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2.6 The Event Horizon

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Assume spacetime M is weakly asymptotically at. De_ne

J􀀀(U)

to be the causal past of a set of points U _ M and

J􀀀(U)

to be the topological closure of J􀀀, i.e. including limit points. De_ne the

boundary of J􀀀 to be

J_􀀀(U) = J􀀀(U) 􀀀 J􀀀(U) (2.170)

The future event horizon of M is

H+ = J_􀀀

􀀀

=+_

(2.171)

i.e. the boundary of the closure of the causal past of =+.

Example Spacetime of a spherically-symmetric collapsing star

.......... .

......... .

..........

......... .

..........

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

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

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

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

􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 �� 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀..........

..........

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

..........

........

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

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

i􀀀

=􀀀

H+

continuation of radial

null geodesic from i+

past endpoint of

null geodesic

generator of H+

Properties of the Future Event Horizon, H+

(i) i0 and =􀀀 are contained in J􀀀 (=+), so they are not part of H+.

(ii) H+ is a null hypersurface.

(iii) No two points of H+ are timelike separated. For nearby points this

follows from (ii) but is also true globally. Suppose that _ and _ were

two such points with _ 2 J􀀀(_). The timelike curve between them

could then be deformed to a nearby timelike curve between _0 and _0

with _0 2 J􀀀 (=+) but _0 62 J􀀀 (=+)

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

_ _

timelike curve?

But _0 2 J􀀀(_) 2 J􀀀 (=+), so we have a contradiction. The timelike

curve between _ and _ cannot exist.

(iv) The null geodesic generators of H+ may have past endpoints in the

sense that the continuation of the geodesic further into the past is no

longer in H+, e.g. at r = 0 for a spherically symmetric star, as shown

in diagram above.

(v) If a generator of H+ had a future endpoint, the future continuation of

the null geodesic beyond a certain point would leave H+. This cannot

happen.

Theorem (Penrose) The generators of H+ have no future endpoints

Proof Consider the causal past J􀀀(S) of some set S.

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

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

J􀀀(S)

Nany null

hypersurface

J_􀀀(S)

boundary of

causal past of S

Consider a point p 2 J_􀀀(S), p 62 S; S. Endpoints of the null geodesic in

J_􀀀(S) through p. Consider also an in_nite sequence of timelike curves fig

from pi 2 neighborhood of p and 2 J􀀀(S) to S s.t. p is the limit point of

fpig on J_􀀀(S).

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

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

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

_ p

The points fqig must have a limit point q on J_􀀀(S). Being the limit of

timelike curves, the curve from p to q cannot be spacelike, but can be null

(lightlike). It cannot be timelike either from property (iii) above, so it is a

segment of the null geodesic generator of N through p. The argument can

now be repeated with p replaced by q to _nd another segment from q to a

point, r 2 N, but further in the future. It must be a segment of the same

generator because otherwise there exists a deformation to a timelike curve

in N separating p and r.

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

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

..........

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

..........

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

......

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

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

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

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

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N

null

timelike

Choosing S = =+, then gives Penrose's Theorem.

Properties (iv) and (v) show that null geodesics may enter H+ but cannot

leave it.

This result may appear inconsistent with time-reversibility, but is not.

The time-reverse statement is that null geodesics may leave but cannot enter

the past event horizon, H􀀀. H􀀀 is de_ned as for H+ with J􀀀 (=+) replaced

by J+ (=􀀀), i.e. the causal future of =􀀀. The time-symmetric Kruskal

spacetime has both a future and a past event horizon.

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

H􀀀

=􀀀

H+

The location of the event horizon H+ generally requires knowledge of the

complete spacetime. Its location cannot be determined by observations over

a _nite time interval.

However if we wait until the black hole settles down to a stationary

spacetime we can invoke:

Theorem (Hawking) The event horizon of a stationary asymptotically at spacetime is a

Killing horizon (but not necessarily of @=@t).

This theorem is the essential input needed in the proof of the uniqueness

theorems for stationary black holes, to be considered later.

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2.6 The Event Horizon

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