The discussion centers on the integrability conditions for differential equations as presented in Philippe G. Ciarlet's book 'An Introduction to Differential Geometry'. The key integrability condition is expressed as ##R^i_{jkl} = 0##, which allows for the existence of a global solution ##F_{lj}##. The conversation highlights the compatibility of non-flat metrics ##g_{ab}## with zero curvature connections, specifically the teleparallel/Weitzenböck connection ##\Gamma^{c}_{ab} = {e^c}_{I} \partial_a {e_b}^{I}##. This connection can coexist with non-zero torsion, challenging the conventional understanding of flatness in metric spaces.
PREREQUISITESMathematicians, physicists, and researchers in differential geometry, particularly those interested in the interplay between curvature, torsion, and metric compatibility in gravitational theories.
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.PeterDonis said:
When you say the result from my calculation is wrong. then you should point out where I have made a mistake.PeterDonis said:
Yes. Now go read my post #58, where I point that out, and then point out something else.Jianbing_Shao said:
Again, go read my post #58. In particular, the second sentence.Jianbing_Shao said:
Correct math is the basis for valid physical models of our world, yes. But only correct math.Jianbing_Shao said:
Then why you don't think that assumption is even consistent with his calculation? can you give me some explaination?PeterDonis said:
So where is my wrong? just the conclusion is wrong？PeterDonis said:
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 said:
In the original definition of teleparalle connection, ##e_a## should be a holonomic frame(coordinate basis field), It is not necessarily to be orthonormal.PeterDonis said:
Please give a reference for this claim.Jianbing_Shao said:
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).Jianbing_Shao said:
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 said:
https://encyclopedia.thefreedictionary.com/TeleparallelismPeterDonis said:
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.PeterDonis said:
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 said:
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.Jianbing_Shao said:
Sorry, this is simply wrong. We can't have a useful discussion if you continue to make wrong claims after repeated corrections.Jianbing_Shao said: