Appendix D: Mathematical Excercises in Schwarzschild Geometry

October 18, 2001; Further Mathematical Check on the fit of the Schwarzchild "Two/Sphere" Cosmological Model with Empirical Evidence

A few weeks ago, on this site, I ran a mathematical check of the empirical validity of the Schwarzschild "Two/Sphere" cosmological model. Using a "best guess" estimate of the universal radius as 10 to the 35th Cm. David checked the mass of a black hole with an event horizon volume at 10 to the minus 33rd of the volume of the universe to see if that mass approximated the mass of the universe.

Actually of course, this singularity, from our frame would APPEAR to be spread thinly thoughout all space, and constitute the "dark energy" which makes up 74% of the cosmic mass. The results- a spectacular 10 to the 49th Kg, confirm the veracity of the Schwarzschild model within the limits of current observation.

The results are re-posted here for those of our readers who missed them due to the recent tragic international events...but after this repost, I have placed an additional check of the Schwarzschild "Two/Sphere" model on this site.

To the best of my knowledge, this is pioneering work, and has never been done before. If anyone knows of similar work, please notify the author.

The results, and conceptual basis for the second mathematical check of the Schwarzschild model immediately follow the first post- with remarks.

OF INTEREST TO THE SCIENTIFIC COMMUNITY; "Universal Mass in Schwarzschild/General Relativity/Quantum modeling"

"POST NUMBER ONE"

Samuel A. (Sam) Cox (oringinally posted at 1:58 pm Wednesday September 5, 2001)

Hi:

What would be the mass of a black hole with an event horizon comprising 10 to the minus 33rd the volume of a sphere with a radius of, say 10 to the 35th Centimeters (our GR universe)? I had a note from a gentleman at Bell South, and suggested this as a possible method of very roughly determining the veracity of the concept that microscopic singularity at a distance in scale, is indeed "dark energy"

Since at scales below 10 to the minus 33rd Centimeter, matter takes singular characteristics, the above mass, from the first paragraph, with appropriate corrections and refinements, should equal 74% of the mass of the universe...

Best Wishes,

Sam Cox

(Response: with many thanks to my friend David from the "Hawking Forum" who has one of the most interesting math sites on the internet at: http://home.aol.com/zcphysicsms/modernrelativity.htm ) re: Universal Mass in Modeling; 1:18 pm Friday September 7, 2001

V0/V = 10-33 = r03/r3

r0 = (10-33) 1/3r

r0 = (10-11)r

r0 = (10-11) (1035cm)

r0 = 1022m

M = r0c2/2G

M = (1022m)(3x108m/s) 2/[2(6.67x10- 11m3/kgs2)]

M = 7x10 to the 48kg

NOTES:

The Planck length is well known. The absolute radius of the universe is NOT, though the radius of the KNOWN universe is 10 to the 28th Cm.

The mass of the universe is estimated, using a variety of methods, to lie between 10 to the 52nd Kg. and 10 to the 61st Kg. I gave David "stab in the dark" figures, yet the result is startling. This is rough modeling at its roughest, yet the result is only a few orders of magnitude low

"POST NUMBER TWO"

5:57PM, Wed. October 10, 2001

This second mathematical check of the empirical veracity of the Schwarzschild "Two/Sphere" cosmological model is based on the following concept. We perceive the "dark energy" to be thinly spread out throughout all space from our frame, because of cosmological time separation from the "Big Bang", while in cosmic reality, because our two-selves are inverted and superimposed in energy (matter) and space, we are suspended over an awesome "Black Hole", only 10 to the minus 33rd-35th Cm away.

Our cosmological time separation prevents our being torn to pieces by this "black hole", but because of our geometric location on the two/sphere (see Hawking index part 2;#38022 featured on the home page) our time, and the way we perceive time with respect to distant parts of the universe should be dilated by the distance in time we can "see" within the universal matrix of singularity...ie our time should be dilated by approximately 10 to the 18th seconds, the light radius available to us out to the "bang".

This concept has profound implications about what photons exactly are, and gives hints about how what we view is topologically related to cosmic reality...(comments afterward)

The gravitational time dilation on an observer hovering over a Schwarzschild black hole with respect to a remote observer is:

t = (1 - 2GM/rc2)-1/2t

You want r = h + 2GM/c2 and so it can be expressed:

t = [h/(h + 2GM/c2)]-1/2t

Plugging in the numbers you are wanting me to consider:

t = {10- 35/[10-35 + 2(6.67x10-11) (1049)/(3x108)2]}- 1/210-45s

t = 4x10- 17s

In these calculations, we are relating time dilation caused by cosmic singularity at our location 10 to the minus 33rd-35th Cm above the event horizon of this black hole of universal mass (+/- 10 to the 49th Kg.) to the "distance" in "space/time" we are able to observe back to the "Big Bang".

To get an appropriate proportion, I suggested to David that we use the shortest possible time as a reference frame (10 to the minus 45th seconds) and see how much this brief interval would dilate at our frame (10 to the minus 33rd-35th Cm.) suspended above the event horizon of the cosmic black hole.

The result is that 10 to the minus 45th second dilates to 10 to the minus 17th seconds. Working with this proportion, we need only subtract 17 from 45= 28 orders of magnitude...ie one second of cosmic time is seen by us to take 10 to the 28th seconds to elapse. Our time grinds away at a very slow pace, making it possible, viewing the universe from our frame, to live in a seemingly stable, if slowly destabilizing, particulate, 4D reality.

15 Billion years is a bit less than 10 to the 18th seconds...5X10 to the 17th seconds, so we seem to be off by 10 orders of magnitude...IF...the universal mass is 10 to the 49th Kg!...which it is probably NOT. The mass of the universe lies somewhere between 10 to the 52nd Kg and 10 to the 61st Kg. If we input 10 to the 60th Kg in the above equations, we get an almost perfect match!

Much work needs to be done in the field to narrow the very wide range of possibilities regarding absolute universal mass and radius, for example. Nevertheless, the results of these two mathematical experiments could actually be programmed into a computer with the results of increasingly accurate field work to assist in determining the configuration of the cosmos!

Best Wishes,

Sam Cox

For further information please consult Schwarzschild Geometry.