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Showing papers on "Introduction to the mathematics of general relativity published in 1975"


Journal ArticleDOI
TL;DR: In this article, a general theory of complex V4 spaces of this type is outlined and examples of nontrivial solutions of all degenerate algebraic types are provided, where Γ = 0 and therefore a fortiori equations Gab = 0 are fulfilled.
Abstract: Complex V4’s are investigated where ΓȦḂ =0 and therefore a fortiori equations Gab=0 are fulfilled. A general theory of spaces of this type is outlined and examples of nontrivial solutions of all degenerate algebraic types are provided.

498 citations


Book
01 Jan 1975
TL;DR: In this article, the authors present a collection of 475 problems with solutions in the fields of special and general relativity, gravitation, relativistic astrophysics, and cosmology, expressed in broad physical terms to enhance their pertinence to readers with diverse backgrounds.
Abstract: Important and useful to every student of relativity, this book is a unique collection of some 475 problems--with solutions--in the fields of special and general relativity, gravitation, relativistic astrophysics, and cosmology. The problems are expressed in broad physical terms to enhance their pertinence to readers with diverse backgrounds. In their solutions, the authors have attempted to convey a mode of approach to these kinds of problems, revealing procedures that can reduce the labor of calculations while avoiding the pitfall of too much or too powerful formalism. Although well suited for individual use, the volume may also be used with one of the modem textbooks in general relativity.

305 citations


Journal ArticleDOI
TL;DR: A singularity-free solution for a static charged fluid sphere in general relativity was obtained in this article, where the solution satisfies physical conditions inside the sphere and satisfies physical properties inside the fluid sphere.
Abstract: A singularity-free solution was obtained for a static charged fluid sphere in general relativity The solution satisfies physical conditions inside the sphere

226 citations


Journal ArticleDOI
TL;DR: In this paper, a simple theorem whose physical interpretation is that an isolated, gravitating body in general relativity moves approximately along a geodesic is obtained, and the theorem is proved.
Abstract: A simple theorem, whose physical interpretation is that an isolated, gravitating body in general relativity moves approximately along a geodesic, is obtained.

109 citations



Journal ArticleDOI
TL;DR: In this paper, a spherically symmetric solution of Yang's vacuum gravitational field equations is given which predicts incorrect values for experimental observations, and it is argued that Yang's field equations must be supplemented by further restrictions on the class of allowable space-times.
Abstract: A static, spherically symmetric solution of Yang's vacuum gravitational field equations is given which predicts incorrect values for experimental observations. It is argued that Yang's field equations must be supplemented by further restrictions on the class of allowable space-times.

56 citations


Journal ArticleDOI
TL;DR: In this article, a study is made of geometric theories of gravitation that are consistent with the local validity of Newtonian dynamics, which involves an analysis of the representations of the Galilean group provided by the curvature tensor of a Newtonian spacetime, and by the contravariant massmomentum tensor.
Abstract: A study is made of geometric theories of gravitation that are consistent with the local validity of Newtonian dynamics. This involves an analysis of the representations of the Galilean group provided by the curvature tensor of a Newtonian spacetime, and by the contravariant mass-momentum tensor. Subject to certain assumptions that are made also in the foundations of general relativity, it is shown that there exists essentially only one such theory that does not place unacceptable restrictions on the mass density of the source. This is the Newtonian theory, generalized by a cosmological term. Any other theory is weaker and is given by a subset of the geometrical equations of the Newtonian theory.

44 citations


Journal ArticleDOI
TL;DR: In this paper, a family of static, axisymmetric, asymptotically flat solutions of the Einstein equations is discussed, and a source with an exterior described by a member of this family initially could have an area smaller than that of a n appropriately defined Schwarzchild surface.

40 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the only solution of this sort in which one of the Petrov scalars is zero is the trivial flat−space one, and that the point at which the four Petrov scales vanish simultaneously (zero curvature tensor) cannot be included as a regular point of a neighborhood over which the scalars are functionally independent.
Abstract: Vacuum Einstein metrics of Petrov type I, general, are considered. It is shown that the only solution of this sort in which one of the Petrov scalars is zero is the trivial flat−space one. Further, it is shown that the point at which the four Petrov scalars vanish simultaneously (zero curvature tensor) cannot be included as a regular point of a neighborhood over which the scalars are functionally independent. In fact, for type I all derivatives of the Petrov scalars must vanish at a point at which the curvature tensor does so that this point cannot be a regular point of any nontrivial analytic solution.

36 citations




Journal ArticleDOI
TL;DR: In this article, a complete set of solutions of the field equations with Tij = (e + p)uiuj − pgij is found, and they divide into three families, first of which contains six types of new solutions with nonzero pressure.
Abstract: The equations of isentropic rotational motion of a perfect fluid are investigated with use of Darboux’ theorem. It is shown that, together with the equation of continuity, they guarantee the existence of four scalar functions on space−time, which constitute a dynamically distinguished set of coordinates. It is assumed that in this coordinate system the metric tensor is constant along the lines tangent to velocity and vorticity fields. Under these assumptions a complete set of solutions of the field equations with Tij = (e + p)uiuj − pgij is found. They divide into three families, first of which contains six types of new solutions with nonzero pressure. The second family contains only the Godel’s solution, and the third one, only the Lanczos’ solution. Symmetry groups, exterior metrics, type of conformal curvature, geometrical and physical properties of the new solutions are investigated. A short review of other models of rotating matter is given.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the worldsheet equation (8) can be derived in general relativity following Einstein's method for the geodesic world line following the conservation law.
Abstract: x 5(x' —z'{s))5(x' —z'(s }) is the mass density which is infinite on the world line z =z (s) of the singularity and zero elsewhere Because of the scalar density character of the 6 function the left-hand side of Eq (2} is a tensor density From Eq (1) follows the conservation law The string equation of motion was further studied by many authors' ' in connection with duality The purpose of this note is to show that the worldsheet equation (8) can be derived in general relativity following Einstein's method for the geodesic world line

Journal ArticleDOI
TL;DR: In this paper, it was proved that a necessary and sufficient condition for a shear-free perfect fluid to be irrotational is that the Weyl tensor be pure electric type.
Abstract: It is proved that a necessary and sufficient condition for a shear‐free perfect fluid to be irrotational is that the Weyl tensor be pure electric type. For shear‐free isentropic flow with unit tangent uα, we find the conservation law ∇α(n1/3iω uα) =0, where i is the relativistic specific enthalpy, n is the conserved particle number density, and ω is the vorticity scalar.

Journal ArticleDOI
TL;DR: In this article, the electromagnetic structure of a one-body solution of the coupled Einstein-Maxwell equations endowed with mass m, charge Q, specific angular momentum a, and a parameter l is examined.
Abstract: We examine the electromagnetic structure of a one-body solution of the coupled Einstein--Maxwell equations endowed with mass m, charge Q, specific angular momentum a, and a parameter l. It is shown how the parameter l, introduced by Newman, Tamburino, and Unti, is related to the magnetic monopole charge distribution of the solution. A relation is presented between the total mass energy of the system and its irreducible mass. The total mass energy can be much smaller than the irreducible mass. A general solution characterized by the four parameters m, Q, a, and l is here introduced. (AIP)

Journal ArticleDOI
TL;DR: In this paper, any metric gravitation theory (including general relativity) is shown to determine transport equations for the connection and curvature of the Lorentz frame bundle P4 defined by the metric g.
Abstract: Any ’’metric gravitation theory’’ (including general relativity) is shown to determine transport equations for the connection and curvature of the Lorentz frame bundle P4 defined by the metric g. Observers are generally defined as curves in P4 which project down to timelike trajectories in space–time. The transport of curvature along an observer trajectory is then given by a Lorentz Lie algebra‐valued current composed of an internal and external part. Einstein’s equations are shown to define one part of the self‐dual limit of the usual Yang–Mills gauge equations, here called a particular form of curvature dynamics. As a consequence, the Yang–Mills‐like energy–momentum tensor, introduced for the Lorentz connection, vanishes identically under Einstein’s vacuum conditions.


Journal ArticleDOI
TL;DR: In this article, it was shown that the conventional correspondence procedure used in the transition from Newtonian to the Einsteinian theory of gravitation is not unique, and a new procedure of correspondence whereby the theory is required to reduce to a special relativistic limit before it reduces to the Newtonian limit is described.
Abstract: It is shown that the conventional correspondence procedure used in the transition from Newtonian to the Einsteinian theory of gravitation is not unique. A new procedure of correspondence whereby the theory is required to reduce to a special relativistic limit before it reduces to the Newtonian limit is described. It is shown that the special relativistic correspondence leads to first and second order requirements, whereas the Newtonian correspondence leads only to first order requirements. The resulting theory is logically plausible and experimentally viable.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the intermediate singularities become localized curvature singularities while the Cauchy horizons are a stable feature of the models and that scalar-wave propagation in these spaces is possible.
Abstract: It has been shown that "intermediate" singularities, where all Riemann tensor invariants are finite, occur in certain cosmological models. Associated with the singularities are Cauchy horizons, across which the matter flows into a stationary region of space-time. We investigate scalar-wave propagation in these spaces. Our results suggest that the intermediate singularities become localized curvature singularities while the Cauchy horizons are a stable feature of the models.


Journal ArticleDOI
TL;DR: In this article, the identifications of the field equations and Bianchi identities of the yA−formalism with the main equations and constraint equations of a metric in a Bondi coordinate system were investigated.
Abstract: This paper applies the ’’yA−formalism’’ of a previous paper (Paper I) to a Lagrangian formulation of the characteristic initial−value problem in general relativity. The essential content of the paper is the respective identifications of the ’’field equations’’ and ’’Bianchi identities’’ of the yA−formalism with the ’’main equations’’ and ’’constraint equations’’ of a metric in a Bondi coordinate system. The identifications are developed in detail in the case of Bondi’s (axially symmetric) radiating metric.

Journal ArticleDOI
TL;DR: In this article, a Lagrangian formalism (the yA−formalism) was developed for use in investigating the Cauchy problem in general relativity, which enables one to define field equations and Bianchi identities for metrics containing arbitrary functions in the case when the arbitrary functions are treated as the field variables.
Abstract: This paper develops a Lagrangian formalism (the yA−formalism) for use in investigating the Cauchy problem in general relativity. In particular it will be used in a subsequent paper to present a Lagrangian formulation of the characteristic initial−value problem in general relativity. The formalism enables one to define ’’field equations’’ and ’’Bianchi identities’’ for metrics containing arbitrary functions in the case when the arbitrary functions are treated as the field variables.




Journal ArticleDOI
TL;DR: In this article, it was shown that a nonstatic charged fluid sphere with nonvanishing pressure undergoing isotropic contraction or expansion can be constructed satisfying certain physical conditions such as both density and pressure of the matter.
Abstract: It is shown that models for a nonstatic charged fluid sphere with nonvanishing pressure undergoing isotropic contraction or expansion can be constructed satisfying certain physical conditions such as both density and pressure of the matter are positive Such spheres, in general, collapse to a point singularity In one case, however, the system may bounce back after reaching a minimum volume, provided a tension is allowed within the distributions

Journal ArticleDOI
TL;DR: In this paper, an interior solution to plane-symmetric Einstein-Maxwell equations is given for zero pressure for the general case, assuming a functional relationship between the coefficients of the metric and the solution can be written in an integral form.
Abstract: For zero pressure we obtain an interior solution to plane-symmetric Einstein-Maxwell equations. For the general case we assume a functional relationship between the coefficients of the metric and show that the solution can be written in an integral form.

Book ChapterDOI
01 Jan 1975
TL;DR: In this paper, the authors highlight the gravity field equations and show that the principle of superposition is valid only approximately for weak fields that permit a linearization of the Einstein equations, particularly the gravitational field in the classical Newtonian limit.
Abstract: Publisher Summary This chapter highlights the gravitational field equations. In a curved space, the parallel displacement of a vector from one given point to another gives different results if the displacement is carried out over different paths. The Einstein equations are nonlinear and are the required equations of the gravitational field—the basic equation of the general theory of relativity. Therefore, for gravitational fields, the principle of superposition is not valid. The principle is valid only approximately for weak fields that permit a linearization of the Einstein equations, particularly the gravitational field in the classical Newtonian limit. The equation of state relates to one another not two but three thermodynamic quantities, for example, the pressure, density, and temperature of the matter. In applications in the theory of gravitation, this point is, however, not important, as the approximate equations of state used here actually do not depend on the temperature. To understand the solution of the Einstein equations for given initial conditions (in the time), the question of the number of quantities for which the initial spatial distribution can be assigned arbitrarily must be considered. The Einstein equations can also be written in an analogous way for the general case of a time-dependent metric.