It is the GRADIENT of the gravitational field that causes objects to curve in space; ie accelerate. Space whose gravitational field (components of its metric) is a constant cause all the effects of gravity except for acceleration.
For instance, a Minkowski metric describes an Inertial System, where only the so-called "mass" of an object is felt.
The Dirac Equation is Relativistically Invariant.
Proof attached.
Proving the Relativistic Energy-Momentum relation, the hard way and the easier way.