Showing papers on "Introduction to the mathematics of general relativity published in 1995"
TL;DR: A new formulation of the Einstein equations is presented that casts them in an explicitly first order, flux-conservative, hyperbolic form, which permits the application to the Einstein equation of advanced numerical methods developed to solve the fluid dynamic equations, for the first time.
Abstract: We present a new formulation of the Einstein equations that casts them in an explicitly first order, flux-conservative, hyperbolic form. We show that this now can be done for a wide class of time slicing conditions, including maximal slicing, making it potentially very useful for numerical relativity. This development permits the application to the Einstein equations of advanced numerical methods developed to solve the fluid dynamic equations, without overly restricting the time slicing, for the first time. The full set of characteristic fields and speeds is explicitly given.
310 citations
TL;DR: In this paper, the energy density of asymptotically flat gravitational fields is defined in the framework of the teleparallel equivalent of general relativity and the ADM energy is obtained upon integration of eg over the whole three dimensional space.
Abstract: In the framework of the teleparallel equivalent of general relativity the energy density eg of asymptotically flat gravitational fields can be naturally and unambiguously defined. Upon integration of eg over the whole three dimensional space the ADM energy is obtained. eg is used to calculate the energy inside a Schwarzschild black hole.
71 citations
TL;DR: In this article, it was shown that the Evans-Rosenquist formulation of the optical-mechanical analogy, so successful in the application to classical problems, also describes the motion of massless particles in the Schwarzschild field of general relativity.
Abstract: We demonstrate that the Evans–Rosenquist formulation of the optical–mechanical analogy, so successful in the application to classical problems, also describes the motion of massless particles in the Schwarzschild field of general relativity. It is possible to obtain the well‐known equations for light orbit and radar echo delay which account for two exclusive tests of Einstein’s field equations. Some remarks including suggestions for future work are also added.
41 citations
TL;DR: In this article, the linearization of the null surface version of general relativity has been explored and compared with the standard linear version of the theory, which allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of null surface theory.
Abstract: Recently there has been developed a reformulation of general relativity (GR)—referred to as the null surface version of GR—where instead of the metric field as the basic variable of the theory, families of three‐surfaces in a four‐manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns it into an Einstein metric. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this null surface theory and compare it with the standard linear GR. This allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory. It furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory.
34 citations
TL;DR: In this paper, a twenty-dimensional space of charged solutions of spin-2 equations is proposed, where each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations.
Abstract: A twenty-dimensional space of charged solutions of spin-2 equations is proposed. The relation with extended (via dilatation) Poincare group is analyzed. Locally, each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations. Globally, the nonsingular metric tensor exists if and only if 10 among the above 20 charges do vanish. The situation is analogous to that in classical electrodynamics, where vanishing of magnetic monopole implies the global existence of the electromagnetic potentials. The notion ofasymptotic conformal Yano-Killing tensor is defined and used as a basic concept to introduce an inertial frame in General Relativity via asymptotic conditions at spatial infinity. The introduced class of asymptotically flat solutions is free of supertranslation ambiguities.
33 citations
TL;DR: In this work, the vacuum Einstein equations are formulated as differential equations for two functions, one complex and one real on a six‐dimensional manifold, M×S2, with M eventually becoming the space–time and the S2 becoming the sphere of null directions over M.
Abstract: We formulate the vacuum Einstein equations as differential equations for two functions, one complex and one real on a six‐dimensional manifold, M×S2, with M eventually becoming the space–time and the S2 becoming the sphere of null directions over M. At the start there is no other further structure available: the structure arising from the two functions. The complex function, referred to as Λ[M×S2], encodes information about a sphere’s worth of surfaces through each point of M. From knowledge of Λ one can define a second rank tensor on M which can be interpreted as a conformal metric, so that the ‘‘surfaces’’ are automatically null or characteristics of this conformal metric. The real function, Ω, plays the role of a conformal factor: it converts the conformal metric into a vacuum Einstein metric. Locally, all Einstein metrics can be obtained in this manner. In this work, we fully develop this ‘‘null surface version of general relativity (GR):’’ we display, discuss and analyze the equations, we show that m...
28 citations
TL;DR: In this paper, the authors proposed the theory of Einstein structured spaces, which includes all types of singularities into a geometrically tractable theoretical scheme, including quasiregular and curvature singularities.
Abstract: To include all types of singularities into a geometrically tractable theoretical scheme we change from Einstein algebras, an algebraic generalization of general relativity, to sheaves of Einstein algebras. The theory of such spaces, called Einstein structured spaces, is developed. Both quasiregular and curvature singularities are studied in some detail. Examples of the closed Friedmann world model and the Schwarzschild spacetime show that Schmidt'sb-boundary is a useful theoretical tool when considered in the category of structured spaces.
27 citations
TL;DR: In this article, the basic notions of general relativity are reviewed and compared with the gauge principle of modern field theories, Motivation for the new spin field is given, several theories are compared, and future directions are given.
Abstract: One of the most exciting prospects in modern formulations of the general theory of relativity is the emergence of a new field which is created by the intrinsic spin of an elementary particle. In this development, this new field permeates space and produces forces and torques on other particles with spin. Thus spin is elevated to a similar status as mass and electric charge; each creates a field that interacts with other like particles. In this article, the basic notions of general relativity are reviewed and compared with the gauge principle of modern field theories. Motivation for the new spin field is given, several theories are compared, and future directions are given.
26 citations
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TL;DR: The physical significance of the change of variables has been questioned by several authors in recent years as mentioned in this paper, who investigated to which extent purely affine, metric-affine, scalar-tensor and purely metric theories can be regarded as physically equivalent to General Relativity, and showed that in general this depends on which metric tensorfield is assumed to represent the true physical space-time geometry.
Abstract: Current generalizations of the classical Einstein-Hilbert Lagrangian formulation of General Relativity are reviewed Some alternative variational principles are known to reproduce Einstein's gravitational equations, and should therefore be regarded as equivalent descriptions of the same physical model, while other variational principles ("Scalar-tensor theories" and "Higher-derivative theories") are commonly presented as truly alternative physical theories Such theories, however, are also known to admit a reformulation which is formally identical to General Relativity (with auxiliary fields) The physical significance of this change of variables has been questioned by several authors in recent years Here, we investigate to which extent purely affine, metric-affine, scalar-tensor and purely metric theories can be regarded as physically equivalent to GR; we show that in general this depends on which metric tensorfield is assumed to represent the true physical space-time geometry For purely metric theories where the Lagrangian is a nonlinear function f(R) of the curvature scalar, we present an argument based on the definition of the physical energy, which leads one to regard the rescaled metric (Einstein frame) as the true physical one As a direct consequence, the physical content of such "alternative" models is reset to coincide with General Relativity, and the "Nonlinear Gravity Theories" become nothing but exotic reformulations of General Relativity in terms of unphysical variables
23 citations
TL;DR: In this paper, the algebraic consequences for the Weyl and Ricci tensors are examined in detail and consideration given to the uniqueness of ua is made concerning the nature of the congruence associated with ua.
Abstract: Purely magnetic space–times, in which the Riemann tensor satisfies Rabcdubud=0 for some unit timelike vector ua, are studied. The algebraic consequences for the Weyl and Ricci tensors are examined in detail and consideration given to the uniqueness of ua. Some remarks concerning the nature of the congruence associated with ua are made.
23 citations
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TL;DR: In this article, it was shown that the two physical degrees of freedom of general relativity are naturally encoded in a quantity closely related to the twist of the pair of null normals to the 2-surfaces.
Abstract: In the (2+2) formulation of general relativity spacetime is foliated by a two-parameter family of spacelike 2-surfaces (instead of the more usual one-parameter family of spacelike 3surfaces). In a partially gauge-fixed setting (double-null gauge), I write down the symplectic structure of general relativity in terms of intrinsic and extrinsic quantities associated with these 2-surfaces. This leads to an identification of the reduced phase space degrees of freedom. In particular, I show that the two physical degrees of freedom of general relativity are naturally encoded in a quantity closely related to the twist of the pair of null normals to the 2-surfaces. By considering the characteristic initial-value problem I establish a canonical transformation between these and the more usually quoted conformal 2-metric (or shear) degrees of freedom. (This paper is based on a talk given at the Fifth Midwest Relativity Conference, Milwaukee, USA.)
TL;DR: In this paper, the authors considered the problem of finding the sources of the Kerr space-time, and a new class of solutions of the Einstein field equations was found, such as half local and singular Kerr-Schild sources.
Abstract: Stationary Kerr–Schild gravitational fields in the interior of elastic–solid bodies are considered. It turns out that the Einstein field equations for such systems split into two parts: a ‘‘differential’’ part which may be written as a linear partial differential equation of the Maxwell type for a suitably defined, three‐dimensional Euclidean vector and an ‘‘algebraic’’ (generally nonlinear) part which allows the calculation of the stresses supporting the bodies. These results are applied to the problem of finding the sources of the Kerr space–time, and a new class of solutions of the Einstein field equations is found. Such solutions describe sources of the Kerr metric composed by an elastic body and a shell distribution of elastic stress energy located on the matching surface. Some explicit examples of ‘‘half local’’ (singular) Kerr–Schild sources of the Kerr space–time are also given.
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TL;DR: Among relativistic theories of gravitation the closest ones to general relativity are the scalar-tensor ones and these with Lagrangians being any function f(R) of the curvature scalar as discussed by the authors.
Abstract: Among relativistic theories of gravitation the closest ones to general relativity are the scalar-tensor ones and these with Lagrangians being any function f(R) of the curvature scalar. A complete chart of relationships between these theories and general relativity can be delineated. These theories are mathematically (locally) equivalent to general relativity plus a minimally coupled self-interacting scalar field. Physically they describe a massless spin-2 field (graviton) and a spin-0 component of gravity. It is shown that these theories are either physically equivalent to general relativity plus the scalar or flat space is classically unstable (or at least suspected of being unstable). In this sense general relativity is universal: it is an isolated point in the space of gravity theories since small deviations from it either carry the same physical content as it or give rise to physically untenable theories.
TL;DR: In this article, a model of elementary particles, regarded as the ultimate constituents of the known particles, is considered in the framework of general relativity, based on that of Lopez, using the Kerr-Newman solution of the Einstein field equations.
Abstract: Classical models of elementary particles, regarded as the ultimate constituents of the known particles, are considered in the framework of general relativity. The work is based on that of Lopez, using the Kerr-Newman solution of the Einstein field equations.
TL;DR: In this article, the field equations of general relativity were derived from a complex Lagrangian of particular simplicity, and it was shown that these field equations are equivalent to the existence of certain covariantly constant vector and bivector-valued two-forms.
Abstract: We derive the field equations of general relativity from a complex Lagrangian of particular simplicity. In this way we exhibit that Einstein’s vacuum equations are equivalent to the existence of certain covariantly constant vector‐ and bivector‐valued two‐forms. We discuss the implications this brings along for the exploration of new (hidden) symmetries as well as for the construction of new solutions. We even engage the equations’ simplicity for the construction of the most general (local) solution of the vacuum equations of general relativity and show that formally the general solution can be expressed in terms of Weyl’s conformal tensor and several arbitrary forms acting as ‘‘constants of integration.’’ In a 3+1 decomposition we demonstrate how the simple structure of the Lagrangian equations continues for the Hamilton equations, and compare the result with Ashtekar’s approach.
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TL;DR: In this article, a survey of recent results on solutions of the Einstein equations with matter is presented and a number of open questions are stated, and some remarks are made on the methods which have been used, and could be used in the future, to study solutions to the problem of general relativity with matter.
Abstract: Recent results on solutions of the Einstein equations with matter are surveyed and a number of open questions are stated. The first group of results presented concern asymptotically flat spacetimes, both stationary and dynamical. Then there is a discussion of solutions of the equations describing matter in special relativity and Newtonian gravitational theory and their relevance for general relativity. Next spatially compact solutions of the Einstein-matter equations are presented. Finally some remarks are made on the methods which have been used, and could be used in the future, to study solutions of the Einstein equations with matter.
TL;DR: In this paper, the center of traceX ∼ CT is defined as an invariant point of the energy-momentum tensor of a spinning particle in General Relativity.
Abstract: The center of traceX
CT
of the energy-momentum tensor of a spinning particle in General Relativity is defined. It is shown to be an invariant point of the particle, and its path is shown to be what is specified by the original side condition of Mathisson,S
αβ
u
β=0.
TL;DR: In this paper, the Einstein-Cartan theory is transformed into the theory of spacetimes with spinless and spinning perfect fluids interacting with the electromagnetic field, and the results are illustrated with two simple examples.
Abstract: In general relativity bounds were given on the possible divergency rates of the elements and of the trace of the tidal-force tensorR
abcdubud along incomplete maximal causal geodesics. These results are transformed into the Einstein-Cartan theory and are illustrated with two simple examples of spacetimes filled with charged spinless and spinning perfect fluids interacting with electromagnetic field.
TL;DR: In this article, the field equations of the New General Relativity NGR, constructed by Hayashi and Shirafuji (1979), have been applied to two geometric structures, given by Robertson (1932), in the domain of cosmology.
Abstract: The field equations of the New General Relativity NGR, constructed by Hayashi and Shirafuji (1979), have been applied to two geometric structures, given by Robertson (1932), in the domain of cosmology. In the first application a family of models, involving two of the parameters characterizing the field equations of the NGR, is obtained. In the second application the models obtained are found to involve one parameter only. The cosmological parameters in both applications are calculated and some cosmological problems are discussed in comparison with the corresponding results of other field theories .
TL;DR: In this article, the equations of motion of a neutral test particle in the field obtained by Wanas (1990) are completely solved and the solution obtained is compared with corresponding solutions in the cases of the Reissner-Nordstrom field, the Kerr field, and the Kerr-Newman field.
Abstract: The equations of motion of a neutral test particle in the field obtained by Wanas (1990) are completely solved. The solution obtained is compared with corresponding solutions in the cases of the Reissner-Nordstrom field, the Kerr field, and the Kerr-Newman field. The comparison shows that the new constant, appeared in the field considered, is connected to electromagnetism and not to spin phenomena.
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27 Feb 1995
TL;DR: It will be shown that the truncation error for the Regge Calculus, as an approximation to Einstein’s equations, varies as O(∆) where ∆ is the typical discretization scale.
Abstract: We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors can not be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.
TL;DR: In this paper, a complex Ricci-flat spacetime is recovered providing some boundary conditions are imposed on two-complex-dimensional surfaces, and a deep relation emerges between complex spacetimes which are not anti-self-dual and 2D surfaces which are totally null.
Abstract: A recent analysis of real general relativity based on multisymplectic techniques has shown that boundary terms may occur in the constraint equations, unless some boundary conditions are imposed. This paper studies the corresponding form of such boundary terms in complex general relativity, where spacetime is a four-complex-dimensional complex-Riemannian manifold. A complex Ricci-flat spacetime is recovered providing some boundary conditions are imposed on two-complex-dimensional surfaces. One then finds that the holomorphic multimomenta should vanish on an arbitrary three-complex-dimensional surface, to avoid having restrictions at this surface on the spinor fields which express the invariance of the theory under holomorphic coordinate transformations. The Hamiltonian constraint of real general relativity is then replaced by a geometric structure linear in the holomorphic multimomenta, and a link with twistor theory is found. Moreover, a deep relation emerges between complex spacetimes which are not anti-self-dual and two-complex-dimensional surfaces which are not totally null.
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Abstract: It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution It is also shown that matter that produces strong force may be negative, in which case there would be no Schwarzschild-like singularity
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TL;DR: In this article, it was shown that strong interaction may also be described by field equations that have the same form as that of general relativity, and it was then shown how such field equations may arise from the coupling of two strong fields.
Abstract: It is observed that, at short range, the field equations of general relativity admit a line element that takes the form of Yukawa potential. The result leads to the possibility that strong interaction may also be described by field equations that have the same form as that of general relativity. It is then shown how such field equations may arise from the coupling of two strong fields.
TL;DR: In this article, the four-dimensional properties of matter were studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to the theory of the 4D cosmological perfect fluid with matter.
Abstract: Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.
TL;DR: The general form of the parallelism tetrad field in the Schwarzschild coordinate system is presented in the framework of Hayashi's gravitation theory with torsion in this article.
Abstract: The general form of the parallelism tetrad field in the Schwarzschild coordinate system is presented in the framework of Hayashi’s gravitation theory with torsion. In the case of the symmetric energy-momentum tensor, we obtain the approximated equation of stellar structure, which, as a limit, is the Oppenheimer-Volkoff equation for hydrostatic equilibrium in general relativity.