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3.3 Cauchy Horizons

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A particle on an ingoing radial geodesic of RN (e.g. surface of collapsing

star) will `hit' the singularity at r = 0, but once in region V

or VI it can accelerate away from the singularity then enter the new

exterior region via the white hole region III'. However, there is no way

to ensure in advance of entering the black hole (e.g. by programming

of rockets) that it will do so because to get to region I' it must cross

a Cauchy horizon, a concept that will now be elaborated.

De_nition A partial Cauchy surface , _, for a spacetime M is a

hypersurface which no causal curve intersects more than once.

De_nition A causal curve is past-inextendable if it has no past endpoint

in M.

De_nition The future domain of dependence, D+(_) of _, is the

set of points p 2 M for which every past-inextendable causal curve

through p intersects _.

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

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

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. . .

past-inextendable

causal curve

non-causal

curve

past light cone

of p

causal curve

but not

past-inextendable

The signi_cance of D+(_) is that the behavior of solutions of hyperbolic

PDE's outside D+(_) is not determined by initial data on _.

The past domain of dependence, D􀀀(_) of _, is de_ned similarly and

_ is said to be a Cauchy surface for M if

D+(_) [ D􀀀(_) = M (3.41)

IfM has a Cauchy surface it is said to be globally hyperbolic. Examples

of globally hyperbolic spacetimes are

1) Spherical, pressure-free collapse (Schwarzschild)

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..........

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

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

...

=􀀀

H+

_1 and _2 are both Cauchy surfaces.

2) Kruskal

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

_1 and _2 are both Cauchy surfaces.

If M is not globally hyperbolic then D+(_) or D􀀀(_) will have a

boundary in M, called the future or past Cauchy horizon.

Examples

(i) Gravitationally-collapsed charged dust ball.

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

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

=􀀀

H+

i􀀀

future Cauchy

horizon at r = r􀀀

singularity

(ii) Maximal analytic extension of RN

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􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 ��􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀�� 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀

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

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

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

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

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i􀀀

D+(_)

future Cauchy

horizon at r = r􀀀

past Cauchy

horizon at r = r􀀀

H+

=􀀀 H􀀀

In example (i) a strange feature of the future Cauchy horizon is that

the entire in_nite history of the external spacetime in region I is in

its causal past, i.e. visible, so signals from I must undergo an in_-

nite blueshift as they approach the Cauchy horizon. For this reason,

the Cauchy horizon usually becomes singular when subjected to any

perturbation, no matter how small. For any physically realistic collapse,

the Cauchy horizon is a singular null hypersurface for which

new physics beyond GR is needed.

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3.3 Cauchy Horizons

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