A post on Einstein and the P≠NP question on the Gödel’s lost letter and P=NP blog invites readers to comment on the similarities and differences between two snappy formulae: Einstein’s E=mc2 and the formula P≠NP which is the title of a recent preprint by Vinay Deolalikar.
The latter is one of the most important open questions in theoretical computer science, for which the Clay Institute is offering a million-dollar prize. I certainly do not intend to discuss Deolalikar’s claimed proof here; see the blog referred to above for serious discussion of this. On the other hand, everyone knows Einstein’s formula, but it is physics where the mathematicians’ concept of proof does not really apply. I thought it might be worth having a brief look at what Einstein did, and the context.
1905 was a very good year for Einstein. He wrote five papers, pubished that year or the next:
- on the photoelectric effect;
- a determination of Avogadro’s number (equivalently, the mass of an
- on Brownian motion;
- introducing his theory of special relativity; and
- “proving” that E=mc2.
English translations of these papers, together with commentary and context, can be found in the book Einstein’s Miraculous Year, edited by John Stachel.
The first paper is regarded as one of the founding documents of quantum theory, a theory which Einstein himself later came to disown. In 1900, Max Planck had explained anomalies in the spectrum of black-body radiation by assuming that energy was transferred in discrete packages or “quanta”. Einstein used this idea to explain the photoelectric effect. When light is shone on certain metals, electrons are emitted. The crucial factor is not the intensity of light, but its frequency; there is a sharp threshold below which no electrons are emitted no matter how intense the light, and above which they are emitted no matter how weak it is. It takes a fixed amount of energy to knock an electron out of the metal; Einstein proposed that if the energy of a quantum is proportional to the frequency (as Planck proposed) then this effect can be explained.
Contrary to popular belief, it was this paper rather than the paper on special relativity for which Einstein was awarded the Nobel Prize.
This paper also contains one of my favourite quotes about probability, which I have tried to drum into cohorts of first-year students:
In calculating entropy by molecular-theoretic methods, the word “probability” is often used in a sense differing from the way the word is defined in probability theory. In particular, “cases of equal probability” are often hypothetically stipulated when the theoretical methods employed are definite enough to permit a deduction rather than a stipulation.
In other words, don’t assume that outcomes are equally likely, especially if you have enough information to show that they are not!
The second paper was Einstein’s dissertation at the University of Zürich, and was based on experimental data on sugar solutions.
Brownian motion is the random motion of small particles in a fluid caused by molecular impacts. In this case, his analysis preceded the availability of detailed experimental data.
I need to say a bit about the fourth paper as background to the fifth. Since Galileo, physicists had known the rule for transforming between frames of reference (essentially frameworks for measuring space and time coordinates) moving with constant velocity relative to one another. The crucial thing is that velocities add (as vectors). In particular, this means that the speed of light should depend on the motion of the frame of reference in which it is measured. (Physicists postulated a mythical “ether” corresponding to a frame of reference at rest; anything else was measured relative to this.)
In the late nineteenth century, Michelson and Morley tried to measure these changes in the speed of light. They found no effect; in other words, the earth is at rest relative to the ether, even though its movement changes with the seasons. Do we have to go back to a geocentric theory? Lorentz proposed a fudge in which measuring rods contract, and clocks run slow, when they move relative to the ether, in just the right amount to ensure that the measured speed of light is constant.
Einstein, starting from the assumptions that no experiment should distinguish between rest and uniform motion, and that the speed of light really is constant (a consequence of Maxwell’s equations), was able to revise Galileo’s transformation formulae to a form which exactly reproduced Lorentz’s length contraction and time dilatation, with a crucial difference: there is no fixed ether, but the effect is caused by relative motion. In other words, if two observers are moving uniformly relative to each other, then each will observe the other’s measuring rods as contracted and clocks as running slow relative to his own.
The paper introducing the famous formula begins from a formula in the special relativity paper. A body with rest mass m moving with velocity v relative to an observer will appear to have mass
M = m(1-v2/c2)-1/2.
Multiplying by c2; and expanding the bracket using the Binomial Theorem (incidentally, one of Newton’s discoveries in his miraculous year of 1666), we get
Mc2 = mc2 + (1/2)mv2 + smaller terms.
Since the second term on the right is the classical kinetic energy of the particle, Einstein interprets this equation as saying that giving kinetic energy E to a particle results in an increase in its mass by an amount E/c2; he then leaps from here (which is clearly just an assumption involving an approximation) to the assertion of the equivalence of mass and energy according to his famous formula.
Einstein knew there was something a bit suspect here. He wrote,
The argument is amusing and seductive; but for all I know, the Lord might be laughing over it and leading me around by the nose.
(If the Lord was laughing, what are we to make of Hiroshima and Nagasaki?)
I am reminded of two encounters of my own with physicists’ views. The first was Mike Green, one of the pioneers of string theory, who was in the physics department of my institution before moving to Cambridge. In those days, it was thought that string theory required 26 dimensions, rather than the currently fashionable 10 or 11. Mike would have lunch with the pure mathematicians, and ask us, “Is it just coincidence that there are 26 sporadic simple groups?” We assured him that it was.
In a lecture on string theory which I attended, he described it as a “perturbative theory”. I was quite shocked: how could the basis of a Theory of Everything be a perturbative theory?
The second incident concerned the only physicist ever to win the Fields Medal, Ed Witten. I was at the ICM in Kyoto when the medal was awarded to him. Vaughan Jones, in his plenary talk, took it upon himself to describe Witten’s work. I don’t recall his exact words, but it was something like this. “Witten wrote down this integral.” [Jones writes an integral on the board.] “He didn’t even say what measure he was integrating with respect to, but I still think it’s a very interesting integral.”
I admit that I found this quite shocking too.