Rindler coordinates/Old

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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