Coordinate Time - Mathematics

Mathematics

For non-inertial observers, and in general relativity, coordinate systems can be chosen more freely. For a clock whose spatial coordinates are constant, the relationship between proper time τ (Greek lowercase tau) and coordinate time t, i.e. the rate of time dilation, is given by

(1)

where g₀₀ is a component of the metric tensor, which incorporates gravitational time dilation (under the assumption that the zeroth component is timelike).

An alternative formulation, correct to the order of terms in 1/c2, gives the relation between proper and coordinate time in terms of more-easily recognizable quantities in dynamics:

(2)

in which:

is a sum of gravitational potentials due to the masses in the neighborhood, based on their distances r_i from the clock. This sum of the terms GM_i/r_i is evaluated approximately, as a sum of Newtonian gravitational potentials (plus any tidal potentials considered), and is represented using the positive astronomical sign convention for gravitational potentials.

Also c is the speed of light, and v is the speed of the clock (in the coordinates of the chosen reference frame) defined by:

(3)

where dx, dy, dz and dt_c are small increments in three orthogonal spacelike coordinates x, y, z and in the coordinate time t_c of the clock's position in the chosen reference frame.

Equation (2) is a fundamental and much-quoted differential equation for the relation between proper time and coordinate time, i.e. for time dilation. A derivation, starting from the Schwarzschild metric, with further reference sources, is given in Time dilation due to gravitation and motion together.