On Saturday, May 9, 2020 at 7:00:20 AM UTC-5, Alan Grayson wrote: > > > > On Saturday, May 9, 2020 at 5:49:07 AM UTC-6, Lawrence Crowell wrote: >> >> On Saturday, May 9, 2020 at 6:22:44 AM UTC-5, Alan Grayson wrote: >>> >>> >>> >>> On Saturday, May 9, 2020 at 4:58:20 AM UTC-6, Lawrence Crowell wrote: >>>> >>>> On Friday, May 8, 2020 at 9:28:02 PM UTC-5, Alan Grayson wrote: >>>>> >>>>> >>>>> >>>>> On Friday, May 8, 2020 at 4:50:55 PM UTC-6, Lawrence Crowell wrote: >>>>>> >>>>>> There is no global meaning to energy conservation. There is the >>>>>> Hamiltonian constraint Nℌ = 0, which just says that for a local region >>>>>> where the lapse function can be parallel translated to the energy is >>>>>> zero. >>>>>> This is for Petrov type O solutions >>>>>> >>>>>> Nℌ = 0 = ½(da/dt)^2 - 4πGρa^2/3c^3 - k >>>>>> >>>>>> Which leads to the FLRW constraint >>>>>> >>>>>> (a’/a)^2 = 8πGρ/3c^3 – k/a^2 a’ = da/dt >>>>>> >>>>>> Where the Hubble parameter H = 70km/s-Mpc is (a’/a)^2 = H^2. This >>>>>> means that in a local region we have energy conservation FAPP. The >>>>>> difficulty is this is not a property of the symmetries of the system so >>>>>> we >>>>>> have no way to extend this globally. >>>>>> >>>>>> What this ultimately means is that all physics is local. It does not >>>>>> mean we have mass-energy locally vanishing. The apparent increase in the >>>>>> kinetic energy due to expansion is well enough compensated for by a >>>>>> decease >>>>>> in gravitational potential energy. >>>>>> >>>>> >>>>> This is the claim, but I haven't seen any proof of it. Do you have >>>>> one? Take a planet. Can you show that Mc^2, where M is the planet's mass, >>>>> is equal to its negative gravitational potential energy? AG >>>>> >>>> > *Sorry; I may have been confused about what you were claiming. I thought > you claimed the total energy of the cosmos is zero, in which case the role > of rest energy cannot be ignored. But apparently you just meant that > kinetic energy and gravitational potential energy sum to zero, which is > apriori plausible. AG * >
The condition Nℌ = 0, for ℌ = ½√(g)[Tr(K^2) - (TrK)^2] - R^(3), means the normal vector or lapse N, with dN = Kdx for K the extrinsic curvature, can be parallel translated to define an extrinsic curvature so that mass-energy is localized. An addition requirement is needed. A manifold with an even or homogeneous distribution of particles is such that a Gaussian surface can’t be found that defines mass-energy on the manifold. This is whether the manifold is open as in ℝ^3 or the sphere S^3. The above Hamiltonian has as its first part is the kinetic energy ½a’^2, a’ = da/dt with a the scale factor and the potential energy part is ℝ^3, or the Ricci scalar curvature of the spatial manifold. LC > >>>> The Hamiltonian constraint above and the FLRW equation are all you >>>> need. It is right there. >>>> >>> >>> If it's so obvious, there'd be no dispute about this. But there >>> definitely is! Bruce, e.g. Can you cite a paper where it's explicitly >>> proven? TIA, AG >>> >> >> Tolman computed some of this early on, His old book Relativity, >> Thermodynamics, and Cosmology. Oxford: Clarendon Press. 1934 is a source. >> There is nothing about physical cosmology that says we will witness some >> horrendous violation of energy conservation locally. It does tell us that >> since GR is a local principle, based on local translations of vectors etc, >> there is then no general symmetry rule for energy conservation. >> >> >> >>>>> >>>>>> There is nothing mysterious going on here. All this means is there is >>>>>> a limitation or horizon to our ability to know if there are global >>>>>> symmetries to the universe. >>>>>> >>>>> >>> >>> There are surely limitations on our observational abilities, but why is >>> a symmetry necessary? Nature seems pretty asymmetric; e.g., imbalance of >>> matter and anti-matter. AG >>> >> >> That has nothing in particular to do with this. >> > > *You're probably right. I was just inquiring why symmetry principles are > so important. AG * > >> >> LC >> >> >>> As such there is no meaning to conservation principles. >>>>>> >>>>> >>>>> We can estimate the volume of the observable universe and its average >>>>> mass-energy density. So it seems we can estimate its total energy. Does >>>>> that energy remain constant or not as the universe expands? This seems >>>>> like >>>>> a reasonable question to ask. AG >>>>> >>>> >>>> Consider a quantum gravitational wave, say at or near the Planck scale, >>>> in the earliest phase of the universe. For that now redshifted or expanded >>>> to the scale of the CMB this means such data, from the near Planck time of >>>> the earliest universe, is around 2 trillion light years away. I have >>>> indicated numerous times how this comes about, This is as far as we can >>>> say >>>> anything about the physics of cosmology. Beyond that scale we are faced >>>> with a fundamental horizon of unobservability. As a result all we can say >>>> is that for any local system energy is conserved, but this conservation >>>> law >>>> is not due to any global symmetry. The Hamiltonian constraint is a >>>> manifestation of a local gauge-like principle of general relativity, and >>>> it >>>> has no global content. On a global level we cab' say anything; it is >>>> unknowable. >>>> >>>> Quantum mechanics is curiously similar. We have nonlocality of a wave, >>>> but we can only infer some things from that by local measurements that >>>> localizes waves or fields. We are not able to ever perform a perfect >>>> observation of a global wave. There is an epistemic horizon in QM that I >>>> think is dual or complementary to that of spacetime or general relativity. >>>> >>>> LC >>>> >>>> >>>>> >>>>>> LC >>>>>> >>>>>> >>>>>> On Friday, May 8, 2020 at 3:00:18 AM UTC-5, Alan Grayson wrote: >>>>>> >>>>>>> If it's not conserved, as seems implied by the red shift due to >>>>>>> expansion, where does it go? TIA, AG >>>>>>> >>>>>> -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/90d8f645-6a10-4c08-8b07-13ba605c4378%40googlegroups.com.
