Showing papers on "Riemann curvature tensor published in 1974"

Conformal energy-momentum tensor in curved spacetime: Adiabatic regularization and renormalization

Abstract: In preparation for an investigation of whether field-theoretic effects helped to make the early universe become isotropic, we seek to determine the physical (divergence-free) energy-momentum tensor through which the geometry of spacetime is influenced by a quantized scalar field with conformal ("new improved") coupling to the metric. The cosmological models studied are the Kasner-like (type I) metrics (homogeneous, spatially flat, nonrotating, but anisotropic), and also the isotropic Robertson-Walker metrics. The methods employed have previously been expounded in the context of a minimally coupled scalar field and a Robertson-Walker metric. Three divergent leading terms are extracted from an adiabatic expansion of the formal expressions for the expectation values of the energy density and pressures. In the Kasner case a slight reshuffling of the leading terms in the energy density displays all divergences to be proportional to either the metric tensor or a second-order curvature tensor which vanishes when the spacetime is isotropic; hence a finite energy-momentum tensor remains after renormalization of the cosmological constant and one other coupling constant in a generalized Einstein equation. In the Robertson-Walker cases, because of conformal flatness, there is no divergence beyond the usual quartically divergent constant vacuum energy; when the mass is not zero, however, a finite renormalization of the gravitational constant is suggested. The correctness of the methods is tested by considering a coordinate system in which flat spacetime assumes the form of a Kasner universe: The adiabatic definition of particle number and vacuum, which is basic to our expansion and renormalization methods, is seen to be consistent with the usual flat-space concepts.

Curvature Functions for Open 2-Manifolds

Abstract: The basic problem posed in [12] is that of describing the set of Gaussian curvature functions which a given 2-dimensional manifold M can possess. In this paper we consider this problem for the case of non-compact M. Other than the Gauss-Bonnet type inequality of Cohn-Vossen [4] (see also [6], [8]), which holds for certain complete metrics on non-compact manifolds, there is no known a priori restriction for a Gaussian curvature function on a non-compact 2-manifold. Indeed, Gromov has recently shown that any non-compact 2-manifold possesses a metric of strictly positive curvature as well as a metric of strictly negative curvature [7]. As the analogue of Question 1 of [12] it therefore seems natural to ask the following:

Teukolsky equation and Penrose wave equation

Abstract: It is shown that Teukolsky's equation can be derived from a second-order wave equation for the Riemann tensor. The derivation is done in a way that emphasizes the modern tensor-analysis content of the Newman-Penrose formalism.

The total absolute curvature of closed curves in Riemannian manifolds

Abstract: This paper is concerned with some extensions of the theorems of Fenchel [3], Milnor [5], Fary [2] on the total absolute curvature of closed curves in euclidean space. We obtain a theorem for closed curves in a complete simply connected riemannian n-manifold with nonpositive sectional curvature, and this leads to more precise results when the curvature is constant. Throughout this paper the summation convention for repeated indices is used, and all indices take the values 1, , n unless stated otherwise.

Space tensors in general relativity. III. The structural equations

Abstract: A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), dT (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis $$(\tilde abla ,\tilde abla T)$$ (M,Γ). The structural equations of $$(\tilde abla ,\tilde abla T)$$ and the corresponding spatial Bianchi identities are discussed. The special case $$(\tilde abla ,\tilde abla T) = (\tilde abla *,\tilde abla T*)$$ is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν4,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.

Weak gravitational fields

Abstract: We consider the set of Ck bounded tensor fields of type (r,s) on R4 in the topology of uniform Ck convergence. For each k≥2, the map sending a metric to its curvature tensor is shown to be analytic at the Minkowski metric. The same is true of the map sending a metric to its Einstein tensor. The well‐known linearized theory of gravitation amounts to studying the directional derivatives of these maps. An iterative method for solving the full field equations along an analytic curve of Einstein tensors passing through zero is proposed.

New commutator identities on the Riemann tensor

Abstract: A number of new generally covariant identities which involve second derivatives of the Riemann tensor are presented. Each of these new identities can be expressed by equating to zero either (a) a particular sum of terms each of which contains an operator of the form (▿μ▿ν − ▿ν▿μ) acting on the Riemann tensor; or (b) a particular sum of terms each of which contains an operator of the form ▿a acting either on the expression (∇μRαβτω+∇βRματω+∇αRβμτω) or on the expression (∇μRαβτω+∇βRματω+∇αRβμτω); or (c) a particular sum of algebraic terms each of which contains no derivatives of the Riemann tensor, but rather is quadratic in the Riemann tensor. Each of the new identities can be expressed in all three of the above‐described forms. Furthermore, each of these new identities can be thought of as an integrability condition derived from the equations that define the Riemann tensor in terms of the Γαβω or the gμν. The requirements of Riquier's existence theorem are used to guide the derivation of the identities. T...

Submanifolds with prescribed mean curvature vector field

Abstract: In a recent paper [2] K. Nomizu has shown that a natural analogue of an n-sphere in an arbitrary Riemannian manifold is an n-dimensional umbilical submanifold with non-zero parallel mean curvature vector, which he calls “extrinsic sphere” sometimes. This note is concerned with the question whether extrinsic spheres have a special topological or differentiable feature.

On the generalized Lagrangian for general relativity and some of its implications.—I

Abstract: The most general scalar Lagrangian leading to Einstein’s field equations (without the cosmological term) is given byL =L0+(−G1/2. denotes the usual scalar density of Einstein’s theory whileF andK are certain invariants quadratic in the components of the curvature tensor. The contribution toL resulting froma=0 and considered in this not eis conformally invariant. The implications of conformal mapping for the Larangian formalism are discussed. Due to the vanishing of the Lagrange derivative of (−g)1/2K, this density may be expressed as the divergence of a 4-component quantity. A general expression for this quantity is determined without use of special co-ordinates. New identities for the Riemann curvature tensor or Weyl’s curvature tensor are derived which are bilinear in the components of such tensors and their duals. Finally, a contribution to the energymomentum complex is obtained containing terms quadratic in the second derivative of the metric.

Vector potential and Riemannian space

Abstract: This paper uncovers the basic reason for the mysterious change of sign from plus to minus in the fourth coordinate of nature's Pythagorean law, usually accepted on empirical grounds, although it destroys the rational basis of a Riemannian geometry. Here we assume a genuine, positive-definite Riemannian space and an action principle which is quadratic in the curvature quantities (and thus scale invariant). The constant σ between the two basic invariants is equated to1/2. Then the matter tensor has the trace zero. In consequence of the constancy of the scalar curvature and the divergence identity of the matter tensor, the perturbation metric has to satisfy a scalar and a vector condition, with a negative sign in the fourth coordinate. These conditions lead to the Lorentz condition and the wave equation for the vector potential. Thus the entire Maxwell-Lorentz type of electrodynamics becomes logically derivable, making no concession to any irrationality.

Asymptotically simple space--time manifolds

Abstract: Asymptotic simplicity is shown to be k‐stable (k≥3) at any Minkowski metric on R4 in both the Whitney fine Ck topology and a coarser topology (in which the Ck twice‐convariant symmetric tensors form a Banach manifold whose connected components consist of tensor field asymptotic to one another at null infinity). This result, together with a sequential method for solving the field equations previously proposed by the authors, allows a fairly straightforward proof that a well‐known result in the linearized theory holds in the full nonlinear theory as well: There are no nontrivial (i.e., non‐Minkowskian) asymptotically simple vacuum metrics on R4 whose conformal curvature tensors result from prescribing zero initial data on past null infinity.

On global embedding

Abstract: A basis vector treatment of tensor calculus in an N ‐dimensional (pseudo‐) Euclidean space is used to obtain new insights into the geometrical properties of curved Riemannian spaces of smaller dimension which are globally embedded in the N ‐space. In particular, it is shown that, in general for a globally embedded hypersurface, (i) partial derivatives of internal basis vectors with respect to internal coordinates must in general be expressed as a linear combination of external as well as internal basis vectors, (ii) there exist two different geometrical expressions, always equal in value, for the intrinsic curvature tensor, (iii) the geodesic equation contains more terms than does the usual one; the extra terms vanish for Schwarzschild metric embeddings. These points are illustrated by the example of a 2‐sphere embedded in Euclidean 3‐space.

Curvature tensors in riemannian manifold—II

Abstract: In a recent paper author and Mishra [1] have studied the recurrent properties of conformal curvature tensor, conharmonic curvature tensor, concircular curvature tensor and projective curvature tensor in a Riemannian manifold. In another paper author and Mishra [2] have defined a new curvature tensor and obtained its relativistic significance. In this paper the geometrical properties of the new tensor in a Riemannian manifold have been discussed.

Representation and differential geometry of the semisimple Lie groups

Abstract: A systematic method is presented whereby any compact Lie group of n‐real parameters is dealt with from an infinitesimal approach with the representative matrix method based on a group of inner automorphisms suggested in a previous paper. The group manifold, defined in terms of a metric of group parameters, is identified as a Riemannian one in which these parameters play a role of n curvilinear coordinates. Riemannian geometry is thus valid in the group manifold, and geometric quantities are explicitly calculated in terms of the symmetric or (0) connection by a straightforward application of the ordinary procedure of tensor analysis. A new and simpler method of computing the invariant volume element is presented within this framework. Furthermore, we discuss in detail the group of inner automorphisms for the calculation of the matrix element of finite rotations for any irreducible representation (abbreviated MEFRIR). It is found that our method works very well and yields right and left vector fields together with a set of 2n equations to be satisfied by the MEFRIR. The global properties of the group may, therefore, be obtained as a solution to these equations. Moreover, it provides not only the generalized Maurer‐Cartan equations, the Lie structure formulas, two parameter groups of point transformations and the adjoint group, but also two additional nonsymmetric (+) and (−) connections with zero curvature, which do not possess any preassigned metric but possess two absolute parallelisms. Thus, our results on differential geometry completely agree with Cartan and Schouten's. A link between differential geometry and representations is presented by the right and left vector fields which are explicitly calculable in terms of the n parameters and through which geometric quantities, e.g., the Riemann tensor, the Ricci tensor, and the scalar curvature, of any connection are explicitly displayed. A theorem relating both the vector fields to the metric tensors is also included. Finally, the l (rank of the group) invariant differential equations to be satisfied by the MEFRIR are cast in the covariant (or Lie derivative) forms in any connection. Examples of the invariant equations are given for SU(2), SO(3), and SU(3). The two invariant equations of the latter can be cast in terms of the eigenvalues of isospin and hypercharge upon carrying out charge and hypercharge quantizations; in this connection, a new nonrelativistic wave equation to be satisfied by SU(3) multiplets as a generalization of the Schrodinger equation of the symmetric top is also proposed.

Kaehlerian manifolds with constant scalar curvature whose Bochner curvature tensor vanishes

Abstract: Publisher Summary This chapter proves the theorem corresponding to that of Ryan, replacing the vanishing of the Weyl conformal curvature tensor in a Riemannian manifold by that of the Bochner curvature tensor in a Kaehlerian manifold. The chapter proves some lemmas that are used in the proof of the theorem. In a Kaehlerian manifold M of dimension n, the scalar curvature is constant, the Bochner curvature tensor vanishes and the Ricci tensor is positive semi-definite. From the method of the proof, it is easily see that the conclusion of the theorem is also valid if the assumptions of compactness and constant scalar curvature are replaced by local homogeneity of M.