Properties of Born rigid congruence

[PeterDonis]

one must rotate tetrad's spacelike vector fields about the 4-velocity

No, this doesn't make sense. The 4-velocity is a timelike vector that is orthogonal to the spacelike vectors. It is not a spacelike vector orthogonal to the plane of rotation, which is what a rotation axis is.

I don't grasp it from a mathematical point: if the generalized Fermi normal coordinates do not work on a worldtube about a given congruence's worldline, on the same ground the construction of coordinate chart involving the timelike KVF shouldn't.

You keep repeating this wrong statement without apparently even noticing the reasons I have already given you for why it is false. I have already pointed out to you the key difference between the 4-velocity field and the KVF. Multiple times. Go back and read my posts again.

is it always true that for a timelike Killing congruence the KVF and the direction of proper acceleration commute at each point along the congruence ?

I don't know, and I don't have time to try either proving this or finding a counterexample.

 

[cianfa72]

No, this doesn't make sense. The 4-velocity is a timelike vector that is orthogonal to the spacelike vectors. It is not a spacelike vector orthogonal to the plane of rotation, which is what a rotation axis is.

Ah ok, you are right. However such rotations leave invariant the timelike coordinate basis vector along the worldline.

I have already pointed out to you the key difference between the 4-velocity field and the KVF. Multiple times. Go back and read my posts again.

Yes that's true, however we don't have a mathematical argument/reason that shows why it doesn't work for generalized Fermi normal coordinates (i.e. Fermi normal coordinates followed from rotations). We know that it works for timelike KVF in special cases like Langevin or similar congruences.

I don't know, and I don't have time to try either proving this or finding a counterexample.

Ok, so for sure we can claim that the timelike KVF of Langevin congruence and the direction of proper acceleration do commute (alike the case of Rindler congruence). However, up to now, we have no proof that this result extend to any timelike Killing congruence.

 

[PAllen]

As I have commented in response to the OP, in general this will not always work. We have seen a counterexample in this thread: the Langevin congruence frame field given in the Insights article I referenced, where $\hat{p}_0$ is the 4-velocity of the worldlines and $\hat{p}_2$ always points radially outward, is not a valid coordinate basis because $\hat{p}_0$ and $\hat{p}_2$ do not commute. Because of the nonzero vorticity of the congruence, $\hat{p}_2$ is not Fermi-Walker transported along the worldlines.

It is possible, however, to use the timelike Killing vector field $K$ whose integral curves are the Langevin congruence worldlines as the timelike basis vector for a coordinate chart in which $\hat{p}_2$ is also a coordinate basis vector, since $K$ does commute with $\hat{p}_2$. This is still "similar" to the Fermi-Walker plus rotation construction you describe since $K$ is still tangent to the worldlines; it's just not a unit tangent to the worldlines.

I disagree. MTW demonstrates this construction working in full GR with arbitrary base world line and arbitrary rotation function. What is generally true is that congruence of resulting constant position lines is not hypersurface orthogonal, and the coverage is only for a tube around the world line. Note, Fermi Normal coordinates , and the rotation generalization, are not based on a starting congruence. Instead they are based on one origin world and one chosen tetrad, which is Fermi-walker transported, then rotated, then used as a basis for the hypersurface. See pages 327 to 332 of MTW.

I can look later for a paper that demonstrates my other claim: that whenever the base world line motion plus rotation are a possible Born rigid motion, then the constant position congruence in generalized Fermi-Normal coordinates is a Born rigid congruence. This result is considered somewhat well known.

 

[cianfa72]

I disagree. MTW demonstrates this construction working in full GR with arbitrary base world line and arbitrary rotation function. What is generally true is that congruence of resulting constant position lines is not hypersurface orthogonal, and the coverage is only for a tube around the world line.

You mean in bold the congruence of worldlines each described by constant spatial coordinates and varying coordinate time in the generalized Fermi normal coordinate chart being built with a coverage only for a worldtube around the picked worldline.

 

[PAllen]

You mean in bold the congruence of worldlines each described by constant spatial coordinates and varying coordinate time in the generalized Fermi normal coordinate chart being built with a coverage only for a worldtube around the picked worldline.

Correct.

 

[cianfa72]

I can look later for a paper that demonstrates my other claim: that whenever the base world line motion plus rotation are a possible Born rigid motion, then the constant position congruence in generalized Fermi-Normal coordinates is a Born rigid congruence. This result is considered somewhat well known.

Sorry, what does it mean that the base worldline motion plus rotation are a possible Born rigid motion ?