Some Exact Solutions of Gravitational Waves Coupled with Fluid Motions

Some Exact Solutions of Gravitational Waves Coupled with Fluid MotionsAuthor(s): Subrahmanyan Chandrasekhar and Basilis C. XanthopoulosSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 402, No. 1823 (Dec. 9, 1985), pp. 205-224Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2397872 .

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Proc. R. Soc. Loncd. A 402, 205-224 (1985)

Printed in Great Britain

Some exact solutions of gravitational waves coupled with fluid motions

BY SUBRAHMANYAN CHANDRASEKHAR1, F.R.S., AND BASILIS C. XANTHOPOULOS2

1 The University of Chicago, Chicago, Illinois 60637, U.S.A. 2 Department of Physics, University of Crete, and Research Centre of Crete,

Iraklion, Greece

(Received 10 June 1985)

Some exact solutions of Einstein's equations are found which represent the interaction of gravitational waves with a perfect fluid in which the velocity of sound equals the velocity of light. These solutions, unlike the solutions representing the collision of impulsive gravitational waves, are bounded by a space-time singularity and have some resemblance to cosmological solutions: every time-like trajectory, extended into the past, encounters the singularity. Moreover, in the generic case, matter may be considered as being created at the singularity.

1. INTRODUCTION

In an earlier paper (Chandrasekhar & Xanthopoulos I985; this paper will be referred to hereafter as Paper I) the analysis of Chandrasekhar & Ferrari (I984),

leading to the Nutku-Halil solution describing the collision of two impulsive gravitational waves, was generalized to the case when the gravitational waves are coupled, in the region of interaction, to sound waves in a perfect fluid in which the energy density, e, is equal to the pressure, p. It was shown that the solution of the coupled Einstein-hydrodynamics equations can, under those circumstances, be reduced to one of supplementing any solution of the vacuum equations with the solution of a pair of equations (decoupled from the rest) derived from the presence of the fluid. Appropriate to the problem of collision and the subsequent development of a space-time singularity, the solution was obtained in a gauge and in a system of coordinates that limited the space-time to a finite duration and to a finite spatial extent normal to the plane wavefronts. In this paper we explore the possibility of other classes of solutions with other choices of gauge and coordinates, which do not so restrict the space-time; and we shall find solutions that are reminiscent of cosmological solutions with initial singularities.

2. THE FREEDOM IN THE CHOICE OF GAUGE AND COORDINATES

We start with the general form of the metric,

ds2 = e2v (dx0)2 - e2#3(dX3)2-e2 (dxl-q2 dx2)2-e 22(dX2)2, (1)

appropriate to space-times with two space-like commuting Killing-vectors, where

[ 205 ]



206 S. Chandrasekhar and B. C. Xanthopoulos

V, #33, I q2, and #2 are functions only of x? and X3, and rewrite it in the form

ds2 = e [(dx0)2 (dx3)2] - dx2)2] (2) d [(dxl A (d - e y +d (dxl' 2 2 VA/

where A = e2(13-v), I = e-f+ 2 and eA = eV+ 2. (3)

The freedom of gauge we have (derived from the freedom to impose any coordinate condition we may desire on /k3 and v) enables us to assume (without any loss of generality) that A _ A (x0). (4)

Also, the freedom in the choice of coordinates we have derives from one of the field equations, [AI(ef), ] - [A i(efl)3] 3=O, (5)

which is valid for the vacuum and also when we have a perfect fluid-source with the equation of state e = p. If we assume (and it would appear that no essential restriction is involved in this assumption) that ef is separable in the variables x? and X3 and is, further, expressible in the form

then equation (5) reduces to efl =f(x3)VA, (6)

a 7

o, O = f, 3, 3/f =constant.

The solution of the equations that we have chosen hitherto are

f= sinx3= 8 and A = Iy2

(8) where d= 1-U2 and jU = cosx3.

The coordinate y so defined is a measure of the time in a suitable unit from a suitable origin. Similarly, jt is a measure of the distance normal to the wavefronts. In the choice of these coordinates we were guided by the fact that in the problem of the collision of impulsive gravitational waves, there is a natural origin of time (y = 0) and place (u = 0) specified by the instant and location of the collision; and the further knowledge that a space-time singularity develops at a definite later time (y = 1) and at a definite place (u = 1). But formally other possibilities exist: we could have chosen f = sinh X3 or cosh X3; (9)

in which case the corresponding solution for A would have been

A = q2+ (10) Therefore, letting

It = coshx3 or sinh x3 (11) we have the possibilities,

A = y2 + 1 and 8 = U2+ 1, (12)

where the alternative definitions of A and 8 can be chosen independently of one another. In all four cases, eA = /(A ) (13)



Gravitational waves coupled with fluid motions 207

We shall find that of the four possible combinations (12) we can consider, the case

A = 2+ 1, 8 = 22+ 1, (14)

does not allow a simple solution within the scope of our present considerations (see ?3 below). Also, it will appear that the two cases,

A = y2+1, 8=#2 -1 and A = y2-1 8 =22+1 (15)

lead to solutions that are essentially the same. The two cases we are left to consider are

A =y2-1 8 =2_1 (case 1), (16)

and A=y2+1, 3= a2-1 (case 2). (17)

We shall find that these two cases provide solutions that have very different interpretations from the ones that have hitherto been considered.

3. THE BASIC EQUATIONS

The equations we have to consider are the same as those of Paper I, ??4, 5. However, since the present choices of coordinates are different, the signs of A, and 8 for the present choices, A = y2 + 1 and 8 = a2 + 1, are the opposite of those of the earlier choice, A = 1- 2 and 8 = 1_12, one has to be careful about the signs in the various equations. We shall, therefore, assemble in this section the basic equations of the problem in forms that are applicable for any combinations of A = y2 + I and 8 = 1a2+ 1.

The definitions that are always valid are

e2(/t3-V) = A, e2# = e2(Wf+/2) = A1 and x = e-+/2. (18)

The equations governing x and q2 are

( -=-x2 [A (q2, )2-(q2 )2] (19)

and (4 q2')- ( q2(2 ) 0. (20)

Defining Z=-y+iq2, (21)

we can combine equations (19) and (20) into the single complex equation

Me (Z) [(AlZ ,-(8Z) A = A(Z )2 - (Z )2. (22)

Now letting

1-E (23) we obtain the Ernst equation

(1-I El2) [(A E, ), -(- 2E [A =(E J)2- (E ")2]. (24)




In terms of a solution of this equation, the solutions for x and q2 are given by

X -I-EI2 and E-E*) IX =-1 E 12 an

-2 11E 12 (25

The foregoing equations, governing X and q2, are the same as those that obtain for a vacuum. The coupling of the gravitational field with the source in the present context - a perfect fluid with the equation of state, e = p - affects only the equations governingV +103. They are (cf. Paper I, equations (27) and (28)):

andy( 3)'+8 1 ^ +z3)z8 = 2(v*-a83e^+aR (26)

2,(v+ jt3) + 2/(V+3), 3 12 [AI Z,= 12+81Z " 12]

+ 4- A- _ 8 +A (27)

Turning to the equations of hydrodynamics, we find that the non-vanishing components, uo and u3, of the four-velocity can be derived from a potential, 0, in the manner (cf. Paper I, equations (31))

and U3A/ = ua 0 ; (28)

and further, that 0 is governed by the hyperbolic equation

A 0 ,q -d 8, /" /= 0. (29)

The reduction of equations (26) and (27) The linearity, in V +JU3, of equations (26) and (27) suggests that we write (as in

Paper I, equations (43) and (44))

V +/k3 = (V +/3)vac. +f, (30)

where (v +1/3)vac is the solution appropriate to the vacuum when e = 0. Since by our choice of coordinates

V/-3 = (V/-3)vac. ' (31) it follows that

v= (V)vac.+f and 3= (#3)vac.+2f1 (32)

We shall now dispense with the distinguishing subscript 'vac.' for v and jt3 and retain for them the meaning that they refer to soluitions appropriate to the vacuum so that for the problem on hand,

V+2f and 3+2f (33)

replace what we have hitherto denoted by v and /3.

From equations (26) and (27) it follows that the equations governing V+#3,

appropriate to the vacuum, are

A (v +#3),, + (v +It3), = X2E (Z, 'Z, ) (34)




1 3,q2 / and 28(^+g3) A+28(^+83) z = '2-[AlZ v12+8jZ 12]+4- - (35)

while the equations governing f are

X f , +f,, = - (36)

and = -7 [A(q 5)2+&8(0q1)2] (37)

We may also note here that in accordance with equation (28) and our present definitions (32) and (33)

e-2'U3- e = C(U2-u2) = -2 (0 - )2f8(o )2] (38)

The metric to which all of the equations in this section apply is

ds2 = +/3 [(de)2_ (dS] \ _A_ 1 (1 -E) dxl +i(l +E) dx 12, (39)

where A = y2 + I and 8 = ,U2 + 1.

In our further considerations in this paper, we shall restrict ourselves to only those cases when the Ernst equation (24) allows a simple linear solution of the form

E = py + iqa, (40)

where p and q are constants. We readily verify that when

a4=112+1 and '=jt2+ 1, (41)

equation (24) does not allow such a solution. We shall not, therefore, consider this case. Equation (24) does, however, allow a solution of the form (40) in the remaining three cases, as we shall see in detail below.

4. THE CASE A =- 2-1 AND 8 = /-2-1

When A = I2 1 and 2 = -1, the Ernst equation (24) allows a solution of the form

E = py + iqu, (42)

where p and q are constants and p 2+q= 1. (43)

The corresponding solutions for X and q2 are (cf. Paper I, equations (26))

X= (1-p)2+q22 -2 and q2 = (1 )2q2+ 2 (44)

The associated solution for Z gives

e (Z7 Z*,) = Oand 2[ I Z,I2+ |Z, 2] (45) I I /J- X212 p 2y2+ q 2#a2




From equations (34) and (35) we now find

ev+/'3 = p2 2 )3+ q( -1) (46)

Combined with the first of equations (18), we have

e 2#c3= (P2Iq2 +)q 82/t2 (47) p2y2+q2I a 2-1

It is now convenient to introduce in place of y and ,u the variables VY and 0 defined by y = cosh Y and ,u = cosh 0, (48)

when the metric takes the form (cf. Paper I, equations (98) and (99))

ds'2 A

sn ef sih (dVf)2 - (dO)2]

(sinh RY sinh 0)2

sinh VY sinh 0 l 1 + E 12 (dx2)2 +1l 1-E 12 (dx')2 -4q(cosh 0) dx dx2}, (49) A

where A= 1E 12-1 = p22 + q2 2-1 -p2 sinh2 f + q2 sinh2 0. (50)

(a) The transformation to null coordinates

We now introduce a system of null coordinates by the definitions (cf. Paper I, equations (54) and (55))

u = coshg and v = sinhC, (51)

where 4 =A(V+0) and C=A(*-0) (52)

By these definitions,

y = cosh L' = cosh(6+ ) = cosh 6 cosh + sinh 6 sinhC = uV(v2+ I)+v V(u2-1),

,u =cosh 0 = cosh (6-) = cosh 6 cosh -sinh 6 sinhC

= uV/ (V2 + 1 )-vV (U2-1 ), (53) and sinh Vf sinh 0 = sinh2 6 cosh2 C-cosh2 6 sinh2 2

= 'u2 - W + 1). (54)

(Other possible definitions of the null-coordinates, apart from (51), are

u= cosh6, v = cosh; u = sinh, v = sinh; (51')

and u = sinh , v = cosh .

Of these three definitions, the first two lead to coordinate singularities at v = + u or - u while the last, as we shall explain towards the end of this section, leads to a metric equivalent to the one we shall find with the definition (51).)

In terms of the variables 6 and ~, equation (29) governing 0 becomes

2q -,?~ - (q g+q )+4/ (q $I-q ) = 0; (55)




or, equivalently,

Y4,6C+sinh f sinh 0 [54 sinh C cosh -q sinh 6 cosh ] = 0. (56)

On the other hand, by our definitions (51),

0b = v, u sinh6, C = 0 v cosh and sinh cosh (57)

Making these substitutions, we find that equation (55) reduces to the form (cf. Paper I, equation (60))

,U,VU+ 2 (+ 1 ) (v - u-U v) = 0 (58)

We may note here, for future reference, the following relations, which one may easily verify:

2(U2 - V2-) Ol = 1[(U2-1) (0b U)2 -(V2+1) (q,V)2], (59)

j (,)2 + 8(o "a)2 = 1[(O, 6)2 + (0, )2]

= 1[(,U2 - U (0 )2 + (V2 + 1 ) (0 V)2], (60)

and -(q V)2( $)2 = C= ()0U 0 V (61)

Returning to equations (36) and (37) governing f, and solving for f and f , we obtain 4

= + a2 1y2 t [A (0q )2 + 8(o $)2]-2/ 0q 4 (62)

and g1-/62{z2_14[A (0,1)2 + 8(o ,)2]-20 ;(3 and =1 -)2 2 {2 0,- (63)

and inserting these expressions in

f u = (f,"+f ,)V(v2+1)+(f "-f ,) UV (64)

and f = (f ,v +f ,) Vv2 + (f" y-f, ) v (U21 ) (65)

(which follow from equation (53)) and simplifying with the aid of equations (59) and (60), we find that we are left with (cf. Paper I, equations (64))

(02 1) 2

and 2(0'v2) V)(66)

Finally, we may note that by making use of equation (61), the expression (38) for e becomes e- 2,3 -f

(U2 -2-1 V 2 - )2 2 2 _V. (67)




(b) The solutions for 0 and f

Letting r = u2+v2 and s = u2-(v2+1), (68)

we find that equation (58) reduces to

,r, r ( ,-1 O'S )= 0. (69)

Equation (69) is clearly separable; and the separable solution, which ensures the positive-definiteness of e and its boundedness for r -x, is given by

0 = C e -rccsIj(ccs) (oc > O), (70)

where ac2 is the separation constant, Ii is the Bessel function of order 1 for a purely imaginary argument in Watson's notation (Watson I 944), and C is a constant.

For 0 given by equation (70)

56 r = -Cx2 e-arjsI1(ccs) and s = +Ca2 e -r sIo(s), (71) and the expression (67) for e gives

4e- 4 = V V(U2-1) V(V2+1) A S1[(0q8)2- r)2]

= 4UvV(uU2-1) (V2+ 1) eA C2 4 e-2ar [IO(s) -Il(xs)], (72)

which is, indeed, positive definite since IO(z) > Ij(z). It is clear that by definition (and by the choice of coordinates), the solution we

have obtained is defined only in the domain

u > 1 and s = u2-(v2+1) > 0. (73)

We shall find in ? (d), below, that a space-time singularity occurs on the rectangular hyperbola u2 = v2+ 1, (74)

which has its vertex at u = 1 and v = 0 and whose asyimptotes are along v = + u. The general solution for 0 can be expressed as an arbitrary linear combination

(as a sum or as an integral) of the fundamental solution (70) for different Cs and as. Thus, restricting ourselves to a sum, we can write

C= CZ C e-air8I(c r S). (75)

The corresponding solution for e is

e=4uvV(u2-)V(v2+1) A

+ 2S Ct2 OC4 e- a2 e- Io(aoj+aS)-I2? r )

+2 > t [Io(ai s) Io(aj s)-I(I, s) I,(ajs)]}. (76)




The solution for f can be found by first noting that in the r and s variables, equations (66) take the form

8 4 2+082] f,r =V ,ro's and f.s=-[(0r )2+(7)2])

With 0$ r and ' s given by equations (71), we can readily verify that the corresponding solution forf is given by

f = -4C2a3 e-27r sIj(acs) IO(as). (78)

For the more general solution (75) for 0, the solution for f is

f = _4 E C2 a3 e- 2,j r 8sI (a.t s) I O(aj s)

-8 E CZ C e- (i+a)r (79) i>j a% aj

(c) The character of the distribution of e in the domain u > 1 and U2 - (V2 + 1) 0 0

From the known asymptotic behaviours,

and Io(z) -> and I,(z)-2z for z- 0 (80)

Io (z)+/(2z)(+8+ )> and I1(z) ->I )(1- + +..) for z->oo,

(81)

we find that the solutions for 0 and f given by equations (70) and (78) have the behaviours

qlca22s2 e-ar (s-0), 23

0 2)ez221 (oo) (82)

and f---2C2a482 e-2ar (S-O0),

2C2L 2 _ V1 (83)

f -,_ 20cz e 2a(2V2+1) = (say) (s -> oo).

IC

From the expression for e given in equation (72) it now follows that

C242232 a e->424u2v2- e-2r (s -> O), (84) A

since f=, u2=1 = v2 and v2 +1 = u2 when s = 0. (85)

The behaviour of A for s3-0 is different for p # 0 and p = 0. Thus, from the definitions (50)-(53) of the various quantities, we find

A 2 = v2 + 1+2uv [uv + (p2 q2) (u2-1) (v2+ 1)1] (86)

so that A = 4p2u2V2 :A 0 (u2 =v2 + 1; p 0). (87)




1.5 (a)

V

to0 u 2.5 1.5

(b)

V

lo0 u 2.5

FIGURE 1. The curves of constant e for a = 0.5 and C= 1.0 for (a) p = 1 and (b) p = 0. The raggedness of the curves in (b), near the critical hyperbola u2 = v2 + 1, is due to errors in computer tracing.




But when p = 0, a more careful examination is necessary; and we find that

(U2 -v2-1 )2 82

4u2 2 (u2-V2-1I-+O (88) From equations (84), (87) and (88) we now conclude that

- 2 S2e-2cr (p # 0),

82 (sO--). (89)

16C2(UV) 4e-2ar (p=O)

It is important to observe that e generically vanishes along the hyperbola, u2 = v2 +1; it diverges only in the exceptional case, p = 0.

Similarly, from the asymptotic behaviour of the various quantities for s -> oo, we find that

2C2e 2 - u(u _) v(v2+ 82 e 2(2v2?1) (S- 00) (90)

TE ~~A

(Note that UV (U2 -1)/A tends to a finite limit for 8 -X 00.)

The behaviour of e predicted for 8 -s 0 and 8 -s oo is exhibited by the curves of constant e illustrated in figure 1 a (for q = 0 and p = 1) and figure 1 b (for q = 1 andp = 0). We are grateful to Mr S. K. Chakrabarti for obtaining computer-traced drawings for these illustrations.

(d) The description of the space-time in a Newman-Penrose formalism and the occurrence of a space-time singularity on U2 = V2 + 1

As in Paper I, ? 10, we can relate the spin coefficients and the Weyl scalars for the problem we are presently considering to the same quantities that pertain to the vacuum. Thus, distinguishing by tildes the quantities that obtain forf #A 0 from the quantities that obtain forf = 0, we find that they are formally related exactly as in Paper I, equations (100)-(104), with the only difference tha.t U must now be defined by VA (p2 sinh2 +q2sinh2 0)(

(sinh V# sinh 0)= (sinh #f sinh o)

instead of by (Paper I, equation (100))

_ VA (2sin2 lfr + q2 sin 2 O92) (sin Rf sin 0)- (sinkb sinO)4 (92)

In particular, we have

TI= e-f(F 0-oDf), '4 = e-f 4+ AAf

and ~'2 =ef9'2+Ae, (93)

where D - ( 1 + 1) and A U (6f-8a), (94) -UV\2 UU\D2 an A i UV2

with U as defined in equation (91).




The spin coefficients and the Weyl scalars for the vacuum appropriate for the metric (49) withf = 0 can be obtained as in Chandrasekhar & Ferrari (I984, ?5). The null-tetrad basis must, however, be defined by

f 0 x1 x2

(Vd)=+,\/ (U, -U, 0, 0 ), V\2

1 (ni) +, /2 (U, +U, 0, 0 ),

V\2

(mi) =- - (0, 0, V&*, +iV&2), V\2

(Mi) I- (0, 0, V, -i V&+), (95) V\2

where u = (sinh ir sinh0)4' V = (sinh ir sinh 0)-, (96)

E+1 and V+ = (E = p cosh r+iq cosh0). (97)

We find that the expressions for the spin coefficients and the Weyl scalars can be directly transcribed from those given by Chandrasekhar & Ferrari (I984, equations (86), (91), (94), and (95); see also the errata in Chandrasekhar & Ferrari I985) by simply replacing the trigonometric functions by the corresponding hyperbolic functions. Thus, we now have

(sinh sii - iq s ( .. 2_1 2sinh (0 + i) sinh (u/)2 - , (98) E T N2*l 2lN3 )4^ (sinh 0 sinh *) 2 J

4(sinh 3 sinh 0) (p sinh Vf + iq sinh 0) s T'4

3 ~~sinh (0-u/rY) 03 1

4 (sinh V sinh 0)2 (p sinh # + iq sinh 0)3 (100)

The occurrence of a space-time singularity on the hyperbola, u2 = v2 +1, can be deduced from the behaviour of TI2 for 8--O0. Since f = 0 for 8= 0 by equation

(83), the term 2 e-f in T has the behaviour

- sinh 2 ur e-f E2 Aih 3 p = 0) t

5 82

(8 0). (101)

8A3sinh2 fr (p0)

Therefore, in accordance with equations (53), (87) and (88),

e V18p2s2 (P #0 ?),

6u4V4 j (S--0). (102) 7 (p=0




Comparison with the behaviour (89) of e shows that V2 e-f dominates e when s-* 0. The behaviours of P2 for s -- 0 is therefore the same as that of 2 e-f. The occurrence of a space-time singularity on the hyperbola, u2 = v2 + 1, is manifest.

Finally, we return to the matter to which reference was made earlier in ?4(a), namely to the different definition, u = sinh 6 and v = cosh (, of the null coordinates that we could have made. It is now clear that the only difference that would have made is to interchange the roles of u and v and restrict the domain of the solution to

v>1 and v2 (u2+1), (103)

instead of to the right of the hyperbola U2 = V2 + 1.

5. THE CASE A = y2+ 1 AND =4,U2-1

When A = ,2+1 and a = #2-1, the Ernst equation (24) allows a solution of the form E = py+iqlt, (104)

where p and q are constants and q2-p 2 = 1. (105)

(Note the difference with the condition (43) in the earlier case.) We find that the corresponding solutions for X, q2, V +jt3, and /3 are:

-p,)2 q )2 - ~ (1-py)2+q21 (106) %VIU = - 1p 2I )2 + q2/0 - I e23 = (1p1 2 q+q q2A 2 - ( 1067)

ev?a3 = (y2 + 1)3 (,t2 _ 1)' e-4 = (2 + 1)1(,21). (107)

Now letting N = sinh /r and It = cosh0, (108)

the metric (39) takes the form (cf. equation (49))

ds2 = A f

[(dV/)2 - (dO)2] (cosh #Y sinh )2

cosh 3b

sinh 0 f { 1 + E 12 (dx2)2 + I1-E 12 (dxl )2 - 4q(cosh 0) dxl dx2}, (109) A

where A = IE12-1 =p2 cosh2 i/+q2 sinh20. (110)

(a) The transformation to null coordinates

We now introduce a system of null coordinates, u and v, in the same manner as in ?4(a) , equations (51):

it = 6 + -, 0= - u = coshf "and v = sinh. (111)

By these definitions, we now have, in contrast to equations (52)-(54),

= (u2 1) (V2 + 1)1 + uv = sinh V, (112) = u(v2+ 1)iv(u2-1)2 = coshO (113)

and cosh Vf sinh 0 = u(u2 1)I-v(v2 +1). (114)




In terms of the (6, C)-variables, equation (55) governing qS (which is valid in the present case, as well) takes the form

0 6 C 2cosh ( + ) sinh - [(cosh 2C) 0, 6-(cosh 26) ] = 0. (115)

In the (u, v)-variables this equation becomes

O, ,+ 2V2 + I 56 2ff2 - I 561 0. (I 16)

,U,V+2[uV(u2 -1)-v/(v2+ 1)] [/v2+ 2V2 1) ]/(U2]-(1)1

With our present definitions, we find that while equations (60) and (61) are unaltered, equation (59) is replaced by

20,,7 O 2[uU22 1 (0 U)2 - (V2 + 1) (?, V)2] (117)

Turning to equations (36) and (37) governing f and solving forf andf we now find (in contrast to equations (62) and (63)):

4 .,z2 {+2+1 [A(q S)2 + 8(o q)2] -2, S S,}' (118)

z

lh = 4P2 + z2 {-z2 _ 1 [3 (11 8)2

+ 8(o U)2] + 2 q ( I 9)

Inserting these expressions in

u = (v+ 1) U [( l)f +fj+v [f, \/(2-jl)f,j (120)

v = (u2-1)i [\Vv2+ 1)f Yv ftl+U [f + V/(v2+ 1)fj] (121)

(which follow from equations (112) and (113)) and simplifying, we find

f = U2u2-1 )4[\/4(u2-1) U) 2 (122)

and fv =(2V2+1)[U(U2-1)-V(V2+1)] ('V)2. (123)

Finally, the expression for e corresponding to equation (67) is

[u= - L8V(u2 - 1 )-vV(V2 + 1 )]2 (fu2 - 1)' (V2+ 1)q2 S qS0. (124)

(b) The solutions for qS and f Letting

r = uV(U2-1) + V(V2 + 1)

(125)

and s = uV((u2-1)-v(v2+ 1),




we find that equations (116), (122), and (123) reduce to

0)r,r- ,s, s SO's) = 0, (126)

8 4~~~~

and f,r=8rVO,s' f,s=- [(, r)2+(0,s)2] (127)

We observe that equations (126) and (127) are the same as equations (69) and (77). Therefore, the solutions for qS and f given in ?4(b) and, in particular, the fundamental solutions (70) and (78) apply equally to the present case with the difference that the variables r and s are now related differently to the null variables u and v. Because of this last difference,

2u2 - 1 2V2+1I O , = ( (U r + r), s) and 5, v = (v2+l) (, r-O,s)~ (128)

and the expression (117) for e gives

6 = C2c4(2 u2-1) (2V2+1) A 2 r [O2(a8)- (2 (129)

(c) The character of the distribution of e in the domain u > 1 and U2 - (V2 + 1) > 0

It is evident that the solutions obtained in ??5 (a), (b) are limited to the domain,

u > 1, any v, and s = u(u2-1)-v(v2+1) > 0, (130)

or since = (u2 + v2) (u2 - v2-1) (131) UV\(u2-1)+vV,(V2+ 1)' 11

it follows that the domain of validity of the solution is to the right of the hyperbola u= (v2+ 1).

Since the asymptotic behaviour (80) and (81) of the Bessel functions apply equally to the present case, the behaviour of qS andf for s - 0 and s -* oo is the same when they are expressed in terms of the variables r and s. Therefore, the behaviour as given in equations (82) and (83) is applicable, as they stand, for s8->0; but for s -* oo the terms in e-Ct(2v2+1) = e-(r-s), (132)

should be replaced by (cf. equations (125))

e- 2aVV/(V2+1) t (1 32')

for example e fvv2?1 =fe (say) (8 -oo). (133)

From the expression for e given in equation (129), it now follows that

- C2a4(u2 + V2)2 - e-4auV(u2) (s- 0). (134) A

8 Vol. 402. A




1.5 (a)

10 U 2.5 1.5

(b)

V

0 1.0 U 2.5

FIGURE 2. The curves of constant e for a = 0.5 and C = 1.0 for (a) p = /0.5 and (b) p = 0. The raggedness of the curves in (b), near the critical hyperbola u2 = v2 + 1, is due to errors in computer tracing.




Since, as may be readily verified, A, as now defined, has the behaviour

A -p2(I +4u2v2) (p 0), (s -0) (135)

) 8s2/(u2 + V2)2 (p 0)

we conclude that

eCb2x4 ) sl e-46UV(u2-1) (p #U 0), (136)

p 2(1 4U2V2se (P A

0),(s -SO). (136) C2x4(U2+V2)4 s-e-4aU(U21) (p -0)

Again, as in the case considered in ?4, e vanishes generically on the hyperbola, U2 = V2+ 1.

Similarly, we find that

02a2 (2u2-1)(2 + 1) e -fo e4aVV(V2+1) (s-o?), ( 137)

where f. is defined in equation (133). (Note that (2u2- 1) (2v2+ 1)/A tends to a

finite limit when s -x oo.) The behaviour of e predicted for s -> 0 and s -x 00 is exhibited by the curves of

constant e illustrated in figure 2a (p # 0) and figure 2b (p = 0).

(d) The description of the space-time in a Newman-Penrose formalism and the occurrence of a space-time singularity on U2 = v2 + 1

The Weyl scalars when f # 0 are related to the Weyl scalars for the vacuum (whenf = 0) by the same formulae (93) and (94). The spin coefficients and the Weyl scalars for the vacuum, appropriate for the metric (109) withf = 0, can be obtained by defining a null-tetrad exactly as in equation (95) with U, V, A, and E now defined by

(cosh if sinh Q)J4 V = (cosh Vt sinh 0)",

E = p sinh f + iq cosh 0 and A = p2 cosh2 i + q2 sinh20. (138)

We find that in place of equations (98)-(100) we now have

- (cosh Vt sinh6) {0p cosh -iq sinh6)2+1A2 cosh (6 + ') cosh (6- ')d T'2 2A3 pcoh -isnh

4 (cosh Vf sinh60)2 ~,(139)

--3 cosh (-Vf) (140) 0 4 (cosh?t sinh 6)l (p cosh?t + iq sinh 6)3

T'4 := + 3 cosh ( + V) (141) 4(coshVt~ sinh60)' (p cosh it+ iq sinh6) 11

(In the Appendix general expressions for the Weyl and the Ricci scalars are given, which are applicable to all cases considered in this paper.)

The occurrence of a space-time singularity on the hyperbola u2 = v2 + 1 (or, equivalently, s = 0) can be deduced from the behaviour of VI2 for s -> 0. Sincef = 0

8-2




for s = 0 (by equation (83)) the term T2 e-f in Tf2 has the behaviour (cf. equations

(101)) -f cosh8Acosh2i/

(101)) Y>2e f-; > p 4 (sO~~~( -> ); (142)

or, by virtue of equations (112) and (135),

1 3 T'2 ef - s- (p #0), I

3(8p2+r) (s- 0). (143)

8(1+4u v2)

Comparison with the behaviour (136) of e shows that T2 e-f dominates e when sO - 0. Therefore, the behaviour of FT2 for s - 0 is the same as that of W2 e-f, i.e. of T2.

The occurrence of a space-time singularity on the hyperbola, u2 = v2 +1, is manifest.

time

= = ' ~~~~~~~~~~~~pace

FIGURE 3. The domain in the (u, v)-plane occupied by the solutions found in the present paper and in Paper I. Solutions for impulsive waves in the region of interaction found in Paper I are represented inside the circle while the solutions found in this paper are external to the four hyperbolae.

It can be readily verified that the case A I- -1 and =zt2 +1 leads to essentially the same space-time as the case A = I2 + 1 and 8 = -1 we have just considered; the only difference is that the hyperbola on which the space-time singularity occurs has to be rotated to a different quadrant (see figure 3).




6. CONCLUDING REMARKS

The principal aim of this paper, in seeking solutions in the alternate gauges and coordinates of Einstein's equations with a perfect-fluid source with the equation of state, c = p, was to supplement the solution obtained in Paper I, in a gauge and in a coordinate system appropriate for treating the collision of impulsive gravitational waves. The solutions that have been obtained enlarge the domain in the (u, v)-plane in which solutions are represented (see figure 3). Besides, the two distinct classes of solutions obtained with the choices (A = q2-1, 8 = -2 -1) and (A = y2 + 1, 2 = -1) are inextensible since their domain of validity is bounded entirely by a space-time singularity. The solutions, in some ways, resemble the standard solutions of cosmology in that any time-like trajectory, extended into the past, encounters a singularity. Also, since e vanishes in the singularity in the generic case, it would appear that a perfect fluid with e = p is generated by null dust at the singularity by analogy with what happens along the null boundaries in the problem of colliding waves considered in Paper I. Finally, the distribution of e that evolves from the singularity and the emergence of density maxima suggest that the solutions may have relevance for the physics of the early universe.

The research reported in this paper has, in part, been supported by grants from the National Science Foundation under grant PHY 84-16691 with the University of Chicago.

REFERENCES

Chandrasekhar, S. & Ferrari, V. I984 Proc. R. Soc. Lond. A 396, 55-74. Chandrasekhar, S. & Ferrari, V. I985 Proc. R. Soc. Lond. A 398, 429. Chandrasekhar, S. & Xanthopoulos, B. C. I985 Proc. R. Soc. Lond. A 402, 37-65. Watson, G. N. I944 A treatise on the theory of Bessel functions. Cambridge University Press.

APPENDIX

The examples of space-times considered in this paper show that a metric of the form V2

ds2 = U2[(di)2-(dO)2]-I E 12-1 1 (1-E) dxl + i(1 + E) dX2 12 (A 1)

is of frequent occurrence in the theory of space-times with two space-like Killing-vectors. For this reason, it will be convenient to have listed the spin coefficients and the Weyl and the Ricci scalars for the general case when U and V are allowed to be any two real functions and E any complex function of VY and 0.

For a metric of the general form (A 1), we can define a null-tetrad basis exactly as in equation (95), where, besides U and V,

E+l = A, A =I E12- 1,}

(A 2)

and S_9S+ +.9 * = 2.

The directional derivatives D and A are also defined as in equation (94).




We find that the non-vanishing spin coefficients and the Weyl and the Ricci scalars are given by:

(E* + E*) (Vv + V0O) UAV2 ' V\UVV2

(E -E) _ (V- V,0) UAV2 '

+ UVV2 2

_____+ 1' e =+ TJA/ 8+ T,2[E(E** + E*)-E*(E ,, + E )

2U2V\2 4UAV\2 '

D , __1+ [E(E* -E*)-E*(E v-E,)]; (A 3) 2 U2V\/2 4UAV\2 I f 1

E? E* + (U )^E*,+2E**9 'o E}'+EO

U4 {(IOn U) +-In-U) + * ( 9 E*O_*(,- )

2 6 U2 [(in U)**(nU ,aE (,*X,@X8E*

(A 4)

and

0 (V ,+ V0^) (U ,+ U0^) V * V0 ,0+2V, ,,9 (E, ,+E ,0) (E**,+E*^) 00- U3VF 2 U2V

E 2 U2A

ElfE 6_ InW- Va (U InU V * + E av-Vv (E#E*v)(E,*f-E,v)

22- U3V 2U2V 2U2A2

-4U2 {(1n U3 -(1n U) 0-[(ln V) f2 +[(1n Vf ]2}- E 4EU22

_ E * ,-E 0a 0a E*[(E ,)2- (E,0)2] E , V ,-E,0E V,Vf T20 2 U2 U22 U22VA

6U =f E, U E, ,1-,6 ,a+12[lnU -l )v A E~~~~~,E*v,~~~~~~~E0E ~ ~ ~ ~ ~ ( 1

It may be noted that the vanishing of the Weyl scalars, F1 and YM3 and of the Ricci scalars, 00 and P02~ iS guaranteed by the existence of two space-like Killing vectors. Also, for the vacuum the expression for ~~2 can be simplified to the form

=-n v 1 + V -, o (E, + EUa ) ( A 6)



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Some Exact Solutions of Gravitational Waves Coupled with Fluid Motions