Near Horizon Coordinates (Rindler space)

The region near the horizon can be explored by replacing r by a coordinate ρ which measures proper distance from the horizon:

 (1.1)

In terms of ρ and t the metric takes the form

 (1.2)

Near the horizon equation 1.1 behaves like

 (1.3)

giving

 (1.4)

Furthermore, if we are interested in a small angular region of the horizon arbitrarily centered at θ = 0 we can replace the angular coordinates by Cartesian coordinates

 (1.5)

Finally, we can introduce a dimensionless time ω

(1.6)

and the metric then takes the form

 (1.7)

It is now evident that ρ and ω are radial and hyperbolic angle variables for an ordinary Minkowski space. Minkowski coordinates T , Z can be defined by

 (1.8)

to get the familiar Minkowski metric

 (1.9)

It should be kept in mind that equation 1.9 is only accurate near r = 2MG, and only for a small angular region. However it clearly demonstrates that the horizon is locally nonsingular, and, for a large black hole, is locally almost indistinguishable from flat space-time. In Figure 1.1 the relation between Minkowski coordinates and the ρ, ω coordinates is shown. The entire Minkowski space is divided into four quadrants labeled I, II, III, and IV. Only one of those regions, namely

Fig. 1.1. Relation between Minkowski and Rindler coordinates.

Region I lies outside the black hole horizon. The horizon itself is the origin T = Z = 0. Note that it is a two-dimensional surface in the four dimensional space-time. This may appear surprising, since originally the horizon was defined by the single constraint r = 2MG, and therefore appears to be a three dimensional surface. However, recall that at the horizon g₀₀ vanishes. Therefore the horizon has no extension or metrical size in the time direction.

        The approximation of the near-horizon region by Minkowski space is called the Rindler approximation. In particular the portion of Minkowski space approximating the exterior region of the black hole, i.e. Region I, is called Rindler space. The time-like coordinate, ω, is called Rindler time. Note that a translation of Rindler time ω → ω + constant is equivalent to a Lorentz boost in Minkowski space.