A proper reference frame in the
theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
is a particular form of
accelerated reference frame
A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are ...
, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in
curved spacetime
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
, as well as in "flat"
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
in which the
spacetime curvature caused by the
energy–momentum tensor Energy–momentum may refer to:
* Four-momentum
*Stress–energy tensor
*Energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also ...
can be disregarded. Since this article considers only flat spacetime—and uses the definition that
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
is the theory of flat spacetime while
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
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)
Accelerations in special relativity (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which ...
, where concepts such as three-acceleration,
four-acceleration In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ap ...
,
proper acceleration
In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily a ...
, hyperbolic motion etc. are defined and related to each other.)
A fundamental property of such a frame is the employment of the
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
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
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
Frenet–Serret formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
, 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
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
s 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 In relativistic physics, the coordinates of a ''hyperbolically accelerated reference frame'' constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating partic ...
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 physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rot ...
.
In the following,
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
indices run over 0,1,2,3,
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
indices over 1,2,3, and bracketed indices are related to tetrad vector fields. The signature of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allo ...
is (-1,1,1,1).
History
Some properties of Kottler-Møller or Rindler coordinates were anticipated by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
(1907)
[ when he discussed the uniformly accelerated reference frame. While introducing the concept of Born rigidity, ]Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
(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
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretic ...
(1910)[ and ]Max von Laue
Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals.
In addition to his scientific endeavors with con ...
(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 Friedrich Kottler (December 10, 1886 – May 11, 1965) was an Austrian theoretical physicist. He was a Privatdozent before he got a professorship in 1923 at the University of Vienna.
Life
In 1938, after the Anschluss, he lost his prof ...
(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
Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He was the first to th ...
(1924).[ The worldlines of constant curvatures were rediscovered by several author, for instance, by Vladimír Petrův (1964),][ as "timelike helices" by ]John Lighton Synge
John Lighton Synge (; 23 March 1897 – 30 March 1995) was an Irish mathematician and physicist, whose seven-decade career included significant periods in Ireland, Canada, and the USA. He was a prolific author and influential mentor, and is cr ...
(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
Christian Møller (22 December 1904 in Hundslev, Als14 January 1980 in Ordrup) was a Danish chemist and physicist who made fundamental contributions to the theory of relativity, theory of gravitation and quantum chemistry. He is known for M ...
(1952)[Møller (1952), §§ 46, 47, 90, 96] 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
C. V. Vishveshwara (6 March 1938 – 16 January 2017) was an Indian scientist and black hole physicist. Specializing in Einstein's General Relativity, he worked extensively on the theory of black holes and made major contributions to this field ...
(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
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
for curves in Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
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
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...
attached to a curved worldline, so in four dimensions this basis can be called a comoving tetrad or vierbein (also called vielbein, moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay t ...
, frame field, local frame, repère mobile in arbitrary dimensions):
Here, is the proper time along the worldline, the timelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
field is called the tangent that corresponds to the four-velocity
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
, the three spacelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differe ...
fields are orthogonal to and are called the principal normal , the binormal and the trinormal . The first curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
corresponds to the magnitude of four-acceleration In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ap ...
(i.e., proper acceleration
In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily a ...
), 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 transport[Kajari & Buser & Feiler & Schleich (2009), section 3] or generalized Fermi transport[Hehl & Lemke & Mielke (1990), section I.6] of tetrad , namely[Misner & Thorne & Wheeler (1973), section 6.8][Iyer and Vishveshwara (1993), section 2.2][Padmanabhan (2010), section 4.9][
where
:
or together in simplified form:
:]four-velocity
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
and \mathbf as four-acceleration In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ap ...
, and "\cdot" indicates the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
and "\wedge" the wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
. The first part (\mathbf\wedge\mathbf)\mathbf_=\mathbf\left(\mathbf\cdot\mathbf_\right)-\mathbf\left(\mathbf\cdot\mathbf_\right) 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 \mathbf consists of an antisymmetric second rank tensor with \omega as the ]angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
four-vector and \epsilon as the Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
. 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 \mathbf_ of a rotating tetrad (such as a Frenet–Serret tetrad) with respect to the non-rotating spacelike fields \mathbf_ of a Fermi–Walker tetrad along the same world line.
Deriving Fermi–Walker tetrads from Frenet–Serret tetrads
Since \mathbf_ and \mathbf_ 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. For instance, the angular velocity vector between the respective spacelike tetrad fields \mathbf_ and \mathbf_ can be given in terms of torsions \kappa_ and \kappa_:][Johns (2005), section 18.18]
Assuming that the curvatures are constant (which is the case in helical
Helical may refer to:
* Helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is ...
motion in flat spacetime, or in the case of stationary axisymmetric
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
spacetimes), one then proceeds by aligning the spacelike Frenet–Serret vectors in the \mathbf_-\mathbf_ plane by constant counter-clockweise rotation, then the resulting intermediary spatial frame \mathbf_ is constantly rotated around the \mathbf_ axis by the angle \Theta=\left, \boldsymbol\\tau, which finally gives the spatial Fermi–Walker frame \mathbf_ (note that the timelike field remains the same):[Bini & Cherubini & Geralico & Jantzen (2008), section 3.2]
For the special case \kappa_=0 and \mathbf_=,0,0,1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math>, it follows \boldsymbol=\left ,0,0,\ \kappa_\right/math> and \Theta=\left, \boldsymbol\\tau=\kappa_\tau and \mathbf_=\mathbf_, therefore () is reduced to a single constant rotation around the \mathbf_-axis:[Mashhoon (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 \mathbf'= ^,x^,x^,x^/math>, and the sequence of such momentary frames which it traverses corresponds to a successive application of Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
s \mathbf=\boldsymbol\mathbf', where \mathbf is an external inertial frame and \boldsymbol the Lorentz transformation matrix. This matrix can be replaced by the proper time dependent tetrads \mathbf_(\tau) defined above, and if \mathbf(\tau) is the time track of the particle indicating its position, the transformation reads:
Then one has to put x^=t'=0 by which \mathbf' is replaced by \mathbf= ^,x^,x^/math> and the timelike field \mathbf_ vanishes, therefore only the spacelike fields \mathbf_ are present anymore. Subsequently, the time in the accelerated frame is identified with the proper time of the accelerated observer by x^=t=\tau. The final transformation has the form[Hehl & Lemke & Mielke (1990), section I.8]
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):[Mashhoon & Münch (2002), section 2, 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):[Petruv (1964)][Synge (1967)][Pauri & Vallisneri (2001), Appendix A][Rosu (2000), section 0.2.3][Louko & Satz (2006), section 5.2]
Case \kappa_=\kappa_=\kappa_=0 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 \kappa_=\alpha, \kappa_=\kappa_=0, where \alpha is the constant proper acceleration
In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily a ...
in the direction of motion, produce hyperbolic motion because the worldline in the Minkowski diagram
A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contracti ...
is a hyperbola:[
The corresponding orthonormal tetrad is identical to an inverted Lorentz transformation matrix with ]hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
s \gamma=\cosh\eta as Lorentz factor and v\gamma=\sinh\eta as proper velocity and \eta=\operatornamev=\alpha\tau as rapidity (since the torsions \kappa_ and \kappa_ are zero, the Frenet–Serret formulas and Fermi–Walker formulas produce the same tetrad):[Kottler (1914a), table I (IIIb); Kottler (1914b), pp. 488-489, 492-493][Synge (1967) p. 35, type III]
Inserted into the transformations () and using the worldline () for \mathbf, the accelerated observer is always located at the origin, so the Kottler-Møller coordinates follow[
:\begin
\beginT & =\left(x+\frac\right)\sinh(\alpha\tau)\\
X & =\left(x+\frac\right)\cosh(\alpha\tau)-\frac\\
Y & =y\\
Z & =z
\end
& \begin\tau & =\frac\operatorname\left(\frac\right)\\
x & =\sqrt-\frac\\
y & =Y\\
z & =Z
\end
\end
which are valid within -1/\alpha, with the metric
:ds^=-(1+\alpha x)^d\tau^+dx^+dy^+dz^.
Alternatively, by setting \mathbf=0 the accelerated observer is located at X=1/\alpha at time \tau=T=0, thus the ]Rindler coordinates In relativistic physics, the coordinates of a ''hyperbolically accelerated reference frame'' constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating partic ...
follow from () and (, ):
:\begin \beginT & =x\sinh(\alpha\tau)\\ X & =x\cosh(\alpha\tau)\\ Y & =y\\ Z & =z \end & \begin\tau & =\frac\operatorname\frac\\ x & =\sqrt\\ y & =Y\\ z & =Z \end \end
which are valid within 0, with the metric
:ds^=-\alpha^x^d\tau^+dx^+dy^+dz^
Uniform circular motion
The curvatures \kappa_^-\kappa_^>0, \kappa_=0 produce uniform circular motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rot ...
, with the worldline[Kottler (1914a), table I (IIb) and § 6 section 3][Nožička (1964), example 1][Synge (1967), section 8][Formiga (2012), section V-b]
where
with h as orbital radius, p_ as coordinate angular velocity, p as proper angular velocity, v as tangential velocity
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quantity ...
, n as proper velocity, \gamma as Lorentz factor, and \theta 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 d_ of ordinary rotating coordinates:
The corresponding non-rotating Fermi–Walker tetrad \mathbf_ on the same worldline can be obtained by solving the Fermi–Walker part of equation (). Alternatively, one can use () together with (), which gives
:\boldsymbol=\left ,0,0,\gamma^p\right\quad\left, \boldsymbol\=\gamma^p,\quad\Theta=\left, \boldsymbol\\tau=\gamma^p_\tau=\gamma p\tau=\gamma\theta
The resulting angle of rotation \Theta together with () can now be inserted into (), by which the Fermi–Walker tetrad follows][
:\begin\mathbf_ & \ =\mathbf_ & \ =\gamma(1,\ -v\sin\theta,\ v\cos\theta,\ 0)\\ \mathbf_ & \ =\mathbf_\cos\Theta-\mathbf_\sin\Theta & \ =\left(-\gamma v\sin\Theta,\ \cos\theta\cos\Theta+\gamma\sin\theta\sin\Theta,\ \sin\theta\cos\Theta-\gamma\cos\theta\sin\Theta,\ 0\right)\\ \mathbf_ & \ =\mathbf_\sin\Theta+\mathbf_\cos\Theta & \ =\left(\gamma v\cos\Theta,\ \cos\theta\sin\Theta-\gamma\sin\theta\cos\Theta,\ \sin\theta\sin\Theta+\gamma\cos\theta\cos\Theta,\ 0\right)\\ \mathbf_ & \ =\mathbf_ & \ =(0,\ 0,\ 0,\ 1) \end
In the following, the Frenet–Serret tetrad is used to formulate the transformation. Inserting () into the transformations () and using the worldline () for \mathbf gives the coordinates][Bini & Lusanna & Mashhoon (2005), Appendix A]
which are valid within (X+h)^+(\gamma Y)^\leqq1/p_^, with the metric
:ds^=-\gamma^\left -(x+h)^p_^-\gamma^p_^y^\right\tau^+2\gamma^p_(x\ dy-y\ dx)d\tau+dx^+dy^+dz^
If an observer resting in the center of the rotating frame is chosen with h=0, the equations reduce to the ordinary rotational transformation
which are valid within 0<\sqrt<1/p_, and the metric
:ds^=-\left -p_^\left(x^+y^\right)\rightt^+2p_(-y\ dx+x\ dy)dt+dx^+dy^+dz^.
The last equations can also be written in rotating cylindrical coordinates (Born coordinates
In relativistic physics, the Born coordinate chart is a coordinate chart for (part of) Minkowski spacetime, the flat spacetime of special relativity. It is often used to analyze the physical experience of observers who ride on a ring or disk rig ...
):
which are valid within 0, and the metric
:ds^=-\left(1-p_^r^\right)dt^+2p_r^dt\ d\phi+dr^+r^d\phi^+dz^
Frames (, , ) can be used to describe the geometry of rotating platforms, including the Ehrenfest paradox
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.
In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity, it discusses an ide ...
and the Sagnac effect
The Sagnac effect, also called Sagnac interference, named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called a ring interferomet ...
.
Catenary
The curvatures \kappa_^-\kappa_^>0, \kappa_=0 produce a catenary, i.e., hyperbolic motion combined with a spacelike translation[Kottler (1914a), table I (IIIa)][Synge (1967), section 6][Louko & Satz (2006), section 5.2.5]
where
where v is the velocity, n the proper velocity, \eta as rapidity, \gamma is the Lorentz factor. The corresponding Frenet–Serret tetrad is:[
:\begin\mathbf_ & =\left(\gamma\cosh\eta,\ \gamma\sinh\eta,\ n,\ 0\right)\\ \mathbf_ & =\left(\sinh\eta,\ \cosh\eta,\ 0,\ 0\right)\\ \mathbf_ & =\left(-n\cosh\eta,\ -n\sinh\eta,\ -\gamma,\ 0\right)\\ \mathbf_ & =\left(0,\ 0,\ 0,\ 1\right) \end
The corresponding non-rotating Fermi–Walker tetrad \mathbf_ on the same worldline can be obtained by solving the Fermi–Walker part of equation ().][ The same result follows from (), which gives
:\boldsymbol=\left ,0,0,na\right\quad\left, \boldsymbol\=na,\quad\Theta=\left, \boldsymbol\\tau=na\tau
which together with () can now be inserted into (), resulting in the Fermi–Walker tetrad
:\begin
\mathbf_ & \ =\mathbf_ & \ =\left(\gamma\cosh\eta,\ \gamma\sinh\eta,\ n,\ 0\right)\\
\mathbf_ & \ =\mathbf_\cos\Theta-\mathbf_\sin\Theta & \ =\left(\sinh\eta\cos\Theta+n\cosh\eta\sin\Theta,\ \cosh\eta\cos\Theta+n\sinh\eta\sin\Theta,\ \gamma\sin\Theta,\ 0\right)\\
\mathbf_ & \ =\mathbf_\sin\Theta+\mathbf_\cos\Theta & \ =\left(\sinh\eta\sin\Theta-n\cosh\eta\cos\Theta,\ \cosh\eta\sin\Theta-n\sinh\eta\cos\Theta,\ -\gamma\cos\Theta\ 0\right)\\
\mathbf_ & \ =\mathbf_ & \ =\left(0,\ 0,\ 0,\ 1\right)
\end
The proper coordinates or Fermi coordinates follow by inserting \mathbf_ or \mathbf_ into ().
]
Semicubical parabola
The curvatures \kappa_^-\kappa_^=0, \kappa_=0 produce a semicubical parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
: y^2 - a^2 x^3 = 0
(with ) in some Cartesian coordinate system.
Solving for leads to the ''explicit form''
: y = \ ...
or cusped motion[Kottler (1914a), table I (IV)][Synge (1967), section 7][Louko & Satz (2006), section 5.2.4]
The corresponding Frenet–Serret tetrad with \theta=a\tau is:[Synge (1967), section 7]
:\begin\mathbf_ & =\left(1+\frac\theta^,\ \theta,\ \frac\theta^,\ 0\right)\\ \mathbf_ & =\left(\theta,\ 1,\ \theta,\ 0\right)\\ \mathbf_ & =\left(-\frac\theta^,\ -\theta,\ 1-\frac\theta^,\ 0\right)\\ \mathbf_ & =\left(0,\ 0,\ 0,\ 1\right) \end
The corresponding non-rotating Fermi–Walker tetrad \mathbf_ on the same worldline can be obtained by solving the Fermi–Walker part of equation ().[ The same result follows from (), which gives
:\boldsymbol=\left ,0,0,a\right\quad\left, \boldsymbol\=a,\quad\Theta=\left, \boldsymbol\\tau=a\tau=\theta
which together with () can now be inserted into (), resulting in the Fermi–Walker tetrad (note that \Theta=\theta in this case):
:\begin
\mathbf_ & \ =\mathbf_ & \ =\left(1+\frac\theta^,\ \theta,\ \frac\theta^,\ 0\right)\\
\mathbf_ & \ =\mathbf_\cos\Theta-\mathbf_\sin\Theta & \ =\left(\theta\cos\theta+\frac\theta^\sin\theta,\ \cos\theta+\theta\sin\theta,\ \theta\cos\theta+\left(\frac\theta^-1\right)\sin\theta,\ 0\right)\\
\mathbf_ & \ =\mathbf_\sin\Theta+\mathbf_\cos\Theta & \ =\left(\theta\sin\theta-\frac\theta^\cos\theta,\ \sin\theta-\theta\cos\theta,\ \theta\sin\theta-\left(\frac\theta^-1\right)\cos\theta,\ 0\right)\\
\mathbf_ & \ =\mathbf_ & \ =\left(0,\ 0,\ 0,\ 1\right)
\end
The proper coordinates or Fermi coordinates follow by inserting \mathbf_ or \mathbf_ into ().
]
General case
The curvatures \kappa_\ne0, \kappa_\ne0, \kappa_\ne0 produce hyperbolic motion combined with uniform circular motion. The worldline is given by[Kottler (1914a), table I (case I)][Synge (1967), section 4][Louko & Satz (2006), section 5.2.6]
where
with v as tangential velocity, n as proper tangential velocity, \eta as rapidity, h as orbital radius, p_ as coordinate angular velocity, p as proper angular velocity, \theta as angle of rotation, \gamma is the Lorentz factor. The Frenet–Serret tetrad is[
:\begin\mathbf_ & =\left(\gamma\cosh\eta,\ \gamma\sinh\eta,\ -n\sin\theta,\ -n\cos\theta\right)\\
\mathbf_ & =\frac\left(\gamma a\sinh\eta,\ \gamma a\cosh\eta,\ -np\cos\theta,\ -np\sin\theta\right)\\
\mathbf_ & =\left(-n\cosh\eta,\ -n\sinh\eta,\ \gamma\sin\theta,\ -\gamma\cos\theta\right)\\
\mathbf_ & =\frac\left(np\sinh\eta,\ np\cosh\eta,\ \gamma a\cos\theta,\ \gamma a\sin\theta\right)
\end
The corresponding non-rotating Fermi–Walker tetrad \mathbf_ 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 \mathbf_ or \mathbf_ 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
:ds^=d\sigma^+\frac(d\nu)^
where
:\begind\nu & =A_d\xi_+A_d\xi_+A_d\xi_+A_d\xi_\\
d\sigma^ & =\sum_^ij\ A_d\xi_d\xi_-\frac\left(A_d\xi_+A_d\xi_+A_d\xi_\right)^
\end
satisfies the condition of Born rigidity when \fracd\sigma^=0. 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) x_=a_+\sum_^a_x_^,\qquad i=1,2,3,4,
where a_ and a_ are functions of proper time \vartheta. By differentiation with respect to \vartheta, and assuming x_ as constant, he obtained
(H2) \frac+q_+\sum_^p_x_^=0
Here, q_ represents the four-velocity of the origin O' of S', and -p_ is a six-vector (i.e., an antisymmetric four-tensor of second order, or ]bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can ...
, having six independent components) representing the angular velocity of S' around O'. As any six-vector, it has two invariants:
:\beginD & =p_p_+p_p_+p_p_,\\
\Delta & =p_^+p_^+p_^+p_^+p_^+p_^,
\end
When x_^ is constant and \vartheta 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 S'. To derive such a group of motion, (H2) can be integrated with arbitrary constant values of q_ and p_. For rotational motions, this results in four groups depending on whether the invariants D or \Delta 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 R_) correspond to hyperbolic motions in R_. The latter have been studied in the 19th century, and were categorized by Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
into loxodromic, elliptic, hyperbolic, and parabolic motions (see also Möbius group
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Pa ...
).
Kottler
Friedrich Kottler Friedrich Kottler (December 10, 1886 – May 11, 1965) was an Austrian theoretical physicist. He was a Privatdozent before he got a professorship in 1923 at the University of Vienna.
Life
In 1938, after the Anschluss, he lost his prof ...
(1912)[ followed Herglotz, and derived the same worldlines of constant curvatures using the following Frenet–Serret formulas in four dimensions, with c^ as comoving tetrad of the worldline, and \frac,\ \frac,\ \frac 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
:\mathbf=\mathbf+\Gamma^c_+\Gamma^c_+\Gamma^c_+\Gamma^c_ (with )
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 can be replaced by who are "rigidly" connected with the tangent, thus
:
The second case is a vector "fixed" in normal space by setting . 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
:X=x+\Gamma^c_+\Gamma^c_+\Gamma^c_+\Gamma^c_,
describes the non-simultaneous coordinates of the points of a body, while the transformation with \Gamma^=0
:X=x+\Gamma^c_+\Gamma^c_+\Gamma^c_,
describes the simultaneous coordinates of the points of a body. These formulas become "generalized Lorentz transformations" by inserting
:\Gamma^=X',\quad\Gamma^=Y',\quad\Gamma^=Z',\quad\Gamma^=ic(T'-\tau)
thus
:X-x=ic(T'-\tau)c_+Z'c_+X'c_+Y'c_
in agreement with (). He introduced the terms "proper coordinates" and "proper frame" (german: Eigenkoordinaten, Eigensystem) 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
:\mathfrak=x+\Delta^c_+\Delta^c_+\Delta^c_,
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:
{, class="wikitable" style="text-align:center"
!Hyperbolic motion
! colspan="2" , Uniform circular motion
, -
, 1914b
, 1914a
, 1921
, -
, \scriptstyle \begin{matrix}\begin{matrix}
c_{1}^{(1)}=0, & & c_{1}^{(2)}=0, & & c_{1}^{(3)}=\frac{1}{i}\sinh u, & & c_{1}^{(4)}=\cosh u,\\
c_{2}^{(1)}=0, & & c_{2}^{(2)}=0, & & c_{2}^{(3)}=\frac{1}{i}\cosh u, & & c_{2}^{(4)}=-\sinh u,\\
c_{3}^{(1)}=1, & & c_{3}^{(2)}=0, & & c_{3}^{(3)}=0, & & c_{3}^{(4)}=0,\\
c_{4}^{(1)}=0, & & c_{4}^{(2)}=1, & & c_{4}^{(3)}=0, & & c_{4}^{(4)}=0,
\end{matrix}\\
\boldsymbol{\downarrow}\\
X=x+\Delta^{(2)}c_{2}+\Delta^{(3)}c_{3}+\Delta^{(4)}c_{4}\\
\boldsymbol{\downarrow}\\
\begin{align}
X & =x_{0}+\mathfrak{X}'\\
Y & =y_{0}+\mathfrak{Y}'\\
Z & =\left(b+\mathfrak{Z}'\right)\cosh\mathfrak{u}\\
cT & =\left(b+\mathfrak{Z}'\right)\sinh\mathfrak{u}
\end{align}\\
\left(\Delta^{(2)}=\mathfrak{X}',\ \Delta^{(3)}=\mathfrak{Y}',\ \Delta^{(4)}=\mathfrak{Z}'\right)\\
\boldsymbol{\downarrow}\\
\begin{align}
\mathfrak{X}' & =X_{0}-x_{0}+q_{x}T\\
\mathfrak{Y}' & =Y_{0}-y_{0}+q_{y}T\\
b+\mathfrak{Z}' & =\sqrt{\left(Z_{0}+q_{x}T\right)^{2}-c^{2}T^{2\\
c\mathfrak{T}' & =b\operatorname{artanh}\frac{cT}{Z_{0}+q_{x}T}
\end{align}\\
\left(X=X_{0}+q_{x}T,\ Y=Y_{0}+q_{y}T,\ Z=Z_{0}+q_{x}T\right)\\
\boldsymbol{\downarrow}\\
dS^{2}=(d\mathfrak{X}')^{2}+(d\mathfrak{Y}')^{2}+(d\mathfrak{Z}')^{2}-c^{2}\left(\frac{b+\mathfrak{Z}'}{b^{2\right)^{2}(d\mathfrak{T}')^{2}
\end{matrix}
, \scriptstyle \begin{matrix}{\begin{matrix}
c_{1}^{(h)}=-\frac{a\omega\sin\omega t}{ic\sqrt{1-\frac{a^{2}\omega^{2{c^{2 ,\ \frac{a\omega\cos\omega t}{ic\sqrt{1-\frac{a^{2}\omega^{2{c^{2 ,\ 0,\ \frac{1}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \\
c_{2}^{(h)}=\cos\omega t,\ \sin\omega t,\ 0\ 0\\
c_{3}^{(h)}=-\frac{\sin\omega t}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 ,\ \frac{\cos\omega t}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 ,\ 0,\ \frac{ia\omega}{c\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \\
c_{4}^{(h)}=0,\ 0,\ 1,\ 0
\end{matrix\\
\boldsymbol{\downarrow}\\
{X^{(h)}=x^{(h)}+\Gamma^{(1)}c_{1}^{(h)}+\Gamma^{(2)}c_{2}^{(h)}+\Gamma^{(3)}c_{3}^{(h)}+\Gamma^{(4)}c_{4}^{(h)\\
\boldsymbol{\downarrow}\\
\begin{align}
X= & a\cos\omega t-\frac{a\omega(T-t)+\Theta'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \sin\omega t+R'\cos\omega t\\
Y= & a\sin\omega t+\frac{a\omega(T-t)+\Theta'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \cos\omega t+R'\sin\omega t\\
Z= & z_{0}+Z'\\
icT= & ict+\frac{(T-\tau)+\frac{a\omega}{c^{2\Theta'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2
\end{align}
\end{matrix}
, \scriptstyle \begin{matrix}\begin{align}
X & =(a+x')\cos\omega t-\frac{y'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \sin\omega t\\
Y & =(a+x')\sin\omega t+\frac{y'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \cos\omega t\\
Z & =b+z'\\
T & =t+\frac{\frac{a\omega}{c^{2y'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 \\
& t'=t\sqrt{1-\frac{a^{2}\omega^{2{c^{2}
\end{align}\\
\boldsymbol{\downarrow}\\
{ \begin{align}
ds^{2}= & dx^{\prime2}+dy^{\prime2}+dz^{\prime2}-2\frac{\omega y'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 dx'dt+2\frac{\omega x'}{\sqrt{1-\frac{a^{2}\omega^{2{c^{2 dy'dt\\
& +\left(-c^{2}+(a+x')^{2}\omega^{2}+\frac{y^{\prime2}\omega^{2{1-\frac{a^{2}\omega^{2{c^{2}\right)dt^{2}
\end{align
\end{matrix}
In (1916a) Kottler gave the general metric for acceleration-relative motions based on the three curvatures
:{ \begin{align}dS^{2}= & d\xi^{\prime2}+d\eta^{\prime2}+d\zeta^{\prime2}-2c\ d\tau'd\xi'\cdot\eta'i/R_{2}+2c\ d\tau'd\eta'\cdot\left(\xi'i/R_{2}-\zeta'i/R_{3}\right)+c\ d\tau'd\zeta'\cdot\eta'i/R_{3}\\
& -c^{2}d\tau^{\prime2}\left left(1-\xi'/R_{1}\right)^{2}+\eta^{\prime2}/R_{2}^{2}+\eta^{\prime}/R_{3}^{2}+\left(\xi'/R_{2}-\zeta'/R_{3}\right)^{2}\right\end{align}
}
In (1916b) he gave it the form:
:{ ds^{2}=dx^{2}+dy^{2}+dz^{2}+2g_{14}dx\ dit+2g_{24}dy\ dit+2g_{34}dz\ dit+g_{44}(dit)^{2
where { g_{14}g_{24}g_{34}g_{44 are free from t, and \frac{\partial g_{i4{\partial x_{k+\frac{\partial g_{k4{\partial x_{i=0, and \frac{\partial g_{i4{\partial x_{k-\frac{\partial g_{k4{\partial x_{i=\text{const.}, and \sqrt{g} linear in xyz.
]
Møller
Møller (1952)[ defined the following transport equation
:\frac{de_{i{d\tau}=\frac{\left(e_{l}\dot{U}_{l}\right)U_{i}-\dot{U}_{i}\left(i_{l}U_{l}\right)}{c^{2
in agreement with Fermi–Walker transport by (, without rotation). The Lorentz transformation into a momentary inertial frame was given by him as
:x_{i}=f_{i}(\tau)+x_{k}^{\prime}\alpha_{ki}(\tau)
in agreement with (). By setting x^{i}=x_{l}^{\prime}, x_{4}^{\prime}=0 and t=\tau, he obtained the transformation into the "relativistic analogue of a rigid reference frame"
:X_{i}=f_{i}(t)+x^{\prime\kappa}\alpha_{\kappa i}(\tau)
in agreement with the Fermi coordinates (), and the metric
:ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}\left +\frac{g_{\kappa}x^{\kappa{c^{2\right{2}
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):
{, class="wikitable" style="text-align:center"
! colspan="2" , Hyperbolic motion
! , Uniform circular motion
, -
, 1943
, 1952
, 1952
, -
, {\scriptstyle \begin{matrix}\begin{align}x & =\frac{1}{g}\left\{ \sqrt{(1+gX)^{2}-g^{2}T^{2-1\right\} \\
y & =Y\\
z & =Z\\
t & =\frac{1}{2g}\ln\frac{1+gX+gT}{1+gX-gT}
\end{align}
\\
\boldsymbol{\downarrow}\\
ds^{2}=dx^{2}+dy^{2}+dz^{2}-(1+gx)^{2}dt^{2}
\end{matrix
, \scriptstyle\begin{matrix}\alpha_{ik}=\left(\begin{matrix}U_{4}/ic & 0 & 0 & iU_{1}/c\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
U_{1}/ic & 0 & 0 & U_{4}/ic
\end{matrix}\right)\\
U_{i}=\left(c\sinh\frac{g\tau}{c},\ 0,0,\ ig\cosh\frac{g\tau}{c}\right)\\
\boldsymbol{\downarrow}\\
X_{i}=\mathbf{f}_{i}(t)+x^{\prime\kappa}\alpha_{\kappa i}(\tau)\\
\boldsymbol{\downarrow}\\
\begin{align}
X & =\frac{c^{2{g}\left(\cosh\frac{gt}{c}-1\right)+x\cosh\frac{gt}{c}\\
Y & =y\\
Z & =z\\
T & =\frac{c}{g}\sinh\frac{gt}{c}+x\frac{\sinh\frac{gt}{c{c}
\end{align}\\
\boldsymbol{\downarrow}\\
ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}\left(1+gx/c^{2}\right)^{2}\\
\\
\end{matrix}
, \scriptstyle\begin{matrix}\alpha_{ik}=\left(\begin{matrix}\cos\alpha\cos\beta+\gamma\sin\alpha\sin\beta & \sin\alpha\cos\beta-\gamma\cos\alpha\sin\beta & 0 & -i\frac{u\gamma}{c}\sin\beta\\
\cos\alpha\sin\beta-\gamma\sin\alpha\cos\beta & \sin\alpha\sin\beta+\gamma\cos\alpha\cos\beta & 0 & i\frac{u\gamma}{c}\cos\beta\\
0 & 0 & 1 & 0\\
i\frac{u\gamma}{c}\sin\alpha & -i\frac{u\gamma}{c}\cos\alpha & 0 & \gamma
\end{matrix}\right)\\
{ \alpha=\omega\gamma\tau},\ { \beta=\gamma\alpha=\omega\gamma^{2}\tau}.
\end{matrix}
]
Worldlines of constant curvatures by Herglotz and Kottler
{, class="wikitable" style="text-align: center;"
, General case
, Uniform rotation
, Catenary
, Semicubical parabola
, Hyperbolic motion
, -
! colspan="5" , Herglotz (1909)
, -
!loxodromic
!elliptic
!hyperbolic
!parabolic
!hyperbolic
\scriptstyle (\alpha=0)
, -
, \scriptstyle \begin{matrix}D\ne0\\
p_{21}=-p_{12}=1\\
p_{34}=-p_{43}=i\\
q_{i}= ,0,0,0\end{matrix}
, \scriptstyle \begin{matrix}D=0,\ \Delta>0\\
p_{21}=-p_{12}=1\\
\\
q_{i}= ,0,0,\delta i\end{matrix}
, \scriptstyle \begin{matrix}D=0,\ \Delta<0\\
p_{34}=-p_{43}=i\\
\\
q_{i}= alpha,0,0,0\end{matrix}
, \scriptstyle \begin{matrix}D=0,\ \Delta=0\\
p_{31}=-p_{13}=1\\
p_{41}=-p_{14}=i\\
q_{i}= ,\beta,0,\delta i\end{matrix}
,
, -
, colspan="5" , Lorentz-Transformations
, -
, \scriptstyle \begin{align}x+iy & =(x'+iy')e^{i\lambda\vartheta}\\
x-iy & =(x'-iy')e^{-i\lambda\vartheta}\\
t-z & =(t'-z')e^{\vartheta}\\
t+z & =(t'+z')e^{-\vartheta}
\end{align}
, \scriptstyle \begin{align}x+iy & =(x'+iy')e^{i\vartheta}\\
x-iy & =(x'-iy')e^{-i\vartheta}\\
z & =z'\\
t & =t'+\delta\vartheta
\end{align}
, \scriptstyle \begin{align}x & =x'+\alpha\vartheta\\
y & =y'\\
t-z & =(t'-z')e^{\vartheta}\\
t+z & =(t'+z')e^{-\vartheta}
\end{align}
, \scriptstyle \begin{align}x & =x'+\vartheta(t'-z')+\frac{1}{2}\delta\vartheta^{2}\\
y & =y'+\beta\vartheta\\
z & =z'+\vartheta x'+\frac{1}{2}\vartheta^{2}(t'-z')+\frac{1}{6}\delta\vartheta^{3}\\
t-z & =t'-z'+\delta\vartheta
\end{align}
, \scriptstyle \begin{align}x & =x'\\
y & =y'\\
t-z & =(t'-z')e^{\vartheta}\\
t+z & =(t'+z')e^{-\vartheta}
\end{align}
, -
, colspan="5" , Trajectories (time)
, -
, \scriptstyle \begin{align}x+iy & =(x_{0}+iy_{0})e^{i\lambda u}\\
x-iy & =(x_{0}-iy_{0})e^{-i\lambda u}\\
z & =\sqrt{z_{0}^{2}+t^{2\\
u & =\lg\frac{\sqrt{z_{0}^{2}+t^{2-t}{z_{0
\end{align}
, \scriptstyle \begin{align}x+iy & =(x_{0}+iy_{0})e^{i\frac{t}{\delta\\
x-iy & =(x_{0}-iy_{0})e^{-i\frac{t}{\delta\\
z & =z_{0}
\end{align}
, \scriptstyle \begin{align}x & =x_{0}+\alpha\lg\frac{\sqrt{z_{0}^{2}+t^{2-t}{z_{0\\
y & =y_{0}\\
z & =\sqrt{z_{0}^{2}+t^{2
\end{align}
, \scriptstyle \begin{align}x & =x_{0}+\frac{1}{2}\delta\vartheta^{2}\\
y & =y_{0}+\beta\vartheta\\
z & =z_{0}+x_{0}\vartheta+\frac{1}{6}\delta\vartheta^{3}\\
t-z & =\delta\vartheta
\end{align}
, \scriptstyle \begin{align}x & =x_{0}\\
y & =y_{0}\\
z & =\sqrt{z_{0}^{2}+t^{2
\end{align}
, -
! colspan="5" , Kottler (1912, 1914)
, -
!hyperspherical curve
!uniform rotation
!catenary
!cubic curve
!hyperbolic motion
, -
, colspan="5" , Curvatures
, -
, \scriptstyle \begin{align}\left(\frac{1}{R_{1\right)^{2} & =\frac{a^{2}\lambda^{4}+b^{2{\left(b^{2}-a^{2}\lambda^{2}\right)^{2\\
\left(\frac{1}{R_{2\right)^{2} & =-\frac{a^{2}b^{2}\lambda^{2}\left(1+\lambda^{2}\right)}{\left(b^{2}-a^{2}\lambda^{2}\right)^{2}\left(a^{2}\lambda^{4}+b^{2}\right)}\\
\left(\frac{1}{R_{3\right)^{2} & =-\frac{\lambda^{2{a^{2}\lambda^{4}+b^{2
\end{align}
, \scriptstyle \begin{align}\left(\frac{1}{R_{1\right)^{2} & =\frac{a^{2}\lambda^{4{\left(1-a^{2}\lambda^{2}\right)^{2\\
\left(\frac{1}{R_{2\right)^{2} & =-\frac{\lambda^{2{\left(1-a^{2}\lambda^{2}\right)^{2\\
\left(\frac{1}{R_{3\right)^{2} & =0
\end{align}
, \scriptstyle \begin{align}\left(\frac{1}{R_{1\right)^{2} & =\frac{b^{2{\left(b^{2}-\alpha^{2}\right)^{2\\
\left(\frac{1}{R_{2\right)^{2} & =-\frac{\alpha^{2{\left(b^{2}-\alpha^{2}\right)^{2\\
\left(\frac{1}{R_{3\right)^{2} & =0
\end{align}
, \scriptstyle \begin{align}\left(\frac{1}{R_{1\right)^{2} & =\frac{\alpha^{2{\left(\alpha^{2}+2x_{0}^{(1)}\right)^{2\\
\left(\frac{1}{R_{2\right)^{2} & =-\frac{\alpha^{2{\left(\alpha^{2}+2x_{0}^{(1)}\right)^{2=-\left(\frac{1}{R_{1\right)^{2}\\
\left(\frac{1}{R_{3\right)^{2} & =0
\end{align}
, \scriptstyle \begin{align}\left(\frac{1}{R_{1\right)^{2} & =\frac{1}{b^{2\\
\left(\frac{1}{R_{2\right)^{2} & =0\\
\left(\frac{1}{R_{3\right)^{2} & =0
\end{align}
, -
, colspan="5" , Trajectory of \scriptstyle S_4
, -
, \scriptstyle \begin{align}x^{(1)} & =a\cos\lambda\left(u-u_{0}\right)\\
x^{(2)} & =a\sin\lambda\left(u-u_{0}\right)\\
x^{(3)} & =b\cos iu\\
x^{(4)} & =b\sin iu
\end{align}
, \scriptstyle \begin{align}x^{(1)} & =a\cos\lambda\left(u-u_{0}\right)\\
x^{(2)} & =a\sin\lambda\left(u-u_{0}\right)\\
x^{(3)} & =x_{0}^{(3)}\\
x^{(4)} & =iu
\end{align}
, \scriptstyle \begin{align}x^{(1)} & =x_{0}^{(1)}+\alpha u\\
x^{(2)} & =x_{0}^{(2)}\\
x^{(3)} & =b\cos iu\\
x^{(4)} & =b\sin iu
\end{align}
, \scriptstyle \begin{align}x^{(1)} & =x_{0}^{(1)}+\frac{1}{2}\alpha u^{2}\\
x^{(2)} & =x_{0}^{(2)}\\
x^{(3)} & =x_{0}^{(3)}+x_{0}^{(1)}u+\frac{1}{6}\alpha u^{3}\\
x^{(4)} & =i\left(x_{0}^{(3)}+x_{0}^{(1)}u+\frac{1}{6}\alpha u^{2}\right)+i\alpha u
\end{align}
, \scriptstyle \begin{align}x^{(1)} & =x_{0}^{(1)}\\
x^{(2)} & =x_{0}^{(2)}\\
x^{(3)} & =b\cos iu\\
x^{(4)} & =b\sin iu
\end{align}
, -
, colspan="5" , Trajectory (time)
, -
, \scriptstyle \begin{align}x & =a\cos\lambda\left(u-u_{0}\right)\\
y & =a\sin\lambda\left(u-u_{0}\right)\\
z & =\sqrt{b^{2}+c^{2}t^{2\\
u & =\ln\frac{-ct+\sqrt{b^{2}+c^{2}t^{2}{b}
\end{align}
, \scriptstyle \begin{align}x & =a\cos\omega_{z}\left(t-t_{0}\right)\\
y & =a\sin\omega_{z}\left(t-t_{0}\right)\\
z & =z_{0}
\end{align}
, \scriptstyle \begin{align}x & =x_{0}+\alpha\ln\frac{-ct+\sqrt{b^{2}+c^{2}t^{2}{b}\\
y & =y_{0}\\
z & =\sqrt{b^{2}+c^{2}t^{2
\end{align}
, \scriptstyle \begin{align}x & =x_{0}+\frac{1}{2}\alpha u^{2}\\
y & =y_{0}\\
z & =z_{0}+x_{0}u+\frac{1}{6}\alpha u^{3}\\
ct & =z_{0}+x_{0}u+\frac{1}{6}\alpha u^{3}+\alpha u\\
& =z+\alpha u
\end{align}
, \scriptstyle \begin{align}x & =x_{0}\\
y & =y_{0}\\
z & =\sqrt{b^{2}+c^{2}t^{2
\end{align}
References
Bibliography
Textbooks
*; First edition 1911, second expanded edition 1913, third expanded edition 1919.
* New edition 2013: Editor: Domenico Giulini, Springer, 2013 .
*
*
*
*
*
*
*
*
*
Journal articles
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Historical sources
[; English translatio]
On the relativity principle and the conclusions drawn from it
at Einstein paper project.
[
]
[{{Citation, author=Lemaître, G., year=1924, title=The motion of a rigid solid according to the relativity principle, journal=Philosophical Magazine , series=Series 6, volume=48, issue=283, pages=164–176, doi=10.1080/14786442408634478]
External links
*Physics FAQ
*Eric Gourgoulhon (2010)
Special relativity from an accelerated observer perspective
Special relativity
Acceleration