A Review About Invariance Induced Gravity: Gravity and Spin from Local Conformal-Affine Symmetry

Abstract

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics.

The Extended Relativity Theory in Clifford Phase Spaces and Modifications of Gravity at the Planck/Hubble Scales

“…A review of the formulation of gravity and spin from a local conformal-affine symmetry point of view can be found in [20]. A geometric approach to f (R) gravity with torsion within the Jet-bundle framework can be found in [21].…”

Cosmological viable mimetic f(R) and f(R, T) theories via Noether symmetry

“…It is remarkable that, when geometry is decoupled from the matter part, in the limit f (R, T ) ≡ f (R), we recover equation of motion of f (R) gravity from (12). Trace of (12) gives us the following equation:…”

The affine structure of gravitational theories: Symplectic groups and geometry

“…One possible way of framing this interlocking is to employ a nonlinear realization of the gauge group G, provided that a closed subgroup H ⊂ G exists. The interlocking requirement is then transformed into the interplay between groups G and one of its closed subgroups H [42]. Let us denote by G a Lie group with elements {g}.…”

“…being the rest described in detail in [47]. Then, we have the concrete possibility to make a nonlinear realization of Sp (8), where due to the partial isomorphism described before, the conformal affine structure of the gravitational theories can be included [42]. As we have pointed out in the Introduction, the nonlinear realization of the standard sp(8) algebra described above does not lead to geometrical invariants as measures, then we need to perform the nonlinear realization with the non-standard CE sp(8) algebra.…”