The Nature of Gravitational Field and its Legitimate Energy–Momentum Tensor

In this paper we show how a gravitational field generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel and nonnull nonmetricity spacetimes) or that we even can dispense all those geometrical structures and simply represent the gravitational field as a field, in the Faraday sense, living in Minkowski spacetime. The explicit Lagrangian density for this theory is given and the field equations (which are a set of four Maxwellʹs-like equations) are shown to be equivalent to Einsteinʹs equations. We also analyze whether the teleparallel formulation can give a mathematical meaning to “Einsteinʹs most happy thought”, i.e. the equivalence principle. Moreover we discuss the Hamiltonian formalism for our theory and its relation to one of the possible concepts for energy of the gravitational field which emerges from it and the concept of ADM energy. One of the main results of the paper is the identification in our theory of a legitimate energy-momentum tensor for the gravitational field expressible through a really nice formula.