Deriving the Relativistic Dispersion Relation (E² = m²c⁴ + p²c²)

The energy-momentum equation is used everywhere — from quantum mechanics to general relativity. But how exactly does one derive it without using linear algebra or calculus?

In my previous post: Deriving the Lorentz factor (γ) of Special Relativity, I worked out a simple way to derive the Lorentz factor — an important component of Special Relativity that determines how much a system deviates from classical behavior under relativistic conditions. Using this factor, there is, in fact, an elegant way to relate three fundamental quantities of nature: Energy, Mass, and Momentum.

In this post, I’ll be showing two methods for deriving the relativistic dispersion relation, a.k.a the energy-momentum equation, using the Lorentz factor and only middle-school algebra. It’s a simple case of rearranging some terms.

First off, there are two methods we can use. The first method involves extrapolating from the concept of relativistic mass whereas the second method uses the expression of the total energy of a relativistic system.

Method 1

Albert Einstein’s special theory of relativity put forth many bold new ideas, but perhaps the boldest of them all is the hypothesis that mass is not a constant quantity but is subject to change during motion. S.R. predicted that during relativistic collisions, momentum was not conserved in all frames of reference. To reconcile this, Einstein hypothesized that the normally invariant mass component of momentum is a changing quantity that was dependent on the speed of the object. The expression proposed for such a changing or “relativistic” quantity of mass was:

Where m stands relativistic/effective mass, m0 is the rest mass, i.e., the mass of the object in its rest frame of reference, v is the velocity of the object, and c is the speed of light. The expression can be derived using S.R.’s coordinate frame transformations — Lorentz transformations, and could be rewritten in terms of the Lorentz factor γ:

Such a hypothesis that went against the preexisting Newtonian world view of invariant mass was, unsurprisingly, controversial at that time. However, it made perfect sense when viewed in the context of the mass-energy equivalence relation E=mc², which establishes that mass and energy can be treated as equivalent, interchangeable quantities. If you think about it, the relation implies that the total mass of an object is dependent not only on its rest mass but its internal energies such as potential and kinetic energies as well.

Since kinetic energy is the energy of motion, it is only logical to assume that the change in kinetic energy during motion will manifest as a change in total mass as well, giving the idea of relativistic mass a bit more credibility. However, when we move one step further and include momentum into the picture, we arrive at a much bigger and significant result.

First, we start with the expression for relativistic mass and do some rearranging and squaring:

Then we multiply throughout with c⁴:

Now, we know that the momentum of an object p is the product of its mass and velocity (mv) and the total energy E of an object as given by the mass-energy equivalence relation is mc². For clarification, m here refers to the effective/relativistic mass of the object in question. Using these two identities,

We arrive at the relativistic dispersion relation, which expresses the total energy of a body in terms of its rest mass and momentum.

The second method, although it works, is a bit more uncomfortable for our friendly foes with the chalk a.k.a the mathematicians.

Method 2

The total energy E of a relativistic system is defined as,

Where m0 is the effective rest mass of the system, v is the velocity of the system, and γ is the same Lorentz factor mentioned in Method 1.

Now, if we rearrange some terms,

We know p = mv, so substituting and rearranging again,

Multiplying and dividing the equation with c² and E² respectively,

Now, recall that the mass-energy equivalence relation is an equivalence that goes both ways, meaning that:

Using this, we can make some clever substitutions: plug in the corresponding relation for the E² in the L.H.S and the m0²c⁴ in the R.H.S respectively. Such an algebraic “sleight of hand” is akin to walking on the mathematical equivalent of eggshells, however,

And there you have it, the same relativistic dispersion relation that we obtained in Method 1. This relation is one of the most important equations in modern physics and plays a governing role in determining the relationship between three fundamental quantities of nature in physical systems.

Thanks for reading! Hope you enjoyed this blog post and learned something new about Einstein’s elegant special theory of relativity.. I apologize for the LaTeX equations being presented as images. Medium doesn’t have LaTeX support which is a bummer (something Quora blogs had). Coming from Quora, Medium was initially a big change but it’s starting to grow on me. A similar post (which I published a year ago) can also be found on my now-defunct Quora blog of the same name: https://qr.ae/TWvAWz.

I plan to continue publishing posts like these regularly and possibly even start a series of posts pertaining to a certain topic. If there are any topics you want me to write about, please mention them in the comments below.

“I… a universe of atoms, an atom in the universe.” — Richard P. Feynman