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3.2 Pressure-Free Collapse to RN

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Consider a spherical dust ball for which each particle of dust has

charge/mass ratio

; jj < 1 (3.37)

where Q is the total charge and M is the total mass. The exterior

metric is M > jQj RN. The trajectory of a particle at the surface is

the same as that of a radially infalling particle of charge/mass ratio

in the RN spacetime. This is not a geodesic because of the additional

electrostatic repulsion. From the result of Question II.4, we see that

the trajectory of a point on the surface obeys

= "2 􀀀 Ve_; (" < 1) (3.38)

where

Ve_ = 1 􀀀

􀀀

1 􀀀 "2_ 2M

􀀀

1 􀀀 2_ Q2

r2 (3.39)

.......... .

......... .

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

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

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

V r0 e_

r0 =

􀀀

1 􀀀 2

(1 􀀀 "2)

􀀀

1 􀀀 2

(1 􀀀 "2)

M (3.40)

The collapse will therefore be halted by the electrostatic repulsion.

All timelike curves that enter r < r+ must continue to r < r􀀀, so

the `bounce' will occur in region V. The dust ball then enters region

III', explodes as a white hole into region I' and then recollapses and

re-expands inde_nitely.

This is illustrated by the following CP diagram

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

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

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

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

........

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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...........

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.. .

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

=􀀀

i􀀀

black hole horizon in I' at r = r+

white hole horizon in I'

at new branch of r = r+

curvature singularity at r = 0

r = r􀀀

centre

of dust

ball at r = 0

future even horizon

at r = r+

timelike

escape path

of criminal

H+

i00

=+0

everywhere

non-sinuglar

r = 0

Notes

i) No singularity is visible from =+, in agreement with cosmic censorship.

ii) Although the dust ball never collapses to zero size and its interior

is completely non-singular, there is nevertheless a singularity

behind H+ on another branch of r = 0, in agreement with the

singularity theorems.

iii) It seems that a criminal could escape justice in universe I by

escaping on a timelike path into universe I'. Is this science _ction?

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3.2 Pressure-Free Collapse to RN

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