Maximally extended Reissner-Nordström manifold with cosmological constant

Abstract

We discuss the maximal analytic extension along an axis where O,+ = constant of the solution to Einstein's field equations with cosmological constant with a charged point-mass source.Einstein's field equations with cosmological conthe cosmological constant may well be important stant, in understanding the global properties of our actual space-time, and (c) because metric (3) provides Rub -% g a p + Agab = -K T ,~ , another example which illustrates that the global have solutions with line elements nature of solutions to nonlinear field equations Taking for Tub the energy-momentum tensor a s s ociated with the electric field surrounding a charged point m a s s , one obtains for @ ( r ) in Eq. (2) Here w i and e a r e the m a s s and charge of the central body. The solution (3) can be obtained by anintegrationprocedure illustrated in textbooksfor the Reissner-Nordstram solution [A = 0 in Eq. (3)11 o r the Schwarzschild solution with o r without cosmological constant [e = 0 in Eq. (3)].2 Solution (3) i s containedin a more general solution mentionedby Carter.3 A metric of the f o r m (2) h a s singularities where @ vanishes o r becomes infinite. However, s o m e of these singularities a r e pseudosingularities, caused by a n inappropriate coordinate system. Graves and rill^ have shown how to extend a metric of the f o r m (2) a c r o s s a pseudosingularity by means of a coordinate transformation of the Kruskal type. They illustrated the procedure f o r the Reissner-, Nordstrijm c a s e with e 2 < m 2 . The resulting maximally extended analytic manifold turned out to be a world infinitely repeated in a timelike direction with a timelike singularity a t r = 0. It is best r e presented by means of a P e n r o s e diagram a s in some m o r e recent t e~t b o o k s .~~' The method was also used by c a r t e r 7 to find the maximal analytic extension of the K e r r solution a s well a s that of the Reissner-Nordstrijm solution f o r the special c a s e of e2=wz2. Here we do the s a m e for the metric (3). This i s an interesting exercise (a) because it i s important, in trying to understand the physical significance of Eq. (1) to study the qualitative properties of all exact solutions in idealized situations, (b) because may depend drastically on s m a l l perturbations of the equations. Since the technique of analytic extension a c r o s s pseudosingularities is well understood,' we quote h e r e only the results. It is convenient and interesting in itself to consider first the c a s e when e = 0 in metric (3), i.e., the c a s e of the Schwarzschild solution with cosmological constant. It will be easy a t the end to piece together the result f o r e = 0 with the result for A = 0 (~e i s s n e r -N o r d s t r i j m ) to obtain the result f o r the general c a s e (3).F o r e = 0, O < A < l/9tn2, @(Y) has two positive roots r, > r,, and one negative root r,. The two positive roots a r e pseudosingularities. The negative unphysical root cannot be reached because of the singularity a t r = 0 and will not concern us further...

The Structure of the Extreme Schwarzschild-de Sitter Space-time

“…Therefore, there is only one null circular geodesic r = 3m (for h = 0) and one spacelike circular orbit (for |h| = 3m). Of course, the same result we get if we start from the Kruskal-type coordinates (5).…”

Reissner–Nordstrøm–de Sitter manifold: photon sphere and maximal analytic extension

“…However, few works can be found on the RNdS spacetime. Some of the early works [5,14,15] shortly discuss the construction of maximally extended RNDS spacetimes. Works on global spacetime solutions and on their constructions that include RNdS cases can also be found for example in [4,11,12].…”

Global structure of Robinson-Trautman radiative space-times with cosmological constant

“…Here r + = (2/ √ Λ) cos(α/3 + 4π/3), with cos α = −3m √ Λ, describes the black-hole horizon, and r ++ = (2/ √ Λ) cos(α/3) is the cosmological horizon -see, e.g., [11] for more details about dependence of parameters on Λ. (Analytic continuation of the Schwarzschild-de Sitter metric is discussed, for example, in [21] and [33]- [36]. )…”

“…The metric is regular for all values r > 0 and, in particular, it describes spherically symmetric Schwarzschild-de Sitter space-time with a naked singularity if f = 1 (i.e., Φ Λ = 0); its conformal diagram is seen in Fig.6. Since the expansion (38) is analogous to (30), we can take over the results (32)- (33) implying that any Robinson-Trautman space-time with 9Λm 2 > 1 approaches smoothly the corresponding Schwarzschild-de Sitter space-time as u → ∞ (û → 0 − ). It contains no horizon (contrary to the cases discussed in the previous sections) so that the metric (36)-(38) need not to be extended pastû = 0; it is already geodesically complete for u > u 0 , as indicated in Fig.7.…”