Schwarzschild Solution in Closed Cosmos ?

[Widdekind]

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'.

 

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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.

Edited February 20, 2011 by Widdekind

[ajb]

The field equations are non-linear. This will give you problems. You cannot simply add two solutions together to get another solutions. I think you will have trouble matching the solutions with the matter content you desire.

[Widdekind]

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:

 

Edited February 22, 2011 by Widdekind