On Caustics and Singularities in General Relativity

Comparing gravitational converging effects with self-focussing and lense effects in ordinary optics, we consider some physical properties of caustics in General Relativity. The resulting relations between geometrical and physical features of caustics are used to discuss the power asymptotes of homogeneous cosmological models in more detail.     

On Singularities of General Relativity and Gravitational Equations of Fourth Order

Comparing Einstein's gravitational equations and equations containing fourth-order derivative corrections we discuss some aspects of singularities of spherically symmetric static vacuum solutions from the point of view, first, of the coupling to non-gravitational matter and, second, of the Einsteinian classical particle programme.     

General Covariance in General Relativity?

The mathematical approach to General Relativity insists that all coordinate systems are equal. However physicists and astrophysicists in fact almost always use preferred coordinate systems not merely to simplify the calculations but also to help define quantities of physical interest. This suggests we should reconsider and perhaps refine the dogma of General Covariance.     

Mass and energy in General Relativity

We consider the Denisov-Solov'ov example which shows that the inertial mass is not well defined in General Relativity. It is shown that the mathematical reason why this is true is a wrong application of the Stokes theorem. Then we discuss the role of the order of asymptotically flatness in the definition of the mass. In conclusion some comments on conservation laws in General Relativity are presented.     

Modified General Relativity and Cosmology

Aspects of the modified general relativity theory of Rastall, Al-Rawaf and Taha are discussed in both the radiation- and matter-dominated flat cosmological models. A nucleosynthesis constraint on the theory's free parameter is obtained and the implication for the age of the Universe is discussed. The consistency of the modified matter-dominated model with the neoclassical cosmological tests is demonstrated.     

Fermat's principle in General Relativity

We derive Fermat's principle from the causal structure of spacetime, as well as from an appropriate variational principle. We show that the latter leads to a particular Hamilton-Jacobi formalism.     

Electromagnetic Duality in General Relativity

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we consider duality relations analogous to those in electromagnetic theory. It turns out that the duality transformation implies the Einstein vacuum equation without the cosmological term. The vacuum equation is invariant under interchange of active and passive electric parts, giving rise to the same vacuum solutions but with the opposite sign for the gravitational constant. Further, by modifying the equation it is possible to construct interesting dual solutions to vacuum as well as to flat spacetimes.     

Electrical Conductivity in General Relativity

The general relativistic kinetic theory including the effect of a stationary gravitational field is applied to the electromagnetic transport processes in conductors. Then it is applied to derive the general relativistic Ohm's law where the gravitomagnetic terms are incorporated. The total electric charge quantity and charge distribution inside conductors carrying conduction current in some relativistic cases are considered. The general relativistic Ohm's law is applied to predict new gravitomagnetic and gyroscopic effects which can, in principle, be used to detect the Lense-Thirring and rotational fields.     

Quantum CTC's in General Relativity

We review different spacetimes that contain nonchronal regions separated from the causal regions by chronology horizons and investigate their connection with some important aspects one would expect to be present in a final theory of quantum gravity, including: stability to classical and quantum metric fluctuations, boundary conditions of the universe and gravitational topological defects corresponding to spacetime kinks.     

Massive photons in General Relativity

In order to avoid a speed-of-light catastrophe in General Relativity with an electromagnetic source, gauge invariance with respect to the electric charge is broken with the photon acquiring mass. The general equations for the Einstein-Maxwell system are derived for the case with massive photons. Nonminimal couplings which might compete with the small minimal photon mass term are included.     

Curvature Diffusions in General Relativity

We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds.     

Ideal stars and General Relativity

We study a system of differential equations that governs the distribution of matter in the theory of General Relativity. The new element in this paper is the use of a dynamical action principle that includes all the degrees of freedom, matter as well as metric. The matter lagrangian defines a relativistic version of non-viscous, isentropic hydrodynamics. The matter fields are a scalar density and a velocity potential; the conventional, four-vector velocity field is replaced by the gradient of the potential and its scale is fixed by one of the Eulerian equations of motion, an innovation that significantly affects the imposition of boundary conditions. If the density is integrable at infinity, then the metric approaches the Schwarzschild metric at large distances. There are stars without boundary and with finite total mass; the metric shows rapid variation in the neighbourhood of the Schwarzschild radius and there is a very small core where a singularity indicates that the gas laws break down. For stars with boundary there emerges a new, critical relation between the radius and the gravitational mass, a consequence of the stronger boundary conditions. Tentative applications are suggested, to certain Red Giants, and to neutron stars, but the investigation reported here was limited to homogeneous polytropes. Comparison with the results of Oppenheimer and Volkoff on neutron cores shows a close agreement of numerical results. However, in the model the boundary of the star is fixed uniquely by the required matching of the interior metric to the external Schwarzschild metric, which is not the case in the traditional approach. There are solutions for which the metric is very close to the Schwarzshild metric everywhere outside the horizon, where the source is concentrated. The Schwarzschild metric is interpreted as the metric of an ideal, limiting configuration of matter, not as the metric of empty space.     

Quantum Gauge General Relativity

Based on gauge principle, a new model on quantum gravity is proposed in the frame work of quantum gauge theory of gravity. The model has local gravitational gauge symmetry, and the field equation of the gravitational gauge field is just the famous Einstein‘s field equation. Because of this reason, this model is called quantum gauge general relativity, which is the consistent unification of quantum theory and general relativity. The model proposed in this paper is a perturbatively renormalizable quantum gravity, which is one of the most important advantage of the quantum gauge general relativity proposed in this paper. Another important advantage of the quantum gauge general relativity is that it can explain both classical tests of gravity and quantum effects of gravitational interactions, such as gravitational phase effects found in COW experiments and gravitational shielding effects found in Podkletnov experiments.     

Quantum Gauge General Relativity

Based on gauge principle, a new model on quantum gravity is proposed in the frame work of quantum gauge theory of gravity. The model has local gravitational gauge symmetry, and the field equation of the gravitational gauge field is just the famous Einstein‘s field equation. Because of this reason, this model is called quantum gauge general relativity, which is the consistent unification of quantum theory and general relativity. The model proposed in this paper is a perturbatively renormalizable quantum gravity, which is one of the most important advantage of the quantum gauge general relativity proposed in this paper. Another important advantage of the quantum gauge general relativity is that it can explain both classical tests of gravity and quantum effects of gravitational interactions, such as gravitational phase effects found in COW experiments and gravitational shielding effects found in Podkletnov experiments.     

Geometrization of Mass in General Relativity

In this paper we will extend the notion of tangent bundle to a Z ₂ graded tangent bundle. This graded bundle has a Lie algebroid structure and we can develop notions semi-Riemannian metrics, Levi-Civita connection, and curvature, on it. In case of space-times manifolds, even part of the tangent bundle is related to space and time structures (gravity) and odd part is related to mass distribution in space-time. In this structure, mass becomes part of the geometry, and Einstein field equation can be reconstructed in a new simpler form. The new field equation is purely geometric.     

On Parametrized General Relativity

A physical framework has been proposed which describes manifestly covariant relativistic evolution using a scalar time . Studies in electromagnetism, measurement, and the nature of time have demonstrated that in this framework, electromagnetism must be formulated in terms of -dependent fields. Such an electromagnetic theory has been developed. Gravitation must also use of -dependent fields, but many references do not take the metric's dependence on fully into account. Others differ markedly from general relativity in their formulation. In contrast, this paper outlines steps towards a -dependent classical intrinsic formulation of gravitation, patterned after general relativity, which we call parametrized general relativity (PGR). Given the existence of a preferred foliation, the Hamiltonian constraint is removed. We find that some nonmetricity in the connection is allowed, unlike in general relativity. Conditions on the allowable nonmetricity are found. Consideration of the initial value problem confirms that the metric signature should normally be O(3, 2) rather than O(4, 1). Following the lead of earlier works, we argue that concatenation (integration over ) is unnecessary for relating parametrized physics to experience, and propose an alternative to it. Finally, we compare and contrast PGR with other relevant gravitational theories.     

Gravitomagnetic and Gravitoelectric Waves in General Relativity

Lower-order terms in expansions of the equations of General Relativity in powers of v/c (post-Newtonian approximations) have long been a source of analogies with em theory. A classic textbook example is the steadily spinning sphere generating a constant dipole gravitomagnetic field, with its associated vector potential B^* ₀ = × (analog of the magnetic field B of a spinning charged sphere). In the nonsteady case there are associated gravitoelectric fields E* = – _t – * also, where * is the gravitational Coulomb potential. The case of a rigid sphere spun up from rest by an external (nongravitational) torque at t = 0 is enlightening, as it demonstrates the generation of B* and E* wave fields propagating outward with the velocity of light c: for large t, B* B^* ₀. In a coordinate system for which the metric tensor is nearly equal to the Minkowski tensor, the three-vector potential obeys an equation isomorphic to the electrodynamic equation, that is, ² = –*j* with j* = –v, where is the mass density, v the three-velocity, and * = 16Gc^–2 = 3.7 × 10^–26 mksu, G being the gravitational constant. Significantly, one can construct a gauge invariant four-vector potential F* = (ic^–14*, ), obeying field equations isomorphic to Maxwell's in the Lorentz gauge F _, = 0. The traveling transient dipole field exerts torques on matter in its path, setting up shear strains that may be measurable for very large momentum transfers, for example, between massive astronomical bodies. A rough calculation suggests that such strains are in principle observable.     

Time Asymmetry and Chaos in General Relativity

In this work the late-time evolution of Bianchi type VIII models is discussed. These cosmological models exhibit a chaotic behaviour towards the initial singularity and our investigations show that towards the future, far from the initial singularity, these models have a non-chaotic evolution, even in the case of vacuum and without inflation. These space-time solutions turn out to exhibit a particular time asymmetry. On the other hand, investigations of the late-time behaviour of type VIII models by another author have the result that chaos continues for ever in the far future and that these solutions have a time symmetric behaviour: this result was obtained using the approximation methods of Belinski, Khalatnikov and Lifshitz (BKL) and we try to find out a possible reason explaining why the different approaches lead to distinct outcomes. It will be shown that, at a heuristic level, the BKL method gives a valid approximation of the late-time evolution of type VIII models, agreeing with the result of our investigations.