god-given units

October 9, 2010

A bigger Ben

Continuing the themes of religion and big construction works from the last post, a while ago the construction of a gargantuan clock in Mecca was reported in various news outlets (like here):

Saudi Arabia hopes the four faces of the new clock, which will loom over Mecca’s Grand Mosque from what is expected to be the world’s second tallest building, will establish Mecca as an alternate time standard to the Greenwich median.

This effort has been ridiculed (i.e. here, here) as a cargo-cultish attempt to steal away the reference point for global time keeping from Greenwich, just by building a bigger clock than the Big Ben in London. This may not be true, what I have read at least leaves open that it is perfectly understood in Mecca that the clock will have merely a powerful symbolic function… but on the other hand, what I have read about the local understanding of the earth magnetic field, isn’t exactly rational either:




Western scientists have challenged such assertions, noting that the Magnetic North Pole is in actual fact on a line of longitude that passes through Canada, the United States, Mexico and Antarctica.

Anyway, before I get sidetracked on magnetic fields, the topic I wish to discuss is that of the arbitrariness of how we measure universal quantities like space and time. Just how arbitrary is our time unit, for example? We have time zones related to Greenwich for purely historical reasons and measure the passage of time in minutes, hours and days that neatly divide up a single earth’s rotation. But if we were to encounter the proverbial alien species, would we be able to draw upon a system of units for time, energy, momentum, distance that makes equal sense to them as to us? The answer to this question turns out to be yes. And whether we should be surprised by this or not, is one of the deeper questions one can ask in physics.


Part of the mystery is that the ingredients for a universal set of measurement units can be found at an even deeper level than, say, the rest mass of some particle, like the electron, which would still allow the freedom of choosing a different particle. We will take our ingredients from the very structure of the physical laws governing the interactions in the universe instead. And we’ll start with gravity’s constant G.

It is known since the days of Newton that all masses attract each other, and nowadays it probably takes a conscious effort to realize just how incredibly radical this idea was: that the motion of the sun, the moon and the stars is dictated by the same force (another concept that wasn’t even formalized before Newton) that makes sure an apple drops to the ground when we let go of it. Newton’s law of gravity is given by F = mMG / r^2, high school physics now. In other words, any massive object feels a force F from the massive objects around it, proportional to both their masses (m, M) and the square of their relative distance r. This square feels natural, it is after all also how the area on a sphere grows as we increase its radius (so any quantity on the surface of the sphere gets diluted in this fashion when the sphere expands). But don’t read to much into this, for there exist forces with different reach. For now, the element of interest in the equation is the gravitational constant G, which has a very specific value and dimension: G = 6.67428 x 10^11 m^3 kg^-1 s^-2. To the best of our knowledge (and confirmed observationally through a variety of means), this value holds true also at the other end of the universe. We’ll take it as the first building block when constructing our set of universal rulers. This one measures some combination of distance (m), mass (kg) and time (s).

The speed of light

It was only in college that I learned to appreciate just how phenomenal Newton’s insight into nature was. He realized for example something special about masses which completely passed us by in high school even though it was right in front of us on the blackboard: The mass that determines the strength of gravitational attraction doesn’t need to be the same as the mass that features in F = m a. The former kind of mass is akin to electric charge -but without the complication of pluses and minuses. It dictates how much a particular particle feel gravity, just as electric charge dictates how much a particle feels the electric force. The second kind of mass is more general. It tells us how much a particle will budge when a force, any force, is applied to it. When the forces F applied to them are equal, a very massive particle will only undergo a small acceleration a compared to a very light particle -which will accelerate a lot to compensate for its small mass in order to keep the product m and a equal to that of its big brother. Particles whose mass tends to zero will basically never sit still. This also applies when, say, electric force is applied instead of gravitational force, which is the reason I called the second kind of mass more general.

Zero mass particles, like photons, don’t move infinitely fast, even though that is suggested by F = m a. It turns out that they move with a fixed speed. And when I say fixed, I really mean fixed. If you carry a flashlight, flip it on so that photons will start leaving the lamp with speed of light c and then decide to run after the emitted photons, the photons will still be moving with the speed of light with respect to you. Not just in a practical sense, although compared to the speed of light (299 792 458 m / s) you’re practically standing still when running at, say 10 m / s (assuming you’re Guinness book of records material). The speed of light with respect to a moving observer stays exactly c. Where normally you’d just subtract velocities (i.e. c minus 10 m / s) to get the relative velocity, this simple subtraction rule ceases to be valid when one of the velocities approaches the speed of light. Instead, it gets replaced by relativity theory. We will not go into that here, but just note that this aspect of nature yields us a second universal constant in the speed of light c

The quantum uncertainty

We now have G and c, both combinations in the dimensions of mass, length and time (although mass only occurs in G). A third constant would give us as many constants as dimensions (which will turn out to be just what we need). In quantum mechanics we can find our third ingredient, the uncertainty constant h. At 6.626068 × 10^-34 m^2 kg / s, this constant sets the uncertainty between a particles position and momentum, as well as the minimum energy for a photon. The former means that we can only measure a particle’s position at the cost of complete ignorance of its momentum and vice versa. The latter means that photons come in quanta, and indeed the development of theory of quantum mechanics was triggered by scientists discovering (almost accidentally) the existence of h. I discuss quantum uncertainty a little bit here as well. For now we have what we need in the existence of h.

God-given units

The set of G, c and h, provides us with what have been called god-given units, which must please Abdel-Baset al-Sayyed (or followers of other monotheistic religions), as long as they ignore that this is meant in the proverbial sense. Much like Einstein’s famous phrase God does not play dice is a statement about (assumed) aspects of the laws of nature, rather than an expression of religious sentiment.

So what then is a natural unit for time? Since G is m^3 kg^-1 s^-2, c is m / s and h is m^2 kg / s, it follows that the square root of h G / c^5 has the dimension of time. To be precise, we’d get:

1 natural time unit = sqrt(h G / c^5) = 1.35138 x 10^-32 seconds

Mass and distance units can be calculated in the same way. As can energy (for Joule is after all equivalent to kg m^2 / s^2, just think of kinetic energy from high-school physics, 1/2 m v^2) and momentum:

1 natural distance unit = sqrt( h G / c^3) = 4.05134 x 10^-24 meters
1 natural mass unit = sqrt( h c / G) = 5.45552 x 10^-19 kilogram
1 natural energy unit = sqrt( h c / G) x c^2 = 0.04903 Joule

The translations of these natural units into our own units is, of course, as arbitrary as our definition of what constitutes a meter or a kilogram. But that is not the point, which is rather that we now have a unique and universal way of expressing our arbitrary choices, even to aliens from beyond the galaxy. And one of the most profound questions about nature one can ask is, is there any reason for this system to exist?. Or in other words, knowing only about G and h, could we reasonably expect a third fundamental constant to exist in order that we can make combinations to express quantities we measure directly? Or, in yet other words, should we expect that the number of fundamental constants in nature equals the number of truly independent dimensions of measurement? Or is the whole thing just a trivial coincidence?