The fundamental assumption of relativity is
*absolute _spatial_ motion does not exist*,
however placing that principle into a succint
mathematical form seems to be difficult. So
that's what I'll try to do.

Beginning with the well known

ds^2 = g_uv dx^u dx^v  , u,v,w={0,1,2,3}.

We can use association provided the covariant
derivative,

g_uv;w = 0.

Then by association,

ds^2 = dx_u dx^u.

Expanding to time and space gives,

ds^2 = dx_0 dx^0 + dx_i dx^i  , i,j={1,2,3}.

The absolute spatial motion I'll define by

dx_i dx^i = Absolute spatial motion.

Absolute spatial motion cannot exist, IOW's
it vanishes, hence,

dx_i dx^i =0.

However, relative spatial motion cannot vanish,
it is always possible, so I'll select dx^i to
be relative spatial motion, so that dx^i >0 generally
and then require

dx_i =0  always,

to insure

dx_i dx^i=0   always,

and is the mathematical description of the
Principle of Relativity. More formally it's
expressed using the covariant 3-velocity,

U_i = dx_i/ds =0.

By using tensor algebra we obtain from that,

g_0i = - g_ij dx^j/dx^0,

and generally,

ds^2 = g_00 dx^0 dx^0 - g_ij dx^i dx^j   , (always).

For an SR application, sub the metric values,

g_00 = g_11 = g_22 = g_33 =1,

g_ij =0 when i =/= j and

g_0i = -dx^i/dx^0,

and find by algebra,

ds^2 = g_00 dx^0 dx^0 - g_ij dx^i dx^j

== dt^2 - dx^2 - dy^2 - dz^2.

The succinct U_i=0 provides Minkowski spacetime,
which embodies the Lorentz transformation, but
done Generally Covariantly.

Moving to General Relativity, the following absolute
derivative vanishes (because U_i=0),

DU_i = U_i;w dx^w =0 .

Using association,

U_i = g_iu U^u =0

therefore,

DU_i = g_iu DU^u =0

and thus,

DU^u =0,  aka the geodesic equation.

We have arrived at the equation for the geodesic
(for ref see Weinberg's, Grav&Cosmo, Eq.(5.1.6))
using the Principle of Relativity given by U_i =0
and g_uv;w =0.

The g_uv;w=0 is the mathematical expression for
the Principle of Equivalence, and as is obvious,
is required to get DU^u=0 from U_i=0.

The geodesic equation is expanded to,

DU^u/ds = dU^u/ds + GAMMA^u_ab U^a U^b = 0

(ref, see Weinberg's Eq. (5.1.7)), and is the equation of
motion in General Relativity.

Up to this point we've used two assumptions

1)      U_i=0

2)      g_uv;w=0

where (1) is a statement of the law of Relativity that excludes
"absolute motion", and (2) is a statement that excludes absolute
acceleration, and at the same time is the Principle of Equivalence,
used to derive the General Relativity geodesic.

I use (1) and (2) in relativity mathematics, is there a reason not to?