The Revival of Newton-Cartan Theory - Comments

[haushofer]

haushofer submitted a new PF Insights post

The Revival of Newton-Cartan Theory

Continue reading the Original PF Insights Post.

 

[klotza]

An interesting historical note is that Newtonian gravity still predicts the deflection of light by stars, except there is a factor-of-two discrepancy between Newtonian gravity and GR's prediction. Deriving this in terms of Newton's law of gravity may be problematic; instead it's done by realizing that Newtonian trajectories are conic sections and that the path of light can be described by a very eccentric hyperbola with v=c. A guy I'm not related to wrote about it in 1920: http://adsabs.harvard.edu/full/1920JRASC..14..285K Do you know if anyone has done this derivation with Newton-Cartan gravity?

Another historical note: as you've touched on, Einstein is often credited with rewriting gravity in terms of spacetime curvature rather than a gravitational force. However, at the same time, Gunnar Nordstrom (who gives us half of the Reissner-Nordstrom metric) also wrote out a new theory of gravity in terms of spacetime curvature (in modern notation, it has R proportional to T). However, his theory was wrong and Einstein's was right.

 

[haushofer]

Yes, there is a lot of history which can be added to the development of GR.

About your question: such a derivation is not needed; Newton-Cartan theory is exactly equivalent to ordinary Newtonian gravity at the level of equations of motion. So it should give the same result as given by the paper you're citing. I've never encountered such a derivation in the NC literature, anyway.

The same goes e.g. for the precession movement of Mercury. That's why the argument sometimes found in the literature, that this precession in GR is due to spacetime curvature, can be a bit misleading. NC describes also gravity as spacetime-curvature, but only in the 'space-time' direction, not on spatial hypersurfaces of constant absolute time (i.e. 'space is flat').

 

[Ibix]

Thank you for that. Reading about Cartan's reformulation of Newtonian gravity has been on my to-do list for a long time and I'm glad to have wet my toes a little.

 

[Dale]

@haushofer thanks for the article! I, for one, would greatly appreciate if you would include any good online resources you have found on the topic.

 

[strangerep]

Typo: "Newtion". (2nd-last paragraph.)

 

[haushofer]

Some resources:

* Misner, Thorne, Wheeler, "Gravitation", chapter 12 (treats NC without a metric formulation, but is a nice first exposure)
* http://arxiv.org/abs/1011.1145, chapter 3 (my own review of the metric formulation of NC)
* http://arxiv.org/abs/gr-qc/9610036 (another review on NC and how it is connected to the Newtonian limit of GR)
* http://arxiv.org/abs/gr-qc/9604054 (another review with applications to cosmology)
* http://ls.poly.edu/~jbain/papers/NewtCartan.pdf (different versions of NC geometry are compared)
* http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.36.938 (an early review)

Hope this gives some nice introductories :)

edit Added to text

 

[Ssnow]

interesting article!

 

[haushofer]

interesting article!

Thank you!

 

[avi rabinowitz]

This Youtube playlist contains videos of lectures I gave in various universities to student members of the SPS, and to researchers who haven't taken General Relativity; it's a mostly-heuristic introduction to the subject which relied to some extent on Newton-Cartan theory and related material in MTW, Synge and others. The lectures are briefly reviewed here (see #10). Here's a guide to the lectures.
Also, I wrote up a brief heuristic introduction to Newton-Cartan theory based on my approach, meant to be suitable even for high-school student who took physics, and certainly for undergraduates.
All of this was my way of making general relativity accessible to those students who studied calculus-based physics, will probably never take GR, want more than is offered in all the purely-qualitative presentations in the many wonderful popular books on the subject, and would love a presentation which leverages what they learned in that calculus-based physics course without requiring them to learn additional difficult math.
The utility of Newton-Cartan theory as an introductory path to GR is discussed in this lengthy preface to my textbook (not yet published) which is available as a pdf file. The textbook and lectures are inter-related.
Note: The playlist above is followed by this one, with some cosmology here. There are more videos which will eventually be added to these playlists
Any comments on this material will be much appreciated, particularly from students who never took GR (or from those few professionals who are savvy in GR but nevertheless both value pedagogy and appreciate heuristic treatments avoiding complicated mathematics) : air1@nyu.edu (Currently associated with BGU rather than NYU.)