I think you need to distinguish the vector-space of velocities and the
vector space of positions.
Brent
On 2/16/2025 2:10 PM, Alan Grayson wrote:
I got into a fairly heated argument with a Ph'D in physics from
Brent's alma mata, the University of Texas at Austin, concerning the
construction of the tangent vector space in GR. For this and other
reasons we are no longer in communication. He insisted on
considerating particle paths of *all* velocities going through some
point P on the spacetime manifold on which the objective is to
construct the tangent vector space at P. I objected since this would
violate one of the basic postulates of GR, which preclude particles
assumed to be exceeding light speed. I was berated for making such a
criticism. Initially I thought these faster than light speed particles
were needed to form a*vector space*, in order to satisfy the linear
additive property of a vector space under the field of real numbers.
But suppose these vectors are constrained to be added
relativistically, so no pair when added, can exceed light speed. Will
this be sufficient to satisfy the linear additive property of vectors
in a vector space, without violating the postulate of GR precluding
faster than light speed particles? TY, AG
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