Coordinate time

In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time.    (1)    (2)    (3) In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time. In the special case of an inertial observer in special relativity, by convention the coordinate time at an event is the same as the proper time measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the Einstein synchronisation convention. A fuller explanation of the concept of coordinate time arises from its relationships with proper time and with clock synchronization. Synchronization, along with the related concept of simultaneity, has to receive careful definition in the framework of general relativity theory, because many of the assumptions inherent in classical mechanics and classical accounts of space and time had to be removed. Specific clock synchronization procedures were defined by Einstein and give rise to a limited concept of simultaneity. Two events are called simultaneous in a chosen reference frame if and only if the chosen coordinate time has the same value for both of them; and this condition allows for the physical possibility and likelihood that they will not be simultaneous from the standpoint of another reference frame. But the coordinate time is not a time that could be measured by a clock located at the place that nominally defines the reference frame, e.g. a clock located at the solar system barycenter would not measure the coordinate time of the barycentric reference frame, and a clock located at the geocenter would not measure the coordinate time of a geocentric reference frame. 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 where g00 is a component of the metric tensor, which incorporates gravitational time dilation (under the convention 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: