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. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it 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 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. In the following, Greek indices run over 0,1,2,3, Latin indices over 1,2,3, and bracketed indices are related to tetrad vector fields. The signature of the metric tensor is (-1,1,1,1).
History
Some properties of Kottler-Møller or Rindler coordinates were anticipated by Albert Einstein (1907) when he discussed the uniformly accelerated reference frame. While introducing the concept of Born rigidity, Max Born (1909) recognized that the formulas for the worldline of hyperbolic motion can be reinterpreted as transformations into a "hyperbolically accelerated reference system". Born himself, as well as Arnold Sommerfeld (1910) and Max von Laue (1911) used this frame to compute the properties of charged particles and their fields (see Acceleration (special relativity)#History and Rindler coordinates#History). In addition, Gustav Herglotz (1909) gave a classification of all Born rigid motions, including uniform rotation and the worldlines of constant curvatures. Friedrich Kottler (1912, 1914) introduced the "generalized Lorentz transformation" for proper reference frames or proper coordinates (German: Eigensystem, Eigenkoordinaten) by using comoving Frenet–Serret tetrads, and applied this formalism to Herglotz' worldlines of constant curvatures, particularly to hyperbolic motion and uniform circular motion. Herglotz' formulas were also simplified and extended by Georges Lemaître (1924). The worldlines of constant curvatures were rediscovered by several author, for instance, by Vladimír Petrův (1964), as "timelike helices" by John Lighton Synge (1967) or as "stationary worldlines" by Letaw (1981). The concept of proper reference frame was later reintroduced and further developed in connection with Fermi–Walker transport in the textbooks by Christian Møller (1952) or Synge (1960). An overview of proper time transformations and alternatives was given by Romain (1963), who cited the contributions of Kottler. In particular, Misner & Thorne & Wheeler (1973) combined Fermi–Walker transport with rotation, which influenced many subsequent authors. Bahram Mashhoon (1990, 2003) analyzed the hypothesis of locality and accelerated motion. The relations between the spacetime Frenet–Serret formulas and Fermi–Walker transport was discussed by Iyer & C. V. Vishveshwara (1993), Johns (2005) or Bini et al. (2008) and others. A detailed representation of "special relativity in general frames" was given by Gourgoulhon (2013).
Comoving tetrads
Spacetime Frenet–Serret equations For the investigation of accelerated motions and curved worldlines, some results of differential geometry can be used. For instance, the Frenet–Serret formulas for curves in Euclidean space have already been extended to arbitrary dimensions in the 19th century, and can be adapted to Minkowski spacetime as well. They describe the transport of an orthonormal basis attached to a curved worldline, so in four dimensions this basis can be called a comoving tetrad or vierbein e ( η ) {\displaystyle \mathbf {e} _{(\eta )}} (also called vielbein, moving frame, frame field, local frame, repère mobile in arbitrary dimensions):
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