Gravitational Energy Momentum Research Articles

The equivalence principle for a plane gravitational wave in torsion-based and non-metricity-based teleparallel equivalents of general relativity

We study the energy–momentum characteristics of the plane “+”-polarized gravitational wave solution of general relativity in the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR) using the previously constructed Noether currents. The current components describe energy–momentum locally measured by an observer if the displacement vector ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} is equal to the observer’s 4-velocity. To determine the non-dynamical connection in these theories, we use the unified “turning off” gravity principle. For a constructive analysis of the values of Noether currents and superpotentials in TEGR and STEGR, we use the concept of “gauges”. The gauge changing can affect the Noether current values. We study under what conditions the Noether current for the freely falling observer is zero. When they are established, the zero result can be interpreted as a correspondence to the equivalence principle, and it is a novelty for gravitational waves in the TEGR and STEGR. We highlight two important cases with positive and zero energy, which reproduce the results of previous works with a different approach for determining gravitational energy–momentum in the TEGR, and give their interpretation.

Tetrad Fields, Reference Frames, and the Gravitational Energy–Momentum in the Teleparallel Equivalent of General Relativity

AbstractThe concept and definitions of the energy–momentum and angular momentum of the gravitational field in the teleparallel equivalent of general relativity (TEGR) are reviewed. The importance of these definitions is justified by three major reasons. First, the TEGR is a well established and widely accepted formulation of the gravitational field, whose basic field strength is the torsion tensor of the Weitzenböck connection. Second, in the phase space of the TEGR there exists an algebra of the Poincaré group. Not only the definitions of the gravitational energy–momentum and 4‐angular momentum satisfy this algebra, but also the first class constraints related to these definitions satisfy the algebra. And third, numerous applications of these definitions lead to physically consistent results. These definitions follow from a well established Hamiltonian formulation, and rely on the idea of localization of the gravitational energy. In this review, the concept of localizability of the gravitational energy is revisited, in light of results obtained in recent years. The behavior of free particles is studied in the space–time of plane fronted gravitational waves (pp‐waves). Free particles are here understood as particles that are not subject to external forces other than the gravitational acceleration due to pp‐waves. Since these particles acquire or loose kinetic energy locally, the transfer of energy from or to the gravitational field must also be localized. This theoretical result is considered an important and definite argument in favor of the localization of the gravitational energy–momentum, and by extension, of the gravitational 4‐angular momentum.

Separation of Variables Method in Solving Partial Differential Equations and Investigating the Relationship between Gravitational Field Tensor and Energy-Momentum Tensor in Einstein's Theory of Gravity

This research delves into the study of partial differential equations (PDEs) and gravitational fields in spacetime. It focuses on solving PDEs using the Separation of Variables method and explores the relationship between the gravitational field tensor and the energy-momentum tensor, leading to the final equation for the gravitational field tensor. The research also investigates Einstein's theory of gravity and the energy-momentum tensor integral, providing the general solution for the gravitational potential and its implications. Additionally, the mean integration of the gravitational wave tensor is analyzed, yielding an expression for the tensor strain of gravitational waves over an infinitely long period. The components of the gravitational wave tensor and their effect on gravitational sources are examined. Furthermore, the propagation of electromagnetic fields in spacetime is studied using the Retarded Green's Function. The primary objectives of this research are to understand and explore mathematical techniques for solving PDEs and analyzing gravitational fields and their interactions in spacetime. The integration of multiple theoretical concepts related to PDEs, gravitational fields, and electromagnetic fields enhances our understanding of fundamental physics principles. This contributes to the advancement of theoretical physics and opens avenues for potential practical applications, such as gravitational wave detection and electromagnetic field propagation in complex media. In conclusion, this research provides valuable insights into fundamental physics principles and fosters a deeper understanding of their interconnections and implications

The energy–momentum complex in non-local gravity

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator [Formula: see text] are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications.

Modified general relativity and dark matter

Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field [Formula: see text], which exists in all Lorentzian spacetimes, to construct a connection-independent symmetric tensor that represents the energy–momentum of the gravitational field. It solves the problem of the nonlocalization of gravitational energy–momentum in GR, preserves the ontology of the Einstein equation, and maintains the equivalence principle. The line element field provides MGR with the extra freedom required to describe dark energy and dark matter. An extended Schwarzschild solution for the matter-free Einstein equation of MGR is developed, from which the Tully–Fisher relation is derived, and the gravitational energy density is calculated. The mass of the invisible matter halo of galaxy NGC 3198 calculated with MGR is identical to the result obtained from GR using a dark matter profile. Although dark matter in MGR is described geometrically, it has an equivalent representation as a particle with the property of a vector boson or a pair of fermions; the geometry of spacetime and the quantum nature of matter are linked together by the unit line element covectors that belong to both the Lorentzian metric and the spin-1 Klein–Gordon wave equation. The three classic tests of GR provide a comparison of the theories in the solar system and several parts of the cosmos. MGR provides the flexibility to describe inflation after the Big Bang and galactic anisotropies.

Eccentric binary black holes: Comparing numerical relativity and small mass-ratio perturbation theory

The modeling of unequal mass binary black hole systems is of high importance to detect and estimate parameters from these systems. Numerical relativity (NR) is well suited to study systems with comparable component masses, ${m}_{1}\ensuremath{\sim}{m}_{2}$, whereas small mass ratio (SMR) perturbation theory applies to binaries where $q={m}_{2}/{m}_{1}\ensuremath{\ll}1$. This work investigates the applicability for NR and SMR as a function of mass ratio for eccentric nonspinning binary black holes. We produce 52 NR simulations with mass ratios between $1:10$ and $1:1$ and initial eccentricities up to 0.8. From these we extract quantities like gravitational wave energy and angular momentum fluxes and periastron advance, and assess their accuracy. To facilitate comparison, we develop tools to map between NR and SMR inspiral evolutions of eccentric binary black holes. We derive post-Newtonian accurate relations between different definitions of eccentricity. Based on these analyses, we introduce a new definition of eccentricity based on the (2,2)-mode of the gravitational radiation, which reduces to the Newtonian definition of eccentricity in the Newtonian limit. From the comparison between NR simulations and SMR results, we quantify the unknown next-to-leading order SMR contributions to the gravitational energy and angular momentum fluxes, and periastron advance. We show that in the comparable mass regime these contributions are subdominant and higher order SMR contributions are negligible.

On the Black Hole Acceleration in the C-Metric Space-Time

We consider the C-metric as a gravitational field configuration that describes an accelerating black hole in the presence of a semi-infinite cosmic string, along the accelerating direction. We adopt the expression for the gravitational energy-momentum developed in the teleparallel equivalent of general relativity (TEGR) and obtain a possible explanation for the acceleration of the black hole. The gravitational energy enclosed by surfaces of constant radius around the black hole is evaluated, and in particular the energy contained within the gravitational horizon is obtained. This energy turns out to be proportional to the square root of the area of the horizon. We find that the gravitational energy of the semi-infinite cosmic string is negative and dominant for large values of the radius of integration. This negative energy may explain the acceleration of the black hole, that moves towards regions of lower gravitational energy along the string.

Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity

An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy–momentum complex in an nth order gravitational Lagrangian L=Lgμν,gμν,i1,gμν,i1i2,gμν,i1i2i3,⋯,gμν,i1i2i3⋯in and then in a gravitational Lagrangian as Lg=(R¯+a0R2+∑k=1pakR□kR)−g. Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether’s theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor ταη becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy–momentum complex in fR gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor ταη to lowest order in the metric perturbation h was calculated. As a practical application, the power per unit solid angle Ω emitted by a localized source carried by a gravitational wave in a direction x^ for a fixed wave number k under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann–Lemaître–Robertson–Walker spacetime, the gravitational and matter energy density in f(R) gravity both in Palatini and metric formalism was proposed. The gravitational energy–momentum pseudo-tensor could be a useful tool to investigate further modes of gravitational radiation beyond two standard modes required by General Relativity and to deal with non-local theories of gravity involving □−k terms.

Generalization of the ADM gravitational energy momentum

In this paper, it is proved that the teleparallel energy momentum generalizes that of the Arnowitt-Deser-Misner (ADM) formalism. In doing so, it is shown that the teleparallel 4-momentum can be made to coincide with that of the ADM approach whenever the ADM 4-momentum is applicable. The only assumptions are the time gauge for the teleparallel frame and the well-known restrictions for the coordinate system used in the calculation of the ADM 4-momentum. Then, examples where the ADM formalism fails to give consist results, but the teleparallel approach does not, are given. The advantages of the teleparallel stress-energy tensor (density) over the pseudotensor of Landau-Lifshitz are exhibited. Finally, the difficulties in identifying the gravitational angular-momentum density is discussed; it is shown that the spatial part of the proposed angular-momentum density ${M}^{ab}$ vanishes when the teleparallel frame satisfies the time-gauge condition.

On the energy of gravitational waves

The energy of gravitational waves is a fundamental problem in gravity theory. The existing descriptions for the energy of gravitational waves, such as the well-known Isaacson energy-momentum tensor, suffer from several defects. Due to the equivalence principle, the gravitational energy-momentum can only be defined quasilocally, being associated with a closed spacelike 2-surface bounding a region. We propose a new approach to derive the energy of gravitational waves directly from the quasilocal gravitational energy. Such an approach is natural and consistent with the quasilocality of gravitational energy-momentum.

Energy contents of stringy charged black hole in general relativity and teleparallel gravity

In the framework of general relativity (GR) and teleparallel equivalent to GR (TEGR), we investigate the energy–momentum localization of a stringy charged black hole by using Bergman–Thompson’s prescription and the Papapetrou’s prescription. Moreover, we investigate the gravitational energy–momentum (GEM) in TEGR. Our results are consistent with the results of previous studies. Finally, we calculate the GEM flux and gravitational center of mass in TEGR.

Third order post-Newtonian gravitational radiation from two-body scattering: Instantaneous energy and angular momentum radiation

We compute the third post-Newtonian (3PN) accurate instantaneous contributions to the radiated gravitational wave energy and angular momentum arising from the hyperbolic passages of nonspinning compact objects. The present computations employ 3PN-accurate instantaneous contributions to the far-zone energy and angular momentum fluxes and the 3PN-accurate Keplerian type parametric solution for compact binaries in hyperbolic orbits.

Gravitational energy–momentum pseudo-tensor in Palatini and metric [formula omitted] gravity

We derive the gravitational energy–momentum pseudo-tensor τμνμ in both Palatini and metric approaches to f(R) gravity. We then obtain the related cosmological gravitational energy density. Considering a flat Friedmann–Lemaître–Robertson–Walker spacetime, the energy density complex of matter and gravitation vanishes in the metric approach, but results non-vanishing in the Palatini formalism. This feature could be relevant in order to physically discriminate between the two approaches.

Noether charges in the geometrical trinity of gravity

The Noether currents are derived in a generic metric-affine theory of gravity, and the holographic nature of the gravitational entropy and energy-momentum is clarified. The main result is the verification of the canonical resolution to the energy-momentum problem in the Noether formalism.

Electromagnetic and gravitational radiation in all dimensions: A classical field theory treatment

How long does a light bulb shine in odd dimensional flat spacetimes, according to a distant observer? This question is non-trivial because electromagnetic and gravitational waves, despite being comprised of massless particles, can develop tails: they travel inside the light cone. To this end, I attempt to close a gap in the literature by first deriving, strictly within classical field theory, the real-time electromagnetic dipole and gravitational quadrupole energy and angular momentum radiation formulas in all relevant dimensions. The even dimensional case, where massless signals travel strictly on the null cone, depends on the time derivatives of the dipoles and quadrupoles solely at retarded time; whereas the odd dimensional ones involve an integral over their retarded histories. Despite the propagation of light inside the null cone, however, I argue that a monochromatic light bulb of some intrinsic duration in odd dimensions remains approximately the same apparent duration to a distant detector, though the tail effect does produce a phase shift and adds to the signal several transitory non-oscillatory inverse square roots in time.

Effective one-body model for extreme-mass-ratio spinning binaries on eccentric equatorial orbits: Testing radiation reaction and waveform

We provide a systematic analysis of the multipolar gravitational waveform, energy and angular momentum fluxes emitted by a nonspinning test particle orbiting a Kerr black hole along equatorial, eccentric orbits. These quantities are computed by numerically solving the Teukolsky equation in the time domain and are then used to test and improve the radiation reaction (and waveform) of an effective-one-body (EOB) model. Eccentricity is incorporated into EOB by replacing the quasi-circular Newtonian (or leading-order) prefactors in the EOB-factorized multipolar waveform (and fluxes) with their generic counterparts. The comparison between numerical and analytical quantities is carried out over a large portion of the parameter space, notably for orbits close to the separatrix and with high eccentricities. The analytical model agrees to $\sim 1\%$ with the numerical data for orbits with moderate eccentricities ($e\lesssim 0.3$) and moderate spins ($\hat{a}\lesssim 0.5$), although this increases up to $\sim 33\%$ for large, positive, black hole spins ($\sim 0.9$) and large eccentricities ($\sim 0.9$). For moderate eccentricities, the new EOB fluxes can be used to drive the dynamics through the nonadiabatic transition from eccentric inspiral to plunge, merger and ringdown, thus providing accurate an description of the merger. We also show that the same technique can be applied to hyperbolic encounters. In general, our approach to radiation reaction for eccentric inspirals should be seen as a first step toward EOB modelization of extreme-mass-ratio-inspirals waveforms for LISA.

The gravitational energy density of the Universe

The teleparallel gravitational energy–momentum tensor density of the Friedmann–Lemaître–Robertson–Walker spacetime is calculated and analyzed: it is decomposed into density, isotropic pressure, non-isotropic pressures, and the heat-flux 4-vector; the antisymmetric part is decomposed into “electric” and “magnetic” components. It is found that the gravitational field obeys a radiation-like equation of state, the antisymmetric part does not contribute to the gravitational energy–momentum; and the total energy density, the non-isotropic pressures and the heat-flux 4-vector vanish for spatially flat universes. Finally, it is verified that the field equations have a well-defined vacuum.

General Relativity, Mental Causation, and Energy Conservation

The conservation of energy and momentum have been viewed as undermining Cartesian mental causation since the 1690s. Modern discussions of the topic tend to use mid-nineteenth century physics, neglecting both locality and Noether’s theorem and its converse. The relevance of General Relativity (GR) has rarely been considered. But a few authors have proposed that the non-localizability of gravitational energy and consequent lack of physically meaningful local conservation laws answers the conservation objection to mental causation: conservation already fails in GR, so there is nothing for minds to violate. This paper is motivated by two ideas. First, one might take seriously the fact that GR formally has an infinity of rigid symmetries of the action and hence, by Noether’s first theorem, an infinity of conserved energies-momenta (thus answering Schrödinger’s 1918 false-negative objection). Second, Sean Carroll has asked (rhetorically) how one should modify the Dirac–Maxwell–Einstein equations to describe mental causation. This paper uses the generalized Bianchi identities to show that General Relativity tends to exclude, not facilitate, such Cartesian mental causation. In the simplest case, Cartesian mental influence must be spatio-temporally constant, and hence 0. The difficulty may diminish for more complicated models. Its persuasiveness is also affected by larger world-view considerations. The new general relativistic objection provides some support for realism about gravitational energy-momentum in GR (taking pseudotensor laws seriously). Such realism also might help to answer an objection to theories of causation involving conserved quantities, because energies-momenta would be conserved even in GR.

Difficulties of teleparallel theories of gravity with local Lorentz symmetry

A brief discussion is made about the relevance of surface terms in the Lagrangian and Hamiltonian formulations of theories of gravity. These surface terms play an important role in the variation of the action integral and in the definition of field quantities such as the gravitational energy-momentum. Then we point out several inconsistencies of a recently proposed formulation of teleparallel theories of gravity with local Lorentz symmetry.

The Gravitational Energy–Momentum Density of Radially Accelerated Observers in Schwarzschild Spacetime

AbstractA hybrid machinery that is useful for calculations in teleparallel theories when the spacetime is spherically symmetric is developed. Using this machinery, the gravitational energy–momentum tensor density of the Schwarzschild spacetime is evaluated in a frame adapted to observers that accelerate in the radial direction. The energy density, the total energy, and the gravitational energy‐momentum flux are obtained. The regularization procedure and the limit where gravity is absent is discussed. It turns out that the regularized energy and energy–momentum flux are consistent in the whole spacetime. The continuity equation for the gravitational energy–momentum also holds for any point outside the black hole. Finally, the static and freely falling cases are discussed. It is found that a static observer measures a negative gravitational energy density, while a freely falling one measures a vanishing density.