100 years of Relativity

[lylfyl]

I thought today was an appropriate day to post this request, since this is the 100th anniversary of Einstein's paper "On the Electrodynamics of Moving Bodies" in which he introduces Special Relativity. (Wiki link)

I've read various layman explanations of special and general relativity. I've enjoyed Hawking's A Brief History of Time, and I swear that I will get through Briane Greene's The Elegant Universe some day. I figured I've read the layman's versions enough time and I thought I'd actually try looking at the math.

Ha.

Can anyone recommend a primer for the Math behind it?

The Lorentz contraction and relativistic gamma I can follow, but I have no idea how the Lorentz transformation works.
Tried looking up four-vectors on Mathworld, but my head exploded. Relatively speaking. . .

 

[Ziggurat]

The Lorentz contraction and relativistic gamma I can follow, but I have no idea how the Lorentz transformation works.
Tried looking up four-vectors on Mathworld, but my head exploded. Relatively speaking. . .

Ouch. Stay away from that. There's no reason to start using that index notation unless you need to work with tensors (generalized matrices), which you need for general relativity but not for special relativity.

Here's what you need to know about a four-vector: it's a vector with three components for space and one component for time. A position four-vector is the coordinates of an event in space-time (it's position and the time it occured - with time multiplied by c to get the units the same). A momentum four-vector has the momentum of the object for the space components and the energy (kinetic PLUS rest energy) as the time component. The reason four-vectors are usefull is because the length of a four-vector is always preserved by Lorenz transformations.

What is the length of a four-vector? Well, the length of an ordinary Euclidean three-dimensional vector is sqrt(x^2 + y^2 + z^2). We can also write it as
s^2 = x^2 + y^2 + z^2
where s is the length. Now for a four-vector, we have:
s^2 = x^2 + y^2 + z^2 - (ct)^2.
This is the metric for flat Minkowski space: notice the minus sign. Everything in special relativity comes back to that one equation. Master that, and you've got special relativity. Of course, that's easier said than done, because while the equation is simple, it is not intuitive, and our intuition often gets in the way.

Now, going back to Euclidean geometry for a second, consider: what are the transformations that preserve distance in Euclidean space (let's ignore translations and restrict ourselves to ones which maintain the origin)? Well, you can invert an axis, but that isn't something you can do continuously. The only class of transformations that preserve distance for Euclidean space are rotations. In Newtonian mechanics, the laws of physics must be rotationally invariant. Now, you can also ask that same question about our four-dimensional metric, and the answer to that is Lorenz transformations. So changing from one reference frame to another is basically just a kind of rotation is this strange non-Euclidean four-dimensional space.

Special relativity is basically just the claim that the laws of physics have to be the same under this new kind of rotation. For example, it is the momentum four-vector of a system, which contains both the momentum and energy of the system, which is conserved, NOT the Newtonian momentum and energy (although they are equivalent in the low-speed limit). Figuring out how to make the various laws of physics Lorenz-invariant can be tricky, but that is still all special relativity consists of.

 

[lylfyl]

Your explanation was helpful. Thank you. Looks like I should be brushing up on Euclidean geometry before I'll get my head completely around this. Baby steps. Baby steps.