Rindler Coordinates - The Rindler Horizon

The Rindler Horizon

The Rindler coordinate chart has a coordinate singularity at x = 0, where the metric tensor (expressed in the Rindler coordinates) has vanishing determinant. This happens because as x → 0 the acceleration of the Rindler observers diverges. As we can see from the figure illustrating the Rindler wedge, the locus x = 0 in the Rindler chart corresponds to the locus T2 = X2, X > 0 in the Cartesian chart, which consists of two null half-planes, each ruled by a null geodesic congruence.

For the moment, we simply consider the Rindler horizon as the boundary of the Rindler coordinates. If we consider the set of accelerating observers who have a constant position in Rindler coordinates, none of them can ever receive light signals from events with T ≥ X (on the diagram, these would be events on or to the left of the line T = X which the upper red horizon lies along; these observers could however receive signals from events with T ≥ X if they stopped their acceleration and crossed this line themselves) nor can they ever send signals to events with T ≤ −X (events on or to the left of the line T = −X which the lower red horizon lies along; in this case the acceleration is actually irrelevant, no observer to the right of this line can send a signal to events with T ≤ −X simply because those events lie outside their own future light cone). Also, if we consider members of this set of accelerating observers closer and closer to the horizon, in the limit as the distance to the horizon approaches zero, the constant proper acceleration experienced by an observer at this distance (which would also be the G-force experienced by such an observer) would approach infinity. Both of these facts would also be true if we were considering a set of observers hovering outside the event horizon of a black hole, each observer hovering at a constant radius in Schwarzschild coordinates. In fact, in the close neighborhood of a black hole, the geometry close to the event horizon can be described in Rindler coordinates. Hawking radiation in the case of an accelerating frame is referred to as Unruh radiation. The connection is the equivalence of acceleration with gravitation.