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 g i k {\displaystyle g_{ik}} , a synchronous frame is defined such that
g 00 = 1 , g 0 α = 0 {\displaystyle g_{00}=1,\quad g_{0\alpha }=0}
where α = 1 , 2 , 3. {\displaystyle \alpha =1,2,3.} 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 ≡ dx0 = 0 in the metric. The reason for this is the different dependence between proper time τ {\displaystyle \tau } and time coordinate x0 ≡ t in different points of space., i.e., c d τ = g 00 d x 0 . {\displaystyle cd\tau ={\sqrt {g_{00}}}dx^{0}.}
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 x α + d x α {\displaystyle x^{\alpha }+dx^{\alpha }} 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 d s = 0 {\displaystyle ds=0} in the above equation, we can solve for dx0 obtaining two roots:
d x 0 ( 1 ) = 1 g 00 ( − g 0 α d x α − ( g 0 α g 0 β − g α β g 00 ) d x α d x β ) , {\displaystyle dx^{0(1)}={\frac {1}{g_{00}}}\left(-g_{0\alpha }\,dx^{\alpha }-{\sqrt {\left(g_{0\alpha }g_{0\beta }-g_{\alpha \beta }g_{00}\right)\,dx^{\alpha }\,dx^{\beta }}}\right),}
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
d x 0 ( 2 ) − d x 0 ( 1 ) = 2 g 00 ( g 0 α g 0 β − g α β g 00 ) d x α d x β . {\displaystyle dx^{0(2)}-dx^{0(1)}={\frac {2}{g_{00}}}{\sqrt {\left(g_{0\alpha }g_{0\beta }-g_{\alpha \beta }g_{00}\right)\,dx^{\alpha }\,dx^{\beta }}}.}
The respective proper time interval is obtained from the above relationship by multiplication by g 00 / c {\displaystyle {\sqrt {g_{00}}}/c} , 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,
y 0 = ( x 0 + d x 0 ( 1 ) ) + ( x 0 + d x 0 ( 2 ) ) 2 = x 0 + 1 2 ( d x 0 ( 2 ) + d x 0 ( 1 ) ) = x 0 + Δ x 0 . {\displaystyle y^{0}={\frac {(x^{0}+dx^{0(1)})+(x^{0}+dx^{0(2)})}{2}}=x^{0}+{\tfrac {1}{2}}\left(dx^{0(2)}+dx^{0(1)}\right)=x^{0}+\Delta x^{0}.}
Substitute here eq. 2 to find the difference in "time" x0 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 g00 and bringing terms to the left hand side
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 g0α 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 g0α become zeros and thus enable a complete synchronization of clocks. To this class are assigned cases where g0α 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 r ′ ϕ ′ , z ′ {\displaystyle r'\,\phi ',\,z'} and time t ′ {\displaystyle t'} . The interval in this frame is given by d s 2 = c 2 d t ′ 2 − d r ′ 2 − r ′ 2 d ϕ ′ 2 − d z ′ 2 . {\displaystyle ds^{2}=c^{2}dt'^{2}-dr'^{2}-r'^{2}d\phi '^{2}-dz'^{2}.} Transforming to a uniformly rotating coordinate system ( r , ϕ , z ) {\displaystyle (r,\phi ,z)} using the relation x 0 / c = t = t ′ , x 1 = r = r ′ , x 2 = ϕ = ϕ ′ − Ω t ′ , x 3 = z = z ′ {\displaystyle x^{0}/c=t=t',\,x^{1}=r=r',\,x^{2}=\phi =\phi '-\Omega t',\,x^{3}=z=z'} modifies the interval to
d s 2 = ( c 2 − Ω 2 r 2 ) d t 2 − 2 Ω r 2 d ϕ d t − d r 2 − r 2 d ϕ 2 − d z 2 . {\displaystyle ds^{2}=(c^{2}-\Omega ^{2}r^{2})dt^{2}-2\Omega r^{2}d\phi dt-dr^{2}-r^{2}d\phi ^{2}-dz^{2}.}
Of course, the rotating frame is valid only for r < c / Ω {\displaystyle r<c/\Omega } since the frame speed would exceed speed of light beyond this radial location. The non-zero components of the metric tensor are g 00 = 1 − Ω 2 r 2 / c 2 , {\displaystyle g_{00}=1-\Omega ^{2}r^{2}/c^{2},} g 02 = − 2 Ω r 2 / c , {\displaystyle g_{02}=-2\Omega r^{2}/c,} g 11 = − 1 , {\displaystyle g_{11}=-1,} g 22 = − r 2 {\displaystyle g_{22}=-r^{2}} and g 33 = − 1. {\displaystyle g_{33}=-1.} Along any open curve, the relation
Δ x 0 = − g 0 α g 00 d x α = Ω r 2 / c 1 − Ω 2 r 2 / c 2 d ϕ {\displaystyle \Delta x^{0}=-{\frac {g_{0\alpha }}{g_{00}}}dx^{\alpha }={\frac {\Omega r^{2}/c}{1-\Omega ^{2}r^{2}/c^{2}}}d\phi }
can be used to synchronize clocks. However, along any closed curve, synchronization is impossible because
∮ Δ x 0 = ∮ d ϕ Ω r 2 / c 1 − Ω 2 r 2 / c 2 ≠ 0. {\displaystyle \oint \Delta x^{0}=\oint {\frac {d\phi \Omega r^{2}/c}{1-\Omega ^{2}r^{2}/c^{2}}}\neq 0.}
For instance, when Ω r / c ≪ 1 {\displaystyle \Omega r/c\ll 1} , we have
∮ Δ x 0 = Ω c ∮ r 2 d ϕ = ± 2 Ω c S {\displaystyle \oint \Delta x^{0}={\frac {\Omega }{c}}\oint r^{2}d\phi =\pm {\frac {2\Omega }{c}}S}
where S {\displaystyle S} 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
d τ = 1 − Ω 2 r 2 / c 2 d t = 1 − Ω 2 r 2 / c 2 d τ a x i s {\displaystyle d\tau ={\sqrt {1-\Omega ^{2}r^{2}/c^{2}}}dt={\sqrt {1-\Omega ^{2}r^{2}/c^{2}}}d\tau _{\mathrm {axis} }}
indicating that time slows down as we move away from the axis. Similarly the spatial element can be calculated to find
d l = [ d r 2 + r 2 d ϕ 2 1 − Ω 2 r 2 / c 2 + d z 2 ] 1 / 2 . {\displaystyle dl=\left[dr^{2}+{\frac {r^{2}d\phi ^{2}}{1-\Omega ^{2}r^{2}/c^{2}}}+dz^{2}\right]^{1/2}.}
At a fixed value of r {\displaystyle r} and z {\displaystyle z} , the spatial element is d l = ( 1 − Ω 2 r 2 / c 2 ) − 1 / 2 r d ϕ {\displaystyle dl=(1-\Omega ^{2}r^{2}/c^{2})^{-1/2}rd\phi } which upon integration over a full circle shows that the ratio of circumference of a circle to its radius is given by
2 π 1 − Ω 2 r 2 / c 2 {\displaystyle {\frac {2\pi }{\sqrt {1-\Omega ^{2}r^{2}/c^{2}}}}}
which is greater than by 2 π {\displaystyle 2\pi } .
Space metric tensor Eq. 3 can be rewritten in the form
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 γ α β {\displaystyle \gamma _{\alpha \beta }} and the metric of the four-dimensional spacetime g i k {\displaystyle g_{ik}} . In general, however, g i k {\displaystyle g_{ik}} depends on x0 so that γ α β {\displaystyle \gamma _{\alpha \beta }} 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 ∫ d l {\textstyle \int dl} along the space curve acquires some definite sense. The tensor − γ α β {\displaystyle -\gamma _{\alpha \beta }} is inverse to the contravariant 3-dimensional tensor g α β {\displaystyle g^{\alpha \beta }} . Indeed, writing equation g i k g k l = δ l i {\displaystyle g^{ik}g_{kl}=\delta _{l}^{i}} in components, one has:
g α β g β γ + g α 0 g 0 γ = δ γ α , {\displaystyle g^{\alpha \beta }g_{\beta \gamma }+g^{\alpha 0}g_{0\gamma }=\delta _{\gamma }^{\alpha },}
g 0 β g β 0 + g 00 g 00 = 1. {\displaystyle g^{0\beta }g_{\beta 0}+g^{00}g_{00}=1.}
Determining g α 0 {\displaystyle g^{\alpha 0}} from the second equation and substituting it in the first proves that
This result can be presented otherwise by saying that g α β {\displaystyle g^{\alpha \beta }} are components of a contravariant 3-dimensional tensor corresponding to metric γ α β {\displaystyle \gamma ^{\alpha \beta }} :
The determinants g and γ {\displaystyle \gamma } composed of elements g i k {\displaystyle g_{ik}} and γ α β {\displaystyle \gamma _{\alpha \beta }} , respectively, are related to each other by the simple relationship:
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 γ α β {\displaystyle \gamma _{\alpha \beta }} , its contravariant components can be written as g α = γ α β g β {\displaystyle g^{\alpha }=\gamma ^{\alpha \beta }g_{\beta }} . Using eq. 11 and the second of eqs. 8, it is easy to see that
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 g0α are zeros. If, in addition, g00 = 1, then the time coordinate x0 = t is the proper time in each space point (with c = 1). A reference frame that satisfies the conditions
is called synchronous frame. The interval element in this system is given by the expression
with the spatial metric tensor components identical (with opposite sign) to the components gαβ:
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, n0 = 1. The respective contravariant components with the conditions eq. 15 are again nα = 0, n0 = 1. The components of the unit normal coincide with those of the four-vector ui = dxi/ds which is tangent to the world line x1, x2, x3 = const. The ui with components uα = 0, u0 = 1 automatically satisfies the geodesic equations:
d u i d s + Γ k l i u k u l = Γ 00 i = 0 , {\displaystyle {\frac {du^{i}}{ds}}+\Gamma _{kl}^{i}u^{k}u^{l}=\Gamma _{00}^{i}=0,}
since, from the conditions eq. 15, the Christoffel symbols Γ 00 α {\displaystyle \Gamma _{00}^{\alpha }} and Γ 00 0 {\displaystyle \Gamma _{00}^{0}} 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
where S is the action. Its complete integral has the form:
Note that the complete integral contains as many arbitrary constants as the number of independent variables which in our case is 4 {\displaystyle 4} . 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 time coordinate, one obtains a synchronous frame; the transformation from the old coordinates to the new ones is given by equations 18b-18c. In fact, it is guaranteed that for such a transformation the time lines will be geodesics and will be normal to the hypersurfaces S = const. The latter point is obvious from the mechanical analogy: the four-vector ∂S/∂xi which is normal to the hypersurface coincides in mechanics with the four-momentum of the particle, and therefore coincides in direction with its four-velocity ui i.e. with the four-vector tangent to the trajectory. Finally the condition g00 = 1 is obviously satisfied, since the derivative −dS/ds of the action along the trajectory is the mass of the particle, which was set equal to 1; therefore |dS/ds| = 1. The gauge conditions eq. 15 do not fix the coordinate system completely and therefore are not a fixed gauge, as the spacelike hypersurface at t = 0 {\displaystyle t=0} can be chosen arbitrarily. One still have the freedom of performing some coordinate transformations containing four arbitrary functions depending on the three spatial variables xα, which are easily worked out in infinitesimal form:
Here, the collections of the four old coordinates (t, xα) and four new coordinates ( t ~ , x ~ α ) {\displaystyle ({\tilde {t}},{\tilde {x}}^{\alpha })} are denoted by the symbols x and x ~ {\displaystyle {\tilde {x}}} , respectively. The functions ξ i ( x ~ ) {\displaystyle \xi ^{i}({\tilde {x}})} together with their first derivatives are infinitesimally small quantities. After such a transformation, the four-dimensional interval takes the form:
where
In the last formula, the g i k ( x ~ ) {\displaystyle g_{ik}({\tilde {x}})} are the same functions gik(x) in which x should simply be replaced by x ~ {\displaystyle {\tilde {x}}} . If one wishes to preserve the gauge eq. 15 also for the new metric tensor g i k (new) ( x ~ ) {\displaystyle g_{ik}^{\text{(new)}}({\tilde {x}})} in the new coordinates x ~ {\displaystyle {\tilde {x}}} , it is necessary to impose the following restrictions on the functions ξ i ( x ´ ) {\displaystyle \xi ^{i}({\acute {x}})} :
The solutions of these equations are:
where f0 and fα are four arbitrary functions depending only on the spatial coordinates x ~ α {\displaystyle {\tilde {x}}^{\alpha }} . For a more elementary geometrical explanation, consider Fig. 2. First, the synchronous time line ξ0 = t can be chosen arbitrarily (Bob's, Carol's, Dana's or any of an infinitely many observers). This makes one arbitrarily chosen function: ξ 0 = f 0 ( x ~ 1 , x