What are alternative formulations of GR?

[Genady]

Historically, after Newtonian formulation of mechanics, alternative formulations were developed, i.e., Lagrangian and Hamiltonian. In QM, after wave mechanics, matrix mechanics was developed. In QFT, there are S-matrix and path integral formulations. Which alternative formulations of GR are known today?

PS. I think, in SR the parallel examples are Einstein and Minkowski formulations.

[Markus Hanke]

Nice question! The major ones that come to mind are (not an exhaustive list):

- The ADM formalism

- The tetrad formalism

- The Spinor formalism

- The Ashtekar formalism

- Of course the Lagrangian formalism

- The Plebanski formulation

- The geometric algebra formulation

It can also be written as a gauge theory, though I must admit that many of the details here are above my pay grade - there seem to be some unresolved issues.

The above is definitely not exhaustive, but it’s all the ones I can think of OTOH.

 

 

 

 

 

[joigus]

Agreed. Extra-nice topic.

46 minutes ago, Markus Hanke said:

- The Spinor formalism

The Bible on this:

https://www.cambridge.org/core/books/spinors-and-spacetime/B66766D4755F13B98F95D0EB6DF26526

https://www.cambridge.org/core/books/spinors-and-spacetime/24388801C4B4BA419851FD4AF667A8F0

'Twistor' is another key word to look for in this concern. Twistors require masslessness, so it's a bit more of an adventurous approach. But very worth taking a look too.

[studiot]

2 hours ago, Genady said:

Historically, after Newtonian formulation of mechanics, alternative formulations were developed, i.e., Lagrangian and Hamiltonian. In QM, after wave mechanics, matrix mechanics was developed. In QFT, there are S-matrix and path integral formulations. Which alternative formulations of GR are known today?

PS. I think, in SR the parallel examples are Einstein and Minkowski formulations.

Also Nice list Markus, +1.

 

Try this

[joigus]

As to Ashtekar, Plebanski. It kind of boils down to successive changes of variables to get from a Lagrangian formulation to a Hamiltonian one, that's amenable to QM.

The logical path is Palatini action -> Plebanski action -> Ashtekar 

Prime crash course (Lee Smolin, Perimeter Institute):

https://pirsa.org/09070000

I do not properly distinguish between Lagrangian, Palatini, Plebanski, Ashtekar. Lagrangian is the focus. The rest are successive ways of reducing the number of "generalised coordinates" until the theory is really more user-friendly.

A particularly illuminating step is when the curvature tensor is reduced to an expansion in a self-dual part and an anti-self-dual part. The content of Einstein's equations being that the anti-self-dual part is identically zero --something like that, I forget lots and lots of details.

Edited March 14 by joigus
minor correction

[Genady]

Thank you for all the information. Keep it coming, please. I will have what to work on after I finish the 500 pages book I'm working through now (that's on QFT).

[Genady]

On 3/14/2023 at 8:32 AM, studiot said:

Also Nice list Markus, +1.

 

Try this

Free PDF of this book downloadable here: Download PDF - Einstein In Matrix Form: Exact Derivation Of The Theory Of Special And General Relativity Without Tensors [PDF] [561fu48ukqq0] (vdoc.pub)