Ask Slashdot: Math Curriculum To Understand General Relativity? 358
First time accepted submitter sjwaste writes "Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade."
Re:A question borne of helplessness... (Score:4, Insightful)
You could have left off the first paragraph and provided an informative response. I was going to post something about MIT's online courseware, too. But you had to preface a useful bit of information with a put-down. Welcome to slashdot where innocent questions are met with derision and insults.
Re:Easier way to learn it (Score:5, Insightful)
+1 on this and all related posts: Relativity is about physics, about beautiful physics, and is not about math.
There are bits of relativity for which Einstein had to go math-shopping: He knew what the physics must look like, he needed to know if the mathematicians had any tools that matched what he wanted to express (they did, Lorentz transformations being one of the most important).
Note: I have a physics degree, which means I have studied more math than anything else. The math is important to express the physics precisely, important to get useful answers to specific questions. But the physics come first. (There's the old trope of the physics prof saying "set C to 1 so you can see the physics happening.)
Read about and try to reproduce Einstein's thought experiments. Start with the one about travelling at the speed of light, and what you would see as you approached C (hint: if you travel at C, photons can only reach you from in front, from along your axis of travel). Think about the "falling in an elevator" experiment. These get you a long way to the principle of equivalence, the principle of relativity, etc.
Only once you have some idea of the physics should you attempt to tackle the math - and by that time, you'll be starting to get a good idea of what the math might look like.
Do not attempt to learn the math first and thereby get to the physics. There lies madness.
Re:Easier way to learn it (Score:4, Insightful)
No, not really. You can get a very superficial understanding of what is going on without any maths, but you are just fooling yourself if you think that understanding is anything remotely like what you understand once you've actually worked with the maths.
Re:What are your goals? (Score:4, Insightful)
If its just a hobby I don't understand why you would want to know the in-depth details since you probably wont be playing with equations most of the time.
On the contrary, if it's a hobby he's probably interested in reading and playing with the various speculative equations for warp drive and time travel - for example, the Alcubierre Drive [wikipedia.org], or Kip Thorne's wormholes [wikipedia.org]. Which has nothing to do with everyday physics, but everything to do with science fiction worldbuilding and geeky entertainment. Certainly that's what I would do if I understood enough of GR to get to the "test the equations" stage.
Re:Easier way to learn it (Score:3, Insightful)
You can not understand the why with math, math only helps you calculating it To. actually understand it you will have to step up one abstraction level to philosophy.
Re:Easier way to learn it (Score:5, Insightful)
But I think it boils down to not only can we not exceed C we can't go slower either. Everything moves at C and the axis of that motion we perceive as time. And everything else we call reality is the contortions required to make that so under all circumstances.
Sir –
I wouldn't quite describe it that way, from the perspective of the epiphany Einstein must have had. I don't think it's that complex, and in any case I think it's more beautiful than that. As a matter of interest, perhaps someone will find the following worth reading.
We have space, and it's where we live. This space is physical but can be represented by representations in our brains and on various media, which representations we call physics.
We make rules in physics to reflect what happens in our space, our reality. Some rules we can see, and they are generally intuitive. For example: Two points - places - are distinct when not the same position, and these points are indivisible (identity). Also, two lines added together make a third line, regardless of the order those lines are added in (commutativity). Three lines can be added in any order to equal the same distance (associativity). Two lines never meet (parallelism). This is the Euclidian space [wikipedia.org], and applying such to our universe is Newtonian physics (aka classical physics).
Suppose though that the physical world in which we live is not Euclidean, contrary to our observations and intuition. Suppose in this world parallel lines in our world meet at infinity. We can call this a Lobachevsky space [wikipedia.org] (also known as a hyperbolic geometry), and its principles formed the essential breakthrough in general relativity.
Once one accepts as axiomatic that we live in a Lobachevskian space, the acceleration of mass becomes governed (for reasons beyond the scope of this note) - otherwise we would violate other rules (e.g. identity). Hence the perception of time slows in lieu of infinite acceleration (imagine two trains travelling at the speed of light towards each other; to each other they would appear to be travelling only at the speed of light - not, as one might expect, twice the speed of light - because time relative to each other slows; contrast a stationary that expects both to pass at the speed of light in opposite directions). This effect is observed and compensated for in our Global Positioning System [ohio-state.edu].
All to say, by changing our perspective from representing our accepted physical world as a Euclidean geometry to something unintuitive, a Lobachevskian geometry, we arrive at the ability to represent and predict what happens in our physical world.
The consequences inherent to the axiomatic perspective of living in Lobachevskian space are commonly and collectively referred to as "general relativity", and they are non-trivial. The underlying premise that commenced that perspective is itself quite simple.
Re:Easier way to learn it (Score:5, Insightful)
Even Einstein himself never claimed to understand the why of GR. GR is all about the math. The vague analogies sometimes bandied about aren't science. They are flights of fancy and completely unproven and were only ever used to try to explain the math to people who didn't understand the long tensor calculus equations. The math itself is the science. There is no way around the equations. GR cannot be explained with natural language. Only with mathematics.
Re:Easier way to learn it (Score:5, Insightful)
Unless you can work out on your own how to put numbers on it, your understanding is imperfect. Being able to run some numbers through an equation doesn't mean you understand it even as well as the guy who doesn't know any maths but knows where to stand to catch the ball.