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 be used to 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 neglected. Since in this article only flat spacetime is considered, and by using the definition that special relativity is the theory of flat spacetime, while general relativity is a theory of gravitation in terms of curved spacetime, the article is consequently concerned with accelerated frames in special relativity. (For the representation of accelerations in inertial frames, see the article Acceleration (special relativity), where concepts such as three-acceleration, four-acceleration, proper acceleration, hyperbolic motion etc. are defined and related to each other.)

A fundamental property of such a frame is the employment of the proper time of the accelerated observer as the time of the frame itself. This is connected with the clock hypothesis (which is experimentally confirmed), according to which the proper time of an accelerated clock is unaffected by acceleration, thus the measured time dilation of the clock only depends on its momentary relative velocity. The related proper reference frames are constructed using concepts like comoving orthonormal tetrads, which can be formulated in terms of spacetime Frenet–Serret formulas, or alternatively using Fermi–Walker transport as a standard of non-rotation. If the coordinates are related to Fermi-Walker transport, the term Fermi coordinates is sometimes used, or proper coordinates in the general case when rotations are also involved. A special class of accelerated observers are those, which follow worldlines whose three curvatures are constant. These motions belong to the class of Born rigid motions, i.e. the motions at which the mutual distance of constituents of an accelerated body or congruence remains unchanged in its proper frame. Two examples are Rindler coordinates or Kottler-Møller coordinates for the proper reference frame of hyperbolic motion, and Born or Langevin coordinates in the case of uniform circular motion.

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