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...
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...
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...
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?
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...
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...
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...
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...
Hello,
I am trying to understand what the differences would be in replacing the symmetry equation:
g_mn = g_nm
with the Hermitian version:
g_mn = (g_nm)*
In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about...
I have read that the electric and magnetic fields are always "perpendicular". Is that true? And if so, does that mean the inner product of the two vectors is zero?
E_x * B_x + E_y * B_y + E_z * B_z = 0 ?
Also, is there any special meaning in electrodynamics to the quantity:
| B |^2...
If the universe is filled with matter, and that matter causes space-time to bend, wouldn't the over-all structure be closed? Meaning, if I fly off in some random direction I would eventually "wrap around" the universe like a person moving across the surface of a sphere?
If so, is this...
I would like to explore writing differential geometry in matrix format and was wondering if any of the experts here knows a good resource for that? I have tried Google and can't find anything definitive.
Thanks in advance!
The electron pairs, acting as bosons, all fall to the lowest energy state, and can't get enough energy (under normal operating conditions), to make the quantum jump to the next energy level, hence as they move through the conductor they don't lose energy. That is, in a nutshell, my assumptions...
Are there any good references out there for writing the equations of GR in matrix format? For example:
ds^2 = g_mn dx_m dx_n -> ds^2 = dx+ g dx
where the matrix version of g_mn (g) would be hermitian, dx+ is the conjugate...
covariant derivative:
Y_n||m = dY_n/dx_m - {n, km}...
I have a question regarding gauge invariance. When a charged field changes phase:
y -> e^it * y
The electromagnetic field adjusts to make the equations work:
A_m -> A_m - idt / dx_m
What I don't understand is why, purely from a physics standpoint, this would happen? That is, is...
Someone here once said to me, via post, that "any compact spacetime must have closed timelike curves". Are there any good references out there on why that is / how that is derived?
As an after thought...
Isn't it true that a particle traveling in one direction in time is equivalent to its...
The analogy always used is to draw two points on the surface of a balloon and then blow air in the balloon. The points move away as the balloon expands. The issue I have with this is: now draw a "meter stick" on the surface of the balloon. It expands too, at the same rate, so that the number...
I am looking for good reading material and references on something. I have tried the google route and can't find anything so I thought I would ask the community of people who know...
I want to learn more about the following scenario: Suppose I start with a 1 dimensional complex space. I want...
I am looking for good reading material and references on something. I have tried the google route and can't find anything so I thought I would ask the community of people who know...
I want to learn more about the following scenario: Suppose I start with a 1 dimensional complex space. I...