alsodon't think you'd get an acoustic hole out -- it relies on the equations being hyperbolic. the schroedinger equation is hyperbolic but i don't think the dirac equation is.
i may very well, however, be dead wrong because it's now about seven years since i looked at the classification of differential equations. if the dirac equation *is* hyperbolic then it's possible we'd get an acoustic hole. i may look into that actually. it'll while away the long winter evenings...
(i may very well, however, be dead wrong because it's now about seven years since i looked at the classification of differential equations. if the dirac equation *is* hyperbolic then it's possible we'd get an acoustic hole. i may look into that actually. it'll while away the long winter evenings...
boris the spider,
Tue 8 Dec 2009, 21:25,
archived)
well as I say,the Schrodinger equation IS the Dirac equation in the non-relativistic limit, so any solution the former has, the latter ought to have as well. I don't know how you'd get a self-interaction term to work though, but there must be some Lorentz covariant form of it, because if it isn't Lorentz covariant, it can't be true, otherwise Special Relativity isn't true.
(
Moon Girl Technologies horrendous beanbag,
Tue 8 Dec 2009, 21:32,
archived)
Doesn't necessarily followThing is the acoustic hole comes along because of the behaviour of fluctuations; at low energies this is fine, but at high energies there's no immediate reason to believe that there isn't something influencing the fluctuations (indeed, there will be) that changes their behaviour enough that now they don't look like they act on a Schwarzschild spacetime. I'd not be surprised to find that you can't write an effective spacetime at all for the high-energy general case, and only at low energies.
The self-interaction term is just coming from a Taylor expansion of the Gibbs free energy. Or it may be the Helmholtz free energy. One of the two. It's also actually not quite a Schroedinger equation because that psi isn't a wavefunction so much as the product of the individual wavefunctions of the atoms in the condensate. So if you follow the kind of arguments made for the low-energy case and put it at high energies you could doubtless write a modification to the Dirac Lagrangian that included an interaction. Just that it would be a bit odd to consider a high-energy Bose-Einstein condensate...
(The self-interaction term is just coming from a Taylor expansion of the Gibbs free energy. Or it may be the Helmholtz free energy. One of the two. It's also actually not quite a Schroedinger equation because that psi isn't a wavefunction so much as the product of the individual wavefunctions of the atoms in the condensate. So if you follow the kind of arguments made for the low-energy case and put it at high energies you could doubtless write a modification to the Dirac Lagrangian that included an interaction. Just that it would be a bit odd to consider a high-energy Bose-Einstein condensate...
boris the spider,
Tue 8 Dec 2009, 21:38,
archived)
I think Special Relativity relies on Lorentz covariance,in fact I believe the two are pretty much the same thing. Having said that, the general case is often difficult and not particularly useful. I couldn't honestly tell you if the Dirac equation has any real world applications outside of particle accelerators. It's just, kind of... nice.
( Moon Girl Technologies horrendous beanbag,
Tue 8 Dec 2009, 21:58,
archived)
yeahlorentz invariance underpins special relativity -- and general, since special relativity is recovered from general in the flat limit. there'll certainly be real-world applications of it; it's the equation governing the behaviour of relativistic electrons, and if it doesn't have some application somewhere in semiconductor physics that has a bearing on modern computers i'd be quite surprised...
( boris the spider,
Tue 8 Dec 2009, 22:05,
archived)
who knows what's going to happenwhen they move on to the 32nm process. Even still, electron drift velocities may be very low even though the signal speed approaches the speed of light.
( Moon Girl Technologies horrendous beanbag,
Tue 8 Dec 2009, 22:11,
archived)
very trueeven so, i'd be surprised if they didn't have to take relativity into account at some point. though i should maybe prepare to be surprised.
( boris the spider,
Tue 8 Dec 2009, 22:17,
archived)