Aha. Well that explains that. Thanks.
Hi Phinds - thanks for your reply. When you say super luminal recession rates are readily demonstrated in SR can you explain this more? I thought that was the whole point, that in SR it is not possible for anything to go faster than the speed of light, no?
calling @PAllen [not phinds since the post being responded to is his not mine]
@resurgance2001, recession is not proper motion. Proper motion cannot exceed c, recession can be any value at all and objects at the outer area of our Observable Universe are currently receding from us at about 3c.
This is a major weakness in how cosmology is presented in a number of textbooks. To clarify, a few definitions are in order:
1) Relative velocity: The tangent vectors of two world lines are compared. This is an unambiguous operation in SR, no matter how far apart in space and time two vectors are, because parallel transport is path independent. This corresponds to the coordinate velocity of one in the instantaneous rest frame of the other. In GR, this operation is only unambiguous if the two world line tangents are coincident, or at least sufficiently close for spacetime to be treated as flat. Otherwise, relative velocity is simply undefinable or at least, inherently ambiguous - vary your parallel transport path and you get wildly different relative velocities. Thus, velocity comparison over large distances is not a possible operation at all in GR. However, two additional observations are worth making:
a) The result of comparison between a lightlike vector and a timelike vector is path independent and the result is always c, with total generality in GR. Thus, the relative speed of a light and any material body is identically c, with no exceptions in GR.
b) Even though relative velocity of distant world lines is fundamentally ambiguous in GR, this ambiguity does not allow superluminal relative velocities because comparison of any timelike vectors over any possible parallel transport path is always less than c. Thus, a precise statement is that the precise relative speed is ambiguous, but still always less than c. Different comparison paths produce different values, but they are all less than c.
2) Celerity: given some foliation of spacetime, the rate of change in proper distance measured on this foliation between one world line and another by the proper time time on one of them. This is slighly more general that the most common definition of celerity in SR, allowing general use in GR. Cosmological recession rates are a particular instance of celerity: the standard cosmological foliation is used, and the world lines are the comoving ones. In SR, there is no upper bound to the value of a celerity, and the same is true in GR. A trivial example in SR is to use the foliation of a solar inertial frame, use the sun's world line as a reference world line, and the world line of a rocket traveling near c relative to the sun as the one whose celerity we measure. Then as the rocket travels one light year in the chosen foliation, its corresponding proper time can be made arbitrarily small, with corresponding arbitrarily large celerity. Of more relevance to cosmology, you can introduce cosmological style coordinates in SR (Milne coordinates) that have many features of FLRW solutions; in fact, this coordinate system is simply the result of the zero density limit of FLRW solutions. The resulting spacetime is simply flat Minkowski space, foliated by hyperbolic slices. Recession rate computed in these coordinates in flat spacetime (pure SR) grows with distance for 'comoving' world lines, with no upper bound, just as in realistic FLRW solutions.
There are many differences between such a fake flat spacetime cosmology and a realistic one, but superluminal recession rate is not one of these differences. The unfortunately common claim that this is a distinction is a fundamental category error: comparing relative velocities in SR with recession rates in an FLRW solution.
An important difference, for example, is that given a congruence of world lines with isotropy and homogeneity and positive expansion scalar in flat spacetime, the foliation of common proper time from initial coordinate singularity (in flat spacetime, it is only coordinate singularity not a true singularity) must have a unique hyperbolic geometry. In curved spacetime, any constant curvature geometry is possible for different mass densities, including flat spatial slices such as our universe appears very close to.
I woul really need to see this in another source, preferable textbook before I can fully follow the argument. I haven't come across the word foliation before, or celerity. I looked both up but I am a slow learner and I usually need to read three or four different accounts of the same thing before I get it. I will look in Schultz and see if I can find it there. But if you can direct me to any other sources that would be much appreciated. Thanks
I do not know if my question is stupid but, what is the general consensus about relic neutrinos? Are they also relativistic particles or could their velocities have dropped well below the speed of light?
Given the almost total lack of interaction between neutrinos and anything else, what would you expect to slow them down?
The expansion of the universe?
Not sure about that but you may be right. Light always arrives traveling at c (but is red shifted due to expansion) but of course light is special.
I cannot be right because I do not know anything. I just pointed out that there might be a way for relic neutrinos to slow down just because they are not massless. But I completely ignore what cosmologists think about this subject.
Well, I don't know anything either but I never let that stop me from thinking I'm right
That's a good one!
Relic neutrinos are at a low temperature, with standard analyses showing this temperature to be lower than CMB. However there is uncertainty due to lack knowledge of neutrino mass. The consensus is certainly that they cannot explain much of the observations that fall under the umbrella of "dark matter".
OK, but that statement still leaves the subject that Phinds and I were discussing unanswered. Relic photons may not be relevant in terms of cosmological expansion at present times but what we were talking about was if expansion could slow them down to non-relativistic velocities. We simply do not know and we would like to. Or, at least, we would like to know if such an scenario is conceivable or it is just plain crazy.
It's not my expertise, especially given that old analyses assuming massless neutrinos are not accurate anymore. It is correct that relic neutrinos have enormously lower energy than would have been measured at emission, for the same reasons as CMB, which you can poetically call expansion of space (I prefer a different mental model, but it is the same equations). I just do not know, when combined with uncertainties about neutrino mass, whether this leaves the possibility of some significant number having non-relativistic velocities. Hopefully, someone else here, who is up to date on the issues can answer.
Thank you very much!
Seems off-topic from this thread, but neutrinos do not travel at the speed of light because they have mass. Also check out 'neutrino decoupling' and 'cosmic neutrino background'.
The question was about neutrinos moving slow compared to light relative to comoving bodies. To even ask this question means there is no implication of any neutrino moving at c.
Further, most hits on your proposed search do not address this question, others are paywalled. However, I did finally turn up the following, which suggests at least some of flavors of relic neutrinos are non-relativistic.
Thanks again for your help.
Interesting article PAllen enjoyed reading it
However, I feel a note of caution is needed, as despite what the dark matter fraternity say; the idea that the Bullet cluster is evidence for dark matter is not unchallenged. A couple of early papers from 2010 and 2011 suggested that the in-fall velocity was too high to support dark matter. A more recent paper by Craig Lage and Glennys R. Farrar published 25 February 2015 in Journal of Cosmology and Astroparticle Physics, concluded “due to the paucity of examples of clusters with such a high mass in simulations, these features of the main cluster cannot presently be used to test ΛCDM.”
Moffat's modified gravity (MOG) addresses the Bullet Cluster without DM for example https://arxiv.org/abs/astro-ph/0702146
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