Why Protons Can't Travel Faster Than This Speed（原声文稿）

The speed of light is known to be the unbreakable barrier that you can only approach to but never cross.

Special relativity doesn't put any limit for how close to the speed of light you can get, given enough energy.

But there is another speed limit that is lower than the speed of light and it is exactly this specific number which applies specifically to protons. protons. This is called the GZK limit.

But why protons can't travel through the universe faster than this particular speed?

And isn't this breaking the principle of relativity?

According to this principle, there is no absolute motion.

And therefore, there is a reference frame in which the proton is at rest, in which the physics should be the same as in any other inertial reference frame.

So why should anything weird happen to the proton beyond a particular speed? And what is this speed relative to?

This is electroscope, a very simple device that you can easily build at home.

It consists of a disk and a rod made of a conductive material like a brass, which has a very thin leaf of conductor at the end of the road, and all of it is protected by some insulator.

This device is able to measure the total charge captured in its conductive part, since if you remove some electrons, then the leaves will be repelled by each other, but if the total charge is zero, then they will hang straight down.

By this device, you are able to measure air ionization by measuring the speed at which this electroscope discharges.

If there are no free charges in the air, then the electroscope will retain its charge.

If there is a lot of free charges in the air, then the electroscope will attract whatever charges it's missing to neutralize itself.

This air ionization was measured already in the 19th century, but the reason for this ionization was unclear until the discovery of radiation in 1896 by Henry Becquerel.

So Becquerel discovered that certain elements, like uranium, can decay emitting particles that can ionize the air.

But were all the elements in the ground?

So it was expected that the ionization of the air is gonna be much greater closer to the ground and therefore the electroscope should discharge much more quickly close to the ground and very slowly above the ground as the ionization should decrease with altitude.

But for every nice theory you need a proof, and proving this was very easy.

And in 1909 Theodor Wolff tried to measure the ionization on the top of the Eiffel Tower, and he in fact did measure lower aeration.

But to his surprise, this was a much lower effect than he expected.

So the aeration was decreasing very slowly with altitude and therefore this aeration couldn't be explained by the radiation from the ground.

Next step did Victor Hess in 1912, who took two electroscopes on a balloon reaching the altitude over 5 km. Kinda dangerous, don't you think so?

So, the ionization initially did go down, but shortly after, to Victor's surprise, it started to go up.

And this was a proof that there is a radiation coming from the sky.

The most logical argument was that it must be coming from the sun.

But you could do this experiment during the night or during the solar eclipse. And there was measured no impact from the sun.

This argument, though, later turned out to be wrong, as there is a massive magnetic field around the Earth, which curves the trajectories of the charged particles coming from the Sun.

So they can have influence even during the night.

Today we know that there is a radiation that's coming from the Sun, but these particles have low energy.

In 1938, French physicists named Pierre Auger detected a correlated radiation event.

Imagine you have a series of detectors that measure radiation.

Pierre Auger noticed that when one of the detectors detected an increase of radiation, then other detectors detected this as well with a certain time delay.

By analyzing this time delay, he found out that this radiation must be coming from a single cosmic ray event hitting the atmosphere.

Then by the density and the energies of the captured particles, he estimated that the energy of the original particle creating these very wide air showers must be of 10 to the 15 electron volts.

The conclusion of this is that there are particles hitting the atmosphere with extremely high energies.

Today we know that 88 % of this cosmic radiation is made from highly energetic protons.

The number of particles per unit of area with a given energy decreases roughly with a third power in energy.

So, if we take protons with kinetic energy of 10 to the 9 electron volts, which is still very high energy, it would be around 10 ,000 particles per square meter per second.

For protons of much higher energy, like 10 to the 16 electron volts, it would be just three particles per square meter per year.

And for the energy of 10 to the 19 electron volts it will be just around one particle per square kilometer per year.

You could extrapolate this analysis further and calculate the probability of finding particles of higher energy.

But the problem is that there is a limit to this rule and after the energy of 6 .2 times 10 to the 19 electron volts, we barely detect anything.

So what exactly happens at this particular energy?

If the Universe was empty, there would be no limit to the speed of proton and it could get arbitrarily close to the speed of light. But the Universe is not empty.

There is this Cosmic Microwave Background Radiation, or CMB, which is a relic from the early stages of the development of the universe.

The temperature of this CMB is around 2 .7 Kelvin, which corresponds to a blackbody radiation that peaks at certain frequency.

And believe it or not, there is a reference frame in which this CMB is stationary.

And therefore we can measure the speed relative to this CMB by measuring the frequency shift.

If you are traveling towards the CMB, then you will measure higher energy of the photons in the direction of motion.

The faster the proton moves, the more energetic the CMB radiation becomes. And soon it begins to interact with it.

At certain velocity, the energy of the CMB photons is gonna be high enough that there is a possibility of electron -positron pair creation in the presence of the proton.

Just on a side note, photon on its own count the K into electron -positron pair due to momentum conservation law.

It can only happen near a third body so that the momentum can be transferred to that body.

The electron and positron are very light compared to the proton.

And therefore this process doesn't drain much energy from the proton, and therefore you can accelerate further and the proton isn't gonna be losing much energy.

But when you reach this speed, then there is enough energy for creating a pion through a delta resonance.

The mass of the pion is quite comparable to the mass of the proton, and therefore it will eat quite a large chunk of energy from the proton after the interaction.

And now, if you accelerate the proton further, it would start losing energy due to these interactions.

To calculate how the energy of the CMB photons change from frame to frame, you can use the relativistic Doppler shift formula.

Here this is the energy of the CMB photon in the frame of the proton. And this is the rest frame energy.

In order for this interaction to happen, the photon must have at least the energy of 135 mega electron volts.

So you can plug that into the equation and solve for beta, which is the velocity of the proton in the units of the speed of light.

There are more ways how you can calculate this limit, for example, using the invariant mass analysis.

But for me, this is the most intuitive one.

You might ask, how often the proton interacts with this CMB?

And you can calculate it using the cross -section of this interaction, which happens to be around 100 micro barn, and the number density of the CMB photons, which is around 411 photons per cubic centimeter.

Cross -section is measured in the units of meters squared, and one barn is 10 to the minus 28 meters squared.

If you add up the numbers, and you can use it as an exercise, you will get that the average distance proton travels per one interaction is roughly 25 .7 million years.

This might seem like a very little, but most of the high energy protons are coming from the deep universe, from distances much greater than 25 million light years.

And therefore it's very likely that they interacted with the CMB and therefore we don't see them.

Yet physicists detected events that exceed this limit.

One example is oh my god particle with the energy of 2 .8 times 10 to the 21 electron volts, which is around 50 joules.

If you think about it a bullet with such energy could kill a human.

and all these energies are captured in a single particle, but it is still a mystery what are the sources of such high energy cosmic rays, because there are no such sources in our galaxy.

The candidates are heavier elements or incorrect measurement.

And just a bonus, it takes 25 .7 million years, on average, per one interaction of the proton with the CMB.

But from the frame of proton, it would take just 3 .7 days due to relativity.

So the proton in his frame travels the distance that we measured to be 25 million years in just a couple of days.