Widdekind

There is always some "slop" in spin direction, even after QM Measurement, since [math] \mu_z^2 = \left( \frac{\hbar}{2} \right)^2 \hbar^2 \times m^2 < \mu^2 = \frac{3 \hbar^2}{4} = \hbar^2 \times m (m+1)[/math]. Thus, when an electron is "spin up", it is still in a super-position state, of many "wobbling off-kilter" spin states, more-or-less spinning "up", but also w/ components in other directions. Is the following figure an apt visualization, of this process, which would explain why, the "spin up" state, is still a super-position of "half spin left, half spin right" ?? fig.1 -- viewed "from the side", a "spin up" state contains components, which are spinning both clockwise, and counter-clockwise, relative to that "sideways" direction

Type Ia SNe, which are-and-have-been quasi-identical "Standard Candles", where-ever-and-when-ever they have ever occurred, do show cosmological time-dilation, in accordance with their red-shift. If, in contrast, more-ancient-and-distant Quasars, at higher red-shift, appear to age as quickly, as more-recent-and-closer Quasars, then that immediately implies (w/o calling into question GR & cosmological expansion) that Quasars are not "Standard Candles", that they evolve over time, and that the more ancient Quasars were more rapidly varying, than the more recent Quasars. Since sizes are inferred, from time-scales multiplied by the speed-of-light; and, assuming that cosmological time-dilation is actually occurring; then, the more ancient Quasars were smaller, and less massive -- completely consistent, with Quasar evolution, and mass accretion:

The Weizsaecker Formula, for the total nuclear binding-energy [math]E_B[/math], of a nucleus, of mass number [math]A[/math], and proton number [math]Z[/math], is (roughly): Now, viewing a NS, as a "single super-sized nucleus", amounts to taking the limit, as [math]A \rightarrow \infty[/math], with [math]Z \approx 0[/math]. Accordingly, the binding-energy-per-nucleon, [math]\frac{E_B}{A} \rightarrow 16 \, MeV \; - \; 24 MeV \approx -8 \, MeV[/math] where only the "volume" & "Pauli" terms have survived. That's a repulsion, of roughly 1% of the rest-mass-energy of the matter making up the NS. By way of comparison, the gravitational binding-energy, for NS of [math]\approx 1.5 M_{\odot}[/math], and [math]\approx 7 \, km[/math], is roughly 20%. It certainly seems like, in NS, both GR & QM play important roles, requiring some kind of "quantum gravity" theory to accurately model (??).

According to this lecture, pp.39-41, when two MS stars merge, they make a single, new, MS star, of the combined mass; and, an anomalous star, in eta+chi Persei, is ~60 Msol, even when the MS turn-off, of those ~14 Myr old star-clusters, is ~30 Msol -- which would make the same a 'Blue Straggler' (BS). Perhaps the merger "mixes up" the stellar materials, thereby (partially) "refreshing" the fusion processes, in the combined composite resultant star? (Otherwise, why would mixing two old, H-depleted, and He-polluted, stars, which were evolving off the MS, with the rest of their stellar siblings, manage to make an effectively "rejuvinated" more massive star?) Edit -- Wikipedia points to 'SX Phoenicis variable' stars, which are BSs, 'below' the MS, of their GC companions. That means, that they are hotter-and-bluer than stars of the same luminosity (and so smaller); and, dimmer than stars of the same temperature (and so smaller). They also have "high space velocities", and reside within the star-dense cores of GCs. Perhaps, then, they represent, if not a full-fledged merger (as above), a "close encounter", which ripped off their outer-envelope? Such a "skinning of a star" would "White-Dwarf-ize" the star, making it bluer-but-smaller, as hotter-but-smaller once-interior layers were now exposed to space. Re-edit -- according to this article, SX Phoenicis variables reside, on the HR diagram, near the MS base, of the Instability Strip: They also have "high amplitude" pulsations, characteristic of full-fledged Cepheids, which can 'puff up' to >5 AU in radius: Such suggests, then, that variable SX-Ph BSs were once full-fledged, large-amplitude Cepheids, who were "caught while bloated", in a near-miss, close-encounter, with another GC companion, which ripped off much of their outer envelope. Note, tho', that the persistence of pulsations, of still-long-period, even after the outer-envelope was ripped away, implies that the pulsations are driven by deeper processes, beneath those "shed and not necessary" outer shell layers.

fig. 1 -- HR Diagram one-zone approximation for star-core pressures Assuming spherical-symmetry, and hydro-static equilibrium, w.h.t. [math]\frac{dP}{dr} = -g® \rho®[/math]. Then, discretizing that differential equation, and taking solely a single 'step', from star surface to star core (one-zone approximation), w.h.t. [math]-\frac{P_c}{R} \approx - \frac{G M}{R^2} \frac{3 M}{4 \pi R^3}[/math], or (ideal gas law) [math]\frac{\rho_c T_c}{\bar{m}} \approx \frac{3 G}{4 \pi k_B} \frac{M^2}{R^4}[/math]. PP [math]\left( 1/2 - 2 M_{\odot} \right)[/math] Observationally (from fig.1), the luminosities, of 'intermediate mass' stars ([math]1/2 - 2 M_{\odot}[/math]), increase with their (photo-spheric) temperatures, approximately as [math]L \propto T^8[/math]. And, since (assuming spherical symmetry) [math]L \propto R^2 \times T^4[/math], that implies that [math]R \propto T^2[/math], for such stars. Meanwhile, (seemingly) separate estimations (mass-luminosity relation) imply, that the luminosities, of such stars, scales as [math]L \propto M^4[/math]. Therefore, in solar-normalized units, w.h.t. [math]\hat{\rho}_c \hat{T}_c \approx \frac{\hat{M}^2}{\hat{R}^4} \approx \frac{\hat{L}^{1/2}}{\hat{L}} \approx \hat{L}^{-1/2} \approx \hat{M}^{-2}[/math] (ignoring effects, of star-core pollution, from fusion products). Next, 'intermediate mass' star fusion is dominated by the proton-proton (PP) process, whose specific power is proportional to [math]T^4[/math]. And, all the power produced by the fusing star-core, is (ultimately) released, at the star-surface, as the star's observed luminosity. Therefore, in solar normalized units, w.h.t. [math]\hat{V}_c \hat{\rho}_c^2 \hat{T}_c^4 \approx \hat{L}[/math]. Now, [math]\hat{V}_c \hat{\rho}_c = \hat{M}_c[/math], so we define a (solar-normalized) 'core mass fraction' s.t. [math]\hat{M}_c \equiv \hat{f}_c \hat{M}[/math]. Then, and dividing through by the mass term, w.h.t. [math]\hat{f}_c \hat{\rho}_c \hat{T}_c^4 \approx \hat{L}^{3/4} \approx \hat{M}^3[/math] (again ignoring effects, of star-core pollution, from fusion products). Then, taking the ratio, of the former to the latter equations, w.h.t. [math]\hat{f}_c \hat{T}_c^3 \approx \hat{L}^{5/4} \approx \hat{M}^5[/math], implying that [math]\hat{T}_c \approx \hat{L}^{5/12} \approx \hat{M}^{5/3}[/math] and [math]\hat{\rho}_c \approx \hat{L}^{- 11/12} \approx \hat{M}^{-11/3}[/math] (assuming the constancy, of the 'core mass fraction'). Approximately, for 'intermediate mass stars', w.h.t.: [math]\hat{L} \approx \hat{M}^4[/math] [math]\hat{T}_c \approx \hat{L}^{1/2}[/math] [math]\hat{\rho}_c \approx \hat{L}^{-1}[/math] [math]\hat{P}_c \approx \hat{L}^{-1/2}[/math] CNO [math]\left( 2 - 20 M_{\odot} \right)[/math] Observationally (from fig.1), the luminosities, of 'high mass' stars ([math]2-20 M_{\odot}[/math]), increase with their (photo-spheric) temperatures, approximately as [math]L \propto T^6[/math]. And, since (assuming spherical symmetry) [math]L \propto R^2 \times T^4[/math], that implies that [math]R \propto T[/math], for such stars. Meanwhile, (seemingly) separate estimations (mass-luminosity relation) imply, that the luminosities, of such stars, scales as [math]L \propto M^{7/2}[/math]. Therefore, in solar-normalized units, w.h.t. [math]\hat{\rho}_c \hat{T}_c \approx \frac{\hat{M}^2}{\hat{R}^4} \approx \frac{\hat{L}^{4/7}}{\hat{L}^{2/3}} \approx \hat{L}^{-2/21} \approx \hat{M}^{-1/3}[/math] (ignoring effects, of star-core pollution, from fusion products). Next, 'high mass' star fusion is dominated by the carbon-nitrogen-oxygen (CNO) process, whose specific power is proportional to [math]T^{17}[/math]. Again, all the power produced by the fusing star-core, is (ultimately) released, at the star-surface, as the star's observed luminosity. Therefore, in solar normalized units, w.h.t. [math]\hat{V}_c \hat{\rho}_c^2 \hat{T}_c^{17} \approx \hat{L}[/math]. Now, again, [math]\hat{V}_c \hat{\rho}_c = \hat{M}_c[/math], so we again define a (solar-normalized) 'core mass fraction' s.t. [math]\hat{M}_c \equiv \hat{f}_c \hat{M}[/math]. Then, again dividing through by the mass term, w.h.t. [math]\hat{f}_c \hat{\rho}_c \hat{T}_c^{17} \approx \hat{L}^{5/7} \approx \hat{M}^{5/2}[/math] (again ignoring effects, of star-core pollution, from fusion products). Then, taking the ratio, of the former to the latter equations, w.h.t. [math]\hat{f}_c \hat{T}_c^{16} \approx \hat{L}^{17/21} \approx \hat{M}^{17/6}[/math], implying that [math]\hat{T}_c \approx \hat{L}^{17/336} \approx \hat{M}^{17/96}[/math] and [math]\hat{\rho}_c \approx \hat{L}^{- 49/336} \approx \hat{M}^{-49/96}[/math] (assuming the constancy, of the 'core mass fraction'). Approximately, for 'high mass stars', w.h.t.: [math]\hat{L} \approx \hat{M}^{3.5}[/math] [math]\hat{T}_c \approx \hat{L}^{0.05}[/math] [math]\hat{\rho}_c \approx \hat{L}^{-0.15}[/math] [math]\hat{P}_c \approx \hat{L}^{-0.10}[/math] Thus, given the extreme temperature-dependence, of CNO power production, CNO-dominated, 'high-mass' star-cores, operate under quasi-isobaric, quasi-isothermal conditions. So, the cores, of 'high mass stars', are all similar, in bulk properties. And, those star-cores all eventually blow away their outer envelopes, producing Planetary-Nebulae (PNe), cocooning proto-White-Dwarves (proto-WDs): Thus, the similarity, of 'high mass star' pre-WD cores, could account, for the ensuing similarity, of the remnant proto-WDs and WDs, as the enveloping PNe drift away & disperse. Note, too, that the transition, from pre-WDs (T > 50k K), through to isolated WDs (T < 30k K), seems to be associated with the transition from full-to-partial-to-neutral He ionization states (HeIII-to-HeII-to-HeI), and the ensuing decrease in plasma opacity, to interior-generated luminosity (cf. CMBR and Recombination, at z~1100). Does the "characteristic new-born WD surface-temperature layer" (~30-50k K) represent an 'opacity barrier', thermally blanketing the star-core there-within contained ? Unstable Variable Stars [math]\left( \approx 2 M_{\odot} \right)[/math] The transition, from fusion by PP to CNO processes, occurs in the cores of stars slightly more massive than our sun, whose core temperatures exceed ~20M K: And, this transition occurs, for stars near the observed 'break', on the HR diagram, of the stellar luminosity function, at masses of [math]\approx 2 M_{\odot}[/math] (see fig.1). And, this region, on the HR diagram, is also associated with the 'Instability Strip', corresponding to variable stars (e.g. RR Lyrae, Cepheids): Note, that the radial velocities, of the observationally-inferred (effective-)photo-spheric expansions (few AU per day) are characteristic of our sun's "space weather" -- i.e., star-wind (SW), including coronal-mass-ejections (CMEs). Thus, might periodic PP-to-CNO 'power-pulses' may generate rhythmic star-spanning 'super-massive-ejections' ? effects of H depletion ? By definition, the 'average effective mass' [math]\bar{m}[/math], linking total mass-density [math]\rho[/math], to total number-density [math]n[/math], is [math]\frac{1}{\bar{m}} \equiv \frac{X}{m_H} + \frac{Y}{4 m_H} + \frac{Z}{\bar{m}_Z}[/math]. Ignoring the effects, of heavier metal pollutants, upon H-to-He fusion processes ([math]Z \approx 0[/math]), then w.h.t. [math]\frac{m_H}{\bar{m}} \approx X + \frac{Y}{4} \approx \frac{1 + 3 X}{4}[/math]. And, only core H produce power. Thus, accounting for dwindling H mass-fractions ([math]\hat{X}(t) \rightarrow 0[/math]), w.h.t. (solar-normalized units): [math]\hat{\rho}_c \hat{T}_c (1 + 3 \hat{X}) \approx \frac{\hat{M}^2}{\hat{R}^4}[/math] (HSE) [math]\hat{f}_c \hat{M} \hat{\rho}_c \hat{X}^2 \hat{T}^{4,17} \approx \hat{L}[/math] (PP,CNO power) [math]\therefore \hat{f}_c \frac{\hat{X}^2}{1+3 \hat{X}} \hat{T}^{3,16} \approx \hat{L} \frac{\hat{R}^4}{\hat{M}^3} \approx \hat{L}^{5/4,17/21}[/math] So, to maintain constant luminosity, and remain on the Main Sequence (MS), w.h.t.: [math]\hat{T}_c \approx \left( \frac{\hat{X}^2}{1+3 \hat{X}} \right)^{-1/3, -1/16}[/math] And so, as the star slowly ages, over the eons, [math]\hat{X} \rightarrow 0[/math], the star's core temperature may rise. And, such a hottening, could advantage star-core CNO processes, perhaps accounting for subsequent star swellings, 'up' off the MS, into the Giant Branch (GB) phases ? Qualitatives Externally-observed characteristics (i.e., mass-luminosity relation, on MS) may reflect interior H-fusion processes (e.g., PP, CNO). What, then, about 'very-low mass stars' ([math]< 1/2 M_{\odot}[/math]), and 'very-high mass stars' ([math]> 20 M_{\odot}[/math]) ?

Please pay particular attention, to the vast difference in size scale. On the scale of billions of light-years, our cosmos is (quasi-)uniform. Therefore, on such size scales, space-time-and-gravity behaves one way. But, on the scale of light-minutes, our solar-system-within-our-cosmos is varying, in its various properties. Therefore, on those size scales, space-time-and-gravity behaves a totally other way. You have asked an interesting question -- if, within gravity-bound (i.e., galactic scale) systems, space is not stretching; then, how far out-into-deep-space, must one venture, before space just-barely-begins-to-stretch ? Logically, there must, indeed, be such a "Hubble Lagrange Point", on the peripheral marches, of galactic-scale, gravity-bound structures. To give a guess, when the local over-density decreases down to ~0, so that the ambient density decreases down to the cosmic average, then space-time might begin to behave as it does on average, i.e., stretch.

According to the cited source, WFs, initially confined to a spatial size-scale [math]l_0[/math], at an (asymptotic) speed, normalized to that of light, of [math]\beta \approx \frac{1}{2} \left( \frac{l_0}{\lambda_C} \right)^{-1}[/math]. This, of course, comports with the HUP, according to which [math]\Delta p \approx \frac{\hbar}{2} \Delta x^{-1}[/math], dividing through by the mass m, and then speed-of-light c, and converting the quantity [math]\frac{\hbar}{c m} \rightarrow \lambda_C[/math]. Thus, when quantum particles are spatially compressed, to their "Compton" size-scales, Relativistic effects begin to become important.

WFC requires "one witness" ? Considering collisional excitations & de-excitations, between atoms & free electrons, in inter-stellar plasmas, full-fledged energy-eigen-state changes can occur, indicating WFC, yet without the emission of any "off-to-infinity" photon, as the QM hallmark "registration signal", e.g., 13.6 eV = 91 nm photon emitted from electrons binding to protons. In the latter case, the generated-and-emitted photon acts as a "third party witness" to WFC; in the former case, by comparison, the free electron plays the role of the photon, albeit persisting through the interaction, during which it either "off-loads", or "on-loads", energy, either to, or from, the atom. So, seemingly, even if "the universe" or "the (classical) environment" can act as a WFC-inducing (large) quantum system; so, then, any (even small) quantum system, can serve as "registrant witness" to the quantum observation/detection/measurement/actualization event.

The Vector Potential, generated by an individual charge, is [math]\vec{A_{q}} \approx \frac{q}{r} \times \vec{v}[/math]. And, the Dirac expression, for the Vector Potential [math]\vec{A}[/math], of a magnetic monopole, in spherical coordinates, is: [math]\vec{A_{m}} \approx \frac{1}{r} \times tan \left( \frac{\theta}{2} \right) \hat{\phi}[/math] (Jackson. Classical Electro-Dynamics, p. 290) Thus, qualitatively, a spherical distribution of charged plasma, rotating differentially, with "slowly spinning poles" and a "fast rotating equator", would have an azimuthal plasma velocity profile, comparable to [math]\vec{v} \approx tan \left( \frac{\theta}{2} \right) \hat{\phi}[/math]. And, that description sounds strikingly similar to a star, e.g. our sun, whose magnetic field is quasi-radial, according to the Parker model: Indeed, such a shear-inducing, differential azimuthal velocity field, sounds completely consistent, with our sun's tacho-cline: Thus, solar-like differential rotation, of charged plasma: is qualitatively consistent, with quasi-radial "mono-polar" magnetic fields:

Please ponder protons & electrons combining, into neutrons & neutrinos: [math]p^{+} + e^{-} \rightarrow n^0 + \nu_e[/math] If you "flipped" this reaction, then neutrinos could "stimulate" the fission, of neutrons, into protons & electrons. Indeed, in the OP, neutrinos could "stimulate" the fission, of deuterium nuclei, into hydrogen nuclei. Moreover, in a standard nuclear fission chain-reaction, neutrons essentially "stimulate" the emission, of more neutrons -- a little like a "neutron laser" effect. Could not neutrinos do something similar? There seems to be a physical parallel, between spontaneous-and-stimulated photon emission, and spontaneous-and-stimulated nuclear fission. In both cases, there seems to be a process parallel to the spontaneous one. If "neutrino-stimulated nuclear fission" is possible, then one could construct a "neutrino detector", by using a short-lived radio-active element. Then, when a high flux of neutrinos -- e.g., from GRBs -- impinged upon the detector, the neutrinos would induce additional fission events, so that the sample would decay more rapidly, than its half-life would indicate. To wit, your "geiger counter" would start clicking faster, with the increase in frequency, due to neutrino-induced fission events.

Purely putting it poetically

To clarify, yes, the space-time fabric, interior to a gravity-bound system of "objects" (e.g., stars, galaxies) is non-expanding. Some of those "objects", e.g. measuring devices, are electro-chemically-bound, by direct atomic & molecular bonds. Evidently, directly "linking" atoms & molecules together, into macroscopic "objects", also imparts a "viscous drag", of that matter, into space-time, that also stops the latter from stretching.

On the HR diagram, the 'Super-Giant Branch' represents an "outgrowth", of super-massive MS stars ([math]\approx 30-60 M_{\odot}[/math]), at (roughly) constant Luminosity. Such suggests, that the fusing-and-radiating star-core is (largely) unaffected, by the outward swelling, of its outer envelope layers, as the star "puffs up". And, the onset of this stellar "super swelling" coincides closely, with (initial) star surface temperatures above the H-ionization threshold (i.e., Tsurf> 1 eV). Moreover, such stellar super-swelling may be much more extensive. For, super-massive stars, like Betelgeuse & Rigel, "are surrounded by shells of expelled gas", associated with stellar pulsations and extreme mass-loss, producing irregularly-varying Luminosities, "common to Super-Giants". In particular, Rigel's variability is "turbulent & energetic", exhibiting the characteristic complexity. So, perhaps all Super-Massive stars "Nebularize", above Tsurf> 1 eV, corresponding to masses [math]> 30 M_{\odot}[/math], swelling into super-sized Super-Giants, after they reach, and as they then evolve back off of, the MS ? Hyper-Novae (HNe) shock-shells always expand, into circum-stellar nebulae ? Super-massive stars, like Betelgeuse ([math]\approx 20 M_{\odot}[/math]), cast off considerable amounts of material, into circum-stellar nebulae: And, HN after-glows could be caused, by expansion of HN-generated shock-fronts, into surrounding materials (Bergstrom & Goobar. Cosmology & Particle Astrophysics, p.232): Indeed, Rigel appears to pulsate, with a period of nearly a month, and an amplitude of up to 0.3 magnitudes (~30%). These pulsations push out expanding shells of off-cast material. And, afore-mentioned Betelgeuse also apparently pulsates, albeit with a longer period, of nearly 6 years; and, a correspondingly larger amplitude, of 1 magnitude (~3x). Such pulsations apparently arise when the: Now, both Betelgeuse & Rigel have comparable masses ([math]\approx 20 M_{\odot}[/math]), even as indicated by their comparable luminosities. But, Betelgeuse has ballooned, to become much bigger. Having traveled (roughly) twice-as-far off of the MS; and, nearly to the "end of the line", on the SGB; then, perhaps Betelgeuse is twice as old, as Rigel -- 10 Myr vs. 5 Myr, say, if such stars are expected to live for the former amount of time ? If so, then Rigel resembles Betelgeuse, in its younger years. And, stellar cycles of pulsation would apparently, then, increase, in both period & amplitude, over star-evolutionary time. And more, even as stellar pulsations can be caused by contraction-induced "fusion flashes", perhaps SNe & HNe can be viewed, as "humongous pulsations", from a proposed star-destroying "final fusion flash", perhaps converting the core completely into iron ?? Perhaps, then, Betelgeuse is a candidate, for a near-future (<1 Myr) HNe & GRB ?? Note, too, that super-massive stars, living fast & furiously, and dying young & spectacularly, probably do not stray far, from their GMC 'hatcheries'. So, then, such super-massive stars, may often be surrounded, not only by their own off-cast nebulae, but also the nebular remnants, of their original SF region:

Planetary Nebula (PNe) consist of an increasingly exposed "pre-WD" star core, surrounded by the increasingly distant, diffuse, and (often) ionized shed outer layers, of said star. After the failure of He fusion, the post-fusion, "pre-WD" star core remnant quickly cools down, from ~100k K, to ~30k K, in ~10 kyr: [math]L = 4 \pi R^2 \times \sigma T^4 = -\frac{dE}{dt}[/math] [math]E \approx N k_B T \approx \frac{M}{m_H} k_B T[/math] [math]\therefore \frac{dT}{dt} \approx -\frac{4 \pi R^2 \, \sigma}{N k_B}T^4[/math] [math]\therefore \frac{1}{T_i^3} - \frac{1}{T_f^3} \approx \frac{4 \pi R^2 \, \sigma}{3 N k_B} \times t[/math] Plugging in characteristic values ([math]T_i \approx 100k \, K[/math], [math]T_f \approx 30k \, K[/math], [math]R \approx 0.01 R_{\odot}[/math], [math]M \approx M_{\odot}[/math]), I compute a 'cooling time scale' of [math]\tau \approx 2 Myr[/math]. This is more than two orders-of-magnitude more, than the observed characteristic cooling time, of [math]\approx 10 Kyr[/math]. This discrepancy could be simply explained, by assuming that only the outer layers of the pre-White Dwarves actually cool down. Er go, their interiors remain a high-temperature, high-pressure mixture of carbon & oxygen, or "molten diamond". Fig. 1 -- stars of [math]\approx 1-8 M_{\odot}[/math] blow off outer layers, gradually generating PNe, surrounding the increasingly exposed star-cores, down to ever deeper once-interior depths. Mass-loss may cease, along with He fusion in the spent star-core, by the time that the now-exposed 'surface' layer, of the central pre-WD, reveals a temperature of [math]\approx 100k K[/math]. As the pre-WD cools down, to a temperature of [math]\approx 30k K[/math] (i.e., [math]\approx 13.6 eV[/math]), its harsh-and-ionizing radiation produces observed PNe. Afterwards, the PN fades, 'decoupling' from the now-comparatively cold WD, which appears as an independent astronomical object. Missing Mass in WD-PN systems ? PNe contain [math]\approx 0.1-1 M_{\odot}[/math] of observed circum-stellar plasmas. And, PNe are apparently generated, by progenitor-stars, of [math]\approx 1-8 M_{\odot}[/math]. But, WD remnants are predicted to mass only [math]\approx 1.4 M_{\odot}[/math]. This leaves up to [math]\approx 6.5 M_{\odot}[/math] of progenitor-star material unaccounted for.

According to the book A Journey into Gravity & Spacetime, by J.A.Wheeler, "mass tells space-time how to curve; and, space-time tells mass how to move". Therefore, if masses are not moving apart -- i.e., they are 'gravitationally bound' -- then space-time is not "telling" them to expand apart. Er go, the "rubber sheet", of the space-time fabric, between gravity-bound objects, is non-expanding. Only the low-density, inter-galactic voids of deep space represent expanding, or stretching, space-time fabric.

Please ponder the Wave Function (WF), of an electron, spiraling around a field line, in a uniform magnetic field [math]\vec{B} = B \hat{z}[/math]. This field corresponds to a vector potential [math]\vec{A} = \hat{\phi} B r / 2[/math]. The SWE, for a time-independent, 'magnetically bound' state, is then: [math]E \Psi = \frac{\left( \vec{p} - q \vec{A} \right)^2}{2 m} \Psi[/math] Now, the vector potential commutes with the momentum operator (since the former's azimuthal-angle component, is independent of that variable). Thus, defining [math]e \equiv -q[/math], w.h.t.: [math]2 m E \Psi = \left( -\hbar^2 \nabla^2 + 2 e A_{\phi} p_{\phi} + (e A)^2 \right) \Psi[/math] And now, if we assume solutions of the form [math]\Psi \equiv R® e^{i n \phi}[/math], then w.h.t.: [math]2 m E = -\hbar^2 \left( \frac{1}{r R} \frac{\partial}{\partial r} \left( r \frac{\partial R}{\partial r} \right) - \frac{n^2}{r^2} \right) + n \hbar e B + \frac{ \left( e B \right)^2}{4}r^2 [/math] [math]2 m E = -\hbar^2 \left( \frac{1}{r R} \frac{\partial R}{\partial r} + \frac{1}{R} \frac{\partial^2 R}{\partial r^2} - \frac{n^2}{r^2} \right) + n \hbar e B + \frac{ \left( e B \right)^2}{4}r^2 [/math] Assuming that [math]E < 0[/math], w.h.t.: [math]\frac{2 m | E |}{\hbar^2} = \left( \frac{1}{r R} \frac{\partial R}{\partial r} + \frac{1}{R} \frac{\partial^2 R}{\partial r^2} - \frac{n^2}{r^2} \right) - n \left( \frac{e B}{\hbar} \right) + \left(\frac{1}{2} \frac{e B}{\hbar} \right)^2 r^2 [/math] [math]\frac{2 m | E |}{\hbar^2} + n \left( \frac{e B}{\hbar} \right) = \left( \frac{1}{R} \frac{\partial^2 R}{\partial r^2} + \frac{1}{r R} \frac{\partial R}{\partial r} - \frac{n^2}{r^2} \right) + \left(\frac{1}{2} \frac{e B}{\hbar} \right)^2 r^2 [/math] [math]\left[ \frac{2 m | E |}{\hbar^2} + n \left( \frac{e B}{\hbar} \right) \right] r^2 R = \left( r^2 \frac{\partial^2 R}{\partial r^2} + r \frac{\partial R}{\partial r} - n^2 R \right) + \left(\frac{1}{2} \frac{e B}{\hbar} \right)^2 r^4 R [/math] Now, defining the 'Larmor energy' [math]E_L \equiv \hbar \omega_L[/math]; and, the 'reduced mass' [math]\mu \equiv m / \hbar^2[/math]; then, w.h.t.: [math]\mu \left( \, 2 \, |E_n| \, + \, n \, E_L \, \right) r^2 R = \left( r^2 \frac{\partial^2 R}{\partial r^2} + r \frac{\partial R}{\partial r} - n^2 R \right) + \left( \frac{\mu \, E_L}{2} \right)^2 r^4 R [/math] or: [math]\left( r^2 \frac{\partial^2 R_n}{\partial r^2} + r \frac{\partial R_n}{\partial r} - n^2 R_n \right) - \mu \left( \, 2 \, |E_n| \, + \, n \, E_L \, \right) r^2 R_n + \left( \frac{\mu \, E_L}{2} \right)^2 r^4 R_n = 0 [/math] where we have explicitly added in subscripts, denoting the dependence on rotation-rate index "n". Now, one could solve, for the resulting Bessel-function-resembling [math]R_n[/math] functions, by laboriously calculating their infinite-sum-of-terms Taylor expansion. For example, [math]R_1® \approx r + \frac{\mu}{8} \left( 2 |E_1| + E_L \right) r^3 + \frac{\mu^2}{192} \left( \left( 2 |E_1| + E_L \right)^2 - 2 E_L^2 \right) r^5 + ...[/math] Note, too, that, as expected, there is no non-rotating, n=0, solution ([math]a^0_k = 0 \, \forall k[/math]). And, [math]|E_1| < E_L/2[/math], if we demand that [math]a^1_5 < 0[/math], in analogy to the 'alternation of terms' characteristic of Bessel functions. A priori, this formula seemingly suggests, that the electro-magnetic attraction, of electrons, to magnetic field lines, produces a spectrum of 'magnetic bound states', loosely analogous to the Hydrogen WFs, from the electro-static potential, produced by protons. These WFs 'clad' magnetic field lines, and phase-rotate around them, like cylindrical 'x-zylo' frisbee footballs. Synchrotron radiation could be viewed, as loosely analogous to Hydrogen de-excitation emissions, to ground state, as cycling electrons "gear shift down" the magnetic-bound-state energy spectrum.

Please ponder a co-moving volume, which winds up expanding to become 1 cubic cm, today, z=0. That co-moving volume today contains ~400 CMBR photons (Maoz. Astrophys. in a Nutshell, eq. 9.25) ([math]n_{\gamma}[/math]), and about a billionth as many baryons ([math]n_e \approx n_B[/math]). Since cosmic re-ionization, back at z ~ 10, our space-time has been filled with a diffuse, and nearly completely ionized, plasma. Thus, if the 'Look-back Time' to the re-ionization epoch is ~12 Gyr ([math]t_{LB_{10}}[/math]), then the number of photon-free-electron Thompson scatterings, occurring in our comoving cubic volume, since that ancient eon, has been (roughly): [math]N = n_{\gamma} \, n_B \, \sigma_T \, c \, t_{LB_{10}} \equiv \eta \, n_{\gamma}^2 \, \sigma_T \, c \, t_{LB_{10}}[/math] where we have employed the photon-to-baryon ratio [math]\eta \equiv n_{\gamma} / n_B[/math]. Plugging in the numerical values, I compute [math]N \approx 1 \times \eta_9[/math] such scatterings, where [math]\eta_9 \equiv \eta \times 10^9[/math]. Thus, in over 10 billion years of crossing cosmological distances, only a few tenths of a percent of the CMBR photons have been lost (to Thompson scattering). What about Cosmic Rays, could CRs be a considerable component of [math]n_B[/math] ?? Or, might metals have significantly larger Compton Scattering cross-sections ?? According to NRAO, the CMBR is by far the most prominent peak in earth-received, cosmologically originating, extra-terrestrial radiation: Metals reside, in the 'Lyman-alpha forest' inter-galactic clouds, at low levels, but metals may have scattering cross-sections that are enormous, in comparison to the Thompson profile, of lone electrons.

Could you "trick" a particle, into Wave-Function-Collapsing, by creating an artificial potential well; allowing the particle to 'relax' into that well; and then 'pulling the plug', and destroying the potential well, leaving the particle 'high and dry', in a spatially confined state ?? In-so-far as particles, to WFC into potential wells, have to be able to sense, that that potential well, is a well, not a general 'downslope gradient'; then, it seems, that particles would have to 'slosh about' some, within the well, w/ the WF scattering of the attractive center, falling backwards after propagating away, re-scattering, self-entangling (?? leading to 'self-entangled' contradictions??), before finally WFC'ing. So, WFC perhaps requires at least one "crossing time" to set up, before occurring.

Chris' calculations look quite correct. What supposedly happened was, 14 Gya, back at redshift z~1100, CMBR anisotropies "fired" photons, towards the earth. These photons have been in flight, for 14 Gyr, crossing the cosmos, to our earth-based detectors, 'here-and-now'. They have been 'inbound', on radial trajectories, fixed to the comoving coordinate grid, as defined back at z~1100. And, that coordinate grid has been expanding, with the Hubble Flow, the entire time. Thus, after 14 Gyr of Hubble Expansion, those photons are 'inbound', on sight-lines, now spread laterally, side-to-side, ~1100x wider, in angular diameter, than those CMBR anisotropies would look, at their present "true proper distance" of 45 Gly, in flat static Minkowskian space-time. Again, if you had some sort of 'super-luminal true sight'; and if those CMBR anisotropies persisted to the present epoch; then those ~200 kpc anisotropies, would today span only 1/1100th of a degree -- to wit, ~3" of arc. Expanding space-time "magnifies", or "lenses", cosmologically distant objects. (This site discusses CMBR angular size scales, seemingly sans the red-shift magnification effect.)

If our space-time fabric was static (i.e., Minkowskian), then light-rays emitted, towards a 'target', from the two lateral limbs of some astronomical object (e.g., towards earth, from the side-limbs, of some CMBR anisotropy), would converge straight in towards that target, like the straight laser beams in games & movies: However, our space-time fabric is 'stretching'. Thus, when some astronomical object 'lets photons fly', 'loosing light' out into the Hubble Flow, the expansion, of the space-time fabric, between those photons, actually forces those photons apart, laterally-side-to-side, even as they stream towards their mutual distant target. Again, if light, from the extreme opposite edges, of some galaxy, is 'let loose' towards earth, then, even as the emitting galaxy, as a gravitationally self-bound object, 'secedes' from the Hubble Flow, its loosed light will stream out into that Hubble Flow, and start to spread apart, side-to-side. Thus, in illustration, "lasers", shot from the "wing-tips" of an "x-wing fighter" would, if they behaved cosmologically, "fan out" far to the sides of the shooting space-ship, before finally, eventually, reconverging towards their cosmologically-distant target: It is for this reason, that the actual observed angular diameters, of distant objects, seen in earth's skies, are actually much much greater, than they would be, in simple, static (i.e., Minkowskian) space-time: [math]\theta = (1 + z) \, \frac{L}{R(t_0) \, w} \approx (1 + z) \, \frac{L}{D_P(t_0)}[/math] (Carroll & Ostlie. Intro. Mod. Astrophys., p.1274) In particular, then, CMBR anisotropies, of [math]\approx 1^{\circ}[/math], do not correspond to ~200 Mpc spatial size scales, as I naively assumed; but, instead, given that the Last-Scattering-Surface resides out at a radius of z~1100, those angular size scales, having been magnified by a factor of one thousand, actually only correspond, to physical size scales, of ~200 Kpc. Thus, CMBR anisotropies represent (pre-)galactic-scaled 'clumps' of matter. At even smaller angular, and physical, size scales, Silk damping seemingly smoothed out any primordial anisotropies. Thus, entering into its matter-dominated epoch, our cosmos was 'seeded', essentially exclusively, with galactic-scale clumps -- with little 'power', on larger size scales, due to global isotropy & uniformity; and with little 'power', on smaller size scales, due to said smoothing. The gradual emergence, of the < 100 Mpc anisotropies, seen today, can probably, then, be attributed to Hierarchical Structure Formation, over the past 14 Gyr.

"Conformal Time", in closed cosmology, is equivalent to "Hyper-Angle" subtended from "Hyper-Center of Space-Time" In any given time-interval dt, photons will always travel a true metric distance, of c x dt. Thus, the "Conformal Time": [math]\Delta \eta \equiv \int_{t_1}^{t_2} \frac{c \, dt'}{R(t')}[/math] can be visualized, for a closed cosmology, as a "hyper-angle", subtended from the "hyper-center of space-time", [math]d \theta = c \, dt / R(t)[/math]: In a closed cosmology, then, co-moving coordinates correspond to "hyper-angular distances", measured "around the circumference" of space-time. Then, by such visualization, it is immediately apparent, why the proper distance, between two contemporaneous points in space, is equal to [math]D_P(t) = R(t) \times \Delta \eta = R \, \Delta \theta[/math]. This can be verified, by noting, from the FRW metric, for a closed cosmology (k = +1), that: [math]\frac{c \, dt'}{R(t')} = \frac{ dw }{ \sqrt{1 - w^2} } = ds[/math] where w, dw, s, ds are all defined relative to the fixed, co-moving, reference-frame, which expands with 'stretching' space-time: Thus, for closed cosmologies, the Conformal Time has an unambiguous geometrical interpretation, as the "hyper-angular" co-moving coordinate.

Gardner & Shostak's Bicosm book is devoted to this topic.

Now (z=0) -- According to D.Maoz' Astrophysics in a Nutshell, our cosmos, today, is isotropically smooth, at large size scales, >100 Mpc. Then (z ~ 1100) -- Judging from the "CMBR Power Spectrum", which shows a peak at an angular size scale of (slightly less than) 1 degree, our cosmos, nearly 14 Gya, was isotropically smooth, at large-for-back-then size scales, >1 degree. QUESTION -- should not these size-scales correspond? Should not "1 degree then" somehow correlate, to "100 Mpc today" ? Naively using online dark-energy-incorporating cosmological calculators (and cf. PF), the proper distance to z~1100 is ~45 Gly, at which absolute distance, ~1 degree represents ~600 Mly (200 Mpc).

Radiation red-shifts, so that its energy, equal to its number density x energy per photon, scales as R-4. Conversely, 'dust-like' (pressure-less) matter, having rest-mass, has a constant energy per particle, and its energy density scales as R-3. Thus, there was once an era of "Radiation Domination", after which "Matter Dominated" the density: http://www.newscientist.com/article/dn13414-universe-submerged-in-a-sea-of-chilled-neutrinos.html During a core-collapse SN, the fusion of protons & electrons, into neutrons, also releases neutrinos: [math]e^{-} + p^{+} \rightarrow n^0 + \nu_e[/math] Therefore, the reverse reaction is surely physically possible: [math]n^0 + \nu_e \rightarrow e^{-} + p^{+} [/math] Now, except for the "invisible neutrino", that reaction looks allot like 11-minute-half-life neutron decay. QUESTION: Is it possible, that neutron decay, is caused by the "harassment" of neutrons, by the Cosmic Neutrino Background ?? And, if so, could all radio-active decay be induced, by interactions, with the CNB ??

Symbio-Genesis, at all biological scales, is a "process", not an "event" ? Over two billion years ago, a then-free-swimming pre-Mitochondrial, proteobacteria, was incorporated, into the proto-Eukaryotic (post-Thermoplasma ?) host cell. This "event" is often described in "instantaneously singular terms", as if full-fledged Symbiosis emerged, full-flower, from some single happenstance. Yet, that newly-proto-Mitochondrial, intra-cellular symbiont, surely had a genome of ~1000 genes, >2 Gya, which has been steadily eroded down -- by "streamlining deletions", and "nuclear incorporations", of the proto-Mitochondrial genome -- to only 37 genes, today. Thus, Eukaryotic Mitochondria have gradually given up genes, as they have "relaxed" into their "climate controlled" intra-cellular "farm", at an average rate, of -1 gene per 2 Myr. Thus, Symbiogenesis has been a gradual process, evolving over geological time-scales, and not some singular 'one-time' event. Now, terrestrial plant Photo-Respiration can be construed, as earth plants behaving according to a 'pan-planetary perspective' -- to wit, 'stewarding' earth-atmospheric Carbon Dioxide concentrations, as if to avoid additional 'snowball earth' global glaciations: Thus, even as Symbiogenesis may be a gradual process, of "relationship building" -- "they meet, shake hands, start to small-talk, continue a deeper conversation, one-thing-leads-to-another, and they begin to get intimate" -- at the level of the single cell, so to, Meta-Symbiogenesis, at the pan-planetary 'super-scale', may also be an (even more) gradual process, occurring 'glacially slowly', yet inexorably, over geological time.