Generalized De Sitter Space

Home , De Sitter space

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It by these g well approximation. the as order investigated of first emb stability been in an has The hole such method. black in Euclidean a solution usual max the considered its by the as derived connec of embedding multiply Minkowskian properties Sitte the six-dimensional de thermal to a the extension using than its By general and more spacetimes are such one argue, we which, equations hspprdaswt oetoprmtrsltost h sp the to solutions two-parameter some with deals paper This .INTRODUCTION I. osj ueird netgcoe Cient´ıficas, Serrano Investigaciones de Superior Consejo etod ´sc Mge aań,Isiuod Matemát de Catalán”, F´ısica Instituto “Miguel de Centro t .Wietefis criticism first the While 0. = eeaie eSte Space Sitter De Generalized er .González-D´ıaz F. Pedro Spebr1,1999) 14, (September 1 qain a ewitnas written be can equations itrslto [4] solution Sitter with otadL 7 aeas osdrdteipratcs of case important the Recently, considered also have topology. [7] connected Li and simply Gott with space Sitter rotations). six and boosts (four vectors Killings ten shows cesfragnrlshrclysmercmetric stat- symmetric then spherically general Assuming a constant. for cosmological icness the Λ with pee eobtain we sphere, ihapstv omlgclcntn Λ constant (1.1) cosmological equations Ricci positive Einstein the a (positive to with curvature solution symmet- a positive is maximally which constant a scalar) as of [5] space indentified ric be can case. space vacuum ter pure the in i.e. (1.3), Eq. in hshstopology has This tn.d itrssai ouinaie hnw set we when arises solution static Sitter’s de stant. mass the where eSte pc en hnta hc sidcdi this in induced is which that then i.e. embedding, being space Sitter de meddin embedded iulzda v-yebli [6], five-hyperboloid a as visualized htw aehtet ecie orsod oade a to corresponds described hitherto have we What rmasihl oetcnclpito iw h eSit- de the view, of point technical more slightly a From d Ω ds 2 2 ubtoso h isagPrytype Ginsparg-Perry the of turbations eial ymti,vcu Einstein vacuum symmetric, herically 2 e aehv lobe investigated. been also have case ted = nrlzdd itrsaecontaining space Sitter de eneralized ds digsaeaetesm sthose as same the are space edding pc edrsaantteeet of effects the against perdures space − = ouin h lblsrcueof structure global The solution. r 2,206Mdi (SPAIN) Madrid 28006 121, e mletnin ecekta the that check we extension, imal csyF´ısica Fundamental, y icas dθ 2 A v − = 2 E 2 = dv + sin + 5 − m ihtems eea erco the of metric general most the with , e 2 A w e R − + A R 2 B = oe bu sa nerto con- integration an as about comes × dt 2 + µν dw 1 = S θdφ − 2 x 4 Λ = + B 2 nainegroup invariance , 2 + 2 − + e n ec h Schwarschild-de the hence and B h erco h ntthree- unit the on metric the dx g y Λ dr µν 2 3 r 2 + 2 2 , + + − z dy 2 r 2 2 = 2 r m d + Ω Λ , 3 2 2 dz , , > SO 2 .I a be can It 0. . (4 , ) and 1), m (1.2) (1.1) (1.5) (1.3) (1.4) 0 = a multiply connected de Sitter space. These authors im- ρ(r) a given function of the radial coordinate r, has been posed the identification symmetry of the universal cover- shown by Stephani [9] to be two, and therefore one should ing of Misner space [5] to the five-hyperboloid visualizing consider a six-dimensional Minkowski space as the most the de Sitter space, so inducing a boost transformation general embedding for it. This would mean that one of in the region w> v covered by the static metric of this the six degrees of freedom of the embedding space should space that makes this| | region multiply connected. In this become frozen in the case of the usual de Sitter metric, way, the event horizon of the de Sitter space becomes so pointing toward some lack of generality for this met- a (Cauchy) chronology horizon which defines the onset ric. In fact, let us take the function ρ(r) to be given of an interior nonchronal region which supports closed by ρ(r) = H2r2, with H = Λ/3, and write the six- timelike curves (CTC’s). The multiply connected de Sit- dimensional− Minkowski metric as ter space can be used to implement new boundary con- p 2 2 2 ditions for the universe according to which, rather than ds = dz0 + dzi , i =1, ..., 5 (2.1) being created from nothing, the universe created itself − [7]. embedding the de Sitter manifold according to e.g. the In this paper we argue that the static de Sitter space following coordinate transformations for all topologies actually corresponds to rather restricted 2 2 2 2 solutions of the Einstein field equations for a spheri- z0 = k 1 H r sinh(t/k), z1 = k 1 H r cosh(t/k) − − cally symmetric empty space which becomes curved only p p (2.2) by the action of quantum vacuum effects. We show that there exist more general solutions to this problem, defined by two cosmological parameters, namely the z2 = f(r) (2.3) usual cosmological constant, and a new constant vacuum term could could be associated to vacuum effects whose backreaction over the spacetime geometry led to higher- z3 = r cos φ sin θ, z4 = r sin φ sin θ, z5 = r cos θ, (2.4) derivative terms in the gravitational action. The rest of the paper can be outlined as follows. In where k is an arbitrary, dimensional parameter to be Sec. II we discuss the physical arguments that lead to fixed later, and the function f(r), defining the new coor- obtaining a two-parameter, generalized de Sitter solu- dinate z2, is given by [10] tion, deriving the precise form of such a new solution. 2 2 2 df ′ 2 k (dρ/dr) +4ρ We investigate then the nature of the event horizons and = (f ) = . (2.5) the global structure of the new solution, and maximally dr − 4(1 + ρ) " # extend the metric beyond the apparent horizons, both in the simply connected and multiply connected cases. Taking for the arbitrary parameter k the inverse of the −1 −1 The thermal properties associated with the existence of surface gravity of de Sitter space, k = κc = H , these event horizons are studied in Sec. III by using we get then from Eq. (2.5) that the function f(r) be- the usual Euclidean procedure, and a new method based comes a constant, f0, for de Sitter space, and hence it on replacing the Kruskal maximal extension by the six- follows that the coordinate z2 no longer is a degree of dimensional embedding Minkowskian space as the back- freedom for this space. Therefore, though the de Sitter ground spacetime where we allow the propagation of a static metric can still be visualized as a six-hyperboloid, massless scalar field. In Sec. IV we investigate the ef- this is reducible to a five-hyperboloid defined for a re- −1 2 fects that different types of Ginsparg-Perry perturbations scaled cosmological constant ΛR =1/ Λ f0 /3 , and [8] have in the generalized Schwarzschild-de Sitter space, the embedding Minkowski metric becomes that− of a five- concluding that this space is stable to the action of such dimensional space. In what follows we interpret this re- perturbations in first order. Finally, we briefly summa- sult as being an indication that the known de Sitter solu- rize and conclude in Sec. V. tion is not the most general solution that corresponds to general relativity with no matter for consistent nonzero quantum-mechanical vacuum fluctuations, for the follow- II. A TWO-PARAMETER VACUUM SOLUTION ing reason. Einstein equations were initially thought of as satisfying the requirement that they should permit flat We start this section by noticing the sense in which the spacetime as a particular solution in the absence of mat- de Sitter space cannot be considered as the most general ter. By introducing the cosmological constant, Λ, Ein- solution to spherically symmetric Einstein equations for stein dropped [3] this requirement out and got a slightly the pure vacuum case. As it was pointed out in the Intro- more general set of equations which do not allow the exis- duction, de Sitter space can be visualized by embedding it tence of globally flat spacetime in the absence of matter, in the five-hyperboloid (1.4), with embedding Minkowski for then these equations become identical to Eq. (1.1). metric (1.5). However, the general embedding class for Motivated by the facts that (i) any cosmological con- metrics of type (1.2), with A = B = ln[1 + ρ(r)] and stant terms should all be associated to quantum fluctu- − ations of vacuum [11], (ii) the gravitational Lagrangian

2 must contain curvature squared corrections whenever it is field equations (2.6) that A = B, and from the equation considered quantum-mechanically [12], and (iii) the phys- for the spherical angle θ, − ical dimensions of the cosmological constant terms should 2 4 be that of the non-constant part of the geometric terms − Λr Qr˜ 2m eA = e B =1 , (2.7) containing the curvature in the gravitational action, we − 3 − 5 − r now go a step further by introducing in spherically symmetric spaces another vacuum term which we assume with m an integration constant playing the role of an to depend on the radial coordinate, r, as Qr˜ 2, where existing mass. Solution (2.7) describes a generalized the constant Q˜ can, in principle, be positive, negative Schwarzschild-de Sitter spacetime. 2 µν In what follows of this section and in the next one or zero. Like the curvature squared terms R , Rµν R , etc., in higher-derivative gravity theories, the new con- we consider the case m = 0. Then, depending on the ˜ sign of the involved constants, we have distinct kinds stant Q ought to have the physical dimensions of a space 4 −4 of solutions. Setting H = Λ /3 and Q = Q˜ /5, if curvature squared, namely (length) . Disregarding the | | | | H2 > 0 and Q>˜ 0, the resulting solution shows an event curvature squared terms in the left-hand-side of the so p modified field equations, we obtain then horizon at 4 4 2 µν µν 2 H +4Q H R = g Λ+ Qr˜ . (2.6) r2 = − ; H 2Q4 p Strictly speaking, one had to have considered field equa- 2 ˜ tions associated with higher-derivative theories with R2 if H > 0 and Q< 0, then there will be two horizons at terms. However, in order to simplify our formulation, we 2 4 4 restrict here ourselves to work in a semiclassical approx- 2 H H 4Q r± = ± 4 − , (2.8) imation where all quantum-mechanical effects are only p2Q taken into account in the right-hand-side of the field with H2 2Q2. Finally when H2 < 0 and Q˜ > 0 (gen- equations. ≥ Solutions obtained from Eq. (2.6) will then depend on eralized anti-de Sitter solution), we obtain also an event two parameters, the cosmological constant Λ and the new horizon at ˜ parameter Q. The physical meaning of the cosmological 2 4 4 2 H + H +4Q constant is explicitly obtained in the quantum field the- rH = 4 . ory context as being the (homogeneous and isotropic) p2Q ground state expectation value of some vacuum stress ˜ In this paper, we will only consider the second of these tensor. As to the physical interpretation of Q, one may particular solutions, i.e. claim it to have an analogous origin as for Λ. Thus, Q˜ ought to be quantum-mechanically associated with some ds2 = 1 H2r2 + Q4r4 dt2 additional component of the vacuum stress tensor which − − would allow its value to be nonvanishing. On the other 2 2 4 4 −1 2 2 2 hand, given the explicit radial dependence assumed for + 1 H r + Q r dr + r dΩ2. (2.9) the additional term which is displayed in Eq. (2.6), and − the existence of a time-like singularity as one approaches This metric can be embedded in the six-dimensional r =+ (see Fig. 1), one could expect that a nonvanish- Minkowski space with metric (2.1) using the coordinate ∞ ing Q˜ would break homogeneity of the spatial sections at transformations very large scales. This seems to raise the interesting pos- 2 2 4 4 ˜ z0 = k 1 H r + Q r sinh(t/k), sibility that corrections coming from Q might be consid- − ered as fluctuations around a homogeneous space, which p eventually could be regarded as the seeds required for the 2 2 4 4 z1 = k 1 H r + Q r cosh(t/k) (2.10) formation of galaxies. Although in order to fix the pre- − cise form and physical meaning of the relation between p the solutions containing the parameter Q˜ and quantum z2 = f(r) (2.11) vacuum fluctuations would require further detailed stud- ies, it is still at this stage possible to conjecture that the generalized de Sitter spaces must describe additional z3 = r cos φ sin θ, z4 = r sin φ sin θ, z5 = r cos θ, (2.12) quantum fluctuations which can be associated with vir- tual charged particles located at r = 0. Note furthermore where the function f(r) again satisfies condition (2.5) that the way in which Q˜ will enter the relevant solutions which is singular at r±. Taking for the constant k the (see e.g. Eq. (2.7)) remainds one of the electric charge inverse of the surface gravity which now becomes in Reissner-Nordstrom black hole solution. 2 2 2 r+ r− 2Q H 2Q Assuming then staticness, for the general spherically κ = − = − , (2.13) c 2 2 4 4 symmetric metric (1.2), we shall again obtain from the r+ H + H 4Q p − p 3 2 2 2 ∗ dr we get by using the equalities Q =1/r+r−, H = (r+ + r = 2 2 2 4 4 2 2 2 2 2 2 4 4 r−)/r r−, and H 4Q = (r r−)/r r−, 1 H r + Q r + − + − + Z − p 3 3 2 2 r+ r− 2 2 2 r+r− r+ + r− 3 +2 r r+r− 1 r+ r 1 r− r ′ 2 r− r+ (f ) = − = 2 2 ln − ln − ,  2 4 2 2 r+ r− r+ r+ + r − r− r− + r − r+r− (r+ r−) gtt − −  (2.18) 

2 and then the advanced and retarded coordinates 2 2 2 r+ 4 2 6 − r− (r+ r ) 2r+ r2 +1 r +4r+r + − − − , V = t + r∗, W = t r∗, h 2 4 2 i  r r− (r+ r−) gtt − + −  so that the metric becomes (2.14) ds2 = 1 H2r2 + Q4r4 dV dW + r2dΩ2, (2.19) − − 2 which only vanishes when Q approaches zero. It appears that all of the degrees of freedom of the six-hyperboloid with the radial coordinate, r, defined by are now being used and that, therefore, metric (2.9) is ∗ 1 more general than the usual de Sitter metric in order to r = (V W ). 2 describe a spherically symmetric empty space which is − subjected to quantum vacuum fluctuations. In the case that Q2 < H2/2, we introduce now the new ′ 2 The function (f ) is positive and nonsingular for r coordinates V ′ and W ′ by means of the definitions ≤ r−, i.e. in the region of the generalized de Sitter space ′ −1 ′ −1 which can be thought of as a four-dimensional membrane V = tan [exp(κcV )] , W = tan [ exp(κcW )] , immersed in a six-dimensional flat space, with the para- − (2.20) metric equations given by and hence we obtain the Kruskal metric for a generalized Q4r4 z2 z2 = + 1 H2r2 + Q4r4 (2.15) de Sitter space with negative constant Q˜, that is 1 − 0 H2 2Q2 − − 2 ds =

2 z = f(r) (2.16) ′ ′ 2 2 4 4 4dV dW 2 2 1 H r + Q r 2 ′ ′ + r dΩ2, 2 2 2 2 − − κc sin(2V ) sin(2W ) z3 + z4 + z5 = r . (2.17) (2.21) The chosen solution H2 > 0, Q<˜ 0 is valid only for H2 2Q2. As H2 approaches 2Q2 one obtains an extreme≥ where κc is the surface gravity defined in Eq. (2.13), and generalized de Sitter space, a case which we shall briefly r is defined implicitly by the relation: r− discuss later on. r− r +r− r + The transformations (2.10) are not unique and cover ′ ′ r+ r + r− + r tan V tan W = − . only the region z1 > z0 of the Minkowski space. There − r+ + r r− r could actually be four| distinct| transformations that to- " − # gether cover completely the Minkowski space, namely (2.22)

2 2 4 4 z0 = k 1 H r + Q r sinh(t/k), The maximal extension is therefore obtained taking met- ± − ric (2.21) as the metric of the largest manifold which p metrics given only either in terms of V or in terms of 2 2 4 4 z1 = k 1 H r + Q r cosh(t/k) W can be isometrically embedded. There will be then a ± − maximal manifold on which metric (2.21) is C2. One can p plus another two with the coordinates z1 and z0 inter- also use the conformal treatment of the inﬁnities to pro- changed (z1 z0). This is exactly the same situation duce the Penrose diagram corresponding to the general- ↔ as that we are going to ﬁnd when extending from metric ized de Sitter space we are dealing with. It is depicted in (2.9) to its associated Kruskal metric [10]. In fact, met- Fig. 1, where the regions labelled I describe the interval ric (2.9) is geodesically incomplete because of the pres- 0

4 another by means of regions II and III, and each region of the Whitman function that corresponds to the prop- III is bounded by a timelike asymptotically flat infinite agation of a massless scalar field in the six-dimensional singularity at r . Minkowskian embedding space [10,15]. So far, we have→ confined ∞ ourselves to the case of a gen- We start with the Kruskal extension of the static met- eralized de Sitter space with Q˜ < 0 for a simply con- ric which does not contain the geodesic incompleteness nected topology. One can also consider this spacetime at the event horizons. In order for this metric to be def- with multiply connected topology on a given restricted inite positive we have to analytically continue the time region of it. This can be accomplished by imposing the coordinate according to the rotation t = iτ. To see how symmetry of the Misner space [5] to the visualizing six- this can be accomplished, we introduce new coordinates dimensional hyperboloid that embeds the generalized de in the metric (2.21), such that

Sitter space, that is by letting the Minkowski coordinates ′ ′ of that hyperboloid satisfy the identiﬁcation property tan V = R + S, tan W = R S. (3.1) −

(z0,z1,z2,z3,z4,z5) Then, the metric (2.21) becomes ↔ 1 H2r2 + Q4r4 dR2 dS2 ds2 = − − + r2dΩ2, (3.2) (z0 cosh(nb)+ z1 sinh(nb),z0 sinh(nb)+ z1 cosh(nb), κ2 (R2 S2) 2 c − where r is defined by z2,z3,z4,z5) , (2.23) r− r− r +r− r + where b is a dimensionless arbitrary period and n is any 2 2 r+ r + r− + r S R = − , (3.3) integer number, n =0, 1, 2, ..., . The boost transforma- − r+ + r r− r ∞ " # tion in the (z0,z1) plane implied by identifications (2.23) − will induce a boost transformation in the generalized de so that the surface r = 0 is defined by S2 R2 = 1, and Sitter space. Hence, since the boost group in the general- the time t results from − ized de Sitter space must be a subgroup of the generalized de Sitter group, the static metric (2.9) should also be in- R S − = exp( 2κct) . (3.4) variant under the symmetry (2.23). One can readily see R + S − − that this symmetry is in fact satisfied in the generalized de Sitter region covered by metric (2.9) and defined by In the region r r−, metric (3.2) becomes definite positive, provided we≤ perform the continuation z1 > z0 , where there are CTC’s, with the boundaries at | | 5 2 z1 = z0 and a given value of i=2 zi being the Cauchy S = iξ, t = iτ. (3.5) horizons± that limit the onset of the nonchronal region P from the causal exterior; that is the chronology horizons Then, from Eq. (3.4) we obtain for the multiply connected generalized de Sitter space 2 2 [7]. As applied to this space, the identifications (2.23) ξ + iR = R + ξ exp( iκcτ) . (3.6) become − − We thus see that thep Euclidean time τ becomes a peri- nb H2 + H4 4Q4 odic variable, with a period 2π/κ . Following then the − c t t + nbκc = t + , (2.24) usual procedure we can deduce that the generalized de ↔ 2Q2 Hp2 2Q2 − Sitter space with Q<˜ 0 is characterized by a background temperature given by which applies only to the regionp defined by r r−. In ≤ the case of an extreme generalized de Sitter space, where 2 2 2 2 2 κc Q H 2Q Q = H /2, identifications (2.24) reduce to t t + ,a T = = − , (3.7) ↔ ∞ result which corresponds to the fact that the temperature 2π π H2 p+ H4 4Q4 of the extreme generalized de Sitter space is zero. − p where it can be noted that in the extreme case Q2 = H2/2, T = 0, so that no thermal bath remains. III. THERMAL RADIATION IN GENERALIZED Although it reproduces the correct results, the Eu- DE SITTER SPACE clidean method is rather mysterious and too much schematic. To see the physics behind the thermal ra- In order to study the thermal properties of the general- diation process, we shall now follow a different proce- ized de Sitter space with Q<˜ 0, which are to be expected dure which relies on performing a calculation in the six- because of the existence of event horizons in this space, dimensional Minkowski space [10,15], which is thereby we shall follow two distinct procedures. We shall first taken to play the role of the maximum extension of the use the conventional Euclidean method [14], and then a static metric, instead of the Kruskal space on which our new method based on calculating the Fourier transform previous discussion was made.

5 As related to the embedding of the generalized de Sit- of residues, we can evaluate the Fourier integral (3.10) to ter space with Q˜ < 0, the Rindler wedge restriction of give the six-dimensional Minkowski space requires the usual 1 coordinate re-definitions F (ω) , (3.12) ∝ 2 2 4 4 2πω (1 H r + Q r ) exp κc 1 z0 = ζ sinh(gη), z1 = ζ cosh(gη), − − h i the proportionality factor being given by a certain prod- with all other zm remaining unchanged. Then, uct of powers of the frequency ω and the trajectory of the 5 accelerated observer ζ. In this approximation, the power ds2 = g2ζ2dη2 + dζ2 + dz2 . (3.8) spectrum is thus Planckian with a temperature given by − m m=2 expression (3.7). Making the same kind of calculation in X the full six-dimensional Minkowski space, without resort- The lines ζ =constant, zm =constant are the trajectories ing to geometric optics approximation, we finally obtain of the uniformly accelerated observers with constant ac- for the power spectrum celeration g; i.e. those observers moving along the hyper- 2 2 2 2 bolas ζ = z1 z0. They will possess an event horizon κc − 1+ 2 located at ζ = 0. The coordinates η, ζ cover only the F (ω) ω , (3.13) 2πω quadrant z1 > z0, such as the coordinates t, r do. One ∝ exp κc 1 can pass from metric (3.8) to the static metric by using − which is not Planckian and only becomes so in the ge- 2 2 4 4 1 H r + Q r ometric optics approximation where ω κc. The pro- η = t, ζ = − , (3.9) ≫ p κc portionality factor in (3.13) also depends on a product of given powers of frequency ω and observer’s trajectory so as the relations for z2, z3, z4 and z5 given in Sec. II, ζ. It is worth noting the dependence of F (ω) on ζ which and the identification g = κ . Because ζ = constant rep- c reflects the feature that the horizons in our spacetime resents a trajectory with constant proper acceleration, we are observer-dependent, such as it happens in de Sitter conclude that a static observer (r, θ, φ constants) in the space. interior generalized de Sitter static geometry is a particular type of uniformly accelerated observer (with acceleration proportional to κc) in the embedding Minkowski IV. GENERALIZED SCHWARZSCHILD-DE space. SITTER SPACE The power spectrum detected by an accelerated de Witt detector which probes the quantum fluctuations of In this section we briefly discuss the different possible a scalar massless field is given by [16] first-order perturbations of the Ginsparg-Perry type [8] +∞ that can occur in both the singly degenerate and dou- ′ F (ω) D(η, η ) exp(iω η)d( η), (3.10) bly degenerate cases corresponding to the generalized −∞ △ △ Z Schwarzschild-de Sitter space. Let us recall that the met- where D(η, η′) is the Whitman function D(z,z′) [14] eval- ric of this space can be defined by ′ uated along the world line of the detector, η = η η . 2m There is now the problem that, although the△ world− line eA = V (r)=1 H2r2 + Q4r4 . (4.1) − − r of the detector is the same in Minkowski and generalized de Sitter spaces, the scalar field is not restricted to the In order to determine the positions of the horizons in the two-dimensional space, but extends over the full space- extreme (degenerate) case, we first obtain the extremals time. In order to keep studying the behaviour of the field of the function rV (r), i.e. we solve for d [rV (r)] /dr = 0, close to the detector world line, we will make our calcu- which gives for Q2 = H2/2 lation in the geometric optics approximation where only short wavelengths are taken into account [10]. Thus, in 0 2 2(3 2) r± = ± . (4.2) this approximation, the Whitman function for the world 5H2 line (2.10) becomes 0 The horizon at r+ corresponds to the singly degenerate D(η, η′)= case (the two cosmological horizons coincide). From the 0 horizon condition V (r+) = 0 we obtain then m = 0, 0 2 i.e. there is no black hole in this case, and r+ exactly κc is the degenerate radius of the two cosmological horizons 2 2 2 4 4 2 . (3.11) − 16π (1 H r + Q r ) sinh [κc( η iǫ)/2] of the generalized de Sitter space studied in the prece- − △ − 0 dent sections. The horizon at r− can be associated with Choosing as the integration contour in the η complex △ the doubly degenerate case (the degenerate cosmological plane the strip 0 Imκc η 2π and using the theorem ≤ △ ≤

6 2 0 2 0 horizon also coincides with the radius of the black hole). H r− H r− 0 T+ = ,T− = ,Tb = 0; Replacing r− into V (r) = 0, we get then 2π − 2π i.e. the black hole with intermediate size would not ra- 8 0 m = m0 = r−. (4.3) 5 diate, but there is a radiation process transfering energy from the outer to the inner horizon of the generalized We consider now the distinct first-order perturbations de Sitter space, until the double degeneracy is restored. of the type first considered by Ginsparg and Perry [8]for Thus, the system appears to be stable to these perturba- the above two degenerate cases. Let us first study per- tions. turbations of the doubly degenerate case where we first We next consider perturbations of the form introduce the ansatz 0 2 r = r− 1 δ cos χ δ ψ − − t = 0 2 2r−H δ 0 2 11Hr−δ m = m0 1+ (4.10) 8 0 1 2 r = r− 1 δ cos χ + δ (4.4) − 13 1 Q4 = H4 2δ2 , 4 − 1 40 Q4 = H4 δ2 , 4 − 13 with the time coordinate ψ as defined by the first of Eqs. (4.4). Hence, one again obtains Eqs. (4.5) and (4.6) for with m0 unperturbed, and 0 < δ 1. In this case, we V (χ) and the perturbed metric, respectively. Thus, the have ≪ horizons at χ = 0, π will correspond to the respective 0 0 radii rh(χ =0)= r−(1 + δ) and rh(χ = π)= r−(1 δ). 4 − V (χ)= δ2 sin2 χ, (4.5) This may describe two distinct situations. Either (a) 5 rh(χ =0)= rb (radius of an evaporating black hole) and so that at first order in δ, rh(χ = π) = r−b (radius of an anti-evaporating balck hole [18]), with the degeneracy of the cosmological hori- 0 2 1 2 2 2 0 2 2 zons preserved at r± = r−, or (b) rh(χ) = r+ = rb and ds = dχ sin χdψ + r− (1+2δ cos χ) dΩ2. 2H2 − rh(χ = π)= r− = r−b, i.e. the degeneracy of the cosmo- (4.6) logical horizons is broken, but that between the horizons of the evaporating (anti-evaporating) black hole and the As fixed by V (χ) = 0, the horizons will appear at χ =0 cosmological outer (inner) horizons is preserved. In case and χ = π, which correspond to the perturbed cosmo- (a) the temperatures of the black holes, as calculated by 2 0 0 0 using Eqs. (4.7) - (4.9), are T±b = H r−/2π, and those logical horizons at r+ = r−(1 + δ) and r− = r−(1 δ), ± respectively. The radius of the black hole keeps being− of the degenerate cosmological horizons are T± = 0. In 2 0 0 case (b), we obtain instead, T±b = T± = H r−/2π. located at rb = r− and it no longer is associated to an ± event horizon. It follows that in any of the two cases, our spacetime is In order to evaluate the thermal properties of the so- stable to the perturbations specified by Eqs. (4.10). perturbed spacetime, we calculate the surface gravity, κ, Another kind of possible perturbations in the doubly in each case, determining then the associated tempera- degenerate case can finally be considered. It would de- ture by means of the simple expression T = κ/2π. In scribe perturbations which always leave the degeneracy order to compute κ, we shall follow Bousso and Hawking between the cosmological horizons unchanged, and is de- [17], i.e. we set fined by the same time perturbation as in Eqs. (4.4) and (4.10), and γψ κ =2π , (4.7) 0 id r = r− (1 + δ cos χ) ψh where 5 2 m = m0 1 δ (4.11) γ = H2r0 (4.8) − 4 ψ − and 4 1 4 − Q = H . 1 4 id ∂ ψ = 2π√gχχ √gψψ . (4.9) h χ=χh ∂χ These perturbations again lead to Eqs. (4.5) and (4.6) χ=χh ! for V (χ) and the perturbed metric. In this case, the tem- 2 0 We then obtain peratures that result are: T± = 0 and T±b = H r−/2π, ±

7 1 so that the spacetime is once again stable to the pertur- ds2 = (1+2δ cosh σ) sinh2 σdψ2 bations. −2H2 Let us consider in what follows of this section the case in which the horizon is at r = r0 , with m = 0. 2 + (1 2δ cosh σ)dσ2 + (1+2δ cosh σ)dΩ2, (4.18) It describes a singly degenerate generalized de Sitter H2 2 4 − − space with negative Q and no black hole being initially present. We introduce then the perturbative parameter and for the degenerate case δ by means of the equation: 1 2 ds2 = sinh2 σdψ2 dσ2 + dΩ2. (4.19) −2H2 − H2 2 4 4 1 2 2 Q = H δ , 0 δ 1. (4.12) 4 − 5 ≤ ≪ Metric (4.18) would represent a wormhole with the throat at the outer horizon. If we Wick rotate the time variable, The singly degenerate case corresponds to the limit δ → that is ψ = iϕ, we finally obtain a metric with Euclidean 0. The new time and radial coordinates, ψ and χ, will signature. In the degenerate case, this will produce two now be given by round two-spheres of different radius. The interpretation of these perturbations becomes ψ 0 t = 0 2 , r = r+ (1 + δ cos χ) . (4.13) again twofold. One can regard them as representing 2r+H δ a breaking of the degeneracy between the cosmological horizons at r±, in which case We then let the perturbation induce creation (or annihi- lation) of an extremely tiny black hole with mass 0 rh(χ =0)= r+(1 + δ)= r+ = rb 0 2 6r+δ m = . (4.14) 0 5 rh(χ = π)= r+(1 δ)= r− = r−b. − The inner and outer horizons of the generalized de Sitter On the other hand, we can also look at these pertur- space lie, respectively, at χ = π and χ = 0, while the bations as preserving that degeneracy, so that rh(χ = 0 balck hole horizon will be at 2m. To first order in δ, from 0) = rb, rh(χ = π)= r−b, and r± = r+ (as evaluated at the above transformations, we obtain the new metric χ = π/2). The calculation of the surface gravity for these pertur- 2 1 2 2 bations follows the same line as for the doubly degenerate ds = 2 (1+2δ cos χ) sin χdψ 2H cases, unless for the parameter γψ which should now be given by

2 2 2 2 0 (1 2δ cos χ)dχ + (1+2δ cos χ)dΩ2. (4.15) γ = H r , (4.20) − − H2 ψ + We notice that, since V (r)=1 H2r2 + Q4r4 becomes instead of Eq. (4.8). We then obtain for the temperatures negative when it is perturbed, i.e.− corresponding to the two horizons rh 2 0 2 2 H r (1+2δ) V (χ)= 4δ (1+2δ cos χ) sin χ, (4.16) T (χ =0)= + (4.21) − 2π the coordinate ψ becomes spacelike and the radial coordinate becomes timelike. In these coordinates, the topol- 2 0 H r+(1 2δ) ogy of the sections at constant ψ is S1 S2, and this T (χ = π)= − , (4.22) − 2π means creation of a handle. For these solutions,× the radius of the two-spheres, r, varies along the S1 coordinate and for χ = π/2, χ, with the minimal (maximal) two-sphere correspond- π H2r0 δ ing to the inner (outer) cosmological horizon, and the T χ = = + . (4.23) ergoregion between the cosmological horizons nesting a 2 − 2π little black hole of radius 2m much smaller than the size 0 When the degeneracy between the cosmological horizons of that ergoregion r+ r− = 2r+δ. At the degenerate is broken by the perturbations, then Tb = T+ = T (χ = case, the metric (4.14)− becomes 0), T−b = T− = T (χ = π), while the temperature (4.22) measures a thermal process by which the vacuum itself 2 1 2 2 2 2 2 ds = sin χdψ dχ + dΩ2. (4.17) anti-radiates. In the case for which the above degeneracy 2H2 − H2 is preserved, we obtain instead, T = T (χ = 0), T− = b b This metric is not the same as the Nairai solution. One T (χ = π), and T± = T (χ = π/2). All of these thermal still can recover a metric with the conventional signa- processes lead to a ﬁnal state in which there is no black ture by rotating χ = iσ, so that the line element (4.14) hole and the generalized de Sitter space becomes ﬁnally becomes again degenerate, so also these perturbations are stable.

8 V. SUMMARY AND CONCLUSIONS Robertson-Walker metric that can be associated with the static metric studied in this work. On the other hand, one In this paper we argue that the most general solution would also investigate the effects that can be expected to the general-relativity field equations that describe an from inserting a positive vacuum additional term in the empty spherically symmetric spacetime must contain two properties of the anti-de Sitter space. We have already constant parameters in order to account for the quantum obtained that in this case, the generalized anti-de Sitter vacuum effects. According to the possible combinations space has an event horizon and, therefore, one should ex- of the signs of such parameters, the general solution splits pect that, contrary to what happens in the usual anti-de into three essentially different static spacetimes, two de- Sitter space, this space has a precise temperature, and scribing generalized de Sitter spaces and one describing that its properties are similar to those of the generalized 4 a generalized anti-de Sitter space. In this paper, we have de Sitter space with positive Q . confined ourselves to the study of the generalized de Sit- ter case (positive cosmological constant) with a negative ACKNOWLEDGMENTS additional constant whose physical dimensions are the same as those of a space curvature squared. Such a solution possesses two event horizons and can be visualized as This research was supported by DGICYT under Re- a six-hyperboloid with embedding Minkowskian metric. search Project No. PB97-1218. It is maximally extensible by defining Kruskal like coordinates, and corresponds to a spacetime which can be made multiply connected inside the inner event horizon by imposing the symmetry of the Misner space. Since the additional vacuum constant term in the right- hand-side of the Einstein equations ought to correspond to an existing curvature squared terms in the left-hand- [1] W. de Sitter, Proc. Kon. Ned. Akad. Wet. 19, 1217 side of such equations, it could be thought that, as cor- (191/); 20, 229 (1917). responding to all other perturbation terms in the gravi- [2] C. Kahn and F. Kahn, Nature 257, 451 (1975). tational Lagrangian of a Lovelock theory [19], one could [3] A. Einstein, Preuss. Akad. Wiss. Berlin Sitzber. 142 have an arbitrary number of additional constants char- (1917) [English transl. in H.A. Lorentz, A. Einstein, H. acterizing quantum fluctuations of vacuum. From this Minkowski and H. Weyl, The Principles of Relativity. A Collection of Original Memoirs on the Special and Gen- point of view, the solution studied in this paper might be eral Theory of Relativity (Dover, New York, USA, 1952)]. regarded as merely being nothing but the second-order [4] W. Rindler, Essential Relativity (Springer-Verlag, New semiclassical approximation from the most general full York, Heidelberg, Berlin, 1977). solution. [5] S.W. Hawking and G.F.R. Ellis, The Large Scale Struc- The thermal properties of the spacetime dealt with in ture of Space-Time (Cambridge University Press, Cam- this work have been studied by using both the conven- bridge, England, 1973). tional Euclidean procedure, and a novel method based on [6] E.S. Schrödinger, Expanding Universe (Cambridge Uni- considering the propagation of a massless scaler field in versity Press, Cambridge, England, 1956). its six-dimensional embedding space, where the thermal [7] J.R. Gott and Li-Xin-Li, Phys. Rev. D58, 023501 (1998) spectrum is computed as the Fourier transform of the [see also P.F. González-D´ıaz, D59, 123513 (1999)]. Whitman equation. We obtain that an observer moving [8] P. Ginsparg and M.J. Perry, Nucl. Phys. B222, 245 along the origin of the radial coordinates will detect a (1983). background thermal bath at a temperature which van- [9] H. Stephani, in Unified Field Theories in More Than ishes only as one approaches the extreme (degenerate) Four Dimensions, edited by V. de Sabbata and E. case. Schmutzer (World Scientific, Singapore, 1983). We have finally studied the stability of the general- [10] M. Beciu and H. Culetu, Mod, Phys. Lett. A14, 1 (1999). ized Schwarzschild-de Sitter spacetime to the allowed [11] Ya. B. Zel’dovich, Zh. Eksp. Teor. Fiz. Pis’ma 6, 883 Ginparg-Perry perturbations in first-order approxima- (1967) [English Transl. Sov. Phys. JETP Lett. 6, 316 tion. It has been obtained that this spacetime perdures (1967)]. to all types of the considered perturbations, and that in [12] B.S. DeWitt, in General Relativity. An Einstein Cen- the case that no balck hole was initially present, pro- tenary Survey, edited by S.W. Hawking and W. Is- cesses are allowed in which black hole pairs are created rael (Cambridge University Press, Cambridge, England, 1979). and then evaporated away. [13] M.D. Kruskal, Phys. Rev. 119, 1743 (1960). There are two ways along which the present work could [14] N.D. Birrell and P.C.W. Davies, Quantum Fields in be continued. On the one hand, it appears appropri- Curved Spaces (Cambridge University Press, Cambridge, ate to investigate the cosmological effects that the ad- England, 1982). ditional vacuum term may have in cosmological mod- [15] M. Bouhmadi and P.F. González-D´ıaz, Thermal Proper- els. One should then consider the global Friedman- ties of de Sitter Space, to be published.

9 [16] D.W. Sciama, P. Candelas and D. Deutsch, Adv. Phys. 30, 327 (1981). [17] R. Bousso and S.W. Hawking, Phys. Rev. D54, 6312 (1996). [18] R. Bousso and S.W. Hawking, Phys. Rev. D57, 2436 (1998). [19] D. Lovelock, J. Math. Phys. 12, 498 (1971).

Legend for Figure

Fig. 1. The global structure of the generalized de Sitter space with negative Q˜.