Einstein's Gravitational Field Equations Research Articles

Collapse of a null fluid

An exact cylindrically symmetric solution of Einstein's gravitational field equations is given for a null fluid imploding radially along an infinite axis. This solution plays an important role in the late stages of collapse of a long cylinder of matter. One might expect that self-gravitational effects due to the increasingly relativistic mass of the collapsing matter would create arbitrarily large gravitational fields. It is shown, however, that in the null-fluid approximation the metric is everywhere regular.

On a quadratic first integral for the charged particle orbits in the charged Kerr solution

Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.

Nonmetrical Specification of Space-Time Sources

The role played by the energy-momentum conservation law in general relativity is examined. It is noted that this law can be interpreted in two ways. It may be thought of as a condition determining the evolution of the energy-momentum tensor density in time, or it may be thought of as a condition determining the metric. In the present paper, the second of these ways of thinking about the energy-momentum conservation law is explored. Einstein's nonvacuum gravitational field equations (which imply the conservation law) are examined. It is shown that given any analytic symmetric contravariant energy-momentum tensor density as a function of the space-time coordinates, a solution to the gravitational field equations always exists. Furthermore, this solution is such that the law of conservation of energy-momentum is satisfied. The proof uses a coordinate transformation method to exploit the covariance of the energy-momentum conservation law. Riquier's existence theorem enters as an important part of the proof, and a general discussion of Riquier's existence theorem from a physical point of view is given. Both the geodesic nature of the trajectories of free particles and the unit magnitude of the velocity 4-vector are discussed. An interpretation of the above-described results is given.

Slow-motion approximations in general relativity I. Decomposition of the field equations

A new (3+1)-dimensional decomposition of the Einstein gravitational field equations is obtained for a general spacetime. The metric is taken in the form $$ds^2 = e^{ - 2u} k_{ab} (dx^a + \xi ^a dt)(dx^b + \xi ^b dt) - c^2 e^{2u} dt^2 $$ and the resulting equations treatkab as the metric in the space-like hypersurfacest=constant. It is shown that this decompostion forms a more convenient starting point for slow motion approximations than does their usual 4-dimensional formulation. This is illustrated by a derivation of the first post-Newtonian approximation to the field equations, the simplicity there resulting fromkab being still flat to this order.

Exact Wave-Type Solution tof−gTheory of Gravity

The well-known plane-fronted waves with parallel rays, exact solutions of the vacuum Einstein gravitational field equations, are generalized to exact solutions of the equations of $f\ensuremath{-}g$ gravity theory proposed by Isham, Salam, and Strathdee in regions free from matter.

Mixture of outgoing and incoming gravitational radiation: change of mass of the source of radiation

It has been fairly well established theoretically in recent works that outward travelling spherical gravitational waves carry away energy from their source. Methods of successive approximations applied to the Einstein gravitational field equations of general relativity have been used for that purpose. The object of this paper is to show, by a double-series approximation method, a similar result for the general case of any mixture of outgoing and incoming radiation.

Some New General Relativistic Dust Metrics Possessing Isometries

In a four-dimensional Lorentzian manifold in which Einstein's gravitational field equations hold with the field produced by pressure free dust, a significant set of new solutions is found. Assuming that the manifold possesses a four parametric group of isometries of type 5 in Bianchi's classification, which act on a three-dimensional negative definite subspace, all metrics are found. The solutions are unusual in that the geodesics which the particles follow are not orthogonal to the three-dimensional negative-definite subspace so that the space will not appear homogeneous to these observers. The isotropic expansion is nonzero for almost all these solutions.

A SOLUTION OF THE EINSTEIN FIELD EQUATIONS

An exact, cylindrically symmetric, time-dependent solution of the Einstein gravitational field equations for empty space is derived. A particular case of the solution has singularities only on the axis of symmetry and may represent a number of particles in an otherwise empty universe.

Two bodies at rest in general relativity

ABSTRACTIt is well known that there is no static axisymmetric two-body solution of Einstein's gravitational field equations, if it is assumed that the bodies are separated in a certain definite sense. In this paper it is shown, by the construction of a complete physically sensible model, that static two-body solutions do exist for systems in which one body is hollow and contains the other. The stability of the particular system described is briefly discussed.

On the Quantum Field Theory of the Interaction between the Graviton and the Matter Field

In order to quantize the gravitational field so as to be consistent with the general relativistic treatment, we mvestigate what physical etfects the quantizable and unquantizable parts of gravttat10nal field b ave, when the matter fields exist. To construct the Hamiltonian, we show that the mteraction Eamiltoman m the mteractton representation can be obtdined by the method extended from that of Yang-Feldman. The role of the non-linear part of gravitational field for such a problem IS also discussed. Only the potenttal, obtamed by elimmatmg the longitudinal and scalar parts of graVItational field with· the atd of the supplerr.entary condition, has an etfect upon the macroscopical phenomena and gives the equatton of two-body problem m general relativity found by Einstein, Infeld and Hotfmarm. On the other hand, while the pure transverse graviton does not contribute to the macro­ ncopical phenomena, 1t gives, m the region of high energy, the large damping etfect upon the mterac· tion between the elementary particles. § l. Introduction and summary Recendy, in connection with the difficulties encountered in the quantum field theory, the effect of the gravitational field upon the interaction between the elementaty patticles has been discussed by some authors. In order to discuss the gravitational field in the theory of quantized field, we must first consider what sort of the classical gravitational equation should be quantized, and secondly examine which patt of the gravitational field quantities can be quantized and what role another patt that cannot be quantized plays in the macroscopical phenomena. In spite of many attempts on the quantization of the gravitational field, there are neither necessaty reasons for the , quantization nor experimental ·evidences for the existence of the gravitational quanta (say, graviton). It will not be reasonable, therefore, to discuss only the quantum chatacter of gravitational field. In other words, the problem whether the field equations or field quantities that we quantize are appropriate, should be considered not merely by looking into the properties of graviton, but also by examining whether the procedure of quantization is compatible with the general relativistic treatment. For example, if we take Einstein's gravitational equation as the one that we quantize, the gravitational field must be quantized in such a way that the patt remaining unquantized can give the same physical effects as in the general relativity. In the linear approximation of Einstein's gravitational field equation, the metric

Isentropic Hydrodynamics in Plane Symmetric Space-Times

Use is made of a form of the stress energy tensor of a perfect fluid, previously derived for special relativity, to show that for irrotational isentropic motions a co-moving coordinate system exists in which both sides of the Einstein gravitational field equations may be expressed in terms of the dependent variables of the self-gravitational problem for a perfect fluid. It is shown that for a space-time with plane symmetry the field equations and the assumption of isentropy imply the conservation of mass. General methods for dealing with these field equations are given for the static and spatially independent cases. Approximate solutions are obtained for other specific cases. The general exact solution is obtained for the incompressible case. Properties of the incompressible case are discussed.

On the Quantization of Einstein's Gravitational Field Equations

Weiss' method of quantization of field theories characterized by first-order Lagrangians can be carried out in a non-metrical space, as was first stated by Bergmann and Brunings. The gravitational equations can be regarded as differential equations for the field variables ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ in an amorphous space and the quantization procedure can be applied to them. The gravitational field equations are written in canonical form, the Hamiltonian being a function of generalized coordinates, momenta, and velocities. This Hamiltonian is obtained using a method developed by Dirac for Lorentz invariant theories.