Time dilation

The time dilation (from Latin .: dilatare, to stretch 'defer') is an effect that by the theory of relativity is described. Time dilation causes all internal processes of a physical system to run more slowly relative to the observer when this system moves relative to the observer. This means that clocks that move relative to the observer also run more slowly than clocks that are stationary relative to the observer. The greater the relative speed , the stronger the effect . The yardstick is the speed of light . The effect of time dilation is that the simultaneity of spatially separated events is a question of the relative speed of the observer, the so-called relativity of simultaneity . The idea of ​​an absolute time had to be given up in view of the time dilation. For systems moving uniformly relative to one another , the special theory of relativity ( Albert Einstein , 1905) explains time dilation and the related length contraction using Minkowski diagrams as geometric effects of four-dimensional space - time . An older, outdated interpretation of these effects was Lorentz's theory of ethers , see also the history of special relativity .

The gravitational time dilation is an effect of general relativity . It describes the effect that a clock, like any other process, runs more slowly in a stronger gravitational field than in a weaker one. Time passes in distant, approximately gravitation-free space (without taking into account the gravitational fields of other celestial bodies) by about a factor of 1 + 7 · 10 ⁻¹⁰  = 1.0000000007 faster than on the earth's surface. More precisely, every observer who is at rest in relation to the gravitational field measures a longer or shorter duration of processes that were triggered in an identical manner in or outside the gravitational field (such as an oscillation of the electric field strength vector of a light beam, which can be used as a time base ). In contrast to time dilation through movement, gravitational time dilation is not contradictory, but coincident: just as the observer located further up in the gravitational field sees the time of the observer located further down, the lower observer sees the time of the observer above correspondingly pass faster.

Time dilation through relative movement

At constant speed

Clear insight without formulas

One of the most important basic assumptions of the theory of relativity is the invariance of the speed of light. This means that all observers measure the same value for the speed of light, no matter how fast it is moving or how fast the light source is moving. Imagine a train moving very fast. For the passengers in the train, the light beams move vertically downwards from the ceiling lighting to the floor. Let's call the starting point A and the destination point B. If we assume that the distance from A to B is 3 m, then the light needs the unimaginably short time of 0.01 µs for this.

Viewed from the stationary embankment, the situation is somewhat different: While the light is moving from A to B, the train travels a little further, say 1 m, so that the path from A to B is no longer exactly vertical, but rather easy tilted forward. This also makes it a little longer, namely 3.15 m. Since the same value for the speed of light applies to all observers, the observer at the embankment calculates a somewhat longer time for this process (0.0105 µs). Because half a nanosecond less has passed on the train than on the embankment, the resting observer concludes that the time on the moving train is running slower.

The general fact that moving clocks go slower from the point of view of a stationary observer is called time dilation.

Explanation

To understand time dilation, it is necessary to be aware of the basic measurement rules and methods for time measurement with stationary and moving clocks.

If two events occur one after the other at the same place in an inertial system, the proper time (time span between the first and second event) can be determined by directly reading the pointer positions of a clock C resting at this place . The proper time displayed by C is invariant , so in all inertial systems it is agreed that C displayed this time span during the process. If the proper time of C is compared with the clocks of relatively moving inertial systems, the following procedure can be used: An observer in the inertial system S sets up two clocks A and B, which are synchronized with light signals . Clock C rests in S ′ and moves from A to B at speed  v , whereby it should be synchronous with A and B at the start time. When it arrives, the "moving" clock C (for which the proper time has passed) slows down compared to the "resting" clock B (for which has passed), using the following formula for time dilation (see derivation ): ${\displaystyle T_{0}}$ ${\displaystyle T_{0}^{'}}$ ${\displaystyle T}$

(1) ${\displaystyle T_{0}^{'}=T/\gamma ,}$

thus clocks A and B run faster

Lorentz factor with in units of ${\displaystyle \gamma (v)}$ ${\displaystyle v}$ ${\displaystyle c}$

(2) ${\displaystyle T=T_{0}^{'}\gamma ,}$

in which

${\displaystyle \gamma ={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}}$

is the Lorentz factor with the speed of light . ${\displaystyle c}$

Now the principle of relativity says that in S ′ the clock C can be regarded as stationary and consequently clocks A and B must go slower than C. At first glance, however, this contradicts the fact that clock C in both inertial systems meets B which follows from the invariance of the proper times of clocks C and B.

However, this can be explained if the relativity of simultaneity is taken into account. The above measurement was based on the assumption that clocks A and B (and thus also C at the start time) are synchronous, which is only the case in S due to the constancy of the speed of light in all inertial systems. In S ′, the synchronization of A and B fails - because the clocks are moving in the negative direction here and B approaches the time signal while A runs away from it. B is therefore detected by the signal first and begins to run earlier than A according to a value to be determined by the Lorentz transformation . If you take into account this procedure of clock B due to the early start (so if you subtract this amount of time from the total time of B), it also results here that the "moving" clock B (for which the proper time has passed) during the way to the " dormant "clock C (for which has passed) runs more slowly according to the following formula: ${\displaystyle x}$ ${\displaystyle T_{0}}$ ${\displaystyle T^{'}}$

(3) ${\displaystyle T_{0}=T'/\gamma ,}$

thus clock C goes faster

(4) ${\displaystyle T'=T_{0}\gamma .}$

The time dilation is therefore - as required by the principle of relativity - symmetrical in all inertial systems: Everyone measures that the other's clock is running slower than his own. This requirement is fulfilled, although in both inertial systems C follows B when they meet and the proper times of C and B are invariant.

Illustration of the scale

At not very high speeds, the time dilation has practically no effect. To illustrate, let's imagine a fictional and simplified space flight from the solar system to the nearest star Proxima Centauri . Here we do not take into account effects that result from the acceleration or deceleration of the spacecraft for the sake of simplicity. The distance is 4.24 light years . Depending on the travel speed, the following values ​​result:

Speed ​​as a percentage of the speed of light	Travel time in the rest system in years	Travel time in the on-board system in years	Ratio on-board system: rest system (rounded)
0.004	106000	105999.2	1: 1.000008
1	424	423.9	1: 1.00005
10	42.4	42.2	1: 1.005
50	8.48	7.34	1: 1.15
90	4.71	2.05	1: 2.29
99	4.28	0.60	1: 7.09
99.99	4.24	0.06	1: 70.7

The reference system in which the earth and Proxima Centauri rest is meant here by “rest system”. The “on-board system” means the spacecraft's own system. The speed used in the first line (0.004% of the speed of light) is roughly that achieved by the fastest manned spacecraft to date ( Apollo capsule ).

Time dilation and length contraction

It can be seen that the time dilation from opposite (measured with stationary, synchronized clocks) ${\displaystyle T_{0}}$ ${\displaystyle T}$

${\displaystyle T=T_{0}\gamma }$

The reciprocal of the contracted length of moving objects (measured by simultaneous determination of the end points using stationary scales) with regard to their rest length : ${\displaystyle L}$ ${\displaystyle L_{0}}$

${\displaystyle L=L_{0}/\gamma }$

This means that the proper time displayed by moving clocks is always shorter than the time span displayed by stationary clocks, whereas the proper length measured by moving scales is always greater than the length of the same object measured by resting scales.

The opposite occurs when the clock and the scale are not in the same inertial system. If the clock moves within the time span along a scale in S (measured from clocks resting there), then its rest length is simply given, whereas the dilated clock shows a lower proper time. Since its proper time is invariant, it will also display this time span in its own rest system, from which it follows that the rod moved in S ′ has the length . The rod is thus shorter by a factor , which corresponds to the length contraction of the moving rod. ${\displaystyle T}$ ${\displaystyle L_{0}=Tv}$ ${\displaystyle T'_{0}=T/\gamma }$ ${\displaystyle L'=T'_{0}v={\frac {T}{\gamma }}v}$ ${\displaystyle \gamma }$

Light clock

Light clocks, left A at rest, right B moving at 25% of the speed of light

The concept of the light clock can be used for a simple explanation of this factor. A light clock consists of two mirrors at a distance that reflect a short flash of light back and forth. Such a light clock was already discussed in the 19th century in the theory of light transit times in the Michelson-Morley experiment , and was first used in 1909 by Gilbert Newton Lewis and Richard C. Tolman as a thought experiment to derive the time dilation . ${\displaystyle d}$

If there is a light clock A, from the point of view of an observer who moves with it (i.e. who is at rest relative to it), a flash of lightning will need the time for the easy path between the mirrors . Each time the light flash hits one of the two mirrors, the light clock is incremented by a time unit that corresponds to the total duration of the light flash . ${\displaystyle T_{0}=d/c}$ ${\displaystyle 2T_{0}}$

If a second light clock B is now moved perpendicular to the line connecting the mirrors with the speed , the light must cover a greater distance between the mirrors from the point of view of the A-observer than with clock A. Assuming that the speed of light is constant, the A- Observer clock B is therefore slower than clock A. The time that the flash of light needs for the simple path between the mirrors results from the Pythagorean theorem ${\displaystyle v}$ ${\displaystyle T'=d'/c}$ ${\displaystyle d'}$

${\displaystyle d'^{2}=d^{2}+(vT')^{2}.}$

By inserting the expressions for and and solving for one finally gets ${\displaystyle d}$ ${\displaystyle d'}$ ${\displaystyle T'}$

${\displaystyle T^{'}={T_{0}}{\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}}$

or with the Lorentz factor

(4) ${\displaystyle T'=T_{0}\gamma }$

and thus

(3) ${\displaystyle T_{0}=T'/\gamma .}$

In contrast, an observer who moves with clock B can also claim to be at rest according to the principle of relativity . This means that his watch B, which is located with him, will show a simple running time of for the light flash. In contrast, the flash of light from the watch A, which is moving from the observer's point of view, will cover a greater distance for him and requires the time ${\displaystyle 2T_{0}^{'}}$

(2) ${\displaystyle T=T'_{0}\gamma }$

and thus applies

(1) ${\displaystyle T'_{0}=T/\gamma .}$

Proper time

Symmetrical Minkowski diagrams , where two clocks fly in opposite directions at the same speed, giving both world lines the same scale.
Above: proper time and time dilation
Below: twin paradox

The relativistic line element is given by ${\displaystyle \mathrm {d} s}$

${\displaystyle \mathrm {d} s^{2}=c^{2}\mathrm {d} t^{2}-\mathrm {d} x^{2}-\mathrm {d} y^{2}-\mathrm {d} z^{2}.}$

The quotient of this relativistic line element or distance and the speed of light is considered the proper time element ${\displaystyle \mathrm {d} s}$ ${\displaystyle c}$

${\displaystyle \mathrm {d} \tau ={\frac {\mathrm {d} s}{c}}.}$

Then follows by inserting and lifting out ${\displaystyle \mathrm {d} t^{2}}$

${\displaystyle \mathrm {d} \tau ={\sqrt {\frac {\mathrm {d} s^{2}}{c^{2}}}}={\frac {\mathrm {d} t}{c}}{\sqrt {c^{2}-\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}-\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}-\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}}.}$

On the one hand there is the relativistic line element and the proper time element ${\displaystyle \mathrm {d} s}$ ${\displaystyle \mathrm {d} \tau }$

${\displaystyle c={\frac {\mathrm {d} s}{\mathrm {d} \tau }},}$

on the other hand, a speed is generally defined as the derivative of the position vector with respect to time : ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {r}}=(x,y,z)}$ ${\displaystyle t}$

${\displaystyle {\vec {v}}={\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}}$

With the square of the speed

${\displaystyle v^{2}=\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}$

finally follows for the element of proper time

${\displaystyle \mathrm {d} \tau =\mathrm {d} t{\sqrt {\frac {c^{2}-v^{2}(t)}{c^{2}}}}=\mathrm {d} t{\sqrt {1-{\frac {v^{2}(t)}{c^{2}}}}}.}$

The differential is therefore only ever times as large as ${\displaystyle \mathrm {d} \tau }$ ${\displaystyle {\sqrt {1-(v^{2}/c^{2})}}}$ ${\displaystyle \mathrm {d} t.}$

For a system moved with the considered particle, the identity of both differentials results because in this system zero is identical. Similar important identities, such as the famous relation E = mc ² from the energy-momentum four-vector , can easily be found in the moving system for other invariants of the Lorentz transformations. ${\displaystyle v^{2}(t)}$

The proper time is obtained if the following is integrated via the proper time element:

${\displaystyle \tau =\int _{0}^{t}{\sqrt {1-{\frac {v^{2}(t')}{c^{2}}}}}\,\mathrm {d} t'}$.

In terms of measurement technology, the proper time corresponds to the above expression . If a clock C records the duration between the events U and W at the respective event point itself, i.e. along the world line of C, the time interval indicated by C is called the proper time between these events (see first Minkowski diagram on the right). Just like the underlying line element , the proper time is also an invariant , because in all inertial systems it is consistently determined that clock C shows exactly this time span between U and W. The invariant proper time is the reference quantity when the time dilation occurs. As already explained above, the rate of clock C is measured slowed down from the point of view of all other moving systems in relation to its own clocks. As a result, clock C will show a shorter time span between the two observation events U and W , whereas the synchronized S clocks show a longer time span according to ${\displaystyle \tau }$ ${\displaystyle T_{0}}$ ${\displaystyle \mathrm {d} s^{2}}$ ${\displaystyle T_{0}}$ ${\displaystyle T'_{0}=UW}$ ${\displaystyle T=Ux}$

${\displaystyle T=T'_{0}\gamma .}$

If, on the other hand, a clock B rests in S and two events U and V take place on its world line, then the time span is identical to the invariant proper time between these events, consequently a longer time span is measured in system S ′ : ${\displaystyle T_{0}=UV}$ ${\displaystyle T'=Ux'}$

${\displaystyle T'=T_{0}\gamma }$

The proper time of a non-accelerated clock located on site during two events is therefore minimal in comparison to the synchronized coordinate time between the same events in all other inertial systems. Because if none of the clocks is accelerated, there is always only one clock and thus only one straight world line, which shows the proper time between two specific events. It is possible that a single event U is on two straight world lines at the same time (namely where the world lines of C and B intersect), but it is geometrically impossible that the second event W on the world line of C also occurs of B's ​​world line, just as it is impossible that the second event V on B's world line is also on C's world line.

However, if one of the clocks is sped up, the world lines can intersect again. Here it results that the straight world line of the unaccelerated clock shows a greater proper time than the compound-curved world line of the accelerated clock, which is the explanation of the twin paradox. So while, as shown above, the proper time between two events on the world line of an unaccelerated clock is minimal compared to the synchronized coordinate times in all other inertial systems, it is maximum compared to the proper times of accelerated clocks, which are also on site for both events were.

Use of proper time with two inertial systems

Symmetrical Minkowski diagram of proper times.

In the inertial system S, A and B are synchronous. The "moving" clock C ticks more slowly and slows down when it arrives at B. In the Minkowski diagram: A = 3 = dg; B = 3 = ef; C = 2 = df.

In the inertial system S ′, A and B are not synchronous due to the relativity of simultaneity, with B taking precedence over A. Although the “moving” clocks A and B tick more slowly here, B's time advantage is sufficient for C to lag behind B at the meeting. In the Minkowski diagram: A = 1.3 = dh; B = 1.7 = ej; B = 3 = ef; C = 2 = df.

Some proper times are shown opposite in a symmetrical Minkowski diagram and other pictures. Clock C (at rest in S ′) meets clock A at d and clock B at f (both at rest in S). The invariant proper time of C between these events is df. The world line of clock A is the ct axis, the world line of clock B drawn through d is parallel to the ct axis, and the world line of clock C is the ct ′ axis. All events that are simultaneous to d are in S on the x-axis, and in S ′ on the x ′ axis. The respective time periods can be determined directly by counting the markings.

In S the proper time df of C is dilated compared to the longer time ef = dg of clocks B and A. Conversely, the invariant proper time of B is also measured dilated in S ′. Because time ef is shorter in relation to time if, because the start event e of clock B was already measured at time i , before clock C had even started to tick. At time d , B has passed time ej , and here too the time dilation results when df in S 'is compared with the remaining time jf in S.

From these geometrical relationships it becomes clear again that the invariant proper time between two specific events (in this case d and f ) on the world line of an unaccelerated clock is shorter than the time measured with synchronized clocks between the same events in all other inertial systems. As shown, this does not contradict the mutual time dilation, because due to the relativity of simultaneity, the starting times of the clocks are measured differently in other inertial systems.

Time dilation through pure acceleration

The current time dilation, possibly also time lapse effect, of the straight acceleration results from the desynchronization of the clocks:

With every change in the relative speed, which is shown here and does not necessarily have to be accompanied by noticeable inertia effects, the relative desynchronization of the clocks of the observed system also changes.

From the subjective point of view of the observer, the clocks in the observed system are desynchronized according to their local spatial distance as a result of the relativity of simultaneity: ${\displaystyle l'=x_{2}'-x_{1}'}$

${\displaystyle \tau _{\Delta }=-\Delta l'v/c^{2}}$

Here, τ _{Δ is} the rate deviation that is read between two local clocks at their own distance . In the same way, the change in desynchronization can also be represented by a changed speed of the observer, whereby the current distance from the observer must be selected: ${\displaystyle l}$

${\displaystyle \tau _{\Delta }=-x'\Delta v/c^{2}}$

The cause of the change in the relative speed of the observer is irrelevant here. The effect is geometrically determined and purely relativistic. As can be deduced from Bell's spaceship paradox , the effect (based on the complete distance x) is, however, asymmetrical for both parties. This in turn provides the explanation for the asymmetry of the twin paradox . This speed change can now be converted into an acceleration when viewed continuously, but this is not a local acceleration a = F / m , but rather the effective change in the relative speed a _eff = d v / d t :

${\displaystyle {\frac {\tau _{\Delta }}{\mathrm {d} t}}={\frac {a_{\mathrm {eff} }x}{c^{2}\gamma }}={\frac {a\cdot x}{\gamma ^{3}c^{2}}}}$

with distance x, Lorentz factor γ, infinitesimal time interval d t, effective acceleration a _eff , relative speed v and speed of light c. The effect is distance and direction dependent. In theory, the value of △ v · x / .DELTA.t or a _eff · x be arbitrarily high short notice and depending on the sign a · x> 0 (maximum, to contribute to a lapse v = c is achieved simultaneity) and a · x < 0 lead to slow motion (maximum time standstill at v = c ) as with the usual dilation. The calculation of a _eff should not be explained in detail here. Since the relativistic addition of speed is to be used here, and because (v · dv / c²) → 0, the following is calculated:

${\displaystyle a_{\mathrm {eff} }={\dfrac {{\dfrac {v+\mathrm {d} v}{1+{\dfrac {v\ \mathrm {d} v}{c^{2}}}}}-v}{\mathrm {d} t}}={\dfrac {\mathrm {d} v}{\left(\gamma ^{2}\left(1+{\dfrac {v\ \mathrm {d} v}{c^{2}}}\right)\right)\mathrm {d} t}}={\frac {a}{\gamma ^{2}}}}$

Contrary to popular belief, the acceleration a = v / t does not cause any other relativistic effects on time that would be comparable to gravity. This results from the fact that the factors of time dilation and Lorentz contraction in gravitation do not depend on the acceleration at all, but exclusively on the energy potential.

Movement with constant acceleration

→ Main article : Hyperbolic movement

If a test body of the mass is accelerated with a constant force to relativistic speeds (greater than one percent of the speed of light), a distinction must be made between the clock of a stationary observer and a clock on board the test body because of the time dilation. If the test body has the speed , it is useful to use the abbreviation ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle t=0}$ ${\displaystyle v_{0}}$

${\displaystyle \gamma _{0}:={\frac {1}{\sqrt {1-v_{0}^{2}/c^{2}}}}}$

to be able to clearly write down the following calculation results. If the test body is accelerated with a constant force , the following applies ${\displaystyle t=0}$ ${\displaystyle F}$

${\displaystyle v(t)={\frac {at+v_{0}\gamma _{0}}{\sqrt {1+{\dfrac {(at+v_{0}\gamma _{0})^{2}}{c^{2}}}}}},}$

where the constant acceleration is calculated according to . This formula can also be used to calculate the proper time that a clock would display in the accelerated system of the test body. For this, only the instantaneous speed has to be included in the integral given above ${\displaystyle a=F/M}$ ${\displaystyle v(t)}$

${\displaystyle \tau =\int _{0}^{t}{\sqrt {1-{\frac {v(t)^{2}}{c^{2}}}}}\mathrm {d} t}$

can be used. The result of this integration is

${\displaystyle \tau ={\frac {c}{a}}\ln \left({\frac {{\sqrt {c^{2}+(at+v_{0}\gamma _{0})^{2}}}+at+v_{0}\gamma _{0}}{(c+v_{0})\gamma _{0}}}\right).}$

The distance covered in the system of the stationary observer is obtained by integrating the speed over time ${\displaystyle x(t)}$ ${\displaystyle v(t)}$

${\displaystyle x(t)={\frac {c^{2}}{a}}\left({\sqrt {1+{\frac {(at+v_{0}\gamma _{0})^{2}}{c^{2}}}}}-\gamma _{0}\right).}$

If the time is replaced by the proper time when the start speed ( ) disappears , the following applies: ${\displaystyle v_{0}=0}$ ${\displaystyle t}$ ${\displaystyle \tau }$

${\displaystyle x={\frac {c^{2}}{a}}\left(\cosh \left({\frac {a\tau }{c}}\right)-1\right)}$

Journey to distant stars

See also : twin paradox

Another example is the movement of a spaceship that takes off from Earth, heads for a distant planet and comes back again. A spaceship takes off from Earth and flies with the initial acceleration of to a star 28  light years away. The acceleration of was chosen because it enables earthly gravity conditions on board a spaceship to be simulated. Halfway through, the spaceship changes the sign of the acceleration and decelerates just as much. After a six-month stay, the spaceship will return to Earth in the same way. The past times are 13 years, 9 months and 16 days for the traveler (measurement with the watch on board). On the other hand, when the spaceship returned, 60 years, 3 months and 5 hours had passed. ${\displaystyle g=9{,}81\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}$ ${\displaystyle 1\ g}$

Much greater differences are found on a trip to the Andromeda Galaxy , which is about 2 million light years away (with the same acceleration and deceleration phases). It is about 4 million years for the earth, while only about 56 years have passed for the traveler.

The spaceship never exceeds the speed of light. The longer it accelerates, the closer it gets to the speed of light, but it will never reach it. From the perspective of the earth, the acceleration decreases despite the constant engine power. In the spaceship, time runs more slowly according to the time dilation. Since both observers and measuring instruments are subject to time dilation in the spaceship, their own time runs quite normally from their point of view, but the path between earth and travel destination is shortened due to the Lorentz contraction. (From the earth's point of view, it remains constant in this example for the sake of simplicity). If you are now in the spaceship and determine your speed relative to the earth taking into account the Lorentz contraction, then you get the same result as when you determine the speed of the spaceship from the earth. In practice, however, it is currently not possible to implement a drive that can achieve such a high acceleration over such a long period of time.

Time dilation by gravity

Gravitational time dilation describes the relative timing of systems that are at rest at different distances from a center of gravity (for example a star or planet) relative to it. It should be noted that the gravitational time dilation is not caused by a mechanical effect on the clocks, but is a property of spacetime itself. Every observer who is at rest relative to the center of gravity measures different process times for identical processes that take place at different distances from the center of gravity, based on his own time base. One effect that is based on gravitational time dilation is gravitational redshift .

The general and metric-dependent formula for the time dilation between two stationary observers ( FIDO ), one of which is outside and the other inside of the gravitational field, is

${\displaystyle {\frac {{\rm {d}}t}{{\rm {d}}\tau }}={\sqrt {g^{tt}}}}$.

In the Schwarzschild metric is

${\displaystyle g^{tt}={\frac {1}{1-r_{\rm {s}}/r}}}$and . ${\displaystyle g_{tt}={\frac {1}{g^{tt}}}}$

In order to obtain the total time dilation of a stationary observer at a great distance from the mass relative to an observer moving in the gravitational field, it is multiplied by the Lorentz factor; in the reference system of the stationary observer it results that the clock of the moving by the factor

${\displaystyle {\frac {{\rm {d}}t_{\infty }}{{\rm {d}}\tau }}={\sqrt {g^{tt}}}\ \gamma }$ so ${\displaystyle \tau ={t_{\infty }}{\frac {1}{{\sqrt {g^{tt}}}\ \gamma }}}$

slows down while the clock of the stationary observer in the system of moving by the factor

${\displaystyle {\frac {{\rm {d}}t}{{\rm {d}}\tau _{\infty }}}={\sqrt {g_{tt}}}\ \gamma }$ so ${\displaystyle \tau _{\infty }={t}{\frac {\sqrt {g^{tt}}}{\gamma }}}$

ticks faster or slower, depending on whether the gravitational or the kinematic component predominates (the gravitational component causes the clock in the gravitational field to tick absolutely more slowly, while the kinematic component leads to a reciprocal, i.e. relative slowing down of the other clock). In the free fall from infinity v = v _e , both effects cancel each other out exactly from the FFO point of view :

${\displaystyle {\sqrt {g^{tt}}}=\gamma _{e}}$

Acceleration and gravitation: the rotating disk

This problem is also known as the honor festival paradox .

According to the equivalence principle of the general theory of relativity, one cannot locally differentiate between a system at rest in a gravitational field and an accelerated system. Therefore the effect of the gravitational time dilation can be explained using the time dilation caused by movement.

If one looks at a disk rotating with constant angular velocity , a point at a distance from the center moves with the velocity ${\displaystyle \omega }$ ${\displaystyle r}$

${\displaystyle v={\frac {r\omega }{\sqrt {1+{\dfrac {(r\omega )^{2}}{c^{2}}}}}}.}$

Accordingly, the proper time becomes at the distance from the center of the disk ${\displaystyle r}$

${\displaystyle \tau ={\frac {t}{\sqrt {1+{\dfrac {(r\omega )^{2}}{c^{2}}}}}}}$

occur. For sufficiently small distances ( ), this expression is approximately the same ${\displaystyle v^{2}\ll c^{2}}$

${\displaystyle \tau =t\left(1-{\frac {(r\omega )^{2}}{2c^{2}}}\right).}$

A rotating object on the disk now experiences the centrifugal force . Due to the equivalence principle, this force can also be interpreted as a gravitational force to which a gravitational potential ${\displaystyle F=m\omega ^{2}r}$

${\displaystyle \varphi =-(r\omega )^{2}}$

heard. But this is precisely the term that occurs in the numerator during time dilation. This results in "small" distances:

${\displaystyle \tau =t{\sqrt {1+{\frac {2\varphi }{c^{2}}}}}}$

(Note: The potential given here does not correspond to the usual centrifugal potential, since here an adjustment is made to the local rotational speed of the disk, whereas with the usual centrifugal potential, conservation of angular momentum applies instead.)

Time dilation in the earth's gravitational field

Time dilation through gravity and orbital velocity

In a weak gravitational field like that of the earth, the gravitation and thus the time dilation can be approximately described by the Newtonian gravitational potential:

${\displaystyle \tau =t_{0}{\sqrt {1+{\frac {2\phi }{c^{2}}}}}}$

Here the time is at potential and Newton's gravitational potential (multiplication with the mass of a body results in its potential energy at a certain location). ${\displaystyle t_{0}}$ ${\displaystyle \phi =0}$ ${\displaystyle \phi }$

On earth (as long as the height is small compared to the earth's radius of approx. 6400 kilometers) the gravitational potential can be approximated by. At a height of 300 kilometers (this is a possible height at which the space shuttles flew), every “second of the earth” passes , which is about one millisecond per year more. This means that an astronaut who would rest 300 kilometers above the earth would age about a millisecond faster each year than someone who was resting on the earth. For a shuttle astronaut at such an altitude, however, the exact value was different because the shuttle was also moving (it circled the earth), which led to an additional effect in the time dilation. ${\displaystyle \phi =gh}$ ${\displaystyle 1+3{,}27\cdot 10^{-11}\,\mathrm {s} }$

If one compares the reduction in the gravitational time dilation caused by the altitude relative to the earth's surface and the time dilation caused by the orbital velocity required for this altitude, it becomes apparent that with an orbit radius of 1.5 times the earth's radius, i.e. at an altitude of half the radius of the earth, the two effects cancel out exactly and therefore the time on such a circular path passes just as fast as on the earth's surface (if one assumes, for the sake of simplicity, that the earth itself does not rotate, it is exactly 1.5 times the radius, taken into account one also the earth's rotation, it is slightly less).

Gravitational time dilation also means that the core of a celestial body is younger than its surface. For the earth, this time difference between the center of the earth and the earth's surface, taking into account the density distribution of the earth in 2016, was given as a classical approximation of 2.49 years.

Experimental evidence

See also : Tests of the special theory of relativity

Relativistic Doppler effect

→ Main article : Ives Stilwell experiment

The first direct proof of time dilation by measuring the relativistic Doppler effect was achieved with the Ives-Stilwell experiment (1939); Further evidence was provided with the Mössbauer rotor experiments (1960s) and modern Ives-Stilwell variants based on saturation spectroscopy , the latter having reduced the possible deviation of the time dilation to . Indirect evidence are variations of the Kennedy-Thorndike experiment , in which the time dilation must be taken into account together with the length contraction . For experiments in which time dilation is observed for the round trip, see the twin paradox . ${\displaystyle 8{,}4\cdot 10^{-8}}$

Lifetime measurement of particles

→ Main article : Time dilation of moving particles

When cosmic rays hit the molecules in the upper layers of the air, muons are formed at heights of 9 to 12 kilometers . They are a main component of secondary cosmic radiation, move towards the earth's surface at almost the speed of light and can only be detected there because of the relativistic time dilation, because without this relativistic effect their average range would only be around 600 m. In addition, tests of the decay times in particle accelerators with pions , muons or kaons were carried out, which also confirmed the time dilation.

Time dilation by gravity

See also : Measurement of the gravitational redshift

The gravitational time dilation was demonstrated in 1960 in the Pound-Rebka experiment by Robert Pound and Glen Rebka . In addition, NASA launched a Scout-D rocket in 1976 with an atomic clock , the frequency of which was compared with a clock of the same type on Earth. This was the most precise experiment to date that was able to successfully measure gravitational redshift.

Comparison between clocks on the plane and on the ground

→ Main article : Hafele-Keating experiment

A clock in a high-flying airplane is subject to two forms of time dilation compared to a clock standing on the ground. On the one hand, the influence of the earth's gravity decreases with altitude. This slows down the clock on the plane less than the clock on the ground. On the other hand, the aircraft moves relative to the clock on the ground. This causes the clock on the plane to slow down. The two effects work in opposite directions. Which of the two effects predominates depends on the altitude difference and the speed of the aircraft.

The first clock comparison between clocks transported in an airplane and those of identical construction that remained on the ground was made in 1971 as part of the Hafele-Keating experiment . For this experiment, the physicist Joseph Hafele and the astronomer Richard Keating flew four atomic clocks east and west in one Airliner around the world. Before and after the flights, the status of the clocks was compared with that of structurally identical atomic clocks operated in the United States Naval Observatory . The resulting shifts confirmed the predictions of the theories of relativity. Since then, measurements in a similar form have been carried out repeatedly with even greater accuracy.

Practical meaning

Time dilation is of practical importance in satellite-based navigation systems such as the American GPS . These are based on the fact that each satellite in the system transmits a very precise radio-controlled time signal determined by an atomic clock . The users' GPS devices receive these signals from several satellites and determine their distance to the various satellites and from this their exact position based on the transit times of the signals. Since the satellites are subject to time dilation due to both gravity and their movement, navigation systems must correct for these effects to improve accuracy.

Web links

Wiktionary: Time dilation  - explanations of meanings, word origins, synonyms, translations

literature

• Albert Einstein: On the electrodynamics of moving bodies. In: Annals of Physics and Chemistry. 17, 1905, pp. 891–921 (as a facsimile (PDF; 2.0 MB); as full text at Wikilivres; and commented and explained at Wikibooks ).
• Thomas Cremer: Problems of interpretation of the special theory of relativity. A historical-didactic analysis (= Physics series. 2). 2nd, revised edition. Harri Deutsch, Thun et al. 1990, ISBN 3-8171-1105-3 (at the same time: Gießen, University, dissertation, 1988).
• Walter Greiner , Johann Rafelski : Special Theory of Relativity (= Theoretical Physics. Vol. 3A). 2nd, revised and expanded edition. Harri Deutsch, Thun et al. 1989, ISBN 3-8171-1063-4 .
• Harald Fritzsch : A formula changes the world. Newton, Einstein and the theory of relativity (= series Piper. 1325). 3rd edition, new edition. Piper, Munich et al. 1990, ISBN 3-492-11325-7 .
• Roland Pabisch: Derivation of the time dilatation effect from fundamental properties of photons. Springer, Vienna et al. 1999, ISBN 3-211-83153-3 .

Individual evidence

1. ↑ ^{a b} Max Born: Einstein's theory of relativity. 7th edition. Springer Verlag, 2003, ISBN 3-540-00470-X .
2. ^ ^{A b} Roman Sexl, Herbert K. Schmidt: Space-Time-Relativity . Vieweg, Braunschweig 1979, ISBN 3-528-17236-3 , pp. 31-35 .
3. ^ ^{A b c d e} Edwin F. Taylor, John Archibald Wheeler : Spacetime Physics: Introduction to Special Relativity . WH Freeman, New York 1992, ISBN 0-7167-2327-1 .
4. ↑ David Halliday, Robert Resnick, Jearl Walker: Fundamentals of Physics, Chapters 33-37 . John Wiley & Son, 2010, ISBN 0-470-54794-4 , pp. 1032 f.
5. ^ Franz Embacher: Lorentz contraction . Retrieved January 1, 2013.
6. ^ Gilbert N. Lewis, Richard C. Tolman: The Principle of Relativity, and Non-Newtonian Mechanics . In: Proceedings of the American Academy of Arts and Sciences . tape 44 , 1909, pp. 709-726 (in the English language Wikisource ).
7. ↑ ^{a b c} Jürgen Freund: Special Theory of Relativity for New Students . vdf Hochschulverlag AG, 2007, ISBN 3-8252-2884-3 , p. 12.
8. ↑ Eckhard Rebhan: Theoretical Physics I . Spektrum, Heidelberg / Berlin 1999, ISBN 3-8274-0246-8 , pp. 782-783.
9. ↑ Torsten Fließbach : Mechanics. 4th edition, Elsevier - Spektrum Akademischer Verlag, 2003, p. 322 f., ISBN 3-8274-1433-4 .
10. ↑ Rolf Sauermost u. a .: Lexicon of natural scientists. Spectrum Academic Publishing House , Heidelberg / Berlin / Oxford 1996, p. 360.
11. ↑ Ulrik I. Uggerhøj et al .: The young center of the earth In: Eur. J. Phys. 37, 035602 (2016).
12. ^ Clifford Will, The Confrontation between General Relativity and Experiment. 2006.
13. ^ J. Hafele, R. Keating: Around-the-World Atomic Clocks: Predicted Relativistic Time Gains . In: Science . 177, No. 4044, July 14, 1972, pp. 166-168. bibcode : 1972Sci ... 177..166H . doi : 10.1126 / science.177.4044.166 . PMID 17779917 . Retrieved September 18, 2006.
14. ^ J. Hafele, R. Keating: Around-the-World Atomic Clocks: Observed Relativistic Time Gains . In: Science . 177, No. 4044, July 14, 1972, pp. 168-170. bibcode : 1972Sci ... 177..168H . doi : 10.1126 / science.177.4044.168 . PMID 17779918 . Retrieved September 18, 2006.
15. ↑ http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html