3,468 reputation
725
bio website williewong.wordpress.com
location
age
visits member for 3 years, 8 months
seen 17 hours ago

Mathematician


Jan
9
comment Einstein's equations as a Dirichlet boundary problem
"Einstein's equation can be recovered by minimizing the Einstein-Hilbert action" false! We look for critical points, but they don't have to be (local) extrema.
Dec
13
awarded  Yearling
Dec
13
comment How is it possible for astronomers to see something 13B light years away?
@Ernie: please feel free to edit that into my post if you can find a turtle-based version. =)
Dec
10
comment Is Einstein's 1916 General Relativity paper a recommended way to start learning about the subject?
@Eduardo: I disagree with "has almost no GR". The entirety of Part 2 is dedicated to it. Note further that I stated "historical perspective", which is in respect to what mtrencseni wrote above "What problem was he solving, where did he come from, what was the thinking at the time..." None of which requires mathematics.
Nov
9
revised Expectation values of $(x,y,z)$ in the $|n\ell m\rangle$ state of hydrogen?
replaced > with \rangle in mathmode
Nov
7
comment Kerr geodesics differential equations in equatorial plane
I don't have the book handy at the moment, but you should (at least, that's where I would) check Chapter 4 of B. O'Neill's The Geometry of Kerr Black Holes
Nov
5
comment Equation of the saddle-like surface with constant negative curvature?
... These embeddings are very wrinkly. For example, if I were to give you a piece of news print and tell you to put it inside an Altoids tin without creasing it, you will find it very difficult. But if I allow you to fold it (or even cut it), you will find it a bit more possible. (But perhaps still difficult due to the physical thickness of the paper.) Basically, by allowing discontinuities of the first or second derivatives (and throwing away the notion of curvature) one gets a lot more freedom in being able to "squeeze surfaces into tight spaces", so to speak.
Nov
5
comment Equation of the saddle-like surface with constant negative curvature?
@Leos: note that I specified smooth as in twice continuously differentiable. This is the least amount of regularity that is required to make sense of curvature. If you instead consider hyperbolic geometry on the level of geodesic properties (which require only one continuous derivative), you can get the Kuiper construction (and later vastly extended by John Nash for many other manifolds). These embeddings, however, do not satisfy your requirement of constant negative curvature, as curvature cannot be defined by the classical formula...
Oct
31
comment Equation of the saddle-like surface with constant negative curvature?
@Leos: yes. The simply connected hyperbolic space $\mathbb{H}^n$ in any dimension $n \geq 2$ has infinite (hyper)volume. In fact the volume enclosed in each geodesic ball is strictly greater than the volume enclosed in the Euclidean ball of the same radius.
Oct
30
comment Equation of the saddle-like surface with constant negative curvature?
As an aside, the saddle is a very poor mental image for a surface of constant Gaussian curvature. Remember that Gauss curvature is the product of the principal curvatures. To have negative curvature one needs to curve "up (positive $z$)" in the $x$ direction and "down" in the $y$ direction. To get constancy the amount of $x$ curvature must be inversely proportional to the amount of $y$ curvature. So once you fixed the ridge of the saddle to be roughly parabolic, you need the front and back end of the saddle to pinch tightly compared to the middle of the saddle to ensure constant curvature.
Oct
30
answered Equation of the saddle-like surface with constant negative curvature?
Oct
30
comment Equation of the saddle-like surface with constant negative curvature?
The edit looks good. I'll comment a bit more on the Euclidean case.
Oct
26
answered Compactness of spacetime: experiment and math
Oct
19
answered Why has the trace of the energy-momentum tensor to vanish for conserved scaling currents to exist?
Oct
17
awarded  Custodian
Oct
17
reviewed Approve suggested edit on Distribution of charge on a hollow metal sphere
Oct
17
comment Einstein tensor in Friedmann equations : where is the missing $c^2$?
Now, whether there are the right numbers of $c^2$ floating around is a different issue; I haven't checked the computations on Wiki myself so can't say anything. But because of the identification of space and time, one cannot use dimensional analysis to check whether the number of $c$s are correct, since $c$ is essentially dimension free.
Oct
17
answered Einstein tensor in Friedmann equations : where is the missing $c^2$?
Oct
16
answered What is the spatial derivative of strain tensor?
Oct
15
comment Killing vector fields
Consider the Lagrangian functional evaluated among the "functions" (in this case the spacetime metric and manifold) that admit a certain symmetry. Not all symmetries can be easily captured at the level of initial conditions (search arXiv for Killing initial data to see related current research).