Recent content by t_r_theta_phi

Many sources give explanations of the Riemann tensor that involve parallel transporting a vector around a loop and finding its deviation when it returns. They then show that this same tensor can be derived by taking the commutator of second covariant derivatives. Is there a way to understand why...

For any given polytope, imagine we know the information about its structure but not any of the actual numbers of vertices, edges, faces, etc. Let's say we want to find the number of type A elements (could be vertices, cells, anything) in terms of the number of type B elements. This is given by...

I see. So 6 vertices and 6 edges can correspond to different structures, so it is not possible to differentiate between the them based on numbers of elements alone. As you mentioned in your example, for the triangular tiling there exists a tiling with double the number of elements that also...

Is there a way to figure out if a 4-polytope is a tiling of the 3-sphere based on only the number of vertices, edges, faces, and cells? Here is a specific example. Say that there are two "polytopes" - one is a tesseract, and one is a disjoint union of two tesseracts. Both will have an Euler...

Let's say there are two 1/2 spin particles, one in state 1/√2 |up> + 1/√2 |down> and the other in the state - 1/√2 |up> - 1/√2 |down> Both particles then have an equal chance of being measured to be in either the up or down states. Is there any physical difference between these or are they...

Light does not always appear to travel at c in curved space-time. To an observer far from the black hole, a beam of light appears to travel slower near the black hole, and its observed velocity approaches zero at the event horizon. So light escaping from just outside the horizon will appear so...

That makes much more sense. Thank you

I am currently reading Gravitational Curvature by Theodore Frankel. In the derivation of Einstein's equations in chapter 3, he states that the gravitational potential energy of a blob of fluid is ∫B½p0U√gVdx where the integral is a volume integral, p0 is the rest energy density and √gvdx is...

What is the metric for the spacetime around an infinitely thin, infinitely long, uniform rod? Could it be written in the form ds2 = A(r)dt2 + B(r)dr2 + C(r)dh2 + r2dθ2 where h is the coordinate along the rod and r is the radial coordinate, or would it be something more complicated?