Widdekind

Thanks again for the clarifications. If another 'qualitative' question be permitted, then, if the Curvature Tensor is 'inverted' (Guv --> -Guv), what does that do, to the curvature? In the rubber-sheet analogy, does that make the 'bowling balls on the trampoline float, instead of sink' ?

Masses in circular motion, about a center of attraction, by pure geometry, are undergoing an acceleration [math]a = \frac{v^2}{r}[/math]. And, when observing distant clusters of galaxies, doppler shifts in the light from the moving galaxies gives a 'velocity dispersion', the spread in (radial, along LOS) velocities, which is assumed equivalent to the "v2" in the above equation. Then, assuming the same spherical symmetry, G M / R2 is assumed to be the gravity force explaining the observed quasi-circular motions. That works well, for spheroidal, "elliptical" galaxies, but seems wildly inaccurate for a non-spherical, disk-dominated, spiral galaxy.

What about rotational motions, like the rotation of a star? Are you saying, that a (hypothetical) non-rotating star, would produce the same curvature, in spacetime, as a (realistic) rotating one ? If the SET 'looks like' velocities factor in, then how is it 'invariant' ? And, again, what about 'non-inertial' velocities, like rotations, which everybody would see the same, and which everybody would identify as different, from a static situation ? What about motions being 'invariant' ? What makes 'thermodynamic motions' immune to the 'invariance' of the SET ?

The fabric of spacetime is, itself, a 'substance'. Could not 'wave-like' distortions, of that fabric, account for the 'Aether' ? Please ponder (and explain ??), that EM waves travel at precisely the same speed, as gravity waves, as if they both were 'similar sorts' of disturbances, of the same underlying spacetime fabric.

According to Wikipedia, the Stress-Energy Tensor is -- that is not frame-dependent ? The article seems to say, that the form of the S.E.T. changes, in an inertial reference frame co-moving w/ the fluid.

Is not 'Virial Mass' calculated from [math]\frac{<v^2>}{R} \equiv \frac{G M_{vir}}{R^2}[/math] (where the velocity dispersion, and radius, are observed quantities) ?

Trilobites, from c.525 Mya, look like 'derived forms' of Praecambridium sigillum & Skania, from c.40 Myr before. Indeed, the former looks like it had three armored lobes. If so, then 'proto-Arthropods' can be identified, in the Ediacaran period. Is this 'obviously false' ?? Fig. 1 -- Praecambridium sigillum Fig. 2 -- Skania fragilis Fig. 3 -- Trilobites

Thanks for your referenced report. If you were to 'invasively intervene', between two interacting charges, and 'observe' their exchange photons... you would become part of the interaction; you would 'absorb the force influence' which 'had been intended for the other charge'; you would change the dynamics of the other charges -- yes ? I understand, that 'virtual' particles are not constrained to be 'Einsteinian' (E2 = m2 + p2), and thusly 'decay' according to [math]\Delta E \, \Delta t \approx \hbar[/math]*. * This gives the impression, that the Einstein relation, is some sort of 'stability criterion', according to which a 'stable vortex forms', as it were, w/ a weather analogy.

Does charge distort spacetime ? And, if so, does it curve spacetime, 'thru hyperspace', like matter, or would it create 'internal compressions & expansions', wholly within spacetime itself ? (Is there a convenient way of translating your advanced description, or citing some source less opaque than Wikipedia's pages?)

Classical Gravity and E&M are strikingly similar, mathematically. And, the former has been extended, relativistically, as the interplay, between mass curving spacetime, and curved spacetime determining the motions of mass. So, is it possible, that E&M can be similarly extended, and similarly interpreted, as charge curving spacetime (albeit, in a 'charge-like', not 'matter-like', way) ?? ...classical Gravity ~ classical EM ............|.........................| relativistic Gravity ~ relativistic EM ?? Is this a 'moon-is-made-of-cheese', 'look-and-laugh' sort of suggestion ?? [math]G_{\mu \nu} \approx T_{mass, \mu \nu} \oplus T_{charge, \mu \nu}[/math] (all with 'appropriate' constants & signs) If this was true -- 'suspension-of-disbelief-for-ten-seconds' -- then wouldn't, given that there's only one spacetime fabric, large-scale charge distributions influence neutral matter motions, through the shared medium, of curved common spacetime ? Is it not true, that perhaps 3/4ths of the galactic-disk ISM is ionized ? So, 'in-the-last-few-seconds-of-suspension-of-disbelief', would not such vast plasma formations modify spacetime curvature 'noticeably' -- perhaps, thereby, explaining the 'gravity anomaly' of galactic disk rotation curves ? If so, then there should be 'gravity anomalies' associated with ionized plasma pockets, w/in the disk ISM: Since plasma pockets are produced by Super-Novas, so that Star Burst Galaxies ought, then, show 'stranger' galactic rotation curves... And, would not Solar flares, and coronal mass ejections, belching bursts of plasma towards this planet, then produce 'gravity anomalies' too?? So, is this suggestion 'obviously' wrong??

The mass stress-energy tensor incorporates information, about not only the mass density in space, but also the motions of that mass. Moving mass gravitates differently, than the same static.

The energy density of magnetic fields is [math]U \approx \frac{B^2}{\mu_0} = \epsilon_0 \, c^2 \, B^2[/math]. For an inter-galactic magnetic field strength of ~10-12 T, this produces a mass-density equivalent of 10-35 km/m3. This does not appear to be a principal component of the critical density.

The observed gravitational anomalies, seen in space (where theory & observation don't yet mesh), seem to vary, based on spatial scale, with one type of anomaly, on ultra-large scales (clusters, cosmic web); and, another on more middling scales (galaxies): If baryonic disk mass (Mb) is related to the disk velocity (Vf), then the disk material dominates its own dynamics. Ultimately, disk dynamics are not determined by, for example, the mass of the central bulge. As a disk is a highly non-spherical object, g = G M / R2 may be highly inaccurate. [math]\Lambda CDM[/math] predicts that [math]log(M_b) = 8.1 + 2.7 \, log(V_f)[/math], whilst observations show that [math]log(M_b) = 6 + 3.7 \, log(V_f)[/math]. These formula become equal, when [math]V_f = 10^{2.1} \approx 130 \, km/s[/math], corresponding to [math]M_b \approx 10^{13.8} \approx 60 \times 10^{12} M_{\odot}[/math]. Below that mass, the 'detection fraction', defined as the ratio of observation-to-theory, is: [math]f_d \equiv \frac{10^6 \, V_f^{3.7}}{10^{8.1} \, V_f^{2.7}} \approx \frac{V_f}{130 \, km/s}[/math] If the phenomena is not an artifact of detection, perhaps progressively more massive galaxies, with presumably stronger gravity, retain progressively more of their expected baryonic component, the lighter galaxies losing the same, to galactic winds, etc. ??

If all EM interactions are conveyed w/ virtual photons, then does that imply that Gravity interactions are conveyed by virtual gravitons ?

In flat spacetime, differential distances & times are measured within the spacetime fabric. In the Schwarzschild spacetime, differential radial distances can be defined within the spacetime fabric, from [math]r \equiv circumference / 2 \pi[/math]. What about 'time', in the FRW cosmology solutions? Is 'time' measured fully within the spacetime fabric, so that the time is the 'integrated length' measured along the hyper-surface of spacetime? Or, is 'time' an 'absolute coordinate', measured along the 'vertical hyperspatial axis' of a closed cosmic spacetime??

From the following ? H = 75 km/s / Mpc = 0.25e-3 c / Mpc 1 Gly = 300 Mpc Thus, at 300 Mpc, the recession velocity is 0.75e-1 c, or ~75 Kly / Myr.

Mass curves spacetime, and curved spacetime tells mass how to move (Wheeler. Journey into Gravity & Spacetime). Thus, when one witnesses an anomalous movement of matter (e.g., Galactic Rotation Curves), one is witnessing an anomalous curvature of spacetime, not an anomalous matter distribution directly. Therefore, is it possible to explain Galactic Rotation Curves, without invoking 'Dark Matter', so much as 'Dark Curvature' ??? Simply as a suggestion, if our Cosmos is closed, and if under-dense, evacuated, voids 'poof outward', wouldn't that extra 'outward swelling' of the spacetime, of voids surrounding some mass concentration (e.g. galactic group), impose a 'pinching off' effect, on the throat of the gravity well, of that mass concentration, thereby increasing the curvature, and the sensation of gravity???

Please ponder the Flamm's Paraboloid interpretation, of the 'hyper-spatial distortion', of the space-time fabric, induced by mass. The FP can account, for the increasing stretching of space, as one approaches the mass. But, from the same Schwarzschild metric, we know, that time is progressively compressed, as one approaches the mass. Thus, how can you 'stack' a series of FP's, which show the spatial stretching, in such a way, that they are closer together near the mass, and farther apart well away from the same?? I interpret Rudy Rucker's fig.137 to indicate the answer -- mass does not simply 'pull down' on the spacetime fabric, but actually induces an increasing, spiral, 'twist' to spacetime. To wit, the FPs are not 'flat', planar, in 2D -- they 'twist' through 3D, as the spacetime fabric is 'pushed in, and hooked down'. Both the 'push in', and 'hook down', increase, as one approaches the central mass: In a cosmological context, mass over-densities, then, 'do what you'd expect' -- to wit, regions of mass over-density, have a globally reduced Radius of Curvature: Note, that mass causes current cosmic time-slices, to warp 'downward', through 3D. So the 'throats' of black holes, can be longer & deeper, than the Radius of Curvature, of the current cosmic time-slice. Thus, when we switch, from viewing spacetime (1+1D), to simply space, at one given time-slice (2+0D), we can visualize the 'extra curvature', of space, 'hooking downward' deep down inside the 'throats' of gravity wells, with color:

Is it 'mere mortal-humanly feasible', to solve for the Schwarzschild metric, when the gravitating central mass, is embedded in a non-vacuum, uniform/isotropic/homogenous background closed cosmic spacetime ?? What is the metric? Naively, simply 'slapping the two together', one could come up with: [math]ds^2 \propto \frac{dr^2}{1-\frac{r}{R_C} - \frac{R_S}{r}}[/math] Such a solution would be 'Schwarzschild-like', near the gravitating body, but meld back into the 'Friedmann-like' background, at great distances. Note that if you could solve for the metric, you'd be able to tell, whether gravity over-densities, in a closed cosmos, 'sag down' radially inward, or 'poof outward'. According to Rudy Rucker (Geometry, Relativity, & Fourth Dimension, p.112+): a Singularity, in infinite, flat, spacetime, deforms the fabric of spacetime 'downwards & backwards', warping the time axis, of the Singularity, 'over backwards', bending the same through 'hyperspace' by 90 degrees, from 'vertical' to (asymptotically) 'horizontal' Singularities, in a closed Cosmic spacetime, 'invaginate inwards', towards the 'hyper-center' of the spacetime fabric as indicated in the following figures: Since, as seen in the first image, an infinite amount of time-line is warped, by the Singularity, observers far from said Singularity, would experience an infinite number of time-slices ('moments') pass, before infalling material was observed to reach the Event Horizon of the Singularity. However, in a finite spacetime, such as a closed Cosmos, only a finite amount of time-line can be warped. Such suggests, that in a finite spacetime, infalling material would be witnessed reaching a Singularity's Event Horizon, in only a finite amount of time. 'Arguing from pretty pictures', 'time', as understood, is a phenomenon inside spacetime -- the spacetime fabric can warp to-and-fro, through 'hyperspace', even curving backwards and 'heading in the opposite direction' through 'hyperspace', w/o affecting the experienced forward flow of time, observed inside the spacetime fabric ?? Is this correct ? Please note, that we are explicitly assuming a non-flat, non-infinite, non-Minkowskian background spacetime fabric -- aren't most black holes studied, in infinite, flat, spacetimes, only ?? Note that these pictures imply, that, to extend the iconic 'rubber sheet' analogy, to a closed Cosmic spacetime, would be akin, to extending that rubber sheet, into a giant, earth-sized 'batman suit', a rubberized coating wrapping around the whole world. 'Bowling balls', placed upon the world-wide rubber skin, would 'invaginate inwards', looking locally like the standard scenario, but globally looking like Rudy Rucker's fig.138.

This site seemingly says something similar -- somehow, although we can see the CMB, from the Big Bang (t=0), and early quasars/AGN (t=1Gyr), there seems to be some sort of data gap in between, corresponding to a period of chaotic mergers amidst malformed proto-galaxies: Perhaps the first stars were still shrouded by neutral primordial gas? Perhaps, then, whole Globular Cluster caliber objects formed, hidden in cognito, shrouded in cosmic cocoons, of as-yet-un-re-ionized IGM ??

Does the Sun mass more than measured (by about 15 kg) ??? The standard Schwarzschild solution, is a modification, of flat Minkowskian spacetime (to which it returns, in the limit of zero central mass). And, flat Minkowskian spacetime is an admissible solution, of the Friedmann equations, in non-curving, empty, & hence flat, spacetime (k = 0, rho = 0). However, in a closed & curved spacetime, the first Friedmann equation demands the inclusion of a curvature term, incorporating the effects of the background spacetime (positive, closed) curvature. Working through the numbers, this slight effect, in our modern universe (H = 75 km/s/Mpc) works out to about 15 kg. To wit, our star contains roughly 15 extra kilograms of mass, whose spacetime curving effects are 'dissipated', by the background Hubble Expansion, of the spacetime, into which our star's 'Schwarzschild-like solution' is 'anchored': [math]H^2 = \left( \frac{\dot{R}_c}{R_c} \right)^2 = \frac{8 \pi G \rho}{3} - \frac{c^2}{R_c^2}[/math] [math]\frac{8 \pi G \rho_{actual}}{3 c^2} = \frac{1}{R_{c,observed}^2} + \frac{1}{D_H^2}[/math] [math]\frac{r_{S,act}}{R_{\odot}^3}= \frac{1}{R_{c,obs}^2} + \frac{1}{D_H^2}[/math] [math]r_{S,act} = r_{S,obs} + \frac{R_{\odot}^3}{D_H^2}[/math] [math]\frac{r_{S,act}}{r_{S,obs}} = 1 + \frac{R_{\odot}^3}{r_{S,obs} D_H^2} \approx 1 + 7.5 \times 10^{-30}[/math] [math]M_{\odot} \rightarrow M_{\odot} + 15 \, kg[/math] I do not see, where I have made any math mistakes -- is this not valid ?? From 'physics intuition', it 'feels plausible'. EDIT: The sun is a gravitationally bound body -- has its spacetime 'seceded' from the Hubble Flow? If so, then the sun's local spacetime is not expanding, and cannot be described as such (no +15kg).

Why don't we see that first 'flash' of pan-cosmic star formation, shown on the above figure, but absent from the HDF & HUDF ? We can see quasars, from ~1 Gyr after the big bang -- why can't we see the 'star burst' from about the same time, especially if it was super-bright, including copious quantities of ultra-massive 1st generation 200 solar-mass stars ??

The interior, of a spherical massive body of uniform density, has a constant radius of curvature. Thus, the interior of such objects satisfies all of the requirements, for the FRW cosmological metric (uniformity, isotropy, homogeneity, constant curvature). Now, the Schwarzschild solution is static -- the downward pull of 'gravity on the rubber sheet', is offset, by the upward pull of 'tension in the trampoline'. So, why can you not extend that interior solution 'around', into a closed 'mini cosmos', where the contractile tensions still exactly offset the oppositely directed 'gravity' ?? For example, our star, with R = 700 K-km, rS = 3 km, has a Radius of Curvature Rc = 340 M-km, or roughly 2.5 AU. Such a 3D hyper-spherical shell would have a hyper-volume of [math]S_3 = 2 \pi^2 R_c^3[/math], and, hence, a total mass of [math]S_3 \times \bar{\rho}_{\odot} \approx 550 \times 10^6 M_{\odot}[/math]. Note that this 'mini cosmos' is a static solution of the first Friedmann equation: [math]H^2 = \left( \frac{\dot{R}_c}{R_c} \right)^2 = \frac{8 \pi G \rho}{3} - \frac{c^2}{R_c^2}[/math] since: [math]\bar{\rho}_{\odot} = \frac{M_{\odot}}{\frac{4 \pi R_{\odot}^3}{3}}[/math] so that: [math]\frac{8 \pi G \rho}{3} = \frac{2 G M_{\odot}}{R_{\odot}^3} = \frac{c^2 \, r_S}{R_{\odot}^3} = \frac{c^2}{R_c^2}[/math] because [math]R_c^2 = R_{\odot}^3 / r_S[/math]. However, this 'mini cosmos' would then have to be 'imploding', according to the second Friedmann equation: [math]\frac{\ddot{R}_c}{R_c} = - \frac{4 \pi G \rho}{3} = -\frac{1}{2} \times \frac{c^2}{R_c^2}[/math] [math]\therefore \ddot{R}_c = - \frac{c^2}{2 R_c} \approx -130,000 \, m/s/s[/math] Why is the interior solution static, in the Schwarzschild case, but rapidly imploding, in the Friedmann case ?? In the former, the 'downward' press of 'gravity' on the rubber sheet statically offsets the 'tension in the trampoline'. Yet, that trampoline is 'mathematically anchored' out at infinite radius -- is it that implicit 'anchor' which keeps spacetime stretched taught, in the former case, and without which, spacetime swiftly shrinks, in the latter case ?? When a rope, of constant tension T, is wrapped once around a spool, the inward 'crushing' tension force is, also, T. By such an analogy, note that the 'inward acceleration', of the Radius of Curvature, is: [math]\ddot{R}_c = - \frac{4 \pi G \rho R_c}{3} \propto - density \times circumference[/math] Thus, in some sense, the 'line tension', induced in spacetime, by the presence of matter, is proportional to the line integral of the mass density, on a 'Great Great Circle', looping 'all the way around spacetime'. Somehow, mass induces contractile tensions, threading through spacetime, that 'make spacetime want to curl & ball up' ('through hyperspace'). Indeed, in turn, matter density is the integrated product, of number density, and mass per particle. In some sense, every bit of matter, is like a 'bear trap', spread open against the spring, and 'slid, flat, into spacetime' (cf. Flatland). 'Mass' measures the 'spring strength' of the 'bear trap'. The more 'bear traps', and the stronger their springs, you have in your giant 'chain', stretching around spacetime, the more 'pec-fly jaw-snapping force' they generate -- and, hence, the greater the spacetime curvature and/or the greater the 'inward deceleration' of a closed cosmic spacetime fabric*. * There seems to be a hyper-spatial directional preference, to the 'pec-fly jaw-snapping' phenomenon of matter. Might that mean, that anti-matter is the same 'bear trap', but set into spacetime 'the other way', 'face down in the dirt' instead of 'face up' ? If so, anti-matter would cause spacetime to 'curl the other way', hyper-spatially speaking. In turn, anti-matter would then have anti-mass (negative gravity w.r.t. regular mass).

It is possible to estimate, the amount of spacetime 'stretch', along any diameter, threading through some star (or other spherical massive body). That spacetime 'stretch' would be experienced, for example, by a probe, sent 'swimming' straight through the sun -- you'd calculate, from the sun's circumference / 2 pi, that the probe would have to travel 2R ~ 1.4 M km... but it might have to actually 'swim' for another ~3 km, that being the Rs of the sun, which is a simple order-of-magnitude estimate, for the effect: That extra 'stretch' of spacetime is meaningful, and is a real effect. The FRW equations, for a uniform density universe, essentially treat the whole cosmos as the 'inside of an ideal star', of uniform density, and constant Radius of Curvature. The spacetime 'stretch', in the static Schwarzschild solution, is essentially the same phenomenon, as the spacetime 'stretch' seen in the Hubble Expansion, although the latter is a non-static solution to the GR equations. I'm simply saying, that the 'stretch' of spacetime is 'real', and measurable, and is meaningful, since 'star probes' would have to swim that extra distance, through the dense star; and, 'space probes' would have to fly that extra distance, between receding galaxies, in this matter filled 'super star' universe.

Thanks! Does that imply, that there was no dust, nor molecular H2 ? If all the monoatomic hydrogen atoms, were in their ground states, photons would need at least 3/4 x 13.7eV to excite them into 'level 2' states -- so, when the CMB cooled below ~10eV ~105K, the thermal background 'decoupled' from the now neutral matter. But, would molecular hydrogen or dust have absorbed some spectra, had they been present, thereby proving their absence ??