4 ms·

I'm not an expert on GR, but basically: energy density and mass density are summed up in the Einstein field equations to get the metric tensor that determines t

I'm not an expert on GR, but basically: energy density and mass density are summed up in the Einstein field equations to get the metric tensor that determines the curvature of spacetime. We can measure the curvature, so we can measure the mass/energy density, in absolute term. This article gives an example of how that would work: https://en.wikipedia.org/wiki/Gravitational_binding_energy#N.. https://en.wikipedia.org/wiki/Gravitational_binding_energy#N....

crdrost 6y ago
So, uh, general relativity makes all of this stuff really complicated. First off, the gravitational energy is not a direct component in the stress-energy tensor that serves as a "source" for gravitational curvature. So it is formally incorrect to say that the gravitational potential energy is a source for gravity. However, the Einstein equations are not linear the way the Maxwell equations are, and you can maybe interpret these nonlinearities as a self-interaction of the gravitational field, causing gravity to be a source of gravity. But the analogy does not seem to be as easy as just saying "here is the gravitational potential energy and by E = mc² ..." [1] Secondly, I cannot reiterate enough that general relativity does not in its usual formulations conserve energy. General relativity does not generally give you a good way over a non-infinitesimal part of spacetime to define a volume and sum up all of the energy in that volume, so it makes it very hard to define energy. And then when you try to define it in some obvious ways, you run into some weird paradoxes -- for example due to the expansion of space light from distant stars is redshifted when it arrives at us; one can either do a complicated argument from classical electromagnetic theory or a weaker but more straightforward argument from quantum theory (just track a collection of photons of definite number!) to conclude that this redshift must correspond to a loss of energy. This broader idea that the entire universe has zero net "energy" for some definition of energy does exist in some sparse literature, notably Krauss's pop-sci book A Universe from Nothing, but generally those treatments have not made a very robust case for this[2] and I don't think if I asked professional cosmologists that they would regard it as an established fact of modern cosmology, moreso than maybe just a way of speculating that maybe the universe is closed or asymptotically flat or so. (Asymptotical flatness is one way to try to give yourself an "out" so that you can again define mass, either by the ADM or Bondi methods[3].) Finally, there is indeed a correlation given by thermodynamics and stellar evolution that due to the virial theorem, "the total internal energy of the star is simply −(1/2) of its gravitational binding energy." [4] [1] Baez says a lot of this more eloquently at http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_g... [2] e.g. https://arxiv.org/abs/1405.6091 https://arxiv.org/abs/1405.6091 as critical of that particular book, some further history at https://en.wikipedia.org/wiki/Zero-energy_universe https://en.wikipedia.org/wiki/Zero-energy_universe . Note that the wikipedia article's claim has a very nice formulation of "could a quantum fluctuation create a star given that it apparently has no net energy?" and like I would view the lack of experimental evidence for this as perhaps experimental disconfirmation, but perhaps there is a more detailed calculation which confirms it as rare or some other excuse that means that you can't do it with a star but only a whole universe. [3] This topic is adequately mentioned at https://en.wikipedia.org/wiki/Mass_in_general_relativity https://en.wikipedia.org/wiki/Mass_in_general_relativity . [4] PDF warning, https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes3.pdf https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes...

crdrost 6y ago

So, uh, general relativity makes all of this stuff really complicated. First off, the gravitational energy is not a direct component in the stress-energy tensor that serves as a "source" for gravitational curvature. So it is formally incorrect to say that the gravitational potential energy is a source for gravity. However, the Einstein equations are not linear the way the Maxwell equations are, and you can maybe interpret these nonlinearities as a self-interaction of the gravitational field, causing gravity to be a source of gravity. But the analogy does not seem to be as easy as just saying "here is the gravitational potential energy and by E = mc² ..." [1] Secondly, I cannot reiterate enough that general relativity does not in its usual formulations conserve energy. General relativity does not generally give you a good way over a non-infinitesimal part of spacetime to define a volume and sum up all of the energy in that volume, so it makes it very hard to define energy. And then when you try to define it in some obvious ways, you run into some weird paradoxes -- for example due to the expansion of space light from distant stars is redshifted when it arrives at us; one can either do a complicated argument from classical electromagnetic theory or a weaker but more straightforward argument from quantum theory (just track a collection of photons of definite number!) to conclude that this redshift must correspond to a loss of energy. This broader idea that the entire universe has zero net "energy" for some definition of energy does exist in some sparse literature, notably Krauss's pop-sci book A Universe from Nothing, but generally those treatments have not made a very robust case for this[2] and I don't think if I asked professional cosmologists that they would regard it as an established fact of modern cosmology, moreso than maybe just a way of speculating that maybe the universe is closed or asymptotically flat or so. (Asymptotical flatness is one way to try to give yourself an "out" so that you can again define mass, either by the ADM or Bondi methods[3].) Finally, there is indeed a correlation given by thermodynamics and stellar evolution that due to the virial theorem, "the total internal energy of the star is simply −(1/2) of its gravitational binding energy." [4] [1] Baez says a lot of this more eloquently at http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_g... [2] e.g. https://arxiv.org/abs/1405.6091 https://arxiv.org/abs/1405.6091 as critical of that particular book, some further history at https://en.wikipedia.org/wiki/Zero-energy_universe https://en.wikipedia.org/wiki/Zero-energy_universe . Note that the wikipedia article's claim has a very nice formulation of "could a quantum fluctuation create a star given that it apparently has no net energy?" and like I would view the lack of experimental evidence for this as perhaps experimental disconfirmation, but perhaps there is a more detailed calculation which confirms it as rare or some other excuse that means that you can't do it with a star but only a whole universe. [3] This topic is adequately mentioned at https://en.wikipedia.org/wiki/Mass_in_general_relativity https://en.wikipedia.org/wiki/Mass_in_general_relativity . [4] PDF warning, https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes3.pdf https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes...