Spherically symmetric perfect fluid solutions of Einstein's equations in noncomoving coordinates

Almost all known spherically symmetric perfect fluid solutions of Einstein's equations have been obtained in comoving coordinates and nearly all are shear-free. In this paper we study two solutions in noncomoving coordinates and show that they contain shear.     

Spherically symmetric conformally flat distributions of charged dust and zero-mass scalar fields

The problem of charged dust distribution in the presence of zero-mass scalar field for a spherically symmetric conformally flat metric has been investigated. Exact solutions are obtained in the comoving coordinate system for the static model as well as for the nonstatic model. It has been shown that in the nonstatic model the electromagnetic field and dust distribution cannot survive when the scalar field is taken to be a function of timet only. Physical interpretation of the solutions has been investigated.     

Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid

Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vector such that _;v + _v; =2g _v are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.This work was supported in part by the United States Atomic Energy Commission under contract number AT(04-3)-34 Project Agreement No. 125. It was completed when one of the authors (A.H.T.) was on sabbatical leave from the University of California, Berkeley and in residence at the Department of Applied Mathematics and Theoretical Physics, Cambridge University.     

Exact solutions of Einstein's field equations for a massive point-particle with scalar point-charge

The exact static and spherically symmetric solution of Einstein's field equations for a massive point-particle with a scalar point-charge as source of a massless scalar field is derived in Schwarzschild coordinates. There exists no longer a Schwarzschild horizon. Only at the point-particle metric and scalar field are singular (naked singularity).     

General relativity as a conformally invariant scalar gauge field theory

The global symmetry implied by the fact that one can multiply all masses with a common constant is made into a local, gauge symmetry. The matter action then becomes Conformally invariant and it seems natural to choose for the corresponding scalar gauge field the action for a conformally invariant (massless) scalar field. The resulting conformally invariant theory turns out to be equivalent to general relativity. Since this means that the usual Einstein-Hilbert action is not, in fact, a true gauge action for the space-time geometry, the full theory ought to be supplied with such a term. Gauge-theoretic arguments and conformal invariance requirements dictate its form.     

Spherically symmetric static solutions of the Einstein-Maxwell equations

We report a new formalism to obtain solutions of Einstein-Maxwell’s equations for static spheres assuming the matter content to be a charged perfect fluid of null-conductivity. Defining three new variablesu=4πεr ²,ν=4πpr ^{2 2} andw=(4π/3)(ρ+ε)r ² whereε, ρ andε denote respectively energy densities of the electric, matter and free gravitational fields whereasp is the fluid pressure, Einstein’s field equations are rewritten in an elegant form. The solutions given by Bonnor [1], Nduka [2], Cooperstock and De la Cruz [3], Mehra [4], Tikekar [5,6], Xingxiang [7], Patino and Rago [8] are all shown to possess simple relations betweenu, v, andw whereas Pant and Sah’s [9] solution for which all the three functions,u, v, andw are constants is a trivial case of the present formalism, We have presented six new solutions with ε = 2ρ. For the first three solutionsw andu are constants withv as a variable whereas the remaining three solutions satisfy the equation of state for isothermal gas;v =kw =-ku where (i)k is an arbitrary constant but not equal to 1 or 1/3 (ii)k = 1 and (iii)k = 1/3. We also obtained a generalization of Cooperstock and De la Cruz’s [3] solution which is regular for 2ρ > ε but singular for 2ρ ≤ ε.     

A nonstatic plane symmetric solution of Einstein's field equations for zero-rest mass scalar fields

An exact solution of Einstein's field equations of general relativity is given for a plane symmetric zero-rest-mass scalar field in the nonstatic case. The solution generalizes Singh's solution, which itself extends Taub's empty space-time.     

A plane symmetric solution of Einstein's field equations of general relativity containing zero-rest-mass scalar fields

An exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gijand the zero-rest-mass scalar field φexhibit plane symmetry in the sense of Taub [9]. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static zero-rest-mass scalar fields are present. The static plane symmetric solutoins of Einstein's field equations in the presence of massive scalar fields, and the difference between the massless and non-massless scalar fields are being investigated, and will be published separately later on. We also hope to discuss non-static plane symmetric solutions of Einstein's field equations in the presence of scalar fields in future.     

On solutions of Einstein's equations with scalar zero-mass source

It is shown that most, if not all, previously known solutions of the gravitational field equations with a zero-mass scalar sourceø can be found without solving the field equations by applying known theorems. Penney's recent solution, characterized in part by having a conformally flat metricds ₂=e ² _ij dx ⁱ dx ^j is shown to be insufficiently general when his vectora _i is null. The problem is reformulated and new solutions of the conformally flat type are found. These are, in general, such that ø and are no longer linearly related.     

On homogeneous solutions of Einstein's field equations

In the first part of this paper we describe a formalism capable of finding all homogeneous solutions of Einstein's field equations with any arbitrary energy-impulse tensor. In the second part we find all homogeneous vacuum solutions.     

Higher-dimensional vacuum solutions of Einstein's field equations

Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.     

Enveloping algebra identities on solutions of conformally invariant wave equations

We study identities in the enveloping algebra of the conformal group, which is the symmetry group of many wave equations: d'Alambert, Weyl, Maxwell, etc. We find all second-order identities for these equations and, in addition, the dimension of the space of nontrivial symmetry operators of any order for the d'Alambert equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 14–18, September, 1991.     

Two simple solutions to Einstein's field equations

Metrics of the formds ²=dx ²+dy ²–dt ²+N ² dz ² are considered and found to contain rotating dust solutions as well as pure radiation fields.     

Meronlike solutions of conformally covariant coupled nonlinear field equations

We derive a new class of solutions of conformally covariant coupled spinor and scalar equations including a nonlinear spinor self-coupling term, for which the energy-momentum density is nonzero, but the total energy is zero (“meronlike” solutions).     

On axially symmetric solutions of Einstein's field equations in matter and an example of static gravitational shielding

A model system, consisting of a thin spherical shell with radiusR and massM and a point massm at a distances>R from the center of the sphere, held fixed by an appropriate strut, is solved to ordermM. The stresses in the shell are not of the canonical Weyl type, and it is argued that the same is true for more realistic situations, e.g., rotating matter. Owing to the nonlinearity of Einstein's field equations, the field of the point mass is shielded from the interior of the shell by a factor lying between 1–3M/R and 1–2M/R, and the field outside the shell explicitly depends onR.     

Generating solutions of Einstein's field equations by typing mistakes

A solution to Einstein's field equations is presented that represents a Petrov type II electromagnetic null field with one Killing vector. This solution generalizes a vacuum solution previously discovered by Hoenselaers. The solution was found by the peculiar method of generalizing a member of this class inadvertently discovered by making a typing error when checking the vacuum solution with the computer algebra system SHEEP.     

Spherically symmetric solutions in Macroscopic Gravity

Schwarzschild’s solution to the Einstein Field Equations was one of the first and most important solutions that lead to the understanding and important experimental tests of Einstein’s theory of General Relativity. However, Schwarzschild’s solution is essentially based on an ideal theory of gravitation, where all inhomogeneities are ignored. Therefore, any generalization of the Schwarzschild solution should take into account the effects of small perturbations that may be present in the gravitational field. The theory of Macroscopic Gravity characterizes the effects of the inhomogeneities through a non-perturbative and covariant averaging procedure. With similar assumptions on the geometry and matter content, a solution to the averaged field equations as dictated by Macroscopic Gravity are derived. The resulting solution provides a possible explanation for the flattening of galactic rotation curves, illustrating that Dark Matter is not real but may only be the result of averaging inhomogeneities in a spherically symmetric background.     

Long-range scalar field in conformally flat space-time

Consequences of a massless scalar field in conformally flat space-time are studied. Then a wide class of solutions of the scalar field is obtained.     

Spherically symmetric configurations of General Relativity in presence of scalar fields: separation of circular orbits

We show that the test body stable circular orbits around the spherically symmetric black hole (BH) configuration can form disjoint structures in presence of a minimally coupled nonlinear scalar field. General conditions for the disjoint structures to exist are formulated. To present examples we construct a two-parametric family of exact solutions to Einstein equations with scalar fields for appropriate self-interaction potentials. For different values of the family parameters the solutions describe either BH or naked singularity (NS). We found numerically regions of the parameters when there exist two disjoint regions of stable circular orbits; such nonconnected structures indeed can exist in case of both BH and NS solutions.