# Mass and E=MC2 (E=MC²)

P.J.Mazumdar

Was sitting in my old high school classroom and walking in was our old school teacher, Denzil Thirpthorpe. There were three Thirpthorpe brothers in our school from the anglo Indian community, all wizened battle hardened old men with a strange accent and stranger surname, confirmed bachelors and all carrying a long cane. In those unenlightened days, the class teacher carrying a cane seemed a most natural thing to us. Neither my parents nor those of any of the other boys ever thought that an occasional caning would warp our souls or alter our personalities inextricably. As far as we were concerned, it was a game more than anything, the smarting was not really that bad and there were two groups of opinion, one who advocated holding the hand limp to minimize the pain and the other who held it firm.

That day there was the same slouching walk, the same stoop but I saw it was not old Denzil but someone else altogether, someone vaguely familiar from old paintings, someone who..

But I had no time to think, for he came straight up to me and barked out:
“Tell me how mass comes to be acquired?”

“Well, err...there are four ways,” I managed falteringly.

“What, four??”

“ Well, maybe more??” I hazarded.

“ More?? Go on then, tell me what they are!”

“Well, firstly, there is the Higg’s way of giving mass right at the beginning of the Big Bang, after the inflationary period”.

“Ok, that’s quite clear. What are the others?”

“Then there is relativistic mass. You know, E=MC². When a body goes very fast, at relativistic speeds, it gets heavier and heavier. When it acquires energy also, it gets heavier.”.

“Hmm, well, ok, we will see about that. And??”

“Then there are electrons that get heavier when going through a crystalline sold, even at non-relativistic speeds. I am not sure about whether that’s from Higg’s”.

“What, a non-Higg’s interaction that gives mass??” There was that cynical smile on his face. “What the fourth?”

“Then there are photons getting heavier when passing through super conductors. That’s definitely not Higgs because photons don’t interact with the Higgs field.”

“What, photons acquiring mass?? Bring out your physics text book at once, let me see where photons can acquire mass!!”

But just at that moment, the dream started changing. I began acquiring mass and became more and more massive, while old Denzil or rather the man who looked like old Denzil began fading till at last only his cane remained...

And mass indeed is becoming curioser and curioser by the day.

The confusion essentially arises from the fact that there has been no proper union between quantum physics and gravity. This gives rise to two definitions of mass, rest mass and relativistic mass.

In Einstein’s theory of gravity, relativistic mass is related, in a limited sense, to the speed of a body, and in a generalized way, to energy in the famous E=MC² equation. There is no doubt about this equation, physics at present has come to depend on this totally and it would not be wrong to say that this is the most powerful equation at present.

In quantum physics, on the other hand, a particle’s rest mass is related to the Higg’s mechanism. By this mechanism, all particles are shown to have drawn their rest mass from the Higg’s field, right at the beginning of the big bang. Quantum fluctuations in this field due to symmetry breaking as the universe cooled a little caused condensation of mass, so to say, on the particles, and the math behind this hold up very well and there is no reason to doubt this.

Let us look at some examples of mass gain talked about generally:

Rest mass: This is probably the firmest area of definition with little controversy. Here the quantum explanation, Higg’s mechanism, is accepted unreservedly. The maths of Higg’s mechanism agrees well with quantum physics. This also was the first use of Higg’s field, though it has since enlarged into cosmology as well, being used for a vital role in the inflationary theory of the big bang. There is no doubt about rest mass from the question of relativity also. The equation E= MC² where E stands for rest energy and M stands for rest mass is without any controversy.

Relativistic mass: this is the definition of mass that is derived from the general theory of relativity. It is counterintuitive, in that it says that a body becomes more massive when it is in motion. On a more general level, it relates energy to mass, with the equation E= MC², possibly the most used equation in physics today. There is no doubt at all that this relation is true when used for rest mass to relate it to rest energy. Weight of a proton for example, is not just the sum of the rest masses of its constituent quarks and leptons. The energy within the particles, due to their interaction and constant whizzing around, accounts for 99% of its weight and only the remaining 1% or so is due to the rest mass of the constituent particles. This is because the energy of the quarks and bosons contribute to the rest energy of the proton.

Problems arise when we try to relate E= MC² for considerations of mass other than rest mass. The general theory of relativity relates mass to energy well through this equation, but how are we to explain it in terms of quantum physics?

The exact way in which this extra mass comes about in case of relativistic mass is not clearly defined. How is E=MC² related to the Higg’s field? If rest mass comes from the Higg’s field, then where does relativistic mass come from? Different observers can have different values for the velocity of an object, and thus get different values for relativistic mass.

The most logical answer is that ‘relativistic mass’ is an ill defined concept and it is best to avoid this. Using the equation E=MC² for anything other than rest mass is not a legitimate use, because we have no way to explain where exactly the extra mass, the difference between relativistic mass and rest mass, comes from.

The equation E=MC² sits uneasily with quantum physics. Although it is virtually the most used equation in high velocity particle physics, where it is married with quantum physics inextricably, it is ultimately derived from the general theory of relativity, and since the general theory and quantum physics has proved impossible to merge till now, this equation also can give anomalies when it is not used carefully.

Some more examples of the difficulties in using E=MC² unreservedly in quantum physics can be seen with these two definitions of mass:

3. An electron moving through a crystalline solid becomes heavy: This is a common definition used in explaining the passage of an electron through a crystalline solid such as a metal. The electromagnetic field of the metal retards the movement of the electron, and hence more energy has to be added to the electron to move it through. This extra energy is smoothly calculated as mass through E=MC² and the electron is said to have acquired mass, or become heavier.

Such an explanation raises a lot of questions. It must be understood that this acquiring of mass by the electron is not the relativistic acquiring of mass that it would have if it went at speeds close to light. This is a different form of mass altogether, since it can be acquired at speeds quite less than light speed. Is this mass then ‘given’ by the metal to the electron, a third way of acquiring mass which does not need the Higg’s field? Or if we want to preserve the Higg’s field, should we say that the metal distorts the Higg’s field in such a way that it can give mass to the electrons within the metal?

The answer is neither; it is due to a careless application of E=MC². What is acquired by the electron is extra energy; relating this to mass through this equation gives an anomalous definition of mass.

This explanation of electrons gaining weight in crystals is important in another way in that an analogy based on this is used to explain the Higg’s mechanism in many popular books and extensively throughout the web. The website of CERN also uses this analogy. But this analogy may well give a wrong understanding of the Higg’s mechanism, and hence of mass, specially now when there is bound to be a huge rise in popular interest in Higg’s due to the super hadron collider of CERN. See here.

4. Photons acquire mass while passing through super conductors: this is perhaps the most distorted application of E=MC².

By the very definition of Higg’s field, photons, the quanta of the electromagnetic field, do not interact with it. Both the general and the special theory of relativity also define photons as being inherently massless. If there is any talk of photons gaining mass or weight, then by this definition, it must be something other than Higg’s field. What mechanism is this? Is E=MC² capable by itself of giving mass to an object, a competitor to the Higg’s field? Is there a fourth mechanism hidden within superconductors that can give mass to photons?

Of course there isn’t. This is another careless application of the term mass through E=MC². Here also, it is the energy state of the photon that is increased, and this is converted illegitimately into units of mass, and described as the photon acquiring mass and getting heavier.

Mass as such is not fully understood. The quantum description of origin of mass, through the Higg’s field, and relativistic mass in General Relativity through E=MC², are not well integrated with each other.

There is no controversy about the first definition here, of rest mass, and it is the only definition of mass which we can accept without much fuss. This is through the Higg’s mechanism.

But in the second definition, controversy already starts. There is no real explanation about how relativistic mass could actually originate, and physicists in general do not use the term ‘relativistic mass’ at all, and the term ‘mass’ for physicists mean ‘rest mass’.

It is not known at all whether E=MC² of the general theory relates to the Higg’s field to account for its increase in mass. There is no answer to this. Indeed, even asking such a question is probably incorrect. And taking E=MC² as something which can give mass to a particle or body, almost as if it is another sort of Higg’s field, is something which cannot be accepted either.

Of course, it must be understood that ignoring E=MC² except for rest mass does not solve all the problems either. In the case of the proton, we have seen that the extra rest energy inside it does contribute to the mass of the proton according to E=MC². In that case, we may well ask why E=MC² does not work when the energy is not rest energy but kinetic energy. When an electron is accelerated and approaches the speed of light, it does tend to acquire more inertia and it requires more and more energy to accelerate it. Also, a photon does have momentum, it is theorized that a solar sail could be used to gather the momentum from photons from the sun to accelerate a spacecraft. A photon is well known to have momentum, although it is massless, which seems contradictory.

Ignoring relativistic mass and other forms of mass does raise a lot of questions and seems a bad idea. It is only that considering these masses is even worse!!

Lev B. Okun in The Concept of Mass (Physics Today -- June 1989, Volume 42, Issue 6) launched a strong attack against the very use of the term ‘relativistic mass’ and showed that it was an unnecessary term. He argued that ‘mass’ should be used only to mean ‘rest mass’ and what is termed as relativistic mass is in fact only the energy state of the particle. He argued that the equation E=MC² stands only for Rest energy=Rest mass × C² and this was Einstein’s meaning too.

"It is not good to introduce the concept of the mass M=m/(1–√v²/c²) of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the 'rest mass' m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion." Einstein 1948.

(This image is taken from L.Okun's article in ‘Physics Today’)

This problem is well recognized in quantum physics and accelerator physics avoids the term relativistic mass altogether. The only term used is rest mass. To express the higher energy of a moving particle, the rest mass is converted into rest energy through E=MC², and any extra energy is added to it. Thus quantum physics talk only about the energy states of a particle and never about its mass, unless it refers to rest mass.

When the word mass is used also to refer to something other than rest mass, the units used for it are energy units like MeV, so that the use of ‘mass’ is redundant here and in fact it is the energy state that is being discussed. In fact, even rest mass is usually described in terms of energy units, which means that it is the rest energy that is being considered. Because ‘rest mass’ is never used in accelerator physics in particular and most of quantum physics in general, the term ‘mass’ in quantum physics invariably means ‘rest mass’.

This is the best way to go about this business. The equation E=MC² needs to be used with a word of caution when used for anything other than rest mass. E=MC² is not an unvanquished hero, shining wherever it goes; in some situations it can instead spread confusion.

Terms like relativistic mass should be avoided altogether if one is to avoid getting into confused characterizations which on closer examination turn out to be contradicting important scientific axioms, as in the case of photons acquiring mass.

If it is not legitimate to discuss relativistic mass, how much more inappropriate is it to discuss electrons gaining weight in solids or photons gaining weight.

Till we understand mass more fully, the term mass should not be used for anything other than rest mass. The value of mass when related to weight (as in grams) should be used only when discussing this is terms of rest mass. Terms like ‘heavy’ should not be used in quantum physics unless it relates to rest mass.

In all other states, it is only the energy state that should be used to describe a particle. The rest mass can be converted into rest energy quite legitimately, and all other forms of energy added to the particle should be added to this rest energy and the state of the particle described only in terms of this energy state. This energy state, expressed in terms of energy units, should never be converted into mass units through E=MC², unless it is the rest state of the body being discussed.