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We present a formulation of Einstein--Maxwell vacuum fields due to plane fronted electromagnetic waves sharing their wave fronts with gravitational waves. This is based on a recent geometrical reconstruction of plane fronted wave fields by the authors which clearly identifies the cases in which the wave fronts collide or do not collide. In the former case our construction suggests an explicit example of a searchlight beam, accompanied by gravitational radiation, which sweeps across the sky. This gravito--electromagnetic searchlight and its properties are described in detail., 7 pages, 2 figures

We construct metric perturbations of two families of isotropic expanding universes describing gravitational waves propagating through these universes. The waves are non--planar and owe their wave front expansion solely to the expansion of the universes. The presence of this radiation leads to a small perturbation of the perfect fluid matter content of the universes by the appearance of an anisotropic stress. We then construct exact models of gravity waves in these universes. In this case the matter content of the models consists of the perfect fluid matter supplemented by anisotropic stress and lightlike matter traveling with the waves. Under appropriate conditions of approximation the lightlike matter can be neglected and the exact models coincide with the perturbative models., Comment: 12 pages

We demonstrate how plane fronted waves with colliding wave fronts are the asymptotic limit of spherical electromagnetic and gravitational waves. In the case of the electromagnetic waves we utilize Bateman's representation of radiative solutions of Maxwell's vacuum field equations. The gravitational case involves a novel form of the radiative Robinson--Trautman solutions of Einstein's vacuum field equations., Comment: 5 pages

Autoparallel curves along with geodesic curves can arise as trajectories of physical test bodies. We explicitly derive autoparallels as effective post-Riemannian geometric constructs, and at the same time we argue \emph{against} postulating autoparallels as fundamental equations of motion for test bodies in alternative gravity theories., 4 pages

Material particles and photons are of paramount importance in constructing and analysing models of gravitational fields in general relativity. Their histories in space-time are congruences of time-like and light-like (or null) world lines respectively. Material particles and photons therefore play a central role in this book and we begin by describing the standard theory of congruences of time-like world lines and of null congruences. In the light-like case we shall only require a knowledge of null geodesic congruences. For use later we describe geodesic deviation equations with the help of Synge’s world function and parallel propagator.

A central question in general relativity is how the components of the Riemann curvature tensor (the gravitational field) can be determined in an operational way. Here we review two different methods which allow for a complete determination of the Riemann curvature tensor. The first method relies on measuring the accelerations of a suitably prepared set of test bodies relative to an observer. The second method utilizes a set of suitably prepared clocks.

I. The Maxwell field of a charged light-like particle with a non-geodesic world line (a light-like analogue of the Lienard–Wiechert field) can be constructed utilising the Minkowskian geometry in the neighbourhood of such a world line. In the process a fundamental question regarding the existence of a special parameter along the world line has to be addressed.

In the absence of external fields a Reissner–Nordstrom particle of mass m and charge e for which m and e2 are small of first order performs run-away motion. Since in particular m2 < e2 there is no event horizon associated with this object and so we refer to it as a Reissner–Nordstrom particle. A Reissner–Nordstrom particle of mass m and magnetic monopole moment g for which m and g2 are small of first order behaves in precisely the same way.

In Minkowskian space-time in which points are labelled with rectangular Cartesian coordinates x, y, z and time t the so-called t-lines correspond to constant values of x, y, z. These world lines constitute a time-like geodesic congruence which is twist-free, shear-free and expansion-free. They play an important role in encoding in a bivector field on Minkowskian space-time the information contained in the electric and magnetic 3-vectors on three dimensional Euclidean space. A striking illustration of this is found in the study of electromagnetic radiation and the derivation of Bateman electromagnetic waves. The counterpart of the latter in the case of gravitational waves in the linear approximation using the gauge invariant and covariant approach demonstrates that the existence of the gravitational waves is due to perturbations in the shear of the t-lines.

The use of bivectors, or skew symmetric tensors, is a powerful research tool in general relativity. Such objects arise naturally in the context of electromagnetism. We therefore introduce them via electromagnetic test fields on arbitrary space-times. All details are provided including an important application to electromagnetic radiation due to Ivor Robinson. This is followed by the extension of bivector theory to gravitational fields. An explicit illustration of their use in relation to Kerr space-time is given.

The equation u ≡ t − z = constant in Minkowskian space-time with line element $$\displaystyle \begin{array}{@{}rcl@{}} ds^2=dt^2-dx^2-dy^2-dz^2=\eta _{ij}\,dx^i\,dx^j\ , \end{array} $$ is an example of a null hyperplane. It represents the history of a 2-plane, parallel to the x, y-plane in three dimensional Euclidean space, moving with the speed of light in the positive z-direction. Thus the family of null hyperplanes u = constant could be the histories of the wave fronts of plane electromagnetic waves travelling in the positive z-direction in Euclidean space. The vector field normal to the hyperplanes is ki = u,i and with ki = ηijkj this has the properties that kiki = 0 and ki,j = 0. Thus ki is a null vector field (and therefore tangent to u = constant) and covariantly constant (i.e. a constant vector field in the coordinates xi = (t, x, y, z) with i = 0, 1, 2, 3). To generalise this notion of null hyperplanes to space-times of non-zero constant curvature, the first obstacle one encounters is the non-existence in such a space-time of covariantly constant vector fields.

A magnetic black hole in Einstein–Maxwell theory is described by the potential 1-form of a magnetic pole and a line element which coincides with the Reissner–Nordstrom line element dependent upon two parameters representing the mass of the magnetic pole and the magnetic monopole moment. When such an object interacts with external fields it behaves differently to an electric black hole. This can be exhibited most easily if one assumes that the magnetic black hole has small mass and small magnetic monopole moment (and is therefore a small magnetic black hole). In this case the equations of motion can be derived approximately with sufficient accuracy to include electromagnetic radiation reaction, external 4-force and tail terms.

By introducing external Maxwell and gravitational fields we modify the Bonnor--Vaidya field of an arbitrarily accelerating charged mass moving rectilinearly in order to satisfy the vacuum Einstein--Maxwell field equations approximately, assuming the charge $e$ and the mass $m$ are small of first order., 14 pages, 2 figures

We construct the field of an arbitrarily accelerating and rotating point mass which specializes to the Kerr solution when the acceleration vanishes and specializes to Kinnersley's arbitrarily accelerating point mass when the rotation vanishes., 9 pages

The Earth’s geoid is one of the most important fundamental concepts to provide a gravity field- related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions within Einstein’s theory of General Relativity is inevitable. Building on the theoretical construction of isochronometric surfaces we define a relativistic gravity potential as a generalization of known (post-)Newtonian notions. It exists for any stationary configuration and rigidly co-rotating observers; it is the same as realized by local plumb lines and determined by the norm of a timelike Killing vector. In a second step, we define the relativistic geoid in terms of this gravity potential in direct analogy to the Newtonian understanding. In the respective limits, it allows to recover well-known results. Comparing the Earth’s Newtonian geoid to its relativistic generalization is a very subtle problem. However, an isometric embedding into Euclidean three-dimensional space can solve it and allows an intrinsic comparison. We show that the leading-order differences are at the mm-level. In the next step, the framework is extended to generalize the normal gravity field as well. We argue that an exact spacetime can be constructed, which allows to recover the Newtonian result in the weak-field limit. Moreover, we comment on the relativistic definition of chronometric height and related concepts.In a stationary spacetime related to the rotating Earth, the aforementioned gravity potential is of course not enough to cover all information on the gravitational field. To obtain more insight, a second scalar function can be constructed, which is genuinely related to gravitomagnetic contributions and vanishes in the static case. Using the kinematic decomposition of an isometric observer congruence, we suggest a potential related to the twist of the worldlines therein. Whilst the first potential is related to clock comparison and the acceleration of freely falling corner cubes, the twist potential is related to the outcome of Sagnac interferometric measurements. The combination of both potentials allows to determine the Earth’s geoid and equip this surface with coordinates in an operational way. Therefore, relativistic geodesy is intimately related to the physics of timelike Killing vector fields.

We describe a framework for a generalized nonlocal gravity theory inspired by Poincar\'e gauge theory. Our theory provides a unified description of previous nonlocal extensions of Einstein's theory of gravitation, in particular it allows for a clear geometrical foundation. Furthermore, it incorporates recent simplifications for the ansatz of the nonlocality, which should allow for a systematic study of the impact of nonlocal concepts on observations., Comment: 5 pages

By extending the framework of the gravitational clock compass we show how a suitably prepared set of clocks can be used to extract information about the gravitational field in the context of General Relativity. Conceptual differences between the extended and the standard clock compass are highlighted. Particular attention is paid to the influence of kinematic quantities on the measurement process and the setup of the compass. Additionally, we present results of simulations of the inference process for the acceleration and the curvature components. Several examples of different strategies for the computation of the posterior probability distributions of the curvature components are discussed. This allows us to anticipate the precision with which physical quantities could be determined in a realistic measurement., Comment: 20 pages, 18 figures

We analyze the structure of a recent nonlocal generalization of Einstein's theory of gravitation by Mashhoon et al. By means of a covariant technique, we derive an expanded version of the nonlocality tensor which constitutes the theory. At the lowest orders of approximation, this leads to a simplification which sheds light on the fundamental structure of the theory and may prove useful in the search for exact solutions of nonlocal gravity., 7 pages

We present an alternative derivation of the gravitational clock compass and show how such a device can be used for the detection of gravitational waves. Explicit compass setups are constructed in special types of space--times, namely for exact plane gravitational waves and for waves moving radially relative to an observer., Comment: 14 pages, 3 figures

How does one measure the gravitational field? We give explicit answers to this fundamental question and show how all components of the curvature tensor, which represents the gravitational field in Einstein’s theory of General Relativity, can be obtained by means of two different methods. The first method relies on the measuring the accelerations of a suitably prepared set of test bodies relative to the observer. The second method utilizes a set of suitably prepared clocks. The methods discussed here form the basis of relativistic (clock) gradiometry and are of direct operational relevance for applications in geodesy.

Within the theory of General Relativity, we study the solution and range of applicability of the standard geodesic deviation equation in highly symmetric spacetimes. In the Schwarzschild spacetime, the solution is used to model satellite orbit constellations and their deviations around a spherically symmetric Earth model. We investigate the spatial shape and orbital elements of perturbations of circular reference curves. In particular, we reconsider the deviation equation in Newtonian gravity and then determine relativistic effects within the theory of General Relativity by comparison. The deviation of nearby satellite orbits, as constructed from exact solutions of the underlying geodesic equation, is compared to the solution of the geodesic deviation equation to assess the accuracy of the latter. Furthermore, we comment on the so-called Shirokov effect in the Schwarzschild spacetime and limitations of the first order deviation approach.

Deviation equation: Second order differential equation for the 4-vector which measures the distance between reference points on neighboring world lines in spacetime manifolds. Relativistic geodesy: Science representing the Earth (or any planet), including the measurement of its gravitational field, in a four-dimensional curved spacetime using differential-geometric methods in the framework of Einstein's theory of gravitation (General Relativity).

On August 22, 2014, the satellites GSAT-0201 and GSAT-0202 of the European GNSS Galileo were unintentionally launched into eccentric orbits. Unexpectedly, this has become a fortunate scientific opportunity since the onboard hydrogen masers allow for a sensitive test of the redshift predicted by the theory of general relativity. In the present Letter we describe an analysis of approximately three years of data from these satellites including three different clocks. For one of these we determine the test parameter quantifying a potential violation of the combined effects of the gravitational redshift and the relativistic Doppler shift. The uncertainty of our result is reduced by more than a factor 4 as compared to the values of Gravity Probe A obtained in 1976., Comment: 6 pages, 3 figures, published in Phys. Rev. Lett. 121, 231102

We show how a suitably prepared set of clocks can be used to determine all components of the gravitational field in General Relativity. We call such an experimental setup a clock compass, in analogy to the usual gravitational compass. Particular attention is paid to the construction of the underlying reference frame. Conceptual differences between the clock compass and the standard gravitational compass, which is based on the measurement of the mutual accelerations between the constituents of a swarm of test bodies, are highlighted. Our results are of direct operational relevance for the setup of networks of clocks, for example in the context of relativistic geodesy., Comment: 12 pages, 6 figures

We derive a generalized deviation equation in Riemann-Cartan spacetime. The equation describes the dynamics of the connecting vector which links events on two general adjacent world lines. Our result is valid for any theory in a Riemann-Cartan background, in particular, it is applicable to a large class of gravitational theories which go beyond the general relativistic framework., Comment: 9 pages, 1 figure

Based on the formalism of General Relativity, we analyze generalizations of concepts used in conventional geodesy. One such concept is the Earth's geoid. We present our definition of the relativistic geoid in terms of the level sets of a time-independent redshift potential. Such a potential exists for any congruence of Killing observers, i.e. for any rigidly moving object associated with a stationary spacetime in the outer region. The level surfaces of the redshift potential foliate the three dimensional space into isochronometric surfaces, which can be determined with the help of standard clocks. Two such clocks on the same surface will show zero redshift when their frequencies are compared. One of these level surfaces, singled out by a suitable convention, defines the relativistic geoid in our framework. At the same time, the redshift potential is also an acceleration potential for the congruence of observers. Hence, the isochrono-metric surfaces are orthogonal to the acceleration of freely falling objects, i.e. they are orthogonal to the local plumb line. Therefore, two independent kinds of measurements can contribute to the determination of the relativistic geoid. It can be shown that clocks, which are connected by optical fiber links, can be used to determine the redshift potential; this gives the operational foundation of our framework. Moreover, we show that our definition reduces to the well-known Newtonian and post-Newtonian notions in the respective limits. To illustrate our framework, we consider analytic examples of spacetimes, for which we calculate the level surfaces of the redshift potential and illustrate their intrinsic geometry by an embedding into flat Euclidean space. We emphasize that our definition of the geoid in terms of relativistic concepts is valid for arbitrarily strong gravitational fields. We do not use any approximation in the sense of weak fields or post-Newtonian expansion schemes. Hence, the definition can also be applied to very compact objects such as neutron stars.

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., Comment: 24 pages, 7 figures; improved figures, references added

Ever since E.Cartan in the 1920s enriched the geometric framework of general relativity (GR) by introducing a {\it torsion} of spacetime, the question arose whether one could find a measurement technique for detecting the presence of a torsion field. Mao et al.(2007) claimed that the rotating quartz balls in the gyroscopes of the Gravity Probe B experiment, falling freely on an orbit around the Earth, should "feel" the torsion. Similarly, March et al.(2011) argue with the precession of the Moon and the Mercury and extend later their considerations to the Lageos satellite.--- A consistent theory of gravity with torsion emerged during the early 1960's as gauge theory of the Poincar\'e group. This Poincar\'e gauge theory of gravity incorporates as simplest viable cases the Einstein-Cartan(-Sciama-Kibble) theory (EC), the teleparallel equivalent GR|| of GR, and GR itself. So far, PG and, in particular, the existence of torsion have {\it not} been experimentally confirmed. However, PG is to be considered as the standard theory of gravity with torsion because of its very convincing gauge structure.--- Since the early 1970s up to today, different groups have shown more or less independently that torsion couples only to the {\it elementary particle spin} and under no circumstances to the orbital angular momentum of test particles. This is established knowledge and we reconfirm this conclusion by discussing the energy-momentum law of PG, which has same form for all versions of PG. Therefore, we conclude that, unfortunately, the investigations of Mao et al. and March et al. do not yield any information on torsion., Comment: 7 pages of latex with 2 figures, title changed, minor corrections inserted, some references added

How do test bodies move in scalar-tensor theories of gravitation? We provide an answer to this question on the basis of a unified multipolar scheme. In particular, we give the explicit equations of motion for pointlike, as well as spinning test bodies, thus extending the well-known general relativistic results of Mathisson, Papapetrou, and Dixon to scalar-tensor theories of gravity. We demonstrate the validity of the equivalence principle for test bodies., Gravitation, Astrophysics, and Cosmology - Proceedings of the Twelfth Asia-Pacific International Conference, Moscow, 28 Jun - 5 July 2015, Eds. V. Melnikov and J.-P. Hsu, World Scientific (Singapore), 2016, pp. 231-235

We present a direct confirmation of the validity of the equivalence principle for unstructured test bodies in scalar tensor gravity. Our analysis is complementary to previous approaches and valid for a large class of scalar-tensor theories of gravitation. A covariant approach is used to derive the equations of motion in a systematic way and allows for the experimental test of scalar-tensor theories by means of extended test bodies., 5 pages, RevTex format

We give an overview of the derivation of multipolar equations of motion of extended test bodies for a wide set of gravitational theories beyond the standard general relativistic framework. The classes of theories covered range from simple generalizations of General Relativity, e.g., encompassing additional scalar fields, to theories with additional geometrical structures which are needed for the description of microstructured matter. Our unified framework even allows to handle theories with nonminimal coupling to matter, and thereby for a systematic test of a very broad range of gravitational theories.

The super-massive 4 million solar mass black hole Sagittarius A* (SgrA*) shows flare emission from the millimeter to the X-ray domain. A detailed analysis of the infrared light curves allows us to address the accretion phenomenon in a statistical way. The analysis shows that the near-infrared flare amplitudes are dominated by a single state power law, with the low states in SgrA* limited by confusion through the unresolved stellar background. There are several dusty objects in the immediate vicinity of SgrA*. The source G2/DSO is one of them. Its nature is unclear. It may be comparable to similar stellar dusty sources in the region or may consist predominantly of gas and dust. In this case a particularly enhanced accretion activity onto SgrA* may be expected in the near future. Here the interpretation of recent data and ongoing observations are discussed.

We derive a generalized deviation equation -- analogous to the well-known geodesic deviation equation -- for test bodies in General Relativity. Our result encompasses and generalizes previous extensions of the standard geodesic deviation equation. We show how the standard as well as a generalized deviation equation can be used to measure the curvature of spacetime by means of a set of test bodies. In particular, we provide exact solutions for the curvature by using the standard deviation equation as well as its next order generalization., Comment: 16 pages, 5 figures, RevTex format

We study conservation laws for gravity theories invariant under general coordinate transformations. The class of models under consideration includes Einstein's general relativity theory as a special case as well as its generalizations to non-Riemannian spacetime geometry and nonminimal coupling. We demonstrate that an arbitrary vector field on the spacetime manifold generates a current density that is conserved under certain conditions, and find the expression of the corresponding superpotential. For a family of models including nonminimal coupling between geometry and matter, we discuss in detail the differential conservation laws and the conserved quantities defined in terms of covariant multipole moments. We show that the equations of motion for the multipole moments of extended microstructured test bodies lead to conserved quantities that are closely related to the conserved currents derived in the field-theoretic framework., Comment: 15 pages, RevTex format. arXiv admin note: text overlap with arXiv:1505.01680

We use recent measurements of the dimensionless coordinate distances from Fanaroff-Riley Type IIb radio galaxies and the X-ray gas mass fractions in clusters to constrain the parameters of a non-standard cosmological model. This work complements our recent analysis of the SN Ia data within a non-Riemannian cosmological model. We use two independent data sets to constrain the new density parameter $\Omega_\psi$, which is related to the non-Riemannian structure of the underlying spacetime and supplements the field equations that are very similar to the usual Friedmann equations of general relativity. Thereby we place an upper limit on the presence of non-Riemannian quantities in the late stages of the universe. The numerical results of this work also apply to several anisotropic cosmological models which, on the level of the field equations, exhibit a similar scaling behavior of the density parameters like our non-Riemannian model.

The motion of test bodies in gravity is tightly linked to the conservation laws. This well-known fact in the context of General Relativity is also valid for gravitational theories which go beyond Einstein's theory. Here we derive the equations of motion for test bodies for a very large class of gravitational theories with a general nonminimal coupling to matter. These equations form the basis for future systematic tests of alternative gravity theories. Our treatment is covariant and generalizes the classic Mathisson-Papapetrou-Dixon result for spinning (extended) test bodies. The equations of motion for structureless test bodies turn out to be surprisingly simple, despite the very general nature of the theories considered., 4 pages, 2 figures, to appear in Phys. Lett. A

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., Comment: 12 pages, 5 figures, RevTex format

We study general metric-affine theories of gravity in which the metric and connection are the two independent fundamental variables. In this framework, we use Lagrange-Noether methods to derive the identities and the conservation laws that correspond to the invariance of the action under general coordinate transformations. The results obtained are applied to generalized models with nonminimal coupling of matter and gravity, with a coupling function that depends arbitrarily on the covariant gravitational field variables., 9 pages, 1 figure, RevTex format. arXiv admin note: text overlap with arXiv:1303.6050