Ricci Rotation Coefficients Research Articles

Ricci coefficients of rotation in a generalized Riemannian space

Ricci coefficients of rotation in a generalized Riemannian space

Stationary tensor fields in relativistic continuum mechanics

A tensor field in relativistic continuum mechanics is said to be stationary iff its Jaumann derivative with respect to the timelike unit flow vector vanishes. The gravitational potentials, geodesic flow are always stationary. Ifgab is the metric tensor, then the 3-dimensional projection operatorgab−uaub is found to be stationary when and only when the first curvature of the streamline vanishes. The necessary and sufficient conditions for the stationary character of the 2-dimensional projection operatorgab−uaub+PaPb are found to beK2+(1/2)(γ123−γ123)=0,γ124=γ123, wherePa is the unit acceleration vector field,K2 is the second curvature or torsion andγABC are the Ricci-coefficients of rotation formed from the relativistic Serret-Frenet formulae for the stream line of a particle in the continuum. The nonstationary character of the relativistic Serret-Frenet tetrad is also established. Further it is shown thatJuJu(gab−uaub)=0 iffK1=0,γ124, whereJu is the Jaumann transport operator.

Equations of radiation gas dynamics in tetrad form for astrophysical applications

This paper deals with the representation of relativistic equations of gas dynamics with due regard to the general relativity theory effects in the form accepted and widely applied in the special relativity theory. With this purpose, a strict formal definition of a non-inertial co-moving reference frame without rotation is carried out on the basis of a tetrad formalism by use of the Fermi—Walker rules of transport of 4-frame. The equations of physical kinetics, relativistic collapse, Einstein's equations, equations of relatiivistic radiation gas dynamics for ideal and dissipative gases, Taub's equations for a shock wave, which allow for radiation and electron-positron pairs, are obtained in this reference frame. On the basis of the local Lorentz transformation and the Ricci rotation coefficients, these equations are written in the laboratory reference frame, in order to illustrate the fact that the general relativity effects can be simply taken into account in the equations having a form accepted in the special relativity theory.

Shear-free homogeneous cosmological model with heat flux

We present the condition of vanishing shear in a spatially homogeneous spacetime in terms of the Ricci rotation co-efficients corresponding to an orthonormal tetrad (ν α. A η α) (whereν α is the unit vector along the time axis and A η α are the three independent reciprocal group vectors). Assuming that the velocity vector can be expanded in the direction ofν α and any one of the A η α’s it is shown that shear-free motion is possible only in case of some special Bianchi types, and these cases are studied assuming the velocity vector to be geodetic and that there may be a nonvanishing heat flux term.

Lanczos-type potentials in non-Abelian gauge theories

Here we investigate a variational approach for obtaining a potential structure for non-Abelian gauge theories by introducing identically satisfied constraints in a variational principle which is based on one that is identically satisfied. This technique for finding potentials for field theories was used by Lanczos (1962) to find his so-called thirdorder «spin» tensor in the context of Einstein's gravitational theory. In the case of general non-Abelian gauge theories, it is found that the Lanczos-type potentials are related to the usual potentials by generalized gauge transformations. This method is applied in the context ofSO3 (orSU2) gauge theory as well as Einstein and Einstein-Cartan gravitational theories. Various recent attempts to formulate gauge theories of gravity based on the Poincare group are discussed. An application of the results of this paper shows that a mixed form of the Ricci coefficients of rotation can be used to display the gauge structure of the space-time curvature in essential agreement with the results of Hehlet al.

On a discription of dynamical properties of the gravitational field in vacuum

AbstractThe PETROV representation of the vacuum field classification is discussed. The introduction of canonical tetrads leads to an energy definition based of the Ricci rotation coefficients.

Metric and ricci tensors for a certain class of space-times ofD type

The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.

A theorem on stress–energy tensors

The equality of the symmetrized Noether stress–energy tensor (Belinfante’s tensor) and the canonical stress–energy tensor (functional derivative of the Lagrangian density with respect to the metric) is established by methods based on the formalism of tetrads and Ricci rotation coefficients. The result holds for any Lagrangian which contains no derivatives of the fields higher than first order.

Relativistic angular momentum for asymptotically flat Einstein-Maxwell manifolds

Treating general relativity as a Yang-Mills theory based on Lorentz invariance of the second kind leads to the derivation of six spin currents, which are linear combinations of the Ricci rotation coefficients, together with a Gauss theorem for the integrated charges. For asymptotically flat Einstein-Maxwell manifolds these charges are calculated by performing two-surface integrals at future null infinity. The gauge dependence of these charges, which arises because the underlying group is non-Abelian, is removed by making a canonical alinement of the asymptotic frames of reference. The Lorentz generators then arise as a field in asymptotic spin space and defined on the set of all outgoing null cones, which is the carrier space of the B.M.S. group. In the absence of outgoing gravitational radiation the Lorentz generators transform under translations as a moment of momentum. However, the points about which moments are taken belong not to the original Riemannian manifold but rather to a four-parameter family of cones isomorphic to Minkowski space-time. From the angular momentum structure there arises a natural twistor structure. Points on the twistor and the centre of mass of the system are defined as world lines in the manifold of cones. The remainder of the paper is devoted to formulating physical laws in cone space. These include a Poynting’s theorem for the radiation of spin.

Static gravitational fields. II. Ricci rotation coefficients

Investigations of static gravitational fields in paper I are continued further. Instead of the system of second order partial differential equations in I, the equivalent first order system involving the Ricci rotation coefficients is dealt with here. The algebraic and the differential dependences among the equations have been sorted out to prove that the system is determinate. Then an attempt towards classifying this system of partial differential equations has been made. Except for ``the potential equation'' which is obviously elliptic, the remaining system is hyperbolic for which the characteristic surfaces have been determined. To obtain some exact solutions the lead of Newman-Penrose is followed in constructing the complex linear combinations of the equations. The class of static universes where ``gravitational lines of force'' are geodesics have been found. In this class one subclass is transformable to the conformastat metric and the remaining one reduces to a new metric involving the gravitational field of an arbitrary number of parallel infinite plates clamped at infinity. The source at infinity corresponds to that of the Newtonian potential φ =(1/2)[(m + 1/2)(x1)2 −(1/2) {m + 1/2 − (m2 − 1/4)1/2} (x2)2 − (1/2) {m + 1/2 + (m2 − 1/4)1/2} (x3)2]. This metric belongs to the nondegenerate Type I of Petrov's classification scheme. Next the class of static universes with ``shearfree lines of force'' is obtained. Here too one subclass goes over to the conformastat metric and the remaining one reduces to a Weyl-type universe.

Gauge fields and gravitational field equations

It is shown that the gravitational field equations in free space have a similar form to the free Yang-Mills field equations, where the group SL (2, C) replaces the group SU(2). The Ricci rotation coefficients take the role of the Yang-Mills like potentials, whereas the Riemann tensor takes the role of the gauge fields.

Tetrad Formulation of Junction Conditions in General Relativity

Explicit and complete junction conditions across internal boundaries are found for the Ricci rotation coefficients and Riemann tensor components in an arbitrary reference tetrad frame; the reference frame may be discontinuous at the boundary. The allowed jumps in inertial and gravitational field dyadics are given for the case of normal boundary motion. When the reference frame is chosen to be comoving with the matter distribution on either side of the boundary, the junction conditions include such immediately physical results as the relativistic Rankine-Hugoniot relations. It is expected that use of such comoving reference frames will be important in finding discontinuous exact solutions.

An Approach to Gravitational Radiation by a Method of Spin Coefficients

A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiation theory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation.

New Approach to Einstein's Empty Space Field Equations

Tetrad formalism is used to derive a set of 36 scalar field equations which correspond to the ordinary field equations Rμν=0. The scalar equations are obtained by beginning with a given Petrov type of empty-space Riemann tensor and applying the Ricci identity to each of the tetrad vectors. The unknowns or field variables become the 24 Ricci rotation coefficients, the number of which can always be reduced by the Bianchi identities and occasionally by tetrad transformations which leave the form of the Riemann tensor invariant. The use of these scalar field equations is illustrated by their application to a degenerate case of Petrov type I. It is believed that by this method all possible solutions of this particular case have been found.

Ricci’s coefficients of rotation

Ricci’s coefficients of rotation