Theory of relativity/Rindler coordinates

Rindler coordinates(1) are coordinates appropriate for an observer undergoing constant proper acceleration (a constant g-force felt) in an otherwise flat spacetime. Given an unprimed inertial frame set of coordinates, one assigns the accelerated frame observer primed coordinates, "Rindler coordinates", given by

where is the constant proper acceleration.


First Rindler solved the equation of motion for a constant proper acceleration problem. If we go to an inertial frame for which a test mass is instantaneously at rest, the four-vector of acceleration will relate to the coordinate acceleration by


, but the length of the four-acceleration is an invariant we're calling the proper acceleration  , so the inertial frame of instantaneous rest coordinate acceleration relates to the proper acceleration by


The coordinate acceleration according to the inertial frame of instantaneous rest relates to the ordinary force by Newton's second law   so we have


The x component of the ordinary force for a particle of velocity   with respect to the unprimed frame transforms between the two according to


And we are considering force along the direction of motion only so


So our equation for proper acceleration becomes


which means that the equation of motion for the case of constant proper acceleration is given by a constant ordinary force constant mass problem given according to whatever inertial frame we use. So we need to solve the equation of motion for a constant ordinary force, constant mass problem.


Integrating and choosing initial conditions so that its initially at zero velocity yields


Using   one can then solve for the time as a function of the proper time


and using


one can solve for the position as a function of time


and then the position as a function of proper time


Having both the position and time now as a function of proper time


We merely make a natural choice of primed coordinates for which   is described by those equations of motion. Rindler's choice was


which does just that.

Rindler MetricEdit

Rindler's spacetime is then just an accelerated frame transformation of the Minkowski metric of special relativity. Doing the transformation


transforms the Minkowski line element




Generalizing beyond Rindler'sEdit

The Lorentz transformation can be written


If we let   and   be functions of the primed time, we can do a transformation given by (2)


where we are doing anti-derivatives with respect to the primed coordinate time.

It turns out that Rindler's choice of coordinates for an observer undergoing constant proper acceleration is the special case of this transformation where   and   are what you would get as a function of the primed time if they were describing constant proper acceleration. Doing this more general transformation for arbitrary time dependent acceleration the line element transforms to


so for arbitrarily time dependent acceleration, with these transformations you still get the Rindler spacetime, only   is now any function of   instead of a constant.

Further, it turns out that for  ,  , and   being any three arbitrary functions of   that the line element given by


Is a zero Riemann tensor exact vacuum solution and so represents a more general accelerated frame transformation of the Minkowski metric of special relativity.

Rindler HorizonEdit

  • The path of constant proper acceleration as observed by an inertial frame

  • The red curve of the image shows the path of something undergoing constant proper acceleration with respect to the unprimed inertial frame coordinates which we found was given by


    Note it asymptotically approaches the line


    The path of a light speed signal lays at 45 degrees. Therefore information from any event in the shaded region will never intersect the path of the accelerated observer so long as he maintains his constant proper acceleration. As such that asymptote constitutes an event horizon from the perspective of the accelerated observer beyond which he can not be reached by any information there. So let's find out where in the accelerated observer's coordinates this asymptote is. Refer to the transformation to Rindler coordinates


    First lets move the constant on the second


    Next lets replace the   from the equation for the asymptote.


    Then square both sides


    Now subtract the top equation from the bottom equation


    Hyperbolic trig identity


    Now the solution to this is


    So we see that the event horizon for the accelerated observer is observed by him as a constant distance coordinate behind him. This time-time element of the contravariant metric tensor in his coordinates is


    Note that the event horizon corresponds to a singularity in this element of the metric tensor for his coordinates.

    Comparing to SchwarzschildEdit

    In comparing the physics an observer of constant proper acceleration observes to that a remote observer from an uncharged nonrotating black hole using Schwarzschild coordinates observes, it is best to choose the remote behavior of the accelerated observer's spatial coordinate scaled by the transformation


    With this choice of distance coordinate for the accelerated observer, the line element becomes


    which is now comperable to the Schwarzschild metric in Schwarzschild coordinates expressed by



    (1) Rindler, W., 1969, Essential Relativity: Special, General, and Cosmological, Van Nostrand, New York

    (2) Relativity equation 3.3.8