How to prove Einstein’s relativity for under $100

As you stand on the surface of the Earth, what is it that you experience? Yes, the surrounding atoms and molecules of the atmosphere collide with your body, as do photons: particles of light. Some of these particles are particularly energetic, and can kick electrons off of the atoms and molecules they’re normally bound to, creating free electrons and ions that can strike you as well. There are ghostly neutrinos and antineutrinos passing through your body, although they rarely interact with you. But there’s more that you experience than you realize.

All throughout the Universe, from stars, black holes, galaxies, and more, cosmic rays are emitted: particles that stream through the Universe at high-energies. They strike Earth’s atmosphere and produce showers of both stable and unstable particles. The ones that live long enough before decaying eventually make their way down to Earth’s surface. Every second, somewhere between 10 and 100 muons — the unstable, heavy cousin of the electron — passes through your body. 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 fact that these muons pass through your body are more than sufficient to prove it.

While cosmic ray showers are common from high-energy particles, it’s mostly photons, muons, neutrinos, and electrons that make it down to Earth’s surface. Almost all of the neutrinos produced via cosmic ray showers are muon neutrinos, but that doesn’t necessarily mean all of the detected neutrinos will be muon neutrinos.

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. That’s right: with just a little bit of chemistry put to good use, your own human eye can serve as a particle detector.

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 homemade cloud chamber, following the instructions of Frances Green from the Institute of Physics. This can be build in a single day out of readily available materials for less than $100.

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. Particles moving through the atmosphere don’t make a visible trail, but particles moving through a 100% pure alcohol vapor do! Alcohol particles act as centers for condensation, and when a charged particle passes through an alcohol vapor (such as ethyl alcohol or isopropyl alcohol), it ionizes a path of those particles. This creates a trail that’s large enough and long-lasting enough for your eyes to easily pick out.

In general, the way you’ll want to go about building your own is as follows:

• Start by obtaining a rectangular aquarium fish tank, one that has good, solid seals around all the edges and will not leak.
• Cut three large pieces of thick, insulating foam of the same size: two with rectangular holes big enough to fit the fish tank inside, and one that remains solid to serve as your base.
• Cut a piece of galvanized steel sheet metal the same size as the insulating foam. Attach black card stock or matte black felt, or spray paint it with matte black paint, for the surface the size of the fish tank.
• Put the metal plate between the two top layers of insulating foam; add a two-sided layer of modeling clay for the tank to fit around. Add water or some of the alcohol solution into the groove so that when you put the tank atop it, no air can get in or out.
• Modify the fish tank by adding a layer of felt or sponge-like material to the tank’s base. Secure it good; it will be upside-down! Once that’s set, you’re ready to put it all together.
• Place some dry ice in the first two layers (solid base and hollow rectangle) of the insulating foam, then put the metal plate (black side up) atop that, then the last layer of insulating foam. Then put the water/alcohol into the clay groove, while simultaneously soaking/saturating the felt/sponge layer in the fishtank with the alcohol solution. (Pro tip: use more alcohol for saturating the felt/sponge layer than you think you ought to; don’t be stingy here!) Flip the fishtank over and put the edges inside the metal grooves, so that you have an airtight seal all around with the alcohol vapor inside.
• Turn off all the lights so it’s in a dark room, shine a bright flashlight (or projector) through the tank, place a warm, heavy object (like a folded towel, fresh out of the dryer) atop the tank, and wait about 10 minutes.

There are also some detailed guides around if you prefer more detailed instructions.

In this 1957 photograph, a National Advisory Council on Aeronautics (NACA, the predecessor to NASA) scientist studies alpha particles in a cloud chamber. Placing the radioactive mantle of a smoke detector, such as the alpha-emitting Am-241, creates a large supply of slow-moving particles that emanate outward from it.

To make sure it’s working, I always recommend tearing apart an old smoke detector and removing the mantle: the metal component that warns you of its radioactive materials inside, typically an isotope of Americium. Because all isotopes of Americium decay, including the Americium-241 used in smoke detectors, they’ll emit particles that are capable of creating these ionization trails. Place this mantle on the bottom of your cloud chamber, once it’s active by following the steps above, you’ll see particles emanate from it in all directions, leaving tracks in your cloud chamber.

Americium, in particular, 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-4 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.

Although there are four major types of particles that can be detected in a cloud chamber, the long and straight tracks are identifiable as cosmic ray muons, particularly if one applies an external magnetic field to the cloud chamber. The results of experiments such as this can be used to prove to demonstrate the validity of special relativity.

If you build a cloud chamber in precisely this fashion, 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 particle-emitting source of any type inside or nearby), you’ll still see tracks: they’ll be mostly vertical and appear as perfectly straight lines.

This isn’t because of radioactivity, but rather because of cosmic rays: high-energy particles that strike the top of Earth’s atmosphere, producing cascades of particles showering down from up high. Most of the cosmic rays that strike Earth’s atmosphere are composed of protons, but arrive moving 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.

The decays of the positively and negatively charged pions, shown here, occur in two stages. First, the quark/antiquark combination exchanges a W boson, producing a muon (or antimuon) and a mu-neutrino (or antineutrino), and then the muon (or antimuon) decays through a W-boson again, producing a neutrino, an antineutrino, and either an electron or positron at the end. This is the key step in making the neutrinos for a neutrino beamline, and requires two separate decays through the weak interaction: first of the pion into a muon, and then of a muon into an electron. The W+ and W- bosons are one another’s antiparticle, but the Z0 is its own antiparticle.

Pions, made of a quark-antiquark combination, are unstable, and they come in three varieties:

• π⁺, a positively charged pion that lives for around 10 nanoseconds,
• π^–, a negatively charged pion that also lives for around 10 nanoseconds,
• and π⁰, a neutral pion that lives for very short periods of time, only about 0.1 femtoseconds.

Although the neutral pions simply decay into two photons, the charged pions decay primarily into muons of the same charge (in addition to neutrinos/antineutrons). Muons are point particles, just like electrons, but have 206 times the electron’s mass and are, themselves, unstable.

Muons aren’t unstable in the same way as the composite pion, however. In fact, muons are 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 it was created, you might think to multiply its lifetime (2.2 microseconds) by the speed of light (300,000 km/s), which yields an answer of 660 meters. But that leads to a puzzle: why do you see them in your cloud chamber?

This illustration of a cosmic ray shower shows some of the possible interactions the cosmic ray can cause. 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, if the energy is great enough, will reach the surface.

Earth’s atmosphere is more than 100 kilometers in height, and even though it’s very sparse at the highest elevations, it still has more than enough particles in it to ensure a quick interaction with any cosmic ray that comes in. These muons get created 100 kilometers away from Earth’s surface (or more) and have a mean lifetime of only 2.2 microseconds. Here’s the puzzle: if muons 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 down here on the Earth’s surface?

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. The phenomenon of time dilation, first derived by Lorentz in the 1890s, would lead Einstein to discover special relativity soon after.

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, is that rapidly moving objects appear 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, but also carries tremendous implications for theories that aren’t relativistically invariant, like Newtonian gravity.

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 internal clock. But it will experience reality as though it hurtles toward Earth’s surface extremely close to the speed of light, causing lengths to contract along its direction of motion. All of a sudden, it isn’t 100 kilometers it has to travel to Earth’s surface; it’s whatever that “proper distance” is contracted down by the Lorentz-FitzGerald contraction.

If a muon moves at 99.999% the speed of light, for example, every 660 meters outside of its reference frame will appear as though it’s just 3 meters in length: a reduction of its proper length by 99.5%. A journey of 100 km down to the surface would appear to be a journey of 450 meters in the muon’s reference frame. According to the muon’s clock, a muon created 100 kilometers up with this speed would experience only 1.5 microseconds of time passing. With that small amount of time experienced, there’s less than a 50/50 chance each muon will decay along that journey.

The number of muons that remain after a certain number of microseconds, with and without the effects of time dilation. Even way back in 1963, which is when this graph was constructed, the data confirms that time dilation works precisely as predicted by Einstein’s relativity.

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 isn’t a paradox, however, because the muon doesn’t experience 4.5 milliseconds; that’s how much time passes in our reference frame. According to the muon, the time it experiences is dilated relative to us, just as lengths are contracted relative to our lengths. The muon 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.

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Without the laws of Einstein’s relativity, this cannot be accounted for!

Within the context of relativity, however, high velocities correspond to high particle energies. The combined 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, between 10 and 100 muons pass through your body each second. In fact, if you hold out your hand and point it to the sky, approximately one muon per second passes just through that modest portion of your body.

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. As a fun coincidence, 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.

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 at home, cheaply and with just a single day’s efforts, that allows 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.