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Register a new userEntropy of Constant Curvature Black Holes in General Relativity
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Jolien D. E. Creighton
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We consider the thermodynamic properties of the constant curvature black hole solution recently found by Banados. We show that it is possible to compute the entropy and the quasilocal thermodynamics of the spacetime using the Einstein-Hilbert action of General Relativity. The constant curvature black hole has some unusual properties which have not been seen in other black hole spacetimes. The entropy of the black hole is not associated with the event horizon; rather it is associated with the region between the event horizon and the observer. Further, surfaces of constant internal energy are not isotherms so the first law of thermodynamics exists only in an integral form. These properties arise from the unusual topology of the Euclidean black hole instanton.
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The recent detections of gravitational waves from binary systems of black holes are in remarkable agreement with the predictions of General Relativity. In this pedagogical mini-review, I will go through the physics of the different phases of the evolution of black hole binary systems, providing a qualitative physical interpretation of each one of them. I will also briefly describe how these phases would be modified if gravitation were described by a theory extending or deforming General Relativity, or if the binary components turned out to be more exotic compact objects than black holes.
In this work, new solutions for regular black holes that have multihorizons are proposed. These are formed by the direct product of solutions already published in the literature, which are described through the coupling of gravity with nonlinear electrodynamics. We analyze the regularity of the spacetime, the electric field, and the energy conditions of each solution. The strong energy condition is always violated within the event horizon in all solutions, while other energy conditions depend on the ratio between extreme charges of isolated solutions. For solutions with four horizons, we present two examples, Bardeen-Culetu and Balart-Culetu. Both solutions are regular, but the first do not satisfy all the energy conditions, except the strong, because it has an extreme charge ratio of 1.57581, great value. The second solution, on the other hand, can satisfy all other energy conditions, except the SEC, and has an extreme charge ratio of 1.09915, a value that allows this feature. Its also proposed a regular solution with up to six horizons, Balart-Culetu-Dymnikova, where, for a given charge value, we can verify that it satisfies all energy conditions, except the strong one. This was possible due to the ratio between extreme charges that are neither too high nor too close. We propose solutions with any number of horizons. We show that points where $-F(r)$ has a non null minimum represent a cusp in the Lagrangian $-L(F)$. We also show an example of multihorizon solution with magnetic charge. Multihorizon solutions may exhibit exotic properties, such as negative energy density, or violation of energy conditions, but which can be circumvented with a selected choice of customized solutions and extreme charge values, resulting in regular black hole solutions that satisfy all energy conditions, less the strong.
We investigate stationary, self-gravitating, magnetised disks (or tori) around black holes. The models are obtained by numerically solving the coupled system of the Einstein equations and the equations of ideal general-relativistic magnetohydrodynamics. The mathematical formulation and numerical aspects of our approach are similar to those reported in previous works modeling stationary self-gravitating perfect-fluid tori, but the inclusion of magnetic fields represents a new ingredient. Following previous studies of purely hydrodynamical configurations, we construct our models assuming Keplerian rotation in the disks and both spinning and spinless black holes. We focus on the case of a toroidal distribution of the magnetic field and build a large set of models corresponding to a wide range of values of the magnetisation parameter, starting with weakly magnetised disks and ending at configurations in which the magnetic pressure dominates over the thermal one. In all our models, the magnetic field affects the equilibrium structure of the torus mainly due to the magnetic pressure. In particular, an increasing contribution of the magnetic field shifts the location of the maximum of the rest-mass density towards inner regions of the disk. The total mass of the system and the angular momentum are affected by the magnetic field in a complex way, that depends on the black hole spin and the location of the inner radius of the disk. The non-linear dynamical stability of the solutions presented in this paper will be reported elsewhere.
Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the construction of explicit symplectic integrators is frequently difficult in general relativity because all variables are inseparable. Moreover, even if two analytically integrable splitting parts exist in a relativistic Hamiltonian, all analytical solutions are not explicit functions of proper time. Naturally, implicit symplectic integrators, such as the midpoint rule, are applicable to this case. In general, these integrators are numerically more expensive to solve than same-order explicit symplectic algorithms. To address this issue, we split the Hamiltonian of Schwarzschild spacetime geometry into four integrable parts with analytical solutions as explicit functions of proper time. In this manner, second- and fourth-order explicit symplectic integrators can be easily made available. The new algorithms are also useful for modeling the chaotic motion of charged particles around a black hole with an external magnetic field. They demonstrate excellent long-term performance in maintaining bounded Hamiltonian errors and saving computational cost when appropriate proper time steps are adopted.
In previous papers, explicit symplectic integrators were designed for nonrotating black holes, such as a Schwarzschild black hole. However, they fail to work in the Kerr spacetime because not all variables can be separable, or not all splitting parts have analytical solutions as explicit functions of proper time. To cope with this difficulty, we introduce a time transformation function to the Hamiltonian of Kerr geometry so as to obtain a time-transformed Hamiltonian consisting of five splitting parts, whose analytical solutions are explicit functions of the new coordinate time. The chosen time transformation function can cause time steps to be adaptive, but it is mainly used to implement the desired splitting of the time transformed Hamiltonian. In this manner, new explicit symplectic algorithms are easily available. Unlike Runge Kutta integrators, the newly proposed algorithms exhibit good long term behavior in the conservation of Hamiltonian quantities when appropriate fixed coordinate time steps are considered. They are better than same order implicit and explicit mixed symplectic algorithms and extended phase space explicit symplectic like methods in computational efficiency. The proposed idea on the construction of explicit symplectic integrators is suitable for not only the Kerr metric but also many other relativistic problems, such as a Kerr black hole immersed in a magnetic field, a Kerr Newman black hole with an external magnetic field, axially symmetric core shell systems, and five dimensional black ring metrics.
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