## Info

r* = r + mlog(r2 — 2mr + e2) + —-; arc tan I -if e2 > m2.

Defining advanced and retarded coordinates v, w by v = t+r*, w=t — r*

the metric (5.26) takes the double null form

1——+-^\dv d«> + r2 (dfl2 + sin2 6 d^2). (5.27) In the case c2 < m2, define new coordinates ?/', to" by t>" = arctan ^exp I ^jj» w" ~ arctan ^ —exp ^—'

Then the metric (5.27) takes the form

1--+ -5164,-cosec 2v" cosec 2w" &v" dw"

where r is defined implicitly by tan v" tan w" = - exp ^t^2 ) r) ^ ~ ^ ~ r-)~"'2

and a = (r+)-2 (r_)2. The maximal extension is obtained by taking (5.28) as the metric g*, and as the maximal manifold on which this metric is C2,

The Penrose diagram of the maximal extension is shown in figure 25. There are an infinite number of asymptotically flat regions, where r > r+; these are denoted by I. These are connected by intermediate regions II and III where r+ > r > r__ and > r > 0 respectively. There is still an irremovable singularity at r = 0 in each region III,

Figure 26. Penrose diagram for the maximally extended Beissner-NordstrOm solution (e* < m*). An infinite chain of asymptotically flat regions I (co > r > r+) are connected by regions II (r+ > r > r_) and III (r_ > r > 0); each region III is bounded by a timelike singularity at r = 0.

Figure 26. Penrose diagram for the maximally extended Beissner-NordstrOm solution (e* < m*). An infinite chain of asymptotically flat regions I (co > r > r+) are connected by regions II (r+ > r > r_) and III (r_ > r > 0); each region III is bounded by a timelike singularity at r = 0.

but unlike in the Schwarzschild solution, it is timelike and so can be avoided by a future-directed timelike curve from a region I which crosses r = r+. Such a curve can pass through regions II, III and II and re-emerge into another asymptotically flat region I. This raises the intriguing possibility that one might be able to travel to other universos by passing through the 'wormholos' made by chargcs. Unfortunately it seems that one would not be able to get back again to our universe to report what one had seen on the other side.

The metric (5.28) is analytic everywhere except at r = where it is degenerate but one can define different coordinates v'" and w'" by v'" — arc t,an (exp (^ZÍr,;^ .

Was this article helpful?

0 0