Rindler coordinates

Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this hyperbolically accelerated frame can be compared to effects arising in a homogeneous gravitational field. For general overview of accelerations in flat spacetime, see Acceleration and Proper reference frame.
In this article, the speed of light is defined by, the inertial coordinates are, and the hyperbolic coordinates are. These hyperbolic coordinates can be separated into two main variants depending on the accelerated observer's position: If the observer is located at time at position , then the hyperbolic coordinates are often called Rindler coordinates with the corresponding Rindler metric. If the observer is located at time at position, then the hyperbolic coordinates are sometimes called Møller coordinates or Kottler–Møller coordinates with the corresponding Kottler–Møller metric. An alternative chart often related to observers in hyperbolic motion is obtained using Radar coordinates which are sometimes called Lass coordinates. Both the Kottler–Møller coordinates as well as Lass coordinates are denoted as Rindler coordinates as well.
Regarding the history, such coordinates were introduced soon after the advent of special relativity, when they were studied alongside the concept of hyperbolic motion: In relation to flat Minkowski spacetime by Albert Einstein, Max Born, Arnold Sommerfeld, Max von Laue, Hendrik Lorentz, Friedrich Kottler, Wolfgang Pauli, Karl Bollert, Stjepan Mohorovičić, Georges Lemaître, Einstein & Nathan Rosen, Christian Møller, Fritz Rohrlich, Harry Lass, and in relation to both flat and curved spacetime of general relativity by Wolfgang Rindler. For details and sources, see .

Characteristics of the Rindler frame

The worldline of a body in hyperbolic motion having constant proper acceleration in the -direction as a function of proper time and rapidity can be given by
where is constant and is variable, with the worldline resembling the hyperbola. Sommerfeld showed that the equations can be reinterpreted by defining as variable and as constant, so that it represents the simultaneous "rest shape" of a body in hyperbolic motion measured by a comoving observer. By using the proper time of the observer as the time of the entire hyperbolically accelerated frame by setting, the transformation formulas between the inertial coordinates and the hyperbolic coordinates are consequently:
with the inverse
Differentiated and inserted into the Minkowski metric
the metric in the hyperbolically accelerated frame follows as
These transformations define the Rindler observer as an observer that is "at rest" in Rindler coordinates, i.e., maintaining constant x, y, z, and only varying t as time passes. The coordinates are valid in the region, which is often called the Rindler wedge, if represents the proper acceleration of the Rindler observer whose proper time is defined to be equal to Rindler coordinate time. To maintain this world line, the observer must accelerate with a constant proper acceleration, with Rindler observers closer to having greater proper acceleration. All the Rindler observers are instantaneously at rest at time in the inertial frame, and at this time a Rindler observer with proper acceleration will be at position , which is also that observer's constant distance from the Rindler horizon in Rindler coordinates. If all Rindler observers set their clocks to zero at, then when defining a Rindler coordinate system we have a choice of which Rindler observer's proper time will be equal to the coordinate time in Rindler coordinates, and this observer's proper acceleration defines the value of above. It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration, so that can be eliminated from the equations.
The above equation has been simplified for. The unsimplified equation is more convenient for finding the Rindler Horizon distance, given an acceleration.
The remainder of the article will follow the convention of setting both and, so units for and will be 1 unit. Be mindful that setting light-second/second² is very different from setting light-year/year². Even if we pick units where, the magnitude of the proper acceleration will depend on our choice of units: for example, if we use units of light-years for distance, and years for time,, this would mean light year/year², equal to about 9.5 meters/second², while if we use units of light-seconds for distance,, and seconds for time,.

Variants of transformation formulas

A more general derivation of the transformation formulas is given, when the corresponding Fermi–Walker tetrad is formulated from which the Fermi coordinates or Proper coordinates can be derived. Depending on the choice of origin of these coordinates, one can derive the metric, the time dilation between the time at the origin and at point, and the coordinate light speed . Instead of Fermi coordinates, also Radar coordinates can be used, which are obtained by determining the distance using light signals, by which metric, time dilation and speed of light do not depend on the coordinates anymore – in particular, the coordinate speed of light remains identical with the speed of light in inertial frames:

at	Transformation, Metric, Time dilation and Coordinate speed of light
	Kottler–Møller coordinates
	---- ----
	Rindler coordinates
	---- ----
	Radar coordinates

	----

The Rindler observers

In the new chart with and, it is natural to take the coframe field
which has the dual frame field
This defines a local Lorentz frame in the tangent space at each event. The integral curves of the timelike unit vector field give a timelike congruence, consisting of the world lines of a family of observers called the Rindler observers. In the Rindler chart, these world lines appear as the vertical coordinate lines. Using the coordinate transformation above, we find that these correspond to hyperbolic arcs in the original Cartesian chart.
As with any timelike congruence in any Lorentzian manifold, this congruence has a kinematic decomposition. In this case, the expansion and vorticity of the congruence of Rindler observers vanish. The vanishing of the expansion tensor implies that each of our observers maintains constant distance to his neighbors. The vanishing of the vorticity tensor implies that the world lines of our observers are not twisting about each other; this is a kind of local absence of "swirling".
The acceleration vector of each observer is given by the covariant derivative
That is, each Rindler observer is accelerating in the direction. Individually speaking, each observer is in fact accelerating with constant magnitude in this direction, so their world lines are the Lorentzian analogs of circles, which are the curves of constant path curvature in the Euclidean geometry.
Because the Rindler observers are vorticity-free, they are also hypersurface orthogonal. The orthogonal spatial hyperslices are ; these appear as horizontal half-planes in the Rindler chart and as half-planes through in the Cartesian chart. Setting in the line element, we see that these have the ordinary Euclidean geometry,. Thus, the spatial coordinates in the Rindler chart have a very simple interpretation consistent with the claim that the Rindler observers are mutually stationary. We will return to this rigidity property of the Rindler observers a bit later in this article.

A "paradoxical" property

Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up. This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force must accelerate a bit harder than the leading endpoint, or else it must ultimately break. This is a manifestation of Lorentz contraction. As the rod accelerates, its velocity increases and its length decreases. Since it is getting shorter, the back end must accelerate harder than the front. Another way to look at it is: the back end must achieve the same change in velocity in a shorter period of time. This leads to a differential equation showing that, at some distance, the acceleration of the trailing end diverges, resulting in the Rindler horizon.
This phenomenon is the basis of a well known "paradox", Bell's spaceship paradox. However, it is a simple consequence of relativistic kinematics. One way to see this is to observe that the magnitude of the acceleration vector is just the path curvature of the corresponding world line. But the world lines of our Rindler observers are the analogs of a family of concentric circles in the Euclidean plane, so we are simply dealing with the Lorentzian analog of a fact familiar to speed skaters: in a family of concentric circles, inner circles must bend faster than the outer ones.