Why no 'Mini Cosmoses' ??

[Widdekind]

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, r_S = 3 km, has a Radius of Curvature R_c = 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 ??

 

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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).

Edited February 15, 2011 by Widdekind

[Widdekind]

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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).

Edited February 15, 2011 by Widdekind