Ask anybody — even somebody with no background in science — to call one thing that Einstein did, and odds are they’ll come again along with his most well-known equation: E = mc². In plain English, it tells us that vitality is the same as mass multiplied by the velocity of sunshine squared, educating us an unlimited quantity in regards to the Universe. This one equation tells us how a lot vitality is inherent to an enormous particle at relaxation, and in addition tells us how a lot vitality is required to create particles (and antiparticles) out of pure vitality. It tells us how a lot vitality is launched in nuclear reactions, and the way a lot vitality comes out of annihilations between matter and antimatter.
However why? Why does vitality must equal mass multiplied by the velocity of sunshine squared? Why couldn’t it have been another method? That’s what Brad Stuart needs to know, writing in to ask:
“Einstein’s equation is amazingly elegant. However is its simplicity actual or solely obvious? Does E = mc² derive instantly from an inherent equivalence between any mass’s vitality and the sq. of the velocity of sunshine (which looks like a fabulous coincidence)? Or does the equation solely exist as a result of its phrases are outlined in a (conveniently) specific method?”
It’s an ideal query. Let’s examine Einstein’s most well-known equation, and see precisely why it couldn’t have been another method.
To start out with, it’s vital to appreciate a couple of issues about vitality. Power, particularly to a non-physicist, is a very tough factor to outline. There are a lot of examples we are able to all give you off the tops of our heads.
There’s potential vitality, which is a few type of saved vitality that may be launched. Examples embody gravitational potential vitality, like lifting a mass as much as a big peak, chemical potential vitality, the place saved vitality in molecules like sugars can bear combustion and be launched, or electrical potential vitality, the place built-up prices in a battery or capacitor might be discharged, releasing vitality.
There’s kinetic vitality, or the vitality inherent to a shifting object on account of its movement.
There’s electrical vitality, which is the kinetic vitality inherent to shifting prices and electrical currents.
There’s nuclear vitality, or the vitality launched by nuclear transitions to extra secure states.
And, in fact, there are numerous different varieties. Power is a kind of issues that all of us “understand it after we see it,” however to a physicist, we wish a extra common definition. The most effective one we’ve got is solely: extracted/extractable vitality is a method of quantifying our means to carry out work.
Work, to a physicist, has a selected definition itself: a drive exerted in the identical route that an object is moved, multiplied by the gap the item strikes in that route. Lifting a barbell as much as a sure peak does work in opposition to the drive of gravity, elevating your gravitational potential vitality; releasing that raised barbell converts that gravitational potential vitality into kinetic vitality; the barbell placing the ground converts that kinetic vitality into a mixture of warmth, mechanical, and sound vitality. Power isn’t created or destroyed in any of those processes, however reasonably transformed from one kind into one other.
The way in which most individuals take into consideration E = mc², once they first find out about it, is when it comes to what we name “dimensional evaluation.” They are saying, “okay, vitality is measured in Joules, and a Joule is a kilogram · meter² per second². So if we need to flip mass into vitality, you simply have to multiply these kilograms by one thing that’s a meter² per second², or a (meter/second)², and there’s a elementary fixed that comes with models of meters/second: the velocity of sunshine, or c.” It’s an inexpensive factor to suppose, however that’s not sufficient.
In any case, you’ll be able to measure any velocity you need in models of meters/second, not simply the velocity of sunshine. As well as, there’s nothing stopping nature from requiring a proportionality fixed — a multiplicative issue like ½, ¾, 2π, and so on. — to make the equation true. If we need to perceive why the equation have to be E = mc², and why no different prospects are allowed, we’ve got to think about a bodily state of affairs that might inform the distinction between numerous interpretations. This theoretical software, referred to as a gedankenexperiment or thought-experiment, was one of many nice concepts that Einstein introduced from his personal head into the scientific mainstream.
What we are able to do is think about that there’s some vitality inherent to a particle on account of its relaxation mass, and extra vitality that it may need on account of its movement: kinetic vitality. We are able to think about beginning a particle off excessive up in a gravitational discipline, as if it began off with a considerable amount of gravitational potential vitality, however at relaxation. Whenever you drop it, that potential vitality converts into kinetic vitality, whereas the remainder mass vitality stays the identical. In the mean time simply previous to impression with the bottom, there shall be no potential vitality left: simply kinetic vitality and the vitality inherent to its relaxation mass, no matter which may be.
Now, with that image in our heads — that there’s some vitality inherent to the remainder mass of a particle and that gravitational potential vitality might be transformed into kinetic vitality (and vice versa) — let’s throw in yet another concept: that each one particles have an antiparticle counterpart, and if ever the 2 of them collide, they will annihilate away into pure vitality.
(Certain, E = mc² tells us the connection between mass and vitality, together with how a lot vitality you must create particle-antiparticle pairs out of nothing, and the way a lot vitality you get out when particle-antiparticle pairs annihilate. However we don’t know that but; we need to set up this have to be the case!)
So let’s think about, now, that as a substitute of getting one particle excessive up in a gravitational discipline, think about that we’ve got each a particle and an antiparticle up excessive in a gravitational discipline, able to fall. Let’s arrange two totally different situations for what might occur, and discover the implications of each.
Situation 1: the particle and antiparticle each fall, and annihilate on the on the spot they might hit the bottom. This is similar state of affairs we simply thought of, besides doubled. Each the particle and antiparticle begin with some quantity of rest-mass vitality. We don’t have to know the quantity, merely that’s no matter that quantity is, it’s equal for the particle and the antiparticle, since all particles have equivalent lots to their antiparticle counterparts.
Now, they each fall, changing their gravitational potential vitality into kinetic vitality, which is along with their rest-mass vitality. Simply as was the case earlier than, the moment earlier than they hit the bottom, all of their vitality is in simply two kinds: their rest-mass vitality and their kinetic vitality. Solely, this time, simply in the meanwhile of impression, they annihilate, reworking into two photons whose mixed vitality should equal no matter that rest-mass vitality plus that kinetic vitality was for each the particle and antiparticle.
For a photon, nevertheless, which has no mass, the vitality is solely given by its momentum multiplied by the velocity of sunshine: E = computer. Regardless of the vitality of each particles was earlier than they hit the bottom, the vitality of these photons should equal that very same complete worth.
Situation 2: the particle and antiparticle each annihilate into pure vitality, after which fall the remainder of the way in which right down to the bottom as photons, with zero relaxation mass. Now, let’s think about an virtually equivalent situation. We begin with the identical particle and antiparticle, excessive up in a gravitational discipline. Solely, this time, after we “launch” them and permit them to fall, they annihilate into photons instantly: the whole lot of their rest-mass vitality will get changed into the vitality of these photons.
Due to what we realized earlier than, which means the whole vitality of these photons, the place each has an vitality of E = computer, should equal the mixed rest-mass vitality of the particle and antiparticle in query.
Now, let’s think about that these photons finally make their method right down to the floor of the world that they’re falling onto, and we measure their energies once they attain the bottom. By the conservation of vitality, they should have a complete vitality that equals the vitality of the photons from the earlier situation. This proves that photons should achieve vitality as they fall in a gravitational discipline, resulting in what we all know as a gravitational blueshift, nevertheless it additionally results in one thing spectacular: the notion that E = mc² is what a particle’s (or antiparticle’s) relaxation mass must be.
There’s just one definition of vitality we are able to use that universally applies to all particles — large and massless, alike — that permits situation #1 and situation #2 to offer us equivalent solutions: E = √(m²c4 + p²c²). Take into consideration what occurs right here below a wide range of situations.
If you’re an enormous particle at relaxation, with no momentum, your vitality is simply √(m²c4), which turns into E = mc².
If you happen to’re a massless particle, you have to be in movement, and your relaxation mass is zero, so your vitality is simply √(p²c²), or E = computer.
If you happen to’re an enormous particle and also you’re shifting sluggish in comparison with the velocity of sunshine, then you’ll be able to approximate your momentum by p = mv, and so your vitality turns into √(m²c4 + m²v²c²). You possibly can rewrite this as E = mc² ·√(1 + v²/c²), as long as v is small in comparison with the velocity of sunshine.
If you happen to don’t acknowledge that final time period, don’t fear. You possibly can carry out what’s recognized, mathematically, as a Taylor collection enlargement, the place the second time period in parentheses is small in comparison with the “1” that makes up the primary time period. If you happen to do, you’ll get that E = mc² ·[1 + ½(v²/c²) + …], the place when you multiply via for the primary two phrases, you get E = mc² + ½mv²: the remainder mass plus the old-school, non-relativistic system for kinetic vitality.
That is completely not the one solution to derive E = mc², however it’s my favourite method to take a look at the issue. Three different methods might be discovered three right here, right here and right here, with some good background right here on how Einstein initially did it himself. If I had to decide on my second favourite solution to derive that E = mc² for an enormous particle at relaxation, it will be to contemplate a photon — which at all times carries vitality and momentum — touring in a stationary field with a mirror on the tip that it’s touring in direction of.
When the photon strikes the mirror, it quickly will get absorbed, and the field (with the absorbed photon) has to achieve a bit little bit of vitality and begin shifting within the route that the photon was shifting: the one solution to preserve each vitality and momentum.
When the photon will get re-emitted, it’s shifting in the other way, and so the field (having misplaced a bit mass from re-emitting that photon) has to maneuver ahead a bit extra rapidly with the intention to preserve vitality and momentum.
By contemplating these three steps, although there are a variety of unknowns, there are a variety of equations that must at all times match up: between all three situations, the whole vitality and the whole momentum have to be equal. If you happen to remedy these equations, there’s just one definition of rest-mass vitality that works out: E = mc².
You possibly can think about that the Universe might have been very totally different from the one we inhabit. Maybe vitality didn’t must be conserved; if this have been the case, E = mc² wouldn’t must be a common system for relaxation mass. Maybe we might violate the conservation of momentum; in that case, our definition for complete vitality — E = √(m²c4 + p²c²) — would not be legitimate. And if Common Relativity weren’t our idea of gravity, or if a photon’s momentum and vitality weren’t associated by E = computer, then E = mc² wouldn’t be a common relationship for large particles.
However in our Universe, vitality is conserved, momentum is conserved, and Common Relativity is our idea of gravitation. Given these info, all one must do is consider the correct experimental setup. Even with out bodily performing the experiment for your self and measuring the outcomes, you’ll be able to derive the one self-consistent reply for the rest-mass vitality of a particle: solely E = mc² does the job. We are able to attempt to think about a Universe the place vitality and mass have another relationship, however it will look very totally different from our personal. It’s not merely a handy definition; it’s the one solution to preserve vitality and momentum with the legal guidelines of physics that we’ve got.