On Monday, December 31, 2018 at 1:40:22 AM UTC, Lawrence Crowell wrote:
>
> On Saturday, December 22, 2018 at 5:46:18 AM UTC-6, [email protected]
> wrote:
>>
>> If the temperature was non uniform when the BB occurred, if it occurred,
>> why would a sudden increase in its volume, aka inflation, erase or wash out
>> those non uniformities? ISTM, it would preserve them. OTOH, if the initial
>> temperature were uniform, would that obviate the need for inflation, or
>> would non uniformities tend to become manifest were it not for inflation?
>> TIA, AG
>>
>
> I wrote the following to a list concerning this video
>
> https://www.youtube.com/watch?v=XBr4GkRnY04
>
> where I found some problems with this video.
>
*I viewed the video. It is also misleading in the claim that the universe
can expand FTL. I meant to say that's a misinterpretation of Hubble's Law.
The universe can expand very slowly, and we can find regions far away which
recede FTL. It's a purely geometric effect of any global expansion. AG*
I wrote the following below.
>
> We are in an inflationary cosmology now. It is accelerated expansion, but
> far less than during the inflationary period. If this accelerated expansion
> is gentle then local regions with matter can gravitationally clump. If the
> accelerated expansion is huge, as with the inflationary period where it was
> 10^{100} times what it is now, then matter or fields in any local region
> can't clump together as they are rendered apart.
>
> I did not at first watch this because of time and the fact that most
> elementary popularizations do not illuminate things for me. So I did take a
> look at this, and there are some problems. The presenter is right in saying
> that space can expand, which is a feature of general relativity where space
> or spacetime has differential maps or dynamics. The definition of the
> Hubble sphere and horizon is a bit confusing.
>
> I have to introduce a bit of general relativity to make some sense of
> this. Relativity, both special for flat spacetime and general, has it
> absolute invariant quantity the proper interval that is a length measured
> by a clock on a local frame. This is a Lorentzian version of the distance
> formula we learn in early college math which in turn is based on
> Pythagorean theorem. The distance s is
>
> s^2 = g_{tt}c^2t^2 - sum_{ij}g_{ij}x_ix_j
>
> where for flat spacetime g_{tt} = 1 and g_{ij} = 1 for i = j and 0
> otherwise. This interval is a time one measures on a clock, and if you are
> sitting in flat space not moving anywhere at the origin of the frame then s
> = ct which has the curious meaning that we are all heading into this fourth
> dimension at the speed of light.
>
> I will write this metric distance according to an infinitesimal distance
> ds, where we can integrate this as s' = ∫ds and ds^2 = g_{ab}dx^adx^b,
> where we sum on the indices a and b (called Einstein convention) and g_{ab}
> is the metric tensor or matrix which for diagonal entries (1, -1,-1, -1)
> the spacetime is flat and globally special relativistic. There is a deep
> subject of conformal invariance, which gets into considerable depths in
> complex variables and differential geometry. This idea is that a space or
> spacetime may be mapped by some function or multiplicative parameter and
> angle measures remain the same. We may think of this metric as similarly
> mapped as g_{ab} → Ω^2g_{ab} such that Ω is a type of expansion factor
> called a conformal factor. This means the invariant interval is
>
> ds^2 = Ω^2[c^2du^2 - sum_{ij}g_{ij}dx_idx_j]
>
> where now I have written the time variable as u instead to t and in a
> sense absorbed g_{tt} into this conformal factor. The reason for this is I
> want to express the du infinitesimal time unit as du = (du/dt)dt, which is
> just a chain rule of elementary calculus. Sorry, but a bit of mathematics
> is just vital. Now let me write this du in such a way that du/dt = Ω^{-1}
> and this conformal factor means the metric interval is
>
> ds^2 = c^2dt^2 - Ω^2 sum_{ij}g_{ij}dx_idx_j.
>
> I now make some isotropy assumptions to make the metric so that g_{ij} = 1
> for i=j and 0 otherwise. I then make another identification of the
> conformal factor with the FLRW or de Sitter expansion factor and so I have
>
> ds^2 = c^2dt^2 - cosh(t sqrt{Λ/3c^2}) sum_i dx_i^2
>
> or
>
> ds^2 = c^2dt^2 - cosh(t sqrt{Λ/3c^2}) (dr^2 + r^2(dθ^2 +sin^2θdφ^2))
>
> where this just expresses the metric of the three dimensional space in
> spherical coordinates. Here Λ is the infamous cosmological constant. So we
> see that a de Sitter spacetime is a time parametrized conformal expansion
> of space that in turn foliates out spacetime. The space or spatial manifold
> at each instance of time is flat and any two spatial surfaces of 3
> dimensions at different times are related to each other by this expansion
> factor. This metric with the spatial manifold restricted to two dimensions
> is the hyperboloid pictured below
>
> [image:
> The-comoving-coordinate-system-of-de-Sitter-space-In-solid-red-there-are-the-E-const.png]
>
>
> Now for the expansion of the universe at late times is approximated by
> cosh(t sqrt{Λ/3}) ≈ exp(t sqrt{Λ/3c^2}). This then segues into a piece I
> wrote for physics stack exchange
> <https://physics.stackexchange.com/questions/257476/how-did-the-universe-shift-from-dark-matter-dominated-to-dark-energy-dominate/257542#257542>
> back
> in 2017 or so. I employ Newtonian mechanics instead of general relativity
> to derive aspects of cosmological dynamics. This captures most of what you
> get from general relativity, and the metric I assumed has k = 0 for the
> curvature of the space and so Newtonian mechanics in that case is entirely
> captures the physics of general relativity. Why this is so points to some
> very deep principles that I will not delve into.
>
> At any rate we see that the cosmological constant is Λ = 8πGρ/c^2, where ρ
> is some vacuum energy density, which gives the definition of a mass density
> with the division by c^2. We appeal to E = mc^2, which everyone should
> know. This now gets us into where there are some funny elements to this
> presentation. The horizon distance is r_h = sqrt{3/Λ} and the particle
> horizon is computed a bit differently. We let a(t) = exp(t sqrt{Λ/3c^2})
> and the particle horizon distance is
>
> r_p = ∫dt/a(t).
>
> For a genuinely constant cosmological horizon we then see this particle
> horizon is just r_p = sqrt{3/Λ}, which is just the Hubble radius for the
> Hubble sphere! Departures between the two will be apparent with this
> integration. For the cosmological constant a function of time Λ = Λ(t) the
> integral for the particle horizon radius will depart from the Hubble sphere
> radius. I will not dive into the reasons for these departures, but for an
> increasing cosmological constant the Hubble sphere contracts and for an
> decreasing one the Hubble sphere expands. For a constant Λ the radius of
> this sphere is constant.
>
> This is where the presentation loses some of its footing. The Hubble
> sphere is not expanding or at least not much. The physical horizons also
> factor in matter in the spacetime that is a small perturbation on the de
> Sitter metric. As galaxies recede away the horizons will adjust slightly
> and asymptote to the ideal value. So this is one problem here with this
> presentation.
>
> If we see a galaxy with its redshift factor z > 1 we are witnessing it
> around 12 to 13 billion years ago or more. To a fair degree of
> approximation it is also at that distance now. However the expansion and
> acceleration of the universe means the photons were not emitted by this
> galaxy at a radius greater than the horizon length. Such a galaxy was much
> closer in the past when it emitted photons we observe, but the expansion of
> space has red shifted these photons so as to reflect the distance this
> galaxy has with respect to us on the Hubble frame of space and temporal
> simultaneity. Photons that galaxy emits now will never reach a detector
> here on Earth or the Milky Way. So there is a problem with the presenter
> saying these photons were emitted beyond this horizon and that this horizon
> keeps expanding to capture photons emitted towards us. The following
> spacetime picture might help as well, though this has a conformal map that
> shrinks or expands everything as time goes to infinity. An additional
> conformal factor is at play to reduce this to a finite region. However, the
> important fact is that anything emitted outside of that teardrop bounded by
> the event horizon does not reach us until time is infinity or in the
> language of conformal relativity I^+ or scri^∞.
>
>
> https://physics.stackexchange.com/questions/257476/how-did-the-universe-shift-from-dark-matter-dominated-to-dark-energy-dominate/257542#257542
>
> [image: cosmological conformal horizons.png]
>
> LC
>
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