Synchronous Frame - Synchronous Frame

Synchronous Frame

As concluded from eq. 17, the condition that allows clock synchronization in different space points is that metric tensor components g_0α 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

(eq. 18)

is called synchronous frame. The interval element in this system is given by the expression

(eq. 19)

with the space metric tensor components identical (with opposite sign) to the components g_αβ:

(eq. 20)

In synchronous frame time, lines are geodesics in the four-dimensional spacetime. These lines are normal to the hypersurfaces t = const. Indeed, the 4-vector normal to this hypersurface n_i = ∂t/∂xi has covariant components n_α = 0, n₀ = 1. The respective contravariant components at conditions eq. 18 are again nα = 0, n0 = 1, that is, coincide with the components of the 4-vector u i tangential to the time lines.

Conversely, these properties can be used to construct synchronous frame in any spacetime. To this end, choose some time-like 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.

Such a construct, and hence, choice of synchronous frame, is always possible. Moreover, such choice is not unique. Metric of type eq. 19 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.

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 (with a unit mass) in a gravitational field is.