A class of solutions of the Einstein-Maxwell equations

A class of solutions of the Einstein-Maxwell equations
Konrad BajeP)
Institute for Theoretical Physics, University of Warsaw, 00-681 Warsaw, Poland
Jetzy Klemens Kowalczyfiskib)
Departamento de Fkica, Centro de Investigacion y de Estudios Avanzados del I.P.N., AP 14-740. Mexico 14,
D.F., Mexico
(Received 6 October 1983; accepted for publication 9 August 1984)
We present all the exact solutions of the Einstein-Maxwell equations for a special case of the
Robinson-Trautman metric form. Gaussian curvature of the angularlike part of the chosen
metric form is equal to zero. The solutions are of the Petrov types D or 11. Eight of them are
probably new.
1. INTRODUCTION
In this paper we present all the exact solutions of the
Einstein-Maxwell equations (without currents)
Gpv = - RgpV + 2(FppFvp + &pvFprFPr),
FL, 1 = 0, F pv,v = 0, (1.1)
assuming only the limitation
ds2=4p2dYd?+2dpdq +p-'(2mp + f)dq2. (1.2)
The meanings of symbols are as follows: Gpv, gpv , and
Fpv are the Einstein, metric, and electromagnetic field tensors,
respectively, R is the cosmological constant, Y is a complex
coordinate, p and q are real coordinates, m is an arbi-
trary real constant, andf=f(Y,?,p,q)
is a disposable real
function that belongs to class C '. Here and below every symbol
with an overbar means a complex conjugate quantity of
the given symbol.
The signature is + + + - .
The metric form (1.2) is a special case of the Robinson-
Trautman one.' The Gaussian curvature of every two-dimensional
surface p,q = const (p # 0) is equal to zero for
space-times (1.2).'Such a geometrical property of space-time
has a special physical interpretation in the case of the Robinson-Trautman
type space-times. Namely, the interpretation
has been given that the sources of fields, that produce the
Riemannian curvature of these space-times, move with the
speed of light. '*' Thus the solutions presented below make it
possible to find explicitly the detailed properties of all such
space-times admitted by the Einstein-Maxwell theory within
the limitation (1.2).
In Sec. I1 some general properties of the solutions are
briefly reviewed. In Sec. I11 a list of the explicit solutions and
Petrov's classification are given.
II. SOME GENERAL PROPERTIES
When solving Eqs. (1.1) under condition (1.2) we used
the known results3 that include expressions ready for inte-
')Present and permanent address: Institute of Geophysics, University of
Warsaw, U1. Pasteura 7,02-093 Warsaw, Poland.
b)On leave of absence from the Institute of Physics, Polish Academy of Sciences,
Warsaw, Poland. Present and permanent address: Institute of
Physics, Polish Academy of Sciences, Al. Lotnik6w 32/46,02-668 Warsaw,
Poland.
grati~n.~ After easy integration of a part of those expressions
one finds that
f= -f/2p4+pA-BB (2.1)
and
-
A =A, A,, = A,y = A,p = 0,
-
C,y = C,y = C,p = 0, A,, = 2CC (2.2)
for B = 0, where C is an arbitrary complex function of q
only, and
-
A =A, A,p = 0, B,y = B,p = 0,
(2.3a)
A,eB -A, yB,y + 4B2B,, = 0,
(2.3b)
A, yA,y - 8A,,BB = 0 (2.3~)
for B #O.
The electromagnetic field is given by the following
equations:
F- = F, = F, = 0, Fyq = C (2.4)
for B = 0, and
Fyp = 0,
-
Fyy = B - B, F, = - Jp-'(B + B),
-
Fyq = Jp-lB,, - V9yB-l
(2.5)
for B #O.
It is well known that R is introduced into the Robinson-
Trautman metrics in a very simple way.5 This causes the
absence ofR in Eqs. (2.2) and (2.3). Thus Eqs. (2.2)-(2.5) are
special cases of the more general equations that have been
given by Robinson et aL6
It is easy to see from Eqs. (1.2) and (2.1) that real function
A is determined with an accuracy up to an arbitrary
additive real constant since m is arbitrary. Thus condition
A = const is equivalent to condition A = 0.
The following transformations:
Y= Y'+a, Y= Y'eia, q=q'+b (2.6)
do not change Eqs. (1.2) and (2.1)-(2.5) for arbitrary constants
a, a, and 6, where a is complex and a and b are real.
In each of our space-times the vector field k p: = determines
a congruence of the principal null curves of the
electromagnetic field since it is a conclusion from Eqs. (1.2),
(2.4), and (2.5) that kIpFvlpkP = 0 and kpk" = 0. The last
equation means that metric form (1.2) is also the Kerr-Schild
1330 J. Math. Phys. 26 (6), June 1985 0022-2488/85/061330-04$02.50 0 1985 American Institute of Physics 1330

metric form since dq = k,,da+. It is easy to prove (see, e.g.,
Refs. 3 and 4) that vector k“ is geodetic, shear-free, rotationfree,
its expansion does not vanish, and that it is a double
Debever-Penrose vector.
111. LIST OF SOLUTIONS AND PETROV’S
CLASSIFICATION
The explicit solutions of Eqs. (2.2) and (2.3) are as follows:
J
”
A=2 CCdq, B=O,
A = 0, B = 4 (Y),
where 4 is an arbitrary analytic function of Y only,
A = 4ab(Y+ F) + k’q, B = be“, (3.3)
A = bb-+ k’q, B = bY, (3.4)
A = - 8a2q-1ey+p , B = ae’, (3.5)
A = - *a’( Y + P)2q-’/3 + bbi( Y - f) + 6b 2q1/3,
B = ~&q’/~, (3.6)
A = 2a2[ - 2x-+ b (Y’ + P’)](x’ - b ’)-’
+ [2ac(Y + 7) - c21(x + b )-I
+ [2agi(Y- P) -$I(x - b I-’,
B = ae” (x’- b ’),
(3.7a)
where the auxiliary real variable x is determined by the equation
q = p’ - tb’X3 + 2b4x,
(3.7b)
A=k%i(lnF-ln Y)+2a2b2q, B=aeib4Y-’,
(3.8)
A= -2a2(1+bz31n~--’(1-bbi)lnP
(3.9)
B = bemqYa- I, aE + 2(a + E) = 0, (3.10)
A = - k 2kq - aEk YaFE, B = aq2akya - 1 9
k -’ = aE + 2(a + E)#O,
A = 2ab (Y -’ + y-’) + 2azq( Y?)-’,
(3.1 1)
B = eu(aqY -’ + by
-I), (3.12)
A = -@-1/3(aY7+
b)’, B = aql/’Y, (3.13)
where a is a complex constant arbitrary up to the appropriate
limitations given in relations (3.10) and (3.1 l), and a, 6, c,
g, and h are arbitrary real constants. When integrating Eqs.
(2.3) one obtains more complicated expressions for A and B
than ours at right-hand sides of Eqs. (3.3H3.13). Those expressions
include more arbitrary constants but they can be
reduced to Eqs. (3.3H3.13) by means of transformations
(2.6). When obtaining solution (3.5) the transformation
Y = cY ’, p = c- %‘, q = cq‘ has also been used. The same
transformation has also been used to get a special case of
solution (3.11), i.e., when Rea = 1 and Ima#O. If a = 1,
then solution (3.1 1) is a special case of solution (3.7).
The above list includes aZZ the exact solutions of Eqs.
(1.1) with limitation (1.2). A proof of this theorem is given in
Appendices A, B, and C.
Solution (3.1) has been given by Robinson and Trautman,’
solution (3.2) by many author^,^*^*' and solutions (3.4),
(34, and (3.13) have been given by Leroy.8 The remaining
solutions seem to be new [special cases of solutions (3.9),
(3.10), and (3.11) have also been given in Ref. 81.
Each of our metrics is conformally flat if and only if
2m +A = B = 0 (or equivalently m =A = B = 0).
When determining the Petrov types of our solutions we
used Lemma 1 from Ref. 3 (cf. Ref. 4) since the premise of
that lemma holds in the case of our metrics. It appears that
only solution (3.1) is of type [2,2] (Penrose’s notation). The
remaining solutions are of type [2,1,1] with possibilities of
further degenerating to type [2,2]. More precisely, for each
one of our solutions we have the following: if it is not conformally
flat, then it is of type [2,2] if and only if all equations
2(2m +A )A,n - 3(A,y)’ = 0,
(3.14a)
(2m+A)B,yy +M,yyB-4A,yB,y=O, (3.14)
3BB,, - 4(B,y)’ = 0
(3.14~)
hold together.
ACKNOWLEDGMENT
One of us (JKK) wishes to thank Professor Jerzy Plebanski
for his hospitality during the stay of the author at the
Centro de Investigacion del I.P.N. in Mexico City.
APPENDIX A INTRODUCTORY REMARKS AND THE
FIRST STEP OF THE PROOF
Our proof of the theorem, saying that Eqs. (2.1) and
(3.1H3.13) areallsolutionsofEqs. (1.1) withlimitation (1.2),
is too long to allow its presentation here in totality. Therefore,
we give here only its scheme, putting emphasis on the
more important items so as to permit anybody interested to
easily carry out the proof himself in detail.
Notation: D, E, F, ... are complex functions of two variables,
Y and q, analytic in Y (or of f and q analytic in P for the
symbol with an overbar). K, L, M,... are disposable complex
functions of one real variable q; a is an arbitrary complex
constant; a and b are arbitrary real constants. An integer
subscript at any symbol does not change the above meaning
of the symbol.
Assuming B = 0 one immediately obtains (3.1) from
(2.2). Thus, henceforth, we put
B #O.
Integrating (2.3b) we get
A,, = - 4z (D + FB,q),
(All
where D is a disposable function.
Assuming B,, = B,, = 0 we easily obtain from (A2)
and (2.3~) that D = a and then solution (3.3).
Let us assume B,, = 0 and B,, #O. If D = 0, then by
(A2),(2.3c), and (Al) we immediately obtain solution (3.2). If
D #O, then by the fact that A,fi = 2,- and by (A2) we get
B,, ID = K, =El #O, and then from (2.3~) we obtain
B,yyz,FK; + 2B,,zK1,, = 0. Assuming B,,, = 0 we easi-
1331 J. Math. Phys., Vol. 26, No. 6, June 1985 K. Bajer and J. K. Kowalczyhski 1331

ly get solution (3.4). Assuming B,,,#O we obtain solution
(3.5) after a short calculation.
Thus all possibilities are exhausted for B,, = 0 and
henceforth, we put
B,, #O. (A31
Assuming B,, = 0 we obtain by the fact that
A,* = z,-,, and by (A2) that B = L,e" and L, = z,, and
then taking (2.3c),,y and (A2) we find that D = YM, + N,.
Now from (2.3~) we obtain among others that M, = aL c2
andL,,,L f = (bL, + aZ)'". Then after simple calculations
we find solution (3.6) forb = 0 and solution (3.7) for b #O.
Thus all possibilities are exhausted for B,, = 0 and
henceforth, we put
B,y #O.
Assumption:
1 +(B,,'B),,=O. (A51
From (A5) we have B = (Y + K2)-'L2 and then by the
fact that A,,-, =x,yy and by (A2) we obtain D
= (Y + K2)-2(z'; 'M2 - K,,,K,L2) + (Y + K2)-'L2,,K2,
whereM, is real. Then from (2.3~) we get M2 = K,,, = 0 and
L,,,z2 = a#O. Putting L, = N, exp iP2, where N2 and P2
are real, we obtain among others (N:),, = a and then after
short calculations we get solution (3.8) for a = 0 and solution
(3.9) for a#O.
Now we begin the more complicated part of the proof.
Assumption:
1 + (BY, 'B I,, #O. (A61
Since A,,-, = 2,- we obtain from (A2) and (A6) that
D = (BE),,, (A74
B,, = (BF),y,
where
E:= YL+K, K=K,
(A7b)
F:=YM+L#O, M=B. (AW
Inequality F #O results from (A3) and (A7b).9 Then the integration
of (A2) gives us
A = - 4[BB(K + LY+ LY + MY?) + N 3,
and by (A2) and (A7)-(A9) we get from (2.3~) that
B,,[i?(Y+J)'+ 3BP(Y+J)]
where
N = R,
(A91
+B [R(Y+J)+B(P+2m,,)] +2N,, =0, (A10)
G = B,,F2 + 3BFM,
H: = 3G + B (2F,, - 4FM),
J: = EF - = (YE + K )( YM + L ) - I,
P = LL - KM = F.
(A131
(A141
Assuming G = 0 we obtain M #O and B = QF-3 from
(A1 1) and then from (A10) we get N,q = H = P = J,, = 0,
which leads via a short calculation to a special case of solution
(3.12) (i.e., if b = c = 0 there).
Henceforth, we put
G #O. (A151
APPENDIX B: CONTINUATION OF ASSUMPTION (A6)
FORP=O
Relations (Al)-(A4) and (A6)-(Al5) hold here. Assumption
P = 0 gives us
--
M#O, J=zM-', A= -4(BBFFM-'+N).
(B1)
Integrating (A10) we find B and then differentiating the
result once with respect to 7 and several times with respect
to Y we get rid of B and the integrals. In consequence we
obtain a product equal to zero of an expression different
from zero and of a polynomial of Y. Thus the coefficients of
the polynomial, which depend on Pand q, are equal to zero.
This procedure should be conducted twice, separately for
Ntq = 0 and N,, #O.
This gives us among others
H = QIG, BFJ,, = Q2G
for N,, = 0, and
J,, = 0, H(H,y - G,y) = 0
for N,, # 0.
Assumption:
(B2)
(B3)
N,, = 0.
(B4)
Assuming J,, #O (eQ,#O) and taking into account
(A11) we find B from the second equation of (B2) and then
substituting such a B into (A10) we obtain a contradictory
result J,q = 0. Thus J,q = 0, which after the first transformation
from (2.6) gives us
F = MY.
(B5)
Assuming Q, = 3 we get from (B2),(A12), and (B5) that
M= - (2q)-' and then from (AlO),(Bl), and (2.3~) we obtain
a special case of solution (3.12)(i.e., if b = c = 0 there).
Assuming Q, #3 we get from (B2) that In B = In Q3
- [M2(5- 3Ql) + 2M,,]M-2(3 - Q,)-' In Yand then by
(Bl) we obtain Q,,, = 0 and an explicit expression of Q3(q)
from (A2), (A7a), and (2.3~). Putting Q, = 1 -a we get
2M,, + M2[aE + 2(a + Z)] = 0 from (A10) and then we
obtain solution (3.10) for M,, = 0 and solution (3.11) for
Mq # 0.
Assumption:
N,, #O.
(B6)
Since J,, = 0 [see (B3)] thus (B5) holds also here.
If we assume H = 0, then it is easy to see that B which
fulfills both (A10) and (A12) contradicts (A6). Thus, H #O
and we obtain H,y = G,, from (B3). Assuming G,,#O and
differentiating (A10) with respect to Ywe get a contradiction
with (A6). Thus we have G,, = H,, = 0.
Inequality H #3G contradicts (A6) by (A12) and (B5).
Thus, H = 3Gand we obtainM = - (2q)-' from (A12) and
(B5). Then integrating (A10) weget from (Bl) and (2.3c), after
a simple calculation, solution (3.12).
APPENDIX C: CONTINUATION OF ASSUMPTION (A6)
FOR P#O
Relations (Al)-(A4) and (A6)-(A15) hold here. Applying
for P #O, the procedure that is described in the second
1332 J. Math. Phys., Vol. 26, No. 6, June 1985 K. Bajer and J. K. Kowalczynski 1332

paragraph of Appendix B, we obtain
(HG-'),, = 0, (C14
P + 2FJ,,, = 0
(CW
for N,q = 0, and among others
(HG-'),y(G,y -H,y)=O
(C2)
for N,q # 0.
Let us assume N,q = 0. Relations (A4),(AlO),(A15),
and (Clb) give us then H #O. Thus, from (Cla) we obtain
H=xG and R #O. Then (A10) and (Clb) give us
where R, = J + 3BPG -I. Substitut-
B = R,(Y + zJvR,
ing such a B into the latter [taking into account (A1 l)] we get
a polynomial of Y equal to zero. Its coefficients (equal to
zero) give us a system of algebraic equations involving L, z,
M, R, R,, and E2. The system appears to be self-contradictory
by the fact that F, P, R #O. Thus we have
NSq #O.
(C3)
Let us assume H = 0 and split our considerations into
two separate cases M = 0 and M # 0. The procedures are the
same in both cases. We integrate (A12) getting B 's and then
after substitution ofthose B 's into (A10) we obtain contradictionswith(A4)forM=
Oby(C3),andwith(A6)forM #Oby
(C3),(A4), and (A15).
Thus we have H #O and assuming H,, = 0 we get
G,, = 0 from (C2). If H #3G, then from (A12) we obtain
contradictions with (A4) for = 2M2 and with (A6) for
M,q # 2M '. Thus we have H = 3G. Integrating (A11) we get
B that after substitution into (A 10) gives us a polynomial of Y
equal to zero. Its coefficients give us a self-contradictory system
of equations depending on ?and q. The contradictions
are caused by (A15) and equation H = 3G for M = 0 and by
(A6) and assumption P #O for M #O. Thus we have
H,y#O.
The most general solution of (C2) is
(C4)
H= GS+ T, (S- l)T=O, (C5)
which by (C4) gives us
G, y s # 0. (C6)
Differentiating (A10) with respect to 7, then dividing it
by B,yG,y and differentiating once again with respect to Y
and we obtain
8, + Z2(B, 'B ), , = 0. (C7)
Assumption:
E2 = 0.
This gives us by (C7)
El = 0.
Integrating (C8) and (C9) we get
2GJ + 3BP = GU, + U,,
HJ + B (P + 2FJ,q) = GV, + V,.
(C9)
(C10)
(C11)
Using (C10) and (Cll) in (A10) and differentiating (A10) with
respect to ?, and then dividing (A10) by c,y and differentiating
once again with respect to Pwe obtain by (A4) an equation
that after integration gives us
J(GUI+ U2 - GJ) = GWI + W,.
(C12)
Now let us use J as a new variable instead of Y. This is
possible since J, = F -'P # 0. In the new variable language
thereisF=P(Z- JM)-'.
Integrating (C10) and (C12) withrespect to J [see(A1 l)]
we obtain two expressions for B. Equating those expressions
and differentiating the obtained equation with respect to J to
get rid of integrals we obtain a polynomial of Jequal to zero.
Its coefficients give us a system of algebraic equations involvingz,
M, U,, U,, W,, and W,. IfM = 0, then the system
gives us U, = W, = 0, which contradicts (A15) by (C12). If
M #O, then the system gives us U,#O by (A15) and (C12)
andthenitgivesus U: =4W1and UlU,=2W,(ifL=0,
then Ul = W, = W, = 0). This leads to a contradiction with
(A4) by (C10) and (C12). Thus, the case E, = 0 is empty.
Assumption:
E2#0.
(C13)
Relations (C7) and (C13) give us (B,? 'B),= = 0. This
equation has only two solutions with respect to Y. The first
one is B = U3euy but it contradicts (A10) by (A4) and (C3).
The second solution is
B = V,( Y + V,)".
(C14)
Substituting such a B into (A10) and taking into account
(A6), which implies V # - 1, we analyze (A10). Simple analysis
gives us
T=O, s= -1, V=l (C15)
by (C5) and (C6). Then substituting B from (C14) into (A12)
and using (A1 l), (C5), and (C15) we obtain a polynomial of Y
equal to zero. Its coefficients give us a system of differential
(with respect to q) equations involving L, M, and V,. Solving
the system with the use of (A8b) we get M= (%)-I,
L = aq-', and V, = 6a. Then we obtain after a short calculation
solution (3.13) by (A7b), (A9), and (2.3~).
This terminates the proof since all possibilities have
been exhausted.
'I. Robinson and A. Trautman, Proc. R. SOC. London Ser. A 265, 463
(1962).
'I. Robinson and A. Trautman, Phys. Rev. Lett. 4,431 (1960).
3J. K. Kowalczyhski and J. F. Plebahski, Int. J. Theor. Phys. 16,357 (1977);
Errata 17,387 (1978).
'Those expressions can be found in Sec. 2 of Ref. 3 and they concern spacetimes
with rotation and nonzero Gaussian curvature of the mentioned twodimensional
surface. It appears, however, that if one puts therefl= 0 (cancellation
of rotation), E = Im rn = 0, and a = 2-"* (notation as in Ref. 3),
then one obtains our metrics (1.2), and then the simplified expressions are
valid in the case considered in the present paper.
5D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of
Einstein's Field Equations (VEB DVW, Berlin, 1980), Theorem 24.7.
61. Robinson, A. Schild, and H. Strauss, Int. J. Theor. Phys. 2,243 (1969).
'G. C. Debney, R. P. Kerr, and A. Schild, J. Math. Phys. 10, 1842 (1969).
*J. Leroy, Bull. CI.Sci. Acad. R. Belg. 62,259 (1976).
m e general solution of Eq. (A7b) is B = fl2)exp J Mdq, where tc, is an
arbitrary analytic function of one complex variable 2 = Y exp J Mdq
+ J (L exp J Mdq)dq, but this solution is not used in the proof.
1333 J. Math. Phys., Vol. 26, No. 6, June 1985 K. Bajer and J. K. Kowalczyhski 1333