[Richard777 (OP)]

No Christoffel symbols, No tensors.

The dot product of two vectors of acceleration represents an “interaction” of acceleration. The vectors act upon the same point in space-time. Each vector is four dimensional.

The expansion of the dot product gives a “scalar field equation”.

If groupings and operators are suitably defined, the scalar field equation will return the Schwarzschild metric, no Christoffel symbols, no tensors.

Is this expansion a valid representation of a scalar field?

The math is attached.

[Bored chemist]

Can a dot product give the Schwarzschild metric?
No
Wrong units.