The Einstein pseudo-tensor and the flux integral for perturbed static space-times

1991 ◽  
Vol 435 (1895) ◽  
pp. 645-657 ◽  
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Second Variation ◽  
Common Procedure ◽  
Static Vacuum ◽  
Linear Perturbations ◽  
Maxwell Space ◽  
The Common ◽  
Static Space

The flux integral for axisymmetric polar perturbations of static vacuum space-times, derived in an earlier paper directly from the relevant linearized Einstein equations, is rederived with the aid of the Einstein pseudo-tensor by a simple algorism. A similar earlier effort with the aid of the Landau–Lifshitz pseudo-tensor failed. The success with the Einstein pseudo-tensor is due to its special distinguishing feature that its second variation retains its divergence-free property provided only the equations governing the static space-time and its linear perturbations are satisfied. When one seeks the corresponding flux integral for Einstein‒Maxwell space-times, the common procedure of including, together with the pseudo-tensor, the energy‒momentum tensor of the prevailing electromagnetic field fails. But, a prescription due to R. Sorkin, of including instead a suitably defined ‘Noether operator’, succeeds.

scholarly journals Semi-Classical Einstein Equations: Descend to the Ground State

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 74
Author(s):  
Zbigniew Haba
Keyword(s):  
Ground State ◽  
Energy Momentum Tensor ◽  
Local Maximum ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Thermal Dissipation ◽  
Stationary Probability Distribution ◽  
Expectation Value ◽  
Warm Inflation ◽  
The Right

The time-dependent cosmological term arises from the energy-momentum tensor calculated in a state different from the ground state. We discuss the expectation value of the energy-momentum tensor on the right hand side of Einstein equations in various (approximate) quantum pure as well as mixed states. We apply the classical slow-roll field evolution as well as the Starobinsky and warm inflation stochastic equations in order to calculate the expectation value. We show that, in the state concentrated at the local maximum of the double-well potential, the expectation value is decreasing exponentially. We confirm the descent of the expectation value in the stochastic inflation model. We calculate the cosmological constant Λ at large time as the expectation value of the energy density with respect to the stationary probability distribution. We show that Λ ≃ γ 4 3 where γ is the thermal dissipation rate.

On the origin of black hole evaporation radiation

1976 ◽  
Vol 351 (1664) ◽  
pp. 129-139 ◽  
Keyword(s):  
Black Hole ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Static Vacuum ◽  
Black Hole Evaporation ◽  
A Value ◽  
Special Case ◽  
Collapse Process

The physical basis underlying the black hole evaporation process is clarified by a calculation of the expectation value of the energy-momentum tensor for a massless scalar field in a completely general two dimensional collapse scenario. It is found that radiation is produced inside the collapsing matter which propagates both inwards and outwards. The ingoing com­ponent eventually emerges from the star after travelling through the centre. The outgoing energy flux appears at infinity as the evaporation radiation discovered by Hawking. At late times, outside the star, the former component fades out exponentially, and the latter component approaches a value which is independent of the details of the collapse process. In the special case of a collapsing hollow, thin shell of matter, all the radiation is produced at the shell. These results are independent of regularization ambiguities, which enter only the static vacuum polariza­tion terms in the energy-momentum tensor. The significance of an earlier remark about black hole explosions is discussed in the light of these results.

scholarly journals Gravitational energy in the framework of embedding and splitting theories

2018 ◽  
Vol 27 (02) ◽  
pp. 1750188 ◽  
Author(s):  
D. A. Grad ◽  
R. V. Ilin ◽  
S. A. Paston ◽  
A. A. Sheykin
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Gravitational Energy ◽  
Einstein Equations ◽  
Field Energy ◽  
Energy Localization ◽  
Embedding Theory ◽  
Isometric Embeddings ◽  
Definition Of ◽  
Noether Current

We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge–Teitelboim approach. For the embedding theory, we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case, the independent definition of gravitational energy–momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory), we consider Noether procedure and the alternative method of energy–momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result, we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed.

scholarly journals MATTER INHERITANCE SYMMETRIES OF SPHERICALLY SYMMETRIC STATIC SPACE–TIMES

2006 ◽  
Vol 21 (12) ◽  
pp. 2645-2657 ◽  
Author(s):  
M. SHARIF
Keyword(s):  
Energy Momentum Tensor ◽  
Complete Classification ◽  
Momentum Tensor ◽  
Conformal Killing ◽  
Spherically Symmetric ◽  
Metric Tensor ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Conformal Killing Vectors ◽  
Static Space

In this paper we discuss matter inheritance collineations by giving a complete classification of spherically symmetric static space–times by their matter inheritance symmetries. It is shown that when the energy–momentum tensor is degenerate, most of the cases yield infinite dimensional matter inheriting symmetries. It is worth mentioning here that two cases provide finite dimensional matter inheriting vectors even for the degenerate case. The nondegenerate case provides finite dimensional matter inheriting symmetries. We obtain different constraints on the energy–momentum tensor in each case. It is interesting to note that if the inheriting factor vanishes, matter inheriting collineations reduce to be matter collineations already available in the literature. This idea of matter inheritance collineations turn out to be the same as homotheties and conformal Killing vectors are for the metric tensor.

scholarly journals Inverse problem: What can we tell about the matter and energy in our Universe from its dimensionality and evolution

2018 ◽  
Vol 27 (07) ◽  
pp. 1841005
Author(s):  
Hanna Makaruk ◽  
James Langenbrunner
Keyword(s):  
Energy Momentum Tensor ◽  
Physical Space ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Time Dimension ◽  
Superstring Theory ◽  
Anisotropic Energy ◽  
The One ◽  
Multidimensional Models ◽  
Additional Space

The most popular theories of everything are various versions of the superstring theory. The theories require existence of additional space dimensions, vibrations of which create the material particles in [Formula: see text] space. The additional space dimensions are understood as being currently smaller than the Planck Length and due to this not directly observable. We search for multidimensional models of the Universe (one time dimension; three isotropic, flat external dimensions, and [Formula: see text]-internal dimensions), which satisfy the multidimensional Einstein equations and which started from the same radius of all of the internal and external dimensions, with an anisotropic energy–momentum tensor. Analytical solution of [Formula: see text]-dimensional Einstein equation in a reparameterized time is reminded and discussed. The energy–momentum tensor is solely responsible for expansion of the external dimensions and shrinking of the internal ones; and to obtain this behavior of the space the tensor needs to fulfill some conditions i.e. the energy–momentum tensor cannot include only radiation, vacuum and baryonic matter. For the behavior of the physical space consistent with the one observed in our Universe, the dark energy and/or dark matter have to exist.

scholarly journals Pseudo-Complex General Relativity

2017 ◽  
Vol 45 ◽  
pp. 1760002 ◽  
Author(s):  
Peter O. Hess
Keyword(s):  
General Relativity ◽  
Dark Energy ◽  
Accretion Disks ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Minimal Length ◽  
Einstein Equations ◽  
Vacuum Fluctuations ◽  
The Right ◽  
Complex General Relativity

The present status of the pseudo-complex General Relativity is presented. The pcGR includes many known theories with a minimal length. Restricting to its simplest form, an energy-momentum tensor is added at the right hand side of the Einstein equations, representing a dark energy, related to vacuum fluctuations. We use a phenomenological ansatz for the density and discuss observable consequences: Quaisperiodic Oscillations (QPO), effects on accretion disks and gravitational waves.

scholarly journals Tractor Beams, Pressor Beams and Stressor Beams in General Relativity

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 271
Author(s):  
Jessica Santiago ◽  
Sebastian Schuster ◽  
Matt Visser
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Energy Conditions ◽  
Frequent Use ◽  
Advanced Civilization

The metrics of general relativity generally fall into two categories: those which are solutions of the Einstein equations for a given source energy-momentum tensor and the “reverse engineered” metrics—metrics bespoke for a certain purpose. Their energy-momentum tensors are then calculated by inserting these into the Einstein equations. This latter approach has found frequent use when confronted with creative input from fiction, wormholes and warp drives being the most famous examples. In this paper, we again take inspiration from fiction and see what general relativity can tell us about the possibility of a gravitationally induced tractor beam. We base our construction on warp drives and show how versatile this ansatz alone proves to be. Not only can we easily find tractor beams (attracting objects), but repulsor/pressor beams are just as attainable, and a generalization to “stressor” beams is seen to present itself quite naturally. We show that all of these metrics would violate various energy conditions. This provides an opportunity to ruminate on the meaning of energy conditions as such and what we can learn about whether an arbitrarily advanced civilization might have access to such beams.

Gravitational waves and the radiative field

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
General Solution ◽  
Gravitational Waves ◽  
Radiation Field ◽  
Gravitational Radiation ◽  
Gravitational Force ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Maxwell Theory

This chapter turns to the gravitational radiation produced by a system of massive objects. The discussion is confined to the linear approximation of general relativity, which is compared with the Maxwell theory of electromagnetism. In the first part of the chapter, the properties of gravitational waves, which are the general solution of the linearized vacuum Einstein equations, are studied. Next, it relates these waves to the energy–momentum tensor of the sources creating them. The chapter then turns to the ‘first quadrupole formula’, giving the gravitational radiation field of these sources when their motion is due to forces other than the gravitational force.

scholarly journals Classical Polymerization of the Schwarzschild Metric

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Babak Vakili
Keyword(s):  
Energy Momentum Tensor ◽  
Vacuum Solution ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Spherically Symmetric ◽  
Schwarzschild Black Hole ◽  
Schwarzschild Metric ◽  
Similarities And Differences

We study a spherically symmetric setup consisting of a Schwarzschild metric as the background geometry in the framework of classical polymerization. This process is an extension of the polymeric representation of quantum mechanics in such a way that a transformation maps classical variables to their polymeric counterpart. We show that the usual Schwarzschild metric can be extracted from a Hamiltonian function which in turn gets modifications due to the classical polymerization. Then, the polymer corrected Schwarzschild metric may be obtained by solving the polymer-Hamiltonian equations of motion. It is shown that while the conventional Schwarzschild space-time is a vacuum solution of the Einstein equations, its polymer-corrected version corresponds to an energy-momentum tensor that exhibits the features of dark energy. We also use the resulting metric to investigate some thermodynamical quantities associated with the Schwarzschild black hole, and in comparison with the standard Schwarzschild metric the similarities and differences are discussed.

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