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Gravitational radiation of binary systems can be studied by using the adiabatic approximation in General Relativity. In this
approach a small astrophysical object follows a trajectory consisting of a chained series of bounded geodesics (orbits) in
the outer region of a Kerr Black Hole, representing the space time created by a bigger object. In our...
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We present the structure of a low angular momentum accretion flows around rotating compact objects incorporating relativistic corrections up to the leading post-Newtonian order. To begin with, we formulate the governing post-Newtonian hydrodynamic equations for the mass and energy-momentum flux without imposing any symmetries. However, for the sake...
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... Since then, extensive investigations have explored various aspects of these orbits that provide insights into their properties and behavior (see for example, Refs. [4,7,8,[15][16][17][18][19][20][21][22][23][24][25][26]). In these studies, one often encounters algebraically complex equations, such as the polynomial equations that govern the radii of spherical orbits. ...
... In Eqs. (17)- (20), E and L are the constants of motion related to the Killing symmetries of the spacetime, which are termed, respectively, as the energy and angular momentum of the particles. Furthermore, ...
In this paper, we present a rotating de Rham-Gabadadze-Tolley black hole with a positive cosmological constant in massive gravity, achieved by applying a modified Newman-Janis algorithm. The black hole exhibits stable orbits of constant radii, prompting a numerical study on the behavior of the solutions to a nonic equation governing the radii of planar orbits around the black hole. Additionally, we investigate the stability of orbits near the event horizon and provide a comprehensive analytical examination of the solutions to the angular equations of motion. This is followed by simulating some spherical particle orbits around the black hole.
... Because the radial equation in our case is completely identical with that of the Kerr spacetime, the energy and azimuthal angular momentum for general spherical orbits can be parameterized by their radius r and the Carter constant of the particle. The results have been derived in Ref. [64] (see Refs. [65][66][67] for different parameterizations) and they read [68] ...
Rotating black holes without equatorial reflection symmetry can naturally arise in effective low-energy theories of fundamental quantum gravity, in particular, when parity-violating interactions are introduced. Adopting a theory-agnostic approach and considering a recently proposed Kerr-like black hole model, we investigate the structure and properties of accretion disk around a rotating black hole without reflection symmetry. In the absence of reflection symmetry, the accretion disk is in general a curved surface in shape, rather than a flat disk lying on the equatorial plane. Furthermore, the parameter that controls the reflection asymmetry would shrink the size of the prograde innermost stable circular orbits, and enhance the efficiency of the black hole in converting rest-mass energy to radiation during accretion. The retrograde innermost stable circular orbits are stretched but the effects are substantially suppressed. In addition, we find that spin measurements based on the gravitational redshift observations of the disk, assuming a Kerr geometry, may overestimate the true spin values if the central object is actually a Kerr-like black hole with conspicuous equatorial reflection asymmetry. The qualitative results that the accretion disk becomes curved and the prograde innermost stable circular orbits are shrunk turn out to be generic in our model when the reflection asymmetry is small.
... 2. For the uniqueness, we refer to Appendix II of [26] for a detailed proof. ...
... 2. Let δ 0 > 0 be obtained by Proposition (26). Define the nonlinear operator ...
We construct a one-parameter family of stationary axisymmetric and asymptotically flat spacetimes solutions to the Einstein-Vlasov system bifurcating from the Kerr spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, axisymmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics for stationary axisymmetric spacetimes close to Kerr, where the geodesic flow is not necessarily integrable. Moreover, the analysis of the Einstein field equations relies on the modified Carter Robinson theory developed by Chodosh and Shlapentokh-Rothman. This provides the first construction of black hole solutions to the Einstein-Vlasov system in the axisymmetric case and generalises the construction already done in the spherically symmetric case.
... 3.3. Timelike future directed geodesics in Kerr spacetime 123 2. For the uniqueness, we refer to Appendix II of [54] for a detailed proof. ...
This thesis is devoted to the construction of stationary black hole solutions to the Einstein-Vlasov system. In a first part, we obtain a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system by perturbing the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. In a second part, we generalise the method of construction to the challenging case of axial symmetry in order to prove the existence of a one-parameter family of stationary and axisymmetric solutions which bifurcate from a Kerr spacetime. The common features to these two constructions are the bifurcation argument from a reference black hole solution and the analysis of trapped geodesics and the associated effective potential energy in spacetimes close to Schwarzschild and close to Kerr. More precisely, the stability results for trapped geodesics in static spherically symmetric spaces and in stationary axisymmetric spaces are the key idea for our construction. The results obtained in our thesis show in particular that Schwarzschild spacetime and Kerr spacetime are not asymptomatically stable as solutions to the Einstein-Vlasov system.
... In particular, he plotted out the parameter space of stable spherical orbits around an extremal Kerr black hole. These spherical orbits have been further studied by various authors over the years (see, e.g., [4,7,8,[15][16][17][18][19][20][21][22][23]). ...
... We remark that these solutions have previously appeared in the literature [4,17,18,20], albeit in different forms. In [17], they were parameterised in terms of r and , while in [18,20], they were parameterised in terms of r and E. In [4], they were parameterised in terms of r and Q, but in a form different from (3.1). ...
... We remark that these solutions have previously appeared in the literature [4,17,18,20], albeit in different forms. In [17], they were parameterised in terms of r and , while in [18,20], they were parameterised in terms of r and E. In [4], they were parameterised in terms of r and Q, but in a form different from (3.1). As we will see in Sect. ...
A special class of orbits known to exist around a Kerr black hole are spherical orbits—orbits with constant coordinate radii that are not necessarily confined to the equatorial plane. Spherical time-like orbits were first studied by Wilkins almost 50 years ago. In the present paper, we perform a systematic and thorough study of these orbits, encompassing and extending previous works on them. We first present simplified forms for the parameters of these orbits. The parameter space of these orbits is then analysed in detail; in particular, we delineate the boundaries between stable and unstable orbits, bound and unbound orbits, and prograde and retrograde orbits. Finally, we provide analytic solutions of the geodesic equations, and illustrate a few representative examples of these orbits.
We classify radial timelike geodesic motion of the exterior nonextremal Kerr spacetime by performing a taxonomy of inequivalent root structures of the first-order radial geodesic equation, using a novel compact notation and by implementing the constraints from polar, time, and azimuthal motion. Four generic root structures with only simple roots give rise to eight nongeneric root structures when either one root becomes coincident with the horizon, one root vanishes, or two roots becomes coincident. We derive the explicit phase space of all such root systems in the basis of energy, angular momentum, and Carter’s constant and classify whether each corresponding radial geodesic motion is allowed or disallowed from the existence of polar, time, and azimuthal motion. The classification of radial motion within the ergoregion for both positive and negative energies leads to six distinguished values of the Kerr angular momentum. The classification of null radial motion and near-horizon extremal Kerr radial motion are obtained as limiting cases and compared with the literature. We explicitly parametrize the separatrix describing root systems with double roots as the union of the following three regions that are described by the same quartic respectively obtained when (1) the pericenter of bound motion becomes a double root, (2) the eccentricity of bound motion becomes zero, and (3) the turning point of unbound motion becomes a double root.
