I SR textbooks discussing accelerated reference frames without delving into GR

[renormalize]

And this is fine if you interpret "torsion" to mean a mathematical aspect of the formulation that has no physical consequences (because all of the different formulations you describe are physically equivalent). But that is not what I was using "torsion" to mean, or what "torsion" is standardly used to mean in the GR literature. "Torsion" in the standard meaning, as described in the MTW reference I gave, physically means that the equivalence principle does not hold. That is a physical difference, not just a different mathematical formulation of the same physical theory.

OK, I believe I understand your distinction. Your point is that adding torsion to a non-flat Levi-Civita connection necessarily violates the equivalence principle. In contrast, teleparallel-ism adds torsion to a pure-gauge (flat) connection and formulates a gravitational theory equivalent to GR (including the equivalence principle) using the torsion tensor in lieu of curvature.

 

[PeterDonis]

Your point is that adding torsion to a non-flat Levi-Civita connection necessarily violates the equivalence principle.

No, that's not my point. It's not a matter of "adding torsion" to any particular connection. It's a matter of whether the covariant derivative that is obtained from the connection satisfies the condition I referenced from MTW or not. If it does, the connection is torsion-free and is consistent with the equivalence principle. If it doesn't, the connection has nonzero torsion and violates the equivalence principle. That is the definition of torsion I have been using and which is standard in the GR literature.

teleparallel-ism adds torsion to a pure-gauge (flat) connection and formulates a gravitational theory equivalent to GR (including the equivalence principle) using the torsion tensor in lieu of curvature.

Here you are using "torsion" in a different sense, as I have already pointed out twice now. In the standard GR language I referenced from MTW, you are obtaining a torsion-free connection by taking a flat connection and adding a thingie to it that you are calling "torsion" (but not in the standard sense of "torsion") that makes the resulting connection obey the torsion-free condition I described above (because it has to in order to obey the equivalence principle). To say that you are "using torsion" to do this is confusing two different senses of the term "torsion". As noted, I have already said this twice before, and nothing you have posted has addressed it at all.

 

[renormalize]

To say that you are "using torsion" to do this is confusing two different senses of the term "torsion". As noted, I have already said this twice before, and nothing you have posted has addressed it at all.

To aid my understanding, can you cite a textbook or reference that clearly defines and contrasts these two distinct senses of torsion?

 

[PeterDonis]

can you cite a textbook or reference that clearly defines and contrasts these two distinct senses of torsion?

I already cited you a textbook (MTW) that gives the standard sense of torsion. Wald, Chapter 3, has a similar discussion (property 5 of Section 3.1 is the same definition of "torsion-free" that MTW gives, though expressed in different notation).

I have no references for the other sense of torsion because I am not the one using it. You are.

 

[Jianbing_Shao]

Who said it wasn't? We have said precisely the opposite in this thread: if the spacetime is flat, parallel transport is not path dependent; if the spacetime is curved, it is. So path dependence is the result of spacetime geometry.

In calculus. the path integral of a vector field with non-zero curl along different paths between two points will get different results. it just means that the movement of a scalar field is path dependent. A vector only contains more components than a scalar. So we can define a path dependent movement in a flat spacetime.
The path dependent integral (path dependent movement of a scalar ) is determined by curl of the vector field, it has nothing to do with the structure of space.

 

[PeterDonis]

In calculus. the path integral of a vector field with non-zero curl along different paths between two points will get different results. it just means that the movement of a scalar field is path dependent.

No, it doesn't. You switched from vector to scalar. They're not the same thing.

A vector only contains more components than a scalar.

A vector has a curl. A scalar doesn't.

So we can define a path dependent movement in a flat spacetime.

You can define a path and vector field dependent "movement" in flat spacetime, sure. For example, in electrodynamics.

What you can't do is use this to infer anything about the spacetime curvature or the path dependence of parallel transport that depends on spacetime curvature. Which means that your posts are irrelevant to this thread and are hijacking it.

 

[PeterDonis]

Thread closed for moderation.