Is There A Fundamental Reason Why E = mc²?

by akoloy


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 power is the same as mass multiplied by the velocity of sunshine squared, educating us an infinite quantity in regards to the Universe. This one equation tells us how a lot power is inherent to an enormous particle at relaxation, and likewise tells us how a lot power is required to create particles (and antiparticles) out of pure power. It tells us how a lot power is launched in nuclear reactions, and the way a lot power comes out of annihilations between matter and antimatter.

But why? Why does power must equal mass multiplied by the velocity of sunshine squared? Why couldn’t it have been every other manner? That’s what Brad Stuart desires to know, writing in to ask:

“Einstein’s equation is amazingly elegant. But is its simplicity real or only apparent? Does E = mc² derive directly from an inherent equivalence between any mass’s energy and the square of the speed of light (which seems like a marvelous coincidence)? Or does the equation only exist because its terms are defined in a (conveniently) particular way?”

It’s a terrific query. Let’s examine Einstein’s most well-known equation, and see precisely why it couldn’t have been every other manner.

To begin with, it’s necessary to comprehend a number of issues about power. Energy, particularly to a non-physicist, is a very difficult factor to outline. There are many examples we will all give you off the tops of our heads.

  • There’s potential power, which is a few type of saved power that may be launched. Examples embody gravitational potential power, like lifting a mass as much as a big peak, chemical potential power, the place saved power in molecules like sugars can bear combustion and be launched, or electrical potential power, the place built-up costs in a battery or capacitor will be discharged, releasing power.
  • There’s kinetic power, or the power inherent to a shifting object as a result of its movement.
  • There’s electrical power, which is the kinetic power inherent to shifting costs and electrical currents.
  • There’s nuclear power, or the power launched by nuclear transitions to extra secure states.

And, after all, there are various different sorts. Energy is a type of issues that all of us “know it when we see it,” however to a physicist, we would like a extra common definition. The greatest one we’ve is solely: extracted/extractable power is a manner of quantifying our capacity to carry out work.

Work, to a physicist, has a specific definition itself: a pressure exerted in the identical course that an object is moved, multiplied by the space the item strikes in that course. Lifting a barbell as much as a sure peak does work towards the pressure of gravity, elevating your gravitational potential power; releasing that raised barbell converts that gravitational potential power into kinetic power; the barbell placing the ground converts that kinetic power into a mix of warmth, mechanical, and sound power. Energy isn’t created or destroyed in any of those processes, however reasonably transformed from one kind into one other.

The manner most individuals take into consideration E = mc², after they first find out about it, is by way of what we name “dimensional analysis.” They say, “okay, energy is measured in Joules, and a Joule is a kilogram · meter² per second². So if we want to turn mass into energy, you just need to multiply those kilograms by something that’s a meter² per second², or a (meter/second)², and there’s a fundamental constant that comes with units of meters/second: the speed of light, or c.” It’s an affordable factor to assume, however that’s not sufficient.

After all, you may measure any velocity you need in items of meters/second, not simply the velocity of sunshine. In addition, 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 to think about a bodily state of affairs that would inform the distinction between varied 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 will do is think about that there’s some power inherent to a particle as a result of its relaxation mass, and extra power that it may need as a result of its movement: kinetic power. We can think about beginning a particle off excessive up in a gravitational area, as if it began off with a considerable amount of gravitational potential power, however at relaxation. When you drop it, that potential power converts into kinetic power, whereas the remaining mass power stays the identical. At the second simply previous to affect with the bottom, there can be no potential power left: simply kinetic power and the power inherent to its relaxation mass, no matter which may be.

Now, with that image in our heads — that there’s some power inherent to the remaining mass of a particle and that gravitational potential power will be transformed into kinetic power (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’ll annihilate away into pure power.

(Sure, E = mc² tells us the connection between mass and power, together with how a lot power it is advisable create particle-antiparticle pairs out of nothing, and the way a lot power you get out when particle-antiparticle pairs annihilate. But 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 an alternative of getting one particle excessive up in a gravitational area, think about that we’ve each a particle and an antiparticle up excessive in a gravitational area, able to fall. Let’s arrange two completely different eventualities for what might occur, and discover the results of each.

Scenario 1: the particle and antiparticle each fall, and annihilate on the prompt they’d hit the bottom. This is similar state of affairs we simply considered, besides doubled. Both the particle and antiparticle begin with some quantity of rest-mass power. 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 similar lots to their antiparticle counterparts.

Now, they each fall, changing their gravitational potential power into kinetic power, which is along with their rest-mass power. Just as was the case earlier than, the moment earlier than they hit the bottom, all of their power is in simply two types: their rest-mass power and their kinetic power. Only, this time, simply in the meanwhile of affect, they annihilate, remodeling into two photons whose mixed power should equal no matter that rest-mass power plus that kinetic power was for each the particle and antiparticle.

For a photon, nonetheless, which has no mass, the power is solely given by its momentum multiplied by the velocity of sunshine: E = laptop. Whatever the power of each particles was earlier than they hit the bottom, the power of these photons should equal that very same complete worth.

Scenario 2: the particle and antiparticle each annihilate into pure power, after which fall the remainder of the way in which all the way down to the bottom as photons, with zero relaxation mass. Now, let’s think about an virtually similar state of affairs. We begin with the identical particle and antiparticle, excessive up in a gravitational area. Only, this time, once we “release” them and permit them to fall, they annihilate into photons instantly: the whole lot of their rest-mass power will get became the power of these photons.

Because of what we realized earlier than, meaning the full power of these photons, the place each has an power of E = laptop, should equal the mixed rest-mass power of the particle and antiparticle in query.

Now, let’s think about that these photons ultimately make their manner all the way down to the floor of the world that they’re falling onto, and we measure their energies after they attain the bottom. By the conservation of power, they should have a complete power that equals the power of the photons from the earlier state of affairs. This proves that photons should acquire power as they fall in a gravitational area, resulting in what we all know as a gravitational blueshift, but it surely additionally results in one thing spectacular: the notion that E = mc² is what a particle’s (or antiparticle’s) relaxation mass needs to be.

There’s just one definition of power we will use that universally applies to all particles — huge and massless, alike — that allows state of affairs #1 and state of affairs #2 to present us similar solutions: E = √(m²c4 + p²c²). Think about what occurs right here below a wide range of circumstances.

  • If you’re a huge particle at relaxation, with no momentum, your power is simply √(m²c4), which turns into E = mc².
  • If you’re a massless particle, you have to be in movement, and your relaxation mass is zero, so your power is simply √(p²c²), or E = laptop.
  • If you’re a large particle and also you’re shifting sluggish in comparison with the velocity of sunshine, then you may approximate your momentum by p = mv, and so your power turns into √(m²c4 + m²v²c²). You can rewrite this as E = mc² · √(1 + v²/c²), as long as v is small in comparison with the velocity of sunshine.

If you don’t acknowledge that final time period, don’t fear. You can carry out what’s identified, mathematically, as a Taylor series expansion, the place the second time period in parentheses is small in comparison with the “1” that makes up the primary time period. If you do, you’ll get that E = mc² · [1 + ½(v²/c²) + …], the place should you multiply by for the primary two phrases, you get E = mc² + ½mv²: the remaining mass plus the old-school, non-relativistic formulation for kinetic power.

This is totally not the one option to derive E = mc², however it’s my favourite manner to have a look at the issue. Three different methods will be discovered three herehere and here, with some good background here on how Einstein initially did it himself. If I had to decide on my second favourite option to derive that E = mc² for a large particle at relaxation, it could be to think about a photon — which at all times carries power and momentum — touring in a stationary field with a mirror on the top that it’s touring in the direction of.

When the photon strikes the mirror, it quickly will get absorbed, and the field (with the absorbed photon) has to achieve just a little little bit of power and begin shifting within the course that the photon was shifting: the one option to preserve each power and momentum.

When the photon will get re-emitted, it’s shifting in the wrong way, and so the field (having misplaced just a little mass from re-emitting that photon) has to maneuver ahead just a little extra rapidly in an effort to preserve power and momentum.

By contemplating these three steps, regardless that there are quite a lot of unknowns, there are quite a lot of equations that must at all times match up: between all three eventualities, the full power and the full momentum have to be equal. If you remedy these equations, there’s just one definition of rest-mass power that works out: E = mc².

You can think about that the Universe might have been very completely different from the one we inhabit. Perhaps power didn’t have to be conserved; if this have been the case, E = mc² wouldn’t have to be a common formulation for relaxation mass. Perhaps we might violate the conservation of momentum; in that case, our definition for complete power — E = √(m²c4 + p²c²) — would not be legitimate. And if General Relativity weren’t our concept of gravity, or if a photon’s momentum and power weren’t associated by E = laptop, then E = mc² wouldn’t be a common relationship for large particles.

But in our Universe, power is conserved, momentum is conserved, and General Relativity is our concept of gravitation. Given these information, 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 may derive the one self-consistent reply for the rest-mass power of a particle: solely E = mc² does the job. We can attempt to think about a Universe the place power and mass have another relationship, however it could look very completely different from our personal. It’s not merely a handy definition; it’s the one option to preserve power and momentum with the legal guidelines of physics that we’ve.


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