L. A. S. Evangelista, A. F. Santos

Gravitoelectromagnetism in the Weyl formalism is investigated through an analysis of the consequences of a restricted gauge symmetry acting on the tensor field $A_{μν}$. The propagator associated with the GEM field is explicitly derived and decomposed within the Barnes--Rivers formalism, revealing contributions from the spin-2, spin-1, and scalar spin-0 sectors. By coupling the theory to conserved sources, it is shown that only the spin-2 and scalar sectors contribute to physical processes, while the spin-1 component decouples. The resulting effective propagator can then be written in a compact metric form closely resembling the graviton propagator of linearized General Relativity. The role of gauge fixing is also analyzed by considering both Lorentz-like and de Donder-type gauge conditions. It is shown that the Lorentz-like gauge is consistent with the restricted gauge symmetry of the theory and leads to gauge-independent physical amplitudes, whereas the de Donder gauge introduces residual gauge-dependent scalar contributions, signaling an incompatibility with the underlying symmetry structure. The gauge symmetry is further extended to the fermionic and electromagnetic sectors through diffeomorphism-like transformations. In both cases, conserved currents are derived and shown to coincide with the corresponding energy-momentum tensors, implying that the GEM field couples to matter in the same manner as in linearized gravity. Finally, the associated Ward identities are verified, providing a nontrivial consistency check of the gauge structure and interaction vertices of the theory.