Moving frames in Lorentzian manifolds R^{1,3} and their symmetry generators

Geometrization is a natural way to solve some physical issues, and could be key to explaining recent problems observed on spacetime dynamics, such as the so-called `Hubble tension' and `impossible early galaxies'. To face this challenge, the present Thesis proposes a dynamical embedding technique that builds Lorentzian manifolds R^{1,3} \subset R_{flat}^{1,4} (spacetimes) in a natural manner by using linearly moving frames and homogeneous spatial sections of positive (k > 0), null (k = 0) or negative (k < 0) curvature. The self-consistency of the approach was proved on three scales: (1) The embedded Lorentzian four-manifold, (2) large-scale objects, and (3) particle fields. Main results are the following:

• Embedded manifold. Hamiltonian (ADM) analysis only guarantees consistent dynamics for the (k=1)-hypercone H^4 \subset R_{flat}^{1,4}, while the Lagrangian density of general relativity needs to subtract the background scalar curvature. Moving frames lead to a fictitious radial inhomogeneity in the manifold, but the value of k=1 displays a scalar curvature that, for every point, is equivalent to that of the standard flat-space FLRW metric. The results of the symmetry properties show that only the angular momenta are global symmetries. The radial inhomogeneity breaks all the non-rotational local symmetries at large distances.
• Large-scale objects. A distorted projection is used to assimilate the inhomogeneity as a fictitious acceleration compatible with the standard model, predicting observed values. This projection is also a key point in object dynamics (e.g. galaxy rotation curves). Specifically, centrifugal acceleration presents a small time-like contribution at large-scale dynamics because of curvature of the manifold, and the distorted projection transfers the fictitious acceleration to the rotation curves at larger scales, imitating the extratropical cyclones.
• Particle fields. For particle-moving frames (e.g. for spinors), we explore small perturbations of the metric decomposition related to the Wilson line and the Kaluza-Klein metric, but in four dimensions. The coordinates are now matrices consisting of generators of the su(1,3) Lie algebra. Under this coloured gravity framework, standard electromagnetism is obtained as a particular abelian case. In general terms, moving frames locally perturb the flat metric \eta as follows: \eta  \to  \eta + u x u, where u represents the velocity at the three scales analyzed: the manifold expansion, the galaxy rotation curves, and the particle dynamics with their symmetry generators.