Abstract
As quotient spaces, Minkowski and de Sitter are fundamental, non-gravitational spacetimes for the construction of physical theories. When general relativity is constructed on a de Sitter spacetime, the usual Riemannian structure is replaced by a more general structure called de Sitter–Cartan geometry. In the contraction limit of an infinite cosmological term, the de Sitter–Cartan spacetime reduces to a singular, flat, conformal invariant four-dimensional cone spacetime, in which our ordinary notions of time interval and space distance are absent. It is shown that such spacetime satisfies all properties, including the Weyl curvature hypothesis, necessary to play the role of the bridging spacetime connecting two aeons in Penrose’s conformal cyclic cosmology.
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Notes
See the “Appendix” for a mathematical definition of transitivity of homogeneous spaces.
It is important to note that \(\eta _{\mu \nu } x^\mu x^\nu \ne 0\) in all other points of the cone. What vanishes in all points of the cone is the quadratic form written in terms of the five-dimensional ambient space coordinates \(\eta _\textit{AB} \chi ^A \chi ^B = 0\), as follows from (2) in the contraction limit \(l \rightarrow 0\).
Gravitational waves could in principle exist in the cone spacetime. There is a problem, however: the field equation for a symmetric second-rank tensor (perturbation of the metric) is not conformal invariant [25]. A possible solution to this puzzle is to interpret a spin 2 as a 1-form assuming values in the translation group (perturbation of the tetrad), in which case its field equation turns out to be conformal invariant [26].
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Acknowledgments
The authors would like to thank an anonymous referee for valuable comments and suggestions. They would like to thank also FAPESP, CAPES and CNPq for partial financial support. A.A. thanks Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela, for financial support.
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Appendix: On the notion of transitivity
Appendix: On the notion of transitivity
Spacetimes with constant sectional curvature are maximally symmetric in the sense that they can lodge the highest possible number of Killing vectors [36]. Their curvature tensor are completely specified by the scalar curvature R, which is constant throughout spacetime. Minkowski M, with vanishing curvature, is the simplest one. Its kinematic group is the Poincaré group \({\mathcal {P}} = {\mathcal {L}} \oslash {{\mathcal {T}}}\), the semi-direct product of Lorentz (\({\mathcal {L}}\)) and the translation (\({{\mathcal {T}}}\)) groups. It is a homogeneous space under the Lorentz group:
The Lorentz subgroup provides an isotropy around a given point of M, and the translation symmetry enforces this isotropy around any other point. This is the meaning of homogeneity: all points of Minkowski spacetime are equivalent under spacetime translations. That is to say, Minkowski is transitive under spacetime translations.
Another example of maximally symmetric spacetime is the de Sitter space dS. It has non-vanishing sectional curvature, and \(\textit{SO}(4,1)\) as kinematic group. Furthermore, it is also homogeneous under the Lorentz group:
Like Minkowski, the Lorentz subgroup provides an isotropy around a given point of dS. The notion of homogeneity, however, is completely different: as one can see from the generators (18) or (42), all points of the de Sitter spacetime are equivalent under a combination of translation and proper conformal transformation—the so-called de Sitter “translations”. That is to say, de Sitter is transitive under a combination of translations and proper conformal transformations: in order to move from one point to any other point of a de Sitter spacetime, one has to perform a de Sitter “translation”. This is the meaning of transitivity.
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Araujo, A., Jennen, H., Pereira, J.G. et al. On the spacetime connecting two aeons in conformal cyclic cosmology. Gen Relativ Gravit 47, 151 (2015). https://doi.org/10.1007/s10714-015-1991-4
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DOI: https://doi.org/10.1007/s10714-015-1991-4