Showing papers on "Introduction to the mathematics of general relativity published in 1986"
TL;DR: In this paper, a number of new exact static and non-static solutions of (2+1) general relativity with scalar field, perfect fluid and magnetic field sources are presented and studied in detail.
Abstract: We extend what is known about the structure of (2+ 1)-dimensional gravitational field theories. The non-existence of any Newtonian limit to these theories is investigated in the presence of Brans-Dicke scalar fields and non-linear curvature terms in the gravita- tional action. A number of new exact static and non-static solutions of (2+1) general relativity with scalar field, perfect fluid and magnetic field sources are presented and studied in detail. Some of these possess a correspondence with (3 + 1) solutions of general relativity through a Kaluza-Klein type reduction and exhibit the 'wedge' structure of (3 + 1)- dimensional solutions describing line sources like vacuum strings. An algebraic classification of (2+ 1) gravitational fields is derived using the Bach-Weyl tensor. The description of the general cosmological solution is given in terms of arbitrary spatial functions independently specified on a spacelike surface of constant time together with a series approximation to spacetime in the vicinity of a general cosmological singularity. Various results and conjectures regarding spacetime singularities are given. Two exact cosmological solutions containing self-interacting scalar fields that produce inflationary behaviour are also found.
122 citations
TL;DR: The algebraic classification of second-order symmetric tensors based on Segre type is used to give a systematic description of energy-momentum tensors in General Relativity as discussed by the authors.
Abstract: The algebraic classification of second-order symmetric tensors based on Segre type is used to give a systematic description of energy-momentum tensors in General Relativity. The uniqueness of the physical interpretation of a given energy-momentum tensor is discussed algebraically and a brief description of their “inheritance of symmetry” properties is also given.
48 citations
42 citations
TL;DR: In this paper, Lagrangians for matter fields are easily formulated on this extended spacetime from simple invariance arguments and stress energy tensors can be derived from them in the usual manner so that four-dimensional spacetime expressions are obtained that are consistent in the relativistic as well as the Newtonian case.
Abstract: This paper is a sequel to earlier ones, (1984), in which, on the one hand, classical field theories were described on a curved Newtonian spacetime, and, on the other hand, the Newtonian gravitation theory was formulated on a five-dimensional spacetime with a metric of signature (++++-) and a covariantly constant vector field. Here the authors show that Lagrangians for matter fields are easily formulated on this extended spacetime from simple invariance arguments and that stress-energy tensors can be derived from them in the usual manner so that four-dimensional spacetime expressions are obtained that are consistent in the relativistic as well as the Newtonian case. In the former the theory is equivalent to general relativity. When the magnitude of the distinguished vector field vanishes equations for the (covariant) Newtonian limit follow. They demonstrate this here explicitly in the case of the Klein-Gordon/Schrodinger and the Dirac field and its covariant non-relativistic analogue, the Levy-Leblond field. In especially the latter example the covariant Newtonian theory simplifies dramatically in this five-dimensional form.
28 citations
TL;DR: In this paper, a formalism for classifying and constructing perfect fluids is developed, where the Ricci tensor and its first covariant derivatives in a comoving frame are expressed as functions of energy density, pressure, their gradients and the kinematic quantities.
Abstract: A formalism for classifying and constructing perfect fluids is developed. The Ricci tensor and its first covariant derivatives in a comoving frame are expressed as functions of energy density, pressure, their gradients and the kinematic quantities. All conformally flat perfect fluids are constructed and thus also classified. In general the third covariant derivative is needed for a complete classification of these metrics.
22 citations
TL;DR: In this article, a formulation of multipole moments generalizing that of Thorne is proposed for the stationary, vacuum region of spacetime surrounding a source of gravity, without assuming asymptotic flatness.
Abstract: A formulation of multipole moments generalizing that of Thorne is proposed for the stationary, vacuum region of spacetime surrounding a source of gravity, without assuming asymptotic flatness. In this formalism, such a region of spacetime is characterized by four sets of moments, the internal mass and current moments (those of the internal source) and the external mass and current moments (those of the external universe), which are read out from a de Donder coordinate expansion of the metric density. These moments uniquely determine the vacuum region of spacetime. The interactions between a gravitating body and an external gravitational field can be described in terms of these moments, in close analogy with Newtonian theory. A derivation, using the vacuum Einstein equation alone, is given of the laws of force and torque for an isolated body acted on by an external field. These laws generalize the results of Thorne and Hartle and of Zhang.
15 citations
TL;DR: In this paper, it was shown that the action of euclidean general relativity is the surface term of a five-dimensional O(5) gauge theory, thus realizing Hawking's conjecture that the functional integral be restricted to compact manifolds without boundary.
12 citations
TL;DR: In this article, the vacuum field equations are integrated in complex relativity in a major subcase of the class whose Weyl tensor is of the type N⊗N, i.e., when the left and right Weyl spinors are each of typeN.
Abstract: Einstein's vacuum field equations are integrated in complex relativity in a major subcase of the class whose Weyl tensor is of the type N⊗N, ie, when the left and right Weyl spinors Ψ and\(\tilde \Psi \) are each of typeN The subcase is the complex equivalent of the real nontwisting case Five separate families of solutions are found Three of these are complexified versions of the two families of plane-fronted waves and the Robinson-Trautman real type-N metrics and two are complex solutions which do not have any real slices of Lorentz signature Before the equations are integrated, the relevant general theory and equations are developed in a tetrad frame which is well suited to the discussion of these and a wider class of complex solutions and is called aleft quarter flat frame The relationship between this frame and the coordinates used and some other frames and coordinates, including the complexified version of the frame often used for real type-N metrics, is discussed
12 citations
TL;DR: In this paper, the distance dependence of gravity is found in Newtonian universes with any number n of space dimensions and two independent derivations given are based either on (i) requiring that a (hyper) spherical mass gravitate as if all its mass were concentrated at its center, or (ii) using the field equations of general relativity with the cosmological constant Λ.
Abstract: The distance dependence of gravity is found in Newtonian universes with any number n of space dimensions. Two independent derivations given are based either on (i) requiring that a (hyper) spherical mass gravitate as if all its mass were concentrated at its center, or (ii) using the field equations of general relativity with the cosmological constant Λ. Both approaches lead to identical results. The gravity field at distance r from a point mass has two parts, one going as r1−n, the other as r, i.e., Hooke’s law. The Hookian field obeys a novel form of Gauss’s (flux) law, and is closely related to Λ. The simple mechanical interpretation which emerges gives insight into the meaning of Λ and helps counteract certain prevalent misconceptions.
11 citations
TL;DR: The relation between Regge's equations of motion for a simplicial manifold and the distributional Einstein tensor of the manifold is discussed in this article, where it is shown that the average of contributions to the tensor on a suitably defined 3-surface can be computed.
Abstract: The relation between Regge's equations of motion for a simplicial manifold and the distributional Einstein tensor of the manifold is discussed. Regge's equations imply that the distributional Einstein tensor vanishes 'on average', the averaging involving integrating contributions to the Einstein tensor on a suitably defined 3-surface.
TL;DR: In this article, the energy-momentum tensor of a fluid composed of spinning particles is presented in a form that is structurally similar to that of an irrotational geodesic flow with the energy flux q a with respect to the fluid velocity u a, isotropic pressure p and the trace-free anisotropic pressure π ab.
01 Jan 1986
TL;DR: In this paper, a new approach to the numerical solution of Einstein's Equations based on the characteristic initial value problem is outlined, which requires integration through caustics and up to singularities.
Abstract: A new approach to the numerical solution of Einstein’s Equations based on the characteristic initial value problem is outlined. Because this approach requires integration through caustics and up to singularities, this topic is described and some novel finite-differencing techniques are introduced for integrating up to singularities of non-linear differential equations. Finally some preliminary results from this new approach are presented.
TL;DR: The covariant definition of gravitational field energy given by Lynden-Bell and Katz is expressed in terms of Israel's theory of surface layers in general relativity in this article, which leads to the same results as those given by Einstein's pseudotensor expression in isotropic coordinates.
Abstract: The covariant definition of gravitational field energy given by Lynden-Bell and Katz is expressed in terms of Israel's theory of surface layers in general relativity In this way an expression, valid for arbitrary radial coordinates, of the gravitational field energy in a static, spherically symmetric space-time, is deduced This expression is applied to the Schwarschild and Reissner-Nordstrom space-times, and leads here to the same results as those given by Einstein's pseudotensor expression in isotropic coordinates
TL;DR: In this article, the real time finite temperature Green functions necessary for perturbations in general relativity are obtained in the scheme of thermo field dynamics, and the one-loop effective potential in the real scalar theory in the general curved space background in the normal coordinate expansion.
Abstract: The real time finite temperature Green functions necessary for perturbations in general relativity are obtained in the scheme of thermo field dynamics. As an application, the author calculates the one-loop effective potential in the real scalar theory in the general curved space background in the normal coordinate expansion. The author also provides a basic formula needed for evaluating the transport coefficients by calculating the imaginary part of a scalar self-energy diagram.
TL;DR: In this article, it was shown that if the metric perturbation satisfies certain falloff and gauge conditions, then the radiated energy calculated using the second-order Einstein tensor equals the secondorder contribution to the Bondi energy flux associated with the perturbations.
Abstract: In general relativity there is a well‐defined prescription for defining a quantity that represents the radiated energy of an exact, asymptotically flat solution of Einstein’s equation. This quantity is called the Bondi energy flux. However, in linearized gravity off a stationary and asymptotically flat background, the second‐order Einstein tensor has been used as a stress‐energy tensor for the perturbed gravitational field, enabling one to calculate the energy radiated away in gravitational radiation. It is natural to ask how this method compares to the exact method for calculating the Bondi energy flux. In this paper, it is shown that if the metric perturbation satisfies certain falloff and gauge conditions, then the radiated energy calculated using the second‐order Einstein tensor equals the second‐order contribution to the Bondi energy flux associated with the perturbation. As an application, the second‐order Einstein tensor is used to demonstrate gravitational superradiance from a Kerr black hole. Als...
TL;DR: In this paper, static and spherically-symmetric solutions of the field equations in the bimetric theory of gravitation were obtained for isotropic and anisotropic distributions of matter when the physical metric admits a one-parameter group of conformal motions.
Abstract: Static and spherically-symmetric solutions of the field equations in the bimetric theory of gravitation are obtained for isotropic and anisotropic distributions of matter when the physical metric admits a one-parameter group of conformal motions. The solutions agree with Einstein's general relativity for physical systems comparable to the size of the Universe, such as the solar system.
01 Jan 1986
TL;DR: Two methods of solving the neutrino transport equation in general relativity are presented in this paper, one for the spherical collapse of stars to black holes and the other for cosmology.
Abstract: Two methods of solving the neutrino transport equation in general relativity are presented. The first method is for the spherical collapse of stars to black holes. It uses a Lagrangian hydrodynamic formulation as a framework. The method is of mediocre accuracy, but it can handle all cases efficiently. The second method of solving the transport equation is for use in cosmology. In this case all velocities are of the same size and an explicit monotonic second order in space differencing is used. This latter method includes the effects of neutrino rest mass.
TL;DR: In this paper, a co-rotating pressure-free matter interacting with an electromagnetic field was studied under the assumption that the gravitational attraction is balanced by the centrifugal force and by magnetic pressure.
Abstract: Rigidly rotating pressure-free matter interacting with an electromagnetic field is studied under the assumption that in the co-rotating frame the gravitational attraction is balanced by the centrifugal force and by magnetic pressure.
TL;DR: The existence of gravity is derived from simple arguments based on Newton's mechanics as discussed by the authors, which provides physical understanding both of Einstein's general theory of relativity and its relation to the special theory.
Abstract: The existence of gravitational force is derived from simple arguments based on Newton’s mechanics The derivation provides physical understanding both of Einstein’s general theory of relativity and its relation to the special theory In particular, the proportionality between mass and energy is a direct consequence of this derivation
01 Jan 1986
TL;DR: Einstein this paper proposed the special theory of relativity, a theory that revolutionized the concepts of space, time, and motion which had been considered well established since the days of Newton, and had been led to special relativity from considerations of symmetry, from the deduction that Maxwell's equations of electromagnetic theory were invariant under Lorentz transformations (and not under Galilean transformations).
Abstract: In 1905 Einstein proposed the special theory of relativity, a theory that revolutionized the concepts of space, time, and motion which had been considered well established since the days of Newton Einstein had been led to special relativity from considerations of symmetry, from the deduction that Maxwell’s equations of electromagnetic theory were invariant under Lorentz transformations (and not under Galilean transformations) Observational considerations did not motivate the theory: the null result of the Michelson—Morley experiment followed as a simple deduction from the theory after it was formulated
01 Jan 1986
TL;DR: In this paper, it is assumed that each point of space-time is a microscopic S3 x S1 (S3 = 3-sphere), and that the space is metrically disturbed, so that an isometry is obtained only when a spatial rotation is linked with half a rotation of the S3.
Abstract: It is assumed that each point of space-time is a microscopic S3 x S1 (S3 = 3-sphere). This space is metrically disturbed, so that an isometry is obtained only when a spatial rotation is linked with half a rotation of the S3. This disturbed space is identified with the physical vacuum having an SO3-spin-structure.
01 Jan 1986
TL;DR: In this article, a non-local approach to the theory of relativity is discussed and a side benefit is that this approach is applicable to Maxwell and Yang-Mills fields on both flat asymptotically flat space times.
Abstract: The main purpose of this work is to discuss a non-local approach to the general theory of relativity. A side benefit is that this approach is applicable to Maxwell and Yang-Mills fields on both flat as well as asymptotically flat space times.
TL;DR: In this paper, the authors investigated the Painleve property of the vacuum Einstein field equations and observed that the field equations possess this property when spacetime admits commuting, non-null two Killing vector fields.
Abstract: The Painleve property of the vacuum Einstein field equations is investigated. It is observed that the field equations possess this property when spacetime admits commuting, nonnull two Killing vector fields.
TL;DR: In this article, it was shown that the decay of the muon to an electron and 2γ is forbidden within the author's generally covariant field theory, in which the state of 2γ in this process must identify with an exact solution for the coupled, nonlinear spinor equations for the electron-positron, corresponding to zero energy, zero momentum and zero angular momentum.
Abstract: It is shown that, within the author’s generally covariant field theory, the decay of the muon to an electron and 2γ is forbidden. This follows from earlier results of this field theory in which the state of 2γ in this process must identify with an exact solution for the coupled, nonlinear spinor equations for the electron-positron, corresponding to zero energy (relative to the asymptotically free state of energy 2m), zero momentum and zero angular momentum. In view of the conservation of energy-momentum, it then follows that the muon cannot decay to e±+2γ.
TL;DR: In this article, an expansion of the gravitational field is developed in powers of c−1 with coefficients depending on the asymptotically retarded time, and the far zone to the source is derived using the quadrupole moment of the source.