Only a masochist would write assembler for modern CPUs, especially for RISC CPUs. The compiler will do a better job of generating optimised assembler than any human could.
@JohnRennie hi, I hope you are alright. In your answer https://physics.stackexchange.com/a/613479/197759 you wrote that the Kerr metric can't be written in a diagonal form. Why not?
@NiharKarve that Cern paper is a bit niche. It describes some unusual metrics and the only reference to the Kerr metric is: Most notably, the Kerr metric, and all its generalizations, cannot be brought into diagonal form in holonomic coordinates due to the fact that its timelike Killing field fails to be hypersurface-orthogonal.
But how do you prove that there is no choice of coordinates in which the Killing field can be made orthogonal? After all, the spacelike hypersurfaces are dependent on the choice of coordinates.
I mean, is that something obviously which normally doesn't need more explanation? For me it would be interesting to figure out why it is not possible. Yes, something like a proof which explain why its not possible
@ACuriousMind I would say that fact should have a proof somewhere somehow
You can prove it can be diagonal by finding the appropriate coordinates, but failure to find suitable coordinates may just mean you haven't looked hard enough.
The only problem is that it almost certainly doesn't describe real black holes inside the event horizon. The solution in the interior region is believed to be unstable to perturbations, though I'm not sure if this has been rigorously proved.
That almost certainly means we cannot jump through a rotating black hole and emerge in another universe - boo! :-)
@undefined @JohnRennie There are plenty of sources that show that the "hypersurface orthogonality" condition is equivalent to the "Frobenius condition" (also just called "integrable" by mathematicians) $\xi \wedge \mathrm{d}\xi = 0$ for the one-form $\xi$ dual to the Killing vector, see e.g heiup.uni-heidelberg.de/reader/download/534/…
you really don't need to go looking for specific coordinates
@antimony In a material the photon interacts with the electrons in the material and becomes entangled with them. So the object propagating in a dielectric is not purely a photon but a mixture of the photon and the electrons.
Where the interaction is strong, as in Bose-Einstein condensates, we get a distinct pseudoparticle called a polariton. For normal materials we probably wouldn't describe the photon-electron interaction as a polariton but rather a slightly perturbed photon.
Anyhow, one effect of the interaction is to give the photon an effective mass so it slows down.
ooh interesting, thanks is it more sensible to view a "new" photon at each interaction? (in which case it is reasonable to think of it travelling in a vacuum in between atoms?) tbh that idea of a vacuum "in between" atoms doesn't feel correct to me, as i could imagine the photon interacts with the electric field of more than one electron at a time?
@ACuriousMind interesting, thanks. the reason i wondered that is because i understood the photon/electron interaction will not be deterministic, therefore i wondered if it is reasonable to view it like a state diagram
maybe i think i need more fundamentals to continue the discussion in a meaningful way
@ACuriousMind one day this will blow my mind because I almost can't stop thinking about this. And I can't get rid of the thought in my head that it still matters to ask whether its a "new" photon or not. I know, its kinda meaningless, because one is like the other.
if you like, you can think of the photon being slowed down by being constantly absorbed and re-emitted with a slightly different direction (like a ball bouncing off of posts in its way), but to what extent this is a useful picture for anything other than pretending we "understand" what's going on it questionable
@undefined Just wait until we have Star Trek-like beamers and you will have to ask if the "you" stepping out of the transporter at the other end is the same "you" that stepped into it ;)
do you think its useful to imagine the photon always travels at the speed of light, but the absorbtion and re-emission process is what creates the delay equal to the reduced velocity of a material refractive index > 1 ?
tbh i didn't feel happy with this idea of photon "always travels at the speed of light" and i read it in a physics stack answer which was downvoted, but it said many things so i wasn't sure which thing was being downvoted
@satan29 Classically it works this way. The oscillating electric field of the light makes the electrons in the dielectric oscillate, then those oscillating charges re-emit an EM field.
The messier concept that John alluded to - that the thing that propagates is not really "the photon" but a messy disturbance made out of a photon and many electrons that bears some resemblance to a free photon - is closer to what the QM formalism really computes when we analyze this state
tbh i'm quite happy to have different layers/models with different approaches
just to find a coherent way to move between them
the mind bender for me was when i read that the scattering angle is not deterministic.
and eg. a ray's refraction angle (re. Snell's law), could be considered the constructive superposition of all possible scattering angle states <-- if i interpreted that QM idea properly
My lecturer said we can raise and lower indices only on tensorial object. Like it is not correct to lower an index on total derivative of a velocity field wrt to tau because it isn’t a tensor.....Is this right?
If what he said is right why do we lower an index on the Christoeffel symbol which are not tensors
we shouldn’t raise or lower index on non tensors (as Caroll asks not to lower indices on Christoffel symbols)....But in Dirac’s book, Dirac has shown that lowering indices on tensors can be done just like that for tensors ...(I don’t think how that make sense, it looks like an abuse of notation unless someone here can clarify what is correct and what is not)
If I lower index on the total derivative of velocity field (not the absolute derivative), I will get the wrong geodesic equation (for contravariant velocity field)...raising and lowering indices on such non tensors will give such contradictions. So I assume it should not be done
I’m still confused about a post I wrote in the chat some days ago regarding the field of a moving point charge (constant velocity). First it seemed natural that the field would be pointing along the current position of the charge, however upon closer thought it seems it should be along the retarded position instead since that is the field that one observes...
Maybe there is no intuition behind it. It shouldn’t violate causality since the charge is moving at a constant velocity and there is no causality to begin with, i.e. changes in velocity.
