Third-order tensor potentials for the Riemann and Weyl tensors

The representations of the Riemann and the Weyl tensors of a four-dimensional Riemannian manifold through covariant derivatives of third-order potentials are examined in detail. The Weyl tensor always admits a completely general representation whereas the Riemann tensor does not. Nevertheless there exists a class of Riemannian manifolds whose Riemann tensors may be calculated in terms of potentials; in this connection, specific examples are exhibited explicitly. The possibility of introducing gauges on the potentials is reexamined in connection with the previous result. New properties of the representations are also discussed.     

The weight of extended bodies in a gravitational field with flat spacetime

Einstein's gravitational field equations in empty space outside a massive plane with infinite extension give a class of solutions describing a field with flat spacetime giving neutral, freely moving particles an acceleration. This points to the necessity of defining the concept gravitational field not simply by the nonvanishing of the Riemann curvature tensor, but by the nonvanishing of certain elements of the Christoffel symbols, called the physical elements, or the nonvanishing of the Riemann curvature tensor. The tidal component of a gravitational field is associated with a nonvanishing Riemann tensor, while the nontidal components are associated with nonvanishing physical elements of the Christoffel symbols. Spacetime in a nontidal gravitational field is flat. Such a field may be separated into a homogeneous and a rotational component. In order to exhibit the physical significance of these components in relation to their transformation properties, coordinate transformations inside a given reference frame are discussed. The mentioned solutions of Einstein's field equations lead to a metric identical to that obtained as a result of a transformation from an inertial frame to a uniformly accelerated frame. The validity of the strong principle of equivalence in extended regions for nontidal gravitational fields is made clear. An exact calculation of the weight of an extended body in a uniform gravitational field, from a global point of view, gives the result that its weight is independent of the position of the scale on the body.     

Characterization of the algebraic symmetry properties of the Riemann tensor by a single tensor equation

It is shown that the algebraic symmetry properties of the Riemann tensor can be expressed by a single tensor equation.     

A note on a paper by Massa and Pagani

Massa and Pagani [1] have given a neat refutal to the conjecture [2] that the Riemann tensor is derivable from a tensor potential. Their method consists of assuming such a relationship does exist and examining the resulting integrability conditions; they show that the existence of such a potential will impose nontrivial restrictions on the Riemann tensor and so conclude that, in general, such a potential cannot exist. Although Massa and Pagani posed the problem and interpreted the conclusion in ordinary tensor notation the actual derivation of the crucial constraint equation was carried out in the language of tensor-valued differential forms, and is quite involved. In this note it is shown that the crucial equation can be obtained quite naturally and easily in ordinary tensor notation.     

Hamiltonian formulation of Riemann ellipsoid dynamics

A Riemann ellipsoid is a classical fluid with an ellipsoidal boundary whose motion depends linearly on position. The Riemann ellipsoid Newtonian equations of motion are proven to form a Hamiltonian dynamical system. The co-adjoint orbits of a Lie group GCM(3) on which the inertia tensor is positive-definite are the reduced phase spaces of Riemann ellipsoids for which conservation of circulation has been exploited fully.     

String corrections to the motion of a particle in the gravitational field of a black hole

The low-energy effective action, quadratic with respect to the Riemann tensor, of string theory is used to study the motion of a test particle in the gravitational field of a black hole.V. I. Ul'yanov-Lenin Kazan State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 83–85, January, 1994.     

On the energy-momentum and spin tensors in the Riemann–Cartan space

General classical theories of material fields in an arbitrary Riemann–Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for canonical energy-momentum and spin tensors are constructed in the cases when a Lagrangian contains (a) an arbitrary set of tensorial material fields and their covariant derivatives up to the second order, as well as (b) the curvature tensor and (c) the torsion tensor with its covariant derivatives up to the second order. A non-trivial manifestly generally covariant generalization of the Belinfante symmetrization procedure, suitable for an arbitrary Riemann–Cartan space, is carried out. A covariant symmetrized energy-momentum tensor is constructed in a general form.     

Third-order tensor potentials for the Riemann and Weyl tensors. II: Singular solutions

The analysis of the admissibility of a potential representation for the Riemann tensor is here continued. As in the preceding paper, the starting point is to regard the relationship between the Riemann tensor and its possible potential as a system of partial differential equations determining the unknown potential. The first result, strengthening a previous conclusion, is that there never exist ordinary solutions. Surprisingly enough, in a four-dimensional Riemannian manifold the existence of singular solutions is established without requiring any integrability condition. Possible applications and generalizations are also suggested.     

双参数曲面的泡泡网格化方法

为将双参数曲面离散成高质量的网格,首先在参数域内利用各向异性的非均匀泡泡布点方法优化布点,然后用各向异性Delaunay三角化方法将参数域网格化,最后用映射法得到双参数曲面的离散网格.参数域中的节点由二阶黎曼度量矩阵控制,该度量矩阵由三维曲面的网格度量矩阵和曲面参数方程的梯度计算得到.数值算例表明,泡泡布点法在参数域上能生成满足度量矩阵要求的节点集,将节点连接成网格并投影回曲面,所得曲面网格具有很高的质量.     

The double duals and gauges of the Lanczos tensor

It is shown that: i) the Weyl tensor can be expressed in terms of the sum of a tensor and its double dual, where the tensor is constructed from the covariant derivatives of the Lanczos tensor, ii) a similar expression does not exist for the Riemann tensor in electromagnetic theory, iii) the electromagnetic field cannot be identified with the differential gauge freedom of the Lanczos tensor, iv) the symmetries of Einstein Maxwell theory and the Lanczos tensor do not prohibit the identification of the electromagnetic field with the algebraic gauge freedom of the Lanczos tensor, these symmetries require a differential equation relating the electromagnetic field tensor to the algebraic gauge vector and this is given.     

Weyl geometry

We develop the properties of Weyl geometry, beginning with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime. We generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, enlarging the symmetry from Poincaré to Weyl, we develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We present a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that it is conformally equivalent to general relativity. This theory is invariant under local dilatations, but not the full conformal group.     

Computer calculation of tensors in Riemann normal coordinates

A new method of calculation is given for arbitrary tensors in Riemann normal coordinates. Inventing a compact notation for an abstract form of tensors which is suitable to a noncommutative algebra system, we carry out the computer calculations to obtain coefficients of the Taylor expansion of tensors in Riemann normal coordinates. Explicit forms are given up to the tenth order for the metric tensor.     

On vacuum metrics of type (3, 1)

A technique is described for constructing solutions of Einstein's equations for empty space, in which the Riemann tensor has a triply degenerate principal null direction with twist.     

On purely gravito-magnetic vacuum space-times

It is shown that there are no purely magnetic, vacuum, spacetime metrics where any one of the following conditions holds: (a) the ratio of any two eigenvalues of the Weyl tensor is constant, (b) both of the Riemann principal null directions, defining the time-like blade, are nonrotating, (c) the shear tensor possesses an eigenvector v ^a which is defined by one of the space-like Riemann principal directions, (d) the vorticity is parallel to v ^a , where v ^a is defined by one of the space-like Riemann principal directions.This revised version was published online in April 2005. The publishing date was inserted.     

Local extensions in singular space-times II

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

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

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.     

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus g > 1, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g > 1. Received: 12 September 1996 / Accepted: 6 January 1997     

On poisson structure and curvature

We consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure. In a simple example we determine a relation between this structure and the Riemann tensor.     

Detection of the gravitomagnetic field using an orbiting superconducting gravity gradiometer: principle and experimental considerations

The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order of 10⁻¹⁰ of the Newtonian terms arising from the mass of the Earth. Due to the dragging of the local inertial frame by the spinning Earth, there are also secular terms, which grow in time. These fields can be detected in principle by a set of orbiting superconducting gravity gradiometers. The Riemann tensor components for various spacecraft orientations have been computed and the principle of detecting the gravitomagnetic tidal force has been published. In this paper, we review the conclusions of the earlier analyses and discuss the feasibility of the gravity gradiometer experiment.