Exact solution of the relativistic magnetohydrodynamic equations in the background of a plane gravitational wave with combined polarization

Gr av. Cosmol. No. 1, 2011 Exact solution of the rela tivistic magnetoh ydro dynamic equations in the bac kground of a plane gra vitational w a v e with com bined p olarization 1 A. A. Agathono v and Y u. G. Ignaty ev Kazan State Pe dago gic al University, Mezhlauk str. 1, Kazan 420021, Russia W e obt ain an exact solution of the self-consisten t relativistic magnetohydrodyn amic eq uations for an anisotropic magnetoactiv e plasma in the bac k ground of a plane gravitational wa ve metric (PGW) with an arbitrary p olarization. It i s sho wn that, in the linear approximation in the gra vitational wa v e amplitude, only the e + p olarization of the PGW in teracts with a magnetoactiv e p lasma. 1 In tro duction In a series of previo us articles by o ne o f the a uthors (see, e.g., [1–3] a theor y o f gr avimagnetic sho ck waves in a ho- mogeneous magnetoac tive plasma has b een develop ed. The es sence of t his phenomenon is that a mag netized plasma in ano malously strong magnetic fields drifts un- der the a ction of gravitational wa v es (GWs) in the GW propaga tion direction under the co ndition that the wav e amplitude is large enough, and, on a certain wa v e fro nt , the plas ma velo city tends to the sp eed of lig ht. Its en- ergy dens it y a nd the intensit y of the frozen-in magnetic field then tend to infinit y . In the subsequent pap er s this effect was proved on the basis of the kinetic theo r y , and the p os s ibilit y of using this mechanism as a n effective to ol for detecting GWs from a strophysical s ources w as also s hown. How ever, in all cited pap ers, a monop olar - ized gravitational wav e w as considered. In the present pap er we consider the action of a GW with c o mbined po larizatio n on a magnetoa ctive plasma. 2 Self-consisten t RMHD equations in a gra vitational field In [1 ], under the assumption that the dyna mic velo city of the plasma ( v i ) is equa l to that of the electr o magnetic field 2 p T ij v j = ε p v i ; f T ij v j = ε f v i , ( v , v ) = 1 , (1) a full self-consistent set of relativistic magnetohydrody- namic equatio ns for a mag netized plasma in arbitrary gravitational field has b een obtained. It c onsists of the 1 T alk given at the Internat ional Conference RUDN-10, June 28 — July 3, 2010, PFU R , M osco w 2 The index “ p ” refers to the pl asma, the i ndex “ f ” to the field, the comma denote a co v ariant deriv ative. The dynamic v elocity of any kind of matter is, by definition, a timelike unit eigenv ector of the energy-momentum tensor of this m atter [4] . Maxwell equa tions of the first group ∗ F ik ,k = 0 (2) with the necessa ry and sufficient co ndition Inv 1 = F ij F ij = 2 H 2 > 0 , (3) Inv 2 = ∗ F ij F ij = 0 , (4) the Maxwell e quations of the seco nd group 3 : F ik ,k = − 4 π J i dr (5) with a spacelike drift cu rr ent J i dr = − 2 F ik p T l k,l F j m F j m , ( J dr , J dr ) < 0 (6) and a conse r v ation law for the tota l energy -momentum of the sys tem T ik ,k = p T ik ,k + f T ik ,k = 0 . (7) The energy- momentum tensor (EMT) of the elec- tromagnetic field, in the case of a coincidence o f the plasma’s and the field’s dynamic velocities (1), is ex- pressed throug h a pa ir of vectors, v and H [1 ]: f T i k = − 1 8 π  ( δ i k − 2 v i v k ) H 2 + 2 H i H k )  . (8) The E MT of a relativistic anis otropic magnetoactive plasma in gr avitational and mag ne tic fields is (see, e.g ., [3]) p T ij = ( ε + p ⊥ ) v i v j − p ⊥ g ij + ( p k − p ⊥ ) h i h j , (9) where h i = H i /H is the spacelike unit vector of the magnetic field ( ( h, h ) = − 1 ); p ⊥ and p k are the plasma pressures in the directions or thogonal and parallel to the magnetic field, res pe ctively . 3 c = G = ~ = 1 2 3 Solving the RMHD equat ions in the PGW metric Consider a solution of the Cauch y problem of the s elf- consistent RMHD e q uations in the background of a v ac- uum gravitational-wav e metric (see, e.g., [5]) 4 : ds 2 = 2 dudv − L 2  cosh 2 γ  e 2 β ( dx 2 ) 2 + 2 e − 2 β ( dx 3 ) 2  − sinh 2 γ dx 2 dx 3  , (10) with homog eneous initial co nditions on the null hyper - surface u = 0 : β ( u ≤ 0) = 0; β ′ ( u ≤ 0) = 0; L ( u ≤ 0) = 1 , (11) W e assume the following: • the plasma is homog eneous and at rest: v v ( u ≤ 0) = v u ( u ≤ 0) = 1 / √ 2; v 2 = v 3 = 0; ε ( u ≤ 0) = 0 ε ; p k ( u ≤ 0) = 0 p k ; p ⊥ ( u ≤ 0) = 0 p ⊥ ; (12) • a homoge neo us magnetic field is directed in the ( x 1 , x 2 ) plane: H 1 ( u ≤ 0) = 0 H cos Ω ; H 2 ( u ≤ 0) = 0 H sin Ω ; H 3 ( u ≤ 0) = 0 , E i ( u ≤ 0) = 0 , (13) where Ω is the angle b e t ween the axis 0 x 1 (the PGW propa g ation direction) a nd the ma gnetic field H . The metric (1 0) admits the g roup of is o metries G 5 , asso ciated with three linear ly independent (at a p oint) Killing vectors ξ i (1) = δ i v ; ξ i (2) = δ i 2 ; ξ i (3) = δ i 3 . (14) Due to their existence in the metric (1 0), all geometr ic ob jects, including the Christoffel symbo ls , the Riemann tensor, the Ricci tens or and consequently the EMT of a magnetoactive pla sma, are automatically co nserved at motions along the Killing directions: L ξ α g ij = 0 ⇒ L ξ α R ij = 0 ⇒ L ξ α T ij = 0 , (15) 4 β ( u ) and γ ( u ) are the ampl i tudes of the p olarizations e + and e × , resp ectiv ely; u = ( t − x 1 ) / √ 2 i s the retarded tim e, v = ( t + x 1 ) / √ 2 is the adv anced time. The PGW amplitudes are arbi trary functions of the retarded time u , and L ( u ) is a bac kground factor of the PGW. where L ξ T ij is a Lie deriv a tive in the dir ection of ξ . W e further require that the E MTs of the pla sma p T ij and the electr o magnetic field f T ij inherit the symmetry s ep- arately . Thus all observed physical quantities P inherit the symmetry of the metric (1 0): L ξ α P = 0 ( α = 1 , 3) , (16) i.e., ta king into account the explicit form of the K illing vectors (14), p = p ( u ) , ε = ε ( u ) , v i = v i ( u ); (17) F ik = F ik ( u ) , H i = H i ( u ) , h i = h i ( u ) . (18) The vector p otential agreeing with the initial condi- tions (13) is A v = A u = A 2 = 0 ; A 3 = 0 H ( x 1 sin Ω − x 2 cos Ω); ( u ≤ 0) . (19) In the presence of a PGW, the v ector p otential b ecomes A 2 = A v = A u = 0 ; A 3 = 0 H  1 √ 2 ( v − ψ ( u )) sin Ω − x 2 cos Ω  , (20) where ψ ( u ) is an ar bitrary function of the r e ta rded time, satisfying the initial condition ψ ( u ≤ 0 ) = u. (21) Thu s the magnetic field freezing-in condition in t he plasma reduces to the t wo equalities v 3 = 0 , 1 √ 2 ( v v ψ ′ − v u ) sin Ω + v 2 cos Ω = 0 . (22) The cov ariant co mpo nents of the vector of magne tic field int ensity r elative to the Maxwell tensor are H v = − 0 H L 2  v v cos Ω + 1 √ 2 v 2 sin Ω  (23) H u = 0 H L 2  v u cos Ω − 1 √ 2 v 2 ψ ′ sin Ω  , (24) H 2 = − 1 √ 2 0 H cosh 2 γ e 2 β sin Ω ( v v ψ ′ + v u ) , (25) H 3 = 1 √ 2 0 H sinh 2 γ sin Ω( v v ψ ′ + v u ) . (26) The magnetic field in tensity squar ed is H 2 = 0 H 2 L 4 ( L 2 ψ ′ cosh 2 γ e 2 β sin 2 Ω + cos 2 Ω) . (27) 3 Using (23)-(27), the norma liz a tion relation for the velocity vector can be written in the equiv a le nt form  v v cos Ω + v 2 1 √ 2 sin Ω  2 = H 2 0 H 2 v 2 v L 4 − sin 2 Ω 2 L 2 cosh 2 γ e 2 β . (28) The comp onents of the drift cur rent are J i dr = − 1 4 π L 2 ∂ u ( L 2 F iu ) . (29) Then, J v dr = J u dr = 0 , (30) J 2 dr = − 0 H sin Ω 2 √ 2 π L 2 cosh 2 γ · γ ′ , (31) J 3 dr = − 0 H sin Ω e 2 β 2 √ 2 π L 2 (sinh 2 γ · γ ′ + co sh 2 γ · β ′ ) . (32) Because o f existence of the isometries (14), we obta in the following in tegrals [1]: L 2 ξ ( α ) i T vi = C a = c o nst ( α = 1 , 3) . (33) W e consider only the c a se of tr ansverse PGW pr op- agation ( Ω = π / 2 ). Then, s ubstituting the e xpressions for the plasma and electro magnetic field EMT into the int egrals (33 ), using the relations (26 )-(28) and also the initial c o nditions (11 ), we bring the integrals of motion to the for m 2 L 2 ( ε + p k ) v 2 v − ( p k − p ⊥ ) 0 H 2 H 2 cosh 2 γ e 2 β = ( 0 ε + 0 p )∆( u ) , (34) L 2 ( ε + p k ) v v v 2 = 0 , (35) L 2 ( ε + p k ) v v v 3 = 0 , (36) where 0 p = 0 p ⊥ , (37) and the so-c a lled governing function of the GMSW is int ro duced: ∆( u ) = 1 − α 2 (cosh 2 γ e 2 β − 1 ) , (38) with the dimensionless p ar ameter α 2 , α 2 = 0 H 2 4 π ( 0 ε + 0 p ) . (39) Solving (3 4) with resp ect to v v , we o btain expres- sions for the co mpo nents of the velo city vector a s func- tions of the sca la rs ε , p k , p ⊥ , ψ ′ and explicit functions of the re tarded time: v 2 v = ( 0 ε 0 p ) 2 L 2 ( ε + p k ) ∆( u ) + ( p k − p ⊥ ) ( ε + p k ) 0 H 2 H 2 cosh 2 γ e 2 β 2 L 2 . (40) F ro m (35), (36) we get: v 2 = v 3 = 0 . (41) W e obtain the comp onent v u from the normaliza tion relation for the velocity vector, using (40) and (41 ): v u = 1 2 v v , (42) and from the freezing -in condition (22) we get the v alue of the der iv ative of p otential ψ ′ : ψ ′ = 1 2 v 2 v . (43) Using it, the sca lar H 2 is determined from the r elation (27): H 2 = 0 H 2 L 2 cosh 2 γ e 2 β 2 v 2 v . (44) F ro m the RMHD set of equations it is p ossible to obtain the following differen tial equation in the P GW metric: L 2 ε ′ v v + ( ε + p k )( L 2 v v ) ′ + 1 2 L 2 ( p k − p ⊥ ) v v (ln H 2 ) ′ = 0 . (45) T o solve this equa tion, it is necessa ry to impo se tw o additional relations b etw een the functions ε , p k , a nd p ⊥ , i.e., a n eq ua tion of state: p k = f ( ε ) ; p ⊥ = g ( ε ) . (46) 4 Barotropic equation of state Consider a baro tr opic equation of s tate of the anisotropic plasma, where the r elations (46) ar e linear: p k = k k ε ; p ⊥ = k ⊥ ε . (47) Equation (45) is eas ily in tegrated under the co nditions (47), and we get one more integral: ε ( √ 2 L 2 v v ) (1+ k k ) H ( k k − k ⊥ ) = 0 ε 0 H ( k k − k ⊥ ) . (48) In the case o f a barotro pic eq uation of state under the conditions (47), s ubstitution of (44) into (40) results in v 2 v = 1 2 0 ε L 2 ε ∆( u ) . (49) 4 Substituting (44) and (49 ) in to (48 ), we o btain a closed equation with res p ect to the v ariable ε , whos e solution gives: ε = 0 ε h ∆ 1+ k ⊥ L 2(1+ k k ) (cosh 2 γ e 2 β ) k k − k ⊥ i − g ⊥ , (50) v v = 1 √ 2  ∆ L ( k k + k ⊥ ) (cosh 2 γ e 2 β ) k k − k ⊥ 2  g ⊥ , (51) H = 0 H  ∆ L (1+ k k ) (cosh 2 γ e 2 β ) − 1 − k k 2  − g ⊥ , (52) where g ⊥ = 1 1 − k ⊥ ∈ [1 , 2] . (53) In particular , fo r an ultrarela tivistic plasma with zer o parallel pres sure, k k → 0 ; k ⊥ → 1 2 (54) we obtain from (50)–(53): v v = 1 √ 2 L ∆ 2 (cosh 2 γ e 2 β ) − 1 / 2 , (55) ε = 0 ε L − 4 ∆ − 3 (cosh 2 γ e 2 β ) , (56) H = 0 H L − 2 ∆ − 2 (cosh 2 γ e 2 β ) . (57) 5 The energy balance equation In [1 ], it has b een shown that the singular state, which exists in a magnetized plasma under the co ndition 2 β 0 α 2 > 1 on the hypers ur face ∆( u ∗ ) = 0 , (58) is r emov ed using the bac k reac tio n of the magnetoa ctive plasma on the GW. That leads to e fficient absor ptio n of GW energy by the plasma and a r estriction o n the GW amplitude. A qua litative analysis of this s itua tion can be carr ied out using a simple mo del of energy ba lance prop osed in [2]. The energy flow of the magnetoactive plasma is dir ected a long the P GW pro pagation dire c - tion, i.e., a long the x 1 axis. Let β ∗ ( u ) and γ ∗ ( u ) b e the v acuum P GW amplitudes. In the WKB approxima- tion, 8 π ε ≪ ω 2 , (59) where ω is the c haracteris tic PGW freq ue nc y a nd ε is the matter energy density , all functions still dep end o n the retarded time only (see [6]). Thus β ( u ) and γ ( u ) are the PGW amplitudes sub ject to absor ption in pla s mas. The lo cal ener gy conserv a tion law should b e satisfied: T 41 ( β , γ )+ g T 41 ( β , γ ) = g T 41 ( β ∗ , γ ∗ ) , (60) where g T 41 ( β , γ ) is the energ y flow of a weak GW in the direction 0 x 1 (see [7]). In the case of tr ansversal P GW pr opaga tion and with a ba rotro pic equa tion of state of an anisotr o pic plasma, us ing the solutions of ma gnetohydrodynamics and Eqs. (5 0), (51), (52) with the dimensionless param- eter α 2 (39), one can obta in the ener gy balance equation in the form 0 H 2 4  ∆ − 4 g ⊥ − 1   1 α 2 + 1  + ( γ ′ ) 2 + ( β ′ ) 2 = ( γ ′ ∗ ) 2 + ( β ′ ∗ ) 2 . (61) Since, in a linear approximation by smallnes s of the a m- plitudes β and γ , the governing function (38) do es not depe nd on the function γ ( u ) , ∆( u ) = 1 − 2 α 2 β + O ( β 2 , γ 2 ) , (62) and the functions β ( u ) , γ ( u ) are a rbitrar y and func- tionally indep endent, then, up to β 2 , γ 2 , the r elation (61) can b e split into tw o independent parts: 2 0 H 2 g ⊥ (1 + α 2 ) β + ( β ′ ) 2 = ( β ′ ∗ ) 2 , (63) ( γ ′ ) 2 = ( γ ′ ∗ ) 2 . (64) Here, acco rding to the meaning o f the lo ca l energy bal- ance equa tion, we consider short gravitational wa v es (59), so we can neglect the squa res o f the PGW ampli- tudes as compa red with the sq ua res of their deriv ativ es with r esp ect to the retarded time. Th us, a ccording to (64), γ ∗ ( u ) = γ ( u ) , (65) i.e., in the linear approximation, a weak gravitational wa ve with the po larization e × do es not interact with a magnetized plas ma. This coincides with the conclus io n of the paper [8 ]. Thus the energy bala nce equation takes the form obtained in [3]: ˙ ∆ 2 + ξ 2 Υ 2 h ∆ − 4 g ⊥ − 1 i = Υ 2 sin 2 ( s ) , (66) where ξ 2 is the so-calle d first p ar ameter of the GMSW [2]: ξ 2 = 0 H 2 4 β 2 0 ω 2 , (67) Υ = 2 α 2 β 0 (68) — the se c ond GMSW p ar ameter . The do t denotes differ- ent iation with resp ect to the dimensionles s time v aria ble s , s = √ 2 ω u. 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