Rindler coordinates/Old

From Wikipedia, the free encyclopedia

This article or section is in need of attention from an expert on the subject.
WikiProject Physics or the Physics Portal may be able to help recruit one.
If a more appropriate WikiProject or portal exists, please adjust this template accordingly.

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

Two-dimensional representation of the Rindler coordinate system with T as the "angular" coordinate and X as the "radial" coordinate.

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

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

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 = X s i n h (T)

x = X c o s h (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 $X < 0$ as a parameter. Quadrants II and IV can be included by using the following alternate relations