Saturday, October 18, 2014

Planck Units: M = R = t

In response to some evocative comments, here is an old post from October 1, 2007! The Planck units point to a quantum nature of space and time. Though this post is celebrating its seventh birthday, this marvelously simple equation is still not common knowledge.
"I perceive the Universe as a single equation, and it is so simple..."

--LT Barclay in STAR TREK TNG: "The Nth Degree"

Posting was light the past week, but some work got done. Today we'll return to the subject of Planck units. Max Planck started as a conservative physicist studying atomic spectra. The ultraviolet catastrophe led him to a "act of desperation," the quantum value h. Planck was also instrumental in getting a patent clerk's first papers published in 1905. If not for Planck, the world might have taken decades to hear of Einstein.

Planck noted that combinations of h, c and G led to this "universal" system of units. At the time he had no way of knowing whether h or c were constant. Some science types get lazy and say that h = c = G =1. They are not equal, or they could be used interchangeably.

We'll use Planck's units to express something more useful. A basic principle states that scale R of the Universe is its age, a timelike separation from the "Big Bang." R and t are related by factor c, the "speed of light."

R = ct

This equation (1) caused the Big Bang. As t increases, the Universe expands.

R/l_pl = ct/l_pl

Now l_pl = ct_pl, so:

R/l_pl = t/t_pl

Expressed in Planck units, equation (1) becomes:

R = t

We can simply express that size of the Universe is related to its age. This may appear more palatable to those used to thinking that c is constant.

The Universe can't expand at the same rate forever, for Mass and Gravity slow it down. We do some calculations, and c is further related to t by:

GM = tc^3

Expressing equation (2) in Planck units:

M/t = c^3/G = m_pl/t_pl

M/m_pl = t/t_pl

The Planckian expression of GM=tc^3 was also noted by bloggers Thomas Dent and Lubos Motl. Using Planck units can be misleading, because they are not all constant. Now we can state both equations (1) and (2) in a single line:

M = R = t

Repeat: This must be the simplest equation ever! It relates everything you want to know about the Universe but were afraid to ask: Mass M, size R, age t, expansion rate and how it slows with time. This shows just how powerful mathematics can be. According to STAR TREK, one line may explain an entire Universe.

Planck is not the only one who started as a conservative physicist. When these equations are worked out, the appeal is hard to deny even for the conservative. Arriving at a simple solution makes all the challenges of science worthwhile. The simplicity may someday be noticed by physicists, possibly in the 24th century. This may be an equation far ahead of its time.

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Blogger Ed said...

Evocative? I'll take that as a compliment. ;)

It occurs to me that the ultraviolet catastrophe and inflation are, mathematically speaking, intimately related problems. And, where quantized light and the introduction of quantum mechanics solved the ultraviolet catastrophe, quantized space solves inflation and leads directly to GM=tc^3.

Quantized space also suggests that gravity and other forces can interact in only a maximum of 12 directions when considering the smallest of scales; at double the Planck length, the number of directions has blossomed to 44. The longer the scale, the more it seems to be a smooth, continuous universe. Does that sound like a possible route to quantum gravity to you?

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You might find Jose Almeida's 4-D optical relativity/ 5-D monogenic (tau as 5th dim.) theory interesting: "Can physics laws be derived from monogenic functions?", which, using Planck units and geometric algebra, treats gravity as a (potentially tensor) refractive index, thus as a varying speed of light, treats proper time as a dimension of the same signature as the spatial dimensions (and time as an affine parameter; opposite to general relativity). Proper time since the big bang is treated as the radius of a hypersphere, with galactic clusters maintaining constant angular separations on the 3-D surface of the hypersphere. He derives explanations of the Hubble relation, galactic rotation, electromagnetism and particle symmetries.

There is visualization software (GAViewer) for the surprisingly useful G(4,1) conformal geometric algebra implicitly used by Almeida, which adds two null dimensions (origin and infinity points) to 3-D algebra, giving easy unified representations of circles/lines/points/point pairs/spheres/planes/meets and joins. CGA tutorial

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