Abstract
The causal properties of the family of Kerr-de Sitter spacetimes are analyzed and compared to those of the Kerr family. First, an inextendible Kerr-de Sitter spacetime is obtained by joining together Carter’s blocks, i.e. suitable four dimensional spacetime regions contained within Killing horizons or within a Killing horizon and an asymptotic de Sitter region. Based on this property, and leaving aside topological identifications, we show that the causal properties of a Kerr-de Sitter spacetime are determined by the causal properties of the individual Carter’s blocks viewed as spacetimes in their own right. We show that any Carter’s block is stably causal except for the blocks that contain the ring singularity. The latter are vicious sets, i.e. any two events within such block can be connected by a future (respectively past) directed timelike curve. This behavior is identical to the causal behavior of the Boyer–Lindquist blocks that contain the Kerr ring singularity. These blocks are also vicious as demonstrated long ago by Carter. On the other hand, while for the case of a naked Kerr singularity the entire spacetime is vicious and thus closed timelike curves pass through any event including events in the asymptotic region, for the case of a Kerr-de Sitter spacetime the cosmological horizons protect the asymptotic de Sitter region from a-causal influences. In that regard, a positive cosmological constant appears to improve the causal behavior of the underlying spacetime.
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Notes
A Carter’s block is the analogue of a Boyer–Lindquist block employed in the description of the Kerr (or Kerr–Newman) spacetimes. Since in this work, the background is a Kerr-de Sitter, and in order to avoid confusion, we use the term Carter’s blocks and further ahead we define precisely these blocks.
Note that the possibility that \(\Delta (r)=0\) admits two pairs of complex conjugate roots is not compatible with \(\Lambda >0\) and \(a^{2}>0\).
It should be mentioned that the rotation axis is also considered as being part of a Carter’s block even though points on this axis are not covered by the Boyer–Lindquist coordinates.
We restrict our attention to the case where \(\Delta (r)=0\) admits four distinct real roots arranged according to \(r_{1}<0<r_{2}<r_{3}<r_{4}\). Once the structure of these spacetimes is understood, it is relatively easy to understand the structure of spacetimes where some of the roots of \(\Delta (r)=0\) coincide.
We denote by \((M_{-\,\infty }, g_{-\,\infty })\), \((M_{1}, g_{1})\), \((M_{2}, g_{2})\), \((M_{3}, g_{3})\), \((M_{4}, g_{4})\) these two dimensional spacetimes and in these spacetimes the coordinate r takes its values respectively in the intervals: \((-\,\infty , r_{1}), (r_{1}, r_{2}), (r_{2},r_{3}), (r_{3},r_{4}), (r_{4},\infty ).\)
For this gluing process, at first each of the two dimensional spacetimes defined by (11) are mapped conformally either into the interior of a diamond configuration or to a half diamond configuration (for details of this mapping see for instance [8,9,10]). The spacetimes in (11) defined on \( (r_{1},r_{2}), (r_{2},r_{3}), (r_{3},r_{4})\) are mapped into the interior of a diamond configuration, while those defined on \((-\,\infty , r_{1})\) and \( (r_{4},\infty )\) are mapped onto a half of a diamond configuration. Each of these five spacetimes can be time oriented so that for any block where \(\Delta (r)>0\), the timelike Killing field \(X=\frac{\partial }{\partial t}\) (or the alternative \({\hat{X}}=-\frac{\partial }{\partial t}\)) can be chosen to provide the future direction, while for any block with \(\Delta (r)<0\) the timelike field \({\hat{X}}=-\frac{\partial }{\partial r}\) (or the alternative \({\hat{X}}=\frac{\partial }{\partial r}\)) provides the future direction.
As far as we are aware, the maximal analytical extension of a four dimension Kerr-de Sitter has not been addressed in the literature before. Often and by analogy to what occurs for the Kerr case, the Carter–Penrose diagram representing the axis of a Kerr-de Sitter is interpreted as representing the maximal analytical extension of the four dimension Kerr de Sitter. Although it is likely that is the case, we are not aware of any detailed work supporting this interpretation. A referee kindly pointed out that some results that are reported in ref. [13], regarding the structure of the \(t=const.\), \(r=0\) equatorial disk of a Kerr-de Sitter offer the opportunity for a distinct extension of the block that contains the ring singularity. Needless to say, issues regarding possible extendability (or extendabilities) of Carter’s blocks deserve further attention.
Even though we believe that the results of ref. [7] hold for all causal geodesics, unfortunately the completeness property of a few families of geodesics needs to be addressed. For instance the completeness property of geodesics hitting the bifurcation spheres, or the completeness property of geodesics through the axis have to be worked out. These issues are under investigation and will be reported elsewhere [12].
The advantage of the extension through the gluing process advocated in ref. [6] lies in the fact that the method does not require an a priori understanding of the behavior of the causal geodesics. Once an extension is obtained, there follows the laborious task of checking whether all causal geodesics in the extended spacetime are indeed complete.
These coordinates are based on the family of principal null congruences admitted by a Kerr-de Sitter metric. For an introduction to these congruences and their role in constructing the Eddington–Finkelstein charts see for instance Sect. V of ref. [7].
Just to stress the formal analogy between the extension of the rotation axis and the full four dimensional Kerr-de Sitter, this \((IKS, {\hat{g}}_{i})\) is the analogue of the two dimensional spacetime \((IEF,g_{i})\) introduced in the treatment of the rotation axis.
The map J, plays the role of the map \(\Phi \) defined in eq. (14), although here the presence of the coordinate singularity on the axis needs to be given special consideration. Nevertheless, it can be shown that this J has a unique analytical extension as an isometry of the entire, i.e. including the axis, \(T_{(r_{i}, r_{i+1})}\) into \(J(T_{(r_{i}, r_{i+1})})\).
To make matters simple we use the same notation as the one employed in Sect. 1.4 of ref. [6].
The maximality property of the diagram in Fig. 2, for the case that the blocks are considered to be four dimensional, ought to be worked out in detail and establishing this property is not a trivial task. For the rest of this section we will assume that the extension shown in Fig. 2 is maximal and we discuss the consequences of this assumption.
The term \(\frac{\Delta (r)}{\rho ^{2}}\) is well defined over the entire domain of validity of the ingoing chart and this coupled with the fact that the left hand side of (24) is an analytic function relative to ingoing coordinates shows that the claim is not based on Boyer–Lindquist coordinates. The latter have been used only as an intermediate step.
Our convention for the surface gravity follows the same conventions as those in Wald’s book ref. [18].
The reader is warned that the \(r=r_{i}\) hypersurfaces defined relative to the the outgoing \((u, \overrightarrow{\varphi }, r,\vartheta )\) coordinates are distinct hypersurfaces from those defined by the ingoing \((v, \overleftarrow{\varphi }, r,\vartheta )\) coordinate system. For simplicity, we have avoided introducing different symbols for the “radial like” coordinate in the two systems.
In the present context, a time machine is a spacetime region that can generate closed timelike curves passing through any point in the spacetime under consideration. Here the region defined in (30) acts as a time machine for the block that contains the ring singularity.
In this section, by the term equatorial plane of the CTM we mean the collection of events coordinatized according to: \((t,r, \frac{\pi }{2}, \varphi )\) with \(-\,\infty<t<\infty \), \(\varphi \) varying in the usual range, while r is negative and is chosen to satisfy the restriction: \(g(\xi _{\varphi },\xi _{\varphi })_{\vartheta =\frac{\pi }{2}}<0\).
It is worth pointing out here an important difference between the curves \(\delta _{\pm \,1}\) introduced above. While both are timelike and future pointing note that \(\delta _{1}(t)>0\) implying that t increases along \(\delta _{1}\) while for the case of \(\delta _{-1}\) we have \(\delta _{-1}(t)<0\), i.e the coordinate t decreases as one moves along \(\delta _{-1}\). It is this property of the curve \(\delta _{-1}\) which is responsible for traveling backward in time. An observer following \(\delta _{-1}\), while moving towards the future, finds as a consequence of \(\delta _{-1}(t)<0\) that the value of the Boyer–Lindquist t steadily reduces.
Although Gödel’s solution [25] seems to be the best known example of a spacetime violating causality, chronologically it is not the first constructed solution of Einstein’s equations that exhibits this property. In 1937, van Stockum [26] published a solution of Einstein’s equations with source a rapidly rotating, infinitely long, dust cylinder and showed that this spacetime admits closed timelike curves. A re-examination of the causality properties of the van Stockum solution has been presented in [27].
In ref. [31], it is asked whether the laws of physics permit the creation of wormholes in a universe whose spatial sections initially are simply connected. If the laws indeed allow the formation of wormholes, then the appearance of closed timelike curves (and also violation of the weak energy condition) is unavoidable. For a proof of the former property see [30], while for the latter see [32].
References
Carter, B.: Phys. Rev. 174, 1559 (1968)
Carter, B.: Commun. Math. Phys. 17, 233 (1970)
Carter, B.: In: DeWitt, C., DeWitt, B.S. (eds.) Black Holes. Gordon and Breach, New York (1973)
Lake, K., Zannias, T.: On the global structure of the Kerr-de Sitter spacetimes. Phys. Rev. D 92, 084003 (2015)
Carter, B.: Phys. Rev. 141, 1242 (1966)
O’Neill, B.: The Geometry of Kerr Black Holes. A.K.Peters, Wellesley (1995). (also available in Dover ed. (2014))
Salazar, J.F., Zannias, T.: Phys. Rev. D 96, 024061 (2017)
Walker, M.: J. Math. Phys. 11, 2280 (1970)
Chrusciel, P.T., Olz, C.R., Szybka, S.J.: Phys. Rev. D 86, 124041 (2012)
Salazar, J.F.: Introduction to Carter–Penrose conformal diagrams, M.Sc. Thesis, IFM-UMSNH (2017)
Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738 (1977)
Salazar, J.F., Zannias, T.: Kruskal coordinates for Kerr-de Sitter and some applications (in preparation)
Manko, V., Garcia-Compean, H.: Phys. Rev. D 90, 047501 (2014)
Stuchlik, Z., Slany, P.: Equatorial circular orbits in the Kerr-de Sitter space–times. Phys. Rev. D 69, 064001 (2004)
Stoghianidis, E., Tsoubelis, D.: Gen. Relativ. Gravit. 19, 12 (1987)
Teo, E.: Gen. Relativ. Gravit. 35, 11 (2003)
Hackmann, E., Lammerzahl, C., Kagramanova, V., Kunz, J.: Phys. Rev. D 81, 044020 (2010)
Wald, R.M.: General Relativity. Chicago University Press, Chicago (1984)
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of the Spacetime. C.U.P., Cambridge (1973)
Chrusciel, P.T.: The Geometry of Black Holes (2015). Report: http://homepage.univie.ac.at/piotr.chrusciel
Galvani, M., de Felice, F.: Gen. Relativ. Gravit. 9, 155 (1978)
de Felice, F., Galvani, M.: Gen. Relativ. Gravit. 10, 335 (1979)
Akcay, S., Matzner, R.A.: Kerr-de Sitter Universe. Class. Quant. Grav. 28, 085012 (2011)
Carter, B.: Phys. Lett. 21, 423 (1966)
Gödel, K.: Rev. Mod. Phys. 21, 447 (1949)
van Stockum, W.J.: Proc. R. Soc. Edinb. 57, 135 (1937)
Tipler, F.J.: Phys. Rev. D 9, 2203 (1974)
Lobo, F.J.: Clas. and Quan. Gravity: Theory, Analysis and Applications, Chapter 9. Nova Science Publishers, Hauppauge (2008)
Smeenk, C., Wuthrich, C.: Time Travel and Time Machines. In: Callender, C. (ed.) Oxford Handbook of Time. Oxford University Press, Oxford (2011)
Hawking, S.W.: Phys. Rev. D 46, 603 (1992)
Morris, M.S., Thorne, K.S., Yurtsever, U.: Phys. Rev. Lett. 61, 1446 (1988)
Tipler, F.J.: Phys. Rev. Lett. 37, 879 (1976)
Ori, A.: Phys. Rev. Lett. 71, 2517 (1993)
Bhattacharya, S., Chakraborty, S., Padmanabhan, T.: Phys. Rev. D 96, 084030 (2017)
Krtous, P., Frolov, V.P., Kubiznák, D.: Living Rev. Relativ. 20, 6 (2017). https://doi.org/10.1007/s41114-017-0009-9
McNutt, D.D. et al.: arXiv:1709.03362 [math.DG]
Hintz, P., Vasy, A.: arXiv:1606.04014 [math.DG]
Acknowledgements
It is a pleasure to thank the members of the relativity group at the Universidad Michoacana for stimulating discussions. Special thanks are due to O. Sarbach for his constructive criticism, J. Felix Salazar for discussions and for his help in drawing the diagrams and F. Astorga for comments on the manuscript. The research was supported in part by CONACYT Network Project 280908 Agujeros Negros y Ondas Gravitatorias and by a CIC Grant from the Universidad Michoacana.
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Appendix
Appendix
In this Appendix, we remind the reader of a few basic notions of causality theory (a more elaborate discussion can be found in [18, 19]). We recall that for any physically relevant spacetime (M, g), besides the standard requirements that M ought to be smooth, connected, Hausdorff and paracompact, it is further required that (M, g) to be time orientable and causally well behaved. Causally well behaved means that (M, g) is minimally causal (resp. chronological) according to the definition:
Definition 1
A time orientable spacetime (M, g) is said to be causal (resp. chronological), if admits no causal (resp. timelike) closed curves.
The absence of closed timelike curves from any physically relevant (M, g) is required to be a stable property of (M, g) in the sense that any small perturbations of the background metric g should not lead to the appearance of closed causal curves. This additional requirement leads to the notion of stable causality according to the definition:
Definition 2
A time orientable spacetime (M, g) is stably causal if there exists a continuous timelike vector field t such that the spacetime \((M,{\hat{g}})\) with \({\hat{g}}=g-\hat{t} \otimes \hat{t}\) possesses no closed timelike curves, (here the covector \(\hat{t}\) is defined by: \(\hat{t}=g(t, )\)).
A very useful criterion guaranteeing that a given (M, g) is stably causal is expressed by the following theorem [18, 19]:
Theorem 1
A time orientable (M, g) is stably causal if and only if there exists a differentiable function \(\tau \) (often referred as the time function) such that \(\nabla \tau \) is a past directed timelike vector field.
Clearly, if (M, g) admits a function \(\tau : M \rightarrow R\) with these properties, then no closed timelike curves can occur, since for any future directed timelike curve \(\gamma \) with tangent vector field X, the inequality \(0<g(X, \nabla \tau )=X(\tau )\) implies that \(\tau \) is strictly increasing along this \(\gamma \). Therefore, under the hypothesis of the theorem, there exist no closed timelike curves in this (M, g). The proof of the converse is more involved but it can be found in [18, 19].
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Zannias, T. On causality violation on a Kerr-de Sitter spacetime. Gen Relativ Gravit 50, 134 (2018). https://doi.org/10.1007/s10714-018-2456-3
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DOI: https://doi.org/10.1007/s10714-018-2456-3