Narrow Results

In this letter we discuss the possibility of treating the spacetime by itself as a kind of deformable body for which we can define a fundamental lattice, just like atoms in crystal lattices. We show three signs pointing in that direction. We simulate the spacetime manifold by a very specific congruence of curves and use the Landau–Raychadhuri equation to study the behavior of such a congruence. The lattice appears because we are forced to associate to each curve of the congruence a sort of fundamental "particle". The world-lines of these particles should be identified with the congruence fulfilling the spacetime manifold. The conclusion is that when describing the deformations of the spacetime the Einstein equations emerge and the spacetime metric should be treated as a secondary (not fundamental) object of the theory.

General solution for braneworld dynamics coupled with the bulk Einstein equation is considered under the Schwarzschild ansatz. They relate the brane metric to the exterior configurations, and establish conditions for the braneworlds to reproduce successful results of the Einstein gravity.

Metrics representing black holes in General Relativity may exhibit naked singularities for certain values of their parameters. This is the case for super-extremal (J² > M > 0) Kerr and super-extremal (|Q| > M > 0) Reissner-Nördstrom spacetimes, and also for the negative mass Schwarzschild spacetime. We review our recent work where we show that these nakedly singular spacetimes are unstable under linear gravitational perturbations, a result that supports the cosmic censorship conjecture, and also that the inner stationary region beyond the inner horizon of a Kerr black hole (J² < M) is linearly unstable.

In this essay we marshal evidence suggesting that Einstein gravity may be an emergent phenomenon, one that is not "fundamental" but rather is an almost automatic low-energy long-distance consequence of a wide class of theories. Specifically, the emergence of a curved spacetime "effective Lorentzian geometry" is a common generic result of linearizing a classical scalar field theory around some nontrivial background. This explains why so many different "analog models" of general relativity have recently been developed based on condensed matter physics; there is something more fundamental going on. Upon quantizing the linearized fluctuations around this background geometry, the one-loop effective action is guaranteed to contain a term proportional to the Einstein–Hilbert action of general relativity, suggesting that while classical physics is responsible for generating an "effective geometry," quantum physics can be argued to induce an "effective dynamics." This physical picture suggests that Einstein gravity is an emergent low-energy long-distance phenomenon that is insensitive to the details of the high-energy short-distance physics.

After criticizing the various existing attempts at extending the concept of the Minkowski space, the following problem is considered: If there is the ultimate theory at all, how should the space–time in it be formulated? The principle of "quantum priority" is proposed, and under this principle, it is argued that the ultimate theory should have the affine symmetry in the framework of quantum gravity. It is shown that the Poincaré symmetry of particle physics is realized as a result of spontaneous breakdown of the affine symmetry.

We study properties of resonant solutions to the scalar wave equation in several axially symmetric spacetimes. We prove that nonaxial resonant modes do not exist neither in the Lanczos dust cylinder, the extreme (2 + 1) dimensional Bañados–Taitelboim–Zanelli (BTZ) spacetime nor in a class of simple rotating wormhole solutions. Moreover, we find unstable solutions to the wave equation in the Lanczos dust cylinder and in the r² < 0 region of the extreme (2 + 1) dimensional BTZ spacetime, two solutions that possess closed timelike curves. Similarities with previous results obtained for the Kerr spacetime are explored.

We would first show that how the two fundamental constants, the velocity of light, $c$, and the so called, cosmological constant, $\mathrm{\Lambda }$, arise from geometric characterization of forcefree state of spacetime. They are the two most fundamental constants as they have been synthesized in the structure of spacetime, no other constant can claim this degree of fundamentalness. Next when spacetime turns inhomogeneous, Einstein gravity arises most naturally from the spacetime geometry without any prescription from outside; i.e. the inverse square law is mandated by the geometry, and not a prescription. It would then suggest that dynamics of a fundamental force should not be prescribed but should perhaps be determined by the geometry of appropriate spacetime manifold.

The general relativity theory is redefined equivalently in almost Kähler variables: symplectic form, θ[g], and canonical symplectic connection, (distorted from the Levi–Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metric g on a manifold V enabled with a necessary type nonholonomic 2 + 2 distribution. Such nonholonomic symplectic variables allow us to formulate the problem of quantizing Einstein gravity in terms of the A-model complexification of almost complex structures on V, generalizing the Gukov–Witten method [1]. Quantizing , we derive a Hilbert space as a space of strings with two A-branes which for the Einstein gravity theory are nonholonomic because of induced nonlinear connection structures. Finally, we speculate on relation of such a method of quantization to curve flows and solitonic hierarchies defined by Einstein metrics on (pseudo) Riemannian spacetimes.

For (2 + 2)-dimensional nonholonomic distributions, the physical information contained into a spacetime (pseudo) Riemannian metric can be encoded equivalently into new types of geometric structures and linear connections constructed as nonholonomic deformations of the Levi–Civita connection. Such deformations and induced geometric/physical objects are completely determined by a prescribed metric tensor. Reformulation of the Einstein equations in nonholonomic variables (tetrads and new connections, for instance, with constant coefficient curvatures and/or Yang–Mills like potentials) reveals hidden geometric and rich quantum structures. It is shown how the Einstein gravity theory can be redefined equivalently as certain gauge models on nonholonomic affine and/or de Sitter frame bundles. We speculate on possible applications of the geometry of nonholonomic distributions with associated nonlinear connections in classical and quantum gravity.

A new framework for perturbative quantum gravity is proposed following the geometry of non-holonomic distributions on (pseudo)Riemannian manifolds. There are considered such distributions and adap-ted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two-loop renormalizable effective actions. We use a key result from our partner work arXiv: 0902.0911 that the classical Einstein gravity theory can be reformulated equivalently as a non-holonomic gauge model in the bundle of affine/de Sitter frames on pseudo-Riemannian space–time. It is proven that (for a class of non-holonomic constraints and splitting of the Levi–Civita connection into a "renormalizable" distinguished connection, on a base background manifold, and a gauge-like distortion tensor, in total space) a non-holonomic differential renormalization procedure for quantum gravitational fields can be elaborated. Calculation labor is reduced to one- and two-loop levels and renormalization group equations for non-holonomic configurations.