Kerr-de Sitter Universe (1011.0479v1)

Abstract: It is now widely accepted that the universe as we understand it is accelerating in expansion and fits the de Sitter model rather well. As such, a realistic assumption of black holes must place them on a de Sitter background and not Minkowski as is typically done in General Relativity. The most astrophysically relevant black hole is the uncharged, rotating Kerr solution, a member of the more general Kerr-Newman metrics. A generalization of the rotating Kerr black hole to a solution of the Einstein's equation with a cosmological constant $\Lambda$ was discovered by Carter \cite{DWDW}. It is typically referred to as the Kerr-de Sitter spacetime. Here, we discuss the horizon structure of this spacetime and its dependence on $\Lambda$. We recall that in a $\La>0$ universe, the term `extremal black hole' refers to a black hole with angular momentum $J > M^2 $. We obtain explicit numerical results for the black hole's maximal spin value and get a distribution of admissible Kerr holes in the ($\Lambda$, spin) parameter space. We look at the conformal structure of the extended spacetime and the embedding of the 3-geometry of the spatial hypersurfaces. In analogy with Reissner-Nordstr\"{o}m -de Sitter spacetime, in particular by considering the Kerr-de Sitter causal structure as a distortion of the Reissner-Nordstr\"{o}m-de Sitter one, we show that spatial sections of the extended spacetime are 3-spheres containing 2-dimensional topologically spherical sections of the horizons of Kerr holes at the poles. Depending on how a $t=$ constant 3-space is defined these holes may be seen as black or white holes (four possible combinations).

Overview of The Kerr-de Sitter Universe

The paper, "The Kerr-de Sitter Universe" by Sarp Akcay and Richard A. Matzner, explores the analysis of the Kerr-de Sitter (KdS) spacetime as a representative model for the accelerating universe described by the de Sitter (dS) cosmology, focusing particularly on the implications of a non-zero cosmological constant, Λ, in the description of black holes.

Key Contributions and Numerical Outcomes

Horizon Structure: The authors explore the horizon structure of the KdS spacetime, deriving explicit numerical results that depict the conditions under which black holes within this framework can exist. They demonstrate that for black holes in a Λ > 0 universe, the spin can exceed the typical Kerr bound, indicating that the black hole spin value may surpass mass (a > M). Notably, if Λ > 1/9M⁻², non-rotating black holes are not permissible.

Conformal and Extended Spacetime: A significant part of this work is dedicated to elucidating the conformal structure and embedding of KdS spacetime. By drawing analogies with the Reissner-Nordström-de Sitter (RNdS) geometry, the authors propose that spatial sections of KdS spacetime can be interpreted as 3-spheres with Kerr black holes at antipodal positions, which opens intriguing discussions on the time-reversal symmetries across null horizons. This builds upon earlier narratives in classical relativity and cosmology regarding neutral and charged black hole pairs.

Implications

Implications for Cosmological Models: The paper advances our understanding of black holes in a universe where dark energy, modeled by a cosmological constant, becomes significant. These insights are pertinent as they explore how the geometry and physics of black holes are influenced by cosmic expansion. This research is particularly relevant given modern cosmological observations that support a ΛCDM model, wherein dark energy dominates the universe's energy density.

Primordial Black Hole Formation: The possibility that rotating black holes must exist in a high Λ universe hints at new constraints on primordial black hole formation scenarios, especially during epochs such as inflation. If spin is indeed required for black holes in such cosmologies, it challenges traditional models that attempt to describe these objects in the early universe.

Future Directions

The results presented can inspire further studies into the nature of black hole extremality in non-flat backgrounds and their stability under quantum disturbances. Additionally, they catalyze discussions on the role of cosmological constants in other potential astrophysical processes, such as accretion dynamics around supermassive black holes.

In continuing this research, future work may also address unresolved questions on the Kerr-de Sitter solution's implications for cosmic censorship. It can also extend into comparative analyses with other solutions in the landscape of general relativity that incorporate additional parameters, like charge or exotic matter constituents.

In summary, Akcay and Matzner's work provides valuable insights into the intersection of black hole physics and cosmology, prompting theoretical considerations and practical investigations that reflect the evolving complexities of our understanding of the universe. This exploration of the Kerr-de Sitter architecture underscores a consistent theme in modern physics: that the marriage of cosmic principles with black hole theory can yield rich theoretical landscapes with profound implications.