that for a suitable choice of and n our results reduce to those found by Holcomb and Tajima Phys. Rev. D. 40, 3809 1989 and later by Holcomb Astrophys. J. 362 ...
PHYSICAL REVIEW D
VOLUME 53, NUMBER 12
15 JUNE 1996
Magnetohydrodynamics in a homogeneous cosmological background A. Sil, N. Banerjee, and S. Chatterjee* Relativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Calcutta 700 032, India ~Received 21 August 1995! Magnetohydrodynamic waves through a plasma with a general equation of state P } r g in the presence of a homogeneous ambient background magnetic field are studied in a general Friedmann-Robertson-Walker background universe with the metric components being an arbitrary power function of time (;t n ). We find that for a suitable choice of g and n our results reduce to those found by Holcomb and Tajima @Phys. Rev. D 40, 3809 ~1989!# and later by Holcomb @Astrophys. J. 362, 381 ~1990!#. Our general approach to this problem makes it possible to realize the existence of an extra term in the equation for the velocity of the fluid, which until now had remained unnoticed. This extra term has no Newtonian analogue. Unlike the usual results in magnetohydrodynamics, transverse waves traveling perpendicular to the background magnetic field are found to exist. @S0556-2821~96!02910-5# PACS number~s!: 04.40.Nr, 52.60.1h, 98.80.Hw
The application of a theory of magnetized plasma is necessary in many astrophysical situations. In extreme cases, when the gravitational field involved is very strong, we need a theory of general relativistic magnetohydrodynamics ~GRMHD!. However, a linearized GRMHD is good enough to understand most of the real situations. The 311 split of the laws of physics in curved spacetime is the most well known formalism among the attempts made to develop the theory of GRMHD. Thorne and Macdonald @1# gave a particularly good review of it. Other important works are due to Evans and Hawley @2#, Sloan and Smarr @3#, Zhang @4#, Holcomb and Tajima @5#, and later Holcomb @6#. Holcomb and Tajima @5# investigated linearized equations of motion for free photons, longitudinal and transverse oscillations, and Alfve´n waves in a plasma at an ultrarelativistic temperature in a radiation-dominated Friedmann-Robertson-Walker ~FRW! universe. In the next paper @6# Holcomb took up the same problem, but this time for a nonrelativistic plasma in a pressureless ‘‘dust’’ universe. Dettman, Frankel, and Kowalenko @7# showed, while solving linearized Vlasov-Maxwell equations for the case of an unmagnetized plasma, that the conformal flatness of the spacetime chosen by Holcomb and Tajima ensures that the electrodynamic equations in a plasma are similar to those of flat spacetime. In many astrophysical situations, the background spacetime is neither pure ‘‘dust’’ nor pure radiation, but some kind of mixture of them. Also the temperature involved may be neither ultrarelativistic nor nonrelativistic. So it appears worthwhile to investigate the magnetohydrodynamics ~MHD! phenomena in a more general background metric and with a more general adiabatic equation of state for the plasma. In an attempt to extend the work of Holcomb and Tajima for a more general background with the metric components being an arbitrary power function of time (;t n ), we have shown in our earlier paper @8# that the more rapid the expansion of the background universe the faster the electric field falls off and the greater the redshift of the electromag-
netic radiation frequency. However, the dielectric constant of the plasma appears to be independent of the parameter n. In a very recent work Gailis et al. @9# discussed quite exhaustively a magnetized plasma in an early universe scenario. They considered plasma oscillations in both the ultrarelativistic and nonrelativistic limits. For the nonrelativistic limit they considered two types of background fluids: namely, radiation and a silent background ~p50!. In both of their works @7,9# their results differ from the results given by Holcomb @6# in the sense that the plasma modes in the nonrelativistic limit do not redshift like a free photon. However, they did not consider Alfve´n waves in their investigation. In the present work we investigate the time dependence of the various MHD equations, in particular the Alfve´n wave velocity in the same general background universe (R;t n ) with a very general adiabatic equation of state for the plasma. We find that the results of Holcomb and Tajima can be recovered from general results by a proper choice of the parameters. Also the appearance of an extra term in the MHD wave equation for g Þ 34 shows the possibility of the existence of transverse waves traveling in a direction perpendicular to the background magnetic field. To start with, we briefly discuss the 311 formalism. We choose a set of fiducial observers ~FIDO’s! at each spacetime point, with respect to which all physical quantities are measured. The familiar three-vectors and scalars are defined by an appropriate form of tensors projected perpendicular and parallel to the FIDO four-velocity. The three-metric of the spacelike slice is
g m n 5g m n 1n m n n and the spatial coordinates x i are propagated along t m5 a n m1 b m.
53
~2!
Here n m is the unit timelike FIDO four-velocity vector normal to the slice and a and b m are the lapse function and shift vector, respectively. The usual four-metric may be written as
*Permanent address: Department of Physics, New Alipore College, Calcutta 700 053, India. 0556-2821/96/53~12!/7369~4!/$10.00
~1!
ds 2 52c 2 a 2 dt 2 1 g i j ~ dx i 1 b i dt !~ dx j 1 b j dt ! . 7369
~3!
© 1996 The American Physical Society
7370
BRIEF REPORTS
In this paper greek indices run from 0 to 3 and latin indices from 1 to 3. To study a MHD system, our primary job is to write down Maxwell’s equations, Ohm’s law, and conservation laws using a 311 split. For the general background metric these equations are derived by Thorne and Macdonald @1# in the hypersurface formalism. We simplify our task by choosing the line element in the standard form of FRW cosmology given by ds 52c dt 1A ~ t !@ dx 1dy 1dz # , 2
2
2
2
2
2
2
t ¹ 3BW 5
4 p 3n 1 ] 3n W !, t Wj 1 ~t E c c ]t
W 3EW 52 t 2n ¹
1 ] 3n W !. ~t B c ]t
tn W. vW 3B c
~6!
~7!
~9!
with similar equations for EW and r, where BW 1 , EW 1 , and r1 are small compared to their corresponding background values. We also assume the equilibrium velocity of the fluid to be zero and hence vW 5 vW 1 .
~10!
It may be noted at this point that, since mass density r5mN and charge density r e 5eN, where m and e are the rest mass and charge of the particle, respectively, the number density N also obeys a similar equation N5N 0 1N 1 ~ xW ,t ! .
] 3n W • vW . ~ t r 1 ! 52t 3n r 0 ¹ 1 ]t
P5K r g ,
~13!
~11!
~14!
so that the pressure gradient becomes
W P5hc 2 ¹ W ¹ s r,
~15!
where c 2s [ g P/ r h is the relativistic sound speed. The quantity h, the relativistic enthalpy, is defined by h[11
U P 21 rc rc2
~16!
and hence the quantity rh plays the role of the inertial mass of the gas @10#. U, P, and r are the internal energy, pressure, and mass density, respectively, as observed in the rest frame of the gas. It must be noted that the adiabatic index g takes the value 34 when the plasma is ultrarelativistic and 35 when it is nonrelativistic. However, in a more realistic situation, when the plasma is neither ultrarelativistic nor nonrelativistic, g may take any intermediate value @11#. So unlike Holcomb and Tajima @5#, we relax the restriction of g5 34 to address a more general situation and we will presently see that gÞ 34 has nontrivial consequences. For our linear analysis the linearized force balance equation becomes
] 5n t 6n Wj 3BW 2t 3n hc 2 ¹ W @ t r 0 h vW 1 # 5 s r1 . ]t c
~8!
We assume that there exists a spatially homogeneous ambient magnetic induction BW 0 throughout the background fluid W and r may of spatially uniform mass density r0 . Both B 0 0 vary in time. The total magnetic induction is assumed to be BW 5BW 0 1BW 1 ~ xW ,t ! ,
~12!
Thus both r0 and re 0 vary in time as t 23n . It then follows from the equation for the conservation of energy that
~5!
The second term on the right-hand side of Eq. ~6! is identified as the displacement current and neglected in our subsequent calculations. The condition for perfect conductivity for the given metric reduces to EW 52
] N 0 3n 1 N 50. ]t t 0
We assume a perfect gas adiabatic equation of state for the plasma,
We shall assume the scale factor to be an arbitrary power function of time in the form A;t n . In linearized MHD it is appropriate to ignore both divergence equations of Maxwell. The electric field is completely determined by the curl equations and Ohm’s law, whereas it can be shown that once ¹ i B i 50 is imposed as the initial condition to the Faraday law ( ] / ] t)(¹ i B i )50, it will continue to hold at later times. We also neglect the displacement current. The curl equations of Maxwell for the given metric in the usual three-vector notations are given by 2n W
The time evolution of the background value N 0 is described by
~4!
for which a51, bi 50, and the three-metric
g i j 5 diag~ A 2 ,A 2 ,A 2 ! .
53
~17!
In what follows, we must be careful about the time dependence of the quantities such as the mass-energy density, relativistic enthalpy, and sound speed. The time dependence of the quantity r 0 h, the inertial mass @10#, can be obtained from Einstein’s field equation for the k50, FRW model. As for our chosen metric, the scale factor varies as t n and both the mass-energy density T 0 0 and the pressure T 1 1 5T 2 2 5T 3 3 vary as t 22 and hence the inertial mass r 0 h;t 22 . Also, since r 0 ;t 23n , from Eqs. ~14! and ~15! we find that hc 2s ;t 23n( g 21) . Thus the time dependence of h and c 2s are now apparent: h;t 3n22 and c 2s ;t 223n g . Taking care of the time dependence of the above quantities, we combine the conservation equations ~13! and ~17! to get a single equation
r 0 ht 2
F
] 3n ~ g 21 ! ] 5n22 t ~t vW 1 ! ]t ]t
G
] 1 W ~¹ W • vW ! . ~18! 5 r e0 t 6n @ t 3n ~ g 21 ! vW 1 # 3BW 0 1t 3n g hc 2S r 0 ¹ 1 c ]t
53
BRIEF REPORTS
From the curl equations ~6! and ~7! and the condition for perfect conductivity ~8! we get 4p ] W 3¹ W 3 vW 3BW , r e0 ~ t n vW 1 ! 3BW 0 52BW 0 3¹ 1 0 c ]t
~19!
where we have considered the current density vector Wj > r e0 vW 1 for our linear analysis. From Eqs. ~18! and ~19! we finally obtain the required equation for the velocity of the fluid
F
] 3n ~ g 21 ! ] 5n22 t ~t vW 1 ! ]t ]t
G
W 3¹ W 3 vW 3 vW 2c 2 ¹ W W W 1t 3n g 24n vW Ai 3¹ 1 Ai si ~ ¹ • v 1 ! 5 ~ 3n g 24n ! M i t 3n g 23n21 vW 1 3 vW Ai ,
~20!
where the time invariant quantities vW Ai , M i , and c 2Si are designed as follows. The time invariant vector vW Ai [ vW A t 3n21 ,
any adiabatic equation of state and also in a very general background metric. It may be noticed that the term involving M i in Eq. ~20! has no Newtonian analogue. It vanishes when the background metric is Minkowskian, i.e., n50. This term has its origin in the Wj 3BW term in Eq. ~17! and hence, in view of Eq. ~8!, is clearly a contribution from the force due to the electric field. But the effect of this part of the force is separately recognizable if one considers the acoustic waves together with the magnetized plasma in a curved background that is not dominated by radiation, i.e., gÞ 34 . To get solutions of Eq. ~20!, we note that the spacedependent part of vW 1 is exp~ikW •rW !. In what follows we shall very briefly discuss some special cases with different choices of the relative orientations of the vectors kW , vW 1 , and BW 0 . Case 1: kW'BW 0 . There may be three possible subcases. Subcase (a): kW i vW 1 and hence vW 1'BW 0 . Equation ~20! reduces for the magnitude of vW 1 to t2
~21!
vW A [
.
~22!
4 p e r e0 hmc2
~23!
A4 pr 0 h
The quantity M i is defined as M i [t 3n21
A
and the time invariant sound speed as c 2si [t 3n g 22 c 2s .
Notice that the right-hand side ~RHS! of Eq. ~20! disappears whenever we consider an ultrarelativistic plasma, i.e., g5 34 , irrespective of the value of n, the expansion rate of the background FRW metric. Holcomb and Tajima @5# considered an ultrarelativistic plasma in a radiation dominated background universe ~n5 21 !. This explains why this term remained unnoticed to them. For g5 34 and n5 12 , Eq. ~20! reduces to the similar equation found by them. Also for a silent background, where pressure is zero, one may set c 2si 50 and show that the factor M i vW 1 3 vW Ai on the RHS of Eq. ~20! may be replaced by ( ] / ] t)(t 5n22 vW 1 ). Thus, for a plasma without pressure, Eq. ~20! reduces to
F
G
] n ] 5n22 W 3¹ W 3 vW 3 vW 50. t ~t vW 1 ! 1 vW Ai 3¹ 1 Ai ]t ]t
2 ~ 3n g 24n ! M i v Ai t 25n13 #v 1 50.
] 2v 1 ]v1 1 @~ 5n22 !~ 3n g 12n23 ! 2 1 ~ 3n g 17n24 ! t ]t ]t 1 ~ 423 g ! nM i v Ai t 25n13 #v 1 50,
~27!
which can be transformed to a Bessel equation of order p 1 for nÞ 53 , where p 21 5
F
G
3n g 23n21 2 . 25n13
~28!
Note that for g. 34 Eq. ~27! is a modified Bessel equation of order p 1 . For n5 21 and g5 34 , the case considered by Holcomb and Tajima, it is interesting to point out that there exists no propagating solution. Since we require a propagating solution, we choose Hankel functions H (2) p 1 to write the solution for the time-dependent part of v 1 for nÞ 53 as
~25!
Holcomb @6# considered a nonrelativistic plasma in a dust dominated background universe ~n5 32 ! for MHD waves. Anticipating no simple solution for magnetosonic modes, he solved the problem only for two limiting cases: the acoustic wave in an unmagnetized plasma and shear Alfve´n waves in a magnetized plasma. This explains why he failed to notice the term involving M i . For n5 32 Eq. ~25! gives the shear Alfve´n wave solution found by Holcomb. Equation ~20! is useful in physically realistic situations and more general in the sense that it includes a plasma with
~26!
For g5 34 and n5 21 , this equation reduces to the same as that obtained by Holcomb and Tajima @5# and becomes readily reducible to Bessel’s equation. Any other choice for g and n makes it very difficult to be integrated. Subcase (b): kW' vW 1 and vW 1'BW 0 . Equation ~20! reduces for the magnitude of vW 1 to t2
~24!
] 2v 1 ]v1 1 @~ 5n22 !~ 3n g 12n23 ! 2 1 ~ 3n g 17n24 ! t ]t ]t 1c 2si k 2 t 23n g 22n14 1 v 2Ai k 2 t 26n14
where vW A , the relativistic Alfve´n wave velocity, is defined as BW 0
7371
v 1 5t 2 ~ 3n g 17n25 ! /2H p 1
FA
G
2 ~ 423 g ! nM i v Ai 2 ~ 25n13 ! /2 t . ~ 25n13 ! ~29!
The asymptotic behavior of the Hankel function reveals that the frequency of oscillation v 1 ;t 25(12n)/2. For n5 53 , Eq. ~27! is a Euler equation with the characteristic equation r ~ r21 ! 1 51 ~ 9 g 11 ! r1 51 @ 9 ~ g 21 ! 23 ~ 3 g 24 ! M i V Ai # 50. ~30! For imaginary roots of Eq. ~30!, r5a6ib, we can write
BRIEF REPORTS
7372
v 1 5t a @ A cos~ b ln t ! 1B sin~ b ln t !# .
~31!
These modes are oscillatory and waves do propagate in such a situation. However, they are not oscillatory in cosmic time but in its logarithm. Subcase (c): kW' vW 1 and vW 1 i BW 0 . Equation ~20! reduces for the magnitude of vW 1 to t2
] 2v 1 ]v1 1 ~ 3n g 17n24 ! t ]t2 ]t ~32!
which is a Euler equation, and the solutions are for g Þ11
1 ~33! 3n
and for g 511
1 . 3n
~34!
Case 2: kW i BW 0 . There may be two possible subcases. Subcase (a): kW i vW 1 and hence vW 1 i BW 0 . Equation ~20! reduces for the magnitude of vW 1 to
] 2v 1 ]v1 1 @~ 5n22 !~ 3n g 12n23 ! 2 1 ~ 3n g 17n24 ! t ]t ]t 1c 2si k 2 t 23n g 22n14 #v 1 50,
~35!
which is a Bessel equation of order p 2 for 3n g 12nÞ4, where
F
G
3n g 23n21 2 . 3n g 12n24
F
G
2c si k t 2 ~ 3n g 12n24 ! /2 . 23n g 22n14 ! ~ ~37!
It is a pure acoustic wave. The asymptotic behavior of the Hankel function reveals that the frequency of oscillation v 2 ;t 2(3n g /2)2n11 . For 3n g 12n54 Eq. ~35! is a Euler equation with the auxiliary equation ~38!
The roots of this equation determine the nature of the solution for v 1 . Propagating solutions exist when the roots are imaginary. Subcase (b): kW' vW 1 and hence vW 1'BW 0 . Equation ~20! reduces for the magnitude of vW 1 to t2
v 1 5t 5n22 ~ C 3 1C 4 ln t !
t2
v 1 5t 2 ~ 3n g 17n25 ! /2H p 2
r ~ r21 ! 15nr1 $ ~ 5n22 ! 1c 2si k 2 % 50.
1 ~ 5n22 !~ 3n g 12n23 !v 1 50,
v 1 5C 1 t 23n g 22n13 1C 2 t 25n12
53
] 2v 1 ]v1 1 @~ 5n22 !~ 3n g 12n23 ! 2 1 ~ 3n g 17n24 ! t ]t ]t 1 v 2Ai k 2 t 26n14 1 ~ 423 g ! nM i v Ai t 25n13 #v 1 50. ~39!
This can be readily integrated for the particular choice n51. For n51 Eq. ~39! can be transformed to a Bessel equation of order p 3 , where p 23 5 14 ~ 3 g 24 ! 2
~40!
and the resulting solutions for the time-dependent part of v 1 is given by v 1 5t 2 ~ 3 g 12 ! /2H p 3 @ 2 A$ v 2Ai k 2 1 ~ 423 g ! M i v Ai % t 21 # .
~41!
The solution for the time-dependent part of v 1 is written as
The asymptotic behavior of the Hankel function reveals that the frequency of oscillation v 3 ;t 22 . This is a pure Alfve´n wave. When g Þ 34 , the newly found term M i modifies the frequency of it.
@1# K. S. Thorne and D. Macdonald, Mon. Not. R. Astron. Soc. 198, 339 ~1982!. @2# C. R. Evans and J. F. Hawley, Astrophys. J. 332, 659 ~1988!. @3# J. H. Sloan and L. Smarr, in Numerical Astrophysics, edited by J. M. Centrella, J. M. Le Blanc, and D. L. Bowers ~Jones and Bartlett, Boston, 1985!. @4# X. H. Zhang, Phys. Rev. D. 39, 2933 ~1989!. @5# K. A. Holcomb and T. Tajima, Phys. Rev. D. 40, 3809 ~1989!. @6# K. A. Holcomb, Astrophys. J. 362, 381 ~1990!.
@7# C. P. Dettman, N. E. Frankel, and V. Kowalenko, Phys Rev. D 48, 5655 ~1993!. @8# A. Banerjee, S. Chatterjee, A. Sil, and N. Banerjee, Phys. Rev. D 50, 1161 ~1994!. @9# R. M. Gailis, C. P. Dettmann, N. E. Frankel, and V. Kowalenko, Phys. Rev. D 50, 3847 ~1994!. @10# C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation ~Freeman, San Francisco, 1970!, p. 632. @11# S. Weinberg, Gravitation and Cosmology ~Wiley, New York, 1972!, p. 51.
p 22 5
~36!
