What are Some Potential Topics for a Master's Thesis in General Relativity?

[Avogadro Number]

Hello,
I was wondering if somebody could help me by suggesting a concrete topic in GR for my master's dissertation. My background is from Mathematics, but I have followed first year masters program courses in Theoretical Physics, including one on General Relativity based on Schutz's book. Ideally, the project could involve

• studying a GR paper,
  
• maybe writing a self-contained exposition of it, or perhaps writing it in coordinate-free language, and
  
• perhaps if there is an open issue mentioned at the end of the paper, then performing numerical simulations to study the open issue, or considering some special cases.

I have tried my own hand at finding such a topic, and while there are glimpses of what might work, I am not sure at all, and I find it all a bit overwhelming! So it would help me a lot if an experienced researcher in the area could suggest a topic which is interesting and would most certainly work. Unfortunately, there is no one at my university who works in general relativity, and so I am shouting out in the www abyss! Thank you!

 

[haushofer]

My PhD-research concerned the Newton-Cartan formalism, which would make a very nice masterthesis-topic also, I think. The Newton-Cartan formalism is a general-covariant description of Newton's theory of gravity, i.e. "Newton's gravity in Einstein's mathematical language of curved spacetime geometry". It involves some subtleties, like degenerate metrical structures (because of the absolute time). Nowadays, the topic is mainly studied for applications in holography. My thesis can be found here,

https://www.rug.nl/research/portal/...ed(fb063f36-42dc-4529-a070-9c801238689a).html

which should be perfectly readable for master students (at least, that was my aim). The Newton-Cartan formalism makes also a great learning tool to think about topics like general covariance and supersymmetry. Hope this helps!

Just a question: what are the research topics of your supervisor? I think that also matters and should give a clue about the direction to be pursued.

 

[Avogadro Number]

@haushofer: Many thanks for your kind help! This sounds very interesting. I had seen Frederic Schuller's lecture :

in which he showed how Newtonian spacetime is already curved if one takes Newton's laws seriously.
I am not familiar with the Newton-Cartan formalism, so it would be great to learn this while doing the dissertation. Thanks again!
About your question: I have a nominal advisor from Physics (the teacher who taught me GR, but who works in particle physics), and a differential geometer from the mathematics department who will be the main supervisor. The latter has left it up to me to find a physics GR topic, and offered supervision with the maths aspects.

 

[haushofer]

To get a first impression on the topic, you can also consult Misner,Thorne&Wheeler; it's one of the few textbooks to treat Newton-Cartan. To sprinkle around some interesting questions I thought about during my research (some have partial answers, some don't):

* What's the precise content/meaning of general covariance/"diffeomorphism invariance" if even Newton's gravity can be described general-covariantly? How's the famous hole-argument recasted in Newton-Cartan formalism?
* What are the possible "Einstein equations" for Newton-Cartan theory? In the literature you'll find the "Einstein equations" which lead to the Poisson equation, but are there more possibilities to impose gravitational dynamics?
* How to impose supersymmetry in order to obtain "Newton-Cartan supergravity?" Algebraically, simple SUSY (N=1) does not lead to interesting non-relativistic theories, because the supercharges decouple from the spacetime translations, so you need to go to N=2. I derived the corresponding D=3 SUGRA theory in my PhD-thesis, but the N=2, D=4 theory is still an open question.
* How about the analogy of the Einstein-Hilbert action for Newton-Cartan theory? The geodesic equation can be obtained by an action, but how about the "Einstein equations" of Newton-Cartan?

Personally, one of the things I liked about my research, is that it also shows a very physical application of central extensions of Lie-algebras. Usually, this topic is presented in a very sophisticated, mathematical way (Chevalley-Eihlenberg cohomologies etc.). One of the nice things I discovered was that the gravitational Newton-potential can be seen as part of the gauge field belonging to the central extension of the Galilei algebra.

Well, this was a lot of blabla, but hopefully you've found yourself at least one possible direction to pursue. I really enjoyed doing it ;)

edit: to give you one final motivation: Newton-Cartan formalism is very approachable and doesn't contain very fancy mathematics, yet it has scrutinizing subtleties. In my opinion, that's a perfect playground to learn new physics without drowning in math. And whenever I gave talks, I often found that people from all kinds of different research areas were intrigued. After all, I claimed to tell them something new about a topic (Newtonian gravity) which most of them covered already at high school.

 

[Avogadro Number]

Thanks again for your reply! I don't know enough yet to pick a particular direction from the multiple avenues you've suggested, but I am sure that I will be able to use your help after I have studied more from your thesis, and from MTW as advised. Would you be willing to be an external supervisor --- I can try to e-mail you maybe --- does your rug e-mail still work?

 

[haushofer]

No, the email given there doesn't work anymore since I left academia after my PhD. Because I'm also in between jobs looking for something new right now on top of that, unfortunately I can't commit myself to supervision. But you can always send me private messages here, and if you have any questions you can always drop them in the GR-subforum here. I'd be happy to help. :)

 

[Avogadro Number]

Okay, I understand of course! Thank you for your kind offer of asking you questions. Good luck with your future endeavours! :)

 

[haushofer]

Thanks! I just reminded myself that I also wrote an Insight article about it a while ago,

https://www.physicsforums.com/insights/revival-Newton-cartan-theory/

It contains useful resources :) Good luck to you too choosing a topic and doing research!

 

[Avogadro Number]

I read it! Very nice and intriguing...thank you again!