Gravitational Energy Momentum Research Articles

Energy and Momentum of Robinson-Trautman Space-Times

Robinson and Trautman space-times are studied in the context of teleparallel equivalent of general relativity (TEGR). These space-times are the simplest class of asymptotically flat geometries admitting gravitational waves. We calculate the total energy for such space-times using two methods, the gravitational energy-momentum and the translational momentum 2-form. The two methods give equal results of these calculations. We show that the value of energy depends on the gravitational mass M, the Gaussian curvature of the surfaces λ(u,θ) and on the function K(u,θ). The total energy reduces to the energies of Schwarzschild’s and Bondi’s space-times under specific forms of the function K(u,θ).

ON THE GRAVITATIONAL ENERGY–MOMENTUM VECTOR IN f(T) THEORIES

This work is devoted to present and analyze an expression for the gravitational energy–momentum vector in the context of f(T) theories through field equations. Such theories are the analogous counterpart of the well-known f(R) theories, except using torsion instead of curvature. We obtain a general expression for the gravitational energy–momentum vector in this framework. Using the hypothesis of the isotropy of spacetime, we find the gravitational energy for a closed universe, since construction of real tetrads that do not constrain the functional form of the Lagrangian density was not possible for an open universe. Thus, we find a vanishing gravitational energy for the tetrad that we have used.

Structure of the integrals of motion for a rotating body in the relativistic theory of gravitation

The energy-momentum tensor for the Kerr metric is calculated within the relativistic theory of gravitation. The external gravitational energy and angular momentum of a collapsing body are found. It is concluded from the analysis of their behavior near the gravitational radius that gravitational collapse is impossible.

Asymptotic states and the definition of the S-matrix in quantum gravity

Viewing gravitational energy–momentum as equal by observation, but different in essence from inertial energy–momentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space M4. The generalized asymptotic free scalar, Dirac and gauge fields in that theory are canonically quantized, the Fock spaces of stationary states are constructed and the gravitational limit—mapping the gravitational energy–momentum onto the inertial energy–momentum to account for their observed equality—is introduced. Next the S-matrix in quantum gravity is defined as the gravitational limit of the transition amplitudes of asymptotic in- to out-states in the gauge theory of volume-preserving diffeomorphisms. The so-defined S-matrix relates in- and out-states of observable particles carrying gravitational equal to inertial energy–momentum. Finally, generalized Lehmann–Symanzik–Zimmermann reduction formulae for scalar, Dirac and gauge fields are established which allow us to express S-matrix elements as the gravitational limit of truncated Fourier-transformed vacuum expectation values of time-ordered products of field operators of the interacting theory. Together with the generating functional of the latter established in Wiesendanger (2011 arXiv:1103.1012) any transition amplitude can in principle be computed consistently to any order in perturbative quantum gravity.

Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of (N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity (TEGR). The fundamental gravitational field variables are the (N+1)-dimensional vector fields, defined globally on a manifold M, and the gravitational field is attributed to the torsion. The form of Lagrangian density is quadratic in torsion tensor. We then give an exact five-dimensional spherically symmetric solution (Schwarzschild (4+1)-dimensions). Finally, we calculate energy and spatial momentum using gravitational energy—momentum tensor and superpotential 2-form.

Conservation of Gravitational Energy-Momentum and Inner Diffeomorphism Group Gauge Invariance

Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M4. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.

Conservation of Gravitational Energy Momentum and Renormalizable Quantum Theory of Gravitation

Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given.

Regularization of Kerr-NUT Spacetimes and Their Thermodynamical Quantities

In the context of the teleparallel equivalent of general relativity (TEGR) theory, continues calculations of the total energy and momentum for Kerr-NUT spacetimes using three different methods, the gravitational energy-momentum, the Riemannian connection 1-form, ${\widetilde{\Gamma}_\alpha}^\beta$ and the Euclidean continuation method, have been achieved. Many local Lorentz transformations, that play the role of regularizing tool, are given to get the commonly known form of energy and momentum. We calculate the thermodynamic quantities of Kerr-NUT spacetime. We also investigate the first law of thermodynamics and quantum statistical relation.

On the Regularization of Kerr-NUT Spacetime. I

Within the framework of teleparallel equivalent of general relativity (TEGR) theory, calculation of the total energy and momentum of Kerr-NUT spacetimes have been employed using two methods of the gravitational energy-momentum, which is coordinate independent, and the Riemannian connection 1-form, ${\widetilde{\Gamma}_\alpha}^\beta$. It has been shown that the two methods give the same an unacceptable result, i.e., divergent value. Therefore, a local Lorentz transformation that plays a role of a regularizing tool, which subtracts the inertial effects without distorting the true gravitational contribution, has been suggested. This transformation keeps the resulting spacetime to be a solution of the equations of motion of TEGR.

Kerr-Taub-NUT General Frame, Energy, and Momentum in Teleparallel Equivalent of General Relativity

A new exact solution describing a general stationary and axisymmetric object of the gravitational field in the framework of teleparallel equivalent of general relativity (TEGR) is derived. The solution is characterized by three parameters “the gravitational massM, the rotationa, and the NUTL.” The vierbein field is axially symmetric, and the associated metric gives the Kerr-Taub-NUT spacetime. Calculation of the total energy using two different methods, the gravitational energy momentum and the Riemannian connection 1-formΓα̃β, is carried out. It is shown that the two methods give the same results of energy and momentum. The value of energy is shown to depend on the massMand the NUT parameterL. IfLis vanishing, then the total energy reduced to the energy of Kerr black hole.

LARGE-SCALE KINEMATICS, ASTROCHEMISTRY, AND MAGNETIC FIELD STUDIES OF MASSIVE STAR-FORMING REGIONS THROUGH HC3N, HNC, AND C2H MAPPINGS

We have mapped 27 massive star-forming regions associated with water masers using three dense gas tracers: HC3N 10-9, HNC 1-0 and C2H 1-0. The FWHM sizes of HNC clumps and C2H clumps are about 1.5 and 1.6 times higher than those of HC3N, respectively, which can be explained by the fact that HC3N traces more dense gas than HNC and C2H. We found evidence for increase in optical depth of C2H with `radius' from center to outer regions in some targets, supporting the chemical model of C2H. The C2H optical depth is found to decline as molecular clouds evolve to later stage, suggesting that C2H might be used as "chemical clock" for molecular clouds. Large-scale kinematic structure of clouds was investigated with three molecular lines. All these sources show significant velocity gradients. The magnitudes of gradient are found to increase towards the inner region, indicating differential rotation of clouds. Both the ratio of rotational to gravitational energy and specific angular momentum seem to decrease toward the inner region, implying obvious angular momentum transfer, which might be caused by magnetic braking. The average magnetic field strength and number density of molecular clouds is derived using the uniformly magnetic sphere model. The derived magnetic field strengths range from 3 to 88 \mu G, with a median value of 13 \mu G. The mass-to-flux ratio of molecular cloud is calculated to be much higher than critical value with derived parameters, which agrees well with numerical simulations.

Energy and spatial momentum of charged rotating frames in tetrad gravity

We compute the total energy and the spatial momentum of four charged rotating (Kerr—Newman) frames by using the gravitational energy—momentum 3-form within the framework of the tetrad formulation of the general relativity theory. We show how the effect of the inertial always makes the total energy divergent. We use a natural regularization method, which yields the physical value for the total energy of the system. We show how the regularization method works on a number of different rotating frames that are related to each other by the local Lorentz transformation. We also show that the inertial has no effect on the spatial momentum components.

Transition from adiabatic inspiral to plunge into a spinning black hole

A test particle of mass mu on a bound geodesic of a Kerr black hole of mass M >> mu will slowly inspiral as gravitational radiation extracts energy and angular momentum from its orbit. This inspiral can be considered adiabatic when the orbital period is much shorter than the timescale on which energy is radiated, and quasi-circular when the radial velocity is much less than the azimuthal velocity. Although the inspiral always remains adiabatic provided mu << M, the quasi-circular approximation breaks down as the particle approaches the innermost stable circular orbit (ISCO). In this paper, we relax the quasi-circular approximation and solve the radial equation of motion explicitly near the ISCO. We use the requirement that the test particle's 4-velocity remain properly normalized to calculate a new contribution to the difference between its energy and angular momentum. This difference determines how a black hole's spin changes following a test-particle merger, and can be extrapolated to help predict the mass and spin of the final black hole produced in finite-mass-ratio black-hole mergers. Our new contribution is particularly important for nearly maximally spinning black holes, as it can affect whether a merger produces a naked singularity.

Nonlinear electrodynamics coupled to teleparallel theory of gravity

Using nonlinear electrodynamics coupled to teleparallel theory of gravity, regular charged spherically symmetric solutions are obtained. The nonlinear theory is reduced to the Maxwell one in the weak limit and the solutions correspond to charged spacetimes. One of the obtained solutions contains an arbitrary function which we call general solution since we can generate from it the other solutions. The metric associated with these spacetimes is the same, i.e., regular charged static spherically symmetric black hole. In calculating the energy content of the general solution using the gravitational energy—momentum within the framework of the teleparallel geometry, we find that the resulting form depends on the arbitrary function. Using the regularized expression of the gravitational energy—momentum we obtain the value of energy.

Gravitational radiation fields in teleparallel equivalent of general relativity and their energies

We derive two new retarded solutions in the teleparallel theory equivalent to general relativity (TEGR). One of these solutions gives a divergent energy. Therefore, we use the regularized expression of the gravitational energy—momentum tensor, which is a coordinate dependent. A detailed analysis of the loss of the mass of Bondi space—time is carried out using the flux of the gravitational energy—momentum.

Teleparallel equivalent theory of (1+1)-dimensional gravity

A theory of (1+1)-dimensional gravity is constructed on the basis of the teleparallel equivalent of general relativity. The fundamental field variables are the tetrad fields eiμ and the gravity is attributed to the torsion. A dilatonic spherically symmetric exact solution of the gravitational field equations characterized by two parameters M and Q is derived. The energy associated with this solution is calculated using the two-dimensional gravitational energy—momentum formula.

Energy contents of some rotating spacetimes in teleparallel gravity

The present work is devoted to elaborate energy-momentum problem using the integral form of the constraint equations in the Hamiltonian formulation of teleparallel equivalent to general relativity. This approach is used to evaluate energy and its related quantities for Lewis-Papapetrou spacetimes, spatially homogeneous rotating spacetimes and their special cases. The results of energy are compared with those already found using different prescriptions. We conclude that energy distribution for these spacetimes do not coincide in general with those already available in literature. We also evaluate gravitational and matter energy-momentum flux.

ENERGY OF SPHERICALLY SYMMETRIC SPACE–TIMES ON REGULARIZING TELEPARALLELISM

We calculate the total energy of an exact spherically symmetric solutions, i.e. Schwarzschild and Reissner–Nordström, using the gravitational energy–momentum 3-form within the tetrad formulation of general relativity. We explain how the effect of the inertial makes the total energy unphysical. Therefore, we use the covariant teleparallel approach which makes the energy always physical one. We also show that the inertial has no effect on the calculation of momentum.

Brane world black holes in teleparallel theory equivalent to general relativity and their Killing vectors, energy, momentum and angular momentum

The energy–momentum tensor, which is coordinate-independent, is used to calculate energy, momentum and angular momentum of two different tetrad fields. Although, the two tetrad fields reproduce the same space-time their energies are different. Therefore, a regularized expression of the gravitational energy–momentum tensor of the teleparallel equivalent of general relativity (TEGR), is used to make the energies of the two tetrad fields equal. The definition of the gravitational energy–momentum is used to investigate the energy within the external event horizon. The components of angular momentum associated with these space–times are calculated. In spite of using a static space–time, we get a non-zero component of angular momentum! Therefore, we derive the Killing vectors associated with these space–times using the definition of the Lie derivative of a second rank tensor in the framework of the TEGR to make the picture more clear.

ENERGY CONTENTS OF GRAVITATIONAL WAVES IN TELEPARALLEL GRAVITY

The conserved quantities, that are, gravitational energy–momentum and its relevant quantities are investigated for cylindrical and spherical gravitational waves in the framework of teleparallel equivalent of General Relativity using the Hamiltonian approach. For both cylindrical and spherical gravitational waves, we obtain definite energy and constant momentum. The constant momentum shows consistency with the results available in General Relativity and teleparallel gravity. The angular momentum for cylindrical and spherical gravitational waves also turn out to be constant. Further, we evaluate their gravitational energy–momentum fluxes and gravitational pressure.