Einstein's E=mc <sup>2</sup> and the mass of the photon.

The accepted meaning of Einstein's E=mc2, the mass of the photon, and some misconceptions about them.
By Chandre Dharmawardana.
Send any comments to chandre.dharma@yahoo.ca

Virtually every one has heard of Einstein, Plank, Niels Bohr, as well as the theory of Relativity and the Quantum Theory. Furthermore, every one has heard of the equation E=mc2 which adorned the triumphant flag that waved in the new physics. The latter has given us not only X-rays, electronics, telecommunication, computers, etc., but also the terrible power of the atomic bomb. Recent discussions of this equation in the popular science magazine Vidusara , among writers named Asoka Amaratunga, Bodhi Dhanapala and Nalin de Silva prompted us to post this commentary. It appears that the currently accepted physics meaning of this equation, which seems to relate the energy E and the mass m of an object is not always well understood, and several misconceptions have been stated in those discussions and elsewhere.

Scientific experiments conducted during different epochs are carried out within the limitations of the accuracy of the tools and instruments available in each era. The laws proposed via such experiments are also only valid within the accessible time-scales and length- scales of each era. It is difficult to measure time to better than a millisecond (one thousandth of a second) with instruments available in most homes even today. In the very early part of the 20th century scientists could measure a millionth of a second using special methods. Each epoch had its minimum accessible time-scale Tm, a minimum length-scale Lm, and similarly a maximum energy-scale Em. Thus, the power of dynamite set this maximum energy-scale for the early 20th century. Accordingly, experiments and the laws of physics of a given epoch had a domain of generality limited by Tm, Lm, and Em. Within a given domain, the laws, once established, become part of engineering practice, and are used without further scrutiny. They are culture neutral, although their applications may not be.

When it becomes possible to extend the time-scales, length-scales and energy-scales, new results emerge, and pave the way for the further maturing of the existing theories, or creating new revolutions in physics. It is the process of more accurate instruments becoming available to do experiments that ultimately generate scientific revolutions. This process is in fact different to that proposed by philosophers like Thomas Kuhn in 1962, and are discussed in detail in Chapter 2, and chapter 7 of a recent book.

The energy-scales, time-scales and length-scales applying to the laws of classical physics do not apply to the behaviour of atoms, or chemical reactions that depend on atomic properties. These involve much shorter length-scales and time-scales. Similarly, it was already clear by the end of the 19th century that highly energetic processes like the motion of the planet Mercury, or the behaviour of ultra-violet light were outside the ambit of classical physics.

In a chemical experiment where 100.3 grams of mercury are reacted with 16.0 grams of oxygen, it is found that 116.3 grams of mercury oxide are formed. The total mass is conserved. Experiments suggested that matter is neither created, nor destroyed. This is enunciated as the law of conservation of mass. By the middle of the 19th century, the experiments of Rumford, Joule and other scientists had shown that heat can be converted to mechanical energy and vice versa. Faraday, Clark Maxwell and others had shown that electricity and magnetism are also forms of energy. They even showed that light is a form of electromagnetic energy ! Different forms of energy could be transformed into one another, but not destroyed or "created" from nothing. This is the law of conservation of energy. That is, within the time-scales, length-scales and energy-scales accessible to that age, the 19th century physicists had enunciated two distinct conservation laws, i.e., for matter, and separately for energy.

Meanwhile very accurate mirrors, prisms, lenses and chronometers had been constructed. The technically talented physicist Fraunhofer succeeded in constructing very high quality diffraction gratings where very fine parallel lines were engraved on accurately flat glass sheets. Thus the time-scales and length-scales that limited the accuracy of experiments could be shortened. Among the new experiments carried out with these more accurate instruments, those by Michelson and Morley proved to be of great importance. They set up experiments to determine if the earth when in motion around the sun had an absolute velocity with respect to the "space" through which it is assumed to move. On examining the results of the experiments that used light pulses, it dawned on Einstein that the idea of an absolute velocity is not necessary. Furthermore, the crucial experiments could be explained if light had the same constant velocity relative to any observer whatever! This universal velocity of light is about 300 million meters per second. It is customary to denote it by the letter c.

Our intuitive feeling is that the world is a three-dimensional "space" having length, width and height, floating in the "river" of time which flows forward from the past to the future, in a regular manner. If the speed of light is to remain the same for every observer however fast he or she is going, Einstein realized that time itself has to slow down for fast movers. Einstein realized that if the velocity of light c is to be the same for all observers, then the world we live in is actually a four dimensional "space-time" involving length, width, height and also time, all being parts of it.

By requiring that the laws of mechanics have the same symmetries as those of electromagnetism, Einstein showed that an object cannot go faster than light because, as its speed increases, its kinetic mass also increases and tends to infinity when its speed tends to c, the speed of light. Einstein's equations show that the kinetic energy of the object re-appears as mass, showing that energy and mass can be converted into each other. Einstein was a pioneer in the unification of the two conservation laws (mass and energy) into a single law for the conservation of mass and energy. Thus we see that when the accessible time-scales Tm are shortened, and the energy-scales Em increased, physical laws that seemed to be unconnected get unified.

Einstein showed that the law that applies to the inter-conversion of energy (E) and mass (m) is E=mc2. Furthermore, he wrote a popular, non-technical book on relativity in 1920, where he explained the proper use of this equation as a conservation law. The famous French physicist Langevin emphasized in many talks that this should not be used to relate the mass of an elementary particle to its energy. At this point the reader may well ask, "what equation relates the energy to the mass for an elementary particle?". The accepted convention for mass is such that:
m=sqrt(E2/c4-p2/c2) or equivalently E2=m2c4+p2c2

is used. Here p is the momentum of the particle, m is its invariant mass while "sqrt" indicates taking the square root. In fact, very soon, Klein and Gordon used this to construct the very first relativistic equation (for bosons) in the quantum theory. The "photon" is the particle associated with light which is a type of electro-magnetic wave. If the above equation is used for a photon with momentum p, however large, since its energy is pc, the mass of the photon is found to be always zero.

Dr. Nalin de Silva, writing in a recent issue of Vidusara wrote as follows (our translation from the Sinhala):
"According to quantum physics, the energy of a photon with a frequency ν has an energy E=hν. In addition, according to special relativity, in general for particles E=mc2. Using these two equations we can assign a mass m=hν/c2 to a photon with a frequency ν."

Since the above is also derived using E=mc2, i.e., an energy conversion relation, we should ask if it ever corresponds to a mass that may be seen in some experiment, say, when a photon is absorbed by a target. Then the increase in mass of the target would indicate a suitable mass for the photon. Except for a hypothetical possibility, Dr. Nalin de Silva's proposed mass cannot be linked with any known experiment. If we hit another elementary particle like an electron with a photon, the photon merely bounces off and nothing happens. This is because a structureless particle cannot absorb the energy of the photon. Even if it could, it cannot at the same time recoil sufficiently (i.e., it cannot satisfy the conservation of energy and momentum at the same time). You need at least two particles, together with the photon, to absorb its energy, where the second particle helps to carry away the momentum p of the photon. A hydrogen atom contains a heavy nucleus and an electron which is held in some energy level of the atom. The light couples to the very light electron. The energy of the photon is absorbed by the electron where the electron goes from one energy state to another, while the nucleus picks up the recoil. The internal energy E1 changes to E2 while absorbing the photon energy E, i.e., with E≅ E2-E1. There is a very very slight correction to all this as a tiny bit of energy goes also to the nucleus as kinetic energy. There are also smaller corrections (e.g., angular momentum issues) that we ignore here. The momentum of the photon (i.e., the "recoil effect") is taken up by the center-of mass M of the atom. That is, the atom acquires a kinetic energy Ekin =p2/(2M). This produces the additional mass given by:
mkin=Ekin/c2= p2/(2Mc2) as required by Einstein's E=mc2.

The electron also acquires a similar mass correction from its energy component. Now, if the photon were a fundamental particle, by definition its mass cannot be whittled into parts like this. Furthermore, this partitioning of the energy can be made entirely arbitrary as we discuss below.

Since the photon momentum p is hν/c, the increase in mass due to the added kinetic energy is mkin=h2ν2/(2Mc4). This is clearly not the mass proposed by Dr. Nalin de Silva, and in fact conforms to the photon mass being zero and it being pure energy which is redistributed into two parts. The mass correction acquired by the electron also does not agree with the value given by Dr. Silva. The one hypothetical instance that we can construe where Dr. Silva's mass might perhaps be observed nearly intact is in, say, graphene (a nano-material made from graphite) where novel elementary excitations varying from normal electrons to massless Dirac Fermions may be engineered ( See the research papers by Dharmawardana, J. of Physics, UK, 10, 386228, (2007) and Phys. Rev. B 75, 075427, (2007)). Hence one may wonder if a "band" electron may absorb the photon, while the momentum is carried away by another (hypothetical) electron or hole in a different energy "valley", where the effective mass of the second particle has to be engineered to be
M≅hν/(2c2). Then mkin≅hν/c2.

Here we have transferred almost all the mass to the "center of mass", and virtually nothing to the band electron, showing that there is no invariant mass that we can attach to the photon. We note this unlikely possibility, following the spirit of science where even maverick suggestions are examined seriously, and discarded if found to be without merit. Not that there are some simplifications in the above discussion. It may be contrasted with the case of the absorption of an electron, which has a true invariant mass which cannot be arbitrarily divided out. Other well known experiments (e.g., using he Mossbauer effect) are discussed in text books, showing that the mass of the photon is zero. A photon always has zero mass, and a constant velocity c.

Deuterium is a heavy form of hydrogen. When two deuterium nuclei are fused together to from a helium nucleus, the total mass is not conserved. A tiny amount of mass is converted into energy. Such nuclear energy is the source of power for "hydrogen" bombs. Nuclear power stations can use isotopes of Uranium, Plutonium etc., and thorium (found in abundance in Sri lanka) as the fuel. New designs of nuclear reactors are exceptionally safe, and produce less waste products. However, no ideal solution for dealing with nuclear waste is currently available. The origin of all this is E=mc2.

Today, our experimental capabilities are such that energies just beyond the Higgs energy have been reached. We know that the invariant mass of a particle is of quantum mechanical origin, caused by the presence of the Higgs field. This field obstructs all particles except the photon and other transverse excitations, and gives them mass.

If we are to discuss any aspect of physics in a deep manner, be it the theory of relativity or not, we need the help of mathematics. This is of course not possible in popular writings. Hence there are many errors in popular discussions and it is futile to try to deal with them. However, it is important to draw attention to errors that could be misleading. The velocity of a photon is a constant c, and the mass of a photon is zero, for all observers.

[The author is a Principal Research Scientist in the Quantum Theory group of the National Research Council of Canada, and a Professor of Theoretical Physics attached to the University of Montreal.]

Click below for an entertaining movie in two parts on U-tube:
See : Einstein's Big Idea (Part 1 of 2) See also: Einstein's Big Idea (Part 2 of 2)

click below for some relevant chapters of a book:
A Physicist's view of matter and mind (World Scientific, 2013)
What do students typically learn in Quantum mechanics courses? see:
Exploring Quantum Physics