E=mc^2 is one of the most famous equations in the world, people who have no understanding of physics or relativity can immediately know that it is the work of Einstein. But what is relativity? What do those three letters mean?
Too understand that, we must first understand the principles behind the idea.
Galilean Relativity (Classical physics)
There are two types of physics that we study today, classical physics, which is ideas and theories postulated before the 1900s, and modern physics, or discoveries made after the 1900s. In classical physics, Galileo proposed that velocities add linearly, and for small velocities, like the ones we experience every day on Earth, that’s true. If a car is driving at 70 miles per hour, and you throw a baseball forward at 30 miles per hour, how fast is the baseball moving? Common sense and Galilean Relativity would tell you that the baseball is moving forward at 100 miles per hour.
But things get weird when we start approaching relativistic speeds, or speeds above 30,000,000 meters per second (c/10). Velocities don’t add linearly at high speeds. Why is that?
Before we get to an explanation of relativistic velocity addition, I first want to make a detour and talk about frames of reference. Above I used an example of a person throwing a baseball out of a moving car. In that, there are 3 frames of reference, the first is to an observer standing on the side of the road. He sees the baseball moving at 100 miles per hour to the left, and the car moving at 70 miles per hour to the left.
The second is someone sitting inside of the car. In their reference frame, the car is not moving, the Earth and baseball are. They see the baseball moving forward from them at 30 miles per hour, and the environment whizzing past them at 70 miles per hour. The net result is the same, the Earth and baseball have a difference in speed of 100 miles per hour.
The last reference frame is the one of the baseball itself. Before it is thrown, the baseball sees the car moving at zero miles per hour, and the earth moving at 70 miles per hour. After it is thrown the baseball sees the car moving away at 30 miles per hour, and itself moving through the earth at 100 miles per hour.
In every scenario, there are reference frames, and one inertial reference frame. An inertial reference frame is a frame that has no acceleration, meaning all of the forces are balanced.
In physics, we always assume that the laws of physics remain the same in all frames. Whether or not that is true remains to be proven (we have no idea because no one has ever traveled at that speed). Another key assumption is that c is always equal to c. There is nothing faster than the speed of light, and light can only travel at the speed of light, no more, no less.
The first confusing concept of the day is time dilation. In all cases of physics, time is always a variable. Imagine a train traveling across your screen, from left to right, with a flashlight and a mirror.
Please excuse my terrible drawing skills here
In the top set, you see the beam of light as someone inside the train. You see the beam go up from the flashlight, and then bouncing off a mirror on the ceiling, and then back down. The light only travels straight up and down.
In the bottom set, you see the beam as someone standing outside the train. You see the light beam go up, and back down, but the light is also moving to the right at speed v, the same speed as the train. For that reason, the light takes a diagonal path.
For an observer of the train, the light travels a distance greater than the height of the train, but the speed of light cannot be greater than c. Specifically, the extra distance traveled means that the observer saw the trip of the light take longer than someone inside the train. The factor by which the trip of light is longer is described as lowercase gamma (γ).
Let’s assume that we have a metal rod, that is exactly a meter long when measured at rest. If the rod is moving at a relative speed .8c, or 80% the speed of light, the rod will measured to be shorter. The length would appear to be shorter because of the relative motion. To itself, the rod will be measured at its normal length, but to an observer in a stationary reference frame, the length will appear shorter. The equation for this is L(contracted)=L/ γ
Lets consider Newton’s second law, F=ma. How is this affected by relativity? Using gamma(γ), which is the ratio of speed relative to c, γ=1/sqrt(1-(v/c)^2), you can calculate momentum to be equivalent to ρ= γ*m*v.
Bringing it all together
Because of the momentum equation, we can calculate energy to be equivalent to KE=m γc^2-mc^2. But hold on, that has more stuff in it than just E=m*c^2! That is because kinetic energy is applied only when a particle is moving. If the velocity is equal to zero, than γ is equal to zero, making the above equation E=m*c^2. This implies that all particles have an inherent “rest energy” giving us the mass-energy equivalency. This means that should a particle be destroyed, all of its mass converts to energy! This is the fundamental theory behind the Manhattan Project, which generated the atomic bomb. By destroying the nucleus of a heavy atom (an isotope of Uranium or Plutonium) in chain reactions, energy equivalent to thousands of tons of TNT is expelled, resulting in destructive force. More recently, the idea has been used to power cities in the form of nuclear energy.
Questions can go in the comments below!