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3.4 Isotropic Coordinates for RN

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Let

r = _ +M +

M2 􀀀 Q2

(3.42)

Then

ds2 = 􀀀

_dt2

r2(_)

􀀀

d_2 + _2d2_

| {z }

at space metric

(3.43)

_ =

_ 􀀀

􀀀

M2 􀀀 Q2

(3.44)

is RN metric in isotropic coordinates (t; _; _; _). As in Q = 0 case,

there are two values _ for every value of r > r+, but _ is complex for

r < r+.

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

..........

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

pM2􀀀Q2

This new metric covers two isometric regions (I&IV) exchanged by the

geometry.

_ !

M2 􀀀 Q2

(3.45)

The _xed points set at _ =

M2 􀀀 Q2=2 (i.e. r = r+) is a minimal

2-sphere of an ER bridge as in the Q = 0 case.

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

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

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

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

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

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

_ = 1

_ =

M2 􀀀 Q2

IV I

_ = 0

minimal 2-sphere on

t = constant hypersurface

The distance to the horizon at r = r+ along a curve of constant t; _; _

from r = R is

s =

Z R

dr q􀀀

1 􀀀 r+

_ 􀀀

1 􀀀 r􀀀

_ (3.46)

! 1 as r+ 􀀀 r􀀀 ! 0; i.e. as M 􀀀 jQj ! 0 (3.47)

so the ER bridge separating regions I & IV becomes in_nitely long in

the limit as jQj ! M. In this limit, the spatial sections look like:

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

..................................... 1 as jQj ! M

iii) M = jQj `Extreme' RN (r_ = M)

ds2 = 􀀀

1 􀀀

dt2 +

dr2

􀀀

1 􀀀 M

_2 + r2d2 (3.48)

This is singular at r = M so de_ne the Regge-Wheeler coordinate

r_ = r + 2M ln

____

r 􀀀M

____

􀀀

r 􀀀M ) dr_ =

1 􀀀 M

(3.49)

and introduce ingoing EF coordinates as before. Then

ds2 = 􀀀

1 􀀀

dv2 + 2dv dr + r2d2 (3.50)

This is non-singular on the null hypersurface r = M.

Proposition r = M is a degenerate (i.e. surface gravity _ = 0)

Killing horizon of the Killing vector _eld k = @=@v.

Proof From the previous calculation l = f@=@v so r = M is a Killing

horizon of k, and k _Dk = 0 when r+ = r􀀀 = M.

Since the orbits of k on r = M are a_nely parameterized they must

go to in_nite a_ne parameter in both directions ) internal 1. This

is the same internal 1 that we _nd down the in_nite ER bridge.

Note that k is null on r = 2M, but timelike everywhere else, so region

II has disappeared and region I now leads directly to region V. The

CP diagram is

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀 􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀􀀀

i􀀀

=􀀀

singularity

at r = 0

internal 1

future event horizon

and Cauchy horizon

orbits of time-translation

Killing vector _eld k

H+

in_nite ER

`bridge'

3.4.1 Nature of Internal 1 in Extreme RN

The asymptotic metric as r ! 1 is Minkowski. To determine the

asymptotic metric as r ! M we introduce the new coordinate _ by

r = M(1 + _) and keep only the leading terms in _, to get

F _ d_ ^ dt (3.51)

ds2 _

􀀀

􀀀_2dt2 +M2_􀀀2d_2_

| {z }

adS2

+ M2d2 | {z }

2-sphere

of radius M

(3.52)

This is the Robinson-Bertotti metric. It is a kind of `Kaluza-Klein'

vacuum in which two directions are compacti_ed and the `e_ective'

spacetime is the two-dimensional `anti-de Sitter' (adS2) spacetime of

constant negative curvature. (See Q.II.7).

3.4.2 Multi Black Hole Solutions

The extreme RN in isotropic coordinates is

ds2 = V 􀀀2dt2 + V 2 􀀀

d_2 + _2d2_

(3.53)

where

V = 1 +

(3.54)

This is a special case of the multi black hole solution

ds2 = V 􀀀2dt2 + V 2d~x _ d~x (3.55)

where d~x _ d~x is the Euclidean 3-metric and V is any solution of r2V = 0.

In particular,

V = 1 +

i=1

Mi __ _~x 􀀀~_xi

___

(3.56)

yields the metric for N extreme black holes of masses Mi at positions _xi.

Note that the `points' _xi are actually minimal 2-spheres. There are no

_-function singularities at x = _xi because the lines of force continue inde_-

nitely into the asymptotically RB regions (`charge without charge').

Note that a static multi black hole solution is possible only when there is

an exact balance between the gravitational attraction and the electrostatic

repulsion. This occurs only for M = jQj.

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3.4 Isotropic Coordinates for RN

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