v2: Section 4 is completely new, and important references have been added. v3: Sub-section 4.1 has been added, with new original calculations. v4: The figure has been amended
The concept of boundary plays an important role in several branches of general relativity, e.g., the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped surfaces. On the other hand, in a branch of mathematics known as geometric measure theory, the usefulness has been discovered long ago of yet another concept, i.e., the reduced boundary of a finite-perimeter set. This paper proposes therefore a definition of finite-perimeter sets and their reduced boundary in general relativity. Moreover, a basic integral formula of geometric measure theory is evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry, for the first time in the literature. This research prepares the ground for a measure-theoretic approach to several concepts in gravitational physics, supplemented by geometric insight. Moreover, such an investigation suggests considering the possibility that the in-out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole.
Exploring the characterization of singular black hole spacetimes, we study the relation between energy density, curvature invariants, and geodesic completeness using a quadratic $f(R)$ gravity theory coupled to an anisotropic fluid. Working in a metric-affine approach, our models and solutions represent minimal extensions of General Relativity (GR) in the sense that they rapidly recover the usual Reissner-Nordström solution from near the inner horizon outwards. The anisotropic fluid helps modify only the innermost geometry. Depending on the values and signs of two parameters on the gravitation…
In an ever-expanding spatially closed universe, the fractional change of the volume is the preeminent intrinsic time interval to describe evolution in General Relativity. The expansion of the universe serves as a subsidiary condition which transforms Einstein's theory from a first class to a second class constrained system when the physical degrees of freedom (d.o.f.) are identified with transverse traceless excitations. The super-Hamiltonian constraint is solved by eliminating the trace of the momentum in terms of the other variables, and spatial diffeomorphism symmetry is tackled explicitly …
Firstly, we present a reformulation of the standard canonical approach to spherically symmetric systems in which the radial gauge is imposed. This is done via the gauge unfixing technique, which serves as the exposition in the context of the radial gauge. Secondly, we apply the same techniques to the full theory, without assuming spherical symmetry, resulting in a reduced phase space description of general relativity. The canonical structure of the theory is analyzed.
The two-body problem in general relativity is reduced to the problem of an effective particle (with an energy-dependent relativistic reduced mass) in an external field. The effective potential is evaluated from the Born diagram of the linearized quantum theory of gravity. It reduces to a Schwarzschild-like potential with two different `Schwarzschild radii'. The results derived in a weak field approximation are expected to be relevant for relativistic velocities.
General relativity is a set of physical and geometric principles, which lead to a set of (Einstein) field equations that determine the gravitational field, and to the geodesic equations that describe light propagation and the motion of particles on the background. But open questions remain, including: What is the scale on which matter and geometry are dynamically coupled in the Einstein equations? Are the field equations valid on small and large scales? What is the largest scale on which matter can be coarse grained while following a geodesic of a solution to Einstein's equations? We address t…
This is the centenary year of general relativity, it is therefore natural to reflect on what perspective we have evolved in 100 years. I wish to share here a novel perspective, and the insights and directions that ensue from it.
In the paraxial regime of Newtonian optics, propagation of an ensemble of rays is represented by a symplectic ABCD transfer matrix defined on a reduced phase space. Here, we present its analogue for general relativity. Starting from simultaneously applied null geodesic actions for two curves, we obtain a geodesic deviation action up to quadratic order. We achieve this by following a preexisting method constructed via Synge's world function. We find the corresponding Hamiltonian function and the reduced phase space coordinates that are composed of the components of the Jacobi fields projected o…
Unlike the relativity theory it seeks to replace, causal set theory has been interpreted to leave space for a substantive, though perhaps 'localized', form of 'becoming'. The possibility of fundamental becoming is nourished by the fact that the analogue of Stein's theorem from special relativity does not hold in causal set theory. Despite this, we find that in many ways, the debate concerning becoming parallels the well-rehearsed lines it follows in the domain of relativity. We present, however, some new twists and challenges. In particular, we show that a novel and exotic notion of becoming i…
General relativity lacks the notion of the speed of gravity. This is inconvenient and the present paper is aimed at filling this gap up. To that end I introduce the concept of the "alternative" and argue that its variety called the "superluminal alternative" describes exactly what one understands by the "superluminal gravitational signal". Another, closely related, object called the "semi-superluminal alternative" corresponds to the situation in which a massive (and therefore gravitating) body reaches its destination sooner than a photon \emph{would}, be the latter sent \emph{instead} of the b…
We discuss what we take to be three possible misconceptions in the foundations of general relativity, relating to: (a) the interpretation of the weak equivalence principle and the relationship between gravity and inertia; (b) the connection between gravitational redshift results and spacetime curvature; and (c) the Einstein equivalence principle and the ability to "transform away" gravity in local inertial coordinate systems.
A new formulation of what may be called the "fundamental theorem of the theory of relativity" is presented and proved in (3+1)-space-time, based on the full classification of special transformations and the corresponding velocity addition laws. A system of axioms is introduced and discussed leading to the result, and a study is made of several variants of that system. In particular the status of the group axiom is investigated with respect to the condition of the two-way isotropy of light. Several issues which are ignored or misunderstood in the literature are emphasized.
The limiting mass of cold white dwarfs was first calculated by E. Stoner in an approximate model of a uniform star and was soon reduced by ~20% in papers by S. Chandrasekhar and L. D. Landau based on an exact solution of the equations for the stellar equilibrium. Here we examine uniform models of white dwarfs taking general relativity effects and the influence of finite temperature into account. Solutions are obtained in the form of finite analytic formulas and, for masses differing by no more than ~20% from the exact solutions, found by numerical integration of the differential equations for …
Detailed knowledge of the different classes of stellar orbits that can be accommodated in a given galactic potential is a prerequisite when building self-consistent models using for instance the Schwazschild technique. Furthermore, observational properties of galaxies depend on what these classes of orbits are and on the presence of chaos in the systems. In the realistic case in which the starting point for modeling is not a gravitational potential, but an observed density distribution, we will require a gravitational theory to make the connection between the stars that we see and the movement…
General relativity in four dimensions can be reformulated as a gauge theory, referred to as Palatini-Cartan-Holst theory. This paper describes its reduced phase space using a geometric method due to Kijowski and Tulczyjew and its relation to that of the Einstein-Hilbert approach.
