Thin-sandwich conjecture and a fifth initial value equation for the gravitational field

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed by H. Friedrich (H. Friedrich, “Cauchy problem for the conformal vacuum field equations in general relativity”, Commun. Math. Phys. 91 (1983), 445-472.), (H. Friedrich, “On the existence of n-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure”, Commun. Math. Phys. 107 (1986), 587-609.), (-,“On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations”, Jour. Diff. Geom. 34 (1991), 275-345) using conformal completions techniques. In this conformal approach one has to 1) construct initial data, which satisfy the general relativistic constraint equations, with appropriate behavior near the conformal boundary, and 2) show a local (and perhaps also a global) existence theorem for the associated evolution problem. In this context solutions of the constraint equations can be found by solving a nonlinear elliptic system of equations, one of which resembles the Yamabe equation (and coincides with this equation in some cases), with the system degenerating near the conformal boundary. In the first part of the talk I (PTC) will describe the existence and boundary regularity results about this system obtained some years ago in collaboration with Helmut Friedrich and Lars Andersson. Some new applications of those techniques are also presented. In the second part of the talk I will describe some new results, obtained in collaboration with Olivier Lengard, concerning the evolution problem.

Ideal observational cosmology

We consider several tensorial wave equations, specifically the equations of Maxwell, Yang–Mills, and Weyl fields, posed on a curved spacetime, and we establish new energy inequalities under certain one-sided geometric conditions. Our conditions restrict the lapse function and deformation tensor of the spacetime foliation, and turn out to be a one-sided and integral generalization of conditions recently proposed by Klainerman and Rodnianski as providing a continuation criterion for Einstein's field equations of general relativity. As we observe it here for the first time, one-sided conditions are sufficient to derive energy inequalities for certain tensorial equations, provided one takes advantage of some algebraic properties enjoyed by the natural energy functionals associated with the equations under consideration. Our method especially applies to the Bel–Robinson energy for Weyl fields, and our inequalities control the growth of the energy in a uniform way, with implied constants depending on the one-sided geometric bounds, only.

Uniqueness of Kottler spacetime and the Besse conjecture

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

In this work, we examine the possible existence of dark fields by showing that an energy‐momentum tensor associated with a dark field can be introduced into Einstein field equations of general relativity, in which the trace of the energy‐momentum of the dark field is identified with the cosmological constant. The introduction of dark fields into Einstein field equations is made possible by establishing field equations for the Ricci curvature tensor, which have similar mathematical structure to Maxwell field equations of electromagnetism. We also establish a system of field equations for the Riemann curvature tensor and investigate the possibility to represent physical systems consisting of dark fields and observable fields as spaces of constant scalar curvature, which are maximally symmetric spaces that admit the maximal number of Killing vectors. As an illustration, we show that if a dark field is considered as a dark fluid, then the pressure associated with the dark field can take negative values if the cosmological constant is assigned with positive values.

It is proved that the combined neutrino gravitational field equations in general relativity admit no nontrivial solutions in a static space-time provided that the energy flow vector of the neutrino field is timelike or null for all observers.

In a preceding paper, (see abstr. A2555 of 1974), the authors obtained a solution of the field equations of general relativity for an elastic sphere of constant density. The present investigation obtains a solution of the charged elastic fluid distribution in general relativity.

The metric tensor of a Riemannian space-time is composed quadratically of timelike and spacelike tetrad fields. A system of first-order field equations for the tetrad fields is postulated, which determines the geometric structure of space-time as well as the second-order dynamics of the fields. The energy-momentum content of the fields automatically corresponds to the geometric structure of the space-time according to the Einstein field equations. The spin contribution of the fields is mutually compensated and therefore spin does not influence the space-time geometry. As a consequence, it is not necessary to include torsion into the general theory of relativity and nevertheless the equivalence principle can be used to consistently transfer energy-momentum tensors from flat to curved space.

The generalized harmonic equations of general relativity are written in $3+1$ form. The result is a system of partial differential equations with first-order time and second-order space derivatives for the spatial metric, extrinsic curvature, lapse function and shift vector, plus fields that represent the time derivatives of the lapse and shift. This allows for a direct comparison between the generalized harmonic and the Arnowitt-Deser-Misner formulations. The $3+1$ generalized harmonic equations are also written in terms of conformal variables and compared to the Baumgarte-Shapiro-Shibata-Nakamura equations with moving puncture gauge conditions.

The core of general relativity are the Einstein field equations, which relate curvature to energy and momentum. This chapter introduces curvature and the energy-momentum tensor to derive tensor equations between them that fulfill the correct Newtonian limit—the Einstein field equation. Finally, the small curvature limit in which the Einstein field equations are linear is worked out. This establishes the contact with the other classical field theory, Maxwell electrodynamics.

R-separable coordinate systems are introduced as a new class of systems in which the equations of general relativity may be solved. The static, axially symmetric vacuum and electrovac Einstein field equations are solved for two such systems, tangent spheres and bispherical. The bispherical is found to be more versatile than any previously used coordinate system in that its eigensolutions can represent the exteriors of single point, double point and line sources. The tangent sphere eigensolutions are found to be generalisations of the Curzon solution. The relatively simple nature of the individual bispherical eigensolutions allows explicit integration of the field equations for a completely general, static, two-body source. The bodies are then charged, according to the Weyl formalism, and the conditions for balance obtained. Finally, it is shown that all vacuum Weyl solutions are either type I or type D.

The covariant wedge products of tetrads in General Relativity are introduced and related field equations are derived. The electromagnetic dual field is treated in detail and it is shown that this field is closely related with one component field, which is eigenfunction of d'Alembertian operator. The formalism is developed to incorporate internal symmetry.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

This book deals with Colombeau solutions to Einstein field equations in general relativity: Gravitational singularities, distributional SAdS BH spacetime-induced vacuum dominance. This book covers key areas of Colombeau nonlinear generalized functions, distributional Riemannian, geometry, distributional schwarzschild geometry, Schwarzschild singularity, Schwarzschild horizon, smooth regularization, nonsmooth regularization, quantum fields, curved spacetime, vacuum fluctuations, vacuum dominance etc. This book contains various materials suitable for students, researchers and academicians of this area.