2.7 Black Holes vs. Naked Singularities

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The singularity at r = 0 that occurs in spherically symmetric collapse is

hidden in the sense that no signal from it can reach =+. This is not true of

the Kruskal spacetime manifold since a signal from r = 0 in the white hole

region can reach =+.

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H􀀀

=+

=􀀀

H+

This singularity is naked. Another example of a naked singularity is the

M < 0 Schwarzschild solution

ds2 = 􀀀

_

1 +

2jMj

r

_

dt2 +

_ 1

1 + 2jMj r

_dr2 + r2d2 (2.172)

This solves Einstein's equations so we have no a priori reason to exclude it.

The CP diagram is

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

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

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

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

i+

i0

i􀀀

=+

singularity

at r = 0

Neither of these examples is relevant to gravitational collapse, but consider

the CP diagram:

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.......... .

......... .

......... .

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

......... .

......... .

......... .

......... .

......... .

.......... .

......... .

......... .

......... .

......... .

......... .

......... .

......... .

......... .

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

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

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

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

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

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

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

.. .

.. ..

.

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

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

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

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

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

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

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

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

i+

=+

_ i0

r = 0

boundary of star

naked singularity

at r = 0

signal from singularity

can reach =+

Cauchy surface

At late times the spacetime is M < 0 Schwarzschild but at earlier times it

is non-singular. Under these circumstances it can be shown that M _ 0 for

physically reasonable matter (the `positive energy' theorem) so the possibility

illustrated by the above CP diagram (formation of a naked singularity

in spherically-symmetric collapse) cannot occur. There remains the possibility

that naked singularities could form in non-spherical collapse. If this

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were to happen the future would eventually cease to be predictable from

data given on an initial spacelike hypersurface (_ in CP diagram above).

There is considerable evidence that this possibility cannot be realized for

physically reasonable matter, which led Penrose to suggest the:

Cosmic Censorship Conjecture `Naked singularities cannot form from

gravitational collapse in an asymptotically at spacetime that is non-singular

on some initial spacelike hypersurface (Cauchy surface).'

Notes

(i) Certain types of `trivial' naked singularities must be excluded.

(ii) Initial, cosmological, singularities are excluded.

(iii) There is no proof. This is the major unsolved problem in classical

G.R.

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