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answered Why is Entropy's Definition Useful?
Jul
21
comment What is “Energy” of a vacuum in the context of quantum theory?
The problem with 'true nothing' is that it is unphysical: Take a volume of space and remove 'everything', and you won't end up with 'nothing', but with the vacuum, which, philosophically speaking, is a 'something'. Krauss goes a step beyond the vacuum and assumes that spacetime as well as the laws of physics as we know them emerge dynamically (eg via symmetry breaking) from the true groundstate of a theory of quantum gravity. This groundstate is what he calls 'nothing'.
Jul
16
comment Hamiltonian Noether's theorem in classical mechanics
the key word to look for would be 'moment map'; see eg ncatlab.org/nlab/show/…
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
But, but, but: [ insert rant about Noether's 2nd theorem, general covariance and the gravitational stress-energy pseudo-tensor ]
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
@brightmagus: the gravitational potential fixes conservation of energy in curved space-time in a similar way to how the centrifugal potential fixes energy conservation in rotating frames of reference; in the latter case, it all works out because there's a globally inertial frame in which energy conservation holds, in the former case, it works out because energy conservation holds trivially in any generally covariant theory; sorry, but that's the best I can do...
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
@brightmagus, it's hard to argue with forces in GR: if you think of dark energy due to a cc as some fluid, you have to wonder why gravitational attraction acts repulsively if you choose the right equation of state (negative pressure), without any gradients being present; I'm unsure what kind of explanation you're looking for beyond "that's what Noether's theorem tells us"; essentially, energy conservation boils down to the first Friedmann equation $\rho+\rho_\Lambda-3\dot a^2/8\pi Ga^2=0$, which many physicists are uncomfortable with, claiming it's not a proper conservation law
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
@brightmagus: Noether's second theorem applies to general relativity and yields a 'trivial' law of energy conservation, where the contribution due to the cosmological constant gets cancelled by the negative potential energy of the gravitational field; this law of conservation of energy can be expressed in terms of a gravitational stress-energy pseudo-tensor, but there's no tensorial expression for gravitational stress-energy as general relativity does not differentiate between gravity and inertia
Jul
12
answered If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
Jul
8
comment What kind of manifold can be the phase space of a Hamiltonian system?
topological restrictions aren't placed by authors, but by existence of a symplectic form; Wikipedia lists orientability and non-trivial de-Rham cohomology $H^2(M)$ (ruling out all spheres except the 2-sphere)
Jul
8
comment Why does a flat universe imply an infinite universe?
For example the surface of a torus is flat but finite slight correction: the surface of a torus can be flat, but the generic 2-dimensional torus embedded in $\mathbb R^3$ isn't; Wikipedia informs me that it cannot be flat if the embedding is at least $\mathcal C^2$, but an explicit $\mathcal C^1$ embedding has been found rather recently (April 2012); see eg gipsa-lab.fr/~francis.lazarus/Hevea/Presse/index-en.html for some pictures
Jul
5
comment Equivalence between Hamiltonian and Lagrangian Mechanics
so we just vary $v,p$ independently and do the integration by parts on $\delta\dot q$ as usual to end up with $\delta S = \int(\dot q-v)\delta p + (\partial L/\partial q - \mathrm dp/\mathrm dt)\delta q + (\partial L/\partial v - p)\delta v + \text{boundary terms}$; I'll have to meditate on that for a bit...
Jul
5
comment Equivalence between Hamiltonian and Lagrangian Mechanics
note that due to its dependence on $\dot q^i$, $L_E$ is not a function on some (extended) phase space, but a functional(?!)
Jul
5
answered What predictions can a quantum gravity theory make?
Jul
4
comment What predictions can a quantum gravity theory make?
Why no mention of BICEP2 - or do you expect their discovery to turn to dust?
Jul
4
comment What are the alternative theories of dark energy? ($w \neq -1$)
Hajdukovic tries to explain dark matter and energy both by gravitational vacuum polarization due to repulsion between virtual matter and anti-matter particles; in arXiv:1201.4594, he calculates a current $w_\text{eff}\approx -0.99$, which will eventually approach $w_\text{eff}=-1/3$
Jul
3
comment Inflation in a closed universe or a stage 1 multiverse?
Just having edited the question, perhaps going with ininite and finite instead of open/non-compact and closed/compact would be even clearer?
Jul
3
revised Inflation in a closed universe or a stage 1 multiverse?
better link, improve terminology: flat -> open
Jul
3
comment Inflation in a closed universe or a stage 1 multiverse?
the difference between infinite and finite but very large is extremely marked as far as Tegmark's musings go: if I remember the one talk I watched about his stuff correctly, he thinks about identifying Everett's (or rather DeWitt's) many worlds with causally disconnected parts of an inflationary universe, which only works if the multiverse is big enough to realize all branches of the (no-longer-quite universal) wavefunction
Jul
3
comment E&M and geometry - a historical perspective
@gn0m0n: sure, no need for gauge theory just to use differential forms; but the question (or at least its title) was concerned with EM and geometry - and geometrically, vector potential and field strength are not just some arbitrary forms, but principal connection and corresponding curvature
Jul
3
answered E&M and geometry - a historical perspective