The Hodograph Transform Between Thermodynamics and Relativity

The Ho dograph T ransform Bet w een Thermo dynamics and Relativit y Leonid P olterovic h ∗ Marc h 31, 2026 Abstract In the con tact-geometric approac h to general relativity , the sky of an ev ent—namely , the set of all incoming light ra ys—forms a Legendrian submanifold of the spherical cotangen t bundle of a Cauch y hypersurface. When the hypersurface is c hosen to be the Minko wski h yp erb oloid, a h yp erbolic version of the ho dograph transform identifies this bundle with a thermo dynamic phase space. W e consider a uniformly accelerating observ er starting on the h yp erboloid and study the evolution of its skies. W e sho w that the asso ciated generating functions, after a suitable rescaling, admit a natural in terpretation as reduced free energies of equilibrium thermo dynamic systems go verned b y the relativistic Doppler effect. F rom this data, we extract an effective temp erature that is prop ortional to the acceleration, in agreemen t with the scaling of the Unruh effect, although the n umerical c onstan t differs from the Unruh v alue. 1 In tro duction It was recently understo o d that the causal structure on the space of Legendrian submani- folds—a deep phenomenon in con tact geometry—appears b oth in general relativity [4] and in thermo dynamics [7]. In this note w e suggest a to y mo del whic h migh t be view ed as an argumen t in fav or of a duality b etw een these tw o sub jects. W e view the unit future h yp erb oloid in the (2 + 1)-dimensional Mink owski space as a Cauc hy hypersurface of the chronological future of the origin, which is iden tified with the h yp erb olic upp er half-plane H . W e consider an observer who starts at an even t A on the h yp erb oloid with constant prop er acceleration a , and in prop er time t arrives at an ev en t B . The sky (i.e. the set of all incoming light ra ys) of the even t B is viewed as a Legendrian submanifold of the spherical cotangent bundle S ∗ H of the h yp erb olic plane H . Next, we apply a v ersion of the hodograph transform H [2, 8, 4] b etw een the jet space J 1 S 1 and the spherical cotangent bundle S ∗ H . The latter can b e viewed as a thermo dynamic phase space, with the directions of ligh t ra ys in Mink o wski space serving as con trol (in tensive) v ariables. ∗ T el Aviv Universit y , Israel 1 Our main result (see Theorem 4.1 b elow) pro vides an explicit expression for the generating function of H − 1 ( S ), the preimage of the sky S of the ev ent B . Namely , it yields a function g t on S 1 whose 1-jet j 1 g t coincides with H − 1 ( S ). The form ulas for g t naturally in volv e quan tities of thermo dynamic flav or, most notably the energy (giv en b y the Doppler factor, i.e. the ratio of photon energies measured by the observ ers at B and A , see Section 2). In the thermodynamic in terpretation of the phase space J 1 S 1 , Legendrian submanifolds represen t equilibrium states, and their generating functions serv e as thermo dynamic p otentials, suc h as free energy . W e therefore in terpret g t as a reduced free energy . After an appropriate Loren tzian time rescaling—motiv ated by the Unruh effect 1 , ac- cording to whic h uniformly accelerating observers exhibit thermal b eha vior—the resulting form ulas allow us to extract an effectiv e temp erature. The corresp onding temp erature scales as ∼ a , in agreemen t with the Unruh temp erature scaling, T unruh = a 2 π , (1) although the n umerical constan t w e obtain differs from the Unruh v alue. Guided by this scaling, we construct a thermo dynamic system whose reduced free energy repro duces the leading b eha vior of g t (see Theorem 5.1 in Section 5). This system is related to a classical rotor (see Section 5.2). The pap er is organized as follows. In Section 2 we discuss the Busemann function of the h yp erb olic plane and its relation to the photon energy measured b y an observer, as w ell as to the Doppler effect. In Section 3 w e introduce the h yp erb olic ho dograph transform, whose prop erties are pro ved in Section 6.1. In Section 4 w e present our mo del of an accelerating observ er and state our main result, Theorem 4.1, which pro vides an expression for the generating function of the sky of the observer under the ho dograph transformation. Section 5 relates this generating function to the free energy of a thermo dynamic system. Con ven tion on units: Unless otherwise stated, all quan tities in this pap er are dimension- less. In particular, w e w ork in normalized geometric units asso ciated with the unit hyper- b oloid, so that prop er time, acceleration, photon energy , and temp erature are all understo o d after division b y the appropriate reference scales. W e explain how to restore ph ysical units in Section 4.5. 2 Busemann F unction as a Logarithmic Energy Profile Let K ⊂ R 1 , 2 b e the unit future hyperb oloid K = { x ∈ R 1 , 2 | ⟨ x, x ⟩ = − 1 , x 0 > 0 } , equipp ed with the hyperb olic metric induced by the Minko wski form ⟨ u, v ⟩ = − u 0 v 0 + u 1 v 1 + u 2 v 2 . 1 See Wikipedia contributors “Unruh effect.” Wikip edia, The F ree Encyclop edia. 2 Denote by ∂ ∞ K the ideal b oundary of K which we identify with the set of future light-lik e ra ys starting at the origin. Note that ∂ ∞ K is diffeomorphic to the circle. Fix a ray q ∈ ∂ ∞ K . Let γ ( t ) be any geo desic ra y in K asymptotic to q , parametrized b y h yp erb olic arclength. W rite d for the h yp erb olic distance. The Busemann function asso ciated with q is defined by b q ( x ) = lim t →∞  d ( x, γ ( t )) − t  , x ∈ K . The Busemann function associated with q is defined only up to an additiv e constant; shifting the origin of the geodesic ray asymptotic to q changes b q b y a constan t. Fixing a reference observ er remov es this ambiguit y . Let us calculate the Busemann function. Fix a reference p oint A ∈ K . T ake an y ray q ∈ ∂ ∞ K . Choose the (unique) n ull v ector ˆ q spanning q ∈ ∂ ∞ K , so that ⟨ A, ˆ q ⟩ = − 1 . (2) One readily c hecks that the curve γ ( t ) := e t 2 ˆ q + e − t 2 (2 A − ˆ q ) , t ≥ 0 is a geo desic ray with the natural parameterization whic h lies on K , satisfies γ (0) = A , and whic h is asymptotic to q as t → + ∞ . In the h yp erb oloid mo del, the hyperb olic distance satisfies cosh( d ( x, y )) = −⟨ x, y ⟩ . Th us d ( x, γ ( t )) = arccosh  − e t 2 ⟨ x, ˆ q ⟩ − e − t 2 ⟨ x, 2 A − ˆ q ⟩  . Using the asymptotic expansion arccosh( z ) = log (2 z ) + o (1) as z → ∞ , w e obtain d ( x, γ ( t )) = t + log ( −⟨ x, ˆ q ⟩ ) + o (1) . Th us, the Busemann function is given b y b q ( x ) = log ( −⟨ x, ˆ q ⟩ ) . (3) In what follo ws we fix once and for all a reference observer A ∈ K . Equality (2) remo v es the scaling am biguity for the choice of a light vector in the given ra y . F ollo wing [10], Section 3.3, put ε x ( q ) := −⟨ x, ˆ q ⟩ . When x ∈ K , it is a unit future– directed timelik e vector and hence can b e in terpreted as an observer. In this case, ε x ( q ) is the energy (proportional to the frequency) of the photon with null momentum ˆ q measured b y the observer represented by x . By construction, ε A ( q ) = 1 for all q . By (3) b q ( x ) = log ε x ( q ) ε A ( q ) = log ε x ( q ) . (4) 3 The dimensionless ratio ε x ( q ) /ε A ( q ) reflects the relativistic Doppler effect. Let us men tion that the quan tities app earing in our form ulas — namely , the photon energy ratios (Doppler factors) — coincide with the redshift factors, which, as shown in [5], are giv en by the ratios of con tact forms on the space of ligh t rays asso ciated to different Cauc hy hypersurfaces. 3 Space of ligh t ra ys and ho dograph transform The space of future ligh t geo desics in Minko wski space can b e identified with the unit co- sphere bundle S ∗ K , which w e identify with S ∗ H . This is a contact manifold. Let us describ e the con tactomorphism H : J 1 S 1 − → S ∗ H , whic h w e call the hyp erb olic ho do gr aph tr ansform . It iden tifies the unit cosphere bundle of the h yp erb olic plane with the 1–jet space of the b oundary circle. The 1–jet space J 1 S 1 = T ∗ S 1 × R carries the standard con tact form α = dz − p dq , where ( q , p, z ) are co ordinates with q ∈ S 1 , p ∈ T ∗ q S 1 , and z ∈ R . Its Reeb vector field is R = ∂ z . The unit cosphere bundle S ∗ H = { ( u, ξ ) ∈ T ∗ H : | ξ | g H = 1 } carries the Liouville con tact form λ . Its Reeb flo w is the geo desic flow of the hyperb olic metric. T o define the ho dograph transform, w e view the Busemann function as a function b q : H → R . Given ( q , p, z ) ∈ J 1 S 1 , w e set H ( q , p, z ) = ( u, db q ( u )) , where u ∈ H is determined by the conditions b q ( u ) = z , ∂ b q ( u ) ∂ q = p. Prop osition 3.1. The map H is a c ontactomorphism, and furthermor e H ∗ λ = α , The ho dograph transformation is well known, see [3, 2, 4]. The hyperb olic version which w e are using is a minor mo dification of the construction in [8]. F or reader’s conv enience, we presen t a pro of of Prop osition 3.1 in Section 6.1, while in Section 6.2 we revise the Euclidean ho dograph transformation from the viewp oint of the Busemann functions. 4 Time s geo desic flow shifts the Busemann function by a constant, b q 7→ b q + s, so in J 1 S 1 this flo w corresp onds to the v ertical translation along the z -axis. Let us no w discuss the action of the in verse ho dograph map on skies. F or u ∈ H , the fib er S ∗ u H is mapped b y H − 1 to the Legendrian graph j 1 f of f ( q ) = b q ( u ). Hence the skies lying on the Cauc hy h yp ersurface corresp ond to the Legendrian graphs of Busemann functions, and their images under the time s maps of the geo desic flo w corresp ond to the Legendrian graphs of b q ( u ) + s . Com bining this consideration with (4), we get the following result. Prop osition 3.2. L et S u,s b e the image of S ∗ u H under time s map of the ge o desic flow. Then H − 1 ( S u,s ) is gener ate d by g s ( q ) = s + log ε u ( q ) . (5) 4 Case study: accelerated motion 4.1 Set up W e view the unit future h yp erb oloid K = { x ∈ R 1 , 2 | ⟨ x, x ⟩ = − 1 , x 0 > 0 } , as a Cauc hy hypersurface of the chronological future of the origin, i.e., of the cone { ( t, x, y ) ∈ R 1 , 2 : t > 0 , x 2 + y 2 < t 2 } . Let A = (1 , 0 , 0) ∈ K b e the reference observer used for the normalization of photon energies. Let n ∈ T A K b e a spatial unit vector: ⟨ n, n ⟩ = 1 , ⟨ A, n ⟩ = 0 . Consider the uniformly accelerated worldline starting at A with proper acceleration a > 0 and initial four–v elo cit y A . It has the explicit form γ ( t ) =  1 + sinh( at ) a  A + cosh( at ) − 1 a n, where t is the prop er time. Denote B = γ ( t ) . The p oint B lies in the in terior of the future light cone. T o describ e the sky of B on the fixed h yp ersurface K , we pro ject B radially to K by normalizing its Mink o wski length: B ′ = B p −⟨ B , B ⟩ . 5 Then ⟨ B ′ , B ′ ⟩ = − 1, so B ′ ∈ K . Geometrically , B ′ is the unique p oint of K lying on the ray R > 0 B . The past ligh t cone of B in tersects K in the hyperb olic circle { u ∈ K | d ( u, B ′ ) = ρ ( t ) } , where ρ ( t ) := 1 2 log  −⟨ B , B ⟩  , (6) and d is the h yp erb olic distance on K . Indeed, a p oint u ∈ K lies on the past light cone of B if and only if ⟨ u − B , u − B ⟩ = 0 . W rite λ := p −⟨ B , B ⟩ > 0 , B = λB ′ . Since u, B ′ ∈ K , w e ha v e ⟨ u, u ⟩ = ⟨ B ′ , B ′ ⟩ = − 1 . Therefore 0 = ⟨ u − B , u − B ⟩ = ⟨ u, u ⟩ − 2 ⟨ u, B ⟩ + ⟨ B , B ⟩ = − 1 − 2 λ ⟨ u, B ′ ⟩ − λ 2 . Hence −⟨ u, B ′ ⟩ = λ + λ − 1 2 . On the other hand, in the hyperb oloid mo del the hyperb olic distance satisfies cosh d ( u, B ′ ) = −⟨ u, B ′ ⟩ . Th us cosh d ( u, B ′ ) = λ + λ − 1 2 = cosh(log λ ) . Since d ( u, B ′ ) ≥ 0 and λ > 1, w e get d ( u, B ′ ) = log λ = 1 2 log  −⟨ B , B ⟩  . This pro ves that the intersection of the past light cone of B with K is precisely { u ∈ K | d ( u, B ′ ) = ρ ( t ) } , where ρ ( t ) = 1 2 log  −⟨ B , B ⟩  . Th us the sky of B on K is the unit conormal lift S B ′ , − ρ ( t ) , where ρ ( t ) = 1 2 log  1 + 2 a sinh( at ) + 2 a 2  cosh( at ) − 1   . (7) 6 4.2 The measured photon energy . Let u t := ˙ γ ( t ) = cosh( at ) A + sinh( at ) n (8) b e the four–v elo cit y of the accelerated observ er. Consider a photon with the n ull momen tum ˆ q . The corresp onding photon energy , as measured by the observer, is E t ( q ) := −⟨ u t , ˆ q ⟩ = cosh( at ) − sinh( at ) ⟨ n, ˆ q ⟩ . (9) F or small t , E t ( q ) = 1 − a ⟨ n, ˆ q ⟩ t + O ( t 2 ) . (10) 4.3 Main theorem Theorem 4.1. (i) The gener ating function g t of H − 1 ( S B ′ , − ρ ( t ) ) is given by g t ( q ) = log ε B ′ ( q ) − ρ ( t ) . (11) (ii) The gener ating function g t ( q ) admits the fol lowing explicit expr ession in terms of E t ( q ) : g t ( q ) = log  1 + 1 a tanh  at 2  (1 + E t ( q ))  − log  1 + 2 a sinh( at ) + 2 a 2 (cosh( at ) − 1)  . (12) (iii) We have the fol lowing smal l t exp ansion: g t ( q ) = − 3 2 t + 1 2 t 2 + 1 2 tE t ( q ) + O ( t 3 ) . (13) The right-hand sides of (11) and (13) are closely related to the relativistic Doppler effect for accelerating observ ers. In Section 5 we in terpret g t in terms of free energy . Pr o of. (i): F ormula (11) follows from Prop osition 3.2 with s = − ρ ( t ). (ii): By part (i), g t ( q ) = log ε B ′ ( q ) − ρ ( t ) . Recall that ε B ′ ( q ) = α ( t ) − β ( t ) ⟨ n, ˆ q ⟩ p D ( t ) , ρ ( t ) = 1 2 log D ( t ) , where α ( t ) = 1 + sinh( at ) a , β ( t ) = cosh( at ) − 1 a , and D ( t ) = 1 + 2 a sinh( at ) + 2 a 2  cosh( at ) − 1  . 7 Hence g t ( q ) = log  α ( t ) − β ( t ) ⟨ n, ˆ q ⟩  − log D ( t ) . No w E t ( q ) = cosh( at ) − sinh( at ) ⟨ n, ˆ q ⟩ , so ⟨ n, ˆ q ⟩ = cosh( at ) − E t ( q ) sinh( at ) . Therefore α ( t ) − β ( t ) ⟨ n, ˆ q ⟩ = 1 + sinh( at ) a − cosh( at ) − 1 a · cosh( at ) − E t ( q ) sinh( at ) . Com bining the terms giv es α ( t ) − β ( t ) ⟨ n, ˆ q ⟩ = 1 + cosh( at ) − 1 a sinh( at )  1 + E t ( q )  . Using cosh( at ) − 1 sinh( at ) = tanh  at 2  , w e obtain α ( t ) − β ( t ) ⟨ n, ˆ q ⟩ = 1 + 1 a tanh  at 2   1 + E t ( q )  . Substituting this in to the formula for g t ( q ) yields g t ( q ) = log  1 + 1 a tanh  at 2  (1 + E t ( q ))  − log  1 + 2 a sinh( at ) + 2 a 2 (cosh( at ) − 1)  , as claimed. (iii): Using the formula prov ed in (ii), g t ( q ) = log  1 + 1 a tanh  at 2  (1 + E t ( q ))  − log  1 + 2 a sinh( at ) + 2 a 2 (cosh( at ) − 1)  . Since 1 a tanh  at 2  = t 2 + O ( t 3 ) , E t ( q ) = 1 + O ( t ) , w e get 1 a tanh  at 2  (1 + E t ( q )) = t 2 (1 + E t ( q )) + O ( t 3 ) . 8 Hence log  1 + 1 a tanh  at 2  (1 + E t ( q ))  = t 2 (1 + E t ( q )) − t 2 8 (1 + E t ( q )) 2 + O ( t 3 ) . Since E t ( q ) = 1 + O ( t ), we hav e (1 + E t ( q )) 2 = 4 + O ( t ) , and therefore log  1 + 1 a tanh  at 2  (1 + E t ( q ))  = t 2 + t 2 E t ( q ) − 1 2 t 2 + O ( t 3 ) . Also, sinh( at ) = at + O ( t 3 ) , cosh( at ) − 1 = a 2 t 2 2 + O ( t 4 ) , so 1 + 2 a sinh( at ) + 2 a 2 (cosh( at ) − 1) = 1 + 2 t + t 2 + O ( t 3 ) , and th us log  1 + 2 a sinh( at ) + 2 a 2 (cosh( at ) − 1)  = 2 t − t 2 + O ( t 3 ) . Subtracting, g t ( q ) =  t 2 + t 2 E t ( q ) − 1 2 t 2  − (2 t − t 2 ) + O ( t 3 ) , whic h gives g t ( q ) = − 3 2 t + 1 2 t 2 + 1 2 tE t ( q ) + O ( t 3 ) . This pro ves (13). Remark 4.2. The use of the h yp erb oloid K as a Cauch y hypersurface is essen tial for the thermo dynamic interpretation. Indeed, the Busemann function b q ( x ) depends on the spacetime ev en t x , while the photon energy ε u ( q ) = −⟨ u, ˆ q ⟩ is defined with respect to an observ er, i.e. a future unit timelike vector u . On the h yp erb oloid K these notions coincide, since ev ery point x ∈ K can be naturally in terpreted as an observ er with velocity x . This yields the identit y b q ( x ) = log ε x ( q ) , whic h underlies the free energy interpretation. In con trast, if one c ho oses a Euclidean Cauc hy hypersurface such as { t = t 0 } , there is no canonical iden tification b et ween points and observ er velocities. As a result, the Busemann function and the photon energy dep end on differen t geometric ob jects, and no natural energy- t yp e term arises on the thermo dynamic side. 9 4.4 Comparison with the Unruh temp erature Deriv ation of the effectiv e in verse temp erature. W e start from (13): g t ( q ) = − 3 2 t + 1 2 t 2 + 1 2 t E t ( q ) + O ( t 3 ) , where E t ( q ) = cosh( at ) − sinh( at ) ⟨ n, ˆ q ⟩ . In tro duce the new time parameter (called a b o ost parameter or the Rindler time) 2 η := at, t = η a , and define e g η ( q ) := g η /a ( q ) , f η ( q ) := e g η ( q ) η = g η /a ( q ) η . (14) Substituting t = η /a into the expansion of g t yields e g η ( q ) = − 3 η 2 a + η 2 2 a 2 + η 2 a E η /a ( q ) + O ( η 3 ) . Dividing b y η , we obtain f η ( q ) = − 3 2 a + η 2 a 2 + 1 2 a E η /a ( q ) + O ( η 2 ) . (15) Hence, up to a q -indep enden t term, f η ( q ) = C ( η ) + 1 2 a E η /a ( q ) + O ( η 2 ) , (16) where C ( η ) does not dep end on q . Recall that the reduced free energy (or the Massieu function, with the opp osite sign) of a thermo dynamic system equals β U − S , where S is the en tropy , and U the in ternal energy . A t this p oint w e make an “ansatz” and in terpret f η as the reduced free energy , and E = E η /a ( q ) as the in ternal energy . W e consider f η as a function of E (cf. Theorem 4.1)(i)) and define the effectiv e inv erse temp erature by β eff := ∂ f η ∂ E     η =0 . Using (16), we obtain β eff = 1 2 a . (17) The assumption that f η is a kind of free energy (or, more generally , a thermo dynamic p oten tial) is motiv ated b y the fact that we view the jet space J 1 S 1 as a thermo dynamic phase space, and the physically meaningful Legendrians as equilibrium submanifolds of a thermo dynamic system. 2 The b o ost parameter η is dimensionless (it is a rapidit y). In physical units one has η = aτ /c , where c is the speed of light. 10 Relation to the b o ost parameter and the Unruh effect. The parameter η = at admits a natural interpretation as the L or entz b o ost p ar ameter along the worldline of a uniformly accelerating observ er. Indeed, the velocity of the observer is giv en by ˙ γ ( t ) = cosh( at ) A + sinh( at ) n, so that at is precisely the rapidit y (hyperb olic angle) of the corresp onding Lorentz transfor- mation. In quantum field theory , the b o ost parameter η plays an imp ortant role in the deriv ation of the Unruh effect. 3 W e tak e this as a black box and use η in the subsequen t considerations. Rewriting the generating function g t in terms of η and introducing the normalized quan- tit y f η therefore allo ws for a direct comparison with the thermo dynamic structure underlying the Unruh effect. F ormula (17) shows that the effectiv e inv erse temp erature has the same a − 1 scaling as the Unruh temp erature (1). Heuristic connection to the Unruh effect. The discussion b elow is based on the sim- plified deriv ation of the Unruh effect [1]. An accelerating observ er may b e viewed as probing the am bien t field b y receiving w av es arriving from all ligh tlike directions q ∈ S 1 . Each suc h direction corresp onds to a mo de of propagation, and the observ er measures its energy through the relativistic Doppler factor E t ( q ) = −⟨ ˙ γ ( t ) , ˆ q ⟩ . F or an inertial observer these energies would b e constan t in time, but along an acceler- ated tra jectory they acquire a nontrivial time dep endence. In particular, the frequencies of the incoming wa ves become time-dep enden t, so that eac h mode is p erceived not as a pure harmonic oscillation, but as a signal with time-dep endent frequency (a so-called “c hirp ed” signal) whose instan taneous frequency evolv es along the tra jectory . The total signal seen by the observer is therefore a sup erp osition ov er all directions q of suc h oscillations with time-v arying frequencies. A natural w ay to analyze this signal is to pass to its sp ectral decomp osition in the observ er’s proper time. In heuristic deriv ations of the Unruh effect, this sp ectral analysis yields a distribution of frequencies of the form N (Ω) ∝ 1 e 2 π Ω c/a − 1 . Here N (Ω) denotes the sp ectral densit y of the signal observed along the tra jectory , that is, the squared mo dulus of its F ourier transform, represen ting the distribution of frequencies in prop er time. This has exactly the Planck form 1 e ℏ Ω / ( kT ) − 1 , and the temperature is then iden tified by matching the exponents. This deriv ation, follo wing [1], yields the Unruh temp erature (1) (in units where ℏ = k = c = 1). 3 More precisely , the Mink o wski v acuum, when restricted to a Rindler w edge, is a thermal (KMS) state with resp ect to the generator of Loren tz b o osts. The associated imaginary-time perio dicity is 2 π in the v ariable η [11]. 11 In our mo del, instead of p erforming a F ourier analysis in time, w e enco de the same Doppler energy profile geometrically . The function g t ( q ) = log ε B ′ ( q ) − ρ ( t ) records, for each direction q , the logarithmic energy of the corresponding mo de. In this sense, g t ma y b e view ed as a generating function that pack ages the directional dependence of the observ ed frequencies. Its small-time expansion in volv es the energies E t ( q ), and, after an appropriate rescaling, it admits an in terpretation as a reduced free energy . The deriv ativ e of this quantit y with resp ect to the energy yields an effectiv e in v erse temp erature, whic h scales linearly with the acceleration, in agreemen t with the Unruh scaling. 4.5 Restoration of ph ysical units So far, w e hav e w ork ed with the unit hyperb oloid and therefore with dimensionless quan tities. In particular, the time v ariable t , the acceleration parameter a , and the function E t ( q ) are all dimensionless. T o restore ph ysical units, fix a length scale R > 0 and replace the unit hyperb oloid by K R = { x ∈ R 1 , 2 : ⟨ x, x ⟩ = − R 2 , x 0 > 0 } . Let τ denote the ph ysical prop er time and α the physical prop er acceleration. The dimen- sionless v ariables used throughout the pap er are then giv en b y t = cτ R , a = αR c 2 , η = at = ατ c . With this iden tification, all formulas of the previous sections remain v alid without mo di- fication. In particular, the function E t ( q ) dep ends on t only through the com bination at = η , so that E t ( q ) = E η /a ( q ) = cosh η − sinh η ⟨ n, ˆ q ⟩ . W e define the ph ysical energy by E η ( q ) := γ E η /a ( q ) = γ  cosh η − sinh η ⟨ n, ˆ q ⟩  , where γ > 0 is a constant with units of energy . With this con ven tion, formula (15) takes the form f η ( q ) = − 3 2 a + η 2 a 2 + 1 2 aγ E η ( q ) + O ( η 2 ) . (18) Since f η is dimensionless, the co efficient of E η has units of inv erse energy and can b e in terpreted as an effectiv e in v erse temp erature: β eff = 1 2 aγ = c 2 2 αR γ . Accordingly , T eff = 2 αR γ c 2 . In particular, the temp erature is prop ortional to the acceleration, in agreement with the Unruh scaling, although the numerical constant dep ends on the choice of the energy scale γ . 12 5 A statistical in terpretation as reduced free energy In this section we in tro duce randomness in the direction of the spatial acceleration v ector and in terpret the resulting generating function in terms of reduced free energy . 5.1 Randomization of the acceleration direction W e mo del the spatial acceleration direction by a random unit vector n ∈ S 1 . Fix a preferred direction n 0 ∈ S 1 and assume that n is distributed according to the von Mises distribution dµ κ ( n ) = 1 2 π I 0 ( κ ) exp  κ ⟨ n, n 0 ⟩  dϕ, κ ≥ 0 , where I 0 is the mo dified Bessel function of the first kind, defined by I 0 ( κ ) = 1 2 π Z 2 π 0 e κ cos θ dθ . W e also recall that I 1 ( κ ) = 1 2 π Z 2 π 0 e κ cos θ cos θ dθ. The parameter κ con trols the concentration: for κ = 0 the distribution is uniform, while for κ → ∞ it concentrates near n 0 . Introduce the b o ost parameter η := at, t = η a . With this notation the four–v elo cit y of the accelerating observ er from our mo del is giv en by u η /a = cosh η A + sinh η n , see (8). Define the Hamiltonian as the photon energy 4 H η ( n, q ) := ε u η/a ( q ) = cosh η − sinh η ⟨ n, ˆ q ⟩ . (19) F or a future use, recall that ⟨ A, ˆ q ⟩ = − 1. W rite A = (1 , 0 , 0) and ˆ q = ( α , ˆ q sp ), where ˆ q sp denotes the spatial comp onent of ˆ q . It follows that α = 1 and | ˆ q sp | 2 = 1. Define the partition function Z η ( q ) := Z S 1 e − β H η ( n,q ) dµ κ ( n ) , where β is the inv erse temp erature. Substituting (19), we obtain Z η ( q ) = e − β cosh η 2 π I 0 ( κ ) Z 2 π 0 exp  κ ⟨ n, n 0 ⟩ + β sinh η ⟨ n, ˆ q ⟩  dϕ. A direct computation yields Z η ( q ) = e − β cosh η I 0 ( R η ( q )) I 0 ( κ ) , (20) 4 T o k eep the units consistent, the expression on the right-hand side of (19) should be m ultiplied b y a constan t, sa y γ , with units of energy . F or simplicity , we tacitly assume γ = 1. 13 where R η ( q ) = q κ 2 + β 2 sinh 2 η + 2 κβ sinh η ⟨ n 0 , ˆ q ⟩ . (21) Indeed, κ ⟨ n, n 0 ⟩ + β sinh η ⟨ n, ˆ q ⟩ = ⟨ n, v η ( q ) ⟩ , where v η ( q ) := κn 0 + β sinh η ˆ q sp and ˆ q sp , as ab ov e, denotes the spatial comp onent of ˆ q . Hence | v η ( q ) | = q κ 2 + β 2 sinh 2 η + 2 κβ sinh η ⟨ n 0 , ˆ q ⟩ = R η ( q ) . W riting v η ( q ) = R η ( q ) m η ( q ) with m η ( q ) ∈ S 1 , w e obtain κ ⟨ n, n 0 ⟩ + β sinh η ⟨ n, ˆ q ⟩ = R η ( q ) ⟨ n, m η ( q ) ⟩ . Therefore Z S 1 exp  κ ⟨ n, n 0 ⟩ + β sinh η ⟨ n, ˆ q ⟩  dϕ = Z S 1 e R η ( q ) ⟨ n,m η ( q ) ⟩ dϕ. Since the measure dϕ is inv ariant under rotations of S 1 , the integral dep ends only on R η ( q ) and not on the direction m η ( q ). Th us w e can rotate co ordinates so that m η ( q ) = (1 , 0), and obtain Z S 1 e R η ( q ) ⟨ n,m η ( q ) ⟩ dϕ = Z 2 π 0 e R η ( q ) cos ϕ dϕ = 2 π I 0 ( R η ( q )) . Substituting this in to the definition of Z η ( q ) yields Z η ( q ) = e − β cosh η I 0 ( R η ( q )) I 0 ( κ ) . Equation (20) follo ws. Define the r e duc e d fr e e ener gy by F η ( q ) := − log Z η ( q ) . Using (20), we obtain F η ( q ) = β cosh η − log I 0 ( R η ( q )) + log I 0 ( κ ) . (22) Theorem 5.1. Assume that the sp atial ac c eler ation dir e ction is distribute d ac c or ding to the von Mises law with p ar ameter κ > 0 . Then, as η → 0 , F η ( q ) = β − β I 1 ( κ ) I 0 ( κ ) η ⟨ n 0 , ˆ q ⟩ + O ( η 2 ) . 14 Pr o of. Set s η := β sinh η = β η + O ( η 3 ) . Then R η ( q ) = q κ 2 + s 2 η + 2 κs η ⟨ n 0 , ˆ q ⟩ = κ + s η ⟨ n 0 , ˆ q ⟩ + O ( η 2 ) . Expanding log I 0 at κ , w e get (by using that I ′ 0 = I 1 ) that log I 0 ( R η ( q )) = log I 0 ( κ ) + I 1 ( κ ) I 0 ( κ ) s η ⟨ n 0 , ˆ q ⟩ + O ( η 2 ) . Substituting in to (22) yields F η ( q ) = β − β I 1 ( κ ) I 0 ( κ ) η ⟨ n 0 , ˆ q ⟩ + O ( η 2 ) . (23) This completes the pro of. Recall that E η /a ( q ) = cosh η − sinh η ⟨ n 0 , ˆ q ⟩ = 1 − η ⟨ n 0 , ˆ q ⟩ + O ( η 2 ) . Corollary 5.2. Put β = 1 / (2 a ) (se e (17) ). As κ → + ∞ , η → 0 , F η ( q ) = β E η /a ( q ) + O  κ − 1 η + η 2  . (24) Thus, if κ − 1 = O ( η ) , the r esc ale d gener ating function f η ( q ) := η − 1 g η /a ( q ) satisfies f η ( q ) = C ( η ) + F η ( q ) + O ( η 2 ) . Pr o of. The corollary immediately follows from (10), (23), (16), (14), and the expansion I 1 ( κ ) I 0 ( κ ) = 1 + O (1 /κ ) , κ → + ∞ . 5.2 Comparison with the classical rotor Consider a classical rotor whose configuration space is the unit circle S 1 (cf. [6]). Let u = (cos θ , sin θ ) b e a fixed unit vector represen ting the direction of an external field. The Hamiltonian of the rotor is H rot ( ϕ ) = − h n ( ϕ ) · u, n ( ϕ ) = (cos ϕ, sin ϕ ) . Here dot stands for the scalar pro duct in R 2 . The constan t h , with units of energy , will b e sp ecified later. A t the inv erse temp erature β > 0 the partition function is Z rot ( h, β ) = Z S 1 e − β H rot ( ϕ ) dϕ 2 π = Z 2 π 0 e ( β h ) cos( ϕ − θ ) dϕ 2 π = I 0 ( β h ) , 15 where I 0 is the mo dified Bessel function. The corresp onding reduced free energy is F rot ( h, β ) = − log Z rot ( h, β ) . W e consider the small–field regime with T fixed. Using the standard expansion I 0 ( x ) = 1 + x 2 4 + O ( x 4 ) , w e obtain F rot ( h, β ) = − β 2 h 2 4 + O ( h 4 ) . Recall (see Theorem 5.1) that the photon free energy is given by F η ( q ) = β − β I 1 ( κ ) I 0 ( κ ) η ⟨ n 0 , ˆ q ⟩ + O ( η 2 ) . Assume that ⟨ n 0 , ˆ q ⟩ > 0 . Cho ose the field strength h = 2 s β − 1 I 1 ( κ ) I 0 ( κ ) η ⟨ n 0 , ˆ q ⟩ . Substituting this in to the rotor free energy gives F η ( q ) = β + F rot ( h, β ) + O ( η 2 ) . (25) Hence, after an appropriate scaling of the field strength, the free energy of the classical rotor captures the leading directional dep endence of the photon free energy , up to a q – indep enden t term. Note that the field strength h dep ends on q ; th us w e are not comparing with a single fixed rotor, but rather with a family of rotor mo dels whose field is tuned to matc h the directional dependence. 6 Ho dograph transform-miscellanea 6.1 Hyp erb olic ho dograph-pro ofs Pro of of Prop osition 3.1: W e use co ordinates ( u, v ) on S ∗ H , with u ∈ H and v b eing a unit cov ector at u . Recall that λ = v du . W rite b q ( u ) as b ( q , u ). The hodograph map has the form H ( q , p, z ) = ( U, V ) with V = ∂ b ∂ u ( q , U ). Since z = b ( q , U ) and p = ∂ b ∂ q ( q , U ), we get dz = p dq + V dU, so that V dU = dz − p dq . This pro v es the prop osition. □ Next, let us pro ve that H is one-to-one and on to. T o this end w e w ork in the unit disc mo del of the hyperb olic plane. Let D = { u ∈ C : | u | < 1 } b e the Poincar ´ e disk with metric g u = 4 (1 − | u | 2 ) 2 | du | 2 . 16 W e iden tify K with the P oincar´ e disk via the standard radial pro jection from the origin onto the plane x 0 = 1. Note that under the identification of K with D , the ideal b oundary ∂ ∞ K corresp onds to the b oundary S 1 = ∂ D . Without loss of generalit y , the mark ed p oint in the definition of the Busemann function is the origin. F or q ∈ S 1 the Busemann function is giv en b y b q ( u ) := log 1 − | u | 2 | u − q | 2 , u ∈ D . Prop osition 6.1. L et S ∗ D = { ( u, ξ ) : ∥ ξ ∥ g ∗ u = 1 } b e the unit c ospher e bund le. Define the map Θ : S ∗ D − → J 1 S 1 , Θ( u, ξ ) = ( q , p, z ) , as fol lows: q = q ( u, ξ ) ∈ S 1 is the unique p oint such that ξ = d u b q , (26) and then z = b q ( u ) , p = ∂ ∂ ϕ b e iϕ ( u )    e iϕ = q , (27) wher e ϕ is an angular c o or dinate on S 1 (henc e p is the c ove ctor c omp onent in the c anonic al trivialization T ∗ S 1 ∼ = S 1 × R ). Then Θ is a diffe omorphism whose inverse is the ho do gr aph map H . Pr o of. Step 1: Θ is well-defined. W e first record tw o standard facts ab out Busemann functions on ( D , g ): (a) Unit sp e e d. F or every q ∈ S 1 , the Busemann function b q satisfies the eikonal equation ∥ d u b q ∥ g ∗ u = 1 , ∀ u ∈ D . (28) (Equiv alen tly , ∥∇ g b q ∥ g = 1.) (b) Uniqueness of q fr om ( u, ξ ) . Fix u ∈ D . The map S 1 ∋ q 7− → d u b q ∈ S ∗ u D is a diffeomorphism. Geometrically , d u b q is the unit co vector p ointing in the direction of the (unique) geo desic ray from u to q ; distinct q giv e distinct forw ard directions, and every forw ard direction determines a unique ideal endp oin t q . Hence for each ( u, ξ ) ∈ S ∗ D there exists a unique q satisfying (26), and then (27) defines ( p, z ) uniquely . Th us Θ is well-defined. Step 2: An explicit in v erse map J 1 S 1 → S ∗ D . W rite q = e iϕ ∈ S 1 and set a := e z > 0. W e shall reconstruct u from ( q , p, z ) and then set ξ = d u b q . By rotational symmetry , it suffices to solve the reconstruction for q = 1, and then rotate bac k. Let R q : D → D b e the rotation R q ( w ) = q w . Since b q ( u ) = b 1 ( ¯ q u ) (b ecause | u − q | = | ¯ q u − 1 | and | u | = | ¯ q u | ), if we set w = ¯ q u then z = b q ( u ) = b 1 ( w ) , p = ∂ ∂ ϕ b e iϕ ( u )    e iϕ = q = ∂ ∂ ϕ b e iϕ ( q w )    ϕ =arg q = ∂ ∂ ϕ b e iϕ ( w )    ϕ =0 . 17 Th us, without loss of generality , we ma y assume that q = 1 and solv e the equation Θ( w , ξ ) = (1 , p, z ) . Once w is determined, the cov ector is uniquely given by ξ = d w b 1 . W rite w = x + iy . Set d := | w − 1 | 2 = ( x − 1) 2 + y 2 , r 2 := | w | 2 = x 2 + y 2 . F rom the definitions, a = e z = 1 − r 2 d , p = ∂ ∂ ϕ b e iϕ ( w )    ϕ =0 = 2 y d . (29) (The last iden tity is a direct differentiation of b e iϕ ( w ) = log 1 − r 2 | w − e iϕ | 2 at ϕ = 0.) F rom (29) w e express y = pd 2 and r 2 = 1 − ad . Also d = | w − 1 | 2 = r 2 − 2 x + 1, hence x = 1 − (1 + a ) d 2 . (30) No w r 2 = x 2 + y 2 together with r 2 = 1 − ad , y = pd 2 , and (30) gives a single p ositive solution d = 4 (1 + a ) 2 + p 2 , (31) and then x = a 2 + p 2 − 1 (1 + a ) 2 + p 2 , y = 2 p (1 + a ) 2 + p 2 . (32) Equiv alen tly , w ( a, p ) = a 2 + p 2 − 1 (1 + a ) 2 + p 2 + i 2 p (1 + a ) 2 + p 2 . (33) Finally , define for general q ∈ S 1 : u ( q , p, z ) := q w ( e z , p ) . (34) This yields a smo oth map G : J 1 S 1 → D , G ( q , p, z ) = u ( q , p, z ) . Moreo ver, from (33) one chec ks | w ( a, p ) | 2 = ( a − 1) 2 + p 2 ( a + 1) 2 + p 2 < 1 , (35) so indeed u ( q , p, z ) ∈ D for all ( q , p, z ) ∈ J 1 S 1 . No w set H ( q , p, z ) :=  u ( q , p, z ) , d u ( q ,p,z ) b q  ∈ S ∗ D . (36) The mem b ership in S ∗ D follo ws from (28). 18 Step 3: Θ and H are in verse to each other. W e show Θ ◦ H = id and H ◦ Θ = id. (i) Θ ◦ H = id J 1 S 1 . T ake ( q , p, z ) ∈ J 1 S 1 and set ( u, ξ ) = H ( q , p, z ). By construction ξ = d u b q , so the first component of Θ( u, ξ ) is exactly q . It remains to c hec k that the resulting ( p ′ , z ′ ) coincide with ( p, z ). But u w as obtained by solving the system (29) with a = e z and p , hence for q = 1 w e ha ve b 1 ( w ) = z and ∂ ϕ b e iϕ ( w ) | ϕ =0 = p . Rotating back (using w = ¯ q u ) yields b q ( u ) = z , ∂ ∂ ϕ b e iϕ ( u )    e iϕ = q = p. Therefore Θ( u, ξ ) = ( q , p, z ). (ii) H ◦ Θ = id S ∗ D . T ake ( u, ξ ) ∈ S ∗ D and let ( q , p, z ) = Θ( u, ξ ). By definition, ξ = d u b q . Applying H pro duces H ( q , p, z ) =  u ( q , p, z ) , d u ( q ,p,z ) b q  . F rom part (i) we kno w that Θ( H ( q , p, z )) = ( q , p, z ), so in particular the u –comp onen t is uniquely determined by ( q , p, z ). Since ( u, ξ ) and H ( q , p, z ) ha v e the same ( q , p, z ) under Θ, their u –components coincide: u ( q , p, z ) = u . Consequen tly d u ( q ,p,z ) b q = d u b q = ξ , and hence H ◦ Θ = id. Th us Θ is bijectiv e with inv erse H . 6.2 Euclidean ho dograph transform and Busemann functions The classical Euclidean ho dograph transform admits a natural interpretation in terms of Busemann functions. Let q ∈ S 1 b e a direction in R 2 and consider the ray γ q ( t ) = tq , t ≥ 0. The asso ciated Busemann function is defined by b q ( x ) = lim t →∞  d ( x, γ q ( t )) − t  . A direct computation sho ws that in Euclidean space b q ( x ) = −⟨ x, q ⟩ . Th us b q is a linear function whose gradient equals − q . Using these functions one can define a map ( x, q ) 7− → j 1 q ( b q ( x )) from R n × S 1 to the 1–jet bundle J 1 S 1 . W riting z = b q ( x ) , p = ∂ q b q ( x ) , this map tak es the form ( x, q ) 7− → ( q , p, z ) . One readily chec ks that this map coincides with the classical Euclidean ho dograph trans- form. Indeed, writing q = (cos φ, sin φ ) and x = ( x 1 , x 2 ), w e hav e b q ( x ) = − ( x 1 cos φ + x 2 sin φ ) . 19 Hence z = b q ( x ) = − ( x 1 cos φ + x 2 sin φ ) , and p = ∂ φ b q ( x ) = x 1 sin φ − x 2 cos φ. This, up to a sign con v ention, coincides with formula (6.2) in [4]. This in terpretation shows that the hodograph transform arises from the family of Buse- mann functions asso ciated with directions q ∈ S 1 . In particular, the Euclidean ho dograph transform should b e viewed as the flat coun terpart of the hyperb olic ho dograph transform used ab ov e. Indeed, replacing the Euclidean distance by the hyperb olic distance in the def- inition of b q pro duces the hyperb olic Busemann functions, and the same jet construction yields the h yp erb olic ho dograph map. 7 Conclusion W e prop ose a dualit y b et ween the Loren tzian geometry of ligh t rays and thermo dynamics, based on the hyperb olic hodograph transform. The ho dograph transform is a contactomor- phism b et ween the spherical cotangent bundle of the Mink o wski h yp erb oloid in R 1 , 2 and the jet space of the circle. The hyperb oloid is a Cauc hy hypersurface of the c hronological future of the origin in Minko wski spacetime, while the jet space is view ed as a thermodynamic phase space. The in tensive v ariables (points of the circle) corresp ond to future light directions in R 1 , 2 , and the thermo dynamic p oten tial (the R -co ordinate on J 1 S 1 = R × T ∗ S 1 ) corresp onds to a reduced free energy . W e follow the tra jectory of an observer starting on K and undergoing uniform acceleration with proper acceleration a . W e trac k the ev olution of its skies and consider their (suitably rescaled) images under the ho dograph map. The corresp onding Legendrians are given b y generating functions, whic h w e interpret as reduced free energies. This yields a dualit y , some facets of whic h are summarized in the following table. Thermo dynamics Relativit y 1. Legendrian submanifold generated b y the reduced free energy (thermo dynamic equi- librium) Legendrian sky of an ev en t 2. Internal energy E Photon energy measured in direction q b y t wo observers – Doppler effect (Busemann function) 3. T emp erature 2 a Unruh temp erature a/ 2 π 4. Entrop y S Dimensionless b o ost parameter at (Rindler time) 5. Reeb c hord b et w een Legendrians in 1 (ul- trafast pro cess) Ligh t ray b etw een even ts 6. Quasistatic thermo dynamic pro cess Deformation of skies along causal paths Some commen ts are in order. The first tw o lines summarize the main constructions of this pap er, see (16) and (9), resp ectively . The third line is explained in Section 4.4. 20 Line 4 deserv es a sp ecial discussion. Recall that in equilibrium thermo dynamics, if f = β F denotes the reduced free energy , then the entrop y is giv en b y s = β ∂ f ∂ β − f . Using the unit-consisten t form (18), f η ( q ) = − 3 2 a + η 2 a 2 + 1 2 aγ E η ( q ) + O ( η 2 ) , w e rewrite it in thermo dynamic form as f η ( q ) = − 3 γ β + 2 η ( γ β ) 2 + β E η ( q ) + O ( η 2 ) , where η is view ed as a parameter and β = 1 2 aγ is the effectiv e in v erse temp erature. A direct computation yields s = 2 η ( γ β ) 2 . Substituting β = (2 aγ ) − 1 , w e obtain s = η 2 a 2 . Th us the entrop y is prop ortional to the b o ost parameter η (often called the Rindler time). This is natural, since entrop y and time enco de the causal structures in thermo dynamics and relativit y , resp ectiv ely (see e.g. [9]). Finally , lines 5 and 6 allude to the classification of thermodynamic processes within the con tact-top ological framew ork prop osed in [7]. As a final remark, all our results extend to higher dimensions. The corresp onding ho do- graph map is a con tactomorphism betw een the spherical cotangent bundle S ∗ H n of the n - dimensional hyperb olic space and the jet bundle of the ( n − 1)-dimensional sphere, J 1 S n − 1 . The ph ysical mo del and its consequences remain intact. Ac knowledgmen t. This pap er was written during the 2026 program “Contact geometry , general relativity and thermodynamics” at the Simons Cen ter for Geometry and Physics at Ston y Bro ok. I thank the Simons Center for the excellen t research atmosphere. I am esp ecially grateful to Shin-itiro Goto for n umerous helpful commen ts and suggestions. I also thank Alessandro Bra v etti, Olaf M ¨ uller, and Miguel S´ anchez Ca ja for useful discussions and v aluable feedback on the manuscript, as w ell as Stefan Nemiro vski for pro viding references on the h yp erb olic ho dograph map. References [1] Alsing PM, Milonni PW. Simplified deriv ation of the Hawking–Unruh temp erature for an accelerated observer in v acuum. American Journal of Physics. 2004 Dec 1;72(12):1524-9. 21 [2] Alv arez P aiv a JC. The symplectic geometry of spaces of geo desics. Rutgers The State Univ ersity of New Jersey , School of Graduate Studies; 1995. [3] Arnold VI. T op ological inv ariants of plane curves and caustics. American Mathematical So c.; 1994. [4] Chernov V, Nemiro vski S. Legendrian links, causalit y , and the Low conjecture. Geo- metric and F unctional Analysis. 2010 F eb;19(5):1320-33. [5] Chernov V, Nemirovski S. Redshift and contact forms. Journal of Geometry and Ph ysics. 2018 Jan 1;123:379-84. [6] Demch uk E, Singh H. Statistical thermo dynamics of hindered rotation from computer sim ulations. Molecular Physics. 2001 Apr 20;99(8):627-36. [7] Ento v M, Poltero vic h L, Ryzhik L. A con tact top ological glossary for non-equilibrium thermo dynamics. arXiv:2501.05955. 2025 Jan 10, to app ear in Letters in Mathematical Ph ysics. [8] F errand E. Sur la structure symplectique de la v ari´ et ´ e des g´ eo d´ esiques d’un espace de Hadamard. Geometriae Dedicata. 1997 No v;68(1):79-89. [9] Hawking S, P enrose R. The Nature of Space and Time, Princeton Univ ersity Press, 1996. [10] Pereira RF. The kinematic space of sp ecial relativity and its hyperb olic geometry (Do c- toral dissertation, Univ ersidade de S˜ ao P aulo). [11] Witten E. Introduction to black hole thermo dynamics. The Europ ean Ph ysical Journal Plus. 2025 Ma y 22;140(5):430. 22