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Kinetic model of GMSW in an anisotropic plasma Yu.G. Ignatev Kazan State PedagogicalUniversity 1 Mezhlauk Str., Kazan 420021,Russia Abstract 1 1 A kinetic model of gravimagnetic shock waves (GMSW) in a locally 0 anisotropicplasmaisinvestigated. Theequationsofadriftapproximation 2 are written, and themomentsof thedistribution function are calculated. n Solutions of the drift equations for a highly anisotropic ultrarelativistic a plasma are found. It is shown that in this case the GMSW essentially J affecttheangularcharacteristicsandtheintensityofthemagneto-brem- 5 sstrahlung of themagnetoactive plasma. ] c 1 Introduction q - r In Ref. [1], from the requirement that the dynamical velocity of plasma be g equalto that of the electromagneticfield due to the Einstein equations and the [ first group of the Maxwell equations, the equations of the relativistic magnetic 1 hydrodynamics (RMHD) of magnetoactive plasma in a gravitational field were v derived. In the same work (see also [2]) a remarkable class of exact solutions 2 0 of the RMHD equations was obtained; it describes the motion of a magnetoac- 1 tive locally isotropic plasma in the field of a plane gravitational wave (PGW) 1 and is called gravimagnetic shock waves (GMSW). In Ref. [3] it was shown . 1 that in pulsar magnetospheres the GMSW are a highly effective detector of the 0 gravitaionalradiation of neutron stars. An observed consequence of the energy 1 transformation from the gravitaional wave to the GMSW energy are the so- 1 called giant pulses, sporadically arising in the radiationof a number of pulsars. : v The estimates made in [3] and [4] allow one to identify the giant pulses in the i pulsar NP 0532 radiation with the gravitational radiation of this pulsar in the X basic quadrupole mode of a neutron star. r a A fundamental importance of GMSW for the gravitational theory leads to thenecessityofamoredetailedandcomprehensiveinvestigatonofthisphenom- ena. As was shownin [1] and [3], a GMSW is realizedin an almost collisionless and nonequilibrium plasma situated in an abnormally strong magnetic field. Under these conditions, as a consequence of strong magneto - bremsstrahlung, the isotropy of the local distribution of plasma electrons is significantly violated,aswasassumedinobtainingthe solutionin[1]. In[5],onthe basisofthe general RMHD equations, a hydrodynamic model of GMSW in an anisotropic plasma was built. Since in the hydrodynamical approach the number of equations obtained is smaller than that of unknown functions, we had to postulate a relation between the longitudinal and transverse components of the plasma pressure. In [5] the simpliest variant of such a (linear) relation was studied. The study revealed a high dependence of the GMSW process on the degree of 1 plasma anisotropy, leading to the necessity of building a dynamical model of anisotropic magnetoactive plasma motion in the gravitaionalradiation field. It is the problem dealt with in the present paper. Throughout the paper a set of units is used where (c=G=~=1). 2 Drift solution to the kinetic equation for an anisotropic plasma 2.1 Field quantities in the plane gravitaional wave metric Let us study a collisionless plasma in the PGW metric [6]: ds2 =2dudv L2[e2β(dx2)2+e 2β(dx3)2] 2dudv A(dx2)2 B(dx3)2,(1) − − ≡ − − where β(u) is an arbitrary function (the PGW amplitude), the function L(u) (the backgroundfactor of the PGW) obeys a second-orderordinary differential equation; u = 1 (t x1) is the retarded time, v = 1 (t+x1) is the advanced √2 − √2 time. The absolute future corresponds to the range T+ : u > 0;v > 0 , the { } absolutepasttoT : u<0;v <0 . Themetric(1)admitsagroupofmotions − { } ,withthecorrespondingthreelinearlyindependentKillingvectorsatapoint: 5 G ξi=δi; ξi=δi; ξi=δi. (2) v 2 3 (1) (2) (3) Let there be no GW for u60, i.e., – β(u) =0; L(u) =1, (3) u60 u60 | | and the homogeneous magnetic field be directed in the plane x1,x2 : { } H =H cosΩ; H =H sinΩ; 1u60 0 2u60 0 | | H =0; E =0, (4) 3u60 iu60 | | where Ω is an angle between the 0x1 axis (PGW propagation direction) and the magnetic field direction H. The conditions (4) correspond to the vector potential A =A =A =0; A =H (x1sinΩ x2cosΩ); (u60). (5) v u 2 3 0 − The electromagnetic field in the PGW metric (1) for an initially homogeneous plasma is described by the vector potential [1]: A =A =A =0; 2 v u 1 A = H x2cosΩ+ H [v ψ(u)]sinΩ, (6) 3 0 0 − √2 − whereψ(u)isanarbitrarydifferentiablefunctionsatisfyingtheinitialcondition: ψ =u. (7) u60 | 2 In this case the only nonzerocomponent ofthe Maxwelltensordepending on ψ is 1 F = H ψ sinΩ. (8) u3 0 ′ −√2 Other nontrivial components of the Maxwell tensor are: 1 F = H cosΩ; F = H sinΩ. (9) 23 0 v3 0 − √2 2.2 Collisionless kinetic equation Thecollisionlesskineticequationforthe8-dimensionaldistributionfunctionfor charged particles of a kind a, F (xi, ) has the form [7]: a i P ∂F ∂ ∂F ∂ a a a a [ ,F ] H + H =0, (10) Ha a ≡ ∂xi ∂ ∂ ∂xi i i P P where =p +e A (11) i i a i P is the generalized momentum of a particle and 1 (xi, )= gij( e A )( e A ) (12) a i i a i j a j H P 2 P − P − is the Hamiltonian function of a charged particle. Theprocessofobtainingequationsinthedriftapproximationonthebasisof the collisionless kinetic equations (10) was described by the author in [8]. Here we only somewhat transform the process for the case of an initially anisotropic distribution. Note,besides,thatinthecaseΩ=π/2theresults[8]areerroneous 6 because they use in the role of an integral of motion, which is the case only 2 P if Ω=π/2. That was mentioned in the work cited above [1]. Let us study the case when the GW propargates perpendicularly to the magnetic field, Ω = π/2, and let us look for solutions of the kinetic equation independent of the variables v,x2 and x3 . As in [8], let us introduce the unit timelike vector vi: ψ 1 v =v =0; v = ′ ; v = (13) 2 3 u r 2 v r2ψ ′ andtransformthekineticequationintotheframeofreference(FR)movingwith the velocity vi: =v ( + ) =v ( ). (14) v u 4 1 u v 4 1 P P P P P −P The Jacobian of this transformation is equal to unity: D( , ) v u P P =1. (15) D( , ) 1 4 P P 3 As a result, we get the equation: ∂F ∂F ∂F eH ∂F a a a a v ( + ) v ( + ) + v P4 P1 ∂u − v′ P4 P1 (cid:18)P1∂ 4 P4∂ 1(cid:19)− √BP3∂ 1 P P P 1 ∂F ∂F + (A 1) 2+(B 1) 2 v a + a =0, (16) 2 − ′P2 − ′P3 u(cid:18)∂ ∂ (cid:19) (cid:2) (cid:3) P4 P1 where: 2 e2β H2 = ∂ A ∂ A =H2ψ (17) −B u 3 v 3 0 ′ L2 is aninvariantofthe electromagneticfield (the magneticfield intensity squared in the FR moving with the velocity vi). On the PGW front the solution (16) must satisfy the initial condition corresponding to a homogeneous anisotropic currentless plasma: 1 F (xi, ) =f (p2,p2)δ( m2), (18) a Pi |u=0 a ⊥ k Ha− 2 a where f is an arbitrary function of its arguments and the following notations a are introduced: p2 =p2+p2; p = p . (19) ⊥ 1 3 k − 2 The kinetic equation (16) has three exact integrals: 1 (xi, )= m2 =Const; Ha Pi 2 a =Const; =Const. (20) 2 3 P P For a complete solutionofthe problem we need one more independent integral. 2.3 Drift approximation We shallsolve Eq. (16) in the drift approximation,when the Larmorfrequency for each kind of charged particles – e H a ω = (21) a m a is much greater than the characteristic frequency ω of the gravitationalwave: ω Λ= 1. (22) ω ≪ a In the zeroth order with respect to the parameter Λ (16) takes the form: ∂F a H =0, (23) ∂ 1 P 4 i.e. in the drift approximation F is independent obviously of . Thus in the a 1 P drift approximation, apart from the above exact integrals, the kinetic equation has also drift (approximate) integrals [8]: Const; u Const. (24) 4 P ≈ ≈ The solution of the kinetic equation corresponding to the drift approximation, which, in the absence of GW, is transformed to an anisotropic and currentless one, can be written in the form 1 F =f 2 2, 2,u δ( m2), (25) a a P4 −P2 P2 Ha− 2 a (cid:0) (cid:1) where f is an arbitrary function of its arguments. In particular, the following a fa can be chosen: 0 fa =fa hµ−⊥2(P42−P22−m2a)+µ−k2P22i , (26) where µ (u) and µ (u) are arbitraryfunctions of their arguments. Thus in the FR(14)kmovingwi⊥ththe velocityvithe driftsolutionlocallycoincideswiththe unperturbed distribution (18). Substituting the obtained distribution function of the zero drift approximation (25) into the kinetic equation (16), it is easy to derive a correction of the first drift approximation, δF Λ 1, for the distribution function determined a − ∼ uptoanadditivecomponentbeinganarbitraryfunctionoftheabovedriftinte- grals(see [8]). However,exactconsequencesofthe collisionlesskinetic equation (16)areconservationlawsforthe numberofeachkindofparticlesandthetotal energy-momentum tensor (EMT) of the plasma and the electromagnetic field [9]. These laws impose certain restrictions on the above additive component and lead to differential equations for the functions µ (u) and µ (u). Due to the symmetry properties, it turns out thakt the first o⊥rder correction tothedistributionfunctioncontributesonlytothecomponentn oftheparticle 3 number current density vector and the components Ta and Ta of the particle v3 u3 EMT.However,thecomponentn3(u,v)doesnotaffectthecontinuityequation, only the above two components of the particle EMT appear in the transport equationofthe totalEMTforthe componenti=3they donotappearinother transport equations. On the other hand, the drift current determined by the firstcorrectionto the distributionfunction canbe obtainedasa consequenceof the energy-momentum conservation laws and the Maxwell equations, without addressing to a solution of the kinetic equation (see [1]). As a result, it turns out that the set of the transport equations and the Maxwell equations split into two subsets, one of which, determined by the zero drift approximation, is self-consistentandclosedandentirelydeterminesthefunctionsψ(u), µ (u)and µ (u). Thecorrectionofthefirstdriftaproximationtothedistributionkfunction ⊥ does not affect the magnetic hydrodynamics equation. 5 3 Derivation of the magnetic hydrodynamics equations 3.1 Algebraic structure of the distribution function moments Returning to the original FR by means of (14), let us introduce the following scalars: p =(p,n); p2 =(vivj gij)p p ; i j k − p =p2 p2 =(vivj gij ninj)p p , (27) i j ⊥ − k − − where ni is the unit spacelike vector in the direction of the magnetic field intensity vector Hi =vj F∗ji: H n = i ; H2 = (H,H). (28) i H − then the distribution function in the zero drift approximation (26) takes the form 0 fa =fa (µ−2p2+µ−2p2). (29) k k ⊥ ⊥ It is not difficult to show that the moments of this ditribution are: ni(x)= f (x,p)pidP ; a a Z P(x) Tij(x)= f (x,p)pipjdP , (30) a a Z P(x) where √ g(2S+1)dp1dp2dp3 dP = − (2π~)3p 4 (S - is the particle spin), and have the following algebraic structure: ni(u)=nvi; (31) a Tij(u)=(ε+P )vivj P gij +(P P )ninj, (32) a ⊥ − ⊥ k− ⊥ where n, ε, P and P are some scalar functions of the variable u; the indices ofthe kindofkparticle⊥s,a, aredroppedinthesescalarsforthe sakeofsimplicity of notations. The EMT track of particles is equal to: T (u)=ε 2P P . (33) a − ⊥− k 6 From(31)itfollowsthatthevelocityvectorviintroducedin(13)coincideswith the plasma kinematic velocity vector. It is not difficult to check the fulfilment of the condition [1]: (v,n)=0, (34) and consequently the vector vi is an eigenvector of the particle EMT, i.e. it is the dynamic plasma velocity vector according to Synge [10]. thus the frame of reference introduced by the relations (14) is comoving the plasma. Note that in the first drift approximation the equation for kinematic and dynamic velocities of particles is violated and therefore a drift current arises. 3.2 Magnetic hydrodynamics equation for an anisotropic plasma in the PGW field In Sec. 3 of Ref. [1] the RMHD equations for a plasma in an arbitrary gravi- p tational field and for an arbitrary structure of the plasma EMT, Tij, with the eigenvector vi were obtained: p Tij vj =εvi, (35) where the invariant ε > 0 is the plasma energy density in the comeoving FR. In this case the following conditions were imposed on the invariants of the electromagnetic field: Fij F∗ij=0; (36) F Fij =2H2 >0. (37) ij In [1] it was shown that, under these conditions, from the conservation law Tij =0 (38) ,j for the whole EMT of the plasma and the electromagnetic field p f Tij =Tij +Tij and the first group of the Maxwell equations F∗ij =0 (39) ,j it follows: 1. The conditions of embedding of the magnetic field in the plasma are: F vj =0 (40) ij (in this case the velocity vector vi also automatically becomes an eigenvector of the electromagnetic field EMT); 7 2. The second group of the Maxwell equations is Fij = 4πJi (41) ,j − dr with the drift current p 2Fik Tl Ji = k,l ; (42) dr − F Fjm jm 3. The differential relations are p vi Tk =0, (43) i,k p Hi Tk =0. (44) i,k It is not difficult to ascertain that the Maxwell tensor determined by the vector potential (6) automatically satisfies the conditions (37) and the velocity vector 13) – the embedding conditions (40). Therefore in this case, due to the EMTconservation(39),whichis anexactconsequenceofthe kinetic equations, the relations (41) — (44) must hold. The isotropic Killing vector (2) gives an exact integral of Eqs. (39): (39): L2Tu =Const. (45) v thus, taking into account Eqs. (13) and (32) and the initial conditions (3) and (7), we get from (45): L2(ε+P )=ψ′(ε0 +P0 )∆(u). (46) ⊥ ⊥ where we have introduced the so-called GMSW governing function (see [1]): ∆(u)=1 α2(e2β 1) (47) − − and the GMSW dimensionless parameter H2 α2 = 0 . (48) 0 0 4π(ε+P ) ⊥ Itcanbefurthershownthattherelation(44)istransformedintoanidentity, and Eq. (43) gives: 1 ψ′ ′ ε +(lnH)(P P ) ln (ε+P )=0. (49) ′ ′ k− ⊥ − 2(cid:18) L4(cid:19) k The EMT conservation equation and the Maxwell equations do not give other independent relations. 8 The missing equations are obtained from the particle number conservation laws which are also an exact consequence of the kinetic equations: ni,i =L−2 L2nvv ′ =0⇒ (cid:0) (cid:1) n (u)L2 = ψ n0 . (50) a ′ a p Thus we get the set of equations (46), (49) and (50) for determining the three unknown scalar functions ψ, µ and µ . It is only necessary to obtain in this way explicitly the scalars n, ε, Pk and P⊥ out of (1+2n) scalars, where n k ⊥ is the particle kind number. Note that there are at least two kinds of charged particles (in the case of interest these are protons and electrons) in the plasma. Thus, there are two kindsofscalarsµ andµ ,andoneparticlenumberconservationlawforagiven kindofparticlesckonnects⊥eachcoupleofthem(50). Thesummedcomponentsof pressureandenergydensityappearinequations(46)and(49). Asaconsequence oftheinitialelectroneutralityoftheplasma,(50),wegetarelationbetweenthe local concentrations of neutrons and protons [8]: n (u)=n (u). (51) e p 3.3 Calculating the moments of the distribution function For calculating the above scalar functions it is necessary to find the moments of the distribution function (30). An easiest way to do it is to use Eqs. (31) and (32) in the comoving FR according to (14), using the property (15) of this transformation and using the spherical coordinates in the momentum space: p = µ pcosθcosφ; 1 ⊥ p = µ pcosθsinφ; (52) 3 ⊥ p = µ psinθ. 2 k In so doing we obtain: n=n µ2µ , (53) 0 ⊥ k where 2S+1 ∞ 0 n = f (p2)p2dp, (54) 0 2π2 Z 0 the integration variable is: p2 =µ−2p2+µ−2p2 . (55) k k ⊥ ⊥ thus we obtain from (53), (50) and (51)): √ψ (µ2µ ) =(µ2µ ) = ′(µ2µ ) . (56) k ⊥ e k ⊥ p L2 k ⊥ 0 9 Further, setting for definiteness µ 6µ , ⊥ k we get the expressions for the components of the plasma pressure and energy density: (2S+1)µ2µ3 ∞ 0 P = ⊥ k f (p2)p2dp k 4π2(µ2 µ2) Z × k− ⊥ 0  m2+µ2p2 m2+µ2⊥p2 lnqm2+µ2kp2+pqµ2k−µ2⊥;(57) q k − p µ2 µ2 m2+µ2p2  q k− ⊥  p ⊥  (2S+1)µ4µ ∞ 0 P = ⊥ k f (p2)p2dp m2+µ2p2 ⊥ 8π2(µ2 µ2) Z ×h−q k − k ⊥ 0 m2+p2(2µ2 µ2) m2+µ2p2+p µ2 µ2 + k− ⊥ lnq k q k− ⊥; (58) p µ2 µ2 m2+µ2p2 q k− ⊥  p ⊥  2S+1 ∞ 0 ε= µ2µ f (p2)p2dp 4π2 ⊥ kZ × 0  m2+µ2p2+ m2+µ2⊥p2 lnqm2+µ2kp2+pqµ2k−µ2⊥ . (59) q k p µ2 µ2 m2+µ2p2  q k− ⊥  p ⊥  Notethatalltheexpressions(57)–(59)havefinitelimitsatµ2 µ2 =0. From k− ⊥ (57) – (59) one can obtain: ε P 2P = − k− ⊥ (2S+1)m2µ2⊥µk ∞lnqm2+µ2kp2+pqµ2k−µ2⊥f0 (p2)pdp. (60) 2π2 µ2 µ2 Z m2+µ2p2 q k− ⊥ 0  p ⊥  Besides, note that in the case µ 6 µ it is essential that the logarithmic k ⊥ functions in the above formulae for the moments are to be substituted by the functions arcsin. 4 Ultrarelativistic plasma 4.1 Energy density and pressure magnetosphere(see,e.g.,[11]), leadstoveryhighvaluesofthekineticenergyof electronsandprotons(upto1018ev). ThusGMSWappeartobealwaysrealized in an ultrarelativistic plasma. 10