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...
Are there any good papers or books that go over our current understanding of differential geometry for 2-dimensional complex spaces? Hermitian vs anti-symmetric metric tensors, dealing with complex conjugates, and defining affine connections?
Yes, I've already hit up Google, so I was hoping...
I am working on some theoretical ideas of my own and am looking for people who would be willing to help work on some of these ideas. Finding the right people can be tough, since many of you already have your own work to deal with and wouldn't necessarily be open to someone else's ideas, but if...
If we assume:
E = mc^{2}
and for photons:
E = hv
Then we can derive an effective mass:
m = \frac{hv}{c^{2}}
And using simple classical gravity obtain:
hv - \frac{GMm}{r} = hv - \frac{GMhv}{c^{2}r} = Constant
You can derive the constant by evaluating the equation above at...
First is right, second is only half right. GR is an extension of the principal of relativity to all frame, not just inertial. This is accomplished by replacing all equations with tensor equations. It was via thought experiment that once can conclude gravity = warped space-time if you assume...
I know that GR deals exclusively with tensors (at least, in every book I have), but how does the same concepts of affine connection etc extend to non-tensor entities? Example would be a spinor, or even a mix of spinor / tensor. Are there different affine connections? Something different yet...
If a certain space-time region has a constant curvature (caused by, say, an even distribution of energy over the region) how would radiation be effected by the curvature? Would it create a red-shift / blue-shift as the radiation moved through the region or would it be un-effected?
Has anyone...
As light moves through a dense material, it would interact with the atoms. Would the light lose energy, on average, in the process? As a result, would we observe a red-shift as it moves through?
If so, what is the relationship between the amount of red-shift (energy lost) to the distance of...
The covariant derivative is different in form for different tensors, depending on their rank.
What about other mathematical entities? The electromagnetic field A is a vector, but it has complex values. Is the covariant derivative different for complex valued vectors? And what about...
Thank you. This makes complete sense. Donuts really just a flat plane like Euclid had, but a different formula for distances. Couldn't you transform one to the other by making time real / vs imaginary?
Ok. Thank you all for your help. One thing I was thinking about was to take the proper time and use that as a "third" dimension so I could visualize things. If you do that, then eucledian space is just a cone. z^2 = x^2 + y^2. Minkowski space would just be the curve z^2 = x^2 - y^2.
So, suppose for visualization there are only two dimensions: ct and x. Now if the metric where Euclidean, we could visualize this space is a simple plane.
What would be the shape of the "plane" when the metric is +1, -1 (Minkowski)?
Is it somehow hyperbolic?
Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)?
I know that it's coordinate dependent, so suppose you have two coordinates: with one being like "latitude", 0 at the bottom pole, and 2R at the northern pole, and the other being like...
Ben
Thank you for the reply. I am aware of what you are talking about, but honestly I'm not concerned with creating a "real only" metric. I guess my question is more of a pure mathematical question than physics, and i was hoping that someone out here could point me to some references that...