Diffeomorphism invariance in the Hamiltonian formulation of General Relativity

It is shown that when the Einstein-Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as the crucial consistency requirement that must be met by any Hamiltonian formulation of General Relativity.     

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.     

Towards a general solution of the Hamiltonian constraints of General Relativity

The present work has a double aim. On the one hand, we call attention on the relationship existing between the Ashtekar formalism and other gauge-theoretical approaches to gravity, in particular the Poincaré Gauge Theory. On the other hand, we study two kinds of solutions for the constraints of General Relativity, consisting of two mutually independent parts, namely a general three-metric-dependent contribution to the extrinsic curvature K _ab in terms of the Cotton–York tensor, and besides it further metric independent contributions, which we analyze in particular in the presence of isotropic three-metrics.     

Events and Observables in Generally Invariant Spacetime Theories

We address the problem of observables in generally invariant spacetime theories such as Einstein’s general relativity. Using the refined notion of an event as a “point-coincidence” between scalar fields that completely characterise a spacetime model, we propose a generalisation of the relational local observables that does not require the existence of four everywhere invertible scalar fields. The collection of all point-coincidences forms in generic situations a four-dimensional manifold, that is naturally identified with the physical spacetime.     

General Relativity on a Null Surface: Hamiltonian Formulation in the Teleparallel Geometry

The Hamiltonian formulation of generalrelativity on a null surface is established in theteleparallel geometry. No particular conditions on thetetrads are imposed, like the time gauge condition. Bymeans of a 3 + 1 decomposition the resultingHamiltonian arises as a completely constrained system.However, it is structurally different from the standardArnowitt–Deser–Misner (adm) typeformulation. In this geometrical framework the basic fieldquantities are tetrads that transform under the globalSO(3, 1) and the torsion tensor.     

Hamiltonian Two-Body System in Special Relativity

We consider an isolated system made of two pointlike bodies interacting at a distance in the nonradiative approximation. Our framework is the covariant and a priori Hamiltonian formalism of “predictive relativistic mechanics”, founded on the equal-time condition. The center of mass is rather a center of energy. Individual energies are separately conserved and the meaning of their positivity is discussed in terms of world-lines. Several results derived decades ago under restrictive assumptions are extended to the general case. Relative motion has a structure similar to that of a nonrelativistic one-body motion in a stationary external potential, but its evolution parameter is generally not a linear function of the center-of-mass time, unless the relative motion is circular (in this latter case the motion is periodic in the center-of-mass time). Finally the case of an extreme mass ratio is investigated. When this ratio tends to zero the heavy body coincides with the center of mass provided that a certain first integral, related to the binding energy, is not too large.     

Quantum General Relativity and the Meaning of (R + R2) Theories

It is argued that, due to the cut-off lengths arising in Quantum General Relativity, R2 corrections of Einstein's theory cannot be interpreted as quantum corrections.     

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.     

Hamiltonian Formalism of de-Sitter Invariant Special Relativity

The Lagrangian of Einstein's special relativity with universal parameter c (SR_c) is invariant under Poincaré transformation, which preserves Lorentz metric η_μν. The SR_c has been extended to be one which is invariant under de Sitter transformation that preserves so-called Beltrami metric B_μν. There are two universal parameters, c and R, in this Special Relativity (denoted as SR_cR). The Lagrangian-Hamiltonian formulism of SR_cR is formulated in this paper. The canonic energy, canonic momenta, and 10 Noether charges corresponding to the space-time's de Sitter symmetry are derived. The canonical quantization of the mechanics for SR_cR-free particle is performed. The physics related to it is discussed.     

Hamiltonian Formalism of de-Sitter Invariant Special Relativity

The Lagrangian of Einstein's special relativity with universal parameter c (SRc) is invariant under Poincaré transformation, which preserves Lorentz metric ημν. The SRc has been extended to be one which is invariant under de Sitter transformation that preserves so-called Beltrami metric Bμν. There are two universal parameters, c and R, in this Special Relativity (denoted as SRcR). The Lagrangian-Hamiltonian formulism of SRcR is formulated in this paper.The canonic energy, canonic momenta, and 10 Noether charges corresponding to the space-time's de Sitter symmetry are derived. The canonical quantization of the mechanics for SRcR-free particle is performed. The physics related to it is discussed.