A What math do I need to know to understand General Relativity

[Felix Quintana]

I'm a 16 year old whose summer goal is two understand general relativity, but I'm lost on what math to have to understand it, I understand topological spaces and a topological manifold. but then it becomes more complicated math, and I know I simply don't understand because of the mathematics.

 

[BvU]

Well, ask on PF and hopefully you shall be assisted...
Preferably specific questions: we can't cough up all kinds of general teaching material; that's what others are for.

 

[Felix Quintana]

Well, ask on PF and hopefully you shall be assisted...
Preferably specific questions: we can't cough up all kinds of general teaching material; that's what others are for.

So that wasn't a suitable enough question?

 

[mathman]

Differential geometry and anything prerequisite are necessary.

 

[Felix Quintana]

Differential geometry and anything prerequisite are necessary.

If you could ever be so kind to let me know what are those prerequisites are. It would be very helpful.

 

[BvU]

So that wasn't a suitable enough question?

There was no question in this thread, so I assume you are referring to another thread ?
There are sites that explain special relativity in as simple as possible terms.

 

[mathman]

If you could ever be so kind to let me know what are those prerequisites are. It would be very helpful.

Calculus and its prerequisites (most high school math).

 

[Felix Quintana]

So linear algebra and differential equations have no need for general relativity?

 

[BvU]

No, but special relativity helps...

 

[Mondayman]

Exploring Black Holes by Wheeler/Taylor is probably the most elementary GR book there is, requiring just calculus and linear algebra. I highly recommend the book Black Holes and Time Warps by Kip Thorne to go along with it.

 

[Felix Quintana]

Exploring Black Holes by Wheeler/Taylor is probably the most elementary GR book there is, requiring just calculus and linear algebra. I highly recommend the book Black Holes and Time Warps by Kip Thorne to go along with it.

I have read most of Kip Thorne, I also agree

 

[micromass]

I'm a 16 year old whose summer goal is two understand general relativity, but I'm lost on what math to have to understand it, I understand topological spaces and a topological manifold. but then it becomes more complicated math, and I know I simply don't understand because of the mathematics.

Please tell us in detail what math and physics you know well.

 

[Felix Quintana]

Please tell us in detail what math and physics you know well.

I know Newtonian well, my special relativity iffy, multivariable calculus, some differential equation as in anything first order but can learn more, and currently working on linear algebra and matrix theory. Topological spaces, manifolds. Dabble in abstract algebra( lost interest after congruency chapter with mod's)

 

[micromass]

I know Newtonian well, my special relativity iffy, multivariable calculus, some differential equation as in anything first order but can learn more, and currently working on linear algebra and matrix theory. Topological spaces, manifolds. Dabble in abstract algebra( lost interest after congruency chapter with mod's)

If I asked you why all singletons in a Hausdorff space are closed, would you be able to answer me?

How deep do you want to go into the math behind GR? Enough to understand most physics books? Enough to understand Wald and Malament? Enough to understand all the math behind it very rigorously?

 

[Felix Quintana]

Yes! Wald is what I would like to understand for now. I am bored with no more physics available for me to take at my school. Just want to move forward, and general relativity is my next stop on the physics side, but I know my math needs to catch up

 

[Cruz Martinez]

Yes! Wald is what I would like to understand for now. I am bored with no more physics available for me to take at my school. Just want to move forward, and general relativity is my next stop on the physics side, but I know my math needs to catch up

So, how do you prove that singletons are closed in a Hausdorff space?

 

[micromass]

Also, can you explain us why people care about the Hausdorff property? Can you explain why we care about compactness? Why do we let manifolds be second countable?

Sorry, but I want to gauge your topology knowledge.

 

[Felix Quintana]

Oh my fault... I meant yes as in the sense yes for understand wald... I don't know any of that stuff... Topology I know is the very basics like charting ithe original into a image by taking open subsets of the original and lowering the dimension... Practically the first few pages of wald's book I understand.

 

[Felix Quintana]

So I know relatively close to nothing on topology... Just the very basic... From a set of videos part of a general relativity course on YouTube by something winter school. I can post the link if you'd like.. I know I sound like some ignorant kid messing with something out of his reach, but I insure you I can do it if I'm guided to what I need to know.. I am at the full mercy of all you highly credible geniuses. Thank you

 

[micromass]

Well, the good news is that there are definitely some GR books that you can go through right now, such as Schutz. The bad news that Wald will definitely have to wait for some time until you brush up on your topology and differential geometry. Whethe you are willing to make this investment is up to you.

 

[PeroK]

Yes! Wald is what I would like to understand for now. I am bored with no more physics available for me to take at my school. Just want to move forward, and general relativity is my next stop on the physics side, but I know my math needs to catch up

Do you fully understand Special Relativity?

I'm learning GR from Hartle's book at the moment. I'd say the biggest prerequisite is to have SR nailed. Also, definitely Lagrangian mechanics. Hartle covers both of these, but really only at a revision level.

Obviously knowing classical gravitation helps as well.

I've reached Chapter 9 (Schwarzschild Geometry) and I'd say I haven't needed Topology or Differential Geometry yet. The Differential Geometry has really been what I would call vector calculus.

Hartle, it seems to me, is quite accessible. I'm judging from experience of it alone, but it certainly isn't a mathematical minefield. The only issue is that he gives few hints and no solutions to his problems. But, then, that's where Physics Forums can help!

 

[ShayanJ]

Somebody with a good background in calculus can start reading Zee's Einstein Gravity in a nutshell. It teaches all the math needed in a pedagogic way. But for people who may still think its still advanced for them, Hartle is a good book to start with.

 

[Felix Quintana]

Do you fully understand Special Relativity?

I'm learning GR from Hartle's book at the moment. I'd say the biggest prerequisite is to have SR nailed. Also, definitely Lagrangian mechanics. Hartle covers both of these, but really only at a revision level.

Obviously knowing classical gravitation helps as well.

I've reached Chapter 9 (Schwarzschild Geometry) and I'd say I haven't needed Topology or Differential Geometry yet. The Differential Geometry has really been what I would call vector calculus.

Hartle, it seems to me, is quite accessible. I'm judging from experience of it alone, but it certainly isn't a mathematical minefield. The only issue is that he gives few hints and no solutions to his problems. But, then, that's where Physics Forums can help!

Tell me more about this Hartle book, What is the book called?

 

[PeroK]

Tell me more about this Hartle book, What is the book called?

http://www.amazon.com/dp/9332535086/?tag=pfamazon01-20

 

[Cruz Martinez]

I'm a 16 year old whose summer goal is two understand general relativity, but I'm lost on what math to have to understand it, I understand topological spaces and a topological manifold. but then it becomes more complicated math, and I know I simply don't understand because of the mathematics.

In addition to the great books that others have recomended to you, I can't keep myself from suggesting you do some pure mathematics aiming at getting a good foundation in the mathematics of GR. This will be very beneficial even if you later on decide you want to do physics only. Trust me, mathematics is beautiful to the extreme, so much that I started trying to learn the math of GR but ended up being more interested in pure mathematics, far more.
There are people on this forum who would really go out of their way to help anyone wanting to learn, have you looked at micromass' signature? I talk from experience when I say he is an excellent teacher.
Anyway, whatever your interests are, I do hope you pursue them. It's great that you have this initiative to learn these subjects.

 

[Felix Quintana]

I've spoke to him, I agree mathematics gives physics a type of elegance, anyways we couldn't work anything out unfortunately but I will push on

 

[mpresic]

I wonder if Einstein himself new what a topological manifold or what topological spaces were. I think, in his day, he was probably more in need of the absolute differential calculus as developed by Levi Civita, among many other mathematical developments. I think you can find various treatments for GR all which emphasize different areas of specialized mathematics. Misner, Thorne, and Wheeler treats GR with differential forms, Hartle, and Caroll give a treatment using tensors. For sure, the more mathematics you know the better, but I know some very detailed differential geometry courses that barely touch on relativity. Learning a lot of differential geometry theory is a big investment if you want to learn the basics of GR. Of course, if you like the mathematical theory, then it is good to pursue it.

 

[dustball]

Pick up the book "The Principle of Relativity" and try to read the Einstein's paper on GR. You will see the math you need. Good luck.

 

[ProfChuck]

I would begin by getting Einsteins book "Relativity, The Special and General Theory" Crown Publishers. It takes you through the entire process by which he established the theories and the math begins with ordinary algebra and goes all the way to tensor calculus which is the form of his field equations.. To get an appreciation of Einsteins reasoning you should first gain an understanding of the equivalence principal and his "thought experiments" regarding elevators and space craft. These show, for example, why a beam of light is bent in a gravitational field and provide an underlying mechanism for gravitational red shift.

 

[dustball]

And there is a great book "Gravitation" by Mister, Thorne and Wheeler that explains all the math along with GR, fun to read, try it.

 

[dustball]

As the general math background for physics, V.I. Arnold's Mathematical Methods of Classical Mechanics is the best.

 

[AgentCachat]

I'm a 16 year old whose summer goal is two understand general relativity, but I'm lost on what math to have to understand it, I understand topological spaces and a topological manifold. but then it becomes more complicated math, and I know I simply don't understand because of the mathematics.

Calculus, basic linear(matrix) algebra, some differential equation familiarity, tensor algebra and analysis. Tensors are a part of differential geometry and are absolutely essential for understanding GR. Shaum's Tensor Calculus may be helpful, not really knowing more details. It includes needed linear algebra, and some other topics like tensor fields on manifolds. Its an outline, but should enable you to reach your goal. Don't be too discouraged. Einstein needed help on some math before he could complete his theory.

 

[dustball]

For SR try The Feynman lectures, vol. 1.

 

[rrogers]

This might seem strange but the most understandable book that I read was "The Large Scale Structure of Space-Time" by Hawkins and Ellis. Of course this was after I had gone through several other books that left an aura of mystery about a lot of things. In my mind that book nailed down a lot! Now I can read "Gravity" (which sort self reflects because it's weight certainly proves Gravity) without a hitch; and the details are just that details.

 

[Kevin McHugh]

Also, can you explain us why people care about the Hausdorff property? Can you explain why we care about compactness? Why do we let manifolds be second countable?

Sorry, but I want to gauge your topology knowledge.

I too am trying understand GR, and have read most of MTW. I can't recall any mention of the Hausdorff property in that tome. How is it necessary to understand curvature?

And to the OP, you can download Schutz for free from this site. It is a very good intro to GR.

 

[micromass]

I too am trying understand GR, and have read most of MTW. I can't recall any mention of the Hausdorff property in that tome. How is it necessary to understand curvature?

A spacetime is by definition Hausdorff. So it is already needed for the very definition of what we're working with. If MTW doesn't need the Hausdorff property, then MTW is just not a rigorous book. That's ok, I'm not saying that physics books need to be mathematically rigorous. But the OP mentioned Wald, and Wald definitely is rigorous (and does cover the Hausdorff property).

 

[Kevin McHugh]

A spacetime is by definition Hausdorff. So it is already needed for the very definition of what we're working with. If MTW doesn't need the Hausdorff property, then MTW is just not a rigorous book. That's ok, I'm not saying that physics books need to be mathematically rigorous. But the OP mentioned Wald, and Wald definitely is rigorous (and does cover the Hausdorff property).

Thanks for that micromass. You do learn something new every day.