Geometrized Newtonian Gravity

[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.

 

[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

 

[atyy]

http://arxiv.org/abs/gr-qc/9604054

 

[WannabeNewton]

Thanks guys!

 

[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!

 

[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 :)

 

[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.

 

[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!

 

[robphy]

http://pubman.mpdl.mpg.de/pubman/item/escidoc:153004:1/component/escidoc:153003/328699.pdf
Examples of Newtonian limits of relativistic spacetimes (Ehlers, 1997)
..and references within.

http://www.google.com/search?q=ehlers+Newtonian

 

[dextercioby]

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

 

[WannabeNewton]

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

 

[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.

 

[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.

This article by Trautman is closer in spirit to Newton-Cartan
(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"
http://www.google.com/search?q="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.

 

[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.

 

[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!

 

[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?).

 

[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.

 

[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.

 

[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.

 

[atyy]

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?).

I think the vector calculus form was also from Gibbs.

 

[WannabeNewton]

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.

I can certainly agree with that

 

[robphy]

I think there has been a flow of influence from mathematics to physics to philosophy...and this flow is not instantaneous.

While more modern mathematical presentations of relativity use forms and bundles, there are still many that use classical tensor calculus and differential geometry... and there continue to be new texts that are more oriented toward physicists (not necessarily theoretical-relativists) that still use it. So, I don't agree with the "faded away" description.

I think the Minnesota conference "Foundations of Space-Time Theories" 1977 http://www.mcps.umn.edu/philosophy/completeVol8.html (and the future works of those there) was influential to the philosophers of science who discuss relativity. So, the tensor [as opposed to form and bundle] language persists today among the more mathematical philosophy-of-physics relativity papers.

It may be also be argued that the classical tensor formulation (whether abstract-index or component) is closer in spirit to traditional references... and closer in spirit to the physics as presented to undergraduates and graduate physics students. I think the typical physicist would feel that the required effort for the mathematical sophistication and precision using forms and bundles doesn't payoff much for understanding "the physics" (or at least the aspects of physics that they are interested in). For the modern mathematical physicist, who is likely interested in more theoretical aspects and "model-building", those sophisticated tools _may_ just be what is necessary to (say) unify the interactions or quantize gravity.

 

[WannabeNewton]

While more modern mathematical presentations of relativity use forms and bundles, there are still many that use classical tensor calculus and differential geometry... and there continue to be new texts that are more oriented toward physicists (not necessarily theoretical-relativists) that still use it. So, I don't agree with the "faded away" description.

Point taken. There certainly are a number of popular texts that still use the classical index based tensor calculus, be it abstract or component-wise. In fact, Choquet-Bruhat rather recently (I believe 2010) came out with a book on GR (mainly on its mathematical aspects) and the book is primarily set in the language of index based tensor calculus.

It may be also be argued that the classical tensor formulation (whether abstract-index or component) is closer in spirit to traditional references... and closer in spirit to the physics as presented to undergraduates and graduate physics students. I think the typical physicist would feel that the required effort for the mathematical sophistication and precision using forms and bundles doesn't payoff much for understanding "the physics" (or at least the aspects of physics that they are interested in).

I would agree. It's just that it is somewhat disappointing to have gotten so used to the classical index based tensor calculus formulations of GR that are presented in so many standard GR texts (e.g. Wald's text and Geroch's notes) and then much more recently seeing it get used extensively in Malament's GR text (which was clearly influenced by Geroch's notes) but not seeing it get used nearly as much in contemporary physics papers on theoretical GR (but as you say it still gets used here and there e.g. Ellis seems to still stick with it: http://arxiv.org/pdf/gr-qc/9812046v5.pdf). There just seems to be a notable discrepancy between its use in various GR texts and its use in literature which is most likely related to what you just said above.