Skip to content
If you’ve ever heard of Albert Einstein, chances are you know at least one equation that he himself is famous for deriving: E = mc2. This simple equation details a relationship between the energy (E) of a system, its rest mass (m), and a fundamental constant that relates the two, the speed of light squared (c2). Despite the fact that this equation is one of the simplest ones you can write down, what it means is dramatic and profound.
At a fundamental level, there is an equivalence between the mass of an object and the inherent energy stored within it. Mass is only one form of energy among many, such as electrical, thermal, or chemical energy, and therefore energy can be transformed from any of these forms into mass, and vice versa. The profound implications of Einstein’s equations touch us in many ways in our day-to-day lives. Here are the five lessons everyone should learn.
This iron-nickel meteorite, examined and photographed by Opportunity, represents the first such object ever found on the Martian surface. If you were to take this object and chop it up into its individual, constituent protons, neutrons, and electrons, you would find that the whole is actually less massive than the sum of its parts.
NASA / JPL / Cornell
1.) Mass is not conserved. When you think about the things that change versus the things that stay the same in this world, mass is one of those quantities we typically hold constant without thinking about it too much. If you take a block of iron and chop it up into a bunch of iron atoms, you fully expect that the whole equals the sum of its parts. That’s an assumption that’s clearly true, but only if mass is conserved.
In the real world, though, according to Einstein, mass is not conserved at all. If you were to take an iron atom, containing 26 protons, 30 neutrons, and 26 electrons, and were to place it on a scale, you’d find some disturbing facts.
- An iron atom with all of its electrons weighs slightly less than an iron nucleus and its electrons do separately,
- An iron nucleus weighs significantly less than 26 protons and 30 neutrons do separately.
- And if you try and fuse an iron nucleus into a heavier one, it will require you to input more energy than you get out.
Iron-56 may be the most tightly-bound nucleus, with the greatest amount of binding energy per nucleon. In order to get there, though, you have to build up element-by-element. Deuterium, the first step up from free protons, has an extremely low binding energy, and thus is easily destroyed by relatively modest-energy collisions.
Each one of these facts is true because mass is just another form of energy. When you create something that’s more energetically stable than the raw ingredients that it’s made from, the process of creation must release enough energy to conserve the total amount of energy in the system.
When you bind an electron to an atom or molecule, or allow those electrons to transition to the lowest-energy state, those binding transitions must give off energy, and that energy must come from somewhere: the mass of the combined ingredients. This is even more severe for nuclear transitions than it is for atomic ones, with the former class typically being about 1000 times more energetic than the latter class.
In fact, leveraging the consequences of E = mc2 is how we get the second valuable lesson out of it.
Countless scientific tests of Einstein’s general theory of relativity have been performed, subjecting the idea to some of the most stringent constraints ever obtained by humanity. Einstein’s first solution was for the weak-field limit around a single mass, like the Sun; he applied these results to our Solar System with dramatic success. We can view this orbit as Earth (or any planet) being in free-fall around the Sun, traveling in a straight-line path in its own frame of reference. All masses and all sources of energy contribute to the curvature of spacetime.
LIGO scientific collaboration / T. Pyle / Caltech / MIT
2.) Energy is conserved, but only if you account for changing masses. Imagine the Earth as it orbits the Sun. Our planet orbits quickly: with an average speed of around 30 km/s, the speed required to keep it in a stable, elliptical orbit at an average distance of 150,000,000 km (93 million miles) from the Sun. If you put the Earth and Sun both on a scale, independently and individually, you would find that they weighed more than the Earth-Sun system as it is right now.
When you have any attractive force that binds two objects together — whether that’s the electric force holding an electron in orbit around a nucleus, the nuclear force holding protons and neutrons together, or the gravitational force holding a planet to a star — the whole is less massive than the individual parts. And the more tightly you bind these objects together, the more energy the binding process emits, and the lower the rest mass of the end product.
Whether in an atom, molecule, or ion, the transitions of electrons from a higher energy level to a lower energy level will result in the emission of radiation at a very particular wavelength. This produces the phenomenon we see as emission lines, and is responsible for the variety of colors we see in a fireworks display. Even atomic transitions such as this must conserve energy, and that means losing mass in the correct proportion to account for the energy of the produced photon.
When you bring a free electron in from a large distance away to bind to a nucleus, it’s a lot like bringing in a free-falling comet from the outer reaches of the Solar System to bind to the Sun: unless it loses energy, it will come in, make a close approach, and slingshot back out again.
However, if there’s some other way for the system to shed energy, things can become more tightly bound. Electrons do bind to nuclei, but only if they emit photons in the process. Comets can enter stable, periodic orbits, but only if another planet steals some of their kinetic energy. And protons and neutrons can bind together in large numbers, producing a much lighter nucleus and emitting high-energy photons (and other particles) in the process. That last scenario is at the heart of perhaps the most valuable and surprising lesson of all.
A composite of 25 images of the Sun, showing solar outburst/activity over a 365 day period. Without the right amount of nuclear fusion, which is made possible through quantum mechanics, none of what we recognize as life on Earth would be possible. Over its history, approximately 0.03% of the mass of the Sun, or around the mass of Saturn, has been converted into energy via E = mc^2.
NASA / Solar Dynamics Observatory / Atmospheric Imaging Assembly / S. Wiessinger; post-processing by E. Siegel
3.) Einstein’s E = mc2 is responsible for why the Sun (like any star) shines. Inside the core of our Sun, where the temperatures rise over a critical temperature of 4,000,000 K (up to nearly four times as large), the nuclear reactions powering our star take place. Protons are fused together under such extreme conditions that they can form a deuteron — a bound state of a proton and neutron — while emitting a positron and a neutrino to conserve energy.
Additional protons and deuterons can then bombard the newly formed particle, fusing these nuclei in a chain reaction until helium-4, with two protons and two neutrons, is created. This process occurs naturally in all main-sequence stars, and is where the Sun gets its energy from.
The proton-proton chain is responsible for producing the vast majority of the Sun’s power. Fusing two He-3 nuclei into He-4 is perhaps the greatest hope for terrestrial nuclear fusion, and a clean, abundant, controllable energy source, but all of these reaction must occur in the Sun.
Borb / Wikimedia Commons
If you were to put this end product of helium-4 on a scale and compare it to the four protons that were used up to create it, you’d find that it was about 0.7% lighter: helium-4 has only 99.3% of the mass of four protons. Even though two of these protons have converted into neutrons, the binding energy is so strong that approximately 28 MeV of energy gets emitted in the process of forming each helium-4 nucleus.
In all cases, the energy that comes out — in all its combined forms — is exactly equal to the energy equivalent of the mass loss between products and reactants. The ultimate example is the case of matter-antimatter annihilation, where a particle and its antiparticle meet and produce two photons of the exact rest energy of the two particles.
Take an electron and a positron and let them annihilate, and you’ll always get two photons of exactly 511 keV of energy out. It’s no coincidence that the rest mass of electrons and positrons are each 511 keV/c2: the same value, just accounting for the conversion of mass into energy by a factor of c2. Einstein’s most famous equation teaches us that any particle-antiparticle annihilation has the potential to be the ultimate energy source: a method to convert the entirety of the mass of your fuel into pure, useful energy.
With particles, though, the story is different. If you take two electrons and smash them together, you’ll get two electrons out, but with enough energy, you might also get a new matter-antimatter pair of particles out, too. In other words, you will have created two new, massive particles where none existed previously: a matter particle (electron, muon, proton, etc.) and an antimatter particle (positron, antimuon, antiproton, etc.).
This is how particle accelerators successfully create the new particles they’re searching for: by providing enough energy to create those particles (and, if necessary, their antiparticle counterparts) from a rearrangement of Einstein’s most famous equation. Given enough free energy, you can create any particle(s) with mass m, so long as there’s enough energy to satisfy the requirement that there’s enough available energy to make that particle via m = E/c2. If you satisfy all the quantum rules and have enough energy to get there, you have no choice but to create new particles.
More than 60 years after Einstein’s death, it’s long past time to bring his famous equation down to Earth. The laws of nature aren’t just for physicists; they’re for every curious person on Earth to experience, appreciate, and enjoy.
Cosmic rays, which are ultra-high energy particles originating from all over the Universe, strike protons in the upper atmosphere and produce showers of new particles. The fast-moving charged particles also emit light due to Cherenkov radiation as they move faster than the speed of light in Earth’s atmosphere, and produce secondary particles that can be detected here on Earth.
Simon Swordy (U. Chicago), NASA
When you hold out your palm and point it towards the sky, what is it that’s interacting with your hand? You might correctly surmise that there are ions, electrons and molecules all colliding with your hand, as the atmosphere is simply unavoidable here on Earth. You might also remember that photons, or particles of light, must be striking you, too.
But there’s something more striking your hand that, without relativity, simply wouldn’t be possible. Every second, approximately one muon — the unstable, heavy cousin of the electron — passes through your outstretched palm. These muons are made in the upper atmosphere, created by cosmic rays. With a mean lifetime of 2.2 microseconds, you might think the ~100+ km journey to your hand would be impossible. Yet relativity makes it so, and the palm of your hand can prove it. Here’s how.
While cosmic ray showers are common from high-energy particles, it’s mostly the muons which make it down to Earth’s surface, where they are detectable with the right setup.
Alberto Izquierdo; courtesy of Francisco Barradas Solas
Individual, subatomic particles are almost always invisible to human eyes, as the wavelengths of light we can see are unaffected by particles passing through our bodies. But if you create a pure vapor made out of 100% alcohol, a charged particle passing through it will leave a trail that can be visually detected by even as primitive an instrument as the human eye.
As a charged particle moves through the alcohol vapor, it ionizes a path of alcohol particles, which act as centers for the condensation of alcohol droplets. The trail that results is both long enough and long-lasting enough that human eyes can see it, and the speed and curvature of the trail (if you apply a magnetic field) can even tell you what type of particle it was.
This principle was first applied in particle physics in the form of a cloud chamber.
A completed cloud chamber can be built in a day out of readily-available materials and for less than $100. You can use it to prove the validity of Einstein’s relativity, if you know what you’re doing!
Instructables user ExperiencingPhysics
Today, a cloud chamber can be built, by anyone with commonly available parts, for a day’s worth of labor and less than $100 in parts. (I’ve published a guide here.) If you put the mantle from a smoke detector inside the cloud chamber, you’ll see particles emanate from it in all directions and leave tracks in your cloud chamber.
That’s because a smoke detector’s mantle contains radioactive elements such as Americium, which decays by emitting α-particles. In physics, α-particles are made up of two protons and two neutrons: they’re the same as a helium nucleus. With the low energies of the decay and the high mass of the α-particles, these particles make slow, curved tracks and can even be occasionally seen bouncing off of the cloud chamber’s bottom. It’s an easy test to see if your cloud chamber is working properly.
For an extra bonus of radioactive tracks, add the mantle of a smoke detector to the bottom of your cloud chamber, and watch the slow-moving particles emanating outward from it. Some will even bounce off the bottom!
If you build a cloud chamber like this, however, those α-particle tracks aren’t the only things you’ll see. In fact, even if you leave the chamber completely evacuated (i.e., you don’t put a source of any type inside or nearby), you’ll still see tracks: they’ll be mostly vertical and appear to be perfectly straight.
This is because of cosmic rays: high-energy particles that strike the top of Earth’s atmosphere, producing cascading particle showers. Most of the cosmic rays are made up of protons, but move with a wide variety of speeds and energies. The higher-energy particles will collide with particles in the upper atmosphere, producing particles like protons, electrons, and photons, but also unstable, short-lived particles like pions. These particle showers are a hallmark of fixed-target particle physics experiments, and they occur naturally from cosmic rays, too.
Although there are four major types of particles that can be detected in a cloud chamber, the long and straight tracks are the cosmic ray muons, which can be used to prove that special relativity is correct.
Wikimedia Commons user Cloudylabs
The thing about pions is that they come in three varieties: positively charged, neutral, and negatively charged. When you make a neutral pion, it just decays into two photons on very short (~10-16 s) timescales. But charged pions live longer (for around 10-8 s) and when they decay, they primarily decay into muons, which are point particles like electrons but have 206 times the mass.
Muons also are unstable, but they’re the longest-lived unstable fundamental particle as far as we know. Owing to their relatively small mass, they live for an astoundingly long 2.2 microseconds, on average. If you were to ask how far a muon could travel once created, you might think to multiply its lifetime (2.2 microseconds) by the speed of light (300,000 km/s), getting an answer of 660 meters. But that leads to a puzzle.
Cosmic ray shower and some of the possible interactions. Note that if a charged pion (left) strikes a nucleus before it decays, it produces a shower, but if it decays first (right), it produces a muon that will reach the surface.
Konrad Bernlöhr of the Max-Planck-Institute at Heidelberg
I told you earlier that if you hold out the palm of your hand, roughly one muon per second passes through it. But if they can only live for 2.2 microseconds, they’re limited by the speed of light, and they’re created in the upper atmosphere (around 100 km up), how is it possible for those muons to reach us?
You might start to think of excuses. You might imagine that some of the cosmic rays have enough energy to continue cascading and producing particle showers during their entire journey to the ground, but that’s not the story the muons tell when we measure their energies: the lowest ones are still created some 30 km up. You might imagine that the 2.2 microseconds is just an average, and maybe the rare muons that live for 3 or 4 times that long will make it down. But when you do the math, only 1-in-1050 muons would survive down to Earth; in reality, nearly 100% of the created muons arrive.
A light-clock, formed by a photon bouncing between two mirrors, will define time for any observer. Although the two observers may not agree with one another on how much time is passing, they will agree on the laws of physics and on the constants of the Universe, such as the speed of light. When relativity is applied correctly, their measurements will be found to be equivalent to one another, as the correct relativistic transformation will allow one observer to understand the observations of the other.
John D. Norton
How can we explain such a discrepancy? Sure, the muons are moving close to the speed of light, but we’re observing them from a reference frame where we’re stationary. We can measure the distance the muons travel, we can measure the time they live for, and even if we give them the benefit of the doubt and say that they’re moving at (rather than near) the speed of light, they shouldn’t even make it for 1 kilometer before decaying.
But this misses one of the key points of relativity! Unstable particles don’t experience time as you, an external observer, measures it. They experience time according to their own onboard clocks, which will run slower the closer they move to the speed of light. Time dilates for them, which means that we will observe them living longer than 2.2 microseconds from our reference frame. The faster they move, the farther we’ll see them travel.
One revolutionary aspect of relativistic motion, put forth by Einstein but previously built up by Lorentz, Fitzgerald, and others, that rapidly moving objects appeared to contract in space and dilate in time. The faster you move relative to someone at rest, the greater your lengths appear to be contracted, while the more time appears to dilate for the outside world. This picture, of relativistic mechanics, replaced the old Newtonian view of classical mechanics, and can explain the lifetime of a cosmic ray muon.
How does this work out for the muon? From its reference frame, time passes normally, so it will only live for 2.2 microseconds according to its own clocks. But it will experience reality as though it hurtles towards Earth’s surface extremely close to the speed of light, causing lengths to contract in its direction of motion.
If a muon moves at 99.999% the speed of light, every 660 meters outside of its reference frame will appear as though it’s just 3 meters in length. A journey of 100 km down to the surface would appear to be a journey of 450 meters in the muon’s reference frame, taking up just 1.5 microseconds of time according to the muon’s clock.
At high enough energies and velocities, relativity becomes important, allowing many more muons to survive than would without the effects of time dilation.
Frisch/Smith, Am. J. of Phys. 31 (5): 342–355 (1963) / Wikimedia Commons user D.H
This teaches us how to reconcile things for the muon: from our reference frame here on Earth, we see the muon travel 100 km in a timespan of about 4.5 milliseconds. This is just fine, because time is dilated for the muon and lengths are contracted for it: it sees itself as traveling 450 meters in 1.5 microseconds, and hence it can remain alive all the way down to its destination of Earth’s surface.
Without the laws of relativity, this cannot be explained! But at high velocities, which correspond to high particle energies, the effects of time dilation and length contraction enable not just a few but most of the created muons to survive. This is why, even all the way down here at the surface of the Earth, one muon per second still appears to pass through your upturned, outstretched hand.
The V-shaped track in the center of the image arises from a muon decaying to an electron and two neutrinos. The high-energy track with a kink in it is evidence of a mid-air particle decay. By colliding positrons and electrons at a specific, tunable energy, muon-antimuon pairs could be produced at will. The necessary energy for making a muon/antimuon pair from high-energy positrons colliding with electrons at rest is almost identical to the energy from electron/positron collisions necessary to create a Z-boson.
The Scottish Science & Technology Roadshow
If you ever doubted relativity, it’s hard to fault you: the theory itself seems so counterintuitive, and its effects are thoroughly outside the realm of our everyday experience. But there is an experimental test you can perform right at home, cheaply and with just a single day’s efforts, that allow you see the effects for yourself.
You can build a cloud chamber, and if you do, you will see those muons. If you installed a magnetic field, you’d see those muon tracks curve according to their charge-to-mass ratio: you’d immediately know they weren’t electrons. On rare occasion, you’d even see a muon decaying in mid-air. And, finally, if you measured their energies, you’d find that they were moving ultra-relativistically, at 99.999%+ the speed of light. If not for relativity, you wouldn’t see a single muon at all.
Time dilation and length contraction are real, and the fact that muons survive, from cosmic ray showers all the way down to Earth, prove it beyond a shadow of a doubt.
Follow me on Twitter. Check out my website or some of my other work here.
Source: How To Prove Einstein’s Relativity In The Palm Of Your Hand