Field Equations In Empty Space Research Articles

An intuitive and simplified formulation of general relativistic gravity in empty space

General relativistic gravity in empty space is formulated using undergraduate calculus, differential equations, and matrix algebra. The key to the formulation involves certain limited assumptions of Newtonian-like behavior in freely falling frames. The formulation results in a field-motion matrix equation that serves essentially the same purpose as Einstein’s field equations in empty space. The limited assumptions of Newtonian-like behavior in freely falling frames are then justified using well-known tensor relationships. To demonstrate its use, the new field-motion equation is used to solve for the Schwarzschild metric.

Generalized transformations and coordinates for static spherically symmetric general relativity.

We examine a static, spherically symmetric solution of the empty space field equations of general relativity with a non-orthogonal line element which gives rise to an opportunity that does not occur in the standard derivations of the Schwarzschild solution. In these derivations, convenient coordinate transformations and dynamical assumptions inevitably lead to the Schwarzschild solution. By relaxing these conditions, a new solution possibility arises and the resulting formalism embraces the Schwarzschild solution as a special case. The new solution avoids the coordinate singularity associated with the Schwarzschild solution and is achieved by obtaining a more suitable coordinate chart. The solution embodies two arbitrary constants, one of which can be identified as the Newtonian gravitational potential using the weak field limit. The additional arbitrary constant gives rise to a situation that allows for generalizations of the Eddington–Finkelstein transformation and the Kruskal–Szekeres coordinates.

Exact Solutions of the Field Equations for Empty Space in the Nash Gravitational Theory

John Nash has proposed a new theory of gravity. We define a Nash-tensor equal to the curvature tensor appearing in the Nash field equations for empty space, and calculate its components for two cases: 1. A static, spherically symmetric space; and 2. The expanding, homogeneous and isotropic space of the Friedmann-Lemaitre-Robertson-Walker (FLRW) universe models. We find the general, exact solution of Nash’s field equations for empty space in the static case. The line element turns out to represent the Schwarzschild-de Sitter spacetime. Also we find the simplest non-trivial solution of the field equations in the cosmological case, which gives the scale factor corresponding to the de Sitter spacetime. Hence empty space in the Nash theory corresponds to a space with Lorentz Invariant Vacuum Energy (LIVE) in the Einstein theory. This suggests that dark energy may be superfluous according to the Nash theory. We also consider a radiation filled universe model in an effort to find out how energy and matter may be incorporated into the Nash theory. A tentative interpretation of the Nash theory as a unified theory of gravity and electromagnetism leads to a very simple form of the field equations in the presence of matter. It should be noted, however, that the Nash theory is still unfinished. A satisfying way of including energy momentum into the theory has yet to be found.

Mixed gravitational field equations on globally hyperbolic spacetimes

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting of M, as exhibited by Bernal and Sánchez (2005 Commun. Math. Phys. 257 43–50). We give sufficient geometric conditions (e.g. is isoparametric and is totally umbilical) for the existence of exact solutions to mixed field equations in free space. We linearize and solve the mixed field equations for empty space, where is the mixed scalar curvature of foliated spacetime (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If gϵ = g0 + ϵγ is a solution to the linearized field equations, then each leaf of is totally geodesic in to order O(ϵ). We derive the equations of motion of a material particle in the gravitational field gμν governed by the mixed field equations . In the weak field (ϵ ≪ 1) and low velocity (‖v‖/c ≪ 1) limit, the motion equations are d2r/dt2 = ∇ϕ + F, where ϕ = (ϵ/2)c2γ00.

Metric‐affine f(R)‐gravity with torsion: an overview

AbstractTorsion and curvature could play a fundamental role in explaining cosmological dynamics. f(R)‐gravity with torsion is an approach aimed to encompass in a comprehensive scheme all the Dark Side of the Universe (Dark Energy and Dark Matter). We discuss the field equations in empty space and in presence of perfect fluid matter taking into account the analogy with the metric‐affine formalism. The result is that the extra curvature and torsion degrees of freedom can be dealt under the standard of an effective scalar field of fully geometric origin. The initial value problem for such theories is also discussed.

Generalized Kerr spacetime with an arbitrary mass quadrupole moment: geometric properties versus particle motion

An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with an arbitrary mass quadrupole moment and is specified by three parameters, the mass M, the angular momentum per unit mass a and the quadrupole parameter q. It reduces to the Kerr spacetime in the limiting case q = 0 and to the Erez–Rosen spacetime when the specific angular momentum a vanishes. The geometrical properties of such a solution are investigated. Causality violations, directional singularities and repulsive effects occur in the region close to the source. Geodesic motion and accelerated motion are studied on the equatorial plane which, due to the reflection symmetry property of the solution, also turns out to be a geodesic plane.

F(R) gravity with torsion: the metric-affine approach

The role of torsion in f(R) gravity is considered in the framework of metric-affine formalism. We discuss the field equations in empty space and in the presence of perfect fluid matter, taking into account the analogy with the Palatini formalism. As a result, the extra curvature and torsion degrees of freedom can be dealt as an effective scalar field of a fully geometric origin. From a cosmological point of view, such a geometric description could account for the whole dark side of the universe.

NEW GEOMETRIC FORMALISM FOR GRAVITY EQUATION IN EMPTY SPACE

In this paper, a complex daor field which can be regarded as the square root of space–time metric is proposed to represent gravity. The locally complexified geometry is set up, and the complex spin connection constructs a bridge between gravity and SU(1, 3) gauge field. Daor field equations in empty space are acquired, which are one-order differential equations and do not conflict with Einstein's gravity theory.

Killing pairs and the empty space field equations

The empty space field equations are investigated for each of the canonical forms obtained previously for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal. It is found that the rational first integral of the geodesic equation, corresponding to the Killing pair, is always necessarily the ration of two linear first integrals.

A class of empty spacetimes admitting a rational first integral of the geodesic equation

The empty space field equations are solved for one of the canonical forms obtained previously by Vaz and Collinson for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal.

The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC αβμσ ,α = 0 whereC αβμσ is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC αβμσ = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.

The field of a particle in general relativity theory

A time-dependent, spherically symmetric solution of the Einstein field equations for empty space is found. It can be regarded as describing the field of a particle in an empty expanding or contracting universe. The co-ordinatest andr have temporal and spatial character, respectively, both outside and inside the Schwarzschild sphere. Fort finite there is a real sngularity atr=0. The motion of a falling particle is investigated. In the limit of a static field one gets the Schwarzschild solution outside of this sphere, while insideg00 = 0, ¦g11¦ = ∞. This suggests that the inside of the Schwarzschild sphere is unphysical. Some speculations are presented on the final state of a star that has undergone gravitational collapse: it may involve a new state of matter, or there may be a three-dimensional timeless space inside it. The relation between the Schwarzschild an the Kruskal solutions is 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.

Cylindrical wave solutions of the field equations representing zero-rest-mass scalar fields and those of lichnerowicz's « total radiation » in a generalized Einstein-Rosen space-time

Lichnerowicz (1960), by using a property of the Riemann tensor, has given the field equations, Rij=ϱwiwj, wiwi=0, ϱ being a scalar and has termed them as representing a state of « total radiation ». Recently Rao (1970) derived these field equations under a geometrical relation which as stated by him imposes on the Rieman tensor a severe restriction, and he obtained a class of exact solutions of the above field equations corresponding to the cylindrically symmetric space-time with two degrees of freedom. The present authors, in this paper, have obtained exact wave solutions of the field equations representing zero-rest-mass scalar field in the generalized Einstein-Rosen space-time, and it has been shown that by a suitable choice of the scalar field, the field equations considered by Lichnerowicz and Rao can be deduced without imposing any condition on the geometrical nature of the Riemann tensor. Finally a method is given by which most general solutions of these field equations, can be generated from those of the empty space field equations. It is further shown that the solutions of Rao (1960) are only a special case of those found in this paper.

Cylindrically symmetric solutions of a scalar–tensor theory of gravitation

The cylindrically symmetric solutions for the Einstein–Rosen metric of a scalar–tensor theory proposed by Dunn have been obtained. A method has been given by which one can obtain, under certain conditions, solutions of this scalar–tensor theory from known solutions of the empty space field equations of Einstein’s theory of gravitation. It is also found that one of the solutions of the scalar–tensor theory is nonsingular in the sense of Bonnor. Further some special solutions are obtained which reduce to the well-known solution of Levi-Civita and a time dependent solution obtained by Misra and Radhakrishna.

Charged Tomimatsu-Sato metrics

We show that the solutions of Einstein-Maxwell field equations can be constructed from the Tomimatsu-Sato solutions of Einstein's field equations for empty space simply by inspection when δ has integral values.

The motion of particles in general relativity beyond the geodetic approximation

A tensorial surface integral relationship which is equivalent to the Einstein field equations for empty space is used to derive the equation of motion of a finite-mass monopole correct to terms proportional to the square of the mass of the particle. The resulting equation of motion is covariant under both continuous transformations of the four co-ordinates and also under singular (mass-dependent) co-ordinate transformations. No infinities occur in the calculations.

Newman-Penrose approach to twisting degenerate metrics

The well known method ofNewman andPenrose is used to find solutions of the Einstein empty space field equations, which are algebraically special and where the degenerate principal null vectors are not hypersurface orthogonal. As is to be expected the method systematically yields the results obtained byKerr. An explanation is given of the complex coordinate transformation technique of generating new metrics from Schwarzschild's; also a generalisation of Kerr and Schild type metrics is investigated.

Canonical Forms of Gravitational Field Equations in Empty Space

Canonical Forms of Gravitational Field Equations in Empty Space

Some solutions of Einstein's equations in general relativity

AbstractA new class of axi-symmetric stationary solutions of Einstein's empty space field equations is obtained. Non-existence of solutions of certain other classes is proved.