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Time dilation is the factor by which an inertial observer measures another observer's clock as going slow.

Time dilation is composed of two factors:

1) a relative factor of [itex]\sqrt{1\ -\ v^2/c^2}[/itex] for Lorentz time dilation, which depends only on the velocity of the clock

2) an absolute factor of [itex]\sqrt{-g\,_{0\,0}}[/itex] for gravitational time dilation, which depends only on the position of the clock.

Time dilation does not depend on the acceleration of the clock.

Lorentz time dilation is mutual for two inertial observers, in the sense that they will each regard the other's clock as running slow by the same factor.

Gravitational time dilation is greater (the clock is slower) where gravity is stronger (and gravitational potential is higher).

Equations

Lorentz (special relativity) time dilation:

[tex]\sqrt{1\ -\ v^2/c^2}[/tex]

Static metric, with gravitational potential U:

[tex]ds^2\ =\ g_{00}dt^2\ +\ g_{ij}dx^idx^j\ \simeq\ - (1\ -\ 2U)dt^2\ +\ g_{ij}dx^idx^j[/tex]

Gravitational time dilation in static metric:

[tex]\sqrt{\frac{g_{00}(clock)}{g_{00}(observer)}}\ \simeq\ \sqrt{\frac{1\ -\ 2U(clock)}{1\ -\ 2U(observer)}}\ \simeq\ 1\ -\ U(clock)\ +\ U(observer)\ =\ 1\ -\ \Delta\,U[/tex]

Schwarzschild (static metric) gravitational potential at distance r from mass M:

[tex]U\ =\ \frac{2GM}{rc^2}\ =\ \frac{2gr}{c^2}[/tex]

Extended explanation

Accelerating observers:

These formulas arenotintended to apply when anacceleratingobserver measures another observer's clock.

But theydoapply when only theclockis accelerating: for example, when an observer on Earth measures a satellite clock.

Time dilation and red-shift:

The Lorentz red-shift or blue-shift for movement directly away from or toward the observer is [itex]\sqrt{(1\ -\ v/c)(1\ +\ v/c)}[/itex]

Gravitational red-shift (for any velocity) is the same as gravitational time dilation.

Static metric:

A static metric isstationary(the coefficients do not depend on t), and has [itex]g_{i\,0}\ =\ g_{0\,i}\ =\ 0,\ \ i\ =\ 1,2,3[/itex] (so there are are no "space-and-time" terms such as dxdt dydt or dzdt).

Simultaneity:

For gravitational time dilation to be meaningful, the spacetime metric must … time coordinate … simultaneity … [hmm … still thinking about this … see p100, Ciufolini & Wheeler … anyone wanting to jump in and finish this, or add anything else, be my guest! ]

Approximations:

At low speeds, [itex]\sqrt{1\,-\,v^2/c^2}[/itex] is approximately [itex]1\ -\ (1/2)(v/c)^2[/itex]

For example, the Earth orbits the Sun at about 18 miles per second (about 64,000 mph), which is about 1/10,000 of the speed of light, and so time dilation, as seen by a non-orbiting observer, would be about 1 - 1/200,000,000.

Near the speed of light, [itex]\sqrt{1\,-\,v^2/c^2},\ =\ \sqrt{(1\,+\,v/c)(1\,-\,v/c)}[/itex] is approximately [itex]\sqrt{2(1\,-\,v/c)}[/itex]

Orbits:

A clock on a satellite in orbit goes slower than a stationary clock on the planet: it has a "slowing-down" SR time dilation depending on its speed, and a smaller "speeding-up" GR time dilation depending on its distance from the planet.

There is no "clock paradox" since the satellite clock is the clock of anacceleratingobserver, not of aninertialobserver (the instantaneous inertial frame of the satellite clock keeps changing).

Circular orbits:

Betweentwo satellites oncircular orbits of the same radius, only their relative speed matters.

If they're on thesameorbit, in the same direction, then their relative speed will be constant, and there will be no time dilationbetweenthem (though both will be slower than a clock on the planet).

Ondifferentorbits (of the same radius), therewillbe time dilation between them (though that dilation will cancel out once every orbit, as can be seen by comparison with any planetary clock).

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