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.Misner & Thorne & Wheeler (1973), p. 163: "Accelerated motion and accelerated observers can be analyzed using special relativity."Koks (2006), p. 234. "It is sometimes said that to describe physics properly in an accelerated frame, special relativity is insufficient, and the full machinery of general relativity is necessary for the job. This is quite wrong. Special relativity is entirely sufficient to derive the physics of an accelerated frame."In some textbooks the same formulas and results for flat spacetime are discussed in the framework of GR, using the historical definition that SR is restricted to inertial frames, while accelerated frames belong to the framework of GR. However, since the results are the same in terms of flat spacetime, it does not affect the content of this article. For instance, Møller (1952) discusses successive Lorentz transformations, successive inertial frames, and tetrad transport (now called Fermi–Walker transport) in §§ 46, 47 related to special relativity, while rigid references frames are discussed in section §§ 90, 96 related to general 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 () 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).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)Møller (1952), §§ 46, 47, 90, 96 or Synge (1960).Synge (1960), §§ 3, 4 An overview of proper time transformations and alternatives was given by Romain (1963),Romain (1963), particularly section VI for the "proper time approach" 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)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)Bini & Cherubini & Geralico & Jantzen (2008), section 3 and others. A detailed representation of "special relativity in general frames" was given by Gourgoulhon (2013).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 (also called vielbein, moving frame, frame field, local frame, repère mobile in arbitrary dimensions):Synge (1960), § 3Iyer and Vishveshwara (1993), section 2.1Formiga & Romero (2006), section 2Gourgoulhon (2013), section 2.7.3
Here, is the proper time along the worldline, the timelike field is called the tangent that corresponds to the four-velocity, the three spacelike fields are orthogonal to and are called the principal normal , the binormal and the trinormal . The first curvature corresponds to the magnitude of four-acceleration (i.e., proper acceleration), the other curvatures and are also called torsion and hypertorsion.
Fermi–Walker transport and proper transport
While the Frenet–Serret tetrad can be rotating or not, it is useful to introduce another formalism in which non-rotational and rotational parts are separated. This can be done using the following equation for proper transportKajari & Buser & Feiler & Schleich (2009), section 3 or generalized Fermi transportHehl & Lemke & Mielke (1990), section I.6 of tetrad , namelyMisner & Thorne & Wheeler (1973), section 6.8Iyer and Vishveshwara (1993), section 2.2Padmanabhan (2010), section 4.9Gourgoulhon (2013), section 3.5.3
where
or together in simplified form:
with as four-velocity and as four-acceleration, and "" indicates the dot product and "" the wedge product. The first part represents Fermi–Walker transport, 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. Thus Fermi–Walker transport can be seen as a standard of non-rotation. The second part consists of an antisymmetric second rank tensor with as the angular velocity four-vector and as the Levi-Civita symbol. It turns out that this rotation matrix only affects the three spacelike tetrad fields, thus it can be interpreted as the spatial rotation of the spacelike fields of a rotating tetrad (such as a Frenet–Serret tetrad) with respect to the non-rotating spacelike fields of a Fermi–Walker tetrad along the same world line.
Deriving Fermi–Walker tetrads from Frenet–Serret tetrads
Since and 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,Johns (2005), section 18.19 which not only works in flat spacetime but for arbitrary spacetimes as well, even though the practical realization can be hard to achieve.Maluf & Faria (2008) For instance, the angular velocity vector between the respective spacelike tetrad fields and can be given in terms of torsions and :Johns (2005), section 18.18Bini & Cherubini & Geralico & Jantzen (2008), section 3.1Gourgoulhon (2013), eq. 3.58
Assuming that the curvatures are constant (which is the case in helical motion in flat spacetime, or in the case of stationary axisymmetric spacetimes), one then proceeds by aligning the spacelike Frenet–Serret vectors in the plane by constant counter-clockweise rotation, then the resulting intermediary spatial frame is constantly rotated around the axis by the angle , which finally gives the spatial Fermi–Walker frame (note that the timelike field remains the same):Bini & Cherubini & Geralico & Jantzen (2008), section 3.2
For the special case and , it follows and and , therefore () is reduced to a single constant rotation around the -axis:Irvine (1964), section VII, eq. 41Bini & Jantzen (2003), Appendix AMashhoon (2003), section 3, eq. 1.17, 1.18
Proper coordinates or Fermi coordinates
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 , where is an external inertial frame and the Lorentz transformation matrix. This matrix can be replaced by the proper time dependent tetrads defined above, and if is the time track of the particle indicating its position, the transformation reads:Møller (1952), § 46
Then one has to put by which is replaced by and the timelike field vanishes, therefore only the spacelike fields are present anymore. Subsequently, the time in the accelerated frame is identified with the proper time of the accelerated observer by . The final transformation has the formMøller (1952), § 96Hehl & Lemke & Mielke (1990), section I.8Mashhoon & Muench (2002), section 2Kopeikin & Efroimsky & Kaplan (2011), section 2.6
These are sometimes called proper coordinates, and the corresponding frame is the proper reference frame. They are also called Fermi coordinates in the case of Fermi–Walker transportSynge (1960), § 10 (even though some authors use this term also in the rotational case). The corresponding metric has the form in Minkowski spacetime (without Riemannian terms):Ni & Zimmermann (1978), including Riemannian termsHehl & Lemke & Mielke (1990), section I.8, without Riemannian termsMarzlin (1994), section 2, including Riemannian termsNikolić (1999), section 2, without Riemannian termsMashhoon & Münch (2002), section 2, without Riemannian termsBini & Jantzen (2002), section 2, including Riemannian termsVoytik (2011), section 2, without Riemannian termsMisner & Thorne & Wheeler (1973), section 13.6, gave the first order approximation to this metric, without Riemannian terms
However, these coordinates are not globally valid, but are restricted to
Proper reference frames for timelike helices