Friday, May 23, 2008

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.

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12 Comments:

Blogger nige said...

This comment has been removed by the author.

12:15 AM  
Anonymous surprised said...

"quantum mechanics predicts that matter will appear via pair production."

Do you mean matter-antimatter?

12:59 PM  
Blogger nige said...

Surprised,

That's obviously what is meant because that's Dirac's prediction from the spinor of his famous equation. He had to modify the Hamiltonian and one consequence is antimatter.

It was Schwinger, however, who found that pair production occurs spontaneously in the vacuum if the electric field strength exceeds a threshold of 1.3*10^18 volts/metre. See equation 359 in Dyson's 1951 Lectures on Advanced Quantum Mechanics, Second Edition, http://arxiv.org/abs/quant-ph/0608140, or equation 8.20 of Luis Alvarez-Gaume and Miguel A. Vazquez-Mozo, Introductory Lectures on Quantum Field Theory, http://arxiv.org/abs/hep-th/0510040

One thing that really annoys me about popular books on the subject is that they claim - falsely - that pairs of fermions are constantly popping into existence and annihilating everywhere in the vacuum, without limit.

Actually, that only occurs with a distance of 32.953 fm from an electron (see http://nige.wordpress.com/2007/06/13/feynman-diagrams-in-loop-quantum-gravity-path-integrals-and-the-relationship-of-leptons-to-quarks/ ).

So all those physicists who state that the entire vacuum is a seething foam of Heisenberg-formula controlled pair-production and annihilation (i.e., looped Feynman diagrams), are talking out of their hats.

It's been known for over fifty years that there is a cut-off on the pair production. It's pair production that allows pairs of short-lived (virtual) fermions to become briefly polarized in a field, which opposes and partially the primary electric field, thereby explaining physically the reason for electric charge renormalization.

If pair-production occurred throughout the vacuum, there would be no infrared cutoff on the low-energy range for running couplings, and the observable electric charge would get for ever smaller as you got further from an electron. This doesn't happen, proving that pair production-annihilation certainly doesn't occur everywhere in the vacuum.

11:52 AM  
Blogger nige said...

Hi Louise,

Sorry for the first comment I made about curved spacetime, which on re-reading was so awfully written.

I made a comment somewhere else at about the same time on Sunday and offended someone there, and I'm sorry for the tacky comment about curvature I also made here.

11:57 AM  
Blogger L. Riofrio said...

Nige's comments are thoughtful and always welcome. Matter seems to have been favoured over antimatter, something which I am working on all the time.

4:13 PM  
Anonymous Peter said...

This hypothesis has some testable predictions. An easy one is the net mass of the observable universe. plugging in 13.7B years for time and accepted values for G and C, I get M of 1.75*10^53. The exact value of M isn't determined to any precision, but based on a quick check, http://en.wikipedia.org/wiki/Observable_universe , the calculated value is within reason.

11:20 AM  
Blogger CarlBrannen said...

Now I understand. However, the equation ρi = (2 πG t^2)^{-1} needs to have the pi squared. Then the 4.5% follows.

12:32 AM  
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No, there is a hemisphere cap to the cylinder, since it was removed from the sphere. Easiest way would be to drop it in a bucket of water with known volume and markings, then find the difference in the water levels.

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