2.5 Asymptopia

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A spacetime (M; g) is asymptotically simple if 9 a manifold (fM; ~g) with

boundary @fM = M and a continuous embedding f(M) : M ! fM s.t.

(i) f(M) = fM 􀀀 @fM

(ii) 9 a smooth function _ on fM with _ > 0 on f(M) and ~g = _2f(g).

(iii) _ = 0 but d_ 6= 0 on @fM.

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

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Example M = Minkowski, fM = 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

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

where fM is the `conformal compacti_cation' of some asymptotically

simple manifold.

Example M = Kruskal, fM its conformal `compacti_cation'.

Note

(i) fM is not actually compact because @fM excludes i_.

(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.

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.. .. .. .. ..

.. .. .. .. ..

................................................................................................................................................................................................................................ ..... ...... ..... ...... ..... ...... ..... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... .......... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... ..... ...... .

i0

=+

=􀀀

In particular they admit vectors that are asymptotic to the Killing vectors

of Minkowski spacetime near i0, which allows a de_nition of total mass, momentum

and angular momentum on spacelike hypersurfaces. The asymptotic

symmetries on =_ are much more complicated (the `BMS' group, which will

not be discussed in this course).

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