Time dilation
At the time dilation ( from Latin: dilatare, spread ',' defer '), is a phenomenon of the theory of relativity. An observer is in a state of uniform motion or he rest in an inertial frame, goes to the special theory of relativity, each moving relative to him clock in his view slowly. However, this phenomenon are not only watches, but the time in the moving system itself, and thus, any process. Time dilation is greater, the greater the relative speed of the clock. They can practically not be observed in everyday life, but only at speeds which are not negligibly small compared to the speed of light. The fact that time is passing each other slowly for all observers, this does not constitute a contradiction, as a closer examination of the relativity of simultaneity shows (see also Special Relativity and Minkowski diagram ).
Such an effect was first described by Joseph Larmor (1897 ) and Hendrik Antoon Lorentz (1899 ) as part of a now outdated ether theory derived. Albert Einstein ( 1905), however, managed to show in the context of special relativity that the altered watches transition not of interference is related by an ether, but with a radical reinterpretation of the concepts of space and time (see also History of special relativity ).
In the gravitational time dilation is a phenomenon of general relativity. With the gravitational time dilation is defined as the effect that a clock - like any other process - in a gravitational field runs slower same as outside. Thus passes the time on the earth's surface by a factor of 7:10 -10 slower than in the distant, approximate gravity-free space. More specifically, each measures against the gravitational field of static observers a longer or shorter expiry time of events that were triggered in an identical manner in or outside the gravitational field (such as an oscillation of the electric field vector of a light beam, which can be used as a time base ). Unlike the time dilation by motion is the gravitational time dilation not mutually While the above located in the gravitational field observer sees run slower the time of the located below the observer sees the lower observers, the time of the upper observer run faster.
- 2.1 Acceleration and Gravitation: the rotating disk
- 2.2 Time dilation in the gravitational field of the earth
- 3.1 Relativistic Doppler Effect
- 3.2 lifetime measurement of particles
- 3.3 Tests of the gravitational time dilation
Time dilation by relative motion
At a constant speed
Explanation
To understand the time dilation, it is necessary to visualize the basic measurement requirements and methods for measuring time with stationary and moving clocks.
Successively If two events occur at the same place in an inertial system, then can be determined by direct reading of the pointer positions of a dormant during these events clock C the operating time ( time between the first and second event). The displayed by C proper time is invariant, ie in all inertial systems is agreed that C indicated this time during the process. If the operating time of C compared with the clocks moving relative inertial frames, can use the following guide: An observer in the inertial frame S introduce two clocks A and B, which are synchronized with light signals (pictures right). Clock C rests in S ' and moves with the velocity v from A to B, where they will be in sync at the start time of A and B. The "moving" clock C ( for which the proper time has elapsed ) goes on arrival compared to the " resting" clock B ( for which has passed ) to, in accordance with the following formula for time dilation (see derivation ):
So go watch A and B faster
In which
Now, says the principle of relativity, that in S ' the clock C can be considered as dormant and therefore must watch A and B go slower than C. At first glance, however, this contradicts the fact that in both inertial clock C when meeting with B pursues what follows from the invariance of the operating times of the clocks C and B.
However, this is explained if the relativity of simultaneity is taken into account. As above measurement was based on the premise that the clocks A and B (and thus the time of starting and C ) are synchronized, which, however, due to the constancy of the velocity of light in all inertial systems only in S is the case. In S ' synchronization fails from A and B - because the clocks move here in the negative x direction and B the time signal comes to meet, while A that runs away. B is thus detected by the first signal and begins pursuant to a run by the Lorentz transformation value to be determined earlier than A. Considering this approach of clock B because of the premature start (ie you pull this amount of time from the total time on B ), also arises here is that the "moving" clock B ( for which the proper time has elapsed ) while the path to " ( which is passed ) resting " clock runs slower C according to the following formula:
Therefore goes faster clock C
So the time dilation drops - as required by the principle of relativity - in all inertial symmetrically: Each measures that the clock of the other running slower than his own. This requirement is fulfilled, although in both inertial frames C to B when meeting pursues and the operating times of C and B are invariant.
Minkowski diagram
The respective operating times are shown opposite in a Minkowski diagram and other images. Clock C (stationary in S ' ) meets at d on clock A and clock B at f on (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 to d simultaneous events in S for the X- axis, and S ' on the x -axis. The respective time periods can be determined directly by counting the marks.
In S the proper time df of C is dilated compared to the longer term ef = dg watches B and A. Conversely, the invariant proper time of B is also in S ' measured dilated. Because time is shorter if ef with respect to time, because the start event was e from Clock B already measured at time i, C before clock had begun to tick at all. At the time d B has the time ej behind, and also here there is time dilation, if df is compared in S ' with the rest of the time in S jf.
From these geometrical conditions is again clear that the invariant proper time between two particular 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 frames. As shown, this is not in contradiction to the reciprocal time dilation, because due to the relativity of simultaneity, the starting times of the clocks in different inertial frames are measured differently.
Time dilation and length contraction
It can be seen that the dilation of oppositely (as measured by the stationary, synchronized clocks )
Reciprocal is the contracted length of moving objects ( measured by simultaneous determination of the endpoints using static scales) with respect to their resting length:
This means that the direction indicated by entrained watch proper time is always less than the time period indicated by the stationary clocks of the same phenomenon, whereas the entrained standards measured by proper length is always greater than the measured length of the stationary scales of the same object.
The reverse occurs when clock and scale not rest in the same inertial frame. Moves namely the clock within the time span along a scale in S (measured by clocks at rest there ), then its rest length is simply given, whereas the dilated clock indicating a lower intrinsic time according to. As their proper time is invariant, it will display the time in their own rest frame, from which it follows that in S ' moving rod has the length. The staff here is so for shorter, which corresponds to the length contraction of the moving rod.
Lightwatch
For a simple explanation of this factor, the concept of light clock can be used. A light clock consists of two mirrors in the distance, reflecting a short light flash back and forth. Such a light clock has already been covered in the 19th century in the theory of light propagation times with the Michelson - Morley experiment, and used as a thought experiment to derive the time dilation for the first time in 1909 by Gilbert Newton Lewis and Richard C. Tolman.
When a light clock A is given, a flash will need time for the easy route between the mirrors from the perspective of a comoving observer with her. At one of the two mirror each incidence of the light flash is registered and each time the light clock is adjusted by a unit of time equal to the total duration of the light flash.
Now, if a second light clock B moves perpendicular to the line connecting the mirror with the speed, so the light from the perspective of the A - observer between the mirrors must have a greater distance to travel than clock A. Assuming the constancy of the speed of light goes for the A - observer clock B therefore slower than clock A. the time it takes for the light flash for the easy way out between the mirrors, resulting on the Pythagorean Theorem
Substituting the expressions for and and solving for we obtain finally
Or with the Lorentz factor:
And thus
Contrast, can also claim to be to be in peace with a comoving clock B observers according to the principle of relativity. That is, its located at his clock B will show a simple duration of the flash of light. In contrast, the flash of light moving from his perspective Clock A is for him to travel a greater distance and requires the following time:
And thus
Proper time
The relativistic metric ds is given by
As a proper time element shall be the quotient of this relativistic line element or distance and the speed of light
By inserting and lifting out followed
On one hand, results with the relativistic line element and proper time element
On the other hand, a velocity is generally defined as the derivative of the position vector with respect to time
With the square of the velocity
Finally, it follows for the element of proper time
The operating time is obtained, when integrated over the operating time element:
At a constant speed is the root factor, and there is.
Measuring Technically, the proper time of the above expression. Draws a clock C, the duration between the events U and W at each event point itself, ie, along the world line of C, on, indicated by C time interval is called the proper time between these events (see first Minkowski diagram right). Just as the line element underlying ds ², the proper time is an invariant, because in all inertial systems is consistently found that clock C shows exactly this time between U and W. The invariant proper time is the reference size when the time dilation occurs. The transmission rate of the clock C from the viewpoint of all other moving system is already explained above with regard to slowing down the own clocks measured. Accordingly clock C between the two observation events is U and W show a shorter period of time, whereas the S- synchronized clocks display a greater amount of time according to:
However, rests a clock B in S, and visit its world line two events U and V instead, then the time period is identical to the invariant proper time between these events, therefore, a greater amount of time is in the system S 'is measured:
The proper time of a located at two events on the ground unaccelerated clock is so minimal compared to the synchronized time coordinates between the same events in all other inertial frames. Because unless the watch is accelerated, so there is only one clock and thus only a straight world line, which indicates the proper time between two particular events. Because it is possible that a single event U on two straight world lines is the same (namely, where the world lines of C and B cut ), but it is geometrically impossible that the second event W on the world line of C on the world line of B, and it is also impossible that the second event V is on the world line of B on the world line of C.
However, if one of the clocks is accelerated, the world lines can intersect again. Here it turns out that the straight world line of the unaccelerated clock displays time with a better sense than the composed - curved world line of the accelerated clock, which is the explanation of the twin paradox. Thus, while, as shown above, the proper time between two events is minimal on the world line of an unaccelerated clock compared to the synchronized coordinate times in all other inertial frames, it is maximum in comparison to the operating times of accelerated clocks were with two events also on site.
Constant acceleration motion
If a test body of mass with a constant force to relativistic speeds speeds (greater than one percent of the speed of light), must be distinguished because of time dilation between the clock of an observer at rest and a clock on board the test body. Does the test body at the speed at which it is appropriate, the abbreviation:
Introduce in order to write down the following calculation results provide a clear overview. If the test body from accelerated by a constant force, it shall
Where the constant acceleration in accordance calculated. Using this formula, the proper time can also be calculated, which would show a clock in the accelerated system of the test body. This requires only the instantaneous velocity in the above mentioned integral
Be used. The result of this integration is
The distance traveled in the system of the observer at rest is obtained by integrating the velocity over time
If the time is still replaced by the proper time at zero starting speed (), applies
Travel to distant stars
Another example would be the movement of a spaceship that takes off from the Earth, a distant planet drives and comes back. A spaceship takes off from the earth, and flies with the constant acceleration of a 28 light-years distant star. The acceleration of was chosen, as this earthly gravity conditions can be simulated on board a spaceship. Halfway the spaceship changes the sign of the acceleration and decelerates. After completing a six-month stay period, the spacecraft returns to Earth in the same way. The past few times result for the traveler to 13 years, 9 months and 16 days ( measurement with on-board clock ). On Earth, 60 years, 3 months and 5 hours on the return of the spaceship, however, passed.
Much larger differences are obtained in a flight to the Andromeda Galaxy, which is about 2 million light years away ( with the same acceleration and deceleration phases). For the earth pass away, about 4 million years, while only about 56 years have passed for the traveler.
The spacecraft never exceeds the speed of light. The longer it accelerates, the closer it comes closer to the true velocity of light, it will, however, never reach this. From the ground perspective, the time corresponding to the time dilation is slower on the spaceship. Since the spacecraft are subject to both observer and measurement instruments of time dilation, the proper time is running as normal from their point of view, however, is shortened due to the Lorentz contraction of the way between Earth and destination. ( From the surface in sight he would stay constant for simplicity in this example). Now, if you're in a space ship and its speed is determined relative to the earth taking into account the Lorentz contraction, then we arrive at the same result as if one determined from the Earth, the speed of the spaceship. The big problem in this example is just that there is currently no drive realized that reaches over such a long time such a high acceleration.
Time dilation due to gravitational
The gravitational time dilation describes the relative timing of systems that lie at different distances a center of gravity (such as a star or planet ) relative to this. It should be noted that the gravitational time dilation does not come about by a mechanical action on the watch, but is a property of spacetime itself. Each relative to the center of gravity stationary observer measures for identical, but at different distances from the center of gravity processes running different expiration times, based on its own time base. An effect that is due to the gravitational time dilation, the gravitational redshift.
Acceleration and gravitation: the rotating disk
This problem is also referred to as Ehrenfest cal paradox.
According to the equivalence principle of general relativity, one can not distinguish between a system at rest in a gravitational field and an accelerated system locally. Therefore, one can explain the effect of time dilation based Gravitationszeitdilatation through movement.
Consider a rotating with constant angular velocity disc, then a point is moving at a distance from the center with the speed
Accordingly, the distance from the center of the disk, the operating time is
. occur For sufficiently small distances (), this expression is approximately
A on the disk befindliches, co-rotating object is now experiencing the centrifugal force. Because of the equivalence principle can interpret this force as the force of gravity to the gravitational potential
Heard. However, this is just the term that occurs in the time dilation in the numerator. Thus, for "small" distances:
(Note: The potential given here does not correspond to the usual centrifugal potential, since an adaptation to the local rotational speed of the disc is made while conservation of angular momentum applies in the conventional centrifugal potential instead)
Time dilation in the gravitational field of the earth
In a weak gravitational field such as the Earth, the gravitational time dilation and therefore can be approximately described by the Newtonian gravitational potential:
Here, the time is at potential and the Newtonian gravitational potential ( multiplied by the mass of a body gives its potential energy at a particular location )
On Earth, the gravitational potential (as long as the amount is small compared to the Earth's radius of about 6400 kilometers ) can be approximated by. In 300 km altitude (which is a typical height, fly in the space shuttle ) thus pass in each " Erdbodensekunde ", which is about one millisecond per year more. That is, an astronaut who would rest in 300 kilometers above the earth (for example, with the support of a rocket engine ), would each year to age for about a millisecond faster than someone who is resting on the ground. It must be noted that this number does not indicate how a shuttle astronaut ages, since the shuttle is additionally moved ( it revolves around the Earth ), which leads to an additional effect of time dilation.
If one compares caused by the reduction in height of the gravitational time dilation relative to the earth's surface and caused by the circular velocity required for this amount time dilation with each other, shows that from at a track radius of 1.5 times the earth's radius, ie at an altitude half the Earth's radius, the two effects cancel exactly and therefore the time on such a circular path as fast passes like on the surface.
Experimental evidence
Relativistic Doppler effect
The first direct evidence of time dilation by measuring the relativistic Doppler effect led to the Ives - Stilwell experiment ( 1939); further evidence were made with the Moessbauer rotor experiments ( 1960 ) and modern Ives - Stilwell variants based on saturation spectroscopy, the latter have reduced the possible deviation of time dilation up to. An indirect proof are variations of the Kennedy - Thorndike experiment in which the time dilation must be considered together with the length contraction. For experiments in which the time dilation for the round trip is observed, see twin paradox.
Lifetime measurement of particles
When cosmic radiation on the molecules of the upper-air height occur in 9 to 12 kilometers muons. They are a major component of the secondary cosmic radiation, move towards the surface with almost the speed of light and can be detected there only because of relativistic time dilation, because without this relativistic effect would their reach are only about 600 m. In addition, tests of the decay times were carried out in particle accelerators with pions, muons or kaons, which also confirmed the time dilation.
Tests of gravitational time dilation
The gravitational time dilation was detected in 1960 in Pound Rebka experiment by Robert Pound and Glen Rebka. In addition, the 1976 NASA launched a Scout D rocket with an atomic clock whose frequency was compared with a clock of the same type on Earth. This was the most precise experiment that could successfully measure the gravitational redshift.