Showing papers on "Introduction to the mathematics of general relativity published in 1994"
TL;DR: The global net of relationships between the nonlinear gravity theories, scalar-tensor theories, and general relativity is clarified, showing that in a sense these are ``canonically conjugated'' to each other.
Abstract: We argue that in a nonlinear gravity theory (the Lagrangian being an arbitrary function of the curvature scalar R), which according to well-known results is dynamically equivalent to a self-gravitating scalar field in general relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as the Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical, in the sense that Minkowski space is unstable due to the existence of negative-energy solutions close to it. To this aim we first clarify the global net of relationships between the nonlinear gravity theories, scalar-tensor theories, and general relativity, showing that in a sense these are ``canonically conjugated'' to each other. We stress that the isomorphisms are in most cases local; in the regions where these are defined, we explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and vice versa. We study energetics for asymptotically flat solutions for those Lagrangians which admit conformal rescaling to the Einstein frame in the vicinity of flat space. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space, and this is determined by the lowest-order terms R+${\mathit{aR}}^{2}$ in the Lagrangian. The proof of this positive-energy theorem relies on the existence of the Einstein frame, since in the (Helmholtz-)Jordan frame the dominant energy condition does not hold and the field variables are unrelated to the total energy of the system. This is why we regard the Jordan frame as unphysical, while the Einstein frame is physical and reveals the physical contents of the theory. The latter should hence be viewed as physically equivalent to a self-interacting general-relativistic scalar field.
467 citations
TL;DR: In this article, it is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed.
Abstract: It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the \warp drive" of science ction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here.
421 citations
TL;DR: In this paper, a symmetric energy-momentum tensor for the gravitational Einstein-Hilbert action is derived and discussed in detail using Noether's theorem and a generalized Belinfante symmetrization procedure in 3+1 dimensions.
Abstract: We discuss general properties of the conservation law associated with a local symmetry. Using Noether's theorem and a generalized Belinfante symmetrization procedure in 3+1 dimensions, a symmetric energy-momentum (pseudo) tensor for the gravitational Einstein-Hilbert action is derived and discussed in detail. In 2+1 dimensions, expressions are obtained for energy and angular momentum arising in the ISO(2,1) gauge-theoretical formulation of Einstein gravity. In addition, an expression for energy in a gauge-theoretical formulation of the string-inspired (1+1)-dimensional gravity is derived and compared with the ADM definition of energy.
107 citations
TL;DR: In this paper, it was shown that given a Newtonian solution, there exist continuous one-parameter families of solutions to the full Einstein's equations, where the parameter being the inverse of the speed of light.
Abstract: We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einstein's equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einstein's equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones). Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.
86 citations
TL;DR: A relativistic generalization of the classical virial theorem is obtained for any stationary and asymptotically flat spacetime as mentioned in this paper, which may be useful as a consistency check of numerical solutions of the Einstein equations.
Abstract: A relativistic generalization of the classical virial theorem is obtained for any stationary and asymptotically flat spacetime. This formulation is derived within the 3+1 formalism of general relativity. It may be useful as a consistency check of numerical solutions of the Einstein equations.
82 citations
TL;DR: A systematic method for solving the Hamilton-Jacobi equation for general relativity with the inclusion of matter fields, using a conformal transformation of the three-metric as well as a line integral in superspace to solve the Hamiltonian constraint.
Abstract: We demonstrate a systematic method for solving the Hamilton-Jacobi equation for general relativity with the inclusion of matter fields. The generating functional is expanded in a series of spatial gradients. Each term is manifestly invariant under reparametrizations of the spatial coordinates ("gauge invariant"). At each order we solve the Hamiltonian constraint using a conformal transformation of the three-metric as well as a line integral in superspace. This gives a recursion relation for the generating functional which then may be solved to arbitrary order simply by functionally differentiating previous orders. At fourth order in spatial gradients we demonstrate solutions for irrotational dust as well as for a scalar field. We explicitly evolve the three-metric to the same order. This method can be used to derive the Zel'dovich approximation for general relativity.
72 citations
TL;DR: In this paper, it was shown that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3) connections.
Abstract: I show in this letter that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3)$ connections. The constraints are simple polynomials in the basic variables. The present framework gives us a new formulation of General Relativity that keeps some of the interesting features of the Ashtekar formulation without the complications associated with the complex character of the latter.
64 citations
TL;DR: A unified approach towards spectral shifts in general relativity brings the cosmological and gravitational redshifts within the same framework as the more familiar Doppler effect was first proposed by Synge [Relativity: The General Theory (North‐Holland, Amsterdam 1960)] and described here in a more simplified form as mentioned in this paper.
Abstract: A unified approach towards spectral shifts in general relativity brings the cosmological and gravitational redshifts within the same framework as the more familiar Doppler effect. This approach was first proposed by Synge [Relativity: The General Theory (North‐Holland, Amsterdam 1960)] and is described here in a more simplified form.
54 citations
TL;DR: In this paper, the evolution equations for the electric and magnetic parts of the Weyl tensor for cold dust from both general relativity and Newtonian gravity were derived for a locally inertial frame at rest in the fluid frame.
Abstract: We derive the evolution equations for the electric and magnetic parts of the Weyl tensor for cold dust from both general relativity and Newtonian gravity. In a locally inertial frame at rest in the fluid frame, the Newtonian equations agree with those of general relativity. We give explicit expressions for the electric and magnetic parts of the Weyl tensor in the Newtonian limit. In general, the magnetic part does not vanish, implying that the Lagrangian evolution of the fluid is not purely local.
52 citations
TL;DR: In this paper, Bertschinger and Hamilton derived equations for the electric and magnetic parts of the Weyl tensor for cold dust for both General Relativity and Newtonian theory.
Abstract: In an interesting recent paper on the growth of inhomogeneity through the effect of gravity [1], Bertschinger and Hamilton derive equations for the electric and magnetic parts of the Weyl tensor for cold dust for both General Relativity and Newtonian theory. Their conclusion is that both in General Relativity and in Newtonian theory, in general the magnetic part of the Weyl tensor does not vanish, implying that the Lagrangian evolution of the fluid is not local. We show here that the `Newtonian' theory discussed by them is in fact not Newtonian theory {\it per se}, but rather a plausible relativistic generalisation of Newtonian theory. Newtonian cosmology itself is highly non-local irrespective of the behaviour of the magnetic part of the Weyl tensor; in this respect the Bertschinger-Hamilton generalisation is a better theory.
43 citations
Posted Content•
TL;DR: In this article, the Maupertuis principle is used to describe the dynamics of a geodesic flow in general relativity and a criterion for local instability of the trajectories may be set up in terms of curvature invariants (e.g. the Ricci scalar).
Abstract: It is tempting to raise the issue of (metric) chaos in general relativity since the Einstein equations are a set of highly nonlinear equations which may exhibit dynamically very complicated solutions for the space-time metric. However, in general relativity it is not easy to construct indicators of chaos which are gauge-invariant. Therefore it is reasonable to start by investigating - at first - the possibility of a gauge-invariant description of local instability. In this paper we examine an approach which aims at describing the dynamics in purely geometrical terms. The dynamics is formulated as a geodesic flow through the Maupertuis principle and a criterion for local instability of the trajectories may be set up in terms of curvature invariants (e.g. the Ricci scalar) of the manifold on which geodesic flow is generated. We discuss the relation of such a criterion for local instability (negativity of the Ricci scalar) to a more standard criterion for local instability and we emphasize that no inferences can be made about global chaotic behavior from such local criteria.
TL;DR: In this article, the authors give a definition of rigid congruences in both General and Special Relativity, and try to make the definition plausible, and apply the definition to the Earth-Sun system in the post-Newtonian approximation.
Abstract: We give a definition of rigid congruences in both General and Special Relativity, and we try to make the definition plausible. To this end we recall Fermat's principle in General Relativity and we show that this principle allows us to reinterpret the “quotient metric” as the quadratic form which defines the optical length in a gravitational field. We apply the definition to the Earth-Sun system in the post-Newtonian approximation. Furthermore we compute the Fermat tensor and the corresponding relative variation of the speed of light in a Michelson-Morley like experiment performed on the Earth's surface. According to all measurements to date, this quantity is extremely small (10 -13 ).
TL;DR: In this article, it is suggested how Bernhard Riemann might have discovered General Relativity soon after 1854 and how today's undergraduate students can be given a glimpse of this before, or independently of, their study of Special Relativity.
Abstract: It is suggested how Bernhard Riemann might have discovered General Relativity soon after 1854 and how today’s undergraduate students can be given a glimpse of this before, or independently of, their study of Special Relativity At the same time, the whole field of relativity theory is briefly surveyed from the space–time point of view
Posted Content•
TL;DR: A theory which claims to describe all the universe is advanced in this paper, which unifies general relativity, quantum field theory, and indeterministic conception Basic entities are: classical metric tensor $g, cosmic reference frame (including cosmic time $t$), operator $T$ of energy-momentum tensor, Hamiltonian $H_t, and state vector $Psi$ Dynamical equations are: the Einstein equation $G[g]=(\Psi,T \Psi)$ ($G$ is the Einstein tensor), the Heisen
Abstract: A theory which claims to describe all the universe is advanced It unifies general relativity, quantum field theory, and indeterministic conception Basic entities are: classical metric tensor $g$, cosmic reference frame (including cosmic time $t$), operator $T$ of energy-momentum tensor, Hamiltonian $H_t$, and state vector $\Psi$ Dynamical equations are: the Einstein equation $G[g]=(\Psi,T\Psi)$ ($G$ is the Einstein tensor), the Heisenberg equation $dT/dt=i[H_t,T]$, and the condition $H_t\Psi_t=\epsilon_t \Psi_t$ arising from the cosmic energy determinacy principle advanced in the theory The last equation describes quantum jump dynamics Quantum jumps lead to the instantaneous transferring of action and information, which, however, neither violates the causality principle, nor contradicts quantum field theory and general relativity The cosmic energy determinacy principle implies the eternal universe, ie, the cyclic one without beginning and ending, the minimal energy in every cycle being finite
TL;DR: In this article, the authors discuss the failure of general relativity to provide a proper Newtonian limit when the spacetime dimensionality is reduced to 2+1 and try to bypass this difficulty by assuming alternative equations for the gravitational field.
Abstract: We discuss the failure of general relativity to provide a proper Newtonian limit when the spacetime dimensionality is reduced to 2+1 and try to bypass this difficulty by assuming alternative equations for the gravitational field. We investigate the properties of spacetimes generated by circularly symmetric matter distributions in two cases: weakening Einstein equations, and by considering the Brans-Dicke theory of gravity. A comparison with the corresponding Newtonian picture is made.
TL;DR: In this article, the authors presented a method of obtaining varieties of new exact solutions representing static balls of perfect fluid in general relativity, which indicated the possibility of constructing a plethora of new physically significant models of relativistic stellar interiors with equations of state fairly applicable to the case of extremely compressed stars.
Abstract: In this paper we present a method of obtaining varieties of new classes of exact solutions representing static balls of perfect fluid in general relativity. A number of previously known classes of solutions has been rediscovered in the process. The method indicates the possibility of constructing a plethora of new physically significant models of relativistic stellar interiors with equations of state fairly applicable to the case of extremely compressed stars. To emphasize our point we have derived two new classes of solutions and discussed their physical importance. From the solutions of these classes we have constructed three causal interiors out of which in two models the outward march of pressure, density, pressure-density ratio and the adiabatic sound speed is monotonically decreasing.
TL;DR: For an asymptotically flat initial-data set in general relativity, the total mass-momentum may be interpreted as a Hermitian quadratic form on the complex, two-dimensional vector space of "asymptotic spinors" as discussed by the authors.
Abstract: For an asymptotically flat initial‐data set in general relativity, the total mass‐momentum may be interpreted as a Hermitian quadratic form on the complex, two‐dimensional vector space of ‘‘asymptotic spinors.’’ A generalization to an arbitrary initial‐data set is obtained. The mass‐momentum is retained as a Hermitian quadratic form, but the space of ‘‘asymptotic spinors’’ on which it is a function is modified. Indeed, the dimension of this space may range from zero to infinity, depending on the initial data. There is given a variety of examples and general properties of this generalized mass‐momentum.
TL;DR: In this article, the Einstein equations for a thin wall with cylindrical symmetry are solved for a collapsing matter shell leading to a naked, without horizon, singularity, and the solution can correspond to a collapsing mass shell.
Abstract: Assuming a continuous ansatz for the metric the Einstein equations for a thin wall with cylindrical symmetry are solved herein. The solution can correspond to a collapsing matter shell leading to a naked, without horizon, singularity.
Posted Content•
TL;DR: In this article, it was shown that general relativity is not a parametrized field theory and that there are essentially no local observables for vacuum spacetimes for canonical gravity.
Abstract: We present 2 recent results on the problems of time and observables in canonical gravity. (1) We cannot use parametrized field theory to solve the problem of time because, strictly speaking, general relativity is not a parametrized field theory. (2) We show that there are essentially no local observables for vacuum spacetimes.
Book•
01 Apr 1994
TL;DR: The Newtonian Universe Waves and their differences from Particles Fields: Space Is Not Empty Probability: What Does It Measure? Special Relativity: Only One Velocity Is Absolute Quantum Theory: New Phenomena, New Principles General Relativity as discussed by the authors.
Abstract: The Newtonian Universe Waves and Their Differences from Particles Fields: Space Is Not Empty Probability: What Does It Measure? Special Relativity: Only One Velocity Is Absolute Quantum Theory: New Phenomena, New Principles General Relativity: Gravity as Field Distortions A Look Down Further Roads Neither Determinism Nor Indeterminism Road to the Stars Appendices Index.
TL;DR: In this article, the authors proposed a method to obtain axisymmetric, dynamic solutions to the Einstein equations that can represent a radiating collapsing body in slow differential rotation, which is a generalization of the semi-numeric approach developed by Herrera, Jimenez, & Ruggeri, in 1980, for the spherically symmetric case.
Abstract: We propose a method to obtain axisymmetric, dynamic solutions to the Einstein equations that can represent a radiating collapsing body in slow differential rotation. The method is a generalization of the semi-numeric approach developed by Herrera, Jimenez, & Ruggeri, in 1980, for the spherically symmetric case. Solutions are properly matched to the exterior Kerr-Vaidya metric, and the values of the physically relevant variables (density, pressure, fluid velocity, and energy flux) are obtained inside the matter distribution. As an example of the method, a model based on Schwarzschild interior homogeneous static solution is presented.
TL;DR: In this article, a general formalism for numerically evolving initial data in general relativity is discussed, in which the complex Ashtekar connection and the Newman-Penrose scalars are taken as the dynamical variables.
Abstract: We discuss a general formalism for numerically evolving initial data in general relativity in which the (complex) Ashtekar connection and the Newman-Penrose scalars are taken as the dynamical variables. In the generic case three gauge constraints and twelve reality conditions must be solved. The analysis is applied to a Petrov type \{1111\} planar spacetime where we find a spatially constant volume element to be an appropriate coordinate gauge choice.
TL;DR: In this paper, it was shown that for time independent fields, the two seemingly different and unrelated Tolman and Moller mass-energy formulae in general relativity are, in fact, completely equivalent.
Abstract: It is shown that, for time independent fields, the two seemingly different and unrelated Tolman and Moller mass-energy formulae in general relativity are, in fact, completely equivalent.
TL;DR: This paper uses GRTensor to evaluate the Ricci and Weyl invariants for the radiating Kerr–Newman metric, which includes, as a special case, all nonvanishing invariants of the Kerr metric—the archetypical black hole solution in general relativity.
Abstract: GRTensor is an interactive PC‐based program for tensor analysis primarily of interest for teaching and research in general relativity. It uses either M A P L E V or M A T H E M A T I C A as its algebraic engine. In this paper we use GRTensor to evaluate the Ricci and Weyl invariants for the radiating Kerr–Newman metric. This includes, as a special case, all nonvanishing invariants of the Kerr metric—the archetypical black hole solution in general relativity.
TL;DR: Bianci-IX metrics have been used to obtain new exact solutions to the GRT equations, which may describe the early stages in the evolution of an expanding and rotating universe as discussed by the authors.
Abstract: Bianci-IX metrics have been used to obtain new exact solutions to the GRT equations, which may describe the early stages in the evolution of an expanding and rotating universe.
TL;DR: In this article, the authors presented some programs constructed to write the Einstein and matter conservation equations under the ADM or 3+1 formalism of general relativity as well as the relativistic Boltzmann equation in a fully covariant treatment.
TL;DR: In this article, the authors proposed an experiment to test general relativity in which the light beam in one arm of a large interferometer runs by means of a small vacuum pipe through the center of the large sphere of water.
Abstract: We propose an experiment to test general relativity in which the light beam in one arm of a large interferometer runs by means of a small vacuum pipe through the center of a large sphere of water. The phase shift caused by the reduction of the propagation velocity of the light in the gravitational field of the sphere will be measurable with near-future technology. This phase shift has a surprisingly large sensitivity to the details of the metric inside the sphere where the energy-momentum tensor is non-zero. The interior metric of general relativity yields results about 25 per cent higher than a comparison metric which has the same Newtonian gravitational field. This allows us to investigate general relativity experimentally with non-zero energy-momentum tensor for the first time.
Dissertation•
01 Jan 1994
TL;DR: In this paper, the authors model spherically symmetric radiating stars dissipating energy in the form of a radial heat flux and obtain the junction conditions necessary for the matching of the exterior Vaidya solution to an interior radiating line element.
Abstract: In this thesis we model spherically symmetric radiating stars dissipating energy in the form of a radial heat flux. We assume that the spacetime for the interior matter distribution is shear-free. The junction conditions necessary for the matching of the exterior Vaidya solution to an interior radiating line element are obtained. In particular we show that the pressure at the boundary of the star is nonvanishing when the star is radiating (Santos 1985). The junction conditions, with a nonvanishing cosmological constant, were obtained. This generalises the results of Santos (1985) and we believe that this is an original result. The Kramer (1992) model is reviewed in detail and extended. The evolution of this model depends on a function of time which has to satisfy a nonlinear second order differential equation. We solve this differential equation in general and thereby completely describe the temporal behaviour of the Kramer model. Graphical representations of the thermodynamical and gravitational variables are generated with the aid of the software package MATHEMATICA Version 2.0 (Wolfram 1991). We also analyse two other techniques to generate exact solutions to the Einstein field equations for modelling radiating stars. In the first case the particle trajectories are assumed to be geodesics. We indicate how the model of Kolassis et al (1988) may be extended by providing an ansatz to solve a second order differential equation. In the second case we review the models of de Oliveira et al (1985, 1986, 1988) where the gravitational potentials are separable functions of the spatial and temporal coordinates.
TL;DR: In this paper, the possibility of a new extension of the general relativistc theory using Finsler geometry was considered, and it was shown that this theory can include the general theory of relativity under a certain special condition.
Abstract: The possibility of a new extension of the general relativistc theory will be considered using Finsler geometry. The extension of Einstein’s general relativity can be expected to regard gravitational, electroweak, and strong interactive fields as geometrical structure of a spacetime based on Finsler geometry. Indeed, it will be shown that this theory can include the general theory of relativity under a certain special condition. In addition, Maxwell’s equations will be expressed using new metric representations of the electromagnetic vector and its tensor. Moreover, it will be suggested that this theory may include metric representations of weak and strong interactive fields.