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7.3 Hawking Radiation

Back

The spacetime associated to gravitational collapse to a black hole cannot

be everywhere stationary so we expect particle creation. But the exterior

spacetime is stationary at late times, so we might expect particle creation

to be just a transient phenomenon determined by details of the collapse.

But the in_nite time dilation at the horizon of a black hole means that

particles created in the collapse can take arbitrarily long to escape - suggests

a possible ux of particles at late times that is due to the existence of

the horizon and independent of the details of the collapse. There is such a

particle ux, and it turns out to be thermal - this is Hawking radiation

We shall consider only a massless scalar _eld _ in a Schwarzschild black

hole spacetime. From Question IV.4 we learn that the positive frequency

outgoing modes of _ have the behaviour

_! _ e􀀀i!u (7.42)

near =+. Consider a geometric optics approximation in which a particle's

worldline is a null ray, , of constant phase u, and trace this ray backwards

in time from =+. The later it reaches =+ the closer it must approach H+

in the exterior spacetime before entering the star.

................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...... ..... ..... ...... ..... ...... ..... ..... ...... ..... ...... ..... ..... ...... ..... ...... ..... ...... ..... ..... ...... ..... ...... ..... ...... ..... ...... ..... ..... ...... ..... ...... .. . ......... .......................................................................................................................................................................................................􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 �� 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 ..............................................................................................................................................................................................

..........

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

..........

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

..........

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

..........

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

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

..........

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

.. ..

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

.. .

.. ..

.. .. .. ..

.. ..

.. .. .. ..

.. .

.. .. .

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

.. .. ..

.. .. .. ..

.. ..

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

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

.. .. .. ..

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

.. .. .. ..

.. .. .. .

.. ..

.. .. .. .. ..

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

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

.. .. .. ..

.. .. .. .

.. ..

.. .. .. .. ..

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

.. .. .. .. ..

.. .. .. .

.. .. .. .. ..

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

.. .. .

.. ..

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

.. .. .. .. .. .

.. ..

.. .. .. .. ..

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

.. .. .. .. ..

.. .. .. .

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

.. .. .. ..

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

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

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

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

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

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

=􀀀

v = 0

continuation into past of null

geodesic generator of H+

H+

U = 􀀀_, u = 􀀀

log _

v = 􀀀_

The ray is one of a family of rays whose limit as t ! 1 is a null geodesic

generator, H, of H+. We can specify by giving its a_ne distance from

H along an ingoing null geodesic through H+

125

..........

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

..........

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

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

..........

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

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

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

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

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

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

􀀀􀀀

􀀀􀀀􀀀

􀀀􀀀

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

􀀀 􀀀

􀀀􀀀􀀀

􀀀 􀀀

􀀀 􀀀 􀀀 􀀀

􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀

................................... .................................................................................................................................................... _

ingoing

null geodesic

The a_ne parameter on this ingoing null geodesic is U, so U = 􀀀_. Equivalently

u = 􀀀

log _ (on near H+) (7.43)

_! _ exp

log _

near H+ (7.44)

This oscillates increasingly rapidly as _ ! 0, so the geometric optics approximation

is justi_ed at late times.

We need to match _! onto a solution of the K-G equation near =􀀀. In

the geometric optics approximation we just parallely-transport n and l back

to =􀀀 along the continuation of H. Let this continuation meet =􀀀 at v = 0.

The continuation of the ray back to =􀀀 will now meet =􀀀 at an a_ne

distance _ along an outgoing null geodesic on =􀀀

... ...... .

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

..... .....

.... ......

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

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

..... .....

.... ......

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

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

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

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

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

􀀀 􀀀􀀀

􀀀􀀀

􀀀􀀀􀀀

􀀀􀀀

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

􀀀􀀀 􀀀

􀀀 􀀀

��􀀀

􀀀􀀀

􀀀 􀀀

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

=􀀀

v = 0

v = 􀀀_

The a_ne parameter on outgoing null geodesics in =􀀀 is v (since ds2 =

du dv + r2d2 on =􀀀), so v = 􀀀_ on so

_! _ exp

log(􀀀v)

(7.45)

126

This is for v < 0. For v > 0 an ingoing null ray from =􀀀 passes through H+

and doesn't reach =+, so _! = _!(v) on =􀀀, where

_!(v) =

0 v > 0

exp

􀀀i!

_ log(􀀀v)

v < 0

(7.46)

Take the Fourier transform,

! =

􀀀1

ei!0v_!(v)dv (7.47)

Z 0

􀀀1

exp

i!0v +

log(􀀀v)

dv (7.48)

Lemma

!(􀀀!0) = 􀀀exp

􀀀

!(!0) for !0 > 0 (7.49)

Proof Choose branch cut in complex v-plane to lie along the real axis

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

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

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

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

􀀀

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

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

.......................................................................................................... branch cut

!0 > 0

!0 < 0

For !0 > 0 rotate contour to the positive imaginary axis and then set v = ix

to get

!(!0) = 􀀀i

exp

􀀀!0x +

log

xe􀀀i_=2

dx (7.50)

= 􀀀exp

__!

_ Z

exp

􀀀!0x +

log(x)

dx (7.51)

127

Since !0 > 0 the integral converges. When !0 < 0 we rotate the contour to

the negative imaginary axis and then set v = 􀀀ix to get

!(!0) = i

exp

!0x +

log

xei_=2

dx (7.52)

= exp

􀀀

_ Z

exp

!0x +

log(x)

dx (7.53)

Hence the result.

Corollary A mode of positive frequency ! on =+, at late times, matches

onto mixed positive and negative modes on =􀀀. We can identify (for positive

!0)

A!!0 = ~_!(!0) (7.54)

B!!0 = ~_!(􀀀!0) = 􀀀e􀀀_!=_~_!(!0) (7.55)

as the Bogoliubov coe_cients. We see that

Bij = 􀀀e􀀀_!i=_Aij (7.56)

But the matrices A and B must satisfy the Bogoliubov relations, e.g.

_ij =

AAy 􀀀 BBy

(7.57)

AikA_jk 􀀀 BikB_jk (7.58)

e_(!i+!j)=_ 􀀀 1

BikB_jk (7.59)

Take i = j to get

BBy

e2_!i=_ 􀀀 1

(7.60)

Now, what we actually need are the inverse Bogoliubov coe_cients corresponding

to a positive frequency mode on =􀀀 matching onto mixed positive

and negative frequency modes on =+. As we saw earlier, the inverse B

coe_cient is

B0 = 􀀀BT (7.61)

The late time particle ux through =+ given a vacuum on =􀀀 is

hNii=+ =

_􀀀

y B0

B_BT

BBT

(7.62)

128

But

􀀀

BBT

ii is real so

hNii=+ =

e2_!i=_ 􀀀 1

(7.63)

This is the Planck distribution for black body radiation at the Hawking

temperature

TH = ~_

(7.64)

We conclude that at late times the black hole radiates away its energy

at this temperature. From Stephan's law

dt ' 􀀀_AT4H

;

_ =

_2k4B

60~3c2

(7.65)

where A is the black hole area. Since

E = Mc2; A =

; kBTH _

~c3

(7.66)

we have

dt _

~c4

G2M2 (7.67)

which gives a lifetime

_ _

~c4

M3 (7.68)

Note The calculation of Hawking radiation assumed no backreaction, i.e.

M was taken to be constant. This is a good approximation when dM=dt _

M, but fails in the _nal stages of evaporation.

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7.3 Hawking Radiation

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