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, 2023 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)

[studiot]

@Genady  @joigus  @Markus Hanke

 

Update.

I have been evaluating this new book by Professor Needham of California, and I have to say I like it and am impressed.

 

Needham favours a move back from 'algebraic geometry' to 'geometric geometry' , as a resullt this book has lots of new results not (readily) available elsewhere and a comprehensive section on GR which is his favourite topic.

The does not mean that he shies away from calculation, quite the reverse. He simply wants the calculations to offer real world meaning as well as algebra.

 

PS Markus I hope all is well with you.

Edited August 24, 2023 by studiot

[joigus]

22 minutes ago, studiot said:

@Genady  @joigus  @Markus Hanke

 

Update.

I have been evaluating this new book by Professor Needham of California, and I have to say I like it and am impressed.

 

Needham favours a move back from 'algebraic geometry' to 'geometric geometry' , as a resullt this book has lots of new results not (readily) available elsewhere and a comprehensive section on GR which is his favourite topic.

The does not mean that he shies away from calculation, quite the reverse. He simply wants the calculations to offer real world meaning as well as algebra.

 

PS Markus I hope all is well with you.

Sounds like an appealing approach to physicists and engineers. Thank you, Studiot.

Although I've been much less active lately, I was also wondering about Markus. I found I'd missed a brief message announcing he was to be away for a while. I hope he's OK too.

[Genady]

1 hour ago, studiot said:

@Genady  @joigus  @Markus Hanke

 

Update.

I have been evaluating this new book by Professor Needham of California, and I have to say I like it and am impressed.

 

Needham favours a move back from 'algebraic geometry' to 'geometric geometry' , as a resullt this book has lots of new results not (readily) available elsewhere and a comprehensive section on GR which is his favourite topic.

The does not mean that he shies away from calculation, quite the reverse. He simply wants the calculations to offer real world meaning as well as algebra.

 

PS Markus I hope all is well with you.

Thank you for this update.

He also has a pleasant style, in my taste.

[Markus Hanke]

Funny coincidence…just as this thread gets posted, do I get back online, after a fashion anyway 

Yes all is well with me, I was just without Internet connectivity for a while. I live in Norway now, and helping to build up a monastery here.

My Internet connection is still basic, so I mightn’t be quite as active as before. But it’s good to be back

And thanks for the book recommendation @studiot, I’ve had this in my library for a while, but haven’t gotten around to reading it. Hopefully soon!

[studiot]

It was really pleasing to find I had messaged the rapid response team.

🙂

Thanks to you all for you comments.

3 hours ago, Genady said:

He also has a pleasant style, in my taste.

Yes I agree, but he seems more than just a pretty face, he offers some interesting ideas.

I think the notion of an absolute measeure or standard of length (ie one without units) that he quietly drops on p15 is worth a discussion all by itself.
This is quite different from a Physicist's idea of what such a measure might be, yet so simple and elegant.

Finally my apologies to those members I omitted, for example @Mordred though I am sure there must be many more, in my rush to get my first post to print.

Of course I present the information to all who might be interested or benefit.

[Genady]

30 minutes ago, studiot said:

I think the notion of an absolute measeure or standard of length (ie one without units) that he quietly drops on p15 is worth a discussion all by itself.
This is quite different from a Physicist's idea of what such a measure might be, yet so simple and elegant.

Here is this notion:

Quote

... natural way of defining the absolute unit of length is in terms of the constant K. Since the radian measure of angle is defined as a ratio of lengths, E is a pure number.  On the other hand, the area A has units of (length)^2. It follows that K must have units of  1/(length)^2, and so there exists a length R such that K can be written as follows: K = +(1/R^2)  in Spherical Geometry; K = −(1/R^2) in Hyperbolic Geometry.

It can be an absolute unit of length in a homogeneous non-Euclidean space, but our physical space is non-homogeneous on smaller scales and might be Euclidean on larger scales, AFAIK.

[Genady]

On 8/24/2023 at 8:53 AM, studiot said:

@Genady  @joigus  @Markus Hanke

 

Update.

I have been evaluating this new book by Professor Needham of California, and I have to say I like it and am impressed.

 

Needham favours a move back from 'algebraic geometry' to 'geometric geometry' , as a resullt this book has lots of new results not (readily) available elsewhere and a comprehensive section on GR which is his favourite topic.

The does not mean that he shies away from calculation, quite the reverse. He simply wants the calculations to offer real world meaning as well as algebra.

 

PS Markus I hope all is well with you.

Did you happen to evaluate his previous book, Visual Complex Analysis? Would like to know your opinion if you did.

[Eise]

On 3/14/2023 at 11:30 AM, Genady said:

In QM, after wave mechanics, matrix mechanics was developed.

<antfuckermode> It was the other way round. Heisenberg came first with his matrix formulation, followed about a year later with Schrödinger's 'wave mechanics'. </antfuckermode>

'Wave mechanics' became more popular in those days than matrix mechanics, because it was more 'visible', and used concepts of the physics of waves, which was of course more familiar to physicists. Heisenberg did not even realise that his 'tables of energy transitions' already existed in mathematics as matrices. I think it was Born who recognised that.

 

[Genady]

1 hour ago, Eise said:

<antfuckermode> It was the other way round. Heisenberg came first with his matrix formulation, followed about a year later with Schrödinger's 'wave mechanics'. </antfuckermode>

'Wave mechanics' became more popular in those days than matrix mechanics, because it was more 'visible', and used concepts of the physics of waves, which was of course more familiar to physicists. Heisenberg did not even realise that his 'tables of energy transitions' already existed in mathematics as matrices. I think it was Born who recognised that.

 

Thank you for the correction.

[studiot]

On 8/26/2023 at 3:38 PM, Genady said:

Did you happen to evaluate his previous book, Visual Complex Analysis? Would like to know your opinion if you did.

OK so before I give any detail a couple of important things you need to know.

There are two editions of VCA, which was first published in 1997.

VDG&F followed in 2021 and was written in the same style and from the same viewpoint.

A second edition of VCA (which I have) was published in 2023 and has some important typographic deficiencies remedied.

The maths text is the same, but the many diagrams now all have explanatory captions, the index has been much expanded and a conventional referencing system added, and the book is in a larger physical size.

 

The two books contain much common material and frequently reference each other.

Between them they develop the author's theme that it is good for understanding to approach fundamental principles in maths from multiple viewpoints.
He expresses how reassuring it is to come to the same result from different routes.
Both books have Feynman's american ability to pull out the essential statements in a clear and obvious form and to highlight and separate them from the block of the text.

I would certainly recommend both as a pair; you really need the 2023 version of VCA though.

 

I already have three books entitled "The Geometry of Complex Numbers", but apart from the usual few diagrams you might find in any work on complex analysis they do not approach it from the geometric viewpoint at all.

VCA certainly achieves this.

VCA covers a lot of ground with good mathematical insights.

But what it is not is a tabulation or treatise on the applications of complex analysis, which is where many readers are coming from.

So if you want CA in the solution of differential equations, it only mentions 2, Schrodinger and Dirac, complex Bessel functions are not treated at all, you will have to go to alfhors for that.

Complex integration is dealt with at a fundamental level, in relation to measure theory. It is not a textbook of complex integration techniques.

(Conformal) mapping again much wanted by engineers is treated at a mathematically fundamental level rather that a catalogue of techniques.

 

As promised the book is not a catalogue of algebraic results and formulae, with the excuse 'we can explain this result geometrically'.

The results are there but arise naturally by considering the geometry ( my preferred way ) and also the topology / continuity.

Also arising from this geometric approach Tristran delves deeply into noneuclidian geometry.

 

Hope this helps, sorry it has taken so long to reply, but I needed to do justice to the books.

 

 

 

[Genady]

3 hours ago, studiot said:

OK so before I give any detail a couple of important things you need to know.

There are two editions of VCA, which was first published in 1997.

VDG&F followed in 2021 and was written in the same style and from the same viewpoint.

A second edition of VCA (which I have) was published in 2023 and has some important typographic deficiencies remedied.

The maths text is the same, but the many diagrams now all have explanatory captions, the index has been much expanded and a conventional referencing system added, and the book is in a larger physical size.

 

The two books contain much common material and frequently reference each other.

Between them they develop the author's theme that it is good for understanding to approach fundamental principles in maths from multiple viewpoints.
He expresses how reassuring it is to come to the same result from different routes.
Both books have Feynman's american ability to pull out the essential statements in a clear and obvious form and to highlight and separate them from the block of the text.

I would certainly recommend both as a pair; you really need the 2023 version of VCA though.

 

I already have three books entitled "The Geometry of Complex Numbers", but apart from the usual few diagrams you might find in any work on complex analysis they do not approach it from the geometric viewpoint at all.

VCA certainly achieves this.

VCA covers a lot of ground with good mathematical insights.

But what it is not is a tabulation or treatise on the applications of complex analysis, which is where many readers are coming from.

So if you want CA in the solution of differential equations, it only mentions 2, Schrodinger and Dirac, complex Bessel functions are not treated at all, you will have to go to alfhors for that.

Complex integration is dealt with at a fundamental level, in relation to measure theory. It is not a textbook of complex integration techniques.

(Conformal) mapping again much wanted by engineers is treated at a mathematically fundamental level rather that a catalogue of techniques.

 

As promised the book is not a catalogue of algebraic results and formulae, with the excuse 'we can explain this result geometrically'.

The results are there but arise naturally by considering the geometry ( my preferred way ) and also the topology / continuity.

Also arising from this geometric approach Tristran delves deeply into noneuclidian geometry.

 

Hope this helps, sorry it has taken so long to reply, but I needed to do justice to the books.

 

 

 

Thank you very much! Your review is very helpful.

Both aspects - what the book offers and what it does not - are positive for me, because I am interested in ideas and fundamentals rather than in techniques.

I am going to have this book, the 2023 edition, and will enjoy studying it.

Cheers.