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We establish a general relation between the canonical energy-momentum tensorof Lagrangian dynamics and the tensor that acts as the source of thegravitational field in Einstein's equations, and we show that there is adiscrepancy between these tensors when there are direct nonminimal couplingsbetween matter and the Riemann tensor. Despite this discrepancy, we give ageneral proof of the exact equality of the gravitational and inertial massesfor any arbitrary system of matter and gravitational fields, even in thepresence of nonminimal second-derivative couplings and-or linear or nonlinearsecond-derivative terms of any kind in the Lagrangian. The gravitational massis defined by the asymptotic Newtonian potential at large distance from thesystem, and the inertial mass is defined by the volume integral of the energydensity determined from the canonical energy-momentum tensor. In theBrans-Dicke scalar field theory, we establish that the nonminimal coupling andlong range of the scalar field leads to an inequality between the gravitationaland inertial masses, and we derive an exact formula for this inequality andconfirm that it is approximately proportional to the gravitational self-energy(the Nordvedt effect), but with a constant of proportionality different fromwhat is claimed in the published literature in calculations based on the PPNscheme. Similar inequalities of gravitational and inertial masses are expectedto occur in other scalar and vector theories.

The Energy-Momentum Tensor in General Relativity and in Alternative Theories of Gravitation, and the Gravitational vs. Inertial Mass