GR for a mathematician and a physicist? What's the difference?

[Matter_Matters]

Have members of the community had the experience of being taught GR both from a mathematical and physics perspective?

I am a trained mathematician ( whatever that means - I still struggle with integral equations :) ) but I have always been drawn to applied mathematical physics subjects and much prefer them, hence my recent jump into the weird and wonderful world of GR. Most graduate books on GR allow students to develop the necessary tools to actually apply tensor analysis and calculus to solve a problem for e.g. deriving the Schwarzschild solution. But, as soon as one steps into the big bad world of GR research papers, the jargon is incomprehensible to those not absolutely and soundly cemented in the world of Differential Geometry.

Manifolds! manifolds! manifolds is all I hear.

Now, I understand that GR is a geometric theory of gravity and hence the need for differential geometry but what ever happened to getting down and dirty with some applied tensorial mathematics?

Summary of questions

Members of the community having experience with being taught GR from both a pure mathematical (diff geom) and an applied sense.

Does publishing in GR require having both the language and skills of applying tensor analysis and calculus but also being able to speak the language of the more pure side of Diff. Geom?

 

[jedishrfu]

I think when General Relatibity first came out tensor methods ruled. Einsteins paper is all tensor-based. However as it matured as a subject more and more, it came to be expressed in Differential Geometry terms using notions like manifolds and differential forms which eventually get reduced to tensor manipulation when you go for a computation.

 

[pervect]

The mathematicians spend a lot of time studying manifolds in great detail. Ideally they start with Zermelo-Fraenel set theory (ZFC) - one needs a theory that handles infinite sets properly, something that's not especially intuitive. Then they build structures on top of ZFC, to generate point set topology. The basic structures are fairly simple, the main concerns are the properties of neighborhoods, abstracted as "open sets" or "open balls". But getting all the necessary proofs from the starting assumptions is far from a trivial task.

Manifolds are a third layer, built on top of point set topology.

Physicists will rush past all these details. Knowing that the 2 dimensional surface of a sphere is enough to regard a mainfold as some sort of "curved surface". Of course one needs more than 2 dimensions to handle a space-time manifold, which has 4 dimiensions, so perhaps one takes a detour to EInstein's "heated slab" as an alternate discussion of what it might be to be a curved surface in a way that's more easily generalized to higher dimensions. See for instance my post https://www.physicsforums.com/threads/rulers-on-a-heated-slab.123922/ which has the quote from Einstein.

With some vague intuitive ideas of what a manifold is, physicists can then rush into the physics. So I think in spite of being a trained mathematician, the physicists approach might be more appealing to you.

There's lots of physics oriented books on GR. Sean Caroll's lecture notes (https://www.preposterousuniverse.com/grnotes/) are not bad, and they're free. Hartle seems popular, some people seem to love it, others - not so much. Misner, Thorne, and Wheeler's "Gravitation" is an ancient, huge, book - but it's got a lot of interesting insights buried in it. I'm sure I'm missing some other popular texts, there are several threads recommending GR textbooks.

If you're already familiar with tensor notation, the following might be helpful. Consider the previously mentioned surface of a sphere as an example of a 2 dimensional manifold. Displacement operations on the surface of the sphere are not vectors. If you start out at some point on a plane, go in one direction in a straight line, make a right turn, go straight again, make a right turn, straight a third time, make another right, and go straight a fourth time, you trace out a square, and wind up where you started.

However, if you try this on a sphere, you won't get back where you started. (Straight lines on a sphere are replaced with geodesics, which for this purpose can be regarded as great circles, the curve of shortest distance connecting two points on the sphere. This is a bit oversimplified, be warned). If displacement operations on the sphere were vectors you would wind up at your starting point, but you don't - so they're not.

The reason displacement operators on a sphere aren't vectors is that they don't commute. If you go north x meters and east y meters, you don't wind up at the same place as if you go east y meters first, then north x meters. (This is also oversimplified!).

However, velocities on the surface of a sphere can be regarded as vectors, by considering that at every point on the sphere there is a flat space, called a tangent space, that's literally tangent to the sphere. And on this tangent space, velocities add and commute just fine, and behave as vectors.

With some idea of what a 2-dimensional manifold is, and that velocities on said manifold can be regarded as vectors in the tangent space,, you can perhaps dive into the stuff you find more interesting, and pick up some of the more rigorous discussion of a manfiold later on. Perhaps not, but - it's worth a try :-).

 

[m4r35n357]

Manifolds! manifolds! manifolds is all I hear.

Have you tried Dirac's book (it has no index entry for "manifold")?

 

[robphy]

Have members of the community had the experience of being taught GR both from a mathematical and physics perspective?

As an undergrad, I learned relativity from "physics" books like Skinner [Dover], Lawden, Landau, and Weinberg...
where there was more emphasis on setting up classical tensorial calculations for explicit evaluation for a specific metric, via christoffel symbols, to obtain some differential equations.
As a grad student, I learned relativity from more mathematical, geometrically-leaning texts like Schutz and Wald (with texts like MTW and Hawking&Ellis for reference).
Admittedly, for me, there was a great divide from one mode of thinking to the other..
until the a-ha event occurred when I sat in on a course by Geroch (whose notes are now available http://www.minkowskiinstitute.org/mip/books/geroch-gr.html ).

Now, I understand that GR is a geometric theory of gravity and hence the need for differential geometry but what ever happened to getting down and dirty with some applied tensorial mathematics?

I guess it depends on what you want to do.
Some aspects of modern GR research are about (say) the space of a specific class of solutions, applications to a specific [toy] model, existence/uniqueness/stability of solutions, numerical aspects, structural aspects in search of quantum gravity, etc... which may or may not require dirty applied tensorial mathematics.
Of course, with new methods and techniques developed over the years, topics of interest have changed since (say) the fifties [before the more abstract geometrical structure was formulated].

If you are looking for a book to learn from,
you might like Tom Moore's GR Workbook http://pages.pomona.edu/~tmoore/grw/

 

[bhobba]

My background is math but learned GR proper from Ohanian:
https://www.amazon.com/dp/B00ADP76ZO/?tag=pfamazon01-20

It takes a very physical viewpoint, treating GR as analogous to Electromagnetism and then only later introduces the geometrical view.

However at the time I posted a lot on sci.physics.relativity and Steve Carlip suggested with my background Wald would be a better choice:
https://www.amazon.com/dp/0226870332/?tag=pfamazon01-20

Much more mathematically rigorous, and very geometrical in its view.

Then I got the generally recognised reference MTW - Gravitation:
https://www.amazon.com/dp/0691177791/?tag=pfamazon01-20

Not as mathematically rigorous as Wald, but IMHO better at developing intuition of why GR is than either of the above.

I have all 3, I like Wald the best, but would recommend everyone gets all 3 and reads them in the order - Ohanian, MTW then Wald. That way you get a good appreciation of a physicists vs mathematicians view.

Thanks
Bill