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2.3 Spherically-Symmetric Pressure Free Collapse

Back

While it is impossible to say with complete con_dence that a real star of mass

M _ 3M_ will collapse to a BH, it is easy to invent idealized, but physically

possible, stars that de_nitely do collapse to black holes. One such `star' is

a spherically-symmetric ball of `dust' (i.e. zero pressure uid). Birkho_'s

theorem implies that the metric outside the star is the Schwarzschild metric.

Choose units for which

G = 1; c = 1: (2.31)

Then

ds2 = 􀀀

1 􀀀

dt2 +

1 􀀀

􀀀1

dr2 + r2d2 (2.32)

where

d2 = d_2 + sin2 _d'2 (metric on a unit 2-sphere) (2.33)

This is valid outside the star but also, by continuity of the metric, at the

surface. If r = R(t) on the surface we have

ds2 = 􀀀

1 􀀀

􀀀

1 􀀀

􀀀1

dt2+R2d2;

(2.34)

On the surface zero pressure and spherical symmetry implies that a point on

the surface follows a radial timelike geodesic, so d2 = 0 and ds2 = 􀀀d_2,

1 =

1 􀀀

􀀀

1 􀀀

􀀀1

_R2

(2.35)

But also, since @=@t is a Killing vector we have conservation of energy:

" = 􀀀g00

1 􀀀

(energy/unit mass) (2.36)

" is constant on the geodesics. Using this in (2.35) gives

1 =

1 􀀀

􀀀

1 􀀀

􀀀1

1 􀀀

􀀀2

"2 (2.37)

2 =

1 􀀀

_2 _

R 􀀀 1 + "2

(2.38)

(" < 1 for gravitationally bound particles).

Rmax

1 􀀀 "2

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

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

R _ _ 2M

= 0 at R = Rmax so we consider collapse to begin with zero velocity at

this radius. R then decreases and approaches R = 2M asymptotically as

t ! 1. So an observer `sees' the star contract at most to R = 2M but no

further.

However from the point of view of an observer on the surface of the star,

the relevant time variable is proper time along a radial geodesic, so use

􀀀1 d

1 􀀀

(2.39)

to rewrite (2.38) as

R 􀀀 1 + "2

= (1 􀀀 "2)

Rmax

R 􀀀 1

(2.40)

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

.......... .

......... .

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

2M Rmax 0

Surface of the star falls from R = Rmax through R = 2M in _nite proper

time. In fact, it falls to R = 0 in proper time

_ =

(1 􀀀 ")3=2

(Exercise) (2.41)

Nothing special happens at R = 2M which suggests that we investigate the

spacetime near R = 2M in coordinates adapted to infalling observers. It is

convenient to choose massless particles.

On radial null geodesics in Schwarzschild spacetime

dt2 =

􀀀

1 􀀀 2Mr

_2 dr2 _ (dr_)2 (2.42)

where

r_ = r + 2M ln

____

r 􀀀 2M

____

(2.43)

is the Regge-Wheeler radial coordinate. As r ranges from 2M to 1, r_

ranges from 􀀀1 to 1. Thus

d(t _ r_) = 0 on radial null geodesics (2.44)

De_ne the ingoing radial null coordinate v by

v = t + r_; 􀀀1 < v < 1 (2.45)

and rewrite the Schwarzschild metric in ingoing Eddington-Finkelstein coordinates

(v; r; _; _).

ds2 =

1 􀀀

􀀀dt2 + dr_2

+ r2d2 (2.46)

= 􀀀

1 􀀀

dv2 + 2dr dv + r2d2 (2.47)

This metric is initially de_ned for r > 2M since the relation v = t + r_(r)

between v and r is only de_ned for r > 2M, but it can now be analytically

continued to all r > 0. Because of the dr dv cross-term the metric in EF

coordinates is non-singular at r = 2M, so the singularity in Schwarzschild

coordinates was really a coordinate singularity. There is nothing at r = 2M

to prevent the star collapsing through r = 2M. This is illustrated by a

Finkelstein diagram, which is a plot of t_ = v 􀀀 r against r:

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

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

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t_ = v 􀀀 r

collapsing

star lines of constant v

increasing v

radial outgoing null

geodesic at r = 2M

surface of the star

light cone

singularity

r = 2M

r = 0

The light cones distort as r ! 2M from r > 2M, so that no future-directed

timelike or null worldline can reach r > 2M from r _ 2M.

Proof When r _ 2M,

2dr dv = 􀀀

􀀀ds2 +

r 􀀀 1

dv2 + r2d2

(2.48)

_ 0 when ds2 _ 0 (2.49)

for all timelike or null worldlines dr dv _ 0. dv > 0 for future-directed

worldlines, so dr _ 0 with equality when r = 2M, d = 0 (i.e. ingoing

radial null geodesics at r = 2M).

2.3.1 Black Holes and White Holes

No signal from the star's surface can escape to in_nity once the surface

has passed through r = 2M. The star has collapsed to a black hole. For

the external observer, the surface never actually reaches r = 2M, but as

r ! 2M the redshift of light leaving the surface increases exponentially fast

and the star e_ectively disappears from view within a time _ MG=c3. The

late time appearance is dominated by photons escaping from the unstable

photon orbit at r = 3M.

The hypersurface r = 2M acts like a one-way membrane. This may seem

paradoxical in view of the time-reversibility of Einstein's equations. De_ne

the outgoing radial null coordinate u by

u = t 􀀀 r_; 􀀀1 < u < 1 (2.50)

and rewrite the Schwarzschild metric in outgoing Eddington-Finkelstein coordinates

(u; r; _; _).

ds2 = 􀀀

1 􀀀

du2 􀀀 2dr du + r2d2 (2.51)

This metric is initially de_ned only for r > 2M but it can be analytically

continued to all r > 0. However the r < 2M region in outgoing EF coordinates

is not the same as the r < 2M region in ingoing EF coordinates. To

see this, note that for r _ 2M

2dr du = 􀀀ds2 +

r 􀀀 1

du2 + r2d2 (2.52)

_ 0 when ds2 _ 0 (2.53)

i.e. dr du _ 0 on timelike or null worldlines. But du > 0 for future-directed

worldlines so dr _ 0, with equality when r = 2M, d = 0, and ds2 = 0. In

this case, a star with a surface at r < 2M must expand and explode through

r = 2M, as illustrated in the following Finkelstein diagram.

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increasing u

lines of constant u

r = 2M

r = 0

surface of star

u + r

singularity

This is a white hole, the time reverse of a black hole. Both black and white

holes are allowed by G.R. because of the time reversibility of Einstein's

equations, but white holes require very special initial conditions near the

singularity, whereas black holes do not, so only black holes can occur in

practice (cf. irreversibility in thermodynamics).

2.3.2 Kruskal-Szekeres Coordinates

The exterior region r > 2M is covered by both ingoing and outgoing

Eddington-Finkelstein coordinates, and we may write the Schwarzschild

metric in terms of (u; v; _; _)

ds2 = 􀀀

1 􀀀

du dv + r2d2 (2.54)

We now introduce the new coordinates (U; V ) de_ned (for r > 2M) by

U = 􀀀e􀀀u=4M; V = ev=4M (2.55)

in terms of which the metric is now

ds2 = 􀀀32M3

e􀀀r=2MdU dV + r2d2 (2.56)

where r(U; V ) is given implicitly by UV = 􀀀er_=2M or

UV = 􀀀

r 􀀀 2M

er=2M (2.57)

We now have the Schwarzschild metric in KS coordinates (U; V; _; _). Initially

the metric is de_ned for U < 0 and V > 0 but it can be extended by

analytic continuation to U > 0 and V < 0. Note that r = 2M corresponds

to UV = 0, i.e. either U = 0 or V = 0. The singularity at r = 0 corresponds

to UV = 1.

It is convenient to plot lines of constant U and V (outgoing or ingoing

radial null geodesics) at 45

, so the spacetime diagram now looks like

... .

.. .. ..

.. .. .. ..

.. .. .. .. ..

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.. .. ..

... .

.. ... .

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

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

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

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

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

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

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

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

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

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

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

r > 2M

singularity

r = 0

singularity

r = 0

I U < 0

V > 0

r = 2M

r < 2M

III

There are four regions of Kruskal spacetime, depending on the signs of U and

V . Regions I and II are also covered by the ingoing Eddington-Finkelstein

coordinates. These are the only regions relevant to gravitational collapse

because the other regions are then replaced by the star's interior, e.g. for

collapse of homogeneous ball of pressure-free uid:

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

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

..........

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

..........

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

..........

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

..........

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

..........

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

..........

...

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

..........

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

..........

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

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

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

... .

.. .. ..

..........

.. .. ..

........

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

......

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

.. .. ..

......

.. .. .. .. ..

......

.. .. .. .. ..

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

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

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

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

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

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

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

II I

r = 2M

r = 0 surface of star

singularity

at r = 0

Similarly, regions I and III are those relevant to a white hole.

Singularities and Geodesic Completeness

A singularity of the metric is a point at which the determinant of either it or

its inverse vanishes. However, a singularity of the metric may be simply due

to a failure of the coordinate system. A simple two-dimensional example is

the origin in plane polar coordiates, and we have seen that the singularity

of the Schwarzschild metric at the Schwarzschild radius is of this type. Such

singularities are removable. If no coordinate system exists for which the

singularity is removable then it is irremovable, i.e. a genuine singularity of

the spacetime. Any singularity for which some scalar constructed from the

curvature tensor blows up as it is approached is irremovable. Such singularities

are called `curvature singularities'. The singularity at r = 0 in the

Schwarzschild metric is an example. Not all irremovable singularities are

`curvature singularities', however. Consider the singularity at the tip of a

cone formed by rolling up a sheet of paper. All curvature invariants remain

_nite as the singularity is approached; in fact, in this two-dimensional example

the curvature tensor is everywhere zero. If we could assign a curvature

to the singular point at the tip of the cone it would have to be in_nite but,

strictly speaking, we cannot include this point as part of the manifold since

there is no coordinate chart that covers it.

We might try to make a virtue of this necessity: by excising the regions

containing irremovable singularities we apparently no longer have to worry

about them. However, this just leaves us with the essentially equivalent

problem of what to do with curves that reach the boundary of the excised

region. There is no problem if this boundary is at in_nity, i.e. at in_nite

a_ne parameter along all curves that reach it from some speci_ed point in

the interior, but otherwise the inability to continue all curves to all values of

their a_ne parameters may be taken as the de_ning feature of a `spacetime

singularity'. Note that the concept of a_ne parameter is not restricted to

geodesics, e.g. the a_ne parameter on a timelike curves is the proper time

on the curve regardless of whether the curve is a geodesic. This is just as

well, since there is no good physical reason why we should consider only

geodesics. Nevertheless, it is virtually always true that the existence of a

singularity as just de_ned can be detected by the incompleteness of some

geodesic, i.e. there is some geodesic that cannot be continued to all values

of its a_ne parameter. For this reason, and because it is simpler, we shall

follow the common practice of de_ning a spacetime singularity in terms of

`geodesic incompleteness'. Thus, a spacetime is non-singular if and only if

all geodesics can be extended to all values of their a_ne parameters, changing

coordinates if necessary.

In the case of the Schwarzschild vacuum solution, a particle on an ingoing

radial geodesics will reach the coordinate singularity at r = 2M at

_nite a_ne parameter but, as we have seen, this geodesic can be continued

into region II by an appropriate change of coordinates. Its continuation

will then approach the curvature singularity at r = 0, coming arbitrarily

close for _nite a_ne parameter. The excision of any region containing

r = 0 will therefore lead to a incompleteness of the geodesic. The vacuum

Schwarzschild solution is therefore singular. The singularity theorems of

Penrose and Hawking show that geodesic incompleteness is a generic feature

of gravitational collapse, and not just a special feature of spherically

symmetric collapse.

Maximal Analytic Extensions

Whenever we encounter a singularity at _nite a_ne parameter along some

geodesic (timelike, null, or spacelike) our _rst task is to identify it as removable

or irremovable. In the former case we can continue through it by

a change of coordinates. By considering all geodesics we can construct in

this way the maximal analytic extension of a given spacetime in which any

geodesic that does not terminate on an irremovable singularity can be extended

to arbitrary values of its a_ne parameter. The Kruskal manifold is

the maximal analytic extension of the Schwarzschild solution, so no more

regions can be found by analytic continuation.

2.3.3 Eternal Black Holes

A black hole formed by gravitational collapse is not time-symmetric because

it will continue to exist into the inde_nite future but did not always exist in

the past, and vice-versa for white holes. However, one can imagine a timesymmetric

eternal black hole that has always existed (it could equally well

be called an eternal white hole, but isn't). In this case there is no matter

covering up part of the Kruskal spacetime and all four regions are relevant.

In region I

= e􀀀t=2M (2.58)

so hypersurfaces of constant Schwarzschild time t are straight lines through

the origin in the Kruskal spacetime.

... .

.. .. ..

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. ..

... .

.. .. ..

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. ..

........ ..

...... ....

..........

...........

.........

..........

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

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

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

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

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

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

U V

t= constant

These hypersurfaces have a part in region I and a part in region IV. Note

that (U; V ) ! (􀀀U;􀀀V ) is an isometry of the metric so that region IV is

isometric to region I.

To understand the geometry of these t = constant hypersurfaces it

is convenient to rewrite the Schwarzschild metric in isotropic coordinates

(t; _; _; _), where _ is the new radial coordinate

r =

1 +

_ (2.59)

Then (Exercise)

ds2 = 􀀀

1 􀀀 M

1 + M

dt2 +

1 +

_4 _

d_2 + _2d2_

| {z }

at 3-space metric

(2.60)

In isotropic coordinates, the t = constant hypersurfaces are conformally at,

but to each value of r there corresponds two values of _

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

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

..........

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

..........

.........

M=2 _

The two values of _ are exchanged by the isometry, _ ! M2=4_ which has

_ = M=2 as its _xed `point', actually a _xed 2-sphere of radius 2M. This

isometry corresponds to the (U; V ) ! (􀀀U;􀀀V ) isometry of the Kruskal

spacetime. The isotropic coordinates cover only regions I and IV since _ is

complex for r < 2M.

.......... .

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

.......... .

......... .

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

......... .

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

.......... .

......... .

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

......... .

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

.......... .

......... .

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

.......... .

......... .

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

.......... .

......... .

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

.......... .

......... .

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

......... .

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

.......... .

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

.......... .

......... .

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

.......... .

......... .

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

......... .

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

.......... .

......... .

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

.......... .

......... .

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

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

.......... .

......... .

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

.......... .

......... .

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

......... .

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

_ ! 1

at space

_ ! 1

at space

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

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

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

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

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

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

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

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

.. .. .. . . .. .

_ = M=2

t = constant

_ complex

As _ ! M=2 from either side the radius of a 2-sphere of constant _ on a

t = constant hypersurface decreases to minimum of 2M at _ = M=2, so

_ = M=2 is a minimal 2-sphere. It is the midpoint of an Einstein-Rosen

bridge connecting spatial sections of regions I and IV.

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

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

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

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

..........

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

..........

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

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

at radius 2M, _ = M=2

Einstein-Rosen bridge

(`throat')

_ = 1

_ = 0

2.3.4 Time translation in the Kruskal Manifold

The time translation t ! t + c, which is an isometry of the Schwarzschild

metric becomes

U ! e􀀀c=4MU; V ! ec=4MV (2.61)

in Kruskal coordinates and extends to an isometry of the entire Kruskal

manifold. The in_nitesimal version

_U = 􀀀

U; _V =

V (2.62)

is generated by the Killing vector _eld

k =

@V 􀀀 U

(2.63)

which equals @=@t in region I. It has the following properties

(i) k2 = 􀀀

􀀀

1 􀀀 2Mr

)

timelike in I & IV

spacelike in II & III

null on r = 2M, i.e. fU = 0g [ fV = 0g

(ii) fU = 0g and fV = 0g are _xed sets on k.

fU = 0g k = @=@v

fV = 0g k = @=@u

where v; u are EF null coordinates.

:_: v is the natural group parameter on fU = 0g. Orbits of k correspond

to 􀀀1 < v < 1, (where v is well-de_ned).

(iii) Each point on the Boyer-Kruskal axis, fU = V = 0g (a 2-sphere) is a

_xed point of k.

The orbits of k are shown below

... .

.. .. ..

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. ..

... .

.. .. ..

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. ..

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

.. ........

..........

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

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

..........

. .........

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

U V

IV I

III

U = 0

_xed points of

k=B-K axis

orbit of k

_xed set

(static observer)

2.3.5 Null Hypersurfaces

Let S(x) be a smooth function of the spacetime coordinates x_ and consider

a family of hypersurfaces S = constant. The vector _elds normal to the

hypersurface are

l = ~ f(x) (g__@_S)

@x_ (2.64)

where ~ f is an arbitrary non-zero function. If l2 = 0 for a particular hypersurface,

N, in the family, then N is said to be a null hypersurface.

Example Schwarzschild in ingoing Eddington-Finkelstein coordinates (r; v; _; _)

and the surface S = r 􀀀 2M.

l = ~ f(r)

1 􀀀

(2.65)

= ~ f(r)

1 􀀀

(2.66)

while

l2 = g__@_S@_S ~ f2 (2.67)

= grr ~ f2 =

1 􀀀

~ f2 (2.68)

so r = 2M is a null hypersurface, and

ljr=2M = ~ f

(2.69)

Properties of Null Hypersurfaces

Let N be a null hypersurface with normal l. A vector t, tangent to N, is

one for which t _ l = 0. But, since N is null, l _ l = 0, so l is itself a tangent

vector, i.e.

l_ =

dx_

(2.70)

for some null curve x_(_) in N.

Proposition The curves x_(_) are geodesics.

Proof Let N be the member S = 0 of the family of (not necessarily null)

hypersurfaces S = constant. Then l_ = ~ f g__@_S and hence

l _ Dl_ =

l_@_ ~ f

g__@_S + ~ fg__l_D_@_S (2.71)

l _ @ ln ~ f

l_ + ~ f g__l_D_@_S (by symmetry of 􀀀) (2.72)

ln ~ f

l_ + l_ ~ fD_

~ f􀀀1l_

(2.73)

ln ~ f

l_ + l_D_l_ 􀀀

@_ ln ~ f

l2 (2.74)

ln ~ f

l_ +

l2;_ 􀀀

@_ ln ~ f

l2 (2.75)

Although l2

= 0 it doesn't follow that l2;_

= 0 unless the whole family

of hypersurfaces S = constant is null. However since l2 is constant on N,

t_@_l2 = 0 for any vector t tangent to N. Thus

@_l2

N / l_ (2.76)

and therefore

l _ Dl_jN / l_ (2.77)

i.e. x_(_) is a geodesic (with tangent l). The function ~ f can be chosen such

that l _ Dl = 0, i.e. so that _ is an a_ne parameter.

De_nition The null geodesics x_(_) with a_ne parameter _, for which

the tangent vectors dx_=d_ are normal to a null hypersurface N, are the

generators of N.

Example N is U = 0 hypersurface of Kruskal spacetime. Normal to U =

constant is

l = 􀀀

~ fr

32M3 er=2M @

(2.78)

ljN = 􀀀

~ fe

16M2

since r = 2M on N (2.79)

Note that l2 _ 0, so l2 and l2;_ both vanish on N; this is because U =

constant is null for any constant, not just zero. thus l _ Dl = 0 if ~ f is

constant. Choose ~ f = 􀀀16M2e􀀀1. Then

l =

(2.80)

is normal to U = 0 and V is an a_ne parameter for the generator of this

null hypersurface.

2.3.6 Killing Horizons

De_nition A null hypersurface N is a Killing horizon of a Killing vector

_eld _ if, on N, _ is normal to N.

Let l be normal to N such that l _ Dl_ = 0 (a_ne parameterization).

Then, since, on N,

_ = fl (2.81)

for some function f, it follows that

_ _ D__ = ___; on N (2.82)

where _ = _ _ @ ln jfj is called the surface gravity.

Formula for surface gravity

Since _ is normal to N, Frobenius' theorem implies that

_[_D___]

= 0 (2.83)

where `[ ]' indicates total anti-symmetry in the enclosed indices, _; _; _.

For a Killing vector _eld _, D___ = D[___] (i.e. symmetric part of D___

vanishes). In this case (2.83) can be written as

__D___ jN

+ (__D___ 􀀀 __D___)jN

= 0 (2.84)

Multiply by D___ to get

__ (D___) (D___ )jN

= 􀀀 2 (D___) __ (D___)jN

(since D___ = D[___])(2.85)

__ (D___) (D___ )jN

= 􀀀 2 (_ _ D__)D___jN

(2.86)

= 􀀀 2__ _ D__jN

(for Killing horizon) (2.87)

= 􀀀 2_2__

(2.88)

Hence, except at points for which _ = 0,

_2 = 􀀀

(D___) (D___ )

____

(2.89)

It will turn out that all points at which _ = 0 are limit points of orbits of _

for which _ 6= 0, so continuity implies that this formula is valid even when

_ = 0 (Note that _ = 0 6) D___ = 0).

Killing Vector Lemma For a Killing vector _eld _

D_D___ = R_

_____ (2.90)

where R_

___ is the Riemann tensor.

Proof: Exercise (Question II.1)

Proposition _ is constant on orbits of _.

Proof Let t be tangent to N. Then, since (2.89) is valid everywhere on N

t _ @_2 = 􀀀 (D___) t_D_D___jN

(2.91)

= 􀀀(D___) t_R _

___ __ (using Lemma) (2.92)

Now, _ is tangent to N (in addition to being normal to it). Choosing t = _

we have

_ _ @_2 = 􀀀(D___ )R________ (2.93)

= 0 (since R____ = 􀀀R____) (2.94)

so _ is constant on orbits of _.

Non-degenerate Killing horizons (_ 6= 0)

Suppose _ 6= 0 on one orbit of _ in N. Then this orbit coincides with only

part of a null generator of N. To see this, choose coordinates on N such

that

_ =

(except at points where _ = 0) (2.95)

i.e. such that the group parameter _ is one of the coordinates. Then if

_ = _(_) on an orbit of _ with an a_ne parameter _

_jorbit =

= fl

8>>><

>>>:

f =

l =

dx_(_)

(2.96)

Now

ln jfj = _ (2.97)

where _ is constant for orbit on N. For such orbits, f = f0e__ for arbitrary

constant f0. Because of freedom to shift _ by a constant we can choose

f0 = __ without loss of generality, i.e.

= __e__ ) _ = _e__ + constant (2.98)

Choose constant = 0

_ = _e__ (2.99)

As _ ranges from 􀀀1 to 1 we cover the _ > 0 or the _ < 0 portion of

the generator of N (geodesic in N with normal l). The bifurcation point

_ = 0 is a _xed point of _, which can be shown to be a 2-sphere, called the

bifurcation 2-sphere, (BK-axis for Kruskal).

Bifurcation

2-sphere, B

Killing horizon N,

of _

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

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

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

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

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

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

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

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

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

..........

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

.... ......

....... ...

......... .

. .........

.... ......

...... ....

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

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

......... .

. .........

... .......

...... ....

......... .

. .........

... .......

..........

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

..........

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

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

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

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

..........

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

..........

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

..........

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

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

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

orbits of _

This is called a bifurcate Killing horizon.

Proposition If N is a bifurcate Killing horizon of _, with bifurcation 2-

sphere, B, then _2 is constant on N.

Proof _2 is constant on each orbit of _. The value of this constant is the

value of _2 at the limit point of the orbit on B, so _2 is constant on N if it

is constant on B. But we saw previously that

t _ @_2 = 􀀀 (D___) t_R _

___ __

(2.100)

= 0 on B since __jB = 0 (2.101)

Since t can be any tangent to B, _2 is constant on B, and hence on N.

Example N is fU = 0g [ fV = 0g of Kruskal spacetime, and _ = k, the

time-translation Killing vector _eld.

On N,

k =

8>>><

>>>:

on fU = 0g

􀀀

on fV = 0g

9>>>=

>>>;

= fl (2.102)

where

f =

8>><

>>:

V on fU = 0g

􀀀

U on fV = 0g

9>>=

>>;

; l =

8>>><

>>>:

on fU = 0g

on fV = 0g

9>>>=

>>>;

(2.103)

Since l is normal to N, N is a Killing horizon of k. Since l _ Dl = 0, the

surface gravity is

_ = k _ @ ln jfj =

8>>><

>>>:

ln jV j on U = 0

􀀀

ln jUj on V = 0

(2.104)

8>><

>>:

on fU = 0g

􀀀

on fV = 0g

(2.105)

So _2 = 1=(4M)2 is indeed a constant on N. Note that orbits of k lie either

entirely in fU = 0g or in fV = 0g or are _xed points on B, which allows a

di_erence of sign in _ on the two branches of N.

[N.B. Reinstating factors of c and G, j_j =

4GM

]

Normalization of _

If N is a Killing horizon of _ with surface gravity _, then it is also a Killing

horizon of c_ with surface gravity c2_ [from formula (2.89) for _] for any

constant c. Thus surface gravity is not a property of N alone, it also depends

on the normalization of _.

There is no natural normalization of _ on N since _2 = 0 there, but in

an asymptotically at spacetime there is a natural normalization at spatial

in_nity, e.g. for the time-translation Killing vector _eld k we choose

k2 ! 􀀀1 as r ! 1 (2.106)

This _xes k, and hence _, up to a sign, and the sign of _ is _xed by requiring

k to be future-directed.

Degenerate Killing Horizon (_ = 0)

In this case, the group parameter on the horizon is also an a_ne parameter,

so there is no bifurcation 2-sphere. More on this case later.

2.3.7 Rindler spacetime

Return to Schwarzschild solution

ds2 = 􀀀

1 􀀀

dt2 +

1 􀀀

􀀀1

dr2 + r2d2 (2.107)

and let

r 􀀀 2M =

(2.108)

Then

1 􀀀

(_x)2

1 + (_x)2

_ =

(2.109)

_ (_x)2 near x = 0 (2.110)

dr2 = (_x)2dx2 (2.111)

so for r _ 2M we have

ds2 _ 􀀀(_x)2dt2 + dx2

| {z }

2-dim Rindler

spacetime

4_2 d2

| {z }

2-sphere of

radius 1=(2_)

(2.112)

so we can expect to learn something about the spacetime near the Killing

horizon at r = 2M by studying the 2-dimensional Rindler spacetime

ds2 = 􀀀(_x)2dt2 + dx2 (x > 0) (2.113)

This metric is singular at x = 0, but this is just a coordinate singularity. To

see this, introduce the Kruskal-type coordinates

U0 = 􀀀xe􀀀_t; V 0 = xe_t (2.114)

in terms of which the Rindler metric becomes

ds2 = 􀀀dU0 dV 0 (2.115)

Now set

U0 = T 􀀀 X; V 0 = T + X (2.116)

to get

ds2 = 􀀀dT2 + dX2 (2.117)

i.e. the Rindler spacetime is just 2-dim Minkowski in unusual coordinates.

Moreover, the Rindler coordinates with x > 0 cover only the U0 < 0; V 0 > 0

region of 2d Minkowski

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

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

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

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

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

.......... .

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

.......... .

......... .

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

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

.......... .

......... .

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

.......... .

......... .

.......... .

......... .

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

..........

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

..........

....

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

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

...

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

.. .. . . .. . .

U0 V 0

x = 0

region covered by x > 0 Rindler

(corresponds to region I of

Kruskal spacetime)

From what we know about the surface r = 2M of Schwarzschild it follows

that the lines U0 = 0; V 0 = 0, i.e. x = 0 of Rindler is a Killing horizon of

k = @=@t with surface gravity __.

Exercise

(i) Show that U0 = 0 and V 0 = 0 are null curves.

(ii) Show that

k = _

V 0 @

@V 0 􀀀 U0 @

@U0

(2.118)

and that kjU0=0 is normal to U0 = 0. (So fU0 = 0g is a Killing horizon).

(iii) (k _ Dk)_jU0=0 = _k_jU0=0 (2.119)

Note that k2 = 􀀀(_x)2 ! 􀀀1 as x ! 1, so there is no natural normalization

of k for Rindler.

i.e. In contrast to Schwarzschild only the fact that _ 6= 0 is a property

of the Killing horizon itself - the actual value of _ depends on an arbitrary

normalization of k | so what is the meaning of the value of _?

Acceleration Horizons

Proposition The proper acceleration of a particle at x = a􀀀1 in Rindler

spacetime (i.e. on an orbit of k) is constant and equal to a.

Proof A particle on a timelike orbit X_(_) of a Killing vector _eld _ has

4-velocity

u_ =

(􀀀_2)1=2 (since u / _ and u _ u = 􀀀1) (2.120)

Its proper 4-acceleration is

a_ = D(_ )u_ = u _ Du_ (2.121)

_ _D__

􀀀_2 +

􀀀

_ _ @_2

2_2 (2.122)

But _ _ @_2 = 2____D___ = 0 for Killing vector _eld, so

a_ =

_ _ D__

􀀀_2 (2.123)

and `proper acceleration' is magnitude jaj of a_.

For Rindler with _ = k we have (Exercise)

a_@_ =

@V 0

V 0

@U0

(2.124)

jaj _ (a_a_g__ )1=2 =

􀀀

U0V 0

_1=2

(2.125)

(2.126)

so for x = a􀀀1 (constant) we have jaj = a, i.e. orbits of k in Rindler are

worldlines of constant proper acceleration. The acceleration increases without

bound as x ! 0, so the Killing horizon at x = 0 is called an acceleration

horizon.

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

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

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

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

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

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

x = 0

worldlines of x = constant

orbits of k = @=@t

in Rindler spacetime

Although the proper acceleration of an x = constant worldline diverges as

x ! 0 its acceleration as measured by another x = constant observer will

remain _nite. Since

d_2 = (_x)2dt2 (for x = a􀀀1, constant) (2.127)

the acceleration as measured by an observer whose proper time is t is

= (_x) _

= _ (2.128)

which has a _nite limit, _, as x ! 0.

In Rindler spacetime such an observer is one with constant proper acceleration

_, but these observers are in no way `special` because the normalization

of t was arbitrary.

t ! _t ) _ ! _􀀀1_; (_ 2 R) (2.129)

For Schwarzschild, however,

d_2 = dt2 )

r = constant ! 1

_; _ constant

(2.130)

i.e. an observer whose proper time is t is one at spatial 1. Thus

surface gravity is the acceleration of a static particle near the horizon as measured at spatial

in_nity

This explains the term `surface gravity' for _.

2.3.8 Surface Gravity and Hawking Temperature

We can study the behaviour of QFT in a black hole spacetime using Euclidean

path integrals. In Minkowski spacetime this involves setting

t = i_ (2.131)

and continuing _ from imaginary to real values. Thus _ is `imaginary time'

here (not proper time on some worldline).

In the black hole spacetime this leads to a continuation of the Schwarzschild

metric to the Euclidean Schwarzschild metric.

ds2

E =

1 􀀀

d_2 +

dr2

􀀀

1 􀀀 2M

_ + r2d2 (2.132)

This is singular at r = 2M. To examine the region near r = 2M we set

r 􀀀 2M =

(2.133)

to get

ds2

E _ (_x)2d_2 + dx2

| {z }

Euclidean Rindler

4_2 d2 (2.134)

Not surprisingly, the metric near r = 2M is the product of the metric on S2

and the Euclidean Rindler spacetime

ds2

E = dx2 + x2d(__)2 (2.135)

This is just E2 in plane polar coordinates if we make the periodic identi_cation

_ _ _ +

(2.136)

i.e. the singularity of Euclidean Schwarzschild at r = 2M (and of Euclidean

Rindler at x = 0) is just a coordinate singularity provided that imaginary

time coordinate _ is periodic with period 2_=_. This means that the Euclidean

functional integral must be taken over _elds _(~x; _) that are periodic

in _ with period 2_=_ [Why this is so is not self-evident, which is presumably

why the Hawking temperature was not _rst found this way. Closer

analysis shows that the non-singularity of the Euclidean metric is required

for equilibrium].

Now, the Euclidean functional integral is

Z =

[D_] e􀀀SE[_] (2.137)

where

SE =

dt (􀀀ipq_ + H) (2.138)

is the Euclidean action. If the functional integral is taken over _elds _ that

are periodic in imaginary time with period ~_ then it can be written as (see

QFT course)

Z = tr e􀀀_H ; (2.139)

which is the partition function for a quantum mechanical system with Hamiltonian

H at temperature T given by _ = (kBT)􀀀1 where kB is Boltzman's

constant.

But we just saw that ~_ = 2_=_ for Schwarzschild, so we deduce that a

QFT can be in equilibrium with a black hole only at the Hawking temperature

TH =

(2.140)

i.e. in units for which ~ = 1, kB = 1

TH =

(2.141)

N.B.

(i) At any other temperature, Euclidean Schwarzschild has a conical singularity

! no equilibrium.

(ii) Equilibrium at Hawking temperature is unstable since if the black hole

absorbs radiation its mass increases and its temperature decreases, i.e.

the black hole has negative speci_c heat.

2.3.9 Tolman Law - Unruh Temperature

Tolman Law The local temperature T of a static self-gravitating system

in thermal equilibrium satis_es

􀀀

􀀀k2_1=2

T = T0 (2.142)

where 􀀀 T0 is constant and k is the timelike Killing vector _eld @=@t. If

! 􀀀1 asymptotically we can identify T0 as the temperature `as seen

from in_nity'. For a Schwarzschild black hole we have

T0 = TH =

(2.143)

Near r = 2M we have, in Rindler coordinates,

(_x)T =

(2.144)

T =

x􀀀1

(2.145)

is the temperature measured by a static observer (on orbit of k) near the

horizon. But x = a􀀀1, constant, for such an observer, where a is proper

acceleration. So

T =

(2.146)

is the local (Unruh) temperature. It is a general feature of quantum mechanics

(Unruh e_ect) that an observer accelerating in Minkowski spacetime

appears to be in a heat bath at the Unruh temperature.

In Rindler spacetime the Tolman law states that

(_x)T = T0 (2.147)

Since T = x􀀀1=(2_) for x = constant, we deduce that T0 = _=(2_), as in

Schwarzschild, but this is now just the temperature of the observer with

constant acceleration _, who is of no particular signi_cance. Note that in

Rindler spacetime

T =

x􀀀1

2_ ! 0 as x ! 1 (2.148)

so the Hawking temperature (i.e. temperature as measured at spatial 1) is

actually zero.

This is expected because Rindler is just Minkowski in unusual coordinates,

there is nothing inside which could radiate. But for a black hole

Tlocal ! TH at in_nity (2.149)

) the black hole must be radiating at this temperature. We shall con_rm

this later.

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