arXiv subjects

Eva Hackmann

Publications and source records attributed to Eva Hackmann.

At least 37 records · Page 2Linked to original sources

arXiv2019

Constraining the charge of the Galactic centre black hole

In this contribution, we summarize our results concerning the observational constraints on the electric charge associated with the Galactic centre black hole - Sgr A*. According to the no-hair theorem, every astrophysical black hole, including supermassive black holes, is characterized by at most three classical, externally observable parameters - mass, spin, and the electric charge. While the mass and the spin have routinely been measured by several methods, the electric charge has usually been neglected, based on the arguments of efficient discharge in astrophysical plasmas. From a theoretical point of view, the black hole can attain charge due to the mass imbalance between protons and electrons in fully ionized plasmas, which yields about $\sim 10^8\,{\rm C}$ for Sgr A*. The second, induction mechanism concerns rotating Kerr black holes embedded in an external magnetic field, which leads to electric field generation due to the twisting of magnetic field lines. This electric field can be associated with the induced Wald charge, for which we calculate the upper limit of $\sim 10^{15}\,{\rm C}$ for Sgr A*. Although the maximum theoretical limit of $\sim 10^{15}\,{\rm C}$ is still 12 orders of magnitude smaller than the extremal charge of Sgr A*, we analyse a few astrophysical consequences of having a black hole with a small charge in the Galactic centre. Two most prominent ones are the effect on the X-ray bremsstrahlung profile and the effect on the position of the innermost stable circular orbit.

astro-ph.GA↗

arXiv2019

The propagation delay in the timing of a pulsar orbiting a supermassive black hole

The observation of a pulsar closely orbiting the galactic center supermassive black hole would open the window for an accurate determination of the black hole parameters and for new tests of General Relativity. An important relativistic effect which has to be taken into account in the timing model is the propagation delay of the pulses in the gravitational field of the black hole. Due to the extreme mass ratio of the pulsar and the supermassive back hole we use the test particle limit to derive an exact analytical formula for the propagation delay in a Schwarzschild spacetime. We then compare this result to the propagation delays derived in the usually employed post-Newtonian approximation, in particular to the Shapiro delay up to the second post-Newtonian order. For edge-on orbits we also consider modifications of the Shapiro delay which take the lensing effects into account. Our results are then used to assess the accuracy of the different orders of the post-Newtonian approximation of the propagation delay. This comparison indicates that for (nearly) edge-on orbits the new exact delay formula should be used.

gr-qc↗

arXiv2019

Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation

Accurate orbit modeling plays a key role in contemporary and future space missions such as GRACE and its successor GRACE-FO, GNSS, and altimetry missions. To fully exploit the technological capabilities and correctly interpret measurements, relativistic orbital effects need to be taken into account. Within the theory of General Relativity, equations of motion for freely falling test objects, such as satellites orbiting the Earth, are given by the geodesic equation. We analyze and compare different solution methods in a spherically symmetric background, i.e. for the Schwarzschild spacetime, as a test bed. We investigate satellite orbits and use direct numerical orbit integration as well as the semi-analytical Lie-series approach. The results are compared to the exact analytical reference solution in terms of elliptic functions. For a set of exemplary orbits, we determine the respective accuracy of the different methods. Within the post-Newtonian approximation of General Relativity, modified orbital equations are obtained by adding relativistic corrections to the Newtonian equations of motion. We analyze the accuracy of this approximation with respect to the general relativistic setting. Therefore, we solve the post-Newtonian equation of motion using the eXtended High Performance Satellite dynamics Simulator. For corresponding initial conditions, we compare orbits in the Schwarzschild spacetime to those in its post-Newtonian approximation. Moreover, we compare the magnitude of relativistic contributions to several typical perturbations of satellite orbits due to, e.g., solar radiation pressure, Earth's albedo, and atmospheric drag. This comparison is done for our test scenarios and for a real GRACE orbit to highlight the importance of relativistic effects in geodetic space missions.

gr-qc↗

arXiv2018

Charged fluid structures around a rotating compact object with a magnetic dipole field

We study stationary, electrically charged fluid structures encircling a rotating compact object with a dipole magnetic field oriented along the rotation axis. This situation is described in an idealized way by the Kerr metric and a magnetic dipole "test" field, that does not affect the spacetime. The self-gravitational and self-electromagnetic field of the fluid are neglected and the fluid is assumed to be non conductive and in rigid motion. Our work generalizes a previous study by Kovář et al.(2016) by taking into account the rotation of the central object. Therefore, we focus on the influence of the rotation onto the existence and position of bound fluid structures. Frame dragging effects allow the existence of polar clouds, which could not be found in non-rotating case. Furthermore counter-rotating equatorial tori become more preferred the faster the central object is spinning.

astro-ph.HE↗

arXiv2017

Relativistic dust accretion of charged particles in Kerr-Newman spacetime

We describe a new analytical model for the accretion of particles from a rotating and charged spherical shell of dilute collisionless plasma onto a rotating and charged black hole. By assuming a continuous injection of particles at the spherical shell and by treating the black hole and a featureless accretion disc located in the equatorial plane as passive sinks of particles we build a stationary accretion model. This may then serve as a toy model for plasma feeding an accretion disc around a charged and rotating black hole. Therefore, our new model is a direct generalization of the analytical accretion model introduced by E. Tejeda, P. A. Taylor, and J. C. Miller (2013). We use our generalized model to analyze the influence of a net charge of the black hole, which will in general be very small, on the accretion of plasma. Within the assumptions of our model we demonstrate that already a vanishingly small charge of the black hole may in general still have a non-negligible effect on the motion of the plasma, as long as the electromagnetic field of the plasma is still negligible. Furthermore, we argue that the inner and outer edges of the forming accretion disc strongly depend on the charge of the accreted plasma. The resulting possible configurations of accretion discs are analyzed in detail.

astro-ph.HE↗

arXiv2017

Redshift and frequency comparison in Schwarzschild spacetime

We derive exact expressions for the relativistic redshift between an Earth-bound observer, that is meant to model a standard clock on the Earth's surface, and various (geodesic) observers in the Schwarzschild spacetime. We assume that the observers exchange radial light signals to compare the frequencies of standard clocks, which they transport along their respective worldlines. We calculate the redshift between an Earth-bound clock and static observers, observers in radial free fall, on circular geodesics, and on arbitrary bound quasi-elliptical orbits. For the latter case, we consider as examples an almost circular orbit, the Schwarzschild analog of Galileo satellites 5 orbits with a moderate eccentricity, and a highly elliptical orbit as special examples. Furthermore, we also use orbits close to a Schwarzschild black hole to highlight the influence of the relativistic perigee precession on the redshift signal. Calculating a post-Newtonian expansion of our results, the total redshift is decomposed into its special relativistic Doppler parts and the gravitational part due to the theory of General Relativity. To investigate the impact of higher order relativistic multipole moments on the gravitational redshift, we consider static observers in a general Weyl spacetime. We give a general expression for the mutual redshift of their standard clocks and consider in particular the effect of the relativistic quadrupole as a modification of the Schwarzschild result.

gr-qc↗

arXiv2017

Definition of the relativistic geoid in terms of isochronometric surfaces

We present a definition of the geoid that is based on the formalism of general relativity without approximations; i.e. it allows for arbitrarily strong gravitational fields. For this reason, it applies not only to the Earth and other planets but also to compact objects such as neutron stars. We define the geoid as a level surface of a time-independent redshift potential. Such a redshift potential exists in any stationary spacetime. Therefore, our geoid is well defined for any rigidly rotating object with constant angular velocity and a fixed rotation axis that is not subject to external forces. Our definition is operational because the level surfaces of a redshift potential can be realized with the help of standard clocks, which may be connected by optical fibers. Therefore, these surfaces are also called isochronometric surfaces. We deliberately base our definition of a relativistic geoid on the use of clocks since we believe that clock geodesy offers the best methods for probing gravitational fields with highest precision in the future. However, we also point out that our definition of the geoid is mathematically equivalent to a definition in terms of an acceleration potential, i.e. that our geoid may also be viewed as a level surface orthogonal to plumb lines. Moreover, we demonstrate that our definition reduces to the known Newtonian and post-Newtonian notions in the appropriate limits. As an illustration, we determine the isochronometric surfaces for rotating observers in axisymmetric static and axisymmetric stationary solutions to Einstein's vacuum field equation, with the Schwarzschild metric, the Erez-Rosen metric, the q-metric and the Kerr metric as particular examples.

gr-qc↗

arXiv2015

Analytical solutions for geodesics in black hole spacetimes

We review the analytical solution methods for the geodesic equations in Kerr-Newman-Taub-NUT-de Sitter spacetimes and its subclasses in terms of elliptic and hyperelliptic functions. A short guide to corresponding literature for general timelike and lightlike motion is also presented.

gr-qc↗

arXiv2015

Geodesic equations and algebro-geometric methods

For an investigation of the physical properties of gravitational fields the observation of massive test particles and light is very useful. The characteristic features of a given space-time may be decoded by studying the complete set of all possible geodesic motions. Such a thorough analysis can be accomplished most effectively by using analytical methods to solve the geodesic equation. In this contribution, the use of elliptic functions and their generalizations for solving the geodesic equation in a wide range of well known space-times, which are part of the general Plebański-Demiański family of solutions, will be presented. In addition, the definition and calculation of observable effects like the perihelion shift will be presented and further applications of the presented methods will be outlined.

gr-qc↗

arXiv2015

Analytical solution methods for geodesic motion

The observation of the motion of particles and light near a gravitating object is until now the only way to explore and to measure the gravitational field. In the case of exact black hole solutions of the Einstein equations the gravitational field is characterized by a small number of parameters which can be read off from the observables related to the orbits of test particles and light rays. Here we review the state of the art of analytical solutions of geodesic equations in various space--times. In particular we consider the four dimensional black hole space--times of Plebański--Demiański type as far as the geodesic equation separates, as well as solutions in higher dimensions, and also solutions with cosmic strings. The mathematical tools used are elliptic and hyperelliptic functions. We present a list of analytic solutions which can be found in the literature.

gr-qc↗

arXiv2015

Geodesic equation in Schwarzschild--(anti-)de Sitter space--times: Analytical solutions and applications

The complete set of analytic solutions of the geodesic equation in a Schwarzschild--(anti-)de Sitter space--time is presented. The solutions are derived from the Jacobi inversion problem restricted to the set of zeros of the theta function, called the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer Anomaly is addressed. The periastron shift and its post--Schwarzschild limit is derived. The developed method can also be applied to the geodesic equation in higher dimensional Schwarzschild space--times.

gr-qc↗

arXiv2015

Complete analytic solution of the geodesic equation in Schwarzschild--(anti) de Sitter space--times

The complete set of analytic solutions of the geodesic equation in a Schwarzschild--(anti) de Sitter space--time is presented. The solutions are derived from the Jacobi inversion problem restricted to the theta--divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The solutions are completely classified by the structure of the zeros of the characteristic polynomial which depends on the energy, angular momentum, and the cosmological constant.

gr-qc↗

arXiv2015

Motion of test particles in a regular black hole space--time

We consider the motion of test particles in the regular black hole space-time given by Ayón-Beato and Garc\'ıa in Phys. Rev. Lett. 80:5056 (1998). The complete set of orbits for neutral and weakly charged test particles is discussed, including for neutral particles the extreme and over-extreme metric. We also derive the analytical solutions for the equation of motion of neutral test particles in a parametric form and consider a post-Schwarzschild expansion of the periastron shift to second order in the charge.

gr-qc↗

arXiv2014

Motion of spinning test bodies in Kerr spacetime

We investigate the motion of spinning test bodies in General Relativity. By means of a multipolar approximation method for extended test bodies we derive the equations of motion, and classify the orbital motion of pole-dipole test bodies in the equatorial plane of the Kerr geometry. An exact expression for the periastron shift of a spinning test body is given. Implications of test body spin corrections are studied and compared with the results obtained by means of other approximation schemes.

gr-qc↗

arXiv2014

Generalized gravitomagnetic clock effect

In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of $2π$. For two clocks on counterrotating equatorial circular orbits around the Earth, the effect is about $10^{-7}$ seconds per revolution, which is quite large. We introduce a general relativistic definition of the gravitomagnetic clock effect which is valid for arbitrary pairs of orbits. This includes rotations in the same direction and different initial conditions, which are crucial if the effect can be detected with existing satellites or with payloads on nondedicated missions. We also derive the post-Newtonian expansion of the general relativistic expression and calculate the effect for the example of a satellite of a Global Navigation Satellite System compared to a geostationary satellite.

gr-qc↗

arXiv2013

Charged particle motion in Kerr-Newmann space-times

The motion of charged test-particles in the gravitational field of a rotating and electromagnetically charged black hole as described by the Kerr-Newman metric is considered. We completely classify the colatitudinal and radial motion on the extended manifold $-\infty \leq r \leq \infty$, including orbits crossing the horizons or $r=0$. Analytical solutions of the equations of motion in terms of elliptic functions are presented which are valid for all types of orbits.

gr-qc↗

arXiv2012

Observables for bound orbital motion in axially symmetric space-times

The periastron shift and the Lense-Thirring effect of bound orbital motion in a general axially symmetric space-time given by Plebański and Demiański are analyzed. We also define a measure for the conicity of the orbit and give analytic expressions for all three observables in terms of hyperelliptic integrals and Lauricella's $F_D$ function. For an interpretation of these analytical expressions, we perform a post-Schwarzschild and a post-Newton expansion of these quantities. This clearly shows the influence of the different space-time parameters on the considered observables and allows to characterize Kerr, Taub-NUT, Schwarzschild-de Sitter, or other space-times.

gr-qc↗

arXiv2010

Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity

The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of a holomorphic hyperelliptic integral. The result of the inversion is defined only locally, and is done using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the $θ$--divisor. For a representation of the hyperelliptic functions the Klein--Weierstraß multivariable sigma function is introduced. It is shown that all parameters needed for the calculations like period matrices and Abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and theta-constants. The cases of genus two and three are considered in detail. The method is exemplified by particle motion associated with a genus three hyperelliptic curve.

gr-qc↗