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 formula_1, a synchronous frame is defined such that
formula_2 
where formula_3 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" ≡ "dx"0 = 0 in the metric. The reason for this is the different dependence between proper time formula_4 and time coordinate "x"0 ≡ "t" in different points of space., i.e., formula_5
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 formula_6 to Alice who is at a very close point "A" with coordinates "x"α 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:
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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 formula_7 in the above equation, we can solve for "dx"0 obtaining two roots:
 formula_8
which correspond to the propagation of the signal in both directions between Alice and Bob. If "x"0 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 "x"0 + "dx"0 (1) and "x"0 + "dx"0 (2). The thick lines on Fig. 1 are the world lines of Alice and Bob with coordinates "x"α and "x"α + "dx"α, respectively, while the red lines are the world lines of the signals. Fig. 1 supposes that "dx"0 (2) is positive and "dx"0 (1) is negative, which, however, is not necessarily the case: "dx"0 (1) and "dx"0 (2) may have the same sign. The fact that in the latter case the value "x"0 (Alice) in the moment of signal arrival at Alice's position may be less than the value "x"0 (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
 formula_9
The respective proper time interval is obtained from the above relationship by multiplication by formula_10, 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 "y"0 and Bob's clock reads "x"0 then via Einstein Synchronization condition,
formula_11
Substitute here eq. 2 to find the difference in "time" "x"0 between two simultaneous events occurring in infinitesimally close points as
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. 4 by "g"00 and bringing terms to the left hand side 
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or, the "covariant differential" "dx"0 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 Δ"x"0 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.
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.
Example: Uniformly rotating frame.
Consider a rest (inertial) frame expressed in cylindrical coordinates formula_12 and time formula_13. The interval in this frame is given by formula_14 Transforming to a uniformly rotating coordinate system formula_15 using the relation formula_16 modifies the interval to
formula_17
Of course, the rotating frame is valid only for formula_18 since the frame speed would exceed speed of light beyond this radial location. The non-zero components of the metric tensor are formula_19 formula_20 formula_21 formula_22 and formula_23 Along any open curve, the relation
formula_24
can be used to synchronize clocks. However, along any closed curve, synchronization is impossible because 
formula_25
For instance, when formula_26, we have
formula_27
where formula_28 is the projected area of the closed curve on a plane perpendicular to the rotation axis (plus or minus sign corresponds to contour traversing in, or opposite to the rotation direction).
The proper time element in the rotating frame is given by
formula_29
indicating that time slows down as we move away from the axis. Similarly the spatial element can be calculated to find
formula_30
At a fixed value of formula_31 and formula_32, the spatial element is formula_33 which upon integration over a full circle shows that the ratio of circumference of a circle to its radius is given by
formula_34
which is greater than by formula_35.
Space metric tensor.
Eq. 3 can be rewritten in the form
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where
is the three-dimensional metric tensor that determines the metric, that is, the geometrical properties of space. Equations eq. 7 give the relationships between the metric of the three-dimensional space formula_36 and the metric of the four-dimensional spacetime formula_1.
In general, however, formula_1 depends on "x"0 so that formula_36 changes with time. Therefore, it doesn't make sense to integrate "dl": this integral depends on the choice of world line between the two points on which it is taken. It follows that in general relativity the distance between two bodies cannot be determined in general; this distance is determined only for infinitesimally close points. Distance can be determined for finite space regions only in such reference frames in which "gik" does not depend on time and therefore the integral formula_40 along the space curve acquires some definite sense.
The tensor formula_41 is inverse to the contravariant 3-dimensional tensor formula_42. Indeed, writing equation formula_43 in components, one has:
 formula_44
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 formula_45
Determining formula_46 from the second equation and substituting it in the first proves that
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This result can be presented otherwise by saying that formula_42 are components of a contravariant 3-dimensional tensor corresponding to metric formula_48:
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The determinants "g" and formula_49 composed of elements formula_1 and formula_36, respectively, are related to each other by the simple relationship:
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In many applications, it is convenient to define a 3-dimensional vector g with covariant components
Considering g as a vector in space with metric formula_36, its contravariant components can be written as formula_53. Using eq. 11 and the second of eqs. 8, it is easy to see that
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From the third of eqs. 8, it follows
Synchronous coordinates.
As concluded from eq. 5, the condition that allows clock synchronization in different space points is that metric tensor components "g"0α are zeros. If, in addition, "g"00 = 1, then the time coordinate "x"0 = "t" is the proper time in each space point (with "c" = 1). A reference frame that satisfies the conditions
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is called "synchronous frame". The interval element in this system is given by the expression
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with the spatial metric tensor components identical (with opposite sign) to the components "g"αβ:
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In synchronous frame time, time lines are normal to the hypersurfaces "t" = const. Indeed, the unit four-vector normal to such a hypersurface "ni" = ∂"t"/∂"xi" has covariant components "nα" = 0, "n"0 = 1. The respective contravariant components with the conditions eq. 15 are again "nα" = 0, "n"0 = 1.
The components of the unit normal coincide with those of the four-vector "ui" = "dxi/ds" which is tangent to the world line "x"1, "x"2, "x"3 = const. The "ui" with components "u"α = 0, "u"0 = 1 automatically satisfies the geodesic equations:
formula_54
since, from the conditions eq. 15, the Christoffel symbols formula_55 and formula_56 vanish identically. Therefore, in the synchronous frame the time lines are geodesics in the spacetime.
These properties can be used to construct synchronous frame in any spacetime (Fig. 2). To this end, choose some spacelike hypersurface as an origin, such that has in every point a normal along the time line (lies inside the light cone with an apex in that point); all interval elements on this hypersurface are space-like. Then draw a family of geodesics normal to this hypersurface. Choose these lines as time coordinate lines and define the time coordinate "t" as the length "s" of the geodesic measured with a beginning at the hypersurface; the result is a synchronous frame.
An analytic transformation to synchronous frame can be done with the use of the Hamilton–Jacobi equation. The principle of this method is based on the fact that particle trajectories in gravitational fields are geodesics. The Hamilton–Jacobi equation for a particle (whose mass is set equal to unity) in a gravitational field is
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where "S" is the action. Its complete integral has the form:
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Note that the complete integral contains as many arbitrary constants as the number of independent variables which in our case is formula_57. In the above equation, these correspond to the three parameters "ξα" and the fourth constant "A" being treated as an arbitrary function of the three "ξα". With such a representation for "S" the equations for the trajectory of the particle can be obtained by equating the derivatives "∂S"/"∂ξα" to zero, i.e. 
For each set of assigned values of the parameters "ξα", the right sides of equations 18a-18c have definite constant values, and the world line determined by these equations is one of the possible trajectories of the particle. Choosing the quantities "ξα", which are constant along the trajectory, as new space coordinates, and the quantity "S" as the new