In 1905, a young patent clerk with emerging crazy hair changed the world of science and physics forever with four revolutionary scientific papers. In Albert Einstein’s so-called miracle year, his revolutionary discoveries included proposing that light was a particle decades before other scientists, confirming the existence of atoms, and establishing relativity as a scientific principle.[1] But his most famous discovery today is not any of these groundbreaking discoveries, but an idea from his fourth and final paper, “Does The Inertia Of A Body Depend Upon Its Energy-Content?” in which he provided the proof for his now famous equation m = L/c2.[2] Don’t know that version? Maybe you’ll recognize the less historically accurate, but more culturally relevant one: E = mc2.
Although it is the most famous of his discoveries, it was only released as a quick follow-up to his third and previous paper, “On the Electrodynamics of Moving Bodies,” which established his theory of special relativity. Special and his later established theory of general relativity are the basis for the full theory of relativity, the basis for all modern astrophysics.[3] But despite the importance of these discoveries, they are criminally misunderstood, mostly acting as a shorthand for complicated and theoretical physics. But to break into physics beyond Newton’s laws, it's essential to understand the world's most famous physicist’s most famous paper.
“I’m travellin at the speed of lighhhhhhttttttt”
e - Energy in Joules
m = Mass in Kilograms
c = The (Constant) Speed of Light
299,792,458 meters per second
This equation at the most basic level establishes the relationship between mass and energy, not as a constant ratio but instead as a conversion rate. Basically, one kilogram of energy can be converted into 299,792,458 joules of energy and a joule of energy is equivalent to a tiny portion of a kilogram. There are three things that are first obvious about this:
Mass and energy are the same things and they can be converted between each other
Just a little bit of mass has a massive amount of energy
The speed of light is FAST
At the most basic level, what are mass and energy? Well, they’re properties of particles. There are 17 (or 25 depending on your count) currently known elementary particles (particles that can’t be subdivided), each of which is, under the laws of Quantum Field Theory, better described as an “excitation in a quantum field.”[4] Some people might be familiar with regarding particles as waves, and this theory essentially says that each type of particle is a part of one big 3D wave (or quantum field) that stretches across the entire universe, and particles are just points where that field spikes up. Take waves on a beach, and stretch them into 3D; the height of the wave (or particle) is how much energy it has.
It’s a bird! It’s a plane! It’s a quantum fluctuation!
There is one specific group of particles that is important in this context:bosons. They have some more complex differences from other particles called fermions, but the main distinction is that boso ns act as “force-carrying particles,” transferring energy from one particle to another.[5] The four fundamental forces (gravity, weak interaction, strong interaction, and electromagnetics) of the universe are the interactions that take place between particles at the atomic level, and each one has a corresponding boson that transfers energy through that particle’s field (the particle for gravity hasn’t been discovered, but it is still believed by many physicists to exist).[6] These bosons can be “absorbed” and later expelled to transfer energy between fermions. There are two main bosons that are crucial to this story.
Photons
Photons are bosons that function in the electromagnetic field. They are completely massless, and travel at the the speed of light. However, they are not completely unique as gluons, the bosons for the strong interaction which holds quarks together in protons and neutrons--- also massless and moving at the speed of light.[7]
Higgs Bosons
The Higgs boson is the most recently discovered particle in the standard model (found only in 2013), but it is crucial to our understanding of the universe. Its the only scalar boson, a technical distinction, but it is more importantly the particle of the Higgs field, which gives all other particles (except photons and gluons) mass. Although they are recent and so hotly debated, François Englert and Thifhelimbilu Daphney Bucher, some of the primary physicists who discovered the Higgs field describe it as “a sea” that covers the whole universe. Particles travel through this sea, and particle interactions slow them down giving massive particles inertia.[8]
We have two particles that fit importantly into this equation, but how did Einstein figure out this relationship? And where does the speed of light come from?
Special Relativity
The aforementioned theory of relativity was computed using the work of James Clerk Maxwell, who’s research constitutes the foundational principles of electromagnetism.[9] He also discovered that the speed of light is constant in all frames of reference. Think about it like this. If a train is moving north at 50 mph and someone is standing on the ground, they’ll see it moving away from them at that speed, but if they’re on a train that is moving in the same direction at the same speed of 50 mph, it will seem to not be moving at all, while the ground beside both trains will seem to be moving away in the southward direction at a speed of 50 mph. For instance, even though the Earth moves at an extremely fast speed through space, relative to other planets, but to us, organisms gravitationally bound to this blue ball, it seems stationary..
The frame of reference, or the properties of an observer, determine the properties of an object: how fast it is moving, in what direction, and much more. Everything is relative except for the speed of light.[10] No matter your direction, no matter how fast, the speed of light is exactly the same. Maxwell’s discovery butted heads with this well-established principle of relativity. Einstein took his rules and tested them with a variety of situations, and poked some holes in the logic; problems that needed to be solved.
Imagine a train moving at 10 mph down a track, and two mirrors facing each other on the sides of one of the cabins (perpendicular to the movement of the train). If light is emitted from one directly toward the other, and it bounces three times, to someone on the train it covers exactly the distance between them, say 5 meters. But to someone outside the train (say it has glass walls), because they also see the train moving relative to themselves the light will move a longer diagonal distance. The hypotenuse of a right triangle can be calculated using one leg (the distance between the mirrors) and the other (the amount the mirror traveled in the time that the light bounced) which amounts to about .0000000166782048 seconds (the time it would take light to travel that time on the train) times the speed of the train (10 meters per hour). This is an insanely small number, but the Pythagorean theorem tells us that the hypotenuse that connects these bases is greater than either of the bases alone. This means that a person outside the train experiences the light to travel a longer distance than the person within the train. But, because the speed of light is constant there’s only one explanation: the time experienced by the person traveling and the person standing on the grass is different.
This was essentially Einstein’s conclusion: the closer an object moves (relative to another) to the speed of light, the more time dilation there is between them.[11] While that was a slow example and thus insignificant, if someone is moving just below the speed of light for a year in their time, you will both experience time that feels the same, but you will age exponentially more than them.
The reason for this, interestingly, is that all objects actually move at the speed of light through spacetime, but not equal amounts through space and time. We’ve established that as the relative speed of an object increases it experiences less time. What this tells us is that if you put speed in spacetime on a line, on one side you have photons and gluons, moving at the fastest speed possible, and on the other, you have a completely stationary object that moves fully through time. But in the middle the two blend, and so all objects move at the same speed, through space and time, but have a different ratio of the two depending on their velocity.[12]
Now, we understand why the speed of light is so important, but how is mass related? Well, the Higgs field gives all affected particles inertia, therefore limiting the motion of particles. But photons and gluons don’t have this property; they are pure energy with no mass and no inertia to slow them down, and so they naturally move at the fastest speed they can through space. With Einstein's theory of relativity, the connections are beginning to show, but to fully understand how the speed of a light, mass, and the energy of particles is related, we need to look at the proof Einstein presents for his revolutionary equation.
Einsteins Proof[2]
Situation A:
Einstein's full proof in “Does The Inertia Of A Body Depend Upon Its Energy-Content?” is obviously complicated, but as long as you don’t look too much into the mathematics, it's actually not too hard to understand. Let's start with another scenario: if you’re sitting down on the couch and you turn on the statically moving light next to you and it emits a single pulse before turning off because, well, the circuit is fried, how much energy does it lose? That amount of energy is E, and it is carried off by massless photons moving away from the light, and the velocity of the bulb doesn’t change because photons were emitted equally on all sides. Thus the light bulb must have lost E energy, right? Now that your experiment is done, you get up to get some coffee and move at constant speed S.
Situation B:
But, what if you are walking at a steady speed when the light turns on. First of all, from your perspective, the emitter has gained energy because you are moving (if you’re walking away, it’s moving relative to you). But something that’s additionally important is how the light changes. I know that I told you the speed of light doesn’t change, and that’s important because of something called the Doppler effect. The energy of a photon and really any particle is determined by its wavelength, a property of the particle/wave. But if you are moving relative to the particle, from your frame of reference the wavelength can change. Essentially, the wavelength looks longer because you’re moving away from the particle. With any normal particle, this would be OK because the speed would change along with the wavelength, but photons are different, their speed is constant, which means that the energy of the particle must have decreased due to the increase in wavelength (let’s call the new energy L). But if the bulb is still moving at the same speed as when you got up to get coffee (in Situation A), then there is some missing energy lost because of the doppler shift. Keep in mind that the difference caused by the Doppler effect and all of these interactions between frames of reference and light-speed movement are dictated by the laws Einstein established in special relativity.
So?:
The equation for energy, in this case, is E (or L) = ½mv2; the velocity of the bulb is the same in both situations, but the energy taken by photons is different and so the energy of the bulb that they took it from (E > L, so in situation, A the energy of the bulb is lesser) is different. The only other variable in the equation is m, or the mass of the bulb, and so it must fulfill the difference in energy. If you simplify the equation here, you get E (the extra energy of the lightbulb or the difference between E and L) is equal to the lightbulb’s mass m (comes from the equation for energy) multiplied by the speed of light squared (comes from the equation that calculates the difference caused by the Doppler effect). E = mc2. It’s not that bad, right? If you want to see the actual equations that I’m referring to take a look to the right and you’ll see the basic progression Einstein poses.
In Conclusion
Eleven years after Einstein released his groundbreaking equation, he would release another revolutionary theory, one with even more important effects on modern physics: general relativity. While special relativity dealt with rarer situations involving light and its physical effects, his new theory focused on gravity and exploring its role in the universe. When comparing them one might argue that the latter is far more important, and physicists would probably agree, but that's not the only metric to be used. The reason why is that, another thirty years after general relativity and at the very end of the most destructive war in human history, Einstein’s legendary equation would be used to fuel the most magnificent and terrible weapon in human history: the atomic bomb.[15]
EXTRA CONTENT: Proofs
The variable L is what was previously mentioned as E (energy of light in Situation A), and what was called L (in Situation B) is modeled by the below equation.
E0 is the energy of the bulb before and E1 after the light is emitted in Situation A. L is the total energy of both photons (if the bulb only emits 2 total to simplify).
If you subtract Situation B from the Situation A equation is the result:
