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. 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 () 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 formula_1 (also called vielbein, moving frame, frame field, local frame, repère mobile in arbitrary dimensions):
Here, formula_2 is the proper time along the worldline, the timelike field formula_3 is called the tangent that corresponds to the four-velocity, the three spacelike fields are orthogonal to formula_3 and are called the principal normal formula_5, the binormal formula_6 and the trinormal formula_7. The first curvature formula_8 corresponds to the magnitude of four-acceleration (i.e., proper acceleration), the other curvatures formula_9 and formula_10 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 transport or generalized Fermi transport of tetrad formula_1, namely
where
or together in simplified form:
with formula_14 as four-velocity and formula_15 as four-acceleration, and "formula_16" indicates the dot product and "formula_17" the wedge product. The first part formula_18 represents Fermi–Walker transport, which is physically realized when the three spacelike tetrad fields don't 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 formula_19 consists of an antisymmetric second rank tensor with formula_20 as the angular velocity four-vector and formula_21 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 formula_22 of a rotating tetrad (such as a Frenet–Serret tetrad) with respect to the non-rotating spacelike fields formula_23 of a Fermi–Walker tetrad along the same world line.
Deriving Fermi–Walker tetrads from Frenet–Serret tetrads.
Since formula_23 and formula_22 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, which not only works in flat spacetime but for arbitrary spacetimes as well, even though the practical realization can be hard to achieve. For instance, the angular velocity vector between the respective spacelike tetrad fields formula_23 and formula_22 can be given in terms of torsions formula_9 and formula_10:
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 formula_30 plane by constant counter-clockweise rotation, then the resulting intermediary spatial frame formula_31 is constantly rotated around the formula_32 axis by the angle formula_33, which finally gives the spatial Fermi–Walker frame formula_23 (note that the timelike field remains the same):
For the special case formula_35 and formula_36, it follows formula_37 and formula_38 and formula_39, therefore () is reduced to a single constant rotation around the formula_7-axis:
Proper coordinates or Fermi coordinates.
In flat spacetime, an accelerated object is at any moment at rest in a momentary inertial frame formula_41, and the sequence of such momentary frames which it traverses corresponds to a successive application of Lorentz transformations formula_42, where formula_43 is an external inertial frame and formula_44 the Lorentz transformation matrix. This matrix can be replaced by the proper time dependent tetrads formula_45 defined above, and if formula_46 is the time track of the particle indicating its position, the transformation reads:
Then one has to put formula_47 by which formula_48 is replaced by formula_49 and the timelike field formula_3 vanishes, therefore only the spacelike fields formula_22 are present anymore. Subsequently, the time in the accelerated frame is identified with the proper time of the accelerated observer by formula_52. The final transformation has the form
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 transport (even though some authors use this term also in the rotational case). The corresponding metric has the form in Minkowski spacetime (without Riemannian terms):
However, these coordinates are not globally valid, but are restricted to
Proper reference frames for timelike helices.
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. 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):
Case formula_53 produces uniform translation without acceleration. The corresponding proper reference frame is therefore given by ordinary Lorentz transformations. The other five types are:
Hyperbolic motion.
The curvatures formula_54 formula_55, where formula_56 is the constant proper acceleration in the direction of motion, produce hyperbolic motion because the worldline in the Minkowski diagram is a hyperbola:
The corresponding orthonormal tetrad is identical to an inverted Lorentz transformation matrix with hyperbolic functions formula_57 as Lorentz factor and formula_58 as proper velocity and formula_59 as rapidity (since the torsions formula_9 and formula_10 are zero, the Frenet–Serret formulas and Fermi–Walker formulas produce the same tetrad):
Inserted into the transformations () and using the worldline () for formula_62, the accelerated observer is always located at the origin, so the Kottler-Møller coordinates follow
which are valid within formula_64, with the metric
Alternatively, by setting formula_66 the accelerated observer is located at formula_67 at time formula_68, thus the Rindler coordinates follow from () and (, ):
which are valid within formula_70, with the metric
Uniform circular motion.
The curvatures formula_72, formula_35 produce uniform circular motion, with the worldline
where
with formula_74 as orbital radius, formula_75 as coordinate angular velocity, formula_76 as proper angular velocity, formula_77 as tangential velocity, formula_78 as proper velocity, formula_79 as Lorentz factor, and formula_80 as angle of rotation. The tetrad can be derived from the Frenet–Serret equations (), or more simply be obtained by a Lorentz transformation of the tetrad formula_81 of ordinary rotating coordinates:
The corresponding non-rotating Fermi–Walker tetrad formula_82 on the same worldline can be obtained by solving the Fermi–Walker part of equation (). Alternatively, one can use () together with (), which gives
The resulting angle of rotation formula_84 together with () can now be inserted into (), by which the Fermi–Walker tetrad follows
In the following, the Frenet–Serret tetrad is used to formulate the transformation. Inserting () into the transformations () and using the worldline () for formula_62 gives the coordinates
which are valid within formula_87, with the metric
If an observer resting in the center of the rotating frame is chosen with formula_89, the equations reduce to the ordinary rotational transformation
which are valid within formula_90, and the metric
The last equations can also be written in rotating cylindrical coordinates (Born coordinates):
which are valid within formula_92, and the metric
Frames (, , ) can be used to describe the geometry of rotating platforms, including the Ehrenfest paradox and the Sagnac effect.
Catenary.
The curvatures formula_94, formula_35 produce a catenary, i.e., hyperbolic motion combined with a spacelike translation
where
where formula_77 is the velocity, formula_78 the proper velocity, formula_98 as rapidity, formula_79 is the Lorentz factor. The corresponding Frenet–Serret tetrad is:
The corresponding non-rotating Fermi–Walker tetrad formula_82 on the same worldline can be obtained by solving the Fermi–Walker part of equation (). The same result follows from (), which gives
which together with () can now be inserted into (), resulting in the Fermi–Walker tetrad
The proper coordinates or Fermi coordinates follow by inserting formula_1 or formula_82 into ().
Semicubical parabola.
The curvatures formula_106, formula_35 produce a semicubical parabola or cusped motion
The corresponding Frenet–Serret tetrad with formula_108 is:
The corresponding non-rotating Fermi–Walker tetrad formula_82 on the same worldline can be obtained by solving the Fermi–Walker part of equation (). The same result follows from (), which gives
which together with () can now be inserted into (), resulting in the Fermi–Walker tetrad (note that formula_112 in this case):
The proper coordinates or Fermi coordinates follow by inserting formula_1 or formula_82 into ().
General case.
The curvatures formula_116, formula_117, formula_118 produce hyperbolic motion combined with uniform circular motion. The worldline is given by
where
with formula_77 as tangential velocity, formula_78 as proper tangential velocity, formula_98 as rapidity, formula_74 as orbital radius, formula_75 as coordinate angular velocity, formula_76 as proper angular velocity, formula_80 as angle of rotation, formula_79 is the Lorentz factor. The Frenet–Serret tetrad is
The corresponding non-rotating Fermi–Walker tetrad formula_82 on the same worldline is as follows: First inserting () into () gives the angular velocity, which together with () can now be inserted into (, left), and finally inserted into (, right) produces the Fermi–Walker tetrad. The proper coordinates or Fermi coordinates follow by inserting formula_1 or formula_82 into () (the resulting expressions are not indicated here because of their length).
Overview of historical formulas.
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.
Herglotz.
Herglotz (1909) argued that the metric
where
satisfies the condition of Born rigidity when formula_133. He pointed out that the motion of a Born rigid body is in general determined by the motion of one of its point (class A), with the exception of those worldlines whose three curvatures are constant, thus representing a helix (class B). For the latter, Herglotz gave the following coordinate transformation corresponding to the trajectories of a family of motions:
(H1) formula_134,
where formula_135 and formula_136 are functions of proper time formula_137. By differentiation with respect to formula_137, and assuming formula_139 as constant, he obtained
(H2) formula_140
Here, formula_141 represents the four-velocity of the origin formula_142 of formula_143, and formula_144 is a six-vector (i.e., an antisymmetric four-tensor of second order, or bivector, having six independent components) representing the angular velocity of formula_143 around formula_142. As any six-vector, it has two invariants:
When formula_148 is constant and formula_137 is variable, any family of motions described by (H1) forms a group and is equivalent to an equidistant family of curves, thus satisfying Born rigidity because they are rigidly connected with formula_143. To derive such a group of motion, (H2) can be integrated with arbitrary constant values of formula_141 and formula_152. For rotational motions, this results in four groups depending on whether the invariants formula_153 or formula_154 are zero or not. These groups correspond to four one-parameter groups of Lorentz transformations, which were already derived by Herglotz in a previous section on the assumption, that Lorentz transformations (being rotations in formula_155) correspond to hyperbolic motions in formula_156. 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.
Friedrich Kottler (1912) followed Herglotz, and derived the same worldlines of constant curvatures using the following Frenet–Serret formulas in four dimensions, with formula_157 as comoving tetrad of the worldline, and formula_158 as the three curvatures
corresponding to (). Kottler pointed out that the tetrad can be seen as a reference frame for such worldlines. Then he gave the transformation for the trajectories
in agreement with (). Kottler also defined a tetrad whose basis vectors are fixed in normal space and therefore do not share any rotation. This case was further differentiated into two cases: If the tangent (i.e., the timelike) tetrad field is constant, then the spacelike tetrads fields formula_162 can be replaced by formula_163 who are "rigidly" connected with the tangent, thus
The second case is a vector "fixed" in normal space by setting formula_165. Kottler pointed out that this corresponds to class B given by Herglotz (which Kottler calls "Born's body of second kind")
and class (A) of Herglotz (which Kottler calls "Born's body of first kind") is given by
which both correspond to formula ().
In (1914a), Kottler showed that the transformation
describes the non-simultaneous coordinates of the points of a body, while the transformation with formula_169
describes the simultaneous coordinates of the points of a body. These formulas become "generalized Lorentz transformations" by inserting
thus
in agreement with (). He introduced the terms "proper coordinates" and "proper frame" () for a system whose time axis coincides with the respective tangent of the worldline. He also showed that the Born rigid body of second kind, whose worldlines are defined by
is particularly suitable for defining a proper frame. Using this formula, he defined the proper frames for hyperbolic motion (free fall) and for uniform circular motion:
In (1916a) Kottler gave the general metric for acceleration-relative motions based on the three curvatures
In (1916b) he gave it the form:
where formula_176 are free from formula_177, and formula_178, and formula_179, and formula_180 linear in formula_181.
Møller.
Møller (1952) defined the following transport equation
in agreement with Fermi–Walker transport by (, without rotation). The Lorentz transformation into a momentary inertial frame was given by him as
in agreement with (). By setting formula_184, formula_185 and formula_186, he obtained the transformation into the "relativistic analogue of a rigid reference frame"
in agreement with the Fermi coordinates (), and the metric
in agreement with the Fermi metric () without rotation. He obtained the Fermi–Walker tetrads and Fermi frames of hyperbolic motion and uniform circular motion (some formulas for hyperbolic motion were already derived by him in 1943):