# Special relativity theory means several realities

An easily comprehensible introduction to the derivation of the effects of the special theory of relativity from the basic postulates is the thought experiment of **Light clock**. A light clock is a device of a given length along which a photon oscillates back and forth:

In this illustration the limits of the light clock are shown in blue. We can imagine two mirrors that are fixed at a given distance from one another. The graphic on the left shows the light clock from the point of view of an observer who is at rest across from it, i.e. from the point of view of yours **Rest system**.

We now ask how **same process** in one **on the other hand moving inertial system** looks, whereby the direction of movement should take place transversely to the direction of travel of the photons. To an observer in this new system, the light clock is moving, and we denote the value of its speed by v. The photons will run along inclined paths in the moving system - this is shown in the right part of the figure above. We can just as well imagine that two light clocks of identical design are available and that we consider one stationary (left) and one moving with velocity v (right).

According to Galilean physics, it would be expected that the photon observed from an inertial system moving transversely to the direction of travel would be *more quickly* than moves in the rest system of the light clock. One could argue, for example, that the speed component is c in the vertical direction and v in the horizontal direction, so that the total amount of the speed is the value (c^{2} + v^{2})^{1/2} would have.

Qualitative reasoning |

Now comes the postulate from the **Constancy of the speed of light** in the game. The photons have in *each* Inertial system the velocity c. This leads to an unpleasant consequence for Galilean physics:

All the photons shown here have the same speed. However, the distance from the lower to the upper mirror is for the photon of the moving light clock *longer* than for that of the dormant light clock, and therefore one passes *greater time span*until it goes from one mirror to the other! A price has to be paid for the constancy of the speed of light: The time that a process lasts in an inertial system is not necessarily the same as the time that it lasts during *same* Process in one *other* Inertial system passes. The moving light clock has a longer period than the stationary one. This means that the process of photon oscillation, when observed from a moving system, is slower than in the rest system of the light clock. Now that such a light clock can be used to measure the duration *any* To measure other processes (it is actually just a special kind of "clock"), this effect is not limited to light clocks, but affects the flow of time in both systems in general. He is often short of the words

**"Moving clocks go slower"**

summarized and is called **Time dilation** ("**Time stretching**"). The time that passes for a clock in its rest system is called **Proper time**.

Quantitative reasoning |

The statement just made that a clock moves more slowly in an inertial system that is moving relative to it than in its rest system can be quantified more precisely. Let us consider the movement of the photon from the lower to the upper mirror. All lengths that occur in this situation are given in the following figure:

Let us analyze the entries in this sketch:

- The duration of the process in the
**Rest system**the light clock is with Dt_{Rest}designated. The distance between the two mirrors is therefore c Dt_{Rest}, since c is the speed of the photon. (We could designate this distance with a letter of our own, but this is not necessary).

- From
**moving system**From a point of view, a time interval elapses during the same process, which we do not know at first and which with Dt_{bew}describe. The path covered by the photon (which also moves in this inertial system with speed c) therefore has the length c Dt_{bew}. We use the value c Dt determined in the rest system as the distance between the two mirrors_{Rest}. During the process, the upper mirror is by the distance v Dt_{bew}advanced (since the light clock in this system moves to the right with the speed v). Overall, these three lengths form a right triangle.

We can now express our argumentation, which was previously only qualitative, in a formula: Dt_{bew} > German_{Rest}. To quantify the size of the effect, we apply Pythagoras' theorem to the right triangle in the right part of the figure above:

(c Dt_{Rest} )^{2} + (from Dt_{bew} )^{2} = (c Dt_{bew} )^{2} | (1) |

and solve after Dt_{bew} on. We obtain

| (2) |

That is the **Formula for that**** Time dilation**, which we also more compact than Dt_{bew} = Dt_{Rest} (1 - v^{2}/ c^{2 })^{-1/2} can write. In words it can be formulated like this:

"A clock moving at speed v is about the factor(1 - v^{2}/ c^{2})^{-1/2} slower than in their rest system. " |

This is not some kind of "apparent effect" or "illusion" - it is that *actual times* affects how they can be measured with (sufficiently accurate) clocks of any design. The length of time that a process takes is not a universal variable, but depends on the state of motion of the observer. "Time" has lost its absolute character - which it had in Galilean physics.

In the course of our argument, we tacitly assumed that the distance between the mirrors is the same in both systems. In the next section, which deals with the Lorentz contraction, we will find that not only time intervals, but also spatial distances depend on the state of motion of the observer. However, this effect only affects lengths *in* the relative direction of movement of the two systems. In other words, we tacitly assumed that lengths *across* to the direction of movement in both systems are the same. This can be derived from the postulates with a little more effort - but we want to leave it with the simplified argumentation presented here.

c as the top speed |

As a by-product, we immediately get another physical result: The formula (2) makes no sense if a velocity v is used, the amount of which is greater than or equal to c. Obviously we have to conclude that one inertial frame is relative to another only with **Sub-light speed** can move. Therefore no material object, which can be at rest with respect to an inertial system, can move at a speed whose magnitude is ³ c. This applies to all particles and bodies that have a non-vanishing mass. (Only "*massless* Particles "like photons are an exception here: they always move exactly at the speed of light - we will say a few words about these particles in the section on energy. Velocities that are greater than c cannot occur for all these objects. From this." it also follows that there is no signal that can transmit information faster than light.

Let us summarize (where v denotes the magnitude of the velocity):

- For the movement of one
**Body**, which has a non-vanishing mass, always holds (with respect to*each*Inertial system)**v**. **A****photon**always moves (in relation to*each*Inertial system) with**v = c**.**A physical one****signal**always moves (in relation to*each*Inertial system) with**v £ c**.

**With this we have one of the most famous statements of the special theory of relativity (the " Prohibition of faster than light speed") justified. It follows directly from the initial postulates. The factor (1 - v^{2}/ c^{2 })^{-1/2} , which is mathematically responsible for this, will appear in the following sections in connection with many other effects. The role of c as the maximum speed is therefore not specifically related to the time dilation, but results from practically all relevant results of the special theory of relativity.**

**A dynamic The reason for the statement v **

**All of this may sound a bit strange at first because it contradicts our everyday conceptions of speeds. What happens, for example, if a particle moves forward at 3/4 the speed of light in a train that is itself traveling at 3/4 the speed of light - shouldn't it then have to move at one and a half times the speed of light in the system of rails? The answer is no, and you can find out what its real speed is in the section on relativistic speed addition.**

**There is a frequently heard argument that can be used against the faster than light speed in the meeting: It leads to the possibility of time travel, i.e. traveling into one's own past. The elaboration of a more precise argumentation is provided as a task in the Tricky section. (In order to be able to solve it, you should already know the basics of what the Lorentz transformation is). **

**We want another one warning pronounce: The prohibition of the "faster than light" applies only to the movement of physical objects such as bodies, particles, photons, etc., ie movements with the help of which information ("from one point to the next") can be transmitted. But it can also other Forms of "speeds" occur that are not subject to any limitation: In lectures, a point of light is sometimes projected onto a screen with the help of a laser pointer. If the laser pointer is rotated quickly, the point of light whizzes across the screen - the speed at which it does this can in principle be greater than c (if the screen is sufficiently far away) because it does not move with it one Object (or the transfer of information) connected "from one point to the next"!**

**In particle physics, " Tachyons"that move at faster than light speed. These are actually hypothetical faster-than-light particles that occur in certain theories and cause problems there. There are two methods of getting rid of these problems: (i)A theory in which tachyons occur is rejected. (ii)The theory is changed in such a way that the tachyons contained in it do not interact with the rest of the world and can therefore be viewed as a purely mathematical structure that has no equivalent in physical reality.**

A refinement: localized clocks |

**In the reasoning presented above, a spatially extensive Construction (consisting of two mirrors and a photon oscillating between them) used as a "clock". Like any clock, it can be used to measure time intervals between events - it directly measures the time that "passes in its rest system". However, this is a difficult term - especially since the universal flow of time (the synchronous beating all Clocks, regardless of their state of motion, as assumed in Galilean physics) has been lost due to the effect of time dilation.**

**The idea that there is a clock is simpler tiny(in the borderline case punctiform) - a kind of "particle" that "ticks" at regular intervals. A clock assumed to be punctiform always has a well-defined location, and only There it can measure time intervals between events. It represents the concept of a timepiece taken along by an observer on his way. A time measurement is then - as is also the case in physical practice - initially one Reading on such a - sufficiently small, localized - watch. (The "global" time, which is valid within an inertial system, only comes about through the synchronization of localized clocks, which are located in different places and are at rest relative to one another. In the introduction we briefly talked about the synchronization of clocks, and in the section Simultaneously we will take up the topic again).**

**However, it is not difficult to filter out a localized clock from the concept of the light clock: We can define it as the point on one of the two mirrors that the photon hits again and again at regular intervals. This is illustrated in the following figure:**

**The "ticking" is represented by the repeated impingement of the photon on the lower mirror, and it is clear that a "clock" constructed in this way moves to the right at speed v. In addition, the distance between the mirrors (and thus the period) can be made as small as desired.**

**Localized clocks can be shown as world lines in spacetime diagrams. Furthermore, they can perform not only straight, uniform, but also accelerated movements. We shall now briefly turn to these two topics. **

Representation in the space-time diagram |

**The world line of a localized clock that moves in a straight line uniformly in a given inertial system can be represented as a straight line in a spacetime diagram:**

**Events are drawn as red dots on the world line, representing the "ticking", two of which are named A and B. Die ZeitDt _{bew}that elapses between them in the given inertial system can be read off directly. (One therefore also speaks of the Coordinate time in relation to the given system). The time Dt_{Rest}between A and Bin the rest system of the clock- i.e. for the watch itself, you can also say: along their world line- passes, is shorter (although the connection between A and B is in the diagram longer is than their vertical distance Dt_{bew}): this is the effect of time dilation. However, the time that passes for a moving watch can be shown in such a diagram Not can simply be read as "length along its world line". If you need it, you can use the formula Dt_{Rest} = Dt_{bew }(1 - v^{2}/ c^{2 })^{1/2} be calculated. **

**Although the effect of time dilation - formula (2) - has only been derived for clocks that move at a constant speed relative to one another, it also applies to accelerated watches. In order to formulate it correctly, we have to **

**fix an inertial system I and****look at a (localized) clock that is moving at an accelerated rate. The movement should not be jerky, so that the clock - viewed from system I - is a well-defined one at all times****Current speed**v (t) has. Obviously, the clock should always move slower than the light: -c

**Now let dt be a small (more precisely: an "infinitesimally small") time interval. Between the times t and t + dt, the time span dt elapses in system I. This is illustrated in the following space-time diagram:**

**Event A takes place in system I at time t, event B at time t + dt. The (smaller) time interval for the accelerated clock is between these two events**

German_{Clock} = dt (1 - v (t)^{2}/ c^{2 })^{1/2} | (3) |

**past. (Under German _{Clock} can we do that of theClock itself displayed Introduce the time interval. As in the case of uniformly moving clocks, the time indicated by the clock itself becomes Proper time called. Of course, we want to assume that the forces caused by the acceleration have no mechanical influence on the speed of the watch. Strictly speaking, statement (3) is about the effect of time dilation for the "momentary rest system" of the clock: If the movement of the clock between events A and B is assumed to be uniform (which is legitimate for very small dt), then results in (3) as a direct application of formula (2). The role played by the constant speed v in (2) is here taken over by the instantaneous speed v (t) measured in system I.**

**The times occurring in formula (3) can be added up by integration: The clock runs between the times t - measured in system I. _{0} and t_{1} an arbitrary accelerated movement (see the figure above), then the time interval t does not pass for it_{1}- t_{0}, but the (smaller) Proper time-Interval**

| (4) |

**This effect could be confirmed experimentally simply by reading the accelerated watch! This statement already contains the effect of the twin paradox. For the special case of a uniformly moving clock (v (t) = const) it is reduced to formula (2). Statements (3) and (4) also apply to movements that are not in a straight line: v (t) ^{2} then means the square of the speed vector (i.e. the square of the amount of the speed vector).**

**tasks** For this:

**Show that from (4) always T**_{Clock}£ t_{1}- t_{0}no matter how the clock moves!**Show that T**_{Clock}= t_{1}- t_{0}exactly when the clock in System I is idle!

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