Synchronous Frame - Simultaneity in General Relativity

Simultaneity in General Relativity

Synchronization of clocks located at different space points means that events happening at different places can be measured as simultaneous if those clocks show the same times. Let us see if this is possible in general relativity (in curved space).

It is obvious that such synchronization should be done by exchange of light signals between points. Consider again propagation of signals between infinitesimally close points A and B in Fig. 1. The clock reading in B which is simultaneous with the moment x0 in A lies in the middle between the moments of sending and receiving the signal in B; this is the moment

Substitute here eq. 5 to find the difference in "time" x0 between two simultaneous events occurring in infinitesimally close points as

(eq. 16)

This relationship allows clock synchronization in any infinitesimally small space volume. By continuing such synchronization further from point A, one can synchronize clocks, that is, determine simultaneity of events along any open line. The synchronization condition can be written in another form by multiplying eq. 16 by g₀₀ and bringing terms to the left hand side

(eq. 17)

or, the "covariant differential" dx₀ between two infinitesimally close points should be zero.

However, it is impossible, in general, to synchronize clocks along a closed contour: starting out along the contour and returning to the starting point one would obtain a Δx0 value different from zero. Thus, unambiguous synchronization of clocks over the whole space is impossible. An exception are reference frames in which all components g_0α are zeros.

Note the inability to synchronize all clocks is a property of the reference frame and not of the spacetime itself. It is always possible in infinitely many ways in any gravitational field to choose the reference frame so that the three g_0α become zeros and thus enable a complete synchronization of clocks. To this class are assigned cases where g_0α can be made zeros by a simple change in the time coordinate which does not involve a choice of a system of objects that define the space coordinates.

In the special relativity theory, too, proper time elapses differently for clocks moving relatively to each other. In general relativity, proper time is different even in the same reference frame at different points of space. This means that the interval of proper time between two events occurring at some space point and the time interval between the events simultaneous with those at another space point are, in general, different from one another.