A new painlevé solution of the ernst equations

Volume 135. number

89

PHYSICS

A NEW PAINLEVE

SOLUTION

LETTERS A

OF THE ERNST

13 March

1989

EQUATIONS

Patrick WILS I Physics Department, Universitaire Instellingdntwerpen. Universileitsplein I. B-2610 Antwerp. Belgium Received 25 November 1988; accepted Communicated by J.P. Vigier

for publication

A new solution of the Ernst equations admits a Bianchi type II three-dimensional

11January

in general relativity isometry group.

Several exact solutions of general relativity have been found in the literature which reduce to Painleve transcendents [ I]. They are found by assuming the metric form ds’= -tj(dv-wdy)‘+r’j--’ +6f-‘e’;‘(dr”+c

dy’

du’)

,

(1)

with f; w and y functions of r and u, and E= _+1 and 6=&l, depending on whether u (e=-1, 6=+1), v(e=+1,6=+1)orr(t=-1,6=--l)isatimelike coordinate, representing cylindrically symmetric, stationary axisymmetric or cosmological metrics respectively. The Einstein-Maxwell equations for this metric reduce to the Ernst equations [ 1] jV2G=

(VG+2$V@).VG,

(2)

fV'@= (VG+2$V@).V@,

(3)

with scalar product and gradient in the flat space metric ds; =r2 dy’+d(dr’+c

du’)

.

operator

defined as

(4)

’ Senior Research Assistant of the National Fund for Scientific (Belgium).

03759601/89/S ( North-Holland

is presented,

which contains

03.50 0 Elsevier Science Publishers Physics Publishing Division )

a Painleve

III function.

The metric

functions of u only, so that the Ernst function @ is independent of r. The combined Einstein-Maxwell equations then force @and w to be linear functions of u. By choosing an appropriate electromagnetic gauge and a translation of the u coordinate, these can then be written as

q?$,

(5)

w=qu,

with p and q constants. The non-vanishing nents of the field equations then give 7 as a of r only, to be determined by the equation prime denoting differentiation with respect 2y’=rf

-‘f

“/2+q’r-‘f

‘/2+p2Tf

-’ .

compofunction (with a to r) (6)

By a global resealing of the metric the arbitrary constant appearing from the integration of this equation can always be put equal to zero. The only remaining equation to be solved is then (y. -If ‘)‘=p2rf

In general, Painleve functions are found by supposing that the Ernst functions G and @,which are combinations of the metric functions fand w, and the electromagnetic potentials and their derivatives, are separable functions of r and u. A new solution can be found by considering f a function of r only, and electromagnetic potentials

Research

1989

-‘_q’r-lf

2)

(7)

which can be transformed to a Painleve III equation [2] withcu=S=O, /3=p2, y= -q’, for W=rf-‘. The Ernst function G is then given by G=cf-p’u’/Z-ieqV,

(8)

with Vdefined by V’=r-‘f ‘=rW’. Solutions of this type which are locally flat near the axis r=O (6=+1), have f(r=O)=l. When q differs from zero, the solution cannot become asymptotically flat at infinity for all u. When q=O the solutions reduce to the well known static cylinB.V.

425

Volume 135. number

PHYSICS

8.9

drically symmetric Einstein-Maxwell metrics [ 3 ] for 6=$1. When q is different from zero, the general solution admits an isometry group of Bianchi type II, with the following Killing vectors.

none of which is hypersurface orthogonal (HSO). The resulting Painleve equation (7) is the same as the one found by Chitre et al. [ 41 for t = - 1. d= + 1, andVandenBerghandWils [5] fort=+1,6=+1. However the solution given in this paper is clearly different, because the third Killing vector is not HSO. In addition, no relation could be found between the new solution and other known Painleve solutions [ 1 1. For instance, the particular case c=O of the Painleve III solution found in ref. [ 11, has exactly the same metric form. and the same vacuum solution as a particular case, but the principal null directions of the electromagnetic field are different. Also the resulting Painleve equation has a=y=O. and cannot be transformed into the form found above. The new metric also differs from those given by Barnes with a GJI [ 61. Take the null tetrad formed by the principal null directions of the electromagnetic field, {.f “‘e-:‘[

&

(6+ 1 )d/ar--

I3 March

LETTERS 4

1989

and eitherp=u=t=O for 6= - 1, or ~=r=cr=O for 6= + 1. So the initial assumptions of Barnes [ 61 are valid. However the further restrictions for the twistfree case, imposed by his choice of tetrad, are not satisfied here. As for the solutions of refs. [ 41 and [ 5 1. a particular solution results by putting both sides of eq. (7) equal to zero. By appropriate coordinate rescalings, q can be put equal to p, so that the simple metric

+&-“l’

exp( zp’rJ”)(dr’+t

du’)

(13)

results. This metric is however not locally flat at the axis, so that it needs to be matched to another solution. In the vacuum case p=O, eq. (7 ) results in the metric functions f= (ar”+br-“) w=qu,

-‘,

4abn’=q’,

2;~=$n’ln

r. (14)

with a, b and n constants. This metric can be reduced to the general GJI on S3 metric ( 11.53) of ref. [ 31, by choosing r-ln r and a=b, d=t= - 1. If in addition n2 = 4, the metric admits a fourth Killing vector, given by

(15)

(6- 1 )a/au]

-[(6-l)~-“‘-(S+l)qur~‘,f.“‘]a/d~ +(s+l)r-‘.T”‘a/a,J}

(10)

The author wishes to thank Professor D.K. Callebaut for continuing interest in this work.

and the vector References

S(l+i)lf’,“e-,[(S-l)a/ar-(s+l)a/au] -i[(s+l)f-“‘+(s-l)qur-‘f’“]a/av

[ 1] P. Wils. Class. Quantum Grav. ( 1988). submitted. [ 21 E.L. Ince. Ordinary dtfferential equations (Dover. New York,

-i(a_l)r-‘.f’“a/aJ))

(11)

and its complex conjugate. coefficients then satisfy p=,/l,

K=U.

N=P,

p-p=T+E=O

426

S=7t.

a=>..

The Newman-Penrose

t=y.

(12)

1956). [3] D. Kramer, H. Stephani, M. MacCallum and E. Herlt. Exact solutions of Einstein’s equations (Cambridge tlniv. Press, Cambridge, 1980). [4] D.M. Chitre, R. Gtiven and Y. Nutku, J. Math. Phys. 16 (1975) 475. [5] N. Van den Bergh and P. Wils, J. Phys. 4 16 ( 1983) 3843. [6] 4. Barnes. J. Phys. 4 I I (1978) 1303.