Showing papers on "Riemann curvature tensor published in 1994"

Hamiltonian formulation of the teleparallel description of general relativity

Abstract: The Hamiltonian formulation of the teleparallel description of Einstein’s general relativity is established. Under a particular gauge fixing the Hamiltonian of the theory is written in terms of first class constraints. The algebra of the Hamiltonian and vector constraints resembles that of the standard Arnowitt–Deser–Misner formulation. This geometrical framework might be relevant as it is known that in manifolds with vanishing curvature tensor but with nonzero torsion tensor it is possible to carry out a simple construction of Becchi–Rouet–Stora–Tyutin‐like operators.

Dimensionally continued black holes.

Abstract: Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with a negative cosmological constant are studied. The action is, in odd dimensions, the Chern-Simons form for the anti-de Sitter group and, in even dimensions, the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. Both actions are special cases of the Lovelock action, and they reduce to the Hilbert action (with a negative cosmological constant) in the lower dimensional cases $\mathcal{D}=3$ and $\mathcal{D}=4$. Exact black hole solutions characterized by mass ($M$) and electric charge ($Q$) are found. In odd dimensions a negative cosmological constant is necessary to obtain a black hole, while in even dimensions both asymptotically flat and asymptotically anti-de Sitter black holes exist. The causal structure is analyzed and the Penrose diagrams are exhibited. The curvature tensor is singular at the origin for all dimensions greater than three. In dimensions of the form $\mathcal{D}=4k,4k\ensuremath{-}1$, the number of horizons may be zero, one, or two, depending on the relative values of $M$ and $Q$, while for a negative mass there is no horizon for any real value of $Q$. In the other cases, $\mathcal{D}=4k+1,4k+2$, both naked and dressed singularities with a positive mass exist. As in three dimensions, in all odd dimensions anti-de Sitter space appears as a "bound state" of mass $M=\ensuremath{-}1$, separated from the continuous spectrum ($M\ensuremath{\ge}0$) by a gap of naked curvature singularities. In even dimensions anti-de Sitter space has zero mass. The analysis is Hamiltonian throughout, considerably simplifying the discussion of the boundary terms in the action and the thermodynamics. The Euclidean black hole has the topology ${R}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{\mathcal{D}\ensuremath{-}2}$. Evaluation of the Euclidean action gives explicit expressions for all the relevant thermodynamical parameters of the system. The entropy, defined as a surface term in the action coming from the horizon, is shown to be a monotonically increasing function of the black hole radius, different from the area for $\mathcal{D}g4$.

The Analysis of Space-Time Singularities

Abstract: The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

Quadratic divergence of geodesics in CAT(0) spaces

Abstract: A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.

Almost Hermitian Geometry

Abstract: An algebraic study is made of the torsion and curvature of almost-Hermitian manifolds with emphasis on the space of curvature tensors orthogonal to those of Kahler metrics.

Supergravity coupled to chiral matter at one loop.

Abstract: We extend earlier calculations of the one-loop contributions to the effective Bose Lagrangian in supergravity coupled to chiral matter. We evaluate all logarithmically divergent contributions for arbitrary background scalar fields and space-time metrics. We show that, with a judicious choice of gauge fixing and of the definition of the action expansion, much of the result can be absorbed into a redefinition of the metric and a renormalization of the Kaehler potential. Most of the remaining terms depend on the curvature of the Kaehler metric. Further simplification occurs in models obtained from superstrings in which the Kaehler Riemann tensor is covariantly constant.

Manifolds of Positive Scalar Curvature

Abstract: We survey the status of the problem of determining which differentiable manifolds (without boundary) have Riemannian metrics of positive scalar curvature. Of course, if the manifold is non-compact, one requires the metric to be complete.

Spectral Geometry and One-loop Divergences on Manifolds with Conical Singularities

Abstract: Geometrical form of the one-loop divergences induced by conical singularities of background manifolds is studied. To this aim the heat kernel asymptotic expansion on spaces having the structure $C_{\alpha}\times \Sigma$ near singular surface $\Sigma$ is analysed. Surface corrections to standard second and third heat coefficients are obtained explicitly in terms of angle $\alpha$ of a cone $C_{\alpha}$ and components of the Riemann tensor. These results are compared to ones to be already known for some particular cases. Physical aspects of the surface divergences are shortly discussed.

Gravity from non-commutative geometry

Abstract: We introduce the linear connection in the non-commutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, define the metric connection and calculate the curvature. We define also the Ricci tensor and the scalar curvature. We find that the latter differs from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant, and apart from that we find no other dynamic fields in the model. Finally we discuss an example solution of flat linear connection, with the non-trivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as confirmed by the standard model, with the scaling factor corresponding to the Weinberg angle.

Addendum: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature

Abstract: is finite. On the other hand the argument at the beginning of Proposition 2.1 in [E] shows that if Al (B) is finite then Q(M, OM) is finite. Jin Zhiren pointed out to me that the Sobolev quotient Q(M, O9M) can be -oo. This is the case if we delete a small geodesic ball on a compact manifold without boundary with negative scalar curvature. More generally the Sobolev quotient is -oo if the first eigenvalue for the conformal Laplacian, with respect to Dirichlet boundary condition, is negative. In order to see that let p1 be the first eigenfunction for the problem

On existence of Kähler metrics with constant scalar curvature

Abstract: Let N be a compact Kahlerian manifold, let Ω be a Kahler class on ΛΓ, and let Ω be the set of Kahler forms representing Ω. On Ω + , consider the functional ΦΩ that assigns to a Kahler form the square of the L -norm of the scalar curvature. A critical point of ΦΩ is called an extremal Kahler metric. Any Kahler metric with constant scalar curvature is extremal. Conversely, the variational appraoch can be used to find metrics with constant scalar curvature. We begin with an existence theorem for extremal metrics. Recall that a Kahler metric is called a generalized Einstein-Kahler metric if the eigenvalues of the Ricci tensor are constant, see [27]. For example, a product of Einstein-Kahler metrics is a generalized Einstein-Kahler metric. If M is homogeneous under the action of a compact Lie group, then every Kahler class on M is represented by a generalized Einstein-Kahler metric.

The wave equation for the Lanczos potential. I

Abstract: The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor C$\_{abcd}$ of Weyl. For this field Lanczos found a tensor potential L$\_{abc}$ with 16 independent components. We can make L$\_{abc}$ have only 10 effective degrees of freedom by imposing the 6 gauge conditions L$ \smallmatrix ab;s \\ s \endsmallmatrix $ = 0. Both fields C$\_{abcd}$, L$\_{abc}$ satisfy wave equations. The wave equation satisfied by L$\_{abc}$ is nonlinear, even in vacuo. However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for L$\_{abc}$ has yet been published. Here, we derive a correct tensor wave equation for L$\_{abc}$ and when it is simplified with the aid of some four-dimensional identities it is equivalent to Illge's wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge's spinor wave equation. A set of analogues of well-known results of classical electromagentic radiation theory can now be given. We indicate how a Green's function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.

Curvature of gravitationally bound mechanical systems.

Abstract: In the present work mathematical aspects of determining the local instability parameters are focused on by using invariant characteristics of the internal Riemannian geometry with the Jacobi metric (in principle, for Hamiltonian dynamical systems with the natural Lagrangian). First, it is shown that the Ricci scalar indeed measures the sectional curvature averaged upon all two‐directions. Second, necessary and sufficient criteria for non‐negativity and of nonpositivity of the sectional curvature for any system with the natural Lagrangian are given. Third, analytical formulas allowing us to compute the separation rate of nearby trajectories are given. Fourth, it is shown that for any collisionless problem of n gravitationally bounded bodies, the sectional curvature in every direction is negative if n tends to infinity.

On Riemannian 3-manifolds with distinct constant Ricci eigenvalues

Abstract: According to Singer [Si] a Riemannian manifold (M, g) is said to be curvature homogeneous if, for every two points, p, q~M, there is a linear isometry F : T p M ~ T q M between the corresponding tangent spaces such that F* Rq=Rp (where R denotes the curvature tensor of type (0, 4)). Note that a (locally) homogeneous Riemannian manifold is automatically curvature homogeneous. Explicit locally nonhomogeneous examples have been constructed by many authors ([Sel, T, Ya, K-T-V1-K-T-V3, K1-K3]; see especially [K-T-V2] and [K-T-V3] for more complete references). For 3-dimensional Riemannian manifolds (M, 9) the following simple criterion holds: (M, g) is curvature homogeneous if and only if all Ricci eigenvalues of (M, g) are constant. Thus the problem to classify all 3-dimensional Riemannian manifolds with prescribed constant Ricci eigenvalues is of considerable interest. This problem was investigated already in 1916 by Bianchi [B] who made a classification under a strong additional hypothesis of "normality". He found only some homogeneous Riemannian spaces as solutions. On the other hand, the following conclusion follows from an observation by Milnor [M] and a result by Sekigawa [Se2]: For a homogeneous Riemannian 3-manifold (M, g), the signature of the Ricci tensor is never equal to ( +, +, ) or (+,0, -). Let us describe shortly what is known about the problem at the present. The case in which all Ricci eigenvalues are equal is trivial we obtain only spaces of constant curvature. The case pl = P2 4= P3 was solved completely by the first author in [K 1] and [K2]. The answer is that the local isometry classes of the corresponding metrics always depend on two arbitrary functions of one variable. (Notice that the local isometry classes of locally homogeneous spaces

Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions

Abstract: The existing refutal, in four-dimensional spacetimes, of the conjecture that the Lanczos tensor can be used as a potential for the Riemann tensor, is derived in a much simpler manner which is valid for dimension n ≥ 4 and any signature.

The regularity of static spherically cylindrically and plane symmetric spacetimes at the origin

Abstract: We find the necessary and sufficient conditions for the regularity of all scalar invariants polynomial in the Riemann tensor at the origin of spherically, cylindrically and plane symmetric static spacetimes under the assumption that the metric functions are sufficiently smooth there. These conditions turn out to be simple enough to allow a check for regularity by inspection.

A complete set of Bianchi identities for tensor fields along the tangent bundle projection

Abstract: The various derivations defined along the tangent bundle projection tau in a series of papers by Martinez, Carinena and Sarlet (1992) are expressed as components of a single linear connection Del on E, the tangent bundle of the evolution space E=R*T M. This connection is equivalent to a system of second-order ordinary differential equations (SODE) on M. Using the linear connection, we calculate the torsion and curvature of (E, Del ), the components of which are expressed in terms of the tensors along tau defined by Martinez et. al. From these, the full set of Bianchi identities are calculated. We also show that the generalized Jacobi equation, defined by several authors, is precisely the horizontal component of the conventional Jacobi equation along geodesics of (E, Del ). Finally, we use this to show that if a Jacobi field of the lift of a SODE solution is a certain lift, then it can be extended to a symmetry of the SODE.

Curvature tensors in dimension four which do not belong to any curvature homogeneous space

Abstract: A six-parameter family is constructed of (algebraic) Riemannian curvature tensors in dimension four which do not belong to any curvature homogeneous space. Also a general method is given for a possible extension of this result.