Showing papers on "Mathematics of general relativity published in 2002"
TL;DR: In this paper, interior perfect fluid solutions for the Reissner-nordstrom metric are studied on the basis of a new classification scheme, which specifies which two of the characteristics of the fluid are given functions and accordingly picks up one of the three main field equations, the other two being universal.
Abstract: Interior perfect fluid solutions for the Reissner-Nordstr\"om metric are studied on the basis of a new classification scheme. It specifies which two of the characteristics of the fluid are given functions and accordingly picks up one of the three main field equations, the other two being universal. General formulas are found for charged de Sitter solutions, the case of a constant energy component of the energy-momentum tensor, the case of known pressure (including charged dust), and the case of a linear equation of state. Explicit new global solutions, mainly in elementary functions, are given as illustrations. The known solutions are briefly reviewed and corrected.
311 citations
TL;DR: In this paper, a complete set of gauge invariant observables, in the context of general relativity coupled with a minimal amount of realistic matter (four particles), have been presented, which have a straightforward and realistic physical interpretation.
Abstract: I present a complete set of gauge invariant observables, in the context of general relativity coupled with a minimal amount of realistic matter (four particles). These observables have a straightforward and realistic physical interpretation. In fact, the technology to measure them is realized by the Global Positioning System: they are defined by the physical reference system determined by GPS readings. The components of the metric tensor in this physical reference system are gauge invariant quantities and, remarkably, their evolution equations are local.
103 citations
TL;DR: In this article, the authors consider a version of Palais' principle of symmetric criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories and investigate the validity of PSC for local gravitational theories built from a metric.
Abstract: We consider a version of Palais' principle of symmetric criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. Given a group action on a space of fields, PSC asserts that for any group-invariant Lagrangian, the equations obtained by restriction of Euler–Lagrange equations to group-invariant fields are equivalent to the Euler–Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric and show that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. These results are illustrated with a variety of examples from general relativity.
48 citations
TL;DR: The application of Regge calculus, a lattice formulation of general relativity, is reviewed in the context of numerical relativity in this paper, where the strengths and weaknesses of the lattice approach are highlighted.
Abstract: The application of Regge calculus, a lattice formulation of general relativity, is reviewed in the context of numerical relativity. Particular emphasis is placed on problems of current computational interest, and the strengths and weaknesses of the lattice approach are highlighted. Several new and illustrative applications are presented, including initial data for the head on collision of two black holes, and the time evolution of vacuum axisymmetric Brill waves.
46 citations
TL;DR: In this paper, the flat space theory of a scalar field in general covariant form is cast in terms of an action integral, and all the fields (the scalar, the spacetime metric, and the auxiliary field) are dynamical in the sense of being varied freely in the action integral.
Abstract: We cast the flat space theory of a scalar field in generally covariant form by introducing an auxiliary field $\lambda$ The resulting theory is couched in terms of an action integral $S$, and all the fields (the scalar, the spacetime metric, and $\lambda$) are dynamical in the sense of being varied freely in $S$ Conservation of energy-momentum emerges as a formal consequence of diffeomorphism invariance, in close analogy with the situation in ordinary general relativity
43 citations
TL;DR: In this article, an exact solution of the Friedmann-Robertson field equation with the appropriate matter density for generic values of the cosmological constant Λ and curvature constant K was derived and a formal expression for the Hubble constant was derived.
Abstract: Strong field (exact) solutions of the gravitational field equations of general relativity in the presence of a cosmological constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres–Szafron family of spacetime line elements with a non-zero cosmological constant. The resulting solution connects, in an intrinsic way, general relativity with the theory of modular forms and elliptic curves. The homogeneous FLRW limit of the above spacetime elements is recovered and we solve exactly the resulting Friedmann–Robertson field equation with the appropriate matter density for generic values of the cosmological constant Λ and curvature constant K. A formal expression for the Hubble constant is derived. The cosmological implications of the resulting nonlinear solutions are systematically investigated. Two particularly interesting solutions: (i) the case of a flat universe K = 0, Λ ≠ 0 and (ii) a case with all three cosmological parameters non-zero, are described by elliptic curves with the property of complex multiplication and absolute modular invariant j = 0 and 1728, respectively. The possibility that all nonlinear solutions of general relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed.
40 citations
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TL;DR: In this paper, the authors define the time travel paradox in physical terms and prove its existence by constructing an explicit example, and argue further that in theories such as general relativity, where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise.
Abstract: We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in theories---such as general relativity---where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise.
39 citations
TL;DR: Barbour, Foster and O Murchadha as discussed by the authors developed a 3-space approach for the formulation of classical bosonic dynamics without time nor a locally Minkowskian structure of spacetime are presupposed.
Abstract: Barbour, Foster and O Murchadha have recently developed a new framework, called here the 3-space approach, for the formulation of classical bosonic dynamics Neither time nor a locally Minkowskian structure of spacetime are presupposed Both arise as emergent features of the world from geodesic-type dynamics on a space of three-dimensional metric–matter configurations In fact gravity, the universal light-cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism It yields relativity—and more—without presupposing relativity This paper completes the recovery of the presently known bosonic sector within the 3-space approach We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang–Mills fields minimally coupled to gravity
34 citations
TL;DR: This Letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature that vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation.
Abstract: This Letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature. While many such invariants exist, this one vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation. In more general regions which incontrovertibly support nontrivial radiation fields, it can be used to extract local, coordinate-independent information partially characterizing that radiation. While a clear, physical interpretation is possible only in such radiation zones, a simple algorithm can be given to extend the definition smoothly to generic regions of space-time.
29 citations
TL;DR: In this paper, a reformulation of the Arnowitt-Deser-Misner equations in general relativity, which has dramatically improved the stability properties of numerical implementations, has a direct analogue in classical electrodynamics.
Abstract: We show that a reformulation of the Arnowitt-Deser-Misner equations in general relativity, which has dramatically improved the stability properties of numerical implementations, has a direct analogue in classical electrodynamics. We numerically integrate both the original and the revised versions of Maxwell's equations, and show that their distinct numerical behavior reflects the properties found in linearized general relativity. Our results shed further light on the stability properties of general relativity, illustrate them in a very transparent context, and may provide a useful framework for further improvement of numerical schemes.
TL;DR: In this article, a curve shortening method was used to obtain the Morse relations for light rays joining an event with a smooth timelike curve in a Lorentzian manifold with boundary.
Abstract: In this paper we use a general version of Fermat’s principle for light rays in general relativity and a curve shortening method to write the Morse relations for light rays joining an event with a smooth timelike curve in a Lorentzian manifold with boundary The Morse relations are obtained under the most general assumptions and one can apply them to have a mathematical description of the gravitational lens effect in a very general context Moreover, Morse relations can be used to check if existing models are corrected
TL;DR: In this article, a new class of exact solutions in spherical symmetry is found and discussed, such that the energy-momentum tensor has two two-dimensional distinct isotropic subspaces.
Abstract: It is known that de Sitter spacetime can be seen as the solution of the field equation for completely isotropic matter. In the present paper, a new class of exact solutions in spherical symmetry is found and discussed, such that the energy–momentum tensor has two two-dimensional distinct isotropic subspaces.
TL;DR: In this paper, a new code for evolving axisymmetric isolated systems in general relativity is described by asymptotically flat space-times which have the property that they admit a conformal extension.
Abstract: We describe in this article a new code for evolving axisymmetric isolated systems in general relativity. Such systems are described by asymptotically flat space-times which have the property that they admit a conformal extension. We are working directly in the extended ``conformal'' manifold and solve numerically Friedrich's conformal field equations, which state that Einstein's equations hold in the physical space-time. Because of the compactness of the conformal space-time the entire space-time can be calculated on a finite numerical grid. We describe in detail the numerical scheme, especially the treatment of the axisymmetry and the boundary.
TL;DR: In this article, the approximate stress energy tensor of the quantized massive scalar field with arbitrary curvature coupling in the spacetime of a charged black hole was constructed and analyzed.
Abstract: Building on general formulas obtained from the approximate renormalized effective action, the approximate stress-energy tensor of the quantized massive scalar field with arbitrary curvature coupling in the spacetime of charged black hole being a solution of coupled equations of nonlinear electrodynamics and general relativity is constructed and analysed. It is shown that in a few limiting cases, the analytical expressions relating obtained tensor to the general renormalized stress-energy tensor evaluated in the geometry of the Reissner-Nordstr\"{o}m black hole could be derived. A detailed numerical analysis with special emphasis put on the minimal coupling is presented and the results are compared with those obtained earlier for the conformally coupled field. Some novel features of the renormalized stress-energy tensor are discussed.
01 Jan 2002
TL;DR: In this article, a regularization procedure that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed.
Abstract: A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density , associated to the Einstein tensor of the regularized metric, rather than the Einstein tensor itself, be a distribution and (ii) the regularized metric be a continuous metric with a discontinuous extrinsic curvature across a non-null hypersurface of codimension one. In this paper, the curvature and Einstein tensors of the geometries associated to point sources in the (2 + 1)-dimensional gravity and the Schwarzschild spacetime are considered. In both examples the regularized metrics are continuous regular metrics, as defined by Geroch and Traschen, with well defined distributional curvature tensors at all the intermediate steps of the calculation. The limit in which the support of these curvature tensors tends to the singular region of the original spacetime is studied and the results are contrasted with the ones obtained in previous works.
TL;DR: In this paper, the consequences of the existence of spacelike Ricci inheritance vectors (SpRIVs) parallel to xa for a model of string cloud and string fluid stress tensor in the context of general relativity are studied.
Abstract: We study the consequences of the existence of spacelike Ricci inheritance vectors (SpRIVs) parallel to xa for a model of string cloud and string fluid stress tensor in the context of general relativity. Necessary and sufficient conditions are derived for a spacetime with a model of string cloud and string fluid stress tensor to admit a SpRIV, and a SpRIV which is also a spacelike conformal Killing vector. Also, some results are obtained.
TL;DR: In this article, an anisotropic, homogeneous Bianchi type-I cosmological micro model is obtained in Barber's modified theory of general relativity, and it is found that this theory leads to Einstein theory as the coupling parameterλ → 0 in micro level (i.e., quantum level) in general.
Abstract: An anisotropic, homogeneous Bianchi type-I cosmological micro model is obtained in Barber's modified theory of general relativity. Some properties of the model are discussed. Further, it is found that this theory leads to Einstein theory as the coupling parameterλ → 0 in micro level (i.e., quantum level) in general.
TL;DR: In this article, a distributional approach for dynamics of light-like hypersurfaces in general relativity is presented, where the authors consider the problem of glueing two different manifolds which admit signature change in the context of Colombeau's theory of generalized functions.
Abstract: Some areas of modern theoretical physics such as modern cosmology contain different manifolds which must be glued together along a common boundary. These boundaries can be spacelike, timelike, or lightlike hypersurfaces. This paper shows how this gluing for different hypersurfaces is possible. Two different approaches are considered and the extent to which these approaches are equivalent are discussed. In particular, we will construct a distributional approach for dynamics of lightlike hypersurfaces in general relativity. Since Einstein’s equations are nonlinear PDEs, for discontinuous metrics such as signature changing metrics, product of distributions are unavoidable. To glue two different manifolds which admit signature change, we consider this problem in the context of Colombeau’s new theory of generalized functions. Some examples are given for clarification.
TL;DR: A short review of the status and applications of computer algebra systems for calculations in relativity and gravitation is given in this article, where the authors also give a short survey of the current state of the art.
Abstract: This paper gives a short review of the status and applications of computer algebra systems for calculations in relativity and gravitation.
TL;DR: GRworkbench as discussed by the authors is a tool for numerical work in General Relativity that facilitates the development of robust and chart-independent numerical algorithms for geodesic tracing on two charts covering the exterior Schwarzschild space-time.
Abstract: We have developed a new tool for numerical work in General Relativity: GRworkbench. We discuss how GRworkbench's implementation of a numerically-amenable analogue to Differential Geometry facilitates the development of robust and chart-independent numerical algorithms. We consider, as an example, geodesic tracing on two charts covering the exterior Schwarzschild space-time.
TL;DR: In this paper, the role of SO(2, 1) symmetry in general relativity is analyzed and cosmological solutions of Einstein field equations invariant with respect to a spacelike Lie algebra r, with 3 ≤ r ≤ 6 and containing so(2.1) as a subalgebra, are also classified.
Abstract: The role of SO(2, 1) symmetry in general relativity is analysed. Cosmological solutions of Einstein field equations invariant with respect to a spacelike Lie algebra r, with 3 ≤ r ≤ 6 and containing so(2, 1) as a subalgebra, are also classified.
TL;DR: In this paper, the authors take Mach's principle to mean that the local properties of a test particle should depend on the global properties of the geometry, and they use a complex wave-like metric and an appropriate redefinition of the energy-momentum tensor within the context of general relativity.
Abstract: We take Mach's principle to mean that the local properties of a test particle should depend on the global properties of the geometry. Using a complex wave-like metric and an appropriate redefinition of the energy-momentum tensor, we show this to be possible in principle within the context of general relativity. We outline implications for higher-dimensional theories of gravity.
01 Sep 2002
TL;DR: In this article, the vector nature of magnetic fields and the geometrical interpretation of gravity introduced by general relativity lead to a special coupling between magnetism and spacetime curvature, which effectively transfers the tension properties of the field into the spacetime fabric, triggering a variety of effects with potentially far-reaching implications.
Abstract: We discuss how the vector nature of magnetic fields and the geometrical interpretation of gravity introduced by general relativity lead to a special coupling between magnetism and spacetime curvature. This magneto-geometrical interaction effectively transfers the tension properties of the field into the spacetime fabric, triggering a variety of effects with potentially far-reaching implications.
TL;DR: In this article, the structure and dynamics of a light-like thin shell in general relativity can be obtained from a distributional approach, which is similar to our approach in this paper.
Abstract: This paper shows how the structure and dynamics of a lightlike thin shell in general relativity can be obtained from a distributional approach.
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TL;DR: In this article, a general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications, where the basic variables are the metric tensor and an additional four-vector.
Abstract: A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations, when supplemented by suitable coordinate conditions, determine the time evolution of all these variables without any constraint. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints hold true. The extended system is well posed when using the natural extension of either harmonic coordinates or the harmonic slicing condition in normal coordinates