Synchronous Frame - Synchronization Over The Whole Space

Synchronization Over The Whole Space

In the special relativity theory, the space distance element dl is defined as the intervals between two very close events that occur at the same moment of time. In the general relativity theory this cannot be done, that is, one cannot define dl by just substituting dx0 = 0 in ds. The reason for this is the different dependence between proper time and time coordinate x0 in different points of space.

To find dl in this case, one can first synchronize time over the whole space in the following way (Fig. 1): Send a light signal from some space point B with coordinates xα + dxα into a very close point A with coordinates xα and then immediately reflect back the signal from A to B. The time necessary for this operation (measured in point B), multiplied by c is, obviously, the doubled distance between the two points.

The squared interval, with separated space and time coordinates, is:

(eq. 4)

where, as usual, a repeated Greek index within a term means summation by values 1, 2, 3. The interval between the events of signal arrival in point A and its immediate reflection back is zero (two events in the same time at the same point). Equation ds2 = 0 solved for dx0 gives two roots:

(eq. 5)

which correspond to the propagation of the signal in both directions between A and B. If x0 is the moment of arrival/reflection of the signal in A, the moments of signal departure from B and its arrival back in B correspond, respectively, to x0 + dx0 (1) and x0 + dx0 (2). The solid lines on Fig. 1 are the world lines of points A and B with coordinates xα and xα + dxα, respectively, while the dashed lines are the world lines of the signals. Fig. 1 supposes that dx0 (2) is positive and dx0 (1) is negative, which, however, is not necessarily the case: dx0 (1) and dx0 (2) may have the same sign. The fact that in the latter case the value x0 (A) in the moment of signal arrival at A may be less than the value x0 (B) in the moment of signal departure from B does not contain a contradiction because clocks in different points of space are not supposed to be synchronized. It is clear that the full "time" interval between departure and arrival of the signal in point B is

The respective proper time interval is obtained from the above relationship according to eq. 1 by multiplication by, and the distance dl between the two points – by additional multiplication by c/2. As a result:

(eq. 6)

This is the required relationship that defines distance through the space coordinate elements.