P.K.Townsend - Black Holes

Since k = @=@t in static coordinates we have k2 ! 1 as r ! 1. So we identify as the surface gravities of N .

Each of the Killing horizons N will have a bifurcation 2-sphere in the neighborhood of which we can introduce the KS-type coordinates

This metric covers four regions of the maximal analytic extension of RN,

.......

These coordinates do not cover r r because of the coordinate singularity at r = r (and U+V + is complex for r < r ), but r =

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r and a similar four regions are covered by the (U V ) KS-type coordinates to this case (Exercise).

This metric covers four regions around U = V = 0.

.

r = constant r < r < r+

Region II is the same as the region II covered by the (U+; V +) coordinates. The other regions are new. Regions V and VI contain the curvature singularity at r = 0, which is timelike because the normal to r = constant is spacelike for > 0, e.g. in r < r .

We know that region II of the diagram is connected to an exterior spacetime in the past (regions I, III, and IV), by time-reversal invariance, region III' must be connected to another exterior region (isometric regions I', II', and IV').

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...... r = constant r < r < r+

Regions I' and IV' are new asymptotically at `exterior' spacetimes. Continuing in this manner we can nd an in nite sequence of them.

Internal In nities

Consider a path of constant r; ; in any region for which < 0, e.g. region II. In ingoing EF coordinates

Since ds2 > 0 the path is spacelike. The distance along it from v = 0 to v = 1 (i.e. to V + = 0 or V = 0) is

So there is an `internal' spatial in nity behind the r = r+ horizon. (Note that one can still reach V = 0 in nite proper time on a timelike path, so the null hypersurfaces V = 0 are part of the spacetime).

If all points at 1, external and internal, are brought to nite a ne parameter by a conformal transformation, one nds the following CP diagram, which can be in nitely extended in both directions:

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