Space is curved, so how do I model it?

[KBeee]

If you want to learn GR, the only realistic way would be to take graduate courses in mathematics or physics (which require either an undergraduate degree in mathematics or physics to understand, or just being very good at math and/or having an excellent professor).

Having learned it in the above way, I don't know whether or not it would be possible to learn from just books. You could, of course, attempt to learn from one of the GR texts that is out there (Sean Carroll has a good one), but those will set you back quite a bit of cash, and would probably be very hard to understand without a strong physics background.

One reason why GR math is so challenging is just due to the large number of equations and parameters. We write these equations in tensor form, so that they are relatively simple to write down, but actually doing calculations with 3rd-rank tensors is very cumbersome and requires you to be very precise about your mathematics (I remember taking an entire weekend to derive the Friedmann equations for the first time from the FRW metric with constant spatial curvature, and that's one of the simplest problems you can do).

Granted, I've never done numerical GR solving, so in principle that might be easier, as you don't have to deal with the cumbersome algebra, but you still have to deal with cumbersome algebraic manipulations regardless.

You're over simplifying - the Universe is beer bottled shaped

 

[K.I.L.E.R]

Tensors are just matrices right?

I can't believe after learning all that stuff in Computer Science I can't stick anything realistic into practice.

 

[KimB]

Tensors are just matrices right?

I can't believe after learning all that stuff in Computer Science I can't stick anything realistic into practice.

Nope. A matrix can be thought of as one special type of 2-index tensor. Tensors can have as many indices as you want. Some common ones are:
The metric: 2 indices
The connection: 3 indices
The Riemann Curvature Tensor: 4 indices

Also, you have to be careful as to whether the tensor is contra-variant or co-variant in each index. You define a tensor by the way in which it is transformed when you change coordinate systems. A co-variant index and a contra-variant index transform differently.

A matrix can be thought of as a 2nd-rank tensor (2 indices) with one index co-variant, and one index contra-variant.

Basically, there's a lot more mathematical rigor involved in dealing with Tensors that requires some learning, and taking derivatives also becomes very different in curved space-time.

 

[Fred]

Which is why you shouldn't use tensors, and instead use differential forms. B/c keeping track of indices is perhaps the most god awful annoying thing ever. I still have flashbacks to computing the full Schwarschild metric from the field equations, and then going back to check and recheck the Riemann tensor, the Ricci tensor and so forth. Horrendous

Oh and in many ways differential forms are more fundamental anyway and of course far more easy to use (not to mention allowing powerful topological methods)

 

[K.I.L.E.R]

Vector calculus covers all this?

 

[NRP]

I agree with Chal and Fred.

Tensor algebra is a bitch. Tensor calculus is . . . a bigger bitch.

Hell, just coming to grips with indicial notation is hard (at least it was for me).

 

[Bigus Dickus]

Oh god, I only got a taste of tensors during my applied elasticity graduate course. I honestly don't know if they are just "useful" for solving elasticity problems or required, but in either case I wasn't too fond of them. Good thing that was a closed-end graduate course with no reason to actually remember how to use tensors afterwards.

 

[KimB]

I think they're rather nice for handling a numer of computations where you inherently have to keep track of a large number of parameters. But that doesn't mean they're easy to deal with.

 

[K.I.L.E.R]

What's better:
Multivariate claculus or tensors?

 

[KimB]

What's better:
Multivariate claculus or tensors?

Better? Well, differential geometry is a superset of multivariate calculus. So it's certainly best to learn multivariate calculus first, because it's much more intuitive and visual.

But once you want to deal with curved space-time, you have to move to differential geometry.

 

[K.I.L.E.R]

I can't believe that I cannot even make a simple game dealing with real life stuff.
My game is stalled until I learn something that I can stick from real life into my game. It has to be exact science and not the stupid approximations that everyone else uses.

Maybe Computer Science is the wrong course for me?

Thanks for the help Chalnoth.

 

[KimB]

It's just not realistic to have uber-realistic simulations in a real-time game. Physicists can't even get numerical simulations run on supercomputers to agree with one another very well.

That said, if you want to do some fairly-realistic simulation that's within reach but still fairly challenging, look into fluid dynamics.

 

[Sobek]

Volume of a Torus?

http://whistleralley.com/torus/torus.htm

 

[Davros]

I dissagree with most people here :
you dont need complex maths or physics - just a lot of lego

 

[nthd]

Roger Penrose's The Road to Reality: A Complete Guide to the Physical Universe should give you some ideas. Try doing the exercises.