7.2 Particle Production in Non-Stationary Spacetimes

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Consider a `sandwich' spacetime M = M􀀀 [M0 [M+

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t

M+ stationary spacetime

M0 time-dependent metric

M􀀀 stationary spacetime

t1

t2

t > t2

t < t1

In M􀀀 we can choose to expand a scalar _eld solution of the Klein-Gordon

equation as

_(x) =

X

i

h

aiui(x) + ayi

u_i (x)

i

in M􀀀 (7.29)

The functions ui(x) solve the KG equation in M􀀀 but not in M, so its

continuation through M0 will lead to some new function i(x) in M+, so

_(x) =

X

i

h

ai i(x) + ayi

_i (x)

i

in M+ (7.30)

Because the inner product ( ; ) was independent of the hypersurface _, the

matrix of inner products will still be as before, i.e. as in (7.9). But, as we

have seen this implies only that

i =

X

j

􀀀

Aijuj + Biju_j

_

(7.31)

for some matrices A and B satisfying (7.16). Thus, in M+

_(x) =

X

i

_

ai i + ayi

_i

_

(7.32)

=

X

i

2

4ai

X

j

􀀀

Aijuj + Biju_j

_

+ ayi

X

j

􀀀

A_iju_j + B_ijuj

_

3

5(7.33)

=

X

i

h

a0iui(x) + a0iyu_i (x)

i

(7.34)

123

where

a0j =

X

i

_

aiAij + ayi

B_ij

_

(7.35)

This is called a Bogoliubov transformation. A and B are the Bogoliubov

coe_cients.

Note that (Exercise)

h

a0i; a0j

i

= 0

h

a0i; a0jy

i

= _ij

9>>=

>>;

, relations (7.25) satis_ed by A & B (7.36)

If B = 0 then (7.16) and (7.25) imply AyA = AAy = 1, i.e. the change of

basis from fuig to f ig is just a unitary transformation which permutes the

annihilation operators but does not change the de_nition of the vacuum.

The particle number operator for the ith mode of k is

Ni = ayi

ai in M􀀀

N0 i = a0iya0i in M+

(7.37)

The state with no particles in M􀀀 is jvaci s.t. ai jvaci = 0 8i. The expected

number of particles in the ith mode in M+ is then

 

N0 i

_

_

 

vac

__

N0 i

__

vac

_

=

D

vac

___a0i ya0i

___

vac

E

(7.38)

=

X

j;k

D

vac

__ _

(

akBki)

_

ayj

B_ji

____ vac

E

(7.39)

=

X

j;k

D

vac

__ _

akayj ___

vac

E

| {z }

_kj

BkiByij (7.40)

=

_

ByB

_

ii

(7.41)

The expected total number of particles is therefore tr

􀀀

ByB

_

. Since ByB is

positive semi-de_nite, this vanishes i_ B = 0.

124

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