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Concept
Rindler coordinates
Summary
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 (special relativity) and Proper reference frame (flat spacetime).
In this article, the speed of light is defined by c = 1, the inertial coordinates are (X, Y, Z, T), and the hyperbolic coordinates are (x, y, z, t). These hyperbolic coordinates can be separated into two main variants depending on the accelerated observer's position: If the observer is located at time T = 0 at position X = 1/α (with α as the constant proper acceleration measured by a comoving accelerometer), then the hyperbolic coordinates are often called Rindler coordinates with the corresponding Rindler metric. If the observer is located at time T = 0 at position X = 0, 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.
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Acceleration (special relativity)
Accelerations in special relativity (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which also leads to more complex definitions of "acceleration". SR as the theory of flat Minkowski spacetime remains valid in the presence of accelerations, because general relativity (GR) is only required when there is curvature of spacetime caused by the energy–momentum tensor (which is mainly determined by mass).
Proper reference frame (flat spacetime)
A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded.
Ehrenfest paradox
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity. In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary.