The line element of a class of solutions which match to the solution of Kerr on an oblate spheroid if the two functions $ F(r)$ and $H(r)$ on which it depends satisfy certain matching conditions is presented. The non vanishing components of the Ricci tensor $R_{μν}$, the Ricci scalar $ R$, the second order curvature invariant $K$, the eigenvalues of the Ricci tensor, the energy density $μ$, the tangential pressure $P_{\perp}$, and the quantity $μ+P_{\perp}$ are calculated. A function $F(r)$ is given for which $R$ and $K$ and therefore the solutions are regular. The function $ H(r) $ should be such that the solution it gives satisfies at least the Weak Energy Conditions (WEC). Several $H(r)$ are given explicitly for which the resulting solutions satisfy the WEC and also the Strong Energy Conditions (SEC) and the graphs of their $μ$, $P_{\perp}$ and $μ+P_{\perp}$ for certain values of their parameters are presented. It is shown that all solutions of the class are anisotropic fluid solutions and that there are no perfect fluid solutions in the class.
We present a simple exact solution for the interior of a rotating star. The interpretation of the stress energy tensor as that of a fluid requires the existence of a high viscosity, which is quite expected for a rotating fluid. In spite of the negative stresses, energy conditions are in fact all satisfied.
We numerically solve Einstein's equations coupled to a scalar field in the interior of Kerr black holes. We find shock waves form near the inner horizon. The shocks grow exponentially in amplitude and need not be axisymmetric. Observers who pass through the shocks experience exponentially large tidal forces and are accelerated exponentially close to the speed of light.
A, recently presented, general procedure to find static and axially symmetric, interior solutions to the Einstein equations, is extended to the stationary case, and applied to find an interior solution for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy conditions for a wide range of values of the parameters, and matches smoothly to the Kerr solution, thereby representing a globally regular model describing a non spherical and rotating source of gravitational field. In the spherically symmetric limit, our model converges to the well known incompre…
The interior volume of black holes as defined by Christodoulou and Rovelli exhibits many surprising features. For example, it increases with time, even under Hawking evaporation. For some black holes, the interior volume is not even a monotonic increasing function of its area, which means one cannot infer how large a black hole is by just looking from the outside. Such a notion of volume, however, turns out to be useful in the context of holography, as it seems to be dual to the complexity of the boundary field theory. In this study, we investigate the properties of the interior volume of 4-di…
A stationary axially symmetric solution describing a rotating anisotropic source for Einstein Field Equations(EFE) is proposed which matches to the exterior Kerr metric. The anisotropic source satisfies all energy conditions - weak, strong, and dominant energy condition - for a wide range of metric's free parameters values. The resultant energy-momentum tensor components, and consequently energy density and pressure profiles, redshift function and angular velocity are singularity free and behave as expected. As rotation parameter goes to zero a spherical source consisting of normal and exotic …
Recent work has argued that the strong cosmic censorship (SCC) conjecture is violated by near-extremal Reissner-Nordström de Sitter (RNdS) black holes but respected by Kerr-de Sitter black holes. It has also been shown that the conjecture is violated by near-extremal BTZ black holes. The latter result relies on a coincidence between "exterior" and "interior" quasinormal frequencies. If this coincidence were to occur also for RNdS or Kerr-dS then it would significantly modify the conclusions of earlier work. In this paper, it is demonstrated that this coincidence does not occur for RNdS or Kerr…
Christodoulou and Rovelli have shown that the maximal interior volume of a Schwarzschild black hole linearly grows with time. Recently, their conclusion has been extended to the Reissner{-}Nordstr$\ddot{\text{o}}$m and Kerr black holes. Meanwhile, the entropy of interior volume in a Schwarzschild black hole has also been calculated. Here, a new method calculating the entropy of interior volume of the black hole is given and it can be used in more general cases. Using this method, the entropy associated with the volume of a Kerr black hole is calculated and it is found that the entropy is propo…
Precise late-time asymptotics of scalar field in the interior of a subextreme Kerr black hole and its application in Strong Cosmic Censorship conjecture
The interior of the Kerr solution is singular and achronological. The classic singularity theorem by Hawking and Penrose relies on chronology, and thus does not apply to the Kerr solution. An improvement of their theorem by Kriele partially removes the requirement of chronology. However, both of these singularity theorems fail to give any information on the type or location of the incomplete geodesics. Here, using recent results of Minguzzi, we prove a new singularity theorem, specifically designed to apply to black holes, which enables locating null incomplete geodesics within the black hole …
We consider solutions of the massless scalar wave equation $\Box_gψ=0$, without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It follows from previous analyses in the Kerr exterior that for solutions $ψ$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon $\mathcal H^+$. Using the derived decay rate, we show that $ψ$ is in fact uniformly bounded, $|ψ|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which $ψ$ in…
We prove that a large class of smooth solutions $ψ$ to the linear wave equation $\Box_gψ=0$ on subextremal rotating Kerr spacetimes which are regular and decaying along the event horizon become singular at the Cauchy horizon. More precisely, we show that assuming appropriate upper and lower bounds on the energy along the event horizon, the solution has infinite (non-degenerate) energy on any spacelike hypersurfaces intersecting the Cauchy horizon transversally. Extrapolating from known results in the Reissner--Nordström case, the assumed upper and lower bounds required for our theorem are conj…
Energy and angular momentum of charged rotating black holes
我们表明，爱因斯坦，托尔曼，朗道和Lifshitz，Papapetrou和温伯格（ETLLPW）的伪张量给出相同的能量分布，线动量和角动量，任何克尔-Schild度规。这一结果推广了Gürses和Gürsey以前的工作，该工作只涉及爱因斯坦、Landau和Lifshitz的伪张量。我们计算这些分布的Kerr-Newman和Bonnor-Vaidya度量，并找到合理的结果。所有计算都是在Kerr-Schild笛卡尔坐标系中进行的，没有任何近似。对于Reissner-Nordström度规，这些定义给出了与彭罗斯准定域质量相同的结果。对于克尔黑洞，所有的能量都被限制在它的内部，而对于克尔-纽曼黑洞，正如预期的那样，能量由它的内部和外部共享。Kerr-Newman黑洞的总能量和角动量分别为$M$和$ M a$（$M$是质量参数，$a$是旋转参数）。Bonnor-Vaidya度规的能量分布与Tod得到的彭罗斯准局域质量相同。
We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and moreover that, if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr-Newman black holes. These results are the analogues of results obtained in [D. Gajic, Linear waves in the interior of extremal black holes I, arXiv:1509.06568] for extremal Reissner-Nordst…
Quantum field effects on a classical background spacetime may be obtained from the semiclassical equations of General Relativity with the expectation value of the stress-energy tensor of the quantum field as a source. This expectation value can be calculated from Hadamard's elementary two-point function, which in practice is given in terms of sums of products of field modes evaluated at two spacetime points. We derive expressions for the two-point function for a massless scalar field in the Unruh state on a Kerr black hole spacetime. Our main result in this paper is a novel expression valid wh…
In this paper, we investigate the Weak Cosmic Censorship Conjecture in nearly extremal Kerr black hole by absorbing particles . In previous work, they ignore the process of black hole absorbing particles. They assume the whole particle can be absorbed and the overspinning occurs. However, it is questionable whether the whole particle can be absorbed. Therefore, we will investigate it in this paper. We consider the absorption process of a particle with finite size. During this absorption process, the black hole's parameters will change. This change will prevent the rest part of particle enter b…
We investigate some important physical aspects of a recently presented interior solution for the Kerr metric. It is shown that, as in the spherically symmetric case, there is a specific limit for the maximal value of the surface potential (degree of compactness), beyond which, unacceptable physical anomalies appear. Such a bound is related to the appearance of negative (repulsive) gravitational acceleration, that is accompanied by the appearance of negative values of the pressure. A detailed discussion on this effect is presented. We also study the possibility of a fragmentation scenario, assu…
Source-free equations of nonlinear electrodynamics minimally coupled to gravity admit regular axially symmetric asymptotically Kerr-Newman solutions which describe charged rotating black holes and electromagnetic spinning solitons (lumps). Asymptotic analysis of solutions shows, for both black holes and solitons, the existence of de Sitter vacuum interior which has the properties of a perfect conductor and ideal diamagnetic and displays superconducting behaviour which can be responsible for practically unlimited life time of an object. Superconducting current flows on the equatorial ring repla…
In the context of Lorentz-Finsler spacetime theories the relativity principle holds at a spacetime point if the indicatrix (observer space) is homogeneous. We point out that in four spacetime dimensions there are just three kinematical models which respect an exact form of the relativity principle and for which all observers agree on the spacetime volume. They have necessarily affine sphere indicatrices. For them every observer which looks at a flash of light emitted by a point would observe, respectively, an expanding (a) sphere, (b) tetrahedron, or (c) cone, with barycenter at the point. The…
An exact rotating anisotropic fluid solution and a family of exact rotating anisotropic fluid solutions are presented which satisfy all energy conditions for certain values of their parameters. The components of the Ricci tensor the eigenvalues of this tensor and the energy-momentum tensor of the solutions are given explicitly. All have the ring singularity of Kerr's solution and in addition the solution one more singularity and some solutions of the family additional singularities.The solution matches to the extremal solution of Kerr on two surfaces, which are thin shells and for proper value…
It is believed that extremal black holes do not emit Hawking radiation as understood by taking extremal limits of non-extremal black holes. However, it is debated whether one can make such conclusion reliably starting from an extremal black hole, as the associated Bogoliubov coefficients which relate ingoing and outgoing field modes do not satisfy the required consistency condition. We address this issue in a canonical approach firstly by presenting an exact canonical derivation of the Hawking effect for non-extremal Kerr black holes. Subsequently, for extremal Kerr black holes we show that th…
We show that the conformal structure for the Riemannian analogues of Kerr black-hole metrics can be given an ambitoric structure. We then discuss the properties of the moment maps. In particular, we observe that the moment map image is not locally convex near the singularity corresponding to the ring singularity in the interior of the black hole. We then proceed to classify regular ambitoric $4$-orbifolds with some completeness assumptions. The tools developed also allow us to prove a partial classification of compact Riemannian $4$-manifolds which admit a Killing $2$-form.
We construct thought experiments involving the perturbations of Kerr-Newman black holes by neutral test fields to evaluate the validity of the weak form of the cosmic censorship conjecture. We first show that neglecting backreaction effects, extremal Kerr-Newman black holes which satisfy the condition $(J^2/M^4)<(1/3)$ can be overspun by scalar fields. This result, which could not be discerned in the previous analyses to first order, is prone to be fixed by employing backreaction effects. However the perturbation of Kerr-Newman black holes by neutrino fields leads to a generic overspinning of …
We consider light propagation in a non-magnetized pressureless plasma around a Kerr black hole. We find the necessary and sufficient condition the plasma electron density has to satisfy to guarantee that the Hamilton-Jacobi equation for the light rays is separable, i.e., that a generalized Carter constant exists. For all cases where this condition is satisfied we determine the photon region, i.e., the region in the spacetime where spherical light rays exist. A spherical light ray is a light ray that stays on a sphere $r = \mathrm{constant}$ (in Boyer-Lindquist coordinates). Based on these resu…
This paper investigates the dynamics of thin-shell in the presence of perfect fluid as well as the scalar field. We formulate the equations of motion using Israel thin-shell formalism by taking the interior and exterior regions of Schwarzschild, Kerr as well as Kerr-Newmann black hole. We find numerical solutions of equations of motion and effective potential to analyze the scalar shell for collapse and expansion. It is found that the rate of collapse and expansion of scalar shell through shell's radius depends on charge and rotation parameters. We conclude that the massive scalar shell leads …
