Thursday, April 30, 2009

Rest Mass Versus Relativistic Mass

I can't think of any other concept in physics in which (i) it is as 'well-known' to the general public as it is to physicists and (ii) it causes as much confusion to the general public as it does to physicists. I'm talking about the concept of "mass" and the infamous Einstein's equation "E = mc^2".

One of the issues surround the concept of mass and the Einstein equation is the idea on what "mass" really means in relativity, and the validity of the concept of "relativistic mass". There have been many articles written to address this issue, but it is obvious that, even today, many media, including textbooks and popular writings, continue to use the term "relativistic mass" to mean an increase in the measured mass when an entity is moving at relativistic speeds. Whether the faulty understanding of such a concept can create a stumbling block in understanding relativity or not is an entirely different issue. But can there be a simpler approach to such a concept without invoking the name "relativistic mass"?

Lev Okun seems to think so. In a highly compact 2-page paper in the Am. Journal of Physics[1], he wrote a very concise explanation of what "mass" is, and why there is really only ONE concept of mass as defined in terms of momentum and energy by what he called the most fundamental equation of relativity theory:

m^2 = (E/c^2)^2 - (p/c)^2,

where E is the energy, p is the momentum. There is nothing new here that someone who has gone through an intro class in relativity/Modern Physics would not have seen. But it is put in such a compact and clear form that it summarizes Special Relativity in almost 1 1/2 pages!

What is as interesting is his commentary on how this issue has been treated in the media and in textbooks.

Unfortunately, sometimes and especially in his popular writings Einstein was careless about the subscript 0 and spoke about the equivalence of mass and energy and omitted the attribute “rest” for the energy. As a result Einstein's equation E0=mc^2 became known in its famous but misleading form E=mc^2. One of the most unfortunate consequences is the concept that the mass of a relativistic body increases with its velocity. This velocity dependent mass is known as “relativistic mass.” Another consequence is the term “rest mass” and the corresponding symbol m0. These confusing concepts and notations prevail in such classic texts as the ones by Born and Feynman. Moreover, in these texts the dependence of mass on velocity is presented as an experimental fact predicted by relativity theory and proving its correctness.

To substantiate the formula m=E/c^2 some authors use the connection between momentum and velocity in Newtonian mechanics, p=mv, forgetting that this relation is valid only when v (is significantly less than) c and that it contradicts the basic equation m^2=(E/c^2)^2−(p/c)^2. Einstein's tolerance of E=mc^2 is related to the fact that he never used in his writings the basic equation of relativity theory. However, in 1948 he forcefully warned against the concept of mass increasing with velocity. Unfortunately this warning was ignored. The formula E=mc^2, the concept relativistic mass, and the term rest mass are widely used even in the recent popular science literature, and thus create serious stumbling blocks for beginners in relativity.


[1] L.B. Okun Am. J. Phys. v.77, p.430 (2009).


Yuri Danoyan said...

L.B.Okun asking your e- mail adres for private correspodence.

a.k.satsangi said...

According to Einstein Theory of Relativity, E=mc^2. According to this relationship of Energy and Mass
1 kg mass of any matter is equivalent to 9 x 10^16 J of energy.

Does it mean that,

Mass of any matter is Condensed Form of Energy and Energy is Diffused Form of Mass of any matter ?

A question may also arise what existed before the creation of the Universe Energy or Mass or both?

A.K.Satsangi said...

Based on E=mc^2, can it be said that mass is the ‘potential state’ of matter and energy is the ‘kinetic state’ of matter and just multiply mass with c^2 you will get huge amount of energy and divide energy by c^2 you get very small amount of mass OR some other factors/ mechanisms are essential for these conversions ?

A.K.Satsangi said...

E=mc^2 is called ‘Einstein’s energy-mass relation’. According to this relation, 1 kg mass of any matter is equivalent to 9x10^16J of energy. This is a huge amount of energy, equal to 2.5x10^10kWh. It is evident that the amount of energy is same irrespective of the matter taken, whether it is carbon, iron, copper or any other including radioactive elements. The amount of energy thus released does not depend on the atomic number, atomic weight, electronic configuration etc. It is the mass of the matter only based on which the amount of energy is calculated. It means that ‘mass’ is the connecting link between energy and matter.

A.K.Satsangi said...

It is written in the Text-Books of Physics that if we give ∆E energy to some matter, then according to E=mc^2, its mass will increase by ∆m, where


Since the value of c is very high, the increase in mass ∆m is very small. For example, if we heat a substance, then the heat-energy given to this substance will increase its mass. But this increase in mass is so small that we cannot measure it even by the most sensitive balance. Similarly, if we compress a spring, its mass will increase, but we cannot confirm this mass-increase by any experiment.

Now the question is whether the change in mass as quoted in these two examples is reversible i.e. when the same substance of example one is cooled down, energy is produced equal to ∆m x c^2 (∆E=∆m x c^2) and in second example when we release the spring , energy is produced equal to ∆m x c^2 and initial mass is retained in both the cases ? Or the above changes are irreversible ?

Unknown said...

Of course this equation E=mc2 assumes you can get each and EVERY atom in said mass to release its energy which requires perfect efficiency. Which never happens in the real world. In a fission reaction you'd probably be lucky to get 10 % of the atoms to release their energy (that's just a guess maybe a physicist could answer that more accurately). But if you COULD release the energy of matter perfectly than 1 lb of anything equates to roughly 10 million tons of TNT.