Unification in five dimensions

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.

Obtaining exact solutions of the spherically symmetric general relativistic gravitational field equations describing the interior structure of an isotropic fluid sphere is a long standing problem in theoretical and mathematical physics. The usual approach to this problem consists mainly in the numerical investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity equations, which describes the hydrostatic stability of the dense stars. In the present paper we introduce an alternative approach for the study of the relativistic fluid sphere, based on the relativistic mass equation, obtained by eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation. Despite its apparent complexity, the relativistic mass equation can be solved exactly by using a power series representation for the mass, and the Cauchy convolution for infinite power series. We obtain exact series solutions for general relativistic dense astrophysical objects described by the linear barotropic and the polytropic equations of state, respectively. For the polytropic case we obtain the exact power series solution corresponding to arbitrary values of the polytropic index $n$. The explicit form of the solution is presented for the polytropic index $n=1$, and for the indexes $n=1/2$ and $n=1/5$, respectively. The case of $n=3$ is also considered. In each case the exact power series solution is compared with the exact numerical solutions, which are reproduced by the power series solutions truncated to seven terms only. The power series representations of the geometric and physical properties of the linear barotropic and polytropic stars are also obtained.

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 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.

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.

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

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.

A modified version of the field equations of general relativity is obtained on relaxing the covariant energy-momentum conservation condition. This introduces a single arbitrary constant and does not appear to upset the successes of general relativity in or outside cosmology. The matter-dominated cosmological model, based on the generalized field equations, is discussed. It is shown to provide more room for consistency with the observational data.

The importance of Einstein’s geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar identity in the absolute parallelism geometry is given. A more general differential identity in the parameterized absolute parallelism geometry is derived. Comparison and interrelationships between the above mentioned identities and their role in constructing field theories are discussed.

November 2015 marks the 100th anniversary of Einstein's theory of general relativity (GR). He published four papers on the subject in November 1915. Remarkable! The first two papers led to the field equations of GR, and the other two laid out vital consequences of the theory, including the anomalous precession of the perihelion of Mercury (unexplained since 1859) and the gravitational bending of light. The latter phenomenon was confirmed famously by Eddington's 1919 observation of a total solar eclipse and the shift of apparent position for stars, the light of which passed very near the sun. Einstein, already famous, became a household name — an early 20th-century superstar.

It is shown that the gravitational field in general relativity has the properties of a parametric manifold, a mathematical structure generalizing the concept of gauge fields. A theory of parametric spinors is developed which contains, as a limiting case, spinor fields of unitary groups in Riemannian manifolds. Transition from the familiar [Formula: see text] spinor formalism to the parametric picture makes it possible to obtain a reparametrization-invariant decomposition of the gravitational equations with respect to a non-null congruence. This new formulation of the relativistic field equations appears to be amenable to canonical quantization, and provides more versatility in the treatment of phase-space variables.

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.

An alternative derivation of general relativity is presented in which the field equations are derived from a gauge invariant quadratic Lagrangian by varying the affine field Gamma bca and keeping the geometry fixed. The field equations show that the Gamma bca can be considered as a Christoffel connection of a tensor gab and if, after the variation, the metric tensor is chosen to be this tensor, the field equations of general relativity with a cosmical constant are recovered.

The thin-sandwich conjecture for Einstein's field equations of general relativity is discussed. Three of the four initial-value equations of general relativity are considered. This system of three equations is shown to have an integrability condition when the spatial metric, its time rate of change, and the lapse function N are given. The nature of this integrability condition is discussed in detail. It is also noted that this integrability condition can be expected to play a significant role in any proof of existence for a thin-sandwich problem in which the spatial metric and its time derivative are chosen arbitrarily. A local proof of existence is given for the first three thin-sandwich equations. The proof allows the spatial metric components, their time rate of change, and the lapse function N to be chosen arbitrarily. No coordinate conditions of any kind are used in the proof.

When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. Because metric lengths are the physically measured lengths, their use makes it possible to understand general relativity directly in terms of physical quantities such as energy and momentum within a curved space–time. The metric form of the second law contains gravitational effects in exactly the same manner as occurs in relativity. Its mathematical derivation uses vectors rather than tensors, and nongravitational forces can occur in this modified second law without a tensor form. Because quantum mechanics is based on Newtonian concepts of energy and momentum, it is shown that when metric lengths replace coordinate lengths in Dirac’s wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation. Metric transformations are also used to describe the Dirac equation for the gravitational central field in a Schwarzschild metric.