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Semiclassical perdurance of de Sitter space
Nuclear Physics B222 (1983) 2,15-268 © North-Holland Publishin£ Company
SEMICLASSICAL PERDURANCE OF DE SrITER SPACE Paul GINSPARG*
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Malcolm J. PERRY t
Department of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA Received 24 January 1983
We investigate the classical and semidassical instabilities of de Sitter space. Due to the presence of a cosmological event horizon and its associated Hawking temperature, de Sitter space might be expected to behave in some respects like flat space at finite temperature. We use the euclidean formulation of quantum gravity to show that de Sitter space does exhibit a semiclassical instability to the nucleation of black holes. We find, however, no analog to the classical instability due to gravitational dumping of thermally excited gravitons.
De Sitter space is the maximally symmetric solution of the vacuum Einstein equations with cosmological constant A. Despite the present limits on the cosmological constant (defined as 8~rG times the vacuum energy density, thus having dimensions of mass squared), there has been a recent revival of interest in de Sitter space prompted by the potential solution of some important cosmological problems in inflationary models of the early universe [1,2]. In such models the universe is assumed to have undergone an early phase with a large effective cosmological constant, A - (10t°-10 II GeV) 2 for GUT era inflation, or A - (1016-10 Is GeV) z for Planck era inflation. A subsequent phase transition would then produce a region of space-time with A < (10 -42 GeV) 2 (the present observed limit) which becomes the space in which we now live. Since de Sitter space plays a key role in these models, we should like to learn more about its properties, particularly its quantum properties. The most straightforward means of quantizing fields on a background de Sitter space leads to the result that geodesic observers observe a thermal spectrum of excited states [3] (we shall shortly discuss this property at greater length). Such 'thermal' properties suggest that we compare and contrast de Sitter space with
* Research supported by Harvard Society of Fellows and by NSF grant No. PHY77-22864. * Research supported by NSF grant No. PHY80-19754. 245
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P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
ordinary fiat space at finite temperature T. Hot fiat space has been shown to exhibit two instabilities [4]. It has a classical instability, the Jeans instability [5], corresponding to gravitational clumping of thermally excited modes on length scales greater than a Jeans length, of order m e l t 2 (rap = G - I / 2 = Planck mass). It also has a semiclassical instability corresponding to spontaneous nucleation of black holes of size - l / T , w i t h rate per unit volume of order exp(-m2/16~rT2). In this paper, we shall examine de Sitter space for signs of analogous instabilities. De Sitter space [6a] is a space of constant curvature and is a solution of the vacuum Einstein equations Rab--'ASab,
(1.1)
with A > 0 (A < 0 is called anti-de Sitter space). The space can be visuali7ed [6b] as a hyperboloid --l)2 + W2 + X2 + y2 + z2 = ~t2,
(1.2)
embedded in E 5 with metric d s 2 = - d v 2 + d w 2 + d x 2 + d y 2 + d z 2,
(1.3)
where here and henceforth a = ~/-~.
(1.4)
The hyperboloid (1.2) has topology R x S 3 and invariance group SO(4, 1). In the limit A --* 0, this group becomes the usual Poincar6 group incorporating the Lorentz group SO(3, 1) and space-time translations. The de Sitter metric is the metric induced by this embedding; it is conveniently exhibited in coordinates t ~ ( - oo, oo), X ~ [0, It], 0 ~ [0, It], ~ E [0,2~r), defined by v = asinh(t/a), w = acosh(t/a)cos
y = acosh(t/a)sin
×sin0cos ~,
z = acosh(t/a)sin
X sin 0 sin ¢p,
X,
x = a c o s h ( t / c t ) s i n XcoS 0,
(i .5)
giving d s 2 = - d t 2 + a 2 c o s h Z ( t / a ) [ d x 2 + sin2x(d02 + sin20 d~ 2 )].
(1.6)
Eq. (1.6) is a k--- + 1 Robertson-Walker metric with spatial sections which are 3-spheres of radius a c o s h ( t / a ) . In these coordinates, de Sitter space appears to
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
247
~ + (t .(:0, ~ • 'n'l2 )
Surfoces of Constont t "<
/ N
J t-O
Lines of Constant X
\
J
'\ \
• -X , .tr
X.O
\ %
~°( t , - 0(:),~9 , - "~Z)
Fig. 1. A Penrose diagram of de Sitter space for constant 0, ¢,. Lines of X" constant are timelike geodesics. The dashed line represents the observer dependent cosmologicalevent horizon of an observer moving along the fine X" 0. The compactified time coordinate v/~ [ - ~r,' ½~r]is defined in sect. 3.
contract in size until t = 0, and then re-expand to infinite size. The coordinates (t, X, 0, ~,) cover the entirety of de Sitter space. Fig. 1 shows a conformal diagram of the space for constant 0, ~ [7]. We observe that the causal past of an observer moving along X = 0 does not coincide with the conformal boundary of de Sitter space, even as the observer moves into the infinite future. There is thus a region of the space with which the observer can never have causal contact. The boundary of this region is called his cosmological event horizon. It is observer dependent since different observers see different boundaries depending on their asymptotic destination [71. Other coordinate systems are frequently used to describe de Sitter space but only (1.5) actually covers the entire spacetime with just one coordinate patch. In coordinates t E ( - oo, oo), ~ ~ ( - oo, oo), .~ ~ ( - oo, oo), ~ ~ ( - oo, oo) defined by
w + v = aC/a,
y
=.~e t/~', A
x.~-.~e I/a,
(1.7)
z-~.~et/a,
the metric (1.3) becomes
d s 2 --- - d r
2 4- e 2 t ' / a ( d . ~ 2 4- d f l 2 4-
d,~2).
(1.8)
This takes the form of a k -- 0 Robertson-Walker metric and has flat spatial sections which exponentially expand. The coordinate patch (1.7) covers only that part of the spacetime with w + v > 0, as indicated in fig. 2a.
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P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
Surfoces of Coc~stont
Linls of Constant
~.0 t=-CO
(a)
,./~-~ t
I
Surfaces of Constant t'
I
Lines of Confront r
(*0
r,O
I
'~
x,Tr/2 (b) Fig. 2. The portion of de Sitter space covered by (a) the k = 0 coordinates (1.7); (b) the static coordinates (I.9). Conventions are the same as in fig. I. In (a), ;2 i ~2 +);2 + ~2.
In coordinates t' ~ ( - oo, o0), r ~ [0, a], 0 ~ [0, Ir], q~e [0,2¢r) defined by
v
=
a ~ l - r2/a 2 sinh(t'/a),
w = a ~ l - r2/a 2 c o s h ( t ' / o ) , x = rcos 0,
y
-
rsin 0cos ¢,
z = rsin 0 sinO, (1.9)
the metric (1.3) becomes
ds 2= - (l -½Ar2)dt'2 + (l - ½ A r 2 ) - ' drz + r2(dOz + sin2Odo2). (l.IO) In this coordinate system, de Sitter space looks static. This is because successive spatial sections are generated by 4 + 1-dimensional Lorentz transformations. These coordinates cover only the portion of the space with w > 0 and x 2 + y 2 + z 2 < et2. This corresponds to the region interior to the particle and event horizons of a
P. Ginsparg, M.J. Perry / Semiclassical perdurance of de Sitter space
249
geodesic observer moving along r - - 0 , as indicated in fig. 2b. The horizons are located at • -Quantizing fields on a background de Sitter space requires a careful definition of the vacuum state. Requiring only that it be de Sitter (SO(4, 1)) invariant leaves a one-parameter degenerate set of such vacua [8]. One prescription for implicitly choosing among these vacua is to work in an imaginary time formalism and analytically continue the resulting euclidean Green functions back to Minkowski space. This choice of vacuum quantization is probably the only one which generates Green functions having only causal poles and no spurious singularities. Well behaved Green functions for fields in de Sitter space have been computed by various authors [9] and they are periodic in imaginary time, with period This is explainable geometrically since, as will be described in sect. 3, the euclidean version of de Sitter space is simply S4. This periodicity suggests that observers in the space will perceive a thermal spectrum characterized by a temperature
ff/A.
2rt~/~-/A.
1
(1.11)
•
This was originally demonstrated in ref. [3] and subsequently confirmed by a Bogoliubov transform technique in ref. [10]. It is a rather curious form of temperature, though, since geodesic observers moving at constant velocity with respect to one another nonetheless perceive the same 'temperature'. This is because the expectation value of the energy-momentum tensor for gravitons [11]
26
( T a b ) = -- 1 3 5 ~ 2
AZg
' ab
(1.12)
and all other properties of the ground state preserve the de Sitter invariance. Moreover, despite having introduced the notion of temperature to the space, we assert that the de Sitter vacuum is still, by definition, a pure quantum state. A mixed state description need only be introduced for the individual local observer who feels compelled to construct a density matrix by integrating out all ground state information beyond his horizon. Since there is no de Sitter invariance breaking heat bath (as Lorentz invariance is broken by ordinary heat baths) and since we are dealing with a pure state, there is no doubt that this temperature can be entirely described in terms of quantum fluctuations. There are nonetheless sufficiently many similarities to ordinary thermal formalisms to suggest determining how the instabilities of de Sitter space compare to the thermally induced instabilities of Minkowski space at finite temperature. In sect. 2, we perform a perturbative stability analysis of de Sitter space along the lines originally proposed by Lifshitz [12]. This type of analysis was later reformulated by Bardeen and by Press and Vishniac [13], to treat carefully coordinate invariance and ensure that physical conclusions are drawn only from gauge invariant
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P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
perturbations. Using the Lifshitz formalism, we probe de Sitter space with perturbations taking a perfect fluid form, on the assumption that any tendency to gravitationally clump would be signalled by an instability in such perturbations. We find that any such effect is cut off by the ambient exponential expansion of the space. Thus there can be no analog in de Sitter space to the instability due to clumping of thermal gravitons displayed by hot flat space. The informed reader who is wondering how de Sitter space evades various singularity theorems [7] should notice that the energy-momentum tensor, including the cosmological term, does not satisfy the strong energy condition [7]. In sect. 3, we use the euclidean formulation of quantum gravity to look for semiclassical instabilities of de Sitter space. We do not pretend to understand what a fully quantized version of gravity would look like, but maintain that the formalism used here properly incorporates quantum mechanical contributions to processes occurring on scales large compared to the Planck length. For de Sitter space we shall have to assume GA ,~: 1 so that the characteristic size of the space is large compared to the Planck length. This is equivalent to requiring that the de Sitter temperature ( 1.1 !) be small compared to the Planck mass. Just as in the hot flat space case [4], we find that the boundary conditions for the euclidean functional integral allow finite action saddle points with non-trivial topology. They are interpreted as a probability of nucleating asymptotically de Sitter black holes with a rate per unit volume of order e x p ( - m 2 / 1 2 ~ r T 2 ) . This result is rather similar to the corresponding result already cited for hot flat space. Despite the semiclassical suppression factor, however, for de Sitter space it is the dominant instability since there are no classical instabilities. The semiclassical instability in hot flat space, by contrast, is never the most important unless the system is put in a box small enough to cut off the Jeans instability. Sect. 3 concludes with an argument that the subsequent minkowskian evolution of these nucleated black holes begins at an intrinsic temperature higher than the cosmological horizon temperature. The black holes thus always evaporate on a time scale small compared to the mean time between nucleations, rather than accreting radiation and growing. In this sense, the space perdures. Finally, an appendix reviews some of the formalism of Kahler geometry useful for computing the negative mode on S2 × S2, our gravitational instanton. Before proceeding to our analysis, we remark briefly on the relationship between stability and positivity of energy. It is true that energy becoming negative represents some form of instability. The converse is not the case, however, as can be seen by the example of hot flat space [4, 5]. Both its classical instability, the Jeans instability, and its semiclassical instability to spontaneous nucleation of black holes, are situations in which the positive energy theorem applies [14]. The issue of positivity of the energy for de Sitter and anti-de Sitter space has been discussed by Abbott and Deser [15] and for anti-de Sitter space by Freedman and Breitenlolmer [16a]. We do not believe the de Sitter space stability analysis of ref. [15] to be entirely complete due both to its restriction to energy considerations and to its need to make assumptions about
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
251
the behavior of perturbations on the horizon. Anti-de Sitter space is rather different from de Sitter space since it possesses a globally time-like Killing vector field but does not have a Cauchy surface. In refs. [15, 16], it is noted that this problem can be circumvented by imposing a suitable boundary condition at infinity. With this condition, semiclassical instabilities can be excluded by the non-existence of any equal energy metric into which the space could decay. Anti-de Sitter space does, nevertheless, possess an ordinary Jeans instability.
2. Perturbative stability In this section we apply the techniques of Lifshitz [12] to the analysis of perturbations of de Sitter space. Details omitted from our brief review of the general formalism may be found in the original references. As mentioned in the introduction, we perform our analysis in the coordinate system (1.5) which covers the entire spacetime*. In general, a k --- + 1 Robertson-Walker metric can be written ds 2 = - d t 2 + a 2 ( t ) [ d x 2 + sin2x(d0 2 + sin2Odq~2)],
(2.1)
where a ( t ) gives the radius of the S 3 spatial sections. Instead of t, we shall prefer the coordinate 71 defined by dt = a d n ,
(2.2)
so that d s 2 = g a b d x a d x b = a 2 ( ~ / ) ( - dl/2 + dx 2 + sin2X (d0 2 + sin20d¢2))
-- a2(~)(-d~ 2 + yopdx a dx p),
(2.3)
where "r=a= a-2g~t~ is the metric for a unit 3-sphere. In the analysis to follow, a 0 coordinate index corresponds to the ~/coordinate, Latin indices (a, b .... ) run from 0 to 3, Greek indices (a, fl .... ) run from 1 to 3, and a prime denotes differentiation with respect to 7. Unless otherwise noted, indices are raised and lowered with respect to the metric gob of eq. (2.3). The components of the unperturbed Ricci tensor R~, are R~ = a - ' ( 2 a 2 + a '2 + aa") 8~,
(2.4a)
R ° = 3a-'(aa" - a'2),
(2.4b)
R °0 -- 0,
(2.40
* After the completion of this work, we learned of recent similar perturbative analyses of de Sitter space stability [17]. While our final conclusionsare consistent, we feel that the k = 1 coordinatesused here give a more complete analysis than that of [17a]. Also, the exact solution to (2.26) (eq. (2.29) of this paper) implicidy givingthe synchronousgauge graviton propagator in de Sitter space, has not to our knowledgeappeared previously.
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P. Ginsparg, M.J. Perry / Semiclassicalperduranee of de Sitter space
and the Pdcci scalar is
R =- R a = 6 a - 3 ( a + a " ) .
(2.5)
The function a(~) is determined by the Einstein equations R~b - ½gobR + gobA ~- 8~rGT~b.
(2.6)
For the de Sitter metric (1.6), the change (2.2) leads to a compactified time coordinate
n=
os-'
(2.7)
with - ½~r < 11~ ½,r. De Sitter space thus takes the form (2.3) with a(n) ~ asecrl.
(2.8)
We now perturb the metric (2.3) by setting gob = g ~ sitter) + h ab"
(2.9)
General coordinate invariance always allows us in what follows to choose a gauge, the 'synchronous gauge', such that hoo = hoa = 0.
(2.10)
The equations which determine the remaining perturbations h°B of the metric tensor are obtained by expanding the Einstein equations (2.6) to leading order in h. The lowest-order perturbation of the Ricci tensor 6R~, is
8Rg;--L
2a2
h"+
h'
(2.11a)
8R°=--~-I ( ~Tah'- uah~'), 2a 2
(2.11b)
2a 2 +
1.__1_ a,,
a'
2a 2ha + a S
,
a
a'h'
2a 3~a
(2.11c)
where h =- h~ = h,ljg aa, [] ~- V'a Va, and the covariant and contravariant derivatives, Va and V a = "ta~ Va, are defined with respect to the unit 3-sphere metric 'Ga of
P. Ginapar$, M.J. Perry / Semiclatsicalperdurance of de Sitter space
253
(2.3). The resultant perturbation of the Ricci scalar is 8R = 1 ( I;'~Yah ~p - Oh) + 1 h" + 3a'h' a2 a3 a 2
2h a2 •
(2.12)
These metric perturbations fall into one of three classes. They may be associated with some energy-momentum tensor of a matter field, they may be gravitational waves, or they may be pure coordinate transformations. In what follows, we need to separate out these three physically distinct possibilities. The energy-momentum tensor which gives rise to metric perturbations is assumed to take the form of a perfect fluid, with pressure 8p, energy density 80, and fluid velocity flow vector u ~ normalized such that u~uo -- - 1. Hence
ro = (sp + 80)u-u + 8pg
(2.13)
and is isotropic in the comoving coordinates specified by u ° = 1/a, u a = O. The pressure and energy density are related by an equation of state 8__pp 8p = cs2 '
(2.14)
where cs is the speed of sound (assumed constant) for the fluid in question. The cases of physical interest are delimited by the values c~ = 0 (pressureless dust) and cs2 ---- ½ (radiation). Since any instability must develop slowly, at least initially, we can assume the fluid is in slow motion with respect to our synchronous gauge coordinates. The velocity four-vector in these coordinates is thus dominated by the time component and we can neglect terms quadratic in the u °. Since uau~ = - 1, we can put u ° = 1/a in what follows. The linearized Einstein equations (2.6) become (Gob = Rob -- ½Rgab), 8G o = _ --'2a --~1 ( Va vahOa _ Oh ) _ a'h__~'+ ha2
- 8¢tG Sp ,
(2.15a)
(2.15b)
8Go _- 2a 2
l(
6Gff=~a2
2°
V ~ v a h ~ + W~W°hVa- W a w a h - D h a ~ + h ~ " + a ha~'14h~
+ 2~8~(E3h_
wyWah~ + 2 h
2a'h'a
h " ) = 8 , r G c ~ SpS~.
)
(2.15c)
A convenient pair of equations for h can be derived from the off-diagonal components of (2.15c) and from eliminating 8p from (2.15a) and the trace of (2.15c). This
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P. Ginsparg,M.J. Perry / Semiclassicalperduranceof de Sitter space
pair is
WvV'#hV,+ V.rvah Ih'- Vavah-t-lh#a+h~"+
h.#'-4h.#=0, (2.16)
2 _tLt
(~TaV~h#-nh-2h)(l+3c2)+-~(2+3c2)+2h"=O.
(2.17)
The density is then determined, by (2.15a), to be
8p =
l [ Va v#h~a - E3h +. 2a'h---~'- 2 h ] 16,rGa 2 a '
(2.18)
and the velocity of the perturbing material, by (2.15b), is 1
8pu.= 16¢rGa(1 +c2) [ V ~ h ' - vah~'].
(2.19)
Since our constant time spatial sections are maximally symmetric 3-spheres, an arbitrary initial hap is decomposible into representations of SO(4). These fourdimensional spherical harmonics are labelled by two integers (I, m) with 1 >t m/> 0 [18]. Useful properties of these functions, including their construction in terms of homogeneous polynomials of cartesian coordinates on flat R4, can be found in [12, 19]. Suppressing the l= 0, 1.... dependence, the (1,0) functions are spacetime scalars Q, the (l, 1) functions are transverse spacetime vectors S,~ ( ~7'~S,~= 0), and the (l, 2) functions are symmetric, transverse and tracefree tensors Ha# (0 = H~ = v,H~). The most general metric perturbation is then h~a = (h(~) Pa# +/~(71)Qa#) + a(aT)S~# + v(~l)H~#,
(2.20)
where Q~a =
½y~Q,
(2.21)
1
P"# = l(t + 2) v,,vaQ + Q,,I~, =
+
7 so.
(2.22) (2.23)
After substitution of (2.20), eqs. (2.16) and (2.17) become differential equations for
P. Ginsparg,M.J. Perry / Semiclassicalperduranceof de Sitterspace
255
X, F, a and J,, l#'+(2+3c2s)l~'tanrt+½(l+3c2s)(l+3)(l-1)(X+t~)=O, h" + 2h'tanT/- ½1(1+ 2)(h + / Q -- O,
(2.24a) (2.24b)
a" + 2a'tan 7/= 0
(2.25)
v" + 2v'tan*l+l(l+ E)v=O,
(2.26)
where we have substituted the de Sitter space form of a(~), eq. (2.8). Finally, eqs. (2.15a, b) give 8p and u ~ as BO = c°s2~7 [-#'tanT1 - ~ ( l + 3 ) ( i - 1 ) ( A + / ~ ) ] Q , 8 ¢rGa 2
(2.27)
cos3~1 [ 1 8pu" = 8rrOa3(1 + c2 ) (~(1+ 3)(•- 1)X' +31(l+ 2)/d)1(1+ 2) waQ
+ ½(t+ 3)(i-
(2.28)
We first consider the perturbations generated by the tensor spherical harmonics H,a. Since eq. (2.26) is a source-free equation and since the perturbation v(*I)H.~ is transverse and tracefree, this perturbation represents gravitational radiation (with no unphysical gauge degrees of freedom). The solution of (2.26) is v(~) = c t [sin(hT) + lcos 17sin((/+ 1)~7)] + ¢2 [cos(h7) + lcos 7/cos((l + 1)~)], (2.29) where the G are here and throughout arbitrary constants. It should be noted that the amplitude of v(~l) does not necessarily vanish as T/--* ½or. To understand best the asymptotic behavior of v, we reexpress it in terms of the uncompactified time coordinate t. As 71~ ½~r, t ~ co, and we find
v(t)
- &
+ A2e
(2.30)
where A I and A 2 axe constants depending on c 1, c a, and 1. Even though r(t) does not vanish as t ~ o0, neither does it grow with t, so we have excluded any unstable growing modes of gravitational radiation from de Sitter space. Any physical effects of the modes (2.31) are in any event vanishingly small since their physical wavelengths increase with the scale factor a, while their amplitude remains constant.
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Physical quantities, involving at least two derivatives of the metric, are thus asymptotically suppressed by powers of 1/a. In this sense, our results are consistent with the "no-hair" theorem for de Sitter space conjectured by Hawking and Moss [20]. These modes are nonetheless potentially interesting in inflationary models of the universe. If excited during the inflationary epoch, they could conceivably survive the phase transition to the present universe before being exponentially damped. They would then give rise to inhomogeneities for which there are strict observational bounds [21]. Before continuing to the vector and scalar perturbations, we point out that the gauge condition (2.10) does not completely specify the choice of coordinates, so there remain unphysical gauge degrees of freedom in ~t, h, and o which can be removed by a coordinate transformation x°--, x° +
For the general metric (2.3), changes parametrized by a t
raft),
(2.31a)
leave the resulting space-time invariant, for arbitrary functions f0, f. of the spatial variables x a. For perturbations of de Sitter space, this reduces to h .a -* h .ij + sin( ~ ) V ,, V lffo + sin( 71)f o'Gl~ + ( V . f ij + V lff, ) .
(2.31b)
We now turn to the perturbations generated by the vector harmonics S.. Letting f, = - o o S . in (2.31) we see that physical perturbations of the form oS.B are generated by ( o ( ~ ) - o 0 ) S . ~ where o0 is an arbitrary constant chosen at our convenience. The solution to (2.25) is o(~1) = c3 + c,(2~ + sin211).
(2.32)
Now (2.27) implies that 8p = 0 (there is no scalar associated with S. since v , S " = 0) and hence (2.28) gives c4 -- 0 for the matter of interest. The most general allowable o(,/) is therefore constant and consequently a pure gauge transformation as is evident by choosing o0 ~-c 3. Thus, perturbations of the form oS, p turn out to be completely irrelevant for de Sitter space. Finally, we examine the perturbations h P ~ + / ~ Q . ~ generated by the scalar harmonics Q. Choosing • Ct
f°=CQ"
f'--- 21(1+2) v o Q ,
(2.33)
P. Ginsparg, M.J. Perry / Semiclassicalper~b.4ranceof de Sitter space
257
in (2.31b), with C and C' being arbitrary constants, shows that
A(~l)=Cl(l+2)sin~+C',
p(~)= -C(l+3)(l-
l)sin~-C'
(2.34)
are pure gauge transformations and can be removed from an arbitrary perturbation without affecting its physical content. This freedom may be used to reduce (2.24a, b) to a pair of first-order equations whose solutions are then straightforwardly expressible, for arbitrary cs2, in terms of hypergeometric functions. Here, we will only give explicit solutions for the physically interesting cases of c2 -- 0, ], and the special case cs2 = - 1 which corresponds to local fluctuations of the cosmological constant. Such fluctuations might be expected to occur in theories whose cosmological constant results from a vacuum energy density before or after a symmetry breaking phase transition. On physical grounds, increasing c2, thereby increasing the pressure holding things apart, makes a system more resistant to gravitational collapse. In fact, analysis of (2.24a, b) shows that ~ a n d / t cannot blow up for any value of t when c2 >1 - 1 . Furthermore, for c2 >/0, 8p and 8pu ~ must vanish at least as fast as e -3t/° and e-4,/=, respectively, as t --* co. Finally, there is a gauge in which ~ = p = ~' = / t ' = ~," = #" = 0 at 7/= ½~r so the gauge invariant effective energy-momentum tensor of the induced gravitational perturbations must also vanish exponentially as t -* co. This is because this tensor, extracted from the second order variation of R°b -- ½gobR + Agob, is quadratic in h and its first two derivatives and thus depends only on ~ and/t and their first two derivatives. The case of pressureless dust, c 2 = 0, is that most likely to exhibit some form of instability. The solution to (2.24eq b) is
/t(~) = c5 [½sin2v; + (v/- ½~r)] + %(1+ 3)(•- l)[cos T/+ (7/- ½~r)sin v/], Z(~l) = -%[½sin271+(~l-½~r)]+c61(l+ 2)[cosTl+(~l-½¢)sin~l],
(2.35)
where we have (arbitrarily) fixed the gauge via (2.34) so that
,(½..) =x(½..)= o The induced energy density and velocity field are, from (2.27) and (2.28), 1
3
•
8p = 8¢~a2"cos ~[2c, sm~ + ( i + 3 ) ( 1 - 1)c6] Q,
SPa==
4¢
1Ga3c°s5~c5!(I +1 2)
v=Q"
(2.36)
(2.37)
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
258
For c s2 = ~, the solution to
(2.24a,b) is
X +/*=CT[cos(~oT1)+sin(~l)I,] + cs [sin(¢o~) + sin(~l)I2] X ' - ~' = c7 [-o~ sin(~0~)cos2~l + (2~02- l)(sin ~1cos ~ cos(~orl) + cos(71) I,)]
+ cs [¢o cos(oa'q)cos2~ + (2~o2- 1)(sin~ cos ~!sin(oa'q) + cos('q) I2) ] (2.38) where ~2-~1(1+2), I, = f d~ (cos(o~)cos ~ + ¢osin(co~l)sin ~),
/2 --- f dn (sin( ,0n )cos n - ,0 cos(,on)sin n).
(2.39)
The constants of integration implicit in Ij and 12 correspond to gauge transformations. The energy density and velocity are determined again by application of (2.27) and (2.28):
8p = (~2 _ 1) cos'(~)(c, +
cg)Q,
(2.40)
8~'Got 2
(~02.______ 1)
r
~7~Q
8pu= = 32~rGa ' °~c o s ' " t cTsin ~°" + cacos ~°"] l ( l + 2)"
(2.41)
T h e s e t w o c a s e s , c s2 = 0 and c s2 = ~, exemplify the general properties mentioned above. By inspection of (2.38) and (2.39), we see that in the gauge for which X --/~ = 0 at ~ = ½~r we also have k' = X" -/~' = #" = 0 at ~ ffi ½7r. Thus any energymomentum associated with the gravitational field asymptotically vanishes. We also find that 80 and 8#u ~' vanish rather quickly as ~! -~ ~r so scalar generated perturbations are completely harmless. The special case, c~ = - 1, implying 8p = - 8 0 , is even simpler. From (2.13), we find ST0~ ffi 0 so that (2.28) requires
(1+ 3)(•- 1)X'+ 1(1+ 2 ) ~ ' - - 0 .
(2.42)
The only solutions to this consistent with (2.24a, b) are identically the gauge transformations (2.34). We thus have a rigidity theorem which excludes local fluctuations in the cosmological constant as physical excitations.
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
259
Thus we have shown that in perturbation theory, there is no analog of the Jeans instability in de Sitter space. Regarded as a function of t, any perturbation which starts at t - - 0 will grow like a power of t initially, but, with the exception of pure gravitational radiation, will be damped exponentially at late times. We conclude that de Sitter space is classically stable.
3. Semi-classical instability The euclidean functional integral is the most convenient means of describing processes in quantum field theories which are classically forbidden, but allowed quantum mechanically [22]. The euclidean form of the functional integral for gravity, first derived by Faddecv and Popov [23], is Z = f Dgob ( x ) e x p [ - I ( g ) + gauge-fixing terms],
(3.1a)
where the euclidean action is given by
I(g)=
16~cjf(R-2a)g'/2d'x.
(3.1b)
The functional integral is evaluated by integrating over all metrics which are positive definite and obey appropriate boundary conditions. This functional integral construction of quantum gravity is poorly understood and its proper definition is the subject of much recent research and speculation. Fortunately, saddle point methods give a means for treating the functional integral and are adequate for a semiclassical stability analysis. To investigate localized instabilities of de Sitter space, we consider spaces which are asymptotically de Sitter. In euclidean space, this boundary condition translates to looking at geodesically complete and non-singular solutions to the euclidean Einstein equations Rab = Agab, with A > 0. These solutions are called gravitational instantons and have, from (3. l b), A V (4)
I=-
8~r'----G-'
(3.2)
with V (4) the four-volume of the instanton. Thus the requirement of finite action restricts consideration to compact instantons (for this reason we have also omitted boundary term corrections from (3.1b)). At present, we know of only four Einstein metrics* satisfying this condition. * By Einstein metric, we mean metrics for which ~'cRob ~ O, implyingRob-- ~go~.
P. Ginsparg, M.J. Perry I Semiclassicalperdurance of de Sitter space
260
The first such metric is the Einstein metric on S 4. It can be obtained by an analytic continuation of the de Sitter metric (1.10) by setting t'ffi it, which gives
ds2 =ffi(l - -]TAr2) d'r 2 -F (1- iliArZ) -I dr 2 4- r2d£/2, d122 = dO 2 + sin20 d~ 2.
(3.3)
The singularity at • -- ( 3 / A ) I/2 is a removable coordinate singularity provided that is identified with period 2~r(3/A) t/~. (3.3) is then recognized as the metric for a four sphere of radius ( 3 / A ) I/2. De Sitter space Green functions are thus periodic in i m a g i n a r y time and by analogy with a similar calculation in the black hole ease [24], we infer that de Sitter space behaves as though it has an intrinsic temperature Tc ffi (1/2~rX]A) I/: [31. Finally, using (3.2) with V (4) = ] ~ r 2 ( ~ ) 4, we see that the metric (3.3) has euclidean action I = - 3~r/AG. The next two metrics falling into the class of interest are the FJnstein metrics on CP2125] and CPz#c'-P"~ [26]. Neither of these metrics, however, has a spin structure: they admit only so-caEed 'generalized' spinors with an extra half unit of charge with respect to bosons [27]. We thus exclude these instantons from our saddle point evaluation of the functional integral. We now proceed to derive the final Einstein metric satisfying our boundary conditions from another asymptotically de Sitter-Lorentz metric. It is known as the Kottler [28] or Sehwarzschild-de Sitter metric, and represents a non-rotating black hole immersed in de Sitter space. Its metric, in static form similar t o (3.3), is ds z_- _ V ( r ) d t z + V - i ( r ) d r 2+ r2dl2 z, l/(r)
2Gm
= 1
~Ar2 '
(3.4)
r
where m represents the mass of the black hole. The physical interpretation of this metric is perhaps more evident in terms of asymptotically fiat coordinates (similar to (1.8) for ordinary de Sitter space)* 1 d8 2 ~ _
m --t'/*
2r I 1 + rn e-t~/~ 2rl
2
dttz + (1 4- ~-~rl m e-tl/a) 4e2tl/a(dr~ + r~d~2),
where
r=r,o',"(,+
m
2
dt; = dt
* We are grateful to L. Alvarez-Gaum~: for b t ~
-
r/a
dr.
ill - ( 2 m l r ) V(r) this form of the metric to our attention.
(3.5)
P. Ginsparg,M.J. Perry / Semiclassicalperdurance of de Sitter space
261
(3.5) clearly reduces to the familiar asymptotically flat coordinates of ordinary Schwarzschild and de Sitter spaces, respectively, in the limits 1 / a 2 = 13A = 0 and m = 0. For 9G2m2A < 1, the function V(r) of (3.4) has zeroes at two positive values of r, denoted r+ and r++, with r + < r . + . In the coordinate system of (3.4) these correspond to a cosmological event horizon at r++ and a black hole event horizon at r+. For 9G2m2A = 1, r + s r++ is a double root and a degenerate horizon is formed just as occurs for the extreme (charge = mass) Reissner-Nordstrom black hole horizon. The surface gravity i¢, defined by 1b V'~!° - r l °, where 1° is the null geodesic Killing vector which generates the horizon [29], vanishes in this limit. A euclidean metric can be constructed from (3.4) just as (3.3) was analytically continued from (1.10) for de Sitter space. Letting t = i~', the metric (3.4) becomes ds 2 = V ( r ) d ~ 2 + V-'(r)dr 2 + r 2 d ~ 2.
(3.6)
Now we must deal with singularities at both r+ and r++. Each is a conical singularity unless ¢ is identified with an appropriate period. Simultaneous removal of both singularities, however, requires the same period at the two singular values of r, which in turn only occurs in the limit as r+ tends towards r++. Since the euclidean region of the metric lies between r ffi r+ and r = r++, it might seem that the euclidean region shrinks to zero. This is not the case, however, and is an artifact of a poor coordinate choice. To see what really happens, let 9G2m2A = 1 - 3e 2,
(3.7)
so that the limit r + ~ r + + corresponds to e---0. We now def'me a new radial coordinate X, and a new time coordinate i/,, by
x=cos-'(a '/~(r-r°) ,
V,=a'/~,,
(3.8)
where r0 - A - I/2(1 -- i~e2) SO that r++. + -- r0 + A - I/2e. The resultant metric, to order e 2, is
d s 2 -- 1 ( 1 - ~COsx- i~2cos2x)dx2 + X1 (l + }ecos X + }e2cosZx)sin2xd~2
+X1 (1 -
2 e c o s x + e2cos2x - ~e2)(dO 2 + sin2Od~2).
(3.9)
When • -- O, there are no conical singularities and the metric is 1
d ~ = X [dx~ + sin~xd~'~ + d°~ + sin~Od~]'
(3.10)
262
P. Ginsparg, M,J. Perry / Semiclassicalperdurance of de Sitter space
which is a metric on S 2 x S 2, each sphere having radius A - ~/2. The Lorentz version of this metric, with topology H 2 × S 2, was discussed by Nariai [30]. The space described by (3.10) evidently has non-zero volume, V c4)= 16~r2/A2, and has action I = - 2 , ¢ / ~ G . With Euler character X = 4, it is also topologically distinct from S 4, which has X = 2. Even without taking the limit e --* 0, the metric (3.9) still has finite action [31]. The &function behavior of the Ricci scalar is integrable, with a 2-dimensional surface of area A and deficit angle 8 (i.e. a polar angle identified with period 2~r - 8 rather than 2,r) giving a contribution of - A S / 8 , r to the euclidean action. The Einstein equations are not, however, satisfied at the conical singularities so the space is not actually an instanton. It is clear, nonetheless, that as m is decreased from ½ G N / 2 to zero, the action decreases, continuously from - 2 , r / A G to - 3 , r / A G . For e small, the corrections to the action for the metric (3.9) come entirely from the conical singularities, which give 2~r
I=
AG
20~r 2
9AGe
- - 4,
+O~e
).
(3.11)
This situation is very similar to the relationship between hot flat space and the euclidean Schwaxzschild space, as discussed in [4]. By analogy, the presence of a non-normaliTable negative mode (corresponding to moving mass out of the space) suggests that there is also a normalizable negative mode in the spectrum of gauge invariant fluctuations about (3.10). By formalizing this statement below, we shall show that the S 2 x S 2 metric is not a local minimum of the euclidean action, but instead contributes an imaginary part to the free energy of de Sitter space, indicating an instability. To determine whether the S 2 x S 2 action is a local minimum or a saddle point, we must examine the spectrum of a differential operator G restricted to transverse tracefree symmetric tensors ~ab, which represent gauge-invariant quantum perturbations of the metric [19]. Acting with eigenvalue h, it is defined by Gab~dq/° = -- Dq~¢d -- 2RacbdeP"b = ~qJ~d"
(3.12)
Spaces for which all of the eigenvalues h of G are positive or zero are minima of the action. Correspondingly, spaces for which G has a negative eigenvalue are saddle points of the action. G turns out to be positive for the euclidean de Sitter space (3.3) but has a single negative mode for the metric (3.10) on S 2 x S 2 (see appendix). In the coordinate system of (3.10), this negative mode takes the form ] diag( 1, sin2 X, - 1, - sin20). ¢~b = ~-
(3.13)
o(S2xS 2) + e Q~b is evidently a metric on $ 2 × S 2 for which the The metric g~b-~-b=b ,
P. Gin~parg, M.J. Perry / Semiclassicalperdurance of de Sitter space
263
spheres have radii A - 1/2(1 + e') I/2. It has the same symmetries as (3.10) but is not a solution of the Einstein equations. Its action is I=
2¢r
2~re'2
AG
AG
"
(3.14)
(It should be noted that it is this perturbation which appears in the gaussian approximation of the functional integral and not the deformation exhibited by (3.9) whose action is (3.11). Had we iinearized that perturbation, it would turn out that it was not square-integrable, and not an eigenfunction of G.) By arguments parallel to those explained in [4], we can estimate the partition function for our system. The zero modes of the S 2 × S 2 instanton can be treated by the standard collective coordinate method (they are harmless in any event since the space is compact). The partition function takes the form Z = c + 3~r/AG.
(quantum corrections) + ½i e + 2,,/A6. (quantum
corrections). (3.15)
The first term arises from the S4 instanton, the second from the S 2 × S 2 instanton, and the factor of ½i multiplying e +2f/A6 arises from the negative mode. The quantum corrections, arising from the gaussian and higher-order contributions, have not been calculated in detail, principally because the non-renormaliTability of the theory causes them to be cut-off dependent. By the usual dilute gas approximation (reviewed in ref. [4]), the result (3.14) is interpreted as a decay rate per unit volume proportional to e x p [ - ~r/AG]. T h e tunneling process resembles a thermally activated process in which the entire space tunnels at once to a degenerate Schwarzschildde Sitter spacetime with two horizons located on top of one another. Higher-order quantum corrections, however, effectively result in one horizon moving inside the other due to the instability represented by the euclidean negative mode. We can then describe the subsequent minkowskian evolution for the metric (3.9) by analogy with the Schwarzschild case. The horizon temperatures of the inner (black hole) and outer (cosmological) event horizons are given by (1/4 ~r) IOV/Orl r - - , . . . . . [3], which equals X3A(r + + - r +)(2 + r ++/r +) and ] A ( r ++- r+X2 + r +/r ++), respectively. Since by definition r++> r+, we see that the Hawking temperature associated with the black hole horizon will always be the greater* [for the metric (3.9), these two temperatures are (1/2¢r)AI/2r(1 + ]e)]. The black hole will then presumably evaporate on a time scale roughly of order t - G2m 3. Since G2m 2 - A -I, t - A - 3 / 2 / G . T h e point is that within the regime of validity of the semiclassical approximation, GA .~ 1, the black hole evaporates away on a time scale much shorter than that required for * This has been previously observed in ref. [15] in which the authors conclude that evaporation is the only possible classical evolution. They do not, however, exhibit the instanton responsible for the decay.
264
P. Ginsparg, M.J. Perry / Semiclassicaiperdurance of de Sitter space
Fig. 3. Pearose diagram representilag the nucleation and subsequent decay of an asymptotically de Sitter black hole. The wavy line is a spacelike singularity and the 45 ° line is the black hole event horizon.
another nucleation. A conformal diagram for the semiclassical nucleation of one such black hole and its subsequent evaporation appears in fig. 3. From a minkowskian standpoint, eq. (3.4) or (3.5), one might worry that the opposite fluctuation, making 3raG > A-t/2, could occur with equal probability. Unfortunately, this metric has a naked space-like singularity and has no real euclidean section. It corresponds to de Sitter space being eaten by a giant black hole. We believe that such behavior should be excluded by a suitable generali7.ation of the (physically plausible but mathematically unproven) cosmic censorship hypothesis to semiclassical processes. In any event, the lack of a real euclidean section for the case at hand prevents its consistent inclusion in our formalism. We conclude that de Sitter space, faithful to its periodic identification in imaginary time, possesses a semiclassical instability analogous to that possessed by hot fiat space. The cosmological event horizon in de Sitter space, however, acts to prevent fluctuations to larger than critical size black holes so there is no subsequent black hole growth. The semiclassical suppression factor in their nucleation rate suggests that these black holes would have no greater cosmological influence than that determined [4] for those semiclassically nucleated, with comparable improbability, from hot flat space. We thank I. Affleck, L. Alvarez-Gaum~, S. Coleman, D. Freedman, G. Gibbons, D. Gross, A. Outlh S. Hawking, D. Page, K. Wiget, and E. Witten for useful conversations. One of us also thanks the Princeton High Energy Theory Group for its hospitality.
AppmUx The easiest way to show that the operator G of (3.12) has a negative mode on S 2 x S 2 is to use the fact that the F~instein metric is IOhler [32]. Defining a complex
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
265
variable z by z --- 2 ei¢ tan½0,
(A.I)
d # 2 = d02 + sin20d¢ 2 = (1 + ~:.7z)-2 d E d z ,
(A.2)
gives the unit 2-sphere metric as
where ~ is the complex conjugate of z. To see that this is Kahler, we recall that any Kahler metric has an associated K~thler potential K such that the metric tensor gze is locally given by g~e = a, OeK.
(A.3)
K = 2log(1 + ~Ez).
(A.4)
For the unit S 2 of (A.2), we find
The generaliTation to higher-dimensional complex spaces is given by g . g = O. OKK,
ds 2 = g . d d z ' d ~ b + c.c.
(A.5)
The Kahler potential for S 2 x S 2, with A = 1, is thus K = 2log[(1 + ~,~z)(1 + ¼~w)],
(A.6)
where z and w axe each complex coordinates for a 2-sphere. The connection coefficients, used to define the covaxiant derivative V,, are Fal'c= gb~ Ocg,~,
rave = O .
(A.7)
The Riemann tensor is given by
R,~,,= 8,a~aea,K-g'/( a,a~aaK)(alaaaeK),
(A.S)
so that for (A.6) we find R , m = - ¼(1 + -}ez) - ( ,
R~,,~. =
- ¼(1 + } ~ w ) -')
(A.9)
(with all other components, not related to the above two by symmetry, vanishing).
266
P. Ginsparg,M.J. Perry / Semiclassicalperdurance of de Sitter space
The differential operator G can now be transcribed into this language. The transverse tracefree fluctuation ~ag can be written
wfB,
(A.IO)
where A and B are real. The condition that ~ be tracefree is then
O=g"f'%f, = 2A(1 + ¼~z) 2 + 2B(1 + ¼~w) 2.
(A.11)
For ~ to be transverse requires
O = g ' i V J b : , = 2 ( 1 + ¼~w)2Cw + 2(1 + ¼~z)2A~ + ~(1 + ¼ez)A, 0 = g#-~7~q~[~-- 2(1 + ¼2z)2~ + 2(1 + ¼~w)2B~ + ~(1 + ¼~w)B,
(A.12)
where A z = O~A, etc. Finally, we write the zz, ww, and z~ components of G#~ = h#~ respectively as - 2 ( 1 + ¼ ~ w ) 2 B ~ - 2(1 + ¼~.z)2Be, - (1 + ¼~w)(wB~+ ~B~)
-(I+½~w)B-2B=XB, (A.13a) -2(1 + ~w)2B~,
- 2(1 + ¼Y.z)2Be~- (1 + ¼~w)(wB w + ~ B ~ )
-(1 + ½
w)a- 2B = xB, (A.13b)
_ 2(I
+ ~.z)2C,,
_
3.
2
2(1 + 4~w) C~w
_
(1 + ¼~w)wC,,,
- (1 + ¼Y.z)Y.Ce - C = XC.
(A.13c)
From (A.11), (A.12), and (A.13), it follows* that the only eigenfunction with h < 0 " Actually, the formalismof this appendix would only allow us to find modes which preserve, the complex structure. A somewhatlengthieranalysisin the original real coordinates,however, can be used to show that (A.14) is indeedthe only negativemode of G.
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
267
which is transverse and tracefree corresponds to A=(l+~z)
-2,
/
B=-(l+,ffw)
-2
,
C=0,
•=-2.
(A.14)
By comparison with (A.2), we see that (A.14) is the mode exhibited in (3.13). For A = 1, this mode has eigenvalue ~, = - 2A.
References [!] A. Guth, Phys. Rev. D23 (1981) 347; A.D. Linde, Phys. Lett. 108B (1982) 389; A. Albrecht and P.J. Stelnhardt, Phys. Rev. Lett. 48 (1982) 1220 [2] UP. Grishchuk and Y& B. Zeldovich, Acad. Sci, USSR Space Research Institute preprint trp-726 (1982); A.A. Starobinsky, Proc. of Nuffield Workshop on the early universe, ed. G. Gibbons, S.W. Hawking and S.T.C. Siklos (Cambridge Univ. Press, 1983), to appear; J. Ellis et al, CERN ref. TH3404 (1982), and references therein [3] G.W. Gibbons and S.W. Hawking, Phys. Re,/. DI5 (1977) 2738 [4] D.J. Gross, M.J. Perry and L.G. Yaffe, Phys. Rev. D25 (1982) 330 [5] J. Jeans, Phil. Trans. Rev. Soc. A199 (1902) 491 [6a] W. de Sitter, Mon. Not. Roy. Astron. Soc. 78 (1917) 3 [6b] E. Schrodinger, Expanding universes (Cambridge Univ. Press, 1956) [7] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge Univ. Press, 1973) [8] D. Gross and E. Witten, unpublished, N.A. Chernikov and E.A. Tagirov, Anal. Inst. Henri Poincar~ 9 (1968) 109; T.S. Bunch and P.C. Davies, Proc. Roy. So(:. A360 (1978) 117 [9] P. Candelas and D. J. Ralne, Phys. Rev. DI5 (1977) 1494; J.S. Dowker and R. Critchley, Phys. Rev. D16 (1977) 3390; I.T. Drummond, Nucl. Phys. B94 (1975) 115, I.T. Drummond and G. Shore, Ann. of Phys. 117 (1979) 89; G.M. Shore, Ann. of Phys. 128 (1980) 376 [10] A.S. Lapedes, J. Math. Phys. 19 (1978) 2289 [I 1] S.M. Christensen and M.J. Duff, Nucl. Phys. BI70 (1979) 480 [12] E.M. Lifshitz, Sov..[. Phys. 10 (1945) !16; E.M. Lifshitz and I.M. Khalatnikov, Adv. in Phys. 12 (1963) 185 [13] J.M. Bardeen, Phys. Rev. D22 (1980) 1882; W.H. Press and E.T. Vishaiac, Ap. J. 236 (1980) 323; W.H. Press and E.T. Vishniac, Ap. J. 239 (1980) 1 [14] G.W. Gibbons, S.W. Hawking, G.T. Horowitz and M.J. Perry, Comm. Math. Phys. 88 (1983) 295 [15] L.F. Abbott and S. Deser, Nucl. Phys. BI95 (1982) 76 [16a] P. Breitenlohner and D.Z. Freedman, Phys. Lett. 115B (1982) 197 [16b] S.J. Avis, C.J. lsham and D. Storey, Phys. Rev. DI8 (1978) 3565 [17a] J.A. Frieman and C.M. Will, Ap. J. 259 (1982) 437 [17b] J.D. Barrow, Proc. Nuffield Workshop on the very early universe, ed. G. Gibbons, S.W. Hawking and S.T.C. Siklos (Cambridge Univ. Press, 1983), to appear [18] M. Hammermesh, Group theory (Addison-Wesley, 1962) [19] G.W. Gibbons and M.J. Perry, Nucl. Phys. 146 (1978) 90 [20] S.W. Hawking and I.G, Moss, Phys. Lett. 110B (1982) 35 [21] P.J.E. Peebles, The large-scale structure of the universe (Princeton Univ. Press, 1980) [22] S. Coleman, Phys. Rev. DI5 (1977) 2929; C.G. Callan and S. Coleman, Phys. Re,/. DI6 (1977) 1762
268 [23] [24] [25] [26] [27]
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L.D. Faddecv and V.N. Popov, Soy. Phys. Usp. 16 (1974) 777 G.W. Gibbons and M.J. Perry, Prec. Roy. Sec. A358 (1978) 467 G.W. Gibbons and C.N. Pope, Comm. Math. Phys. 61 (1978) 239 D.N. Page, Phys. Lett, 79B (1978) 235 S.W. Hawking and C.N. Pope, Phys. Lett. 73B (1978) 42; C.J. Islam, Prec. Roy. Soc. A364 (1978) 591; CJ. Isham and C.N. Pope, Phys. Lctt. ll4B (1982) 137 [28] F. Kottler, Ann. der Phys. 56 (1918) 401 [29] B. Carter, Black holes (Les Houches, 1972), ed. C. DcWitt and B.S. DeWitt (Gordon and Breach, 1973) [30] H. Nariai, Science Reports of the Tohoku Univ. 35 (1951) 62 [311 G.W. Gibbons and M.J. Perry, Phys. Rev. D22 (1980) 313 [32] E.J. Flaherty, Hermitian and K~thler geometry in relativity (Springer Verlag, 1976); L. Alvarez-Ganm~ and D.Z. Freedman, A simple introduction to complex manifolds, in Unification of the fundamental particle interactions, ed. S. Ferrara, J. Ellis and P. van Nieuwenhuizen (Plenum, 1980)
SEMICLASSICAL PERDURANCE OF DE SrITER SPACE Paul GINSPARG*
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Malcolm J. PERRY t
Department of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA Received 24 January 1983
We investigate the classical and semidassical instabilities of de Sitter space. Due to the presence of a cosmological event horizon and its associated Hawking temperature, de Sitter space might be expected to behave in some respects like flat space at finite temperature. We use the euclidean formulation of quantum gravity to show that de Sitter space does exhibit a semiclassical instability to the nucleation of black holes. We find, however, no analog to the classical instability due to gravitational dumping of thermally excited gravitons.
De Sitter space is the maximally symmetric solution of the vacuum Einstein equations with cosmological constant A. Despite the present limits on the cosmological constant (defined as 8~rG times the vacuum energy density, thus having dimensions of mass squared), there has been a recent revival of interest in de Sitter space prompted by the potential solution of some important cosmological problems in inflationary models of the early universe [1,2]. In such models the universe is assumed to have undergone an early phase with a large effective cosmological constant, A - (10t°-10 II GeV) 2 for GUT era inflation, or A - (1016-10 Is GeV) z for Planck era inflation. A subsequent phase transition would then produce a region of space-time with A < (10 -42 GeV) 2 (the present observed limit) which becomes the space in which we now live. Since de Sitter space plays a key role in these models, we should like to learn more about its properties, particularly its quantum properties. The most straightforward means of quantizing fields on a background de Sitter space leads to the result that geodesic observers observe a thermal spectrum of excited states [3] (we shall shortly discuss this property at greater length). Such 'thermal' properties suggest that we compare and contrast de Sitter space with
* Research supported by Harvard Society of Fellows and by NSF grant No. PHY77-22864. * Research supported by NSF grant No. PHY80-19754. 245
246
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
ordinary fiat space at finite temperature T. Hot fiat space has been shown to exhibit two instabilities [4]. It has a classical instability, the Jeans instability [5], corresponding to gravitational clumping of thermally excited modes on length scales greater than a Jeans length, of order m e l t 2 (rap = G - I / 2 = Planck mass). It also has a semiclassical instability corresponding to spontaneous nucleation of black holes of size - l / T , w i t h rate per unit volume of order exp(-m2/16~rT2). In this paper, we shall examine de Sitter space for signs of analogous instabilities. De Sitter space [6a] is a space of constant curvature and is a solution of the vacuum Einstein equations Rab--'ASab,
(1.1)
with A > 0 (A < 0 is called anti-de Sitter space). The space can be visuali7ed [6b] as a hyperboloid --l)2 + W2 + X2 + y2 + z2 = ~t2,
(1.2)
embedded in E 5 with metric d s 2 = - d v 2 + d w 2 + d x 2 + d y 2 + d z 2,
(1.3)
where here and henceforth a = ~/-~.
(1.4)
The hyperboloid (1.2) has topology R x S 3 and invariance group SO(4, 1). In the limit A --* 0, this group becomes the usual Poincar6 group incorporating the Lorentz group SO(3, 1) and space-time translations. The de Sitter metric is the metric induced by this embedding; it is conveniently exhibited in coordinates t ~ ( - oo, oo), X ~ [0, It], 0 ~ [0, It], ~ E [0,2~r), defined by v = asinh(t/a), w = acosh(t/a)cos
y = acosh(t/a)sin
×sin0cos ~,
z = acosh(t/a)sin
X sin 0 sin ¢p,
X,
x = a c o s h ( t / c t ) s i n XcoS 0,
(i .5)
giving d s 2 = - d t 2 + a 2 c o s h Z ( t / a ) [ d x 2 + sin2x(d02 + sin20 d~ 2 )].
(1.6)
Eq. (1.6) is a k--- + 1 Robertson-Walker metric with spatial sections which are 3-spheres of radius a c o s h ( t / a ) . In these coordinates, de Sitter space appears to
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
247
~ + (t .(:0, ~ • 'n'l2 )
Surfoces of Constont t "<
/ N
J t-O
Lines of Constant X
\
J
'\ \
• -X , .tr
X.O
\ %
~°( t , - 0(:),~9 , - "~Z)
Fig. 1. A Penrose diagram of de Sitter space for constant 0, ¢,. Lines of X" constant are timelike geodesics. The dashed line represents the observer dependent cosmologicalevent horizon of an observer moving along the fine X" 0. The compactified time coordinate v/~ [ - ~r,' ½~r]is defined in sect. 3.
contract in size until t = 0, and then re-expand to infinite size. The coordinates (t, X, 0, ~,) cover the entirety of de Sitter space. Fig. 1 shows a conformal diagram of the space for constant 0, ~ [7]. We observe that the causal past of an observer moving along X = 0 does not coincide with the conformal boundary of de Sitter space, even as the observer moves into the infinite future. There is thus a region of the space with which the observer can never have causal contact. The boundary of this region is called his cosmological event horizon. It is observer dependent since different observers see different boundaries depending on their asymptotic destination [71. Other coordinate systems are frequently used to describe de Sitter space but only (1.5) actually covers the entire spacetime with just one coordinate patch. In coordinates t E ( - oo, oo), ~ ~ ( - oo, oo), .~ ~ ( - oo, oo), ~ ~ ( - oo, oo) defined by
w + v = aC/a,
y
=.~e t/~', A
x.~-.~e I/a,
(1.7)
z-~.~et/a,
the metric (1.3) becomes
d s 2 --- - d r
2 4- e 2 t ' / a ( d . ~ 2 4- d f l 2 4-
d,~2).
(1.8)
This takes the form of a k -- 0 Robertson-Walker metric and has flat spatial sections which exponentially expand. The coordinate patch (1.7) covers only that part of the spacetime with w + v > 0, as indicated in fig. 2a.
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P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
Surfoces of Coc~stont
Linls of Constant
~.0 t=-CO
(a)
,./~-~ t
I
Surfaces of Constant t'
I
Lines of Confront r
(*0
r,O
I
'~
x,Tr/2 (b) Fig. 2. The portion of de Sitter space covered by (a) the k = 0 coordinates (1.7); (b) the static coordinates (I.9). Conventions are the same as in fig. I. In (a), ;2 i ~2 +);2 + ~2.
In coordinates t' ~ ( - oo, o0), r ~ [0, a], 0 ~ [0, Ir], q~e [0,2¢r) defined by
v
=
a ~ l - r2/a 2 sinh(t'/a),
w = a ~ l - r2/a 2 c o s h ( t ' / o ) , x = rcos 0,
y
-
rsin 0cos ¢,
z = rsin 0 sinO, (1.9)
the metric (1.3) becomes
ds 2= - (l -½Ar2)dt'2 + (l - ½ A r 2 ) - ' drz + r2(dOz + sin2Odo2). (l.IO) In this coordinate system, de Sitter space looks static. This is because successive spatial sections are generated by 4 + 1-dimensional Lorentz transformations. These coordinates cover only the portion of the space with w > 0 and x 2 + y 2 + z 2 < et2. This corresponds to the region interior to the particle and event horizons of a
P. Ginsparg, M.J. Perry / Semiclassical perdurance of de Sitter space
249
geodesic observer moving along r - - 0 , as indicated in fig. 2b. The horizons are located at • -Quantizing fields on a background de Sitter space requires a careful definition of the vacuum state. Requiring only that it be de Sitter (SO(4, 1)) invariant leaves a one-parameter degenerate set of such vacua [8]. One prescription for implicitly choosing among these vacua is to work in an imaginary time formalism and analytically continue the resulting euclidean Green functions back to Minkowski space. This choice of vacuum quantization is probably the only one which generates Green functions having only causal poles and no spurious singularities. Well behaved Green functions for fields in de Sitter space have been computed by various authors [9] and they are periodic in imaginary time, with period This is explainable geometrically since, as will be described in sect. 3, the euclidean version of de Sitter space is simply S4. This periodicity suggests that observers in the space will perceive a thermal spectrum characterized by a temperature
ff/A.
2rt~/~-/A.
1
(1.11)
•
This was originally demonstrated in ref. [3] and subsequently confirmed by a Bogoliubov transform technique in ref. [10]. It is a rather curious form of temperature, though, since geodesic observers moving at constant velocity with respect to one another nonetheless perceive the same 'temperature'. This is because the expectation value of the energy-momentum tensor for gravitons [11]
26
( T a b ) = -- 1 3 5 ~ 2
AZg
' ab
(1.12)
and all other properties of the ground state preserve the de Sitter invariance. Moreover, despite having introduced the notion of temperature to the space, we assert that the de Sitter vacuum is still, by definition, a pure quantum state. A mixed state description need only be introduced for the individual local observer who feels compelled to construct a density matrix by integrating out all ground state information beyond his horizon. Since there is no de Sitter invariance breaking heat bath (as Lorentz invariance is broken by ordinary heat baths) and since we are dealing with a pure state, there is no doubt that this temperature can be entirely described in terms of quantum fluctuations. There are nonetheless sufficiently many similarities to ordinary thermal formalisms to suggest determining how the instabilities of de Sitter space compare to the thermally induced instabilities of Minkowski space at finite temperature. In sect. 2, we perform a perturbative stability analysis of de Sitter space along the lines originally proposed by Lifshitz [12]. This type of analysis was later reformulated by Bardeen and by Press and Vishniac [13], to treat carefully coordinate invariance and ensure that physical conclusions are drawn only from gauge invariant
250
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
perturbations. Using the Lifshitz formalism, we probe de Sitter space with perturbations taking a perfect fluid form, on the assumption that any tendency to gravitationally clump would be signalled by an instability in such perturbations. We find that any such effect is cut off by the ambient exponential expansion of the space. Thus there can be no analog in de Sitter space to the instability due to clumping of thermal gravitons displayed by hot flat space. The informed reader who is wondering how de Sitter space evades various singularity theorems [7] should notice that the energy-momentum tensor, including the cosmological term, does not satisfy the strong energy condition [7]. In sect. 3, we use the euclidean formulation of quantum gravity to look for semiclassical instabilities of de Sitter space. We do not pretend to understand what a fully quantized version of gravity would look like, but maintain that the formalism used here properly incorporates quantum mechanical contributions to processes occurring on scales large compared to the Planck length. For de Sitter space we shall have to assume GA ,~: 1 so that the characteristic size of the space is large compared to the Planck length. This is equivalent to requiring that the de Sitter temperature ( 1.1 !) be small compared to the Planck mass. Just as in the hot flat space case [4], we find that the boundary conditions for the euclidean functional integral allow finite action saddle points with non-trivial topology. They are interpreted as a probability of nucleating asymptotically de Sitter black holes with a rate per unit volume of order e x p ( - m 2 / 1 2 ~ r T 2 ) . This result is rather similar to the corresponding result already cited for hot flat space. Despite the semiclassical suppression factor, however, for de Sitter space it is the dominant instability since there are no classical instabilities. The semiclassical instability in hot flat space, by contrast, is never the most important unless the system is put in a box small enough to cut off the Jeans instability. Sect. 3 concludes with an argument that the subsequent minkowskian evolution of these nucleated black holes begins at an intrinsic temperature higher than the cosmological horizon temperature. The black holes thus always evaporate on a time scale small compared to the mean time between nucleations, rather than accreting radiation and growing. In this sense, the space perdures. Finally, an appendix reviews some of the formalism of Kahler geometry useful for computing the negative mode on S2 × S2, our gravitational instanton. Before proceeding to our analysis, we remark briefly on the relationship between stability and positivity of energy. It is true that energy becoming negative represents some form of instability. The converse is not the case, however, as can be seen by the example of hot flat space [4, 5]. Both its classical instability, the Jeans instability, and its semiclassical instability to spontaneous nucleation of black holes, are situations in which the positive energy theorem applies [14]. The issue of positivity of the energy for de Sitter and anti-de Sitter space has been discussed by Abbott and Deser [15] and for anti-de Sitter space by Freedman and Breitenlolmer [16a]. We do not believe the de Sitter space stability analysis of ref. [15] to be entirely complete due both to its restriction to energy considerations and to its need to make assumptions about
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
251
the behavior of perturbations on the horizon. Anti-de Sitter space is rather different from de Sitter space since it possesses a globally time-like Killing vector field but does not have a Cauchy surface. In refs. [15, 16], it is noted that this problem can be circumvented by imposing a suitable boundary condition at infinity. With this condition, semiclassical instabilities can be excluded by the non-existence of any equal energy metric into which the space could decay. Anti-de Sitter space does, nevertheless, possess an ordinary Jeans instability.
2. Perturbative stability In this section we apply the techniques of Lifshitz [12] to the analysis of perturbations of de Sitter space. Details omitted from our brief review of the general formalism may be found in the original references. As mentioned in the introduction, we perform our analysis in the coordinate system (1.5) which covers the entire spacetime*. In general, a k --- + 1 Robertson-Walker metric can be written ds 2 = - d t 2 + a 2 ( t ) [ d x 2 + sin2x(d0 2 + sin2Odq~2)],
(2.1)
where a ( t ) gives the radius of the S 3 spatial sections. Instead of t, we shall prefer the coordinate 71 defined by dt = a d n ,
(2.2)
so that d s 2 = g a b d x a d x b = a 2 ( ~ / ) ( - dl/2 + dx 2 + sin2X (d0 2 + sin20d¢2))
-- a2(~)(-d~ 2 + yopdx a dx p),
(2.3)
where "r=a= a-2g~t~ is the metric for a unit 3-sphere. In the analysis to follow, a 0 coordinate index corresponds to the ~/coordinate, Latin indices (a, b .... ) run from 0 to 3, Greek indices (a, fl .... ) run from 1 to 3, and a prime denotes differentiation with respect to 7. Unless otherwise noted, indices are raised and lowered with respect to the metric gob of eq. (2.3). The components of the unperturbed Ricci tensor R~, are R~ = a - ' ( 2 a 2 + a '2 + aa") 8~,
(2.4a)
R ° = 3a-'(aa" - a'2),
(2.4b)
R °0 -- 0,
(2.40
* After the completion of this work, we learned of recent similar perturbative analyses of de Sitter space stability [17]. While our final conclusionsare consistent, we feel that the k = 1 coordinatesused here give a more complete analysis than that of [17a]. Also, the exact solution to (2.26) (eq. (2.29) of this paper) implicidy givingthe synchronousgauge graviton propagator in de Sitter space, has not to our knowledgeappeared previously.
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P. Ginsparg, M.J. Perry / Semiclassicalperduranee of de Sitter space
and the Pdcci scalar is
R =- R a = 6 a - 3 ( a + a " ) .
(2.5)
The function a(~) is determined by the Einstein equations R~b - ½gobR + gobA ~- 8~rGT~b.
(2.6)
For the de Sitter metric (1.6), the change (2.2) leads to a compactified time coordinate
n=
os-'
(2.7)
with - ½~r < 11~ ½,r. De Sitter space thus takes the form (2.3) with a(n) ~ asecrl.
(2.8)
We now perturb the metric (2.3) by setting gob = g ~ sitter) + h ab"
(2.9)
General coordinate invariance always allows us in what follows to choose a gauge, the 'synchronous gauge', such that hoo = hoa = 0.
(2.10)
The equations which determine the remaining perturbations h°B of the metric tensor are obtained by expanding the Einstein equations (2.6) to leading order in h. The lowest-order perturbation of the Ricci tensor 6R~, is
8Rg;--L
2a2
h"+
h'
(2.11a)
8R°=--~-I ( ~Tah'- uah~'), 2a 2
(2.11b)
2a 2 +
1.__1_ a,,
a'
2a 2ha + a S
,
a
a'h'
2a 3~a
(2.11c)
where h =- h~ = h,ljg aa, [] ~- V'a Va, and the covariant and contravariant derivatives, Va and V a = "ta~ Va, are defined with respect to the unit 3-sphere metric 'Ga of
P. Ginapar$, M.J. Perry / Semiclatsicalperdurance of de Sitter space
253
(2.3). The resultant perturbation of the Ricci scalar is 8R = 1 ( I;'~Yah ~p - Oh) + 1 h" + 3a'h' a2 a3 a 2
2h a2 •
(2.12)
These metric perturbations fall into one of three classes. They may be associated with some energy-momentum tensor of a matter field, they may be gravitational waves, or they may be pure coordinate transformations. In what follows, we need to separate out these three physically distinct possibilities. The energy-momentum tensor which gives rise to metric perturbations is assumed to take the form of a perfect fluid, with pressure 8p, energy density 80, and fluid velocity flow vector u ~ normalized such that u~uo -- - 1. Hence
ro = (sp + 80)u-u + 8pg
(2.13)
and is isotropic in the comoving coordinates specified by u ° = 1/a, u a = O. The pressure and energy density are related by an equation of state 8__pp 8p = cs2 '
(2.14)
where cs is the speed of sound (assumed constant) for the fluid in question. The cases of physical interest are delimited by the values c~ = 0 (pressureless dust) and cs2 ---- ½ (radiation). Since any instability must develop slowly, at least initially, we can assume the fluid is in slow motion with respect to our synchronous gauge coordinates. The velocity four-vector in these coordinates is thus dominated by the time component and we can neglect terms quadratic in the u °. Since uau~ = - 1, we can put u ° = 1/a in what follows. The linearized Einstein equations (2.6) become (Gob = Rob -- ½Rgab), 8G o = _ --'2a --~1 ( Va vahOa _ Oh ) _ a'h__~'+ ha2
- 8¢tG Sp ,
(2.15a)
(2.15b)
8Go _- 2a 2
l(
6Gff=~a2
2°
V ~ v a h ~ + W~W°hVa- W a w a h - D h a ~ + h ~ " + a ha~'14h~
+ 2~8~(E3h_
wyWah~ + 2 h
2a'h'a
h " ) = 8 , r G c ~ SpS~.
)
(2.15c)
A convenient pair of equations for h can be derived from the off-diagonal components of (2.15c) and from eliminating 8p from (2.15a) and the trace of (2.15c). This
254
P. Ginsparg,M.J. Perry / Semiclassicalperduranceof de Sitter space
pair is
WvV'#hV,+ V.rvah Ih'- Vavah-t-lh#a+h~"+
h.#'-4h.#=0, (2.16)
2 _tLt
(~TaV~h#-nh-2h)(l+3c2)+-~(2+3c2)+2h"=O.
(2.17)
The density is then determined, by (2.15a), to be
8p =
l [ Va v#h~a - E3h +. 2a'h---~'- 2 h ] 16,rGa 2 a '
(2.18)
and the velocity of the perturbing material, by (2.15b), is 1
8pu.= 16¢rGa(1 +c2) [ V ~ h ' - vah~'].
(2.19)
Since our constant time spatial sections are maximally symmetric 3-spheres, an arbitrary initial hap is decomposible into representations of SO(4). These fourdimensional spherical harmonics are labelled by two integers (I, m) with 1 >t m/> 0 [18]. Useful properties of these functions, including their construction in terms of homogeneous polynomials of cartesian coordinates on flat R4, can be found in [12, 19]. Suppressing the l= 0, 1.... dependence, the (1,0) functions are spacetime scalars Q, the (l, 1) functions are transverse spacetime vectors S,~ ( ~7'~S,~= 0), and the (l, 2) functions are symmetric, transverse and tracefree tensors Ha# (0 = H~ = v,H~). The most general metric perturbation is then h~a = (h(~) Pa# +/~(71)Qa#) + a(aT)S~# + v(~l)H~#,
(2.20)
where Q~a =
½y~Q,
(2.21)
1
P"# = l(t + 2) v,,vaQ + Q,,I~, =
+
7 so.
(2.22) (2.23)
After substitution of (2.20), eqs. (2.16) and (2.17) become differential equations for
P. Ginsparg,M.J. Perry / Semiclassicalperduranceof de Sitterspace
255
X, F, a and J,, l#'+(2+3c2s)l~'tanrt+½(l+3c2s)(l+3)(l-1)(X+t~)=O, h" + 2h'tanT/- ½1(1+ 2)(h + / Q -- O,
(2.24a) (2.24b)
a" + 2a'tan 7/= 0
(2.25)
v" + 2v'tan*l+l(l+ E)v=O,
(2.26)
where we have substituted the de Sitter space form of a(~), eq. (2.8). Finally, eqs. (2.15a, b) give 8p and u ~ as BO = c°s2~7 [-#'tanT1 - ~ ( l + 3 ) ( i - 1 ) ( A + / ~ ) ] Q , 8 ¢rGa 2
(2.27)
cos3~1 [ 1 8pu" = 8rrOa3(1 + c2 ) (~(1+ 3)(•- 1)X' +31(l+ 2)/d)1(1+ 2) waQ
+ ½(t+ 3)(i-
(2.28)
We first consider the perturbations generated by the tensor spherical harmonics H,a. Since eq. (2.26) is a source-free equation and since the perturbation v(*I)H.~ is transverse and tracefree, this perturbation represents gravitational radiation (with no unphysical gauge degrees of freedom). The solution of (2.26) is v(~) = c t [sin(hT) + lcos 17sin((/+ 1)~7)] + ¢2 [cos(h7) + lcos 7/cos((l + 1)~)], (2.29) where the G are here and throughout arbitrary constants. It should be noted that the amplitude of v(~l) does not necessarily vanish as T/--* ½or. To understand best the asymptotic behavior of v, we reexpress it in terms of the uncompactified time coordinate t. As 71~ ½~r, t ~ co, and we find
v(t)
- &
+ A2e
(2.30)
where A I and A 2 axe constants depending on c 1, c a, and 1. Even though r(t) does not vanish as t ~ o0, neither does it grow with t, so we have excluded any unstable growing modes of gravitational radiation from de Sitter space. Any physical effects of the modes (2.31) are in any event vanishingly small since their physical wavelengths increase with the scale factor a, while their amplitude remains constant.
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P. Ginspar$, M.J. Perry / Semiclassicalperdurance of de Sitter space
Physical quantities, involving at least two derivatives of the metric, are thus asymptotically suppressed by powers of 1/a. In this sense, our results are consistent with the "no-hair" theorem for de Sitter space conjectured by Hawking and Moss [20]. These modes are nonetheless potentially interesting in inflationary models of the universe. If excited during the inflationary epoch, they could conceivably survive the phase transition to the present universe before being exponentially damped. They would then give rise to inhomogeneities for which there are strict observational bounds [21]. Before continuing to the vector and scalar perturbations, we point out that the gauge condition (2.10) does not completely specify the choice of coordinates, so there remain unphysical gauge degrees of freedom in ~t, h, and o which can be removed by a coordinate transformation x°--, x° +
For the general metric (2.3), changes parametrized by a t
raft),
(2.31a)
leave the resulting space-time invariant, for arbitrary functions f0, f. of the spatial variables x a. For perturbations of de Sitter space, this reduces to h .a -* h .ij + sin( ~ ) V ,, V lffo + sin( 71)f o'Gl~ + ( V . f ij + V lff, ) .
(2.31b)
We now turn to the perturbations generated by the vector harmonics S.. Letting f, = - o o S . in (2.31) we see that physical perturbations of the form oS.B are generated by ( o ( ~ ) - o 0 ) S . ~ where o0 is an arbitrary constant chosen at our convenience. The solution to (2.25) is o(~1) = c3 + c,(2~ + sin211).
(2.32)
Now (2.27) implies that 8p = 0 (there is no scalar associated with S. since v , S " = 0) and hence (2.28) gives c4 -- 0 for the matter of interest. The most general allowable o(,/) is therefore constant and consequently a pure gauge transformation as is evident by choosing o0 ~-c 3. Thus, perturbations of the form oS, p turn out to be completely irrelevant for de Sitter space. Finally, we examine the perturbations h P ~ + / ~ Q . ~ generated by the scalar harmonics Q. Choosing • Ct
f°=CQ"
f'--- 21(1+2) v o Q ,
(2.33)
P. Ginsparg, M.J. Perry / Semiclassicalper~b.4ranceof de Sitter space
257
in (2.31b), with C and C' being arbitrary constants, shows that
A(~l)=Cl(l+2)sin~+C',
p(~)= -C(l+3)(l-
l)sin~-C'
(2.34)
are pure gauge transformations and can be removed from an arbitrary perturbation without affecting its physical content. This freedom may be used to reduce (2.24a, b) to a pair of first-order equations whose solutions are then straightforwardly expressible, for arbitrary cs2, in terms of hypergeometric functions. Here, we will only give explicit solutions for the physically interesting cases of c2 -- 0, ], and the special case cs2 = - 1 which corresponds to local fluctuations of the cosmological constant. Such fluctuations might be expected to occur in theories whose cosmological constant results from a vacuum energy density before or after a symmetry breaking phase transition. On physical grounds, increasing c2, thereby increasing the pressure holding things apart, makes a system more resistant to gravitational collapse. In fact, analysis of (2.24a, b) shows that ~ a n d / t cannot blow up for any value of t when c2 >1 - 1 . Furthermore, for c2 >/0, 8p and 8pu ~ must vanish at least as fast as e -3t/° and e-4,/=, respectively, as t --* co. Finally, there is a gauge in which ~ = p = ~' = / t ' = ~," = #" = 0 at 7/= ½~r so the gauge invariant effective energy-momentum tensor of the induced gravitational perturbations must also vanish exponentially as t -* co. This is because this tensor, extracted from the second order variation of R°b -- ½gobR + Agob, is quadratic in h and its first two derivatives and thus depends only on ~ and/t and their first two derivatives. The case of pressureless dust, c 2 = 0, is that most likely to exhibit some form of instability. The solution to (2.24eq b) is
/t(~) = c5 [½sin2v; + (v/- ½~r)] + %(1+ 3)(•- l)[cos T/+ (7/- ½~r)sin v/], Z(~l) = -%[½sin271+(~l-½~r)]+c61(l+ 2)[cosTl+(~l-½¢)sin~l],
(2.35)
where we have (arbitrarily) fixed the gauge via (2.34) so that
,(½..) =x(½..)= o The induced energy density and velocity field are, from (2.27) and (2.28), 1
3
•
8p = 8¢~a2"cos ~[2c, sm~ + ( i + 3 ) ( 1 - 1)c6] Q,
SPa==
4¢
1Ga3c°s5~c5!(I +1 2)
v=Q"
(2.36)
(2.37)
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
258
For c s2 = ~, the solution to
(2.24a,b) is
X +/*=CT[cos(~oT1)+sin(~l)I,] + cs [sin(¢o~) + sin(~l)I2] X ' - ~' = c7 [-o~ sin(~0~)cos2~l + (2~02- l)(sin ~1cos ~ cos(~orl) + cos(71) I,)]
+ cs [¢o cos(oa'q)cos2~ + (2~o2- 1)(sin~ cos ~!sin(oa'q) + cos('q) I2) ] (2.38) where ~2-~1(1+2), I, = f d~ (cos(o~)cos ~ + ¢osin(co~l)sin ~),
/2 --- f dn (sin( ,0n )cos n - ,0 cos(,on)sin n).
(2.39)
The constants of integration implicit in Ij and 12 correspond to gauge transformations. The energy density and velocity are determined again by application of (2.27) and (2.28):
8p = (~2 _ 1) cos'(~)(c, +
cg)Q,
(2.40)
8~'Got 2
(~02.______ 1)
r
~7~Q
8pu= = 32~rGa ' °~c o s ' " t cTsin ~°" + cacos ~°"] l ( l + 2)"
(2.41)
T h e s e t w o c a s e s , c s2 = 0 and c s2 = ~, exemplify the general properties mentioned above. By inspection of (2.38) and (2.39), we see that in the gauge for which X --/~ = 0 at ~ = ½~r we also have k' = X" -/~' = #" = 0 at ~ ffi ½7r. Thus any energymomentum associated with the gravitational field asymptotically vanishes. We also find that 80 and 8#u ~' vanish rather quickly as ~! -~ ~r so scalar generated perturbations are completely harmless. The special case, c~ = - 1, implying 8p = - 8 0 , is even simpler. From (2.13), we find ST0~ ffi 0 so that (2.28) requires
(1+ 3)(•- 1)X'+ 1(1+ 2 ) ~ ' - - 0 .
(2.42)
The only solutions to this consistent with (2.24a, b) are identically the gauge transformations (2.34). We thus have a rigidity theorem which excludes local fluctuations in the cosmological constant as physical excitations.
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
259
Thus we have shown that in perturbation theory, there is no analog of the Jeans instability in de Sitter space. Regarded as a function of t, any perturbation which starts at t - - 0 will grow like a power of t initially, but, with the exception of pure gravitational radiation, will be damped exponentially at late times. We conclude that de Sitter space is classically stable.
3. Semi-classical instability The euclidean functional integral is the most convenient means of describing processes in quantum field theories which are classically forbidden, but allowed quantum mechanically [22]. The euclidean form of the functional integral for gravity, first derived by Faddecv and Popov [23], is Z = f Dgob ( x ) e x p [ - I ( g ) + gauge-fixing terms],
(3.1a)
where the euclidean action is given by
I(g)=
16~cjf(R-2a)g'/2d'x.
(3.1b)
The functional integral is evaluated by integrating over all metrics which are positive definite and obey appropriate boundary conditions. This functional integral construction of quantum gravity is poorly understood and its proper definition is the subject of much recent research and speculation. Fortunately, saddle point methods give a means for treating the functional integral and are adequate for a semiclassical stability analysis. To investigate localized instabilities of de Sitter space, we consider spaces which are asymptotically de Sitter. In euclidean space, this boundary condition translates to looking at geodesically complete and non-singular solutions to the euclidean Einstein equations Rab = Agab, with A > 0. These solutions are called gravitational instantons and have, from (3. l b), A V (4)
I=-
8~r'----G-'
(3.2)
with V (4) the four-volume of the instanton. Thus the requirement of finite action restricts consideration to compact instantons (for this reason we have also omitted boundary term corrections from (3.1b)). At present, we know of only four Einstein metrics* satisfying this condition. * By Einstein metric, we mean metrics for which ~'cRob ~ O, implyingRob-- ~go~.
P. Ginsparg, M.J. Perry I Semiclassicalperdurance of de Sitter space
260
The first such metric is the Einstein metric on S 4. It can be obtained by an analytic continuation of the de Sitter metric (1.10) by setting t'ffi it, which gives
ds2 =ffi(l - -]TAr2) d'r 2 -F (1- iliArZ) -I dr 2 4- r2d£/2, d122 = dO 2 + sin20 d~ 2.
(3.3)
The singularity at • -- ( 3 / A ) I/2 is a removable coordinate singularity provided that is identified with period 2~r(3/A) t/~. (3.3) is then recognized as the metric for a four sphere of radius ( 3 / A ) I/2. De Sitter space Green functions are thus periodic in i m a g i n a r y time and by analogy with a similar calculation in the black hole ease [24], we infer that de Sitter space behaves as though it has an intrinsic temperature Tc ffi (1/2~rX]A) I/: [31. Finally, using (3.2) with V (4) = ] ~ r 2 ( ~ ) 4, we see that the metric (3.3) has euclidean action I = - 3~r/AG. The next two metrics falling into the class of interest are the FJnstein metrics on CP2125] and CPz#c'-P"~ [26]. Neither of these metrics, however, has a spin structure: they admit only so-caEed 'generalized' spinors with an extra half unit of charge with respect to bosons [27]. We thus exclude these instantons from our saddle point evaluation of the functional integral. We now proceed to derive the final Einstein metric satisfying our boundary conditions from another asymptotically de Sitter-Lorentz metric. It is known as the Kottler [28] or Sehwarzschild-de Sitter metric, and represents a non-rotating black hole immersed in de Sitter space. Its metric, in static form similar t o (3.3), is ds z_- _ V ( r ) d t z + V - i ( r ) d r 2+ r2dl2 z, l/(r)
2Gm
= 1
~Ar2 '
(3.4)
r
where m represents the mass of the black hole. The physical interpretation of this metric is perhaps more evident in terms of asymptotically fiat coordinates (similar to (1.8) for ordinary de Sitter space)* 1 d8 2 ~ _
m --t'/*
2r I 1 + rn e-t~/~ 2rl
2
dttz + (1 4- ~-~rl m e-tl/a) 4e2tl/a(dr~ + r~d~2),
where
r=r,o',"(,+
m
2
dt; = dt
* We are grateful to L. Alvarez-Gaum~: for b t ~
-
r/a
dr.
ill - ( 2 m l r ) V(r) this form of the metric to our attention.
(3.5)
P. Ginsparg,M.J. Perry / Semiclassicalperdurance of de Sitter space
261
(3.5) clearly reduces to the familiar asymptotically flat coordinates of ordinary Schwarzschild and de Sitter spaces, respectively, in the limits 1 / a 2 = 13A = 0 and m = 0. For 9G2m2A < 1, the function V(r) of (3.4) has zeroes at two positive values of r, denoted r+ and r++, with r + < r . + . In the coordinate system of (3.4) these correspond to a cosmological event horizon at r++ and a black hole event horizon at r+. For 9G2m2A = 1, r + s r++ is a double root and a degenerate horizon is formed just as occurs for the extreme (charge = mass) Reissner-Nordstrom black hole horizon. The surface gravity i¢, defined by 1b V'~!° - r l °, where 1° is the null geodesic Killing vector which generates the horizon [29], vanishes in this limit. A euclidean metric can be constructed from (3.4) just as (3.3) was analytically continued from (1.10) for de Sitter space. Letting t = i~', the metric (3.4) becomes ds 2 = V ( r ) d ~ 2 + V-'(r)dr 2 + r 2 d ~ 2.
(3.6)
Now we must deal with singularities at both r+ and r++. Each is a conical singularity unless ¢ is identified with an appropriate period. Simultaneous removal of both singularities, however, requires the same period at the two singular values of r, which in turn only occurs in the limit as r+ tends towards r++. Since the euclidean region of the metric lies between r ffi r+ and r = r++, it might seem that the euclidean region shrinks to zero. This is not the case, however, and is an artifact of a poor coordinate choice. To see what really happens, let 9G2m2A = 1 - 3e 2,
(3.7)
so that the limit r + ~ r + + corresponds to e---0. We now def'me a new radial coordinate X, and a new time coordinate i/,, by
x=cos-'(a '/~(r-r°) ,
V,=a'/~,,
(3.8)
where r0 - A - I/2(1 -- i~e2) SO that r++. + -- r0 + A - I/2e. The resultant metric, to order e 2, is
d s 2 -- 1 ( 1 - ~COsx- i~2cos2x)dx2 + X1 (l + }ecos X + }e2cosZx)sin2xd~2
+X1 (1 -
2 e c o s x + e2cos2x - ~e2)(dO 2 + sin2Od~2).
(3.9)
When • -- O, there are no conical singularities and the metric is 1
d ~ = X [dx~ + sin~xd~'~ + d°~ + sin~Od~]'
(3.10)
262
P. Ginsparg, M,J. Perry / Semiclassicalperdurance of de Sitter space
which is a metric on S 2 x S 2, each sphere having radius A - ~/2. The Lorentz version of this metric, with topology H 2 × S 2, was discussed by Nariai [30]. The space described by (3.10) evidently has non-zero volume, V c4)= 16~r2/A2, and has action I = - 2 , ¢ / ~ G . With Euler character X = 4, it is also topologically distinct from S 4, which has X = 2. Even without taking the limit e --* 0, the metric (3.9) still has finite action [31]. The &function behavior of the Ricci scalar is integrable, with a 2-dimensional surface of area A and deficit angle 8 (i.e. a polar angle identified with period 2~r - 8 rather than 2,r) giving a contribution of - A S / 8 , r to the euclidean action. The Einstein equations are not, however, satisfied at the conical singularities so the space is not actually an instanton. It is clear, nonetheless, that as m is decreased from ½ G N / 2 to zero, the action decreases, continuously from - 2 , r / A G to - 3 , r / A G . For e small, the corrections to the action for the metric (3.9) come entirely from the conical singularities, which give 2~r
I=
AG
20~r 2
9AGe
- - 4,
+O~e
).
(3.11)
This situation is very similar to the relationship between hot flat space and the euclidean Schwaxzschild space, as discussed in [4]. By analogy, the presence of a non-normaliTable negative mode (corresponding to moving mass out of the space) suggests that there is also a normalizable negative mode in the spectrum of gauge invariant fluctuations about (3.10). By formalizing this statement below, we shall show that the S 2 x S 2 metric is not a local minimum of the euclidean action, but instead contributes an imaginary part to the free energy of de Sitter space, indicating an instability. To determine whether the S 2 x S 2 action is a local minimum or a saddle point, we must examine the spectrum of a differential operator G restricted to transverse tracefree symmetric tensors ~ab, which represent gauge-invariant quantum perturbations of the metric [19]. Acting with eigenvalue h, it is defined by Gab~dq/° = -- Dq~¢d -- 2RacbdeP"b = ~qJ~d"
(3.12)
Spaces for which all of the eigenvalues h of G are positive or zero are minima of the action. Correspondingly, spaces for which G has a negative eigenvalue are saddle points of the action. G turns out to be positive for the euclidean de Sitter space (3.3) but has a single negative mode for the metric (3.10) on S 2 x S 2 (see appendix). In the coordinate system of (3.10), this negative mode takes the form ] diag( 1, sin2 X, - 1, - sin20). ¢~b = ~-
(3.13)
o(S2xS 2) + e Q~b is evidently a metric on $ 2 × S 2 for which the The metric g~b-~-b=b ,
P. Gin~parg, M.J. Perry / Semiclassicalperdurance of de Sitter space
263
spheres have radii A - 1/2(1 + e') I/2. It has the same symmetries as (3.10) but is not a solution of the Einstein equations. Its action is I=
2¢r
2~re'2
AG
AG
"
(3.14)
(It should be noted that it is this perturbation which appears in the gaussian approximation of the functional integral and not the deformation exhibited by (3.9) whose action is (3.11). Had we iinearized that perturbation, it would turn out that it was not square-integrable, and not an eigenfunction of G.) By arguments parallel to those explained in [4], we can estimate the partition function for our system. The zero modes of the S 2 × S 2 instanton can be treated by the standard collective coordinate method (they are harmless in any event since the space is compact). The partition function takes the form Z = c + 3~r/AG.
(quantum corrections) + ½i e + 2,,/A6. (quantum
corrections). (3.15)
The first term arises from the S4 instanton, the second from the S 2 × S 2 instanton, and the factor of ½i multiplying e +2f/A6 arises from the negative mode. The quantum corrections, arising from the gaussian and higher-order contributions, have not been calculated in detail, principally because the non-renormaliTability of the theory causes them to be cut-off dependent. By the usual dilute gas approximation (reviewed in ref. [4]), the result (3.14) is interpreted as a decay rate per unit volume proportional to e x p [ - ~r/AG]. T h e tunneling process resembles a thermally activated process in which the entire space tunnels at once to a degenerate Schwarzschildde Sitter spacetime with two horizons located on top of one another. Higher-order quantum corrections, however, effectively result in one horizon moving inside the other due to the instability represented by the euclidean negative mode. We can then describe the subsequent minkowskian evolution for the metric (3.9) by analogy with the Schwarzschild case. The horizon temperatures of the inner (black hole) and outer (cosmological) event horizons are given by (1/4 ~r) IOV/Orl r - - , . . . . . [3], which equals X3A(r + + - r +)(2 + r ++/r +) and ] A ( r ++- r+X2 + r +/r ++), respectively. Since by definition r++> r+, we see that the Hawking temperature associated with the black hole horizon will always be the greater* [for the metric (3.9), these two temperatures are (1/2¢r)AI/2r(1 + ]e)]. The black hole will then presumably evaporate on a time scale roughly of order t - G2m 3. Since G2m 2 - A -I, t - A - 3 / 2 / G . T h e point is that within the regime of validity of the semiclassical approximation, GA .~ 1, the black hole evaporates away on a time scale much shorter than that required for * This has been previously observed in ref. [15] in which the authors conclude that evaporation is the only possible classical evolution. They do not, however, exhibit the instanton responsible for the decay.
264
P. Ginsparg, M.J. Perry / Semiclassicaiperdurance of de Sitter space
Fig. 3. Pearose diagram representilag the nucleation and subsequent decay of an asymptotically de Sitter black hole. The wavy line is a spacelike singularity and the 45 ° line is the black hole event horizon.
another nucleation. A conformal diagram for the semiclassical nucleation of one such black hole and its subsequent evaporation appears in fig. 3. From a minkowskian standpoint, eq. (3.4) or (3.5), one might worry that the opposite fluctuation, making 3raG > A-t/2, could occur with equal probability. Unfortunately, this metric has a naked space-like singularity and has no real euclidean section. It corresponds to de Sitter space being eaten by a giant black hole. We believe that such behavior should be excluded by a suitable generali7.ation of the (physically plausible but mathematically unproven) cosmic censorship hypothesis to semiclassical processes. In any event, the lack of a real euclidean section for the case at hand prevents its consistent inclusion in our formalism. We conclude that de Sitter space, faithful to its periodic identification in imaginary time, possesses a semiclassical instability analogous to that possessed by hot fiat space. The cosmological event horizon in de Sitter space, however, acts to prevent fluctuations to larger than critical size black holes so there is no subsequent black hole growth. The semiclassical suppression factor in their nucleation rate suggests that these black holes would have no greater cosmological influence than that determined [4] for those semiclassically nucleated, with comparable improbability, from hot flat space. We thank I. Affleck, L. Alvarez-Gaum~, S. Coleman, D. Freedman, G. Gibbons, D. Gross, A. Outlh S. Hawking, D. Page, K. Wiget, and E. Witten for useful conversations. One of us also thanks the Princeton High Energy Theory Group for its hospitality.
AppmUx The easiest way to show that the operator G of (3.12) has a negative mode on S 2 x S 2 is to use the fact that the F~instein metric is IOhler [32]. Defining a complex
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
265
variable z by z --- 2 ei¢ tan½0,
(A.I)
d # 2 = d02 + sin20d¢ 2 = (1 + ~:.7z)-2 d E d z ,
(A.2)
gives the unit 2-sphere metric as
where ~ is the complex conjugate of z. To see that this is Kahler, we recall that any Kahler metric has an associated K~thler potential K such that the metric tensor gze is locally given by g~e = a, OeK.
(A.3)
K = 2log(1 + ~Ez).
(A.4)
For the unit S 2 of (A.2), we find
The generaliTation to higher-dimensional complex spaces is given by g . g = O. OKK,
ds 2 = g . d d z ' d ~ b + c.c.
(A.5)
The Kahler potential for S 2 x S 2, with A = 1, is thus K = 2log[(1 + ~,~z)(1 + ¼~w)],
(A.6)
where z and w axe each complex coordinates for a 2-sphere. The connection coefficients, used to define the covaxiant derivative V,, are Fal'c= gb~ Ocg,~,
rave = O .
(A.7)
The Riemann tensor is given by
R,~,,= 8,a~aea,K-g'/( a,a~aaK)(alaaaeK),
(A.S)
so that for (A.6) we find R , m = - ¼(1 + -}ez) - ( ,
R~,,~. =
- ¼(1 + } ~ w ) -')
(A.9)
(with all other components, not related to the above two by symmetry, vanishing).
266
P. Ginsparg,M.J. Perry / Semiclassicalperdurance of de Sitter space
The differential operator G can now be transcribed into this language. The transverse tracefree fluctuation ~ag can be written
wfB,
(A.IO)
where A and B are real. The condition that ~ be tracefree is then
O=g"f'%f, = 2A(1 + ¼~z) 2 + 2B(1 + ¼~w) 2.
(A.11)
For ~ to be transverse requires
O = g ' i V J b : , = 2 ( 1 + ¼~w)2Cw + 2(1 + ¼~z)2A~ + ~(1 + ¼ez)A, 0 = g#-~7~q~[~-- 2(1 + ¼2z)2~ + 2(1 + ¼~w)2B~ + ~(1 + ¼~w)B,
(A.12)
where A z = O~A, etc. Finally, we write the zz, ww, and z~ components of G#~ = h#~ respectively as - 2 ( 1 + ¼ ~ w ) 2 B ~ - 2(1 + ¼~.z)2Be, - (1 + ¼~w)(wB~+ ~B~)
-(I+½~w)B-2B=XB, (A.13a) -2(1 + ~w)2B~,
- 2(1 + ¼Y.z)2Be~- (1 + ¼~w)(wB w + ~ B ~ )
-(1 + ½
w)a- 2B = xB, (A.13b)
_ 2(I
+ ~.z)2C,,
_
3.
2
2(1 + 4~w) C~w
_
(1 + ¼~w)wC,,,
- (1 + ¼Y.z)Y.Ce - C = XC.
(A.13c)
From (A.11), (A.12), and (A.13), it follows* that the only eigenfunction with h < 0 " Actually, the formalismof this appendix would only allow us to find modes which preserve, the complex structure. A somewhatlengthieranalysisin the original real coordinates,however, can be used to show that (A.14) is indeedthe only negativemode of G.
P. Ginsparg, M.J. Perry / Semiclassicalperdurance of de Sitter space
267
which is transverse and tracefree corresponds to A=(l+~z)
-2,
/
B=-(l+,ffw)
-2
,
C=0,
•=-2.
(A.14)
By comparison with (A.2), we see that (A.14) is the mode exhibited in (3.13). For A = 1, this mode has eigenvalue ~, = - 2A.
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