General Relativity and Flat Space. I

Within the framework of the general theory of relativity, it is proposed to introduce at each point of space-time a Euclidean metric tensor ${\gamma }_{\mu \nu }$ in addition to the usual Riemannian metric tensor $g_{\mu \nu }$. In this way one imparts tensor character to quantities which in the usual form of the theory do not have it. For example, one can obtain a gravitational energy-momentum density tensor in place of the usual pseudo-tensor. Furthermore one can impose four additional covariant conditions on the gravitational field and thus restrict the form of the solution for the field corresponding to a given physical situation.