# Does the time depend on the speed

Why ...... does time run slower at high speed? |

### An attempt to explain Einstein's time dilation in a simple way

### Overview

**Basics of the special theory of relativity****The Doppler Effect****An example of time dilation****Time dilation, considered in more detail**

You've probably heard of them **Vacuum speed of light c = 299 792 458 m / s** plays a special role in physics. In addition to visible light, radio waves, infrared, ultraviolet, X-rays and gamma rays also propagate in a vacuum at the speed of light. It is also known that no material object (apart from the spaceship Enterprise?) Can reach or even exceed the speed of light. When approaching the speed of light, unusual phenomena occur, which the famous physicist Albert Einstein predicted as early as 1905 in his special theory of relativity.

One conclusion from Einstein's theory is particularly startling:

**If you are moving almost at the speed of light, time will go a lot slower.**

In the following I would like to try to explain this fact as simply (but still correctly) using an example.

### 1. Basics of the special theory of relativity

Physics uses a time scale and a spatial one to describe a process **Coordinate system** (for example a Cartesian coordinate system with three mutually perpendicular axes). The time zero point can be set arbitrarily (for example on the birth of Christ). As far as the spatial coordinate system is concerned, the origin and the axis directions can be chosen as desired. It is also possible to have a **"moving" coordinate system** to describe the movement of trees (!) from the perspective of a car.

If coordinate systems or reference systems are mentioned in the following, they are always so-called **Inertial systems** In other words, these are reference systems in which force-free bodies remain at rest according to the law of inertia or move with constant speed and direction. A typical example of an inertial system is the reference system of a non-propelled spaceship in which the astronauts are in a state of weightlessness. (In a non-inertial system, on the other hand, there would be inertial forces, such as when a car suddenly brakes, or centrifugal forces, such as in the looping roller coaster.)

The special theory of relativity essentially deals with the following question:

**To what extent can physical quantities have different values in coordinate systems that are moving in relation to one another?**

For the speed, the dependency on the reference system is obvious: If the coordinate system used is linked to a parked car, for example, the trees on the roadside have a speed of 0 km / h. If you look at the same trees in the reference system of a pedestrian or a moving car, then these trees (!) Move, for example, at 5 km / h or 100 km / h. (Conceivable quote from Einstein: "When does Ulm stop at this train?")

On the other hand, it seems unthinkable to "common sense" that length, time or mass could depend on the coordinate system used. Albert Einstein was the first to have the revolutionary idea that these quantities could also change with the reference system. In his deliberations, Einstein started from two principles, the **Relativity Principle** and the principle of the **Constancy of the speed of light**:

Relativity Principle |

The laws of nature have the same form in two different reference systems that move uniformly against each other (that is, with constant speed and constant direction of movement); such frames of reference are therefore equal. |

This principle of relativity is already the basis of classical (Newtonian) mechanics. An example is the attempt to determine in a train compartment (without a window, with total sound insulation, perfect suspension, absolutely level track, etc.) through physical measurements whether the train is moving or how high its speed is. According to Newtonian mechanics, this is impossible. On the other hand, the train is accelerating or cornering, for example when a ball starts to move on the floor of the railroad car. If one uses two different reference systems in Newtonian mechanics, which move uniformly against each other, then one obtains different values for speed measurements, while those for acceleration measurements are the same.

It is important to understand the principle of relativity that the absence of external forces (e.g. gravitational forces) is assumed. In general, different reference systems on earth that move against each other cannot be regarded as having equal rights. If, on the other hand, one considers two rockets that fly past each other without propulsion at a great distance from the nearest celestial body, then the coordinate systems associated with these rockets are equal. Each of the two rocket crews can, for example, claim that their own spaceship is moving at 100 km / s while the other rocket stands still; Another observer could say with equal rights that the first rocket is moving at 20 km / s in one direction and the second at 80 km / s in the opposite direction.

**Conclusion: Movement is a relative (i.e. dependent on the reference system used) term. There is no such thing as an absolute space.**

The second fundamental principle for the special theory of relativity was predicted by James Clerk Maxwell in his Theory of Electrodynamics (1861 to 1864) and confirmed experimentally for the first time in 1881 by Albert Abraham Michelson:

This principle has surprising consequences. If an observer moves towards a sound source at half the speed of sound and measures the speed of the incoming sound in his reference system, he receives one and a half times the normal value. On the other hand, for an observer who races towards a light source at half the speed of light, the speed of the incoming light in his reference system only has the normal value.

### 2. The Doppler Effect

Everyone has already observed when an ambulance drives by that the sound of the siren suddenly becomes deeper the moment the car drives by. This phenomenon is called **Doppler effect** (after the Austrian physicist Christian Doppler) and is typical for the propagation of waves, for example sound or light waves. How does this effect come about?

The sound source sends out wave fronts at a certain time interval (for example 0.0020 s), which propagate in all directions at about 330 m / s. If the sound source moves towards the observer, the wavefronts transmitted later have to cover a shorter path to the observer and therefore need less time to get to the observer. As a result, the wavefronts arrive at the observer at a shorter time interval (for example 0.0018 s). We perceive this faster succession of the wave fronts as an increase in the tone.

If the sound source moves away from the observer, the wavefronts sent out later need more time to reach the observer due to the longer path. In this way, the time interval between the wave fronts is increased (for example to 0.0022 s). In this case we perceive a deeper tone. In the numerical example mentioned last, the is **Extension factor** for the time interval between the wavefronts k = 0.0022 s / 0.0020 s = 1.1.

A corresponding effect also occurs with light waves. If the light source approaches the observer, the wave fronts arrive at the observer at a shorter time interval (violet shift). In the opposite case, a redshift is observed. (A famous example is the redshift in the light of the galaxies moving away from our Milky Way. This redshift is considered evidence of the Big Bang theory.)

If the speed of the light source relative to the observer becomes similar to the speed of light, the Doppler effect can have an extreme effect. This should be explained in more detail using the following example:

example 1 |

In the year 2500 an unmanned space probe starts with the speed v = 0.8 c (that is with 0.8 times the speed of light) to the distant Andromeda Nebula. Every year the earth station sends a radio signal (at the speed of light) to the space probe. Every time such a signal arrives, the on-board computer registers the time (measured with the space probe's clock, of course) and sends a response signal to earth as confirmation. |

Seen from the spacecraft, the earth behaves like a light signal source moving away from the spacecraft. We already know that the arrival times of the signals recorded by the on-board computer must be more than a year apart. Now it should be calculated exactly how big this time interval is. So we want that **Extension factor** know k for the time interval between the signals.

The signals sent by the earth every year (earth time!) Arrive at the space probe every k years (space probe time!). The size k depends on the speed v. So far we only know that k has to be greater than 1.

We now consider the propagation of the response signals from the perspective of the earth. Seen in this way, the space probe is a light signal source that sends its signals every k years and moves away from the earth at a speed of v = 0.8 c. According to the principle of relativity, the reference systems of the earth and space probe are equal. The same principle must therefore apply to the response signals of the space probe as before for the original signals originating from the earth: They must arrive (according to earth time!) At a time interval k times as long as they were sent by the space probe (according to Space probe time!), I.e. at an interval of k · k years. However, this time interval can be read directly from the time-distance diagram (t-s diagram): It is exactly 9 years. With this we can give the size of k. From k · k = 9 it follows immediately that k = 3.

A corresponding consideration for an aircraft with 0.8 times the speed of sound, which receives sound signals and immediately sends them back, would be wrong: the speed of sound would have the normal value in the system of the ground station, while in the system of the aircraft only 0.2 times this speed. (There is a principle of the constancy of the speed of light, but no principle of the constancy of the speed of sound.) A different extension factor k would therefore apply to the response signals than to the original signals.

Result of example 1 |

If a signal source, which regularly emits light signals at a certain time interval, moves away from the observer at a speed of v = 0.8 c, the observer registers the signals at a time interval three times as long. Accordingly, one can consider that in the case of a signal source that comes closer to the observer with v = 0.8 c, the time interval between the individual signals is shortened to a third of the original value. |

A different extension factor k would of course have resulted for a different speed value v. With a little math, the following formula can be derived:

k ... extension factor for the time interval between the signals v ... speed of the signal source relative to the observer c ... vacuum speed of lightThe above result is also obtained from this formula for v = 0.8 c:

### 3. An example of time dilation

Example 2 |

On January 1st, 3000 some astronauts interested in literature set off from Earth to a planet 4 light-years away to enjoy a Vogon poetry reading there (for more information on Vogon poetry, see Douglas Adams). The cruising speed is v = 0.8 c, so that the return flight takes 5 years each (earth time, not spaceship time!). The ground station on earth sends a New Year's greeting on January 1st of the following years. |

Using the Doppler effect, to which these signals are also subject, we can find out how much time passes for the space travelers during the flight. We use a t-s diagram that shows the movement of the spaceship and the radio signals in the earth's reference system.

As you can see, the crew only received the congratulations from January 1, 3001 at the turning point.

First of all, the outward flight should be viewed from the perspective of the spaceship crew. The earth as a signal source moves away with v = 0.8 c, so that its congratulatory signals do not arrive every year, but because of the Doppler effect (as shown above) every three years. It is therefore clear that the spaceship clock shows January 1, 3003 at the turning point. From the point of view of the earth, however, the reversal will take place on January 1st, 3005 (see diagram)! This shows for the first time that the duration of a process can be different in different reference systems.

The return flight should also be viewed in the spaceship's frame of reference. The earth as a signal source is now approaching the spaceship with v = 0.8 c, so that its congratulatory signals do not arrive every year, but this time every third year. Now you just have to read the number of third-year intervals in the diagram: There are nine such time periods; accordingly (as expected) the return flight after spaceship time takes exactly as long as the outward flight, namely three years.

Result of example 2 |

If the spaceship flies with the speed v = 0.8 c to the planet 4 light years away and back again, only 6 years pass for the crew, while on earth the time advances by 10 years. |

The technical term for this "time expansion" is called **Time dilation**.

**Conclusion: Time is a relative term (i.e. it depends on the reference system used). There is no such thing as an absolute time.**

### 4. Time dilation, considered in more detail

We have come to the end of our deliberations, at least almost. However, one important detail remains to be clarified. In our last example, the moving clock in the spaceship goes slower than the clocks on earth. On the other hand, we know that movement is a relative concept. We can therefore also take the position that the spaceship stands still all the time and the earth moves back and forth. Is not that a contradiction?

In reality, the reference systems of the earth and the spaceship are not the same. On the one hand, the spaceship has to accelerate strongly when turning back. The associated coordinate system is therefore not an inertial system. In addition, a single clock in the spaceship is compared with several clocks that are stationary relative to the earth, namely clocks on earth and at the travel destination.

After this clarification, we are now in a position to formulate our findings precisely:

Time dilation |

A clock U that moves relative to an inertial system S moves more slowly than the synchronized clocks resting in system S. |

You can try out how much time stretching has an effect at different speeds with an HTML5 app on this subject.

Of course there is a formula for time dilation. It reads as follows:

t '... time indicated by the moving clock U t ... time indicated by the clocks of the reference system S v ... speed of the clock U relative to the reference system S c ... vacuum speed of lightThe application of this formula to the outward or return flight of the spaceship confirms the result of our considerations using the t-s diagram:

You can try out this formula with a small calculator (implemented as an HTML5 app).

Not only watches are subject to time dilation, but also, in general, all processes such as breathing, heartbeat, movement and aging. The astronauts do not feel anything of the time dilation on their space flight until their return, since all processes are affected in the same way.

Even if we justified the time dilation with two thought experiments, there may still be a little mistrust of the result. We are too used to the misconception of an absolutely valid time. In addition, it would be conceivable that something is wrong with the fundamentals of our considerations (principle of relativity and constancy of the speed of light). For this reason, attempts have been made to prove the time dilation experimentally.

- When the particles of cosmic rays hit the uppermost layers of the earth's atmosphere, muons are formed. These are elementary particles that are similar to electrons, but on average only have a very short lifespan of 2.2 microseconds (2.2 millionths of a second). Although these muons move almost at the speed of light, without time dilation they would come an average of just 660 m and would then decay. So you would have practically no chance of ever reaching the surface of the earth. In reality, however, muons hit the earth in huge numbers - as a result of time dilation!
- Time dilation could also be checked in a more direct way. In 1971 the American physicists Hafele and Keating carried out measurements with atomic clocks on board airliners. You were able to confirm the formula mentioned above - within the scope of the measurement accuracy.

In the distant future, time dilation may open up fantastic opportunities for interstellar spaceflight. Such a space flight would lead the astronauts into the future like a time machine (but not back!). Whether the astronauts will find their way around when they return to a planet on which centuries or millennia have passed since their departure is another question.

### Walter Fendt, May 10, 1997

Last change: January 11, 2021

URL: www.walter-fendt.de/zd/index.htmlThis page was created from a lesson held in 1995 for the then class 9b of the Paul-Klee-Gymnasium Gersthofen. Many thanks to Mr. Norbert Feist for his contributions to the correct representation of the Hafele-Keating experiment! The two simulations come from the "Apps for Physics", a collection of currently 54 programs on various topics in physics.

**swell**

- Müller, Leitner, Dilg, Mràz: Advanced physics course, 2nd semester (Ehrenwirth)
- Povh, Rith, Scholz, Zetsche: Particles and Cores (Springer)
- Hafele, Keating: Around-the-World Atomic Clocks: Observed Relativistic Time Gains (Science, Vol. 177)
- Hafele, Keating: Around-the-World Atomic Clocks: Predicted Relativistic Time Gains (Science, Vol. 177)
- Meyer's Great Universal Lexicon
- Adams, Douglas: The Hitchhiker's Guide to the Galaxy (Ullstein)

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