Geometrized Newtonian Gravity

1. Jul 30, 2013

WannabeNewton

Hi guys! It is well known that the usual force based formulation of Newtonian gravity can be recast in a purely geometric form much like general relativity. This was originally done by Cartan and his theory is known as Newton-Cartan theory. Now I've tried to read up on rigorous formulations of geometrized Newtonian gravity but all I can find are philosophy of physics sources (not to say that this is a bad thing). For example, the entirety of chapter 4 of Malament's text "Topic in the Foundations of General Relativity and Newtonian Gravitation Theory" is dedicated to giving a truly rigorous formulation of geometrized Newtonian gravity but there is a clear philosophical underpinning in the physics so presented.

If I try to find online resources I keep ending up at philosophy of physics. Does anyone know of a physics text/resource that deals rigorously with geometrized Newtonian gravity? I understand that this will not exactly be common because it has no direct use in physics and is really just around in order to juxtapose the foundations of general relativity with that of Newtonian gravity while on a level playing field (i.e. treating them as two different geometric theories of gravity) but I would really like a resource/text that has zero philosophical underpinnings (i.e. a physics text). Thanks in advance.

2. Jul 30, 2013

Daverz

MTR covers it (chapter 12). Penrose also discusses it in his Road to Reality.

http://www.whfreeman.com/Catalog/product/gravitation-firstedition-misner/tableofcontents [Broken]

Last edited by a moderator: May 6, 2017
3. Jul 30, 2013

atyy

4. Jul 30, 2013

WannabeNewton

Thanks guys!

5. Jul 30, 2013

atyy

The only problem is that the Feynman lectures are full of philosophy ....

He tried to disguise it by ranting against philosophers. But he still mentioned frogs and composers in the same breath!

6. Jul 30, 2013

WannabeNewton

Feynman is quite the hypocrite, there's no denying that!

I never knew MTW had a section on Newton-Cartan theory. I guess it's not all that surprising since they have like everything in that book.

That paper written by Straumann makes me wish that he had included that material in his GR text :)

Last edited: Jul 30, 2013
7. Jul 30, 2013

robphy

I'd be surprised if Malament didn't have references to papers/articles by Jurgen Ehlers and by Andrzej Trautman.
I think Graham S. Hall has some articles on Newton-Cartan connections.

8. Jul 30, 2013

WannabeNewton

Thanks robphy. He does give a single Ehlers reference and a single Trautman reference but I unfortunately can't access them through my university. I'll check out Graham thanks!

9. Jul 30, 2013

robphy

10. Jul 30, 2013

dextercioby

Another reference would be Trautman's article on arxiv: gr-qc/0606062v1.

11. Jul 30, 2013

WannabeNewton

Thank you very much dexter and robphy! That's perfect :)

12. Jul 30, 2013

WannabeNewton

robphy I'm curious as to your opinion on this: which of the mathematical approaches to classical metric theories of gravity (general relativity, Newton-Cartan theory etc.) do you personally prefer? If you take a look at atyy's (Newton-Cartan theory) and dexter's (Einstein-Cartan theory) links for example, the language is primarily of a modern nature through the use of connection forms and bundles but if you take a look at for example Malament's GR text, Wald's GR text, and Geroch's notes the language is primarily classical tensor calculus/differential geometry cast in the abstract index notation (e.g. using a certain derivative operator $\nabla_{a}$ to basically write down the theory) so I was wondering which of these languages you preferred personally.

13. Jul 30, 2013

robphy

Note that the "Cartan" aspect of the gr-qc/0606062v1 article (on Einstein-Cartan)
is about including torsion with a Lorentzian-signature metric, rather than using a degenerate metric for Newtonian spacetimes.

(1965 "Comparison of Newtonian and relativistic theories of space-time")
http://bazhum.icm.edu.pl/bazhum/dow...2726eb/full-text/match6130940075236899637.pdf
from a google search of Trautman's "Sur la theorie Newtonienne de la gravitation"

From Trautman's homepage,
http://www.fuw.edu.pl/~amt/CompofNewt.pdf (1966 "Comparison of Newtonian and relativistic theories of space-time") has the same title as above... but slightly different text.

14. Jul 30, 2013

robphy

Since my worldline took me through Chicago, my preference and comfort-level is for abstract-index tensor calculus and differential geometry. When there is a compelling reason to use forms and bundles, I would try to learn more about them as a second language.

15. Jul 30, 2013

WannabeNewton

Thank you for the further links robphy. I think I'll go through the Trautman papers first, then the Ehlers paper, and finish off with Straumann's paper.

I personally also prefer the abstract index tensor calculus/differential geometry mainly because I find it very elegant and since I primarily learned GR from resources like Wald's text and Geroch's notes, I have (like you) become comfortable with and very fond of the abstract index tensor calculus/differential geometry. Whenever I read contemporary papers on GR that don't have to do with experimental results/coordinate based calculations, I always see the more modern language of forms and bundles being used and it seems the language of abstract index tensor calculus is only present in contemporary papers on GR within the domain of philosophy of physics. Oh well!

16. Jul 30, 2013

robphy

It may just be the evolution of geometrical-physics thought and expression...
like Maxwell Equations first as a coupled-set of PDEs
then as a coupled-set of vector-calculus expressions (Heaviside?)
to a coupled-set of tensor-calculus expressions (Minkowski?).

17. Jul 30, 2013

dextercioby

I think that the 'langauge' of fiber bundles is the appropriate description of all theories of non-quantum physics. And if we can find a way (perhaps already found) to 'marry' this modern formulation of diff. geom. to the hardcore functional analysis of quantum physics, then at least until the newer theories of SUSY/SUGRA/Strings/LQG one should be happy of knowing the most rigorous formulation of the well-established physics.

18. Jul 30, 2013

WannabeNewton

I don't disagree at all dexter. I'm just curious as to why the classical language that you see in e.g. Wald's text has faded away from contemporary physics but still gets used rather extensively in philosophy of physics. For example in the following links http://philsci-archive.pitt.edu/4939/1/NCTPG_paper_for_archive.pdf and http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf you can see the same language that pervades Wald's text being used extensively. I'm just curious as to why it still gets used a lot in philosophy of physics even though it doesn't in physics itself.

19. Jul 30, 2013

dextercioby

The language of fiber bundles is more abstract, it basically has a narrow audience (some physicists and some mathematicians, a.k.a geometers). I think the philosophy of science should be addressed to a wider audience and the difficulty of mathematics within should be kept to a decent level. After all, really, philosophy is about words and meanings. The same goes for history of science.

20. Jul 30, 2013

atyy

I think the vector calculus form was also from Gibbs.