arXiv subjects

Volker Perlick

Publications and source records attributed to Volker Perlick.

At least 55 records · Page 3Linked to original sources

arXiv2015

Schwarzschild radial perturbations in Eddington-Finkelstein and Painlevé-Gullstrand coordinates

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 extend the treatment of the massless case to Painlevé-Gullstrand coordinates, which are associated with radially infalling observers.

gr-qc↗

arXiv2015

Isofrequency pairing of spinning particles in Schwarzschild-de Sitter spacetime

It has been established in Schwarzschild spacetime (and more generally in Kerr spacetime) that pairs of geometrically different timelike geodesics with the same radial and azimuthal frequencies exist in the strong field regime. The occurrence of this socalled isofrequency pairing is of relevance in view of gravitational-wave observations. In this paper we generalize the results on isofrequency pairing in two directions. Firstly, we allow for a (positive) cosmological constant, i.e., we replace the Schwarzschild spacetime with the Schwarzschild-de Sitter spacetime. Secondly, we consider not only spinless test-particles (i.e., timelike geodesics) but also test-particles with spin. In the latter case we restrict to the case that the motion is in the equatorial plane with the spin perpendicular to this plane. We find that the cosmological constant as well as the spin have distinct impacts on the description of bound motion in the frequency domain.

gr-qc↗

arXiv2015

On analytic solutions of wave equations in regular coordinate systems on Schwarzschild background

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 generalized Riemann scheme we collect properties of all the singular points and construct analytic (local) solutions in terms of the standard confluent Heun function HeunC, Frobenius and asymptotic Thome series. We study the Eddington-Finkelstein case in detail and obtain a solution that is regular at the black hole horizon. This solution satisfies causal boundary conditions, i.e., it describes purely ingoing radiation at $r=2M$. To construct solutions on the entire open interval $r \, \in \; ]0,\infty[ \,$, we give an analytic continuation of local solutions around the horizon. Black hole scattering and quasi-normal modes are briefly considered as possible applications and we use semi-analytically calculated graybody factors together with the Damour-Ruffini method to reconstruct the power spectrum of Hawking radiation emitted by the black hole.

gr-qc↗

arXiv2015

Photon Regions and Shadows of Accelerated Black Holes

In an earlier paper we have analytically determined the photon regions and the shadows of black holes of the Plebański class of metrics which are also known as the Kerr--Newman--NUT--(anti-)deSitter metrics. These metrics are characterized by six parameters: mass, spin, electric and magnetic charge, gravitomagnetic NUT charge, and the cosmological constant. Here we extend this analysis to the Plebański--Demiański class of metrics which contains, in addition to these six parameters, the so-called acceleration parameter. All these metrics are axially symmetric and stationary type D solutions to the Einstein--Maxwell equations with a cosmological constant. We derive analytical formulas for the photon regions (i.e., for the regions that contain spherical lightlike geodesics) and for the boundary curve of the shadow as it is seen by an observer at Boyer--Lindquist coordinates $(r_O, \vartheta _O)$ in the domain of outer communication. Whereas all relevant formulas are derived for the whole Plebański--Demiański class, we concentrate on the accelerated Kerr metric (i.e., only mass, spin and acceleration parameter are different from zero) when discussing the influence of the acceleration parameter on the photon region and on the shadow in terms of pictures. The accelerated Kerr metric is also known as the rotating $C$-metric. We discuss how our analytical formulas can be used for calculating the horizontal and vertical angular diameters of the shadow and we estimate these values for the black holes at the center of our Galaxy and at the center of M87.

gr-qc↗

arXiv2014

Testing Gravity with Pulsars in the SKA Era

The Square Kilometre Array (SKA) will use pulsars to enable precise measurements of strong gravity effects in pulsar systems, which yield tests of gravitational theories that cannot be carried out anywhere else. The Galactic census of pulsars will discover dozens of relativistic pulsar systems, possibly including pulsar -- black hole binaries which can be used to test the "cosmic censorship conjecture" and the "no-hair theorem". Also, the SKA's remarkable sensitivity will vastly improve the timing precision of millisecond pulsars, allowing probes of potential deviations from general relativity (GR). Aspects of gravitation to be explored include tests of strong equivalence principles, gravitational dipole radiation, extra field components of gravitation, gravitomagnetism, and spacetime symmetries.

astro-ph.HE↗

arXiv2014

Finsler-type modification of the Coulomb law

Finsler geometry is a natural generalization of pseudo-Riemannian geometry. It can be motivated e.g. by a modified version of the Ehlers-Pirani-Schild axiomatic approach to space-time theory. Also, some scenarios of quantum gravity suggest a modified dispersion relation which could be phrased in terms of Finsler geometry. On a Finslerian spacetime, the Universality of Free Fall is still satisfied but Local Lorentz Invariance is violated in a way not covered by standard Lorentz Invariance Violation schemes. In this paper we consider a Finslerian modification of Maxwell's equations. The corrections to the Coulomb potential and to the hydrogen energy levels are computed. We find that the Finsler metric corrections yield a splitting of the energy levels. Experimental data provide bounds for the Finsler parameters.

gr-qc↗

arXiv2014

On the self-force in electrodynamics and implications for gravity

We consider the motion of charged point particles on Minkowski spacetime. The questions of whether the self-force is finite and whether mass renormalisation is necessary are discussed within three theories: In the standard Maxwell vacuum theory, in the non-linear Born-Infeld theory and in the higher-order Bopp-Podolsky theory. In a final section we comment on possible implications for the theory of the self-force in gravity.

gr-qc↗

arXiv2014

Photon Regions and Shadows of Kerr-Newman-NUT Black Holes with a Cosmological Constant

We consider the Plebański class of electrovacuum solutions to the Einstein equations with a cosmological constant. These space-times, which are also known as the Kerr-Newman-NUT-(anti-)de Sitter space-times, are characterized by a mass $m$, a spin $a$, a parameter $β$ that comprises electric and magnetic charge, a NUT parameter $\ell$ and a cosmological constant $Λ$. Based on a detailed discussion of the photon regions in these space-times (i.e., of the regions in which spherical lightlike geodesics exist), we derive an analytical formula for the shadow of a Kerr-Newman-NUT-(anti-)de Sitter black hole, for an observer at given Boyer-Lindquist coordinates $(r_O, \vartheta_O)$ in the domain of outer communication. We visualize the photon regions and the shadows for various values of the parameters.

gr-qc↗

arXiv2012

Observable effects in a class of spherically symmetric static Finsler spacetimes

After some introductory discussion of the definition of Finsler spacetimes and their symmetries, we consider a class of spherically symmetric and static Finsler spacetimes which are small perturbations of the Schwarzschild spacetime. The deviations from the Schwarzschild spacetime are encoded in three perturbation functions $ϕ_0(r)$, $ϕ_1(r)$ and $ϕ_2(r)$ which have the following interpretations: $ϕ_0$ perturbs the time function, $ϕ_1$ perturbs the radial length measurement and $ϕ_2$ introduces a spatial anisotropy which is a genuine Finsler feature. We work out the equations of motion for freely falling particles and for light rays, i.e. the timelike and lightlike geodesics, in this class of spacetimes, and we discuss the bounds placed on the perturbation functions by observations in the Solar system.

gr-qc↗

arXiv2011

On the hyperbolicity of Maxwell's equations with a local constitutive law

Maxwell's equations are considered in metric-free form, with a local but otherwise arbitrary constitutive law. After splitting Maxwell's equations into evolution equations and constraints, we derive the characteristic equation and we discuss its properties in detail. We present several results that are relevant for the question of whether the evolution equations are hyperbolic, strongly hyperbolic or symmmetric hyperbolic. In particular, we give a convenient characterisation of all constitutive laws for which the evolution equations are symmetric hyperbolic. The latter property is sufficient, but not necessary, for well-posedness of the initial-value problem. By way of example, we illustrate our results with the constitutive laws of biisotropic media and of Born-Infeld theory.

gr-qc↗

arXiv2010

Gravitational Lensing from a Spacetime Perspective

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.

gr-qc↗

arXiv2010

On the initial-value problem of the Maxwell-Lorentz equations

We consider the Maxwell-Lorentz equations, i.e., the equation of motion of a charged dust coupled to Maxwell's equations, on an arbitrary general-relativistic spacetime. We decompose this system of equations into evolution equations and constraints, and we demonstrate that the evolution equations are strongly hyperbolic. This result guarantees that the initial-value problem of the Maxwell-Lorentz equations is well-posed. We illustrate this general result with a discussion of spherically symmetric solutions on Minkowski spacetime.

gr-qc↗

arXiv2008

On timelike surfaces in Lorentzian manifolds

We discuss the geometry of timelike surfaces (two-dimensional submanifolds) in a Lorentzian manifold and its interpretation in terms of general relativity. A classification of such surfaces is presented which distinguishes four cases of special algebraic properties of the second fundamental form from the generic case. In the physical interpretation a timelike surface can be viewed as the worldsheet of a ``track'', and timelike curves in this surface can be viewed as the worldlines of observers who are bound to the track, like someone sitting in a roller-coaster car. With this interpretation, our classification turns out to be closely related to (i) the visual appearance of the track, (ii) gyroscopic transport along the track, and (iii) inertial forces perpendicular to the track. We illustrate our general results with timelike surfaces in the Kerr-Newman spacetime.

gr-qc↗

arXiv2007

On the generalized Jacobi equation

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.

gr-qc↗

arXiv2007

On the radar method in general-relativistic spacetimes

If a clock, mathematically modeled by a parametrized timelike curve in a general-relativistic spacetime, is given, the radar method assigns a time and a distance to every event which is sufficiently close to the clock. Several geometric aspects of this method are reviewed and their physical interpretation is discussed.

gr-qc↗

arXiv2007

Theoretical gravitational lensing. Beyond the weak-field small-angle approximation

An overview is given on those theoretical gravitational lensing results that can be formulated in a spacetime setting, without assuming that the gravitational fields are weak and that the bending angles are small. The first part is devoted to analytical methods considering spacetimes in which the equations for light rays (lightlike geodesics) is completely integrable. This includes spherically symmetric static spacetimes, the Kerr spacetime and plane gravitational waves. The second part is devoted to qualitative methods which give some information on lensing properties without actually solving the equation for lightlike geodesics. This includes Morse theory, methods from differential topology and bifurcation theory.

gr-qc↗

arXiv2005

On totally umbilic submanifolds of semi-Riemannian manifolds

The notion of being totally umbilic is considered for non-degenerate and degenerate submanifolds of semi-Riemanian manifolds. After some remarks on the general case, timelike and lightlike totally umbilic submanifolds of Lorentzian manifolds are discussed, along with their physical interpretation in view of general relativity. In particular, the mathematical notion of totally umbilic submanifolds is linked to the notions of photon surfaces and of null strings which have been used in the physics literature.

gr-qc↗

arXiv2005

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

Consider, in the domain of outer communication of a Kerr-Newman black hole, a point (observation event) and a timelike curve (worldline of light source). Assume that the worldline of the source (i) has no past end-point, (ii) does not intersect the caustic of the past light-cone of the observation event, and (iii) goes neither to the horizon nor to infinity in the past. We prove that then for infinitely many positive integers k there is a past-pointing lightlike geodesic of (Morse) index k from the observation event to the worldline of the source, hence an observer at the observation event sees infinitely many images of the source. Moreover, we demonstrate that all lightlike geodesics from an event to a timelike curve in the domain of outer communication are confined to a certain spherical shell. Our characterization of this spherical shell shows that in the Kerr-Newman spacetime the occurrence of infinitely many images is intimately related to the occurrence of centrifugal-plus-Coriolis force reversal.

gr-qc↗