Speaker
Description
We show that because in a curved spacetime parallel transportations of (r,s)- tensors with r+s>0
depend on paths, one cannot add up (r,s)-tensors at different points to get a definite sum
(r,s)-tensor when r+s>0. However, when restricted to an infinitesimal spacetime region, one still
can add up (r,s)-tensors at different points to get a definite sum (r,s)-tensors, if neglecting
higher order infinitesimals. Due to these sound facts from geometry, we cannot talk about the sum
of (r,s)- tensors distributing on a finite or infinite hypersurface, nor the net increase of (r,s)-
tensors in a finite or infinite spacetime region, if r+s>0; but we still can talk about the sum of
(r,s)-tensors distributing on an infinitesimal hypersurface element, or the net increase of
(r,s)-tensor in an infinitesimal spacetime region, when neglecting higher order infinitesimals.
Therefore, denoting by 𝐽 the flux density (r+1,s)- tensor field of (r,s)- tensor Q, the
conservation law of Q in curved spacetime can only be “the covariant divergence of 𝐽 vanishes
everywhere”. It reads, “the net increase of tensor Q in any infinitesimal 4-dimensional
neighborhood is zero”.
In particular, matter energy-momentum P is a (1,0)-tensor, denote its flux density field by T. The
conservation law for P in GR cannot be anything else but “the covariant divergence of T vanishes
everywhere”. It reads, the net increase of matter energy-momentum in any infinitesimal spacetime
region is zero. This means matter fields and matter particles exchange energy-momentum with each
other, but not with anything else including gravitational field.
Force, or interaction in physics, always means exchange of energy-momentum. Now that the
gravitational field does not exchange energy-momentum with matter fields and matter particles. So,
gravitational field does not carry energy- momentum, it is not a force field, and gravity is not a
natural force.
