The Rindler Observers
In the new chart, it is natural to take the coframe field
which has the dual frame field
This defines a local Lorentz frame in the tangent space at each event (in the region covered by our Rindler chart, namely the Rindler wedge). The integral curves of the timelike unit vector field give a timelike congruence, consisting of the world lines of a family of observers called the Rindler observers. In the Rindler chart, these world lines appear as the vertical coordinate lines . Using the coordinate transformation above, we find that these correspond to hyperbolic arcs in the original Cartesian chart.
As with any timelike congruence in any Lorentzian manifold, this congruence has a kinematic decomposition (see Raychaudhuri equation). In this case, the expansion and vorticity of the congruence of Rindler observers vanish. The vanishing of the expansion tensor implies that each of our observers maintains constant distance to his neighbors. The vanishing of the vorticity tensor implies that the world lines of our observers are not twisting about each other; this is a kind of local absence of "swirling".
The acceleration vector of each observer is given by the covariant derivative
That is, each Rindler observer is accelerating in the direction. Individually speaking, each observer is in fact accelerating with constant magnitude in this direction, so their world lines are the Lorentzian analogs of circles, which are the curves of constant path curvature in the Euclidean geometry.
Because the Rindler observers are vorticity-free, they are also hypersurface orthogonal. The orthogonal spatial hyperslices are ; these appear as horizontal half-planes in the Rindler chart and as half-planes through in the Cartesian chart (see the figure above). Setting in the line element, we see that these have the ordinary Euclidean geometry, . Thus, the spatial coordinates in the Rindler chart have a very simple interpretation consistent with the claim that the Rindler observers are mutually stationary. We will return to this rigidity property of the Rindler observers a bit later in this article.
Read more about this topic: Rindler Coordinates
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