Suggestions for graduate studies in General Relativity

In summary, the individual is seeking recommendations for references to gain a better understanding of the mathematical foundations of GR and topics of Numerical Relativity before starting their PhD. They have taken courses in GR, Cosmology, Black Holes & Gravitational Waves, and an introductory course in Numerical Relativity, but are concerned about not having a strong enough understanding to have new ideas and prove results. They have received a recommendation for a book on mathematical foundations of GR and lecture notes on GR, but are also considering other references. They are unsure if they will have time to read everything before starting their PhD and are also wondering if it is common to continue studying fundamental aspects of the theory while working on a PhD. They also mention some concepts
  • #1
jpmferreira
6
3
Hello everyone, I am a couple of months away of starting my PhD in Numerical Relativity and I am seeking recommendations for references to get a batter understanding on the mathematical foundations of GR and topics of Numerical Relativity.

So far I have had an undergraduate introductory course on GR, two courses in Cosmology (which was the topic of my master's thesis), a master's course on Black Holes & Gravitational Waves and a very introductory course to Numerical Relativity.

What is motivating this thread is my concern of not being able to understand GR well enough to be able to have new ideas and be able to proof the results that I'm after, which is crucial given that I wish to remain in academia.
This is true for both for its mathematical structure and its physical interpretation.
Therefore, I'm after references which do not neglect the mathematical foundations of GR, but also shouldn't introduce it in an excessively abstract way, while discussing the physics behind it.

I asked a professor of mine and he recommended me the book "Modern Geometry - Methods and Applications: Part I" for the studies of the mathematical foundations of GR.
With respect to GR itself, I was pointed towards Sean Carrol's lecture notes on GR.
On Numerical Relativity, I remember briefly looking at Alcubierre's booking on the 3+1 decomposition, which felt like skipped a couple of steps in the demonstrations.

Do you find these references suitable for my studies?
Are there any others I should consider?

I don't believe I have the time to read through the whole thing before starting my PhD, so I'll keep them close during my project.

Thank you in advance!
 
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  • #2
Honestly, I don't see the problem. You only have two months, and you have a recommendation. OK, it may not be perfect, but starting on even an imperfect book now is likely to be more usefull than burning up a lot of your time looking for a book that will be differently imperfect.
 
  • #3
jpmferreira said:
What is motivating this thread is my concern of not being able to understand GR well enough
How did you do in the courses you mention?

jpmferreira said:
With respect to GR itself, I was pointed towards Sean Carrol's lecture notes on GR.
I would expect pretty much everything in those notes to have been covered in the introductory course on GR that you say you took.
 
  • #4
This might err on the side of mathiness, but maybe try Introduction to General Relativity, Black Holes & Cosmology by Yvonne Choquet
 
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  • #6
Vanadium 50 said:
Honestly, I don't see the problem. You only have two months, and you have a recommendation. OK, it may not be perfect, but starting on even an imperfect book now is likely to be more usefull than burning up a lot of your time looking for a book that will be differently imperfect.
That's a good point, I always tend to neglect the time it takes to choose a path and often jump between references without reading them entirely...
Do you feel like it would be possible to keep reading topics on more fundamental aspects of the theory in parallel with the PhD? Or is it something that is not usually done in favor of gaining better insight of the state-of-the-art?
 
  • #7
PeterDonis said:
How did you do in the courses you mention?
Fairly well.
Looking back at some of the more advanced exercises I still have some trouble seeing the way to solve it.
The same happens with some concepts, which were introduced in those courses but not developed much further (e.g.: killing vectors/fields, lie derivative, gauge transformations), that I would like to feel more confident with.

PeterDonis said:
I would expect pretty much everything in those notes to have been covered in the introductory course on GR that you say you took.
Would you replace this reference with some other that is slightly more advanced?
 
  • #8
jpmferreira said:
On Numerical Relativity, I remember briefly looking at Alcubierre's booking on the 3+1 decomposition, which felt like skipped a couple of steps in the demonstrations.

There also are two very good books by Baumgarte and Shapiro:

1) "Numerical Relativity: Starting from Scratch", which is introductory and quite short;

2) "Numerical Relativity: Solving Einstein's Equations on the Computer", which is more advanced and much longer.

Check them out on Amazon, including using the "Look Inside" feature.
 
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  • #9
jpmferreira said:
Would you replace this reference with some other that is slightly more advanced?
There's no harm in looking through Carroll to test my prediction that you will find that your previous course already covered the material. I don't know of any reference that I would say is "slightly more advanced" than Carroll; the classic textbooks, MTW and Wald, I would say are quite a bit more advanced. I think it is useful to at least look through them, but since their focus is not on numerical relativity I don't think you need to master all the material in them (which, particularly for MTW, is a lot).

If you are planning to specialize in numerical relativity, I would say you want to focus on more advanced references in that particular specialty. It doesn't look like your professor gave you any such references; you might want to ask specifically for that.
 
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  • #10
PeterDonis said:
If you are planning to specialize in numerical relativity, I would say you want to focus on more advanced references in that particular specialty. It doesn't look like your professor gave you any such references; you might want to ask specifically for that.
My professor, along with the rest of the institute I'm currently in, doesn't do any Numerical Relativity, most of our focus is on Cosmology.
So I only asked him for references in the fundamentals behind GR, given that I also wish to improve my understanding of it.
 
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  • #11
jpmferreira said:
My professor, along with the rest of the institute I'm currently in, doesn't do any Numerical Relativity
But you are planning to get a PhD from them in Numerical Relativity? That seems odd.
 
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  • #12
I would say that you should expect a PhD program to consume most of your time. I'd also say that if you feel you need to study something else independently because the program doesn't provide it, you should ask if this is really the program you want.
 
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  • #13
I read the original post to mean that the author is looking for something to read before the first year of grad school at a different school.
 
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  • #15
PeterDonis said:
But you are planning to get a PhD from them in Numerical Relativity? That seems odd.
I'm switching institute.
I've applied and got accepted, but the project will only begin in September.
This is just for the meantime, while I'm also improving on my knowledge of programming languages and techniques.
 
  • #16
jpmferreira said:
I'm switching institute.
Ah, I see. Have you reached out to anyone at the new institute you are going to to see what references they recommend?
 
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  • #17
PeterDonis said:
Ah, I see. Have you reached out to anyone at the new institute you are going to to see what references they recommend?
Not yet, I am planning to do so in a conference that will take place shortly.
That is why I had no reference on Numerical Relativity so far when I first made this post, but one on the mathematics behind GR.
 
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  • #18
https://www.amazon.ca/dp/0521537800/ is something I'd suggest, along with the numerical books mentioned above.

For some other resources: https://einsteintoolkit.org/ is something good to learn, and be familiar with.

And if you're new new, here is a intro step by step guide for GW, which is kind of the butter field of numerical relativity right now (although, there are a lot of other areas to explore, one I find cool are black hole kicks): https://gwosc.org/s/events/GW150914/LOSC_Event_tutorial_GW150914.html
 
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  • #19
You could also look for papers and lecture notes from faculty at the next institute… then look at their references.
 
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