel [ 45 ], Khakshournia–Mansouri [ 48 ], and Manzano–Mars [ 57 ]); in a distributional sense, the former constitute solutions to the v acuum Einstein equations Ric g = 0 , while the latter often exhibit timelike nonnegative Ricci curv ature Ric g ≥ 0 dep ending on the matter of the junction hypersurface. In particular, all these examples will b e cov ered by our results. Moreov er, Lipschitz spacetimes are the sharp threshold of regularit y that av oids bubbling issues, cf. Chrusćiel–Grant [ 28 ]. They are closed cone structures in the sense of Minguzzi [ 61 ] and regularly lo calizable Lorentzian length spaces in the sense of Kunzinger–Sämann [ 52 ]. COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 3 The imp ortance of a systematic comparison theory in Lorentzian geometry , guided b y its inﬂuen tial Riemannian precedent (cf. e.g. Cheeger–Ebin [ 26 ]), was emphasized b y Ehrlic h [ 30 ]. It is naturally link ed to timelik e Ricci curv ature bounds, whic h corresp ond to energy conditions after Penrose [ 67 ] and Hawking [ 44 ]. F or instance, it is a cornerstone for the Lorentzian splitting theorems of Esc henburg [ 33 ], Gallo wa y [ 35 ], and Newman [ 65 ], whic h are understo od as rigidity statemen ts for the v acuum Einstein equations and ultimately led to the op en splitting conjecture of Bartnik [ 4 ]. How ev er, b esides the Hawking incompleteness theorem of Calisti et al. [ 17 ], comparison theorems ha ve remained unknown for Lipschitz spacetimes. Cen tral issues are the lack of an exp onen tial map and the mere av ailability of distributional curv ature tensors, whic h mak e it diﬃcult to carry out the usual Jacobi ﬁeld computations. Moreov er, as p ointed out by Kunzinger–Ob erguggen berger– Vic kers [ 50 ], Braun–Calisti [ 11 ], and Calisti et al. [ 17 ], a related op en problem is the compatibility of distributional timelike Ricci curv ature b ounds for nonsmo oth spacetimes with the synthetic ones introduced by Cav alletti–Mondino [ 24 ]. These motiv ations naturally lead to the main contributions of this pap er w e outlined ab ov e and detail b elo w, together with relev an t literature. 1.2. Sharp timelik e Bonnet–Myers inequality. The Bonnet–Myers theorem is a classical result in Riemannian geometry . It states a p ositiv e low er bound on the Ricci curv ature implies an upp er b ound on the diameter of the Riemannian manifold in question. Its analog in Lorentzian geometry is in terpreted as an incompleteness theorem: under lo w er b oundedness of the Ricci curv ature in timelike directions by a p ositiv e constan t, it implies a sharp upp er b ound on the timelike diameter of the considered spacetime ( M , g ) . In the globally hyperb olic case, the classical smooth version found e.g. in Beem–Ehrlic h–Easley [ 5 ] w as generalized by Graf [ 39 ] when g is C 1 , 1 and Braun–Calisti [ 11 ] if g is C 1 . W e generalize their results as follows. Theorem 1.1 (Timelik e Bonnet–Myers diameter estimate, Theorem 5.1 ) . W e ﬁx K > 0 . In addition, assume ( M , g ) is a glob al ly hyp erb olic Lipschitz sp ac etime such that Ric g ≥ K timelike distributional ly, cf. Deﬁnition 2.7 . Then diam ( M , g ) ≤ π r dim M − 1 K . As for all of our results that in volv e a curv ature hypothesis, a weigh ted version of Theorem 1.1 holds as well. Although not stated explicitly therein, Theorem 1.1 can b e deduced from the Ha wking incompleteness theorem of Calisti et al. [ 17 ] combined with the choice of a suitable hypersurface by an argument of Graf [ 39 , Rem. 4.3]. W e give a diﬀerent pro of which adds new insights, cf. Theorem 1.2 b elo w. An ingredien t we frequently use is Calisti et al. ’s setup of an appro ximation of g b y smo oth metrics ( g n ) n ∈ N suc h that the “excess” of the g n -timelik e Ricci curv ature compared to K tends to zero lo cally in L p as n → ∞ for every p ∈ [1 , ∞ ) [ 17 ], as recalled in Lemmas 2.11 and 2.16 ; we call such a sequence “go o d approximation” of g , cf. Deﬁnition 2.13 . W e com bine this with the lo calization technique, cf. § 4.2 , pioneered in the Lorentzian case by Cav alletti–Mondino [ 24 ] and developed further b y themselves [ 25 ] and Braun–McCann [ 14 ], to reduce Theorem 1.1 to a family of one-dimensional problems along a geo desic foliation. Using this synthetic tool, w e are able to in vok e the w ell-established theory of Riemannian manifolds with Ricci curv ature in L p in the semiclassical range p ∈ ( dim M / 2 , ∞ ) for Schrödinger op erators, whose comparison theory was initiated by P etersen–W ei [ 69 ]. Bonnet– My ers-type theorems in this setting w ere shown b y Petersen–Sprouse [ 70 ], Sprouse [ 76 ], and Aubry [ 3 ]; later extensions to Kato-type b ounds w ere giv en by Carron 4 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL [ 19 ], Rose [ 73 ], and Carron–Rose [ 20 ]. More recently , comparison theory for metric measure spaces with integral curv ature bounds in the sense of Ketterer [ 46 ] was dev elop ed by Caputo–Nobili–Rossi [ 18 ]. Our new link of the preceding Riemannian theories to Lorentzian geometry was anticipated by Braun–McCann [ 14 ]. In fact, a byproduct of our pro of of Theorem 1.1 is the subsequent Lorentzian coun terpart of P etersen–Sprouse’s qualitative diameter estimate prov ed 30 years ago for Riemannian manifolds [ 70 ]. They also anticipated the connection of their result with general relativity (where it is reasonable to exp ect incompleteness under small quantum ﬂuctuations, which can b e mo deled b y a small curv ature excess in an integral sense, as expressed in [ 70 ]), which the follo wing theorem settles. Theorem 1.2 (Timelik e Petersen–Sprouse diameter estimate, Corollary A.3 ) . L et K > 0 , N ∈ [2 , ∞ ) , and p ∈ ( N / 2 , ∞ ) . Then for every ε > 0 , ther e is a c onstant ξ K,N ,p,ε > 0 with the fol lowing pr op erty. L et ( M , g , m ) form a glob al ly hyp erbolic Lipschitz sp ac etime with dim M ≤ N and let k : M → R b e a c ontinuous function with Ric g ≥ k in al l timelike dir e ctions such that for every nonempty, op en, and pr e c omp act W ⊂ M , 1 v ol g [ W ] Z W   ( k − K ) −   p dv ol g ≤ ξ K,N ,p,ε . Then we have diam ( M , g ) ≤ π r N − 1 K + ε. Loren tzian counterparts of the quan titative Riemannian diameter estimates of A ubry [ 3 ] are established as well, cf. Theorem A.2 . Despite the hypothesized smo othness of the metric tensor (whic h we b eliev e can b e relaxed), it is interesting to read Theorem 1.2 in the context of curren t quests for precompactness in some Lorentz–Gromo v–Hausdorﬀ top ology (cf. e.g. the review of Ca v alletti–Mondino [ 23 ]), as it implies a uniform and almost sharp diameter bound across a family of spacetimes with uniformly small curv ature excess. 1.3. Analytic and synthetic spacetime geometry . The energy condition “ Ric g ≥ K in all timelike directions” of Penrose [ 67 ] and Ha wking [ 44 ], where K ∈ R , pro- vides a natural connection of general relativit y and Lorentzian comparison theory . W e now outline t wo w a ys to deﬁne it on a Lipschitz spacetime ( M , g ) . T o simplify the presen tation, in the remainder we only state our main results (and adjacent deﬁnitions) in the case when K is zero. The usual analytic wa y to deﬁne Ricci curv ature (and its timelik e lo wer b ound- edness) on Lipschitz spacetimes — in fact, Gero c h–T rasc hen regularity H 1 , 2 loc ∩ C 0 suﬃces [ 37 ] — is the theory of tensor distributions. W e refer to Grosser–Kunzinger– Ob erguggen berger–Steinbauer [ 42 ], Graf [ 40 ], and § 2.2 for details. Throughout our w ork, the reader should think of ( M , g ) as a distributional solution to the Einstein equations with timelike energy-momentum tensor b ounded from b elo w, suc h as those mentioned in § 1.1 . On the other hand, current researc h dev otes signiﬁcant attention to synthetic form ulations of timelike Ricci curv ature low er b ounds, in tro duced by Cav alletti– Mondino [ 24 ] inspired by preceding breakthroughs by McCann [ 58 ] and Mondino– Suhr [ 64 ]. It is foreshadow ed by the w ell-developed theory of so-called CD metric measure spaces by Sturm [ 77 , 78 ] and Lott–Villani [ 56 ]. In a nutshell, the synthetic theory from [ 24 ] stipulates conv exit y prop erties of an entrop y functional along geo desics in the Lorentz–W asserstein space introduced by Eckstein–Miller [ 29 ]. An adv antage of this approach is that it makes sense on “metric measure spacetimes” without a notion of Ricci curv ature or ev en a diﬀerentiable structure, such as COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 5 Busemann’s timelik e spaces [ 16 ], Kunzinger–Sämann’s Lorentzian length spaces [ 52 ], or Minguzzi–Suhr’s b ounded Loren tzian metric spaces [ 60 ]. Among several adv antages p oin ted out e.g. by Cav alletti–Mondino’s review [ 23 ], let us note for no w that several comparison theorems hold in this abstract setting, cf. Cav alletti– Mondino [ 24 , 25 ], Beran et al. [ 6 ], Braun [ 10 ], and § 1.4 b elo w. It is therefore natural to ask how analytic and synthetic timelike Ricci curv ature b ounds are related. This question was inspired by pioneering works of McCann [ 58 ] and Mondino–Suhr [ 64 ], pro ving the equiv alence of b oth approaches for smo oth g . If g is C 1 , the implication from analytic to synthetic theory was accomplished b y Braun–Calisti [ 11 ]. In Riemannian signature, an analogous implication w as sho wn by Kunzinger–Ob erguggen b erger–Vic k ers [ 50 ] in regularity C 1 and later Mondino–R yb orz [ 63 ] in Gero ch–T rasc hen regularity H 1 , 2 loc ∩ C 0 [ 37 ] and Kunzinger– Ohan yan–V ardabasso [ 51 ] (based on a stability result of Ketterer [ 47 ]) in Lipschitz regularit y; in fact, [ 63 ] characterizes the Riemannian manifolds in Gero ch–T rasc hen regularit y satisfying the CD condition of Lott–Sturm–Villani. Our second main theorem bridges analytic and synthetic theory for Lipsc hitz spacetimes by deriving the so-called timelike me asur e c ontr action pr op erty , brieﬂy TMCP , from distributional timelike Ricci curv ature b ounds. Thu s, w e reduce the kno wn regularit y for this implication from C 1 to lo cal Lipsc hitz contin uit y . The TMCP was introduced by Cav alletti–Mondino [ 24 ] and later adapted by Braun [ 8 ]. An adv antage of the formulation of [ 8 ], which relies on a diﬀerent choice of entrop y ( 1.1 ) and on which we fo cus, is that it implies the comparison theorems indicated ab o v e in sharp form, without requiring timelike non branc hing hypotheses; how ever, our arguments can b e used to show the TMCP of [ 24 ] as well. T o formulate our second main result, given q ∈ (0 , 1) let ℓ q b e the q -Loren tz– W asserstein distance ( 3.4 ) induced b y the canonical time separation l of ( M , g ) . It is deﬁned on the space P ( M ) of Borel probabilit y measures on M . As deﬁned b y McCann [ 58 ], a curv e µ : [0 , 1] → P ( M ) will b e called q -ge o desic if it is aﬃnely parametrized with respect to ℓ q , cf. Deﬁnition 3.14 . F or N ∈ (1 , ∞ ) , deﬁne the N -R ényi entr opy S N ( · | v ol g ) : P ( M ) → R − ∪ {−∞} with resp ect to v ol g b y S N ( µ | v ol g ) := − Z M h d µ ac dv ol g i 1 − 1 / N dv ol g , (1.1) where µ ac is the absolutely contin uous part in the Leb esgue decomp osition of the argumen t µ with resp ect to v ol g . Deﬁnition 1.3 (Timelike measure contraction prop ert y , Deﬁnition 3.26 ) . The glob al ly hyp erb olic weighte d Lipschitz sp ac etime ( M , g , vol g ) is said to ob ey the p ast timelike me asur e c ontr action pr op erty, brieﬂy TMCP − (0 , dim M ) , if for every o ∈ M and every c omp actly supp orte d, v ol g -absolutely c ontinuous µ 1 ∈ P ( M ) c onc entr ate d on I + ( o ) , ther e exist • an exp onent q ∈ (0 , 1) and • a q -ge o desic µ : [0 , 1] → P ( M ) fr om µ 0 := δ o to µ 1 such that for every N ′ ∈ [dim M , ∞ ) and every t ∈ [0 , 1] , S N ′ ( µ t | vol g ) ≤ t S N ′ ( µ 1 | vol g ) . The TMCP do es not depend on the transp ort exp onen t q , cf. § 3.4 . Moreo v er, the ab ov e condition N ′ ∈ [dim M , ∞ ) ensures dimensional consistency . Theorem 1.4 (Timelik e measure con traction prop ert y , Theorem 6.1 ) . W e assume that ( M , g ) is a glob al ly hyp erb olic Lipschitz sp ac etime with Ric g ≥ 0 timelike dis- tributional ly. Then the glob al ly hyp erb olic weighte d Lipschitz sp ac etime ( M , g , vol g ) satisﬁes TMCP − (0 , dim M ) . 6 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL The “future” analog of the past TMCP (where the Dirac mass lies in the chrono- logical future of the absolutely contin uous distribution) is established as w ell. R emark 1.5 (Improv emen ts of Theorem 1 .4 ) . In fact, we show a slightly stronger statemen t in Prop osition 6.2 , whic h impro v es Theorem 1.4 to the timelik e curv ature- dimension condition recalled in § 3.3 pro vided ( M , g ) is timelike nonbranc hing after Ca v alletti–Mondino [ 24 ] by results of Braun [ 8 ]. Under such nonbranc hing h yp otheses, one also gets isop erimetric-type inequalities à la Cav alletti–Mondino [ 25 ], exact representation formulas and sharp comparison estimates for d’Alem b ertians of Lorentz distance functions from general spacelike submanifolds à la Braun [ 10 ], and a volume incompleteness theorem, cf. García-Heveling [ 36 ] and Braun [ 10 ]. Since the exp onen tial map is not ev en well-deﬁned, it is not clear that Lipschitz spacetimes are timelik e non branc hing. How ev er, the Riemannian results of Mondino– R yb orz [ 63 ] in ev en lo wer regularity and Deng’s result ab out nonbranc hing of ﬁnite-dimensional RCD spaces in the sense of Gigli [ 38 ] suggest the hypotheses of Theorem 1.4 actually imply timelik e nonbranching, hence the abov e results; see Ca v aletti–Mondino [ 24 , Rem. 5.15]. ■ Conceptually , our pro of is based on Calisti et al. ’s go o d appro ximation [ 17 ] and the strong stability prop erties of the TMCP realized b y Cav alletti–Mondino [ 24 ] (because it is a conv exit y inequalit y). Ho wev er, in con trast to the work of Braun–Calisti [ 11 ] (which relies on approximation results of Graf [ 40 ]), the timelike lo wer b ounds of the approximating Ricci tensors are not lo cally uniformly small relativ e to the bound prescrib ed b y g , but only small in an in tegral sense. W e will address this challenge b y com bining Braun–McCann’s recent theory of metric measure spacetimes with v ariable timelike low er Ricci curv ature b ounds [ 14 ] and estimates developed by Ketterer [ 47 ] in Riemannian signature. 1.4. Applications to comparison theory . Lastly , w e establish sev eral sharp comparison theorems as a consequence of Theorem 1.4 : timelik e Brunn–Minko wski, timelik e Bishop–Gromov, and d’Alembert comparison 1 . When g is smo oth, compar- ison theory was systematically developed b y Eschen burg [ 33 ], Ehrlich–Jung–Kim [ 31 ], Ehrlich–Sánc hez [ 32 ], T reude [ 79 ], and T reude–Gran t [ 80 ]. On nonsmo oth spacetimes, the facts we prov e for Lipschitz spacetimes were only kno wn in regularity C 1 , 1 b y Graf [ 39 ] and C 1 b y Braun–Calisti [ 11 ]. As established by Ca v alletti–Mondino [ 24 ] and later Braun [ 8 ], the ﬁrst tw o comparison theorems w e will obtain are standard consequences of the TMCP from Theorem 1.4 . Giv en X 0 , X 1 ⊂ M , let G ( X 0 , X 1 ) b e the set of all timelik e aﬃnely parametrized maximizing geo desics starting in X 0 and terminating in X 1 . Given t ∈ [0 , 1] , let e t : C 0 ([0 , 1]; M ) → M denote the ev aluation map e t ( γ ) := γ t . Theorem 1.6 (Timelike Brunn–Mink o wski inequalit y , Theorem 6.3 ) . A ssume ( M , g ) is a glob al ly hyp erb olic Lipschitz sp ac etime with Ric g ≥ 0 timelike distributional ly. Then for every o ∈ M , every c omp act set X 1 ⊂ I + ( o ) , and every t ∈ [0 , 1] , v ol ∗ g [ e t ( G ( { o } , X 1 ))] 1 / dim M ≥ t v ol g [ X 1 ] 1 / dim M , wher e vol ∗ g denotes the outer me asur e induc e d by v ol g . Next, let E ⊂ I + ( o ) ∪ { o } b e compact and star-shap ed with resp ect to a p oin t o ∈ M . Given r > 0 , let v ( r ) denote the volume of the intersection of E with the 1 W e note that p er se, the timelik e Bonnet–Myers inequality is another standard implication of synthetic timelike Ricci curv ature b ounds, cf. Ca v alletti–Mondino [ 24 ]. Ho wev er, in our w ork Theorem 1.1 is a prerequisite for Theorem 1.4 , not a consequence: the estimates of Ketterer [ 47 ] we use, cf. Lemma 3.4 , necessitate an upp er bound on the length of the geo desics in question. Consequently , w e m ust ensure ad hoc longer geodesics do not arise. COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 7 h yp erboloid of all p oin ts in I + ( o ) with distance at most r to o ; moreov er, let s ( r ) denote the area of the intersection of E with the hypersurface of all p oin ts in I + ( o ) with distance exactly r to o . W e refer to § 6.2 for details ab out these notions. Theorem 1.7 (Timelik e Bishop–Gromo v inequalit y , Theorem 6.4 ) . L et ( M , g ) b e a glob al ly hyp erb olic Lipschitz sp ac etime with Ric g ≥ 0 timelike distributional ly. Given o ∈ M , ﬁx a c omp act set E ⊂ I + ( o ) ∪ { o } that is star-shap e d with r esp e ct to o . Then for every r, R > 0 with r < R , s ( r ) s ( R ) ≥ r dim M − 1 R dim M − 1 , v ( r ) v ( R ) ≥ r dim M R dim M . Our last ma jor results are d’Alembert comparison theorems. They were estab- lished in the more abstract setting of TMCP metric measure spacetimes by Beran et al. [ 6 ]. How ev er, rather than verifying compatibility with the diﬀeren tial calculus dev elop ed in [ 6 ] and its coun terparts for Lipsc hitz spacetimes — whic h w ould render the results b elo w direct applications of [ 6 ] — we instead provide a self-contained and streamlined pro of based on the arguments of [ 6 ]; see § B . Let l o denote the future distance function ( 2.7 ) from a p oin t o ∈ M . It is lo cally Lipsc hitz contin uous on I + ( o ) , a result which is a sp ecial case of Prop osition 2.19 w e establish in § 4.1 . Theorem 1.8 (D’Alembert comparison theorem for p ow ers of Lorentz distance functions, Theorem 6.5 ) . A ssume ( M , g ) is a glob al ly hyp erb olic Lipschitz sp ac etime. Supp ose that Ric g ≥ 0 timelike distributional ly. L et q ∈ (0 , 1) and let q ′ < 0 b e its c onjugate exp onent. Then for every o ∈ M , the q ′ -d’A lemb ert c omp arison estimate div h    ∇ l q o q    q ′ − 2 ∇ l q o q i ≤ dim M (1.2) holds distributional ly on I + ( o ) , me aning that for every Lipschitz c ontinuous function ϕ : I + ( o ) → R + with c omp act supp ort, − Z M d ϕ h ∇ l q o q i    ∇ l q o q    q ′ − 2 dv ol g ≤ dim M Z M ϕ dvol g . The inequality ( 1.2 ) should b e compared to the analogous classical comparison estimate for the (tw o-)Laplacian of the squared distance ov er tw o on Riemannian manifolds, cf. Cheeger–Ebin [ 26 ]. Theorem 1.9 (D’Alembert comparison theorem for Lorentz distance functions, Theorem 6.6 ) . L et ( M , g ) b e a glob al ly hyp erb olic Lipschitz sp ac etime with Ric g ≥ 0 timelike distributional ly. Then for every o ∈ M , the d’Alemb ert c omp arison estimate div ∇ l o ≤ dim M − 1 l o (1.3) holds distributional ly on I + ( o ) , me aning that for every Lipschitz c ontinuous function ϕ : I + ( o ) → R + with c omp act supp ort, − Z M d ϕ ( ∇ l o ) dvol g ≤ (dim M − 1) Z M ϕ l o dv ol g . W e exp ect these comparison theorems to b e inﬂuential. First, they imply the formal left-hand sides of ( 1.2 ) and ( 1.3 ) deﬁne generalized signed Radon measures; see Corollaries 6.8 and 6.9 . This holds despite the nonsmo othness of the functions in volv ed, which already arises even when g is smo oth; cf. § 4.1 . Second, the ab o ve w eak form ulations of ( 1.2 ) and ( 1.3 ) extend across the future timelike cut lo cus of 8 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL o , an extension that was not known even in the smo oth setting prior to the w ork of Beran et al. [ 6 ], whic h is no w established in Lipschitz regularity . Third, they ha ve recen tly led to a signiﬁcantly simpler and unexp ected “elliptic” pro of of the Esc henburg–Gallo wa y–Newman splitting theorem mentioned ab o v e, as well as to its extension to regularity C 1 b y Braun et al. [ 13 , 12 ]. Our d’Alembert comparison theorems op en the do or to extending the splitting theorem to Lipsc hitz spacetimes, whic h w e lea ve for future work. 1.5. Organization. In § 2 , we review basic notions of the Lorentzian structure of (w eigh ted) Lipschitz spacetimes. W e further discuss how to create a suitable smo oth appro ximation of the nonsmo oth metric tensor and discuss its prop erties regarding time separation and curv ature. In § 3 , we outline the basics of Lorentzian optimal transp ort. In § 4 , w e collect results for smo oth spacetimes that are necessary for our main results. Then § 5 contains the pro of of Theorem 1.1 . W e provide the pro ofs for Theorem 1.4 in § 6.1 and discuss its main applications in § 6.2 . Finally , w e establish Theorem 1.2 in § A . 2. Anal ytic timelike Ricci cur v a ture lower bounds 2.1. Lipsc hitz spacetime geometry. In this subsection, we collect basics ab out spacetimes with lo cally Lipschitz contin uous metric tensors. F or general causality theory , we refer to Minguzzi’s review [ 62 ] and to Chrusćiel–Gran t [ 28 ] and Sämann [ 74 ] for spacetimes with merely contin uous metric tensors. Throughout the sequel, M is a top ological manifold that is smo oth, Hausdorﬀ, second-coun table, and connected, with dimension dim M at least 2 . Let g b e a ﬁxed lo cally Lipschitz con tinu ous Lorentzian metric, i.e. a metric tensor of Lorentzian signature + , − , . . . , − whose co eﬃcients in charts are lo cally Lipschitz con tinuous; w e deﬁne Lorentzian metrics to ha v e other regularities analogously . (In §§ A and 4 , w e will in fact assume smo othness of g .) In particular, our sign conv en tion for timelik e vectors v ∈ T M is g ( v , v ) > 0 , in whic h case we will o ccasionally write | v | := p g ( v , v ) . W e assume ( M , g ) is a Lipschitz sp ac etime , i.e. g is locally Lipschitz con tinuous and there exists a (tacitly ﬁxed) contin uous timelike vector ﬁeld Z on M ; if g is smo oth, we simply call ( M , g ) sp ac etime . Given o ∈ M , let T + o b e the set of all future-directed timelik e v ectors in T o M ; moreo ver, let T + denote the disjoin t union of T + o o ver all o ∈ M . Unless stated otherwise, all tangent vectors, curves, v ector ﬁelds, etc. will tacitly b e assumed to b e future-directed with resp ect to Z . Throughout, we ﬁx a tacit complete Riemannian metric r on M , assumed to b e smo oth. Its existence is guaran teed by Nomizu–Ozeki [ 66 ]. F or v ∈ T M , we write | v | r := p r ( v, v ) . The induced length functional will b e written Len r , while the induced metric will b e denoted d r . W e will use the standard signs ≤ and ≪ to denote causality and chronology of ( M , g ) . In addition, w e use the standard letters J and I to denote their induced sets; for instance, I + ( o ) ⊂ M denotes the chronological future of a p oin t o ∈ M . The time sep ar ation function l : M 2 → R + ∪ {−∞ , ∞} is given b y l ( x, y ) := sup { Len g ( γ ) : γ : [0 , 1] → M smo oth causal curve with γ 0 = x and γ 1 = y } , (2.1) where we adopt the conv en tion sup ∅ := −∞ and abbreviate Len g ( γ ) := Z [0 , 1] p g ( ˙ γ t , ˙ γ t ) d t. It ob eys the customary reverse triangle inequality (where we adopt the con ven tion −∞ + z := z + ( −∞ ) := −∞ for ev ery z ∈ R ∪ {−∞ , ∞} ). Giv en L > 0 , we will say a curv e γ : [0 , L ] → M is pr op er time p ar ametrize d if l ( γ s , γ t ) = t − s for COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 9 ev ery s, t ∈ [0 , L ] with s < t . W e say a causal curve γ : [0 , 1] → M is aﬃnely p ar ametrize d provided l ( γ s , γ t ) = ( t − s ) l ( γ 0 , γ 1 ) for every s, t ∈ [0 , 1] with s < t . A causal curve attaining the supremum ( 2.1 ) and solving the geo desic equation in the sense of Filipp o v [ 34 ] will b e called maximizing ge o desic . Lange–Lytchak–Sämann [ 54 ] show that on Lipsc hitz spacetimes, every causal maximizer of ( 2.1 ) admits a reparametrization into a C 1 , 1 -curv e which is a maximizing geo desic in the ab ov e sense. (If g is even of regularity C 1 , Graf [ 40 ] had previously prov ed reparametrizability of causal maximizers into C 2 -curv es solving the geo desic equation in the classical sense.) F urthermore, by the clause after [ 54 , Prop. 1.4], the C 1 , 1 -norm of that reparametrization is b ounded from ab o v e by the C 0 , 1 -norm of g and the r -length of γ (where the occurring norms can b e understo od e.g. as uniform norms with resp ect to r , cf. ( 2.3 ) ). In particular, maximizing geo desics (and maximizers) hav e a deﬁnite causal character, which means they cannot hav e b oth timelike and lightlik e subsegmen ts. This fact conﬁrms an earlier result of Graf–Ling [ 41 ]. Convention 2.1 (Multiple metric tensors) . If multiple metric tensors are in volv ed, to a void confusion or ambiguit y we will o ccasionally tag a quan tity induced by the corresp onding metric tensor g with a subscript. F or instance, w e write ≤ g instead of ≤ , I + g ( o ) instead of I + ( o ) (where o ∈ M ), l g instead of l , etc. ■ Deﬁnition 2.2 (Global hyperb olicit y) . W e wil l c al l the Lipschitz sp ac etime ( M , g ) globally hyperb olic if the fol lowing pr op erties hold. (i) F or every c omp act set C ⊂ M , ther e exists c > 0 such that the r -length of al l smo oth c ausal curves whose image is c ontaine d in C is b ounde d fr om ab ove by c . (ii) F or every x, y ∈ M , the c ausal diamond J ( x, y ) ⊂ M is c omp act. The prop erty from the ﬁrst item is called nontotal imprisonment . Sev eral equiv alent characterizations of the preceding (classical) notion of global h yp erbolicity , even when g is merely contin uous, are given in the work of Sämann [ 74 ]. A few consequences are worth mentioning as well, cf. Sämann [ 74 , Prop. 3.3, Cor. 3.4] and Minguzzi [ 61 , Prop. 2.26, Thm. 2.55] for details. Theorem 2.3 (Properties of globally hyperb olic Lipsc hitz spacetimes) . L et ( M , g ) b e a glob al ly hyp erb olic Lipschitz sp ac etime. Then the fol lowing hold. (i) The c ausal r elation ≤ is close d. (ii) F or al l c omp act X , Y ⊂ M , the c ausal emer ald J ( X , Y ) ⊂ M is c omp act. (iii) The p ositive p art l + of the time sep ar ation function is ﬁnite and c ontinuous. (iv) A l l p oints x, y ∈ M with x ≤ y ar e c onne cte d by a maximizing ge o desic. Finally , let v ol g denote the volume measure induced by g . W e will ﬁx a Borel measure m suc h that the negative logarithmic density − log d m / d v ol g is lo cally Lipsc hitz contin uous. W e call m smo oth if − log d m / d v ol g is. W e will say a Borel subset of M has measure zero if it is negligible with resp ect to some (hence every) smo oth measure on M . W e will deﬁne a function to b e lo cally in L ∞ analogously , without referring to a sp eciﬁc choice of measure. Deﬁnition 2.4 (W eighted Lipsc hitz spacetime) . W e wil l c al l the triple ( M , g , m ) globally hyperb olic weigh ted Lipschitz spacetime if a. ( M , g ) is a glob al ly hyp erb olic Lipschitz sp ac etime and b. m forms a Bor el me asur e on M such that the ne gative lo garithmic density − log d m / dv ol g is lo c al ly Lipschitz c ontinuous. 2.2. Regularization. Since g has merely lo cally Lipsc hitz con tinuous co eﬃcien ts in almost all of our work, fundamental geometric quantities like curv ature or the exp onen tial map are not classically deﬁned. T o work with these, we will regularize 10 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL g in suc h a wa y that p ossible distributional timelike Ricci curv ature b ounds are “almost” preserved in an L p -sense. Such approximations ha v e b een constructed by Calisti et al. [ 17 ], whose approac h w e extend to the weigh ted case. W e also refer to Grosser–Kunzinger–Ob erguggen berger–Steinbauer [ 42 ] and Graf [ 40 ] for more details and background ab out regularization of distributional tensor ﬁelds. 2.2.1. T ensor distributions. Let V ol ( M ) denote the volume bund le of M ; if M is orien table (whic h w e do not assume), this is simply the vector bundle Λ dim M T ∗ M of top-degree diﬀerential forms. Let Γ c ( V ol ( M )) denote the space of smo oth and compactly supp orted sections of V ol ( M ) , whose elemen ts are called volume densities . Given a volume density µ ∈ Γ c ( V ol ( M )) and an op en set U ⊂ M , the in tegral R U µ is deﬁned in analogy to the integration of top-degree diﬀerential forms on orientable manifolds. W e say µ is nonnegativ e if R U µ ≥ 0 for every op en set U ⊂ M . The set D ′ ( M ) of scalar distributions is deﬁned as the top ological dual space of Γ c ( V ol ( M )) with dualit y pairing ⟨ · | · ⟩ . The space C ∞ ( M ) em b eds into D ′ ( M ) b y iden tifying ψ ∈ C ∞ ( M ) with the assignmen t ⟨ ψ | µ ⟩ := R M ψ µ . Deﬁnition 2.5 (Comparison of distributions) . Given u, v ∈ D ′ ( M ) , we write u ≥ v if ⟨ u | µ ⟩ ≥ ⟨ v | µ ⟩ for every nonne gative volume density µ ∈ Γ c (V ol( M )) . In fact, every nonnegative scalar distribution (i.e. every u ∈ D ′ ( M ) satisfying u ≥ 0 in the ab o v e sense) is a measure. Since w e will deal with distributional (metric tensors and) curv ature, we also recall the space of (0 , 2) -tensor distributions deﬁned by D ′ (( T ∗ ) ⊗ 2 M ) := D ′ ( M ) ⊗ C ∞ ( M ) Γ(( T ∗ ) ⊗ 2 M ) , where ⊗ C ∞ ( M ) designates the classical tensor pro duct ov er the ring C ∞ ( M ) . Like their smo oth counterparts, (0 , 2) -tensor distributions admit lo cal co ordinate rep- resen tations on each chart. By pushing forw ard and pulling back the resp ectiv e co eﬃcien ts, such distributions can lo cally b e seen as a vector-v alued distribution on an op en subset of R dim M , cf. Graf [ 40 , Prop. 3.1]. In particular, since g is lo cally Lipsc hitz contin uous and letting f : M → R b e lo cally Lipschitz con tin uous, each directional deriv ativ e of f and the Christoﬀel symb ols , lo cally deﬁned by Γ k ij := g kl 2 h ∂ g j l ∂ x i + ∂ g il ∂ x j − ∂ g ij ∂ x l i , are all lo cally in L ∞ . Given a smo oth vector ﬁeld X ∈ X ( M ) , say with supp ort in a single chart, the lo cal form ulas Ric g ( X, X ) := h ∂ Γ m ij ∂ x m − ∂ Γ m im ∂ x j + Γ m ij Γ k km − Γ m ik Γ k j m i X i X j , Hess g f ( X , X ) := h ∂ 2 f ∂ x i ∂ x j − Γ k ij ∂ f ∂ x k i X i X j , (d f ⊗ d f )( X, X ) := ∂ f ∂ x i ∂ f ∂ x j X i X j (2.2) yield w ell-deﬁned ob jects in D ′ ( M ) . They induce (0 , 2) -tensor distributions Ric g , Hess g f , and d f ⊗ d f , resp ectiv ely . Recall the reference measure m on M , for which we assume − log d m / d v ol g is lo cally Lipschitz contin uous. Deﬁnition 2.6 (Distributional Bakry– Émery –Ricci tensor) . F or N ∈ [ dim M , ∞ ) we deﬁne the distributional N -Bakry– Émery –Ricci tensor Ric g , m ,N ∈ D ′ (( T ∗ ) ⊗ 2 M ) COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 11 thr ough the formula Ric g , m ,N :=            Ric g − Hess g log d m dv ol g − 1 N − dim M d log d m dv ol g ⊗ d log d m dv ol g if N > dim M , Ric g otherwise . Deﬁnition 2.7 (Bounds on distributional Bakry– Émery –Ricci tensor) . F or K ∈ R and N ∈ [dim M , ∞ ) , we say Ric g , m ,N ≥ K timelike distributionally if a. in the c ase N > dim M , every timelike ve ctor ﬁeld X ∈ X ( M ) ob eys Ric g , m ,N ( X, X ) ≥ K g ( X, X ) in the sense of Deﬁnition 2.5 , b. otherwise, the density d m / d v ol g is c onstant and every timelike ve ctor ﬁeld X ∈ X ( M ) ob eys Ric g ( X, X ) ≥ K g ( X, X ) in the sense of Deﬁnition 2.5 . R emark 2.8 (Scaling) . Note that the tensor distribution Ric g , m ,N (and its timelike distributional low er b oundedness) are unchanged by scaling the reference measure m by a p ositiv e constan t. ■ 2.2.2. Go o d appr oximation. Let { ρ ε : ε > 0 } b e a family of standard molliﬁers on R dim M . Using a partition of unit y and chart wise conv olution of lo cal co eﬃcien ts, giv en T ∈ D ′ (( T ∗ ) ⊗ 2 M ) one can deﬁne its con v olution T ∗ M ρ ε , which b ecomes an element of Γ(( T ∗ ) ⊗ 2 M ) for every ε > 0 , cf. e.g. Graf [ 40 , §3.3] for details; an analogous statement is true for the regularization u ∗ M ρ ε of a scalar distribution u ∈ D ′ ( M ) . Man y standard prop erties of conv olutions of distributions on R dim M transfer to regularizations of such distributions on M . The ﬁrst is c onver genc e . Recall r is the background Riemannian metric on M ; w e denote its induced Levi-Civita connection and v olume measure by r ∇ and v ol r , resp ectiv ely . F or a given T ∈ Γ(( T ∗ ) ⊗ 2 M ) and i ∈ N 0 , let | r ∇ i T | : M → R + b e the p oin t wise op erator norm of r ∇ i T ; here, r ∇ i denotes the i -fold application of r ∇ , suc h that r ∇ 0 T := T . F or a compact subset C ⊂ M and p ∈ [1 , ∞ ) , we set   r ∇ i T   L p ,C := h Z C   r ∇ i T   p dv ol r i p ,   r ∇ i T   ∞ ,C := sup {   r ∇ i T   ( x ) : x ∈ C } . (2.3) The following prop erties addressing the regularizations of the lo cally Lipschitz con tinuous metric g were established by Calisti et al. [ 17 , Lem. 3.2]. Lemma 2.9 (Basic prop erties of regularizations of g ) . L et C ⊂ M b e c omp act. Then the fol lowing pr op erties hold. (i) W e have the uniform Lipschitz estimate sup {   r ∇ ( g ∗ M ρ ε )   ∞ ,C : ε > 0 } < ∞ . (ii) F or every p ∈ [1 , ∞ ) , we have g ∗ M ρ ε → g lo c al ly in W 1 ,p as ε → 0 + , viz. lim ε → 0 +    ( g ∗ M ρ ε ) − g   L p ,C +   r ∇ ( g ∗ M ρ ε ) − r ∇ g   L p ,C  = 0 . R emark 2.10 (Ab out the choice of v ol r ) . The c hoice of v ol r as reference measure in ( 2.3 ) is merely cosmetic yet inessential for our arguments to follo w. It may and will b e replaced by a conv erging sequence of weigh ted measures whose densities (with resp ect to metric tensors with Lorentzian signature) are p ositive, con tinuous, and 12 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL con verge locally uniformly , cf. Lemma 2.11 . In particular, all adjacen t statements ab out lo cal conv ergence of tensor distributions remain unaltered. ■ The second is pr eservation of nonne gativity . Namely , if u, v ∈ D ′ ( M ) satisfy u ≥ v in the distributional sense of Deﬁnition 2.5 , then u ∗ M ρ ε ≥ v ∗ M ρ ε holds p oin t wise on M for every ε > 0 . There are tw o issues with these “naive” regularizations. First, given ε > 0 the causal structures from g and g ∗ M ρ ε are in general unrelated. Second, let X ∈ X ( M ) b e a timelik e vector ﬁeld. If Ric g , m ,N ≥ K timelik e distributionally for some K ∈ R and N ∈ [ dim M , ∞ ) , w e obtain Ric g , m ,N ( X, X ) ∗ M ρ ε ≥ K g ( X, X ) ∗ M ρ ε b y the preceding paragraph. As the form ulas ( 2.2 ) dep end on g and − log d m / d v ol g in a nonlinear manner, it is not true in general that the latter inequality implies a geometrically more tangible condition of the form Ric g ε , m ε ,N ( X, X ) ≥ K g ε ( X, X ) for an y smo oth regularizations { g ε : ε > 0 } and { m ε : ε > 0 } of g and m , respectively . The setup of a net { ˇ g ε : ε > 0 } of smo oth metric tensors of Lorentzian signature that approximates g , resp ects its causal structure, and preserves its distributional curv ature bounds in an L p -sense is one of the main results of Calisti et al. [ 17 ]. Their approac h w as inspired by prior works of Kunzinger–Stein bauer–Sto jk o vić–Vic k ers [ 53 ] (assuming g is lo cally C 1 , 1 ) and Graf [ 40 ] (assuming g is of class C 1 ). The curv ature asp ect will b e discussed b elow; for no w, w e fo cus on causality . Recall for tw o rough Lorentzian metrics g 1 and g 2 on M , we write g 1 ≺ g 2 if for every v ∈ T M \ { 0 } , g 1 ( v , v ) ≥ 0 implies g 2 ( v , v ) > 0 ; more pictorially , g 1 has strictly narrow er lightcones than g 2 . Moreov er, recall the lo cal norms from ( 2.3 ). Lemma 2.11 (Existence of go o d approximation [ 17 , Lem. 2.4]) . Ther e is a family { ˇ g ε : ε > 0 } of smo oth L or entzian metrics, time-orientable by the same c ontinuous ve ctor ﬁeld as g , with the fol lowing pr op erties. (i) F or every ε > 0 , we have ˇ g ε ≺ g . (ii) F or every p ∈ [1 , ∞ ) , we have ˇ g ε → g and ˇ g − 1 ε → g − 1 lo c al ly uniformly and lo c al ly in W 1 ,p as ε → 0 + , i.e. for every c omp act set C ⊂ M , lim ε → 0 +   ˇ g ε − g   ∞ ,C = 0 , lim ε → 0 +    ˇ g ε − g   L p ,C +   r ∇ ˇ g ε − r ∇ g   L p ,C  = 0 and analo gously for the inverses. (iii) F or every p ∈ [1 , ∞ ) , we have ˇ g ε − g ∗ M ρ ε → 0 and ˇ g − 1 ε − ( g ∗ M ρ ε ) − 1 → 0 lo c al ly in C ∞ as ε → 0 + , i.e. for al l i ∈ N 0 and every c omp act set C ⊂ M , lim ε → 0 +   r ∇ i ˇ g ε − r ∇ i ( g ∗ M ρ ε )   ∞ ,C = 0 and analo gously for the inverses. (iv) F or every c omp act set C ⊂ M , ther e exists a se quenc e ( ε n ) n ∈ N in (0 , ∞ ) de cr e asing to zer o, dep ending on C , with ˇ g ε n ≺ ˇ g ε n +1 for every n ∈ N . R emark 2.12 (Inheritance of global hyperb olicit y) . In the ab o v e lemma, ˇ g ε inherits global hyperb olicit y from g for every ε > 0 as a consequence of the ﬁrst item, cf. Sämann [ 74 , Thms. 5.7, 5.9]. ■ Deﬁnition 2.13 (Goo d approximation) . Given a c omp act set C ⊂ M , we c al l two se quenc es ( g n ) n ∈ N and ( m n ) n ∈ N of L or entzian metrics and smo oth me asur es on M go od approximation of g and m on C , r esp e ctively, if for every n ∈ N , g n = ˇ g ε n , log d m n dv ol g n = h log d m dv ol g i ∗ M ρ ε n , COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 13 wher e the family { ˇ g ε : ε > 0 } satisﬁes the c onclusions of L emma 2.11 and ( ε n ) n ∈ N is the se quenc e fr om the last statement ther ein. 2.2.3. Pr op erties of go o d appr oximation. W e ﬁx go o d approximations ( g n ) n ∈ N and ( m n ) n ∈ N of g and m on a compact set C ⊂ M , respectively . Their ﬁrst property concerns the p ositiv e parts of their induced time separation functions, cf. Calisti et al. [ 17 , Lem. 5.5] for the pro of. Lemma 2.14 (Lo cally uniform conv ergence of time separations functions) . The se quenc e ( l g n , + ) n ∈ N c onver ges lo c al ly uniformly to l g , + . The retention of timelike distributional curv ature b ounds from g and m is more delicate. Kunzinger–Steinbauer–Stojko vić–Vick ers [ 53 ] (when g is C 1 , 1 ) and Graf [ 40 ] (when g is C 1 ) previously obtained a low er b ound for the curv ature tensors induced b y { ˇ g ε : ε > 0 } whose diﬀerence to that of the timelike distributional curv ature b ound originally given by g is lo cally uniformly small. Their considerations carry o v er to the weigh ted case, cf. Braun–Calisti [ 11 , Lem. 2.8]. In our case where g and − log d m / dv ol g are merely lo cally Lipschitz contin uous, one cannot exp ect to retain this lo cally uniform error. Ho w ev er, one may still hop e for a very rough uniform lo w er b ound and, for more precise estimates, smallness in L p . This is the con ten t of Lemmas 2.15 and 2.16 summarizing Calisti et al. ’s results [ 17 ]. W e ﬁrst in tro duce some notation. W e ﬁx N ∈ [ dim M , ∞ ) . Let V ⊂ T +   C b e compact. W e say that V is ev en tually uniformly timelike if there are λ > 0 and n 0 ∈ N suc h that for ev ery n ∈ N with n ≥ n 0 , we hav e g n ( v , v ) ≥ λ 2 for ev ery v ∈ V . With this deﬁnition, we can already state the follo wing straightforw ard generalization of Calisti et al. ’s result [ 17 , Prop. 5.8] from the unw eighted case. Lemma 2.15 (Rough uniform lo w er b ound) . A ssume K ∈ R and N ∈ [ dim M , ∞ ) satisfy Ric g , m ,N ≥ K timelike distributional ly. L et V ⊂ T +   C b e c omp act and uni- formly timelike, wher e C ⊂ M is c omp act. Then ther e exist ρ ∈ R + and n 0 ∈ N such that for every n ∈ N with n ≥ n 0 and every v ∈ V , Ric g n , m n ,N ( v , v ) ≥ − ρ g n ( v , v ) . T o obtain ﬁner estimates, assume again V is as ab o v e. Then for every n ∈ N with n ≥ n 0 , the function k V ,g n , m n ,N : C → R given by k V ,g n , m n ,N ( x ) := inf n Ric g n , m n ,N ( v , v ) g n ( v , v ) : v ∈ V ∩ T x M o (2.4) is contin uous. In particular, for ev ery n ∈ N with n ≥ n 0 , every x ∈ C , and every v ∈ V ∩ T x M , we obtain Ric g n , m n ,N ( v , v ) ≥ k V ,g n , m n ,N ( x ) g n ( v , v ) . With an analogous pro of as Calisti et al. [ 17 , Lem. 5.9] (where we c ho ose the sequence of v ector ﬁelds therein to minimize ( 2.4 ) ), the follo wing statement is then readily established. It is easily seen to hold in the weigh ted case by applying the F riedrichs-t ype lemma [ 17 , Lem. 3.3] to the w eigh ted quan tities from ( 2.3 ) and noting the densities of the inv olv ed reference measures conv erge lo cally uniformly . Lemma 2.16 ( L p -con vergence of curv ature b ounds) . Supp ose t her e ar e K ∈ R and N ∈ [ dim M , ∞ ) with Ric g , m ,N ≥ K timelike distributional ly. L et V ⊂ T +   C b e c omp act and uniformly timelike, with C ⊂ M c omp act. Then for every p ∈ [1 , ∞ ) , lim n →∞ Z C   ( k V ,g n , m n ,N − K ) −   p d m n = 0 . 14 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL 2.2.4. Construction of c omp act and uniformly timelike sets. Finally , w e identify suﬃcien t conditions under which a set of tangen t v ectors as ab ov e can actually b e constructed. A v arian t of the follo wing result implicitly appears in the pro of of Calisti et al. ’s Hawking incompleteness theorem for Lipschitz spacetimes [ 17 , Thm. 5.1] (for prop er time instead of aﬃnely parametrized maximizing geodesics). Recall r is the ﬁxed background Riemannian metric on M . Lemma 2.17 (Bounds on Lorentzian and Riemannian lengths) . L et X, Y ⊂ M b e c omp act sets with X × Y ⊂ I g . W e deﬁne C := J g ( X, Y ) . L et ( g n ) n ∈ N b e a go o d appr oximation of g on C ac c or ding to Deﬁnition 2.13 . Then ther e exist c onstants c > 0 , λ > 0 , and n 0 ∈ N with the fol lowing pr op erty. F or every n ∈ N with n ≥ n 0 , we have X × Y ⊂ I g n and e ach tangent ve ctor of e ach timelike aﬃnely p ar ametrize d g n -maximizing ge o desic γ : [0 , 1] → M that starts in X and terminates in Y b elongs to the set V := { v ∈ T +   C : g n ( v , v ) ≥ λ 2 for every n ∈ N with n ≥ n 0 , | v | r ≤ c } , which is c omp act and eventual ly uniformly timelike. Pr o of. It is clear that V is compact as a closed subset of a compact subset of T M . Moreo ver, it is even tually uniformly timelike by deﬁnition. W e start with some general considerations. Given any n 0 ∈ N , abbreviate the class of curves in question by G ≥ n 0 . By nontotal imprisonment of ( M , g ) , implied b y global h yp erbolicity , there is a constan t L > 0 suc h that the r -length of ev ery g -causal curv e is b ounded from ab o v e by L . In particular, since every γ ∈ G n 0 is g -timelik e, w e obtain Len r ( γ ) ≤ L . Since γ solv es the geo desic equation with respect to g n for some n ∈ N with n ≥ n 0 , the results from Lange–Lytchak–Sämann [ 54 ] recalled in § 2.1 imply the C 1 , 1 -norm of γ on the compact set C with resp ect to r is b ounded in terms of the C 0 , 1 -norm of g n on C and Len r ( γ ) . While the latter is b ounded from ab ov e by L as already inferred, the former is b ounded in terms of the C 0 , 1 -norm of g on C (whic h is ﬁnite by assumption on g ) by Lemma 2.11 and Calisti et al. [ 17 , Lem. 3.2]. In particular, this implies the existence of a constant c > 0 such that for every γ ∈ G ≥ n 0 , we hav e | ˙ γ | r ≤ c on [0 , 1] . Our h yp othesis on X and Y , global h yp erbolicity of ( M , g ) , and con tinuit y of l + imply the existence of λ > 0 suc h that inf { l g ( x, y ) : x ∈ X, y ∈ Y } ≥ 3 λ. By Lemma 2.14 and compactness of X × Y , there exists a num b er n 0 ∈ N 0 suc h that for every n ∈ N with n ≥ n 0 , inf { l g n ( x, y ) : x ∈ X, y ∈ Y } ≥ 2 λ. (2.5) On the other hand, by Lemma 2.11 again, up to increasing n 0 , every n, m ∈ N with n, m ≥ n 0 satisﬁes   g n − g m   ∞ ,C ≤ 3 λ 2 c 2 . (2.6) W e will prov e the claim of the lemma for the ab o ve choices of c , λ , and n 0 . Let γ ∈ G ≥ n 0 b e a timelike aﬃnely parametrized g n -maximizing geo desic, where n ∈ N with n ≥ n 0 , and let t ∈ [0 , 1] . Recall | ˙ γ t | r ≤ c . Aﬃne parametrization with resp ect to g n , ( 2.5 ), and ( 2.6 ) yield, for every m ∈ N with m ≥ n 0 , g m ( ˙ γ t , ˙ γ t ) = g m ( ˙ γ t , ˙ γ t ) − g n ( ˙ γ t , ˙ γ t ) + g n ( ˙ γ t , ˙ γ t ) ≥ −   g n − g m   ∞ ,C   ˙ γ t   2 r + 4 λ 2 ≥ λ 2 . This is the desired statement. □ COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 15 With a similar pro of, the following is veriﬁed. Lemma 2.18 (Uniform precompactness) . W e r etain the hyp otheses and the notation fr om L emma 2.17 . Then the set of al l timelike aﬃnely p ar ametrize d g -maximizing ge o desics γ : [0 , 1] → M that start in X and end in Y is e qui-Lipschitz c ontinuous with r esp e ct to the uniform top olo gy on C 0 ([0 , 1]; C ) induc e d by r . 2.3. Regularit y of time separation function. W e now use the previous consid- erations to establish some key properties of l on our globally hyperb olic weigh ted Lipsc hitz spacetime ( M , g , m ) . Prop osition 2.19 (Lo cal equi-Lipschitz con tin uit y) . L et ( g n ) n ∈ N b e a go o d ap- pr oximation of g as in Deﬁnition 2.13 . Then ther e is n 0 ∈ N such that the class { l g n : n ∈ N with n ≥ n 0 } is lo c al ly e qui-Lipschitz c ontinous on I g ; that is, for ev- ery x, y ∈ M with x ≪ g y , ther e exist open neighb orho o ds U x , U y ⊂ M of x and y , r esp e ctively, and c 0 > 0 such that for every x ′ , x ′′ ∈ U x and every y ′ , y ′′ ∈ U y , sup {   l g n ( x ′ , y ′ ) − l g n ( x ′′ , y ′′ )   : n ∈ N with n ≥ n 0 } ≤ c 0  d r ( x ′ , x ′′ ) + d r ( y ′ , y ′′ )  . In p articular, l g is lo c al ly Lipschitz c ontinuous on I g . The pro of of this prop osition is partly based on well-kno wn prop erties of time separation functions in the smo oth setting; we recapitulate the necessary material in § 4.1 b elow, to where we thus defer the pro of of Prop osition 2.19 . Deﬁne the Lorentz distance function l o : I + ( o ) ∪ { o } → R + from o by l o ( x ) := l ( o, x ) . (2.7) Corollary 2.20 (Eik onal equation) . The gr adient ∇ l o of l o , wher ever it exists on I + ( o ) , is p ast-dir e cte d and satisﬁes g ( ∇ l o , ∇ l o ) = 1 m I + ( o ) -a.e. Pr o of. By Prop osition 2.19 and Rademacher’s theorem, l o is diﬀerentiable m -a.e. on I + ( o ) , thus the statement makes sense. W e ﬁrst claim g ( ∇ l o , ∇ l o ) ≥ 1 at ev ery diﬀerentiabilit y p oin t x ∈ I + ( o ) of l o . Let v ∈ T + x . Moreov er, let r exp denote the exp onen tial map induced by the bac kground Riemannian metric r . Since g is contin uous, for every suﬃcien tly small s > 0 the curv e γ s : [ 0 , 1] → M deﬁned through γ s t := r exp x ( tsv ) is timelike; in particular, r exp x ( sv ) ∈ I + ( x ) . By the rev erse triangle inequalit y and the deﬁnition of l , l o ( r exp x ( sv )) − l o ( x ) s ≥ l ( x, r exp x ( sv )) s ≥ 1 s Z [0 , 1] p g ( ˙ γ s t , ˙ γ s t ) d t. By lo cal Lipschitz con tinuit y of g and a T aylor expansion, uniformly in t ∈ [0 , 1] , p g ( ˙ γ s t , ˙ γ s t ) = s p g ( v , v ) + o( s ) as s → 0 + . Consequen tly , d l o ( v ) = lim s → 0 + l o ( r exp x ( sv )) − l o ( x ) s ≥ p g ( v , v ) . (2.8) By the arbitrariness of v , this giv es d l o ( v ) ≥ 0 for ev ery causal vector v ∈ T x M . In other w ords, d l o b elongs to the dual cone of the set of causal vectors in T x M . This cone is precisely the image of the set of past-directed causal vectors in T x M under the musical isomorphism ♭ . This implies ∇ l o is a past-directed causal v ector in T x M . In turn, the well-kno wn duality formula p g ( ∇ l o , ∇ l o ) = inf { d l o ( v ) : v ∈ T + x with g ( v , v ) ≥ 1 } (2.9) com bined with ( 2.8 ) giv e the desired estimate. 16 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL It remains to show g ( ∇ l o , ∇ l o ) ≤ 1 at the ab o v e p oin t x . Let γ : [0 , l o ( x )] → M b e a timelike prop er time parametrized maximizing geo desic from o to x . By the results from Lange–Lytc hak–Sämann [ 54 ] recalled in § 2.1 , γ is a C 1 , 1 -curv e. By prop er time parametrization, w e ha v e g ( ˙ γ t , ˙ γ t ) = 1 for every t ∈ [0 , l o ( x )] . Thus, using the duality formula ( 2.9 ) and prop er time parametrization, p g ( ∇ l o , ∇ l o ) ≤ d l o ( ˙ γ l o ( x ) ) = lim s → 0 + l o ( γ l o ( x ) ) − l o ( γ l o ( x ) − s ) s = 1 , whic h is the desired inequality . □ 3. Synthetic timelike Ricci cur v a ture lower bounds 3.1. One-dimensional considerations. 3.1.1. Distortion c o eﬃcients. F or L > 0 , let κ : [0 , L ] → R b e a contin uous function. The induced gener alize d sine function sin κ : [0 , L ] → R is deﬁned as the unique solution to the ODE u ′′ + κ u = 0 with u (0) = 0 and u ′ (0) = 1 . F or instance, if κ is constan t, then for every θ ∈ [0 , L ] , sin κ ( θ ) =            sin( √ κ θ ) √ κ if κ > 0 , θ if κ = 0 , sinh( √ − κ θ ) √ − κ otherwise . (3.1) Let π κ ∈ (0 , ∞ ] denote the ﬁrst p ositiv e zero of sin κ (with the con ven tion π κ := ∞ if sin κ v anishes only at zero). F or instance, if κ is constant, π κ =    π √ κ if κ > 0 , ∞ otherwise . (3.2) Deﬁnition 3.1 ( σ -Distortion co eﬃcients) . Given t ∈ [0 , 1] and θ ∈ [0 , L ] , we set σ ( t ) κ ( θ ) :=        t if θ = 0 , sin κ ( t θ ) sin κ ( θ ) if θ ∈ (0 , π κ ) , ∞ otherwise . F or instance, if κ is zero, then σ ( t ) κ ( θ ) = t for ev ery t ∈ [0 , 1] and ev ery θ ∈ [0 , L ] . Note that if σ ( t ) κ ( θ ) < ∞ for θ ∈ [0 , L ] and some (hence every) t ∈ [0 , 1] , then the function v : [0 , 1] → R + deﬁned by v ( t ) := σ ( t ) κ ( θ ) solv es v ′′ ( t ) + κ ( t θ ) θ 2 v ( t ) = 0 for every t ∈ [0 , 1] as well as v (0) = 0 and v (1) = 1 . By taking a geometric mean, we no w recapitulate another family of distortion co eﬃcien ts. T o this aim, we ﬁx N ∈ (1 , ∞ ) . W e stipulate the inﬁnity conv en tions 0 · ∞ := 0 and α · ∞ := ∞ α := ∞ for every α > 0 . Deﬁnition 3.2 ( τ -Distortion co eﬃcien ts) . Given t ∈ [0 , 1] and θ ∈ [0 , L ] , we set τ ( t ) κ,N ( θ ) := t 1 / N σ ( t ) κ/ ( N − 1) ( θ ) 1 − 1 / N . R emark 3.3 (Basic prop erties) . F or every t ∈ [0 , 1] and ev ery θ ∈ [0 , L ] , the follo wing statemen ts hold. • If κ ′ : [ 0 , L ] → M satisﬁes κ ′ ≤ κ on [0 , L ] , then σ ( t ) κ ′ / N ( θ ) ≤ σ ( t ) κ/ N ( θ ) , τ ( t ) κ ′ / N ( θ ) ≤ τ ( t ) κ/ N ( θ ) . COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 17 • W e hav e the relation τ ( t ) κ,N ( θ ) ≥ σ ( t ) κ/ N ( θ ) . • If ( κ n ) n ∈ N is a sequence of con tinuous functions κ n : [0 , L ] → R suc h that κ ≤ liminf n →∞ κ n p oin t wise on [0 , 1] , then σ ( t ) κ/ N ( θ ) ≤ liminf n →∞ σ ( t ) κ n / N ( θ ) , τ ( t ) κ/ N ( θ ) ≤ liminf n →∞ τ ( t ) κ n / N ( θ ) . F or details, we refer to Ketterer [ 46 , §3]. ■ 3.1.2. A defe ct estimate. F or later purp oses, we will need to quan tify if a function is “close” to a num ber K ∈ R in an integral sense, then the corresp onding distortion co eﬃcien ts are “close” as well. F or the estimate from Lemma 3.4 , given K ∈ R , N ∈ [2 , ∞ ) , p ∈ ( N / 2 , ∞ ) , and η ∈ (0 , π K/ ( N − 1) / 2) , we deﬁne tw o constants Ω K,N ,p,η := ( C N ,p if K ≤ 0 , max { C N ,p , sin K/ ( N − 1) ( η ) N +1 − 4 p } otherwise , Λ K,N ,η := ( D K,N ,η if K > 0 , 1 otherwise , (3.3) where C N ,p := (2 p − 1) p h N − 1 2 p − N i p − 1 , D K,N ,η := 1 + max n sin K/ ( N − 1) ( r ) 1 − N : r ∈ h π K/ ( N − 1) 2 , π K/ ( N − 1) − η io . The following estimate is established by Ketterer [ 47 , Cor. 4.7]. Lemma 3.4 (Defect of distortion coeﬃcients) . L et us ﬁx K ∈ R , N ∈ [2 , ∞ ) , p ∈ ( N / 2 , ∞ ) , and η ∈ (0 , π K/ ( N − 1) / 2) . Mor e over, let κ : [0 , L ] → R b e c ontinuous, wher e L > 0 . Then for every t ∈ (0 , 1) and every θ ∈ (0 , min { L, π K/ ( N − 1) − η } ) , τ ( t ) K,N ( θ ) − τ ( t ) κ,N ( θ ) ≤  Λ K,N ,η Ω 1 / (2 p − 1) K,N ,p,η  1 / N h Z [ t, 1] τ ( r ) κ,N ( θ ) N d r i 2( p − 1) / N (2 p − 1) × h t θ 2 p Z [0 , 1] ( κ ( r θ ) − K ) − σ ( r ) κ/ ( N − 1) ( θ ) N − 1 d r i 1 / N (2 p − 1) . 3.1.3. CD densities and diameter estimates. Let I ⊂ R b e an interv al, κ : I → R b e contin uous, and N ∈ (1 , ∞ ) . Let γ : [0 , 1] → I b e a straight line; in particular, it has constant sp eed | ˙ γ | . W e deﬁne the function κ + γ : [ 0 , | ˙ γ | ] → R b y κ + γ := κ ◦ ¯ γ , where ¯ γ : [0 , | ˙ γ | ] → I designates the unit sp eed reparametrization of γ . In addition, w e deﬁne κ − γ : [ 0 , | ˙ γ | ] → I b y κ − γ := κ ◦ ¯ γ ◦ T , where T : [0 , | ˙ γ | ] → [0 , | ˙ γ | ] means the orien tation rev ersal T ( r ) := | ˙ γ | − r . Deﬁnition 3.5 (CD density) . A function h : I → R + is c al le d CD ( κ, N ) density if for every str aight line γ : [0 , 1] → I and every t ∈ [0 , 1] , h ( γ t ) 1 / ( N − 1) ≥ σ (1 − t ) κ − γ / ( N − 1) ( | ˙ γ | ) h ( γ 0 ) 1 / ( N − 1) + σ ( t ) κ + γ / ( N − 1) ( | ˙ γ | ) h ( γ 1 ) 1 / ( N − 1) . R emark 3.6 (Basic prop erties) . Let h : I → R + b e a CD ( κ, N ) densit y . Then the follo wing statemen ts hold. 18 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL • It is is lo cally semiconcav e (hence lo cally Lipschitz contin uous) on I ◦ . In particular, its righ t-deriv ative exists everywhere on I \ { sup I } and is righ t- con tinuous; analogously for the left-deriv ative on I \ { inf I } . Moreo ver, h can b e contin uously extended to the closure of I . • Either h v anishes identically on I or it is p ositiv e on I ◦ . • If h is p ositiv e on I ◦ , it is t wice diﬀerentiable L 1 I -a.e., and at ev ery twice diﬀeren tiability p oin t in I ◦ , (log h ) ′′ + 1 N − 1   (log h ) ′   2 = ( N − 1) ( h 1 / ( N − 1) ) ′′ h 1 / ( N − 1) ≤ − κ. F or details, we refer to Ca v alletti–Milman [ 21 , §A]. ■ The following prop erty justiﬁes the name “CD density” . Prop osition 3.7 (Curv ature-dimension condition [ 46 , Prop. 3.8]) . L et h : I → R b e lo c al ly L 1 I -inte gr able. Then the top olo gic al ly one-dimensional metric me asur e sp ac e (cl I , | · − · | , h L 1 I ) ob eys the curvatur e-dimension c ondition CD ( κ, N ) of K etter er [ 46 ] if and only if h has an L 1 I -version that is a CD ( κ, N ) density. A family of facts w e recapitulate now are diameter estimates. F or Riemannian manifolds, the next Theorem 3.8 was pro v en b y Aubry [ 3 , Thm. 1.2]. The version w e rep ort b elo w states the extension of Aubry’s result to essentially non branching CD metric measure spaces by Caputo–Nobili–Rossi [ 18 , Thm. 1.2] for CD densities. Their extension hypothesizes a “noncollapsing” of the reference measure which, in the case of CD densities, forces the reference measure to v anish identically . Ho w ever, this additional h yp othesis is only needed in the proof of [ 18 , Prop. 4.3] to deal with nonsmo oth densities. In our setting, all relev an t densities will b e smo oth; in particular, the noncollapsing hypothesis can b e disp ensed, cf. [ 3 , Lem. 3.1]. Theorem 3.8 (One-dimensional diameter estimate) . W e let K > 0 , N ∈ [2 , ∞ ) , and p ∈ ( N / 2 , ∞ ) . Ther e exists a c onstant C K,N ,p > 1 with the fol lowing pr op erty. L et h : I → R b e a smo oth CD ( κ, N ) pr ob ability density on an interval I ⊂ R , wher e κ : I → R is c ontinuous, with Z I   ( κ − K ) −   p h d L 1 ≤ 1 C K,N ,p . Then we have diam ( I , | · − · | ) ≤ π r N − 1 K h 1 + C K,N ,p h Z I   ( κ − K ) −   p h d L 1 i 1 / 5 i . Corollary 3.9 (Qualitativ e one-dimensional diameter estimate) . W e let K > 0 , N ∈ [2 , ∞ ) , and p ∈ ( N / 2 , ∞ ) . Then for every ε > 0 , ther e exists δ K,N ,p,ε > 0 with the fol lowing pr op erty. L et h : I → R b e a smo oth CD ( κ, N ) pr ob ability density on an interval I ⊂ R , wher e κ : I → R is c ontinuous, with Z I   ( κ − K ) −   p h d L 1 ≤ δ K,N ,p,ε . Then we have diam ( I , | · − · | ) ≤ π r N − 1 K + ε. R emark 3.10 (Explicit integral deﬁcit) . In the previous corollary , one can take δ K,N ,p,ε = min nh ε π C K,N ,p r K N − 1 i 5 , 1 C K,N ,p o . This is clear from Theorem 3.8 . ■ COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 19 3.2. Lorentzian optimal transp ort. F or background ab out the material recalled here, we refer to Eckstein–Miller [ 29 ], McCann [ 58 ], and Ca v alletti–Mondino [ 24 ]. Let P ( M ) denote the space of all Borel probability measures on M . It is endo w ed with the narro w top ology induced by conv ergence against b ounded and contin uous test functions. The subscript ♯ will denote the usual push-forward op eration. Giv en µ, ν ∈ P ( M ) , let Π( µ, ν ) denote the set of all couplings of µ and ν . W e will call a coupling π ∈ Π( µ, ν ) c ausal (whic h is equiv alen t to spt π ⊂ J thanks to Theorem 2.3 ) if it is concentrated on the causal relation J and chr onolo gic al if it is concentrated on the c hronological relation I . Note Π( µ, ν ) is never empty , as it alw ays con tains the pro duct measure µ ⊗ ν ; how ev er, in general µ and ν ma y not admit any causal coupling. 3.2.1. q -L or entz–W asserstein distanc e. Throughout the sequel, w e ﬁx q ∈ (0 , 1) . De- ﬁne the q -L or entz–W asserstein distanc e ℓ q : P ( M ) 2 → R + ∪ {−∞ , ∞} , introduced b y Ec kstein–Miller [ 29 ], through ℓ q ( µ, ν ) := sup nh Z M 2 l ( x, y ) q d π ( x, y ) i 1 /q : π ∈ Π( µ, ν ) o . (3.4) where we adopt the conv ention ( −∞ ) q := ( −∞ ) 1 /q := −∞ . It inherits the reverse triangle inequality from l . W e call a coupling π ∈ Π( µ, ν ) ℓ q -optimal if it is causal and it attains the suprem um in ( 3.4 ) ; in particular, we exclude the cost −∞ but include the cost ∞ in our notion of ℓ q -optimalit y . By a standard argument and the prop erties of l from Theorem 2.3 , if µ and ν ha ve compact supp ort and admit a causal coupling, they also admit an ℓ q -optimal coupling (which has ﬁnite cost). As we will only deal with transp ort in timelike directions, the following notion in tro duced by Cav alletti–Mondino [ 24 , Def. 2.18] will b e conv enien t. Deﬁnition 3.11 (Timelike q -dualizabilit y) . A p air ( µ, ν ) ∈ P ( M ) 2 is c al le d timelik e q -dualizable if it admits a chr onolo gic al ℓ q -optimal c oupling. Note timelik e q -dualizabilit y of µ and ν in the sense of the previous deﬁnition is stronger than requiring ℓ q ( µ, ν ) > 0 . On the other hand, it is clear by the ab ov e discussions that ev ery pair ( µ, ν ) ∈ P ( M ) 2 of t wo compactly supp orted measures satisfying spt µ × spt ν ⊂ I is timelike q -dualizable. R emark 3.12 (Nice cost function) . If µ, ν ∈ P ( M ) are compactly supported and satisfy spt µ × spt ν ⊂ I , we clearly hav e ℓ q ( µ, ν ) = sup nh Z M 2 l + ( x, y ) q d π ( x, y ) i 1 /q : π ∈ Π( µ, ν ) o In other words, in this case the ℓ q -optimal transp ort problem from µ to ν can b e translated into an optimal transp ort problem with a b ounded and contin uous cost function, making standard to ols more accessible than for p ossibly degenerate cost functions such as l q ; cf. e.g. the pro of of Prop osition 3.21 b elo w. ■ The follo wing stabilit y and compactness prop ert y shown by Cav alletti–Mondino [ 24 , Lem. 2.11] will b e used later. Lemma 3.13 (Static stability of optimal couplings) . A ssume ( µ i ) i ∈ N and ( ν i ) i ∈ N ar e se quenc es that c onver ge narr ow ly to µ, ν ∈ P ( M ) , r esp e ctively. L et ( π i ) i ∈ N b e a se quenc e of ℓ q -optimal c ouplings π i of µ i and ν i . Then { π i : i ∈ N } is narr ow ly pr e c omp act in P ( M 2 ) and every narr ow limit p oint π of ( π i ) i ∈ N that is c onc entr ate d on the chr onolo gic al r elation I is ℓ q -optimal. 20 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL 3.2.2. q -Ge o desics and ℓ q -optimal dynamic al c ouplings. W e go on with the following notion introduced by McCann [ 58 , Def. 1.1]. Deﬁnition 3.14 (Geo desics of probability measures) . A curve µ : [0 , 1] → P ( M ) is c al le d q -geo desic if ℓ q ( µ 0 , µ 1 ) ∈ (0 , ∞ ) and for every s, t ∈ [0 , 1] with s < t , ℓ q ( µ s , µ t ) = ( t − s ) ℓ q ( µ 0 , µ 1 ) . W e will later need the follo wing stabilit y and compactness prop ert y . It is prov en b y Braun–McCann [ 14 , Thm. 2.69, Rem. 2.70]; note that Hypothesis 2.42 therein is satisﬁed by [ 14 , Ex. 2.44] since we work on a Lipschitz spacetime. Lemma 3.15 (Static stabilit y and compactness of geodesics) . L et ( µ i ) i ∈ N b e a given se quenc e of q -ge o desics. F or a c omp act set C ⊂ M , assume µ i t is supp orte d in C for every i ∈ N and every t ∈ [0 , 1] . L et µ 0 , µ 1 ∈ P ( M ) b e narr ow limit p oints of ( µ i 0 ) i ∈ N and ( µ i 1 ) i ∈ N , r esp e ctively, and assume 0 < ℓ q ( µ 0 , µ 1 ) ≤ liminf n →∞ ℓ q ( µ i 0 , µ i 1 ) . Then ther e is a q -ge o desic µ : [0 , 1] → M fr om µ 0 to µ 1 such that up to a nonr elab ele d subse quenc e, ( µ i t ) i ∈ N c onver ges narr ow ly to µ t for every t ∈ [0 , 1] ∩ Q . Mor e over, if every ℓ q -optimal c oupling of µ 0 and µ 1 is chr onolo gic al, then any q -ge o desic µ as in the pr evious statement is narr ow ly c ontinuous. A tigh tly linked concept are optimal dynamical couplings, which we review now. Giv en t ∈ [0 , 1] , let e t : C 0 ([0 , 1]; M ) → M b e the ev aluation map e t ( γ ) := γ t . Deﬁnition 3.16 (Optimal dynamical couplings) . Given µ 0 , µ 1 ∈ P ( M ) , we c al l π ∈ P ( C 0 ([0 , 1]; M )) an ℓ q -optimal dynamical coupling of µ 0 and µ 1 if a. it is c onc entr ate d on timelike aﬃnely p ar ametrize d maximizing ge o desics and b. ( e 0 , e 1 ) ♯ π is an ℓ q -optimal c oupling of µ 0 and µ 1 . R emark 3.17 (Geodesy from optimal dynamical couplings) . Given an ℓ q -optimal dynamical coupling π of µ 0 , µ 1 ∈ P ( M ) , it is not hard to chec k (cf. e.g. Cav alletti– Mondino [ 24 , Rem. 2.32]) that if ℓ q ( µ 0 , µ 1 ) < ∞ , the curv e µ : [0 , 1] → P ( M ) giv en b y µ t := ( e t ) ♯ π is a q -geo desic. ■ Con versely , there are recent “lifting theorems” stating general conditions under whic h a q -geo desic is in fact represented b y an ℓ q -optimal dynamical coupling in the sense of the previous remark. W e will use the following result by Braun–McCann [ 14 , Prop. 2.67], but also refer to Beran et al. [ 6 , Thm. 2.43, Cor. 2.46]. Lemma 3.18 (Lifting theorem) . A ssume µ 0 , µ 1 ∈ P ( M ) ar e c omp actly supp orte d and satisfy spt µ 0 × spt µ 1 ⊂ I . L et µ : [0 , 1] → M b e a q -ge o desic c onne cting µ 0 and µ 1 . Then µ is r epr esente d by an ℓ q -optimal dynamic al c oupling π of µ 0 and µ 1 , i.e. for every t ∈ [0 , 1] , we have µ t = ( e t ) ♯ π . In the smo oth framework, the follo wing stronger characterization by McCann [ 58 , Lem. 4.4, Thm. 5.8, Cor. 5.9] holds. Finer structural prop erties, in particular with regards to the Monge–Amp ère and Rayc haudh uri equations, are summarized in § 4.3 b elow. Theorem 3.19 (Existence and uniqueness) . L et ( M , g , m ) b e a glob al ly hyp erb olic weighte d sp ac etime. L et µ 0 , µ 1 ∈ P ( M ) b e c omp actly supp orte d and m -absolutely c ontinuous with spt µ 0 × spt µ 1 ⊂ I . Then ther e exist an op en neighb orho o d U ⊂ M of spt µ 0 and a lo c al ly semic onvex ( henc e lo c al ly Lipschitz c ontinuous ) function COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 21 φ : U → R with m -a.e. timelike gr adient ∇ φ such that for the m -a.e. ( henc e µ 0 -a.e. ) wel l-deﬁne d map Ψ : [0 , 1] × U → M given by Ψ t ( x ) := exp x ( − t |∇ φ | q ′ − 2 ∇ φ ) , wher e q ′ < 0 is the c onjugate exp onent of q , we have the fol lowing pr op erties. (i) The function φ is a K antor ovich p otential for µ 0 and µ 1 . (ii) The me asur e (Id , Ψ 1 ) ♯ µ 0 is the unique ( ne c essarily chr onolo gic al ) ℓ q -optimal c oupling of µ 0 and µ 1 ; in p articular, µ 1 = (Ψ 1 ) ♯ µ 0 . (iii) The curve µ : [0 , 1] → M deﬁne d by µ t := (Ψ t ) ♯ µ 0 c onsists only of c omp actly supp orte d, m -absolutely continuous me asur es. F urthermor e, it deﬁnes the unique q -ge o desic fr om µ 0 to µ 1 . (iv) Deﬁne F : U → C 0 ([0 , 1]; M ) by F ( x ) t := Ψ t ( x ) . Then the me asur e F ♯ µ 0 is the unique ℓ q -optimal dynamic al c oupling of µ 0 and µ 1 . W e refer to McCann [ 58 , §4] and Cav alletti–Mondino [ 24 , §2.4] for the notion of Kan torovic h p oten tials and their duality theory in the Lorentzian framework. R emark 3.20 (Inv ertibility) . In fact, since µ 0 and µ 1 are assumed to be m -absolutely con tinuous, the results of McCann [ 58 ] quoted ab o ve also show for every t ∈ [0 , 1] , the transp ort map Ψ t is inv ertible on spt µ 0 and its inv erse is the restriction of a lo cally semiconv ex (hence lo cally Lipschitz contin uous) nonrelab eled map Ψ − 1 t deﬁned on an op en neigh b orhoo d of spt µ t . ■ In addition to Lemmas 3.13 and 3.15 , w e will need stabilit y and compactness prop erties along the go o d approximations from Deﬁnition 2.13 . Similar static facts are established by Braun [ 8 , Prop. B.11] and Braun–McCann [ 14 , Prop. 2.67]. Prop osition 3.21 (V arying stability and compactness of optimal dynamical cou- plings) . L et ( g n ) n ∈ N b e a go o d appr oximation of g ac c or ding to Deﬁnition 2.13 . L et X, Y ⊂ M b e c omp act sets such that X × Y ⊂ I g . L et ( µ n 0 ) n ∈ N and ( µ n 1 ) n ∈ N form se quenc es supp orte d in X and Y , r esp e ctively. Final ly, let ( π n ) n ∈ N c onstitute a se quenc e such that π n is an ℓ g n ,q -optimal dynamic al c oupling of µ n 0 and µ n 1 . Then { π n : n ∈ N } is narr ow ly pr e c omp act in P ( C 0 ([0 , 1]; M )) and every narr ow limit π of ( π n ) n ∈ N is an ℓ g ,q -optimal dynamic al c oupling of µ 0 and µ 1 ; in p articular, lim n →∞ ℓ g n ,q ( µ n 0 , µ n 1 ) = ℓ g ,q ( µ 0 , µ 1 ) . (3.5) Pr o of. By deﬁnition, π n is concen trated on timelike aﬃnely parametrized maximiz- ing g n -geo desics for ev ery n ∈ N . These b elong to an even tually n -indep enden t compact subset of C 0 ([0 , 1]; M ) b y Lemma 2.17 and the Arzelà–Ascoli theorem. By Prokhoro v’s theorem, this implies narrow compactness of { π n : n ∈ N } . No w let π b e any narrow limit p oint of ( π n ) n ∈ N . Since the in volv ed ev aluation maps are contin uous, ( e 0 , e 1 ) ♯ π is a (necessarily chronological) coupling of µ 0 and µ 1 . W e claim it is ℓ g ,q -optimal. Since X × Y ⊂ I g with compact inclusion and l g , + is contin uous by Theorem 2.3 , Lemma 2.14 implies there exists n 0 ∈ N suc h that X × Y ⊂ I g n for ev ery n ∈ N with n ≥ n 0 . Combin ing this with Remark 3.12 , Lemma 2.14 again, and uniform b oundedness of the in v olv ed cost functions, we can apply well-kno wn stability properties for optimal transp orts with uniformly con vergen t contin uous cost functions, cf. e.g. Villani [ 81 , Thm. 5.20], to infer that ( e 0 , e 1 ) ♯ π is an ℓ g ,q -optimal coupling of µ 0 and µ 1 , as desired. Next, we show ( 3.5 ) . As l g , + is contin uous, hence b ounded on X × Y , Villani [ 81 , Lem. 4.3] and the narrow conv ergence of ( π n ) n ∈ N to π imply lim n →∞ Z l g , + ( γ 0 , γ 1 ) q d π n ( γ ) = Z l g , + ( γ 0 , γ 1 ) q d π ( γ ) . 22 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL On the other hand, Lemma 2.14 implies that given ε > 0 there exists n 1 ∈ N such that for every n ∈ N with n ≥ n 1 , we hav e sup {   l g n ( x, y ) q − l g ( x, y ) q   : x ∈ X, y ∈ Y } ≤ ε. This implies limsup n →∞   ℓ g n ,q ( µ n 0 , µ n 1 ) q − ℓ g ,q ( µ 0 , µ 1 ) q   ≤ limsup n →∞ Z   l g n ( γ 0 , γ 1 ) q − l g ( γ 0 , γ 1 ) q   d π n ( γ ) + limsup n →∞    Z l g , + ( γ 0 , γ 1 ) q d π n ( γ ) − Z l g , + ( γ 0 , γ 1 ) q d π ( γ )    ≤ ε. The arbitrariness of ε yields the desired statement. It remains to establish π is concen trated on timelike aﬃnely parametrized maxi- mizing geo desics. By Braun–McCann [ 14 , Cor. 2.65], it suﬃces to show the curve µ : [0 , 1] → P ( M ) deﬁned b y µ t := ( e t ) ♯ π forms a q -geo desic (for the q -Loren tz– W asserstein distance induced b y g ). W e only sketc h the argument. Given n ∈ N , deﬁne µ n : [ 0 , 1] → P ( M ) by µ n t := ( e t ) ♯ π n . Let s, t ∈ [0 , 1] with s < t . W e claim limsup n →∞ ℓ g n ,q ( µ n s , µ n t ) ≤ ℓ g ,q ( µ s , µ t ) . (3.6) Indeed, giv en n ∈ N , the rev erse triangle inequalit y satisﬁed b y ℓ g n ,q and ℓ g n ,q - optimalit y of ( e 0 , e 1 ) ♯ π n imply that ( e s , e t ) ♯ π n is an ℓ g n ,q -optimal coupling of its marginals. Since its marginal sequences are tight by the ﬁrst paragraph, the latter con v erges narro wly to a coupling π s,t of µ s and µ t , up to a nonrelab eled subsequence. Since the prop erty of admitting a causal coupling is closed in the narrow top ology , cf. Braun–McCann [ 14 , Thm. B.5], π s,t is a causal coupling of µ s and µ t ; a priori, ho wev er, we do not kno w if it is ℓ g ,q -optimal. As in the previous paragraph, using basic semicon tin uity prop erties of cost functionals, cf. e.g. Villani [ 81 , Lem. 4.3], it is not diﬃcult to show the intermediate estimate in limsup n →∞ ℓ g n ,q ( µ n s , µ n t ) q = limsup n →∞ Z l g n ( γ s , γ t ) q d π n ≤ Z M 2 l q g d π s,t ≤ ℓ g ,q ( µ s , µ t ) q , as desired. Using the reverse triangle inequalit y for ℓ g ,q , ( 3.6 ) thrice, and ( 3.5 ) giv es ℓ g ,q ( µ 0 , µ 1 ) ≥ ℓ g ,q ( µ 0 , µ s ) + ℓ g ,q ( µ s , µ t ) + ℓ g ,q ( µ t , µ 1 ) ≥ limsup n →∞  ℓ g n ,q ( µ n 0 , µ n s ) + ℓ g n ,q ( µ n s , µ n t ) + ℓ g n ,q ( µ n t , µ n 1 )  =  s + ( t − s ) + (1 − t )  limsup n →∞ ℓ g n ,q ( µ n 0 , µ n 1 ) = ℓ g ,q ( µ 0 , µ 1 ) . This forces equality in the simple consequence ( t − s ) ℓ g ,q ( µ 0 , µ 1 ) = ( t − s ) limsup n →∞ ℓ g n ,q ( µ n 0 , µ n 1 ) ≤ ℓ g ,q ( µ s , µ t ) of ( 3.6 ), which terminates the pro of. □ 3.3. V ariable timelike curv ature-dimension condition. Now we deﬁne the v ariable timelik e curv ature-dimension condition, brieﬂy TCD condition, we will w ork with in this article. Inspired by the works of McCann [ 58 ] and Mondino–Suhr [ 64 ], for constant timelike Ricci curv ature b ounds it was in tro duced on general metric measure spacetimes by Ca v alletti–Mondino [ 24 ]. A v ariable version of their TCD condition was given by Braun–McCann [ 14 ]. The deﬁnitions we adopt in the next tw o subsections diﬀer from [ 14 ] and follo w the prop osal of Braun [ 8 ] for constant timelike Ricci curv ature b ounds. F or metric COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 23 measure spaces, an analogous deﬁnition was given by Ketterer [ 46 ]. The adv antage of this formulation is that as in [ 8 ], it will yield the geometric inequalities for the globally h yp erbolic weigh ted spacetime ( M , g , m ) w e will obtain later in sharp form a priori, without relying on nonbranc hing of timelike maximizing geo desics (which is unclear even if g is C 1 -regular). Let m b e a Radon measure on M , such as the one from Deﬁnition 2.4 . Given N ∈ (1 , ∞ ) , let us deﬁne the N -R ényi entr opy S N ( · | m ) : P ( M ) → R − ∪ {−∞} with resp ect to m b y S N ( µ | m ) := − Z M h d µ ac d m i − 1 / N d µ = − Z M h d µ ac d m i 1 − 1 / N d m , where µ ac denotes the absolutely contin uous part in the Leb esgue decomp osition of its input µ with resp ect to m . If µ has compact supp ort, Jensen’s inequality gives S N ( µ | m ) ≥ − m [spt µ ] 1 / N . Lemma 3.22 (Join t low er semicontin uit y [ 56 , Thm. B.33]) . Given a c omp act set C ⊂ M , let ( µ n ) n ∈ N and ( m n ) n ∈ N b e two se quences in P ( M ) with supp ort in C . Then al l narr ow limit p oints µ, m ∈ P ( M ) of these se quenc es, r esp e ctively, satisfy S N ( µ | m ) ≤ liminf n →∞ S N ( µ n | m n ) . Let γ : [0 , 1] → M b e a timelike aﬃnely parametrized maximizing geodesic; in particular, it has constant sp eed | ˙ γ | := l ( γ 0 , γ 1 ) . Let ¯ γ : [0 , | ˙ γ | ] → M denote its prop er time reparametrization. As in § 3.1.3 , giv en a con tinuous function k : M → R w e deﬁne k ± γ : [ 0 , | ˙ γ | ] → R b y k + γ := k ◦ ¯ γ and k − γ := k ◦ ¯ γ ◦ T , where T : [0 , | ˙ γ | ] → [0 , | ˙ γ | ] means the function T ( r ) := | ˙ γ | − r . Lastly , τ ( t ) k ± γ ,N means the τ -distortion co eﬃcien ts from Deﬁnition 3.2 induced b y k ± γ , where N ∈ (1 , ∞ ) and t ∈ [0 , 1] . Deﬁnition 3.23 (V ariable timelike curv ature-dimension condition) . F or q ∈ (0 , 1) , a c ontinuous function k : M → R , and N ∈ (1 , ∞ ) , we say ( M , g , m ) satisﬁes the timelik e curv ature-dimension condition , brieﬂy TCD q ( k , N ) , if for every timelike q -dualizable p air ( µ 0 , µ 1 ) ∈ P ( M ) 2 of c omp actly supp orte d, m -absolutely c ontinuous me asur es, ther e exist • a q -ge o desic µ : [0 , 1] → P ( M ) fr om µ 0 to µ 1 and • an ℓ q -optimal dynamic al c oupling π of µ 0 and µ 1 such that for every N ′ ∈ [ N , ∞ ) and every t ∈ [0 , 1] , S N ′ ( µ t | m ) ≤ − Z τ (1 − t ) k − γ ,N ′ ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N ′ d π ( γ ) − Z τ ( t ) k + γ ,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ d π ( γ ) . The pro of of the following result is p erformed as in Braun–Ohta [ 15 , Thm. 5.9]. F or more precise estimates, we refer to § 6.1 b elo w. See also Braun [ 8 , Thm. 3.35] for a similar pathwise version of Cav alletti–Mondino’s TCD condition [ 24 ]. Theorem 3.24 (Essen tial pathwise semiconcavit y for TCD) . A ssume ( M , g , m ) is a glob al ly hyp erb olic weighte d sp ac etime. L et µ 0 , µ 1 ∈ P ( M ) b e two given c omp actly supp orte d, m -absolutely c ontinuous me asur es with spt µ 0 × spt µ 1 ⊂ I . L et µ and π b e their unique q -ge o desic and ℓ q -optimal dynamic al c oupling fr om The or em 3.19 , r esp e ctively, wher e q ∈ (0 , 1) . Supp ose that ther e exist a c ontinuous function k : M → R and N ∈ [ dim M , ∞ ) such that Ric g , m ,N ( ˙ γ , ˙ γ ) ≥ ( k ◦ γ ) g ( ˙ γ , ˙ γ ) on [0 , 1] for every timelike aﬃnely p ar ametrize d maximizing ge o desic γ : [0 , 1] → M starting in spt µ 0 and ending in spt µ 1 . Then π -a.e. γ ob eys the fol lowing ine quality for every N ′ ∈ 24 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL [ N , ∞ ) and every t ∈ [0 , 1] : d µ t d m ( γ t ) − 1 / N ′ ≥ τ (1 − t ) k − γ ,N ′ ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N ′ + τ ( t ) k + γ ,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ . As opp osed to the previous theorem, the following is not needed in the sequel, but it clariﬁes the name “timelike curv ature-dimension condition” . Corollary 3.25 (TCD from timelike Ricci curv ature b ounds) . A ssume ( M , g , m ) is a glob al ly hyp erb olic weighte d sp ac etime. L et k : M → R b e c ontinuous and let N ∈ [dim M , ∞ ) . Then the fol lowing statements ar e e quivalent. (i) W e have Ric g , m ,N ≥ k in al l timelike dir e ctions. (ii) The triple ( M , g , m ) satisﬁes TCD q ( k , N ) for some q ∈ (0 , 1) . (iii) The triple ( M , g , m ) satisﬁes TCD q ( k , N ) for every q ∈ (0 , 1) . The extension to merely timelik e q -dualizable endp oints in this claim is argued as in McCann [ 58 , Thm. 7.4] and Braun [ 8 , Thm. 3.35]. 3.4. V ariable timelike measure contraction prop ert y. Deﬁnition 3.26 (V ariable timelik e measure contraction prop ert y) . Given a c on- tinuous function k : M → R and N ∈ [ dim M , ∞ ) , we say ( M , g , m ) sat isﬁes the past timelik e measure con traction property , brieﬂy TMCP − ( k , N ) , if for every p oint o ∈ M and every c omp actly supp orte d, m -absolutely c ontinuous µ 1 ∈ P ( M ) that is c onc entr ate d on I + ( o ) , ther e exist • an exp onent q ∈ (0 , 1) , • a q -ge o desic µ : [0 , 1] → P ( M ) fr om µ 0 := δ o to µ 1 , and • an ℓ q -optimal dynamic al c oupling π of µ 0 and µ 1 such that for every N ′ ∈ [ N , ∞ ) and every t ∈ [0 , 1] , S N ′ ( µ t | m ) ≤ − Z τ ( t ) k + γ ,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ d π ( γ ) . (3.7) Mor e over, we say ( M , g , m ) ob eys a. the future timelike measure contraction prop ert y , brieﬂy TMCP + ( k , N ) , if the sp ac etime ( M , g ) endowe d with opp osite time orientation yet the same r efer enc e me asur e m satisﬁes TMCP − ( k , N ) , and b. the timelike measure contraction prop ert y , brieﬂy TMCP ( k , N ) , if it ob eys b oth TMCP − ( k , N ) and TMCP + ( k , N ) . If k is constant, ( 3.7 ) b ecomes S N ′ ( µ t | m ) ≤ − Z M τ ( t ) k,N ◦ l o h d µ 1 d m i 1 − 1 / N ′ d m , where l o is from ( 2.7 ) . This follows as | ˙ γ | = l o ( γ 1 ) for π -a.e. γ and as ( e 0 , e 1 ) ♯ π = δ o ⊗ µ 1 is the only (necessarily chronological and ℓ q -optimal) coupling of µ 0 and µ 1 . The TMCP do es not dep end on the transp ort exp onen t q , cf. [ 24 , Rem. 3.8]. In addition, since every p oin t in M has an arbitrarily close p oin t in its chronological past by strong causality , one can equiv alen tly require { o } × spt µ 1 ⊂ I instead of the weak er prop erty µ 1 [ I + ( o )] = 1 in the previous deﬁnition. Via Lemma 3.22 (appro ximating Dirac masses with uniform distributions), it is not diﬃcult to prov e that TCD q ( k , N ) implies TMCP ( k , N ) for every q ∈ (0 , 1) , ev ery contin uous k : M → R , and every N ∈ [ dim M , ∞ ) , cf. Cav alletti–Mondino [ 24 , Prop. 3.12] and Braun–McCann [ 14 , Prop. A.3]; see the pro of of Theorem 6.1 b elo w. In general, how ever, the TMCP is strictly weak er than the TCD condition, cf. Ca v alletti–Mondino [ 24 , Rem. A.3]. Y et, it has the same consequences we detail in § 6.2 as the TCD condition. COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 25 The following TMCP version of Theorem 3.24 is prov en following the lines of Braun–Oh ta [ 15 , Lem. 5.4]. Theorem 3.27 (Essential pathwise semiconcavit y for TMCP) . A ssume ( M , g , m ) is a glob al ly hyp erb olic weighte d sp ac etime. Mor e over, let o ∈ M and let µ 1 ∈ P ( M ) b e a c omp actly supp orte d, m -absolutely c ontinuous me asur es with { o } × spt µ 1 ⊂ I . L et µ and π c onstitute the unique q -ge o desic and ℓ q -optimal dynamic al c oupling fr om The or em 3.19 that c onne ct µ 0 := δ o and µ 1 , r esp e ctively, wher e q ∈ (0 , 1) . Supp ose that ther e exist a c ontinuous function k : M → R and N ∈ [ dim M , ∞ ) such that Ric g , m ,N ( ˙ γ , ˙ γ ) ≥ ( k ◦ γ ) g ( ˙ γ , ˙ γ ) on [0 , 1] for every timelike aﬃnely p ar ametrize d maximizing ge o desic γ : [0 , 1] → M starting in o and ending in spt µ 1 . Then π -a.e. γ ob eys the fol lowing ine quality for every N ′ ∈ [ N , ∞ ) and every t ∈ [0 , 1] : d µ t d m ( γ t ) − 1 / N ′ ≥ τ ( t ) k + γ ,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ . 4. Smooth considera tions No w we collect some materials from the smo oth case. T o this aim, in this section w e assume ( M , g ) is a globally hyperb olic spacetime (in particular, g is smo oth ) endo wed with a smo oth measure m on M , except for the pro of of Prop osition 2.19 at the end of § 4.1 . W e will abbreviate these hypotheses b y calling ( M , g , m ) globally h yp erbolic weigh ted spacetime. 4.1. Timelik e cut lo cus. Let o ∈ M b e a given point. F or details ab out the facts to follow, we refer to T reude [ 79 ] and T reude–Grant [ 80 ] (where the timelike cut lo cus is called “causal cut lo cus”). Their discussion for suitable submanifolds includes p oints [ 79 , p. 117], the only relev an t case for us. F or v ∈ T + o , we let γ v : I v → M denote the unique timelike geo desic with initial v elo cit y v suc h that I v is the maximal domain of deﬁnition of γ v relativ e to R + . Then the o -futur e cut function s + o : T + o → R + ∪ {∞} deﬁned by s + o ( v ) := sup { t ∈ I v : l o (( γ v ) t ) = Len g ( γ v   [0 ,t ] ) } (4.1) has no zeros [ 79 , Cor. 3.2.23] and is low er semicontin uous [ 79 , Prop. 3.2.29]. Deﬁnition 4.1 (Timelik e cut lo cus) . The set TC + ( o ) := { exp o ( s + o ( v ) v ) ∈ I + ( o ) : v ∈ T + o , s + o ( v ) ∈ I v } is c al le d future timelike cut lo cus of o . An y elemen t of TC + ( o ) will b e called futur e timelike cut p oint of o . Prop osition 4.2 (Characterization of timelik e cut lo cus [ 79 , Prop. 3.2.28]) . A p oint y ∈ I + ( o ) b elongs to TC + ( o ) if and only if either ther e exists mor e than one timelike maximizing ge o desic fr om o to y or y is the ﬁrst fo c al p oint of o along a timelike maximizing ge o desic. In fact, the set of all y ∈ I + ( o ) for whic h there exists more than one timelike maximizing geo desic from o to y is dense in TC + ( o ) [ 79 , Prop. 3.2.30]. Recall we say a Borel measurable subset of M has measure zero if it is negligible with resp ect to some (hence every) smo oth measure on M . Theorem 4.3 (Prop erties of future before timelike cut lo cus [ 79 , Thm. 3.2.31, Prop. 3.2.32]) . W e deﬁne J + tan ( o ) := { tv ∈ T o M : v ∈ T + o , t ∈ [0 , s + o ( v )) } and let I + tan ( o ) denote the interior of J + tan ( o ) . Then the fol lowing statements hold. 26 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL (i) The “futur e b efor e timelike cut lo cus” I + ( o ) := exp o ( I + tan ( o )) is op en and diﬀe omorphic to I + tan ( o ) thr ough exp o . (ii) W e have I + ( o ) = I + ( o ) \ TC + ( o ) . (iii) The set TC + ( o ) is close d and has me asur e zer o. (iv) The set I + ( o ) is the lar gest op en subset of I + ( o ) with the pr op erty that e ach of its p oints c an b e c onne cte d to o by a unique timelike maximizing ge o desic. Recall the Lorentz distance function l o from ( 2.7 ). Theorem 4.4 (Smo othness of distance function [ 80 , Prop. 2.7]) . The r estriction of the L or entz distanc e function ( 2.7 ) to I + ( o ) is smo oth. Mor e over, for every y ∈ I + ( o ) we have −∇ l o ( y ) = ( ˙ γ exp − 1 o ( y ) ) l o ( y ) . In other wor ds, γ exp − 1 o ( y ) c oincides with the unique ne gative gr adient ﬂow of l o that starts in o and p asses thr ough y . In fact, as established b y Andersson–Gallow a y–Ho ward [ 2 , Prop. 3.1] and later McCann [ 58 , Prop. 3.4, Thm. 3.5], the Lorentz distance function ( 2.7 ) is lo cally semicon vex (hence lo cally Lipschitz con tin uous) on all of I + ( o ) and fails to b e semiconca ve precisely on TC + ( o ) . In our case, we can at least obtain lo cal Lipschitz contin uity as follows. Lo cal semicon vexit y seems harder, since the pro ofs of [ 2 , 58 ] quoted ab o ve (and Braun et al. [ 12 , Prop. 13] in the C 1 -case) are unclear to b e “stable” if the metric tensor is merely lo cally Lipschitz con tinuous. Pr o of of Pr op osition 2.19 . W e ﬁrst assume g is smo oth and show a b ound on the lo cal Lipschitz constant of l on I in terms of the l -distance of the neighborho ods in question and the lo cal C 0 -norm of g . W e recall r is the complete bac kground Riemannian metric on M . Given x, y ∈ M with x ≪ g y , let U x , U y ⊂ M b e small geo desically con vex Riemannian balls with resp ect to r . Without restriction, we ma y and will assume cl U x × cl U y ⊂ I . Let c > 0 and λ > 0 b e from Lemma 2.17 applied to a constant sequence of metric tensors, X := cl U x , and Y := cl U y . By the triangle inequality and a symmetric argument it suﬃces to pro v e there is c 0 > 0 suc h that for every o ∈ U x and every y ′ , y ′′ ∈ U y , sup {   l o ( y ′ ) − l o ( y ′′ )   : n ∈ N with n ≥ n 0 } ≤ c 0 d r ( y ′ , y ′′ ) . Giv en y ′ , y ′′ ∈ U y \ TC + ( o ) , let γ : [0 , 1] → M b e the unique aﬃnely parametrized maximizing r -geo desic from y ′ to y ′′ ; it do es not leav e I + ( o ) . Since the curve γ is Lipsc hitz con tinuous, the set of its intersections with TC + ( o ) has one-dimensional Leb esgue measure zero. Then by Theorem 4.4 and Lemma 2.17 (exchanging prop er time by aﬃne parametrization), we obtain p r ( ∇ l o , ∇ l o ) ≤ c λ γ ♯ ( L 1 [0 , 1]) -a.e. (4.2) On the other hand, again by Theorem 4.4 , l o ( y ′ ) − l o ( y ′′ ) = Z [0 , 1] d l o ( ˙ γ t ) d t = Z [0 , 1] g ( ∇ l o , ˙ γ t ) d t. Since the co eﬃcien ts of g are uniformly b ounded on U x and by Lemma 2.11 , we use the usual Cauch y–Sch w arz inequality and lo cal equiv alence of r to any other Riemannian metric (here a pull-back of the Euclidean metric) to ﬁnd a constant c 1 > 0 dep ending on the C 0 -norms of g and r on cl U x suc h that Z [0 , 1] g ( ∇ l o , ˙ γ t ) d t ≤ c 1 Z [0 , 1] p r ( ∇ l o , ∇ l o ) p r ( ˙ γ , ˙ γ ) d  γ ♯ ( L 1 [0 , 1])  COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 27 ≤ c 0 d r ( y ′ , y ′′ ) , where w e used ( 4.2 ) and geo desy of γ with resp ect to r and w e set c 0 := c 1 c/λ . As TC + ( o ) has measure zero, the consequential inequality   l o ( y ′ ) − l o ( y ′′ )   ≤ c 0 d r ( y ′ , y ′′ ) easily extends to arbitrary y ′ , y ′′ ∈ U y b y appro ximation. If g is merely locally Lipsc hitz contin uous, we inv ok e its goo d approximation ( g n ) n ∈ N from Deﬁnition 2.13 . Using lo cally uniform conv ergence of that sequence com bined with Lemma 2.17 , the previous argumen ts thus show that there exist constan ts n 0 ∈ N and c 0 > 0 suc h that for every n ∈ N with n ≥ n 0 , ev ery o ∈ U x , and every y ′ , y ′′ ∈ U y , we hav e the desired equi-Lipschitz estimate   l g n ,o ( y ′ ) − l g n ,o ( y ′′ )   ≤ c 0 d r ( y ′ , y ′′ ) . The last statement follows since on every compact subset of I g , l g is the uniform limit of the equi-Lipschitz contin uous functions ( l g n ) n ∈ N b y Lemma 2.11 . □ 4.2. Localization. The paradigm of lo calization (also called “needle decomp osi- tion”) originates in con v ex geometry . In Loren tzian geometry , even for p ossibly nonsmo oth metric measure spacetimes, it was introduced b y Cav alletti–Mondino [ 24 ] and then expanded by themselves [ 25 ] and Braun–McCann [ 14 ]. In a nutshell, the strategy is as follows. A set in question, such as the chronological future of a giv en p oint o ∈ M , is foliated into timelik e prop er time parametrized maximizing geo desics (called “rays”) starting at o . The reference measure disintegrates as m I + ( o ) = Z M α m α d q ( α ) , where • q is a probability measure on a set Q ⊂ I + ( o ) which “lab els the ra ys”, • for q -a.e. α ∈ Q , M α forms a timelike proper time parametrized maximizing geo desic starting at o , and • for q -a.e. α ∈ Q , the conditional measure m α is concentrated on M α . As p ointed out by Cav alletti–Mondino [ 24 , Rem. 5.4], this result is tightly linked to the area formula. (A nonexp ert can think of a “nonstraight F ubini theorem” .) By prop er time parametrization, for q -a.e. α ∈ Q the ray M α b ecomes isometric (through l ) to an interv al [0 , b α ) ⊂ R + , where b α ∈ (0 , ∞ ] , where each interv al will b e endow ed with the Euclidean distance | · − · | . By this identiﬁcation and with a sligh t abuse of notation, ( M α , | · − · | , m α ) b ecomes a top ologically one-dimensional metric measure space. (It can b e shown m α is absolutely contin uous with resp ect to the one-dimensional Lebesgue measure on M α with Radon–Nik o dým densit y h α for q -a.e. α ∈ Q .) The crucial p oin t of the ab o v e paradigm is that am bien t timelik e Ricci curv ature b ounds for ( M , g , m ) turn into Ricci curv ature b ounds for ( M α , | · − · | , m α ) after Sturm [ 78 ] and Lott–Villani [ 56 ] for q -a.e. α ∈ Q . W e will clarify tw o asp ects ab out the results of Cav alletti–Mondino [ 24 , 25 ] and Braun–McCann [ 14 ], cf. Theorem 4.6 . First, since we assume g and m to b e smo oth, w e will sho w the conditional density h α is smo oth for q -a.e. α ∈ Q . On Riemannian manifolds, corresp onding results are obtained by Klartag [ 49 ]. Second, the w orks cited ab ov e hypothesize ambien t timelike Ricci curv ature b ounds in every timelike direction; in fact, it suﬃces to stipulate these only in all “relev an t” directions, namely those tangen tial to the rays. Our results b elo w are not trimmed for optimality , yet will suﬃce for our purp oses. 28 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL 4.2.1. F r amework. Let o ∈ M b e a given p oin t. Recall its future timelike cut lo cus TC + ( o ) ⊂ I + ( o ) from § 4.1 as well as the Lorentz distance function l o from ( 2.7 ) . In addition, ﬁx an op en Riemannian ball U ⊂ I + ( o ) with compact inclusion. Then con tinuit y of l + implies that ϑ > 0 , where 2 ϑ := inf { l o ( y ) : y ∈ cl U } > 0 . Reminiscen t of Theorem 4.3 , deﬁne H + tan ( o, U ) := { tv ∈ T o M : v ∈ exp − 1 o ( U ) , t ∈ [0 , s + o ( v )) } and let G + tan ( o, U ) denote the interior of H + tan ( o, U ) . The set G + ( o, U ) := exp o ( G + tan ( o, U )) (4.3) is the “thin ice cone” consisting of o and all p oints in I + ( o ) that lie in the relative in terior of a (necessarily uniquely determined) timelike prop er time parametrized maximizing geo desic from o that meets U . In particular, G + ( o, U ) ∩ TC + ( o ) = ∅ thanks to Prop osition 4.2 . Also, by Theorem 4.3 the set G + ( o, U ) \ { o } is op en and the inclusion U ⊂ G + ( o, U ) holds up to an m -negligible set, for m [ U \ G + ( o, U )] ≤ m [ U ∩ TC + ( o )] ≤ m [TC + ( o )] = 0 . 4.2.2. Smo oth disinte gr ation. Fix r ∈ (0 , ϑ ) and set Σ r := { l o = r } ∩ G + ( o, U ) . (4.4) As r is a regular v alue of the restriction of l o to G + ( o, U ) (b ecause g ( ∇ l o , ∇ l o ) = 1 there), Σ r is a hypersurface. It is clearly precompact and achronal. As its normal v ector ﬁeld −∇ l o   Σ r , cf. Theorem 4.4 , is timelike, the hypersurface Σ r is spacelik e. Let µ r b e the restriction of m to Σ r . Employing that g restricts to a Riemannian metric on Σ r , up to the implicit exp onen tial weigh t we see that µ r corresp onds to the induced Riemannian volume measure on Σ r . Let Φ : W → M form the maximally deﬁned (jointly smo oth) negative gradient ﬂo w of l o , where W ⊂ R × G + ( o, U ) is op en, deﬁned by Φ s ( y ) := exp y ( − s ∇ l o ) . Recall b y Theorem 4.4 , for every y ∈ G + ( o, U ) the translation Φ •− l o ( y ) ( y ) restricts to the unique timelike geodesic that starts in o and passes through y on a suitable subin terv al of R + con taining zero. Let b : G + ( o, U ) → (0 , ∞ ) b e deﬁned b y b y := sup { t ∈ R + : Φ t − l o ( y ) ( y ) ∈ G + ( o, U ) } . whic h is clearly p ositiv e and ﬁnite. Since G + ( o, U ) do es not contain future timelike cut p oints of o , w e eviden tly ha ve b ≤ s + o ◦ exp − 1 o , where the o -future cut function on the right-hand side is from ( 4.1 ) . In particular, w e see Φ •− l o ( y ) ( y ) is maximizing on all of [0 , b y ) . By assumption on m , its densit y d m / dv ol g is smooth. Then the induced w eighted Jacobian J m Φ • : W → (0 , ∞ ) is given by J m Φ s ( y ) := d m dv ol g ( y ) − 1 h d m dv ol g ◦ Φ s ( y ) i   det DΦ s ( y )   , (4.5) where the last factor denotes the Jacobian of the map Φ s : Σ r → Σ r + s with resp ect to the Riemannian volume measures induced by g on Σ r and Σ r + s . With these preparations, the subsequent area form ula is standard. W e refer to T reude [ 79 , §§1.3, A.2] for comprehensive pro ofs. The formulas provided therein directly generalize to the weigh ted case by mo difying the test function. COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 29 Prop osition 4.5 (Area formula) . L et ψ : G + ( o, U ) → R b e c ontinuous and c om- p actly supp orte d. Then we have Z G + ( o,U ) ψ d m = Z Σ r Z [0 ,b y ) ψ ◦ Φ t − r ( y ) J m Φ t − r ( y ) d t d µ r ( y ) . 4.2.3. Smo oth CD disinte gr ation. No w we inv ok e curv ature properties along the individual rays given by Prop osition 4.5 . Recall a map F : X → Y b et w een metrizable spaces X and Y is called universal ly me asur able if it is µ -measurable for every Borel probability measure µ on X . Let M ( M ) denote the space of ﬁnite Borel measures on M , which b ecomes a metric space when endow ed with the total v ariation distance. Theorem 4.6 (Smo oth CD disin tegration of m ) . L et ( M , g , m ) b e a glob al ly hy- p erb olic weighte d sp ac etime. Given a p oint o ∈ M and an op en Riemannian b al l U ⊂ I + ( o ) with c omp act inclusion, let G + ( o, U ) ⊂ I + ( o ) ∪ { o } b e deﬁne d by ( 4.3 ) . L et N ∈ [ dim M , ∞ ) and assume ther e is a c ontinuous function k : G + ( o, U ) → R with Ric g , m ,N ( ˙ γ , ˙ γ ) ≥ ( k ◦ γ ) g ( ˙ γ , ˙ γ ) on [0 , 1] for every timelike aﬃnely p ar ametrize d maximizing ge o desic γ : [0 , 1] → M starting in o and interse cting U . Then ther e ar e • a sp ac elike hyp ersurfac e Q ⊂ G + ( o, U ) , • a pr ob ability me asur e q on Q , and • a c ontinuous map m • : Q → M ( M ) satisfying the fol lowing pr op erties. (i) W e have m G + ( o, U ) = Z Q m α d q ( α ) . (ii) F or q -a.e. α ∈ Q , the c onditional me asur e m α is c onc entr ate d on the timelike pr op er time p ar ametrize d maximizing ge o desic Φ •− l o ( α ) ( α )   [0 ,b α ) . (iii) F or q -a.e. α ∈ Q , the c onditional me asur e m α is absolutely c ontinuous with r esp e ct to the one-dimensional L eb esgue me asur e on M α . L etting f α b e its R adon–Niko dým density, f α ◦ Φ •− l o ( α ) ( α ) is a smo oth CD ( k ◦ Φ •− l o ( α ) ( α ) , N ) density on (0 , b α ) in the sense of Deﬁnition 3.5 . Pr o of. W e fo cus on the construction and only sketc h the pro of. Given r ∈ (0 , ϑ ) as ab o v e, we set Q := Σ r from ( 4.4 ) and q := µ r [Σ r ] − 1 µ r . Moreov er, for α ∈ Q w e deﬁne m α as the push-forward of the measure J m , •− r ( α ) L 1 [0 , b α ) under the map Φ •− r ( α ) . By Prop osition 4.5 , this directly yields the claimed disin tegration form ula and the second statement. The last statement follows from standard Jacobi ﬁeld computations, cf. T reude [ 79 , §1.4] and T reude–Grant [ 80 , §3], in combination with Remark 3.6 . Alternatively , this follows by arguing as in the pro ofs of Cav alletti– Mondino [ 25 , Thm. 3.2] and Braun–McCann [ 14 , Thm. 6.37], observing that the am bient timelik e Ricci curv ature b ounds hypothesized therein are only required in the directions tangential to the rays (which corresp ond to the ﬂow tra jectories of Φ in our situation). □ Eviden tly , by multiplying the measures in question with suitable constants, this yields a disintegration theorem for the probability measure n := m [ G + ( o, U )] − 1 m G + ( o, U ) . Recall P ( M ) is the space of Borel probability measures on M , endow ed with the narro w top ology (which is metrizable, cf. Ambrosio–Gigli–Sa v aré [ 1 , Rem. 5.1.1]). Corollary 4.7 (Smo oth CD disintegration of n ) . W e r etain the hyp otheses and the notation fr om The or em 4.6 . Then ther e exist • a sp ac elike hyp ersurfac e Q ⊂ G + ( o, U ) , 30 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL • a pr ob ability me asur e q on Q , and • a c ontinuous map n • : Q → P ( M ) satisfying the fol lowing pr op erties. (i) W e have n = Z Q n α d q ( α ) . (ii) F or q -a.e. α ∈ Q , the c onditional me asur e n α is c onc entr ate d on the timelike pr op er time p ar ametrize d maximizing ge o desic Φ •− l o ( α ) ( α )   [0 ,b α ) . (iii) F or q -a.e. α ∈ Q , the c onditional me asur e n α is absolutely c ontinuous with r esp e ct to the one-dimensional L eb esgue me asur e on M α . L etting h α b e its R adon–Niko dým density, h α ◦ Φ •− l o ( α ) ( α ) is a smo oth CD ( k ◦ Φ •− l o ( α ) ( α ) , N ) density on (0 , b α ) in the sense of Deﬁnition 3.5 . W e will call a triple ( Q, q , n • ) as given b y the previous corollary smo oth CD ( k , N ) disinte gr ation of n . 4.3. Displacemen t conv exity with defect. F or the pro of of our main technical result, Theorem 4.10 , ﬁner structural prop erties of a certain q -geo desic µ : [0 , 1] → M through a globally hyperb olic weigh ted spacetime ( M , g , m ) , where q ∈ (0 , 1) , are needed and recalled now. They were obtained by McCann [ 58 ] and Braun–Ohta [ 15 ], whose results even hold in the Lorentz–Finsler framew ork. Moreo ver, it will b e conv enien t to represent µ as push-forw ard of an intermediate measure µ t , where t ∈ (0 , 1) , instead of µ 0 as done in Theorem 3.19 . Let t ∈ (0 , 1) . Assume µ 0 , µ 1 ∈ P ( M ) are compactly supp orted, m -absolutely con tin uous, and satisfy spt µ 0 × spt µ 1 ⊂ I . Then the pair ( µ 0 , µ 1 ) is q -separated in the sense of McCann [ 58 , Def. 4.1] by [ 58 , Lem. 4.4], whic h (unlik e the chronology h yp othesis on spt µ 0 and spt µ 1 ) propagates to the interior of the unique q -geo desic µ : [0 , 1] → P ( M ) connecting µ 0 and µ 1 from Theorem 3.19 [ 58 , Prop. 5.5]. Let Ψ b e the transp ort map from Theorem 3.19 . By this theorem and Remark 3.20 , the map Φ t : [ 0 , 1] × spt µ t → M deﬁned by Φ t r := Ψ r ◦ Ψ − 1 t (4.6) is well-deﬁned; in particular, Φ t t = Id   spt µ t . (4.7) Moreo ver, deﬁning G t : spt µ t → C 0 ([0 , 1]; M ) by G t ( x ) r := Φ t r ( x ) , (4.8) w e see ( G t ) ♯ µ t equals the unique ℓ q -optimal dynamical coupling π of µ 0 and µ 1 . On the other hand, there exist both an op en neighborho o d U ⊂ M of spt µ t and a lo cally semiconv ex (hence lo cally Lipsc hitz con tin uous) Kan torovic h p oten tial φ t : U → R for µ t and µ 1 . It can be computed from φ using the Hopf–Lax ev olution, cf. McCann [ 58 , Rem. 5.6]. Again by uniqueness, for every r ∈ [ 0 , 1] we hav e exp • ( − ( r − t ) |∇ φ t | q ′ − 2 ∇ φ t ) = Φ t r µ t -a.e. An explicit relation b et w een the Radon–Nikodým densities of µ t and µ 1 with resp ect to m , coming from the change of v ariables formula and ultimately leading to the Jacobi ﬁeld computations linking optimal transp ort and timelik e Ricci curv ature, is the Monge–Ampère equation. T o this aim, giv en a p oin t x ∈ spt µ t let | det DΦ t ( x ) | denote the Jacobian of Φ t ( x ) asso ciated with v ol g along the timelike aﬃnely parametrized maximizing geo desic G t ( x ) ; here, D is understo od as the COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 31 appro ximate deriv ative from McCann [ 58 , Def. 3.8]. Reminiscent of ( 4.5 ) , let us deﬁne the corresp onding Jacobian with resp ect to m b y J m Φ t r ( x ) := d m dv ol g ( x ) − 1 h d m dv ol g ◦ Φ t r ( x ) i   det DΦ t r ( x )   . Prop osition 4.8 (Monge–Amp ère iden tity [ 58 , Cor. 5.11]) . W e have d µ t d m = J m Φ t 1 d µ 1 d m ◦ Φ t 1 µ t -a.e. Moreo ver, the computations from Step 4 in the pro of of Braun–Ohta [ 15 , Thm. 5.9] using an orthonormal frame around eac h point of the transport geo desic in question show given r ∈ [0 , 1] and x ∈ spt µ t , the ab o v e transp ort Jacobian from µ t to µ r admits a factorization of the form J m Φ t r ( x ) = L t r ( x )   det B t r ( x )   d m dv ol g ( x ) − 1 d m dv ol g ◦ Φ t r ( x ) , where B t r ( x ) is the matrix representation of the appro ximate deriv ative DΦ t r on the orthogonal complement of the tangent vector at G t ( x ) r relativ e to g and L t r ( x ) := J m Φ t r ( x ) Y t r ( x ) , describ es the distortion in the tangential direction at G t ( x ) r , where Y t r ( x ) :=   det B t r ( x )   d m dv ol g ( x ) − 1 d m dv ol g ◦ Φ t r ( x ) . (4.9) Note that by ( 4.7 ) and with the ab o v e normalization, w e ha ve L t t ( x ) = 1 . (4.10) Lemma 4.9 (Concavit y inequalities [ 15 , Thm. 5.9]) . The fol lowing statements hold for every x ∈ spt µ t . (i) The tangential p art L t ( x ) is c onc ave, i.e. for every r ∈ [0 , 1] , L t r ≥ (1 − r ) L t 0 + r L t 1 . (ii) Given a c ontinuous function k : M → R , deﬁne k ± γ : [ 0 , | ˙ γ | ] → R as b efor e Deﬁnition 3.23 , wher e γ := G t ( x ) . In addition, given N ∈ [ dim M , ∞ ) , we assume Ric g , m ,N ( ˙ γ , ˙ γ ) ≥ ( k ◦ γ ) g ( ˙ γ , ˙ γ ) on [0 , 1] . Then the ortho gonal p art Y t ( x ) fr om ( 4.9 ) satisﬁes the fol lowing ine quality for every r ∈ [0 , 1] : Y t r ( x ) 1 / ( N − 1) ≥ σ (1 − r ) k − γ / ( N − 1) ( | ˙ γ | ) Y t 0 ( x ) 1 / ( N − 1) + σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) Y t 1 ( x ) 1 / ( N − 1) . W e are in a p osition to prov e our main technical ingredient. Theorem 4.10 (Displacemen t conv exit y with defect) . A ssume that ( M , g , m ) is a glob al ly hyp erb olic weighte d sp ac etime. W e supp ose µ 0 , µ 1 ∈ P ( M ) ar e c omp actly supp orte d and m -absolutely c ontinuous. L et q ∈ (0 , 1) , K ∈ R , and p ∈ ( N / 2 , ∞ ) . Supp ose ther e exist λ > 0 and η ∈ (0 , π K/ ( N − 1) / 2) with λ ≤ l ◦ ( e 0 , e 1 ) ≤ π K/ ( N − 1) − η π -a.e. , (4.11) wher e π is the unique ℓ q -optimal dynamic al c oupling r epr esenting the q -ge o desic µ : [0 , 1] → P ( M ) given by µ 0 to µ 1 fr om The or em 3.19 and π K/ ( N − 1) is fr om ( 3.2 ) . L et a given c ontinuous function k : M → R and N ∈ [ dim M , ∞ ) have the pr op erty that Ric g , m ,N ( ˙ γ , ˙ γ ) ≥ ( k ◦ γ ) g ( ˙ γ , ˙ γ ) on [0 , 1] for every timelike aﬃnely p ar ametrize d maximizing ge o desic γ : [0 , 1] → M starting in spt µ 0 and terminating in spt µ 1 . 32 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL L astly, let L > 0 b e an upp er b ound on the Lipschitz c onstant of l on spt µ 0 × spt µ 1 , cf. Pr op osition 2.19 . Then for every t ∈ (0 , 1) , S N ( µ t | m ) ≤ − Z τ (1 − t ) K,N ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N d π ( γ ) − Z τ ( t ) K,N ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N d π ( γ ) + 2  Λ K,N ,η Ω 1 / (2 p − 1) K,N ,p,η  1 / N m [ C ] 2( p − 1) / N (2 p − 1) ×  diam ( C , g ) 2 p − q +1 λ q − 1 L  1 / N (2 p − 1) × h Z C   ( k − K ) −   p d m i 1 / N (2 p − 1) , wher e Λ K,N ,η and Ω K,N ,p,η ar e fr om ( 3.3 ) and we set C := J (spt µ 0 , spt µ 1 ) . Pr o of. By Theorem 3.24 , π -a.e. γ satisﬁes d µ t d m ( γ t ) − 1 / N ≥ τ (1 − t ) k − γ ,N ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N + τ ( t ) k + γ ,N ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N . (4.12) Our ob jectiv e will b e to replace the v ariable k distortion co eﬃcien ts by the constan t K distortion coeﬃcients after in tegration against π , up to an explicit error. In the following, we concentrate on replacing the distortion co eﬃcien t τ ( t ) k + γ ,N ( | ˙ γ | ) b y τ ( t ) K,N ( | ˙ γ | ) ; analogously , τ (1 − t ) k − γ ,N ( | ˙ γ | ) is replaced by τ (1 − t ) K,N ( | ˙ γ | ) . By Lemma 3.4 and ( 4.11 ), π -a.e. γ satisﬁes τ ( t ) K,N ( | ˙ γ | ) − τ ( t ) k + γ ,N ( | ˙ γ | ) ≤  Λ K,N ,η Ω 1 / (2 p − 1) K,N ,p,η  1 / N h Z [ t, 1] τ ( r ) k + γ ,N ( | ˙ γ | ) N d r i 2( p − 1) / N (2 p − 1) × h t | ˙ γ | 2 p Z [0 , 1]   ( k ( γ r ) − K ) −   σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) N − 1 d r i 1 / N (2 p − 1) . Multiplying this inequalit y by d µ 1 / d m ( γ 1 ) − 1 / N and applying Jensen’s inequality ﬁrst and Hölder’s inequality second, Z τ ( t ) K,N ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N d π ( γ ) − Z τ ( t ) k + γ ,N ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N d π ( γ ) ≤  Λ K,N ,η Ω 1 / (2 p − 1) K,N ,p,η  1 / N Z d π ( γ ) h d µ 1 d m ( γ 1 ) − 1 / N × h Z [ t, 1] τ ( r ) k + γ ,N ( | ˙ γ | ) N d r i 2( p − 1) / N (2 p − 1) × h t | ˙ γ | 2 p Z [0 , 1]   ( k ( γ r ) − K ) −   σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) N − 1 d r i 1 / N (2 p − 1) i ≤  Λ K,N ,η Ω 1 / (2 p − 1) K,N ,p,η  1 / N h Z d π ( γ ) d µ 1 d m ( γ 1 ) − 1 × h Z [ t, 1] τ ( r ) k + γ ,N ( | ˙ γ | ) N d r i 2( p − 1) / (2 p − 1) × h t | ˙ γ | 2 p Z [0 , 1]   ( k ( γ r ) − K ) −   σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) N − 1 d r i 1 / (2 p − 1) i 1 / N ≤  Λ K,N ,η Ω 1 / (2 p − 1) K,N ,p,η  1 / N h Z Z [ t, 1] τ ( r ) k + γ ,N ( | ˙ γ | ) N d µ 1 d m ( γ 1 ) − 1 d r d π ( γ ) | {z } A i 2( p − 1) / N (2 p − 1) COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 33 × h t Z d π ( γ ) | ˙ γ | 2 p h Z [0 , 1] d r   ( k ( γ r ) − K ) −   p × σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) N − 1 d µ 1 d m ( γ 1 ) − 1 i | {z } B i 1 / N (2 p − 1) . In the sequel, we will estimate the integrals A and B separately . In particular, note carefully the factor t σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) N − 1 app earing in B do es not summarize to the N -th p o w er of a single τ -distortion co eﬃcient. Finding an upp er b ound for A is simple. In deed, using ( 4.12 ) , F ubini’s theorem, and the fact that spt µ r ⊂ C for every r ∈ [ t, 1] , A = Z Z [ t, 1] τ ( r ) k + γ ,N ( | ˙ γ | ) N d µ 1 d m ( γ 1 ) − 1 d r d π ( γ ) ≤ Z [ t, 1] Z d µ r d m ( γ r ) − 1 d π ( γ ) d r = Z [ t, 1] Z M h d µ r d m i − 1 d µ r d r ≤ m [ C ] . In particular, the contribution from A is b ounded. T o ﬁnd an upp er b ound for B , we ﬁrst rewrite the integral with resp ect to π as an in tegral with resp ect to µ t . T o this aim, we recall the transp ort maps Ψ 1 , Ψ t , and Φ t 1 from Theorem 3.19 and ( 4.6 ); in particular, Ψ 1 = Φ t 1 ◦ Ψ t µ 0 -a.e. (4.13) Giv en a p oin t x ∈ spt µ t , let the timelike aﬃnely parametrized maximizing geo desic γ x : [0 , 1] → M b e deﬁned b y γ x := G t ( x ) , where G t ( x ) is from ( 4.8 ) . Com bining the identit y µ 1 = (Ψ 1 ) ♯ µ 0 from Theorem 3.19 , ( 4.13 ), and (Ψ t ) ♯ µ 0 = ( e t ) ♯ π , B = t Z d π ( γ ) | ˙ γ | 2 p h Z [0 , 1] d r   ( k ( γ r ) − K ) −   p × σ ( r ) k + γ / ( N − 1) ( | ˙ γ | ) N − 1 h d µ 1 d m ◦ Φ t 1 i ( γ t ) − 1 i = t Z M d µ t | ˙ γ • | 2 p h Z [0 , 1] d r   ( k ( γ • r ) − K ) −   p × σ ( r ) k + γ • / ( N − 1) ( | ˙ γ • | ) N − 1 h d µ 1 d m ◦ Φ t 1 i − 1 i . Let L t and B t b e as ab ov e; more precisely , they are giv en by decomp osing the ℓ q - optimal transp ort from µ t to µ 1 in to a tangential and an orthogonal part, resp ectiv ely . The Monge–Amp ère-type identit y from Prop osition 4.8 implies B = Z M d µ t | ˙ γ • | 2 p hh Z [0 , 1] d r   ( k ( γ • r ) − K ) −   p × σ ( r ) k + γ • / ( N − 1) ( | ˙ γ • | ) N − 1   det B t 1   d m dv ol g ◦ Φ t 1 i × t L t 1 h d m dv ol g i − 1 h d µ t d m i − 1 i . Noting that Φ t 1 ( x ) = γ x 1 for µ t -a.e. x ∈ M by ( 4.6 ) yields B ≤ Z M d µ t | ˙ γ • | 2 p hh Z [0 , 1]   ( k ( γ • r ) − K ) −   p   det B t r   d m dv ol g ( γ • r ) d r i 34 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL × L t t h d m dv ol g i − 1 h d µ t d m i − 1 i = Z M h Z [0 , 1]   ( k ( γ • r ) − K ) −   p   det B t r   d m dv ol g ( γ • r ) d r i | ˙ γ • | 2 p dv ol g | {z } B ′ ; here, we used the relations σ ( r ) k • γ / ( N − 1) ( | ˙ γ | ) N − 1   det B t 1   d m dv ol g ◦ Φ t 1 ≤   det B t r   d m dv ol g ( γ • r ) , t L t, 1 1 ≤ L t, 1 t = 1 implied by Lemma 4.9 and the normalization ( 4.10 ). A standard application of the coarea form ula, cf. e.g. Ketterer [ 47 , Lem. 5.2], no w allo ws us to disinte grate v ol g along the lev el sets of the Kantoro vic h p oten tial φ t for the ℓ q -optimal transp ort from µ t to µ 1 ; more precisely , there exist b oth an L 1 -measurable set Q ⊂ R con tained in the image of φ t, 1 and, for L 1 -a.e. a ∈ Q , an H dim M − 1 -measurable subset Σ a ⊂ { φ t = a } such that B ′ = Z Q Z Σ a h Z [0 , 1]   ( k ( γ • r ) − K ) −   p   det B t r   d m dv ol g ( γ • r ) d r i | ˙ γ • | 2 p |∇ φ t | d H dim M − 1 d a = Z Q d a Z Σ a d H dim M − 1 | ˙ γ • | 2 p − q h Z [0 , 1] d r   ( k ( γ • r ) − K ) −   p ×   det B t r   d m dv ol g ( γ • r ) | ˙ γ • | i ≤ diam ( C, g ) 2 p − q Z Q d a Z Σ a d H dim M − 1 h Z [0 , 1] d r   ( k ( γ • r ) − K ) −   p ×   det B t r   d m dv ol g ( γ • r ) | ˙ γ • | i ; here, H dim M − 1 designates the ( dim M − 1) -dimensional Hausdorﬀ measure and w e ha v e used the identit y | ˙ γ • | = |∇ φ t | q ′ − 1 µ t -a.e. implied by Theorem 3.19 in the second equality . W e abbreviate the iterated in tegral from the last line ab o v e by B ′′ . No w a standard application of the area form ula, cf. e.g. Ketterer [ 47 , Prop. 5.6], sho ws that L 1 -a.e. a ∈ Q satisﬁes Z [0 , 1] Z Σ a   ( k ( γ • r ) − K ) −   p   det B t r   | ˙ γ • | d m dv ol g ( γ • r ) d H dim M − 1 d r ≤ Z C   ( k − K ) −   p d m , and consequently B ′′ ≤ L 1 [ Q ] Z C   ( k − K ) −   p d m . It remains to ﬁnd an upp er b ound for L 1 [ Q ] . By construction of Q , L 1 [ Q ] ≤ L 1  φ t (spt µ t )  . On the other hand, using µ is a q -geo desic that connects its endp oin ts µ 0 and µ 1 whic h satisfy spt µ 0 × spt µ 1 ⊂ I , the results of McCann [ 58 , Thm. 4.3, Lem. 4.4, Prop. 5.5] imply that the Lipsc hitz constan t of φ t on spt µ t is no larger than the Lipsc hitz constan t of l q /q on spt µ 0 × spt µ 1 . In turn, b y ( 4.11 ) the latter is no larger than λ q − 1 times the Lipschitz constan t of l on spt µ 0 × spt µ 1 , where we used COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 35 q ∈ (0 , 1) . Since L 1 [ φ t (spt µ t )] is no larger than the oscillation of φ t on spt µ t , by our choice of L this easily implies L 1  φ t (spt µ t )  ≤ diam ( C, g ) λ q − 1 L. This concludes the pro of. □ 5. Sharp timelike Bonnet–Myers inequality W e now prov e our ﬁrst main result. Besides the go od approximation of Lipschitz con tinuous metric tensors from § 2.2 and the lo calization paradigm from § 4.2 , our argumen t relies on the qualitative diameter estimate from Corollary 3.9 . Theorem 5.1 (Sharp timelike Bonnet–My ers diameter estimate) . L et K > 0 and N ∈ [ dim M , ∞ ) . A ssume that ( M , g , m ) is a glob al ly hyp erb olic weighte d Lipschitz sp ac etime such that Ric g , m ,N ≥ K timelike distributional ly. Then diam ( M , g ) ≤ π r N − 1 K . Pr o of. Assume to the contrary there exist ε > 0 and o, x ∈ M with l o ( x ) ≥ π r N − 1 K + 4 ε. Since l + is contin uous, there is an op en Riemannian ball U ⊂ M of x with inf { l g ,o ( y ) : y ∈ U } ≥ π r N − 1 K + 3 ε. Let ( g n ) n ∈ N and ( m n ) n ∈ N b e ﬁxed go od approximations of g and m on the compact set C := J g ( o, cl U ) , resp ectively . Recall the induced weigh ted spacetime ( M , g n , m n ) is globally hyperb olic for ev ery n ∈ N . By Lemma 2.14 , we ma y and will assume there exists n 1 ∈ N such that for every n ∈ N with n ≥ n 1 , inf { l g n ,o ( y ) : y ∈ U } ≥ π r N − 1 K + 2 ε. (5.1) In the sequel, for n as ab o v e w e abbreviate G n := G + g n ( o, U ) , where the right-hand side is from ( 4.3 ) , deﬁned with resp ect to g n . By Theorem 4.3 and narrow conv ergence of ( m n ) n ∈ N to m , liminf n →∞ m n [ G n ] ≥ liminf n →∞ m n [ U \ TC + g n ( o )] = liminf n →∞ m n [ U ] = m [ U ] > 0 . On the other hand, since G n ⊂ J g ( o, cl U ) for ev ery n ∈ N , limsup n →∞ m n [ G n ] ≤ limsup n →∞ m n [ C ] = m [ C ] < ∞ . In particular, we may and will assume ( m n [ G n ]) n ∈ N con verges in (0 , ∞ ) , up to a nonrelab eled subsequence. In turn, there exist constan ts w > 0 and n 2 ∈ N with n 2 ≥ n 1 suc h that for every n ∈ N with n ≥ n 2 , m n [ G n ] ≥ w . (5.2) Up to increasing the v alue of n 2 , we deﬁne k V ,g n , m n ,N : M → R b y ( 2.4 ) , where V ⊂ T +   C is from Lemma 2.17 , for ev ery n ∈ N with n ≥ n 2 . F or p ∈ ( N / 2 , ∞ ) , let δ K,N ,p,ε > 0 b e as in Corollary 3.9 . Moreo ver, giv en η ∈ (0 , 1) , by Lemma 2.16 there exist δ > 0 such that √ δ ≤ min { δ K,N ,p,ε , η } (5.3) 36 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL and n 3 ∈ N with n 3 ≥ n 2 suc h that for every n ∈ N with n ≥ n 3 , Z G n   ( k V ,g n , m n ,N − K ) −   p d m n ≤ w δ. (5.4) F or every such n , deﬁning n n := m n [ G n ] − 1 m n G n , the relations ( 5.4 ) and ( 5.2 ) imply Z G n   ( k V ,g n , m n ,N − K ) −   p d n n ≤ δ. (5.5) Giv en n as ab o ve, Lemma 2.17 combines with Corollary 4.7 to ensure there is a smo oth CD ( k V ,g n , m n ,N , N ) disintegration ( Q n , q n , n n, • ) of n n . W e now use this to ol to seek the desired contradiction. First, note that by construction of G n and the disin tegration, the ray M n,α in tersects U for q n -a.e. α ∈ Q n . On the other hand, let R n b e the set of all α ∈ Q n suc h that Z M n,α   ( k V ,g n , m n ,N − K ) −   p d n n,α ≤ √ δ . By Marko v’s inequality , the disintegration Corollary 4.7 , ( 5.5 ), and ( 5.3 ), q n [ Q n \ R n ] ≤ 1 √ δ Z Q n Z M n,α   ( k V ,g n , m n ,N − K ) −   p d n n,α d q n ( α ) = 1 √ δ Z G n   ( k V ,g n , m n ,N − K ) −   p d n n ≤ η . Therefore, b y ( 5.3 ) again together with Corollary 3.9 , the diameter of q n -a.e. ray passing through the set R n (of q n -measure at least 1 − η ) is b ounded from ab o ve b y π p ( N − 1) /K + ε . On the other hand, as q n -a.e. ra y intersects U b y construction, w e see the diameter of q n -a.e. ray is b ounded from b elo w b y π p ( N − 1) /K + 2 ε b y ( 5.1 ) . As the in tersection of the tw o sets in question has q n -measure at least 1 − η > 0 , hence is nonempty , we obtain the sought contradiction. □ 6. Main resul ts and applica tions 6.1. F rom analytic to syn thetic. Theorem 6.1 (Timelik e measure contraction prop ert y) . L et ( M , g , m ) b e a glob al ly hyp erb olic weighte d Lipschitz sp ac etime. A ssume K ∈ R and N ∈ [ dim M , ∞ ) satisfy Ric g , m ,N ≥ K timelike distributional ly. Then ( M , g , m ) ob eys TMCP ( K , N ) . In fact, the main w ork in showing Theorem 6.1 will b e inv ested in the pro of of the following somewhat stronger statement. Prop osition 6.2 (Displacement semiconv exity under chronological supp ort) . W e r etain the hyp otheses and the notation fr om The or em 6.1 . F or every q ∈ (0 , 1) and every c omp actly supp orte d, m -absolutely c ontinuous me asur es µ 0 , µ 1 ∈ P ( M ) with spt µ 0 × spt µ 1 ⊂ I , ther e exist • a q -ge o desic µ : [0 , 1] → P ( M ) fr om µ 0 to µ 1 and • an ℓ q -optimal dynamic al c oupling π of µ 0 and µ 1 such that for every N ′ ∈ [ N , ∞ ) and every t ∈ [0 , 1] , S N ′ ( µ t | m ) ≤ − Z τ (1 − t ) K,N ′ ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N ′ d π ( γ ) − Z τ ( t ) K,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ d π ( γ ) . COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 37 Pr o of. Let ( g n ) n ∈ N and ( m n ) n ∈ N b e ﬁxed go od approximations of g and m on the compact set C := J g ( spt µ 0 , spt µ 1 ) as in Deﬁnition 2.13 . Note that by construction, for every i ∈ { 0 , 1 } and every n ∈ N we hav e d µ i d m n = a n d µ i d m , (6.1) where ( a n ) n ∈ N is the sequence of contin uous functions a n := d m / d m n con verging uniformly to one on C ; in particular, µ 0 and µ 1 are m n -absolutely contin uous and compactly supported. By Lemma 2.17 , we may and will also assume spt µ 0 × spt µ 1 ⊂ I g n without restriction. Consider the corresp onding sequence ( π n ) n ∈ N , where π n is the unique ℓ g n ,q -optimal dynamical coupling of µ 0 and µ 1 pro vided b y Theorem 3.19 . By Proposition 3.21 , we may and will assume that it con v erges narro wly to an ℓ g ,q -optimal dynamical coupling π of µ 0 and µ 1 , up to a nonrelab eled subsequence. In the sequel, given t ∈ [0 , 1] we write µ t := ( e t ) ♯ π and µ n t := ( e t ) ♯ π n , where n ∈ N . The idea now is to take the limit of the inequalit y from Theorem 4.10 along this conv erging sequence, which requires some further preparations. W e will only address the case when K is p ositiv e; the other cases are cov ered analogously (with simpler argumen ts, since w e do not ha v e to take into accoun t diameter b ounds in this complementary situation). By Lemma 2.17 , we may and will assume the existence of λ > 0 such that for every n ∈ N , l g n ◦ ( e 0 , e 1 ) ≥ λ π n -a.e. On the other hand, let us ﬁx K ′ ∈ (0 , K ) . By Theorem 5.1 and since Ric g , m ,N ≥ K timelik e distributionally , there exists η ∈ (0 , π K/ ( N − 1) / 2) such that diam ( J g (spt µ 0 , spt µ 1 ) , g ) ≤ π r N − 1 K ′ − 2 η . By Lemma 2.14 , we may and will assume every n ∈ N satisﬁes diam ( J g n (spt µ 0 , spt µ 1 ) , g n ) ≤ π r N − 1 K ′ − η , whic h in turn implies l g n ◦ ( e 0 , e 1 ) ≤ π r N − 1 K ′ − η π n -a.e. Lastly , given n ∈ N let V b e the set of tangen t vectors from Lemma 2.17 ; giv en n ∈ N , let k V ,g n , m n ,N ′ b e the contin uous function from ( 2.4 ) , where N ′ ∈ [ N , ∞ ) is ﬁxed. By Lemma 2.17 again, we may and will assume for every n ∈ N , the Bakry– Émery –Ricci tensor Ric g n , m n ,N ′ is b ounded from b elow b y k V ,g n , m n ,N ′ along every timelik e aﬃnely parametrized maximizing g n -geo desic from spt µ 0 to spt µ 1 . Lastly , let L > 0 b e a uniform upp er b ound the Lipschitz constant of l g n on spt µ 0 × spt µ 1 for ev ery n ∈ N , as giv en by Prop osition 2.19 . Thus, given p ∈ ( N ′ / 2 , ∞ ) and t ∈ (0 , 1) , these considerations combine with Theorem 4.10 (and the dimensional consistency of its hypotheses) to imply S N ′ ( µ n t | m n ) ≤ − Z τ (1 − t ) K ′ ,N ′ ( | ˙ γ | ) d µ 0 d m n ( γ 0 ) − 1 / N ′ d π n ( γ ) − Z τ ( t ) K ′ ,N ′ ( | ˙ γ | ) d µ 1 d m n ( γ 1 ) − 1 / N ′ d π n ( γ ) + 2  Λ K ′ ,N ′ ,η Ω 1 / (2 p − 1) K ′ ,N ′ ,p,η  1 / N ′ m n [ C ] 2( p − 1) / N ′ (2 p − 1) ×  diam ( C , g n ) 2 p − q +1 λ q − 1 L  1 / N (2 p − 1) × h Z C   ( k V ,g n , m n ,N ′ − K ) −   p d m n i 1 / N ′ (2 p − 1) , 38 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL where Λ K ′ ,N ′ ,η and Ω K ′ ,N ′ ,p,η are from ( 3.3 ) and we write C := J g (spt µ 0 , spt µ 1 ) ; moreo ver, here we hav e employ ed the inclusion J g n (spt µ 0 , spt µ 1 ) ⊂ C implied by g n ≺ g for every n ∈ N . No w w e send n → ∞ . First, given t ∈ (0 , 1) , Lemma 3.22 implies S N ′ ( µ t | m ) ≤ liminf n →∞ S N ′ ( µ n t | m n ) . Second, ( 6.1 ) and the pro of of Sturm [ 78 , Lem. 3.3] easily yield Z τ (1 − t ) K ′ ,N ′ ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N ′ d π ( γ ) ≤ liminf n →∞ Z τ (1 − t ) K ′ ,N ′ ( | ˙ γ | ) d µ 0 d m n ( γ 0 ) − 1 / N ′ d π n ( γ ) , Z τ ( t ) K ′ ,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ d π ( γ ) ≤ liminf n →∞ Z τ ( t ) K ′ ,N ′ ( | ˙ γ | ) d µ 1 d m n ( γ 1 ) − 1 / N ′ d π n ( γ ) . Third, since the sequence ( a n ) n ∈ N from ( 6.1 ) is uniformly b ounded on C , so is ( m n [ C ]) n ∈ N . F ourth, b y Lemma 2.14 the sequence ( diam ( C, g n )) n ∈ N is b ounded. Fifth, using Lemma 2.16 we obtain lim n →∞ Z C   ( k V ,g n , m n ,N ′ − K ) −   p d m n = 0 . In summary , this entails S N ′ ( µ t | m ) ≤ − Z τ (1 − t ) K ′ ,N ′ ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N ′ d π ( γ ) − Z τ ( t ) K ′ ,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ d π ( γ ) . Sending K ′ → K and using Levi’s monotone conv ergence theorem with Remark 3.3 , S N ′ ( µ t | m ) ≤ − Z τ (1 − t ) K,N ′ ( | ˙ γ | ) d µ 0 d m ( γ 0 ) − 1 / N ′ d π ( γ ) − Z τ ( t ) K,N ′ ( | ˙ γ | ) d µ 1 d m ( γ 1 ) − 1 / N ′ d π ( γ ) , whic h is the desired inequality . □ Pr o of of The or em 6.1 . The result in question follows by combining Prop osition 6.2 and [ 8 , Prop. 4.9]. Indeed, while the latter assumes the so-called “weak timelike curv ature-dimension condition”, its pro of only uses displacement semicon v exit y of the Rényi entrop y betw een mass distributions satisfying the hypotheses from Prop osition 6.2 . The idea is to shrink one of the m -absolutely contin uous masses from the ab o ve prop osition to a Dirac mass and using similar stabilit y prop erties at the level of geo desics of probability measures as in the ab o v e pro of. □ 6.2. Applications of Theorem 6.1 . As well-kno wn, the TMCP established in Theorem 6.1 (and the prop ert y from Proposition 6.2 ) has tw o standard consequences: the timelike Brunn–Minko wski inequality as w ell as the timelike Bishop–Gromo v inequalit y . (The third usual consequence, the timelike Bonnet–Myers inequality , w as shown in Theorem 5.1 and used to pro v e Theorem 6.1 .) W e refer to Cav alletti– Mondino [ 24 , Props. 3.4, 3.5] and Braun [ 8 , Prop. 3.11, Thm. 3.16] for the pro ofs. In the follo wing, giv en X 0 , X 1 ⊂ M , let G ( X 0 , X 1 ) denote the set of all timelik e aﬃnely parametrized maximizing geo desics that start in X 0 and terminate in X 1 . Moreo ver, recall the deﬁnition of the ev aluation map e t from § 3.2.2 . COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 39 Theorem 6.3 (Timelik e Brunn–Mink o wski inequality) . L et ( M , g , m ) b e a glob al ly hyp erb olic weighte d Lipschitz sp ac etime. A ssume that K ∈ R and N ∈ [ dim M , ∞ ) satisfy Ric g , m ,N ≥ K timelike distributional ly. Then the fol lowing claims hold. (i) F or every o ∈ M , every pr e c omp act Bor el set X 1 ⊂ M with { o } × X 1 ⊂ I , and every t ∈ [0 , 1] , m ∗ [ e t ( G ( { o } , X 1 ))] 1 / N ≥ inf { τ ( t ) K,N ( | ˙ γ | ) : γ ∈ G ( { o } , X 1 ) } m [ X 1 ] 1 / N , wher e m ∗ denotes the outer me asur e induc e d by m . (ii) F or every pr e c omp act Bor el sets X 0 , X 1 ⊂ M such that cl X 0 × cl X 1 ⊂ I and every t ∈ [0 , 1] , m ∗ [ e t ( G ( X 0 , X 1 ))] 1 / N ≥ inf { τ (1 − t ) K,N ( | ˙ γ | ) : γ ∈ G ( X 0 , X 1 ) } m [ X 0 ] 1 / N + inf { τ ( t ) K,N ( | ˙ γ | ) : γ ∈ G ( X 0 , X 1 ) } m [ X 1 ] 1 / N . T o form ulate the timelike Bishop–Gromo v inequality , let o ∈ M . Given r > 0 , let B g ( o, r ) := { y ∈ I + ( o ) : l o ( y ) < r } ∪ { o } b e the future r -“ball” centered at o , which typically has inﬁnite m -measure. Thus, w e ﬁx a compact set E ⊂ I + ( o ) ∪ { o } that is star-shap e d with resp ect to o ; that is, ev ery timelik e aﬃnely parametrized maximizing geo desic γ : [0 , 1] → M that starts in o and ends in E do es not leav e E . W e deﬁne the induced v olume function v : R + → R + and the induced area function s : R + → R + ∪ {∞} by v ( r ) := m [ E ∩ cl B g ( o, r )] , s ( r ) := limsup δ → 0 + v ( r + δ ) − v ( r ) δ = limsup δ → 0 + 1 δ m [ E ∩ ((cl B g ( o, r + δ )) \ B g ( o, r ))] . Moreo ver, giv en κ ∈ R w e recall the generalized sine function sin κ from ( 3.1 ) and its ﬁrst p ositive ro ot π κ from ( 3.2 ), resp ectiv ely . Theorem 6.4 (Timelike Bishop–Gromov inequalit y) . L et ( M , g , m ) b e a glob al ly hyp erb olic weighte d Lipschitz sp ac etime satisfying Ric g , m ,N ≥ K timelike distribu- tional ly for some K ∈ R and N ∈ [ dim M , ∞ ) . L et o ∈ M and let E ⊂ I + ( o ) ∪ { o } b e c omp act and star-shap e d with r esp e ct to o . Then for every r , R ∈ (0 , π K/ ( N − 1) ) such that r < R , s ( r ) s ( R ) ≥ sin K/ ( N − 1) ( r ) N − 1 sin K/ ( N − 1) ( R ) N − 1 , v ( r ) v ( R ) ≥ Z [0 ,r ] sin K/ ( N − 1) ( t ) N − 1 d t Z [0 ,R ] sin K/ ( N − 1) ( t ) N − 1 d t . Our ﬁnal application of Theorem 6.1 are tw o d’Alembert comparison theorems, one for distance functions from p oints and one for appropriate p o w ers. In a muc h less smo oth framew ork, these were established b y Beran et al. [ 6 ]. How ever, the compatibilit y of the abstract ﬁrst-order quan tities they introduce with the classical notions on our given Lipschitz spacetime is unclear (cf. [ 6 , §A.1] for the smo oth framew ork). T o av oid in terrupting the ﬂow of ideas, w e defer the self-contained 40 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL (and more streamlined) pro ofs of Theorems 6.5 and 6.6 , based on the arguments of Beran et al. [ 6 ], to § B . T o formulate the comparison theorems, whose pro of is analogous to Beran et al. [ 6 , Thm. 5.24], given K ∈ R , N ∈ (1 , ∞ ) , and θ ∈ [0 , π K/ ( N − 1) ) we set ˜ τ K,N ( θ ) := d d t     1 τ ( t ) K,N ( θ ) =            1 N + θ N p K ( N − 1) cot h θ r K N − 1 i if K > 0 , 1 if K = 0 , 1 N + θ N p − K ( N − 1) coth h θ r − K N − 1 i otherwise . Moreo ver, in view of Theorem 5.1 , given o ∈ M w e deﬁne the op en set I + K,N ( o ) := I + ( o ) ∩ { l o < π K/ ( N − 1) } . (6.2) Theorem 6.5 (D’Alembert comparison theorem for p ow ers of Lorentz distance functions) . A ssume ( M , g , m ) is a glob al ly hyp erb olic weighte d Lipschitz sp ac etime. Supp ose that K ∈ R and N ∈ [ dim M , ∞ ) ob ey Ric g , m ,N ≥ K timelike distribution- al ly. L et q ∈ (0 , 1) and let q ′ < 0 b e its c onjugate exp onent. Then for every o ∈ M and every Lipschitz c ontinuous function ϕ : I + K,N ( o ) → R + with c omp act supp ort, − Z M d ϕ h ∇ l q o q i    ∇ l q o q    q ′ − 2 d m ≤ N Z M ˜ τ K,N ◦ l o ϕ d m . Theorem 6.6 (D’Alem b ert comparison theorem for Lorentz distance functions) . A ssume ( M , g , m ) is a glob al ly hyp erb olic weighte d Lipschitz sp ac etime. Supp ose that K ∈ R and N ∈ [ dim M , ∞ ) satisfy Ric g , m ,N ≥ K timelike distributional ly. L et us ﬁx q ′ < 0 . Then for every o ∈ M and every Lipschitz c ontinuous function ϕ : I + K,N ( o ) → R + with c omp act supp ort, − Z M d ϕ ( ∇ l o ) |∇ l o | q ′ − 2 d m ≤ Z M ϕ N ˜ τ K,N ◦ l o − 1 l o d m . In fact, like the right-hand sides, the left-hand sides in the ab o v e conclusions are indep enden t of q ′ b y Corollary 2.20 (and thus hold for switched signs of q and q ′ , resp ectiv ely). Nevertheless, making this contribution visible has made surprising elliptic metho ds accessible in Loren tzian geometry , as realized by Beran et al. [ 6 ]; w e refer to the reviews of McCann [ 59 ] and Braun [ 9 ]. As in Beran et al. [ 6 ], whose results w ere reﬁned by Braun [ 10 ], these tw o theorems imply the Lorentz distance functions in question (and their appropriate p ow ers) admit a distributional d’Alembertian in the sense of Deﬁnition 6.7 . Corollaries 6.8 and 6.9 b elo w follow from [ 6 , Prop. 5.31]; the basic idea is to apply Riesz–Marko v– Kakutani’s representation theorem, using Theorems 6.5 and 6.6 to generate the needed nonnegative functionals. F or an op en set U ⊂ M , let Lip c ( U ) denote the set of all Lipsc hitz contin uous functions ϕ : U → R with compact supp ort. A Radon functional on U is a linear map T : U → R that is contin uous with resp ect to the top ology induced by uniform con vergence on compact subsets of U . W e will call such a Radon functional T nonnegativ e if T ( ϕ ) ≥ 0 whenev er its argumen t ϕ ∈ Lip c ( U ) is nonnegativ e. Recall b y the Riesz–Mark ov–Kakutani represen tation theorem, ev ery nonnegativ e Radon functional is given by (integration against) a Radon measure. The diﬀerence of tw o Radon measures alwa ys makes sense as a Radon functional, but in general not as a signed Radon measure (as b oth p ositiv e and negative parts may b e inﬁnite, which prev ents the resulting ob ject from b eing σ -additiv e). COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 41 Deﬁnition 6.7 (Distributional d’Alembertian) . Given q ′ < 0 , we wil l say a lo c al ly Lipschitz c ontinuous function u : U → R with m U -a.e. timelike gr adient lies in the domain of the distributional q ′ -d’Alem b ertian , symb olic al ly u ∈ D ( □ g , m ,q ′ U ) , if ther e is a R adon functional T : Lip c ( U ) → R such that for every ϕ ∈ Lip c ( U ) , − Z U d ϕ ( ∇ u ) |∇ u | q ′ − 2 d m = T ( ϕ ) . Of course, if a map T as ab o ve exists, it is unique. Corollary 6.8 (Distributional d’Alembertian for p ow ers of Lorentz distance func- tions) . A ssume K ∈ R and N ∈ [ dim M , ∞ ) ar e such that the glob al ly hyp erb olic weighte d Lipschitz sp ac etime ( M , g , m ) satisﬁes Ric g , m ,N ≥ K timelike distribution- al ly. L et q ∈ (0 , 1) with c onjugate exp onent q ′ < 0 . Then for every o ∈ M , we have l q o /q ∈ D ( □ g , m ,q ′ I + K,N ( o )) , wher e the set I + K,N ( o ) is fr om ( 6.2 ) . Mor e pr e cisely, the unique R adon functional T : Lip c ( I + K,N ( o )) → R that c ertiﬁes the pr evious inclusion admits the de c omp osition − T = µ T − ν T , wher e µ T is a R adon me asur e on I + K,N ( o ) and ν T := N ˜ τ K,N ◦ l o m I + K,N ( o ) . In p articular, the singular p art of T with r esp e ct to m , deﬁne d as the m -singular p art of the signe d R adon me asur e − µ T , is nonp ositive. Corollary 6.9 (Distributional d’Alembertian for Lorentz distance functions) . W e assume K ∈ R and N ∈ [ dim M , ∞ ) ar e such that the glob al ly hyp erb olic weighte d Lipschitz sp ac etime ( M , g , m ) satisﬁes Ric g , m ,N ≥ K timelike distributional ly. Then for every o ∈ M and every q ′ < 0 , we have l o ∈ D ( □ g , m ,q ′ I + K,N ( o )) . Mor e pr e cisely, the unique R adon functional S : Lip c ( I + K,N ( o )) → R that c ertiﬁes the pr evious inclusion admits the de c omp osition − S = µ S − ν S , wher e µ S is a R adon me asur e on I + K,N ( o ) and ν S := N ˜ τ K,N ◦ l o − 1 l o m I + K,N ( o ) . In p articular, the singular p art of S with r esp e ct to m , deﬁne d as the m -singular p art of the signe d R adon me asur e − µ S , is nonp ositive. Appendix A. Fur ther sharp timelike diameter estima tes The pro of of Theorem 5.1 indicates a diameter estimate in the st yle of corre- sp onding results for Riemannian manifolds by P etersen–Sprouse [ 70 ] and Aubry [ 3 ]. In this app endix, we conﬁrm its Loren tzian coun terpart an ticipated in [ 70 , pp. 271–272] almost 30 years ago. Let ( M , g , m ) b e a given globally hyperb olic weigh ted spacetime; in particular, w e will only consider smo oth metric tensors and reference measures in this section. Deﬁnition A.1 (Curv ature deﬁcit) . L et K ∈ R and p ∈ [1 , ∞ ) . The L p -deﬁcit of an m -me asur able function k : M → R with r esp e ct to K is deﬁne d by def k,K,p ( M , m ) := sup n 1 m [ W ] Z W   ( k − K ) −   p d m : W ⊂ M nonempty, op en, and pr e c omp act o . 42 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL In the previous deﬁnition, of course def k,K,p ( M , m ) = 0 if and only if k ≥ K m -a.e. (which holds ev erywhere if k is upp er semicon tin uous). The normalization in this deﬁnition ensures the v alue of def k,K,p ( M , m ) do es not c hange if w e scale m b y a positive constan t; the lo calization to ﬁnite m -measure subsets will allow us to include the case when m is inﬁnite. Theorem A.2 (Sharp timelike Aubry diameter estimate) . L et us ﬁx K > 0 , N ∈ [2 , ∞ ) , p ∈ ( N / 2 , ∞ ) , and θ > 0 . Then the c onstant C K,N ,p > 1 fr om The or em 3.8 has the fol lowing pr op erty. L et ( M , g , m ) b e a glob al ly hyp erb olic weighte d sp ac e- time ( M , g , m ) satisfying Ric g , m ,N ≥ k in al l timelike dir e ctions, wher e the function k : M → R is c ontinuous, with def k,K,p ( M , m ) ≤ 1 C 1+ θ K,N ,p . Then we have diam ( M , g ) ≤ π r N − 1 K  1 + C K,N ,p def k,K,p ( M , m ) 1 / 5(1+ θ )  . Pr o of. Let ε > 0 satisfy def k,K,p ( M , m ) ≤ h ε π C K,N ,p r K N − 1 i 5(1+ θ ) ≤ 1 C 1+ θ K,N ,p . (A.1) By the arbitrariness of ε , the claim will b e established if we show diam ( M , g ) ≤ π r N − 1 K + ε. (A.2) Assume ( A.2 ) do es not hold. As in the pro of of Theorem 5.1 , there exist η > 0 , o ∈ M , and an op en Riemannian ball U ⊂ M such that inf { l o ( y ) : y ∈ U } ≥ π r N − 1 K + ε + η . (A.3) Consider the measure n := m [ G ] − 1 m G, where the set G := G + ( o, U ) is from ( 4.3 ) . Since G is a comp etitor in the deﬁnition of def k,K,p ( M , m ) , where precompactness is implied by global hyperb olicit y of ( M , g ) , ( A.1 ) yields Z G   ( k − K ) −   p d n ≤ h ε π C K,N ,p r K N − 1 i 5(1+ θ ) . (A.4) Corollary 4.7 together with the hypothesized lo wer b oundedness of Ric g , m ,N b y k in all timelik e directions ensures there exists a smo oth CD ( k , N ) disin tegration ( Q, q , n • ) of n . By construction of G + ( o, U ) and the disintegration, M α in tersects U for q -a.e. α ∈ Q . On the other hand, let R b e the set of all α ∈ Q with Z M α   ( k − K ) −   p d n α ≤ h ε π C K,N ,p r K N − 1 i 5 . Thanks to the second inequalit y from ( A.1 ) and our choice of C K,N ,p , the righ t-hand side corresp onds to the num ber δ K,N ,p,ε from Remark 3.10 , which lies in (0 , 1) by the second inequalit y from ( A.1 ) since C K,N ,p > 1 . By Mark o v’s inequality , the disin tegration Corollary 4.7 , and ( A.4 ), q [ Q \ R ] ≤ 1 δ K,N ,p,ε Z Q Z M α   ( k − K ) −   p d n α d q ( α ) COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 43 = 1 δ K,N ,p,ε Z G   ( k − K ) −   p d n ≤ δ θ K,N ,p,ε . Th us, by Corollary 3.9 , the diameter of q -a.e. ray passing through the set R (of q -measure at least 1 − δ θ K,N ,p,ε ) is b ounded from ab ov e by π p ( N − 1) /K + ε . On the other hand, as q -a.e. ra y intersects U b y construction, we see the diameter of q - a.e. ra y is b ounded from b elo w b y π p ( N − 1) /K + ε + η b y ( A.3 ) . The in tersection of the tw o sets in question has q -measure b ounded from b elo w by 1 − δ θ K,N ,p,ε > 0 ; in particular, it is nonempty . This is the desired contradiction. □ Corollary A.3 (Timelike Petersen–Sprouse diameter estimate) . L et K > 0 , N ∈ [2 , ∞ ) , and p ∈ ( N / 2 , ∞ ) . Then for every ε > 0 , ther e exists ξ K,N ,p,ε ∈ (0 , δ K,N ,p,ε ) , wher e δ K,N ,p,ε is fr om R emark 3.10 , with the fol lowing pr op erty. L et ( M , g , m ) b e a glob al ly hyp erb olic weighte d sp ac etime and let k : M → R b e a c ontinuous function with Ric g , m ,N ≥ k in al l timelike dir e ctions such that def k,K,p ( M , m ) ≤ ξ K,N ,p,ε . Then we have diam ( M , g ) ≤ π r N − 1 K + ε. Appendix B. Det ails about Theorems 6.5 and 6.6 B.1. Goo d geo desics. One feature that is logically implied by the TMCP as sho wn by Braun [ 7 , Thm. 4.11], later adapted by Beran et al. [ 6 , Thm. 5.12], is the existence of “go od geo desics”, viz. geo desics of probabilit y measures with uniformly b ounded densities. This fact will b e needed in the pro of of Prop osition B.4 b elow. Instead of taking the abstract results from [ 7 , 6 ] as a blackbox, we will instead give a more direct pro of based on the go od appro ximations from Deﬁnition 2.13 and the “a priori estimate” from Lemma 2.15 . Prop osition B.1 (A priori estimates for limit optimal dynamical plans) . A ssume the glob al ly hyp erb olic weighte d Lipschitz sp ac etime ( M , g , m ) satisﬁes Ric g , m ,N ≥ K timelike distributional ly, wher e K ∈ R and N ∈ [ dim M , ∞ ) . L et ( g n ) n ∈ N and ( m n ) n ∈ N b e go o d appr oximations of g and m , r esp e ctively. Mor e over, let o ∈ M and let µ 1 ∈ P ( M ) b e c omp actly supp orte d and m -absolutely c ontinuous with d µ 1 / d m ∈ L ∞ ( M , m ) and { o } × spt µ 1 ⊂ I g . Given q ∈ (0 , 1) , let ( π n ) n ∈ N form a se quenc e of ℓ g n ,q -optimal dynamic al c ouplings π n of µ 0 := δ o and µ 1 . Then ther e exist c onstants ϑ ∈ R + and ρ ∈ R + such that every narr ow limit p oint π of ( π n ) n ∈ N has the pr op erty that for every t ∈ (0 , 1] , ( e t ) ♯ π is m -absolutely c ontinuous and    d( e t ) ♯ π d m    L ∞ ( M , m ) ≤ 1 t N e ϑ √ ρN/ ( N − 1)    d µ 1 d m    L ∞ ( M , m ) . Pr o of. Let n 0 ∈ N b e the maxim um of the integers provided by Lemmas 2.15 and 2.17 , and let n ∈ N suc h that n ≥ n 0 . By Lemma 2.17 , Theorem 3.19 , and m utual absolute con tin uity of m n and m , π n necessarily coincides with the unique ℓ g n ,q - optimal dynamical coupling of µ 0 and µ 1 ; in particular, b y Prop osition 3.21 , π is an ℓ g ,q -optimal dynamical coupling of µ 0 and µ 1 . On the other hand, b y Lemma 2.17 there is a constant ρ ∈ R + suc h that for every n ∈ N with n ≥ n 0 , every timelike aﬃnely parametrized g n -maximizing geodesic γ : [0 , 1] → M that starts in o and ends in spt µ 1 , and every t ∈ [0 , 1] , Ric g n , m n ,N ( ˙ γ t , ˙ γ t ) ≥ − ρ g n ( ˙ γ t , ˙ γ t ) . 44 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL F rom this, we ﬁrst deriv e some a priori estimates. W e abbreviate ϑ := sup { l g n ,o ( y ) : n ∈ N with n ≥ n 0 , y ∈ spt µ 1 } . By Theorem 3.27 and monotonicity prop erties of the inv olved distortion co eﬃcients, π n -a.e. γ satisﬁes the following inequality for every t ∈ [0 , 1] : d( e t ) ♯ π n d m n ( γ t ) − 1 / N ≥ τ ( t ) − ρ,N ◦ l g n ,o ( γ 1 ) h d µ 1 d m n i ( γ 1 ) − 1 / N ≥ τ ( t ) − ρ,N ( ϑ )    d µ 1 d m n    − 1 / N L ∞ ( M , m n ) ≥ a − 1 / N τ ( t ) − ρ,N ( ϑ )    d µ 1 d m    − 1 / N L ∞ ( M , m ) , (B.1) where the constant a := sup n d m d m n ( x ) : n ∈ N with n ≥ n 0 , x ∈ C o is clearly p ositiv e and ﬁnite by the setup of the go od approximation ( m n ) n ∈ N of m . By basic estimates for the inv olv ed distortion co eﬃcients, cf. e.g. Ca v alletti–Mondino [ 22 , Rem. 2.3], we know every θ ∈ R + satisﬁes τ ( t ) − ρ,N ( θ ) ≥ t e − (1 − t ) θ √ ρ/ N ( N − 1) . Com bining this with ( B.1 ), w e infer    d( e t ) ♯ π n d m    L ∞ ( M , m ) ≤ b    d( e t ) ♯ π n d m n    L ∞ ( M , m n ) ≤ 1 t N e ϑ √ ρN/ ( N − 1)    d µ 1 d m    L ∞ ( M , m ) , (B.2) where the constant b := sup n d m n d m ( x ) : n ∈ N with n ≥ n 0 , x ∈ C o is clearly p ositive and ﬁnite by construction of ( m n ) n ∈ N again and Lemma 2.14 . W e now turn to the conclusion. Given t ∈ (0 , 1] , we set ξ := 1 t N e ϑ √ ρN/ ( N − 1)    d µ 1 d m    L ∞ ( M , m ) and deﬁne the “mass excess functional” F ξ : P ( C ) → R + b y F ξ ( µ ) := Z C    h d µ ac d m − ξ i +    d m + µ sing [ C ] , where µ ac and µ sing denote the absolutely contin uous part and the singular part of µ with resp ect to m , resp ectively . By ( B.2 ) and mutual absolute contin uity of the in volv ed reference measures, every n ∈ N with n ≥ n 0 ob eys F ξ (( e t ) ♯ π n ) = 0 . Since F ξ is narrowly low er semicontin uous, cf. e.g. Ra jala [ 71 , Lem. 3.6], we get F ξ (( e t ) ♯ π ) = 0 . The arbitrariness of t establishes the second statement. □ By taking the narrow limit from this prop osition to b e the sp eciﬁc optimal dynamical coupling constructed in the pro of of Theorem 6.1 , the following holds. COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 45 Theorem B.2 (Existence of go od geo desics) . A ssume that the glob al ly hyp erb olic weighte d Lipschitz sp ac etime ( M , g , m ) satisﬁes Ric g , m ,N ≥ K timelike distribution- al ly, wher e K ∈ R and N ∈ [ dim M , ∞ ) . L et o ∈ M . F urthermor e, let µ 1 ∈ P ( M ) b e c omp actly supp orte d and m -absolutely c ontinuous such that d µ 1 / d m is m -essential ly b ounde d and { o } × spt µ 1 ⊂ I . Then for every q ∈ (0 , 1) , ther e exist • an ℓ q -optimal dynamic al c oupling π fr om µ 0 := δ o to µ 1 and • c onstants ϑ ∈ R + and ρ ∈ R + satisfying the fol lowing pr op erties for every t ∈ (0 , 1] . (i) W e have S N (( e t ) ♯ π | m ) ≤ − Z M τ ( t ) K,N ◦ l o h d µ 1 d m i 1 − 1 / N d m . (ii) The me asur e ( e t ) ♯ π is m -absolutely c ontinuous with    d( e t ) ♯ π d m    L ∞ ( M , m ) ≤ 1 t N e ϑ √ ρN/ ( N − 1)    d µ 1 d m    L ∞ ( M , m ) . Ev ery optimal dynamical coupling that ob eys the conclusions of the previous theorem will b e informally called go o d ge o desic . B.2. A Brenier–McCann theorem. A key p oin t is an analog of Beran et al. [ 6 , Thm. 5.19], sp eciﬁed to the simpler case of p ow ers of Lorentz distance functions. It mirrors Theorem 3.19 , but do es not rely on the exp onen tial map, which do es not exist in general on Lipschitz spacetimes. Prop osition B.3 (Brenier–McCann theorem) . W e ﬁx an exp onent q ∈ (0 , 1) . L et o ∈ M and let µ 1 ∈ P ( M ) b e m -absolutely c ontinuous with c omp act supp ort in I + ( o ) . L et π b e an ℓ q -optimal dynamic al c oupling fr om µ 0 := δ o to µ 1 . Then π -a.e. γ ob eys    ∇ l q o q    ( γ 1 ) = l ( γ 0 , γ 1 ) q − 1 . The pro of is a direct application of Prop osition 2.19 (the claim makes sense by the hypothesized absolute contin uit y), the chain rule, and Corollary 2.20 . B.3. Pro ofs of Theorems 6.5 and 6.6 . Prop osition B.4 (Horizontal vs. v ertical diﬀeren tiation) . L et o ∈ M . F urthermor e, let µ 1 ∈ P ( M ) b e c omp actly supp orte d in I + ( o ) and m -absolutely c ontinuous with d µ 1 / d m m -essential ly b ounde d. Given q ∈ (0 , 1) , let π b e an ℓ q -optimal dynamic al c oupling that deﬁnes a go o d ge o desic fr om µ 0 to µ 1 . Final ly, we let f : M → R b e a Lipschitz c ontinuous function that is supp orte d in I + ( o ) . Then, denoting by q ′ < 0 the c onjugate exp onent of q , lim t → 1 − Z f ( γ 1 ) − f ( γ t ) 1 − t d π ( γ ) = Z M d f ( ∇ l o ) |∇ l o | q ′ − 2 d( e 1 ) ♯ π . In a nutshell, this result mimics the fact from the smooth framework that the in tegrand on the righ t-hand side can b e computed in t w o w ays: either b y considering “v ertical” p erturbations (by slightly p erturbing the function l o in question) or b y taking the “horizontal” deriv ativ e of f in the direction of the transport rays set up by π . W e refer to the review of Braun [ 9 ] for more consequences of this simple realization. The following somewhat lengthy pro of, following the lines of Beran et al. [ 6 , Thms. 4.15, 4.16], can b e skipp ed at ﬁrst reading. 46 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL Pr o of of Pr op osition B.4 . W e ﬁrst claim lim t → 1 − Z l o ( γ 1 ) q − l o ( γ t ) q q (1 − t ) d π ( γ ) = Z M 1 q ′    ∇ l q o q    q ′ d( e 1 ) ♯ π + Z l ( γ 0 , γ 1 ) q q d π ( γ ) . (B.3) First recall π -a.e. γ satisﬁes γ 0 = o . Hence, given t ∈ [0 , 1] , l o ( γ 1 ) q − l o ( γ t ) q = l ( γ 0 , γ 1 ) q − l ( γ 0 , γ t ) q =  1 − t q ] l ( γ 0 , γ 1 ) q . Therefore, Leb esgue’s dominated con vergence theorem, conjugacy of q and q ′ , and Prop osition B.3 yield the desired identit y lim t → 1 − Z l o ( γ 1 ) q − l o ( γ t ) q q (1 − t ) d π ( γ ) = Z l ( γ 0 , γ 1 ) q d π ( γ ) = Z M 1 q ′    ∇ l q o q    q ′ d( e 1 ) ♯ π + Z l ( γ 0 , γ 1 ) q q d π ( γ ) . Let U ⊂ M constitute a ﬁxed precompact op en neigh b orhoo d of spt f whic h is compactly contained in I + ( o ) . W e claim the existence of ϑ > 0 and ε 0 > 0 such that for every ε ∈ R \ { 0 } such that | ε | ≤ ε 0 , we hav e g h ∇ l q o q + ε ∇ f , ∇ l q o q + ε ∇ f i ≥ ϑ m U -a.e. (B.4) T o this aim, we will abbreviate ϑ 1 := inf { l o ( y ) : y ∈ cl U } , ϑ 2 := sup { l o ( y ) : y ∈ cl U } , whic h are p ositiv e and ﬁnite constants by our choice of U and contin uit y of l + , cf. Theorem 2.3 . Recall from Prop osition 2.19 that l o is lo cally Lipschitz contin uous on I + ( o ) . Thanks to the chain rule and Corollary 2.20 , at every diﬀerentiabilit y p oin t of l o in I + ( o ) \ U , we easily obtain the relations g h ∇ l q o q + ε ∇ f , ∇ l q o q + ε ∇ f i = g h ∇ l q o q , ∇ l q o q i = l 2 q − 2 o g ( ∇ l o , ∇ l o ) ≥ ϑ 1 . irresp ectiv e of the v alue of ε . On the other hand, again by Prop osition 2.19 , the c hain rule, Corollary 2.20 , the hypothesized Lipschitz contin uit y of f , and uniform b oundedness of the co eﬃcien ts of g in the compact set cl U , there exist constan ts c 1 , c 2 > 0 such that at every diﬀerentiabilit y p oin t of l o in U , g h ∇ l q o q + ε ∇ f , ∇ l q o q + ε ∇ f i = l 2 q − 2 o g ( ∇ l o , ∇ l o ) + 2 ε l q − 1 o g ( ∇ l o , ∇ f ) + ε 2 g ( ∇ f , ∇ f ) ≥ ϑ 2 q − 2 2 − 2 ε c 1 − c 2 ε 2 . The right-hand side is uniformly p ositiv e when ε is suﬃciently small, as claimed. Next, we claim that given ε as ab o v e, liminf t → 1 − Z l q o ( γ 1 ) + q ε f ( γ 1 ) − l q o ( γ t ) − q ε f ( γ t ) q (1 − t ) d π ( γ ) ≥ Z M 1 q ′    ∇ l q o q + ε ∇ f    q ′ d( e 1 ) ♯ π + Z l ( γ 0 , γ 1 ) q q d π ( γ ) . (B.5) By Lemma 2.18 and Arzelà–Ascoli’s theorem, the supp ort of π is equicontin uous. In turn, there is δ ∈ (0 , 1) such that π -a.e. γ ob eys γ t ∈ U for every t ∈ [1 − δ, 1] . COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 47 F or such a t , the fundamental theorem of calculus (applicable by Prop osition 2.19 and Theorem B.2 ) and the reverse Cauch y–Sch w arz and Y oung inequalities imply Z l q o ( γ 1 ) + q ε f ( γ 1 ) − l q o ( γ t ) − q ε f ( γ t ) q d π ( γ ) ≥ Z Z [ t, 1] d h l q o q + ε f i ( ˙ γ s ) d s d π ( γ ) ≥ Z Z [ t, 1] 1 q ′    ∇ l q o q + ε ∇ f    q ′ d( e s ) ♯ π d s + Z Z [ t, 1]   ˙ γ s   q q d π ( γ ) = Z Z [ t, 1] 1 q ′    ∇ l q o q + ε ∇ f    q ′ d( e s ) ♯ π d m d m d s + (1 − t ) Z l ( γ 0 , γ 1 ) q q d π ( γ ) . (B.6) It suﬃces to prov e the identit y lim t → 1 − 1 1 − t Z Z [ t, 1]    ∇ l q o q + ε ∇ f    q ′ d( e s ) ♯ π d m d m d s = Z M    ∇ l q o q + ε ∇ f    q ′ d( e 1 ) ♯ π , (B.7) since then ( B.5 ) follo ws b y dividing ( B.6 ) b y 1 − t and sending t → 1 − . On the one hand, the density d( e s ) ♯ π / d m is m -essen tially bounded uniformly in s ∈ [1 − δ, 1] thanks to Theorem B.2 . On the other hand, by ( B.4 ) , negativity of q ′ , and our c hoice of δ , the in tegrand |∇ l q o /q + ε ∇ f | q ′ b elongs to L 1 ( M , m ) . The narrow conv ergence of ( e s ) ♯ π → ( e 1 ) ♯ π as s → 1 − easily implies lim t → 1 − 1 1 − t Z [ t, 1] d( e s ) ♯ π d m d s = d µ 1 d m against functions in L 1 ( M , m ) . This establishes ( B.7 ). Finally , subtracting the identit y ( B.3 ) from the inequality ( B.5 ) yields liminf t → 1 − ε Z f ( γ 1 ) − g ( γ t ) 1 − t d π ( γ ) ≥ Z M 1 q ′ h    ∇ l q o q + ε ∇ f    q ′ −    ∇ l q o q    q ′ i d( e 1 ) ♯ π . W e divide this inequality by ε and then send ε → 0 , whic h creates t wo inequalities dep ending on the sign of ε . On the one hand, employing m -essen tial b oundedness of d( e 1 ) ♯ π / d m , Prop osition 2.19 , and Leb esgue’s dominated conv ergence theorem, liminf t → 1 − ε Z f ( γ 1 ) − g ( γ t ) 1 − t d π ( γ ) ≥ lim ε → 0 + Z M 1 q ′ ε h    ∇ l q o q + ε ∇ f    q ′ −    ∇ l q o q    q ′ i d( e 1 ) ♯ π = Z M d f h ∇ l q o q i    ∇ l q o q    q ′ − 2 d( e 1 ) ♯ π . On the other hand, an analogous argument implies limsup t → 1 − ε Z f ( γ 1 ) − g ( γ t ) 1 − t d π ( γ ) ≥ lim ε → 0 − Z M 1 q ′ ε h    ∇ l q o q + ε ∇ f    q ′ −    ∇ l q o q    q ′ i d( e 1 ) ♯ π = Z M d f h ∇ l q o q i    ∇ l q o q    q ′ − 2 d( e 1 ) ♯ π . Com bining the previous estimates yields the claim. □ 48 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL No w w e are in a p osition to establish Theorems 6.5 and 6.6 . Pr o of of The or em 6.5 . Giv en α > 0 , let µ 0 , µ 1 ∈ P ( M ) b e deﬁned b y µ 0 := δ o , µ 1 :=  c α ( ϕ + α )  N/ ( N − 1) m spt ϕ, where c α > 0 is a normalization constant. Evidently , w e hav e spt µ 0 × spt µ 1 ⊂ I . By Theorem B.2 , there exists an ℓ q -optimal dynamical coupling π of µ 0 and µ 1 that deﬁnes a go o d geo desic b etw een these marginals; in particular, given t ∈ (0 , 1] , S N ( µ t | m ) ≤ − Z M τ ( t ) K,N ◦ l o h d µ 1 d m i 1 − 1 / N d m . A simple algebraic manipulation then yields S N ( µ 1 | m ) − S N ( µ t | m ) 1 − t ≥ − Z 1 − τ ( t ) K,N ◦ l o 1 − t h d µ 1 d m i 1 − 1 / N d m . (B.8) W e now send t → 1 − in this iden tit y and start with the right-hand side. As ϕ has compact supp ort in I + K,N ( o ) and π is concentrated on timelik e aﬃnely parametrized maximizing geodesics, by Theorem 5.1 and contin uit y of l + , the argumen t l o is uniformly b ounded (aw a y from π K/ ( N − 1) ) on the supp ort of π . On the other hand, b y construction of µ 1 , the function (d µ 1 / d m ) 1 − 1 / N is b ounded on the supp ort of ϕ . Smo othness of the inv olved τ -distortion co eﬃcien ts on [0 , π K/ ( N − 1) ) and Leb esgue’s dominated conv ergence theorem thus imply − liminf t → 1 − Z 1 − τ ( t ) K,N ◦ l o 1 − t h d µ 1 d m i 1 − 1 / N d m = − Z M ˜ τ K,N ◦ l o h d µ 1 d m i 1 − 1 / N d m = − c α Z spt ϕ ˜ τ K,N ◦ l o ( ϕ + α ) d m . No w w e address the left-hand side. Deﬁne s N : R + → R − b y s N ( r ) := − r 1 − 1 / N . Using conv exit y of s N , we obtain limsup t → 1 − S N ( µ 1 | m ) − S N ( µ t | m ) 1 − t = limsup t → 1 − Z M 1 1 − t h s N ◦ d µ 1 d m − s N ◦ d µ t d m i d m ≤ limsup t → 1 − Z M 1 1 − t s ′ N ◦ d µ 1 d m h d µ 1 d m − d µ t d m i d m = limsup t → 1 − Z 1 1 − t h s ′ N ◦ d µ 1 d m ( γ 1 ) − s ′ N ◦ d µ 1 d m ( γ t ) i d π ( γ ) . Prop osition B.4 implies limsup t → 1 − Z 1 1 − t h s ′ N ◦ d µ 1 d m ( γ 1 ) − s ′ N ◦ d µ 1 d m ( γ t ) i d π ( γ ) ≤ Z M d h s ′ N ◦ d µ 1 d m ih ∇ l q o q i    ∇ l q o q    q ′ − 2 d( e 1 ) ♯ π = Z M d h s ′ N ◦ d µ 1 d m ih ∇ l q o q i    ∇ l q o q    q ′ − 2 d µ 1 d m d m = c α N Z M d ϕ h ∇ l q o q i    ∇ l q o q    q ′ − 2 d m , COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 49 where the last identit y follows from a direct computation and the c hain rule. Can- celing c α in the inequality resulting from these expansions of ( B.8 ) yields − Z M d ϕ h ∇ l q o q i    ∇ l q o q    q ′ − 2 d m ≤ N Z spt ϕ ˜ τ K,N ◦ l o ( ϕ + α ) d m . Sending α → 0 + together with the m -in tegrabilit y of ˜ τ K,N ◦ l o on spt ϕ as discussed ab o v e entails the desired statement. □ Pr o of of The or em 6.6 . Let q ∈ (0 , 1) b e the conjugate exp onen t of q ′ . By Prop osi- tion 2.19 and as ϕ has compact support aw a y from o , the function ϕ/l o is again Lipsc hitz con tinuous, nonnegativ e, and has compact supp ort. By Theorem 6.5 , Z M N ˜ τ K,N ◦ l o l o ϕ d m ≥ − Z M d ϕ l o h ∇ l q o q i    ∇ l q o q    q ′ − 2 d m . (B.9) As identities of (co)tangent vectors, the chain rule yields d ϕ l o = 1 l o d ϕ − ϕ l 2 o d l o m I + ( o ) -a.e. , ∇ l q o q = l q − 1 o ∇ l o m I + ( o ) -a.e. In turn, a direct computation, conjugacy of q and q ′ , and Corollary 2.20 yield d ϕ l o h ∇ l q o q i    ∇ l q o q    q ′ − 2 = d ϕ ( ∇ l o ) − ϕ l o |∇ l o | 2 = d ϕ ( ∇ l o ) |∇ l o | q ′ − 2 − ϕ l o m I + ( o ) -a.e. Inserting this into ( B.9 ) and rearranging terms yields the claim. □ References [1] L. Ambrosio, N. Gigli, and G. Sav aré, Gradient ﬂows in metric sp ac es and in the sp ac e of pr obability me asures , Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser V erlag, Basel, 2008. MR 2401600 [2] L. Andersson, G. J. Gallow ay, and R. How ard, The cosmolo gic al time function , Classical Quantum Gravit y 15 (1998), no. 2, 309–322. MR 1606594 [3] E. Aubry, Finiteness of π 1 and ge ometric ine qualities in almost positive Ric ci curvatur e , Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 4, 675–695. MR 2191529 [4] R. Bartnik, R emarks on c osmolo gical sp ac etimes and c o nstant me an curvatur e surfac es , Comm. Math. Phys. 117 (1988), no. 4, 615–624. MR 953823 [5] J. K. Beem, P . E. Ehrlich, and K. L. Easley, Glob al Lor entzian ge ometry , Second, Monographs and T extb ooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker, Inc., New Y ork, 1996. MR 1384756 [6] T. Beran, M. Braun, M. Calisti, N. Gigli, R. J. McCann, A. Ohan yan, F. Rott, and C. Sämann, A nonline ar d’Alemb ert c omp arison the orem and c ausal diﬀer ential c alculus on metric me asur e sp ac etimes , Preprin t, [7] M. Braun, Go od geo desics satisfying the timelike curvatur e-dimension c ondition , Nonlinear Anal. 229 (2023), Paper No. 113205, 30 pp. MR 4528587 [8] , Rényi’s entropy on Lor entzian sp ac es. Timelike curvatur e-dimension c onditions , J. Math. Pures Appl. (9) 177 (2023), 46–128. MR 4629751 [9] , New p ersp e ctives on the d’Alembertian fr om gener al r elativity. An invitation , Indag. Math., to appear. [10] , Exact d’Alemb ertian for Lor entz distanc e functions , Preprin t, [11] M. Braun and M. Calisti, Timelike Ric ci b ounds for low r e gularity sp ac etimes by optimal tr ansp ort , Comm un. Contemp. Math. 26 (2024), no. 9, P aper No. 2350049, 30. MR 4793142 [12] M. Braun, N. Gigli, R. J. McCann, A. Ohany an, and C. Sämann, A Lor entzian splitting the or em for c ontinuously diﬀer entiable metrics and weights , Preprin t, [13] , An el liptic pr o of of the splitting the orems fr om Lor entzian ge ometry , Preprint, [14] M. Braun and R. J. McCann, Causal c onver genc e c onditions thr ough variable timelike Ric ci curvatur e b ounds , Mem. Eur. Math. So c., to appear. 50 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL [15] M. Braun and S. Ohta, Optimal tr ansp ort and timelike lower Ric ci curvatur e b ounds on Finsler sp acetimes , T rans. Amer. Math. Soc. 377 (2024), no. 5, 3529–3576. MR 4744787 [16] H. Busemann, Timelike sp ac es , Dissertationes Math. (Rozprawy Mat.) 53 (1967), 52. MR 220238 [17] M. Calisti, M. Graf, E. Hafemann, M. Kunzinger, and R. Steinbauer, Hawking’s singularity the or em for Lipschitz Lor entzian metrics , Comm. Math. Phys. 406 (2025), no. 9, P aper No. 207, 31. MR 4940197 [18] E. Caputo, F. Nobili, and T. Rossi, Comp arison estimates on nonsmo oth sp ac es with inte gr able Ric ci lower b ounds via lo c alization , Preprin t, [19] G. Carron, Ge ometric ine qualities for manifolds with Ric ci curvatur e in the Kato class , Ann. Inst. F ourier (Grenoble) 69 (2019), no. 7, 3095–3167. MR 4286831 [20] G. Carron and C. Rose, Ge ometric and sp e ctr al estimates b ase d on sp e ctr al Ric ci curvatur e assumptions , Journal für die reine und angewandte Mathematik (Crelles Journal) 2021 (2021), no. 772, 121–145. [21] F. Ca v alletti and E. Milman, The glob alization theor em for the curvatur e-dimension c ondition , Inv ent. Math. 226 (2021), no. 1, 1–137. MR 4309491 [22] F. Cav alletti and A. Mondino, Optimal maps in essential ly non-br anching spac es , Commun. Contemp. Math. 19 (2017), no. 6, 1750007, 27. MR 3691502 [23] , A r eview of Lor entzian synthetic the ory of timelike Ric ci curvatur e bounds , Gen. Relativity Gravitation 54 (2022), no. 11, Pa p er No. 137, 39 pp. MR 4504922 [24] , Optimal tr ansp ort in Lor entzian synthetic sp ac es, synthetic timelike Ric ci curvature lower b ounds and applications , Camb. J. Math. 12 (2024), no. 2, 417–534. MR 4779676 [25] , A sharp isop erimetric-typ e ine quality for Lor entzian spac es satisfying timelike Ric ci lower b ounds , Preprint, [26] J. Cheeger and D. G. Ebin, Comp arison the or ems in Riemannian ge ometry , North-Holland Mathematical Library , v ol. V ol. 9, North-Holland Publishing Co., Amsterdam-Oxford; Ameri- can Elsevier Publishing Co., Inc., New Y ork, 1975. MR 458335 [27] Y. Cho quet-Bruhat, Esp ac es-temps einsteiniens génér aux, cho cs gr avitationnels , Ann. Inst. H. Poincaré Sect. A (N.S.) 8 (1968), 327–338. MR 233576 [28] P . T. Chruściel and J. D. E. Grant, On Lorentzian c ausality with c ontinuous metrics , Classical Quantum Gravit y 29 (2012), no. 14, 145001, 32. MR 2949547 [29] M. Eckstein and T. Miller, Causality for nonlo c al phenomena , Ann. Henri P oincaré 18 (2017), no. 9, 3049–3096. MR 3685983 [30] P . E. Ehrlic h, Comp arison the ory in Lorentzian ge ometry , Lecture notes, Isaac Newton Institute, 2005. [31] P . E. Ehrlic h, Y.-T. Jung, and S.-B. Kim, V olume comp arison the or ems for Lorentzian manifolds , Geom. Dedicata 73 (1998), no. 1, 39–56. MR 1651891 [32] P . E. Ehrlic h and M. Sánc hez, Some semi-Riemannian volume c omp arison the or ems , T ohoku Math. J. (2) 52 (2000), no. 3, 331–348. MR 1772801 [33] J.-H. Esc henburg, The splitting the orem for sp ace-times with str ong ener gy c ondition , J. Diﬀerential Geom. 27 (1988), no. 3, 477–491. MR 940 115 [34] A. F. Filippov, Diﬀer ential e quations with disc ontinuous righthand sides , Mathematics and its Applications (Soviet Series), v ol. 18, Kluw er A cademic Publishers Group, Dordrech t, 1988. T ranslated from the R ussian. MR 1028776 [35] G. J. Gallow ay, The Lorentzian splitting the or em without the completeness assumption , J. Diﬀerential Geom. 29 (1989), no. 2, 373–387. MR 982 181 [36] L. García-Heveling, V olume singularities in gener al r elativity , Lett. Math. Ph ys. 114 (2024), no. 3, Paper No. 71. MR 4751750 [37] R. Geroch and J. T raschen, Strings and other distributional sour c es in gener al r elativity , Phys. Rev. D (3) 36 (1987), no. 4, 1017–1031. MR 904839 [38] N. Gigli, On the diﬀer ential structur e of metric me asure spac es and applic ations , Mem. Amer. Math. So c. 236 (2015), no. 1113, vi+91 pp. MR 3381131 [39] M. Graf, V olume comp arison for C 1 , 1 -metrics , Ann. Global Anal. Geom. 50 (2016), no. 3, 209–235. MR 3554372 [40] , Singularity the or ems for C 1 -Lor entzian metrics , Comm. Math. Ph ys. 378 (2020), no. 2, 1417–1450. MR 4134950 [41] M. Graf and E. Ling, Maximizers in Lipschitz sp ac etimes ar e either timelike or nul l , Classical Quantum Gravit y 35 (2018), no. 8, 087001, 6. MR 3789523 [42] M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer, Ge ometric the ory of gener alized functions with applic ations to gener al r elativity , Mathematics and its Applications, vol. 537, Kluwer Academic Publishers, Dordrec ht, 2001. MR 1883263 COMP ARISON THEOR Y FOR LIPSCHITZ SP ACETIMES 51 [43] S. W. Hawking and G. F. R. Ellis, The large scale structure of sp ac e-time , Cam bridge Monographs on Mathematical Ph ysics, vol. No. 1, Cambridge Universit y Press, London-New Y ork, 1973. MR 424186 [44] S. W. Ha wking, The o c curr enc e of singularities in c osmolo gy. III. Causality and singularities , Proc. Ro y . Soc. London Ser. A. Math. Phys. Sci. 300 (1967), no. 1461, 187–201. [45] W. Israel, Singular hyp ersurfac es and thin shel ls in gener al r elativity , Il Nuo vo Cimen to B (1965–1970) 44 (1966), no. 1, 1–14. [46] C. Ketterer, On the ge ometry of metric me asur e sp ac es with variable curvatur e b ounds , J. Geom. Anal. 27 (2017), no. 3, 1951–1994. MR 3667417 [47] , Stability of metric me asur e sp ac es with inte gr al Ric ci curvature b ounds , J. F unct. Anal. 281 (2021), no. 8, Paper No. 109142, 48. MR 4271791 [48] S. Khakshournia and R. Mansouri, The art of gluing spac e-time manifolds—metho ds and applic ations , SpringerBriefs in Physics, Springer, Cham, [2023] © 2023. MR 4697132 [49] B. Klartag, Ne e d le de comp ositions in Riemannian ge ometry , Mem. Amer. Math. Soc. 249 (2017), no. 1180, v+77. MR 3709716 [50] M. Kunzinger, M. Oberguggenberger, and J. A. Vic k ers, Synthetic versus distributional lower Ric ci curvature b ounds , Pro c. Roy . So c. Edinburgh Sect. A 154 (2024), no. 5, 1406–1430. MR 4806282 [51] M. Kunzinger, A. Ohany an, and A. V ardabasso, Ric ci curvatur e b ounds and rigidity for non-smo oth Riemannian and semi-Riemannian metrics , Manuscripta Math. 176 (2025), no. 4, Paper No. 53, 39. MR 4936063 [52] M. Kunzinger and C. Sämann, L or entzian length spac es , Ann. Global Anal. Geom. 54 (2018), no. 3, 399–447. MR 3867652 [53] M. Kunzinger, R. Stein bauer, M. Stojković, and J. A. Vic k ers, Hawking’s singularity the or em for C 1 , 1 -metrics , Classical Quantum Gravit y 32 (2015), no. 7, 075012, 19. MR 3322151 [54] C. Lange, A. Lytc hak, and C. Sämann, Lor entz me ets Lipschitz , A dv. Theor. Math. Phys. 25 (2021), no. 8, 2141–2170. MR 4489478 [55] A. Lichnerowicz, Thé ories relativistes de la gravitation et de l’électr omagnétisme. Relativité génér ale et thé ories unitair es , Masson et Cie, P aris, 1955. MR 71917 [56] J. Lott and C. Villani, R ic ci curvature for metric-me asur e sp aces via optimal tr ansp ort , Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619 [57] M. Manzano and M. Mars, A bstract formulation of the sp ac etime matching pr oblem and nul l thin shel ls , Ph ys. Rev. D 109 (2024), no. 4, Paper No. 044050, 28. MR 4718174 [58] R. J. McCann, Displacement c onvexity of Boltzmann ’s entr opy char acterizes the str ong ener gy c ondition fr om gener al r elativity , Cam b. J. Math. 8 (2020), no. 3, 609–681. MR 4192570 [59] , Tr ading linearity for el lipticity: a nonsmo oth appr o ach to Einstein ’s the ory of gr avity and the Lorentzian splitting the or ems , Pro ceedings of the F orward F rom the Fields Medal 2024, to appear. [60] E. Minguzzi and S. Suhr, L or entzian metric spac es and their Gr omov–Hausdorﬀ c onver gence , Lett. Math. Phys. 114 (2024), no. 3, P ap er No. 73. MR 4752400 [61] E. Minguzzi, Causality the ory for closed c one structur es with applic ations , Rev. Math. Ph ys. 31 (2019), no. 5, 1930001, 139. MR 3955368 [62] , L or entzian c ausality theory , Living Reviews in Relativity 22 (2019), no. 3, 202 pp. [63] A. Mondino and V. Ryborz, On the e quivalence of distributional and synthetic Ric ci curvature lower b ounds , J. F unct. Anal. 289 (2025), no. 8, Paper No. 111035, 82. MR 4903405 [64] A. Mondino and S. Suhr, A n optimal tr ansp ort formulation of the Einstein e quations of gener al r elativity , J. Eur. Math. So c. (JEMS) 25 (2023), no. 3, 933–994. MR 4577957 [65] R. P . A. C. Newman, A pr o of of the splitting c onje ctur e of S.-T. Yau , J. Diﬀerential Geom. 31 (1990), no. 1, 163–184. MR 1030669 [66] K. Nomizu and H. Ozeki, The existenc e of complete Riemannian metrics , Pro c. Amer. Math. Soc. 12 (1961), 889–891. MR 133785 [67] R. Penrose, Gravitational col lapse and sp ac e-time singularities , Phys. Rev. Lett. 14 (1965), 57–59. MR 172678 [68] , The ge ometry of impulsive gr avitational waves , General relativit y (pap ers in honour of J. L. Synge), 1972, pp. 101–115. MR 503490 [69] P . Petersen and G. W ei, R elative volume c omp arison with inte gr al curvatur e b ounds , Geom. F unct. Anal. 7 (1997), no. 6, 1031–1045. MR 1487753 [70] P . Petersen and C. Sprouse, Inte gral curvatur e b ounds, distanc e estimates and applic ations , J. Diﬀerential Geom. 50 (1998), no. 2, 269–298. MR 168 4981 [71] T. Ra jala, Interp olated me asur es with b ounde d density in metric spac es satisfying the curvatur e- dimension c onditions of Sturm , J. F unct. Anal. 263 (2012), no. 4, 896–924. MR 2927398 [72] A. D. Rendall, The or ems on existenc e and glob al dynamics for the Einstein equations , Living Rev. Relativ. 5 (2002), 2002–6, 62. MR 1932418 52 MA THIAS BRAUN AND MAR T A SÁLAMO CAND AL [73] C. Rose, A lmost positive Ric ci curvatur e in Kato sense—an extension of Myers’ the or em , Math. Res. Lett. 28 (2021), no. 6, 1841–1849. MR 4477675 [74] C. Sämann, Glob al hyp erb olicity for sp ac etimes with continuous metrics , Ann. Henri Poincaré 17 (2016), no. 6, 1429–1455. MR 3500220 [75] C. Sämann and R. Steinbauer, On ge o desics in low r e gularity , J. Ph ys. Conf. Ser. 968 (2018), 012010, 14. MR 3919953 [76] C. Sprouse, Inte gr al curvatur e b ounds and b ounde d diameter , Comm. Anal. Geom. 8 (2000), no. 3, 531–543. MR 1775137 [77] K.-T. Sturm, On the ge ometry of metric me asur e sp ac es. I , A cta Math. 196 (2006), no. 1, 65–131. MR 2237206 [78] , On the geometry of metric measur e spac es. II , Acta Math. 196 (2006), no. 1, 133–177. MR 2237207 [79] J.-H. T reude, Ric ci curvatur e c omp arison in Riemannian and Lor entzian ge ometry , Master’s Thesis, 2011. [80] J.-H. T reude and J. D. E. Grant, V olume c omp arison for hypersurfac es in Lor entzian manifolds and singularity the orems , Ann. Global Anal. Geom. 43 (2013), no. 3, 233–251. MR 3027611 [81] C. Villani, Optimal tr ansport , Grundlehren der mathematischen Wissenschaften [F undamental Principles of Mathematical Sciences], v ol. 338, Springer-V erlag, Berlin, 2009. Old and new. MR 2459454 Institute of Ma thema tics, EPFL, 1015 Lausanne, Switzerland Email addr ess : mathias.braun@epfl.ch F akul t ä t für Ma thema tik, University of Vienna, 1090 Vienna, Austria Email addr ess : marta.salamo.candal@univie.ac.at

Comparison theory for Lipschitz spacetimes

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment