Simply Relativity, Cosmic Coincidences, & A Mathematical Mystery

After several busy weeks of travel and actual work, I finally have returned to this blog. Since I haven't posted in a while, I will post three at once...

1. I will start with a link to my webpages on the Bondi k-calculus. For a lot of people, the special theory of relativity is a complete mystery, with assumptions that the theory is just speculation or that it requires mathematics so complicated that no ordinary person could hope to understant it . This is not true.
The Bondi k-calculus is a very simple method of looking at special relativity, and requires no math past grade school. It is also accurate - many advanced mathematics and physics courses in university teach this method.

Having said all that, it is also true that there are possible violations of the theory of relativity. At the end of this posting I will give two of the possibilities that are currently being explored.

2. The Cosmic Coincidence. This mystery is well known by physicists, but it is not well known among the general public.
As we understand it, everything in the universe fits into three categories. There are photons, which form light and radio waves and x-rays. There is matter, both the ordinary stuff of stars and planets as well as an unknown form of dark matter. And there is dark energy, a mysterious substance that fills the entire universe causing it to expand.
Now comes the problem. The density of photons decreases with the fourth power of the size of the universe, matter density decreases with the third power of the size of the universe, and dark energy (probably) stays constant. Since the universe is rapidly growing, and has been for 14 billion years, that means the three densities must be radically different.
Wrong! Dark energy and matter densities are about 2 to 1, while photons are maybe a factor of 10 less dense. In the future matter and photons densities will drop to almost zero. In the past they were billions of times larger than the dark energy density.
So why do we happen to live at the one point in history at which the three values are almost equal?

3. Now for something completely different. A mathematical mystery! (This result was published last year in a paper by Sondow, and has been summarized as part of John Baez's weekly posting )

The result is simply three formulae that give the values of pi , the natural exponent e, and Euler's constant γ.

       2  1/1      22   1/2     23 × 4  1/3       24 × 44    1/4
e =  ( - )     ( ----- )     ( ------- )     ( ------------ )     ...
     1         1 × 3          1 × 33          1 × 36 × 5

π     2  1/2      22    1/2^2      23 × 4   1/2^3          24 × 44      1/2^4
- = ( - )     ( ------ )       ( --------- )           ( ------------- )     ...
2     1         1 × 3              1 × 33                  1 × 36 × 5

 γ     2  1/2     22   1/3     23 × 4   1/4      24 × 44    1/5
e  = ( - )     ( ----- )     ( ------- )     ( ----------- )     ...
     1         1 × 3         1 × 33           1 × 36 × 5

Three completely different fundamental mathematical constants, with three nearly identical formula!


Answers to part 1:
What are the exceptions to relativity? There could be several, but these two are popular in the literature right now.
- Einstein assumes that all light travels at the 'speed of light' with no possible variation. In reality, it is possible that high energy light (x-rays, γ-rays) might travel at a different speed than mid-energy light (visible light) and than low energy light (radio waves, microwaves). This would introduce some changes to the theory, although experiments looking for this effect require the speed difference to be miniscule.
- The theory of relativity also assumes empty space, or at the least that there is just bits of stuff in an empty background. But the Higgs mechanism which provides masses for all matter requires space to be filled with a form of vacuum energy, so this assumption is not true. While the Higgs mechanism does not affect relativity, it is possible that there is a related mechanism (sometimes called the Bumblebee model) in which the particles are spinning, and their contribution to the vacuum energy (which would require the spins to align like in a magnet) would include a preferred direction along the common axis of their spin. Again, experiments require this effect to be tiny.