Non-Riemannian gravity and the Einstein - Proca system

Abstract

We argue that all Einstein-Maxwell or Einstein-Proca solutions to general relativity may be used to construct a large class of solutions (involving torsion and non-metricity) to theories of non-Riemannian gravitation that have been recently discussed in the literature. -2 - IntroductionNon-Riemannian geometries feature in a number of theoretical descriptions of the interactions between fields and gravitation. Since the early pioneering work by Weyl, Cartan, Schroedinger and others such geometries have often provided a succinct and elegant guide towards the search for unification of the forces of nature [1]. In recent times interactions with supergravity have been encoded into torsion fields induced by spinors and dilatonic interactions from low energy effective string theories have been encoded into connections that are not metric-compatible [2], [3], [4], [5]. However theories in which the non-Riemannian geometrical fields are dynamical in the absence of matter are more elusive to interpret. It has been suggested that they may play an important role in certain astrophysical contexts [6]. Part of the difficulty in interpreting such fields is that there is little experimental guidance available for the construction of a viable theory that can compete effectively with general relativity in domains that are currently accessible to observation. In such circumstances one must be guided by the classical solutions admitted by theoretical models that admit dynamical non-Riemannian structures [7], [8], [9], [10], [11]. A number of recent papers have pursued this approach and have found static spherically symmetric solutions to particular models [12], [13]. In [6] it was pointed out that in a particularly simple model all EinsteinMaxwell solutions to general relativity could be used to generate dynamic non-Riemannian geometries and a tentative interpretation was offered for the matter couplings in such a model. Particular solutions have also been found to more complex models, provided the coupling constants in the action are correlated [12], [13], [14], [15]. It is the purpose of this note to point out that, if such correlations are maintained then solutions may be generated from all Einstein-Maxwell solutions of general relativity. Furthermore the correlations may be discarded and solutions generated from all Einstein-Proca solutions of general relativity with or without the inclusion of a cosmological term in the action.

Nonlinear Realization of the Local Conform-Affine Symmetry Group for Gravity in the Composite Fiber Bundle Formalism

“…An arbitrary element p ∈ P can be written in terms of the null section with the aid of (45), (46) and (48) as…”

Nonlinear connections and spinor geometry

“…This way we define the short exact sequence 19) where ρ c (y, a) = ρ c (y). If λ is oriented, real γ-bundle π : E λ → M n of rank n, with base M n , the complex spinor structure, spin structure, on λ is given by the principal bundle P with structural group Spin c (m) and isomorphism λ ≈ P × Spin c (n) R n (see (5.19)).…”

“…Various types of Finsler-like structures can be parametrized by generic off-diagonal metrics, which cannot be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection). Such structures may be defined as exact solutions of gravitational field equations in the Einstein gravity and its generalizations [75,79,80,94,95,96,97,98,99,100,102,103,104,105,109,110,111], for instance, in the metric-affine [19,23,56] Riemann-Cartan gravity [24,25]. Finsler-like configurations are considered in locally anisotropic thermodynamics, kinetics, related stochastic processes [85,96,107,108], and (super-) string theory [84,87,90,91,92].…”

A class of colliding waves in metric-affine gravity, nonmetricity and torsion shock waves

“…We assume that nonmetricity and torsion are represented by a triplet of one-forms [10,11], i.e., the Weyl covector (the dilation piece), the traceless covector piece of the nonmetricity (a proper shear piece) and the torsion trace, so that…”