The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by a regular Carter tetrad and the Dirac spinors and matrices are defined in a chiral Newman-Penrose dyad representation. Applying Chandrasekhar's mode ansatz, the Dirac equation is separated into radial and angular systems of ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are derived, and the decay of the associated errors is analyzed. Moreover, specific aspects of the angular eigenfunctions and eigenvalues are discussed. Finally, as an application, the scattering of massive Dirac particles by the gravitational field of a rotating Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates and for the construction of an integral spectral representation of the Dirac propagator that yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon.
The Eddington-Finkelstein metric is obtained from the Schwarzschild metric by a change of the time variable. It is well known that a test mass falling into a black hole does not reach the event horizon for any finite value of the Schwarzschild time variable $t$. By contrast, we show that the event horizon is reached for a finite value of the Eddington-Finkelstein time variable $t'$. Then we study in Eddington-Finkelstein time the fate of a massive particle traversing an Einstein-Rosen bridge and obtain a different conclusion than recent proposals in the literature: we show that the particle re…
In a previous paper we have considered the Regge-Wheeler equation for fields of spin $s=0$, $1$ or $2$ on the Schwarzschild spacetime in coordinates that are regular at the horizon. In particular, we have constructed in Eddington-Finkelstein coordinates exact solutions in terms of series that are regular at the horizon and converge on the entire open domain from the central singularity to infinity. Here we extend this earlier work in two different directions. Firstly, we consider in Eddington-Finkelstein coordinates a massive scalar field that can serve as a dark matter candidate. Secondly, we…
We study backreaction of accreting matter onto a spherically symmetric black hole in a perturbative way, when accretion is in a quasi-steady state. General expressions for corrections to the metric coefficients are found in the Eddington-Finkelstein coordinates. It is shown that near the horizon of a black hole, independently of the form of the energy-momentum tensor, the leading corrections to the metric are of the Vaidya form. The relation to other solutions is discussed and particular examples are presented.
We present an approximate time-dependent metric in ingoing Eddington-Finkelstein coordinates for an evaporating nonrotating black hole as a first-order perturbation of the Schwarzschild metric, using the linearized back reaction from a realistic approximation to the stress-energy tensor for the Hawking radiation in the Unruh quantum state.
We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive a functional analytic integral representation of the associated propagator using the spectral theorem for unbounded self-adjoint operators, Stone's formula, and quantities arising in the analysis of Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon. In the derivation, we first write the Dirac equation in Hamiltonian for…
In the paper we analyze the quantum-mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field. We consider the Schwarzschild metrics in the spherical, isotropic and harmonic coordinates, and the Eddington-Finkelstein, Painleve-Gullstrand, Lemaitre-Finkelstein, Kruskal metrics. The scope of the analysis includes domains of the wave functions of Dirac's equation, hermiticity of Hamiltonians, and the possibility of existence of stationary bound states of spin-half particles. The constraint on the domain of the wave functions of the Hamiltonian in a Schwarzschild…
In this report, we discuss the dynamics of photons in a five-dimensional Schwarzschild black hole. We find that the corresponding Eddington-Finkelstein-like coordinate admits two time-coordinate singularities for a particular condition. This condition is due to the existence of a non-zero angular momentum related to the extra spatial dimension. By introducing a new Regge-Wheeler tortoise coordinate, both singularities can be removed from the Eddington-Finkelstein-like coordinate. It is identified that the additional singularity is related to the photon sphere in a hyperplane, and we analyze it…
We present a solution for rotating black holes in massive gravity. We first give a solution of massive gravity with one dynamical metric. Both metrics of this solution are expressed in the advanced Eddington-Finkelstein-like coordinates: the physical metric has the original Kerr line element, while the fiducial metric is flat, but written in a rotating Eddington-Finkelstein form. For the bi-gravity theory we give an analogue of this solution: the two metrics have the original Kerr form, but, in general, different black hole masses. The generalisation of the solution to include the electric cha…
The Construction and Application of Penrose Diagrams, with a Focus on the Maximally Analytically Extended Schwarzschild Spacetime
Argelander Institute for Astronomy - University of Bonn（波恩大学阿尔兰德天文研究所）；Lichtenberg Group for History and Philosophy of Physics - Faculty of Philosophy, University of Bonn（波恩大学哲学学院利希滕贝格物理历史与哲学研究组）
A one-fold infinity of explicit quasi-stationary regular line elements for the Schwarzschild geometry is obtained directly from the vacuum Einstein equations. The class includes the familiar Eddington-Finkelstein coordinates, and the coordinates discussed recently by Kraus and Wilczek.
The propagation of (massless) scalar, electromagnetic and gravitational waves on fixed Schwarzschild background spacetime is described by the general time-dependent Regge-Wheeler equation. We transform this wave equation to usual Schwarzschild, Eddington-Finkelstein, Painleve-Gullstrand and Kruskal-Szekeres coordinates. In the first three cases, but not in the last one, it is possible to separate a harmonic time-dependence. Then the resulting radial equations belong to the class of confluent Heun equations, i.e., we can identify one irregular and two regular singularities. Using the generalize…
It is known that the solutions of pure classical 5D gravity with $AdS_5$ asymptotics can describe strongly coupled large N dynamics in a universal sector of 4D conformal gauge theories. We show that when the boundary metric is flat we can uniquely specify the solution by the boundary stress tensor. We also show that in the Fefferman-Graham coordinates all these solutions have an integer Taylor series expansion in the radial coordinate (i.e. no $log$ terms). Specifying an arbitrary stress tensor can lead to two types of pathologies, it can either destroy the asymptotic AdS boundary condition or…
We perform a numerical free evolution of a selfgravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in Eddington-Finkelstein coordinates. The simplicity of the system allow to display and deal with the typical gauge instability present in these coordinates. The numerical evolution is performed with a standard method of lines fourth order in space and time. The time algorithm is Runge-Kutta while the space discrete derivative is symmetric (non-dissipative). The constraints are pres…
We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is a…
The Schwarzschild-deSitter metric is the known solution of Einstein field equations with cosmological constant term for vacuum spherically symmetric space around a point mass M. Recently it has been reported that in a $Lamda$-dominant world the Schwarzschild type coordinate systems are disqualified by redshift-magnitude test as a proper frame of reference(gr-qc/9812092). We derive the solution in a FRW type coordinate system which is qualified according to the mentioned test. Asymptotically it approachs to the non-static form of deSitter metric. The obtained metric is transformable to Schwarzs…
We propose a consistent setup for a holographic dual of Bjorken flow of strongly coupled large-Nc N=4 SYM-theory plasma. We employ Eddington-Finkelstein type coordinates for the dual geometry, and we propose a late-time expansion there. We construct the dual geometry order by order, and we show that the transport coefficients are determined by the regularity of the geometry. We also show that the dual geometry has an apparent horizon hence an event horizon, which covers the singularity at the origin. We prove that the dual geometry is regular at all orders under an appropriate choice of the tr…
We study the end stages of gravitational collapse of the thin shell of matter in ingoing Eddington-Finkelstein coordinates. We use the functional Schrodinger formalism to capture quantum effects in the near singularity limit. We find that that the equations of motion which govern the behavior of the collapsing shell near the classical singularity become strongly non-local. This reinforces previous arguments that quantum gravity in the strong field regime might be non-local. We managed to solve the non-local equation of motion for the dust shell case, and found an explicit form of the wavefunct…
Despite the fact that the Schwarzschild and Kerr solutions for the Einstein equations, when written in standard Schwarzschild and Boyer-Lindquist coordinates, present coordinate singularities, all numerical studies of accretion flows onto collapsed objects have been widely using them over the years. This approach introduces conceptual and practical complications in places where a smooth solution should be guaranteed, i.e., at the gravitational radius. In the present paper, we propose an alternative way of solving the general relativistic hydrodynamic equations in background (fixed) black hole …
We revisit our previous proposed conjecture -- horizon creates a local instability which acts as the source of quantum temperature of black hole. It is found that a chargesless massless particle moving along the null trajectory in Eddington-Finkelstein (EF) coordinates feels instability in the vicinity of the horizon. Such instability is observer independent for this particle motion. Moreover, an observer associated to EF coordinates finds the local Hamiltonian as $xp$ where $p$ is the canonical momentum corresponding the coordinate $x$. Finally, using this Hamiltonian we notice that at the qu…
In this paper, we consider the relativistic Hamilton-Jacobi (HJ) equation and study the Hawking radiation (HR) of scalar particles from uncharged Grumiller black hole (GBH) which is affordable for testing in astrophysics. GBH is also known as Rindler modified Schwarzschild BH. Our aim is not only to investigate the effect of the Rindler parameter a on the Hawking temperature ($T_{H}$), but to examine whether there is any discrepancy between the computed horizon temperature and the standard $T_{H}$ as well. For this purpose, in addition to its naive coordinate system, we study on the three regu…
A new form of the Kerr-Newman solution is presented. The solution involves a time coordinate which represents the local proper time for a charged massive particle released from rest at spatial infinity. The chosen coordinates ensure that the solution is well-behaved at horizons and enable an intuitive description of many physical phenomena. If the charge of the particle $e = 0$, the coordinates reduce to Doran coordinates for the Kerr solution with the replacement $M \to M - Q^2/(2r)$, where $M$ and $Q$ are the mass and charge of the black hole, respectively. Such coordinates are valid only fo…
Quantum fluxes and $\langle\hatΦ^2\rangle$ for a non-minimally coupled scalar field: ringdown and tail on approaching the polar Kerr inner horizon
Technion（以色列理工学院）；Max Planck Institute for Mathematics in Sciences (MiS)（马克斯·普朗克科学促进研究所数学研究所）；Princeton University（普林斯顿大学）；Institut für Theoretische Physik, Universität Leipzig（莱比锡大学理论物理研究所）；University College Dublin（都柏林大学学院）；Centro Brasileiro de Pesquisas Físicas (CBPF)（巴西物理学研究中心）
