Covariant gauges and point sources in general relativity

The tree graph contributions to the vacuum expectation value of the quantized gravitational field produced by a point mass source are found to diverge. These divergences can be removed only by giving the source a finite extension, and it is first necessary to analyze the corresponding classical situation before making a comparison with the quantum theory. In this paper, the model for such an extended particle is taken to be a spherical shell of initially static pressure-free dust. Without solving the Einstein equations explicity, a coordinate-independent mass renormalization formula can be derived, valid at the moment of time symmetry, which relates the total mass of the system to the bare mass of the source and its invariant radius. The equations are then solved for various choices of coordinate systems, allowing the invariant radius of the shell to be expressed in terms of its coordinate dependent extension. The results are in agreement with those obtained previously by Arnowitt, Deser, and Misner. The work of these authors is generalized to include coordinate frames for which the metric is discontinuous across the shell. Aside from any intrinsic interest, such a generalization is necessary since the most convenient coordinate system for the quantum calculations, namely the covariant de Donder (harmonic) gauge, falls into this category. By expanding the total mass of the source in terms of its bare mass and harmonic coordinate extension, the classical Schwarzschild solution may be cast into a form which facilitates a direct comparison with the quantum theory in the de Donder gauge.     

Lagrangians forn point masses at the second post-Newtonian approximation of general relativity

We study the effect of an infinitesimal coordinate transformation on the Lagrangian and on the metric functional of a system ofn point masses. We show how to compute the Lagrangians ofn point masses at the second postNewtonian approximation of general relativity in different coordinate systems. The Lagrangians are shown to depend on the accelerations except in a special class of coordinates. This class includes the coordinates associated with the canonical formalism of Arnowitt, Deser, and Misner, but excludes most other coordinate systems used in the literature (notably the harmonic one).     

Covariant change of signature in classical relativity

This paper gives a covariant formalism enabling investigation of the possibility of change of signature in classical General Relativity, when the geometry is that of a Robertson-Walker universe. It is shown that such changes are compatible with the Einstein field equations, both in the case of a barotropic fluid and of a scalar field. A criterion is given for when such a change of signature should take place in the scalar field case. Some examples show the kind of resulting exact solutions of the field equations.     

Energy-momentum complex for nonlinear gravitational Lagrangians in the first-order formalism

It has been recently shown that there is universality of Einstein equations, in the first-order (Palatini) formalism, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations. In this paper the energy-density flow for nonlinear gravitational Lagrangians is investigated in this formalism. It is shown that in the generic case the energy-momentum complex does not depend on the Lagrangian and is in fact equal to the Komar complex, known in the purely metric formalism for the standard linear Hilbert Lagrangian.On leave from the Institute of Theoretical Physics, University of Wrocaw, pl. Maxa Borna 9, 50-204 Wrocaw, Poland     

Hedgehog structures in general quark-meson Lagrangians

We investigate stationarity properties of hedgehog configurations in effective quarkmeson Lagrangians involving scalar and vector boson fields with arbitrary self-couplings. The Lagrangians are assumed to be in theSU(2)-sector and to be invariant under Lorentz-transformations, isorotations and time reversal. Restricting ourselves to the mean field solution based on valence quarks in sphericals-orbits and bilinear couplings the quark-hedgehog is shown to generate simply structured meson fields and to correspond to a stationary solution of the equations of motion. In most practical cases this is even of minimal energy.     

Kinks in general relativity

The problem of classifying topologically distinct general relativistic metrics is discussed. For a wide class of parallelizable space-time manifolds it is shown that a certain integer-valued topological invariant n always exists, and that quantization when n is odd will lead to spinor wave functionals.     

Time in general relativity

In order to distinguish between physical and coordinate effects in an arbitrary gravitational field, the space coordinate system and the clock rates must be specified operationallya priori. Once this is done, it is no longer possible to set up an initial surface arbitrarily, since this operation must be consistent with certain physical experiments, whose results depend upon the particular physical situation. A method is given for setting up the initial surface, and the time evolution of the system is discussed.NASA Predoctoral Fellow.     

Conventionalism and general relativity

We argue that the geometry of spacetime is a convention that can be freely chosen by the scientist; no experiment can ever determine this geometry of spacetime, only the behavior of matter in space and time. General relativity is then rewritten in terms of an arbitrary conventional geometry of spacetime in which particle trajectories are determined by forces in that geometry, and the forces determined by fields produced by sources in that geometry. As an example, we consider radial trajectories in the field of a single particle expressed in the spacetime of special relativity.     

Singularities in general relativity

The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.Read on 15 May 1970 at the Gwatt Seminar on the Bearings of Topology upon General Relativity     

Tachyons in general relativity

The tachyonic version of the Schwarzschild (bradyonic) gravitational field within the framework of extended relativity is considered. The metric of a tachyonic black hole is obtained through superluminal transformations from a bradyonic metric. The extended space-time manifold of this geometry which includes both black and white tachyonic holes is analysed, and the differences between the tachyonic and bradyonic versions are noted. It is shown that the meanings of black holes, tachyons and bradyons depend on the character of the reference frame and are not absolute.     

On general and restricted covariance in general relativity

We reconsider the principle of general covariance and give a rigorous formulation of a principle ofrestricted covariance. We give a number of examples of preferred coordinate systems, considered in the literature, and in each case demonstrate the applicability of the notion of restricted covariance proposed.     

Energy and momentum in general relativity

In general relativity, conservation of energy and momentum is expressed by an equation of the form /x= 0, where –gT represents the total energy, momentum, and stress. This equation arises from the divergence formula dV _v = (/x ^v )d ⁴ d. Here we show that this formula fails to account properly for the system of basis vectors e(x). We obtain the (invariant) divergence formula e dV _v = e (/x ^v + )d ⁴ d. Conservation of energy and momentum is therefore expressed by the covariant equation (/x ^v ) + = 0. We go on to calculate the variation of the action under uniform displacements in space-time. This calculation yields the covariant equation of conservation, as well as the fully symmetric energy tensor . Finally, we discuss the transfer of energy and momentum, within the context of Einstein's theory of gravitation.     

Covariant formulation of the weak conservation law in the general theory of relativity without the energy-momentum of the gravitational field

A systematic covariant formulation is given of the weak conservation law in the general theory of relativity for arbitrary coordinate transformations without introducing the unsatisfactory definition of the energy-momentum of the gravitational field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 13–16 July, 1979.     

On isoperimetric surfaces in general relativity,II

We determine the optimal way to enclose volume in a class of domains inside certain Friedmann–Robertson–Walker metrics. The method employed is an adaptation of the Bray–Morgan isoperimetric comparison procedure to the Lorentzian setting. We also make some remarks on isoperimetric comparison in the Riemannian setting, for rotationally-symmetric space-like slices in non-vacuum space-times.     

Spinors and torsion in general relativity

Conformal rescalings of spinors are considered, in which the factor Ω, inε _AB ?Ωε _AB, is allowed to be complex. It is argued that such rescalings naturally lead to the presence of torsion in the space-time derivative▽ _a. It is further shown that, in standard general relativity, a circularly polarized gravitational wave produces a (nonlocal) rotation effect along rays intersecting it similar to, and apparently consistent with, the local torsion of the Einstein-Cartan-Sciama-Kibble theory. The results of these deliberations are suggestive rather than conclusive.     

Junction conditions in general relativity

We examine the three sets of junction conditions commonly used in general relativity: those due to Darmois, to O'Brien and Synge, and to Lichnerowicz. We show that those due to Darmois and Lichnerowicz are equivalent. The O'Brien and Synge set is stronger than the other two and is unsatisfactory in that it may rule out physically plausible junctions. We conclude that the Darmois set is the most convenient and reliable.     

Nongeodesic motion in general relativity

The equations of motion of a spinning body in the gravitational field of a much larger mass are found using both the Corinaldesi-Papapetrou spin supplementary condition (SSC) and the Pirani SSC. These equations of motion are compared with our previous result derived from Gupta's quantum theory of Gravitation. It is found that the spin-dependent terms differ in each of the above three results due to a different location of the center of mass of the spinning body. As expected, these terms are not affected by the choice of either Schwarzschild or isotropic coordinates. Finally, for the presently planned Stanford gyroscope experiment, we find the maximum secular displacement of the orbit of the gyro with respect to the orbit of its non-rotating housing to be of the order of (10⁻⁷ cm/year)t, a result much smaller than Schiff's result which is proportional to time squared.     

Algebraic computing in general relativity

The purpose of this paper is to bring to the attention of potential users the existence of algebraic computing systems, and to illustrate their use by reviewing a number of problems for which such a system has been successfully used in General Relativity. In addition, some remarks are included which may be of help in the future design of these systems.