The timeless equivalence of energy and matter — and why it matters
Einstein is commonly cited as having come up with the famous principle of equivalence of energy and mass, i.e. E=mc² (in which E=Energy, m=mass and c=speed of light in vacuum).
His 3-page paper titled “Does the inertia of a body depend upon its energy content?” as submitted to the “Annalen der Physik” in 1905 (in German) is said to be the source of this equation and the broader theory of relativity. However, where does it show that specific famous equation?
As it turns out, Émilie du Châtelet (1706–1749) translated Newton’s Principia into French, and commented on the more generic E=½mv² formula. Einstein was certainly not the first scientist to relate matter and energy.
“inertia”
According to the Internet, “inertia” in this context means
A property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force.
Einstein originally used the German word “Trägheid” — which also translates to “slowness”
“mass”
Einstein only refers to “mass” in the concluding paragraph of his paper:
In the above equation, “K” refers to Kinetic energy; K₀ and K₁ are different observations from 2 different coordinates. An Erg is a unit of energy.
In later papers Einstein refers to “inertial” mass versus “gravitational” mass.
“massless” photons
In my mind I struggle with the concept of “massless” photons. Photons clearly have non-zero energy, so if energy and mass are equivalent, how can they have no mass at the same time? As it turns out, to understand this we need to make a distinction between “rest mass” m₀ and “relativistic mass” m: Photons have zero rest mass, but non-zero relativistic mass. The former seems to be what the formula is referring to.
But this leads to another confusion: If rest mass and relativistic mass are not the same thing, why do we call both “mass”? That is highly confusing to say the least.
The Energy-Momentum relationship highlights this difference:
In this equation, the left side “pc” represents momentum (i.e. movement) while the right side accounts for rest mass. Since photons have no rest mass, the equation simplifies to E=pc, or — using de Broglie’s matter-waves
Similarly, we can see that E=mc² is really about energy (E₀) and mass (m₀) at rest — a timeless static system without any context of spacetime. Simplified, but unrealistic and only part of the story — and not relevant to photons.
Redshift and phase velocity (propagation speed) of light
Given the above, we can now look into redshift (the phenomenon where visible light frequencies shift towards the redder part of the spectrum, i.e. longer wave lengths) and phase velocity / propagation speed:
As the wave length gets longer, the speed of propagation increases. At the same time, the relativistic energy/mass (E=hc/λ for vacuum, substitute c with vₚ for other materials) decreases, trending towards 0 in the limit — though of course, no photon can go faster than “the” speed of light c.
Do photons always travel at constant speed?
Most scientists believe/assume that photons always travel at the speed of light in vacuum. Some challenge this assumption. In practice on earth, due to local circumstances one could make a case that every photon is unique.
When small differences matter
For centuries, Newton’s laws of mechanical movements sufficed to explain ordinary everyday motions. However, as measurements became more accurate and the observable universe of science evolved, there came a point where the old and aging laws stopped matching the experimental results.
Relativistic kinetic energy can be expressed as:
For human scale velocities (v⋘c) this evaluates to almost 0, i.e. negligible. But as Philip W. Anderson once wrote: “More is different”
In conclusion
“Does the inertia of a body depend upon its energy content?”
The tendency of an object to resist changes in its state of motion varies with mass. Mass is that quantity that is solely dependent upon the inertia of an object. The more inertia that an object has, the more mass that it has. A more massive object has a greater tendency to resist changes in its state of motion.
The famous E=mc² formula derived from Einstein’s work relates mass to energy. It expresses how an object’s mass is reduced when energy is extracted from it — like a human loosing weight through exercise.
So yes: Inertia depends on energy content, more energy = more mass = slower.