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.
A special class of accelerated observers follow worldlines whose three curvatures are constant.
Some properties of Kottler-Møller or Rindler coordinates were anticipated by Albert Einstein (1907)[H 1] when he discussed the uniformly accelerated reference frame.
In addition, Gustav Herglotz (1909)[H 5] gave a classification of all Born rigid motions, including uniform rotation and the worldlines of constant curvatures.
Friedrich Kottler (1912, 1914)[H 6] 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.
[6] 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)[7] or Synge (1960).
[8] An overview of proper time transformations and alternatives was given by Romain (1963),[9] who cited the contributions of Kottler.
In particular, Misner & Thorne & Wheeler (1973)[10] combined Fermi–Walker transport with rotation, which influenced many subsequent authors.
The relations between the spacetime Frenet–Serret formulas and Fermi–Walker transport was discussed by Iyer & C. V. Vishveshwara (1993),[12] Johns (2005)[13] or Bini et al. (2008)[14] and others.
[15] 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.
represents Fermi–Walker transport,[13] which is physically realized when the three spacelike tetrad fields do not change their orientation with respect to the motion of a system of three gyroscopes.
on the same worldline are connected by a rotation matrix, it is possible to construct non-rotating Fermi–Walker tetrads using rotating Frenet–Serret tetrads,[24][25] which not only works in flat spacetime but for arbitrary spacetimes as well, even though the practical realization can be hard to achieve.
[26] For instance, the angular velocity vector between the respective spacelike tetrad fields
plane by constant counter-clockweise rotation, then the resulting intermediary spatial frame
-axis:[29][30][31][24] In flat spacetime, an accelerated object is at any moment at rest in a momentary inertial frame
, and the sequence of such momentary frames which it traverses corresponds to a successive application of Lorentz transformations
is the time track of the particle indicating its position, the transformation reads:[32] Then one has to put
The corresponding metric has the form in Minkowski spacetime (without Riemannian terms):[39][40][41][42][43][44][45][46] However, these coordinates are not globally valid, but are restricted to[43] In case all three Frenet–Serret curvatures are constant, the corresponding worldlines are identical to those that follow from the Killing motions in flat spacetime.
They are of particular interest since the corresponding proper frames and congruences satisfy the condition of Born rigidity, that is, the spacetime distance of two neighbouring worldlines is constant.
[47][48] These motions correspond to "timelike helices" or "stationary worldlines", and can be classified into six principal types: two with zero torsions (uniform translation, hyperbolic motion) and four with non-zero torsions (uniform rotation, catenary, semicubical parabola, general case):[49][50][4][5][6][51][52][53][54] Case
is the constant proper acceleration in the direction of motion, produce hyperbolic motion because the worldline in the Minkowski diagram is a hyperbola:[55][56][57][58][59][60] The corresponding orthonormal tetrad is identical to an inverted Lorentz transformation matrix with hyperbolic functions
, the accelerated observer is always located at the origin, so the Kottler-Møller coordinates follow[67][68][62][69][70] which are valid within
, and the metric Frames (6d, 6e, 6f) can be used to describe the geometry of rotating platforms, including the Ehrenfest paradox and the Sagnac effect.
In addition to the things described in the previous #History section, the contributions of Herglotz, Kottler, and Møller are described in more detail, since these authors gave extensive classifications of accelerated motion in flat spacetime.
For the latter, Herglotz gave the following coordinate transformation corresponding to the trajectories of a family of motions:
is a six-vector (i.e., an antisymmetric four-tensor of second order, or bivector, having six independent components) representing the angular velocity of
The latter have been studied in the 19th century, and were categorized by Felix Klein into loxodromic, elliptic, hyperbolic, and parabolic motions (see also Möbius group).
Kottler also defined a tetrad whose basis vectors are fixed in normal space and therefore do not share any rotation.
who are "rigidly" connected with the tangent, thus The second case is a vector "fixed" in normal space by setting