Rindler coordinates/Old

The Rindler coordinate system describes a uniformly accelerating frame of reference in Minkowski space. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion.

Minkowski space is the topologically trivial flat pseudo Riemannian manifold with Lorentzian signature. This is a coordinate-free description of it. One possible coordinatization of it (the standard one) is the Cartesian coordinate system

ds² = dt² − dx² − dy² − dz²

It is possible to use another coordinate system with the coordinates T, X, Y, and Z. These two coordinate systems are related according to

t = Xsinh(T)

x = Xcosh(T)

y = Y

z = Z

for X > 0.

In this coordinate system, the metric takes on the following form:

ds² = X²dT² − dX² − dY² − dZ²

This coordinate system does not cover the whole of Minkowski spacetime but rather a wedge (called a Rindler wedge or Rindler space). If we define this wedge as quadrant I, then the coordinate system can be extended to include quandrant III by simply allowing X < 0 as a parameter. Quadrants II and IV can be included by using the following alternate relations

t = Xcosh(T)

x = Xsinh(T),

in which case the metric becomes

ds² = − X²dT² + dX² − dY² − dZ²

Furthermore, defining a variable R where

2R − 1 = x² − t²

results in a single expression for the metric for all quadrants

ds² = (2R−1)dT² − (2R−1)⁻¹dR² − dY² − dZ².

Rindler coordinates are analogous to cylindrical coordinates via a Wick rotation. See also Unruh effect

Observers in an Accelerated Reference Frame

Truncated Minkowski space with T as the "angular" coordinate and R as the "radial" coordinate.

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