Synchronous frame

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.

To find dl in this case, time can be synchronized over two infinitesimally neighboring points in the following way (Fig.

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.

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.

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.

4 by g00 and bringing terms to the left hand side or, the "covariant differential" dx0 between two infinitesimally close points should be zero.

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.

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

5, the condition that allows clock synchronization in different space points is that metric tensor components g0α are zeros.

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: since, from the conditions eq.

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.

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.

First, the synchronous time line ξ0 = t can be chosen arbitrarily (Bob's, Carol's, Dana's or any of an infinitely many observers).

When discussing general solutions gαβ of the field equations in synchronous gauges, it is necessary to keep in mind that the gravitational potentials gαβ contain, among all possible arbitrary functional parameters present in them, four arbitrary functions of 3-space just representing the gauge freedom and therefore of no direct physical significance.

Synchronous coordinates are generally considered the most efficient reference system for doing calculations, and are used in many modern cosmology codes, such as CMBFAST.

Introduction of a synchronous frame allows one to separate the operations of space and time differentiation in the Einstein field equations.

This does not apply to operations of shifting indices in the space components of the four-tensors Rik, Tik.

are the three-dimensional Christoffel symbols constructed from γαβ: where the comma denotes partial derivative by the respective coordinate.

25, the components Rik = gilRlk of the Ricci tensor can be written in the form: Dots on top denote time differentiation, semicolons (";") denote covariant differentiation which in this case is performed with respect to the three-dimensional metric γαβ with three-dimensional Christoffel symbols

For other equations of state a similar situation can occur only in special cases when the pressure gradient vanishes in all or in certain directions.

of the metric tensor is the absolute value of the triple product of the row vectors in the matrix

31, containing the stress–energy tensors of matter and electromagnetic field, is a positive number because of the strong energy condition.

is a real symmetric matrix, the eigenvectors form an orthonormal basis defining a rectangular parallelepiped whose length, width, and height are the magnitudes of the three eigenvalues.

Continuing with the obliteration and equating the width to zero, one is left with a line of size length, a 1-dimensional space.

An analogy from geometrical optics is comparison of the singularity with caustics, such as the bright pattern in Fig.

The light rays are an analogue of the time lines of the free-falling observers localized on the synchronized hypersurface.

The glass of water is an analogue of the Einstein equations or the agent(s) behind them that bend the time lines to form the caustics pattern (the singularity).

One can distinguish an overlap of two-, one-, or zero-dimensional spaces, i.e., intermingling of surfaces and lines, some converging to a point (cusp) such as the arrowhead formation in the centre of the caustics pattern.

Figure 1. Synchronization of clocks in curved space through light signals.
Figure 2. A synchronous frame built with the choice of the timelike hypersurface t = const (teal color). Only one spatial coordinate x 1 = x is shown. The four observers have the same proper times x 0 = t which are normal to the hypersurface in their locally flat spacetimes (shown by the light cones ). The unit vector n 0 = u 0 = 1 is shown in yellow. There are no spatial velocity components ( u α = 0) so the common proper time is a geodesic line with a beginning at the hypersurface and a positive direction (red arrows).
Figure 3.