E=mc² (And The Rest of the Equation)

Appearing in the background of television shows scrawled on chalkboards and in the introduction of Bill Nye The Science Guy, Einstein’s equation E=mc² is one of the most widely known equations in the world. Many people may not fully understand the meaning behind Einstein’s most famous equation, but they remember it due to its brevity. Today, we’d like to introduce you to the equation’s relativistic form: E²=(mc²)² + (pc)²

Before Einstein, physicists presumed there was a substance known as the “aether,” permeating the known universe. Scientists hypothesized that the aether possessed the necessary properties to explain the extreme speed at which light waves propagate through space. However, multiple experiments proved that the aether did not exist, leaving a large gap in our knowledge of the universe. Einstein’s theory of special relativity, developed in 1905, sought to explain how particles behave when moving extremely fast in a universe without aether.

Einstein’s theory of special relativity makes two postulates:

1. the speed of light is constant in space, no matter the speed of its source,

2. the laws of physics remain consistent between different non accelerating observers.

From these two suggestions, Einstein was able to perform thought experiments and develop new mathematics, presenting us with an improved understanding of how particles move in our Universe. The universe has many strange properties that defy our day to day experiences. As humans, we simply do not move quickly enough to easily observe the effects of special relativity.

The full equation for the total energy of a particle in our universe is:

E2=(mc2)2 + (pc)2

Let’s break down the equation into its variables: E is the total energy, m is the mass of the particle in question, p is the momentum of the particle in question, and c is the speed of light, a staggering 186,000 miles per second.  In an everyday situation, objects are not moving anywhere near the speed of light, making the momentum p so small that (pc²)² is essentially 0.

Often in science fiction,  a group of explorers are traveling to a distant star in a spaceship, frozen in some sort of “cryosleep.” In order for the explorers to arrive in a timely manner, the ship is traveling at or near the speed of light. Einstein's theory of special relativity states that the speed of light is the speed limit for the entire universe. Because light is a massless particle, it can travel at its speed, c. So how difficult is it for something with mass to travel close to the speed of light? Rather than a spaceship, let’s start a little smaller with In Focus Astronomy’s resident Australian Shepherd mix, Kepler. Before we begin our calculation, we have to express Einstein’s equation in different terms.

A relatively short derivation transforms the previously stated equation into an expression of the kinetic energy of the particle. The relativistic kinetic energy for a particle moving extremely fast is:

Ek=mγc²-mc²

The multiplicative effect of the greek letter gamma (γ), officially known as the Lorentz factor, corresponds to a strange effect; when objects approach speeds close to the speed of light, their masses appear to increase.

Now that we understand relativistic kinetic energy, let’s see how much energy it would take to get Kepler, who has a mass m of about 23 kilograms, to 99% the speed of light.

The Lorentz factor for Kepler is about 7.14. If we plug these numbers into our equation for relativistic kinetic energy, we determine it would require roughly 12,700,000,000,000,000,000 joules for Kepler to move at 99% of the speed of light. This energy expenditure is close to the yearly energy consumption by the entire United States (circa 2009).

Einstein’s thought experiments and mathematics revealed a non-obvious quality of the universe - that the laws of the conservation of mass and the conservation of energy were actually intertwined. For reference, the conservation of energy states that energy can neither be created nor destroyed; it can only change forms. Einstein’s postulates were groundbreaking as they revealed that energy and mass are the same and could be converted back and forth between each other. Your computer, your stove, you, every piece of matter in the universe is energy given form and substance.