Rindler coordinates/Old

} on their talk page(s)." 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 coordinatizationof it (the standard one) is the Cartesian coordinate system

:$ds^2\; ,\; =dt^2-dx^2-dy^2-dz^2$

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:

:$ds^2\; ,\; =X^2dT^2-dX^2-dY^2-dZ^2$

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 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

:$ds^2\; ,\; =-X^2dT^2+dX^2-dY^2-dZ^2$

Furthermore, defining a variable $R$ where

:$2R-1=x^2-t^2$

results in a single expression for the metric for all quadrants

:$ds^2=(2R-1)dT^2-(2R-1)^\{-1\}dR^2-dY^2-dZ^2$.

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

Observers in an Accelerated Reference Frame

Further reading

* "Relativity: Special, General and Cosmological" by Wolfgang Rindler ISBN 0-19-850835-2