# How is the gravitational time dilation formula obtained

## Time dilation

In the Time dilation (from Latin: dilatars 'Expand', 'postpone') is a phenomenon of the theory of relativity. If an observer is in a state of uniform movement or if he is at rest in an inertial system, according to the special theory of relativity, every clock moved relative to him goes slower from his point of view. However, not only clocks are subject to this phenomenon, but the time in the moving system itself and thus any process. The time dilation is greater, the greater the relative speed of the clock. It can practically not be observed in everyday life, but only at speeds that are not negligibly small compared to the speed of light. The fact that each other's time passes more slowly for all observers does not represent a contradiction, as a closer look at the relativity of simultaneity shows (see also the special theory of relativity and Minkowski diagram).

Such an effect was first derived by Joseph Larmor (1897) and Hendrik Antoon Lorentz (1899) in the context of a now outdated theory of ethers. Albert Einstein (1905) succeeded in showing, however, within the framework of the special theory of relativity, that the changed clock rate is not related to an influence by an ether, but to a radical reinterpretation of the concepts of space and time (see also the history of the special theory of relativity).

In the gravitational time dilation it is a phenomenon of general relativity. Gravitational time dilation refers to the effect that a clock - like any other process - runs more slowly in a gravitational field than outside it. The time on the earth's surface passes by a factor of 7 · 10−10 slower than in distant, approximately gravitation-free space. 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 mutually exclusive: While 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 upper observer pass correspondingly faster.

### At constant speed

#### Explanation Time dilation: 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. Time dilation symmetry: 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 here as well.

To understand time dilation, it is necessary to be aware of the basic measuring rules and methods for measuring time with stationary and moving clocks. If the relative speed between the observer (or his clock) and the observed phenomenon is zero (i.e. everything rests in the same inertial system), then the "proper time" \$ ​​T_0 \ \$ can be determined simply by reading the pointer position of a clock near the phenomenon. However, if the relative speed is> 0, 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 (Fig. 1). Let there be a clock C at rest in S ’, which moves with the speedv moved from A to B and which (from S's point of view) should be synchronous with A and B at the start time. The time dilation now has the effect that the "moving" clock C (for which the time span \$ T_0 ^ {'} \ \$ has passed) when it arrives opposite the "resting" clock B (for which \$ T \ \$ has passed) according to the following Following formula:

(1) \$ T_0 ^ {'} = T \ cdot \ sqrt {1- \ frac {v ^ 2} {c ^ 2}}. \$

Now the principle of relativity says that in S ’the clock C can be regarded as stationary and consequently the clocks A and B must run more slowly than C. At first glance, however, this contradicts the observed fact that C follows up when it meets B. However, this can be explained if one takes into account the relativity of simultaneity. Because the above measurement was based on the assumption that clocks A and B 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 x-direction and B is approaching the time signal while A is running 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 (i.e. you subtract this amount of time from the total time displayed by B), the result is that the "moving" clock B (for which the time period \$ T_0 \ \$ has passed) during the way to the "resting" clock C (for which \$ T ^ {'} \ \$ has passed) runs more slowly according to the following formula:

(2) \$ T_0 = T ^ {'} \ cdot \ sqrt {1- \ frac {v ^ 2} {c ^ 2}}. \$

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. It can be seen that the time dilation of moving clocks is reciprocal to the Lorentz contraction of moving lengths. This means that the proper time displayed by moving clocks is always smaller than the time span displayed by non moving clocks for the same phenomenon, whereas the proper length measured by moving scales is always greater than the length measured by non-moving observers.

#### Light clock Light clock, resting on the left, moving at 25% of the speed of light on the right

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 \$ d \ \$ that reflect a short flash of light back and forth. This thought experiment was first discussed in 1909 by Gilbert Newton Lewis and Richard C. Tolman.

If there is a light clock A, from the point of view of an observer who moves with it, a flash of lightning will need the time \$ T_ {0} = d / c \ \$ for the simple 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 \$ 2T_ {0} \ \$.

Now a second light clock B becomes perpendicular to the line connecting the mirrors with the speed \$ v \ \$emotional, so from the point of view of the A-observer the light has to travel a greater distance between the mirrors than at clock A. Assuming the constancy of the speed of light, clock B runs slower than clock A for the A-observer. The time \$ T '= d '/ c \ \$, which the flash of light needs for the simple path \$ d' \ \$ between the mirrors, results from the Pythagorean theorem

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

By inserting the expressions for \$ d \ \$ and \$ d '\ \$ and solving for \$ T' \ \$ one finally obtains

\$ T ^ {'} = \ frac {T_0} {\ sqrt {1 - \ frac {v ^ 2} {c ^ 2}}} \$

and thus

(2) \$ T_0 = T ^ {'} \ cdot \ sqrt {1- \ frac {v ^ 2} {c ^ 2}}. \$

On the other hand, an observer who moves with clock B can also claim to be at rest according to the principle of relativity. This means that the clock B he is in will show a simple running time of \$ 2T_0 ^ {'} \ \$ for the flash of light. On the other hand, the flash of light becomes the one from his point of view moved Clock A has to cover a longer distance for him and needs the following time:

\$ T = \ frac {T_0 ^ {'}} {\ sqrt {1 - \ frac {v ^ 2} {c ^ 2}}} \$

and thus

(1) \$ T_0 ^ {'} = T \ cdot \ sqrt {1- \ frac {v ^ 2} {c ^ 2}}. \$.

#### Journey to distant stars

Another example would be the movement of a spaceship that takes off from Earth, heads for a distant planet, and comes back again. A spaceship starts from the earth and flies with the constant acceleration of \$ g = 9 {,} 81 \, \ frac {\ mathrm {m}} {\ mathrm {s} ^ 2} \$ to a star 28 light years away. The acceleration of \$ 1g \$ 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 with \$ 1g \$. After completing a six-month stay, the spaceship will return to Earth in the same way. The past times result for the traveler at 13 years, 9 months and 16 days (measurement with the watch on board). On the other hand, when the spaceship returned, 60 years, 3 months and 5 hours had passed.

Much greater differences are found in a flight to the Andromeda Nebula, 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 point of view of the earth, time on the spaceship 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 one is now in the spaceship and determines its speed relative to the earth taking into account the Lorentz contraction, then one arrives at the same result as when one determines the speed of the spaceship from the earth. The big problem with this example is that it is currently not possible to implement a drive that can achieve such high acceleration over such a long period of time.



### General time dilation

The relativistic line element ds is defined by

\$ \ mathrm ds ^ 2 = c ^ 2 \ mathrm dt ^ 2- \ mathrm dx ^ 2- \ mathrm dy ^ 2- \ mathrm dz ^ 2. \$

The quotient of this relativistic line element or distance \$ \ mathrm d s \$ and the speed of light \$ c \$ is valid as the proper time element

\$ \ mathrm d \ tau = \ frac {\ mathrm ds} {c}. \$

By inserting and lifting out \$ \ mathrm dt ^ 2 \$ then follows

\$ \ mathrm d \ tau = \ sqrt \ frac {\ mathrm ds ^ 2} {c ^ 2} = \ frac {\ mathrm dt} {c} \ sqrt {c ^ 2 - \ left (\ frac {\ mathrm dx } {\ mathrm dt} \ right) ^ 2 - \ left (\ frac {\ mathrm dy} {\ mathrm dt} \ right) ^ 2 - \ left (\ frac {\ mathrm dz} {\ mathrm dt} \ right ) ^ 2} \$

On the one hand, with the relativistic line element \$ \ mathrm ds \$ and the proper time element \$ \ mathrm d \ tau \$

\$ c = \ frac {\ mathrm ds} {\ mathrm d \ tau}, \$

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

\$ \ vec {v} = \ frac {\ mathrm d \ vec {r}} {\ mathrm dt}. \$

With the square of the speed

\$ v ^ 2 = \ left (\ frac {\ mathrm dx} {\ mathrm dt} \ right) ^ 2 + \ left (\ frac {\ mathrm dy} {\ mathrm dt} \ right) ^ 2 + \ left ( \ frac {\ mathrm dz} {\ mathrm dt} \ right) ^ 2 \$

finally follows for the element of proper time

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

The Proper time\$ \ tau \$ is the time that passes in the moving frame of reference. The proper time element is integrated in order to maintain the size

\$ \ tau = \ int_0 ^ t \ sqrt {1 - \ frac {v (t) ^ 2} {c ^ 2}} \ mathrm dt \$.

At constant speed \$ v \$ the root factor is \$ 1 / \ gamma \$, and this results in \$ \ tau = \ tfrac {t} {\ gamma} \$.

If the time \$ t \$ has passed in the rest system, then only the smaller time \$ \ tfrac {t} {\ gamma} \$ has passed in the moving system. Conversely, \$ t = \ gamma \ tau \$ also applies.

If the time \$ \ tau \$ has passed in the moving system, the greater time \$ \ gamma \ tau \$ has passed in the rest system.

### Movement with constant acceleration

If a test body of mass \$ M \$ is accelerated to relativistic speeds (greater than one percent of the speed of light) with a constant force \$ F \$, 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 \$ v_0 \$ at \$ t = 0 \$, it is useful to use the abbreviation:

\$ \ 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 from \$ t = 0 \$ with a constant force \$ F \$, then the following applies

\$ v (t) = \ frac {a t + v_0 \ gamma_0} {\ sqrt {1+ \ frac {(a t + v_0 \ gamma_0) ^ 2} {c ^ 2}}}, \$

where the constant acceleration is calculated according to \$ a = F / M \$. 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 momentary speed \$ v (t) \$ has to be included in the integral given above

\$ \ tau = \ int_0 ^ t \ sqrt {1 - \ frac {v (t) ^ 2} {c ^ 2}} \ mathrm dt \$

can be used. The result of this integration is

\$ \ 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 \$ x (t) \$ in the system of the observer at rest is obtained by integrating the speed \$ v (t) \$ over time

\$ x (t) = \ frac {c ^ 2} {a} \ left (\ sqrt {1 + \ frac {(a t + v_0 \ gamma_0) ^ 2} {c ^ 2}} - \ gamma_0 \ right). \$

If the time \$ t \$ is replaced by the proper time \$ \ tau \$ when the starting speed is vanishing (\$ v_0 = 0 \$), the following applies

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

### 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 space-time itself. Every observer who is at rest relative to the center of gravity measures different process times for identical processes taking 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.

### Acceleration and gravity: the rotating disk

This problem is also known as the Ehrenfest 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 one can explain the effect of the gravitational time dilation using the time dilation caused by movement.

If we consider a disk rotating with constant angular velocity \$ \ omega \$, then a point at a distance \$ r \$ from the center moves with the velocity

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

Correspondingly, the proper time becomes at a distance \$ r \$ from the center of the disk

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

occur. For sufficiently small distances (\$ v ^ 2 \ ll c ^ 2 \$) this expression is approximate

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

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

\$ \ varphi = - (r \ omega) ^ 2/2 \$

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

\$ \ tau = t \ left (1 + \ frac {\ varphi} {c ^ 2} \ right) \$

(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

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:

\$ \ tau = t_0 \ sqrt {1+ \ frac {2 \ phi} {c ^ 2}} \$

Here \$ t_0 \$ is the time at potential \$ \ phi = 0 \$ and \$ \ phi \$ the Newtonian gravitational potential (multiplication with the mass of a body results in its potential energy at a certain location)

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 \$ \ phi = gh \$. At an altitude of 300 kilometers (this is a typical altitude at which space shuttles fly), \$ 1 + 3 {,} 27 \ cdot 10 ^ {- 11} \, \ mathrm {s} \$, that goes by in every "earth second" is about a millisecond more per year. This means that an astronaut who would rest 300 kilometers above the earth (for example with the help of a rocket engine) would age about a millisecond faster than someone who was resting on the earth. It should be noted here that this number does not indicate how a shuttle astronaut ages, as the shuttle also moves (it circles around the earth), which leads to an additional effect in the time dilation.

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 quickly as on the earth's surface.

### Relativistic Doppler effect

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 \$ 8 {,} 4 \ times10 ^ {- 8} \$ . 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.

When cosmic rays hit the molecules in the upper layers of the air, muons are created at heights of 9 to 12 kilometers. They are one of the main components of the secondary cosmic rays, 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 range would only be about 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.

### Tests of gravitational time dilation

The gravitational time dilation was demonstrated in 1960 in the Pound-Rebka experiment by Robert Pound and Glen Rebka. In addition, in 1976 NASA launched a Scout-D rocket 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 the gravitational redshift.

### literature

• Albert Einstein: On the electrodynamics of moving bodies. In: Annals of Physics and Chemistry. 17, 1905, pp. 891–921 (as facsimile (PDF); as full text at Wikilivres; and commented and explained at Wikibooks)
• Thomas Cremer: Interpretation problems of the special theory of relativity. Harri Deutsch, 1990
• Walter Greiner, Johann Rafelski: Special theory of relativity. Harri Deutsch, 1989
• Harald Fritzsch: E = mc². A formula changes the world. Piper, 1990
• Roland Pabisch: Derivation of the time dilatation effect from fundamental properties of photons. Springer, Vienna 1999, ISBN 3-211-83153-3

### Individual evidence

1. ↑ Max Born: Einstein's theory of relativity. 7th edition. Springer Verlag, 2003, ISBN 3-540-00470-X.
2. ^ Roman Sexl, Herbert K. Schmidt: Space-time relativity. Vieweg, Braunschweig 1979, ISBN 3-528-17236-3, pp. 31-35.
3. ↑ Gilbert N. Lewis, Richard C. Tolman: The Principle of Relativity, and Non-Newtonian Mechanics. In: Proceedings of the American Academy of Arts and Sciences. 44, 1909, pp. 709-726 (in the English language Wikisource).
4. ↑ Rolf Sauermost and others: Lexicon of natural scientists. Spectrum Academic Publishing House, Heidelberg / Berlin / Oxford 1996, p. 360
5. ↑ Torsten Fließbach: mechanics. 4th edition, Elsevier - Spektrum Akademischer Verlag, 2003, p. 322 f., ISBN 3-8274-1433-4
6. ↑ Jürgen Freund: Special theory of relativity for first-year students
7. ↑ Clifford Will: The Confrontation between General Relativity and Experiment. 2006