Where Can I Find Detailed Derivation of Teleparallel Gravity Field Equations?

[ShivamM]

Where can I find detailed derivation of Field Equation of Teleparallel Gravity from variation of Action ?

 

[fresh_42]

Where can I find detailed derivation of Field Equation of Teleparallel Gravity from variation of Action ?

Have you studied the references given on Wikipedia?
https://en.wikipedia.org/wiki/Teleparallelism

 

[ShivamM]

Have you studied the references given on Wikipedia?
https://en.wikipedia.org/wiki/Teleparallelism

Yes , I did so. Some of papers have differences even in field equations... Which is what made me derive them myself but I am getting it wrong.

 

[PeterDonis]

Which is what made me derive them myself but I am getting it wrong.

So what are you asking? It looks like you've already found what you were asking for in the OP.

 

[Mordred]

Which model of tellaparallel gravity ? There is numerous variations

 

[PeterDonis]

Which model of tellaparallel gravity ? There is numerous variations

Which have different field equations.

 

[jedishrfu]

Teleparallel gravity theory may not be mainstream, but it's based on Einstein's work in trying to unify EM with gravitation, and there are arXiv papers on the subject.

The old saying, "What goes around comes around," implies that someone may yet find an interesting application for this theory in other areas of physics.

This one in particular, is quite a large paper :

https://arxiv.org/pdf/2106.13793

Springer has a book on the subject by Hohmann, publ in 2013:

https://link.springer.com/book/10.1007/978-94-007-5143-9

 

[jedishrfu]

There is also a Quanta magazine article on one cosmologist attempting to incorporate dark matter and dark energy into the teleparallel gravity equations:

https://www.quantamagazine.org/in-m...-cracks-in-einsteins-gravity-theory-20220223/

Here's some papers that she has authored or coauthored with others:

https://scholar.google.com/scholar?...la-Rivera&hl=en&as_sdt=0&as_vis=1&oi=scholart

 

[ShivamM]

So what are you asking? It looks like you've already found what you were asking for in the OP.

I tried the variation myself , but the nature of Vierbeins is such that you can get the solution in whatever form you like. There is a sign problem in field equation in various papers , which I am trying to understand. So I need a guide to derive the equations in order to get the full understanding of what I am working on.

 

[Mordred]

I would recommend choosing one specific model using tellaparralel gravity and then posting the reference and equations your having issue with. In in this regard we would have a better chance of helping you. As is we have no idea how to offer assistance without the needed detail
Edit : I can only suspect the sign convention your having trouble with is the antisymmetric relations involving the torsion components but without further detail that's just a wild guess

 

[PeterDonis]

There is a sign problem in field equation in various papers

Which could be due to different sign conventions, as it is in standard GR. We can't know that unless you give us references to the specific papers you have looked at and the specific questions you have about them.

I need a guide to derive the equations

If you are reading papers in the field, they should provide this. If different papers are providing different guidance, again, we can't know why unless you give us references to the specific papers and tell us what differences you see.

As has already been said, "teleparallel gravity" is not one single model or field equation or derivation; there are different ones in the literature. So asking for a guide to derive "the" field equations is an unanswerable question. There is no such thing as "the" field equations.

 

[ShivamM]

In the book by Aldrovandi and Pereria, derivation using the Euler - Lagrange Equation is given in appendix C, page 197. In the last steps of the calculation of energy momentum current, there is some algebraic manipulation involved, which I am not being able to do. Earlier, I tried doing it by taking delta variation, which led to field equations in which the coefficient of Torsion scalar was 1/2, which should be 1/4.

 

[Mordred]

I don't have his book nor choose to buy it but I have read some of his articles. I know
The authors use the Weitzenbock connection and their tellaparallel gravity replaces curvature with torsion. If I'm correct then it sounds like the detail your missing to get the 1/4 term is the spin Laplacian shown here
https://en.m.wikipedia.org/wiki/Weitzenböck_identity

That's the only thing I can think of with the details provided.
If so then you will likely find the following details on the Weitzenbock connection useful to your problem

https://www.math.ucla.edu/~petersen/BLWformulas.pdf

All the above however is just a guess as the details provided are far too limited