# Why is Einstein's mass-energy relation usually written as $E=mc^2$, and not $\Delta E=\Delta m c^2$?

Why is Einstein's mass-energy relation usually written as $E=mc^2$, and not $\Delta E=\Delta m c^2$?

When you calculate the energy $\Delta E$ released during nuclear fission, you take the difference $\Delta m$ between the mass of the particles before and after nuclear fission to find it.

Of course, the second formula follows from the first, but in most contexts the second one is used. Which brings me to the question: why did the form $E=mc^2$ become so much more popular than $\Delta E=\Delta m c^2$?

Is there a historical reason?

• $E=mc^2$ tells you that exactly what the full energy content is of a particle (at rest) with mass $m$. I don't think your proposed alternative is an improvement... Therefore, I also don't think one can really find a ''reason'', except for the fact that this is simply correct. – Danu Jul 5 '15 at 8:54
• I think this would be better for physics.SE, but briefly, your second sentence is incorrect. You actually have to work with the mass difference This is not true. – Ben Crowell Jul 5 '15 at 13:15

Einstein is the one who removed the increments from the formula back in 1905, and one could say that the "historical reason" is his influence on theoretical physics. But his position was amply justified by subsequent experimental evidence. To explain, let us assume for now that energy and mass are considered in a frame where the object is stationary relative to the observer, and write $$E_0=m_0c^2$$ to reflect that (there is a separate issue with $$E=mc^2$$ that complicates the answer, I will address it below).

Einstein's original calculation in Does the inertia of a body depend on its energy content (1905) only considered mass changes effected by emission and absorption of electromagnetic radiation, which means that, indeed, he only derived $$\Delta E_0=\Delta m_0c^2$$, and only for electromagnetic processes. As pointed out by a number of authors, mathematically we only get $$E_0-K=(m_0-q)c^2$$, where $$K, q$$ are some constants. $$K$$ can be set to zero by calibration, but not $$q$$:

"Setting $$q = 0$$ involves a hypothesis concerning the nature of matter, because it rules out the possibility that there exists matter that has mass, but which is such that some of its mass can never be “converted” into energy... Einstein's argument allows for the possibility that once a body's energy store has been entirely used up (and subtracted from the mass using the mass-energy equivalence relation) the remainder is not zero":

Thus, Einstein made two conjectures in his derivation of $$E_0=m_0c^2$$.

1) Effect of energy transfer on mass is the same for all processes, as it is for electromagnetic ones.

2) There is no matter with residual mass, which never participates in energy transfers.

The first conjecture is, of course, a logical consequence of special relativity (and its predecessor, the hypothesis of molecular forces, introduced by Lorentz). It was introduced the same year in another paper, where Einstein interpreted relations derived from Maxwell's electrodynamics as kinematic relations that apply to all objects uniformly, and is vindicated by overwhelming evidence accumulated since then. But if that was not enough, Einstein gave another derivation in 1935, which is manifestly based on special relativity.

The second conjecture is also supported by evidence, but it is independent of special relativity and the case for it is not as strong, although so far we have not encountered any matter with residual mass. The most direct evidence is annihilation experiments in particle physics, where massive particles are entirely reduced to radiation, but there are types of matter (exotic, dark) that we are scarcely familiar with, so residual mass is not entirely ruled out.

In general, to get $$E = mc^2$$, one has to introduce the so-called "relativistic mass" $$m:=\frac{m_0}{\sqrt{1-v^2/c^2}}$$, which is ironically named because it is not Lorentz invariant (depends on velocity, and hence on reference frame), and therefore can not represent a physical property in relativity. Then $$m_0$$, which is in fact invariant, is called "rest mass". Experts in relativity, starting with Einstein himself, consider velocity dependent mass an artificial construction, that should be dispensed with, and call the "rest mass" just mass denoted $$m$$. In recent decades, this began to be reflected in textbooks. However, then $$E = mc^2$$ becomes false and should also be dispensed with, but it is hard to overcome its entrenched status in popular culture, see When and why did the concept of relativistic mass become outdated?