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

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

x/t=\coth{T}

x^2-t^2=X^2

y=Y

z=Z

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

ds^2=X^2dT^2-dX^2-dY^2-dZ^2

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

X=0

which correspond to the event horizon. It is possible to extend the wedge, by symmetry, to the left quadrant if we don't restrict

X

, resulting in time running "backwards" within that quadrant. The singularity can then be eliminated by substituting the coordinate

X

with the coordinate

R

where

2R-1=X^2

with the metric now taking the form

ds^2=(2R-1)dT^2-(2R-1)^{-1}dR^2-dY^2-dZ^2

Translations along

T

are described by a Killing vector, meaning it is an isometry of Minkowski space and a Lorentz boost. See also Unruh effect

Rindler Coordinates

Further Reading