Formulation Of General Relativity Research Articles

Hamiltonian formulation of general relativity in terms of Dirac spinors

The Dirac spinors andα matrices are used in combination with the Arnowitt-Deser-Misner formalism in order to obtain yet another formulation of Hamiltonian general relativity, together with a new form of the Gauss-Codazzi equations. The relation with Ashtekar's variables is analyzed; it is shown, for instance, that theα matrices are equivalent to the “electric” field variable. The “electric” and “magnetic” decomposition of the gravitational field is also studie using Dirac matrices.

A formulation of the virial theorem in general relativity

A relativistic generalization of the classical virial theorem is obtained for any stationary and asymptotically flat spacetime. This formulation is derived within the 3+1 formalism of general relativity. It may be useful as a consistency check of numerical solutions of the Einstein equations.

Quasilocal gravitational energy.

A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.

Gauge-invariant cosmological perturbation theory

The linear perturbation theory of Friedman universes is treated in a gauge-invariant manner (i.e., invariant under linearized coordinate transformations). After presenting some mathematical tools in Chapter 1, the perturbation equations are derived with the help of the 3+1 formalism of general relativity in Chapter 2. Bardeens perturbation theory [4] is thereby extended to the treatment of collisionless matter. In Chapter 3 we solve the perturbation equations for perfect fuids explicitely. The results are applied to the discussion of the Sachs-Wolfe effet and of the traditional baryonic scenario of galaxy formation. In Chapter 4 numerical solutions of the gauge invariant perturbation equations are presented for a universe consisting of collisionless particles of mass mx ≠ 0, massless neutrinos and radiation. In this scenarios the masses of the first objects to collapse are of the order of m³pl/m²x. Within the gauge-invariant treatment, the perturbation amplitudes exhibit no growth on scales which are larger than the present horison size.

Commutative asymptotic limit of a quasi-SU(2) formulation of general relativity

The notion of local quasi-gauge bundle structure is introduced. We show that general relativity can be recast in a local quasi-SU(2)-bundle framework. In the limit of weak asymptotic gravitational field, this geometrical setup gives rise to spin-2 tensor fields sourcing global charges. If such charges are available, it is shown that the asymptotic geometrical framework is that of aU(1) gauge bundle overS2, the commutative geometry of the (Dirac) magnetic monopole.

On the symplectic formalism for general relativity

A covariant formalism for the hamiltonian formulation of general relativity in arbitrary dimensions is presented. Specifically, a presymplectic form on the solution space for the vacuum equations in n dimensions is given. The basic variables are taken to be a soldering form and a torsion-free connection on an SO ( p , q )-bundle over the space-time manifold M . It is shown how the present formalism is related to the standard ADM-formalism.

Degenerate metrics and an empty black hole

Ashtekar's formulation of general relativity is polynomial, and admits degenerate metrics. The author considers 'metrics' which are everywhere degenerate on the initial data hypersurface. For a special class of them, the author solves the constraint and evolution equations completely. Some 'neighbours' of the theory are briefly considered. The author then considers spherically symmetric spacetimes. A general form of the stationary solutions, in terms of three arbitrary functions, without gauge fixing, is given for both degenerate and non-degenerate metrics. It is shown that the Schwarzschild solution may be joined smoothly to a degenerate solution across the event horizon, yielding a 'black hole' with a degenerate interior. Some comments on matter couplings are made.

Gravity, holonomy and light-cone cuts

We present an outline of a D'Adhemar-like formulation for General Relativity, which puts emphasis on the light-cones as one of the basic variables. Two important aspects of this reformulation are: (1) it is possible (in a perturbative sense) to construct the general radiative solution for GR, i.e. one can find a (series of) integral relationships between the field and the free-characteristic data, and (2) as the light-cone itself now becomes a basic variable, one can clearly see the difficulties with the conventional view towards the problem of quantum gravity in that there is no natural way to include it in a quantization procedure.

Numerical solution of the general relativistic Boltzmann equation for massive and massless particles

The ADM formalism of general relativity is applied to write the Einstein and matter equations for a spherically symmetric spacetime whose gravitational sources are taken to be represented by ordinary matter, described by a perfect fluid approximation, and either massless or massive weakly interacting particles, for which a kinetic-theory description is employed. The Boltzmann equation, which is solved directly, is written in a form which is particularly amenable to numerical treatment, and for which massive particles can be accommodated as easily as massless ones. Some of the numerical methods used to solve these equations are described and the results of several code tests are discussed. The behavior of massive and massless particles is compared, and results are presented from a code for the cooling of a crude model of a 'neutron star' by emission of massive particles. 33 refs.

Role of general relativity in accretion disk dynamics

In this we briefly review the discussions on accretion dynamics, the standard scenario and the ones including the effects of electromagnetic fields. The emphasis throughout is to show the relevance of general relativistic formalism in discussing the dynamics of magnetofluid around compact objects.

Point to string transformation, or framedependent notion of ?locality?: An attempt to finitize field theories including gravitation

A novel approach has been made to the divergence problem in local field theories, in which the notion of “locality” is still retained but loses its absolute meaning, just like “simultaneity”. The basic idea is to introduce a pure-imaginary elementary length into 3-dimensional space, while keeping “time” structureless so as to retain the unitarity of theS-matrix. Consequently, light becomes dispersive at sufficiently short wavelengths, and Lorentz transformation becomes a point-to-string transformation. When reformulated to meet the new Lorentz invariance, all the localfield (in the above sense) theories in a flat space become finite,while retaining their conventional form. This has been demonstrated by the derivation of finitized Coulomb potential and correct high-momentum behavior of quantum-electrodynamic coupling constant. For diagrams including gravitons, evaluation of the superficial degrees of divergence shows that only a restricted number of 1-(and 2-) loop diagrams might be divergent, while those of more than 3 loops are definitely convergent, thus indicating possible renormalizability (or something better) of quantum gravity in Einstein's formalism of general relativity. Since 4-dimensional simple supergravity removes 1-and 2-loop divergence, a combination of the theory and the present one might lead to a more interesting result.

On the application of computer algebra to velocity dominated approximations

We demonstrate the implementation of a computer algebra system to perform calculations in a variable frame formalism in general relativity in SHEEP. We present the results of the application of such a program to the velocity dominated approximations to solutions near space-like singularities.

Fifth force, sixth force, and all that: a theoretical (classical) comment

In the recent literature, a few claims appeared about possible deviations from the ordinary gravitational laws (both at the terrestrial and the galactic level). The experimental evidence does not seem to be conclusive; nor it is clear if new forces are showing up, or if we have to accept actual deviations from Newton or Einstein gravitation (in the latter case, the validity of the very equivalence principle might be on the stage). In such a situation, the attempts by various authors at explaining the «new effects» juston the basis of the ordinary theory of general relativity (for instance, in terms of quantum gravity) can be regarded as logically questionable. In this pedagogically oriented paper we approach the problem within the classical realm, by exploring whether the possible new effects can be accounted for throughminimal modifications of the standard formulation of general relativity: in particular, through exploitation and extension of the role of the cosmological constant.

A new approach to the theory with variable gravitational constant

A theory with variable gravitational constant, based on the bimetric formulation of general relativity, is suggested. It is in agreement with the Newtonian and post-Newtonian approximations of the theory of gravitation when the theory parameter |ζ|≫4×10−6. The statical spherical-symmetric distribution of gravitating masses is investigated. It is shown that the total action (including the own gravitational field of the system)S=−Mτ, where τ is the time in the remote system of reference, relative to which the celestial body is in rest. In the Brans-Dicke theory |ζ|≫1 andS≠−Mτ. The Tolman formula for the massM in the Einstein theory is also valid in the suggestedGeneralized Bimetric Theory of Gravitation. External analytical and internal numerical solutions of the field equations are found for the case of incompressible matter. It is shown that static and supermassive configurations may exist if −0.13≲ζ<0.

A new approach to Bianchi V models

Spatially homogeneous perfect fluid space-times pertaining to the Bianchi V class are considered from the point of view of the evolution (3+1) formalism of general relativity. It is shown that the metric can be written in diagonal form in the nontilted case. The necessary and sufficient condition for the metric of a Bianchi V model to be diagonalizable is given.

The precessional frequency of a gyroscope in the quaternionic formulation of general relativity

The precessional frequency of a gyroscope in a reference frame that orbits about a gravitational body is compared between Einstein's tensor formulation of general relativity and the author's quaternion generalization—obtained from a factorization of the tensor form. The difference in predictions then suggests an experiment that could choose which of these formulations of general relativity is more valid in the analysis of gyroscopic motion.

Cosmological implications of the Gibbs ensemble in parametrized relativistic classical mechanics.

Earlier investigators have shown that the free energy of an ideal gas in the ultrarelativistic (UR) regime of parametrized relativistic classical mechanics (PRCM) is different from the free energy calculated with use of the standard theory of relativistic classical mechanics (RCM). These differences do not become significant until very high temperatures are reached. Physical examples where the UR regime may be reached include the interior of stars, relativistic gas dynamics, and the first minute after the Big Bang. Of these, it is shown in the present paper that negligible differences are expected at realistic temperatures for relativistic gas dynamics and stellar models. This demonstration requires an evaluation of the PRCM and RCM free energies over a wide range of temperatures extending from the nonrelativistic limit to the UR limit. The greatest differences between PRCM and RCM are manifested in the density parameter of cosmological models. To study this effect, a general relativistic formalism is derived from variational principles within the context of PRCM. Scale factors and the age of Friedmann universes are calculated.

Variational techniques on classes of Lagrangians

We present canonical procedures for the manipulation of whole classes of Lagrangians that share the same transformation law and functional dependence but are otherwise arbitrary in functional form, and for the derivation therefrom of generalized conserved quantities. The techniques are demonstrated on the class of scalar density LagrangiansL=L G+L EM, whereL G is a function of the metric and its first and second derivatives andL EM is a function of the metric and a vector potential and its first derivative, which generate the Einstein-Maxwell equations (without cosmological constant). These procedures should be of interest to those studying alternate formulations of general relativity, those deriving new field theories, and others working with general of modified Lagrangians.

Gravitation, the general theory of relativity, and alternative theories

The main stages in the construction of the theory of gravitation and prospects for its further development are discussed. The main attention is devoted to comparing the properties of the relativistic gravitational field and other physical fields. Two equivalent formulations of the general theory of relativity—the geometrical and the field— are considered in detail. It is explained why some of the field theories of gravitation developed in flat space-time are not different theories of the relativistic gravitational field but merely other formulations of general relativity.

Constraint algebra in general relativity

We study the algebra of the Poisson brackets, of the constraints in the ADM formulation of general relativity and we derive it from the gauge algebra. We discuss the gauge transformations in the canonical framework and we obtain a direct geometrical interpretation of the constraint algebra. In the same way the constraint commutator algebra is derived from the gauge algebra in the quantum theory.