3,607 reputation
1411
bio website
location
age
visits member for 2 years, 1 month
seen 4 hours ago

1d
revised Do there exist functions $\phi$ and $A$ such that $\vec E$ satisfies the Helmholtz Theorem $\vec E = -\nabla \phi + \nabla \times \vec A$?
fixed some math about the symbol for reals
2d
comment What do people actually mean by “rolling without slipping”?
@NeuroFuzzy Interesting. I suspect this comment section isn't the place for this discussion, though. Perhaps you could ask a question about rolling on curved surfaces, present your work, and ask for references that might verify or refute your idea? I'd be happy to follow that question if you link to it here.
2d
comment What do people actually mean by “rolling without slipping”?
@NeuroFuzzy Sure, if you define the overall travel of the wheel with respect to the center of the wheel. If instead you measure the overall distance traveled by whatever point is in contact with the ground, I think the bold statement still holds. Regardless, travel of a wheel along a curved surface is more involved, and I think it's good you point that out.
Nov
7
comment Can Gauss' and Ampere's Laws be written in terms of the divergence of an energy four-vector?
Are you not familiar with the special relativistic form of the EM fields, the faraday bivector $F$? Of how Maxwell's equations in vacuum can be written $\nabla \wedge F = -\mu_0 J$ and $\nabla \cdot F = 0$?
Oct
29
comment Is $E^2=(mc^2)^2+(pc)^2$ correct, or is $E=mc^2$ the correct one?
@user929304 It's a purely geometrical phenomenon. It'd be like rotating a vector and asking it to trace out an ellipse instead of a circle. In Minkowski space, boosting a four-velocity traces out a hyperbola. Hyperbolas have asymptotes--lightlike rays, in this case.
Oct
19
awarded  Yearling
Sep
25
comment Four-momentum, four-velocity, energy
@Kennan You just wrote $\eta_{ab} = e_{\hat a} \cdot e_{\hat b}$. Clearly the $p^{\hat 0}$ term comes from $p^{\hat 0} e_{\hat 0} \cdot e_{\hat 0} = p^{\hat 0} \eta_{\hat 0 \hat 0}$. What is the value of that metric component, $\eta_{\hat 0 \hat 0}$?
Sep
25
comment Four-momentum, four-velocity, energy
What's the signature of your metric?
Aug
29
answered When can I use $\wedge$ instead of curl?
Aug
23
comment Why does the second Weyl scalar describe electromagnetic radiation?
I could interpret that to mean something like telling you about the total mass causing the curvature (see the Schwarzschild case, for instance, which is characterized only by the mass), the way the Coulomb field is characterized by only the total charge of the point charge.
Aug
9
answered Is relative velocity invariant under special relativity?
Jul
24
comment Gradient is covariant or contravariant?
$\nabla \varphi = \sum_i g^i \partial_i \varphi$, where each $g^i$ is a cotangent basis vector. $g^i \cdot g_i = 1$ always, while $g^i \cdot g^j = g^{ij}$ gives you metrical components.
Jul
24
comment Gradient is covariant or contravariant?
Obviously you can expand any such quantity in terms of either the tangent basis or the cotangent basis, but expanding the quantity $\nabla \varphi$ that I defined above would give you metrical terms if you expanded in terms of the tangent basis--it is naturally written in terms of the cotangent basis.
Jul
23
comment Gradient is covariant or contravariant?
Sorry, I don't understand what you're asking me to do.
Jul
22
comment Gradient is covariant or contravariant?
The Cartesian bases are the same whether they're covariant or contravariant, so I don't see how you can draw a conclusion from that.
Jul
21
answered Gradient is covariant or contravariant?
Jul
21
comment Metric tensor in special and general relativity
It's common in differential geometry to identify vectors with directional derivatives. It's kind of a definition of last resort: if your manifold were a vector space, you could take $\partial \vec x/\partial x^\alpha$ and get a vector because $\vec x$ is an element of a vector space. When positions are no longer elements of vector spaces, that notion breaks down. The directional derivatives themselves still obey the vector space structure, though.
Jul
5
revised Tensors in special relativity
some subject-verb agreement
Jul
5
answered Tensors in special relativity
Jul
3
comment E&M and geometry - a historical perspective
You might also find geometric calculus, based upon clifford algebra, interesting here. It manages to take those free space equations for the EM field and marry them into one equation.