Communications
in
Commun. math.
Phys.
54,
1—19
(1977)
Mathematical
Physics
© by
Springer-Verlag
1977
The
Back Reaction
Effect
in
Particle Creation
in
Curved Spacetime*
Robert
M.
Wald**
Enrico Fermi Institute, University
of
Chicago, Chicago,
IL
60637,
USA
Abstract.
The
problem
of
determining
the
changes
in the
gravitational
field
caused
by
particle creation
is
investigated
in the
context
of the
semiclassical
approximation, where
the
gravitational
field
(i.e., spacetime geometry)
is
treated
classically
and an
effective
stress
energy
is
assigned
to the
created particles which
acts
as a
source
of the
gravitational
field.
An
axiomatic approach
is
taken.
We
list
five
conditions which
the
renormalized stress-energy operator
T
μv
should
satisfy
in
order
to
give
a
reasonable semiclassical theory.
It is
proven that these
conditions
uniquely determine
T
μv
,
i.e.
there
is at
most
one
renormalized stress-
energy
operator which satisfies
all the
conditions.
We
investigate existence
by
examining
an
explicit "point-splitting" type prescription
for
renormalizing
T
μv
.
Modulo some standard assumptions which
are
made
in
defining
the
prescrip-
tion
for
T
μv
,
it is
shown that this prescription satisfies
at
least
four
of the
five
axioms.
I.
Introduction
In
the
past
several years,
a
considerable amount
of
progress
has
been made
in our
understanding
of
quantum processes occurring
in a
strong gravitational
field.
A
satisfactory
quantum theory
of the
gravitational
field
itself
still
does
not
exist
[1].
However,
the
framework
of a
semiclassical theory describing other quantum
fields
present
in a
strong gravitational
field
does exist
and has
been used
to
investigate
particle creation
effects.
In
this theory
the
gravitational
field
is
described
in an
entirely
classical manner
as
curvature
in the
geometry
of
spacetime,
in
accordance
with
the
notions
of
general relativity.
The
fields
(e.g.,
a
scalar, Dirac,
or
Maxwell
field)
which
are
present
in
spacetime
are
described
in
accordance with
the
principles
of
quantum
field
theory.
It is not
believed that this theoretical framework
can
provide
an
exact description
of
nature, since
it
cannot
be
entirely consistent
to
have
*
Supported
in
part
by the
National Science Foundation under grant
MPS
74-17456-A01
and by the
Sloan Foundation
**
Sloan Foundation
fellow
2
R. M.
Wald
quantum
fields
(described
in
probabilistic terms) interact with
a
classical gravi-
tational
field
(with
definite,
determined values). Rather, this semiclassical theory
is
viewed
as an
approximation
to the
true—as
yet
unknown—quantum
theory
of
gravitation
interacting with other
fields.
Such
a
semiclassical
framework
is
analogous
to the
situation
in
atomic physics where,
for the
description
of a
wide
range
of
phenomena,
it is a
good approximation
to
describe
the
electromagnetic
field
in an
entirely classical manner while treating
the
electrons quantum
mechanically.
On
dimensional grounds
it is
generally believed that quantum
effects
of
gravity should
be
important
at
least when
the
spacetime curvature becomes
comparable
to the
Planck length
(ftG/c
3
)
1/2
«10"
33
cm.
However,
for
less extreme
spacetime curvature,
one
hopes that
the
semiclassical approximation
will
be
valid
at
least
in
many situations.
If
the
gravitational
field
has
suitable asymptotic behavior
in the
past
and
future,
a
description
of
the
quantum
fields
in
terms
of
particles
will
be
possible
in
these
asymptotic
regimes.
One may
then
ask
about particle
creation:
If the
field
is
initially
in
the
vacuum state,
how
many particles
will
be
present
at
late times? More
generally,
what
is the
S-matrix?
It
turns
out
that
a few
simple assumptions within
the
semiclassical
framework
described above uniquely lead
to an
expression
for the
S-matrix
in a
manner which
is
very
nearly
free
of any
mathematical
difficulties
[2].
Thus,
one can
make
well
defined,
unambiguous predictions concerning particle
creation
in a
strong gravitational
field.
The
most remarkable application
of
these ideas
is, of
course,
Hawking's
discovery
[3]
that particle creation near
a
Schwarzschild black hole
will
result
in a
steady
rate
of
emission
of
particles with
an
exactly thermal spectrum
[2,4].
This
result
is
particularly striking
in
view
of the
analogies
that
had
previously been
discovered
between black hole physics
and
thermodynamics
[5,6].
In the
absence
of
any
experimental
or
observational confirmation
of the
predictions
of the
semiclassical theory,
it is the
beauty
of
Hawking's result
as
well
as the
simplicity,
naturalness,
and
good
mathematical behavior
of the
theory which
gives
one
confidence
that this approach
is on the
right track.
In the
particle creation calculations
referred
to
above,
the
spacetime geometry
(i.e.,
gravitational
field)
is
taken
to be
that
appropriate
to
some classical physical
situation, e.g.,
the
gravitational collapse
of a
body
to
form
a
black hole.
The
particle
creation
is
then calculated
in
this
fixed
spacetime geometry. However,
on
physical
grounds
it is
clear that
the
quantum particle creation must have some
"back
reaction"
effect
on the
spacetime geometry.
In
particular,
for the
case
of a
black
hole,
the
particle creation calculations show
a
flux
of
energy coming
from
the
black
hole.
By
conservation
of
energy
one
would expect this energy
flux to be
balanced
by
a
decrease
in the
mass
of the
black hole
(i.e.,
a
decrease
in the
energy
of the
gravitational
field).
The
determination
of the
nature
and
magnitude
of the
"back
reaction"
effect
is of
great interest
and
importance
in its own right,
particularly
in
the
cosmological
context where
the
"back
reaction"
of
particle creation
may
have
an
important
effect
on the
dynamics
of the
universe.
It is
also needed
to
check
the
validity
of the
particle creation calculations, since
if the
effect
of the
"back
reaction"
is
large,
it
must
be
taken into account
in
these calculations.
In
what
framework
can one
analyze this
"back
reaction"
effect
? It is
conceivable
that
one
will
need
a
complete quantum theory
of
gravitation
in
order
to
describe
it,
Back Reaction
Effect
3
since
to
describe
it in the
semiclassical framework involves having
a
quantum
field
act as a
source
for a
classical
field.
If, for
example,
the
quantum
field
source
has a
probability
of
\
of
being
"very
small"
and a
probability
of
^
of
being
"very
large",
it
would
not
seem reasonable
to try to
describe
the
gravitational
field
to
which
it
gives
rise
as a
"medium
sized"
classical
field.
Thus,
in
particular, Hawking
has
expressed
the
view
that
"back
reaction"
can be
described only
in the
context
of
quantum
gravity. However, there
is
clearly
some
domain
of
validity
to
associating
a
classical
gravitational
field
to a
quantum source.
After
all, ordinary matter
is, of
course,
in
reality
of a
quantum nature
but it
certainly makes sense
to
assign
it a
classical
gravitational
field.
More generally,
if the
gravitational
field
is not so
strong that
the
effects
of
quantum gravity should
be of
direct importance,
it
seems reasonable
to
expect
the
approximation
of a
classical gravitational
field
to be
valid whenever
the
expected quantum
fluctuations in the
source
are
negligible compared with
the
expected value
of the
source
itself.
It is, of
course,
not
obvious whether this domain
of
validity extends
to
cases where particle creation
effects
are
important. However,
the
example
of
particle creation near
a
black hole suggests that this
may be the
case
since
at
least
at
large distances
from
the
black hole
the
created particles
are
thermally
distributed
and
hence should
satisfy
the
above
criterion. Further
indication
that
the
semiclassical approximation
may
have
a
wide range
of
validity
in
treating problems
of an
essentially quantum nature comes
from
the
example
of
atomic
physics:
In
atoms with many electrons
it is a
very
good
approximation
to
treat
the
electric
field
classically even though
the
electrons which give rise
to
this
field
must
be
treated quantum mechanically
in
general,
the
"radiative
corrections"
to
this approximation
are
negligible.
For the
remainder
of
this
paper,
we
shall
assume that
the
semiclassical approach
to the
"back
reaction"
effect
has a
nontrivial
range
of
validity.
In
classical general relativity,
the
source
of the
gravitational
field
is the
stress-
energy tensor
T
μv
of the
fields
present
in
spacetime.
The
gravitation
field
(described
by the
spacetime metric)
is
related
to
T
μv
via
Einstein's equation,
G
μv
=
*πT
μv
(1.1)
where
G
μv
is the
Einstein tensor.
In
quantum theory, observables
are
described
as
operators acting
on the
Hubert space
of
states
of the
system. Hence,
in
quantum
theory,
the
stress-energy tensor should become
an
operator.
A
natural procedure
for
treating
the
"back
reaction"
effect
in the
semiclassical approximation then
suggests itself:
we
require that
the
classical Einstein tensor
be set
equal
to the
expected value
of the
stress energy tensor
in the
given quantum state,
<V
=
8π<7;
v
>.
(1.2)
More precisely,
the
structure
of the
theory
is as
follows:
For
each (suitably well
behaved) classical spacetime geometry, there should exist
a
stress-energy operator
for
each
field
of
interest.
A
spacetime together with
a
quantum state
of the
field
which
satisfies
Equation (1.2)
is
considered
to be a
solution
of
this semiclassical
Einstein
theory. This solution
is to be
taken seriously
if the
characteristic radii
of
curvature
of the
spacetime
are
much
greater
than
the
Planck length
and if the
expected
fluctuations in
T
μv
in
this state
are
negligible compared with
<T
μv
>.
In the
4 R. M.
Wald
limit
where
the
field
can be
described classically
(i.e.,
a
large number
of
appropriately distributed particles
and
negligible particle creation),
the
theory
should reduce
to
classical general relativity.
It
is
natural
to
postulate that
the
stress-energy operator
is
given
in
terms
of the
quantum
field
operator
by the
same formula
by
which
the
classical stress-energy
tensor
is
related
to the
classical
field.
It is
here, however, that
a
serious problem
arises.
The
quantum
field
operator does
not
exist
as an
operator
defined
at
each
point
of
spacetime;
only
"smeared"
fields
make sense mathematically, i.e.,
the
field
is
an
operator valued distribution
on
spacetime. When
one
performs operations
which
are
linear
in the
field,
this distinction
is
basically
just
a
technical point
for
example,
it
leads
to no
difficulties
in the
derivation
of the
S-matrix
[2].
However,
nonlinear
operations
on
distributions, such
as
taking products, have
no
obvious
mathematical meaning. Since
the
stress-energy
is
quadratic
in the
field,
the
formula
for
the
stress-energy
operator
involves
a
product
of
distributions
and
hence must
be
viewed
as
only
a
formal expression. Therefore,
it is not
surprising that when
one
naively
attempts
to
calculate expectation values
of the
stress-energy operator,
one
gets
infinite
answers. Thus, some sort
of
renormalization prescription must
be
given.
In
flat
spacetime there
is a
completely satisfactory solution
to
this
re-
normalization problem: normal ordering.
One can
view this prescription
as
renormalizing
the
energy
of the
vacuum state
to
zero. However,
in
curved
spacetime, when particle creation
takes
place there
is no
invariant vacuum state
and
thus
there
is no
natural analogue
of
normal ordering. Furthermore, even
if no
particle creation occurs (e.g.,
in a
stationary spacetime)
it is not at all
clear that
normal ordering
is
correct, since vacuum polarization
effects
may
cause
the
stress-
energy
of the
vacuum state
to be
nonzero. Thus,
the
problem
of
renormalizing
the
stress-energy tensor
in
curved spacetime
is a
nontrivial one,
as has
been
further
demonstrated
by the
considerable amount
of
effort
that
has
gone into attempts
to
solve
it.
A
number
of
proposals
for
renormalizing
T
μv
are
discussed
by
DeWitt
[7].
More
recently,
dimensional regularization
[8] and
zeta-function
[9]
techniques have
been developed
and
further
work
has
been done
on the
"point-splitting"
method
[10,11].
However,
all the
prescriptions that have been given thus
far are
either
applicable only
to a
very restricted class
of
spacetimes (e.g.,
the
"adiabatic
regularization" scheme
of
Parker
and
Fulling
[12])
or
have ambiguities (e.g.,
the
direction dependent terms
in the
"point-splitting" approach),
or, at the
very least,
have features which
are ad
hoc.
The
extent
to
which
the
different
procedures
agree
or
disagree
has not
been
fully
investigated.
In
this
paper,
we
shall
take
a
different
approach
to the
stress-energy
renormalization problem. Rather than develop
a
particular renormalization
scheme,
we
shall list conditions that
the
renormalized stress-energy operator should
satisfy
in
order
to
give
a
reasonable semiclassical theory
of
"back
reaction"
within
the
framework described above. These
five
axioms
are
stated
in
Section
II. In the
absence
of any
likelihood
of
experimental
or
observational verifications
of
"back
reaction" predictions, conditions such
as
these
are the
only available criteria
for
deciding whether
any
given renormalization prescription
is
likely
to be the
correct
one.
Back
Reaction
Effect
5
The
main result
of the
paper
is the
following:
the
axioms stated
in
Section
II
uniquely
determine
a
renormalized stress-energy operator.
In
other words, there
can
be at
most
one
stress-energy operator which
satisfies
all the
axioms. This result
is
proved
in
Section III. However,
the
proof
does
not
establish existence
of
such
an
operator.
The
question
of
existence
is
investigated
in
Section
IV by
considering
an
explicit
"point-splitting" type prescription
for
renormalizing
T
μv
.
The
discussion here
is
based
on
several technical assumptions
for
which complete proofs have
not
been
given,
so the
results have more
the
flavor
of
plausibility arguments than theorems.
We
show that this prescription yields
a
stress-energy operator which
satisfies
at
least
the
first
four
axioms. However,
the
last axiom
is
very
difficult
to
check since
it
requires
a
detailed knowledge
of how
certain quantities change under variations
of
the
spacetime geometry. Thus, while existence
of the
renormalized stress-energy
operator
is not
shown,
we do
establish (modulo
the
technical assumptions) that
the
first
four
axioms
are
self-consistent. Some concluding remarks
are
made
in
Section
V.
II.
Axioms
for
T
μv
In
this section
we
shall state
and
discuss
the
precise mathematical conditions which
the
renormalized stress-energy operator should
satisfy.
Our
first
task
is to
make
more precise
the
class
of
spacetimes
on
which
T
μv
should
be
defined
and
what type
of
mathematical object
T
μv
is.
We
shall
take
the
class
of
spacetimes
to
consist
of
C
00
spacetimes which
are
sufficiently
well behaved asymptotically
in the
past
and
future
to
admit asymptotic
notions
of
positive
and
negative
frequency
solutions
1
.
This permits
one to
characterize
the
states
of the
quantum
field
in
terms
of
"in"
and
"out" Fock spaces.
We
shall
further
require that
the
spacetime
be
such that
the
condition,
tr(E*E)
<
oo,
described
in
Reference
[2],
is
satisfied,
so
that
the
^-matrix
relating
the
"in"
and
"out" states really does exist.
We do not
mean
to
suggest that this
is the
only class
of
spacetimes
on
which
T
μv
should
be
definable
however,
T
μv
should
be
defined
at
least
on
this class.
What type
of
mathematical object should
T
μv
be? It is
clearly demanding
too
much
to
require that
for
each point
p in
spacetime,
T
μv
(p)
be an
operator mapping
the
Hubert
space
3F
of
quantum states into itself; neither
the
field
operator
nor the
normal ordered stress-energy operator
in flat
spacetime exist
in
this sense.
For the
purposes
of
this paper,
it
will
be
important that
T
μv
be
defined
at
each point
of
spacetime
but it
will only
be
necessary that
the
matrix elements
of
T
μ
v
be
well
defined.
Thus,
it
will
suffice
to
require that
the
renormalized
T
μv
be a
bilinear
map
defined
on a
dense domain
of
vectors
in
JF
x
J^.
(Here
&
denotes
the
dual
of the
Hubert space
of
states
3F
and
^~(2,
0)
denotes
the
vector space
of
2-co
variant
index
symmetric tensor
fields
on
spacetime.) This
is a
reasonable requirement since both
1
This
includes
many nonasymptotically
flat
spacetimes where
the
Feynman propagator con-
structions
of
e.g.,
Rumpf
[13], Hartle
and
Hawking [14],
or
Candelas
and
Raine [15]
are
available