Widdekind

In the CoM frame, for two colliding particles, having equal masses; if both particles impinge upon the point of impact, and if both are "deflected to the right" (say); then did not the virtual particle, effectively carry a momentum impulse, transverse to its direction of travel, from emitter to absorber ?

As observed by AJB in a recent blog entry, Raa = gabRba = R Taa = gabTba = T R = -T Now, this site seems to say, that diag(gab) = (1, -1, -1, -1); and, that T = diag(Taa) = p ( c2 - vx2 - vy2 - vz2) = p (c2 - v2). So, T > 0 ? And so, R < 0 ?? In trying to calculate Rab, from the definition: I thought I found, that many terms cancelled, when one of the dummy indexes was set equal, to one of the formal indexes; that led me to an "einstein strike notation" that simplified the expressions, e.g. with a piece like: gba,b(gab,a - gba,a) - gba,b(gab,a-gba,a) wherein a dummy index a, b runs over every index, except a, b.

Aromaticity is an example of electron charge delocalization, into lower-energy bonding orbitals.

And, CP formation involves lattice distortions, i.e. phonons (SD 2007); which lattice distortions dramatically influence tunneling (SD 2010). Altering lattice spacing influences SC, e.g. by doping (SD 2010); and by applied laser excitations (SD 2011). Applied pressure can be considered a "static phonon loading"; and, iron-based high-temperature SCs can be "pressured into" SC states: as can C-60 molecules: Indeed, pressure is commonly applied, to "tune" materials into SC states (SD 2011, cp. SD 2008). Germanium hydride becomes SC at 2Mbar (SD 2008). And, in manganite Electron localization impedes SC (SD 2011); Mott localization prevents SC, and can be effected, by increasing the lattice spacing, affecting overlap integrals (SD 2010). Lattice spacing, influenced by atomic magnetic moments, influences SC (SD 2009). The pseudogap phase exhibits properties similar to those of the gap-phase, suggesting similarity of electronic wave-functions, i.e. individual/paired de-localized bonding states ?? Tunneling SC (SD 2011, SD 2009). CPs usually form in stationary state, i.e. k=0 (SD 2008). Confining electrons, into 2D planes, may enhance CP formation (SD 2008).

Where would a ~MeV solar neutrino get 80 GeV, to interact Weakly, with human neutrino detectors on earth, e.g. SNO, Super-K ? How can "the books balance", without "borrowing" ?

As a general rule, do all 'particles', emerging from any forceful interaction, emerge into a co-mingled entangled state, until one (or more) of the particles, undergoes a subsequent interaction ?? I.e. the "natural state" of quanta, is to be entangled, in between interactions ?? E.g. the mean-free path, through inter-stellar space, is ~10 Glyr for photons; and <1 lyr for H atoms (assuming [math]d \sim 1 \aa[/math]). Converted into times, t=d/v, yields ~10 Gyr, and ~10 Kyr, respectively. If so, then photons, emitted by processes within our galaxy, could possibly arrive at earth, in still-entangled states; and, the fraction of still-entangled photons could possibly provide range-to-source information. Could astute measurement techniques, cp. "weak measurement", distinguish between still-entangled, and already-collapsed, photons ?? Human scientists have generated entangled matter-energy, electron-photon, states, from artificial atoms (Quantum Dots), that "can emit photons one at a time, such that photons are entangled with [the artificial quantum atom]": Indeed: And, stimulated Rubidium atoms emit photons, with which they are entangled: However, when quantum systems are entangled, all of their past interactions are reversible: What constitutes a "strong light-matter interaction" ?? Would an isolated atom, in deep space, emit coherent & entangled photons ?? For, information can be transferred, to-and-from energy-and-matter, with sufficient precision to reconstitute the original information; but without, seemingly, involving entanglement (SD 2007). Decay product particles emerge entangled ?

I understand, that "while decoherence typically happens very swiftly, it is not quite instantaneous" (dickau); and, that "decoherence" and wave-function "collapse" occur, in chemical reactions: If so, then the "contractive collapse", of the quantum 'particle' "field" -- cp. "Erwin Schrodinger thought that the quantum state of a particle -- in the form of its wave-function -- was a real field, as a classical electro-magnetic field is real" (mermin) -- occurs "quickly" but not "instantly" ??

I see no conflict, between what you said, and what Prof. Ozawa reportedly formulated, i.e. "there are two sources of uncertainty, one intrinsic to the 'particle'; and the other induced by the experimenter" ?

How do particles interact Weakly, via super-massive Weak bosons, of 80-90 GeV, without "borrowing" momentum-and-energy ?

I understand, that a "Cooper pair" is a composite, bosonic, state, of two electrons. Vaguely, the electrons have "opposite spins", so that the CP is S=0; and, the electrons have "opposite momenta", so that the CP is p=0 (in an isolated, non-conducting, ground state, i.e. the CP BEC "just sitting there" at rest). Yet, the electrons are "entangled", so that they really represent only one single conjoined, co-mingled, composite, combined, quantum state. Is the wave-function, of a CP, at rest, representable by the following wave function, indexed by the individual electron momenta k; and individual electron spin s=1/2: [math]\Psi_{CP, k} \approx |+k +s\rangle_1 |-k -s \rangle_2 \; + \; |+k -s\rangle_1 |-k +s \rangle_2 \; + \; |-k +s\rangle_1 |+k -s \rangle_2 \; + \; |-k -s\rangle_1 |+k +s \rangle_2[/math] i.e. a "mixed up mixture" of product-states, each of which individually represents a pair of electrons having opposite linear & angular momenta; but whose combination represents a bosonic CP in a "stationary standing wave" state ? In particular, if a CP were to exit a super-conductor, and enter into a conventional resistive-and-dissipative conductor, so that the wave-function of the CP were to "collapse" (after a propagating through a characteristic "coherence length"); then would the constituent electrons "emerge" into a product state, i.e. each electron defined by oppositely directed linear & angular momenta ? ADDENDUM: I understand, that the constituent electrons, still obey the FD statistics, of the "Fermi sea", of individual electron states, in the (SC) material. Does that mean, that the bosonic "basis states", of CPs, must be indexed by their individual electron momentum [math]\pm\vec{k}[/math] values; and that those momentum basis states range, in momenta (i.e. momentum phase space), from [math]0 \le \frac{\hbar^2 k^2}{2 m_e} \le E_F[/math] ? And, if "entangled" states arise, from "intimate" interactions; then CPs emerge, from "oppositely moving & spinning" electrons, in the (SC) material, "meeting mingling & merging" ?? And, CPs, arising from momentum eigenstates, represent spatially de-localized electron pairs ?? CPs apparently form, from non-valence f,d orbital electrons: Is that because valence s orbital electrons, having energy states into which they can easily scatter, easily do so scatter ?? Could such "near-Fermi-energy" electrons be "suppressed", e.g. by forcing additional electrons into the material, to induce a net negative charge ?? SC "energy gap" emerges from sub-valence electron de-localization, into lower energy bonding orbitals ?? As potential super-conductors are cooled, to their threshold temperatures, an energy gap emerges, near the Fermi energy. Lower lying "sub-valence" momentum states, below the "surface" of the "Fermi sea", decrease in energy. Is that an indication, of the de-localization, of tightly-bound, sub-valence, f/d orbital electrons, from "highly localized" states, into spread-out "bonding orbitals", approximated by the in-phase superposition, of the isolated atomic orbitals, associated with each & every lattice site ? If "interior", strongly localized electrons, "tunneled away" from their native nuclei, into de-localized bonding orbitals; then would they not "drop down" in energy, into a lower energy configuration, thereby "opening up" a gap / pseudogap ? Indeed, the "pseudo-gap" phase is associated with tunneling: Now, I understand, that the electrons, in CPs, are "entangled"; and, that for once-separate wave-functions to become "entangled", they must first interact "intimately". If so, then electron de-localization, from "deep interior" f/d orbitals, to "spread out" bonding orbitals, must precede interactions, with other similarly-spreading electrons ("meeting"); must precede intimate interactions ("mingling"); must precede CP formation ("marrying"). So, could the pseudo-gap phase represent de-localized individual electrons, not yet paired up, into CPs ?? And, the energy "benefit", from forming de-localized bonding orbitals, increases with the overlap integral, between localized orbitals, on neighboring nuclei. So, could the operating temperature regime, of SCs, be increased, by increasing the operating pressure regime, i.e. compress the material, decrease the lattice spacing, increase the overlap integral, and energetically "incentivize" delocalization into bonding orbitals, within which electrons would "meet their mates", and then form CPs ?? Indeed, the pseudo-gap phase competes ("cheaper singles apartments"), against the gap phase, for electrons ("cheaper doubles apartments"): And, CP formation involves lattice distortions, i.e. phonons (SD 2007); which lattice distortions dramatically influence tunneling (SD 2010).

Does not the article agree with you ? Prof. Ozawa distinguishes "intrinsic uncertainty", from "observation uncertainty": I understand, from the article, that "observation uncertainty" can be reduced arbitrarily; whilst "intrinsic uncertainty" is only minimizable, per HUP. If a quantum 'particle' is likened to a "school of fish", or "flock of birds", then the spatial extent of the "herd" represents intrinsic uncertainty, and is only minimizable; whereas observation uncertainty, would be like how much the "herd" is perturbed, by the presence, of the experimenter, e.g. the "flock of birds takes off in a new direction" (strong measurement) vs. "a few birds hop out of the way, whilst most of the flock keeps on feeding" (weak measurement).

I understand, that when quantum 'particles' have, locally, negative energy, i.e. [math]E(\vec{x}) < V(\vec{x})[/math]; then their wave-functions decay away exponentially in time [math]\Psi \propto e^{-\frac{V-E}{\hbar}t}[/math]. So, I understand, that "virtual" particles, being "borrowed" into existence, from the zero-energy vacuum, on a "Heisenberg energy loan", propagate whilst decaying away exponentially quickly. Qualitatively, therefore, I understand that "virtual" particles can only propagate a distance [math]d \approx c \Delta t \approx \hbar / E[/math], before that exponential decay, of their wave-functions, "drives" the particles out of existence [math]\Psi \rightarrow 0[/math]. You said, "virtual particles...conserve energy-and-momentum, at every vertex, in a Feynmann diagram" ? I understand, that, "in the ultimate end", every interaction winds up, after-the-fact, having conserved energy-and-momentum. However, I understand, that, during interactions -- i.e. as they are "in progress", and the virtual particle, mediating that interaction, is propagating, across the FD, from the emitter ("emission vertex"), to the absorber ("absorption vertex") -- force-carrying quanta can be, and are, "borrowed" into existence. That "energy debt", on the force-carrier, explains the short range, of the residual Strong force, between nucleons, mediated by massive pions [math]\left( E \approx 140 MeV\right)[/math]. By analogy, if the force-carrier is likened to a USPS mail delivery truck, which carriers an "letter", i.e. interaction information, from one particle (emitter) to another (absorber); then a massive pion, "borrowed" into existence, is like a USPS mail truck, "borrowed" from the "garage", which has to be "brought back before noon". Accordingly, the "letter", representing transferred momentum-and-energy, can only be transmitted over short ranges, before the carrier is "due back". But, momentum-and-energy, is strictly conserved, amongst the emitter, the actual amount of force transmitted, and the absorber: [math]E_e \longrightarrow E_e' + \delta E \longrightarrow E_e' + \left( E_a + \delta E \right)[/math] When you say "energy-and-momentum is conserved at every vertex", I understand, that you are referring to the quantities [math]E_e \longrightarrow E_e' + \delta E[/math] (emission vertex); and [math]\delta E + E_a \longrightarrow E_a'[/math] (absorption vertex). But, that interaction energy [math]\delta E[/math] may be "carried", "conveyed", i.e. mediated, by a massive boson, "borrowed on a Heisenberg energy loan", of mediator mass-energy [math]\Delta E \ne \delta E[/math] having nothing, directly, to do with the amount of interaction energy [math]\delta E[/math]. The latter is strictly conserved, as you said; the former is "borrowed", as the OP said (as I understand): [math]mediator \; mass-energy \; \ne \; interaction \; energy[/math] [math]\Delta E \ne \delta E[/math]

According to the Scientific American article Solving the Solar Neutrino Problem (to point to a specific, concrete, reference), neutrinos can affect crucial nuclear reactions, e.g. breaking apart D nuclei: Now, before BBN began, "hot" photons broke apart D nuclei, until T < 0.1 MeV, thereby delaying BBN. I understand, that in a plot of theoretical BBN elemental abundances, versus the baryon-to-photon ratio [math]\eta[/math], that [math]\eta \leftrightarrow \rho_B \sim \rho_B \left( T_{BBN} \; \tau_{BBN} \right) \propto \eta = const.[/math] i.e. the "Lawson triple product", where the temperature, and duration, of BBN are assumed, by assuming that BBN occurred, in a radiation-dominated epoch, i.e. a(t) = a0 t1/2; and by assuming that the temperature was dominated by red-shifting photons T(t) ~ a(t)-1. Thus, the "Lawson's triple product" [math]\rho_B \; T_{BBN} \; \tau_{BBN} \sim \frac{1}{a^3} \frac{1}{a} a^2 = a^{-2}[/math] Naively, and qualitatively, I understand, that if there were more baryons, i.e. [math]\rho_B \rightarrow \rho_c[/math]; then, to be consistent with observations (of abundances) & calculations (of [math]\eta[/math]), those baryons must have undergone BBN, later, for less time, at colder temperatures, after our universe had expanded more. Could "deuterium breakup", caused by neutrinos, have further delayed BBN, so that instead of BBN occurring from +100-1000s, at temperatures 0.1-0.03 MeV, BBN occurred only later, for less time, in colder conditions ?? Naively, [math]\Lambda = \int d\Lambda = \int \rho T dt = 2 \rho_0 T_0 t_0 \int a^{-3} a^{-1} a da = \rho_0 T_0 t_0 \left(\frac{1}{a_i^2} - \frac{1}{a_f^2} \right) \propto \eta = const.[/math] I understand, that the only "free parameters" (without invoking new physics), are [math]\rho_0[/math] & [math]a_i[/math], i.e. comparing various scenarios, [math]\rho_0 \left(\frac{1}{a_i^2} - \frac{1}{a_f^2} \right) = \rho_0' \left(\frac{1}{a_i'^2} - \frac{1}{a_f^2} \right)[/math] [math]\left( \frac{1}{x^2} - 1 \right) = \frac{\rho_0'}{\rho_0} \left( \frac{1}{x'^2} - 1 \right)[/math] where [math]x \equiv a_i / a_f \approx 1/3[/math] from the temperature data. If so, then "demanding" that the baryon density be ~20x higher, i.e. close to critical density, would "require": [math]10 \approx 20 \left( \frac{1}{x'^2} - 1 \right)[/math] [math]\frac{3}{2} \approx \frac{1}{x'^2}[/math] [math]x' \approx \frac{4}{5}[/math] i.e. BBN could only have occurred, from T = 0.04-0.03 MeV, corresponding to t = 700-1000s, i.e. a delay of ~10 minutes. Could a dense "fog" of neutrinos, breaking up deuterium nuclei, have accomplished such a delay ??

Soon after the Big Bang, light elements were synthesized, from primordial protons & neutrons, by "neutron capture" reactions. According to BBN theory: According to direct observations of space, there are (relatively) high levels of surviving D; and (relatively) low levels of Li: Thus, the following nuclear reactions occurred: but not: QUESTIONS: Qualitatively, is the Lawson criterion applicable, i.e. the amount of fusion that occurred, when the primordial plasma was "singed" in BBN, was proportional, to the product, of baryon density during BBN, multiplied by the time of BBN, i.e. [math]n \tau[/math] ? (I understand, that the temperature regime, is plausibly constrained, 300 MK < T < 1000 MK.) And so, if the baryon density was higher, then the time for fusion must have been lower ?? Could "inflation", or something similar, have "stretched out space", more swiftly than currently conceived, so that a hypothetically higher baryon density, e.g. [math]\rho \rightarrow \rho_c[/math], would have had less time to "cook" ?

If a satellite orbits closer to a parent body, than the "geo-synchronous" radius; then the satellite will orbit faster than the host planet spins, so that the satellite will "orbit ahead" of the tides that it generates into the central planet. Thereby, i.e. by "tugging spin-forward" on the tidal bulge it generates into the planet, the satellite will deposit its orbital energy, into "spinning up" the planet, even as the satellite gradually de-orbits itself. Er go, if the Roche radius, is less than the synchronous radius, then the bodies will gradually converge & coalesce. For two equal-density planets, the Roche radius is 1.25 - 2.50 R. Meanwhile, the synchronous radius is [math]\omega_{orbit} = \omega_{planet}[/math] [math]\frac{G M}{R^3} \approx \omega^2[/math] Naively, the Roche radius is so close to the planet that said planet would have to be spinning, at near break-up rotation rate, for the synchronous radius to lie within the Roche "disruption radius". If so, then "by the time" some body was orbiting close enough to some planet, to be tidally distorted; then it would also be "orbiting ahead of its tides", and gradually spiraling in towards the other body. Thus, the "Roche world" scenario seems plausible -- except that large planets are not "rigid", and would not "bump into each other", and simply touch at the point of initial contact. Rather, they would "smush" & "squish" together.

mass "hyper-inertia" curves space-time ? Mass, "tied into" the fabric of space-time, was "flung outwards", by the Big Bang. Mass could possibly have "hyper-inertia", tending to retain a constant "outwards hyper-momentum", until restrained by "elastic" space-time fabric: estimating "hyper-thickness" of space-time, from first-generation quark masses ? According to QM, 'particles' confined into 1D square potential wells, of finite energy "depth", have a finite number of bound states. And, the most energetic bound state tends to have an energy comparable to the depth of well. So, up-type quarks (mtop ~ 180 GeV) possibly experience a dramatically deeper well, than down-type quarks (mdown ~ 5 GeV). Yet, the "ground" states, of both types of quarks, i.e. u & d, have nearly identical masses, near 5 MeV. Now, in classically forbidden regions, where the particle energy, is less than the local potential energy, i.e. E < V, wave-functions decay exponentially. So, whether confined within a well of 5 GeV, or 200 GeV, in the low energy limit E << V, all bound particles are experiencing an effectively infinite square well potential. And, for infinite square well potentials, the wave-functions are driven to zero, at the potential "walls", i.e. the bound states are standing waves, with wave-lengths that are (essentially) integer multiples of the potential well width. If so, then the de Broglie wave-length, of u & d quarks, i.e. 10-14m = 10 fm, would presumably be approximately the width of the well. Such a hypothetical hyper-spatial thickness, would be 10x the spatial breadth, of nucleons ("quarks are taller than they are wide"). Naively, a hyper-thickness of our space-time fabric, hypothesized to be "thin", should be "small", in comparison to spatial size scales. Using "dressed" quark masses, of mesons & baryons, which are 10-100x larger, would reduce the estimated hyper-thickness, to fractions of a fm.

Please permit me to follow along: [math]g = -\frac{G M_{<r}}{r^2} \approx -\frac{G M}{R^2} \equiv g_0[/math] [math]G \int 4 \pi r^2 \rho® dr \approx r^2 \frac{G M}{R^2}[/math] [math]G \left( 4 \pi r^2 \rho® \right) \approx 2 r \frac{G M}{R^2}[/math] [math]\rho® \approx \frac{1}{2 \pi r} \frac{4 \pi \bar{\rho} R^3}{3 R^2} = \frac{2}{3} \bar{\rho} \frac{R}{r}[/math] [math]\rho(x) \approx \frac{2}{3} \frac{\bar{\rho}}{x}[/math] If so, then, from HSE, w.ht.: [math]\frac{dP®}{dr} = - g \rho®[/math] [math]\frac{1}{R}\frac{dP}{dx} \approx -g_0 \frac{2}{3} \frac{\bar{\rho}}{x}[/math] [math]\int dP \approx -\frac{2}{3} \left( \frac{G M \bar{\rho}}{R} \right) \int \frac{dx}{x}[/math] [math]P - P_0 \approx -\frac{2}{3} \left( \frac{G M \bar{\rho}}{R} \right) \ln(x)[/math] where we have included the surface pressure [math]P_0[/math] in case the hypothetical world has an appreciable atmosphere. Let [math]B \equiv \frac{2}{3} \left( \frac{G M \bar{\rho}}{R} \right)[/math] then w.h.t.: [math]\frac{\Delta P}{B} \approx -\ln\left(\frac{2}{3} \frac{\bar{\rho}}{\rho}\right)[/math] [math]e^{-\frac{\Delta P}{B}} \approx \frac{2}{3} \frac{\bar{\rho}}{\rho}[/math] [math]\rho \approx \left( \frac{2}{3} \bar{\rho} \right) e^{\frac{\Delta P}{B}} \approx \left( \frac{2}{3} \bar{\rho} \right) \left( 1 + \frac{\Delta P}{B} + \frac{1}{2} \left( \frac{\Delta P}{B} \right)^2 + ...\right)[/math] Thus, to first order in applied pressure, [math]B[/math] is essentially a "Bulk Modulus", of the planet. For our earth, [math]B \approx 230 GPa = 2.3 Mbar[/math], closely comparable to rock, steel, & diamond, i.e. the constituent materials from which our rocky world is composed. Therefore, this simple model explains, and accurately predicts, the BM, of common rocky materials, i.e. human-measured BM represent the lowest order "linear regime", of density increase, due to applied pressures. To date, human civil engineers have not applied multi-mega-bar pressures, to terrestrial structures, and so have yet to perceive higher-order "corrections". Such higher-order effects would plausibly be prominent, for rocky materials, residing deep within the core of Jupiter, wherein the pressures are predicted to be [math]\ge 3 TPa = 30 Mbar[/math] -- and wherein this simple model presumably is inapplicable, predicting, as it does, exponentially large densities. Still, such suggests, that the matter in the cores of large planets (& BDs ?) is dramatically degenerate.

Intuitively, "reifying" space-time, as a real "thing", with real "substance", within which all matter-and-energy resides, like "insects encased in amber", seems plausible. Thank you very much for the clear-if-qualitative explanation. Also thank AJB very much for the Sean M. Carroll reference.

From the Virial Theorem, [math]K = -\frac{1}{2} U[/math] [math]\frac{M k_B T}{\mu} \approx \frac{3}{10} \frac{G M^2}{R}[/math] [math]\therefore R \; T = \frac{3}{10} \frac{G \mu}{k_B} M[/math] Now, the Luminosity, radiated away as heat: [math]L = 4 \pi \sigma R^2 T^4 = \frac{4 \pi \sigma}{R^2} \left( R \; T \right)^4[/math] is balanced by the release, of GPE: [math]L = -\frac{dU}{dt} = \frac{3}{5} \frac{G M^2}{R^2}\dot{R}[/math] Er go, [math]\frac{4 \pi \sigma}{R^2} \left( R \; T \right)^4 = \frac{3}{5} \frac{G M^2}{R^2}\dot{R}[/math] [math]4 \pi \sigma \left( \frac{3}{10} \frac{G \mu}{k_B} M \right)^4 = \frac{3}{5} G M^2 \dot{R}[/math] [math]\frac{2 \pi \sigma}{G} \left( \frac{3}{10} \right)^3 \left( \frac{G \mu}{k_B} \right)^4 M^2 = \dot{R}[/math] Assuming primordial gas composition (X = 3/4, Y = 1/4), so that the average particle mass is ~0.6 mH, w.h.t.: [math]\dot{R} \approx 100 km/s \times \left( \frac{M}{M_{\odot}} \right)^2[/math] [math]\approx \frac{1}{3} 10^{-3} c \times \left( \frac{M}{M_{\odot}} \right)^2[/math] If so, then the "implosion speed" of collapse [math]\dot{R} \rightarrow c[/math] near [math]M \rightarrow 50 M_{\odot}[/math]. Are such speeds plausible ? Such massive proto-stars collapse, on the MS, in ~104yrs: And, Molecular Cloud 'cores' are typically <1 lyr across ; however, cloud collapse occurs isothermally (Sterzik 2003, Sterzik 2003). Perhaps isothermal collapse accounts for the slower observed collapse speeds ?

GR says A "ball of particles" vaguely resembles a star. Er go, as stars plunge in towards SMBH, at the centers of galaxies, they would plausibly become tidally compressed ? If so, then such tidal compression could possibly "squeeze the star" so as to induce faster fusion, and spark a Nova-like or Supernova-like explosion ?

space-time "wants" to be flat ? I understand, that, quantum mechanically, Naively, if concentrated mass-energy curves space-time; but if concentrated mass-energy "seeks" to spread out through space-and-time; then, according to a crude combination of QM & GR, space-time "likes" being flat, and "hates" being curved ?? I.e. space-time "seeks" to "un-dent" & "un-dimple", a little like a self-re-forming car body, which "desire to un-dent" drives QM wave-functions to disperse ?? The "stage" (space-time) is as ontologically real, as the "actors" (fermions & bosons) ? Space-time is a "mesh", a fabric-like "weave", within which matter & energy is embedded, and into whose "weave" matter & energy is coupled, i.e. "hooked in" ? J.A.Wheeler wrote (Journey into Gravity & Spacetime) that matter "grips" space-time, giving a vague impression, of "rock-climbers" being "roped in" to a vast cargo-net-like "rigging", which rigging is distorted, as particles traverse it, cp. Gravity Probe B results, indicating that the "mesh" of space-time can be "twirled" by our spinning world ? If our earth is 5 billion years old; then by how many degrees, has our earth "twirled" the local space-time fabric ? In animations, earth is depicted orbiting our sun. So, does the fabric of space-time "un-twirl", once earth "spins past" some particular place ??

I understand, that quantum wave-packets have calculable "centroids" or "expectation values"; and, that those "centroids" evolve through time, according to the Classical equations of motion, whilst obeying Classical conservation laws. I.e. our macroscopic notions of "Classicality" arise from the "average" behavior, of quantum wave-packets. In analogy, wave-packets are like "schools of fish" or "flocks of birds", characterizable by an average "center of mass", and an average "overall momentum"; but which are also spread out through physical space (xyz) ("many many birds here & there"), and momentum space (pxpypz) ("some birds flying fast left, others flying slow right"). For example, for a fermion (F), composed of a "spin up" component (+), and a "spin down" component (-): [math]\tilde{\Psi}_F(\vec{x}) = \Psi_{+}(\vec{x})|+\rangle + \Psi_{-}(\vec{x})|-\rangle[/math] ("flock of sea-gulls & flock of crows") the "expected (linear) momentum": [math]\langle \vec{p} \rangle \equiv \int d^3 x \tilde{\Psi}_F^{*}(\vec{x}) \left( -\imath \hbar \vec{\bigtriangledown} \right) \tilde{\Psi}_F(\vec{x})[/math] [math] = \left( \int d^3 x \Psi_{+}^{*}(\vec{x}) \left( -\imath \hbar \vec{\bigtriangledown} \right) \Psi_{+}(\vec{x}) \right) + \left( \int d^3 x \Psi_{-}^{*}(\vec{x}) \left( -\imath \hbar \vec{\bigtriangledown} \right) \Psi_{-}(\vec{x}) \right) [/math] [math] = \langle \vec{p}_{+} \rangle + \langle \vec{p}_{-} \rangle[/math] ("sea-gulls' momentum + crows' momentum") and the "expected angular (spin) momentum": [math]\langle s_z \rangle \equiv \int d^3 x \tilde{\Psi}_F^{*}(\vec{x}) \left( \hat{s}_z \right) \tilde{\Psi}_F(\vec{x})[/math] [math] = \frac{\hbar}{2} \left( \int d^3 x \Psi_{+}^{*}(\vec{x}) \Psi_{+}(\vec{x}) \right) - \frac{\hbar}{2} \left( \int d^3 x \Psi_{-}^{*}(\vec{x}) \Psi_{-}(\vec{x}) \right) [/math] [math] = \frac{\hbar}{2} \left( \langle _{+} \rangle - \langle _{-} \rangle \right)[/math] ("sea-gulls' whirling clockwise - crows' whirling counter-clockwise") are well-defined quantities, whose evolution through time obeys Classical equations of motion Thus, via their expectation values, quantum wave-packets (of fermions) can be corresponded to Classical 'particles', having four degrees of freedom (pxpypz,sz). Naively, a system of [math]N[/math] fermions, has [math]4N[/math] dof. And, such a system can undergo [math]\left( \begin{array}{c} N \\ 2 \end{array} \right) = \frac{N (N-1)}{2}[/math] interactions, between pairs of particles. Now, each interaction, between wave-packets co-mingles, i.e. entangles, those wave-packets ("two flocks of birds collide, and get all mixed up"). And, each interaction imposes a conservation law, e.g. conservation of momentum [math]\left( \langle \vec{p}_1 \rangle + \langle \vec{p}_2 \rangle = \langle \vec{p}_1 \rangle' + \langle \vec{p}_1 \rangle' \right)[/math]; and spin [math]\left( \langle s_{z,1} \rangle + \langle s_{z,2} \rangle = \langle s_{z,1} \rangle' + \langle s_{z,2} \rangle' \right)[/math], from before to after the interaction (denoted by single quotes). So, if the number of interactions, i.e. number of constraints, equals the number of dof; then the system is "uniquely determined"; and the system must adopt a unique set of values, for those dof ("number of equations = number of unknowns"). Naively, this occurs, for [math]N \ge 9[/math], i.e. up to eight fermions can be mutually co-entangled, and still manifest quantum indeterminacy, e.g. at least four non-determined dof, corresponding to the linear momentum, and angular momentum, of the entangled "ensemble" of fermions ("the momentum & spin, of the 'super-flock', of all the now-co-mingled flocks"). I understand, that interactions "whittle away" remaining, non-determined, dof, until the system becomes uniquely specified, at which point quantum "decoherence" and wave-function "collapse" occur:

I understand that Now, in the beta-decay, of cobalt-60 nuclei, the electrons are preferentially ejected anti-parallel to the nuclear spin. And, for a down-quark, in a neutron, to decay, into an up-quark, in a newly-restructured proton, presumably requires that the down-cum-up quark undergo a "spin flip". Er go, to conserve angular momentum, when the emitting down-cum-up quark undergoes a [math]\Delta S = 1[/math] transition, presumably requires that the electron & anti-neutrino emerge with parallel spins. But, beta-decay is a Weak interaction, which only couples to LH leptons, and RH anti-leptons. So, if the electron & anti-neutrino are spin-parallel, then they are presumably momentum anti-parallel. Er go, the anti-neutrinos presumably emerge, from the decay, in the direction of the nuclear spin, opposite the direction of the electrons, i.e. "they fly apart", "[math]\bar{\nu}[/math] up, [math]e[/math] down": Now, according to the Lepton mixing matrix, the "electron-like" lepton quantum states, that couple to, and are generated by, Weak interactions, are mixtures, of the two lowest-mass lepton generations, vaguely [math]|l_e\rangle \approx 0.9 |l_1\rangle + 0.5 |l_2\rangle[/math]. Therefore, I understand, that what emerges from the beta-decay, of a cobalt-60 nucleus, are a "Weak electron" [math]|e_W\rangle \approx 0.9 |e\rangle + 0.5 |\mu\rangle[/math]; and a "Weak anti-electron-neutrino" [math]|\bar{\nu}_e\rangle \approx 0.9 |\bar{\nu}_1\rangle + 0.5 |\bar{\nu}_2\rangle[/math]. And, I vaguely understand, that, arising from a common [math]W^{-}[/math] boson, the electron & anti-neutrino emerge in an entangled state: [math]\approx 0.9 |e\rangle|\bar{\nu}_1\rangle + 0.5 |\mu\rangle |\bar{\nu}_2\rangle[/math] If so, then when the "Weak electron" interacts with the environment, e.g. electro-magnetically (EM); the "Weak electron" will undergo wave-function collapse, into an eigenstate, of the EM interaction, i.e. a "physical" or "canonical" or "mass" eigenstate. If so, then approximately 3/4ths of the time, the "Weak electron" would collapse into an electron; and approximately 1/4th of the time, into a muon. Meanwhile, the entangled anti-neutrino would collapse, into corresponding-and-complimentary states. Is this true ? If so, could the decay of cobalt-60, coupled with a quantum 'measurement' of the emerging electrons, generate neutrino beams, in "physical mass" eigenstates, not "Weak flavor" eigenstates ?

Following your equations, accurately-if-imprecisely, gravity is approximately constant down to the core (>90% accuracy). Such would imply a local density, decreasing as r-1. Might other terrestrial-like planets possess similar density profiles ? Could "g = constant above the core" be a generally-if-vaguely applicable principle ?

I do not understand -- "momentum flux" is an input, into the SET; yet "relativistic energy" is not ? I understand, that p42 = E2 - p2 = m2(normalized units). So, m2 is an input; and p is an input; but not E ? Momentum flux is "invariant", due to time dilation effects ?