Einstein Tensor Research Articles

An Isotropic Metric

An isotropic metric for a black hole and a better vacuum condition \nabla^2 V_G =0 are presented which yield distinct terms for the energy densities of ordinary matter and gravitational fields in the Einstein tensor (G^44 =-g^2 (2\nabla^2 V_G +(\nabla V_G)^2)) This model resolves an inconsistency between electromagnetism and gravity in the calculation of field energy. Resolution of this inconsistency suggests a slight modification of the Einstein equation to gG^\mu\nu = 8\pi G T^\mu\nu.

General relativity and collineations

In order to get information from the field equations in General Relativity one makes simplifying assumptions which restrict spacetime metric. These assumptions can be geometric and physical. Collineations are the geometric assumptions which concern the ‘symmetries’ of the metric defined by the Lie symmetries of either the metric itself or the tensors defined form the metric. Because the metric fixes the Einstein tensor, and through Einstein field equations, the energy momentum tensor Tab, the effects of collineations extent to all parts of General Relativity. The purpose of this article is to give a brief overview of certain effects of collineations on the Geometry, the Kinematics and the Physics of General Relativity.

The strong equivalence principle from a gravitational gauge structure

Gravitational self-interactions are assumed to be determined by the covariant derivative acting on the Riemann–Christoffel field strength. Once imposed on a metric theory, this Yang–Mills gauge constraint extends the equality of gravitational mass and inertial mass to compact bodies with non-negligible gravitational binding energy. Applied to generalized Brans–Dicke theories, it singles out Einstein's tensor theory and Nordström scalar theory for gravity but also suggests a way to implement a minimal violation of the strong equivalence principle.

Gravitational amplitudes in black hole evaporation: the effect of non-commutative geometry

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in non-commutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy–momentum tensor as a source on the right-hand side. The present paper, relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, considers from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes is shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited. Approximate formulae for the particle emission rate are also obtained within this framework.

Eigenvalue Monotonicity for the Ricci-Hamilton Flow

In this short note, we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow.

The gravitational field of a point charge and finiteness of self-energy

Singularities in the metric of classical solutions of the Einstein equations (the Schwarzschild, Kerr, Reissner-Nordstroem, and Kerr-Newman solutions) give rise to generalized functions in the Einstein tensor. A technique based on the limiting sequence of solutions is used to analyze these functions, which can have a more complex behavior than the Dirac δ function. We show that the solutions will satisfy the Einstein equations everywhere if the energy-momentum tensor has an appropriate singular addition of nonelectromagnetic origin. When this addition term is included, the total energy turns out to be finite and equal to mc2, while the angular momentum for the Kerr and Kerr-Newman solutions is mca. Since the Reissner-Nordstroem and Kerr-Newman solutions correspond to a point charge in classical electrodynamics, the result allows us to take a fresh look at the divergence of the self-energy of a point charge.

Symmetries at stationary Killing horizons

It has often been suggested (especially by S. Carlip) that spacetime symmetries in the neighborhood of a black hole horizon may be relevant to a statistical understanding of the Bekenstein–Hawking entropy. A prime candidate for this type of symmetry is that one which is exhibited by the Einstein tensor. More precisely, it is now known that this tensor takes on a strongly constrained (block-diagonal) form as it approaches any stationary, non-extremal Killing horizon. Presently, exploiting the geometrical properties of such horizons, we provide a particularly elegant argument that substantiates this highly symmetric form for the Einstein tensor. It is, however, duly noted that, on account of a “loophole,” the argument does fall just short of attaining the status of a rigorous proof.

Curvature tensors on distorted Killing horizons and their algebraic classification

We consider generic static spacetimes with Killing horizons and study properties of curvature tensors in the horizon limit. It is determined that the Weyl, Ricci, Riemann and Einstein tensors are algebraically special and mutually aligned on the horizon. It is also pointed out that results obtained in the tetrad adjusted to a static observer in general differ from those obtained in a free-falling frame. This is connected to the fact that a static observer becomes null on the horizon. It is also shown that finiteness of the Kretschmann scalar on the horizon is compatible with the divergence of the Weyl component Ψ3 or Ψ4 in the freely falling frame. Furthermore, finiteness of these components is compatible with divergence of curvature invariants constructed from second derivatives of the Riemann tensor. We call the objects with finite Kretschmann scalar but infinite Ψ4 or Ψ3 ‘truly naked black holes’. In the (ultra)extremal versions of these objects the structure of the Einstein tensor on the horizon changes due to extra terms as compared to the usual horizons, the null energy condition being violated at some portions of the horizon surface. The demand to rule out such divergences leads to the constancy of the factor that governs the leading term in the asymptotics of the lapse function and in this sense represents a formal analogue of the zeroth law of mechanics of non-extremal black holes. In doing so, all extra terms in the Einstein tensor automatically vanish.

Curvature tensors of singular spaces

A sequence of tensor-valued measures of certain singular spaces (e.g., subanalytic or convex sets) is constructed. The first three terms can be interpreted as scalar curvature, Einstein tensor and (modified) Riemann tensor. It is shown that these measures are independent of the ambient space, i.e., they are intrinsic. In contrast to this, there exists no intrinsic tensor-valued measure corresponding to the Ricci tensor.

Matching conditions for a brane of arbitrary codimension

We present matching conditions for distributional sources of arbitrary codimension in the context of Lovelock gravity. Then we give examples, treating maximally symmetric distributional p-branes, embedded in flat, de Sitter and anti-de Sitter spacetime. Unlike Einstein theory, distributional defects of locally smooth geometry and codimension greater than 2 are demonstrated to exist in Lovelock theories. The form of the matching conditions depends on the parity of the brane codimension. For odd codimension, the matching conditions involve discontinuities of Chern-Simons forms and are thus similar to junction conditions for hypersurfaces. For even codimension, the bulk Lovelock densities induce intrinsic Lovelock densities on the brane. In particular, this results in the appearance of the induced Einstein tensor for p>2. For the matching conditions we present, the effect of the bulk is reduced to an overall topological solid angle defect which sets the Planck scale on the brane and to extrinsic curvature terms. Moreover, for topological matching conditions and constant solid angle deficit, we find that the equations of motion are obtained from an exact p+1 dimensional action, which reduces to an induced Lovelock theory for large codimension. In essence, this signifies that the distributional part of the Lovelock bulk equations can naturally give rise to induced gravity terms on a brane of even co-dimension. We relate our findings to recent results on codimension 2 branes.

Einstein Equations and Hilbert Action: What is missing on page 8 of the proofs for Hilbert's First Communication on the Foundations of Physics?

The history of the publication of the gravitational field equations of general relativity in November 1915 by Einstein and Hilbert is briefly reviewed. An analysis of the internal structure and logic of Hilbert's theory as expounded in extant proofs and in the published version of his relevant paper is given with respect to the specific question what information would have been found on a missing piece of Hilbert's proofs. The existing texts suggest that the missing piece contained the explicit form of the Riemann curvature scalar in terms of the Ricci tensor as a specification of the axiomatically underdetermined Lagrangian in Hilbert's action integral. An alternative reading that the missing piece of the proofs already may have contained the Einstein tensor, i.e. an explicit calculation of the gravitational part of Hilbert's Lagrangian is argued to be highly implausible.

Quantum physical relevance of the Einstein tensor

Taking quantum physics as well as large scale astronomical observations into account, a spacetime metric is introduced, such that the nonlinear part of the Einstein tensor contains effects of order $\hbar$.

Double forms, curvature structures and the $(p,q)$-curvatures

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

POINT CHARGE SELF-ENERGY IN THE GENERAL RELATIVITY

Singularities in the metric of the classical solutions to the Einstein equations (Schwarzschild, Kerr, Reissner – Nordström and Kerr – Newman solutions) lead to appearance of generalized functions in the Einstein tensor that are not usually taken into consideration. The generalized functions can be of a more complex nature than the Dirac δ-function. To study them, a technique has been used based on a limiting solution sequence. The solutions are shown to satisfy the Einstein equations everywhere, if the energy-momentum tensor has a relevant singular addition of non-electromagnetic origin. When the addition is included, the total energy proves finite and equal to mc2, while for the Kerr and Kerr–Newman solutions the angular momentum is mca. As the Reissner–Nordström and Kerr–Newman solutions correspond to the point charge in the classical electrodynamics, the result obtained allows us to view the point charge self-energy divergence problem in a new fashion.

Nonanalytic metric in the presence of gravitomagnetic monopoles

The nonanalytic property of metric resulting from the presence of gravitomagnetic monopoles is considered. The curvature tensors, dual curvature tensors, dual Einstein tensor (and hence the gravitational field equation of gravitomagnetic matter) expressed in terms of nonanalytic metric are analyzed. It is shown that the spinor gravitomagnetic monopole may be one of the potential origins of the cosmological constant. An alternative approach to the cosmological constant problem is thus proposed based on the concept of gravitomagnetic monopole.

The supersymmetric components of the Riemann–Einstein tensor as nine dimensional spheres in ten dimensional space

In eight dimensional superspace, the number of independent components of the Riemann–Einstein tensor is R(8) = 336. The paper shows that these components could be given a geometrical interpretation as a hyperbolic tesselation via the (k = 11 − 4 = 7) modular group Γ(7). In addition the components may be viewed as 336 particles-like states resembling 9-dimensional spheres in 10 dimensional space. Finally a relation to the 528 killing vector fields in 32 dimensional super and maximally symmetric space related to 11 dimensional P-Branes with 528 = (336)(11/7) states is established.

Determining the number of Higgs particles starting from general relativity and various other field theories

The Riemann–Einstein tensor of general relativity as well as chiral and super-symmetry are utilized to develop various extended versions of the standard model of high energy physics. Based on these models, it is possible to predict that few new elementary particles conjectured to be the Higgs are likely to be found experimentally at an energy scale which is just above that of the electroweak. Connections to the massless states of different super-string theories as well as super-gravity are also discussed.

The supersymmetric components of the Riemann–Einstein tensor as nine dimensional spheres in ten dimensional space

Abstract In eight dimensional superspace, the number of independent components of the Riemann–Einstein tensor is R (8) = 336. The paper shows that these components could be given a geometrical interpretation as a hyperbolic tesselation via the ( k = 11 − 4 = 7) modular group Γ (7). In addition the components may be viewed as 336 particles-like states resembling 9-dimensional spheres in 10 dimensional space. Finally a relation to the 528 killing vector fields in 32 dimensional super and maximally symmetric space related to 11 dimensional P-Branes with 528 = (336)(11/7) states is established.

Determining the number of Higgs particles starting from general relativity and various other field theories

The Riemann–Einstein tensor of general relativity as well as chiral and super-symmetry are utilized to develop various extended versions of the standard model of high energy physics. Based on these models, it is possible to predict that few new elementary particles conjectured to be the Higgs are likely to be found experimentally at an energy scale which is just above that of the electroweak. Connections to the massless states of different super-string theories as well as super-gravity are also discussed.

Einstein boundary conditions for the Einstein equations in the conformal-traceless decomposition

In relation to the BSSN formulation of the Einstein equations, we write down the boundary conditions that result from the vanishing of the projection of the Einstein tensor normally to a timelike hypersurface. Furthermore, by setting up a well-posed system of propagation equations for the constraints, we show explicitly that there are three constraints that are incoming at the boundary surface and that the boundary equations are linearly related to them. This indicates that such boundary conditions play a role in enforcing the propagation of the constraints in the region interior to the boundary. Additionally, we examine the related problem for a strongly hyperbolic first-order reduction of the BSSN equations and determine the characteristic fields that are prescribed by the three boundary conditions, as well as those that are left arbitrary.