Kerr &#X2014; Newman Metric in Cosmological Background

J. Astrophys. Astr. (1982) 3, 63–67

Kerr – Newman Metric in Cosmological Background

L. Κ. Patel and Hiren Β. Trivedi Department of Mathematics, Gujarat University, Ahmedabad 380009

Received 1981 November 21; accepted 1982 February 5

Abstract. A new solution of Einstein-Maxwell field equations is pre- sented. The material content of the field described by this solution is a perfect fluid plus sourceless electromagnetic fields. The metric of the solution is explicitly written. This metric is examined as a possible representation of Kerr-Newman metric embedded in Einstein static universe. The Kerr-Newman metric in the background of Robertson- Walker universe is also briefly described.

Key words: Kerr-Newman metric—cosmology—Einstein universe— Robertson-Walker universe

1. Introduction

The Schwarzschild exterior metric and the well-known Kerr (1963) metric go over asymptotically to a flat space. Therefore these solutions can be interpreted as gravitational fields due to isolated bodies (without and with angular momentum respectively). The charged versions of these two solutions are described by the well-known Nordstrom metric and the Kerr-Newman (1965) metric respectively. These charged versions are also described under flat background. These solutions have been proved of great interest in the gravitational theory and its applications to astrophysics. The Kerr metric in the cosmological background has been discussed by Vaidya (1977). Because of the potential use of Kerr-Newman black holes in relativity, it would be worthwhile to obtain the Kerr-Newman metric in the cosmological background. The geometry of Einstein universe is described by the metric

(1)

where R is a constant. Vaidya (1977) has given a transformation from (x, y, z) to (r, α, β) which transforms the metric (1) into the form 64 L. Κ. Patel and Η. Β. Trivedi

(2) where u = t – r, a is a constant and

The Kerr-Newman metric can be expressed in the form

(3)

The constants m, e and a appearing in the metric (3) are interpreted as the mass, the charge and the angular momentum per unit mass respectively. When e = 0, the metric (3) reduces to Kerr metric in the form given by Vaidya (1974). In the next section we shall discuss the metric (3) in the background of Einstein static universe. The last section is devoted to a brief discussion of the Kerr-Newman metric in the background of Robertson-Walker universe.

2. Kerr-Newman metric in the background of Einstein universe

In this section we shall use the following field equations corresponding to the perfect fluid distribution plus source-free electromagnetic fields

(4)

(5)

(6)

(7)

The symbols occurring in the above equations have their usual meanings. We now give in a nutshell the Kerr-Newman metric in the background of Einstein universe. The detailed calculations are lengthy but straightforward. For the sake of brevity, these calculations are not given here. Kerr-Newman metric in cosmological background 65

The metric obtained by us can be expressed in the form

(8) where m, a and e are constants and M2 , µ and γ are given by

(9)

(10)

(11)

Here R is a constant.

After lengthy computations, we have verified that the metric (8) alongwith (9), (10) and (11) satisfies the field equations (4), (5), (6) and (7). The final expressions for the pressure p, the density ρ and the electromagnetic four-potential At are given by

(12)

(13)

(14) where

A = (eR/M2) sin (r/R) cos (r/R) and

(16)

The functions Μ2, µ and γ are given by Equations (9), (10) and (11) respectively. Here we have denoted the coordinates as

A–5 66 L. K. Patel and H. B. Trived

When m = e = 0, the metric (8) reduces to the metric (2) of Einstein universe. When R tends to infinity the metric (8) reduces to the metric (3). Therefore in the vicinity of the source the metric (8) reduces to Kerr-Newman metric. Thus the metric (8) describes the Kerr-Newman metric in the cosmological background of the Einstein Universe.

3. Kerr-Newman metric in the background of Robertson-Walker universe

The present section is devoted to a very brief discussion of the metric describing the field of a charged rotating source in the background of Robertson-Walker universe. The Kerr-Newman metric in the background of Robertson-Walker universe turns out to be

(17)

where Μ, µ and γ are given by Equations (9), (10) and (11) respectively. Here F is an arbitrary function of time t. The detailed calculations are lengthy and tedious and so they are not given here. If we put m = e = 0 in metric (17) we recover the Robertson-Walker metric representing the closed universe. If we put F = 0 in metric (17), we obtain the Einstein-Kerr-Newman metric (8). Thus the metric (17) represents the Kerr-Newman metric in the background of Robertson-Walker universe. There is one qualitative difference between the solution (17) and the solution discussed in the previous section. The resultant effect of the isotropic expansion of the cosmic fluid and the presence of the rotating charged Kerr source is that the cosmic fluid in the vicinity of the source exhibits anisotropy in pressure (see Vaidya 1977). At the point in the 3-space, if we choose three mutually orthogonal infinitesimal vectors θ 1, θ 2, θ 3 defined by

(18) then the pressure in θ 2 and θ 3 directions are equal(say p) and the pressure in θ 1 direc- tion is q ≠ p. This is an unpalatable feature of this solution. However in the case F = 0 we have p = q. The expressions for p and q are lengthy and therefore not reported here. Ken-Newman metric in cosmological background 67

Here it should be noted that when e = 0, the electromagnetic field disappears and our results reduce to those obtained by Vaidya (1977). We shall now study the event horizon of the Kerr-Newman black hole embedded in an expanding universe. Adopting the method described by Vaidya (1977), we get the event horizon of the Kerr-Newman black hole immersed in an expanding universe as the spheroid with r parameter given by

i. e.

(19)

This event horizon exists if the restriction

is satisfied. The substitution e = 0 in the above discussion gives us the event horizon of the Kerr black hole immersed in an expanding universe. When R → ∞ and F = 0, we recover the usual horizon

(20) of Kerr-Newman black hole. In a similar way, the modifications of other electromagnetic effects of the Kerr-Newman black hole in an expanding universe can be studied.

Acknowledgements

The authors are highly indebted to Professor P. C. Vaidya for many helpful discuss- ions and are also thankful to the referee for his constructive comments.

References

Kerr, R. P. 1963, Phys. Rev. Lett., 11, 237. Newman, E. T., Couch, E., Chinnapared, K., Exton, Α., Prakash, Α., Torrence, R. 1965, Phys. Rev. Lett., 15, 231. Vaidya, P. C. 1974, Proc. Camb. phil. Soc. math. phys. Sci., 75, 383. Vaidya, P. C. 1977, Pramana, 8, 512.