A synchronous frame is a reference frame in which the time coordinate defines proper time for all co-moving observers. It is built by choosing some constant time hypersurface as an origin, such that has in every point a normal along the time line and a light cone with an apex in that point can be constructed; all interval elements on this hypersurface are space-like. A family of geodesics normal to this hypersurface are drawn and defined as the time coordinates with a beginning at the hypersurface. In terms of metric-tensor components , a synchronous frame is defined such that
where Such a construct, and hence, choice of synchronous frame, is always possible though it is not unique. It allows any transformation of space coordinates that does not depend on time and, additionally, a transformation brought about by the arbitrary choice of hypersurface used for this geometric construct.
Synchronization in an arbitrary frame of reference
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. In special relativity, the space distance element dl is defined as the intervals between two very close events that occur at the same moment of time. In general relativity this cannot be done, that is, one cannot define dl by just substituting dt ≡ dx0 = 0 in the metric. The reason for this is the different dependence between proper time and time coordinate x0 ≡ t in different points of space., i.e.,
To find dl in this case, time can be synchronized over two infinitesimally neighboring points in the following way (Fig. 1): Bob sends a light signal from some space point B with coordinates to Alice who is at a very close point A with coordinates xα and then Alice immediately reflects the signal back to Bob. The time necessary for this operation (measured by Bob), multiplied by c is, obviously, the doubled distance between Alice and Bob.
The line element, with separated space and time coordinates, is:
where a repeated Greek index within a term means summation by values 1, 2, 3. The interval between the events of signal arrival and its immediate reflection back at point A is zero (two events, arrival and reflection are happening at the same point in space and time). For light signals, the space-time interval is zero and thus setting in the above equation, we can solve for dx0 obtaining two roots:
which correspond to the propagation of the signal in both directions between Alice and Bob. If x0 is the moment of arrival/reflection of the signal to/from Alice in Bob's clock then, the moments of signal departure from Bob and its arrival back to Bob correspond, respectively, to x0 + dx0 (1) and x0 + dx0 (2). The thick lines on Fig. 1 are the world lines of Alice and Bob with coordinates xα and xα + dxα, respectively, while the red 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 (Alice) in the moment of signal arrival at Alice's position may be less than the value x0 (Bob) in the moment of signal departure from Bob 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 Bob's place is
The respective proper time interval is obtained from the above relationship by multiplication by , and the distance dl between the two points – by additional multiplication by c/2. As a result:
This is the required relationship that defines distance through the space coordinate elements.
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 of reflection in A lies in the middle between the moments of sending and receiving the signal in B; in this moment if Alice's clock reads y0 and Bob's clock reads x0 then via Einstein Synchronization condition,
Substitute here to find the difference in "time" x0 between two simultaneous events occurring in infinitesimally close points as