Genady Posted March 14 Share Posted March 14 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. 1 Link to comment Share on other sites More sharing options...

Markus Hanke Posted March 14 Share Posted March 14 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. 3 Link to comment Share on other sites More sharing options...

joigus Posted March 14 Share Posted March 14 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. 2 Link to comment Share on other sites More sharing options...

studiot Posted March 14 Share Posted March 14 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 1 Link to comment Share on other sites More sharing options...

joigus Posted March 14 Share Posted March 14 (edited) 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 1 Link to comment Share on other sites More sharing options...

Genady Posted March 14 Author Share Posted March 14 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). Link to comment Share on other sites More sharing options...

Genady Posted March 15 Author Share Posted March 15 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) Link to comment Share on other sites More sharing options...

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