How to prove Einstein's theory of relativity for 10,000 rubles

When you stand on the surface of the Earth, you experience the collisions of surrounding atoms and molecules of the atmosphere with your body. Photons, particles of light, do the same. Some of these particles are particularly energetic and can knock electrons off the atoms and molecules they are normally associated with, creating free electrons and ions that can also collide with you. Phantom neutrinos and antineutrinos pass through your body, although they rarely interact with you. But there's a lot more going on with your body than you think.

Throughout the Universe, cosmic rays are emitted from stars, black holes, galaxies, etc. – particles rushing through the Universe at high energies. They enter the Earth's atmosphere and cause showers of both stable and unstable particles. Those that live long enough before breaking up eventually end up on the Earth's surface. Every second, between 10 and 100 muons—the unstable, heavy cousins of the electron—pass through your body. With an average lifetime of 2.2 microseconds, you would think that they could not travel the entire thickness of the atmosphere, ~100-odd kilometers, from space to your hand. However, the theory of relativity says this happens, and the fact that these muons are passing through your body is more than enough to prove it right.

Although showers of cosmic rays are a common phenomenon for high-energy particles, it is mainly photons, muons, neutrinos and electrons that reach the Earth's surface. Almost all neutrinos produced by cosmic ray showers are muonic neutrinos, but this does not mean that all detected neutrinos will be muonic since they oscillate. Despite the high energies of these particles, they show no signs of baryon (or lepton) number violation. — Although showers of cosmic rays are a common phenomenon for high-energy particles, it is mainly photons, muons, neutrinos and electrons that reach the Earth's surface. Almost all neutrinos produced by cosmic ray showers are muonic neutrinos, but this does not mean that all detected neutrinos will be muonic since they oscillate. Despite the high energies of these particles, they show no signs of baryon (or leptonic) number violation.

Individual subatomic particles are almost always invisible to the human eye because the wavelengths of light we can see are unaffected by particles passing through our body. But if you create pure steam consisting of 100% alcohol, then a charged particle passing through it will leave a trace that can be visually detected even by such a primitive device as the human eye. That's right: with a little chemistry, your own human eye can serve as a particle detector.

When a charged particle passes through the alcohol vapor, it ionizes a path of alcohol particles that serve as centers for the condensation of alcohol droplets. The resulting trail is long enough for the human eye to see, and the speed and curvature of the trail (if a magnetic field is applied) can even determine what type of particle it was.

This principle was first applied in particle physics using a fog chamber (a cloud chamber).

Homemade fog chamber following instructions from Francis Green of the Institute of Physics. It can be built in one day using readily available materials for less than $100.

Today, anyone can build a fog chamber from commonly available materials with one day's labor and less than $100 in parts. Particles moving in the atmosphere leave no visible trace, but particles moving in the vapor of 100% pure alcohol do! The alcohol particles serve as condensation centers, and when a charged particle passes through an alcohol vapor (such as ethyl or isopropyl alcohol), it ionizes a path of these particles. The result is a trail that is large and long enough for your eyes to easily notice.

Building a fog chamber is very simple and requires just a few simple materials and steps:

Materials:

A transparent plastic or glass bath (for example, an aquarium) with a durable lid (plastic or metal);
Felt;
Isopropyl alcohol (90% or more. You can find it at the pharmacy or order it from a chemical supply company. Wear safety glasses when working with alcohol);
Dry ice (frozen carbon dioxide. Often used in fish markets and grocery stores to refrigerate food. Wear heavy gloves when handling dry ice).

Steps:

Cut a piece of felt so that it is the size of the bottom of the aquarium. Glue it inside the aquarium (on the bottom where sand and fake treasure chests are usually found).
Once the felt is secured, wet it with isopropyl alcohol until it is saturated. Drain off excess alcohol.
Place the lid on dry ice so that it lies flat. You may want to place the dry ice in a container or box to make it more stable.
Turn the container upside down so that the bottom of the container, covered with felt, is on top, and place the neck of the container on the lid.
Wait about 10 minutes… then turn off the light and shine a flashlight into the container.

As a particle rushes through your fog chamber, it collides with atmospheric molecules and knocks off some of their electrons, turning the molecules into charged ions. The atmospheric alcohol is attracted to these ions and sticks to them, forming tiny droplets.

The trails they leave behind are similar to the contrails of an airplane—long, spindly lines marking the particle's path through a fog chamber.

There are many different types of particles that can pass through your fog chamber. It may be hard to notice, but you can actually differentiate between particle types by the trails they leave behind.

Short, thick tracks

Sorry, but this is not a cosmic ray. When you see short, thick trails, it is a radon atom in the atmosphere spitting out an alpha particle (a bunch of two protons and two neutrons). Radon is a naturally occurring radioactive element, but it is found in such low concentrations in the air that it is less radioactive than peanut butter. The alpha particles emitted from radon atoms are bulky and low-energy, so they leave short, fatty trails.

Long, straight tracks

Congratulations! You have muons! Muons are the heavier cousins of the electron and are formed when a cosmic ray collides with atmospheric molecules high in the atmosphere. Because of their mass, muons force their way through the air and leave clean, straight trails.

Zigzags and curls

If your trail resembles that of a lost tourist in a foreign city, you have an electron or a positron (the electron's anti-matter twin) in front of you. Electrons and positrons are created when cosmic rays crash into atmospheric molecules. Electrons and positrons are light particles, and when they collide with air molecules, they fly apart, leaving zigzags and curls.

Branching traces

If your driveway forks, congratulations! You have just seen a particle decay. Many particles are unstable and break up into more stable ones. If your track suddenly splits into two, you've seen physics in action!

In this 1957 photograph, a National Advisory Council for Aeronautics (NACA, predecessor to NASA) scientist studies alpha particles in a fog chamber. Placing the radioactive mantle of a smoke detector, such as the Am-241 alpha emitter, creates a large supply of slow-moving particles that fly out of it.

To make sure it's working, I always recommend taking your old smoke detector apart and removing the mantle: the metal component that alerts you to the presence of radioactive materials inside, usually an isotope of americium. As all isotopes of americium decay, including americium-241 used in smoke detectors, they will emit particles capable of creating these ionization signatures. Place this mantle at the bottom of your fog chamber and once it is activated by following the steps above, you will see particles coming from it in all directions and leaving trails in your fog chamber.

Americium, in particular, decays, emitting alpha particles. In physics, alpha particles consist of two protons and two neutrons: this is the same as the helium-4 nucleus. Due to the low decay energy and large mass of α particles, these particles travel a slow, tortuous path and can sometimes be seen bouncing off the bottom of the fog chamber. This is a simple test to check if your fog camera is working properly.

Although there are four main types of particles that can be detected in a fog chamber, long and straight tracks can be identified as cosmic ray muons, especially if an external magnetic field is applied to the fog chamber. The results of experiments like this can be used to prove the validity of special relativity.

However, if you build a fog chamber in this way, you will see not only tracks of alpha particles. In fact, even if you leave the chamber empty (that is, do not place any particle-emitting sources inside or nearby), you will still see tracks: they will be mostly vertical and will appear as perfectly straight lines.

This is not due to radioactivity, but rather to cosmic rays: high-energy particles that strike the upper part of the Earth's atmosphere, creating cascades of particles falling from above to below. Most cosmic rays striking the Earth's atmosphere are made up of protons, but they travel at widely varying speeds and energies. Higher-energy particles collide with particles in the upper atmosphere, producing protons, electrons and photons, as well as unstable, short-lived particles such as pions.

These particle showers are the hallmark of fixed-target particle physics experiments, and they also arise naturally from cosmic rays.

The decays of positively and negatively charged pions shown here occur in two stages. First, a combination of quark and antiquark is exchanged with a W boson, resulting in a muon (or antimuon) and a mu neutrino (or antineutrino), and then the muon (or antimuon) decays again through the W boson, resulting in a neutrino, antineutrino and at the end either an electron or a positron. This is a key step in creating neutrinos for a neutrino beam, which requires two separate decays through the weak interaction: first a pion into a muon, and then a muon into an electron. The W+ and W- bosons are each other's antiparticles, and Z0 is its own antiparticle.

Pions consisting of a quark-antiquark combination are unstable and come in three types:

π+ is a positively charged pion that lives for about 10 nanoseconds,
π-, a negatively charged pion, which also lives for about 10 nanoseconds,
and π0 is a neutral pion that lives for a very short time, only about 0.1 femtoseconds.

While neutral pions simply decay into two photons, charged pions decay primarily into muons of the same charge (in addition to neutrinos/antineutrons). Muons are point particles, like electrons, but their mass is 206 times that of an electron, and they themselves are unstable.

However, muons are not as unstable as a compound pion. In fact, muons are the longest-lived unstable fundamental particle as far as we know. Due to their relatively small mass, they live on average for an amazingly long time – 2.2 microseconds.

If you were to ask how far a muon can travel once it is created, you might think that you would multiply its lifetime (2.2 microseconds) by the speed of light (300,000 km/s), which would give the answer 660 meters. But then the question arises: why do you see them in your fog chamber?

This illustration of a cosmic ray shower shows some of the possible interactions that cosmic rays can cause. Note that if a charged pion (left) strikes the nucleus before it decays, a shower is formed, but if it decays first (right), a muon is produced, which, if the energy is high enough, will reach the surface.

The Earth's atmosphere is more than 100 kilometers high, and although it is very thin at the highest altitudes, it still contains more than enough particles to ensure rapid interaction with any incoming cosmic ray. These muons are created at a distance of 100 kilometers from the Earth's surface (or more) and have an average lifetime of just 2.2 microseconds. Here's the mystery: if muons can only last 2.2 microseconds, are limited by the speed of light, and are created in the upper atmosphere (at an altitude of about 100 km), then how can these muons reach us here on the surface of the Earth?

You may start making excuses. You might imagine that some cosmic rays have enough energy to keep cascading and creating showers of particles all their way to earth, but that's not the story that muons tell when we measure their energy: the lowest ones are still being created at an altitude of about 30 km. You might think that 2.2 microseconds is just an average value, and perhaps rare muons, living 3-4 times longer, will have time to go down. But if you count, then only 1 out of 10^50 muons should reach the Earth's surface without decaying; in reality, almost 100% of muons reach the Earth.

The “light clock” will look different to observers moving at different relative speeds, but this is due to the constancy of the speed of light. Einstein's law of special relativity governs the transformations of time and distance between different observers. However, for each observer, time will pass at the same speed while he is in his frame of reference: one second per second, although when after the experiment they bring their clocks together, they will find that they no longer coincide.

How can this discrepancy be explained? Of course, muons move at a speed close to the speed of light, but we observe them from a frame of reference in which we are stationary. We can measure the distance that muons travel, we can measure their lifetime, and even if we give them the benefit of the doubt and say that they travel at the speed of light (and not near it), they should not even travel 1 kilometer before they decay .

But this misses one of the key points of the theory of relativity!

Unstable particles do not experience time the way you, an outside observer, measure it. They sense time using their own internal clock, which will tick slower the closer they get to the speed of light. Time slows down for them, which means we will see them living longer than the 2.2 microseconds in our frame of reference. The faster they move, the further their apparent path will be.

One of the revolutionary aspects of relativistic motion, proposed by Einstein but previously created by Lorentz, Fitzgerald and others, is that fast-moving objects appear to contract in space and expand in time. The faster you move relative to someone at rest, the more your body length shortens, while time appears to expand for the world around you. This picture of relativistic mechanics replaces the old Newtonian idea of classical mechanics, but also carries enormous implications for theories that are not relativistically invariant, such as Newtonian gravity. — One of the revolutionary aspects of relativistic motion, proposed by Einstein but previously created by Lorentz, Fitzgerald and others, is that fast-moving objects appear to contract in space and expand in time. The faster you move relative to someone at rest, the more your body length shortens, while time appears to expand for the world around you. This picture of relativistic mechanics replaces the old Newtonian picture of classical mechanics, but also carries enormous implications for theories that are not relativistically invariant, such as Newtonian gravity.

How does this work for the muon?

In his frame of reference, time passes normally, so according to his internal clock, he will live only 2.2 microseconds. But it will perceive reality as if it is rushing towards the surface of the Earth at a speed extremely close to the speed of light, causing the length to compress along the direction of movement. Suddenly, he will have to travel not 100 kilometers to the surface of the Earth, but the entire “proper distance” reduced as a result of the Lorentz-Fitzgerald contraction.

If, for example, a muon moves at 99.999% of the speed of light, then every 660 meters outside its frame of reference will appear as if it is only 3 meters long: its own length is reduced by 99.5%. Traveling 100 km down to the surface would look like traveling 450 meters in the muon frame. According to the muon's own clock, a muon that appeared at an altitude of 100 km and flew at that speed would have lived for only 1.5 microseconds of time. With such a small amount of time lived, the probability that each muon will decay during this journey is less than 1/2.

The number of muons remaining after a certain number of microseconds, with and without the effect of time dilation. Even as far back as 1963, when this graph was created, data confirms that time dilation works exactly as Einstein's relativity predicts.

This allows us to understand how to reconcile the situation with the muon: from our frame of reference on Earth, we see that the muon travels 100 km in about 4.5 milliseconds. However, this is not a paradox, because the muon does not sense that 4.5 milliseconds have passed; so much time passes only in our frame of reference. According to the muon, the time it lives is stretched relative to us, just as length is shortened relative to our length. From the muon's point of view, it traveled 450 meters in 1.5 microseconds, and therefore it can remain itself all the way to its final destination – the surface of the Earth.

Without Einstein's laws of relativity this cannot be explained!

However, in the context of the theory of relativity, high velocities correspond to high particle energies. The combined effect of time dilation and length contraction allows not just a few, but the majority of the muons created to survive. That's why even here on the surface of the Earth, between 10 and 100 muons pass through your body every second. In fact, if you extend your arm and point it toward the sky, about one muon per second will pass through this modest part of your body.

The V-shaped trail in the center of the image is due to the decay of a muon into an electron and two neutrinos. A high-energy track with a break indicates the decay of a particle in the air. The collision of positrons and electrons at specific, tunable energies allows the creation of muon-antimuon pairs at will. An interesting coincidence: the energy required to create a muon/antimuon pair from high-energy positrons colliding with electrons at rest is almost identical to the energy of electron-positron collisions required to create a Z boson.

If you've ever doubted the theory of relativity, you can't be blamed: the theory itself seems so counterintuitive, and its effects are beyond our everyday experience. But there is an experimental test you can do at home, inexpensively and in just one day, that will allow you to see the effects firsthand.

You can build a fog chamber, and if you do, you'll see these muons. If you add a magnetic field to the setup, you will see how the muon tracks will be curved according to their charge-to-mass ratio: you will immediately understand that they are not electrons. In rare cases, a muon could even be seen decaying in the air. And finally, if you were to measure their energy, you would find that they move ultra-relativistically, at 99.999% the speed of light. If it weren't for relativity, you wouldn't see a single muon at all.

Time dilation and length contraction are real, and the fact that muons survive showers of cosmic rays all the way to Earth proves this beyond a shadow of a doubt.