# Time dilation

Time dilation, according to Albert Einstein's special theory of relativity, is the slowing-down of the passage of time as experienced by people or objects moving in relation to an observer. Gravitational time dilation is the slowing down of the passage of time anywhere in the gravitational field.

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## Velocity time dilation

When one accelerates towards the speed of light, time slows down with respect to the rest of the Universe. That is, a stationary observer would see the traveling objects slow down their activity. For them, time passes slower. The effect is of course symmetrical: an observer fixed on the "moving" object sees the "stationary observer" slowing down. See twin paradox.

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 (still 1/10 of the speed of light), that it becomes important.

The formula for determining time dilation involves the Lorentz factor and is:

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

Where T0 is the passage of time measured by a stationary observer and T1 is this passage of time measured by an observer travelling at velocity v.

v (%c)length due to length contraction"relativistic mass"time due to time dilation
01.0001.0001.000
100.9951.0050.995
500.8671.1550.867
900.4362.2940.436
990.1417.0890.141
99.90.04522.3660.045
99.9990.00448224.6580.00448

Notice how dramatically the time dilation effect increases 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. [Note: the concept of increasing "relativistic mass" really refers to the increasing relativistic momentum divided by the (constant) rest mass.]

## Gravitational time dilation

Gravitational time dilation is a verified effect of general relativity, and 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:
$t_0 = \frac{t_f}{ \sqrt{1 - \left( \frac{C_h}{C_0} \right)}}$
Where t0 is time for the object undergoing dilation, tf is time for an observer outside the system, Ch is the circumference of the event horizon, and C0 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

In other words, when our orbiter's orbital circumference is merely one meter longer than the circumference of 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

Time dilation could make it possible to travel "into the future": if we could accelerate a starship enough, one year aboard the ship might correspond to ten years outside. 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.