Re: Construction of the tangent vector space in GR

[Brent Meeker]

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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[Alan Grayson]

On Sunday, February 16, 2025 at 8:20:08 PM UTC-7 Brent Meeker wrote:

I think you need to distinguish the vector-space of velocities and the vector space of positions.

Brent

I'm pretty sure it's the vector space of velocities which defines the vector space on the tangent space at each point on the manifold. If we define addition as relativistic, it should restrict all those vectors when added, to velocities less than light speed. Do you agree? AG 

[Brent Meeker]

On 2/16/2025 7:28 PM, Alan Grayson wrote:

On Sunday, February 16, 2025 at 8:20:08 PM UTC-7 Brent Meeker wrote:

I think you need to distinguish the vector-space of velocities and the vector space of positions.

Brent

I'm pretty sure it's the vector space of velocities which defines the vector space on the tangent space at each point on the manifold. If we define addition as relativistic, it should restrict all those vectors when added, to velocities less than light speed. Do you agree? AG

That's true of the vector space of velocities, but there's on such restriction on the vector space of 4-vectors.

Brent

[Alan Grayson]

What's that restriction? Do we really have to define the tangent space vectors as 4-vectors? I suppose so since the spacetime manifold has dim 4. AG 

[Alan Grayson]

How is velocity defined on the 4d manifold of spacetime? AG 

[Alan Grayson]

I goggled it, and the answer is the 4d velocity is alway light speed in every coordinate system. But this is no help, since it doesn't explain how velocities are added in the tangent vector space and, it would seem, should take on different values less than c. Looks like I have a problem; I don't know how to define the tangent space on spacetime. AG

[Alan Grayson]

I'm pretty sure that the tangent space on a manifold is defined by the usual 3d velocities at a point P on the manifold, and not the 4d velocity on spacetime. This is the only way to define a vector space at P, which has different velocity vectors. The 4d velocity has only one velocity, c, and can't be used to define the tangent space at P. AG