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:

According to Yusuf al-Qaradawi, an Egyptian cleric known around the Muslim world for his popular television show “Sharia and Life”, Mecca has a greater claim to being the prime meridian because it is “in perfect alignment with the magnetic north.”

This claim that the holy city is a “zero magnetism zone” has won support from some Arab scientists like Abdel-Baset al-Sayyed of the Egyptian National Research Centre who says that there is no magnetic force in Mecca.

“That’s why if someone travels to Mecca or lives there, he lives longer, is healthier and is less affected by the earth’s gravity,” he said. “You get charged with energy.”

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?


When time becomes a loop

June 10, 2010

I’ve just read an article on the Huffington Post, titled What Happens When You Die? Evidence Suggests Time Simply Reboots, by a guy named Robert Lanza. I had never heard of him, but that is completely my fault, for according to the author description I just linked to he is one of the greatest scientists currently walking on the face of the planet… seriously:

Robert Lanza was taken under the wing of scientific giants such as psychologist B.F. Skinner, immunologist Jonas Salk, and heart transplant pioneer Christiaan Barnard. His mentors described him as a ‘genius,’ a ‘renegade thinker,’ even likening him to Einstein himself [..] Robert Lanza is considered one of the leading scientists in the world

Anyway, these are just words and sarcasm is easy, and the rest of his resume actually does list some impressive achievements in the field of stem cell research.

But that makes the nonsense written by Lanza, M.D., Scientist, Theoretician, only more baffling. It is rather disappointing that nowhere in the article Lanza even coherently addresses the question (nor the answer!) posed in the title of his piece. Instead we get a rambling anecdote that is too boring and lengthy to repeat here, followed by the following horrendous paragraph of gibberish:

Before he died, Einstein said “Now Besso [an old friend] has departed from this strange world a little ahead of me. That means nothing. People like us … know that the distinction between past, present and future is only a stubbornly persistent illusion.” In fact, it was Einstein’s theory of relativity that showed that space and time are indeed relative to the observer. Quantum theory ended the classical view that particles exist if we don’t perceive them. But if the world is observer-created, we shouldn’t be surprised that it’s destroyed with each of us. Nor should we be surprised that space and time vanish, and with them all Newtonian conceptions of order and prediction.

Aside from doctor Lanza, it would be very difficult to find a Scientist, Theoretician who does not at least roll his eyes upon reading such tripe. There is so much misunderstanding packed together here in a few sentences that it is almost a pity that no abuse of the concept of entropy has been thrown in for good measure.

the theory of relativity

Space and time being relative to the observer in the theory of relativity only means that the spatial and temporal relation between different physical objects depends on their relative motion. Such physical objects can be anything, elementary particles, galaxies, scientists, theoreticians. Any object A moving towards an object B will be, as far as B is concerned, shortened in length. For B, object A will also be perceived to pass through time more slowly. For example, this is the reason we can receive cosmic rays on earth. These are caused by particles that are accelerated in distant galaxies to enormous speeds and are extremely short-lived before falling apart -short lived in relation to anything moving along at the same velocity, that is. It is due to the relative velocity between earth and the cosmic ray particle that the two manage to meet each other before the latter falls apart.

Nowhere in the above the observer plays an active role. Although profound in its implication for the structure and interconnection of space and time (now mathematically merged in a single concept spacetime), the special theory of relativity is essentially all about coordinate transformations between different reference frames. Nothing really changes when moving from one frame of reference to another, in the same way that the world is not destroyed and made whole again but different when I choose to use spherical coordinates instead of x,y,z when charting the walk between my home and my workplace.

In the general theory of relativity a complication is added. Again something which has profound implications for the structure of spacetime, but again nothing that elevates the observer to a special status. According to general relativity the structure of space time is altered by the presence of matter (or energy), so in a very real sense everyone makes a dent in spacetime. But this has nothing to do with observing or experiencing spacetime, the dent would be just as big if one is, in the words of Lanza, rotting in the ground. When performing scientific calculations, the bookkeeping of the local structure of spacetime is done via a mathematical tool called a metric (a four-by-four matrix). An abstraction of a ruler, a set square and a watch mixed into one, this stores the local connections between angles and space and time intervals. Cosmology and the big bang are all about the values of the metric becoming zero as one goes further into the past

Actually relativity theory has not rendered time an illusion either. Although purely within the framework of relativity theory the difference between time and space can almost be reduced to a single minus sign in the metric mentioned above -which is perhaps what prompted Einstein’s remark. But within physics as a whole the arrow of time firmly points in one direction: the future. Unfortunately we are running out of space on this blog entry and not even Lanza mentioned entropy, so lets not pursue this further but move on to quantum mechanics.

quantum mechanics

Perhaps no single topic in physics has been hijacked as much for the purpose of New Age mumbo jumbo as quantum mechanics. The idea that a measurement alters that what is measured has proven irresistible. However, when you get down to it, it is really not this aspect of quantum mechanics that is so weird. Of course a measurement alters that what is measured, for a measurement is precisely that: an interaction between the means of the measurement and the measured object. When we touch a wall to feel what it is like, we have altered it, but in practice we won’t notice that because a macroscopic object like a wall is far less delicate than a single particle that has been specially isolated from its surroundings in a laboratory. Prod the latter and you are bound to make an impact. But that is because of the prodding, both the particle and the wall couldn’t care less about the conciousness and thoughts of the Scientist, Experimenter doing the prodding.

There is a a genuinely difficult and counterintuitive aspect to quantum mechanics however, but it does not mean that particles cease to exist when they are not perceived. It is this: like space and time in relativity’s spacetime, certain properties of particles are interconnected in a way that is counterintuitive to us because we are used to the blunted relationships between material objects on a macroscopic level, like the wall, with the properties of the individual particles making up things remaining invisible to us. This interconnectedness means that when we measure for example the position of a particle, we necessarily change its momentum as well. It is not possible to measure both position and momentum of a particle simultaneously; once we exactly manage to narrow down its position at a given moment in time we will be clueless by default about where it will be directly afterwards. We could keep running after the particle like a toddler Lanza running after dragonflies with his little net, but there will always be a fundamental uncertainty between position and momentum. An uncertainty that, again, has nothing to do with the mind of the observer, but everything with the akwardness in choice of the quantity that we choose to measure. Sometimes interactions between particles and other particles, or measurement apparatuses or whatever, will probe properties of a particle toward which it happened to be attuned at a given time. Then nothing changes for the particle. But if, for example, a particle is left alone after its position has been measured, it will gradually spread out again to occupy the empty space between its stated position and surrounding particles that repel it. When a new measurement of position is made, the particle may be forced in any particular position within this space. Which position will be random (the fact of this randomness is again a truly profound change in our understanding of reality, and arguably one of the things that drove Einstein to his deathbed in the first place). The particle did however not cease to exist in any way other than that it became less attuned to the ruler used to measure its position. In technical terms, all its quantum numbers, like mass, electric charge etc. remain solidly conserved.

To summarize a long story, Lanza may be a good scientist in his own field but when it comes to physics he is so clueless it is painful to read. I shudder at the thought of all those people tricked into reading his book biocentrism, for which I can only assume his Huffington Post piece was a badly thought out advertisement.

And what happens when time reboots and becomes a loop… Orbital knows.

[update:Seems I am far from the only one baffled by Lanza, Scientist, Theoretician, see Pharyngula]