Narrow Results

We present several theories of four-dimensional gravity in the Plebanski formulation, in which the tetrads and the connections are the independent dynamical variables. We consider the relation between different versions of gravitational theories: Einsteinian, "topological," "mirror" gravities and gravity with torsion. We assume that our world, in which we live, is described by the self-dual left-handed gravity, and propose that if the Mirror World exists in Nature, then the "mirror gravity" is the right-handed antiself-dual gravity. In this connection, we give a brief review of gravi-weak unification models. In accordance with cosmological measurements, we consider the Universe with broken mirror parity. We also discuss the problems of cosmological constant and communication between visible and mirror worlds. Investigating a special version of the Riemann–Cartan space–time, which has torsion as an additional geometric property, we have shown that in the Plebanski formulation the ordinary and dual "topological" sectors of gravity, as well as the gravity with torsion, are equivalent. Equations of motion are obtained. In this context, we have also discussed a "pure connection gravity" — a diffeomorphism-invariant gauge theory of gravity. Loop Quantum Gravity is also briefly reviewed.

The names tetrad, tetrads, cotetrads have been used with many different meanings in the physics literature, not all of them equivalent from the mathematical point of view. In this paper, we introduce unambiguous definitions for each of those terms, and show how the old miscellanea made many authors introduce in their formalism an ambiguous statement called the "tetrad postulate," which has been the source of much misunderstanding, as we show explicitly by examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields θ^a, a = 0, 1, 2, 3 is a worthy enterprise, we derive the equation of motion of each θ^a using modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit us to show that each θ^a satisfies a well-defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principle. We compare the cotetrad wave equation satisfied by each θ^a with some others appearing in the literature, and which are unfortunately in error.

This paper applies the first-order Seiberg–Witten map to evaluate the first-order non-commutative Kerr tetrad. The classical tetrad is taken to follow the locally non-rotating frame prescription. We also evaluate the tiny effect of non-commutativity on the efficiency of the Penrose process of rotational energy extraction from a black hole.

Here we discuss a limit of the gravitational constant G goes to zero for the Einstein gravity. It is convenient to use for this a tetradic approach (using tetrads and spin-connection) and the Palatini formalism. Also some external field B is entered, which is 2-form. In this limit an equation for the spin-connection is obtained. Such an equation was obtained earlier in the MacDowell-Mansouri-Stelle-West gravity.