Rindler Coordinates - Minkowski Observers

Minkowski Observers

It is worthwhile to also introduce an alternative frame, given in the Minkowski chart by the natural choice

Transforming these vector fields using the coordinate transformation given above, we find that in the Rindler chart (in the Rinder wedge) this frame becomes

\begin{align} \vec{f}_0 &= \frac{1}{x}\cosh(t) \, \partial_t - \sinh(t) \, \partial_x\\ \vec{f}_1 &= -\frac{1}{x}\sinh(t) \, \partial_t + \cosh(t) \, \partial_x\\ \vec{f}_2 &= \partial_y, \; \vec{f}_3 = \partial_z \end{align}

Computing the kinematic decomposition of the timelike congruence defined by the timelike unit vector field, we find that the expansion and vorticity again vanishes, and in addition the acceleration vector vanishes, . In other words, this is a geodesic congruence; the corresponding observers are in a state of inertial motion. In the original Cartesian chart, these observers, whom we will call Minkowski observers, are at rest.

In the Rindler chart, the world lines of the Minkowski observers appear as hyperbolic secant curves asymptotic to the coordinate plane . Specifically, in Rindler coordinates, the world line of the Minkowski observer passing through the event is

\begin{align} t &= \operatorname{arctanh}\left(\frac{s}{x_0}\right),\; -x_0 < s < x_0\\ x &= \sqrt{x_0^2-s^2},\; -x_0 < s < x_0\\ y &= y_0\\ z &= z_0 \end{align}

where is the proper time of this Minkowski observer. Note that only a small portion of his history is covered by the Rindler chart! This shows explicitly why the Rindler chart is not geodesically complete; timelike geodesics run outside the region covered by the chart in finite proper time. Of course, we already knew that the Rindler chart cannot be geodesically complete, because it covers only a portion of the original Cartesian chart, which is a geodesically complete chart.

In the case depicted in the figure, and we have drawn (correctly scaled and boosted) the light cones at .

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