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*=*mc*^{2} 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

atom); - on Brownian motion;
- introducing his theory of special relativity; and
- “proving” that
*E*=*mc*^{2}.

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-*v*^{2}/*c*^{2})^{-1/2}.

Multiplying by *c*^{2}; and expanding the bracket using the Binomial Theorem (incidentally, one of Newton’s discoveries in *his* miraculous year of 1666), we get

*Mc*^{2} = *mc*^{2} + (1/2)*mv*^{2} + 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*/*c*^{2}; 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.

Read Terry Tao’s account of Einstein’s derivation of his equation here.

You seem to suggest that the formula is an approximate one. I don’t remember the derivation I saw as a student but I don’t remember use of approximations.

I just had a glance at Terrence Tao’s account but it is a mixture of physical intuition and mathematics and I am too lazy/too busy/lack the motivation/incapace to untangle it.

The question is: do both your and Terry Tao’s derivation involve approximation?

This is based both on what Einstein wrote and what I remember as a student.

I think the correct spin is that the classical formula for kinetic energy is the approximation, and that

E=mc^{2}is to be thought of as the correct formula for the energy of a body of massm; then for low velocities, the increase in energy approximately agrees with the classical formula for kinetic energy.But, as I said, this business of approximation makes me uneasy!

This page is primarily concerned with Einstein and E=mc squared. So it might be appropriate to offer here my version of E=mc squared for the 21st century (inspired by The Moon Is New) –

“The Moon Is New” (a book by John Dobson – Berbeo Publishing, 2008) has the potential to completely change our understanding of the universe. On page 14, it’s stated that “Einstein’s equation (E=mc squared) says that mass and energy are the same thing …” and “The c squared is just how many ergs are equal to one gram” (making the equation E=m). In pages 38-40, the book asks “… how many centimeters (are) equal to a second. That ratio, what is known in the trade as the speed of light, is about 30 billion centimeters to a second.” This question, and these pages, could lead to us saying “space and time are the same thing.” But as the book tells us on p. 38, “… time is the opposite of space in the geometry of this world …” and “… the space and time separations between (any) two events are equal and the total space-time separation is, therefore, zero.”

Suppose a star we are viewing is at a distance of 100 light years (this can be represented as +100). Since we see nothing as it presently is but as it was when the light left it, we are seeing the star as it was 100 years ago (represented as the opposite of space i.e. as -100). Repeated experimental verification of Einstein’s Relativity theory confirms its statement that space and time can never exist separately but form what is known as space-time. The space-time distance between us and the star is therefore 100 + (-100) i.e. 100-100 i.e. 0 and there is actually zero separation between us and the star’s gravity, heat etc.

So saying space and time are equivalent (“equal” or “the same thing”) is incomplete and, to be accurate, we need to say space-time separation is equal (and zero). This possibly explains cosmic unification and because the inverse-square law of famous English scientist Isaac Newton (1642-1727) says the force between two particles is infinite if the distance of separation goes to zero; also possibly explains the existence of an all-powerful, and super-intelligent (since those particles could be brain particles), God.

Is it also incomplete to say mass and energy are the same thing? Sure, we can add c squared to E=m. But we can think differently and think of E=m as 10=10 exponent 1. To make the equation totally complete, we must add something without altering the meaning e.g. by writing 10=10 exponent 1+0. Now we have E=m exponent 1+0. Where do we find 1’s and 0’s? In the binary language used by computers. Does this mean the Underlying Existence spoken of in the book is energy as the book suggests – but to be more specific, the energy of a computer (perhaps a quantum supercomputer) processing?

Maybe this quantum supercomputer resides in the same place as the purported Big Bang. Carl Sagan said there is no centre to the universe where the Big Bang could have taken place and initiated expansion. Therefore, the Big Bang (and for our purposes, the quantum supercomputer) would exist outside space and time in what we might call hyperspace. Page 34 suggests “… the rest mass of the proton (is) just the energy represented by its separation … from all the rest of the matter in the … universe.” Since that separation is zero, the universe must be unified with each of its constituent subatomic particles and those particles must follow the rules of fractal geometry being similarly composed of space and time and hyperspace.