5 ms·

A couple little things I won't turn into literal ETA3, just in case you've already been reading the long post above. :) (2) below is also mostly for me, I thin

A couple little things I won't turn into literal ETA3, just in case you've already been reading the long post above. :) (2) below is also mostly for me, I think. 1. Thanks, Michael Weiss. This is essentially the same argument I'm making, only terser. http://math.ucr.edu/home/baez/physics/Relativity/GR/expanding_universe.html http://math.ucr.edu/home/baez/physics/Relativity/GR/expandin... 2. One important thing about the expansion of the universe is that people hear "expansion of space" and reasonably think space has physical properties. https://arxiv.org/abs/0707.0380 https://arxiv.org/abs/0707.0380 is a good rant about that, and https://arxiv.org/abs/0809.4573 https://arxiv.org/abs/0809.4573 is another interesting take. There's a sort of Betteridge effect going on, in that none of the authors think that the concept of expanding space is awful, but they're instead capturing the "gotchas" that befall even working relativists stray into what philosophers call manifold substantivalism. (I hope that doesn't attract house philosophers, no offence. I don't think my text is all honey!)

danbruc 8y ago
I read your response like five minutes after you posted it as I just replied to cbzbc at that time. Thanks a lot for the effort! It's already pretty late here, so a real response will have to wait till tomorrow, but I guess I was able to get quite a few things from your responses. Especially probably somewhat related to your second point in the last comment, namely that I kind of really want to avoid the math and use analogies but without realizing or knowing well enough where those analogies break. Not because I don't like math but because as a layman I mostly consume it passively when reading or watching lectures. In consequence I can follow a presentation but doing calculations myself would be a very different thing and so I also lack mathematical intuition when it comes to the math used in physics. And as I am already typing, I will just add the first question I got when reading your response because I already typed it out. In the second paragraph - not counting the first sentence - you write »[...] \eta is the chosen background field, h is the perturbation field, and g is the "true" metric.« When you say »chosen«, are you only referring to the choice between signatures (1,3) and (3,1) or do you want to allow eta to be something other than the Minkowski metric there? Probably not to important overall but the first thing that I marked.
- raattgift 8y ago
  > When you say "chosen" I mean that neither nature nor theory requires one to choose Minkowski as the background spacetime, although it is the most popular choice in perturbative General Relativity. I've been avoiding sign conventions and coordinates deliberately; I think they would be more distracting here, and of course the choices made have no physical implications. (Using \eta without it necessarily meaning Minkowski spacetime is arguably an abuse of notation, if that's what you mean. :D)
delta-v 8y ago
Could I interpret it this way? In the EFE, if we move the cosmological constant factor to the right handside, where the stress–energy tensor is, then stress–energy tensor and the cosmological constant factor have opposite signs and stress–energy tensor can overcome the cosmological constant factor. When we say EM-forces/Strong/Weak forces "cancel out" the expansion of the universe, it isn't because of those forces are holding matters together. But they contribute to the stress–energy tensor and "offset-ed" the effect of the cosmological constant factor. Even without those forces, as long as the stress–energy tensor is greater than the cosmological constant factor, the solution of EFE will be non expanding. Does this sound right to you? Thanks.
- raattgift 8y ago
  Well, so you write down an initial values surface with that approach and let the matter evolve: either you get something physically plausible or you don't. We have some guides as to what is physically plausible, and of course evidence that any of them is wrong would be very interesting. More theoretically, the LHS doesn't care about how you arrange the RHS at all. More finely, the metric does not care how you compose the metric. If you do it perturbatively, the leading term could be Minkowski or Gödel for all it cares, as long as the sub-leading terms then present a useful background for small perturbations that remain small (or better, vanish) rather than explode into large perturbations. Inside the solar system, Schwarzschild gets that right, and nobody has shown you need to correct for the global behaviour of the cosmic expansion. > ... if we move the cosmological constant factor to the right handside .. Before the 1920s (!) there was a discussion about where to put an everywhere isotropic tension in the EFEs. https://arxiv.org/abs/1211.6338 https://arxiv.org/abs/1211.6338 In section 3, the author quotes Einstein's response about how to determine the density. The part after, "Later, he finishes with..." seems especially germane, and cf. section 4 especially its last sentence (wherein "dark energy" in that context means a source field with a potential and dynamics and a coupling to the other source fields and a set of initial values). [Heh, typo in the second sentence of the last paragraph of §3, "vale" instead of "value".] > (matter forces) contribute to the stress-energy tensor Ok, but if you hold that the CC acts isotropically everywhere you can think about decomposing the curvature tensor into its Ricci (and Weyl) components. Weyl tensor encodes the stretch-squash astigmatism: a spherical volume in the presence of a nonvanishing Weyl start to look more like a U.S. football. Ricci scalar encodes the isotropic change in volume. For a volume, nonvanishing Weyl with a vanishing Ricci is volume-preserving; a vanishing Weyl and a nonvanishing Ricci is a contraction or expansion. The cosmological expansion at cosmological scales clearly is Ricci scalar rather than Weyl, since matter in the large remains isotropic and homogeneous. Shuffling \Lambda over to the source side does not change that observation: we don't see broken cosmological-scale isotropy, so we must still have a vanishing Weyl and a nonvanishing Ricci scalar on the LHS. How does that happen? Presumably in the metric, since it's the obvious choice of dynamic field (although of course you could insist on making the RHS as complicated as you like, and there have been cases where having the metric maximally static with gravitational dynamics shuffled into "fake" matter is useful -- the original Hawking Radiation paper did just that, for instance; and of course you might want a quintessence-style field instead of a CC, and bite the bullet on recovering observations through that field's behaviour). Ultimately what we care about is that we have the correct EOMs, and for test particles the EOMs in our labs and in our experimental platforms elsewhere in the solar system, and observed for astrophysical objects throughout our galaxy and the cluster it's in, are nowhere near the EOMs of FLRW. They are much nearer to Schwarzschild.

danbruc 8y ago

I read your response like five minutes after you posted it as I just replied to cbzbc at that time. Thanks a lot for the effort! It's already pretty late here, so a real response will have to wait till tomorrow, but I guess I was able to get quite a few things from your responses. Especially probably somewhat related to your second point in the last comment, namely that I kind of really want to avoid the math and use analogies but without realizing or knowing well enough where those analogies break. Not because I don't like math but because as a layman I mostly consume it passively when reading or watching lectures. In consequence I can follow a presentation but doing calculations myself would be a very different thing and so I also lack mathematical intuition when it comes to the math used in physics. And as I am already typing, I will just add the first question I got when reading your response because I already typed it out. In the second paragraph - not counting the first sentence - you write »[...] \eta is the chosen background field, h is the perturbation field, and g is the "true" metric.« When you say »chosen«, are you only referring to the choice between signatures (1,3) and (3,1) or do you want to allow eta to be something other than the Minkowski metric there? Probably not to important overall but the first thing that I marked.

raattgift 8y ago
> When you say "chosen" I mean that neither nature nor theory requires one to choose Minkowski as the background spacetime, although it is the most popular choice in perturbative General Relativity. I've been avoiding sign conventions and coordinates deliberately; I think they would be more distracting here, and of course the choices made have no physical implications. (Using \eta without it necessarily meaning Minkowski spacetime is arguably an abuse of notation, if that's what you mean. :D)

delta-v 8y ago

Could I interpret it this way? In the EFE, if we move the cosmological constant factor to the right handside, where the stress–energy tensor is, then stress–energy tensor and the cosmological constant factor have opposite signs and stress–energy tensor can overcome the cosmological constant factor. When we say EM-forces/Strong/Weak forces "cancel out" the expansion of the universe, it isn't because of those forces are holding matters together. But they contribute to the stress–energy tensor and "offset-ed" the effect of the cosmological constant factor. Even without those forces, as long as the stress–energy tensor is greater than the cosmological constant factor, the solution of EFE will be non expanding. Does this sound right to you? Thanks.

raattgift 8y ago
Well, so you write down an initial values surface with that approach and let the matter evolve: either you get something physically plausible or you don't. We have some guides as to what is physically plausible, and of course evidence that any of them is wrong would be very interesting. More theoretically, the LHS doesn't care about how you arrange the RHS at all. More finely, the metric does not care how you compose the metric. If you do it perturbatively, the leading term could be Minkowski or Gödel for all it cares, as long as the sub-leading terms then present a useful background for small perturbations that remain small (or better, vanish) rather than explode into large perturbations. Inside the solar system, Schwarzschild gets that right, and nobody has shown you need to correct for the global behaviour of the cosmic expansion. > ... if we move the cosmological constant factor to the right handside .. Before the 1920s (!) there was a discussion about where to put an everywhere isotropic tension in the EFEs. https://arxiv.org/abs/1211.6338 https://arxiv.org/abs/1211.6338 In section 3, the author quotes Einstein's response about how to determine the density. The part after, "Later, he finishes with..." seems especially germane, and cf. section 4 especially its last sentence (wherein "dark energy" in that context means a source field with a potential and dynamics and a coupling to the other source fields and a set of initial values). [Heh, typo in the second sentence of the last paragraph of §3, "vale" instead of "value".] > (matter forces) contribute to the stress-energy tensor Ok, but if you hold that the CC acts isotropically everywhere you can think about decomposing the curvature tensor into its Ricci (and Weyl) components. Weyl tensor encodes the stretch-squash astigmatism: a spherical volume in the presence of a nonvanishing Weyl start to look more like a U.S. football. Ricci scalar encodes the isotropic change in volume. For a volume, nonvanishing Weyl with a vanishing Ricci is volume-preserving; a vanishing Weyl and a nonvanishing Ricci is a contraction or expansion. The cosmological expansion at cosmological scales clearly is Ricci scalar rather than Weyl, since matter in the large remains isotropic and homogeneous. Shuffling \Lambda over to the source side does not change that observation: we don't see broken cosmological-scale isotropy, so we must still have a vanishing Weyl and a nonvanishing Ricci scalar on the LHS. How does that happen? Presumably in the metric, since it's the obvious choice of dynamic field (although of course you could insist on making the RHS as complicated as you like, and there have been cases where having the metric maximally static with gravitational dynamics shuffled into "fake" matter is useful -- the original Hawking Radiation paper did just that, for instance; and of course you might want a quintessence-style field instead of a CC, and bite the bullet on recovering observations through that field's behaviour). Ultimately what we care about is that we have the correct EOMs, and for test particles the EOMs in our labs and in our experimental platforms elsewhere in the solar system, and observed for astrophysical objects throughout our galaxy and the cluster it's in, are nowhere near the EOMs of FLRW. They are much nearer to Schwarzschild.