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