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

[PeterDonis]

does it means in all cases they can not be equal?

They're not equal in any spacetime other than the flat Minkowski metric, since the whole point of the teleparallel connection is to put the effects of gravity into the torsion instead of the curvature.

if you point out the falses in my calculations, I'll be gald to accept it.

I can't even understand your calculations. The reason I know they're wrong is that they're giving you obviously wrong answers.

isn't it natural that the cuavature of the Levi-Civita connection is zero?

Not if the spacetime geometry is anything other than the flat Minkowski metric. If your math tells you that this is "natural", then your math is wrong. I can't understand what math you think you're doing, so I can't tell you exactly how your math is wrong, but since it's leading you to make this wrong claim, repeatedly, it's wrong somewhere.

 

[PeterDonis]

Is there any book tell us that a metric field can also be compatible with zero-curvature teleparallel connection?

You were given a link to a Wikipedia article that shows this way back in post #2 of this thread.

 

[Jianbing_Shao]

I can't even understand your calculations. The reason I know they're wrong is that they're giving you obviously wrong answers.

Are you serious to say so? If my answers is obviously wrong, then it is very easy to point out where is my wrong. In fact the mathematical technics I used is very easy.

ergospherical in post #10 derived the compatibility between teleparallel connection and a metric field.

1. The teleparallel connection ${\Gamma^c}_{ab} = {e^c}_{I} \partial_a {e_b}^{I}$ is flat in the sense that it has vanishing Riemann, $R= 0$.
2. Metrics of the form $g_{ab} = \eta_{IJ} {e_a}^{I} {e_b}^{J}$ are compatible with the teleparallel connection. This is what follows from the definition of the teleparallel connection, $\partial_a {e_b}^{I} = {\Gamma^c}_{ab} {e_c}^{I}$, since
\begin{align*}
\partial_c g_{ab} &= \eta_{IJ}\left[ (\partial_c {e_a}^{I} ) {e_b}^{J} + {e_a}^{I} (\partial_c {e_b}^{J})\right] \\
&= \eta_{IJ} \left[ {\Gamma}^d_{ca} {e_d}^I {e_b}^J + {\Gamma}^d_{cb} {e_a}^I {e_d}^J\right] \\
&= {\Gamma^d}_{ca} g_{db}+ {\Gamma^d}_{cb} g_{ad} \\
\end{align*}
which is the metric compatibility equation $\nabla_c g_{ab} = 0$.

Can you find any problems in his calculation?

 

[PeterDonis]

If my answers is obviously wrong, then it is very easy to point out where is my wrong.

Not if your reasoning doesn't make sense to me. Your answer is obviously wrong because your answer is that the Levi-Civita connection has zero curvature, and that the Levi-Civita connection is the same as the teleparallel connection, which is wrong for any spacetime that isn't flat Minkowski spacetime. I don't have to know exactly where your math is wrong to know that those answers are wrong; those answers contradict everything in the literature about those two connections.

 

[PeterDonis]

ergospherical in post #10 derived the compatibility between teleparallel connection and a metric field.

@ergospherical is not making the two wrong claims you are making. He's not claiming that the Levi-Civita connection has zero curvature, and he's not claiming that the Levi-Civita connection is the same as the teleparallel connection. Nor does his calculation in post #10 show that either of those things are true (which is good, since they're not). So his calculation in post #10 is irrelevant to the wrong claims you are making.

 

[Jianbing_Shao]

@ergospherical is not making the two wrong claims you are making. He's not claiming that the Levi-Civita connection has zero curvature, and he's not claiming that the Levi-Civita connection is the same as the teleparallel connection. Nor does his calculation in post #10 show that either of those things are true (which is good, since they're not). So his calculation in post #10 is irrelevant to the wrong claims you are making.

Don't hurry,
if we start from the definition of Levi-Civita connection:
$\Gamma_{abc} = \frac{1}{2}(g_{ab,c} + g_{ac,b} - g_{bc,a})$
Then we can use the definition of metric field
$g_{ab} = {e_a}^I {e_b}^J \eta_{IJ}$
and we demand that $e_a$ is a coordinate basis field;
$[e_a , e_b]=0$
Then you can find that:
$\Gamma_{abc} = \frac{1}{2}(g_{ab,c} + g_{ac,b} - g_{bc,a})= e_{aI} {{e_b}^I}_{,c}$

The calculation is just a reverse of ergospherical's.
You can check if I was wrong. But if my calculation is right. then how to explain the result?

 

[PeterDonis]

The calculation is just a reverse of ergospherical's.

Then it can't possibly support the two wrong claims you're making. I never said @ergospherical's calculation was wrong. I only said it doesn't support what you're claiming.

 

[PeterDonis]

and we demand that $e_a$ is a coordinate basis field

@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.

 

[Jianbing_Shao]

Then it can't possibly support the two wrong claims you're making. I never said @ergospherical's calculation was wrong. I only said it doesn't support what you're claiming.

So how to explain my result? if you think my explaination is wrong. then tell me the right one to explain the result I gave.

 

[PeterDonis]

So how to explain my result? if you think my explaination is wrong. then tell me the right one to explain the result I gave.

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.

 

[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?