Time dilation
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Template:Expert Time dilation is the observation of a remote clock running at a different rate from an equivalent local clock. It is a fundamental prediction of Albert Einstein's theories of relativity, and has been experimentally confirmed. In the relativistic framework, the effect is manifested in two ways:
- In special relativity, clocks which are moving with respect to an inertial observer are found to be running slow.
- In general relativity, clocks at lower potentials in a gravitational field are found to be running slow. This is called gravitational time dilation (see also gravitational red shift).
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Temporal coordinate systems and clock synchronization
To understand time dilation, one needs to first realize how it is determined to exist by an observer.
In relativity, temporal coordinate systems are set up with the aid of the Einstein synchronization procedure. In essence, an observer with a clock sends a light signal out towards at time t1 according to his clock. At a distant event, that light signal is reflected back to, and arrives at back at the observer at time t2 according to his clock. Since the light travels the same path at the same rate going both out and back for the observer in this scenario, the coordinate time of the event of the photon being reflected for the observer tE is tE = (t1 + t2) / 2. In this way, a single observer's clock can be used to define temporal coordinates which are good anywhere in the universe.
Time dilation occurs with respect to temporal coordinate systems set up in this manner. Suppose that two identical clocks (one of which belongs to the observer) were at coordinate time t0 calibrated to read t0, and allowed to tick at their own rate after that. In the theory of relativity, if the second clock is in motion or at a lower gravitational potential, when it arrives at the temporal coordinate of t0 + δ, it will show a time of less than t0 + δ. This is the essense of time dilation.
So time dilation is an effect where another clock is being viewed as running slow by an observer. No observer ever considers their own clock time to be time dilated, but may find that another observer sees it as being time dilated.
Velocity time dilation
Because dilation is relative, a measurement of relative time must regard one clock as being "stationary" in spacetime, and that clock is the basis of a temporal coordinate system where time is represented as synchronized with the stationary clock. The traveler's "moving" clock is in motion with respect to this stationary clock. In the special theory of relativity, the moving clock is found to be ticking slow with respect to the temporal coordinate system of the stationary clock. For example, if the moving clock has a speed of 86.6% of the speed of light, then it will be found to have only 1 second of elapsed proper time for every 2 seconds of coordinate time for the stationary clock that it passes. This effect is symmetrical: In a temporal coordinate system synchronized with the "moving" clock, it is the "stationary" clock that is running slow. (A misunderstanding of this symmetry leads to the so-called twin paradox.)
A legitimate question is how special relativity can be self-consistent if clock A is time dilated with respect to clock B and clock B is also time dilated with respect to clock A. In fact, special relativity is consistent because of other effects. The relativity of simultaneity, another effect of the Lorentz transformations, affects how the volumes of simultaneous time are placed with respect to each other by observers who are in motion with respect to each other. Because the volumes are tilted with respect to each other (as illustrated in the twin paradox article), each observer can treat the other clock as being slow without relativity being self-contradictory.
It is important to note that this effect is extremely small at ordinary speeds, and can be safely ignored for all ordinary situations. It is only when an object approaches speeds on the order of 30,000 km/s (1/10 of the speed of light) that it becomes important.
The formula for determining time dilation involves the Lorentz factor and is:
where T0 is the passage of time measured between two ticks on a clock by a stationary observer and T1 is the passage of time between these same two ticks, but measured by an observer traveling at velocity v with respect to the clock.
The Space-time geometry of velocity time dilation
The green dots and red dots in the animation represent spaceships. The ships of the green fleet have no velocity relative to each other, so for the clocks onboard the individual ships the same amount of time elapses relative to each other, and they can set up a procedure to maintain a synchronized standard fleet time. The ships of the "red fleet" are moving with a velocity of 0.866 of the speed of light with respect to the green fleet.
The blue dots represent pulses of light. One cycle of lightpulses between two green ships takes two seconds of "green time", one second for each leg.
As seen from the perspective of the reds, the transit time of the light pulses they exchange among each other is one second of "red time" for each leg. As seen from the perspective of the greens, the red ships' cycle of exchanging light pulses travels a diagonal path that is two light-seconds long. (As seen from the green perspective the reds travel 1.73 (
) light-seconds of distance for every two seconds of green time.)
One of the red ships emits a light pulse towards the greens every second of red time. These pulses are received by ships of the green fleet with two-second intervals as measured in green time. Not shown in the animation is that all aspects of physics are proportionally involved. The lightpulses that are emitted by the reds at a particular frequency as measured in red time are received at a lower frequency as measured by the detectors of the green fleet that measure against green time, and vice versa.
The animation cycles between the green perspective and the red perspective, to emphasize the symmetry. As there is no such thing as absolute motion in relativity (as is also the case for Newtonian mechanics), both the green and the red fleet are entitled to consider themselves as "non-moving" in their own frame of reference.
Time dilation and space flight
Time dilation would make it possible to travel "into the future", to where for example one year of travel might correspond to ten years at home. Indeed, a constant 1g acceleration would permit humans to circumnavigate the known universe (with a radius of some 15 billion light years) in under a subjective lifetime. A more likely use of this effect would be to enable humans to travel to nearby stars without spending their entire lives aboard the ship. However, any such use of this effect would require an entirely new method of propulsion. A further problem with relativistic travel is that the interstellar medium would turn into a stream of cosmic rays that would destroy the ship unless stark radiation protection measures were taken.
Current space flight technology has fundamental theoretical limits based on the practical problem that an increasing amount of energy is required for propulsion as a craft approaches the speed of light. The likelihood of collision with small space debris and other particulate material is another practical limitation. As a result, time dilation is not currently a major factor in space travel.
Experimental confirmations
Time dilation, of one form or another, has been observed numerous occasions since it was initially predicted. Indeed, these observations form a core test of both special and general relativity. They include:
- Rossi and Hall, in 1941, compared the population of cosmic-ray produced muons at the top of Mt. Washington to that observed at sea level. Although the travel time for the muons from the top of the mountain to the base is several muon half-lives, the muon sample at the base was only moderately reduced due to the time-dilation because of their high-speed relative to the experimenters. That is to say, the muons are decaying about 10 times slower than they would in a rest frame.
- Pound, Rebka in 1959 measured the gravitational red shift of photons falling in the Earth's gravitational field. The results were within 10% of general relativity. Later Pound and Snider (in 1964) improve this to 1%.
- Hefele and Keating, in 1971, flew cesium clocks east and west around the planet, and compared their drifts to a clock that remained at the US Naval Observatory. To within experimental error, the results were consistent with all modeled effects in special and general relativity. In 2005, the National Physical Laboratory in the United Kingdom, report their limited replication of this experiment. While the NPL used a shorter baseline, it was with more precise equipment, and the reported results are again consistent with relativity.
- The Global Positioning System can be considered a continuously operating experiment in both special and general relativity. The on-orbit clocks are corrected for both special and general relativistic time-dilation effects so they appear to run at the same (average) rate as clocks at the surface of the Earth. In addition, but not directly time-dilation related, general relativistic correction terms are built into the model of motion that the satellites broadcast to receivers -- uncorrected, these eccentric terms would amount to a 12-hour, approximately 7 metre, oscillation in the pseudo-ranges measured by a receiver.
Etiology
The effect is due to the constancy of the speed of light in all reference frames.
Consider a simple clock consisting of two mirrors, between which a photon is bouncing. The separation of the mirrors is L, and the clock ticks once each time it hits a given mirror.
In the a frame where the clock is at rest, the photon traces out a simple linear path. (See diagram at left.) The period of the clock is going to be twice the separation divided by the speed of light.
In a frame where the clock is moving, the photon will trace out a longer, triangular, path. (See diagram at right.) Since the second postulate of special relativity states that the speed of light is constant in all frames, this necessarily leads to a lengthening of the period of this clock from the moving observer's perspective. That is to say, in a frame moving relative to the clock, the clock appears to be running slower. Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity.
References
- B. Rossi and D. B. Hall, Phys. Rev., 59, 223 (1941).
See also
- Length contraction
- Proper time
- Transverse Doppler effect
- Special Relativity
- General Relativity
- Lorentz transformation
- Hafele-Keating experiment
- Pound-Rebka experiment
- Four-vector
External links
- Time Dilation Demonstration Applet
- UK National Physical Laboratory reports replication of Hefele-Keating experimentTemplate:Relativity-stub
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