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seen Mar 14 at 14:18

Mar
2
comment Relation between Vector space $V$ and its dual $V^{*}$
@Ben: well, if you do not make the distinction between row- and column vectors, you can do that, but I'd advise against it (vectors are traditionally represented as column vectors, and covectors as row vectors, ie $n\times 1$ and $1\times n$ matrices)
Mar
2
comment Relation between Vector space $V$ and its dual $V^{*}$
@Ben: personally, I thinks this is misleading, in particular the Riemannian case, both in terms of differential geometry (vectors are equivalence classes of curves $\mathbb R\to M$, whereas covectors are equivalence classes of functions $M\to \mathbb R$) and physics (velocities vs momenta)
Mar
2
comment Relation between Vector space $V$ and its dual $V^{*}$
@Ben: it's abuse of notation, and I do not believe I've seen any literature going quite that far; the basic idea is that given a non-degenerate bilinear form (resp. a metric tensor in case of (pseudo-)Riemannian geometry), $V$ and $V^*$ become canonically isomorphic and can be 'identified'; if you prefer, you could introduce a new space $V^\blacktriangle$ isomorphic to both, and a single geometric object from $V^\blacktriangle$ can be represented by elements of $V$ and $V^*$ both; the culmination of that idea is that there are only (4-)vectors with co- and contravariant components
Feb
26
comment Is the accelerated expansion of the universe consistent with conservation of energy?
quoting said article: “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” [...] There’s nothing incorrect about that way of thinking about it
Feb
25
comment Is it valid to apply Einstein's Relativity to scenarios involving expansion of space?
another relevant paper I just googled: arXiv:0804.3595
Feb
25
comment Is it valid to apply Einstein's Relativity to scenarios involving expansion of space?
@СимонТыран: ultimately, Einstein ;) I do not have any convenient links lying around that make exactly the point I was trying to make, but you could look at arXiv:0808.1081 and meditate over the beautiful graphic in this answer by Pulsar
Feb
25
comment Is it valid to apply Einstein's Relativity to scenarios involving expansion of space?
@СимонТыран: the same thing it does when you cross a black hole's event horizon
Feb
25
comment Is it valid to apply Einstein's Relativity to scenarios involving expansion of space?
@СимонТыран: time will stand still - but not at the Hubble sphere, but the cosmological event horizon
Feb
25
comment Intuition as to why the orientation (of a 3D object) is not a conserved quantity?
@DonHatch: see edit
Feb
16
comment Why does it take a projectile as long to get to its apex as it does to hit the ground?
@DavidHammen: I'm slightly ashamed to admit it, but I actually did have to draw some pictures to convince myself that this is valid reasoning ;)
Feb
15
comment Why does it take a projectile as long to get to its apex as it does to hit the ground?
note to self: verify that ascent and descent actually do not take the same amount of time once you add drag...
Feb
15
comment Does a symmetry necessarily leave the action invariant?
if we want to apply Noether's theorem, the infinitesimal transformation needs to leave the action invariant (up to a divergence term)
Feb
8
comment It's established that universal energy is not constant. But is the net change positive or negative?
in some sense, the total energy of a homogeneous isotropic universe is well-defined: FLRW spacetime comes with a privileged set of observers, making Pirani's expression for energy well-defined; that energy is trivially 0 in any finite volume as the density vanishes identically, yielding a vanishing total energy as limit
Feb
8
comment Why does heat added to a system at a lower temperature cause higher entropy increase?
$\frac QT = \Delta S = \Delta(k \ln\Omega) \approx k \frac 1\Omega \Delta\Omega \Rightarrow \Delta\Omega\approx \Omega \frac Q{kT}$, ie transferring heat into a system opens up a new number of microstates $\Delta\Omega$ proportional to the number of existing ones and the number of degrees of freedom we can excite given by heat $Q$ divided by average energy per degree of freedom $kT$; cf physics.stackexchange.com/questions/33372/…
Feb
5
comment Gravity: Is there curved space besides curved spacetime?
@AlfredCentauri: but I am, and I suspect Moonraker does so as well
Feb
5
comment Gravity: Is there curved space besides curved spacetime?
@AlfredCentauri: I'm talking about general spacetimes. How would you define spatial geometry in a Schwarzschild spacetime? Aribtrary non-homogeneous, non-insotropic spacetimes? Preferred spatial slicings may exist on a case-by-case basis, but in general, I do not believe there's such a thing as spatial geometry
Feb
5
comment Gravity: Is there curved space besides curved spacetime?
@AlfredCentauri: in GR, there isn't really such a thing as spatial geometry - a more appropriate way to think about it is as the matter distribution being layered
Feb
1
comment A question on Lagrangian dynamics an the velocity phase space
I didn't see anything wrong with your description, but I don't have the time to re-read more carefully right now
Feb
1
comment Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?
cf Piron's theorem
Feb
1
comment Can a metric in General Relativity, Supergravity, String Theory, etc., be asymmetric?
it's effectively "usual sort of metric plus an extra field" anyway. Einstein actually addressed that criticism: symmetry of the tensors is non-essential for the formalism; instead, he demands 'transposition symmetry' of the laws and links it to charge conjugation