How Limited is the Impact of General Relativity in Theoretical Cosmology?

[WannabeNewton]

Tensors and Differential Geometry is the main obstacle to me (and to many other students as well, I am sure). The problem is that nowhere is given a clear (in normal simple language) and explanation of GR ( in particular, Einstein Field Equations). I as a beginner, can only see the a lot of complicated formulas which are not accompanied by conventional thinking. For example, Walter Lewin's lectures; when he talked about Mechanics he made me "see through equations"; not just stare at them as I do now; but to really understand the logic.

Well you're also comparing, at the level of Lewin, a theory based on a very simple set of DEs to the EFEs which are non - linear and MUCH more complicated in terms of the entities involved. Things aren't going to be as "visual" per say. But GR does have a very clear and incredibly elegant explanation with its formulation using riemannian manifolds. Many textbooks explain it well; Hartle's text was mentioned above and my personal favorite, Carroll's text.

 

[WannabeNewton]

This. Restricting the discussion to only physicists (i.e. not engineers), it seems ridiculous to have someone not learn ANY GR. With the pedagogy of Hartle, you can tackle the subject easily in 3rd year of undergraduate or earlier, with no more difficulty than an upper division classical mechanics course. So in terms of educational difficulty, there's really no barrier.

Funny you should say that because I have seen many unis where a very good amount of general relativity is added on to the intermediate classical mechanics classes (the Taylor level ones).

 

[Nabeshin]

Well you're also comparing, at the level of Lewin, a theory based on a very simple set of DEs to the EFEs which are non - linear and MUCH more complicated in terms of the entities involved. Things aren't going to be as "visual" per say. But GR does have a very clear and incredibly elegant explanation with its formulation using riemannian manifolds. Many textbooks explain it well; Hartle's text was mentioned above and my personal favorite, Carroll's text.

I agree, Hartle's motivation seems pretty good to me. After all, he takes a 'physics first' approach and although it takes 22 (or 24, I can't remember) chapters to get to the EFE, once you do get there I feel like it's very well, perhaps overmotivated.

Also the tensor issue is something a student shouldn't really have -- they should encounter these objects already in upper division E&M and classical mechanics courses, although perhaps not quite so many of them. It's perhaps frustrating algebraically, similar to keeping track of minus signs, but really shouldn't impede the understanding very much.

 

[Nabeshin]

Funny you should say that because I have seen many unis where a very good amount of general relativity is added on to the intermediate classical mechanics classes (the Taylor level ones).

Really? Is this just in the sense of adding an extra term from the post-Newtonian expansion into the Lagrangian from the Kepler problem? Or calculus of variation on the Einstein-Hilbert action? I'm curious!

 

[WannabeNewton]

Also the tensor issue is something a student shouldn't really have -- they should encounter these objects already in upper division E&M and classical mechanics courses, although perhaps not quite so many of them. It's perhaps frustrating algebraically, similar to keeping track of minus signs, but really shouldn't impede the understanding very much.

Indeed and even if they haven't had that exposure, going back to Hartle's text, there are copious amounts of worked examples as well as problems to get the person acquainted with the necessary algebra and calculus of tensors. From some people I know at cornell and carnegie mellon I know there are one semester courses on GR that use the very same text. I'm sure other places do as well for undergraduate courses.

 

[WannabeNewton]

Really? Is this just in the sense of adding an extra term from the post-Newtonian expansion into the Lagrangian from the Kepler problem? Or calculus of variation on the Einstein-Hilbert action? I'm curious!

From what I've seen they don't go into the Einstein lagrangian; its more like, after the general curriculum is done you get an intro to GR like you would any other undergraduate GR class but probably more brief (this is for the two semester intermediate mech courses at least those are the ones I've seen this done).

 

[George Jones]

From what I've seen they don't go into the Einstein lagrangian; its more like, after the general curriculum is done you get an intro to GR like you would any other undergraduate GR class but probably more brief (this is for the two semester intermediate mech courses at least those are the ones I've seen this done).

I don't see how this is possible. Even one semester is not enough time to do anything more than a brief introduction to GR. From a thread comparing quantum mechanics and GR:

In my opinion, students could find physics courses in general relativity easier than courses in quantum mechanics. I think that students become more familiar with quantum mechanics because they spend more time studying it.

For example, when I was a student, I:

saw bits of special relativity stuck here and there into a few courses;

did not have the opportunity to take any lecture courses in general relativity;

was required to take three semesters of quantum mechanics as an undergrad and two semesters of advanced quantum mechanics as a grad student;

was required to take two semesters of linear algebra, which gives the flavour of much of the mathematics of quantum mechanics;

was not required to take any maths courses that give the flavour of the mathematics used in general relativity.

Because of the importance and widespread applicability of quantum mechanics, my programme offered much more opportunity to learn quantum mechanics than to learn relativity.

If physics students spent as much time studying general relativity and its mathematical background (say 4 or 5 semesters) as they spend studying quantum mechanics and its mathematical background, then general relativity would be understood by possibly millions of people. I understand why students spend much less time studying relativity than they spend studying quantum theory, and I am not necessarily saying that students should spend more time studying relativity (see the post above by Haelfix), but I do think that this time difference is a big part of the reason that general relativity still has a bit of a reputation.

Fortunately, there are many more good technical books on general relativity (pedagogical, advanced, physical, mathematical, etc.) available now than were available 25 years ago.

 

[WannabeNewton]

I don't see how this is possible. Even one semester is not enough time to do anything more than a brief introduction to GR. From a thread comparing quantum mechanics and GR:

It isn't any more than a brief intro to GR indeed. It isn't a substitute for an actual GR class, I was just agreeing with Nabeshin's comment that the Hartle level undergrad GR class shouldn't be much harder than the intermediate mechanics class and that it is even introduced in such courses.

 

[twofish-quant]

* GR is essential in making GPS work;

* GR is essential in understanding cosmology;

However, the amount of GR that you need to understand cosmology is pretty minimal. You need a tiny bit of GR to derive the Friedman equations and maybe a little more to derive perturbation, but most people just use the "cookbook equations."

* GR is essential in understanding neutron stars and black holes.

It's actually not. For most astrophysical work in black holes and neutron stars, you use "cookbook" equations. And even sometimes that isn't necessary. For most neutron stars computations, you don't need or want to use GR, because that unnecessarily complicates the calculations and you miss the things you really are interested in. You can show that the impact of GR on neutron stars is minimal, and once you run your calculations using Newtonian gravity.

The reason that GR isn't taught very much in comparison to QM is that for all but a few specialist problems, it's not that necessary, and you can get by with either ignoring it, or having a specialist give you "cookbook" equations.

 

[twofish-quant]

For people like those working scientists who deal with neutron stars and black holes, and cosmology, GR is required to explain observed data.

In most situations, it's not. If we had very good measurements of black holes and neutron stars then you need GR. However, with some exceptions (i.e. binary pulsars), the black hole, neutron star, or cosmology data isn't good enough for GR to matter, and there are about a dozen things that matter more than GR.

For my Ph.D. supernova code, I had a GR module which I used to confirm that GR didn't make a difference. Once I confirmed that, then I ran everything else using Newtonian gravity, because it was a waste of CPU cycles which could be used to calculate something else.

 

[romsofia]

Wow, I'm surprised that people didn't take GR. You'd think that people would want to learn gravity to the degree that they learn electromagnetism, as both are introduced in introductory course (well, one is a full semester, while one is probably only a chapter :P).

As others have said, if you want to study GR, go with Carroll's text.

Good luck!

 

[f95toli]

Wow, I'm surprised that people didn't take GR. You'd think that people would want to learn gravity to the degree that they learn electromagnetism

It not a question about what you want to know . It is what you need to know. A good grasp of EM is absolutely essential to every working physicist, simply because EM is used in almost every field of physics. The same can not be said of GR.

Remember that something like 80% of alll physicists work in solid-state physics or one of its subfields; and that the vast majority are experimentalists. Theoretical cosmology is a tiny field.

 

[lsaldana]

I also recommend Hartle as a starting point in learning GR; that is if you're an undergrad or wish to start learning the theory without learning the full-fledged diff geo behind it first. I thought it was a great book, very intuitive and easy to follow. Personally, I took a GR course as a junior in college and then a Cosmology course as a senior. I believe this is just the right time to start. I plan to take further units in grad school. GR is a beautiful theory and differential geometry alone makes it quite interesting. If you have time though, I recommend learning some diff geo first. It is also quite a beautiful field of mathematics.

 

[mfb]

It not a question about what you want to know . It is what you need to know. A good grasp of EM is absolutely essential to every working physicist, simply because EM is used in almost every field of physics. The same can not be said of GR.

This might be true for the concepts of electromagnetism, but knowledge is not the main goal of a degree in physics. You learn how to work in a scientific context, you learn how to learn new things, and how to solve problems. Those things are important. The science you need directly for work is so special that you have to learn it separately anyway.

Concrete example: Which knowledge do you need for a master thesis in experimental particle physics at the LHC? You should know which particles you study, and the relevant decay channels - the basic part (which you learn in courses) can be learned in 10 minutes, a feeling for those things needs practice. You need some basic concepts of special relativity, but probably not in calculations. You need an idea how your particles (or their decay products) behave in the detector - this is very specific, and probably not part of usual courses.

In principle you are right, this could have been done without knowledge of GR. But in practice, what would have happened without knowledge of GR is that the satellites would have been launched with no way of compensating for the clock rate difference, and once that difference was observed, the whole thing would have had to be scrapped and re-done. So knowledge of GR certainly had a large practical effect in this case.

Fix the receivers, if you cannot fix the satellites?

 

[PeterDonis]

Fix the receivers, if you cannot fix the satellites?

In principle it would be possible to apply the corrections in the receivers (although that would have its own problems--millions of copies of the algorithm to convert GPS satellite time to UTC time, and many more opportunities for receiver makers to screw something up). But the real problem is that, in this alternate universe where nobody knows about GR, even *after* the frequency shift is observed, nobody has a theory to explain it, so nobody knows how it behaves. It's not as though people would instantly be able to figure out the correct GR equation for the shift just by observing it, if GR as a theory was unknown.

 

[GRstudent]

Can we say that Hartle is soft of "Walter Lewin" in GR?

 

[twofish-quant]

Remember that something like 80% of alll physicists work in solid-state physics or one of its subfields; and that the vast majority are experimentalists. Theoretical cosmology is a tiny field.

And the use of GR in theoretical cosmology is surprisingly limited. Someone works out the equations, and the rest is cookbook, and the symmetry conditions of the universe vastly reduces the complexity of the equations.

There are people that specialize on applying GR to cosmology, but that's a small subfield of a small subfield. What most people who are GR specialists do is to try to reduce the equations so that "mere mortals" can use it. You have people writing pages and pages of tensor equations and the punchline is that under conditions X, Y, and Z, you probably can get away with just using Newtonian physics.