How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while...
z and z* are the same single complex variable, one the conjugate of the other. I'm wondering what would happen if we gave the plane of the single complex coordinate z a metric? Essentially, break up z into it's component parts x and y and give the x-y plane a non-Euclidean metric. If we...
Are there any scenarios where the Cauchy-Riemann equations aren't true? And if so, would there really be any difference between C^1 and R^2 in those cases?
The function: f(z, z*) = z* z doesn't solve the Cauchy-Riemann equations yet I would think is quite useful.
Couldn't we add a metric...
Suppose you have two people who are in identical orbits around a large star. The only difference between them is the direction they are traveling. At a certain point where they meet ( there are two such points ), they sync clocks.
After a few orbits, they meet again and again compare clocks...
I made a mistake in my question. I know that functions have an unlimited number of degrees of freedom, I meant to ask how many functions are involved in a generic coordinate transform in R^4. My guess is 16, since there are two indexes in the transformation matrix, each running over 4 values...
In general relativity, what are the total number of unknowns for a generic coordinate transform? Is it just 4 * 4 = 16? Is there a way to break those down into combinations of types, such as boosts, rotations, reflections (parity?), etc, or is it just left wide open from an interpretive...
Ok cool. Are there any decent references you know of so I can read about this? Surely the closed nature of the space would manifest somewhere in the equations...
If we assumed an empty space, but also assumed space dimensions are closed ( repeat after some distance D ), what would the metric tensor look like? Is this just equivalent to a space with a constant curvature R? If so, how does R relate to D? Would the time dimension also necessarily be...
Is the Dirac Equation generally covariant and if not, what is the accepted version that is?
For general coordinate changes beyond just Lorentz, how do spinous transform?
Sure that would make sense since curvature means the space is filled with non-zero energy ( Energy Momentum tensor non-zero ). That, I would guess, acts like an interaction and thus particles can be created as the interaction occurs.
Yes, and like I said, it's not relevant to my question. Thank you however for the commentary. I'll make a note to not use relativistic mass anymore.
Semantics. Ok, sounds like the answer to my question is that particle number in invariant under coordinate change. So, despite the extra...
Yes, it happens in one particular coordinate system. I never said it happened in all coordinate systems. This particular coordinate system is used, however, to show what the event horizon is and to do an easy calculation for it. It is HARDLY just an "artifact".
Fair enough, but I think it...
Well, actually that's not true. m changes when it's in motion, as does E. Regardless of that, you're not answering my question. Since the energy of the particle in the moving reference frame is significantly higher, do particles appear? Is the particle number conserved under coordinate...
Because E=mc^2, when we increase the energy of a system we can introduce new particles. What about accomplishing this simply via a change of coordinates via Special (or even General) Relativity?
If I have a system of one electron sitting still (E = E0), then I change to a coordinate system...
The event horizon is a place where the metric tensor contains an infinity. Thus, there are no null geodesics (light paths) that cross this "line". This gets quite sticky, as infinities pose a whole host of mathematical problems that most physicists just choose to ignore (not all). I am of the...