## Monday, October 01, 2007

### M = R = t

"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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riqie arneberg said...

While R=t may express the size of the visible universe, it by no means applies to the greater, possibly infinite universe.

12:11 AM
Anonymous said...

but...
If M = R = t (1)
and GM = tc^3 (2)
then substituting (1) into (2)
G = c^3 ???

7:42 AM
L. Riofrio said...

No, try again. As mentioned, Planck units can be confusing.

7:59 AM
brainunderdevelopment said...

9:09 AM
Kea said...

Thanks for taking the time to explain this simple equation to us again. As you say, children interested in cosmology will find it both interesting and easy to understand. And it helps us think about mass operators in particle physics, too, which are useful for deriving particle masses.

10:01 AM
alex kaplan said...

If it will be proved that Planck units are real physical quantitis,I won't surprise if you will receive Nobel Prize for this

11:00 PM
alex kaplan said...

I am not joking,it is really ingenious.

1:42 AM
Robert said...

The simplicity of M=R=t seems to follow from the satisfaction of GM = tc^3 and R = ct by the Planck units themselves, as quickly verified:

m_pl = (h_bar c/G)^1/2
l_pl = (h_bar G/c^3)^1/2
t_pl = (h_bar G/c^5)^1/2

m_pl G = (h_bar cG^2/G)^1/2 = (h_bar cG)^1/2
c^3 t_pl = (h_bar c^6 G/c^5)^1/2 = (h_bar cG)^1/2

m_pl G = c^3 t_pl

c t_pl = (h_bar c^2 G/c^5)^1/2= l_pl

so the units represent the state of the universe at t_pl. Did Planck know of the space/time equation?

Thanks for the lessons and space program info.

6:08 PM
name said...

actually, that's brilliant. Thank you. I'm going to pass that on to a couple of people.

9:21 AM
Anonymous said...

Just for fun, here is the formula for the universe. I think the most intriquing aspect of it is that the fundamental rule is not "a single equation" as everyone wants to discover, but a set of concurrent inequalities that sometimes produce an equation.

Always not equal. Never Infinity. Sometimes equal. Sometime not False.

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5:03 AM
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5:52 PM
Thomas J Mauro said...

SO the universe is not infinite? Will not expand forever? Will retract back to a singularity once all the energy is sucked into multiple singularities then crunch back down to one single point for another big bang? Or is there a multiverse and we live in a membrane wave that can collide with another existing membrane wave thus creating extreme reactions and causing a BIG BANG. IDK you tell me.

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