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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