A sufficient condition for integrability of equation ##\nabla g=0##

[Jianbing_Shao]

@ergospherical's calculation in post #10 doesn't make that assumption. Indeed, I don't think that assumption is even consistent with his calculation. So your apparent belief that you are just reversing his calculation appears to be wrong.

The difference between me and ergospherical is that he didn't demand the basis field is a coordinate basis, so they assert that there exist some extra terms between a teleparallel connection and Levi-Civita connection. and I demand that the basis should be coordinate basis field, then a teleparallel connection equals a Levi-Civita connection.

 

[Jianbing_Shao]

So far I haven't seen you give any valid "explanation" of anything. I just see you continuing to show math that leads you to results that are obviously wrong, and refusing to acknowledge that fact. Math that gives obviously wrong results can't be a valid explanation of anything.

If you're looking for a suggestion about one step in your latest "explanation" that looks wrong, see my post #58.

When you say the result from my calculation is wrong. then you should point out where I have made a mistake.
If you are serious , you must know that math is exactly the base of explaination of our world.

 

[PeterDonis]

The difference between me and ergospherical is that he didn't demand the basis field is a coordinate basis

Yes. Now go read my post #58, where I point that out, and then point out something else.

When you say the result from my calculation is wrong. then you should point out where I have made a mistake.

Again, go read my post #58. In particular, the second sentence.

 

[PeterDonis]

you must know that math is exactly the base of explaination of our world.

Correct math is the basis for valid physical models of our world, yes. But only correct math.

 

[Jianbing_Shao]

@ergospherical's calculation in post #10 doesn't make that assumption. Indeed, I don't think that assumption is even consistent with his calculation. So your apparent belief that you are just reversing his calculation appears to be wrong.

Then why you don't think that assumption is even consistent with his calculation? can you give me some explaination?

 

[Jianbing_Shao]

Correct math is the basis for valid physical models of our world, yes. But only correct math.

So where is my wrong? just the conclusion is wrong？

 

[PeterDonis]

why you don't think that assumption is even consistent with his calculation?

Because the orthonormal tetrad fields $e_a$ that appear in his calculation don't commute in any spacetime other than flat Minkowski spacetime, which means they can't form a coordinate basis. You can see this in the example he gave for the 2-sphere; the two vector fields of the basis he gave don't commute.

 

[Jianbing_Shao]

Because the orthonormal tetrad fields $e_a$ that appear in his calculation don't commute in any spacetime other than flat Minkowski spacetime, which means they can't form a coordinate basis. You can see this in the example he gave for the 2-sphere; the two vector fields of the basis he gave don't commute.

In the original definition of teleparalle connection, $e_a$ should be a holonomic frame(coordinate basis field), It is not necessarily to be orthonormal.
If you demand $e_a$ is orthonormal, and at the same time it is changing, then the basis field is obviously a non-coordinate basis field, It just can prove that in ergospherical's derivation $e_a$ is not orthonormal.
It seemed that you didn't believe a metric field can be compatible with a zero-curvature connection. so you think $e_a$ only can be orthonormal, but if $e_a$ is only a holonomic frame. can we find any problems in ergospherical's derivation? If we can't, then how to explain the result?

 

[PeterDonis]

In the original definition of teleparallel connection, $e_a$ should be a holonomic frame(coordinate basis field)

Please give a reference for this claim.

It seemed that you didn't believe a metric field can be compatible with a zero-curvature connection

I have never said any such thing. All I have said is that the Levi-Civita connection has nonzero curvature in any spacetime other than flat Minkowski spacetime. But that in no way rules out the possibility of there being some different connection, such as the teleparallel connection, which is also compatible with the same metric and has zero curvature--but nonzero torsion (whereas the Levi-Civita connection has zero torsion).

so you think $e_a$ only can be orthonormal

That has nothing to do with what connections are compatible with what metrics. It has to do with doing the correct math for the teleparallel connection. At this point I have nothing further to say on that since you keep repeating obviously wrong claims and I can't understand what math you think you're doing. I'll let @ergospherical take another try if he wants to.

 

[Jianbing_Shao]

Please give a reference for this claim.

https://encyclopedia.thefreedictionary.com/Teleparallelism

So I just wonder why you say $e_a$ is orthonormal.

I have never said any such thing. All I have said is that the Levi-Civita connection has nonzero curvature in any spacetime other than flat Minkowski spacetime.

But although GR gave some examples, but no one has proved such a conclusion in mathematics, No one have proved that for all non-flat metric field all the corresponding compatible Levi-Civita connections only have non-zero curvature. I didn't find such a proof of this conclusion. So I just need a counter-example then the conclusion is not right.

I have never said any such thing. All I have said is that the Levi-Civita connection has nonzero curvature in any spacetime other than flat Minkowski spacetime. But that in no way rules out the possibility of there being some different connection, such as the teleparallel connection, which is also compatible with the same metric

So if your conclusion is right, then logically In all cases a telleparallel connection can not be equal to Levi-Civita connection. Can you prve it?

 

[PeterDonis]

https://encyclopedia.thefreedictionary.com/Teleparallelism
View attachment 365466

The tetrad in that expression is not the holonomic frame $\partial_\mu$. It's the expressions $X_i = h^\mu_i \partial_\mu$. The $X_i$ are what correspond to the $e_a$ in @ergospherical's post.

 

[PeterDonis]

although GR gave some examples, but no one has proved such a conclusion in mathematics,

Sorry, this is simply wrong. We can't have a useful discussion if you continue to make wrong claims after repeated corrections.

Thread closed.