Commun.
math. Phys.
43,
199—220
(1975)
© by
Springer-Verlag
1975
Particle
Creation
by
Black Holes
S.
W.
Hawking
Department
of
Applied Mathematics
and
Theoretical Physics, University
of
Cambridge,
Cambridge, England
Received
April
12,
1975
Abstract.
In the
classical theory black holes
can
only absorb
and not
emit particles. However
it
is
shown that quantum mechanical
effects
cause black holes
to
create
and
emit particles
as if
they
were
hot
bodies with
temperature
;^10~
6
——
°K
where
K is the
surface gravity
of the
black
2πk
\
M ,
hole.
This thermal emission leads
to a
slow decrease
in the
mass
of the
black hole
and to its
eventual
disappearance:
any
primordial black hole
of
mass less than about
10
15
g
would have evaporated
by
now.
Although these quantum
effects
violate
the
classical
law
that
the
area
of the
event horizon
of a
black hole cannot decrease, there remains
a
Generalized Second
Law:
S+^A
never decreases where
S
is
the
entropy
of
matter outside black holes
and A is the sum of the
surface areas
of the
event horizons.
This
shows that gravitational collapse converts
the
baryons
and
leptons
in the
collapsing body into
entropy.
It is
tempting
to
speculate that this might
be the
reason
why the
Universe contains
so
much
entropy
per
baryon.
1.
Although
there
has
been
a lot of
work
in the
last
fifteen
years (see
[1,
2] for
recent reviews),
I
think
it
would
be
fair
to say
that
we do not yet
have
a
fully
satisfactory
and
consistent quantum theory
of
gravity.
At the
moment classical
General Relativity still provides
the
most successful description
of
gravity.
In
classical General Relativity
one has a
classical metric which obeys
the
Einstein
equations,
the
right hand side
of
which
is
supposed
to be the
energy momentum
tensor
of the
classical matter fields. However, although
it may be
reasonable
to
ignore quantum gravitational
effects
on the
grounds that these
are
likely
to be
small,
we
know that quantum mechanics plays
a
vital role
in the
behaviour
of
the
matter
fields.
One
therefore
has the
problem
of
defining
a
consistent scheme
in
which
the
space-time metric
is
treated classically
but is
coupled
to the
matter
fields
which
are
treated quantum mechanically. Presumably such
a
scheme would
be
only
an
approximation
to a
deeper theory
(still
to be
found)
in
which space-
time
itself
was
quantized. However
one
would
hope
that
it
would
be a
very
good
approximation
for
most purposes except near space-time singularities.
The
approximation
I
shall
use in
this
paper
is
that
the
matter
fields,
such
as
scalar, electro-magnetic,
or
neutrino
fields,
obey
the
usual wave equations with
the
Minkowski
metric replaced
by a
classical space-time metric
g
ab
.
This metric
satisfies
the
Einstein equations where
the
source
on the
right hand side
is
taken
to be the
expectation value
of
some suitably defined energy momentum operator
for
the
matter fields.
In
this theory
of
quantum
mechanics
in
curved
space-time
there
is a
problem
in
interpreting
the
field
operators
in
terms
of
annihilation
and
creation operators.
In
flat
space-time
the
standard procedure
is to
decompose
200 S. W.
Hawking
the
field
into positive
and
negative
frequency
components.
For
example,
if
φ
is
a
massless Hermitian scalar
field
obeying
the
equation
φ.
ab
η
ab
=
^
one
expresses
φ as
where
the
{/J
are a
complete orthonormal
family
of
complex valued solutions
of
the
wave equation
fι.
ab
η
ab
=
^
which contain only positive frequencies with
respect
to the
usual Minkowski time coordinate.
The
operators
α
t
and
αj
are
interpreted
as the
annihilation
and
creation operators respectively
for
particles
in
the
zth
state.
The
vacuum state
|0> is
defined
to be the
state
from
which
one
cannot
annihilate
any
particles,
i.e.
fl
.|0> = 0 for all
z.
In
curved space-time
one can
also consider
a
Hermitian scalar
field
operator
φ
which obeys
the
co
variant
wave equation
φ.
ab
g
ab
=
0.
However
one
cannot
decompose into
its
positive
and
negative
frequency
parts
as
positive
and
negative
frequencies
have
no
invariant_meaning
in
curved space-time.
One
could
still
require
that
the
{/)}
and the
{/J
together formed
a
complete basis
for
solutions
of
the
wave equations with
Ws(fJj
ta
-?jfι
.JdΣ
a
=
δ
ίJ
(1.2)
where
S is a
suitable surface. However condition
(1.2)
does
not
uniquely
fix the
subspace
of the
space
of all
solutions which
is
spanned
by the
{/)}
and
therefore
does
not
determine
the
splitting
of the
operator
φ
into annihilation
and
creation
parts.
In a
region
of
space-time which
was
flat
or
asymptotically
flat,
the
appro-
priate criterion
for
choosing
the
{/•}
is
that they should contain only positive
frequencies
with respect
to the
Minkowski time coordinate. However
if one has
a
space-time which contains
an
initial
flat
region
(1)
followed
by a
region
of
curvature
(2)
then
a
final
flat
region
(3),
the
basis
{f
u
}
which contains only positive
frequencies
on
region
(1)
will
not be the
same
as the
basis
{/
3l
}
which contains
only
positive frequencies
on
region
(3).
This means that
the
initial
vacuum state
lOj),
the
state which satisfies
α
lί
|0
1
)
= 0 for
each initial annihilation operator
α
lί9
will
not be the
same
as the
final
vacuum state
|0
3
>
i.e.
α
3ί
|0
1
>φO.
One can
interpret
this
as
implying that
the
time dependent metric
or
gravitational
field
has
caused
the
creation
of a
certain number
of
particles
of the
scalar
field.
Although
it is
obvious what
the
subspace spanned
by the
{/)}
is for an
asympto-
tically
flat
region,
it is not
uniquely
defined
for a
general point
of a
curved space-
time. Consider
an
observer with velocity vector
υ
a
at a
point
p. Let B be the
least
upper bound
\R
abcd
\
in any
orthonormal tetrad whose timelike vector coincides
with
iΛ
In a
neighbourhood
U of p the
observer
can set up a
local inertial
co-
ordinate system (such
as
normal coordinates) with coordinate radius
of the
order
of
B~*.
He can
then choose
a
family
{/)}
which
satisfy
equation
(1.2)
and
which
in
the
neighbourhood
U are
approximately positive
frequency
with respect
to
the
time coordinate
in U. For
modes
f
i
whose characteristic
frequency
ω
is
high
compared
to
B^,
this leaves
an
indeterminacy between
f
i
and its
complex
con-
jugate
fa of the
order
of the
exponential
of
some multiple
of —
ωB~*.
The
indeter-
minacy
between
the
annihilation operator
a
t
and the
creation operator
a]
for the
Particle
Creation
by
Black Holes
201
mode
is
thus exponentially small. However,
the
ambiguity between
the
a
i
and
the
a\
is
virtually complete
for
modes
for
which
ω
<
B*.
This ambiguity introduces
an
uncertainty
of
±i
in the
number
operator
a\a
i
for the
mode.
The
density
of
modes
per
unit volume
in the
frequency
interval
ω to ω +
dω
is of the
order
of
ω
2
dω
for ω
greater
than
the
rest
mass
m of the
field
in
question. Thus
the un-
certainty
in the
local energy density caused
by the
ambiguity
in
defining
modes
of
wavelength longer than
the
local radius
of
curvature
B~^,
is of
order
B
2
in
units
in
which
G = c
—
h
=
i.
Because
the
ambiguity
is
exponentially small
for
wavelengths short compared
to the
radius
of
curvature
B~^,
the
total uncertainty
in
the
local energy density
is of
order
B
2
.
This uncertainty
can be
thought
of as
corresponding
to the
local energy density
of
particles created
by the
gravitational
field.
The
uncertainty
in the
curvature produced
via the
Einstein equations
by
this
uncertainty
in the
energy density
is
small compared
to the
total curvature
of
space-time provided that
B is
small compared
to
one, i.e.
the
radius
of
curvature
B~^
is
large
compared
to the
Planck length
10~
33
cm.
One
would therefore
expect that
the
scheme
of
treating
the
matter
fields
quantum mechanically
on a
classical curved space-time background would
be a
good
approximation, except
in
regions where
the
curvature
was
comparable
to the
Planck value
of
10
66
cm"
2
.
From
the
classical singularity theorems
[3-6],
one
would expect such high
cur-
vatures
to
occur
in
collapsing stars
and,
in the
past,
at the
beginning
of the
present
expansion phase
of the
universe.
In the
former
case,
one
would expect
the
regions
of
high curvature
to be
hidden
from
us by an
event horizon
[7].
Thus,
as far as
we are
concerned,
the
classical geometry-quantum matter treatment should
be
valid apart
from
the
first
10~
43
s of the
universe.
The
view
is
sometimes expressed
that
this treatment will
break
down when
the
radius
of
curvature
is
comparable
to
the
Compton wavelength
~10~
13
cm
of an
elementary particle such
as a
proton. However
the
Compton wavelength
of a
zero rest
mass
particle such
as
a
photon
or a
neutrino
is
infinite,
but we do not
have
any
problem
in
dealing
with
electromagnetic
or
neutrino radiation
in
curved space-time.
All
that
hap-
pens when
the
radius
of
curvature
of
space-time
is
smaller than
the
Compton
wavelength
of a
given species
of
particle
is
that
one
gets
an
indeterminacy
in the
particle number
or, in
other
words,
particle creation. However,
as was
shown
above,
the
energy density
of the
created particles
is
small locally compared
to the
curvature
which created them.
Even
though
the
effects
of
particle creation
may be
negligible locally,
I
shall
show
in
this
paper
that they
can add up to
have
a
significant
influence
on
black
holes
over
the
lifetime
of the
universe
~10
17
s
or
10
60
units
of
Planck time.
It
seems that
the
gravitational
field
of a
black hole will create particles
and
emit
them
to
infinity
at
just
the
rate that
one
would expect
if the
black hole were
an
ordinary
body with
a
temperature
in
geometric units
of
;c/2π,
where
K is the
"surface
gravity"
of the
black hole
[8].
In
ordinary units this temperature
is of
the
order
of
lO^M"
1
°K,
where
M is the
mass,
in
grams
of the
black hole.
For
a
black hole
of
solar
mass
(10
33
g)
this temperature
is
much lower than
the 3 °K
temperature
of the
cosmic microwave background. Thus black holes
of
this size
would
be
absorbing
radiation
faster than they emitted
it and
would
be
increasing
in
mass. However,
in
addition
to
black
holes formed
by
stellar collapse, there
might also
be
much smaller
black
holes which were formed
by
density
fluctua-
202 S. W.
Hawking
tions
in the
early universe
[9,
10].
These small black holes, being
at a
higher
temperature, would
radiate
more than they absorbed. They would therefore pre-
sumably decrease
in
mass.
As
they
got
smaller, they would
get
hotter
and so
would radiate
faster.
As the
temperature rose,
it
would exceed
the
rest mass
of
particles such
as the
electron
and the
muon
and the
black hole would begin
to
emit them also. When
the
temperature
got up to
about
10
12
°K or
when
the
mass
got
down
to
about
10
14
g the
number
of
different
species
of
particles being emitted
might
be so
great
[11]
that
the
black hole radiated away
all its
remaining rest
mass
on a
strong interaction time scale
of the
order
of
10"
23
s.
This would pro-
duce
an
explosion with
an
energy
of
10
35
ergs. Even
if the
number
of
species
of
particle emitted
did not
increase very much,
the
black hole would radiate away
all
its
mass
in the
order
of
10"
28
M
3
s. In the
last tenth
of a
second
the
energy
released would
be of the
order
of
10
30
ergs.
As
the
mass
of the
black hole decreased,
the
area
of the
event horizon would
have
to go
down, thus violating
the law
that, classically,
the
area cannot decrease
[7,
12].
This violation must, presumably,
be
caused
by a
flux
of
negative energy
across
the
event horizon which balances
the
positive energy
flux
emitted
to
infinity.
One
might picture this negative energy
flux in the
following
way. Just
outside
the
event horizon there will
be
virtual pairs
of
particles,
one
with negative
energy
and one
with positive energy.
The
negative particle
is in a
region which
is
classically forbidden
but it can
tunnel through
the
event horizon
to the
region
inside
the
black hole where
the
Killing vector which represents time translations
is
spacelike.
In
this region
the
particle
can
exist
as a
real particle with
a
timelike
momentum vector even though
its
energy relative
to
infinity
as
measured
by the
time translation Killing vector
is
negative.
The
other
particle
of the
pair, having
a
positive energy,
can
escape
to
infinity
where
it
constitutes
a
part
of the
thermal
emission described above.
The
probability
of the
negative energy particle tun-
nelling
through
the
horizon
is
governed
by the
surface
gravity
K
since this quantity
measures
the
gradient
of the
magnitude
of the
Killing vector
or, in
other words,
how
fast
the
Killing vector
is
becoming spacelike. Instead
of
thinking
of
negative
energy
particles tunnelling through
the
horizon
in the
positive sense
of
time
one
could regard them
as
positive energy particles crossing
the
horizon
on
past-
directed world-lines
and
then being scattered
on to
future-directed world-lines
by
the
gravitational
field.
It
should
be
emphasized that these pictures
of the
mecha-
nism responsible
for the
thermal emission
and
area
decrease
are
heuristic only
and
should
not be
taken
too
literally.
It
should
not be
thought unreasonable that
a
black hole, which
is an
excited state
of the
gravitational
field,
should decay
quantum mechanically
and
that, because
of
quantum
fluctuation of the
metric,
energy
should
be
able
to
tunnel
out of the
potential
well
of a
black hole. This
particle creation
is
directly analogous
to
that caused
by a
deep potential
well
in
flat
space-time
[18].
The
real justification
of the
thermal emission
is the
mathe-
matical
derivation given
in
Section
(2) for the
case
of an
uncharged non-rotating
black hole.
The
effects
of
angular momentum
and
charge
are
considered
in
Section (3).
In
Section
(4) it is
shown
that
any
renormalization
of the
energy-
momentum
tensor
with suitable
properties
must give
a
negative energy
flow
down
the
black hole
and
consequent decrease
in the
area
of the
event horizon.
This negative energy
flow is
non-observable locally.
Particle
Creation
by
Black Holes
203
The
decrease
in
area
of the
event horizon
is
caused
by a
violation
of the
weak
energy
condition
[5—7,
12]
which arises
from
the
indeterminacy
of
particle num-
ber and
energy density
in a
curved space-time. However,
as was
shown above,
this
indeterminacy
is
small, being
of the
order
of B
2
where
B is the
magnitude
of
the
curvature tensor. Thus
it can
have
a
diverging
effection
a
null
surface like
the
event horizon which
has
very small convergence
or
divergence
but it can not
untrap
a
strongly converging trapped
surface
until
B
becomes
of the
order
of
one.
Therefore
one
would
not
expect
the
negative energy density
to
cause
a
breakdown
of the
classical singularity theorems
until
the
radius
of
curvature
of
space-time became
10"
33
cm.
Perhaps
the
strongest reason
for
believing that black holes
can
create
and
emit
particles
at a
steady rate
is
that
the
predicted rate
is
just that
of the
thermal
emission
of a
body with
the
temperature
κ/2π.
There
are
independent, thermo-
dynamic,
grounds
for
regarding some multiple
of the
surface gravity
as
having
a
close relation
to
temperature. There
is an
obvious analogy with
the
second
law
of
thermodynamics
in the law
that, classically,
the
area
of the
event horizon
can
never decrease
and
that when
two
black holes collide
and
merge together,
the
area
of the
final
event horizon
is
greater than
the sum of the
areas
of the two
original
horizons
[7,
12].
There
is
also
an
analogy
to the
first
law of
thermo-
dynamics
in the
result that
two
neighbouring
black
hole equilibrium states
are
related
by [8]
-
8π
where
M,
Ω,
and J are
respectively
the
mass, angular velocity
and
angular
mo-
mentum
of the
black hole
and A is the
area
of the
event horizon. Comparing this
to
one
sees that
if
some multiple
of A is
regarded
as
being analogous
to
entropy,
then
some multiple
of K is
analogous
to
temperature.
The
surface gravity
is
also
analogous
to
temperature
in
that
it is
constant over
the
event horizon
in
equi-
librium.
Beckenstein [19] suggested that
A and K
were
not
merely analogous
to
entropy
and
temperature respectively
but
that,
in
some sense, they actually were
the
entropy
and
temperature
of the
black hole. Although
the
ordinary second
law of
thermodynamics
is
transcended
in
that entropy
can be
lost down black
holes,
the flow of
entropy
across
the
event horizon would always cause some
increase
in the
area
of the
horizon. Beckenstein therefore suggested [20]
a
Gen-
eralized Second
Law:
Entropy
4-
some multiple
(unspecified)
oϊA
never decreases.
However
he did not
suggest that
a
black hole could emit particles
as
well
as
absorb them. Without such emission
the
Generalized Second
Law
would
be
violated
by for
example,
a
black hole immersed
in
black
body radiation
at a
lower
temperature than that
of the
black
hole.
On the
other hand,
if one
accepts that
black holes
do
emit particles
at a
steady rate,
the
identification
oϊκ/2π
with tem-
perature
and
\A
with entropy
is
established
and a
Generalized Second
Law
confirmed.