Rindler coordinates

The Rindler coordinate system describes a uniformly accelerating frame of reference in Minkowski space. In special relativity, a uniformly accelarating 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

<math>ds^2=dt^2-dx^2-dy^2-dz^2<math>

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

<math>x/t=\coth{T}<math>

<math>x^2-t^2=X^2<math>

<math>y=Y<math>

<math>z=Z<math>

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

<math>ds^2=X^2dT^2-dX^2-dY^2-dZ^2<math>

Rindler coordinates are analogous to cylindrical coordinates via a Wick rotation.

This coordinate system does not cover the whole of Minkowski spacetime but rather a wedge (called a Rindler wedge or Rindler space). There is a coordinate singularity at <math>X=0<math> which correspond to the event horizon. It is possible to extend the wedge, by symmetry, to the left quadrant if we don't restrict <math>X<math>, resulting in time running "backwards" within that quadrant. The singularity can then be eliminated by substituting the coordinate <math>X<math> with the coordinate <math>R<math> where

<math>2R-1=X^2<math>

with the metric now taking the form

<math>ds^2=(2R-1)dT^2-(2R-1)^{-1}dR^2-dY^2-dZ^2<math>.

Translations along <math>T<math> are described by a Killing vector, meaning it is an isometry of Minkowski space and a Lorentz boost.

See also Unruh effect

Further Reading

Relativity: Special, General and Cosmological by Wolfgang Rindler ISBN 0198508352