Einstein's Sphere

Everyone should read a book called "RELATIVITY: A Simple Explanation that Anyone Can Understand" by Albert Einstein. No one understood the subject better than Einstein himself. He used methods, like imaginary time, that have been abandoned by today's "relativity" books. In paperback this book is also much less expensive than the heavy book the college forced you to buy! In his writings, Einstein even mentioned changing the speed of light.

In Chapter 31 Einstein attempts to imagine an entire Universe! Like Pythagoras 2500 years before, Einstein is motivated by a search for harmony. He follows what would be called the Cosmological Principle: The Universe looks the same no matter what direction one looks, and every bit resembles every other bit. He rejects a flat Universe, for his General Relativity shows that Space/Time is curved. He rejects the idea of boundaries and considers the Universe “finite yet unbounded”. The obvious analogy is a sphere.

This 4-dimensional spherical Space has a finite volume given by:

V = 2 $\pi$^2 R^3

Where R is radius, with dimensions of length. (If anyone can't abide by this, please complain to Einstein.)

Here Einstein found a conflict. The very gravity which causes Space/Time to be curved would cause the sphere to collapse. Here we could add, "Unless it were already expanding." An expanding Universe it would have been one of history's great predictions. Instead Einstein introduced a fudge factor, a repulsive "cosmological constant" preventing the Universe from collapsing. When Edwin Hubble's observations showed that the Universe is expanding, Einstein would call the cosmological constant his greatest blunder.

We can express the expanding Universe simply:

R = ct

Again R has dimensions of length and c has dimensions of distance/time. For an expanding Universe, it is axiomatic that R is some multiple of t.

The Universe can't expand at the same rate c forever, for gravity slows it down. We do some math and get:

GM = tc^3

Where GM combines mass of the Universe with its gravitational constant.

Together these simple expressions form a solution to the Einstein-Friedmann equations with stable density:

$\rho$f = (6 $\pi$ G t^2)^{-1}

Here we encounter an interesting difference. If an initial mass M is distributed among this spherical volume V, we get an initial density $\rho$i of:

$\rho$i = M/V where M = (tc^3)/G

$\rho$i = (2 $\pi$G t^2)^{-1}

In addition to expanding and slowing the Universe, GM = tc^3 drives it toward the stable density.

The difference is made up by the matter we are made of. When the Universe is "underweight," quantum mechanics predicts that matter will appear via pair production. The amount of this matter is the difference between $\rho$i and $\rho$f, or 4.507034%. This unique prediction is precisely matched by the Wilkinson Microwave Anistropy Probe.

The Albrecht/Maguiejo paper can be a help here. Refer to their equation (10). Here R (not a) is scale and e is the deviation from critical density.

e = $\Omega$ - 1

edot = (1 + e)e(Rdot/R)(1 + 3w) + 2(cdot/c)e

Now we have w = 0,
(Rdot/R) = (2/3t) and (cdot/c) = (-1/3t)

edot = e^2 (2/3t)

Is it not reassuring that the other terms cancel? When t is low e is large and a large edot drives density toward a critical value. Today, when t is billions of years and e is very nearly zero, little mass is being created. Just as scale R began in a Bang and has been slowing since, the amount of mass creation has also been slowing and is now nearly zero.

Can you believe some people act as if this isn't a Theory? It is painful to say that new ideas take time for scientists to figure out. Papers on this subject (including Albrecht and Maguiejo's) face great difficulties in publishing. Arthur Eddington joked in the 1920's that only 3 people understood Relativity, though he couldn't think of the third one! We are doing much better.