Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics (Cambridge Monographs on Mathematical Physics)

(5.16)

Let M" = F(l, vx, vy, vz) in Minkowski coordinates. Then a = F(vx — X) and the characteristic equation (5.12) is written (1 - c s 2 ) r > x - X)2 - c2s{\ - 1 1 ) = 0,

(5.17)

where c2 = p'e, cs>0 being the local speed of sound (corresponding to vx — vy = vz = 0 in the characteristic equation).

110

Relativistic fluids and magneto-fluids

The right eigenvectors read explicitly - c2{X + To) c2{\+Tvxa) d=

(5.18)

-C2TVM

_ (e + p)a

-

Hence, from (5.14), - — = dcp

nc2saTvy,

(5.19) (5.20)

d
whence we obtain the Riemann invariant — = constant.

(5.21)

If in the fluid flow there is a point with vanishing speed, v* = v* = v* — 0, then from (5.19)-(5.20) we obtain (5.22)

vy = v z = 0

throughout. Since we can always assume the existence of a zero speed point for a wave propagating into a uniform constant state (by a suitable Lorentz transformation), which is the case of physical interest, henceforth we shall assume (5.22) to hold. Then the characteristic equation (5.17) yields

vx(l-c2)r2±cs

vx±cs

(5.23)

which is the usual relativistic addition formula for velocities. The C ± characteristics are those lines with slopes /l ± , respectively. From (5.18) it follows that (5.24)

dcp de= d
( i±vxcs'

(5.25)

5. Relativistic simple waves

111

hence

which integrates immediately yielding

J+ = ^]n(\^)± \l-vj

f-^-d^constant, J(e + p)

(5.27)

where J+ are Riemann invariants (J± associated with the C+ characteristics) (Taub, 1948). A right-propagating simple wave has J_ = constant

(5.28)

throughout space-time. It can be obtained by assigning vx as an arbitrary v -f~ c function of cp = x -t9 1 + v xc s <5 29)

-

where/is an arbitrary smooth function specifying the initial wave profile. Then the relationship between e and vx is obtained by J_ = const, that is, by (5.30)

with p = p(e, So). Similarly, a left-propagating simple wave has J+ = constant

(5.31)

throughout space-time. The solution can be obtained as

and the relationship between e and v is given by J + = const., that is, de = - iln with

\l-vxj

+ const.,

(5.33)

p = p(e,S0).

It is interesting to notice that the Riemann invariants are not Lorentz

112

Relativistic fluids and magneto-fluids

scalars. In fact, under a Lorentz transformation with x-velocity /?, one has

hence e + p)

* \l-v

that is,

J'±=J±+iln(f^4).

(5.34)

However, the simple wave condition J + = const, is a definition of a Lorentz invariant because j? = const. In the next section we shall rederive the Riemann invariants in the Lagrangian framework.

5.3. One dimensional relativistic fluid dynamics in Lagrangian coordinates Let u be the fluid's four-velocity vector, assumed to be a smooth vector field in a domain Q of space-time Ji. The integral curves of u represent the world lines of the fluid particles and their differential equations are ^ = n"(x*) as

(5.35)

in an arbitrary coordinate system (xa), where s is the proper time along these world lines. The solutions of (5.35) (which will exist provided that Q is sufficiently small) can be written as x" = *"(£'»,

(5.36)

where £', i= 1,2,3 are three parameters such that

are the parametric equations of a hypersurface Z, the initial hypersurface on which the particles are located at proper time s = 0. The four coordinates (
5. Relativistic simple waves

113

they are like the Lagrangian coordinates of Newtonian continuum mechanics. In relativity they are an example of the more general comoving coordinates. Under the transformation xfi = £', x'° = x'°(l;\ s) the equation of the initial hypersurface s = 0 becomes x'° = xfO(x'\0). The xffl are general comoving coordinates. In these coordinates the line element of space-time is written d52 = ^ v d ^ d x " . ri

(5.37) 2

2

Along a world line x = const, we have ds = g'oo (dx'°) and therefore the components of the four-vector u in these coordinates are 3

"

(5.38)

In the following we shall consider two differential sets of coordinates in Minkowski space: the Eulerian coordinates of the inertial frame which we denote by X" = (T,X,Y,Z) and the comoving coordinates xx = (t, x, y, z). We shall assume that at the initial time T = 0 and t = 0 and the two coordinates coincide, that is, T{0,x,y,z) = 0, X(0,x,y,z) = x, Y(0,x,y,z) = y, Z(0,x,y,z) = z.

(5.39)

Let g^ be the components of the metric tensor in the comoving coordinates. From the transformation formulas dX' we obtain explicitly

- TtTx + XtXx + YtYx + ZtZx = gou -TtTy + XtXy + XtYy + Z,Zy = 002,

114

Relativistic fluids and magneto-fluids -TtTz + XtXz +Y,YZ + ZtZz = 0O3, - TxTy + XxXy +YxYy + ZxZy = g12, - TXTZ + XXXZ + YXYZ + ZXZZ = g13, - TyTz + XyX2 + YyYz + ZyZz = g23.

(5.40)

Let u" = r(l,vx,vy,vz) be the components of the fluid's four-velocity in the inertial frame, with F = (l — v\ — vy — vz)~112. From the definition we have

x, V

dX

dr

dY _Y, dT T;

zt

dZ

~df

(5.41)

T,

because x, y9 z are constant for a given fluid particle. Henceforth we shall assume one dimensional flow according to the following definition: all the physical variables and the components of the metric are functions of the comoving coordinates (x, t). From (5.40) we have

and, by differentiating with respect to y, T 1

—T — 0 ty

L

yt

V

5

hence — Ty = 0 with the initial condition dt Ty(0, x, y,z) = 0 which then yields

Ty(t9x,y,z) = 0. Similarly we obtain Tz(t, x, y, z) = 0, whence T=T(t9x). By analogous reasoning we obtain

X=X(t9x)9

(5.42)

5. Relativistic simple waves

115

Y=Y(t,x) + y, Z = Z(*,x) + z,

(5.43)

with f (0, x) = 0, X(0, x) = x, 7(0, x) = Z(0, x) = 0. From (5.40) we then get 022=033 9l2=%>

=

1

>

Q02=%

9l3 = Zx>

003 = Z t ,

023=0.

Let us write # 0 1 = ^/l^oolM^O- Then the metric reads ds 2 = ( v ^ o T d t + ftdx)2 + ton - h2)dx2 + dy 2 + dz 2 + 2g02 dydt + 2#O3 dzd* + 2g12 dxdy + 2g13 dxdz.

(5.44)

By a well-known theorem on differential forms, the differential form y/\gOo\ dt + hdx has an integrating factor, that is, there exists a differentiable function A(x, t) # 0, such that Vl^ool dt + hdx = A(x,t)dt, with t' a new time variable, t' = t'(x, t). It is easy to check that the transformation (x,y,zj)o(x,y,z,tf) is invertible and that the new coordinates (x, y9 z, t') are still comoving. By relabeling the coordinates and the metric coefficients, finally we obtain for the line element ds 2 = 0Oo dt2 + g11 dx2 + dy 2 + dz 2 + 2g02 dt dy + 2g03 dt dz + 2g12 dxdy + 2g13 dxdz.

(5.45)

Now we shall restrict ourselves to one dimensional flow along the x axis, that is, vy — vz — 0. It follows that Yt = Zt = 0, hence g02 = g03 — 0. Also, from g12= Yx, we have dtg12 = dtYx = dxYt = 0. Then, from the initial condition g12(0,x) = 0 (in the reference state t = 0 the three-metric is Euclidean) it follows that g12(t9x) = 0. Similarly, g13(t,x) = 0. With these restrictions the line element becomes d52 = 0oo dt 2 + 0! x dx 2 + dy 2 + dz 2 .

(5.46)

The transformations from Eulerian to comoving coordinates reduce to T=T(x,t), X = X(x,t), Y = y, Z = z.

(5.47)

116

Relativistic fluids and magneto-fluids

Let us consider the equation of mass conservation V>n") = -)= da(^\i\pu«) = 0,

(5.48)

y/\G\

which in comoving coordinates is written

whence

J\F\^L= = f(x).

(5.49)

It follows that Py/\dll\ = (y/\9li\p)t = O =/(*)•

Because, at t = 0, the two coordinate systems coincide, gli(x90)=l hence/(x) = po(x), the initial value for the mass density. Then

By introducing a mass coordinate \i instead of x, defined by K' {Po>%

(5.51)

we can write the line element in the form ds 2 = a2 dt2 + -2 dp2 + dy2 + dz 2 ,

(5.52)

where we have set a2 = — g00 (for continuity g00 < 0 because at the initial time t = 0, g00 = — 1). The choice of the mass coordinate fi is especially useful for numerical calculations. Then from (5.40) we obtain

r

P

(5.53)

together with the inverse transformation

r

ttxx = - - p '

tT = -

a

=-

PTvx,

fxx = pT.

(5.54)

5. Relativistic simple waves

111

The mass conservation equation (5.48) written in inertial coordinates (Xa) is ^

^

0,

(5.55)

F 2 = l + U 2.

(5.56)

where U = Tvx,

After some manipulations, using (5.53), equation (5.55) yields A=-ap 2 -^.

(5.57)

Because the coordinate transformations (5.47) are assumed to be sufficiently smooth (at least of class C 2), we have Y

— Y

and from equation (5.53) we obtain Vt = pT%.

(5.58)

The energy-momentum conservation equations VaTa/* = 0

(5.59)

are explicitly written as

^ 7 t ) l ^ f\g\

dxp

= 0

2 dx"

In Lagrangian coordinates, for fi = 0 we obtain (5.61)

\P Jt

and for /?= 1,

a^-^j.

(5.62)

It is easily checked that the Riemann tensor of the metric (5.52) reduces to one component, K 1010 (apart from those obtained from it by symmetry), which vanishes as a consequence of equation (5.57) and (5.58). 5.4. Simple waves in Lagrangian coordinates The field equations can be taken to be, by suitably arranging equations (5.57)-(5.62), = 0,

(5.63)

118

Relativistic fluids and magneto-fluids (5.64) _

(5.65)

=0.

\PJt

Equation (5.61) is not an evolution equation and decouples from the others. By assuming an equation of state

equations (5.63)—(5.65) are, in matrix form,

where Y =

"l

0

0

0

1

0

0

, and

(5.66)

ap2

r

-?

0

/

/

0

0

0

0

(5.67)

The system can also be written in normal form (5.68) where

afPp /

/

0

0

-f o

o

a

(5.69)

We look for solutions in the form of simple waves Y = Y(^). Then (5.68)-(5.69) becomes

(si - XI)Y' = 0,

(5.70)

5. Relativistic simple waves where X =

119

-. The eigenvalues are PpP + PsP 1

ii=O,

f

(5.71)

and the corresponding right eigenvectors are, with OL a factor, 1 0 ,

=a

d2.3 = a

+/

(5.72)

The case 1 = 0 represents the material waves (where p and U are constant). The cases X # 0 correspond to the acoustic waves. The simple waves are the solutions of the system dJ7

(5.73) 1/2

(5.74) 1/2

d

=

+

" r

\ppp2

(5.75)

From (5.74)-(5.75) it follows that

+P

hence, from the first law of thermodynamics, (5.76) and, therefore, simple waves are isentropic. Then one can write p = p(e), and it follows that

Ps =

120

Relativistic fluids and magneto-fluids

with c] = p'e. Substituting back into (5.76)-(5.77) yields

%=±Y^

(5J7)

and d0

1 csp

Then, from (5.73)—(5.78) one obtains ^ = ± — ,

(5.79)

I n ( r + U) ± ^ - ^ = J±= const.

(5.80)

which, upon integrating, gives

These J± are easily seen to coincide with the Riemann invariants introduced in Section 2, equation (5.27) (Lanza et al., 1985). 5.5. Evaluation of the Riemann invariants for relativistic acoustic simple waves We shall calculate the Riemann invariants for the various state equations discussed in Section 2.2. In the case of a barotropic fluid with state equation p = (y-l)e,

(5.81)

with 1 < y < 2, we have for the speed of sound c2s=y-\.

(5.82)

Then

and, therefore, right- and left-propagating simple waves are characterized, respectively, by ^

)

,

(5.83)

where e = e0 at vx = 0 (Fig. 5.1). Notice that, for right (left)-propagating simple waves, e -> oo as vx -• 1

5. Relativistic simple waves

121

The solutions we have found provide an interesting way to understand the nonlinear development of acoustic waves in the early universe. The content of the universe consists of three components: ordinary matter (baryon component), radiation (photon component), and possible dark matter (whose precise form is still under debate). Considering only the matter-radiation fluid, it is necessary to distinguish two kinds of perturbations. Under the first kind, both the radiation and matter can be perturbed together in such a way that the ratio of the photon number density to the baryon density within the perturbation is the same as in the unperturbed medium. These perturbations are referred to as the adiabatic modes because in this case the entropy per baryon remains constant. For the second kind, the matter distribution can be perturbed, leaving the photon density unchanged. The temperature within such a perturbation is the same as in the unperturbed medium, and therefore these perturbations are referred to as isothermal modes or entropy fluctuations. Prior to recombination the speed of sound in the matter-radiation mixture is essentially that appropriate for a photon gas, that is, cs ^ c/^/3, and the corresponding Jeans length is only slightly smaller than the cosmic horizon distance (Peebles, 1980). After recombination the speed of sound is that appropriate for a cool hydrogen gas and, therefore, the Jeans Fig. 5.1. Progressive simple waves in a barotropicfluidwith y = 4/3 (radiation gas).

0.2

0.4

0.6

0.8

122

Relativistic fluids and magneto-fluids

length scale drops by several orders of magnitude from its value before recombination. The evolution of an adiabatic recombination, in the linear regime, can be depicted as follows. Let k be the wavelength of the perturbation. We start following the evolution of the perturbation from the time when it first enters the horizon size, k«kH (otherwise, in general relativity, ambiguities may arise; Peebles, 1980). The perturbation will grow in amplitude as long as k> kj. Adiabatic perturbations with wavelengths smaller than the Jeans length immediately prior to recombination will start to oscillate when k < k5 and will continue to do so until trec. If they survive after recombination they will begin to grow because they would once again satisfy the condition k > kj. Adiabatic density fluctuations on sufficiently small scales are damped by the action of viscosity and heat conduction. If the damping time scale is shorter than the cosmic expansion time scale, the wave will be effectively damped before the universe has time to expand by an appreciable factor. This process selects a preferred damping scale kD as the shortest wavelength whose damping time scale equals the cosmic expansion time scale. The exact value of kD is determined by Thomson scattering of radiation off free electrons (Peebles, 1971). Only adiabatic fluctuations with k > kD can therefore survive after recombination. The linear evolution of isothermal fluctuations is much simpler. During the prerecombination period the amplitude of isothermal waves remains constant since their growth is inhibited by the drag due to Thomson scattering. After recombination they will grow in amplitude if k > k3. Nonlinear effects could be important in the evolution of adiabatic perturbations. A weakly nonlinear density perturbation with a sinusoidal profile will tend to break and form shocks. If the breaking time tb (which is a function of the perturbation amplitude and wavelength) is much shorter than the cosmic expansion time t, then the wave will break, form shocks, and dissipate. This nonlinear process imposes an upper limit on the amplitude of adiabatic perturbations at the equipartition time (Peebles, 1980). Precise calculations using relativistic fluid dynamics in a cosmological model have been performed by Liang (1977b), Anile, Miller, and Motta (1983), and by Carioli and Motta (1984). In some cases the process of shock formation can be studied by using the simple wave solutions we have found. For this purpose it is necessary to extend the above solutions from special relativity to the case when the background space-time is a cosmological model. This can be achieved by applying the following result (Anile and Greco, 1978).

5. Relativistic simple waves PROPOSITION

5.2.

123

Let T^v be a traceless symmetric tensor satisfying VaTa/? = 0

with respect to the conformally flat metric

(where rjap denotes the Minkowski metric). If one defines then T*p satisfies

Proof. By direct computation, one has for the Christoffel symbols of gaP, hence VaTa/* = dj^

+ 3 f aTa/*.

Q.E.D.

This proposition can be applied to a radiation fluid in a spatially flat Robertson-Walker Universe with the metric As2 = -dt2

+ K2(f)<5y dxl dx\

(5.84)

where t is the cosmic time and R(t) is the universe expansion factor. By introducing the conformal time x defined by

d

(5 85)

i

-

the above metric can be written in the conformally flat form

ds2 = exp (\j/) {-di2 + Stj dxl dxj), with

For a test-radiation fluid with energy-momentum tensor T** = %eu«u*+

e

-g**

the transformation Ta/* = exp(3^)T a/? amounts to e = RAe,

ua = Ru*.

(5.86)

124

Relativistic fluids and magneto-fluids

Let ua = f(i,tJ,o,o),

f=(i-t;2r1/2.

Then the simple wave solution (5.83) with y = (4/3) corresponds to (5.87) We assume that the test fluid described by Ta/? moves only in the x direction. The peculiar velocity v of the test fluid with respect to the fundamental observers of the Robertson-Walker Universes [which in the coordinates

(0)

(T,X1)

have four-velocity u a = (jR, 0,0,0)] is defined by

r=-ua(u\

r = (i-v2y1/2.

Hence

and it follows that v = v. Therefore, the above solutions can be written, in terms of quantities physically defined in the Robertson-Walker Universe, as ,,88, where e0 = -—; (e0 = constant) is the energy density of the background K cosmological fluid. These equations are the same as in special relativity. The only difference is that T is the conformal time and x is a comoving coordinate. These solutions had already been found by Liang (1977a) by an ad hoc procedure, not as a special case of the general result shown above. Analogous solutions may be found for test-radiation fluid dynamics in a nonspatially flat Robertson-Walker model. These solutions would necessarily be more complicated because the transformations needed in order to put the nonspatially flat RobertsonWalker Universes into a conformally flat form are quite involved (Hawking and Ellis, 1973). In the case of a polytropic fluid with state equation (5.89)

5. Relativistic simple waves

125

we have for the local speed of sound (5.90) Then from the adiabaticity condition wa3aS = 0 and the first law of thermodynamics it follows that, along the flow lines (by a suitable choice of the additive constant in the internal energy) (5.91) whence yp(y-i)

(5.92)

From this we see that

A straightforward calculation shows that

-An

(e + p)

l-

Fig. 5.2. Progressive simple waves in a polytropic fluid with y = 4/3; upper curve, cs = 0.3, lower curve cs = 0.01.

0.440 ^.= 0.3 c ^

—

.

- ~

j

0.330

y

0.220 -

0.110 -

c

0.000 0.0

1

0.2

1

0.4

1

0.6

1

i

0.8

i

i

126

Relativistic fluids and magneto-fluids

hence

/y-1

= ln

/y-1

/y - 1 V Therefore, right(left)-propagating simple waves are characterized by

(5.93) +1 where cs = cs(vx = 0). Notice that cs -> ^/y — 1 as i;x -* + 1 for right- and left-propagating simple waves, respectively (Fig. 5.2). We remark that, like the situation for acoustic simple waves in Newtonian fluid dynamics, from (5.93) and the condition cs > 0, we can draw the following limitation on the fluid's velocity in the direction opposite that of the wave propagation. 1-

p; \ 2/Vy - 1

(5.94) 1+

/y + 1 + c&

for a right-propagating wave. A similar limitation holds for leftpropagating waves. Finally, we consider the case of a Synge gas given by the state equations (2.22)-(2.23). The speed of sound is obtained as follows. From

with z = z(p, S), one has

Now, from (2.23), it follows that

5. Relativistic simple waves

127

hence

Finally, one obtains

c?=-

G'/G

iY

where z =

and G(z) = —^— (Fig. 5.3). K2(z) k BT Because simple waves are isentropic, we obtain

and, therefore, the Riemann invariants are 1/2

dz.

Fig. 5.3. Speed of sound in a relativistic gas (Synge's state equation) as a function of the parameter z = m/kBT in the region 0.1 < z < 1.

0.48 -

0.46 -

0.44 -

0.42 -

0.40

128

Relativistic fluids and magneto-fluids

In this case it is not possible to analytically invert the condition J+ = const, and therefore one must turn to numerical integration (Lanza et al., 1985). A simple wave is determined by first specifying an appropriate profile for the three-velocity vx and then obtaining the corresponding values of z by inverting J+ = const. From the numerical viewpoint it is more convenient to solve equation (5.26) for vx as a function of z (Fig. 5.4). The critical time for breaking can be obtained from (5.10) for an initially sinusoidal profile (5.10') .

2TC

v = i; o sin—-x.

a

For a barotropic fluid one obtains (Muscato, 1988a)

2TI {2[1 + SvUy - 1)] 1/2 - 4v20(y - 1) - 2} 1 ' 2

2- y

Notice that tB-+co as y->2 (Fig. 5.5). For a polytropic gas one must evaluate tB numerically and one obtains (Muscato, 1988a) the results given in Figs. 5.6-5.7. The next section will be devoted to the relativistic theory of isentropic

Fig. 5.4. Progressive simple waves in a Synge gas {vx as a function of the parameter z = m/kBT in the range 0.1 < z < 1).

-0.2 -

-0.4 -

-0.6 -

-0.8

-1.0

5. Relativistic simple waves

129

Fig. 5.5. Critical time in a barotropic fluid. Plot of (t) = 2ntBcs0/d versus (v) = vo/csO for several values of y. The lower curve corresponds to y = 4/3, the upper curve to y = 2, and the intermediate one to y = 1.9.

19952 -

398.1 (f)

7.943 -

0.158 -

0.003

Fig. 5.6. Critical time for a polytropic fluid with y = 5/3 in both classical and relativistic fluid dynamics. Quantities defined as in Fig. 5.5. The lower curve corresponds to the nonrelativistic case, for which d nvo(y + 1) (Jeffrey, 1976), the intermediate one corresponds to c0 = 0.4, and the upper one corresponds to c 0 = 0.8.

74.13

19.49 (t)

5.129 -

1.349 -

0.355

Relativistic fluids and magneto-fluids

130

flow, a preliminary step for developing the relativistic theory of steady supersonic flow. 5.6. General properties of isentropic flow In the case of isentropic flow, S = const., the momentum conservation equation (2.10) can be written as waVaw" + /zaMVa In / = 0

(5.95)

since Equation (5.95) admits an elegant geometric interpretation (Lichnerowicz, 1955). PROPOSITION

5.3.

The flow lines defined by ds

are the geodesies of the metric g^v — f2g^v. Fig. 5.7. Critical time for a polytropic fluid with y = 4/3 in relativistic fluid dynamics. The lower curve corresponds to c0 = 0.4; the intermediate one to c0 = 0.8. The upper curve corresponds to a barotropic fluid with y = 4/3. The quantities plotted are defined as in Fig. 5.5.

83.18

21.88 •

it) 5.754 -

1.513 -

0.398

5. Relativistic simple waves

131

Furthermore equation (5.95) is equivalent to c"Vacp = O,

(5.96)

where ca = fua and V is the Riemannian connection of the metric g^. Proof. The Christoffel symbols of the metric g^ = f2gilx

are

where T^y are the Christoffel symbols of g^v. Therefore,

and the flow lines are geodesies. Furthermore,

= - f2 j f,«up + fu*upfa = 0.

Q.E.D.

The relativistic vorticity tensor Qa/? is defined by ^

= VacrV,

(5.97)

Let co be the 1-form a> = cadx*.

Then Qaj8 can also be written as Q = dco. In the Newtonian limit, Qa/? reduces to the usual vorticity tensor. In fact, / = 1 + e + p/p and in the Newtonian limit we can set / = 1. Since g^ = f2g^ one has

£"V = / ~ V V and therefore fVr-1.

(5.98)

c * V a = 0.

(5.99)

It follows that

132

Relativistic fluids and magneto-fluids

Then, from (5.96), one has (5.100) It can be proved that Qa/3 satisfies the relativistic analog of the Helmoltz theorem (Lichnerowicz, 1955, 1967), which in local coordinates is written, c"V,Qa/? + (Vac«)Q^ + ( V ^ R , = 0. The flow is said to be irrotational if Q = 0. In this case, in a simply connected domain W of Jt, there exists a function oeQ){W\ the potential function, such that

In this case equation (5.98) can be written

£.>

+

/>=0.

,5.0,,

Furthermore, the continuity equation VJipu*) = 0 gives

with / given by (5.101) and p as a function of / through the state equation. In the next section we shall consider a particular example of isentropic and irrotational flow: one dimensional unsteady isentropic flow. 5.7. Unsteady one dimensional and isentropic flow in special relativity For one dimensional motion we have, in Minkowski coordinates (t, x, y, z),

and the motion occurs on the x-axis. It is easy to see that Clafi = 0. In fact, Q 0 2 = Q03 = ^23 = ^12 = ^13 = 0. Furthermore, from (5.100) it follows that

whence O 0 1 = 0 . Therefore the flow is irrotational.

5. Relativistic simple waves

133

Equation (5.101) then can be written 2

/ -
2

2

+
(5.102)

where

at = - / r ,

ax = fTv.

Now we apply the hodograph method (Liang, 1977a). We perform a Legendre transformation by introducing the function x (5.103) We have

dx = T(t -Xv)df

+ T3f(tv - x) dv.

By introducing the new variable

dx = F 2 dv, v = tanh T, T = cosh T, hence d^ = (t cosh T — x sinh T) d / + /(^ sinh T — x cosh T) dr. Therefore

1 = 7 {iff cosh T - xT sinh T), x

= 7 (Z/Sinhi - x T coshr).

The mass conservation equation is 3 t (pr) + 5 , 0 9 ^ = 0.

(5.105)

By expressing it in terms of the variables ( / , T) we finally obtain cU2Xff + fXf-Xrr = 0,

(5.106)

which is a linear equation. Sometimes it is more convenient to use the variable v = lnf. Then equation (5.106) reads cs2Xw + ( l - c s 2 ) X v - ^ = 0.

(5.107)

Note that for a barotropic fluid cs = constant and the latter equation is very simple to solve (Liang, 1977a). In this way the problem of determining the isentropic one dimensional flow corresponding to given initial and boundary conditions is reduced to an initial-boundary value problem for a linear equation. However, the

134

Relativistic fluids and magneto-fluids

applicability of the method is limited by the fact that expressing the physical initial-boundary conditions in terms of the variables v, T, and x is not an easy task. In the next section we shall treat another example of isentropic irrotational flow of great physical interest, the case of stationary potential flow in two dimensions. 5.8. Stationary potential two dimensional flow in special relativity Aflowis said to be potential if it is isentropic, S = constant, and irrotational, In special relativity a flow is stationary if, in an inertial system (x\ t)9 all the physical quantities are independent of time t. Let uti = T(l9vl9v29v3) be the four-velocity in this coordinate system. Then for a steady potential flow the requirement Qa/? = 0 yields / r = / 0 = const.,

(5.108)

with/ 0 the value o f / a t the stagnation point (v = 0) and djvi-divj = 0,

(5.109)

which states that the vector field y=(vl9v29v3) is irrotational in [R3. Equation (5.108) represents the relativistic Bernoulli theorem. In the Newtonian limit it is easily seen to reduce to 1 2

P

\v + 8 + - = const. P For stationary flow equation (5.95) reads TVidiiTvj) + 5,.In/ + r ^ i ^ l n / - 0. By contracting with vj9 using the thermodynamic relationship

which holds for isentropic flow and the mass conservation equation, yields ^(vjdjirv^^d^Tv,).

(5.110)

In the case of two dimensional flow, v3 = 0, v1 = vx, v2 = vy9 equations

5. Relativistic simple waves

135

(5.109) and (5.110) yield, after some manipulations, y

dy

1\

J/2

y

dx

3^

A

(c

1 1 1 o\

dy

dvY dvv - ^ +- - ^ 0 , oy ox

(5.111b)

where

1

csus

L

cscs

are the proper Mach numbers for flows in the x and y directions, with

rCs = (i-c s 2 r 1 / 2 . The system (5.111a)-(5.111b) can be written in the form (Konigl, 1980) where is the field vector and s/°, srf1 are given by 2 /0_{,J x-l

MxJiy\ 1 )

0

^(JtxJ{y, \ -1

Ji)-\ 0

It is easily seen that for supersonic flow

the system (5.111) is hyperbolic. The characteristic curves # ± , are given by % = X ±,

(5.112)

where X± =

--*->^~2-» 1-

.

(5.113)

Then the system (5.11) can be written in characteristic form as T I 02 .

dvx)l±

1\l/2

\-M)

where c ± are the characteristics in the hodograph plane (vX9 vy).

(5.114)

136

Relativistic fluids and magneto-fluids

It is easy to check the orthogonality relations between the %> + and c_ and between the ^ _ and c+ families of characteristics, = -1.

(5.115)

By introducing polar coordinates in the hodograph space (vx,vy); vx = v cos 9, vy = v sin 9, and also the relativistic mach angle v defined by

sinv = —-, Jt

0
(5.116)

equation (5.112) becomes

-/-)

= tan(9±v).

(5.117)

From equations (5.117) and (5.115) it follows that the normal to the c_(c+) characteristic lies at an angle v( —v) with respect to the velocity vector v. Also, equation (5.114) can be written (d9)c> = + — {J( 2-

1)1/2,

(5.118)

which allows a ready determination of the Riemann invariants, because the relationship between M and v is provided by the Bernoulli equation (5.108) and the state equation. For a relativistic gas we must use the Synge state equations (2.22)-(2.23). Then it is convenient to express / and cs as a function of the variable z = m/kBT and use equation (5.108) in order to express v as a function of z. However, the final integral must be computed numerically. It is possible to obtain analytical results in the nonrelativistic and ultrarelativistic limits. In the ultrarelativistic limit we have r2 C

1

s — 3

and equation (5.118) yields for the Riemann invariants J± = 9 + £ M 0 = const,

(5.119)

where E(Jf) = ^ 3 arctan In the nonrelativistic limit,

—-— - a r c t a n ( ^

2

- 1)1/2.

(5.120)

5. Relativistic simple waves

137

and from (5.108) it follows that c2 = c2-

\v2

where cs is the speed of sound at the stagnation point (v = 0), and equation (5.120) then yields j ± = 3 + E(M) = const,

(5.121)

where

M

p

)

(5.122)

By employing standard methods of classical fluid dynamics (Landau and Lifshitz, 1959a) one can derive the relativistic analogs of the Chaplygin and the Euler-Tricomi equations (Konigl, 1980). The theory developed in this section could be used in order to analyze shock formation from simple waves in relativistic steady supersonic flow. Applications could be envisaged to the problem of head-on collisions of heavy ions (Sobel et al, 1975; Clare and Strottman, 1986) and to several astrophysical problems where relativistic supersonic flow might occur (stellar winds, accretion onto neutron stars and black holes, jets in extragalactic radio sources). In many situations magnetic fields cannot be neglected and one needs a theory of relativistic magnetohydrodynamical simple waves. This will be the subject of the next section. 5.9. Simple waves in relativistic magneto-fluid dynamics We introduce Minkowski coordinates (x 1, x 2 , x 3 , t) in Minkowski space M. Then the Maxwell equations V a (u a ^ - frV) = 0 yield, for one dimensional flow, x = x 1 , dx(u1b°-u°b1)

= 0i

1

= 0,

1

dt{u°b -u b°) whence

u°bl - ulb° = J X = const,

(5.123)

which is a first integral of one dimensional flow. By writing u« = r(l,vx,vy,vz),

(5.124)

138

Relativistic fluids and magneto-fluids

from ujf = 0 it follows that

and Jx can be written as Jx = T{(1 - (i;,) 2 )^ - VxVyby -

VxVzbz}.

In the nonrelativistic limit J x~bx, which is an integral of motion for the relativistic one dimensional flow. In order to discuss simple waves we recall that the field vector is

and the characteristic equation is a2A2N4 = 0,

(5.125)

where a = u>a,

A = Ea2- B2,

F o r the phase cp we have

B = bx-Xb°,

G=l-X2,

(5.126)

w h e r e cp0 = — X, cpl = 1. First of all, we treat the case of magnetoacoustic simple waves. We distinguish two cases: (i)4>l. A material wave can coincide with an Alfven wave iff a = 0 is a solution to A = 0, which implies, because E > 0, B = 0. Similarly, a material wave can coincide with a magnetoacoustic wave iff B2G = 0, which implies (since G > 0 under the assumption e'p < 1) B = 0. In the following (except when stated otherwise) we shall assume that B ^ 0, and in this way we avoid the cases when a material wave can coincide with an Alfven or a magnetoacoustic wave. Also, an Alfven wave can coincide with a magnetoacoustic wave iff A = 0 is a solution to N4 = 0, which implies

Therefore, in order to exclude this case, we shall also assume that a2-G\b\2*0.

5. Relativistic simple waves

139

We shall impose the conditions B ^ 0, a2 — G\b\2 # 0 at some given initial point and by continuity they will hold in a neighorhood of such a point. The extent of this neighborhood can be ascertained only by the integration of the full set of equations. Under the above assumptions it has been shown (Section 2.4) that the equation N 4 = 0 admits four real and distinct roots for X{\k\ < 1). The corresponding right eigenvectors are given by equation (2.96). (ii) e'p = 1. In this case iV4 = — AG and since we exclude Alfven waves we must have G = 0. The corresponding right eigenvectors are formally the same as in the previous case. In both cases one has the obvious Riemann invariant jo=z s = const. and by taking p as an independent variable the equations defining magnetoacoustic simple waves are (5.127)

Ea2A'

dp

d/> a _d a + 4 ~dp ~ Ea2A'

(5.128)

From the above equations the following results of general character can be obtained (Anile and Muscato, 1988). 5.4. Under the assumption £ =£ 0, A # 0 , and e'p>\ the quantity \b\2 is an increasing function of p for fast magnetoacoustic simple waves, whereas it is a decreasing function of p for slow ones.

PROPOSITION

Proof. It is easy to show, from equations (5.127)—(5.128), that 2

dp=

e

'p\b\2/i-B2/r,a2.

(5.129)

Let vz be the normal speed of propagation, defined as in equation (2.44). Then (5.129) can be written as: 2 dp

v

"

L

-vl).

(5.130)

Under the above assumptions, we have 0 < r Es <(e p )~ 1/2 < r E / <: 1, where /;ls, vu are the slow and fast magnetoacoustic speeds, respectively, as defined in Proposition 2.4. Hence the statement follows. Q.E.D.

140

Relativistic fluids and magneto-fluids

Remark 1. For e'p=l, for the root corresponding to vzf=l, \b\2 — 2p = constant.

we obtain

Remark 2. We notice that for a fast magnetoacoustic wave A > 0, whereas for a slow one A < 0. In fact,

where v%A is the local Alfven speed, satisfying i;Zs < v^A< Remark 3. Let q = p + ^\b\2 be the total pressure (gas pressure + magnetic pressure). Then, from equation (5.130),

dp

1 -1>!

and for e'p>l9 dq/dp > 0; hence q is always a monotonically increasing function of p. Under the assumptions a #= 0, B ^ 0, we shall consider solutions with a stagnation point, where vx = vy = i;z = 0 at a given pressure p 0 . Hence, fl(p0) = _ A(p0), 5(p 0 ) = bx(p0\ and therefore a and B will maintain their initial signs. One can always choose the reference frame such that bx(p0) > 0 and consider only progressive waves, for which X > 0. With these choices, one always has a < 0, B > 0. Another important result of general nature can be obtained concerning the behavior of the eigenvalue I along the solution X = X(p). By substituting

into N4 = 0, we obtain N4(U,A) = 0, which holds identically for a chosen root X = X(p). Hence dN4 duA duA dp

dN4 dX _ dX dp

which can be written as SN4

dN^dX dX V^O, dp

(5.131)

5. Relativistic simple waves

141

with

dp ~{du«

+

db«

d

jEa2A

+

dp '

In Section 4.4 it has been shown that [equation (4.74)] SNJdp = a2{a2K1 + GK2),

(5.133)

where K1=rie; + ( K2=

-*;\b

We proceed with the following propositions. PROPOSITION 5.5. Under assumptions B ^ 0, a2 - G|b\2 ^ 0, e'p > 1 and compressibility hypothesis W = - e'p + 2 ^ ( 4 - 1)/^/ > 0

(P^^/ condition),

one has SNJdp # 0. Proof. In equation (5.133) we substitute a2 = Gu|/(1 —1;|), hence dp

1 — vi

If dNJdp = 0 at some point, then v\ must satisfy N 4 = 0. Now where

: n{e' - I K ~(n + e'\b\2 - bl)vl(\ - vl) + b2(l - v2).

The compatibility between N4 = 0 and SNJdp = 0 is then the equation:

y = x1|fo|4 + x2|fo|2 + x 3 = o, with X1=-nW2-3e'pW{e'p-\)2, X2=-

rj2W2 - 3nW{e'p - 1) - 9e'p{e'p - I) 3 + b2lr,W2+ 3W(e'p - 1)],

From assumption J B / O , a 2 - G | f o | 2 ^ 0 it follows that b2<\b\2 by using this inequality, after some manipulations, we get Y<-3(e'p-l)W[(e'p-l)\b\2-2Y<0.

and

Q.E.D.

142

Relativistic fluids and magneto-fluids

PROPOSITION 5.6. Under assumptions B ^ 0, a2 - G|fo|2 # 0, e'p > 1 and the compressibility hypothesis W > 0, one has — >0 /or progressive and retrograde waves, respectively. Proof. Under our hypothesis the roots of N 4 = 0 are all distinct, hence

dNJdk # 0. p0?

It follows that dNJdX has the same sign as at the stagnation point Vx = vy = vz = 0. A simple calculation shows that: = ±2X0[(ri + ep\b\2 + B2)-4ep^

where ,

(rl + e'\b\2 +

B2)±l(r1-e'p\b\

The choice of the sign ± corresponds to fast and slow magnetoacoustic waves, respectively. Furthermore, since we shall deal with progressive waves, Xo > 0. It follows that dNJdk is positive for fast magnetoacoustic waves, and negative for slow ones. Because dNJdp ^ 0, it follows that BNJSp has the same sign at the stagnation point. One finds, after lengthy calculations, that

It can be seen that the sign of (SN4/3p)P0 is negative for the fast magnetoacoustic waves and positive for the slow ones. Then the statement follows from dA/dp = - (3NJ3p)/(dNJd2) > 0.

Q.E.D.

Remarks. For e'p = 1, from (5.133), one has dNJdp = 0, which corresponds to the exceptional case. Equations (5.127)—(5.128) can be written explicitly as follows. In the case e'p > 1 (5.134a)

5. Relativistic simple waves

143

= <x1Tvy + a2br

dp

(5.134b)

= oc1Tvz + oc2bz,

(5.134c)

— = pxTvx + )82&x - (efp - l)Ba2/GA,

(5.134d)

^ ^ I X dp

(5.134e)

+ jSA,

d

^ = plTvz + p2bz, dp

(5.134f)

where ax = - a\e' a\epp - l)/AG,

aB(epp a 2 = aB(e'

\)/nA,

In the case e'p=]., under assumption B^O, a2 — G|/?| 2 #0, one has two roots A = ± l and equations (5.127)—(5.128) admit the following invariants, besides Ju

Js = !>* + (t?A + ^ A + vzbz)~]2p, J6 = ———--

with X

l9

K2 constants.

Now we turn our attention to the case e'p> 1. From equations (5.134) we obtain

-£- [T(vybz - vjb,)-] = (e'p - 1 ) ~ [T(vybz - vzby)l

dp At the stagnation point p 0 we have:

(5.135)

A

hence, by the uniqueness theorem for the initial value problem, equation (5.135) yields the following invariant: J2 = vybz-vzby

= 0.

(5.136)

144

Relativistic fluids and magneto-fluids

Similarly, it can be seen that dp\bj

bz

whence another invariant, h = by/bz.

(5.137)

In general, equations (5.134) are too complicated to be integrated analytically. In the following we shall treat some special cases which can be brought to quadratures. First of all, we treat the case of a longitudinal magnetic field,

From equations (5.134) we then have \y = vz = 0. The characteristic equation NA = 0 admits the following four solutions: Xm = (Vx + Xj/(1 + VxXm) where

(I l i 2 )= ±(e'py1/2,

m= 1,2,3,4,

(I 3 , 4 )= ±(\b\2/E)1/2.

The roots I3A coincide with the Alfven waves and shall not be considered here. The simple waves for the acoustic speed are given by the following invariants: J1=bJT9

(5.138) 1-v

The invariants J+ coincide with those of equation (5.27) obtained in the case of fluid dynamics. The other case we shall consider is when the fluid's motion is purely longitudinal

and the magnetic field at the stagnation point is purely transverse

Then from the invariant J1 it follows that bx = 0 throughout the flow, hence b° = 0 and B = 0. In this case we consider the following simple

5. Relativistic simple waves

145

root of N4 = 0, corresponding to the fast magnetoacoustic waves, with From (5.134e)-(5.134f) we obtain the following invariants JA = \nby—

I ev

I

e

—dp, J5 = lnbz— )n

—dp. n

(5.139)

Finally, equation (5.134a) gives the invariants: J± = i l n [(1 + t;x)/(l - «g] + f1-[<,(!, + e'p\b\2)IE]1'2 dp. For a fluid for which p = p(p, S)

J^ and equation (5.139) gives by/p = const.,

bjp = const.,

which are analogous to the corresponding nonrelativistic invariants (Cabannes, 1970). Remark 5. In the case bx = by = bz = O equations (5.134) reduce to the purely fluid equations: dp

T2r,{v, - A ) ' I Vy

dv, dp dvz dp

2

r r,(v•,-A)' I T2ri(v •

-X) 2

(e'p - )T vx ± Wv ~ Another case of interest is when, at the stagnation point, (bx)0 = 0. It follows that J1 = 0. For plane polarized waves we can choose by = vy = 0, hence from the

146

Relativistic fluids and magneto-fluids

invariant J1 we obtain bx = vxvzbz/d,

3=

1-vi

For the magnetoacoustic waves we have

a=

with

r e'p\b\2-(b2)2/F2.

A = rj +

Finally, equations (5.134) yield, in this case, (e'-l)a3 {h)x-\\ ATG

do, dp

(e'p-l)a3Xvz ATG

dvz dp dp

=

~

(5.140a)

(e'p-l)aB rjAT

BYvS(e'p, - D * 4 .

a L

-b°vz),

(5.140b)

p i ) B n , K ,«,-!)£

\

From (5.140a), for the fast wave we have that because a < 0 and A > 0, dvjdp > 0. Since we require a stagnation point p 0 at which vx(p0) = vz(p0) = 0, from equation (5.140b) and the uniqueness theorem for the initial value problem we have vz = 0, and equation (5.140c) reduces to dbz_ dp

f

b z rj9

p

which is the transverse case studied above. The system of equations (5.134) has been integrated numerically (Muscato, 1988b). For the sake of definiteness we choose the field linearly polarized,

From the invariant Jz we then have, assuming bz ^ 0,

It is convenient to write the system (5.134) in dimensionless form, by introducing the variables bx = bx/Jl9

bz = bz/Ju

rj = ri/p0,

5. Relativistic simple waves A = Ea2 - LB2,

E = rj/p0 + L | 5 | 2 ,

B = (l+ b°a)/T9

\b\2 = \b\2/Jl

L =

147

J2/Po,

where p0 is the initial pressure. Therefore the system (5.134) can be written (omitting the bar)

dvzJe'p-\) dp

Ar

a[_a2lvJG + (bz -

(5.141b)

b°vz)BLIn\,

AG

dp

Now we investigate the signs of the derivatives dvjdp, dvjdp, dbz/dp. At the stagnation point (vx = vy = vz = 0) we have dvx dp"

a(e'p - 1 ) (Y\ + L){v 0

(efp -

dvz dp

A

0

2

A

- vl) + L(l + b2) Grj

\)La

r\A

dbz ^ - { e ' t dp" 0 nAV2A P

2

>i -

2

vA).

For fast magnetoacoustic waves a<0,A>Q,B>0 bz>0, (dvx/dp)o<0,

and for an initial datum

(dbz/dp)o>0.

The sign of (dvx/dp)0 depends on the initial datum. For slow magnetoacoustic waves (dvx/dp)0 > 0. From equations (5.141) we see that for vz > 0, bz > 0 one has dbz/dp > 0, hence bz is a monotonically increasing function of p for the fast wave. When starting to integrate system (5.141), we calculate the four roots /I1?/l2, A3, A4 and we choose one to be followed during the integration. A suitable algorithm ensures that we are always following the same root; the accuracy of the numerical code has been tested against the exact solutions previously found. The agreement has been found to be extremely satisfactory. The results of the calculations are plotted in Figs. 5.8-5.11 for various values of the parameter L.

148

Relativistic fluids and magneto-fluids

Fig. 5.8a. Progressive fast magnetoacoustic simple waves, vx as a function of the parameter z = m/kBT in the ultrarelativistic region, corresponding to an initial propagation speed (at the stagnation point) Xo = 0.57, for the values L = 1 and 100 of the parameter L = Jj/p0- The two curves (L = 1 and L = 100) graphically coincide.

-0.2

-0.4 -

-0.6

-0.8

Fig. 5.8b. Same as in Fig. 5.8a but with vz as a function of z for L = 1 and L = 100.

8.e-5 -

6.e-5 -

4.e-5 -

2.e-5 -

5. Relativistic simple waves

149

Fig. 5.8c. Same as in Fig. 5.8a but with bz as a function of the parameter z for L = 1 and L = 100 (the two curves graphically coincide).

0.2 -

0.0 0.0

0.2

0.4

0.8

Fig. 5.9a. Progressive fast magnetoacoustic simple waves, vx as a function of the parameter z in the intermediate regime for the values L = 1 and L = 100. The initial propagation speed is k0 = 0.94.

-.08 -

-.16 -

-.24 -

-.32

-.40

150

Relativistic fluids and magneto-fluids

Fig. 5.9b. Same as in Fig. 5.9a but vz as a function of the parameter z for L = 1 and L= 100.

0.80-

0.60

0.40

0.20 -

0.00

36

Fig. 5.9c. Same as in Fig. 5.9a but bz as a function of the parameter z for L = 1 and L=100.

-0.2

-0.6 -

-1.0 20

5. Relativistic simple waves

151

Fig. 5.10a. Progressive slow magnetoacoustic simple waves, vx as a function of the parameter z in the intermediate relativistic regime for the values L = 1 and L = 100 (the two curves graphically coincide). The initial propagation speed is Xo = 0.18.

-.04

-.08 -

-.12 -

-.16 -

-.20

Fig. 5.10b. Same as in Fig. 5.10a but with vz as a function of the parameter z, for L= 1 and L= 100 (the two curves graphically coincide).

-.04-

-.08 -

-.12 -

-.16 -

-.20

Relativistic fluids and magneto-fluids

152

Fig. 5.10c. Same as in Fig. 5.10a but with bz as a function of the parameter z for L = l and L= 100.

1.08

1.06 -

1.04 -

1.02-

1.00

Fig. 5.11a. Progressive fast magnetoacoustic simple wave, v x as a function of the parameter z in the nonrelativistic regime (z » 1) for the values L = 1 and L = 100 (in the Figs. 5.11a-5.11c the two curves graphically coincide). The initial propagation speed is Ao = 0.99.

-0.8

-.16 -

-.24 -

1000

1800

2600

z

3400

4200

5. Relativistic simple waves

153

Fig. 5.11b. Same as in Fig. 5.11a but with vz as a function of the parameter z for L = l and L= 100.

0.4 -

0.2 -

0.0 1000

Fig. 5.1 lc. Same as in Fig. 5.1 la but with bz as a function of the parameter z for L = 1 and L = 100.

-0.2 -

-0.6 -

-1.0 1000

154

Relativistic fluids and magneto-fluids

Notice that the velocity vx is not a monotone function of z, at striking variance with the nonrelativistic results and with the relativistic case with zero magnetic field. A similar result is found for the ultrarelativistic state equation (e = 3p) but in this case for the component vz (Muscato, 1987). This phenomenon is peculiar of the relativistic regime and can be explained as follows. In a relativistic framework one has the limitation v2x + v2z
for the speeds vx,vz. Therefore, vx and vz cannot both be monotone functions of p, otherwise the above inequality would be violated. It follows that at least one of the variables vx or vz must show a nonmonotone behavior. This phenomenon could have important consequences for the relativistic MFD Riemann problem. The critical time for breaking can also be computed from equations (5.10)-(5.10') (Muscato, 1988a) and the results are plotted in Figs. 5.125.13. Finally, we investigate Alfven simple waves. Because Alfven waves correspond to multiple roots of the characteristic equation, the methods employed in the previous sections are not directly applicable. In this case it is convenient to resort to a general method due to Boillat (1982a). We start with the conservation equations dJ*A = 0, A = 0,1,2,..., 8,

(5.142)

with We look for one dimensional solutions of the kind

with cp = x — X(cp)t.

The equations (5.142) yield

Now, in Section 4.4 we have seen that the Alfven waves are exceptional,

Hence, we obtain the following invariants: JXA _ xpA

= const ?

4 = o, 1,2,..., 8.

(5.143)

5. Relativistic simple waves

155

Fig. 5.12. Barotropic RMFD with y = 4/3. Plot of (T) versus (v). The lower curve corresponds to W=0, the intermediate one to W= 10, and the upper one to W=l00.

1584.9

J

251.2 -

en 39.8 -

0.0

Fig. 5.13. Polytropic RMFD and nonrelativistic MFD with y = 5/3, c = 0.4. Plot of (T) versus (v). The lower curve corresponds to W = 1 and the intermediate and upper ones to W = 10, 100, respectively.

199.5 -

39.81 (7)

7.943

1.584 -

0.316

156

Relativistic fluids and magneto-fluids

From these we obtain, besides the already known invariants J x and 5,

J4. = by —

J5 = bz- BTvJa, J 7 = ux — Au ,

which correspond closely to the nonrelativistic analogs (Cabannes, 1970). The solutions which have been found in this section could be very useful for checking numerical codes for the full set of partial differential equations of relativistic magneto-fluid dynamics (Sloan and Smarr, 1986).

6 Relativistic geometrical optics

6.0. Introduction

In the previous chapters we have studied waves which can be represented either as exact solutions of the field equations (as in the case of simple waves) or as propagating surfaces of discontinuities, for which exact transport laws can be obtained. These classes of waves, although very important for testing the mathematical structure of the theory, have a limited range of applicability. In particular, simple waves are restricted to one dimensional propagation into a constant state whereas propagating surfaces can model only impulsive waves. This leaves out the vast class of harmonic waves propagating into an arbitrary nonuniform state, which comprises most of the interesting applications. In order to treat the latter case it is necessary, in general, to resort to perturbation methods. Most of the perturbation methods used for this purpose are based on the so-called geometrical optics approximation, or variants thereof such as the high-frequency expansion or the method of multiple scales. The idea underlying these methods is that there are at least two widely different length (or time) scales in the problem, the length L characteristic of the variation of the background state into which the wave is propagating, and the mean wavelength X of the wavetrain, with the ordering X «L. One then introduces a parameter a = XjL into the problem (usually through the appearance of one or several rapidly varying phase functions and also through other stretched variables, depending on e) and seeks asymptotic solutions of thefieldequations as appropriate power series in e. One finds that one can define a class of curves, the rays along which the various terms of the asymptotic expansion propagate according to transport laws. It is apparent that the asymptotic expansions which are used are valid only in a restricted class of coordinates (in order to convince oneself of this it suffices to think of coordinate transformations depending on the parameter e which could destroy the ordering of the asymptotic expansion). However, this is not a hindrance to using these methods for relativistic problems because the existence of a restricted class of coordinates enjoying a definite set of properties is in itself an invariant statement. However, in

158

Relativistic fluids and magneto-fluids

concrete examples it may be very difficult to prove the existence of such a class of preferred coordinates (de Arajuro, 1986). In this chapter we shall introduce the basic concepts underlying the perturbation methods referred to as the high-frequency expansion and the two-timing methods, within a relativistic framework. First we shall expound the high-frequency expansion method by applying it to Maxwell's equations in vacuo and in a linear isotropic medium, in general relativity (notice that these equations are linear). Then we shall introduce the twotiming method both in the linear and in the weakly nonlinear case by applying it to Maxwell's equations in the presence of a cold plasma. The high-frequency expansion method in a nonlinear framework (also called the method of asymptotic waves) will be the subject of the next chapter. The theory developed in this chapter is motivated by applications to laboratory plasma physics and astrophysics. The interaction of an electromagnetic wave with a plasma (a topic treated in Sections 6.3 and 6.4 in the cold plasma approximation and under a simplified hypothesis) is definitely a relativistic problem and therefore should be treated in a covariant framework (although for some specific problems it might be more convenient to use a noncovariant formalism; Shukla et al., 1986). The covariant theory of the interaction of an electromagnetic wave with a plasma could be relevant for several laboratory plasma problems (for example, the stimulated emission by intense relativistic electron beams; Miller, 1985) and for diverse astrophysical problems. One of the most interesting applications to fundamental astrophysical theory is in the derivation of the radiative transfer equation. When considering the transfer of radiation in cosmology, as in the case of the microwave background radiation or the light emitted from quasars, or the radiation emitted from nearby a neutron star or black hole, one needs to consider the relativistic radiative transfer equation (Ellis, 1971; Weinberg, 1972; Thorne, 1981). In its turn the relativistic radiative transfer equation can be obtained either from relativistic kinetic theory (starting from the relativistic Boltzmann equation for photons; Ehlers, 1971) or from relativistic geometrical optics (Ellis, 1971; Anile and Breuer, 1974; Anile, 1976) or, precisely, from the propagation laws for the amplitude and polarization of the wave. The latter approach is more satisfactory when dealing with polarized radiation or with radiation propagating through a dispersive medium. In fact, in the kinetic approach, in these cases one has to postulate the propagation laws for the photons (Bicak and Hadrava, 1975) or resort to geometrical optics itself. Instead, as will be seen in Sections 6.2 and 6.3, relativistic geometrical optics is capable of deducing the propagation laws

6. Relativistic geometrical optics

159

for the amplitude and polarization directly from Maxwell's equations. The plan of the chapter is the following. In Section 6.1 we investigate high-frequency waves for Maxwell's equations in vacuo in an arbitrary space-time, following an earlier treatment by Ehlers (1967). In this way we prove that, to the main order in £, the electromagnetic field satisfies the laws of geometrical optics in general relativity, that is, the area-intensity law and the parallel propagation law for the polarization (Synge, 1960). In Section 6.2, following earlier treatments by Ehlers (1967) and Anile and Moschetti (1979), we investigate high-frequency waves for Maxwell's equations in a linear isotropic refractive medium. Proceeding as in the previous section, we obtain the corresponding laws of geometrical optics, that is, the modified area-intensity law (taking into account the refractive index) and the modified propagation law (differing from parallel propagation by a term representing the interaction of the polarization with the vorticity of the medium). In Section 6.3 we deal with the two-timing method. This is necessary if one wants to incorporate dispersive effects, as in the case of waves in plasmas. We discuss linear locally plane electromagnetic waves propagating in a cold relativistic plasma and base our treatment on that of Anile and Pantano (1977, 1979). Further extensions of the theory to cases where the plasma is magnetized can be found in the articles by Breuer and Ehlers (1980, 1981). By applying the two-timing method we are able to obtain the laws of geometrical optics in general relativistic dispersive media, which had been postulated by Synge (1960), without deriving them from Maxwell's equations. Moreover, we obtain also a propagation law for the polarization which differs from the parallel propagation law by a term representing the interaction with the plasma vorticity. Such a term is analogous to that occurring in the case of propagation in a refractive medium. In Section 6.4 we consider the same problem but modify the perturbation method (by introducing several phase functions) in order to treat the weakly nonlinear modulation of a locally plane harmonic wave. The calculations are rather lengthy and the final results are the following: (i) the wave complex amplitude satisfies a propagation law along the rays which is of the form of a generalized nonlinear Schrodinger equation; (ii) the polarization vector of the wave satisfies the same transport law as in the linear case. We remark that in all cases treated in this chapter the only way a background gravitationalfieldcan affect propagating radiation is via its effect on the rays (they can be bent or focussed by space-time

160

Relativistic fluids and magneto-fluids

curvature). This is obviously consistent with the usual interpretation of the equivalence principle which in the approximation we are dealing with (geometrical optics) amounts to stating that a background gravitational field can have no local effect.

6.1. Geometrical optics in general relativity Maxwell's equations in vacuo are equations (4.81)—(4.82) in the absence of charges, J a = 0, and they are written
(6.1)

V a F«' = 0,

(6.2)

where F a v is the electromagnetic field tensor. A high-frequency wave is defined by the following formal asymptotic expansion (6.3) n=0

where cp(x) is a real function to be determined, called the phase, and co is a real parameter, co > 0 (related to the "frequency" of the wave). Obviously this expansion is meaningful only in a selected class of coordinate systems. We substitute the expansion (6.3) into equations (6.1)—(6.2) and proceed by equating to zero term wise the coefficients of the resulting formal series in 1/co. Then, to the zeroth order, we obtain, writing /a = Va
(6.4a)

laFaP = 0.

(6.4b)

(0)

To the nth order we have

tf.J^

+ V.F^O.

(6.5b)

From (6.4a)-(6.4b), because F^ ^ 0, we obtain the compatibility relation iai" = 0.

(6.6)

6. Relativistic geometrical optics

161

This equation shows that the hypersurfaces of constant phase are null hypersurfaces (coinciding with the characteristic hypersurfaces), in analogy with the weak discontinuity case (Section 4.5). Let rf be a vector such that rc°7a= 1. Then by contracting (6.4a) with na, we see that Fap has the following expression

&* = *«/,-/„*„

(6-7)

where \j/a, which is determined up to the "gauge" transformation iAa->*Aa+A/a (AeR), is such that This result is also analogous to equation (4.100) of the weak discontinuity case. The rays are introduced as the bicharacteristic curves, that is, xa = xa(i) such that

£-'• To the first order, equations (6.5a)-(6.5b) read i(l Ffiy + lfiFyx + lyF p) + ljy^y(1)

(1)

(1)

V^f)

+ Z , ( V ^ . - V> v )

+ ' y ( V « ^ - V ^ J = 0. p

a

(6.9a) a

ila F* + (V> )/^ + il/ W/ - (VJW

a

- / V>^ - 0.

(6.9b)

Let ^ = F pl\ Zp = l«(V^p - V ^ a ) . (1)

(D

Then equation (6.9a) by contraction with /a yields, i\l/p + Zp = kip, k real scalar, u

(6.10)

(1)

whence Now Zp = laWail/p +
fc'/^O,

(6.11)

with k'=-k-VJ*. Equation (6.11) provides transport equations for both the amplitude and the polarization of the wave. In fact, by contracting with the complex conjugate ij/fi9 we obtain H7JiA| 2 + |iAI2Va'a = 0, which is the transport equation for the amplitude.

(6.12)

162

Relativistic fluids and magneto-fluids

This equation can also be written as a conservation law, Va(/a|iAI2) = 0.

(6.13)

The transport equation for the polarization can be obtained as follows. Let u* be the four-velocity of an arbitrary observer, uflufl= — 1. The frequency Q of the wave relative to this observer is defined by (Synge, 1960)

Let us introduce, at least locally, an orthonormal tetrad i u*

v

a

ea

e a\

where va is the wave's propagation unit vector in the observer's rest frame,

and e a , ea are two spacelike unit vectors, orthogonal to each other (1)

(2)

and to wa, va. By exploiting the "gauge" freedom ^ a -> \//a + Xla we can always make <M" = 0.

(6.14) a

a

It follows that \^a lies in the two-space spanned by e , e , hence we can write

r = W\
(6.15)

where e* is the polarization vector, ea = c o s 0 e a + s i n 0 e a , (1)

(6.16)

(2)

0 being the polarization angle. At each point, ea, ea are defined up to an orthogonal trans(1)

(2)

formation. We exploit this arbitrariness in the definition of ea, ea by (1)

(2)

choosing them to satisfy

Then from (6.11), (6.12), and (6.17) it follows that i"V«0 = O,

(6.18)

6. Relativistic geometrical optics

163

that is, the polarization angle with respect to the basis e a, ea is propagated (1) (2)

in a parallel manner. The conservation law (6.13) has a simple geometric and physical interpretation. Let L be the null hypersurface given by (p(x*) = c = const. Let $ be a bicharacteristic tube and o a cross section of such a tube and denote by do its area element. Then one can state the following theorem. THEOREM 6.1.

For all cross sections one has \ij/\2da = const.

Proof. We choose coordinates adapted to E, x° = 0 n = O ,

g1A = 0, A = 2,3

for the metric coefficients. In these coordinates the transport equation (6.13) is written

where whence A cross section of a tube of rays is given by

Now the line element in these coordinates is ds 2 = Therefore,

164

Relativistic fluids and magneto-fluids

and it is easy to check that \g\ = \detgAB\. It follows that |i/,|2d(j=

f

f

\\l/\2\g\dx2dx3=

\F(x2,x3)dx2dx3

= const.

Q.E.D.

This result has the following physical meaning. The energy-momentum tensor for the electromagnetic field in vacuo is

By substituting (6.3) into (6.19) we obtain a formal series in 1/co. Now we average out to zero the rapidly oscillating terms containing factors of the kind e~2kt)(p, according to the definition 0) C2*/«

T.

This averaging is to be interpreted as "ensemble" averaging. Physically it corresponds to averaging over a time scale which is long compared to the period of the wave, but much shorter than the typical time scale of change of the wave's amplitude and of the background gravitational field. The averaged zeroth order energy-momentum tensor is J / « j ' .

(0)

(6.20)

87l/i 0

It follows that the averaged energy-density as measured by the observer with four-velocity ua is (0)

>a/!

^ ^ | ,

(6.21)

TCU

whereas the averaged energy-flux in the wave's propagation direction v is 2

< (0)

Q2.

(6.22)

Therefore, for all observers measuring the same frequency Q = constant, Theorem 6.1 expresses that =

< d
for all cross sections of a bicharacteristic tube.

6. Relativistic geometrical optics

165

This is the standard area-intensity law of general relativistic geometrical optics (Friedlander, 1975). The higher order corrections to the geometrical optics field FaP as well (0)

as the resulting modifications of the area-intensity law have been studied by Anile (1976) using the spinor formalism. The propagation law (6.13) for the amplitude and (6.18) for the polarization are at the basis of the derivation of the relativistic radiative transfer equation in vacuo (Ellis, 1971; Anile and Breuer, 1974; Anile, 1976). In the next section we shall treat geometrical optics for radiation propagating through a refractive medium. 6.2 Geometrical optics in general relativistic refractive media Maxwell's equations in current-free refractive media are equations (2.46)(2.47) with Ja = 0, and they read V.F, y + V,F ya + V y F a , = 0,

(6.23)

VJafi = 0,

(6.24)

where F a/? , 1^ are the electromagnetic field tensor and induction tensor, respectively. As is usual in many applications we shall assume a linear and isotropic refractive medium. The constitutive relations between FaP and Iap are equations (2.50)—(2.51) and can be written in the form (Pichon, 1965) Iafi = h%F,*v>

(6.25)

with the susceptibility tensor xip given by

Z3 = J W9} ~ Gfal) ~ )—^- foXty - 9}u>ua - gWufi + 0guX), (6.26) where N is the refractive index,

N2 = fi(l+4nx), u* is the medium's four-velocity, fi is the magnetic permeability, and x is the electric susceptibility. We look for high-frequency waves of the form

f co-jX

(6.27a)

166

Relativistic fluids and magneto-fluids

/a, = Re{e^f; co-»I X

(6.27b)

Then, to the zeroth order, we obtain laF)fi7 + lpF)ya + lyFf0 = 0,

(6.28a)

/ a / a / * = 0,

(6.28b)

ila Fp + Va / a ^ = 0.

(6.29b)

(0)

and to he nth order

From (6.28b), by using the constitutive relation (6.26) we obtain ?Pv-l»FflXufi) = 0.

(6.30)

Let Yp = F^l*. Then equation (6.30) is rewritten Ye + (N2 - l)(Quv f,v - yvMvM
(6.30')

By contracting with M^ it follows that, since N # 0,

Then, from (6.30'), Fv/?fcv = 0,

(6.32)

where fcv = - (p - (N2 — l)Qwv).

(6.33)

Notice thatfevis tangent to Z, because kvlv = 0. Now let na be a vector such that By contracting equation (6.28a) with na we obtain F ao = i//aln — ij/nla,

(6.34a)

with il/a determined up to the gauge transformation (6.34b)

6. Relativistic geometrical optics

167

Now we prove that for a nontrivial solution we must have /°7a ^ 0. In fact, if /°7a = 0, we can always choose the gauge transformation such that ^ a satisfies

Then, from the condition (6.31) it follows that, since

hence Yp = 0. Therefore, from equation (6.30) we have \j/p = 0. Having established that lja # 0, we can choose ^ a , by applying the gauge transformation, such that \jjj* = 0,

(6.34c)

and then equation (6.31) yields il/au" = 0.

(6.34d)

Finally, from (6.30') we obtain the compatibility relation ^

-

1

.

(6.35)

By the way in which it has been obtained it is apparent that equation (6.35) coincides with the characteristic equation for the system of Maxwell's equations (6.23)-(6.26) (Pichon, 1965). It is also easy to see that

To the first order, equations (6.29a)- (6.29b) give 7

y

(D

HW

+ VVAikxF«>(i)

.(AT2 — 1

^) + '«(

FaH a u

(i)

"

^ - ,

Let

Then equation (6.37a), contracted with fea, yields (i)

) = o,

(6.37a)

yfw) = o.

(6.37b)

168

Relativistic fluids and magneto-fluids

with k an arbitrary scalar, which, after substitution into equation (6.37b), gives - 2kaWJp + klfi + ^a(Vafc' (N2 - \) - i- F*%uy = 0. ft (i) ^

(6.37c)

From this equation we derive the amplitude transport law a

) = 0.

(6.38) a

The rays are the curves tangent to k bicharacteristics. Let

{u\v\

and coincide with the

e\ e«}

C

(1)

(2)

J

be the orthonormal frame as introduced in the previous section, but with /a NQ Then, from equation (6.37c) one obtains the transport law for the polarization, 2/c«Va0 + ^ V e , , e , , = O,

(6.39)

where co^v is the vorticity tensor of the vector field wa, defined by with

K= and 6 is defined by (1)

(2)

A physical interpretation of the conservation law (6.38) can be obtained as follows. Let us assume u* is hypersurface orthogonal and that u = dr, where T is a scalar function. Let Z be the hypersurface cp(x) = c, ceU, ox = {XEH:T(X) = const.}. Then it is possible to prove the following result. THEOREM 6.2.

One has

ff i1

It

, JGX = const.

6. Relativistic geometrical optics

169

Proof. Let (y°, x') be normal coordinates based on £ (Synge, 1960), which is given by y° = 0. In these coordinates /« = (L, 0,0,0), /« = (1,0,0,0), where L=1J^ and the metric is given by ds(dj;) Now, on Z the coordinates xl are arbitrary. Let t = T(0,Xl). We choose the coordinates xl on D in such a way that they are normal coordinates based on the family of two-surfaces t = const. On £ the vector

has components wa = (0,1,0,0) and

In these coordinates the metric is ds 2 = Udy0)2

+ - ( d y 1 ) 2 + £y d / d / ,

with w = wawa. The metric on the two-surfaces t = const, is given by d 5 2 i,= Also, (,,0,0). Let |^| 1/2 be the surface element induced on oz by the metric gu. Then (6.38) reads,

which proves the theorem.

Q.E.D.

Now the Minkowski Ta/? energy-momentum tensor for the electromag(M)

netic field in matter is given by equation (2.59).

170

Relativistic fluids and magneto-fluids

By proceeding as in the previous section we obtain, at the lowest order,

=-!-MW (Af)

(6.40)

57C

which gives an energy-density ^ 2 | < A | on ft

2

(6.41)

and an energy flux, in the wave's propagation direction, M2.

(6.42)

Therefore, for all observers measuring the same frequency Q = const., one has

J>-L

d>dat = const,

(6.43)

which is the general-relativistic formulation of the area-flux law of geometrical optics in isotropic refractive media (Anile and Moschetti, 1979). By using this law and equation (6.39) for the polarization one could derive the transfer equation for radiation propagating through refractive media. However, in a plasma, refractive properties are usually associated with dispersion. In the next section we will consider a very simple example for a dispersive medium: a cold electron fluid, which is a model of notable interest for some astrophysical and laboratory plasma.

6.3. Electromagnetic waves in a cold relativistic plasma: linear theory Maxwell's equations in vacuo in the presence of charges and currents are obtained from equation (4.81)-(4.82) and are written V ^ y + V ^ y a + VyFa/, = 0, *9

(6.44) (6.45)

with fi0 the vacuum magnetic permeability. We make the following assumptions for the medium (Madore, 1974): (i) The medium consists of two noninteracting components, the ion and electron components;

6. Relativistic geometrical optics

111

(ii) the energy-momentum tensor for the electron component is that of a pressureless perfect fluid (dust), i.e., Taf} = nmuaup,

(6.46)

a

where n, u are the electron number density and four-velocity of the electrons; (iii) because of the larger proton mass the ions are considered as a fixed background. The total current Ja is given by Ja=-47ce(mi a-wI.tiJ)

(6.47)

where e is the electron charge (absolute value), ni9 uf are the ion number density and four-velocity, respectively. Then to Maxwell's equations (6.44)-(6.45) we must add the equation of motion for the electron fluid

VpTap=

-enFa\9

which is equivalent to the electron number conservation equation Va(nwa) = 0

(6.48)

and the electron momentum equation uflWMa= --F^u..

(6.49)

Now we linearize equations (6.44)-(6.45) and (6.48)-(6.49) around an unperturbed state with FaP = 0. Let Fa/?, n, ua denote the perturbations to the electromagnetic field, number density, and four-velocity, respectively. Then the linearized equations give V ^ +V ^ +V/^^0, V ^ = - 4nfioe(Au* + mT), a

(6.50) (6.51)

Va(nu + m n = 0,

(6.52)

u*%ua + w ^ w " = - - F ° " V

(6.53)

M a = 0,

(6.54)

the latter arising from the normalization condition wawa = — 1. Now we analyze the system (6.50)-(6.54) by using the two-timing method (Whitham, 1974; Jeffrey and Kawahara, 1982). We assume that,

172

Relativistic fluids and magneto-fluids

in a selected class of coordinates (xa), the quantities F a/? , n, ua are of the following form:

pfi (

10()\

#

(

i ( ) V etc.,

with © a function to be determined and e > 0 a real parameter. The unperturbed background fields gafi9 n, ua are assumed to vary on the slow scale, that is, they are of the form gap — Ga/?(exM), n = Nyex*1), etc. It is convenient to introduce the auxiliary variables X*1 = 8xfi, cp = -©(ex1"). For fixed e > 0 the X* can be interpreted as "slow" space-time a coordinates. The parameter 8 measures the ratio of the fast length scale to the slow one. If we w r i t e / =f(Xa, cp) and put J

*

dXa'

J

d« dx«

dX*

dcp'

then where a

is the normal to the wavefront cp = const. For the connection coefficients r^y(x/x,e) we have 1

with

Py -

£1

/Jy

f^ = | G a 5 ( G ^ y + G ^ + G ^ ) ,

whence T^y = 0(8). Henceforth we shall use a semicolon in order to indicate the covariant derivative with respect to the slow variables X", that is

In terms of slow and fast variables equations (6.50)-(6.54) can be rewritten ljfiy

+ lpFya + lyfafi + s(FPy;a + Fyoc;p + Fafiiy) = 0, a

(6.55)

a

(6.56)

su»u\ + fiu"^ = - - F*^

(6.57)

);a = 0.

(6.58)

(nw + nw ),

6. Relativistic geometrical optics

173

A locally plane wave is defined by the following formal asymptotic expansion (Anile and Pantano, 1977)

Substituting into equations (6.55)-(6.58) and termwise equating to zero the coefficients of the resulting formal power series in e, we obtain at the zeroth order

a

iQMa + - F a " w

(6 6o)

-

=0,

( 6 - 61 )

Qn +n/ a w a = 0,

(6.62)

m(0)

(0)

a

(0)

"

(0)

where Q = u^ is the local frequency of the wave relative to an observer at rest with the medium. For the higher orders one gets

) fq/r, + &«, + fa = 0' (6-63) ft u* + n u * ) = 0, (q+D

(«+D

(q)

(6.64)

(«+D

iQ ua +u>lu*+utlu«+-

F^uu = 0,

(6.65)

iQ n +m/ a wa + ( n u a + n« a ). a = 0.

(6.66)

(«+i) (+l)

(«)

(«) (g+l)

m («+D

{q)

M

(q) '

By proceeding as in the previous section, from equation (6.59) we have Ffp = ^Jp-^pL

with \j/a determined up to the transformation

(6.67)

174

Relativistic fluids and magneto-fluids

Since Q = laua^Owe can always choose A such that iAX = 0.

(6.68)

Then equations (6.60)-(6.61) are rewritten il/i//* - iFiil/Hp) + 4nfi0e( n u« + n u«) = 0,

(6.69)

iu" + -il/" = 0. (0)

(6.70)

m

From equations (6.62) and (6.69)-(6.70) we obtain m

)

(0)

Now we assume that m since we are not interested in electrostatic waves. Therefore we have / u a = 0, / \ba == 0

(6.72)

A=0.

(6.73)

a

(0)

and from (6.62), (0)

Then, from equation (6.69), because ^ # 0, we have

where Qp is the plasma frequency, which is the dispersion relation for electromagnetic plasma waves (Stix, 1962). Now we derive the transport equation. Equations (6.63)-(6.66) for q = 0 become

iV.fr + h£r + /,£*) + - ^ . + ( ^ + 1 ^ , = 0, fi

l

0,

(6.76)

= 0,

(6.77)

n +m/ a w a + n.awa + nw a a = 0.

(6.78)

(0)' P

(1)

(1)

z'Q Ma + u VuM + u»ii*+-F^uu (i)

(6.75)

(0)

(1)

(0)'^

'

(1)

' (0)

md)

M

(0)'

6. Relativistic geometrical optics

175

From equation (6.75), by contracting with P, we obtain iFfP = Ke ~
(6.79)

with Z a determined up to the transformation Z a ^ Z a + A/a. Since /°7a ^ 0, we can choose Z a in such a way that Z a / a = 0.

(6.80)

Using equation (6.80) in equations (6.76)-(6.77) and also gives ;/

n u* + 4 ^ 0 ^ wa = 0,

(6.81)

Q wa + - 1 / ^ 1 4 + - ^ ^ ; X - - ^ w J a + - Z a Q = 0.

(6.82)

/

/

%

Using (6.82) in (6.81) gives 47l/ g2n

^

(^X, + ^M;X - z V " ) ^ o. (6.83)

By contracting equation (6.83) with ip, we get lpM?P + M%

= 0,

(6.84)

which is the transport law for the amplitude. Proceeding as in the previous section, we introduce the orthonormal frame (iAva, e\ e% (1) (2)

with

and e", ea two orthonormal spacelike vectors, orthogonal to both ua and I* (1) (2)

and such that e^ e a.u = 0. (2)

(1) •**

Then we can write eacos6+ (1)

^ a sin0), (2)

(6.85)

176

Relativistic fluids and magneto-fluids

and by substituting into equation (6.83) and contracting with e a , we obtain the polarization angle transport equation

where co^ is the electron fluid vorticity. An alternative approach which leads to equivalent results is based on the averaged Lagrangian formalism (Dougherty, 1970). The propagation equation (6.84) for the amplitude can be interpreted in terms of the area-amplitude law, at least in some cases (Moschetti, 1987). The main result which has been obtained is the derivation from Maxwell's equations of the propagation laws (6.84)-(6.86) for the amplitude and polarization of an electromagnetic wave in a cold plasma. These laws form the basis of relativistic geometrical optics in a dispersive medium. They could be used in order to obtain the relativistic transfer equation for polarized radiation in a dispersive medium (the effect of a background magnetic field could easily be taken into account; Anile and Pantano, 1979; Breuer and Ehlers, 1980,1981). In the next section we shall reconsider the problem we have treated, trying to take into account weakly nonlinear effects. 6.4. Electromagnetic waves in a cold relativistic plasma: weakly nonlinear analysis We start with equations (6.44)-(6.45) and (6.48)-(6.49), which we rewrite as = 0,

(6.87)

A

(6.88)

V a (nu a ) = 0,

(6.89)

^VX=

(6.90)

--F^Uu-

In order to study nonlinear effects it is necessary to extend the two-timing method introduced in the latter section and to allow several scales. Let us introduce the following quantities Xa = s2xa

(very slow variables),

Xa = ex* (slow variables),

6. Relativistic geometrical optics

111

and the phase functions (to be determined later)

where s > 0 is a real parameter related to the slow modulation of the waveform. We shall assume that in a selected class of coordinates (xa) we have the following asymptotic expansions Fap= I ^ >

(6.91a)

0=1

(0)

^°

(a)

u>= u"+ £ e'u",

(6.91b)

a=l

(0)

*

(a)

n+Xe""»

(6.91c)

with the functional dependence >^

(0)

(0)

n = w£ = n £ pr a ), ^

(

^

w " = wf = wf (Z a ), ^

u"= 2" '^(Z^^OexpO^),

(6.92a) (6.92b)

(6.92c)

p = - oo (p)

n'=

+

f («(X«, $<«)cxp (ip^).

(6.92d)

Notice that the reality condition implies (a)

(a)

(p)

(-p)

n = n , etc.

We shall also assume for the metric,

and then, for the Christoffel symbols one has

It is convenient to define the covariant derivative W& with respect to the very slow variables Xa, by using the quantities f £v.

178

Relativistic fluids and magneto-fluids

Equations (6.92b)-(6.92d) are equivalent to assuming a periodic depen(a)

(a)

(a)

dence on cp for Fafi9 u11, n, together with suitable regularity properties. We define the vector field la by l.-^-8fi

(6-93)

and denote by dd the derivative with respect to Xa, and by comma a the derivative with respect to Xa. In order to investigate nonlinear effects we need to keep track of the expansion up to the third order in s. We have (i)

(1)

aFp P

(P)

/

(2)

exp(ipq)) + 82Y[ y iplaFny + p\(p/

(

•exp (ipcp) + £3 X p \

O)

(1)

(p)

(p)

5F

'

a

^)J~( d£

ipla Fpy + VfiFpy

(2)

0(e 4 ),

(6.94)

(1)

d

(6.95)

(P)

p

\

(P)

•exp (ipcp) + e3 X dn + ^— {-y-. ) exp (ipcp) + 0(£ 4 ),

(6.96)

6. Relativistic geometrical optics

179

By substituting the above expansions into equations (6.87)-(6.90) we obtain, to the first order in e,

( 6

(P)

\

(p)

(0) (1)

(1)

ip^V'V = - - « > « " , m

(P) (0)

(1)

lau"n

(0)

9 7 )

(6.98)

(p)

P (0)

-

(6.99)

(P)

(1)

+ n / a t * a = O. () ()

(6.100)

The normalization condition

up to the third order gives (0) (0)

u^u^= (0)

(0)

(1)

(2)

V'

-1,

(1)

(1)

+^^a

(6.101)

(0)

Let Q = u °7a be the local frequency of the wave (as measured by the (0)

observer moving with the background four-velocity ua). Equations (6.97)(6.100) can be analyzed as in the previous section. We shall treat the modulation of the lowest harmonic wave, p = 1. Hence, we shall put (1)

(i)

(i)

(P)

(P)

(P)

i^v = 0,

n=0,

u" = 0,

for|p|#l. From equation (6.97) we have (i)

(6-102)

180

Relativistic fluids and magneto-fluids

and, as in the previous section, we can always choose the gauge such that il/au" = 0.

(6.103)

(1) (0)

Then equations (6.98)-(6.99) are rewritten as \U*$*(i)

ila(il/ph) + 4nfi0e({nnu)ci + Wu*) = 0, (i)

(i)

\

(1)

ia

G.

+

/a

(i)

(6.104)

)

0.

(6.105)

Proceeding as in the previous section we obtain (-Q2 + Q^a/a

(6.106)

= a

We also exclude electrostatic waves and therefore we assume and hence ual = 0 , (i)

n =0, (i)

\l/aL = 0.

(6.107)

(i)

Equation (6.104) then yields (Mfi + n2p)il/a = 0

(6.108)

(p)

and therefore we obtain the dispersion relation l% + Q2p = 0.

(6.109)

Note that the above dispersion relation implies Q2>Q2.

(6.110)

To the second order in e, equations (6.87)-(6.90) give (2)

(1) y

(P)

(1)

(2)

(2)

(P)

(P) (1)

\

dF ay

6. Relativistic geometrical optics

181

J1) (0)

(0)

(0)

(0)

_ /

(2)

(i)

(i)

(2) (0)

(0)

\

_l_ V f*t* u

exp yipcp) = 0.

(6.114)

r (p-r) w /

We shall construct the asymptotic solution such that (2)

and this choice, on the basis of our ordering, can be interpreted as assuming that the background plasma is unmagnetized. For p = 0, equation (6.111) is satisfied identically and equation (6.113) yields (0)

(0)

(0)

(0)

u Vdn + nv&u

=0.

(6.115)

From equation (6.114) we obtain (0)

(0)

(6.116)

which shows that the unperturbed neutral plasma moves along geodesic lines, and from equation (6.112) we have (0)(2)

(0) (2)

(0)

(0)

nu*+uan=0.

(6.117)

The normalization condition (6.101), for p = O^gives e2

(0) (2)

—

m2 (i) (i)"

*«>)

and by using it in conjunction with equation (6.117) we get

4^^ mz

(0)

(

(i)

^ =1 ^ ^

(0)

2

m (i) oo

(

(i)

u«.

() (6.118b)

182

Relativistic fluids and magneto-fluids

For p = 2, contracting equation (6.111) with la gives (2)

(2)

(2)

*>=<M,-iM.

<2>

(2)

(

(2)

(2)

and il/a can be chosen such that (2) (2)

^aw (2)

a

= 0.

(6.120)

From equation (6.114) for p = 2, after some manipulations we obtain (2)

ffi (2)

e

2 ie >u )l + i mill t

K (2)

l

l l \

(i) (i)

(6.121)

Using equation (6.121) in equation (6.112) for p = 2 and taking into account the normalization conditions (6.101) gives (0)

4QN2^«

<6'122)

and, finally, (2) 1/(0) 4/ a Q\( 2 )/°) a u" = - ( M + — 2 - j n / n .

(6.123)

For p = 1, equations (6.111)—(6.114) read (2)

(2) \

(1)

CD

Fya

dFa

U)

,

g(/t)

dFfiy

(1)

(1)

5(»+K;)=o,

,,,25,

6. Relativistic geometrical optics

183

The general solution of equation (6.124) is

(6.128) (2)

and i//a can be chosen such that (2)

*A°7a = 0.

(6.129)

(i)

The normalization condition (6.101), for p = 1, gives i

u\iu'l = 0.

(6.130)

(i) (0)

By contracting equation (6.125) with ua we obtain (i)

Contracting equation (6.127) with la gives e (0) (2>a

(2>a

whence (i)

which, after substituting into equation (6.126), finally yields {

n=0.

(6.132)

(i)

Equations (6.125)-(6.127) give

from which, by taking into account equation (6.128) and contracting with

184

Relativistic fluids and magneto-fluids

(0)

u a, it follows that

and (1)

Q

=

Consistently we shall construct asymptotic solutions such that 0.

(6.134)

The latter equation means that the four phases £(M) are constant along the curves tangent to /M (the rays). To the third order in e, from the field equations (6.87)-(6.90) we obtain, for p = 1, (2)

(3)

(3)

(3)

\

(2)

u

Pin)

(\Jv K }

tuy*

u ;

. Mn)

(2) nfP

+

(1)

(l)

(1)

(i)

(i)

(i)

y^ FPy + WsFy<x + V.F a « = 0, r

^(A;)_1_^_ + d ^

(3)

/(O)(3)

(0)

(6.135)

(3)\

ilo FaP + 47ijUoe( w M + ua n \ + A a = 0

(6.136a)

with (2)

^ T + ^ «^), (1) (0) (3)

^ (0) ( 3 )

(i)

m

a

u + -uuF«fl M

d)

(-1)

(6.136b)

(2)J

+ Am = 0

(6.137a)

with

L)

(2) ( - D

(2)^

m V ( - D (2)

(2) (D a

(2)

(D

(or
(2r(-i>

/

(6.137b)

6. Relativistic geometrical optics

185

i(n{n + {nCua) + C = 0 \

with

(i)

(6.138a)

(i) /

C - ( H - V (n + nV (6.138b) The normalization conditions (6.101) give (3) (0)

u«ua

=0

(6.139)

because (1) (2)

(1) (2)

(1) (0)

(1) (2)

uaua=

(3)

(3)

(i)

(i)

uaua

By substituting ua and n from equations (6.137)—(6.138) into equation (6.136) we obtain

(0)

(0)

? M a QC + iQA* - 4nfioe n Am = 0.

(6.140)

Equation (6.135) is written explicitly as (2)

(2)

+ cyclic permutation = 0

(6.141)

and gives (2) (3)

(3)

d*!'

(3)

(2) 8

^

(6.142) (3)

with \j/a determined up to the transformation (3)

(3)

(D*

(I)"

186

Relativistic fluids and magneto-fluids (3)

We can choose \jj a such that (i)

(3)

iAa/a = 0.

(6.143)

(i)

Then equation (6.140) can be written (2)

(1)

(2)

(1)

"

(0)

(0)

^ n 4 ; - 4nfioe u

(6.144) After lengthy and tedious calculations equation (6.144) simplifies considerably and yields (1)

(1)

i 2

(1)

(D

(0)

+ terms parallel to la and u a = 0.

(6.145)

We can write \j/ a in the form (i)


(6.146)

(i)

(0)

a complex function and ^a a real unit vector orthogonal to /a and to wa. Furthermore, without loss of generality we can assume that ea is independent oi^{tl)(\jj absorbing all the dependence on these variables). By contracting equation (6.145) with ea gives

(6.147)

6. Relativistic geometrical optics

187

which is a generalized nonlinear Schrodinger equation (Asano, 1974). We can introduce the orthonormal frame (0)

u\v\ e\ e* (1) (2)

a

a

and e , e are two orthonormal spacelike vectors orthogonal both to la (1) (2) (0)

and to ua and such that (2)

^(1)

Then we can write

e*= eclcos9+ e a sin0, (1)

(2)

with 6 the polarization angle. It is then easy to see that one obtains the same equation for the polarization angle as equation (6.86). The theory developed in this section could be easily extended to cover the case of a magnetized background plasma (Anile and Carbonaro, 1988).

7 Relativistic asymptotic waves

7.0. Introduction

One of the most useful perturbation methods for dealing with nonlinear waves is that of asymptotic and approximate waves, which is a fruitful extension of the high-frequency method of the linear theory. In its full generality, for arbitrary quasi-linear systems, the method has been developed by Choquet-Bruhat in a series of articles (Choquet-Bruhat, 1969a, 1969b, 1973) where applications to relativistic fluid dynamics, to Einstein's equations in vacuo, and to the Einstein-Maxwell system are also presented. Further applications have considered the Einstein equations coupled with a scalar field (Choquet-Bruhat and Taub, 1977), relativistic cosmology (Anile, 1977), relativistic magneto-fluid dynamics (Anile and Greco, 1978), and supergravity theory (Choquet-Bruhat and Greco, 1983). A different but substantially equivalent approach is that of the averaged Lagrangian, originally due to Whitham (1974). Extensions of the averaged Lagrangian approach to the relativistic framework have been made, among others, by Dougherty (1970; 1974), Dewar (1977), and Achterberg (1983) for relativistic plasmas and by MacCallum and Taub (1973), Taub (1978), and de Arajuro (1986) for gravitational waves in vacuo. The method of asymptotic waves is potentially relevant for several problems in relativistic astrophysics and plasma physics. In Chapter 5 we studied the nonlinear evolution of a simple wave. In many situations one deals with waves which cannot be considered as simple waves (for instance, a pulse propagating down a density gradient). In these cases, in general, the only way by which the nonlinear evolution can be studied is by means of perturbation methods. The method of asymptotic waves is the most suitable one in order to study the nonlinear evolution of nondispersive (hyperbolic) waves. This method has also been applied in order to derive necessary nonlinear stability conditions for problems in Newtonian fluid dynamics and plasma physics, such as in the case of an imploding spherical gas shell (Chin et al., 1986). A relativistic covariant formulation of the method could be useful for tackling analogous problems in relativistic fluid dynamics and plasma physics.

7. Relativistic asymptotic waves

189

In this chapter we shall limit ourselves to treat the method of asymptotic and approximate waves. The plan of the chapter is the following. In Section 7.1, following Choquet-Bruhat (1969a) we introduce the concepts of asymptotic and approximate waves. We derive the transport equation for the wave amplitude and discuss the nonlinear distortion of the wave profile and its breaking. In Section 7.2 we treat asymptotic waves in relativistic fluid dynamics. We derive the transport equation and, for plane waves propagating into a constant state, obtain explicit expressions for the nonlinear distortion of the profile and for the critical time for breaking. In Section 7.3 we discuss asymptotic magnetoacoustic waves in relativistic magneto-fluid dynamics. In Section 7.4 we treat asymptotic and approximate waves for Einstein's equations in vacuo. It is found that, in order for approximate waves of order 1 to exist, the background metric cannot be arbitrary, but must satisfy some requirements. These constraints can be interpreted as implying that the background space-time is curved by the presence of gravitational radiation (which is endowed with positive energy density). The method of asymptotic waves thus permits us to introduce the concept of a gravitational wave in a satisfactory way, describing the propagation of a high-frequency gravitational pulse in terms of geometrical optics concepts. This method could be of great interest when calculating the propagation through the universe of gravitational radiation emitted by astrophysical sources (rapidly rotating neutron stars at their birth, asymmetric gravitational collapse, etc.). The content of this section is entirely based on Choquet-Bruhat's (1969b) analysis.

7.1. Asymptotic waves for quasi-linear systems In the manifold M in a local chart (xa) we consider the following system of partial differential equations dUB LA{V) = dA\x, U) —-j. + aA(x9 U) = 0 ex

(7.1)

for the field of unknowns U = T(Ul9..., UN)9 A9B=1929...9N. A 00 We assume srf \x, U) to be C in x and analytic in U in an open domain & of the kind | U - U o | < K in UN, and to have a power series representation in 9 x, U) = stf + stH(Uc - Uc0) + \s/&D(Vc

- UCO)(UD - ! / £ ) + . . . , (7.2)

//2>A{^. T T\ i%) [X, \J) =

/mA _i_ tf2>A(jjC TjC\ _j_ tl&o -\-i%fc\U — U Q J - | - - - - ,

(H T \ {'•*)

190

Relativistic fluids and magneto-fluids

where we have put

Now we look for solutions in the form of formal series of the kind

U= Jo)"«U(x,a q= 0

(7.4)

<«>

where coe(R+, { = axp(x), cp is a differentiable function, and U = U 0(x) is (0)

a solution of the system (7.1) independent of £. The formal series (7.4) is said to be an asymptotic wave for the system (7.1) if, when substituted into these equations, taking into account the developments (7.2)-(7.3), the resulting series

has all the coefficients F = 0 vanishing, Vx,£ (Choquet-Bruhat, 1969a). (q)

The truncated expansion

F= £ co-*F(x,Z) is said to be an approximate wave of order m— 1, if the following condition holds: where || || is a suitable norm for the space of solutions of L(U) = 0. If || || is the supremum norm, in order for U to be an approximate wave of order m—1 it is sufficient that the functions U g = 0, l,...,m (q)

belong to ^ , be bounded together with their derivatives with respect to (x, £), and verify F = 0, for q < m. Now we shall construct asymptotic (q)

waves. Let

s-Gf); •*•(£).•

Then dU

(7.5)

By substituting (7.4) into (7.2)-(7.3) and taking into account (7.5) we obtain for the coefficients F (q)

FB = s/fUAdxcp,

(-1)

0A

(0)

A

(7.6)

^

V

7. Relativistic asymptotic waves

191

FB = jj**( UA8xq> + dx UA) + s/% Uc UAdx(p + @B

(0)

0

A

XT

\l)

A

*f

AC

(1)

\0)

A (

(1)

AA

V

(4-D

K

0

(7.7) '

(«)

(4-1)

C


(2)

*

%{Uc(U*di

4)

AA

C

X

(0)

+ U (U

(4-2)

@*Uc. (7.8) C V (fl) ' Let us assume U t o b e a solution of t h e system (7.1) independent (0)

of I Then

, 5^ 0

(0)

It follows that, FB = 0, (-D

B

0

F B = J ^ 5 A 1 7 ^ 5 ^ = 0, (0)

B

0

A

o

B

A

(i)

(7.9) y

^

}

A

F = j* /dtf> U + s* /dx U + J*%

(1)

(2)

0

C

(1)

0

(7.10) B

A

A

F* = s/ /dx

(4)

(g+i)

0

3CA^

(4)

0

(4)

(1)

+ {3c(A^

(1)

®y (°)

(«)

(4)

where / B depends only on L/'4, C/^4, dx UA, for p < q. (q)

(P)

(P)

(P)

From (7.9) we have, for nontrivial solutions U ^ 0, ^ 5 , < p ) = 0,

(7.12)

which says that (p satisfies the characteristic equation in the unperturbed state U. (0)

First of all, we shall consider the case when cp is a simple root of the characteristic equation (7.12). Let R = (RA), L = (LA) be the right and left eigenvectors of the matrix <srfBAxdxq), defined up to a factor, corresponding to the zero eigenvalue,

Then, from (7.9) we obtain (7.13)

192

Relativistic fluids and magneto-fluids

and by integrating with respect to £, U = n(x,QR + V1(x),

(7.14)

where H(x, £) is a function to be determined and Vx(x) arises from the integration. The compatibility condition for (7.10), considered as a system of linear equations in U, is LnFB = 0, and it yields B

0

A

(2)

"(1)

A

B

(l)

0

AC A

^ (1)

BK

(1)

AC

A

(0)

C

1

having set (pA = dkcp, whence the transport equation,

Kxdji + Arnn + MU + PU + e = o,

(7.15)

where XA = L B ^ 5 A ^ ,

(7.16)

N = LB^B/CRARC9

(7.17)

M = L5{^f5A^ + j/^3A U^c

+

@BRC^

(7.18)

P =LB^caA^^,

(7.19)

fi = LB{AB/8,VA + (^5JaA U^1 + @DV%

(7.20)

with P = 0 and Q = 0 if Vx = 0. By repeating the proof of Proposition (4.2) we have that dA K" = k—, dcpa

(7.21)

with k a normalization factor. Similarly, by repeating the proof of Proposition (4.2), N

k

dA R

C

(722)

Equation (7.15) can be integrated by the method of characteristics. The bicharacteristic system is d

d**

d£

-dn

(723)

The hypersurface (JO = const., containing the point y", is generated by

7. Relativistic asymptotic waves

193

the "rays" xk = x% / ) ,

xA(0, / ) = ya.

(7.24)

Let £ be the initial manifold defined by

s(/) = o, {n(x, mx=y = » W , n\

(7-25)

£

with 5 and W1 given functions. Then, following the standard procedure expounded in Section 4.1, the solution Il(x, £) satisfying the initial conditions xa(0, / ) = / ,

n(0, / , rj) = W,{y\ rj\

«0, / , i/) = rj

is given in parametric form as x« = x% / ) ,

(7.26)

n = n(r,/,iy),

(7.27)

f = $(f, / , ^/),

where s(/) = 0.

(7.28)

In order to obtain U(xa, £) one must invert equations (7.26)-(7.28) in a neighborhood of t = 0. This is possible, locally, if the manifold s(ya) = 0 is differentiable and transverse to the rays (i.e., each ray meets s{ya) = 0 only once). n(t, y", rj) and £{t9 ya, rj) are given by

= exp| - r

1

| ( ' r ' ) | J { =^+

Jo

(7.29)

{iV(x(T, /)n(r, /,./) + P(X(T, / ) ) } dr.

(7.30)

For the choice \ 1 = 0 (convenient in many initial-value problems), one has n=W1(y9riyb{t9y)9

O(r,y) = e x p ( \

y\ Also, one has

Jo

V = f AT* dr. Jo

Mdt), /

(7.31) (7.32)

194

Relativistic fluids and magneto-fluids

and

Therefore, for t sufficiently small, —-(£, y, rj)^O and one can solve for rj = r\{U y, &

This is possible until a "time" t* such that

The critical time tc is defined as the infimum tc=Mt&t,y,r,) = 0. (y,ri)

or]

In the exceptional case, N = 0 and *F = 0, therefore £ = r\ and no breaking occurs. Also, in this case, Wx{y9 r\) = Wx(y9 £) and therefore the initial profile is not distorted. It is now possible to determine the higher order terms. U satisfies the (2)

equation F = 0, and therefore it is of the form (i)

U=K 2 (x,£)R + V 2(x,£),

(7.33)

where V2(x9 £) is an arbitrary function and V2(x, ^) is a solution of the system O, (7.34) where

' ^ f

X

' S i

+ *f )

,

(7.35)

By integration of (7.33) we obtain U = n 2 (x,£)R + I I 2 ( x , a

(7.36)

with I l 2 a solution of the linear system.

•afSVA + gNO.

(7.37)

In order to detemine H2{x, <J) we consider the equation FB = s/B/(px UA + s/B/dk UA + d%
0A ^ A ( 3 )

0

x

A

\l)

X)

ACY

\l)

(2)

*}U{ + fB = 0.

(7.38)

7. Relativistic asymptotic waves

195

The compatibility condition LBFB = 0 then yields Kxdjl2

+ N(UU2 + rill2) + MU2 + PU2 + Q2 = 0,

(7.39)

with Q 2 a known function of x, U, and U2 (x, £). The linear equation (7.39) can be solved by the method of characteristics. Notice that the projection of the bicharacteristics on Jt x* = xa(t9 y\ coincide with those of equation (7.15). This procedure can then be extended to all higher orders. In this way we have constructed an asymptotic wave for the system (7.1). An approximate wave of order 1 is obtained as

U = UQ + 1 Ui(x> Q + _L u 2 ( x ? Q

(7.40)

in a neighborhood of s(ya) = 0, with \J1 = H(x, £)R given by the previous formulas, with W^y^rj) bounded together with its first derivatives with respect to / , £. Furthermore, U 2 must be such that (7.33)-(7.34) hold, and U2 must be bounded together with its derivatives with respect to x a, £. For this it is sufficient that Ui be bounded together with its derivatives with respect to xa, €, as can be seen from the expression (7.35). Similarly, as in Section 4.2, it is possible to give an expressive formula for M when the system (7.1) is hyperbolic. Let us assume propagation into a constant state, U o = const. Then M reduces to M = L^f

dRA — cp^ + LBmRc,

(7-41)

because R = R(
$J£

ZRc.

(7.42)

Notice that in the linear (iV = 0) and homogeneous case ( ^ = 0), for propagation into a constant state, the transport equation (7.15) reduces (for the natural choice Vx = 0) to (7.43) x

with A = -—. 8 Equation (7.43) represents the standard area-intensity conservation law of geometrical optics and linear geometrical acoustics.

196

Relativistic fluids and magneto-fluids

Now we consider the case of multiple roots. Let cp(xfl) = 0 be a multiple root of the characteristic equation (7.12), with multiplicity r. Then, as shown in Section 4.2, A((pa) = {A((pa)yA((p(X),

(7.44)

where A((pa) is not divisible by ^4(<pa), and A((pa) is of first degree in cpa. Let R;, TLj be the corresponding right and left eigenvectors, i = 1,2,..., r. Then from (7.9) we have

U

— TT (Y F\H (1) —

•••-••iv'*'' ^>/

(1 d^

i

\'''^f

[where we have put equal to zero V (1)(x), for the sake of simplicity]. Proceeding as in Section 4.2, we obtain LjA^iaRf

= ktj —,

(7.46)

with kij a nonsingular matrix. The compatibility conditions LjBFB = 0 then yield the following coupled (i)

transport system, kijA^Jlj 4- NijjUjtlj + MijUj = 0,

(7.47)

where

UJ

RfR<j,

M y = {^BAd,RAj + (^BAxcdxUA + ®c)Rf}UBJ.

(7.48) (7.49)

Also, proceeding as in Section 4.2, one has dA NtJj = ktjWeR%

(7.50)

and the exceptionality condition then reads

7.2. Asymptotic waves in relativistic fluid dynamics The characteristic equation, the propagation speeds, and the right and left eigenvectors have been studied in Section 2.3 whereas the propagation of weak discontinuities has been considered in Section 4.3.

7. Relativistic asymptotic waves

197

We treat the cape of acoustic waves, corresponding to the roots of J^(u^(pa)2-p'eh^cpacpp

= O.

(7.51)

Because we have chosen V\ = 0, the transport equation reads s£*djl + M m + MU = 0,

(7.52)

where 3

^

(7.53)

(7<54)

<0)

where

)}(^)0 +i D ^

(7-55) (7.56)

Let us consider the case of special relativity and propagation into a constant state. In Minkowski coordinates (xa) we look for plane waves, that is, cp(xa) is of the form
lx = const.

(7.57)

We can always choose se ua = (1,0,0,0), and therefore the characteristic equation (7.31) yields 3

{

} = 0.

(7.58)

We can always take J2 = / 3 = 0, l0 > 0 and then, for a rightward progressive wave, (p(x)

= \l1\(csx°-x1\

(7.59)

1/2

where cs = (p'e) .

The acoustic ray tangent vector is ^-cJ/J,

j/^^l/il,

J2 = J3 = 0.

(7.60)

198

Relativistic fluids and magneto-fluids

Also, I /? I

f

^ + nl

1

(7.61)

where M = 0. Hence the transport equation is dU

3U

cf|/i

( P e ) 0 Jn—= 0. (7.62)

The rays are given by x1=cat + y19 x° = t + y°, x2 = y29 x3 = y3.

(7.63)

We take as the initial manifold y° = 0 and as the initial data 11(0,

y\t])=W1(y\t1).

It follows that

C

^f^

^t^J

(7.64)

An interesting case arises when the original perturbation has a sinusoidal profile, ^i(/^)=^osin^7.

(7.65)

Then equation (7.64) is rewritten (7.66) where

J^^f^]

(7.67)

Equation (7.66) can be inverted for \t\ < 1, and its solution is (7.68) where Jq is the Bessel function of order q. Therefore, we see that an initial sinusoidal profile is subsequently distorted by the creation of the higher order harmonics, .

1 )

7. Relativistic asymptotic waves

199

Equation (7.66) ceases to be invertible when — = 1-Ecosrj = 0 drj and therefore the critical time tc is found to be tc =

F

————=rC

(7.70) JO

s

Now, from the definition of n and the right eigenvectors, we have, since r = 1 + O(l/co2l

For the chosen initial sinusoidal profile we can write v = v0 sin r], v0 > 0, hence

and therefore tc =

F

=r.

(7.70')

It is easy to check that equation (7.70') is the vo^0 limit of the exact breaking time tB for the corresponding simple wave with the same initial profile [for a barotropic fluid, equation (5.95)].

7.3. Asymptotic waves in relativistic magneto-fluid dynamics We treat the case of magnetoacoustic waves, corresponding to the roots of equation (4.67) jV4 = n(e'p - 1)« 4 - (rj + e'p\b\2)a2G + B2G = 0.

(7.71)

The transport equation reads, 4N a d a n + SN4Utl + Mil = 0,

(7.72)

200

Relativistic fluids and magneto-fluids

where "

1

-

and SN4 is given by equation (4.74). The general expression for M is rather complicated and can be found in the article by Anile and Greco (1978). We consider the case of special relativity and propagation into a constant state. From the right eigenvectors (2.86) we have dp = Ea2A, hence -p

=UEa2A

and in terms of p the transport equation (7.70') is written ^

C(

Pa

i/> p =0,

(1)

(7.72')

with K1 and K2 given by equations (4.75)-(4.76). In Minkowski coordinates (xa) we look for plane waves cp{x) = x%,

la = const.

(7.73)

a

By choosing u = (1,0,0,0) and taking the unperturbed magnetic field along the x 1-direction, ba = (0,fc,0,0), the characteristic equation reads, e'P(rj + \b\2)(l0)4 -\l\2(r, + e'p\b\2 + |ft| 2 cos 2 6)l2 + |b| 2 |/| 4 cos 2 0 = O,

(7.74)

where 11\2 = (/J 2 -h (/2)2 + (/3)2 and 9 is the angle between the magnetic field b* and the spatial direction defined by /a, |/ x | = |/|cos#. It is convenient to introduce the following quantities A = rj + (e'p +cos2 6)\b\2, F={A2-4e'p(ri + \b\2)\b\2cos26}1/2, 1 Then the solutions to equation (7.74) are (/0)2 = |/|2
(7.75)

7. Relativistic asymptotic waves

201

where the double sign refers to the fast and slow magnetoacoustic waves, respectively. Then

JV°=±!/ 0 |/| 2 F, $ 2

1 2

N =±l2\l\ l\b\

H

)

2

g

(

±

N3 = ^I3\l\2[\b\2

p\\

1

2

\ \ ) l

2

cos 0-g±(ri

+ e'p\b\ )l

cos2 9-g±(rj

+ e'p\b\2)l

{1J6)

For propagation orthogonal to the magnetic field we have /x = 0, which implies g~ =0 (nonexistence of the slow magnetoacoustic wave). Then, because we have chosen / 3 = 0 and 12= —1/| < 0, the transport equation reads

J 4 + ^P>=* ox1

with

2(0

(7-77)

+ 4I&I 2 ? 72 \b\2)K2 + (r, + e'p\b\2)(K1 - K2)

For propagation parallel to the magnetic field, 8 = 0, l2 = l3 = 0, /j = —1/|, hence

F=\n-{e'p-\)\b\2\
e^

v-

E

The transport equation reads, for the fast magnetoacoustic waves,

a> with

-

t"

p

,

n

which coincides with the purely fluid-dynamical one.

,7,8,

202

Relativistic fluids and magneto-fluids 7.4. Asymptotic and approximate waves for Einstein's equations in vacuo

We consider Einstein's equations in vacuo, which in local coordinates (xa) read (Misner et al., 1973) r)Tk

F)Tk

r°py being the Christoffel symbols of the metric gafi. Let £ = coq)(x). We look for a solution of (7.79) in the form of a finite development (Choquet-Bruhat, 1969a) o

1I

1 2

0

where 0AAi is a differentiable hyperbolic metric. 1

2

We shall impose the condition that g^(x,^% gx^A) functions of x a , £ in an open subset Q of space-time.

are bounded

The inverse matrix gktl of the matrix gkfl, defined by

is given as a power series in 1/co.

(7.8

where gkli is the inverse matrix of gafi9 g^g^ = 5»\ also, (7.82) P

and ^AAt obeys 0

P,

Qx^

1

(p-1),

2

(p-2),

+ gxa 9 X» + 9x« 9 "" = 0,

(7.83)

so that gr" is given recursively by

Therefore, the series (7.81) is asymptotic for co -* oo provided gfA|i and gkli

7. Relativistic asymptotic waves

203

are bounded functions of x, £. Let us write

Then .0

1.1

1.

12.

_

VX

CO

_ . .

0)

*

1 . 2

. .....

(7-84)

CO*

For the Christoffel symbols of gap one obtains

where (7.86) 0

with T ^ the Christoffel symbols of the background metric g^, and ^^a + Xlt

S

d

d

(7-87) Also, the Ricci tensor Rafi has the following development R^ =
-.

(7.88)

CO

The metric gktl will represent an approximate wave of order zero if -

1

0

-1

Rap = 0 with Rap bounded, and an approximate wave of order 1 if RaP = 0,

0

Rap = 0, and coR^p is bounded VxeQ, V£. Let us consider now the following transformation of coordinates in Q,

204

Relativistic fluids and magneto-fluids

("gauge" transformations) xa = xa + -^*F a (x, £).

(7.89)

CO

One has to the first order in 1/co

co

r

\co*

hence _°

1 i CO

1 CO

with o

o

Safi = 9afi + Uk^Vp

+ i^'>A

(7.90)

The components of the Ricci tensor in the new coordinates are Raft = R*fi + ^(Ra^'^fi

+ Rfi^Va)

+'"

(7-91)

and therefore

(7-92)

««/»=*U

It follows that the equation / ^ = 0, as well as the set of equations o

-i

Rap = 0, Rap = 0 are invariant under "gauge" transformations. -i

One obtains for JR^, - 1 0

0

(7.94)

which yields -i

= 0. (7.95)

7. Relativistic asymptotic waves

205

Now there are two cases: (i) g^VxV^O.

(7.96)

Then g"^ = O^cp^ + 6p(pa, where 0a is an arbitrary covariant vector. Hence g"^ can be annulled by a gauge transformation of the kind (7.89). It follows that g'ap = Ka/? = constant with respect to £, whence g^ = K ^ ^

>^

i

+ X a/ j, Ka/3 is constant with respect to £. Since gafi must be bounded V£, it follows that Kafi = 0 and therefore gaP cannot depend on £. We conclude that this type of perturbation is not physical. o (ii) g ^(px^Pfi = 0 (gravitational waves),

(7.97)

which means that cp(x) = constant is a characteristic hypersurface for the o wave operator of the metric gafi. Then = 0,

(7.98)

which is written as Rafi=-U(Pa%

+
(7.99)

with

Equation (7.99) implies Q>1 = 0. In fact, in a local chart adapted to the hypersurface Z: cp = const., that is, x° = cp and xl are local coordinates on 2 , we have <pa = (1,0,0,0) and equation (7.99) yields, % = 0 and $>• = 0. Because gkfl must be bounded with respect to £ we see that we must have

Therefore, for an approximate wave of order zero, equations (7.97) and o

I

(7.101) must hold. Furthermore, we must have g^g^ bounded together with their partial derivatives (up to the second order) with respect to xa and £

206

Relativistic fluids and magneto-fluids

In adapted coordinates the condition (7.97) reads goo = 0,

(7.102)

and (7.101) yields (7.103)

0..1

o By raising indexes with the background metric gap, equations (7.103) can be rewritten in the form .1

(pJgij = O, where

0

(pa = gaP(pp9

(7.104)

and <700 = 0,

(7.105)

which expresses the statement that the hypersurface Z: cp(x) = const, is also o 1I a characteristic hypersurface for the metric gaP = gafi + —gap(x, £). Now we look for approximate waves of order 1, that is, verifying the -i

o

equations R aP = 0 and Rap = 0. One has

Now, since (px(px = 0, we have k , = 0,

(7.107)

where XaP

= I v + IKp + IHaP + IVaP + VaP,

(7.108) \

(7.109)

(7.110)

7. Relativistic asymptotic waves

207

> I

1,

1

rf)

(7.111)

tfUr

(7.112) _

0

where V denotes the Riemannian connection of the metric g^ and (7.113) _ o with Rap the Ricci tensor of the metric gap. In adapted coordinates, cpa = (1,0,0,0), and we have Zy = 0,

(7.114)

which, by using (7.103)-(7.105) yields

[j - Tkihg'kj ~ f J^w j - kl'P^k + * y = 0.

(7.115)

Equations (7.115) are ordinary differential equations for g'tj along the rays given by (7.116) which are tangent curves to X. The system (7.115) can be written in the form dU — = A(x)U + B(x),

(7.117)

208

Relativistic fluids and magneto-fluids

with U the matrix with elements gftj(x, £), A(x) a matrix, and B(x) the matrix Rij9 on the rays dxx dt

"*

Let S denote an initial hypersurface transverse to the rays, x(0, y) = yeS. Then the rays are given by the solutions of (7.116) x = x(t,y). The solution of (7.117) is then U(t9y9 <J) = ®(t9y)mt,y)

+ ®(y,«),

(7.H8)

where = exp Jo

o with 0(y, f) an arbitrary function such that From (7.118), by integrating with respect to £, we obtain for the matrix V I

with components gtj V(t9 y, Q = 9(t9 y)W(t9 y)£ + O(t9 y)X(y, {),

(7.119)

where x(^, €) is a primitive of @(y, £). Therefore, in order to have gtj bounded V^ we must have ^(r, y) = 0, that is, B = 0, which means Ru = 09

(7.120)

which in general coordinates is written (1A2\) with ma an arbitrary vector. Now we turn to the remaining equations. By performing an appropriate "gauge" transformation of the kind (7.90) we can set

Then the equation

7. Relativistic asymptotic waves

209

gives

A

°(

k

k)

ROi = o,

(7.122)

2

for which it is apparent that in order to have g'i} bounded V£, one must require ROi = 0.

(7.123)

In general coordinates the latter equation is written R*fi = *

(7.124)

with T an arbitrary scalar function. Furthermore, gtj must have primitives with respect to ^ which are bounded V£. Finally, we consider the last equation +- yijg\j

- g'ijgirjch + R00 = o.

(7.125)

Now, if a(xM, ^) is any uniformly bounded function of £, we have

1 lim hm —

fr

a\x^,

r-^oo i Jo

that is, the average of a\x11, £) with respect to ^ vanishes. Therefore, by averaging equation (7.125) with respect to £ and assuming that 0fj-, gtj, g[j, gftj are uniformly bounded, we obtain

im I f T fiWy + *o

lim whence

->oo i Jo \

T = - lim -

i Jo

F r o m gf/A/x = - gx<xg^g'^

y^g'iAZ.

we have

1.1

g'^g'Xfl = - gx*g

0011

'

hence 1 . .1

O.,O,,1

1

(7.127)

210

Relativistic fluids and magneto-fluids

It follows that we must have T>0.

(7.128)

Then T can be interpreted as energy loss due to gravitational radiation. Also, the quantity

can be interpreted as gravitational radiation energy density. It is easily seen that e satisfies the conservation law = 0.

(7.129)

In fact, by contracting equation (7.115) with g'ij one obtains

which is equivalent to equation (7.129) in adapted coordinates. There are other approaches which are also fruitful for discussing gravitational radiation, in particular those based on a direct application of the Wentzel, Kramers, and BriUouin method to Einstein's equations (Brill and Hartle, 1964; Isaacson, 1968; Anile, 1976). One of the great advantages of the method of asymptotic waves applied to discuss gravitational radiation is that it introduces, in the high-frequency limit, the local concept of gravitational energy density.

8 Relativistic shock waves

8.0. Introduction

From the results of the previous chapters we have seen that, under suitable assumptions (the compressibility hypothesis) a relativistic compressive nonlinear acoustic or magnetoacoustic pulse steepens and degenerates into a shock wave. Therefore, shocks are common occurrences in nonlinear wave motion and this chapter is devoted to laying out the basic theory of relativistic shock waves. Relativistic shocks are a very important feature in several models of phenomena occurring in astrophysics, plasma physics, and nuclear physics and in the following we shall briefly touch upon some examples. Supernovas represent one of the mostfiercephenomena occurring in the universe. A star suddenly increases its luminosity by many orders of magnitude such that at its maximum its light can outshine the total light from its parent galaxy. This phenomenon is suggestive of an explosion taking place in the star. Several mechanisms have been proposed in order to explain the source of energy driving the explosion (Carbon detonation, neutrino energy deposition, gravitational collapse and bounce, etc.). In the case of massive stars, in the range between 8 and 100 solar masses, which are thought to be progenitors of type II supernovas, one of the most viable mechanisms for producing an explosion is gravitational collapse and bounce (Van Riper, 1979). At the end of stellar evolution the star will develop a core composed mainly of nuclei near the iron peak and free electrons, with a mass close to the Chandrasekhar limit of about 1.4 solar masses. Due to electron capture and partial dissociation of nuclei, a dynamical instability sets in and the core starts collapsing until hydrodynamical stability is regained. At this point the core overshoots the equilibrium configuration and rebounds acting as a piston. This causes a shock to form outside the inner core, which propagates outward reaching relativistic speeds. If the shock is sufficiently strong it will be able to expel the bulk of the star and leave behind a hot compact remnant which will become a neutron star. This is one of the most viable models, but much work remains to be done

212

Relativistic fluids and magneto-fluids

before it can be considered to be on a firm basis. In particular, the precise conditions under which the shock forms at some point with exactly the necessary strength to expel the bulk of the star but still leave behind a remnant remain to be investigated in detail. We remark that the relativistic shock propagates into a medium with a changing equation of state (having the complexity of nuclear matter). Therefore, a simple analysis of the jump conditions for a polytropic or perfect fluid is not adequate and a deep understanding of this problem calls on the full theoretical description of relativistic shocks in a medium with an arbitrary equation of state. The analysis of the jump conditions under very general assumptions on the state equation will be performed in this chapter. Another complication which might arise is that a significant magnetic field might be present. In fact, one can show that the relative importance of a magneticfieldcan grow during the collapse. Let M denote the ratio of the magnetic energy density to the total matter energy density e. Then, in the case of shear-free collapse, it is possible to prove that (Yodzsis, 1976) -3p)de dt 3e(e + p) dt Therefore, for a collapse one has de/dt > 0 and, assuming that e>3p (the restriction e > 3p definitely holds in the early stages of the collapse) one obtains that 01 increases. In the area of laboratory plasma physics, magnetoacoustic shock waves with speeds of up to 4 x 108 cm/sec have been achieved (Taussig, 1973) with the Columbia University Plasma Laboratory Electromagnetic HighEnergy Shock Tube (Gross, 1971). The shock tube is 3 m long, has an outer diameter of 23 cm, and consists of coaxial concentric cylinders with a width of 5 cm between the tubes. The tube length was chosen in order to provide sufficient time for the shock wave to evolve to a steady state. The shock tube is initially filled with room temperature hydrogen or deuterium at a pressure ranging between 10 m Torr and 200 m Torr. Also, there is a magnetic azimuthal field (bias) in the shock tube test region which is typically 0.7 Teslas. In some experiments the speed of sound in the preshock gas is typically 1.3 x 105 cm/sec and the Alfven speed is 2.7 x 107 cm/sec. The shock thickness, ranging from 3 to 80 cm, is greatly reduced by the transverse magnetic field with respect to the purely gas dynamicalfield.The mean free path in the pre-shock gas is about 2 cm and 140 cm in the post-shock plasma. Therefore, the shock wave can be considered a collisional one. The interpretation of the experiments has been done in a Newtonian framework (Liberman and Velikovich, 1985) and this might just be adequate for the attained speeds. However, the relativistic

8. Relativistic shock waves

213

evolutionary conditions (which will be treated in Section 8.5) seem to be different from the Newtonian ones and therefore a relativistic approach is in the end necessary for a proper theoretical understanding of these phenomena. In the field of nuclear physics, high-energy collisions among heavy ions can be modeled by using fluid dynamical concepts. The rationale for using hydrodynamical concepts is twofold. One reason is that the collision of two heavy nuclei excites a very large number of degrees of freedom (compared to those usually excited in few-body nuclear reactions) and therefore a statistical thermodynamical and hydrodynamical approach seems appropriate. The second reason is that, due to the lack of a detailed understanding of the nuclear dynamics, a fluid dynamical approach (which essentially relies on the basic conservation laws of energy-momentum) seems a safe first step. For a fluid dynamical description to be appropriate the following criteria must be satisfied, at least approximately: (l)many degrees of freedom are excited; (2) there is a short mean free path and mean stopping length (the average distance it takes for a nucleon to dissipate its kinetic energy); (3) the reaction time is sufficiently short in order to ensure local thermodynamical equilibrium and; (4) there is a sufficiently short de Broglie wavelength so that a semiclassical particle description is adequate. Simple estimates of the nuclear parameters show that the above conditions are satisfied, at least for near central collisions of heavy nuclei at a moderate bombarding energy ( > 200 MeV) (Amsden, Harlow, and Nix, 1977). When a nondissipative hydrodynamical description applies and relativistic effects are not negligible, nuclear matter is described by the equations of relativistic fluid dynamics and all the details of nuclear interactions are incorporated in the state equation, giving the pressure p a s a function of nucleon density n and specific entropy S, p = p(n, S). A general constraint on the state equation can be obtained from the relativistic causality principle, which in this case states that the adiabatic speed of sound must not exceed the speed of light (Osnes and Strottman, 1986; Clare and Strottman, 1986, and references therein). In the collision of two heavy nuclei the relative speed of the two nuclei is supersonic. Drawing an analogy with conventional gas dynamics, one expects that a relativistic shock wave might be produced (Sobel et al, 1975). Also, some current models under investigation predict that relativistic shocks (or relativistic detonation and deflagration waves) might be related to the phase transition from a nuclear matter to a quark-gluon plasma (Barz et al., 1985; Clare and Strottman, 1986). Relativistic shock waves have been the subject of early investigations in

214

Relativistic fluids and magneto-fluids

relativisticfluiddynamics and magneto-fluid dynamics. In relativistic fluid dynamics the pioneering work is that of Taub (1948), where the relativistic form of the jump conditions is established (the relativistic analog of the Rankine-Hugoniot relations). A detailed analysis of the thermodynamic properties of relativistic shock waves, patterned after Landau and Lifshitz's treatment of classical shock waves (Landau and Lifshitz, 1959a), is due to Thorne (1973). In Thome's article the shock adiabat is named the Taub adiabat, a terminology which we adopt here. The compressibility assumption used in the analysis of the Taub adiabat [the relativistic counterpart of the classical Weyl conditions (Courant and Friedrichs, 1976)] has been discussed by Israel (1960) in the framework of relativistic fluid dynamics. Explicit solutions of the jump conditions have been obtained for special equations of state. Liang (1977a) and Anile, Miller, and Motta (1983) have treated the case of barotropic and polytropicfluids.The case of the Synge gas has been treated numerically by Fujimara and Kennel (1979), by Lanza, Miller, and Motta (1985), and in an analytical way by Majorana (1987). Shock waves in relativistic magneto-fluid dynamics have been investigated extensively and in a rigorous mathematical way by Lichnerowicz (1967, 1971, 1976). Explicit expressions for transverse magnetoacoustic shocks have been provided by Taussig (1973) for the Synge equation of state. The case of an oblique magnetic field has been treated by Majorana and Anile (1987). Detonation and deflagration waves in relativistic magneto-fluid dynamics have been investigated by Coll (1976) under fairly general thermodynamic assumptions. In relativistic fluid dynamics with combustion, the case of special state equations of interest for nuclear physics and cosmology has been treated, among others, by Steinhardt (1982), Gyulassy etal. (1984), Miller and Pantano, (1984), and Cleymans, Gavai, and Suhonen (1986). The plan of the chapter is the following. In Section 8.1, first of all, we derive the jump conditions for relativistic fluid dynamics by using the methods introduced in Section 3.1. Then the jump conditions for shock waves are used in order to obtain the Taub adiabat. Weak shock waves are studied in detail and the role of the compressibility assumptions is brought to evidence. Shocks of arbitrary strength are investigated by performing a detailed analysis of the properties of the Taub adiabat. We adopt the mathematically rigorous approach of Lichnerowicz (1976), but we apply it to the simpler case of relativistic fluid dynamics (as opposed to magnetofluid dynamics). This has the advantages that the calculations are simpler and the interpretation of the results is not hindered by a tedious and cumbersome analysis. In particular, the timelike character of the shock

8. Relativistic shock waves

215

hypersurface under the relativistic causality assumption is established. Also, the relationship between the compressibility assumptions and the entropy growth across a shock wave is discussed. Finally, the existence of shock wave solutions is established. In Section 8.2 explicit solutions of the jump conditions are found for various state equations. In particular, barotropic and polytropic fluids and the Synge equation of state are treated. In Section 8.3 detonation and deflagration waves are studied for relativisticfluiddynamics. The analysis is based mainly upon Coil's article (1976). The detonation adiabat is defined and its general properties are discussed. In particular, the Jouguet points are introduced and their significance is investigated. Finally, we treat in detail the case of the "bag model" state equation which may be relevant for nuclear matter theory as well as for cosmology. In Section 8.4 we treat shock waves in relativistic magneto-fluid dynamics. Owing to the availability of Lichnerowicz's excellent monographs (Lichnerowicz, 1967, 1971) our treatment will be cursory. First of all, we derive the jump conditions and from these the shock adiabat, which we call the Lichnerowicz adiabat. The properties of the Lichnerowicz adiabat and the theorems guaranteeing the existence of shock waves are simply stated without proofs. The proofs can be reconstructed by applying the techniques introduced in Section 8.1 or can be found in Lichnerowicz (1967, 1971, 1976). At the end we solve the jump conditions for a Synge gas. Finally, in Section 8.5 we introduce the concept of evolutionary shock. The Lax conditions (Jeffrey, 1976; Liberman and Velikovich, 1985) are then recalled and applied to relativistic fluid dynamics. For the case of relativistic magneto-fluid dynamics we refer to Lichnerowicz's work (Lichnerowicz, 1971). Also, some comments are made on the relativistic shock tube problem.

8.1. The jump conditions in relativistic fluid dynamics

Thefieldequations of relativisticfluiddynamics in the form of conservation laws have been discussed in Section 2.2 and are V>tt") = 0,

(8.1)

1 v

(8.2)

VAI7 " = 0, with

216

Relativistic fluids and magneto-fluids

A shock wave is an oriented hypersurface X in space-time Jt, across which the field variables p, wa, p, / suffer a jump discontinuity, i.e., pwM and TMV are regularly discontinuous. Let Q be a neighborhood of Z, which is described by the equation

Equation (8.1)—(8.2) hold in O in the sense of distributions, that is, for the distributions associated with the regularly discontinuous tensor fields pwM, Let f^v be the tensor distribution associated with T^v. Then by equation (3.45),

where /a = Va
(8.3)

Similarly, from equation (8.1) one obtains

d>"m = 0.

(8.4)

This latter equation yields the invariant m, p-utlll.

(8.5)

The case m = 0 corresponds to a slip-discontinuity. In this case equation (8.3) implies

Furthermore, under the condition p + p _ > 0, from m = 0 it follows that

and the tangential components of [[w a ]] are undetermined. The case m ^ 0 corresponds to shock waves and will be discussed in the following. Then equation (8.3) gives m/_ wv_ + p _ /v = m/+ u\ + p + /v.

(8.6)

By contracting with w_v,w + v, respectively, and using equation (8.5), equation (8.6) yields +T

with T = / / p the dynamical volume.

+

) = 0,

(8.7)

8. Relativistic shock waves

217

Equation (8.7) is the Taub adiabat, which is the relativistic generalization of the classical Hugoniot adiabat. General properties of relativistic shock waves are expressed in the following propositions. Let II be the half-plane (T, p\ T > 0, and denote by Z = (T, p) a point in the half-plane. Let 34f(Z,Z+) be the Taub function (the relativistic analog of the classical Hugoniot function) defined on II by 2

+)=-fl+f

+ (p+-p)(T++T).

(8.8)

Then

and the Taub adiabat passing through Z+ is given by

Remark 1. One has = 2fTdS + (T - T + )dp - (p -p PROPOSITION 8.1.

+

)dz.

(8.9)

Under the following assumptions on the state equation

T = T(P,S):

Ts^O,

(8.10a)

T;<0

(8.10b)

the hypersurface Z is timelike. Proof. Consider the straight line t£\m2(r — T + ) = (p+ — p)/a/a, connecting the states Z_, Z + . On 5£ one has / Z* dt = — Ardp, mz and therefore, on S£,

^

^ W

(8.11)

Now, since r' # 0, x' < 0, if A r2 < 0, it follows that — # 0 along if. m2 dp m d

218

Relativistic fluids and magneto-fluids dJ>?

dS

However, on if, from (8.9), —— = 2 / T — . Because Jff vanishes at both dp dp ends of if, JtT(Z+9Z+) = Jf (Z_,Z + ) = 0, there exists Z*eif, such that (Z*) = 0, which is a contradiction. It follows that one must have Lla > 0. Q.E.D. Remark 2. In the nonrelativistic limit, the dynamical volume T = — reduces P to the specific volume V=l/p.

In this case I -— ) = ( — )( - — 1 . \dS Jp \cpJ\dTjp

Therefore, in general ( — ) # 0 and for materials for which the coefficient \ d s JP of thermal dilation is positive, ( —— I > 0. \ d s JP Remark 3. One has

therefore iff

e'p>\.

Remark 4. Let P(Q = {u%)2-p'eh»\lx.

(8.13)

Then P(/J = 0 is the characteristic equation for acoustic waves. It can be immediatly checked that / V1

P(l )e' 2

m hence equation (8.11) can be written as dp

m2

(8.14)

Henceforth, the inequalities (8.10a)-(8.10b) shall be assumed, except when stated otherwise. Then IJ* > 0 and it is convenient to normalize by defining

"•-WF

(815)

8. Relativistic shock waves

219

and substituting na in place of /a in the jump conditions (8.3)-(8.4), which gives, after some manipulations, E

^

(8.16)

T+ - T _

where m = p+u\nfl = p-utn^ m — m/lj*. First of all, the case of weak shocks will be treated. The Taub adiabat passing through the point Z + is and the isentropic (Poisson) adiabat through the same point Z + is S(T,p) = S(Z + ) = const. 8.2. The Taub and isentropic adiabats through Z+ are tangent at Z+ and have a second order contact. PROPOSITION

Proof. Expanding / I — / + around /+ along the Taub adiabat gives fl -fl=(p-

-

P +

){2T++(T'P)+(P-

where we have kept the terms of order (p_ - p + )3 and neglected the contributions in (5_ - S + ) of order greater than 1, by anticipating the result. Similarly, one obtains by a direct expansion of / ! - / + , fl

-f2+=2{f+T+(S--S+

having used the first law of thermodynamics d / = TdS + -dp and the P df2 equality ^— = 2T. dp Equating the two expressions yields p

-~p+)3

+ 0{{p

-"p+)4)-

Q E D

' - (8.17a)

Remarks. Let us = Tscs, with r s = (l -c^)~ 1 / 2 the "proper speed of sound." Then the common tangent to the Taub and isentropic adiabats has

220

Relativistic fluids and magneto-fluids

a negative slope, given by -p2+(u2).

(p'x)+ =

This is easily seen by using equation (8.12). PROPOSITION 8.3. Under the assumptions (8.1Ob)-(8.1Oc) and ip>0

(compressibility condition), S.-S+>0,

(8.17b) (8.17c)

the following inequalities hold, for weak shocks, P->P

,

(8.17d)

/->/+,

(8.17e)

P->P +,

(8.17f)

u.
(8.17g)

u+>(us)+,

(8.17h)

+

where u+ and w_ are defined by u+ = r(v + )

with v± the absolute values of the three-velocities of the fluid with respect to the shock front, and T(v + ), T(v_) the respective Lor entz factors. Proof Equation (8.17d) follows immediately from equation (8.17a)(8.17c). Then (8.17e) follows from (8.7). From x'p < 0 one has

hence (8.17f). From (8.17j) it is easily seen that u\=m2lp\,

ul=m2/p2i9

(8.17k)

whence (8.17g) follows. Equation (8.17h)-(8.17i) can be proved geometrically, from the fact that, as a consequence of x"v > 0, the Taub adiabat is concave.

8. Relativistic shock waves

221

The slope of the chord connecting Z + and Z_ is

-m2=-ulpl=-ulpl. Now, from Fig. 8.1, one obtains |slope if | > |slopea|, |slopeJS?|<|slopefe|, 2 2

which, because TP= - l/p u , imply (8.17h)-(8.17i), respectively. Q.E.D. Remark 6. For a fluid for which T ^ < 0 , the requirement [[<S]] > 0 and equation (8.17a) yield and from (8.7),

P-
+

,

Also, from z'v < 0, T_ > T + , hence

P-