Showing papers on "Riemann curvature tensor published in 1973"

Conformal Energy-Momentum Tensor in Riemannian Space-Time

Abstract: We consider the scalar field with quartic self-interaction in Riemannian space-time. Identities are proved which connect the modified energy-momentum tensors of Callan, Coleman, and Jackiw in different conformally related space-times. We consider the quantized scalar field in a conformally flat metric, and show that our identities relate the matrix elements of the modified energy-momentum tensor to corresponding matrix elements in Minkowski space. We show further that when the mass can be neglected in the conformal wave equation there is no gravitationally induced particle creation in conformally flat space-times, thus generalizing a result proved earlier in the free-field case. The influence of additional fields and interactions on that result is briefly discussed.

The symmetries of Kerr black holes

Abstract: The Kerr solution describes, in Einstein's theory, the gravitational field of a rotating black hole. The axial symmetry and stationarity of the solution are shown here to arise in a simple way from properties of the curvature tensor.

Defect densities in directional media, mainly liquid crystals

Abstract: It is possible to define a local frame of reference at each point of a distorted liquid crystal, transformed from a fixed frame in the perfect crystal by a pure rotation. In the local frame the director n has constant components. A curvature tensor Kij is used to describe rotations between neighbouring frames; it obeys the compatibility condition θ = Rot K − M = 0, where M is a tensor formed from the minors of K. M takes into account the non-commutativity of small rotations. The deformation can therefore be described by three types of defect densities. αij = δij K ii − K ji is the source of translation dislocations, similar to the Nye tensor introduced in multiple glide. In a liquid crystal, in contradistinction to a solid, ’glide’ may involve stretching of the lattice. M ij is related to tilt and twist boundaries terminating in the medium, and is analogous to a disclination density. When disclination lines are present, θ ij = 0. There is, moreover, a vectorial density Λi = K ijnj, related to dis...

Curvature tensors and covariant derivatives

Abstract: The problems considered here are of two types.(i) What are implications of vanishing k-th covariant derivatives of curvature tensors?(ii) Under what conditions on curvature tensors, does the k-th covariant derivative ∇kT=0 for a tensor T mean ∇T=0?

Local extensions in singular space-times. II

Abstract: Previous results of the author are corrected by reformulating them in space-times whose Riemann tensor satisfies a Holder condition.

Model for a one-kink metric.

Abstract: It is well known that the metric tensor of general relativity may be associated with a conserved quantity, which is an integer known as the 'kink number'. This paper introduces a model for a metric corresponding to a single stationary kink centred at the origin. The Christoffel symbols and the energy-momentum tensor are deduced, and by investigating the asymptotic properties of the mass density, an inverse square law is seen to arise.

Estimates for solutions of quasilinear elliptic equations connected with problems of geometry "in the large"

Abstract: The following questions are presented in this paper: A geometric method for obtaining two-sided estimates for general quasilinear elliptic equations and its applications to problems of the calculus of variations and the problem of recovering a hypersurface from its mean curvature in spaces of constant curvature. Estimates of the modulus of the gradient for a hypersurface with boundary in a Riemannian space by means of its mean curvature and the metric tensor of the space. Estimates of the modulus of the gradient of a hypersurface depending on the distance of a point from the boundary and its mean curvature in Euclidean space. Estimates of these three types are of independent interest and play a fundamental role in the proofs of existence theorems for a hypersurface with prescribed mean curvature in Riemannian spaces. Bibliography: 3 items.

Spaces admitting gravitational fields

Abstract: We start with a modern version of Einstein's definition of a gravitational field. Tensors of curvature type and the curvature product of symmetric tensors are defined. The interaction tensor is defined as the curvature product of the fundamental tensor and the energy‐momentum tensor. The tensor W obtained by coupling the Riemann tensor and the interaction tensor is used to obtain a characterization of gravitational fields. The linear transformation of the space of second‐order differential forms, induced by W, is used to give a new definition of a gravitational field. The field equations are expressed in terms of the gravitational sectional curvature function f. Thorpe's theorem characterizing Einstein spaces is obtained as a corollary. New formulations of the field equations are used to solve the problem of classification of gravitational fields. The mathematical foundations of the theory of classification are examined and a geometric interpretation of classification is obtained by using the critical poi...

The Ontology of the Curvature of Empty Space in the Geometrodynamics of Clifford and Wheeler

Abstract: For nearly two decades before 1972, Professor John Wheeler pursued a research program in physics that was predicated on a monistic ontology which W. K. Clifford had envisioned in 1870 and which Wheeler (1962b, p. 225) epitomized in the following words: “There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the bending of space. Physics is geometry.” In an address to a 1960 Philosophy Congress (Wheeler, 1962a), he began with a qualitative synopsis of the protean role of curvature in endowing the one presumed ultimate substance, empty curved space, with a sufficient plurality of attributes to account for the observed diversity of the world. Said he: ... Is space-time only an arena within which fields and particles move about as “physical” and “foreign” entities? Or is the four-dimensional continuum all there is? Is curved empty geometry a kind of magic building material out of which everything in the physical world is made: (1) slow curvature in one region of space describes a gravitational field; (2) a rippled geometry with a different type of curvature somewhere else describes an electromagnetic field; (3) a knotted-up region of high curvature describes a concentration of charge and mass-energy that moves like a particle? Are fields and particles foreign entities immersed in geometry, or are they nothing but geometry? It would be difficult to name any issue more central to the plan of physics than this: whether space-time is only an arena, or whether it is everything [p. 361].

The Large Scale Structure of Space–Time: Space–time singularities

Abstract: In this chapter, we use the results of chapters 4 and 6 to establish some basic results about space–time singularities. The astrophysical and cosmological implications of these results are considered in the next chapters. In §8.1, we discuss the problem of defining singularities in space–time. We adopt b-incompleteness, a generalization of the idea of geodesic incompleteness, as an indication that singular points have been cut out of space–time, and characterize two possible ways in which b-incompleteness can be associated with some form of curvature singularity. In §8.2, four theorems are given which prove the existence of incompleteness under a wide variety of situations. In §8.3 we give Schmidt's construction of the b-boundary which represents the singular points of space–time. In §8.4 we prove that the singularities predicted by at least one of the the theorems cannot be just a discontinuity in the curvature tensor. We also show that there is not only one incomplete geodesic, but a three-parameter family of them. In §8.5 we discuss the situation in which the incomplete curves are totally or partially imprisoned in a compact region of space–time. This is shown to be related to non-Hausdorff behaviour of the b-boundary. We show that in a generic space–time, an observer travelling on one of these incomplete curves would experience infinite curvature forces. We also show that the kind of behaviour which occurs in Taub–NUT space cannot happen if there is some matter present.

General Relativity, Geometrodynamics and Ontology

Abstract: For nearly two decades before 1972, Professor John Wheeler pursued a research program in physics that was predicated on a monistic ontology which W. K. Clifford had envisioned in 1870 and which Wheeler (1962b, p. 225) epitomized in the following words: “There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the bending of space. Physics is geometry.” In an address to a 1960 Philosophy Congress (Wheeler, 1962a), he began with a qualitative synopsis of the protean role of curvature in endowing the one presumed ultimate substance, empty curved space, with a sufficient plurality of attributes to account for the observed diversity of the world. Said he (1962a): … Is space-time only an arena within which fields and particles move about as ‘physical’ and ‘foreign’ entities? Or is the four-dimensional continuum all there is? Is curved empty geometry a kind of magic building material out of which everything in the physical world is made: (1) slow curvature in one region of space describes a gravitational field; (2) a rippled geometry with a different type of curvature somewhere else describes an electromagnetic field; (3) a knotted-up region of high curvature describes a concentration of charge and mass-energy that moves like a particle? Are fields and particles foreign entities immersed in geometry, or are they nothing but geometry? It would be difficult to name any issue more central to the plan of physics than this: whether space-time is only an arena, or whether it is everything (p. 361).

Explizite Darstellungsformeln GREENscher Funktionen von kovarianten Wellengleichungen im schwachen Gravitationsfeld. II. Linearisierte EINSTEINsche Gravitationsgleichungen

Abstract: The propagation equations for small perturbations of a background gravitational field satisfying the EINSTEIN equations are considered For the perturbation potential the covariantly generalized EINSTEIN-HILBERT gauge is chosen With the aid of the method used in [10], bitensor GREEN's functions for the propagation equations in a weak vacuum field are given explicitly The tail term is obtained to be an integral of the first-order RIEMANN curvature tensor As an application of the formulae, GREEN's functions for perturbations of the SCHWARZSCHILD metric are calculated to first order in the mass parameter

The Large Scale Structure of Space–Time: The physical significance of curvature

Abstract: In this chapter we consider the effect of space–time curvature on families of timelike and null curves These could represent flow lines of fluids or the histories of photons In §41 and §42 we derive the formulae for the rate of change of vorticity, shear and expansion of such families of curves; the equation for the rate of change of expansion (Raychaudhuri's equation) plays a central role in the proofs of the singularity theorems of chapter 8 In §43 we discuss the general inequalities on the energy–momentum tensor which imply that the gravitational effect of matter is always to tend to cause convergence of timelike and of null curves A consequence of these energy conditions is, as is seen in §44, that conjugate or focal points will occur in families of non-rotating timelike or null geodesics in general space–times In §45 it is shown that the existence of conjugate points implies the existence of variations of curves between two points which take a null geodesic into a timelike curve, or a timelike geodesic into a longer timelike curve Timelike curves In chapter 3 we saw that if the metric was static there was a relation between the magnitude of the timelike Killing vector and the Newtonian potential One was able to tell whether a body was in a gravitational field by whether, if released from rest, it would accelerate with respect to the static frame defined by the Killing vector

Lie triple systems and affinely connected spaces

Abstract: In this paper the author examines certain properties of affinely connected spaces whose curvature tensor satisfies precise conditions of an algebraic character. From the basic results obtained follow sufficiency conditions under which a compact Riemannian space is symmetric.

Normal-dominated singularities in static space-times

Abstract: We define “normal-dominated” singularities of static solutions of the Einstein equations and show that a uniquely and invariantly defined structure can be assigned to these singularities. We find for the general solution that the dominant term of the Riemann tensor near the singularity is of Petrov Type N. Except for one special class of solutions, it seems that in general the shear of the null geodesics blows up at the same rate as their convergence near the singularity, in contradistinction to the “elementary singularity” of Newman and Posadas. We compute the structure for a variety of known static solutions as well as the stationary Kerr-Newman metrics.

The six dimensional Gauss Bonnet integrand

Abstract: Sufficient conditions are given for a compact 6dimensional Kahler manifold with nonnegative (nonpositive) curvature to have nonnegative (nonpositive) Euler characteristic.