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## Natural units

December 31st, 2010

As part of my not-very-successful quest to keep abreast of the latest developments in science, I just finished Einstein’s Relativity:The Special and the General Theory which was, as you’ll recall was a big hit when it came out around 1915. Right towards the end, you get the famous formula E=mc^2. Reading this, I recalled someone pointing out that, in a sensible system of units, c (the speed of light in a vacuum) would be set at 1, so the equation would just say Energy=Mass.

That got me thinking about how you would design units from scratch, in the light of our current knowledge of the universe. The designers of the metric system, at the time of the French Revolution, used the circumference of the earth as the basis for their measure of distance and volume, then used that, along with the density of water for the measure of mass. For time, they stuck with the Babylonian second ( there were, I think, some failed proposals to use a decimal system).

Given our understanding of the Big Bang, it seems to me that it would be more natural to base an absolute unit of time on the period elapsed since then. If we (arbitrarily) set the time from the Big Bang to the present equal to 1, then the current size of the universe[1] would also be 1. It would obviously also make sense to make the total mass-energy of the universe equal to 1, which would then give us a tractable basis for talking about the uniformity or otherwise of its distribution.

At this point the limitations of my physical knowledge start to bite. I’m pretty sure we need additional constants to calibrate things like electrical charge, but I don’t know how many, or whether there is any prospect of deriving them all from a unified theory.

All of this would of course be totally impractical for everyday purposes, since we would have to multiply everything by 10^-15 or so to get down to human scale. But we manage OK with joules and pascals, which need to be multiplied up by many powers of ten to be of any use (I recall a letter in the paper from an Imperial fan saying something like “a joule is defined as the energy expended in a housefly’s dying breath”). Anyway, it would be a lot of fun (for appropriate values of “fun”).

fn1. That is, the furthest distance from which light could be received at any point.

Update Footnote 1 and the linked text are wrong. I should have checked Wikipedia first “The age of the universe is estimated to be 13.7 billion years. While it is commonly understood that nothing can accelerate to velocities equal to or greater than light, it is a common misconception that the radius of the observable universe must therefore amount to only 13.7 billion light-years. This reasoning makes sense only if the universe is the flat spacetime of special relativity; in the real universe, spacetime is highly curved on cosmological scales, which means that 3-space (which is roughly flat) is expanding, as evidenced by Hubble’s law. Distances obtained as the speed of light multiplied by a cosmological time interval have no direct physical significance.

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1. December 31st, 2010 at 13:00 | #1

Problem is we don’t know the size of the universe or its total mass-energy, making it a non-starter for reference units.

The modern SI system with almost all units now redefined in terms of fundamental constants is a good one.

2. December 31st, 2010 at 13:10 | #2

Why not just set all our units as fractions of infinity? Then we wouldn’t have to adjust them, no matter what scientific developments occur. 😉

3. December 31st, 2010 at 13:34 | #3

And then of course is the use of “Jack’s Correction Factor” of old, in which if the result you want is B, but you have been lumbered with a non-zero result of A, then multiply A by Jack’s Correction Factor, namely J := (B/A). This gives A –> J * A = B. For the special case of A actually being equal , use the additive version A –> A + J = B, where in this case J := B.

4. December 31st, 2010 at 14:53 | #4

Hi John, there are a few different conventions used. See this wikipedia page http://en.wikipedia.org/wiki/Natural_units#Summary_table

Some of the fundamental constants used are

c=1 (the speed of light with units of velocity)

hbar=1 (Planck’s constant with units of energy *time)

e = convention dependent (charge of an electron/proton with units of electric charge)

G = convention dependent (Newton’s gravitational constant with units of distance^3/(time^2*mass))

m_e (alternatively m_p) = convention dependent (mass of electron/proton).

From these and a few others, you can build up secondary units like the Planck length, and Planck time.

5. December 31st, 2010 at 15:09 | #5

http://en.wikipedia.org/wiki/Planck_units

It’s quite a nice convention because it relies only on the properties of free space and not on any arbitrarily chosen particle (such as the electron or proton).

It uses only the following

gravitational constant
Planck’s constant
speed of light
Coulomb constant
Boltzman’s constant

and builds up everything from there.

6. December 31st, 2010 at 16:53 | #6

The cubit is the only acceptable unit of length – everything else is derivative!

7. December 31st, 2010 at 21:08 | #7

Before you worry about what your going to count how about we sort out the base. Surly something a little up market from the number of fingers would be nice. Base 8,16,32 or 64 should be on the table for selection.

8. December 31st, 2010 at 21:55 | #8

I am not sure which edition you are reading (The Folio Society was selling one recently).

But the whole point of relativity is that no absolute standard applies.

Length units are determined by the time taken for energy to move – a length per second.

Time is determined by the distance energy moves – in a period – a time-unit to move a metre.

If space-time was denser at the big bang, then these units all change.

What ever length unit we dream-up now – the universe, at some point in time, was smaller.

9. January 1st, 2011 at 02:38 | #9

I think the real issue is whether Si units are base 10 or base 11.

And maybe scale helps as well.

10. January 1st, 2011 at 08:01 | #10

Well I thought my NYE was quiet by the standards set in my misspent youth. But Pr Q’s sedentary excursion into units of measurement makes my stroll down to the local park to watch the fireworks display looks like a Rolling Stones tour.

Happy NY to all the riotous souls lurking out there in blogspace.

11. January 1st, 2011 at 08:02 | #11

@Chris Warren
It was the Folio edition.

I was aware that there is no absolute standard, so the idea was to fix the time relative to today’s age of the universe, on the (special relativity) assumption that this, combined with the speed of light, would give you the radius of the observable universe. But, as noted in the footnote that was all wrong.

12. January 1st, 2011 at 08:13 | #12

Pr Q updated:

in the real universe, spacetime is highly curved on cosmological scales, which means that 3-space (which is roughly flat) is expanding, as evidenced by Hubble’s law. Distances obtained as the speed of light multiplied by a cosmological time interval have no direct physical significance.”

Another, simpler way of saying this is that space is expanding owing to inflation. So time is not absolutely correlated to distance, as measured by light years.

We have now reached the outer limits of my cosmological knowledge.

13. January 1st, 2011 at 09:30 | #13

I would like to see a year of 13 x 28 day months with the first of each month starting on Sunday. Two days are interpolated as required namely Yule Day and Leap Year Day to make 365 and 366 day years. Yule day and Leap year day are not actual weekdays (not Sunday to Saturday).

By law, interest on debt would not be charged for Yule Day and Leap Year Day. 🙂

Middle Earth calendars are a lot of fun too.

http://en.wikipedia.org/wiki/Middle-earth_calendar

However, I am intrigued that middle earth has a periodicity of 365.25 days to orbit the sun. That is the same as the modern world and seems wrong given the ancientness of Middle Earth.

14. January 1st, 2011 at 09:56 | #14

OT: Whilst we are on the subject of measurement and milestones, one would hope that blogospherical pundits are preparing the decennial reviews. FWIW here is mine, which rivals Pr Q for its retrospective bearishness.

Towards the end of the last millenia I was bullish about the US in particular and the RoW in general. End of Cold War + Internet + PRC + Globalisation = global progress. Prospects for utilising the power of digital technology in medical biotechnology seemed bright.

But since then pretty much everything the US government has touched or has failed to touch has crumbled into dust. So most of the hopes of the new millenium were dashed by the end of its first decade, during which time:

The Cold War morphed into the Global War on Terror/Iraq War after 911
The internet revolution got mixed up with finance in the midst of the dot.com bubble.
The PRC ran its industrial economy on planet cooking dirty coal as Kyoto stalled.
Globalisation became the GFC after Wall Street decided to go long on the Greater Fool

In short, in the global scheme of things, everything that could go wrong did go wrong. Of course given what happened to the CIS at the beginning of the nineties was a harbinger of doom. Shoulda seen it coming.

It would be easy and convenient to blame Bush & the REPs for the lion’s share of global woes. And they do deserve a full measure of blame.

But the bottom line is that most every sentient citizen of the developed world sinned, by omission if not commission, in blowing the millennial opportunity. Except the guys running the PRC which maintains its smooth and steady ascent to global superiority, on track if not ahead of schedule.

The UN in general, and the USA in particular, seems incapable of getting their global act together. Maybe we should think seriously about handing over some global decision making powers to the grey technocrats of the the CCP politburo. They have a solid record getting things done.

(Only half-joking.)

15. January 1st, 2011 at 09:57 | #15

Standard Units of Measurement were one area where the noughties saw genuine progress. I mean the Friedman Unit (FU) which has general application beyond its specific origin in Iraq. It refers to the six month time horizon of pundity beyond which no pundit can be held accountable for failed predictions.

I propose that all bloggers be required to submit their predictions on a quarterly period in order to not exceed the FU of unaccountability.

16. January 1st, 2011 at 10:04 | #16

So John, to repeat, the best natural unit of time is the Planck time given by: (hbar * G/c^5)^(1/2),
where hbar is the reduced Planck’s constant, G is Newton’s gravitational constant, and c is the speed of light. It is about 5.3*10^(-44) seconds

The best natural unit of distance is the Planck distance, which is how far light travels in flat space during the Planck time. It is about 1.6*10^(-35) metres

17. January 1st, 2011 at 14:11 | #17

I note from the wikipedia article that Planck units are available for mass and charge as well as time and distance. Obviously these are preferable SI because they are natural and not anthropomorphic or subject to evolutionary decay.

But they are so small and odd that they are a bit unwieldy for human use. Any suggestions for scaling them up to more anthropomorphic levels and rounder numbers?

18. January 1st, 2011 at 14:53 | #18

@Jack Strocchi
VF Jack (Friedman Unit aka FU) – flexible deadline being a unit of time six months in the future. Sounds like the long run arrival date. Sure this guy isnt related to Uncle Milt?

19. January 2nd, 2011 at 10:34 | #19

Some of these constants might not be so constant, for example, the in the detected variability fine structure constant. Choose units carefully.

20. January 2nd, 2011 at 14:43 | #20

@Jack Strocchi
Good point. I guess you’d just use the prefixes for 10^44 and 10^35, whatever they are.

Jim Birch,
That’s true, but these units are as “constant” as we can make them.

21. January 2nd, 2011 at 17:19 | #21

Then I strongly recommend: The Grand Design, New Answers to the Ultimate Questions of Life. Authors Stephen Hawking and Leonard Mlodinow, Bantam Press, ISBN 9780593058299

22. January 10th, 2011 at 19:38 | #22

John as per Sam’s comment an reference to the Wiki page on Natural units (upthread), physicists actually use the convention c=1 etc, so many advanced texts render such things as E=Mc^2 as
E=M

Similarly, the planck length (and corresponding units) are used as the basis for the lowest level of granularity in measurements.

Makes everything a lot easier, which you can get an appreciation of if you ever have the misfortune to read a (very) old text using Imperial units – completely unreadable these days.

Robespierre introduced Decimal Time (aka French Revolutionary Time) in 1793 but it was discarded two years later on the introduction of metric weights and measures – no-one could get the hang of using it.

And according to the Wiki page http://en.wikipedia.org/wiki/Decimal_time China apparently used Decimal Time as an alternative system for centuries (!)