Horizons and the cosmological constant
aa r X i v : . [ g r- q c ] J a n Horizons and the cosmological constant
Krzysztof A. Meissner
Institute of Theoretical Physics, Dept. of Physics,University of Warsaw, Ho˙za 69, 00-681 Warsaw, Poland
Abstract
A new solution of the Einstein equations for the point mass immersed in the de SitterUniverse is presented. The properties of the metric are very different from both theSchwarzschild black hole and the de Sitter Universe: it is everywhere smooth, light canpropagate outward through the horizon, there is an antitrapped surface enclosing thepoint mass and there is necessarily an initial singularity. The solution for any positivecosmological constant is qualitatively different from the Schwarzschild solution andis not its continuous deformation.
1. Introduction
There is an extensive literature on the solutions of the Einstein equations de-scribing a point mass immersed in the expanding Universe (see [1] and the recentreview [2] and references therein). The first exact solution for a point mass in the deSitter Universe was constructed as early as 1918 and is known as the Kottler metric[3]: d s = − (cid:18) − H r − GMr (cid:19) d t + d r − H r − GMr + r dΩ (1)where H is related to the cosmological constant Λ by Λ = 3 H / (8 πG ). For H = 0the metric is equal to the standard Schwarzschild metric [4]d s = − (cid:18) − GMr (cid:19) d t + d r − GMr + r dΩ (2)However, for M = 0 the metric requires a diffeomorphism to transform it to theusual de Sitter metric [5] d s = − d t + e Ht (cid:0) d r + r dΩ (cid:1) (3)It is the purpose of this paper to present a new solution that in the above mentionedlimits goes over to the standard Schwarzschild (at least outside of the black hole1orizon) or de Sitter metrics. We will show that in the limit of small cosmologicalconstant the metric indeed goes to the Schwarzschild metric but only outside theSchwarzschild horizon – inside it is everywhere singular.
2. The metric
The metric that solves R µν − g µν R = − πG Λ g µν (4)with required properties readsd s = − f ( t, r )d t + e Ht d r f ( t, r ) + e Ht r dΩ (5)where Λ = 3 H / (8 πG ), f ( t, r ) = h ( t, r ) + p h ( t, r ) + H r e Ht (6)and h ( t, r ) = 12 (cid:16) − H r e Ht − r S re Ht (cid:17) (7)where r S = 2 GM . We see that if H = 0 (however small) f ( t, r ) never vanishes. It isimportant to note that in the limit H → t, ρ ) =( t, re Ht ) (the metric is then independent of time):d s = − h (0 , ρ )d t − Hρ d t d ρf (0 , ρ ) + d ρ f (0 , ρ ) + ρ dΩ (8)with the inverse metric in these coordinates g tt = − f (0 , ρ ) , g tρ = − Hρf (0 , ρ ) , g ρρ = 2 h (0 , ρ ) (9)This form shows that there are 4 Killing vectors (1 connected with time translationsand 3 with rotations) as in the Schwarzschild case. In general there are two radii ρ , ρ solving the equation h (0 , ρ i ) = 0 (10)2nd defining two null surfaces. The second null vector is in both cases directedoutside the surfaces therefore the inner horizon is of different type than the usualblack hole horizon.
3. Singularities
We calculate the Riemann tensor for the metric (5) R = R = H + r S r e Ht R = R = R = R = H − r S r e Ht (11)with all other components (except with permuted indices) vanishing. Therefore theRicci tensor and the curvature scalar read R µν = 3 H δ µν , R = 12 H (12)The only singularity of the scalars constructed from the Riemann tensor like R µν ρσ R ρσµν occurs for r → t → −∞ .We now analyze the metric (5). For fixed tf ( t, r ) r →∞ −→ , f ( t, r ) r → −→ H r e Ht r S − re Ht (13)while for fixed r f ( t, r ) t →∞ −→ , f ( t, r ) t →−∞ −→ H r e Ht r S − re Ht (14)Therefore the physical radial distance to r → ∞ is infinite as well as the physicaltime interval to t → ∞ . The physical radial distance to the origin r = 0 is alsoinfinite (although the area is 4 πr e Ht i.e finite and small) Z ǫ e Ht d r p f ( t, r ) ∼ r r S H ǫe Ht (15)so the space is geodesically complete. However the physical time interval from t → −∞ to some T (within the applicability of (14)) Z T −∞ d t p f ( t, r ) ∼ s r e HT r S (16)3s finite. Therefore in the presence of a point mass and the cosmological constantthere must be an initial singularity in contradistinction to the usual de Sitter Uni-verse.The topology of the spacetime described by (5) is R × S .
4. Propagation of light
Since the g µν components are nowhere vanishing and nowhere singular the co-ordinates seem to cover the whole spacetime. To check what happens at the innerhorizon (corresponding to r = 2 GM in the Schwarzschild case) we have to find thebehaviour of light both inside and outside. To do it we solve the equation for thepropagation of light i.e. stemming from d s = 0 and directed radially outwards i.e.satisfying e Ht d r ( t )d t = f ( t, r ( t )) (17)To simplify the discussion we assume in what follows that r S H ≪ r S re Ht ≈ t = 0 r < r S or r > r S .In the latter case ( r > r S ) the solution reads r ( t ) = r + 1 H (cid:0) − e − Ht (cid:1) + r S ln r H r H − e − Ht + O ( r S H ) (20)so that after infinite time the comoving coordinate reaches the de Sitter horizon r ( ∞ ) ≈ r + r S ln( r H ) + 1 H (21)A different situation arises if r < r S . Then f ( t, r ) is positive but extremelysmall so that r ( t ) increases very slowly until it reaches the horizon and time growsto the value corresponding to (19) i.e. T = 1 H ln (cid:18) r S r (cid:19) (22)4hen the light gets outside and the trajectory is given by (20) but with t → t + T .Therefore the total comoving distance is approximately given by r ( ∞ ) ≈ r r S H (23)i.e. less than the outside de Sitter horizon by a factor r r S .It is important to check the dependence on time of the area enclosed by theoutgoing or incoming lightd A d t = 8 πre Ht (cid:0) Hr ± f ( t, r ) e − Ht (cid:1) (24)where A = 4 πr e Ht . Since Hr > f ( t, r ) e − Ht for sufficiently small r it turns outthat there is an antitrapped surface enclosing some region around r = 0. This isconsistent with the fact that antitrapped surfaces must have a singularity in thepast as we have seen is the case for the metric (5).
4. Conclusions
In the paper we have shown that the solution of the Einstein equations (5)for a point mass immersed in the universe with the positive cosmological constanthas very special properties: the metric is everywhere smooth, light can propagateoutward through the horizon, there is an antitrapped surface enclosing the pointmass and there is necessarily an initial singularity. Although with extremely smallvalue of H such an object for all practical purposes looks like a usual black hole theconceptual difference resulting from the fact that there is no horizon for the outwardpropagation of light can be far-reaching – first, one should rethink a notion of a blackhole entropy as proportional to the area of the horizon and second, there seems tobe no information loss even classically since the communication of the inside withthe outside is extremely weak but nonvanishing. It is also interesting to note that inthe presence of such objects there is necessarily an initial singularity in distinctionto the pure de Sitter universe and there is no continuous deformation connectingΛ > Acknowledgements
The author is grateful to D. Christodoulou, J. Lewandowski,H. Nicolai and A. Trautman for discussions and to Max Planck Institute in Potsdamfor hospitality. The work was partially supported by the EU grant MRTN-CT-2006-035863 and the Polish grant N202 081 32/1844.5 eferences [1] A. Krasi´nski,
Inhomogeneous Cosmological Models , Cambridge Monographson Mathematical Physics 1998 (Cambridge University Press, Cambridge).[2] M. Carrera, D. Giulini,
On the influence of global cosmological expansion onthe dynamics and kinematics of local systems , arXiv:0810.2712 [gr-qc] [3] F. Kottler, Ann. Phys. (Leipzig) 56 (1918) 401.[4] K. Schwarzschild, ¨Uber das Gravitationsfeld eines Massenpunktes nach derEinsteinschen Theorie , Sitzungsberichte der K¨oniglich-Preussischen Akademieder Wissenschaften, Berlin (1916) pp 189.[5] W. de Sitter, On the relativity of inertia: Remarks concerning Einstein’s lat-est hypothesis , Proc. Kon. Ned. Acad. Wet. 19: (1917) 1217-1225;