# Special theory of relativity

The founder of the theory of relativity Albert Einstein around 1905

The special theory of relativity (short SRT ) is a physical theory about the movement of bodies and fields in space and time . It extends the Galilean principle of relativity, originally discovered in mechanics , to a special principle of relativity . According to the special relativity principle, not only the laws of mechanics but all laws of physics have the same form in all inertial systems . This applies u. a. for the laws of electromagnetism , which is why the speed of light in a vacuum has the same value in every inertial system. It follows from the principle of relativity that lengths and durations depend on the state of motion of the viewer and that there is no absolute space and no absolute time; this is shown in the Lorentz contraction and the time dilation . Another important consequence of the SRT is the equivalence of mass and energy .

The article on the electrodynamics of moving bodies , which Albert Einstein published in 1905 based on preliminary work by Hendrik Antoon Lorentz and Henri Poincaré , is regarded as the birth of the special theory of relativity . Since the theory deals with the description of frames of reference moving relative to one another and with the relativity of durations and lengths, it soon became known as "relativity theory". In 1915 it was renamed Special Theory of Relativity by Einstein when he published his General Theory of Relativity (ART). Unlike the SRT, this also includes gravity .

The SRT was able to explain the result of the Michelson-Morley experiment and was later confirmed by the Kennedy-Thorndike experiment and a number of other tests .

## introduction

The laws of classical mechanics have the special property of being equally valid in every inertial system ( principle of relativity ). An inertial system is a reference system in which every force-free body moves in a straight line uniformly or remains in a state of rest. This fact makes it possible, even in the ICE at full speed, for. B. to drink a coffee without the speed of 300 km / h having any effect. The transformations (conversion formulas) that are used to convert from one inertial system to another in classical mechanics are called Galilean transformations , and the property that the laws do not depend on the inertial system (i.e. do not change in a Galilean transformation) is called Galilean invariance . The formulas for a Galilean transformation follow directly from the classical idea of ​​a three-dimensional Euclidean space on which all events are based and an independent (one-dimensional) time.

At the end of the 19th century, however, it was recognized that the Maxwell equations , which very successfully describe the electrical, magnetic and optical phenomena, are not Galileo-invariant. This means that the equations change in their form when a Galilei transformation is carried out in a system that moves relative to the original system. In particular, the speed of light would depend on the frame of reference if the Galileo invariance was considered to be fundamental. The Maxwell equations would therefore only be valid in a single reference system, and by measuring the speed of light it should be possible to determine one's own speed in relation to this system. The most famous experiment attempted to measure the speed of the earth against this excellent system is the Michelson-Morley experiment . However, no experiment could prove relative motion.

The other solution to the problem is the postulate that Maxwell's equations hold unchanged in every frame of reference and that instead the Galileo invariance is not universally valid. The Lorentz invariance then takes the place of the Galileo invariance . This postulate has far-reaching effects on the understanding of space and time, because the Lorentz transformations , which leave the Maxwell equations unchanged, are not pure transformations of space (like the Galileo transformations), but change space and time together. At the same time, the basic equations of classical mechanics also have to be reformulated because they are not Lorentz-invariant. For low speeds, however, Galilean transformations and Lorentz transformations are so similar that the differences cannot be measured. The validity of classical mechanics therefore does not contradict the new theory at low speeds.

The special theory of relativity thus provides an expanded understanding of space and time, as a result of which electrodynamics no longer depends on the reference system. Their predictions have been successfully tested many times and confirmed with high accuracy.

## Lorentz transformations

The immutability of the physical laws under Lorentz transformations is the central claim of special relativity. Therefore, the physical effects of the Lorentz transformations are clearly explained in this section.

Since the laws of electrodynamics apply equally in every frame of reference, their prediction of a constant vacuum speed of light also applies in particular. The light is therefore equally fast in every frame of reference. This follows directly from Lorentz invariance, and it is often considered the most important property of the Lorentz transformations that they leave the speed of light unchanged.

### Einstein's thought experiment

Graphic illustration of the thought experiment

In order to illustrate the various aspects of the Lorentz transformations, a thought experiment is used that goes back to Albert Einstein: A train travels through a train station at the speed . There are various observers on the platform and in the train whose observations and measurements are to be compared. They have clocks and rules as well as flashing lights with which light signals can be exchanged. We call the front end of the train in the direction of travel “start of train”, the other we call “end of train”. The beginning of the train first reaches the end of the platform that we refer to as the “rear”. Later it arrives at the "front" end. ${\ displaystyle -v}$

For the train passenger, it looks as if he is resting and the platform is moving at the speed opposite to the direction of travel of the train. According to the principle of relativity, his point of view is just as correct as that of the observer standing at the train station. Both reference systems are inertial systems and are therefore physically equivalent. ${\ displaystyle + v}$

It is very important to note that any observer can only make direct statements about events that take place directly at his place. However, if he wants to know when an event took place in another place, he can only rely on light signals that were sent from this place. Using the distance and the time of flight, he can then infer the time of the event, because the speed of light is the same in all inertial systems.

### simultaneity

One of the major difficulties in understanding the effects of the Lorentz transformations is the notion of simultaneity. In order to understand it, it is therefore important to realize that the simultaneity of events in different places is not defined a priori. The speed of light is used to define simultaneity, since this is the same in all frames of reference. The light signals from two simultaneous events will reach an observer at different times if the events occur at different distances from the observer. However, if an observer is equidistant from two events and light signals from them reach him at the same time, then the two events themselves are called simultaneous .

This definition of simultaneity appears clearly understandable, but together with the Lorentz invariance leads to a paradoxical effect: The simultaneity of two events in different places depends on the state of motion of the observer.

This fact can be understood directly with the thought experiment described at the beginning :

There is a lamp in the middle of the platform. For an observer standing on the platform, it is immediately clear: When the lamp is switched on, the light reaches both ends of the platform at the same time: it has to cover the same path in both directions. Let us now consider the situation from the perspective of a passenger of the train: The platform is now moving backwards at a constant speed v. However, the light also has the speed c in both directions compared to the train. At the time of transmission, both ends of the platform are equidistant from the lamp. Thus, the front end of the platform comes towards the light beam, so that the light moving forward travels a shorter distance until it reaches this end of the platform. Conversely, the rear end of the platform moves in the direction of the light trailing it, so that the light here has to travel a somewhat longer distance before it has reached this end. Therefore, the light will reach the front end of the platform earlier than the rear end, and thus the two ends of the platform will not be reached at the same time.

The observer at the platform and the observer in the train do not agree on the question of whether the two events “the light reaches the front end of the platform” and “the light reaches the rear end of the platform” are simultaneous. However, since both observers move uniformly, neither of the two systems is excellent: The points of view of the two observers are therefore equivalent. Simultaneity is actually different for both observers.

The simultaneity of events, the location of which only changes perpendicular to the direction of movement, is the same in both reference systems: If the lamp hangs halfway up the train, the light becomes the lower for both the observer on the platform and the observer in the train - and reach the top of the train.

### Lorentz contraction

Thought experiment on Lorentz contraction

The relativity of simultaneity results in another, equally paradoxical effect:

Assuming that the beginning of the train (see Einstein's thought experiment ) triggers a flash of light when passing the front end of the platform and the end of the train triggers a similar flash of light when passing the rear end of the platform.

The observer in the middle of the platform sees both flashes of light at the same time as the train passes through. From this, the observer, when he knows that he is in the middle of the platform and what triggered the two flashes of light, concludes that the train and the platform are the same length.

For the observer in the middle of the train, however, the situation is very different: the flash of light from the beginning of the train reaches him earlier than the flash of light from the rear end of the train, because it drives towards the front light flash and at the same time moves away from the rear light flash. Since the “rear” event (the end of the train passes the rear end of the platform) occurs later for him than the “front” (the start of the train passes the front end of the platform), he concludes that the train is longer than the platform because, after all, the end of the train was still there never arrived at the platform when the beginning of the train has already left it.

Thus, the platform is shorter for the observer in the train and the train is longer than for the observer on the platform.

The principle of relativity says again that both are right: if the (moving) platform is shortened from the perspective of the train driver, then the (moving) train must also be shortened from the perspective of the platform observer. The Lorentz contraction is only valid in the direction of movement, since the simultaneity of the events in both reference systems corresponds perpendicular to the direction of movement. Both observers are so z. B. agree on the height of the contact wire .

An indirect proof of the length contraction also results from the problem of the electromagnetic field of an electrical point charge moving at high speed. The electric field of this object is simply the Coulomb field of the charge when it is vanishing or slow compared to the speed of light . H. with even radial directional distribution. With increasing approach to the speed of light, however - because of the contraction of the distance in the direction of movement - the electric fields increasingly concentrate in the transverse directions of movement. In addition to the electric fields, there are also (asymptotically equally strong) magnetic fields that circle the axis of movement.

### Time dilation

Just as distances between observers in different inertial systems are determined differently, the relative speed of the inertial systems must also be taken into account when comparing time spans: The observer in the train (see Einstein's thought experiment ) is at the rear end of the train and there is a clock at each end of the platform . The clock at the front end of the platform is started when the beginning of the train passes it and the clock at the rear end of the platform when the end of the train passes it. Since the train is just as long for the observer on the platform as the platform, the clocks are started simultaneously according to his concept of simultaneity. The clock at the front end of the platform is stopped when the rear end of the train passes it.

The observer on the train starts his clock when he passes the rear end of the platform, i.e. simultaneously with the start of the local platform clock, and stops it when he passes the front end of the platform, simultaneously with the stopping of the local platform clock. According to his concept of simultaneity, the clock at the front end of the platform moves ahead of the clock at the rear end of the platform and thus also moves ahead of his clock, since according to his concept of length the train is longer than the platform. The length of time that he measures for his journey from the back to the front end of the platform is therefore shorter than the time displayed by the clock at the front end of the platform when he passes it.

The observer on the platform can see from the displays of the clocks that the observer on the train is measuring a shorter period of time than himself. Since, according to his concept of simultaneity, the start and stop times of the observer's clock on the train and the clock at the front end of the platform are the same according to his concept of simultaneity, the periods of time are also of the same length. So he comes to the conclusion that the watch of the observer on the train is too slow. According to the concept of simultaneity of the observer in the train, however, the starting times of the clocks do not coincide, so that he does not make this observation.

This view can also be reversed by adding a clock at the beginning and at the end of the train and, according to the observer's concept of simultaneity, starting the train at the same time when the beginning of the train passes the front end of the platform. From the point of view of the observer on the train, it then emerges that the time on the platform passes more slowly than on the train.

Again, it cannot be decided which of the two observers is right. Both observers move without acceleration relative to each other and are therefore equal. Time spans are different for both observers, and for both observers time passes the fastest in their respective rest system , while it passes more slowly in all relatively moving systems. This effect is called time dilation . The time that every observer reads on his own watch is called proper time . This time, measured with a "carried clock", always results in the shortest possible, unchangeable value among all time spans that are measured for two causally connected events in inertial systems moving relative to one another. In contrast, all other values ​​are “dilated over time”.

In concrete terms: The wristwatches that are carried “tick” faster for the train passengers (ie they show a longer time) than similar station clocks that the train rushes past at speed v. When the speed of the train increases, the (usually very small) dilation of the time shown by the station clock increases, while the time measured from the train (the proper time) always remains the same. In contrast to this time dilation, a scale that moves along with the train and whose length has the value L from the perspective of the train passengers appears shortened when viewed from the station clock ( length contraction , see above). The effects are, however, extremely small: an interval Δτ of the proper time is only slightly smaller compared to the time span Δt displayed by the station clock (more precisely, given a constant relative speed, the following applies: where the train speed is (for example 80 km / h) , c, on the other hand, is the extremely much higher speed of light (~ 1 billion km / h). ${\ displaystyle \ Delta \ tau = \ Delta t \ cdot {\ sqrt {1- (v / c) ^ {2}}},}$${\ displaystyle v}$

Incidentally, the proper time is the invariant that determines the coordinate change given above ( Lorentz transformationLorentz invariant).

A direct consequence of time dilation is that the elapsed time depends on the path chosen. Suppose someone gets on the train and drives to the next station. There he changes to a train that goes back to the starting point. Another observer has been waiting there on the platform in the meantime. When they return, they compare their watches. From the point of view of the observer who remained at the station, the traveler has now experienced a time dilation both on the outward journey and on the return journey. So the traveller's watch is now slowing down from the perspective of the waiting person. From the point of view of the traveler, however, the waiting person experiences a time dilation both on the way there and on the way back, so that at first glance the watch of the person waiting has to follow from the perspective of the traveler. This paradox is called the twin paradox . In this case, however, the situation is actually not symmetrical, since the traveler has switched, i.e. has changed the reference system that was moved with him. In contrast to the observer on the platform, the traveler does not remain in a single inertial system during the entire journey, so the traveler’s clock is actually slow.

This paradox has actually been demonstrated in experiments to test the special theory of relativity. In the Hafele-Keating experiment , for example, the measured time spans of two atomic clocks were compared, one of which was circling the earth in an airplane, while the second remained at the take-off and destination airports. The “lagging” watch showed a slight but precisely measurable rate increase.

If the conductor walks forwards at a constant speed on the train (see Einstein's thought experiment ), his speed for an observer on the platform is given according to classical mechanics as the sum of the running speed and the speed of the train. In the theory of relativity, such a simple addition does not give the correct result. Viewed from the platform, the time the conductor z. B. from one car to the next takes longer than for the train traveler because of the time dilation. In addition, the car itself is shortened to Lorentz when viewed from the platform. In addition, the conductor walks forward, so the event “reaching the next car” takes place further up the train: Due to the relativity of simultaneity, this means that the event takes place later for the observer on the platform than for the train passenger. Overall, all these effects result in the fact that the speed difference between the conductor and the train is lower for the observer on the platform than for the observer on the train. In other words: the conductor is traveling more slowly when viewed from the platform than would result from the addition of the speed of the train and the speed of the conductor when viewed from the train. The formula used to calculate this speed is called the relativistic addition theorem for speeds .

The extreme case occurs when looking at a beam of light running forward. In this case, the slowing effect is so strong that the light beam has the speed of light again from the platform. The constancy of the speed of light is the basis of the theory of relativity. This also ensures that the conductor always moves more slowly than the speed of light from the point of view of the observer on the platform, provided that his speed in the rest system of the train is less than the speed of light: Assume the conductor holds a flashlight on a mirror at the end of the car and walks slower than the light. Then, viewed from the train, the light beam is reflected and hits the conductor before it reaches the end of the car. If his speed from the platform were perceived as faster than light, the conductor would reach the end of the car in front of the light beam and thus the meeting with the light beam would not take place. The fact that such a meeting takes place is, however, independent of the observer and thus a contradiction arises. So the relativistic addition of two speeds below the speed of light always gives a result below the speed of light.

Now the conductor can not only run forwards on the train, but also backwards. In this case, the event “reaching the next car” takes place further back in the train and thus “premature” for the platform observer relative to the train passenger, while the other effects still have a “slowing effect”. The effects cancel each other out when the conductor runs backwards in the train at the same speed as the train is traveling: In this case, the theory of relativity also comes to the conclusion that the conductor is at rest relative to the platform. For higher speeds to the rear, the observer on the platform now sees a higher speed than he would expect according to classic mechanics. This goes up to the extreme case of the light beam directed backwards, which in turn travels exactly at the speed of light when viewed from the platform.

### Momentum, mass and energy

Collision of two balls with change of direction of movement by 90 °

In the train station (see Einstein's thought experiment ) there is also a game room with pool tables. On a happening, passing by as the train, just following, described from the perspective of the observer on the platform: two billiard balls, each of which has the same absolute speed as the train, but perpendicular to the track move, come completely elastic together in a way offset so that they move parallel to the track after the impact, the red in the direction of the train (and resting in its reference system) the blue in the opposite direction.

In classical mechanics, the momentum of an object is defined as the product of the object's mass and speed. The total impulse, which results from simply adding the individual impulses, is a conserved quantity . In fact, the impulse defined in this way from the platform point of view is retained in the above impact: Since the balls move at opposite speeds both before and after the impact, the impulse defined in this way is zero before and after the impact.

Viewed from the train, the balls roll diagonally towards each other before the impact: parallel to the track, both have the speed of the platform (since they move with the platform), and perpendicular to the track they have opposite speeds (this component is based on movement of the balls relative to the platform perpendicular to the train). The total momentum of the two balls perpendicular to the track is zero, parallel to the track the total momentum is twice the ball mass times the platform speed.

After the impact, the red ball now has the speed - and thus the momentum - zero (from the platform perspective, it was traveling at the train speed in the direction of the train), so the blue ball now has to carry the entire momentum. In order to determine the speed of the blue ball, however, the relativistic speed addition considered in the previous section must now be used, and - as explained above - this ball now has a speed lower than twice the platform speed (= train speed). This makes it clear that the classic conservation of momentum is no longer valid. To restore the law of conservation, the relativistic momentum is used, which increases more than linearly with velocity. For the same reason, the kinetic energy must also increase faster at high speeds than it does according to classical mechanics.

The equivalence of mass and energy means that the rest energy of every particle, body or physical system is proportional to its mass . The factor that connects these two quantities is the square of the speed of light: ${\ displaystyle E _ {\ text {calm}}}$ ${\ displaystyle m}$

${\ displaystyle E _ {\ text {calm}} = m \ c ^ {2}}$

Because the rest energy can be read from the mass, one understands why, in the case of radioactive decay or nuclear fission, the daughter particles together have less mass than the starting nucleus: Part of the initial rest energy has been converted into kinetic energy of the daughter particles and possibly into other radiation.

The equivalence of mass and energy is experimentally confirmed with high accuracy:

${\ displaystyle {\ frac {m \, c ^ {2}} {E _ {\ text {rest}}}} - 1 \, \ leq \, (1 {,} 4 \ pm 4 {,} 4) \ cdot 10 ^ {- 7}}$

### Relativistic mass and rest mass

You sort through

${\ displaystyle E (v) = m _ {\ text {relativistic}} \ c ^ {2}}$

mathematically assigns a speed-dependent mass to the speed-dependent energy of a particle or body in motion , so it is called a relativistic mass . It is not a fixed property of the particle independent of the reference system, but depends on its speed (or that of the observer). In the rest system it corresponds to the mass , which is therefore sometimes also referred to as rest mass or invariant mass. With a sufficiently close approach to the speed of light, it becomes arbitrarily large. With the relativistic mass, the relativistic impulse is written as “mass times speed” as in Newton's mechanics. The fact that the momentum of a particle can increase indefinitely, while its speed is limited by the speed of light, is caused in this picture by the correspondingly increasing relativistic mass. In the range of relativistic speeds, a particle reacts to a force perpendicular to its direction of flight in such a way that, according to Newtonian mechanics, one would have to ascribe the relativistic mass to it. For a force in the direction of the velocity one would have to take a different mass, and for other directions the acceleration is not even parallel to the force. ${\ displaystyle E (v)}$${\ displaystyle m _ {\ text {relativistic}}}$${\ displaystyle m _ {\ text {relativistic}}}$${\ displaystyle m}$${\ displaystyle m _ {\ text {relativistic}}}$

The concept of relativistic mass is therefore avoided in today's (2017) physics for these and other reasons. Rather, as in Newton's physics, mass is a property of the particle, body or physical system that is independent of the reference system. This means that there is no distinction between “mass” and “rest mass”. Both are names for the same term.

### From space and time to space-time

In view of the relativistic effects explained above, the question arises of how these effects are to be interpreted. If you look at time as the fourth dimension, you can look at the four-dimensional spacetime together with the three dimensions of space , which does not result in the four-dimensional Euclidean space , but the so-called Minkowski space.The difference results from a mathematical peculiarity of the metric ( better: pseudo-metric) of the Minkowski space - it can have both signs. This gives the difference between rotations in four-dimensional Euclidean space and the “rhombohedral” coordinate transformations of four-dimensional space-time given below. At the same time it follows that in the theory of relativity a difference between space-like and time-like or - in the case of time-like - between "past" and "future" can remain, depending on the sign of the metric of the point under consideration in Minkowski space or after the sign of its time coordinate (see also: light cone ). ${\ displaystyle \ mathbb {R} ^ {4}}$ ${\ displaystyle \ mathbb {M} ^ {4}.}$${\ displaystyle \ mathbb {R} ^ {4}}$

The movement of an observer becomes a curve in this four-dimensional space-time (the so-called world line of the observer) and can be represented in Minkowski diagrams . It can be seen that the current change in the reference system is always accompanied by a "tilting" of the time axis (both classical-mechanical and relativistic). This describes the "relativity of equality": While the observer on the train determines that z. For example, if his suitcase stays in the same place the whole time in the baggage net, it is clear to the observer on the platform that the same suitcase is moving with the train, i.e. not staying in the same place. What distinguishes the Minkowski space of the theory of relativity from Newton's space and time is the fact that for frames of reference that are moving to one another, the SIMPLICITY is also relative, as described above. This leads to the fact that according to the theory of relativity (in contrast to classical mechanics) the position axis is tilted together with the time axis.

Comparison of rotation (left) and the "rhombohedral" change of reference system described in the text (right)

A well-known movement in which two coordinate axes are changed is rotation in space. The picture opposite illustrates the difference between the known rotation and the specified change of the reference system : While both axes are rotated in the same direction when rotating in space , when changing the reference system , the location axis and time axis are rotated in opposite directions: The original square creates a rhombus with the same area , where the condition of equality of area corresponds to the constancy of the speed of light. The long diagonal (an angular symmetry of the axes, the so-called 1st median) remains unchanged. But it describes precisely the path of light, its increase is the speed of light. The immutability of these diagonals when there is a change in the frame of reference means that the speed of light is constant.

From these considerations it follows that it makes sense to view space and time as a unit, just as length, width and height form a unit, namely three-dimensional space. The four-dimensional unit of space and time is called space-time . It is therefore no longer possible to specify a very specific direction as the time direction independently of the observer , just as there is no clear ( observer-independent ) "front" in space. So run z. B. both the black time axis and the yellow "rotated" time axis in time direction. However, in contrast to normal space, it is not possible in spacetime to turn the direction of time up to the direction of space or even to “turn around” time, i.e. to swap past and future. Due to the constancy of the diagonals, the areas bounded by the diagonals are always transformed into themselves. This corresponds to the equality of area of ​​the network segments shown.

If you take a closer look at the rotation (left picture) you can see that each coordinate square is converted into a square of the same size (the rotated square at the top right of the origin is hatched in the picture). In addition, the intersection of the rotated y-axis (yellow line) with the intersection of the rotated first parallel lines of the x-axis (light brown line) is the same distance from the origin as the unrotated intersection. The y-value of this intersection, however, is smaller than for the unrotated intersection. This leads to the phenomenon of foreshortening when the line is viewed from the x-direction.

If you now look at the picture on the right, you can see that the coordinate square is also transferred to an area of ​​the same size, but the new area is no longer a square, but rhombohedral . This has the effect that the point of intersection of the "rotated" time axis (yellow) with the next parallel line of the rotated spatial axis (light brown) is higher , i.e. later , than in the unrotated case. Let us now assume that the spatial axes are "set" with every tick of a clock, so you can see that the clock in the "rotated" coordinate system, i.e. the clock moved relative to the observer, apparently moves more slowly (more time of the observer passes between two ticks ). The analogy to rotation also makes it clear that this is also only a "perspective" effect. This easily explains the apparent contradiction that both observers see each other's clock running slower. The foreshortening of the perspective is also mutually perceived without this leading to contradictions.

An essential difference between the change of the reference system and the rotation, however, is that instead of a shortening of the variable “time”, an extension (stretching: time dilatation ) is perceived. This can be clearly seen in the comparison above: When rotating in space, the intersection of the yellow and light brown lines moves downwards ( shortened perspective ), but when the reference system changes, it moves upwards .

## Effects

The effects mentioned, which can only be understood with the Lorentz transformation, can partly be observed directly. In particular, the time dilation has been confirmed by many experiments (see e.g. time dilation of moving particles ). In the following some effects are shown, for which the connection with Lorentz transformations is not so obvious.

### Aberration

If an observer moves faster and faster, the side rays of light come towards him more and more from the front, like raindrops. The angle at which a beam of light strikes a moving observer changes. Originally, this phenomenon, the aberration of light, was explained with Newton's corpuscle theory of light just as with raindrops. In the special theory of relativity, the classical is now replaced by the relativistic addition of speed. From this it follows that a moving observer would observe a different aberration angle according to the corpuscle theory than according to the special relativity theory and would measure different speeds of light of the incident light depending on the speed of movement.

After observing that light spreads like a wave ( undulation theory ), one could no longer understand the aberration. In Newtonian physics, with a light wave the wave fronts would not change when the observer moved. Only in the special theory of relativity do the wavefronts change due to the relativity of simultaneity just like particle trajectories, and aberration becomes understandable, whether it occurs with waves or with particles.

### Doppler effect

In the case of waves that propagate in a carrier medium, such as sound waves , there is a change in the measured frequency when the source or the receiver moves in relation to the carrier medium . The effect is different depending on whether the source or the receiver is moved in relation to the carrier medium. In general, the frequency increases when the source and receiver move towards each other, because the receiver then perceives more wave crests in the same time. Correspondingly, the frequency decreases when the source and receiver move apart. This frequency shift is called the Doppler effect . With sound waves, the receiver can be faster than the waves and escape them entirely; accordingly, the source can run ahead of its own signal, which leads to a sonic boom.

With light waves in a vacuum, no relative movement to the carrier medium can be measured, since the vacuum speed of light is the same in all inertial systems. The Doppler effect of light can only depend on the relative speed of the source and receiver, that is, there is no difference between the movement of the source and the receiver. Since a relative movement is not possible faster than the speed of light in a vacuum, there is no phenomenon analogous to the sonic boom for light in a vacuum. In media such as water, in which the speed of light propagation is slower than in a vacuum, there is a phenomenon similar to the sonic boom, the Cherenkov effect .

It is clear that time dilation has an influence on the frequencies measured by two observers who are moving relative to one another. Therefore, a Doppler effect also occurs with light when the observer moves perpendicular to the direction in which the source lies. This effect is called the transverse Doppler effect . The definition of the angle of incidence depends on the observer due to the aberration. Therefore, depending on the reference system in which the light is incident perpendicularly, an increase in frequency ( blue shift ) or decrease ( red shift ) occurs:

• From the point of view of the rest system of the receiver, the time in the system of the source passes more slowly due to the time dilation. This means that it measures a lower frequency in its system than an observer who is at rest relative to the source, i.e. it measures a redshift. The observer who is at rest in relation to the source explains the effect that the receiver does not move perpendicular to the direction of the source at the time of reception, but away from the source. The light beam hits the receiver from behind, which explains the redshift.
• From the point of view of the source's rest system, time passes more slowly in the recipient's rest system. The receiver therefore measures a higher frequency, i.e. a blue shift, when the light hits the receiver in the rest system of the source perpendicular to the direction of movement. The recipient explains this blue shift differently, because from his point of view, the light beam hits him not at a right angle , but at an angle from the front. So he will explain the blue shift by approaching the source.

### Lorentz force

Illustration of the Lorentz force

The theory of relativity does not only become relevant at very high speeds. The Lorentz force offers an example of how fundamental differences compared to classical physics can arise in the explanation of known effects even at very low speeds.

To do this, one looks at a single negative electrical test charge at a certain distance from a wire that is electrically neutral overall, but consists of a positively charged, rigid basic material (the atomic cores) and many negatively charged, mobile electrons. In the initial situation, the test charge rests and no current flows in the wire. Therefore, neither an electric nor a magnetic force acts on the test charge. If the test charge now moves outside and the electrons inside the wire move at the same speed along the wire, a current flows in the wire. This creates a magnetic field; because it moves, it exerts the Lorentz force on the test charge, which pulls it radially towards the wire. This is the description in the frame of reference in which the positive base material of the wire rests.

The same force acts in the frame of reference, which is moved with the negative charge, but has to be explained quite differently. It cannot be a Lorentz force, because the speed of the test charge is zero. However, the positively charged basic material of the wire moves and now appears shortened by the Lorentz contraction. This gives it an increased charge density, while the electrons in the wire rest in this reference system and therefore have the same charge density as in the initial situation. The total charge density in the wire shows an excess of positive charge. It exerts an electrostatic force on the static negative test charge , which pulls it radially towards the wire. This is the description in the moving reference system.

Both descriptions lead to the same predictions about the force that acts on the test charge. This could not be explained without taking into account the Lorentz contraction; The wire would then remain electrically neutral in both reference systems. From the point of view of the moving reference system, the moving positive basic material of the wire would mean a flow of current that generates a magnetic field, but this would have no effect on the static test charge.

This observation shows that magnetic fields and electrical fields are partially converted into one another by Lorentz transformations. This makes it possible to attribute the Lorentz force to electrostatic attraction. This effect has measurable effects even at low speeds - the mean electron speed in the direction of the wire is typically less than one millimeter per second when a current flows, i.e. much less than the speed of light.

### Indirect effects

Many direct effects are not obvious because they would usually only occur when approaching the speed of light. But there are many indirect effects, including the following:

All of these effects can be seen as indirect confirmations of the special theory of relativity.

## Relation to other theories

### Classic mechanics

The special theory of relativity takes the place of the dynamic laws of classical mechanics . However, the laws of classical mechanics have been confirmed very precisely over the centuries. However, speeds were always considered that were much smaller than the speed of light. For such small velocities, the special theory of relativity should deliver the same results as classical mechanics. This means that the Lorentz transformations must result in the Galilei transformations for very small velocities. From this it follows immediately that the momentum, the kinetic energy and all other quantities also assume the known classical values ​​for small velocities.

If in the above thought experiments the train is traveling much slower than the speed of light, the difference between the observer's notions of simultaneity is very small. This leads to the other relativistic effects becoming so small that they can hardly be observed. So if the time dilation is so small that it goes unnoticed, the Lorentz transformation apparently only transforms the spatial coordinates. If the length contraction also goes unnoticed, exactly the Galileo transformations remain.

This illustrates that the special theory of relativity delivers the same results as classical mechanics for very small speeds. The fact that the predictions of an old, proven theory must also be derivable in a new theory is called the correspondence principle. The special theory of relativity thus fulfills the principle of correspondence with regard to classical mechanics. In the case of non-mechanical, electromagnetic processes, this is not always the case, as illustrated by the explanation of the Lorentz force.

A speed of 0.1c (10% the speed of light) is often used as a rule of thumb in physics classes; Up to this value, calculations according to classical physics are considered acceptable, at higher speeds one has to calculate relativistically. Ultimately, however, the specific problem decides at which speeds relativistic calculations must be made.

### general theory of relativity

In spatial areas in which the effect of gravity is negligible (i.e. in particular far away from large masses), the SRT can describe all types of movements (contrary to a common misconception, also accelerated movements ). In contrast, if gravitation effects are taken into account, the general theory of relativity takes the place of the special theory of relativity. In this respect, a correspondence principle must also be fulfilled here, since the predictions of the special theory of relativity are very precisely confirmed experimentally.

In contrast to the special theory of relativity, spacetime in the general theory of relativity is curved and the theory must therefore be formulated strictly locally. For large distances there may therefore be deviations from the statements of the special theory of relativity. By taking into account gravity , the special theory of relativity is only valid for small distances, especially in the vicinity of large masses, more generally in the vicinity of large energies.

A particularly illustrative effect that shows the limit of the validity of the special theory of relativity is the Shapiro delay : for light that is sent close to a body with great mass, such as the sun, an observer who is further away from the massive body measures away, a smaller speed than the expected vacuum speed of light. In contrast, an observer directly at the light beam measures the "correct" speed of light. Obviously, the laws of the special theory of relativity, like the constancy of the speed of light, only apply in small areas. In the general theory of relativity this becomes clear from the fact that spacetime is a so-called Lorentz manifold or a Riemann space , which, however, can be described locally by a Minkowski space - that is the flat spacetime of the special theory of relativity - at each spacetime point .

### Quantum theory

In contrast to the general theory of relativity , where it is still unclear how it can be merged with quantum physics to form a theory of quantum gravity , special-relativistic quantum theories belong to the standard tools of modern physics. In fact, many experimental results cannot be understood at all if one does not take into account both the principles of quantum theory and the space-time understanding of the special theory of relativity.

In the semi-classical Bohr-Sommerfeld atomic model , it is only possible to explain the fine structure of atomic energy levels when the special theory of relativity is included .

Paul Dirac developed a wave equation , the Dirac equation , which describes the behavior of electrons taking into account the special theory of relativity in quantum mechanics . This equation leads to the description of the spin , a property of the electron that can only be determined by non-relativistic quantum mechanics, but not explained, and to the prediction of the positron as the antiparticle of the electron. As in the semi-classical models, the fine structure cannot be explained by non-relativistic quantum mechanics.

However: The very existence of antiparticles shows that when special relativity theory and quantum theory are combined, a relativistic version of the usual quantum mechanics cannot simply emerge. Instead, a theory is needed in which the number of particles is variable - particles can be destroyed and created (the simplest example: the pairing of particles and antiparticles). This is achieved by (relativistic) quantum field theories , such as quantum electrodynamics as a special relativistic theory of electromagnetic interaction and quantum chromodynamics as a description of the strong force that holds the building blocks of atomic nuclei together.

In the form of the standard model of elementary particle physics , relativistic quantum field theories form the backbone of today's physics of the smallest particles. The predictions of the Standard Model can be tested with high precision on particle accelerators , and the combination of special relativity and quantum theory is one of the most rigorously tested theories in modern physics.

### Ether theories

The special theory of relativity is often understood in the literature as a counter-theory to the ether . Most theories of ether are incompatible with the special theory of relativity and are refuted by the experimental confirmations of the special theory of relativity.

An exception is Lorentz's theory of ether , which was developed by Hendrik Antoon Lorentz and Henri Poincaré before and at the same time as the special theory of relativity. This theory is identical in its predictions to the special theory of relativity, but assumes that there is an absolutely stationary reference system, which cannot be distinguished from any other reference system by any observation. This theory is now considered out of date because the postulate of the unobservable rest system violates the principle of economy . In addition, it is still unclear whether Lorentz's theory of ethers is compatible with the general theory of relativity.

## literature

Commons : Special Theory of Relativity  - collection of images, videos and audio files
Wikibooks: Special Theory of Relativity  - Learning and Teaching Materials

Relativistic Effects