Non-minimal couplings of electromagnetism to gravity in non-Riemannian geometries

Although Einstein's theory of relativity is one of the most successful physical theories of the 20th century, the need to develop alternative or modified theories has emerged due to its incompatibility with quantum mechanics and its inability to fully explain the concepts of dark matter and dark energy. There exist numerous possibilities for constructing modified gravity theories in both Riemannian and non-Riemannian geometries. Moreover, in regions near astrophysical objects such as black holes and neutron stars, where electromagnetic interactions are extremely intense, it is likely that non-minimal couplings between gravity and electromagnetism may arise. In this thesis, we specifically investigate the non-minimal coupling of the electromagnetic field to gravity within a non-Riemannian geometry, which corresponds to the symmetric teleparallel equivalent of Einstein's theory of gravitation and is defined solely by the metric gradient tensor with Rab = 0, Ta = 0, Qab ̸= 0. We consider this non-minimal model in the form Y (Q)F, that is, by coupling a function Y (Q), derived from the symmetric teleparallel Lagrangian, with the Maxwell invariant F in a non-minimal way. We obtain the field equations for the coframe, the connection, and the electromagnetic field using the variational method. By utilizing the coincident gauge symmetry present in the model, we obtain the metric gradient purely in terms of the metric. Subsequently, we seek solutions to these field equations under a spherically symmetric static metric, assuming that the Maxwell tensor has only an electric field component. As a result of the calculations, solutions are obtained under the condition that the coupling constants are chosen specifically for the symmetric teleparallel equivalent geometry. These solutions appear as a subclass of those found in Riemannian geometry. Finally, the singularity structure of these solutions and their application to the rotational velocities of stars in galaxies have been examined.