Einstein Energy-momentum Complex Research Articles

KINEMATICS RENORMALIZATION OF ENERGY IN THE THEORY OF GRAVITY

In this paper, we study the change in the energy-momentum of a gravitational field when the general reference frame is changed. From a mathematical point of view, the general reference frame is described by the affine reference field on the space-time manifold, and the transitions between them form the gauge general linear group . Under the transformations of the group the Lagrangian of the theory, obtains a divergent term: its canonical transformation takes place. Consequently, the gravitational equations do not change, however, the total energy of the system and its superpotential are changes. This phenomenon is called the kinematic renormalization of energy and can be used to deprive the theory of singularities. The paper also examines individual cases of the selection of reference frames, in particular the so-called holonomic reference frames used in the general theory of relativity, as well as non-holonomic pseudoorthonormal reference frames. In the case of holonomic reference frames, it is possible to associate with them coordinate systems whose coordinate rays will set the reference frames. In contrast to the Euler coordinates , which number the points of space-time and whose change does not lead to a change in any physical quantities, the Lagrangian coordinates carry the physical content, since their change leads to a change in the physical reference frame, as a result of which certain physical quantities change, such as the energy-momentum of the system, and so on. In the case of coordinate transformations there are gauge translations in space-time in a fixed general reference frame belonging to the group . In the case of Lagrange coordinates transformations the transformations of holonomic reference frames occur, and such transformations belong to another group, namely the group , forming its subgroup, which is isomorphic to the group of diffeomorphisms due to restriction by holonomic reference frames. In the case of the identification of the Lagrangian coordinates with the Eulerian , the energy-momentum tensor becomes Einstein energy-momentum complex of gravitational field, and its superpotential becomes Freud superpotential. Obtained results open up wide opportunities for depriving the theory of gravity from singularities by choosing the appropriate general reference frames.

Einstein Energy-Momentum Complex for a Phantom Black Hole Metric

We calculate the energy distribution associated with a static spherically symmetric non-singular phantom black hole metric in Einstein's prescription in general relativity. As required for the Einstein energy-momentum complex, we perform the calculations in quasi-Cartesian coordinates. We also calculate the momentum components and obtain a zero value, as expected from the geometry of the metric.

Testing $$R_h=ct$$ R h = c t cosmology from fundamental considerations: Is the Friedmann universe intrinsically flat

Recently Melia and Shevchuk (Mon Not R Astron Soc 419:2579,2012) (MS) have proposed the so-called \(R_h =ct\) cosmology where the “Gravitational Horizon” of the universe \(R_h\) is equal to the distance travelled by light since “Big Bang”. Here we would like to see whether the basic claim \(R_h=ct\) is correct or not because MS have not given any cogent derivation for the same. Essentially we will compare the twin expressions for the Einstein energy momentum complex (EMC) of the Friedmann universe obtained by using an appropriate superpotential and also by a direct method. To enable a meaningful comparison of the twin expressions, both are computed by using the same quasi-Cartesian coordinates. We however do not claim that Einstein EMC is superior to many other routes of defining EM of a self-gravitating system. In fact, for static isolated spherical syatems, the idea of a coordinate independent field energy of Lynden-Bell and Katz (Mon Not R Astron Soc 213:21, 1985) might be quite physically significant. Yet, here, we use Einstein EMC because (i) our system is non-static and not isolated one (ii) our primary aim is not find any absolute value of EM, and, finally, (iii) only Einstein pseudo-tensor offers equivalent twin expressions for EM which one can be equated irrespective of any physical significance. Following such comparison of equivalent twin expressions of Einstein energy, we find an exact proof as to why Friedmann universe must be spatially flat even though, mathematically one can conceive of curved spaces in any dimension. Additionally, it follows that, apparently, the scale factor \(S(t) \propto t\) as insisted by \(R_h=ct\) proposition. Nonetheless, because of close similarity of this form, \(S(t) \propto t\), with the (vacuum) Milne metric, and also because of implied unphysical equation of state, \(R_h=ct\) cosmology is unlikely to represent the physical universe.

Quasi-Local Energy Distribution and Thermodynamics of Reissner—Nordström Black Hole Surrounded by Quintessence

We investigate quasi-local energy distribution and thermodynamics of the Reissner—Nordström black hole space-time surrounded by quintessence. We use the quasi-local energy distribution from Einstein energy-momentum complex. We plot the variation of the energies, temperature and heat capacity with the state parameter related to the quintessence ωq. We show that due to the presence of quintessence, the total energy of the outer region as well as the temperature and heat capacity decreases with the increase of the density of quintessence, while the total energy of the black hole region increases.

ON ENERGY DISTRIBUTION OF TWO SPACE–TIMES WITH PLANAR AND CYLINDRICAL SYMMETRIES

Considering encouraging Virbhadra results about energy distribution of nonstatic spherically symmetric metrics in the Kerr–Schild class, it would be interesting to study some space–times with other symmetries. Using Møller and Einstein energy–momentum complexes in static plane-symmetric and cylindrically symmetric solutions to Einstein–Maxwell equations in 3+1 dimensions, energy (due to matter and fields including gravity) distribution is studied. Energy expressions are obtained finite and well-defined. Our results support the Cooperstock hypothesis about localized energy.

Quasi-Local Energy Distribution of the Modified Reissner–Nordström Black Hole

We discuss the energy distribution of the modified Reissner–Nordström black hole. It is suggested that the quasi-local energy distribution from Einstein energy-momentum complex should be appropriate. For the Reissner–Nordström spacetime, it can be seen that the energies of the outer and inner regions are exactly equivalent.

Energy of the universe in Bianchi-type I models in Møller’s tetrad theory of gravity

In this paper, using the energy definition in Moller's tetrad theory of grav- ity we calculate the total energy of the universe in Bianchi-type I cosmological models which includes both the matter and gravitational fields. The total energy is found to be zero and this result agrees with a previous works of Banerjee-Sen who investigated this problem using the general relativity version of the Einstein energy-momentum complex and Xulu who investigated same problem using the general relativity versions of the Landau-lifshitz, Papapetrou and Weinberg's energy-momentum complexes. The result that total energy of the universe in Bianchi-type I universes is zero supports the viewpoint of Tryon.

Bergmann–Thomson Energy-Momentum Complex for Solutions More General than the Kerr–Schild Class

In a very well-known paper, Virbhadra’s research group proved that the Weinberg, Papapetrou, Landau and Lifhsitz, and Einstein energy-momentum complexes “coincide” for all metrics of Kerr–Schild class. A few years later, Virbhadra clarified that this “coincidence” in fact holds for metrics more general than the Kerr–Schild class. In the present paper, this study is extended for the Bergmann–Thomson complex and it is proved that this complex also “coincides” with those complexes for a more general than the Kerr–Schild class metric.

Energy distribution of a stringy charged black hole

The energy distribution associated with a stringy charged black hole is studied using M{\o}ller's energy-momentum complex. Our result is reasonable and it differs from that known in literature using Einstein's energy-momentum complex.

Energy-momentum and angular momentum of Gödel universes

We discuss the Einstein energy-momentum complex and the Bergmann-Thomson angular momentum complex in general relativity and calculate them for space-time homogeneous Goedel universes. The calculations are performed for a dust acausal model and for a scalar-field causal model. It is shown that the Einstein pseudotensor is traceless, not symmetric, the gravitational energy is "density" is negative and the gravitational Poynting vector vanishes. Significantly, the total (gravitational and matter) energy "density" fro the acausal model is zero while for the casual model it is negative.The Bergmann-Thomson angular momentum complex does not vanish for both G\"odel models.

ENERGY DISTRIBUTION IN 2D STRINGY BLACK HOLE BACKGROUNDS

We utilize Møller's and Einstein's energy–momentum complexes in order to explicitly evaluate the energy distributions associated with the two-dimensional "Schwarzschild" and "Reissner–Nordström" black hole backgrounds. While Møller's prescription provides meaningful physical results, Einstein's prescription fails to do so in the aforementioned gravitational backgrounds. These results hold for all two-dimensional static black hole geometries. The results obtained within this context are exploited in order that Seifert's hypothesis would be investigated.

Total Energy of the Bianchi Type I Universes

Using the symmetric energy-momentum complexes of Landau and Lifshitz,Papapetrou, and Weinberg, we obtain the energy of the universe in anisotropicBianchi type I cosmological models. The energy (due to matter plus field) isfound to be zero and this agrees with a previous result of Banerjee and Sen, whoinvestigated this problem using the Einstein energy-momentum complex. Ourresult supports the importance of the energy-momentum complexes andcontradicts the prevailing folklore that different energy-momentum complexescould give different and hence unacceptable energy distribution in a givenspace-time. The result that the total energy of the universe in these models is zerosupports the viewpoint of Tryon. Rosen computed the total energy of the closedhomogeneous isotropic universe and found it to be zero, which agrees with thestudies of Tryon.

Hamiltonian dynamics of purely affine fields (EINSTEIN-SCHRÖDINGER Theory)

The Lagrangian of the general-relativistic affine field theory of the non-symmetric connection field Γikl is Schrödinger scalar density \documentclass{article}\pagestyle{empty}\begin{document}$ H = \frac{2}{\lambda }\sqrt { - \det [R} _{ik} ] $\end{document}, and the field variables (canonical coordinates) are Einstein's affine tensors Umnl = Γlmn - δnlΓrmn. The field equations are the Einstein-Schrödinger equations The minors give by definiton gmn = λ−1 Rmn, and λ becomes the cosmological constant. The Hamiltonian density is the V00-component of the Einstein energy-momentum complex and the tensor-density components are the canonically conjugated momentum densities of the field coordinates Ulmn. The canonical equations are , and we have no constraints. The affine field theory is invariant with respect to all transformations which preserve the Levi-Civita parallelism (Einstein's unified T-A group), and the field equations possess transposition invariance: With Ũlmn = Ulnm we get R̃mn = Rnm, g̃mn = gnm, and Ñmnl = Unml. The symmetry conditions Γimn = Γinm reduce the space to the general-relativistic Einstein spaces with Rik = Rki. The equation Rik = λgik yields Γikl = {ikl}, and the pathes of test particles define geodesic world lines of the Einstein spaces.

The equation of motion and gravitational radiation of an ideal fluid sphere moving in a gravitational background

On the basis of an approximation method developed in a previous paper the motion of an ideal fluid sphere in a weak gravitational background\(\begin{array}{*{20}c} {(in)} & {(in)} \\ {g_{\mu v} = \eta _{\mu v} + \gamma _{\mu v} } & {} \\ \end{array} \) is investigated. The sphere is assumed to be small in the sense that its radius is small compared with the change of the background\(\begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu v} } \\ \end{array} \). Furthermore the deformations of the sphere when accelerated by the background are assumed to be small compared with the extension of the sphere in the absence of acceleration. In the lowest mixed order (mixed of the background\(\begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu v} } \\ \end{array} \) and the retarded potentials of the sphere in lowest order) the equation of motion is yielded by integrating Einstein's conservation law of energy and momentum over the world-tube of the sphere. One obtains an equation of motion for the center of the sphere that is identical with the geodesic line linearized in\(\begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu v} } \\ \end{array} \). In the case of a static background of a localized matter distribution it is shown that Einstein's energy-momentum complex formed with the retarded potentials from the accelerated motion of the sphere in lowest order (lowest mixed order) leads to an outgoing radiation of gravitational energy. All radiation terms can be expressed in terms of the background and the world-line of the center of the sphere.