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 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 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 Jan 31 comment Can a metric in General Relativity, Supergravity, String Theory, etc., be asymmetric? @innisfree: the field in question is no longer the metric tensor representing gravity, but the field, from which you need to derive all other objects of the theory; I'd have to look up how you get to the metric (there were approaches where the metric is just the symmetric part, but I don't remember if this particular theory is one of them)