The energy problem in Einstein's theory of gravitation (Dedicated to the memory of V. A. Fock)

The review is devoted to a discussion of the definition and properties of energy in Einstein's theory of gravitation. Asymptotically flat space-time is defined in terms of admissible asymptotically Cartesian coordinates and a corresponding group of coordinate transformations. A Lagrange function is introduced on such a space-time, and a generalized Hamiltonian formulation of the theory of gravitation is constructed in accordance with Dirac's method. The energy is defined as the generator of displacement with respect to the asymptotic time. It is shown that the total energy of the gravitational field and the matter fields with normal energy-momentum tensor is positive and vanishes only in the absence of matter fields and gravitational waves. The proof follows Witten's proof but contains a number of corrections and improvements. Various standard criticisms of the energy concept in general relativity are discussed and shown to be without substance.