Generalized De Sitter Space Pedro F. Gonz´alez-D´ıaz Centro de F´ısica “Miguel Catal´an”, Instituto de Matem´aticas y F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano 121, 28006 Madrid (SPAIN) (September 14, 1999) This paper deals with some two-parameter solutions to the spherically symmetric, vacuum Einstein equations which, we argue, are more general than the de Sitter solution. The global structure of one such spacetimes and its extension to the multiply connected case have also been investigated. By using a six-dimensional Minkowskian embedding as its maximal extension, we check that the thermal properties of the considered solution in such an embedding space are the same as those derived by the usual Euclidean method. The stability of the generalized de Sitter space containing a black hole has been investigated as well by introducing perturbations of the Ginsparg-Perry type in first order approximation. It has been obtained that such a space perdures against the effects of these perturbations. PACS numbers: 04.20.Jb, 04.62.+v, 04.70.Dy I. INTRODUCTION equations can be written as µν µν In some respects de Sitter space is to cosmology what R =Λg , (1.1) hydrogen atom is to atomic physics. In most practical sit- with Λ the cosmological constant. Assuming then stat- uations they both become the first example where any re- icness for a general spherically symmetric metric spective new formalism, description or theory is brought to be checked out. Actually, since the very moment it 2 A 2 B 2 2 2 ds = e dt + e dr + r dΩ2, (1.2) was discovered one finds it difficult to encounter a paper − 2 2 2 2 on cosmology in which no mention is made of de Sitter with dΩ2 = dθ + sin θdφ the metric on the unit three- universe. Soon after its publication [1], Einstein sent a sphere, we obtain A = B and hence the Schwarschild-de series of letters to de Sitter [2] in which he did not ex- Sitter solution [4] − press entire approval to the de Sitter cosmology. Two main criticisms were raised by Einstein. Firstly, since it Λr2 2m eA = e−B =1 , (1.3) could only apply without any ’world material’, that is − 3 − r stars, he found the de Sitter metric to make no physical meaning, and secondly, Einstein argued that whereas one where the mass m comes about as an integration con- can always put a rigid circular hoop into de Sitter uni- stant. de Sitter’s static solution arises when we set m =0 verse at any sufficiently early, though nonzero time, there in Eq. (1.3), i.e. in the pure vacuum case. arXiv:gr-qc/9909094v1 30 Sep 1999 is no place in it at time t = 0. While the first criticism From a slightly more technical point of view, the de Sit- leaves de Sitter universe only as a good approximation ter space can be indentified [5] as a maximally symmet- for sufficiently tiny matter density (which actually is the ric space of constant positive curvature (positive Ricci real case), the second one is nothing but an expression of scalar) which is a solution to the Einstein equations (1.1) the Einstein disappeal to the existence of the big bang, with a positive cosmological constant Λ > 0. It can be an attitude which evidently overcomes de Sitter scenario. visualized as a five-hyperboloid [6], It is rather ironic however that, in spite of the Einstein 3 reservations, de Sitter obtained his cosmological solution v2 + w2 + x2 + y2 + z2 = , (1.4) by just introducing the cosmological term first considered − Λ by Einstein himself in the general-relativity field equa- embedded in E5, with the most general metric of the tions for a spherically symmetric spacetime without any de Sitter space being then that which is induced in this matter, that is by considering the simplest possible sca- embedding, i.e. nario meeting the essentials of Einstein ideas. The key differential point between de Sitter and Einstein [3] uni- ds2 = dv2 + dw2 + dx2 + dy2 + dz2. (1.5) verses is that in the latter a given matter density was − explicitly included, setting a preferred value of the cos- This has topology R S4, invariance group SO(4, 1), and mological constant so that the resulting solution repre- shows ten Killings vectors× (four boosts and six rotations). sented a static universe. On the contrary, in de Sitter What we have hitherto described corresponds to a de space the value of the cosmological constant is completely Sitter space with simply connected topology. Recently, arbitrary. In the absence of any matter, the Einstein Gott and Li [7] have also considered the important case of 1 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 prob- lem, 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.