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2.4 Carter-Penrose Diagrams

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2.4.1 Conformal Compacti_cation

A black hole is a \region of spacetime from which no signal can escape to

in_nity" (Penrose). This is unsatisfactory because `in_nity' is not part of

the spacetime. However the `de_nition' concerns the causal structure of

spacetime which is unchanged by conformal compacti_cation

ds2 ! d~s2 = _2(~r; t)ds2; _ 6= 0 (2.150)

We can choose _ in such a way that all points at 1 in the original metric

are at _nite a_ne parameter in the new metric. For this to happen we must

choose _ s.t.

_(~r; t) ! 0 as j~rj ! 1 and/or jtj ! 1 (2.151)

In this case `in_nity' can be identi_ed as those points (~r; t) for which _(~r; t) =

0. These points are not part of the original spacetime but they can be added

to it to yield a conformal compacti_cation of the spacetime.

Example 1

Minkowski space

ds2 = 􀀀dt2 + dr2 + r2d2 (2.152)

Let

u = t 􀀀 r

v = t + r

! ds2 = 􀀀du dv +

(u 􀀀 v)2

d2 (2.153)

Now set

u = tan ~U 􀀀_=2 < ~U < _=2

v = tan ~ V 􀀀_=2 < ~ V < _=2

with ~ V _ ~U

since r _ 0

(2.154)

In these coordinates,

ds2 =

2 cos ~U cos ~ V

􀀀2 h

􀀀4d~U d ~ V + sin2

~ V 􀀀 ~U

(2.155)

To approach 1 in this metric we must take

___

! _=2 or

___

~ V

___

! _=2, so by

choosing

_ = 2 cos ~U cos ~ V (2.156)

we bring these points to _nite a_ne parameter in the new metric

d~s2 = _ds2 = 􀀀4d~U d ~ V + sin2

~ V 􀀀 ~U

d2 (2.157)

We can now add the `points at in_nity'. Taking the restriction ~ V _ ~U into

account, these are

= 􀀀_=2

~ V = _=2

u ! 􀀀1

v ! 1

r ! 1

t _nite

spatial 1, i0

= __=2

~ V = __=2

u ! _1

v ! _1

t ! _1

r _nite

past and future

temporal 1, i_

= 􀀀_=2

j ~ V j 6= _=2

u ! 􀀀1

v _nite

r ! 1

t ! 􀀀1

r + t _nite

;

past null 1

=􀀀

j~U j 6= _=2

~ V = _=2

u _nite

v ! 1

r ! 1

t ! 1

r 􀀀 t _nite

9= ;

future null 1

Minkowski spacetime is conformally embedded in the new spacetime with

metric d~s2 with boundary at _ = 0.

Introducing the new time and space coordinates _; _ by

_ = ~ V + ~U ; _ = ~ V 􀀀 ~U (2.158)

we have

d~s2 = _ds2 = 􀀀d_2 + d_2 + sin2 _d2

_ = cos _ + cos_

(2.159)

_ is an angular variable which must be identi_ed modulo 2_, _ _ _ + 2_.

If no other restriction is placed on the ranges of _ and _, then this metric

d~s2 is that of the Einstein Static Universe, of topology R (time) _ S3 (space).

The 2-spheres of constant _ 6= 0; _ have radius jsin_j (the points _ = 0; _

are the poles of a 3-sphere). If we represent each 2-sphere of constant _ as

a point the E.S.U. can be drawn as a cylinder.

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

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

0 _

But compacti_ed Minkowski spacetime is conformal to the triangular region

􀀀 _ _ _ _ _; 0 _ _ _ _ (2.160)

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

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

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

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

.. .. .. .. ..

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

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

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

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

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

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

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

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

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

0 _

􀀀_ _ 􀀀 _ = _ ,

_ + _ = _ ,

~ V = _=2; =+

= 􀀀_=2; =􀀀

Flatten the cylinder to get the Carter-Penrose diagram of Minkowski spacetime.

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

.......... .

......... .

.......... .

......... .

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

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

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

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

􀀀􀀀 􀀀􀀀 􀀀􀀀 􀀀􀀀 􀀀􀀀 􀀀􀀀􀀀 􀀀 􀀀􀀀􀀀 􀀀􀀀 􀀀 􀀀

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

􀀀􀀀 􀀀􀀀 􀀀 􀀀 􀀀􀀀 􀀀􀀀􀀀 􀀀 􀀀􀀀􀀀 􀀀􀀀 􀀀􀀀 􀀀􀀀

􀀀 􀀀􀀀􀀀 􀀀􀀀 􀀀 􀀀 􀀀􀀀 􀀀􀀀 􀀀􀀀 􀀀 ..........

..........

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

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

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

..... .....

.... ......

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

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

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

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

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

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

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

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

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

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

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

~ V

=􀀀

i􀀀

r = 0

timelike geodesic

radial null

geodesic

t = constant

hypersurface

Each point represents a 2-sphere, except points on r = 0 and i0; i_. Light

rays travel at 45

from =􀀀 through r = 0 and then out to =+. [=_ are null

hypersurfaces].

Spatial sections of the compacti_ed spacetime are topologically S3 because

of the addition of the point i0. Thus, they are not only compact, but

also have no boundary. This is not true of the whole spacetime. Asymptotically

it is possible to identify points on the boundary of compacti_ed

spacetime to obtain a compact manifold without boundary (the group U(2);

see Question I.6). More generally, this is not possible because i_ are singular

points that cannot be added (see Example 3: Kruskal).

Example 2: Rindler Spacetime

ds2 = 􀀀dU0 dV 0 (2.161)

Let

U0 = tan ~U

V 0 = tan ~ V

􀀀_=2 < ~U < _=2

􀀀_=2 < ~ V < _=2

(2.162)

Then

ds2 = 􀀀

cos ~U cos ~ V

􀀀2

d~U d ~ V (2.163)

= _􀀀2d~s2

_ = cos ~U cos ~V

(2.164)

i.e. conformally compacti_ed spacetime with metric d~s2 = 􀀀d~U d ~ V is same

as before but with the above _nite ranges for coordinates ~U ; ~V .

The points at in_nity are those for which _ = 0,

___

= _=2,

___

~ V

___

= _=2.

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

.......... .

......... .

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

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

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

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

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

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

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

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

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

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

i􀀀

(i0)R

=􀀀

X = 0

x = 0

orbit of k = @=@t

orbit of

K = @=@T

(Rindler time coordinate)

Region covered by

Rindler coordinates

(i0)L

Similar to 4-dim Minkowski, but i0 is now two points.

Example 3: Kruskal Spacetime

ds2 = 􀀀

1 􀀀

du dv + r2d2 in region I (2.165)

Let _

u = tan ~U

v = tan ~ V

􀀀_=2 < ~U < _=2

􀀀_=2 < ~ V < _=2

(2.166)

Then

ds2 =

2 cos ~U cos ~ V

􀀀2

􀀀4

1 􀀀

d~U d ~ V + r2 cos2 ~U cos2 ~ V d2

(2.167)

Using the fact that

r_ =

(v 􀀀 u) =

sin

~ V 􀀀 ~U

2 cos ~U cos ~ V

(2.168)

we have

d~s2 = _2ds2 = 􀀀4

1 􀀀

d~U d ~ V +

_ r

sin2

~ V �� ~U

d2(2.169)

Kruskal is an example of an asymptotically at spacetime. It approaches

the metric of compacti_ed Minkowski spacetime as r ! 1 (with or without

_xing t) so i0, and =_ can be added as before. Near r = 2M we can

introduce KS-type coordinates to pass through the horizon. In this way one

can deduce that the CP diagram for the Kruskal spacetime is

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

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

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

. .

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

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

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

. . . . . ....

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

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

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

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

i􀀀

=􀀀

III

singularity at

r = 0

r = 2M

r constant > 2M

r constant < 2M

Note

(i) All r = constant hypersurfaces meet at i+ including the r = 0 hypersurface,

which is singular, so i+ is a singular point. Similarly for i􀀀,

so these points cannot be added.

(ii) We can adjust _ so that r = 0 is represented by a straight line.

In the case of a collapsing star, only that part of the CP diagram of

Kruskal that is exterior to the star is relevant. The details of the interior region

depend on the physics of the star. For pressure-free, spherical collapse,

all parts of the star not initially at r = 0 reach the singularity at r = 0

simultaneously, so the CP diagram is

.......... .

......... .

..........

......... .

..........

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

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􀀀 􀀀 ........ ................ ............... ............... ............... ................ .............................................................................................................................................................................................................................................................................................................................

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

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

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

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

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

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

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

.................. . . . . . .. . . . . . . r = 0

i􀀀

r = 2M =+

=􀀀

singularity at r = 0

surface of star

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