Time dilation

Time dilation is a consequence of Albert Einstein's theories of relativity. It occurs when one observer finds a clock which is measuring proper time ticking slow with respect to the relative time defined by his own clock (which is also measuring proper time). In special relativity, this occurs for clocks which are moving with respect to an inertial observer. In general relativity this occurs for clocks at lower potentials in a gravitational field, and is called gravitational time dilation.

This effect is commonly thought of as being time slowing down for the time dilated clock. This is not the case. Locally, one's proper time always passes at the same rate. Instead what is slowed down is how that proper time passage is perceived by another observer.

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Temporal coordinate systems and clock synchonization

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 essense, an observer with a clock sends a light signal out towards an event at time t1 according to his clock. At the event, that light signal is reflected back to the observer, and arrives at time t2 according to his clock. Since the light travels the same path at the same rate going both out and back for an observer in this scenario, the coordinate time for the event of the photon's begin reflected for the observer is tE = (t1 + t2) / 2.

Time dilation occurs with respect to a temporal coordinate system set up in this manner. Suppose that two identical clocks were at time t0 callibrated to read 0, and allowed to tick at their own rate after that. Time dilation means that when a clock which is in motion or at a different gravitational potential gets to the temporal coordinate of t0 + δ, it has a recorded an elapsed time of less than δ.

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 synchonized 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 illustated 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:

[itex]T_1 = \frac{T_0}{\sqrt {1- \left(\frac{v^2}{c^2}\right)}}[itex]

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 travelling at velocity v with respect to the clock.

v (%c)length due to length contractiontime due to time dilation
01.0001.000
100.9951.005
500.8671.155
900.4362.294
990.1417.089
99.90.04522.366
99.9990.00448224.658

Note the dramatic increase in the time dilation effect as v approaches c. Taken to the extreme, an observer travelling at the speed of light (which, according to special relativity, is impossible for any object with a non-zero rest mass) would be frozen with respect to the outside world. Massless particles (which travel at the speed of light, and have finite energy) include photons and gluons. Recently it was determined that neutrinos have a mass, unlike previously thought.

The Spacetime geometry of velocity time dilation

Missing image
Time_dilation02.gif
Time dilation in transversal motion.

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 lightpulses 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 lightpulses travels a diagonal path that is two lightseconds long. (As seen from the green perspective the reds travel 1.73 ([itex]\sqrt{3}[itex]) lightseconds of distance for every two seconds of green time.)

One of the red ships emits a lightpulse 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.

Gravitational time dilation

Gravitational time dilation occurs in accelerated frames of reference in special relativity and in the curved spacetimes of general relativity. At lower potentials than the observer in a gravitational field, clocks will be found to be running slower than the observer's. Unlike velocity time dilation, gravitational time dilation is not a symmetrical effect: At higher potentials than the observer, clocks be found to be running faster than the observer's.

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks that traveled aboard the airplanes upon return were slightly fast with respect to clocks on the ground. The effect is significant enough that the Global Positioning System needs to correct for its effect on clocks aboard artificial satellites, providing a further experimental confirmation of the effect.

An extreme example of gravitational time dilation occurs near a black hole. A clock falling towards the event horizon would appear (to observers far away) to slow down to a halt as it approached the horizon. A small and sturdy enough clock could conceivably cross the horizon without suffering adverse effects at the horizon, but to far away observers it would "freeze" and be flattened out on the horizon.

Time dilation around a black hole may be described using the following equation:

[itex]t_0 = \frac{t_f}{ \sqrt{1 - \left( \frac{C_h}{C_0} \right)}} [itex]

where [itex]t_0[itex] is time for the object undergoing dilation, [itex]t_f[itex] is time for an observer outside the system, [itex]C_h[itex] is the circumference of the event horizon, and [itex]C_0[itex] is the circumference of the object's orbit about the black hole.

The following chart details the effects of time dilation caused by a black hole (with a circumference of its event horizon of 10,000 km) for an entity orbiting that black hole, relative to an outside observer. For each day that passes for the stalwart black hole orbiters, we can determine the amount of time that would pass for an outside observer.

Circumference of orbitTime experienced by outside
observer per orbiter day
20,000 km1.41 days
15,000 km1.73 days
12,000 km2.44 days
11,000 km3.32 days
10,500 km4.50 days
10,250 km6.40 days
10,050 km14.18 days
10,025 km20.02 days
10,005 km44.73 days
10,000.75 km115.47 days
10,000.50 km141.42 days
10,000.25 km200.00 days
10,000.125 km282.84 days
10,000.050 km447.21 days
10,000.001 km3162.28 days

According to this chart, when an orbiter's orbital circumference is one meter wider than the black hole's event horizon, about eight years and nine months will pass for the outside observer per orbiter day. If the observer could somehow watch the action going on inside the orbiter, she would perceive everything as occurring at a staggeringly slow pace, while the orbiter crew would feel time passing normally. If the crew could watch the life of the outside observer, it would appear to be passing by at a very fast pace, while the observer would feel time passing normally.

Time dilation and space flight

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 likelihod of collision with small space debris and other particlulate material is another practical limitation.

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.

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