# Time dilation can be considered a fact

## The principles of special relativity

The special theory of relativity influences our modern everyday life more than you might think. Navigation systems, clocks and particles in giant particle accelerators, for example, obey their principles. The physics behind it can also be understood without mathematics.

The starting point of the special theory of relativity is the principle that the physical laws in all inertial systems have the same shape. An inertial system is understood to be a system that is at rest or that moves in a straight line and at constant speed. A jumbo jet on a transatlantic flight in calm weather and with darkened windows gives a good impression of such an inertial system. If the passenger interprets the noise of the turbines as the noise of an air conditioning system, they will hardly be able to tell whether the aircraft is on the ground or whether it is flying at 900 kilometers per hour. A passenger can devise all kinds of mechanical, electrical, or optical experiments to measure the speed of the jet. He will not succeed as long as these experiments are restricted to the interior of the closed "box" aircraft and no signals are received from outside. But as soon as the aircraft takes off, flies a curve or lands, you can feel this immediately inside. In these cases the airplane is an accelerated frame of reference, and there are different laws than in an inertial frame. Even in a carousel or a space rocket taking off, the passenger will notice other laws. Such accelerated frames of reference are the subject of general relativity.

Another example of inertial systems are moving walkways, which can be found in many airports, for example. The moving walkway and the stationary airport hall form two inertial systems that move relative to each other. Let us assume that a woman on the moving sidewalk is in a particularly hurry and is walking towards her destination. For an observer outside the conveyor belt, it moves at a speed that results from the sum of the conveyor belt and walking speed. If the traveler were to walk as fast as the moving walkway, only in the opposite direction, they would literally step on the spot for an outside observer.

Physics in the airport hall

This simple addition or subtraction of speeds works fine in everyday life, but it has limits. Suppose the woman stops and takes a forward flash picture with her camera. Then she turns around and takes a second picture in the opposite direction. Will the two flashes of light also be perceived as having different speeds by an observer standing next to the moving walkway? According to the previous example, the flashlight flying forward should be faster by the speed of the escalator, while the one flying backwards should be correspondingly slower. However, this naive approach proves to be wrong: Both flashes of light are exactly the same speed.

The two scientists Albert Abraham Michelson and Edward Williams Morley carried out a similar experiment at the end of the 19th century. The two researchers wanted to determine whether light waves, just like water or sound waves, need a medium in order to propagate. The earth served them as a kind of spaceship on their orbit around the sun in a hypothetical medium called “ether”. The big surprise was that light always has exactly the same speed regardless of the direction of movement of the earth relative to this aether. The speed of light, usually abbreviated to the letter c, is therefore independent of the reference system and is always 299,792,458 meters per second in empty space - exactly.

Albert Einstein took the constancy of the speed of light as an opportunity to question the existence of an absolute and generally valid time. He realized that the so-called Lorentz transformation describes the facts correctly. The resulting relativistic formulas for adding speeds ensure that the speed of light in any inertial system is always 299,792,458 meters per second. The main difference to simple addition: Different time scales apply to objects moving against each other. These are largely determined by the so-called Lorentz factor, which depends solely on the relative speed of the objects.

### Farewell to an absolute time

A direct consequence of the constancy of the speed of light is the time expansion or time dilation: viewed from the laboratory system, a clock “ticks” more slowly in a moving frame of reference. Einstein illustrated the stretching of time through one of his famous thought experiments: the light clock. This clock consists of a flash lamp, a mirror and a detector. The flash lamp emits a short flash of light in a vertical direction, which runs to a mirror, is reflected there and then runs back to the detector, where its arrival is indicated by a ticking. The back and forth running of the light flash corresponds to the back and forth swinging of the pendulum in a grandfather clock.

Einstein's thought experiment

With the light clock at rest, the light path is given by the distance from the lamp to the mirror and back to the detector. When the light clock moves horizontally, the vertical distance to the mirror remains unchanged, but by the time the light has reached the mirror, the mirror has already flown, and by the time the reflected light reaches the detector, it has flown further. The light follows a longer triangular path (see graphic). If one now makes - following Albert Einstein - the fundamental assumption that the light in both systems has exactly the same speed c If viewed from the laboratory, the flash of light needs more time to get to the detector. The ticking noises are therefore heard less often, and the time cycle of the moving clock appears to the resting observer to be lengthened by the Lorentz factor. Moving clocks therefore run more slowly. The most important conclusion from this thought experiment, however, is that absolute time no longer exists in the theory of relativity. This is a fundamental break with the view of the natural sciences and philosophy prior to the formulation of the theory of relativity. Up until the end of the 19th century, it was taken for granted that there was an absolute and universally valid time.

Since the speeds of everyday bodies such as airplanes or satellites are far below the speed of light, the slowing down of moving clocks is extremely low here. With high-precision atomic clocks, however, it can be easily measured and is of great importance for the global positioning system GPS. For example, if the GPS satellite moves at a speed of around four kilometers per second relative to the receiver, the determined GPS position would be a good kilometer off the actual position after twelve hours without corresponding corrections.

### Elementary particles and the theory of relativity

Time dilation plays a very important role for short-lived, mass-laden and high-energy elementary particles that are generated, for example, in particle collisions. The number of such unstable particles decreases exponentially and has dropped to half after the so-called half-life. This half-life is defined for particles at rest, and this has interesting consequences: If a particle moves quickly through the laboratory system, a researcher measures a half-life in it that has been extended by the Lorentz factor.

The best known example are the muons - heavy relatives of electrons - in cosmic rays. They are generated around twenty to thirty kilometers above the earth as secondary products when high-energy protons from space collide with atomic nuclei in the outer atmosphere. Muons decay into one electron and two neutrinos with a half-life of around 1.52 microseconds. In this short time, they could only travel 457 meters even at the speed of light. The probability that one of these particles will “survive” a distance of twenty kilometers, which corresponds to 44 half-lives, would be negligible.

However, contrary to what has just been estimated, scientists observe that numerous muons from cosmic radiation reach the earth's surface. The solution to the riddle lies in extending the service life through time dilation. The particles are created with high energy and therefore high speeds. On average, this increases their half-life by a factor of 16, and the flight distance that half of the muons survive is 7,310 meters. The distance of 20,000 meters therefore corresponds to only 2.7 half-lives, and the probability of covering this distance without decay is now 15 percent.

Muon storage ring

What is the situation in the rest system of the moving particles? There is no increase in half-life here, but there is a contraction in length. This phenomenon is closely linked to time dilation: a moving length scale is perceived as being shortened by a stationary observer, namely by the Lorentz factor. From the point of view of the fast muons, the 20-kilometer route acts like a scale that flies towards the particles. The flight distance is apparently perceived as being shortened and is only 1250 meters. The probability of passing the shortened way to earth without decay is again 15 percent. So we get the same physical statement in both reference systems.

Despite this symmetry between the two inertial systems, a serious problem arises. To see this, we compare a group 1 of muons resting in the laboratory system with a group 2 of fast moving muons. Viewed from the laboratory system, the clocks in the moving system run more slowly - the muons of group 2 should therefore "live" longer than the muons of group 1. From the point of view of the moving system, it is exactly the opposite: Now the laboratory system is moving at high speed, its own Clocks are correspondingly slowed down and consequently the muons of group 1 should live longer than the muons of group 2. These two points of view, which are perfectly legitimate in the theory of relativity, obviously lead to contradicting statements.

What is right now? That is not so easy to say, because the two groups of particles fly further and further away from each other, and it is no longer possible to check which ones actually decay, those that move in the laboratory or those that are dormant in the laboratory. This is the famous "twin paradox". However, there is an experiment with which a decision is possible and this is based on the fact that the time stretching also occurs in accelerated coordinate systems. At the Brookhaven National Laboratory in the US state of New York and at the research center CERN near Geneva, circular storage rings for high-energy muons have been built. In a storage ring, the particles keep returning to their starting point, and one can easily compare the lifetimes of the stationary and fast moving particles.

Comparison of resting and moving muons

In their experiment, the scientists compared ten thousand stationary particles in the laboratory with ten thousand rapidly moving particles in the storage ring. The measurement showed that after twenty microseconds not a single muon at rest is left, but there are still 8,600 moving muons. Einstein's twin paradox is realized here in a spectacular way: Most of the rapidly moving muons still exist, while the particles at rest have all decayed. This clear statement in favor of the fast muons results from the fact that the two reference systems are not equal in this experiment: The laboratory system is an inertial system, the moving system in the storage ring, on the other hand, is similar to a carousel and represents an accelerated system. For this reason, there is between the two Systems no longer have symmetry.

### Theory of Relativity and Particle Accelerator

Another and very important consequence of the theory of relativity is the relativistic increase in mass: The mass of a fast moving particle is greater by the Lorentz factor than that of a stationary particle. If one approaches the speed of light, this factor and with it the "moving mass" grows to infinity. This is the reason why particles with a rest mass other than zero can never exactly reach the speed of light; one would need an infinite amount of energy for this. The speed of massive particles is therefore always less than c. This is different with light quanta, also called photons; they have no rest mass and always move at the speed of light.

The relativistic mass increase of electrons and protons as well as the fact that their maximum speed never exceeds the speed of light play an outstanding role in circular accelerators. Electrons or protons are accelerated here in circular tunnels and held on their path by magnets. The strength of the magnetic fields must be precisely matched to the moving mass of the accelerated particles in order to keep them on the foreseen circular path.

ATLAS component

At the HERA proton accelerator in Hamburg, the particles were injected with an energy of 40 gigaelectron volts (GeV) and accelerated to 920 GeV. (To reach the energy of 920 GeV, the protons have to go through a huge accelerating voltage totaling 920 billion volts. A high-voltage overhead line has far less than a million volts). At the point of injection, the protons have a speed of 99.97 percent of the speed of light, at the maximum energy it is 99.99995 percent. If you calculate the required magnetic fields on the one hand with classical and on the other hand with relativistic equations, the difference is enormous: The non-relativistic formula results in a field of 0.05 Tesla. According to the relativistic equation, on the other hand, the field in the superconducting magnets must have a value of 0.23 Tesla at 40 GeV and 5.2 Tesla at 920 GeV. Exactly these field values ​​were set during the operation of HERA, and the machine did indeed work perfectly. Although the speed of the protons only increases insignificantly between 40 and 920 GeV, the magnetic field must increase 23 times. This increase is exactly proportional to the increase in the moving mass.

The example of HERA shows that the term accelerator is actually inappropriate for relativistic energies. The speed hardly changes in the HERA-Ring, the acceleration is negligibly small. Instead, the energy supply is primarily used to increase the moving mass. These machines should actually be called “mass boosters”, but the name accelerator (eng. accelerator, French accelerateur, Dutch faster) has become so natural that it can no longer be changed.