PH YSICAL RK VIEW D VO L UMK 15, 5 UMBER 10 l5 MA Y 1977

Cosmological event horizons, thermodynamics, and particle creation

G. %'. Gibbons' and S. %. Hawking
D.A. M. T.P., University of Cambridge, Silver Street, Cambridge, United Kingdom
(Received 4 March 1976)
It is shown that the close connection between event horizons and thermodynamics which has been found in
the case of black holes can be extended to cosmological models with a repulsive cosmological constant. An
observer in these models will have an event horizon whose area can be interpreted as the entropy or lack of
information of the observer about the regions which he cannot see. Associated with the event horizon is a
surface gravity v which enters a classical "first law of event horizons*' in a manner similar to that in which
temperature occurs in the first law of thermodynamics. It is shown that this similarity is more than an
analogy: An observer with a particle detector will indeed observe a background of thermal radiation coming
apparently from the cosmological event horizon. If the observer absorbs some of this radiation, he will gain
energy and entropy at the expense of the region beyond his ken and the event horizon will shrink. The
derivation of these results involves abandoning the idea that particles should be defined in an observer-
independent manner. They also suggest that one has to use something like the Everett-Wheeler interpretation
of quantum mechanics because the back reaction and hence the spacetime metric itself appear to be observer-
dependent, if one assumes, as seems reasonable, that the detection of a particle is accompanied by a change in
the gravitational field.

I. INTRODUCTION particles of indefinitely small mass. However,

when quantum mechanics is taken into account, one
The aim of this payer is to extend to cosmologi- would expect that in order to obtain gravitational
cal event horizons some of the ideas of thermo- collapse the energies of the particle would have to
dynamics and particle creation which have recently be restricted by the requirement that their wave-
been successfully applied to black-hole event length be less than the size of the black hole. It
horizons. In a black hole the inward-directed would therefore seem reasonable to postulate that
gravitational field yroduced by a collapsing body is the number of internal configurations is finite. In
so strong that light emitted from the body is drag- this case one could associate with the black hole an
ged back and does not reach an observer at a large entropy S~ which would be the logarithm of this
distance. There is thus a region of spacetime number of possible internal configurations. '~
which is not visible to an external observer. The For this to be consistent the black hole would have
boundary of the region is called the event horizon to emit thermal radiation like a body with a tem-
of the black hole. Event horizons of a different perature
kind occur in cosmological models with a repul-
sive A term. The effect of this term is to cause Q2
the universe to expand so rapidly that for each ob-
server there are regions from which light can The mechanism by which this thermal radiation
never reach him. We shall call the boundary of arises can be understood in terms of pair creation
this region the cosmological event horizon of the in the gravitational potential well of the black hole.
observer. Inside the black hole there are particle states
The "no hair" theorems (Israel, ' Muller sum which have negative energy with respect to an ex-
Hagen et al. ,' Carter, ' Hawking, ' Robinson' ') ternal stationary observer. It is therefore ener-
imply that a black hole formed in a gravitational getically possible for a pair of particles to be
collapse will rapidly settle down to a quasistation- spontaneously created near the event horizon. One
ary state characterized by only three yarameters, particle has positive energy and escapes to infinity,
the mass MH, the angular momentum J» and the the other yarticle has negative energy and falls
charge Q„. A black hole of a given M„, J„,Q„ into the black hole, thereby reducing its mass.
therefore has a large number of possible unobserv- The existence of the event horizon would prevent
able internal configurations which reflect the dif- this happening classically but it is possible quan-
ferent possible initial configurations of the body tum-mechanically because one or other of the
that collapsed to produce the hole. In purely clas- yarticles can tunnel through the event horizon. An
sical theory this number of internal configurations equivalent way of looking at the pair creation is
would be infinite because one could make a given to regard the positive- and negative-energy par-
black hole out of an infinitely large number of ticles as being the same particle which tunnels

2738
 COSMOLOGICAL EVENT HORIZONS, THERMODYNAMICS, AND. . . 2789

out from the black hole on a spacelike or past- event horizon as a measure of one's lack of know-
directed timelike world line and is scattered onto ledge about the rest of the universe beyond one' s
a future-directed world line (Hartle and Hawking'0). ken. If one absorbs the thermal. radiation, one
When one calculates the rate of particle emission gains energy and entropy at the expense of this
by this process it turns out to be exactly what one region and so, by the first law mentioned above,
would expect from a body with a temperature the area of the horizon will go down. As the area
T„= h(2&Ac) 'z„, where v„ is the surface gravity decreases, the temperature of the cosmological
of the black hole and is related to M~, JH, and radiation goes down (unlike the black-hole case),
Qz by the formulas so the cosmological event horizon is stable. On
the other hand, if the observer chooses not to
~„= (r. —y )c'y, ', absorb any radiation, there is no change in area
r = c '[GMa (G'M' J'M-'c' GQ')'~'] of the horizon. This is another illustration of the
fact that the concept of particle production and
y '= r '+G 2J M"'e' the back reaction associated with it seem not to
A. ~= 4gro'. be uniquely defined but to be dependent upon the
measurements that one wishes to consider.
A H is the area of the event horizon of the black The plan of the paper is as follows. In Sec. II
hole.
Combi:~ing this quantum- mechanical argument solutions found by Carter. "
we describe the black-hole asymptotically de Sitter
In Sec. III we derive
with the thermodynamic argument above, one the classical laws governing both cosmol. ogical and
finds that the total number of internal configura- black-hole event horizons. In Sec. IV we discuss
tions is indeed finite and that the entropy is given particl. e creation in de Sitter space. W'e abandon
by the concept of particles as being observer-inde-
pendent and consider instead what an observer
S„=(4G8') 'kc'A„. moving on a timelike geodesic and equipped with
Cosmological models with a repulsive A term a particle detector would actually measure. W' e
which expand forever approach de Sitter space find that he would detect an isotropic background

future infinity is spacelike. ""

asymptotically at large times. In de Sitter space
This means that
of thermal radiation with a temperature (2w) '~c
where ~~ = A'~'3 '~' is the surface gravity of the
for each observer moving on a timelike world line cosmological event horizon of the observer. Any
there is an event horizon separating the region of other observer moving on a timel. ike geodesic wi. ll
spacetime which the observer can never see from also see isotropic radiation with the same tem-
the region that he can see if he waits long enough. perature even though he is moving relative to the
In other words, the event horizon is the boundary first observer. This shows that they are not ob-
of the past of the observer's world line. Such a serving the same particles: Particles are observ-
cosmological event horizon has many formal simi- er-dependent. In Sec. V we extend these results
larities with a black-hole event horizon. As we to asymptotically de Sitter spaces containing black
shall show in Sec. III it obeys laws very similar holes. The implications are considered in Sec. VI.
to the zeroth, first, and second laws of black- It seems necessary to adopt something like the
hole mechanics in the classical theory. It also " Everett-Wheeler interpretation of quantum mech-
bounds the region in which particles can have nega- anics because the back reaction and hence the
tive energy with respect to the observer. One spacetime metric mill be observer-dependent, if
might therefore expect that particle creation with one assumes, as seems reasonable, that the de-
a thermal spectrum would also occur in these tection of a particle is accompanied by a change
cosmological models. In Secs. IV and V we shal. l. in the gravitational field.
show that this is indeed the case: An observer We shall adopt units in which 6 =@=0=c= 1. %'e
wi11 detect thermal radiation with a characteristic shall use a metric with signature +2 and our con-
mavelength of the order of the Hubble radius. This ventions for the Riemann and the Ricci tensors are
would correspond to a temperature of less than
10 "'K so that it is not of much practical signifi- Va. t'g;c]
&
2 ~~d ~c Vd

cance. It is, however, important conceptually be- R+ Rg Qc ~

cause it shows that thermodynamic arguments can

be applied to the universe as a whole and that the
II. EXACT SOLUTIONS WITH COSMOLOGICAL EV'ENT
cl. ose relationship between event horizons, gravi-
HORIZONS
tational fields, and thermodynamics that was
found for black holes has a wider validity. In this section we shall give some examples of
One can regard the area of the cosmological event horizons in exact solutions of the Einstein
2740 G. W. GIBBONS AND S. W. HAWKING

equations UV =- ~(S)

g,~-~g~A+Ag, ~=8nT, . ~ (2 1) r=V ~&, V=O =-1

=0
We shall consider only the case of A positive (cor-
responding to repulsion&. Models with negative A
do not, in general, have event horizons.
The simplest example is de Sitter space which
is a solution of the field equations with T„=O.
One can write the metric in the static form UV=-1 X,u=o
r =0
ds'=-(I -Ar'3 ')dt'+dr'(I -A~'3 ') '
+x'(d H'+ sin'Hdg') . UV = 1(5)
(2. 2) r =co
This metric has an apparent singularity at r FIG. 1. Kruskal diagram of the (r, t) plane of de
= 3'/'A '/'. This singularity caused considerable Sitter space. In this figure null geodesies are at + 45
discussion when the metric was first discov- to the vertical. The dashed curves r =0 are the anti-
ered. "'" However, it was soon realized that it podal origins of polar coordinates on a three-sphere.
arose simply from a bad choice of coordinates The solid curves r = ~ are past and future infinity &
and &+, respectively. The lines r =3 ~ A are the
and that there are other coordinate systems in
past and future event horizons of observers at the ori-
which the metric can be analytically extended to
gin.
a geodesically complete space of constant curva-
ture with topology A'&S'. For a detailed descrip-
tion of these coordinate systems the reader is to make the origin of polar coordinates, r =0, and
referred to Refs. 12 and 19. For our purposes it future and past infinity, 8' and 8,
straight lines,
will be convenient to express the de Sitter metric Also shown are some orbits of the Killing vector
in "Kruskal coordinates": K=8/st. Because de Sitter space is invariant
' under the ten-parameter de Sitter group, SO(4, 1),
ds' = 3A '(U V-1)
E will not be unique. Any timelike geodesic can
x[ 4dUdV -+(UV +I)'(d 'H+si n' Hdg') j be chosen as the origin of polar coordinates and
the surfaces U=O and V=O in such coordinates
(2. 3) will be the past and future event horizons of an
where observer moving on this geodesic. If one normal-
izes E to have unit magnitude at the origin, one
~ =3'~'A-'~'(UV+I)(I -UV) (2. 4) can define a "surface gravity" for the horizon by
exp(2A'~23-'~'t) = —VU-'. {2.5) E,.~K =a~K, {2.6)
The structure of this space is shown in Fig. l. In
this diagram radial null geodesics are at +45' to '
the vertical. The dashed curves UV=-& are time- r=m, 8
like and represent the origin of polar coordinates
and the antipodal point on a three-sphere. The
solid curves VV=+1 are spacelike and represent
past and future infinity 8 and 8', respectively.
In region I (U&0, V&0, UV&-1) the Killing vec-
tor K=8/Bt is timelike and future-directed. How-
ever, in region IV (U&0, V&0, UV &—1, K is still
timelike but past-directed, while in regions II and
III (0& UV&1) K is spacelike. The Killing vector K
is null on the two surfaces U=O, V=(f. These are
respectively the future and past event horizons for
any observer whose world line remains in region
I; in particular for any observer moving along a
curve of constant r in region I.
By applying a suitable conformal transformation
one can make the Kruskal diagram finite and con- r =oo, 8
vert it to the Penrose-Carter form (Fig. 2). Radi- FIG. 2. The Penrose-Carter diagram of de Sitter
al null geodesics are still. +45' to the vertical but space. The dotted curves are orbits of the Killing vec-
the freedom of the conformal factor has been used tor.
 COSMOLOGICAL EVENT HORIZONS, THERMODYNAMICS, AND. . . 2741

on the horizon. This gives left-hand side of the diagrari there is another a
Aj /23 -1/2 asymptotically flat region IV. The Killing vector
C (2.7}
K = 8/&t is now uniquely defined by the condition
The area of the cosmological horizon is that it be timelike and of unit magnitude near 8'
and ~ . It is timelike and future-directed in re-
Ac = 12''A (2. &)
gion I, timelike and past-directed iri region IV,
One can also construct solutions which general- and spacelike in regions II and ID. The Killing

"
ize the Kerr-Newman family to the case when A
is nonzero. ' The simplest of these is the
Schwarzschild-de Sitter metric. When A = 0 the
vector E is null on the horizons which have area
A„=16aM'. The surface gravity, defined by (2. 6},
is ~„=(4M) '.
unique spherically symmetric vacuum spacetime The Schwarzschild solution is usual. ly interpreted
is the SchwarzschiM solution. The metric of this as a black hole of mass M in an asymptotically flat
can be written in static form: space. There is a straightforward generalization
ds'=-(1 —2M' ')dt'+dr'(I -2M' ') ' to the case of nonzero A which represents a black
hole in asymptotically de Sitter space. The metric
+r'(d 8'+ sin'8d P') . (2.9} can be written in the static form
As is now well known, the apparent singularities ds'=-(I -2M' ' —Ar'3 ')dt'
at ~= 2M correspond to a horizon and can be re- ')-'
moved by changing to Kruskal coordinates in which
+d~'(I 2Mr-'-A~'3
the metric has the form +r'(d 8' + sin'8dg') . (2. 13)
sd'=- 23M' 'exp( 2'M -'r)dUd V If A & 0 and 9A M' & 1, the factor (1 —2M' ' -Ax'3 ')
+x'(d 8'+ sin'8d p'), (2. 10) is zero at two positive values of x. The smaller of
of these values, which we shall denote by x„can
where be regarded as the position of the black-hole event
UV=(1 2'M-'r} exp(2 'M 'x} (2. 11) horizon, while the larger value x„represents the
position of the cosmological event horizon for ob-
and servers on world lines of constant r between r,
UV '=-exp(-2 'M 't). . (2. 12) and x„. By using Kruskal coordinates as above
one can remove the apparent singularities in the
The Penrose-Carter diagram of the Schwarzschild metric at x, and r„. One has to employ separate
solution is shown in Fig. 3. The wavy lines marked coordinate patches at r, and x„.
We shall not
~ = 0 are the past and future singularities. Region give the expressions in full because they are rath-
I is asymptotically flat and is bounded on the right er messy; however, the general structure can be
by past and future null infinity 8 and O'. It is seen from the Penrose-Carter diagram shown in
bounded on the left by the surfaces U=O and V=O, Fig. 4. Instead of having two regions (I and Iv) in
r =2M. These are future and past event horizons which the Killing vector K = &/&t is timelike, there
for observers who remain outside x= 2M. On the are now an infinite sequence of such regions, also
labeled I and IV depending upon whether K is,
future- or past-directed. There are also infinite
r= 0 sequences of x=O singularities and spacelike in-
finities ~' and ~ . The surfaces x =~, and x =~„
are black-hole and cosmological event horizons
for observers moving on world lines of constant

r=0 r=Q

r=Q r=Q
FIG. 3. The Penrose-Carter diagram of the Schwarzs-
child solution. The wavy lines and the top and bottom FIG. 4. The Penrose-Carter diagram for Schwarzs-
are the future and past singularities. The diagonal child-de Sitter space. There is an infinite sequence of
lines bounding the diagram on the right-hand side are singularities r = 0 and spacelike infinities r = ~. The
the past and future null infinity of asymptotically flat Killing vector &= 8/8t is timelike and future-directed
space. The region IV on the left-hand-side is another in regions I, timelike and past-directed in regions IV
asymptotically flat space. and spacelike in the others.
2742 G. W. GIBBONS AND S. W. HAWKING 15

x between r, and x„. There are apparent singularities in the metric

The Killing vector K = &/St is uniquely defined at the values of r for which 6„=0. As before,
by the conditions that it be null on both the black- these correspond to horizons and can be removed
hole and the cosmological horizons and that its by using appropriate coordinate patches. The Pen-
magnitude should tend to A' '3 ' 'r as z tends to rose-Carter diagram of the symmetry axis (8=0)
infinity. One can define black-hole and cosmolo-
logical surface gravities aH and ~~ by that 4, has 4 distinct roots: r, r,
of these spaces is shown in Fig. 5 for the case
r „and
As before, r„and r, can be regarded as
K, .~E = gK, (2. 14)
the cosmological and black-hole event horizons,
on the horizons. These are given by respectively. In addition, however, there is now
an inner black-hole horizon at r =x . Passing
a„=A6 '~, '(r„-r, )(r, -r ), (2. 15 a)
through this, one comes to the ring singularity at
zc=A6 'x, , '(r„-r, )(r„— r ), (2. 15b) r =0, on the other side of which there is another
cosmological horizon at x=r and another infin-
where x = x is the negative root of
ity. The diagram shown is the simplest one to
3~ -6M -Ar' = 0. (2. 16) draw but it is not simply connected; one can take
covering spaces. Alternatively one can identify
The areas of the two horizons are
regions in this diagram.
&„=4ay, ' (2. 17) The Killing vector K= &/Q is uniquely defined by
the condition that its orbits should be closed
and
curves with parameter length 2w. The other Kiii-
(2. 18)
If one keep~ A constant and increases M, r, will
increase and x„will decrease. One can under-
stand this in the following way. When M=0 the
gravitational potential g(s/at, 8/st) is 1 Ar'3 -'.
The introduction of a mass M at the origin pro-
duces an additional potential of -2Mr '. Horizons
occur at the two values of r at which g(S/St, S/St)
vanishes. Thus as M increases, the black-hole
horizon r, increases and the cosmological hori-
zon r„decreases. When 9AM' =1 the two hori-
zons coincide. The surface gravity E can be
thought of as the gravitational field or gradient
of the potential at the horizons. As M increases
both ~H and v~ decrease.
The Kerr —Newman-de Sitter space can be ex-
pressed in Boyer-I indquist-type coordinates
as20, 21

ds = p (b, „dr +b, e 'de )

+p '= ' AI eatd(r' a+')dQ]'-

'p '(dt-asin'&dP)', (2. 19)
where
p2 = y'2 + g2 COS2 g (2. 20)
FIG. 5. The Penrose-Carter diagram of the symme-
& =(r +a')(1 Ay'3 '} -2Mr+Q', — (2. 21) try axis of the Kerr —Newman —de Sitter solution for
Ee =1+Aa 3 cos (2. 22) the case that Q has four distinct real roots. The in-
finities r =+~ and r =- ~ are not joined together. The
==1+ha'3 '. (2. 23) external cosmological horizon occurs at r =r++ the ex-
terior black-hole horizon at r = r+, the inner black-hole
The electromagnetic vector potentials, is given by horizon at r =r . The open circles mark where the ring
A, =Qrp '= '(6', -a sin'Ã~}. (2. 24) singularity occurs, although this is not on the symmetry
axis. On the other side of the ring at negative values of
Note that our A has the opposite sign to that in r there is another cosmological horizon at r =r and
Ref. 21. another infinity.
 COSMOLOGICAL EVENT HORIZONS, THERMODYNAMICS, AND. ..
i. e. , I J'(3) is
ing vector K = 8/at is not so specially picked out.
One can add different constants multiples of Jf to development
(A) A
8'(S). "contained
We shall
in the future Cauchy
also assume that
K to obtain Killing vectors which are null on the I J'(&), the portion of the event horizon to
(&) &
different horizons and one can then define surface the future of S, is contained in D'(S). Such an
gravities as before. We shall be interested only event horizon will be said to be predictable. The
in those for the r„x„horizons. They are event horizon will be generated by null geodesic
z„=A6 '= '(r, r -)(r, r)-(r„r-, }(r,'+a') ', segments which have no future end points but
which have past end points if and where they in-
(2.25) tersect other generators. "
In another paper" it
ec=A6 '. '(r„r, )-(r„r)(r-„r)(r-,'+a'} '. is shown that the generators of a predictable event
horizon cannot be converging if the Einstein equa-
(2.26) tions hold (with or without cosmological constant),
provided that the energy-momentum tensor sat-
The areas of these horizons are
isfies, the strong energy condition T,„u'u'
W „=4~(r, '+ a'), (2.21) -3 1';u'u, for any timelike vector u„ i. e. , pro-
A c = 4m(r„'+a') . (2. 28)
vided that p, +P; ~0, p, +L',
P; ~0, where )j. is
the energy density and P& are the principal pres-
sures. This gives immediately the following re-
III. CLASSICAL PROPERTIES OF EVENT HORIZONS
sult, which, because of the very suggestive anal-
In this section we shall generalize a number of ogy with thermodynamics, we call:
results about black-hole event horizons in the The second law of event horizons: The area of
classical theory to spacetimes which are not any connected two surfa-ce in a Predictable event
asymptotically flat and may have a nonzero cosmo- horizon cannot decxease saith time. The area may
logical constant, and to event horizons which are be infinite if the two-di. mensional cross section is
not black-hole horizons. The event horizon of a not compact. However, in the examples in Sec. II,
black hole in asymptotically flat spacetimes is the natural two-sections are compact and have con-
normal. ly defined as the boundary of the region stant area.
from which one can reach future null infinity, ~', In the case of gravitational collapse in asymp-
along a future-directed timelike or null curve. In totically flat spacetimes one expects the space- .
other words it is J (8') [or equivalently I (&')], time eventually to settle down to a quasistationary
where an overdot indicates the boundary and J state because all the avail. able energy will either
is the causal past (I is the chronological past). fall through the event horizon of the black hole
However, one can also define the black-hole hori- (thereby increasing its area} or be radiated away
zon as I (A. ), the boundary of the past of a time- to infinity. In a similar way one would expect that
like curve A. which has a future end point at future where the intersection of I (A) with a spacelike
timelike infinity, i'in Fig. 3. One can think of A. surface S had compact closure (which we shall as-
as the world line of an observer who remains out- sume henceforth}, there would only be a finite
side the black hole and who does not accelerate amount of energy available to be radiated through
away to infinity. The event horizon is the bound- the cosmological event horizon of the observer
ary of the region of spacetime that he can see if and that therefore this spacetime would eventually
he waits long enough. It is this definition of event approach a stationary state. One is thus lead to
hori. zon that we shall extend to more general consider solutions in which there is a Killing vec-
spacetimes which are not asymptotically flat. tor K which is timelike in at least some region of
Let ~ be a future inextensible timelike curve I (A) ll J'(8). Such solutions would represent the
representing an observer's world line. For our asymptotic future l. imit of general spacetimes with
considerations of particle creation in the next sec- predictable event horizons.
tion we shall require that the observer have an in- Several results about stationary empty asymp-
definitely long time in which to detect particles. totically flat black-hol, e solutions can be general-
We shall therefore assume that A, has infinite prop- ized to stationary solutions of the Einstein equa-
er length in the future direction. This means that tions, with cosmological constant, which contain
it does not run into a singularity. The past of A. , predictable event horizons. The first such theo-
I (X), is a terminal indecomposable past set, or rem is that the null geodesic generators of each
TIP in the language of Geroch, Kronheimer, and connected component of the. event horizon must
Penrose. ' It represents all the events that the ob- coincide with orbjts of some Killing vector. 4' '
server can ever see. We shall assume that what These Killing vectors may not coincide with the
the observer sees at late times can be predicted original Killing vector K and may be different for
(classically at least) from a spacelike surface 3, different components of the horizon. In either of
 G. W. GIBBONS AND S. W. HA WKING

these two cases there are at least two Killing vec- temperature proportional to its surface gravity.
tors. One can chose a linear combination K whose One can also generalize the first law of black
orbits are spacelike closed curves in I (&} A J (I). holes. %e shall do this for stationary axisymme-
One coul. d interpret this as implying that the solu- tric solutions with no electromagnetic field and
tion is axisymmetric as wel, l as being stationary, where I (A} & J'(S) consists of two components,
though we have not been able to prove that there a black-hole event horizon and a cosmological
is necessarily any axis on which K vanishes. event horizon. Let K be the Killing vector which
Let E be the Killing vector which coincides with is null. on the cosmological event horizon. The
the generators of one component of the event hori- orbits of K will constitute the stationary frame
zon. If K is not hypersurface orthogonal and if which appears to be nonrotating with respect to
then space is empty or contains only an electro- distant objects near the cosmological event hori-
magentic field, one can apply a generalized zon. In the general case the normalization of K
Lichnerowicz theorem"" to show that K must be is somewhat arbitrary but we shall assume that
spacelike in some "ergoregion" of I (A). One can some particular normalization has been chosen.
then apply energy extraction arguments"'" or The Killing vector K which coincides with the gen-
the results of Hajicek" to show that this ergore- erators of the bl. ack-hole horizon can be expressed
gion contains another component of the event hori- in the form
zon whose generators do not coincide with the or- K=K'+0 K, (3.1)
bits of K. It therefore follows that either K is
hypersurface orthogonal (in which case the solu- where OH is the angular velocity of the black-hole
tion is static) or that there are at least two Kill- horizon relative to the cosmological horizon in
ing vectors (in which case the solution is axis- the units of time defined by the normalization of
symmetric as well as stationary). If there is K and E is the uniquely defined axial Killing vec-
only a cosmological horizon and no black-hole tor whose orbits are cl.osed curves with parame-
horizon, then the solution is necessarily static. ter length 2r.
One would expect that in the static vacuum case For any Killing vector field $' one has
one could generalize Israel's theorem" to prove (a;b It a Tb
(3.2)
that the space was spherically symmetric. One 4

could then generalize Birkhoff's theorem to in- Choose a three-surface 8 which is tangent to K,
clude a cosmological constant and show that the and integrate (3.2) over it with )=K. On using
space was necessarily the Schwarzschild-de Sitter Einstein's equations this gives
space described in Sec. II. In the case that there
was only a cosmological event horizon, it would (8rr) f(C
H
d'E+(8"')rr'
C
lC"'dE„JT"Cr,dE, , =
be de Sitter space. In the stationary axisymme-
(3.3}
tric case one would expect that one could general-
ize and extend the results of Carter and Robin- where the three-surface integral on the right-hand
son"' to show that vacuum solutions were mem- side is taken over the portions of 8 between the
bers of the Kerr-de Sitter family described in black-hole and cosmological horizons and the
Sec. II. If there is matter present it mill distort two-surface integrals marked H and C are taken
the spacetime from the Schwarzschild-de Sitter over the intersections of S with the respective
or Kerr-de Sitter solution just as matter around horizons, the orientation being given by the direc-
a black hole in asymptotically flat space will dis- tion out of I (A. ). One can interpret the right-hand
tort the spacetime away from the Schwarzschild side of (3.3) as the angular momentum of the
or Kerr solution. matter between the two horizons. One can there-
The proof given in Ref. 13 of the zeroth law of fore regard the second term on the left-hand side
black holes can be generalized immediately to of (3.3) as being the total angular momentum, Zc,
the case of nonzero cosmological constant. One contained in the cosmological horizon, and the
thus has: first on the left-hand side term as the negative of
The zeroth laze of event horizons: The surface the angular momentum of the black hole, JH.
gravity of a connected component of the event hori- One can also apply Eq. (3.2) to the Killing vec-
zon I (A. ) is constant over that component. This tor K to obtain
is analogous to the zeroth law of nonrelativistic
thermodynamics which states that the tempera- (4lr) ' ll"'dE. , +(4lr) 'fir"'dE
ture is constant over a body in thermal equili-
brium. We shall show in Secs. IV and V that
quantum effects cause each component of the
= JiE(T., ', T;4 ,)A r)E, + f A(4rr) AdE=. ',
event horizon to radiate thermally with a (3.4)
 COSMOLOGICAL EVENT HORIZONS, THERMODYNAMICS, AND. .. 2745

One can regard the terms on the right-hand side ions of particles that were observer-independent
of Eq. (3.4) as representing respectively the (posi- and invariant under the de Sitter group. Under
tive) contribution of the matter and the (negative) these conditions only two answers are possible
contribution of the A term to the mass within the for the rate of particle creation per unit volume,
cosmological horizon. One can therefore regard zero or infinity, because if there is nonzero pro-
the second term on the left-hand side as the (nega- duction of particles with a certain energy, then
tive) mass Mc within the cosmological horizon by de Sitter group invariance there must be the
and the first term on the Left-hand side as the same rate of creation of particles with all other
negative of the (positive) mass Ms of the black energies. It is therefore not surprising that the
hole. As in Ref. 13, one can express M„and M~ authors mentioned above chose their definitions
of particles to get the zero answer.
An observer-independent definition of particles
M„= K„A„(4ir) '+2Q„J„, (3.5)
is, however, not relevant to what a given observ-
Mc —-KcAc(4i(') (3.6) er would measure with a particle detector. This
depends not only on the spacetime and the quantum
One therefore has the Smarr-type" formulas
state of the system, but also on the observer's
M, =-x, A, (4v}-' world line. For example, Unruh" has shown that
in Minkowski space in the normal vacuum state
=«„A«(4«) '«2()«Z„«J2(T., T;(! ,)K-'«():. accelerated observers can detect and absorb par-
ticles. To a nonaccelerating observer such an
' AK, dZ'. absorption will appear to be emission from the
+(4)(') (3.7)
accelerated observer's detector. In a similar
manner, an observer at a constant distance from
One can take the differential of the mass formu-
a black hoke will detect a steady flux of particles
la in a manner similar to that in Ref. 13. One ob-
coming out from the hole with a thermal spectrum
tains:
while an observer who falls into the hole will not
The first lacu of event horizons.
see many particles.
A feature common to the examples of a uniform-
l 5T~E'dZ = —z, 5Ac(8tr) K„t')A„(8z) --Q„5J„,
(3.8)
ly accelerated observer in Minkowski space and
an observer at constant distance from the black
hole is that both observers have event horizons
where &T„ is the variation in the matter energy-
which prevent them from seeing the whole of the
momentum tensor between the horizons in a gauge
spacetime and from measuring the complete quan-
in which 5E'= 6E'=0.
tum state of the system. It is this loss of informa-
From this law one sees that if one regards the
tion about the quantum state which is responsible
area of a horizon as being proportional to the en- for the thermal radiation that the observers see.
tropy beyond that horizon, then the corresponding Because any observer in de Sitter space also has
surface gravity is proportionaL to the effective
an event horizon, one would expect that such an ob-
temperature of that horizon, that is, the tempera-
server would also detect thermal radiation. %e
ture at which that horizon would be in thermal
shall show that this is indeed the case. This can
equilibrium and therefore the temperature at
be done either by the frequency-mixing method in
which that horizon radiates. In the next section
which the thermal radiation from black holes was
we shall show that the factor of proportionality "'"
between temperature and surface gravity is (2i(') '.
This means that the entropy is & the area. In the
first derived,
of Hartle and Hawking. "
or by the path-integral method
We shall adopt the Latter
approach because it is more elegant and gives a
case of the cosmological horizon in de Sitter
space the entropy is SmA '&10" because A & 10 . " clearer intuitive picture of what is happening.
The same results can, however, be obtained by

IV. PARTICLE CREATION IN DE SITTER SPACE

the former method.
As in the method of Hartle and Hawking, " we
construct the propagator for a scalar field of
In this section we shall calculate particle crea- mass m by the path integral
tion in solutions of the Einstein equations with
positive cosmological constant. The simplest ex-
G(x, x'} = lim dWF(W, x, x') exp[-(im'W+eW ')],
"
(I

ample is de Sitter space and particle production 0 0

"" "
in this situation has been studied by Nachtmann,
Tagirov, Candelas and Raine, and Dowker and (4 &)
Critchley, among others. They all used definit- where
2746 G. W. G IBBO N S AN 0 S. W. HA WKING 15

Ref. 10 me define the complexified horizon by Ax'

)l())', x, x') = f()x{w] exp 4 0 ){i,i)dw (4.2) =3, 8, Q real. On the complexified horizon X, Y,
and Z are real and either Z'=S=p-'~'3'~'V, U=O
and the integral is taken over all paths x(w) from or T=-S=A '~'3'~'U, V=O. By Eq. (4. 7) a com-
x to x'. plex null geodesic from a real point (T', S', X', Y', Z')
As in the Hartle and Hawking paper, ' this path on the hyperboloid can intersect the complex hori-
integral can be given a well-defined meaning by zon only on the real sections T=+S real. If the
analtyically continuing the parameter W to nega- point (T', S', X', Y', Z') is in region I (S&lTl) the
tive imaginary values and analytically continuing propagator G(x', x) will. have a singularity on the
the coordinates to a region where the metric is past horizon at the point mhere the past-directed
positive-definite. A convenient way of doing this null geodesic from x' intersects the horizon. As
is to embed de Sitter space as the hyperboloid shown in Ref. 10, the e convergence factor in (4. 1}
—T'+S'+X'+ F'+Z'=3A ' (4.3) will displace the pole slightly below the real axis
in the complex plane on the complexified past hori-
in the five-dimensional. space with a Lorentz me- zon. The propagator G(x', x) is therefore analytic
tric: in the upper half U plane on the past horizon. Sim-
ds =-dT +QS +/X +d Y +dZ (4.4) ilarly, it will be analytic in the lower Vplane on
the future horizon.
Taking T to be i w (T real}, we obtain a sphere in The propagator G(x', x) satisfies the wave equa-
five-dimensional Euclidean space. On this sphere tion
the function F satisfies the diffusion equation
(o;-m')G(x', x) =-5(x, x) (4. 13)
Q2F (4. 5} Thus if x' is a fixed point in region I, the value
G(x', x) for a point in region II will be determined
where 0 = i W and 2' is the Laplacian on the four- by the values of G(x', x) on a characteristic Cauchy
sphere. Because the four-sphere is compact there surface for region II consisting of the section of
is a unique solution of (4. 5} for the initial condition the U=O horizon for real V&0 and the section of
the V=O horizon for real U&0. The coordinates
Z(O, x, x') = 5(x, x'), (4. 6)
x and t of the point x are related to U and V by
where 5(x, x') is the Dirac 5 function on the four- 82 KQg (4. 14)
sphere. One can then define the propagator '
G(x, x') from (4. 1) by analytically continuing the r= (1+UV)(1 —UV) 'Kc (4. 15)
solution for F back to real values of the parame- If one holds r fixed at a real value but lets t = 7+ia,
ter W and real coordinates x and x'. Because the then
function F is analytic for finite points x and x',
U= lUl exp(-ioKc), (4. 16)
any singularities which occur in G(x, x') must
come from the end points of the integration in V= I Vl exp(+ivKc). (4. 17)
(4. 1). As shown in Ref. 10, there will be singu-
larities in G(x, x') when, and only when, x and x' For a fixed value of the metric (2.3) of de Sitter
o'

can be joined by a null geodesic. This mill be the space remains real and unchanged. Thus the val-
case if and only. if ue of G(x', x) at a complex coordinate t of the point
,

x but real r, 8, Q can be obtained by solving the

(T T')' = (S -S')'+-(X-X')'+ (Y'- 1")'+(Z -Z')'. Klein-Gordon equation with real coefficients and
(4. 7) mith initial data on the Cauchy surface V=O,
U= IUI exp(-tKco') and U=O) V= IVI exp(+iKc(T).
The coordinates, T, S, X, F, Z can be related to Because G(x', x} is analytic in the upper half U
the static coordinates t, r, 8, Q used in Sec. 11 by plane on V=O and the lomer half Vplane on U=O,
T=(A3 ' r')')'sinhA'~'3 -'~ t (4.8) the data and hence the solution will be regular pro-
vided that
S= (A3 ' r')'~'coshA'~'3 -'~'t (4.9)
(4. 18)
X=r sin8cos{t), (4. 10)
The operator
Y=r sin8sing, (4. 11)
Z =~ cos8. (4. 12)
(4.19}

The horizons Ax'=3 are the intersection of the commutes with the Klein-Gordon operator H„'- m'
hyperplanes T = ~S.with the hyperboloid. As in and is zero when acting on the initial data for a
15 COSMOLOGICAL EVENT HORIZONS, THKRMOD YNAMICS, AND. ..
satisfying (4. 18). Thus the solution G(x', x) de- surface which completely surrounds the observ-
termined by the initial data will be analytic in the er's world line. If the observer detects a par-
coordinates t of the point x for o satisfying Eq. ticle, it must have crossed 6' in some mode k&
(4. 18). which is a solution of the Klein-Gordon equation
This is the basic result which enables us to with unit Klein-Gordon norm over the hypersur-
show that an observer moving on a timelike geo- face O'. The amplitude for the observer to detect
desic in de Sitter space will detect thermal radi- such a particle will be
a, tion.
The propagator we have defined appears to be
similar to that constructed by other authors. ' " (4.20)

However, our use of the propaga, tor will be dif- where the volume integral in x' is taken over the
ferent: Instead of trying to obtain some observer- volume of the particle detector and the surface
independent measure of particle creation, we shall integral in x is taken over 6'.
be concerned with what an observer moving on a The hypersurface 6' can be taken to be a space-
timelike geodesic in de Sitter space would mea- like surface of large constant r in the past in
sure with a particle detector which is confined to region III and a spacelike surface of large con-
a small tube around his world line. Without loss stant r in the future in region II. In the limit that
of generality we can take the observer's world r tends to infinity these surfaces tend to past in-
line to be at the origin of polar coordinates in finity 5 and future infinity d, respectively. We
region I. Within the world tube of the particle shall assume that there were no particles present
detector the spacetime can be taken as flat. on the surface in the distant past. Thus the only
The results we shall obtain are independent of contribution to the amplitude (4.20) comes from
the detailed nature of the particle detector. How- the surface in the future. One can interpret this
ever, for explicitness we shall consider a particle as the spontaneous creation of a pair of particles,
model of a detector similar to that discussed by one with positive and one with negative energy
Unruh" for uniformly accelerated observers in with respect to the Killing vector K = a/St. The
flat space. This will consist of some system such particle with positive energy propagates to the
as an atom which can be described by a nonrela- observer and is detected. The particle with neg-
tivistic Schrodinger equation ative energy crosses the event horizon into region
II where K is spaeelike. It ean exist there as a
real particle with timelike four-momentum. .
Equivalently, one can regard the world lines of
where t' is the proper time along the observer's the two particles as being the world line of a
world line, B, is the Hamiltonian of the undis- single particle which tunnels through the event
turbed particle detector and g(t)C is a coupling horizon out of region II and is detected by the
term to the scalar field P. The undisturbed par- observer.
ticle detector will have energy levels E, and Suppose the detector is sensitive to particles of
wave functions 4', (R')e 's~', where R' represents a certain energy E. In this case the positive-
the spatial position of a point in the detector. f
frequency-response function (t) will be propor-
By first-order perturbation theory the ampli- tional to e '~'. By the stationarity of the metric,
tude to excite the detector from energy level E, the propagator G(x', x) can depend on the coordi-
to a higher-energy level EJ is proportional to nates t' and t only through their difference. This
means that the amplitude (4.20) will be zero ex-
dt' d'R'4 gal+, exp[
& i(E& —E-~) t']. cept for modes k& of the form X(r, 8, y) e ' '. H
one takes out a 6 function which arises from the
In other words, the ctetector responds to compo- integral over t —t', the amplitude for detection
nents of field Q which are positive frequency along is proportional to
the observer's world line with respect to his
proper time. By superimposing detector levels (4.21)
with different energies one can obtain a detector
response function of a form where R' and R denote respectively (r', O', Q') and
(r, 8, (t)) and the radial and angular integrals over
f (t')&(R) the functions h and X have been factored out.
where f(t') is a purely positive-frequency func- Using the result derived above that G(x', x) is
tion of the observer's proper time t' and h is zero analytic in a strip of width me~ ' below the real
outside some value of z' corresponding to the t axis, one can displace the contour in (4.21)
radius of the particle detector. Let 6' be a three- down my~ ' to obtain
2748 G. W. GIBBONS AND S. W. HAWKING

ii (R', B)=exp( wE~-s )f'de i' 'G(O, R', t-ivy ', H), (4.22)

By Eqs. (4.16) and (4. 17) the point (t-isxc, r, 8, cp) is the point in region III obtained by reflecting in the
origin of the U, V plane. Thus
/amplitude for particle with energy)
for particle of energy E to propagate
~ ~

amplitude
=exp mE-xc ' E to propagate from region III and
from region II and be absorbed by observer
gabe absorbed by observer

(4.23)
By time-reversal invariance the latter amplitude is equal to the amplitude for the observer s detector in
an excited state to emit a particle with energy E which travels to region II. Therefore
probability for detector to
~ ~

a particle from region II

~
absorb, =exp 2@Ex-c ' probability for detector to emit
a particle to region II
(4.24)

This is just the condition for the detector to be in coordinates T, S, T', S', or alternatively U, V, O', V'
thermal equilibrium at a temperature except when x and x' can be joined by null geo-
desics. On the other hand, the static-time co-
T =(2m) 'xc-=(12) "'v 'A'" (4.25) ordinate t is a multivalued function of T and S or
U and V, being defined only up to an integral mul-
The observer will therefore measure an isotropic
tiple of 2n'ice '. Thus the propagator G(x', x) is
background of thermal radiation with the above
temperature. Because all timelike geodesics are
a periodic function of t with period 2mi~~ .
This
behavior is characteristic of what are known as
"thermal Green's functions. '"' These may be de-
equivalent under the de Sitter group, any other
observer will also see an isotropic background
fined (for interacting fields as well as the non-
with the same temperature even though he is
interacting case considered here) as the expecta-
moving relative to the first observer. This is
tion value of the time-ordered product of the field
yet another illustration of the fact that different
operators, where the expectation value is taken
observers have different definitions of particles.
not in the vacuum state but over a grand canonical
It would seem that one cannot, as some authors
ensemble at some temperature T =P '. Thus
have attempted, construct a unique observer-
independent renormalized energy-momentum ten- Gr(x', x) =iTr[e 8
"gy(x)y(x))/Tre 8",
sor which can be put on the right-hand side of
the classical Einstein equations. This subject (4.26)
will be dealt with in another paper. " where 8 denotes Wick time-ordering and H is the
Another way in which one can derive the result Hamiltonian in the observer's static frame. P is
that a freely moving observer in de Sitter space the quantum field operator and Tr denotes the
will see thermal radiation is to note that the trace taken over a complete set of states of the
propagator G(x, x') is an analytic function of the system. Therefore

-iGr (R', I', R, t) = Tr [e 8 "8 P(R, t) rp(R, t ') ] /T re 8"

=Tr[e 8 "8 rp(R', f'}e "e "Q(R, i)]/Tre
=Tr[e '%y(R', t'+iP)y(R', i}]/Tre '"
=-iGr(R', t'+iP; R, i). (4.27)
I

Since G(x', x) that we have defined by a path integral is

Q(R, t}=e "@(R,t-iP}e ". (4.28)
the same as the thermal propagator Gr(x', x) for
a grand canonical ensemble at temperature T
Thus the thermal propagator is periodic in t —t' T = (2x) 'xc in the observer's static frame. Thus
with period i T '. One would expect Gr(x', x) to to the observer it will seem as if he is in a bath
have singularities when x and x' can be connected of blackbody radiation at the above temperature.
by a null geodesic and these singularities would It is interesting to note that a similar result was
be repeated periodically in the complex t'- t found for two-dimensional de Sitter space by
plane. It therefore seems that the propagator Figari, Hoegh-Krohn, and Nappi'4 although they
15 COSMOLOGICAL EVENT HORIZONS, THERMODYNAMICS, AND. ..
did not appreciate its significance in terms of There are, however, certain problems in show-
particle creation. ing that this is the case. These difficulties arise
The correspondence between G(x', x) and the from the fact that when one has two or more sets
thermal Green's function is the same as that of horizons with different surface gravities one

by Gibbons and Perry.

"
which has been pointed out in the black-hole case
As in their paper, one
has to introduce separate Kruskal-type coordi-
nate patches to cover each set of horizons. The
can argue that because the free-field propagator coordinates of one patch will be real analytic. func-
G(x, x) is identical with the free-field thermal tions of the coordinates of the next patch in some
propagator Gr(x', x), any n-point interacting overlap region between the horizons in the real
Green's function G which can be constructed by manifold. However, branch cuts arise if one
perturbation theory from G in a renormalizable continues the coordinates to complex values. To
field theory will be identical to the n-point inter- see this, let U„V, be Kruskal coordinates i.n a
acting thermal Green's function constructed from patch covering a pair of intersecting horizons
G~ in a similar manner. This means that the re- with a surface gravity z, and let U„V, be a neigh-
sult that an observer will think himself to be boring coordinate patch covering horizons with
immersed in blackbody radiation at temperature surface gravity K, . In the overlap region one has
7 = ~(2m) ' will be true not only in the free-field
case that we have treated but also for fields with VU-&—
1 1 (5 l)
mutual interactions and self-interactions. In 1 2K2t
V2 U2 (5.2)
particular, one would expect it to be true for the
gravitational field, though this is, of course, not Thus
repormalizable, at least in the ordinary sense.
It is more difficult to formulate the propagator (5.3)
for higher-spin fields in terms of a path integral.
However, it seems reasonable to define the prop- where I'= gz, '. There is thus a branch cut in
agators for such fields as solutions of the relevant the relation between the two coordinate patches if
inhomogeneous wave equation with the boundary K24 K~,
conditions that the propagator from a point x' in One way of dealing with this problem would be to
region I is an analytic function of x in the upper imagine perfectly reflecting walls betweeri each
half U plane and lower half V plane on the com- -black-hole horizon and each cosmological horizon.
plexified horizon. With this definition one ob- These walls would divide the manifold up into a
tains thermal radiation just as in the scalar case. number of separate regions each of which could
be covered by a single Kruskal-coordinate patch.
U. PARTICLE CREATION IN BLACK-HOLE In each region one could construct a propagator
DE SITTER SPACES as before but with perfectly reflecting boundary
conditions at the walls. By arguments similar
For the reasons given in Sec. III one would ex- to those given in the previous section, these prop-
pect that a solution of Einstein's equations with agators will have the appropriate periodic and
positive cosmological constant which contained analytic properties to be thermal Green's functions
a black hole would settle down eventually to one with temperatures given by the surface gravities
of the Kerr-Newman-de Sitter solutions described of the horizons contained within each region. Thus
in Sec. II. We shall therefore consider what would an observer on the black-hole side of a wall will
be seen by an observer in such a solution. Con- see thermal radiation with the black-hole tempera-
sider first the Schwarzschild-de Sitter solution. ture, while an observer on the cosmological side
Suppose the observer moves along a world line A, of the wall will see radiation with the cosmological
of constant r, 8, and Q in region l of Fig. .. 4. The temperature. One would expect that, if the walls
world line A, coincides with an orbit of the static were removed, an observer would see a n'. ixture
.
Killing vector K = s/st. Let qr' =g(K, K) on h. One of radiation as described above.
would expect that the observer would see thermal Another way of dealing with the problem would
radiation with a temperature Tc = (2m/) 'ac coming be to define the paopagator G(x', x) to be a solution
from all directions except that of the black hole of the inhomogeneous wave equation on the real
and thermal radiaf, ion of temperature T s= (2m@) 'xs manifold which was such that if the point were
coming from the black hole. The factor g appears extended to complex values of a Krushal-type-
in order to normalize the static Killing vector to coordinate patch covering one set of intersecting
have unit magnitude at the observer. The varia- horizons, it would be analytic on the complexified
tion of g with r can be interpreted as the normal horizon in the upper half or lower half U or V
red-sbifting of temperature. plane depending on whether the point x was re-
2750 G. W. GIBBONS AND S. W. HA W KIN G

spectively to the future or the past of V =0 or gion II~ is analytic in a strip of width n~ below
U=O. Then, using a similar argument to that in the real axis of the complex t plane. Similarly,
the previous section about the dependence of the the propagator G(x', x) between a point x' in re-
propagator on initial data on the complexified gion I and a point x in region IIH will be analytic
horizon, one can show that the propagator G(x', x) in a strip of width wc~ . Using these results one
between a point x in region I and a point x in re- can show that

(probability of a particle of energy E, ) (probability of a particle of energy E, )

relative to the observer, propagating ~=expj-(E2~gzc ')] relative to the observer, propagating l,
( from g' to observer (from observer top' ~

(5.4)

and similarly the probability of propagating from VI. IMPLICATIONS AND CONCLUSIONS
the future singularity of the black hole will be
related by the appropriate factor to the probability We have shown that the close connection be-
for a similar particle to propagate from the ob- tween event horizons and thermodynamics has a
server into the black hole. These results estab- wider validity than the ordinary black-hole situa-
lish the picture described at the beginning of this tions in which if was first discovered. As observer
section. in a cosmological model with a positive cosmo-
One can derive similar results for the Kerr- logical constant will have an event horizon whose
de Sitter spaces. There is an additional complica- area can be interpreted as the entropy or lack of
tion in this case because there is a relative angular information that the observer has about the regions
velocity between the black hole and the cosmologi- of the universe that he cannot see. When the solu-
cal horizon. An observer in region I who is at a tion has settled down to a stationary state, the
constant distance r from the black hole and who is event horizon will have associated with if a surface
nonrotating with respect to distant stars will gravity z which plays a role similar to tempera-
move on an orbit of the Killing vector K which is ture in the classical first law of event horizons
null on the cosmological horizon. For such an derived in Sec. III. As mas shown in Sec. IV. ,
observer the probability of a particle of energy E, .
this similarity is more than an analogy' The ob-
relative to the observer, propagating to him from server will detect an isotropic background of
beyond the future cosmological horizon will be thermal radiation with temperature (2a) 'a coming,
exp[- (2mgEv~ ')] times the probability for a sim- apparently, from the event horizon. This result
ilar particle to propagate from the observer to was obtained by considering what an observer
beyond the cosmological horizon. The probabili- with a particle detector would actually measure
ties for emission and absorption by the black hole rather than by trying to define particles in an
will be similarly related except that in this case observer-independent manner. An illustration of
the energy E mill be replaced by E —nQH, where the observer dependence of the concept of particle
n is the aximuthal quantum number or angular mo- is the result that the thermal radiation in de Sitter
mentum of the particle about the axis of rotation of space appears isotropic and at the same tempera-
the black hole and Q~ is the angular, velocity of the ture to every geodesic observer. If particles had
black-hole horizon relative to the cosmological an observer-independent existence and if the radi-
horizon. As in the ordinary black-hole case, the ation appeared isotropic to one geodesic observer,
black hole will exhibit superradiance for modes it would not appear isotropic to any other geodesic
for which E& nQ~. In the case that the observer observer. Indeed, as an observer approached the
is moving on the orbit of a Killing vector K which first observer's future event horizon the radiation
is rotating with respect to the cosmological hori- would diverge. It seems clear that this observer
zon, one again gets similar results for the radia- dependence of particle creation holds in fhe case
tion from the cosmological and black-hole hori- of black holes as well: An observer at constant
zons with E replaced by E -nQ~ and E-nQ~, re- distance from a black hole will observe a steady
spectively. %here Q~ and Q~ are the angular emission of thermal radiation but an observer
velocities of the cosmological and black-hole hori- falling info a black hole will not observe any di-
zons relative to the observers frame and are de- vergence in the radiafion g, s he approaches the
fined by the requirement that K+ Q~K and K+Q~K first-observer's event horizon.
should be null on the cosmological and black-hole A consequence of the observer dependence of
horizons. particle creation would seem to be that the back
 COSMOLOGICAL EVENT HORIZONS, THERMODYNAMICS, AND. .. 2751

reaction must be observer-dependent also, if one law of event horizons that the area of the cosmo-
assumes, as seems reasonable, that the mass of logical event horizon will be l, ess than it appeared
the detector increases when it absorbs a particle to be before. One can interpret this as a reduc-
and. therefore the gravitational field changes.
This will be discussed further in another paper, " tion in the entropy of the universe beyond the
event horizon caused by the propagation of some
but we remark here that it involves the abandoning radiation from this region to the observer. Un-
of the concept of an observer-independent metric like the black-hole case, the surface gravity of
for spacetime and the adoption of something like the cosmological horizon decreases as the horizon
the Everett-%heeler interpretation of quantum shrinks. There is thus no danger of the observer's
mechanics. '6 The latter viewpoint seems to be cosmological event horizon shrinking catastroph-
required anyway when dealing with the quantum ically around him because of his absorbing
mechanics of the whole universe rather than an too much thermal radiation. He has, however, to
isolated system. be careful that he does not absorb so much radia-
If a geodesic observer in de Sitter space chooses tion that his particle detector undergoes gravita-
not to absorb any of the thermal radiation, his tional collapse to produce a black hole. If this
energy and entropy do not change and so one would were to happen, the black hole would always have
not expect any change in the solution. However, a higher temperature than the surrounding uni-
if he does absorb some of the radiation, his en- verse and so would radiate energy faster than it
ergy and hence his gravitational mass will in- absorbs it. It would therefore evaporate, leaving
crease. If the solution now settles dowg again to the universe as it was before the observer began
a new stationary state, it follows from the first to absorb radiation.

~Present address: Max-Planck-Institute fur Physik and 9E. Schr6dinger, Expanding Universes (Cambridge Univ.
Astrophysik, 8 Munchen 40, Postfach 401212, West Press, New York, 1956).
Germany. Telephone: 327001. B. Carter, Commun. Math. Phys. 17, 233 (1970).
W. Israel, Phys. Rev. 164, 1776 (1967). B. Carter, in Les Astre Occlus (Gordon and Breach,
H. Muller zum Hagen et al. , Gen. Relativ. Gravit. 4, New York, 1973).
53 (1973). R. Geroch, E. H. Kronheimer, and R. Penrose, Proc.
3B. Carter, Phys. Rev. Lett. 26, 331 (1970). R. Soc. London A327, 545 (1972).
4S. W. Hawking, Commun. Math. Phys. 25, 152 (1972). 238. W. Hawking, in preparation.
5D. C. Robinson, Phys. Rev. Lett. 34, 905 (1975). 24S. W. Hawking, Commun. Math. Phys. 25, 152 (1972).
D. C. Robinson, Phys. Bev. D 10, 458 (1974). P. Hajicek, Phys. Rev. D 7, 2311 (1973)'.
J. Bekenstein, Phys. Rev. D 7, 2333 (1973). 2~L. Smarr, Phys. Rev. Lett. 30, 71 (1973); 30, 521(E)
J. Bekenstein, Phys. Rev. D 9, 3292 (1974). (1973).
9S. W. Hawking, Phys. Rev. D 13, 191 (1976). 270. Nachtmann, Commun. Math. 'Phys. 6, 1 (1967).
~
J. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188 28E A. Tagirov, Ann. Phys. (N. Y.) 76, 561 (1973).
(1976). P. Candelas and D. Raine, Phys. Rev. D 12, 965
R. Penrose, in Relativity, Groups and Topology, (1975).
edited by C. DeWitt and B. DeWitt (Gordon and Breach, J. S. Dowker and R. Critchley, Phys. Rev. D 13, 224
New York, 1964). {1976).
'28. W. Hawking and G. F. R. Ellis, Large Scale Struc- 3'S. W. Hawking, Nature 248, 30 (1974).
ture of Spacetime (Cambridge Univ. Press, New York, ~28. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
1973). ~~A. L. Fetter and J. P. Walecka, Quantum Theory of
~3J. Bardeen, B. Carter, and S. W. Hawking, Commun. Many Particle Systems (McGraw-Hill, New York,
Math Phys. 31, 162 (1973). 1971).
'4W. Unruh, Phys. Rev. D 14, 870 (1976). 34R. Figari, R. Hoegh-Krohn, and C. Nappi, Commun.
~~A. Ashtekar and A. Magnon, Proc. R. Soc. London Math. Phys. 44, 265 {1975).
A346, 375 (1975). G. W. Gibbons and M. J. Perry, Phys. Rev. Lett.
'68. W. Hawking, in preparation. 36, 985 (1976).
J. D. North, The Measure of the Universe (Oxford 36The Many Worlds Interpretation of Quantum Mechan-
Univ. Press, New York, 1965). ics edited by B. S. DeWitt and
¹ Graham (Princeton
' C. Kahn and F. Kahn, Nature 257, 451 (1975). Univ. Press, Princeton, ¹ J., 1973).