Recent content by marschmellow

Great answer. Thank you so much!

Not much else to say other than the title. In the Schwarzschild spacetime, the radial coordinate r didn't represent radial distance, but it at least represented the thing that determines the area of a sphere centered on the large mass. It doesn't seem like that interpretation can be given to the...

At some point I'll check to see if I can make your functions equivalent to Wikipedia's, but in the meantime, do you have any idea what's up with Wikipedia? It seems to clearly be wrong. Also, I thought the matrices had to be actual inverses, as in their product is the Kronecker Delta, not the...

Sorry to bump an old thread, but is it possible to diagonalize the metric every where except on the axis of rotation? In other words, is the problem that no general diagonalization exists, or that it is simply a non-diagonalizable matrix? The second seems improbable to me, since the metric is...

Question 1: How does the metric relate to the light cone at a particular point? Obviously, the metric should determine the shape of the light cone, but is the converse true? Does the orientation and width of the light cone tell you everything about the metric? I'm guessing not due to the...

Thanks for your responses! Even if I don't comment on anything specifically, it was all helpful to hear. Thanks, that was really helpful. I just want to ask one follow-up question. Is there a word for a vector that specifies the time derivative of one object using another object's clock? If I...

Hi all, I'm in a modern physics course and am surprised by how confused I am about special relativity. I know multilinear algebra quite well, and so I thought it would be pretty easy, but I'm having some conceptual difficulties. Any help would be greatly appreciated! 1. Why is a special...

Yep, you got it.

I believe it's the Levi-Civita symbol, and it can be made into a tensor density. Permutations will show up in the most random of places, including in real life. It's probably been the so-far most useful topic I learned in algebra.

Okay great that helped clear things up. I still have more questions, however. What does the configuration probability distribution mean? A configuration is a microstate, correct? What does it mean for a state "equilibrium" to have a certain microstate? Don't microstates rapidly change in...

The title says it all. It looks like the configuration probability only depends on where you want to go, not what state you are in now. Yet when I watch simulations, there is clearly a dependence on the previous state. Is there something pretty basic I'm misunderstanding about configuration...

I'm just learning this stuff myself, so please correct me if I'm wrong. The velocity vector is a coordinate-independent object that is an element of the tangent space of the point you're considering. But like all vector spaces, there is not a unique choice of basis--one usually picks the basis...

If sounds like you're thinking of the geometry on a sphere--I've heard that geometry be referred to as "Riemannian" geometry, but I've also heard "Riemannian" refer to a much more general class of geometries. In the geometry on a sphere, the geodesics ("straight" lines) are great circles of the...

Does anyone know exactly what kind of mathematical object a unit (like meters, coulombs, etc.) is? Or what kind of algebraic structure units are elements of?

Thank you for your responses, everybody. They collectively both cleared up some confusion and led me in the right direction for what to read next.