P.K.Townsend - Black Holes

We can choose in such a way that all points at 1 in the original metric are at nite a ne parameter in the new metric. For this to happen we must choose s.t.

In this case `in nity' can be identi ed as those points (~r; t) for which (~r; t) = 0. These points are not part of the original spacetime but they can be added to it to yield a conformal compacti cation of the spacetime.

Example 1

Minkowski space

we bring these points to nite a ne parameter in the new metric

41

Minkowski spacetime is conformally embedded in the new spacetime with metric ds~2 with boundary at = 0.

Introducing the new time and space coordinates ; by

is an angular variable which must be identi ed modulo 2 , + 2 . If no other restriction is placed on the ranges of and , then this metric ds~2 is that of the Einstein Static Universe, of topology R (time) S3 (space).

The 2-spheres of constant 6= 0; have radius jsin j (the points = 0; are the poles of a 3-sphere). If we represent each 2-sphere of constant as a point the E.S.U. can be drawn as a cylinder.

42

.

.

....

.

But compacti ed Minkowski spacetime is conformal to the triangular region

Flatten the cylinder to get the Carter-Penrose diagram of Minkowski spacetime.

43

i

Spatial sections of the compacti ed spacetime are topologically S3 because of the addition of the point i0. Thus, they are not only compact, but also have no boundary. This is not true of the whole spacetime. Asymptotically it is possible to identify points on the boundary of compacti ed spacetime to obtain a compact manifold without boundary (the group U(2); see Question I.6). More generally, this is not possible because i are singular points that cannot be added (see Example 3: Kruskal).

44

45

Then

Using the fact that

Kruskal is an example of an asymptotically at spacetime. It approaches the metric of compacti ed Minkowski spacetime as r ! 1 (with or withoutxing t) so i0, and = can be added as before. Near r = 2M we can introduce KS-type coordinates to pass through the horizon. In this way one can deduce that the CP diagram for the Kruskal spacetime is

Note

46

(i)All r = constant hypersurfaces meet at i+ including the r = 0 hypersurface, which is singular, so i+ is a singular point. Similarly for i , so these points cannot be added.

(ii)We can adjust so that r = 0 is represented by a straight line.

In the case of a collapsing star, only that part of the CP diagram of Kruskal that is exterior to the star is relevant. The details of the interior region depend on the physics of the star. For pressure-free, spherical collapse, all parts of the star not initially at r = 0 reach the singularity at r = 0 simultaneously, so the CP diagram is

2.5Asymptopia

6f

(iii)= 0 but d = 0 on @M.

(iv)Every null geodesic in M acquires 2 endpoints on @M.

47

f

Example M = Minkowski, M = compacti ed Minkowski.

Condition (iv) excludes black hole spacetime. This motivates the following de nition:

A weakly asymptotically simple spacetime (M; g) is one for which 9 an

f open set U M that is isometric to an open neighborhood of @M,

f

where M is the `conformal compacti cation' of some asymptotically simple manifold.

(ii) M is not asymptotically simple because geodesics that enter r < 2M cannot end on =+.

Asymptotic atness

An asymptotically at spacetime is one that is both weakly asymptotically simple and is asymptotically empty in the sense that

(v) R = 0 in an open neighborhood of @M in M.

This excludes, for example, anti-de Sitter space. It also excludes spacetimes with long range electromagnetic elds that we don't wish to exclude so condition (v) requires modi cation to deal with electromagnetic elds.

Asymptotically at spacetimes have the same type of structure for = and i0 as Minkowski spacetime.

48