He is somewhat curious as to what may cause your confusion in
muttering that under gravitation,

** d[p]/dTau = 0

Where

** [p] = momentum 4-vector
** Tau = mythical proper time

The closest thing is the set of geodesic equations. The geodesic
equations are not unique to GR. They exist even when GR is false.

** d^2q^n/dt^2 = - G^n_ij dq^i/dt dq^j/dt

Where

** q^n = coordinate
** G^n_ij = Arrangement of the connection coefficients

One of the two possible arrangements of the connection coefficients is
represented by the Christoffel symbols of the 2nd kind. From the
geodesic equations, you can follow Ricci’s footstep and define an
operator that the result when the operator is applied to the change in
coordinate is null:

** D(dq^n/dt)/dt = 0

Where

** D/dt = covariant derivative

From that, you are able to morph the operator into your own:

** D/dt = d/dTau

Then, toss in the rest mass, you get what you have cooked in your own
pot of alchemy. shrug

Is this where your confusion lies? shrug