The motion of black holes

The European Physical Journal C, 2020

The advanced state of cosmological observations constantly tests the alternative theories of gravity that originate from Einstein’s theory. However, this is not restricted to modifications to general relativity. In this sense, we work in the context of Weyl’s theory, more specifically, on a particular black hole solution for a charged massive source, which is confronted with the classical test of the geodetic precession, to obtain information about the parameters associated with this theory. To fully assess this spacetime, the complete geodesic structure for massive test particles is presented.

General Relativity and Gravitation, 2012

Physical Review D

The fate of black hole thermodynamics under Weyl transformations is investigated by going back to the laws of black hole mechanics. It is shown that the transformed surface gravity, that one would identify with the black hole temperature in the conformal frame, as well as the black hole entropy, that one would identify with the horizon area, cannot be invariant. It is also shown that the conformally invariant surface gravity, attributed to the so-called "conformal Killing horizon", cannot represent the black hole temperature in the conformal frame. Finally, using familiar thought experiments, we find that the effect a Weyl transformation should have on black hole thermodynamics becomes even subtler than what is suggested by the laws of black hole mechanics.

2017

The origin of gravitational theory of relativity includes the discussion of Sir. Isaac Newton and Einstein’s theory of relativity. A brief derivation is introduced about the spherically symmetric solution of Einstein’s field equation, the Schwarzschild solution, and the axisymmetric solution, the Kerr solution. We have also presented the generalizations of both of the solutions followed by inter-related different types of coordinate systems like: EddingtonFinkelstein coordinate system and Kruskal coordinate system. The notion of black hole formation, rotational and non-rotational black hole structure is also shown with mathematical calculations and few graphical figures computation.

2015

We show that with a change of sign into two strategic components of the metric tensor in Kerr geometry, a generator for the Lanczos potential of this rotating black hole, can be obtained. 2010 Mathematical Subject Classification: 41A10, 41A55, 42B05, 97N50.

arXiv: General Relativity and Quantum Cosmology, 2019

We construct theoretical investigation of the black hole shadow for rotating charged black hole in an asymptotically flat, axisymmetric, and stationary spacetime with Weyl corrections. This spacetime is characterized by mass ($M$), charge parameter ($q$), rotation parameter ($a$), and Weyl coupling constant ($\alpha$). We derive photon geodesics around the black hole and compute expressions for impact parameters with the help of photon spherical orbits conditions. We show how the presence of coupling constant $\alpha$ affects the shapes of black hole shadow from the usual Kerr-Newman black hole. A comparison with the standard Kerr and Kerr-Newman black hole is also include to observe the potential deviation from them. We find that the radius of black hole shadow decreases and the distortion in the shape of shadow increases with an increase in charge $q$ for both positive and negative values of coupling constant $\alpha$. We further extend our study by considering the plasma environm...

The European Physical Journal C

We consider numerical black hole solutions in the Weyl conformal geometry and its associated conformally invariant Weyl quadratic gravity. In this model, Einstein gravity (with a positive cosmological constant) is recovered in the spontaneously broken phase of Weyl gravity after the Weyl gauge field ($$\omega _{\mu }$$ ω μ ) becomes massive through a Stueckelberg mechanism and it decouples. As a first step in our investigations, we write down the conformally invariant gravitational action, containing a scalar degree of freedom and the Weyl vector. The field equations are derived from the variational principle in the absence of matter. By adopting a static spherically symmetric geometry, the vacuum field equations for the gravitational, scalar, and Weyl fields are obtained. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, we obtain their solutions numerically. We detect the formation of a black hole from the...

Turkish journal of physics, 2024

This work investigates the role of the Weyl tensor in the formation of a black hole. We discuss the development of the Weyl tensor and prove its existence in spacetime during the gravitational collapse of cosmic objects, utilizing the Riemannian curvature tensor, Ricci tensor, Kulkarni-Nomizu product, and Schouten tensor. By decomposing the Weyl tensor, we use theorems and proofs that satisfy the exact solutions of the Einstein field equations. We observe that the Riemann curvature tensor and Weyl tensor share the same symmetric identities, as trW (δ, .)σ = 0 such that W δσγτ = 0 when Riemannian curvature tensor, R δσγτ = 0. Additionally, the Riemann curvature and Weyl scalar tensor invariants are conformally related to each other, as R δσγτ R δσγτ = W δσγτ W δσγτ = 48(GM) 2 r 6 in the Schwarzschild metric. From the Einstein field equations, the Ricci tensor is R στ = 0 ; consequently, the stress-energy tensor, T στ = 0 , indicating that the Einstein field equation is empty space. However, in the Schwarzschild black hole solution, the Ricci tensor vanishes, but the Weyl tensor does not. Additionally, it seems that divergence occurs around the event horizon in a stagnant and uncharged Schwarzschild black hole with proper acceleration. Furthermore, the investigation into the existence of the Weyl tensor in the Schwarzschild black hole reveals its presence. We also explore the Reissner-Nordström, Kerr, and Kerr-Newman black holes by examining the coupling between the Einstein-Maxwell field equations and the Weyl tensor, utilizing small Weyl corrections. We obtain the metric that reduces to the Kerr-Newman black hole solution in Boyer-Lindquist coordinates when α = 0. The same metric equation obtained reduces to Kerr black hole solutions when the electric charge q = 0 and the coupling parameter α = 0. Furthermore, when the parameter of the charged rotating black hole a vanishes, we obtain solutions for the static and spherically symmetric black hole with Weyl corrections. When the terms a = q = 0 , the obtained metric reduces to the Schwarzschild black hole solution.

Physical Review D

The behavior of black holes horizon and wormholes under the Weyl conformal transformation is investigated. First, a shorter, but more general, derivation of the Weyl transformation of the simple prescription for detecting horizons and wormholes given recently in the literature for spherically symmetric spacetimes is provided. The derivation allows for a simple and intuitive way to understand why and when horizons and wormholes might arise in the conformal frame even if they were absent in the original frame. Then, the conformal behavior of black holes horizon and wormholes in more general spacetimes, based on more "sophisticated" definitions, is provided. The study shows that black holes and wormholes might always arise in the new frame even if they were absent in the original frame. Moreover, it is shown that some of the definitions found in the literature might be transformed into one another under such transformations. Worked-out examples are given.

2012

A solution is obtained for the interior structure of an uncharged rotating black hole that accretes a collisionless fluid. The solution is conformally stationary, axisymmetric, and conformally separable, possessing a conformal Killing tensor. The solution holds approximately if the accretion rate is small but finite, becoming more accurate as the accretion rate tends to zero. Hyper-relativistic counterstreaming between collisionless ingoing and outgoing streams drives inflation at (just above) the inner horizon, followed by collapse. As ingoing and outgoing streams approach the inner horizon, they focus into twin narrow beams directed along the ingoing and outgoing principal null directions, regardless of the initial angular motions of the streams. The radial energy-momentum of the counterstreaming beams gravitationally accelerates the streams even faster along the principal directions, leading to exponential growth in the streaming density and pressure, and in the Weyl curvature and mass function. At exponentially large density and curvature, inflation stalls, and the spacetime collapses. As the spacetime collapses, the angular motions of the freely-falling streams grow. When the angular motion has become comparable to the radial motion, which happens when the conformal factor has shrunk to an exponentially tiny scale, conformal separability breaks down, and the solution fails. The condition of conformal separability prescribes the form of the ingoing and outgoing accretion flows incident on the inner horizon. The dominant radial part of the solution holds provided that the densities of ingoing and outgoing streams incident on the inner horizon are uniform, independent of latitude; that is, the accretion flow is "monopole." The subdominant angular part of the solution requires a special non-radial pattern of angular motion of streams incident on the inner horizon. The prescribed angular pattern cannot be achieved if the collisionless streams fall freely from outside the horizon, so the streams must be considered as delivered ad hoc to just above the inner horizon.