Einstein tensor

Irregularities in the metric tensor of a signature-changing space-time suggest that field equations on such space-times might be regarded as distributional. We review the formalism of tensor distributions on differentiable manifolds, and examine to what extent rigorous meaning can be given to field equations in the presence of signature-change, in particular those involving covariant derivatives. We find that, for both continuous and discontinuous signature-change, covariant differentiation can be defined on a class of tensor distributions wide enough to be physically interesting.

Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally, a derivation of Newtonian Gravity from Einstein's Equations is given.

Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally, a derivation of Newtonian Gravity from Einstein's Equations is given.

Let X be a smooth manifold of dimension 1 + n endowed with a Lorentzian metric g. The energy tensor of a 2-form F is locally defined as \documentclass[12pt]{minimal}\begin{document}$T_{ab}\break := \, - \left( {F_a}^{i} F_{bi} - \frac{1}{4} \, F^{ij} F_{ij} g_{ab} \right)$\end{document}Tab:=−FaiFbi−14FijFijgab. In this paper we characterize this tensor as the only 2-covariant natural tensor associated to a Lorentzian metric and a 2-form that is independent of the unit of scale and satisfies certain condition on its divergence. This characterization is motivated on physical grounds, and can be used to justify the Einstein–Maxwell field equations. More generally, we characterize in a similar manner the energy tensor associated to a differential form of arbitrary order k. Finally, we develop a generalized theory of electromagnetism where charged particles are not punctual, but of an arbitrary fixed dimension p. In this theory, the electromagnetic field F is a differential form of order...

In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of Wn are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds related by a conformal mapping preserving the Einstein tensor with a gradient covector field. Then, we prove that a Weyl manifold Wn and a flat Weyl manifoldWn, which are in a conformal correspondence preserving the Einstein tensor are Einstein-Weyl manifolds. Moreover, we show that an isotropic Weyl manifold is an Einstein-Weyl manifold with zero scalar curvature and we obtain that a Weyl manifold Wn and an isotropic Weyl manifold related by the conformal mapping preserving the Einstein tensor are Einstein-Weyl manifolds.

We consider a hybrid bimetric model where, in addition to the ordinary metric tensor that determines geometry, an informational metric is introduced to describe the reference frame of an observer. We note that the local information metric being Minkowskian explains one of the key aspects of the Einstein equivalence principle. Our approach has the potential to justify the three-dimensional nature of physical space and address the gravitational energy puzzle. Furthermore, it appears to be free of ghost instabilities in the matter sector, as the second metric tensor couples exclusively to the observer and is non-dynamical.

In the Alcubierre warp-drive spacetime, we investigate the following scalar curvature invariants: the scalar I, derived from a quadratic contraction of the Weyl tensor, the trace R of the Ricci tensor, and the quadratic r1 and cubic r2 invariants from the trace-adjusted Ricci tensor. In four-dimensional spacetime the trace-adjusted Einstein and Ricci tensors are identical, and their unadjusted traces are oppositely signed yet equal in absolute value. This allows us to express these Ricci invariants using Einstein’s curvature tensor, facilitating a direct interpretation of the energy-momentum tensor. We present detailed plots illustrating the distribution of these invariants. Our findings underscore the requirement for four distinct layers of an anisotropic stress-energy tensor to create the warp bubble. Additionally, we delve into the Kretschmann quadratic invariant decomposition. We provide a critical analysis of the work by Mattingly et al., particularly their underrepresentation ...

Bernard Riemann was the first to define curvature tensor. Most of the curvature tensors are defined with the help of Riemann curvature tensor, Ricci tensor and metric tensor. It has been observed that different combinations of Ricci tensor and metric tensor in the defined tensors lead to some of the different geometrical and physical properties.

A theoretical model is presented that describes the propagation of sound in biased piezoelectric crystals of any kind of symmetry. The symmetry relations for the higher order material constants of trigonal 3m crystals, are calculated and listed in the appendix. The example of Lithium Niobate is highlighted, under influence of a bias pressure. The change of slowness (inverse velocity), because of this pressure, is calculated for every direction. Also the influence of stress on the acoustic polarization and the energy flow is outlined. Furthermore, the difference between the case where piezoelectricity is included and the case where it is omitted, is discussed. The description of changing slowness surfaces, because of a bias field, is not limited to homogeneous plane waves, but also inhomogeneous plane waves are taken into account.

Recently, pulsar timing array (PTA) collaborations announced evidence for an isotropic stochastic gravitational wave (GW) background. The origin of the PTA signal can be astrophysical or cosmological. In the latter case, the so-called secondary scalar-induced GW scenario is one of the viable explanations, but it has a potentially serious issue of the overproduction of primordial black holes (PBHs) due to the enhanced curvature perturbation. In this letter, we present a new interpretation of the PTA signal. Namely, it is originated from an extra spectator tensor field that exists on top of the metric tensor perturbation. As the energy density of the extra tensor field is always subdominant, it cannot lead to the formation of PBHs. Thus our primordial-tensor-induced scenario is free from the PBH overproduction issue.

In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ≤ D < 4k. In D = 2k + 1 this identity implies that all solutions of pure kth order Lovelock gravity are 'Riemann-Lovelock' flat. It is verified that the static, spherically symmetric solutions of these theories, which are missing solid angle space times, indeed satisfy this flatness property. This generalizes results from Einstein gravity in D = 3, which corresponds to the k = 1 case. We speculate about some possible further consequences of Riemann-Lovelock curvature.

This paper proposes a toy model where, in the Einstein equations, the right-hand side is modified by the addition of a term proportional to the symmetrized partial contraction of the Ricci tensor with the energy-momentum tensor, while the left-hand side remains equal to the Einstein tensor. Bearing in mind the existence of a natural length scale given by the Planck length, dimensional analysis shows that such a term yields a correction linear in to the classical term, that is instead just proportional to the energy-momentum tensor. One then obtains an effective energy-momentum tensor that consists of three contributions: pure energy part, mechanical stress and thermal part. The pure energy part has the appropriate property for dealing with the dark sector of modern relativistic cosmology. Such a theory coincides with general relativity in vacuum, and the resulting field equations are here solved for a Dunn and Tupper metric, for departures from an interior Schwarzschild solution as well as for a Friedmann-Lemaitre-Robertson-Walker universe.

In a recent comment Sneddon discussed the set of fourteen algebraic invariants of the Riemann curvature tensor in four dimensions. The focus was rectification of an error (in the form of lack of independence) in an earlier construction and the presentation of a corrected set suitable for application. Several authors who have worked on this problem were mentioned. The comment, however, did not mention the work of Narlikar and Karmarkar who presented a set of invariants well before the earliest work cited in the comment. The original publication by Narlikar and Karmarkar may not be readily available so we list and make a few comments on their set.

A real version of the Newman-Penrose formalism is developed for (2+ l)dimensional space-times. The complete algebraic classification of the (Ricci) curvature is given. The field equations of Deser, Jackiw, and Templeton, expressing balance between the Einstein and Bach tensors, are reformulated in triad terms. Two exact solutions are obtained, one characterized by a null geodesic eigencongruence of the Ricci tensor, and a second for which all the polynomial curvature invariants are constant.

Many differential geometer studied different types of manifolds with a semi-symmetric metric connection. In this paper, we have considered a Riemannian manifold (M n , g), (n > 2), equipped with a semi-symmetric metric connection and studied the properties of the curvature tensor, the conformal curvature tensor, the Weyl projective curvature tensor and the conharmonic curvature tensor. We have also studied the Einstein spaces and recurrent space with respect to semi-symmetric metric connection and obtained certain results related to them.

Resumen El siguiente artículo establece a los espacios de Einstein tipo N dentro de la clasificación estándar de Petrov. Para ello utiliza la descomposición tensorial del tensor de curvatura intrínseca o tensor de Weyl. Una vez hecho esto e identificando que componente espinorial del tensor se debe de conservar, establece una tetrada nula a partir de la cual se anteponen las condiciones diferenciales que deben de cumplir las funciones métricas de las cuales se componen sus cuatro covectores nulos.

An infinite family of new exact solutions of the Einstein vacuum equations for static and axially symmetric spacetimes is presented. All the metric functions of the solutions are explicitly computed and the obtained expressions are simply written in terms of oblate spheroidal coordinates. Furthermore, the solutions are asymptotically flat and regular everywhere, as it is shown by computing all the curvature scalars. These solutions describe an infinite family of thin dust disks with a central inner edge, whose energy densities are everywhere positive and well behaved, in such a way that their energy-momentum tensor are in fully agreement with all the energy conditions. Now, although the disks are of infinite extension, all of them have finite mass. The superposition of the first member of this family with a Schwarzschild black hole was presented previously [G. A. González and A. C. Gutiérrez-Piñeres, arXiv: 0811.3002v1 (2008)], whereas that in a subsequent paper a detailed analysis of the corresponding superposition for the full family will be presented.

We define a quarter symmetric metric connection in an almost − r paracontact Riemannian manifold and we consider submanifolds of an almost − r paracontact Riemannian manifold endowed with a quarter symmetric metric connection. We also obtain Gauss and Codazzi equations, Weingarten equation and curvature tensor for submanifolds of an almost − r paracontact Riemannian manifold endowed with a quarter symmetric metric connection.

We examine theories of gravity which include finitely many coupled scalar fields with arbitrary couplings to the curvature (wavemaps). We show that the most general scalar-tensor σ-model action is conformally equivalent to general relativity with a minimally coupled wavemap with a particular target metric. Inflation on the source manifold is then shown to occur in a novel way due to the combined effect of arbitrary curvature couplings and wavemap self-interactions. A new interpretation of the conformal equivalence theorem proved for such 'wavemap-tensor' theories through brane-bulk dynamics is also discussed.

In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor σ is small and the Yamabe invariant of the manifold is positive. The method we describe is however much simpler than the original method and allows easy extensions to several other problems. Some non-existence results are also considered.

We present the evolution of the full set of Einstein equations during preheating after inflation. We study a generic supersymmetric model of hybrid inflation, integrating fields and metric fluctuations in a 3-dimensional lattice. We take initial conditions consistent with Eintein's constraint equations. The induced preheating of the metric fluctuations is not large enough to backreact onto the fields, but preheating of the scalar modes does affect the evolution of vector and tensor modes. In particular, they do enhance the induced stochastic background of gravitational waves during preheating, giving an energy density in general an order of magnitude larger than that obtained by evolving the tensors fluctuations in an homogeneous background metric. This enhancement can improve the expectations for detection by planned gravitational waves observatories.

The difference tensor R • C − C • R of a semi-Riemannian manifold (M, g), dim M ≥ 4, formed by its Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0, 6)-Tachibana tensors Q(A, T), where A is a symmetric (0, 2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

The aim of present paper is to study the generalized B-curvature tensor of a normal paracontact metric manifold satisfying the conditions generalized B-flatness, generalized B-semi-symmetric, B. Z = 0, B.S = 0 and B.P = 0, where B, Z, P, S denotes the generalized B-curvature tensor, concircular curvature tensor, projective curvature tensor and Ricci tensor, respectively.

The study deals with curvature tensors on Semi-Riemannian and Generalized Sasakian space forms admitting semi-symmetric metric connection. More specifically, the study shall be to investigate the geometry of Semi-Riemannian and generalized Sasakian space forms, when they are  8 W flat,  8 W symmetric,  8 W semisymmetric and  8 W Recurrent and compared to results of projectively semi-symmetric, Weyl semi-symmetric and concircularly semi-symmetric on these spaces. Further, the conditions that admit a second order parallel symmetric tensor on functions of such spaces, shall be studied.

The Israel junction conditions of a thin shell in the context of Einstein-Cartan gravity are revisited. It is shown that with a choice of the torsion discontinuity taken to be orthogonal to the hypersurface and consistent with the antisymmetric properties of torsion tensor and contorsion tensor, the generalized asymmetric surface energy-momentum tensor turns out to be tangent to the hypersurface while the resulting generalized Israel junction conditions are modified. The main differences to previous works are mentioned.

In this study, we consider a manifold equipped with semi symmetric metric connection whose the torsion tensor satisfies a special condition. We investigate some properties of the Ricci tensor and the curvature tensor of this manifold. We obtain a necessary and sufficient condition for the mixed generalized quasi-constant curvature of this manifold. Finally, we prove that if the manifold mentioned above is conformally flat, then it is a mixed generalized quasi-Einstein manifold and we prove that if the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whose the special torsion tensor is independent from orientation chosen, then this manifold is of a mixed generalized quasi constant curvature.

The object of the present paper is to study three-dimensional general- ized (k; ) contact metric manifolds with recurrent Ricci tensor and harmonic curvature tensor. Ricci symmetric generalized (k; ) contact metric manifolds of dimension three are also considered. Each sections are followed by examples to illustrate the obtained results. 2000 Mathematics Subject Classication : 53C15, 53C40.

This paper presents a study of a general relativistic perfect fluid space-time admitting various types of curvature restrictions on energy-momentum tensors and brings out the conditions for which fluids of the space-time are sometimes phantom barrier and some other times quintessence barrier. The existence of a space-time where fluids behave as phantom barrier is ensured by an example.

In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3-curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-?-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3- ?-Einstein manifold is ?-Einstain. Moreover, we define the notion of (CHR)3- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an ...

We consider M-projective curvature tensor on Lorentzian α-Sasakian manifolds and show that M-projectively semisymmetric Lorentzian α-Sasakian manifold is M-projectively flat. Further, it is shown that, M-projectively semisymmetric Lorentzian α-Sasakian manifold is semisymmetric. Finally, some results related to energy momentum tensor satisfying the Einstein field equation with cosmological constant of the M-projectively semisymmetric Lorentzian α-Sasakian space-times are investigated.

In Newtonian mechanics, space and time are separate but in General, Relativity is unified. It is considered that the space in the weak-field approximation is quasi-static and it arises from a perfect field whose particles have very small velocity in comparison to light velocity in this coordinate system and the metric is a gravitational potential tensor of rank two which implies the field of empty space. If each point of an area in N-dimensional space there existed a corresponding definite tensor, where the components of the tensor are the function of space and space acts as the strong or weak gravitational field.

An independent derivation is given of equations first derived by Rainich which show how, under certain circumstances, the combined theory of gravitation and electromagnetism of Einstein and Maxwell can be unified and described exclusively in terms of geometry. Some algebraic relations are presented between the Ricci tensor, the electromagnetic field tensor, and their principal null vectors. It is shown that in regions of space-time where the two invariants of the electromagnetic field both vanish, the unified theory cannot apply. Either such regions do not exist in nature or their description in terms of pure geometry has yet to be found. Advantage is taken of the correspondences between tensors and spinors to carry out most of the present calculations in spinor space.

1+3 covariant approach to cosmological perturbation theory often employs the electric part (E ab), the magnetic part (H ab) of the Weyl tensor or the shear tensor (σ ab) in a phenomenological description of gravitational waves. The Cotton-York tensor is rarely mentioned in connection with gravitational waves in this approach. This tensor acts as a source for the magnetic part of the Weyl tensor which should not be neglected in studies of gravitational waves in the 1+3 formalism. The tensor is only mentioned in connection with studies of 'silent model' but even there the connection with gravitational waves is not exhaustively explored. In this study, we demonstrate that the Cotton-York tensor encodes contributions from both electric, magnetic part of the Weyl tensor and in directly from the shear tensor. In our opinion, this makes the Cotton-York tensor arguably the natural choice for linear gravitational waves in the 1+3 covariant formalism. The tensor is cumbersome to work with but that should negate its usefulness. It is conceivable that the tensor would equally be useful in the metric approach, although we have not demonstrated this in the current study. We contend that the use of only one of the Weyl tensor or the shear tensor, although phenomenologically correct, leads to loss of information. Such information is vital particularly when examining the contribution of gravitational waves to the anisotropy of an almost-Friedmann-Lamitre-Robertson-Walker (FLRW) universe. The recourse to this loss is the use Cotton-York tensor.

We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically and intuitively motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the source matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter fields must be divergence free, when spacetime is 4 dimensional. Moreover, if the total source matter fields are assumed to be divergence free, then either the spacetime is of dimension 4 or the spacetime has constant scalar curvature.

In the literature one often finds the claim that there is no such thing as an energy-momentum tensor for the gravitational field, and consequently, that the total energy-momentum conservation can only be defined in terms of a gravitational energy-momentum pseudo-tensor. Nevertheless, by relaxing the assumption that gravitational energy-momentum tensor should only depend on first derivatives of the metric, the Einstein equation leads to a trivial result that gravitational energy-momentum tensor is essentially the Einstein tensor. We discuss various peculiarities of such a definition of energy-momentum are argue that all these peculiarities have a sensible physical interpretation.

This article deals with the investigation of perfect fluid GRW spacetimes. It is shown that in a GRW perfect fluid spacetime, the Weyl tensor is divergence free and in dimension 4, a perfect fluid GRW spacetime is a RW spacetime. Moreover, we prove that if a GRW perfect fluid spacetime admits a second order symmetric parallel tensor, then either the state equation of the perfect fluid GRW spacetime is characterized by p = 3−n n−1 μ , or the tensor is a constant multiple of the metric tensor. We also characterize the perfect fluid GRW spacetimes whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons and gradient m-quasi Einstein solitons, respectively.

We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields&#39; energy momentum stress tensor other than symmetry. We give a physical motivation for this choice

Bekenstein's theory of relativistic gravity is conventionally written as a bi-metric theory. The two metrics are related by a disformal transformation defined by a dynamical vector field and a scalar field. In this comment we show that the theory can be rewritten as Vector-Tensor theory akin to Einstein-Aether theories with non-canonical kinetic terms. We discuss some of the implications of this equivalence.

Aim: The possibility of the geometrization of the stress-energy-tensor and the problems associated with such an undertaking is reviewed again. Methods: The usual tensor calculus rules were used. Results: The stress-energy-tensor was geometrized. The stress-energy-tensor of the electromagnetic field was geometrized too. A method how to calculate the value of the anti-cosmological constant Lambda is developed. Conclusion: Finally, Einstein&#39;s field equations are geometrized completely.

In this paper N(k)-Mixed Quasi Einstein Manifolds(N (k) − (MQE) n)are introduced and the existence of these manifolds is proved. We give hyper surfaces of Euclidean spaces as examples of N (k) − (MQE) n and semi symmetric, ricci symmetric and ricci recurrent N (k) − (MQE) n manifolds are studied.

In this paper, we study ?-Ricci solitons on N(k)-contact metric manifolds. At first we consider ?-Ricci solitons on N(k)-contact metric manifolds with harmonic curvature tensor. Then we study ?-Ricci solitons onN(k)-contact metric manifolds with harmonic Weyl tensor. Moreover, we consider ?-Ricci soliton on N(k)-contact metric manifolds with ?-parallel Ricci tensor. Also ?-Ricci soliton on N(k)-contact metric manifolds satisfying some curvature restrictions under projective curvature tensor have been considered. Finally, the existence of an ?-Ricci soliton on a 3-dimensional N(k)-contact metric manifold is ensured by a proper example.

We present a scalar-tensor theory of gravity on a torsion-free and metric compatible Lyra manifold. This is obtained by generalizing the concept of physical reference frame by considering a scale function defined over the manifold. The choice of a specific frame induces a local base, naturally non-holonomic, whose structure constants give rise to extra terms in the expression of the connection coefficients and in the expression for the covariant derivative. In the Lyra manifold, transformations between reference frames involving both coordinates and scale change the transformation law of tensor fields, when compared to those of the Riemann manifold. From a direct generalization of the Einstein-Hilbert minimal action coupled with a matter term, it was possible to build a Lyra invariant action, which gives rise to the associated Lyra Scalar-Tensor theory of gravity (LyST), with field equations for gµν and φ. These equations have a well-defined Newtonian limit, from which it can be seen that both metric and scale play a role in the description gravitational interaction. We present a spherically symmetric solution for the LyST gravity field equations. It dependent on two parameters m and rL, whose physical meaning is carefully investigated. We highlight the properties of LyST spherically symmetric line element and compare it to Schwarzchild solution.