Eddington-Finkelstein Coordinates Research Articles

In horizon penetrating coordinates: Kerr black hole metric perturbation, construction and completion

We investigate the Teukolsky equation in horizon-penetrating coordinates to study the behavior of perturbation waves crossing the outer horizon. For this purpose, we use the null ingoing/outgoing Eddington–Finkelstein coordinates. The first derivative of the radial equation is a Fuchsian differential equation with an additional regular singularity to the ones the radial one has. The radial functions satisfy the physical boundary conditions without imposing any regularity conditions. We also observe that the Hertz-Weyl scalar equations preserve their angular and radial signatures in these coordinates. Using the angular equation, we construct the metric perturbation for a circularly orbiting perturber around a black hole in Kerr spacetime in a horizon-penetrating setting. Furthermore, we completed the missing metric pieces due to the mass M and angular momentum J perturbations. We also provide an explicit formula for the metric perturbation as a function of the radial part, its derivative, and the angular part of the solution to the Teukolsky equation. Finally, we discuss the importance of the extra singularity in the radial derivative for the convergence of the metric expansion.

Shadows of Kerr–Vaidya-like black holes

In this work, we study the shadow boundary curves of rotating time-dependent black hole solutions which have well-defined Kerr and Vaidya limits. These solutions are constructed by applying the Newman–Janis algorithm to a spherically symmetric seed metric conformal to the Vaidya solution with a mass function that is linear in Eddington–Finkelstein coordinates. Equipped with a conformal Killing vector field, this class of solution exhibits separability of null geodesics, thus allowing one to develop an analytic formula for the boundary curve of its shadow. We find a simple power law describing the dependence of the mean radius and asymmetry factor of the shadow on the accretion rate. Applicability of our model to recent Event Horizon Telescope observations of M87 and Sgr A is also discussed.

Thermodynamic studies of a rotating polytropic black hole: Outer and interior regions

In this letter, considering the metric of a rotating polytropic black hole in the Boyer-Lindquist coordinates, at first, we derive the thermodynamic parameters such as entropy S, Helmholtz free energy F, internal energy U and Gibbs free energy G and study its dependence on the outer horizon by depicting suitable graphs. Then after reconstruction of the metric of the same in the Eddington-Finkelstein coordinates, we establish the interior volume of the black hole. We further analyze the variations of the interior volume with the small change of the advanced time with respect to the radius. Here we show the existence of a certain value of the radius for which this variation becomes maximum. Moreover, we show the dependence of this maximum value of the radius on the mass of the black hole. We derive the differential form of the interior volume for this limit of the radius and hence the maximal interior volume of the said black hole. Finally, we analyze the same thermodynamic parameters inside the black hole and present a comparative study between the parameters in the outer and interior regions of the black hole.

Forces in Schwarzschild, Vaidya and generalized Vaidya spacetimes

In this paper we investigate how the leading term in the geodesic equation in Schwarzschild spacetime changes under the coordinate transformation to Eddington-Finkelstein coordinates. This term corresponds to the Newton force of attraction. Also we consider this term when we add the energy-momentum tensor of the form of the null dust and the null perfect fluid into right-hand side of the Einstein equation. We estimate the value of this force in Vaidya spacetime when the naked singularity formation occurs. Also we give conditions in generalized Vaidya spacetime when this force of attraction is replaced by the force of repulsion.

Backreaction of mass and angular momentum accretion on black holes: General formulation of metric perturbations and application to the Blandford–Znajek process

Abstract We study the metric backreaction of mass and angular momentum accretion on black holes. We first develop the formalism of monopole and dipole linear gravitational perturbations around Schwarzschild black holes in Eddington–Finkelstein coordinates against generic time-dependent matter. We derive the relation between the time dependence of the mass and angular momentum of the black hole and the energy–momentum tensors of accreting matter. As a concrete example, we apply our formalism to the Blandford–Znajek process around slowly rotating black holes. We find that the time dependence of the monopole and dipole perturbations can be interpreted as a slowly rotating Kerr metric with time-dependent mass and spin parameters, which are determined from the energy and angular momentum extraction rates of the Blandford–Znajek process. We also show that the Komar angular momentum and the area of the apparent horizon are decreasing and increasing in time, respectively, while they are consistent with the Blandford–Znajek argument of energy extraction in terms of black hole mechanics if we regard the time-dependent mass parameter as the energy of the black hole.

A New Black Hole Solution in Conformal Dilaton Gravity on a Warped Spacetime

An exact time-dependent solution of a black hole is found in a conformally invariant gravity model on a warped Randall-Sundrum spacetime, by writing the metric . Here, represents the “un-physical” spacetime and ω the dilaton field, which will be treated on equal footing as any renormalizable scalar field. In the case of a five-dimensional warped spacetime, we thereafter write . The dilaton field can be used to describe the different notion the in-going and outside observers have of the Hawking radiation by using different conformal gauge freedom. The disagreement about the interior of the black hole is explained by the antipodal map of points on the horizon. The free parameters of the solution can be chosen in such a way that is singular-free and topologically regular, even for ω → 0 . It is remarkable that the 5D and 4D effective field equations for the metric components and dilaton fields can be written in general dimension n = 4,5. From the exact energy-momentum tensor in Eddington-Finkelstein coordinates, we are able to determine the gravitational wave contribution in the process of evaporation of the black hole. It is conjectured that, in context of quantization procedures in the vicinity of the horizon, unitarity problems only occur in the bulk at large extra-dimension scale. The subtraction point in an effective theory will be in the UV only in the bulk, because the use of a large extra dimension results in a fundamental Planck scale comparable with the electroweak scale.

A numerical stability analysis for the Einstein–Vlasov system

We investigate stability issues for steady states of the spherically symmetric Einstein–Vlasov system numerically in Schwarzschild, maximal areal, and Eddington–Finkelstein coordinates. Across all coordinate systems we confirm the conjecture that the first binding energy maximum along a one-parameter family of steady states signals the onset of instability. Beyond this maximum perturbed solutions either collapse to a black hole, form heteroclinic orbits, or eventually fully disperse. Contrary to earlier research, we find that a negative binding energy does not necessarily correspond to fully dispersing solutions. We also comment on the so-called turning point principle from the viewpoint of our numerical results. The physical reliability of the latter is strengthened by obtaining consistent results in the three different coordinate systems and by the systematic use of dynamically accessible perturbations.

Fermionic pole-skipping in holography

We examine thermal Green’s functions of fermionic operators in quantum field theories with gravity duals. The calculations are performed on the gravity side using ingoing Eddington-Finkelstein coordinates. We find that at negative imaginary Matsubara frequencies and special values of the wavenumber, there are multiple solutions to the bulk equations of motion that are ingoing at the horizon and thus the boundary Green’s function is not uniquely defined. At these points in Fourier space a line of poles and a line of zeros of the correlator intersect. We analyze these ‘pole-skipping’ points in three-dimensional asymptotically anti-de Sitter spacetimes where exact Green’s functions are known. We then generalize the procedure to higher-dimensional spacetimes and derive the generic form the boundary correlator takes near the pole-skipping points in momentum space. We also discuss the special case of a fermion with half-integer mass in the BTZ background. We discuss the implications and possible generalizations of the results.

Relativistic Gravitation Based on Symmetry

We present a Relativistic Newtonian Dynamics ( R N D ) for motion of objects in a gravitational field generated by a moving source. As in General Relativity ( G R ), we assume that objects move by a geodesic with respect to some metric, which is defined by the field. This metric is defined on flat lab spacetime and is derived using only symmetry, the fact that the field propagates with the speed of light, and the Newtonian limit. For a field of a single source, the influenced direction of the field at spacetime point x is defined as the direction from x to the to the position of the source at the retarded time. The metric depends only on this direction and the strength of the field at x. We show that for a static source, the R N D metric is of the same form as the Whitehead metric, and the Schwarzschild metric in Eddington–Finkelstein coordinates. Motion predicted under this model passes all classical tests of G R . Moreover, in this model, the total time for a round trip of light is as predicted by G R , but velocities of light and object and time dilation differ from the G R predictions. For example, light rays propagating toward the massive object do not slow down. The new time dilation prediction could be observed by measuring the relativistic redshift for stars near a black hole and for sungrazing comets. Terrestrial experiments to test speed of light predictions and the relativistic redshift are proposed. The R N D model is similar to Whitehead’s gravitation model for a static field, but its proposed extension to the non-static case is different. This extension uses a complex four-potential description of fields propagating with the speed of light.

On the Dynamical 4D BTZ Black Hole Solution in Conformally Invariant Gravity

We review the (2+1)-dimensional Ba\u{n}ados-Teitelboim-Zanelli black hole solution in conformally invariant gravity, uplifted to (3+1)-dimensional spacetime. As matter content we use a scalar-gauge field. The metric is written as $g_{\mu\nu}=\omega^2\tilde g_{\mu\nu}$, where the {\it dilaton field} $\omega$ contains all the scale dependencies and where $\tilde g_{\mu\nu}$ represents the "un-physical" spacetime. A numerical solution is presented and shows how the dilaton can be treated on equal footing with the scalar field. The location of the apparent horizon and ergo-surface depends critically on the parameters and initial values of the model. It is not a hard task to find suitable initial parameters in order to obtain a regular and {\it singular free} $g_{\mu\nu}$ out of a BTZ-type solution for $\tilde g_{\mu\nu}$. In the vacuum situation, an {\it exact} time-dependent solution in the Eddington-Finkelstein coordinates is found, which is valid for the (2+1)-dimensional BTZ spacetime as well as for the uplifted (3+1)-dimensional BTZ spacetime. While $\tilde g_{\mu\nu}$ resembles the standard BTZ solution with its horizons, $g_{\mu\nu}$ is {\it flat}. The dilaton field becomes an infinitesimal renormalizable quantum field, which switches on and off Hawking radiation. This solution can be used to investigate the small distance scale of the model and the black hole complementarity issues. It can also be used to describe the problem how to map the quantum states of the outgoing radiation as seen by a distant observer and the ingoing by a local observer in a one-to-one way. The two observers will use a different conformal gauge. A possible connection is made with the antipodal identification and unitarity issues.

Ingoing Eddington-Finkelstein metric of an evaporating black hole

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.

Inflying perspectives of reduced phase space

There is widespread disagreement about how the general covariance of a theory affects its quantization. Without a complete quantum theory of gravity, one can examine quantum consequences of coordinate choices only in highly idealized `toy' models. In this work, we extend our previous analysis of a self-gravitating shell model [1], and demonstrate that coordinate freedom can be retained in a reduced phase space description of the system. We first consider a family of coordinate systems discussed by Martel and Poisson [2], which have time coordinates that coincide with the proper times of ingoing and outgoing geodesics (for concreteness, we only consider the former). Included in this family are Painlev\'e-Gullstrand coordinates, related to a network of infalling observers that are asymptotically at rest, and Eddington-Finkelstein coordinates, related to a network of infalling observers that travel at the speed of light. We then introduce "inflying" coordinates - a hybrid coordinate system that allows the infalling observers to be arbitrarily boosted from one member of the aforementioned family to another. We perform a phase space reduction using inflying coordinates with an unspecified boosting function, resulting in a reduced theory with residual coordinate freedom. Finally, we discuss quantization, and comment on the utility of the reduced system for the study of coordinate effects and the role of observers in quantum gravity.

Two-dimensional fluids and their holographic duals

We describe the dynamics of two-dimensional relativistic and Carrollian fluids. These are mapped holographically to three-dimensional locally anti-de Sitter and locally Minkowski spacetimes, respectively. To this end, we use Eddington–Finkelstein coordinates, and grant general curved two-dimensional geometries as hosts for hydrodynamics. This requires to handle the conformal anomaly, and the expressions obtained for the reconstructed bulk metrics incorporate non-conformal-fluid data. We also analyze the freedom of choosing arbitrarily the hydrodynamic frame for the description of relativistic fluids, and propose an invariant entropy current compatible with classical and extended irreversible thermodynamics. This local freedom breaks down in the dual gravitational picture, and fluid/gravity correspondence turns out to be sensitive to dissipation processes: the fluid heat current is a necessary ingredient for reconstructing all Bañados asymptotically anti-de Sitter solutions. The same feature emerges for Carrollian fluids, which enjoy a residual frame invariance, and their Barnich–Troessaert locally Minkowski duals. These statements are proven by computing the algebra of surface conserved charges in the fluid-reconstructed bulk three-dimensional spacetimes.

Vaidya spacetimes, black-bounces, and traversable wormholes

We consider a non-static evolving version of the regular ‘black-bounce’/traversable wormhole geometry recently introduced in Simpson and Visser (2019 J. Cosmol. Astropart. Phys. JCAP02(2019)042). We first re-write the static metric using Eddington–Finkelstein coordinates, and then allow the mass parameter m to depend on the null time coordinate (à la Vaidya). The spacetime metric isHere denotes suitably defined null time coordinates; representing time, while, (at least for ), we allow . This spacetime is still simple enough to be tractable, and neatly interpolates between Vaidya spacetime, a black-bounce, and a traversable wormhole. We show how this metric can be used to describe several physical situations of particular interest, including a growing black-bounce, a wormhole to black-bounce transition, and the opposite black-bounce to wormhole transition.

Hawking-like radiation and the density matrix for an infalling observer during gravitational collapse

We study time-dependant Hawking-like radiation as seen by an infalling observer during gravitational collapse of a thin shell. We calculate the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in Eddington-Finkelstein coordinates. We solve the equations for the whole process from the beginning of the collapse till the moment when the collapsing shell reaches zero radius. The radiation distribution is not thermal in the whole frequency regime, but it is approximately thermal for the wavelengths of the order of the Schwarzschild radius of the collapsing shell. After the Schwarzschild radius is crossed, the temperature increases without limits as the singularity is approached. We also calculate the density matrix associated with this radiation. It turns out that the off-diagonal correlation terms to the diagonal Hawking's leading order terms are very important. While the trace of the diagonal (Hawking's) density matrix squared decreases during the evolution, the trace of the total density matrix squared remains unity at all times and all frequencies.

Study of a Scalar Field on the Maximally Extended Schwarzschild Spacetime

In this paper we present preliminary results about the investigation of scalar perturbations both inside and outside a Schwarzschild black hole. We express the wave equation in Ingoing Eddington-Finkelstein coordinates (explicitly showing its separability), write local series expansions for mode solutions and use them to then numerically calculate scalar perturbations inside and outside the future event horizon. Our main motivation is to calculate, in the future, the two-point quantum correlator in Schwarzschild spacetime, with points on opposite sides of the horizon.

Static and dynamic hairy planar black holes

We consider Einstein gravity in general dimensions, coupled to a scalar field either minimally or nonminimally, together with a generic scalar potential. By making appropriate choices of the scalar potential, we obtain large classes of new scalar hairy black holes that are asymptotic to planar anti--de Sitter spacetimes. For some classes of solutions, we can promote the scalar charge to be dependent on the advanced or retarded times in the Eddington-Finkelstein coordinates and obtain exact dynamic solutions. In particular, one class of the collapse solutions describes the evolution from the AdS vacua to some stable black hole states, driven by a conformally massless scalar. It is an explicit demonstration of nonlinear instability of the AdS vacuum that is stable at the linear level.

New look at black holes: Existence of universal horizons

In this paper, we study the existence of universal horizons in a given static spacetime, and find that the test khronon field can be solved explicitly when its velocity becomes infinitely large, at which point the universal horizon coincides with the sound horizon of the khronon. Choosing the timelike coordinate aligned with the khronon, the static metric takes a simple form, from which it can be seen clearly that the metric is free of singularity at the Killing horizon, but becomes singular at the universal horizon. Applying such developed formulas to three well-known black hole solutions, the Schwarzschild, Schwarzschild anti-de Sitter, and Reissner-Nordstr\"om, we find that in all these solutions universal horizons exist and are always inside the Killing horizons. In particular, in the Eddington-Finkelstein and Painleve-Gullstrand coordinates, in which the metrics are not singular when crossing both of the Killing and universal horizons, the peeling-off behavior of the khronon is found only at the universal horizons, whereby we show that the values of surface gravity of the universal horizons calculated from the peeling-off behavior of the khronon match with those obtained from the covariant definition given recently by Cropp, Liberati, Mohd and Visser.

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.

Stability analysis of black holes in massive gravity: A unified treatment

We consider the analytic solutions of massive (bi)gravity which can be written in a simple form using advanced Eddington-Finkelstein coordinates. We analyse the stability of these solutions against radial perturbations. First we recover the previously obtained result on the instability of the bidiagonal bi-Schwarzschild solutions. In the non-bidiagonal case (which contains, in particular, the Schwarzschild solution with Minkowski fiducial metric) we show that generically there are physical spherically symmetric perturbations, but no unstable modes.