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.[1] [2] [3] (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).
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).[4] 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)[5] or Synge (1960).[6] An overview of proper time transformations and alternatives was given by Romain (1963),[7] 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)[8] 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)[9] and others. A detailed representation of "special relativity in general frames" was given by Gourgoulhon (2013).[10]
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)
Here,
\tau
e(0)
e(0)
e(1)
e(2)
e(3)
\kappa1
\kappa2
\kappa3
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[15] or generalized Fermi transport[16] of tetrad
e(η)
where
\vartheta\mu\nu=\underset{Fermi–Walker
or together in simplified form:
de(η) | |
d\tau |
=-\left[(U\wedgeA)e(η)+R ⋅ e(η)\right]
with
U
A
⋅
\wedge
(U\wedgeA)e(η)=A\left(U ⋅ e(η)\right)-U\left(A ⋅ e(η)\right)
R
\omega
\epsilon
e(i)
f(i)
Since
f(i)
e(i)
f(i)
e(i)
\kappa2
\kappa3
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
e(1)-e(3)
h(i)
h(3)
\Theta=\left|\boldsymbol{\omega}\right|\tau
f(i)
For the special case
\kappa3=0
e(3)=[0,0,0,1]
\boldsymbol{\omega}=\left[0,0,0, \kappa2\right]
\Theta=\left|\boldsymbol{\omega}\right|\tau=\kappa2\tau
h(i)=e(i)
e(3)
In flat spacetime, an accelerated object is at any moment at rest in a momentary inertial frame
x'=[x\prime0,x\prime1,x\prime2,x\prime3]
X=\boldsymbol{Λ}x'
X
\boldsymbol{Λ}
e(\nu)(\tau)
q(\tau)
Then one has to put
x\prime0=t'=0
x'
r=[x1,x2,x3]
e(0)
e(i)
x0=t=\tau
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[35] (even though some authors use this term also in the rotational case). The corresponding metric has the form in Minkowski spacetime (without Riemannian terms):[36] [37] [38] [39] [40] [41] [42] [43]
However, these coordinates are not globally valid, but are restricted to
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.[44] [45] 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):[46] [47] [48] [49] [50] [51] [52] [53]
Case
\kappa1=\kappa2=\kappa3=0
See also: Hyperbolic motion (relativity) and Rindler coordinates.
The curvatures
\kappa1=\alpha,
\kappa2=\kappa3=0
\alpha
The corresponding orthonormal tetrad is identical to an inverted Lorentz transformation matrix with hyperbolic functions
\gamma=\coshη
v\gamma=\sinhη
η=\operatorname{artanh}v=\alpha\tau
\kappa2
\kappa3
Inserted into the transformations and using the worldline for
q
\begin{array}{c|c} \begin{align}T&=\left(x+
1 | |
\alpha |
\right)\sinh(\alpha\tau)\\ X&=\left(x+
1 | \right)\cosh(\alpha\tau)- | |
\alpha |
1 | |
\alpha |
\\ Y&=y\\ Z&=z \end{align} &\begin{align}\tau&=
1 | \operatorname{artanh}\left( | |
\alpha |
T | |||
|
\right)\\ x&=\sqrt{\left(X+
1 | |
\alpha |
\right)2-T2
which are valid within
-1/\alpha<X<infty
ds2=-(1+\alphax){}2d\tau2+dx2+dy2+dz2
Alternatively, by setting
q=0
X=1/\alpha
\tau=T=0
\begin{array}{c|c}\begin{align}T&=x\sinh(\alpha\tau)\ X&=x\cosh(\alpha\tau)\ Y&=y\ Z&=z\end{align}&\begin{align}\tau&=
1 | \operatorname{artanh} | |
\alpha |
T | |
X |
\ x&=\sqrt{X2-T2
which are valid within
0<X<infty
ds2=-\alpha2x2d\tau2+dx2+dy2+dz2
The curvatures
2 | |
\kappa | |
2 |
2 | |
-\kappa | |
1 |
>0
\kappa3=0
where
with
h
p0
p
v
n
\gamma
\theta
d(\nu)
The corresponding non-rotating Fermi–Walker tetrad
f(η)
\boldsymbol{\omega}=\left[0,0,0,\gamma2p\right], \left|\boldsymbol{\omega}\right|=\gamma2p, \Theta=\left|\boldsymbol{\omega}\right|\tau=\gamma2p0\tau=\gammap\tau=\gamma\theta
The resulting angle of rotation
\Theta
\begin{alignat}{1}f(0)& =e(0)& =\gamma(1, -v\sin\theta, v\cos\theta, 0)\ f(1)& =e(1)\cos\Theta-e(2)\sin\Theta& =\left(-\gammav\sin\Theta, \cos\theta\cos\Theta+\gamma\sin\theta\sin\Theta, \sin\theta\cos\Theta-\gamma\cos\theta\sin\Theta, 0\right)\ f(2)& =e(1)\sin\Theta+e(2)\cos\Theta& =\left(\gammav\cos\Theta, \cos\theta\sin\Theta-\gamma\sin\theta\cos\Theta, \sin\theta\sin\Theta+\gamma\cos\theta\cos\Theta, 0\right)\ f(3)& =e(3)& =(0, 0, 0, 1)\end{alignat}
In the following, the Frenet–Serret tetrad is used to formulate the transformation. Inserting into the transformations and using the worldline for
q
which are valid within
(X+h)2+(\gammaY)2
2 | |
\leqq1/p | |
0 |
ds2=-\gamma2\left[1-(x+h)2
2 | |
p | |
0 |
-\gamma2
2 | |
p | |
0 |
y2\right]d\tau2+2\gamma2p0(x dy-y dx)d\tau+dx2+dy2+dz2
If an observer resting in the center of the rotating frame is chosen with
h=0
which are valid within
0<\sqrt{X2+Y2
ds2
2 | |
=-\left[1-p | |
0 |
\left(x2+y2\right)\right]dt2+2p0(-y dx+x dy)dt+dx2+dy2+dz2
The last equations can also be written in rotating cylindrical coordinates (Born coordinates):[91] [92] [93] [94] [95]
which are valid within
0<r<1/p0
ds2
2 | |
=-\left(1-p | |
0 |
r2\right)dt2+2p0r2dt d\phi+dr2+r2d\phi2+dz2
Frames can be used to describe the geometry of rotating platforms, including the Ehrenfest paradox and the Sagnac effect.
The curvatures
2 | |
\kappa | |
1 |
2 | |
-\kappa | |
2 |
>0
\kappa3=0
where
where
v
n
η
\gamma
\begin{align}e(0)&=\left(\gamma\coshη, \gamma\sinhη, n, 0\right)\ e(1)&=\left(\sinhη, \coshη, 0, 0\right)\ e(2)&=\left(-n\coshη, -n\sinhη, -\gamma, 0\right)\ e(3)&=\left(0, 0, 0, 1\right)\end{align}
The corresponding non-rotating Fermi–Walker tetrad
f(η)
\boldsymbol{\omega}=\left[0,0,0,na\right], \left|\boldsymbol{\omega}\right|=na, \Theta=\left|\boldsymbol{\omega}\right|\tau=na\tau
which together with can now be inserted into, resulting in the Fermi–Walker tetrad
\begin{alignat}{1} f(0)& =e(0)& =\left(\gamma\coshη, \gamma\sinhη, n, 0\right)\\ f(1)& =e(1)\cos\Theta-e(2)\sin\Theta& =\left(\sinhη\cos\Theta+n\coshη\sin\Theta, \coshη\cos\Theta+n\sinhη\sin\Theta, \gamma\sin\Theta, 0\right)\\ f(2)& =e(1)\sin\Theta+e(2)\cos\Theta& =\left(\sinhη\sin\Theta-n\coshη\cos\Theta, \coshη\sin\Theta-n\sinhη\cos\Theta, -\gamma\cos\Theta 0\right)\\ f(3)& =e(3)& =\left(0, 0, 0, 1\right) \end{alignat}
The proper coordinates or Fermi coordinates follow by inserting
e(η)
f(η)
The curvatures
2 | |
\kappa | |
1 |
2 | |
-\kappa | |
2 |
=0
\kappa3=0
The corresponding Frenet–Serret tetrad with
\theta=a\tau
\begin{align}e(0)&=\left(1+
1 | |
2 |
\theta2, \theta,
1 | |
2 |
\theta2, 0\right)\ e(1)&=\left(\theta, 1, \theta, 0\right)\ e(2)&=\left(-
1 | |
2 |
\theta2, -\theta, 1-
1 | |
2 |
\theta2, 0\right)\ e(3)&=\left(0, 0, 0, 1\right)\end{align}
The corresponding non-rotating Fermi–Walker tetrad
f(η)
\boldsymbol{\omega}=\left[0,0,0,a\right], \left|\boldsymbol{\omega}\right|=a, \Theta=\left|\boldsymbol{\omega}\right|\tau=a\tau=\theta
which together with can now be inserted into, resulting in the Fermi–Walker tetrad (note that
\Theta=\theta
\begin{alignat}{1} f(0)& =e(0)& =\left(1+
1 | |
2 |
\theta2, \theta,
1 | |
2 |
\theta2, 0\right)\\ f(1)& =e(1)\cos\Theta-e(2)\sin\Theta& =\left(\theta\cos\theta+
1 | |
2 |
\theta2\sin\theta, \cos\theta+\theta\sin\theta, \theta\cos\theta+\left(
1 | |
2 |
\theta2-1\right)\sin\theta, 0\right)\\ f(2)& =e(1)\sin\Theta+e(2)\cos\Theta& =\left(\theta\sin\theta-
1 | |
2 |
\theta2\cos\theta, \sin\theta-\theta\cos\theta, \theta\sin\theta-\left(
1 | |
2 |
\theta2-1\right)\cos\theta, 0\right)\\ f(3)& =e(3)& =\left(0, 0, 0, 1\right) \end{alignat}
The proper coordinates or Fermi coordinates follow by inserting
e(η)
f(η)
The curvatures
\kappa1\ne0
\kappa2\ne0
\kappa3\ne0
where
with
v
n
η
h
p0
p
\theta
\gamma
\begin{align}e(0)&=\left(\gamma\coshη, \gamma\sinhη, -n\sin\theta, -n\cos\theta\right)\\ e(1)&=
1 | |
\kappa1 |
\left(\gammaa\sinhη, \gammaa\coshη, -np\cos\theta, -np\sin\theta\right)\\ e(2)&=\left(-n\coshη, -n\sinhη, \gamma\sin\theta, -\gamma\cos\theta\right)\\ e(3)&=
1 | |
\kappa1 |
\left(np\sinhη, np\coshη, \gammaa\cos\theta, \gammaa\sin\theta\right) \end{align}
The corresponding non-rotating Fermi–Walker tetrad
f(η)
e(η)
f(η)
In addition to the things described in the previous
Herglotz (1909) argued that the metric
ds2=d\sigma2+
1 | |
A44 |
(d\nu)2
where
\begin{aligned}d\nu&=A14d\xi1+A24d\xi2+A34d\xi3+A44d\xi4\\ d\sigma2&
3 | |
=\sum | |
1 |
ij Aijd\xiid\xij-
1 | |
A44 |
\left(A14d\xi1+A24d\xi2+A34d\xi3\right)2\end{aligned}
satisfies the condition of Born rigidity when
\partial | |
\partial\tau |
d\sigma2=0
(H1)
xi=ai
4 | |
+\sum | |
1 |
aij
\prime | |
x | |
j |
, i=1,2,3,4
where
ai
aij
\vartheta
\vartheta
xi
(H2)
| |||||||
d\vartheta |
+qi
4 | |
+\sum | |
1 |
pij
\prime | |
x | |
j |
=0
Here,
qi
O'
S'
-pij
S'
O'
\begin{align}D&=p23p14+p31p24+p12p34,\\ \Delta&
2 | |
=p | |
23 |
2 | |
+p | |
31 |
2 | |
+p | |
12 |
2 | |
+p | |
14 |
2 | |
+p | |
24 |
2 | |
+p | |
34 |
, \end{align}
When
\prime | |
x | |
j |
\vartheta
S'
qi
pij
D
\Delta
R4
R3
Friedrich Kottler (1912) followed Herglotz, and derived the same worldlines of constant curvatures using the following Frenet–Serret formulas in four dimensions, with
c(\alpha)
1 | , | |
R1 |
1 | , | |
R2 |
1 | |
R3 |
{\begin{matrix}
| =\\ | |||||||
ds |
| =\\ | |||||||
ds |
| =\\ | |||||||
ds |
| |||||||
ds |
= \end{matrix}\left.\begin{matrix}*&
| |||||||
R1 |
&*&*\\ -
| |||||||
R1 |
&*&
| |||||||
R2 |
&*\\ *&-
| |||||||
R2 |
&*&
| |||||||
R3 |
\\ *&*&-
| |||||||
R3 |
&* \end{matrix}\alpha=1,2,3,4\right\}}
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
y=x+\Gamma(1)c1+\Gamma(2)c2+\Gamma(3)c3+\Gamma(4)c4
{h=1,2,3,4}
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
{
(h) | |
c | |
2 |
(h) | |
,c | |
3 |
(h) | |
,c | |
4 |
(h) | |
{b | |
2 |
(h) | |
,b | |
3 |
(h) | |
,b | |
4 |
{
(1) | |
y=x+η | |
0 |
c1
(2) | |
+η | |
0 |
b2
(3) | |
+η | |
0 |
b3
(4) | |
+η | |
0 |
b4
The second case is a vector "fixed" in normal space by setting
{η(1)=0}
{
(2) | |
y=x+η | |
0 |
b2
(3) | |
+η | |
0 |
b3
(4) | |
+η | |
0 |
b4
and class (A) of Herglotz (which Kottler calls "Born's body of first kind") is given by
{y=x+\Gamma(2)c2+\Gamma(3)c3+\Gamma(4)c4
which both correspond to formula .
----In (1914a), Kottler showed that the transformation
X=x+\Gamma(1)c1+\Gamma(2)c2+\Gamma(3)c3+\Gamma(4)c4
describes the non-simultaneous coordinates of the points of a body, while the transformation with
\Gamma(1)=0
X=x+\Gamma(2)c2+\Gamma(3)c3+\Gamma(4)c4
describes the simultaneous coordinates of the points of a body. These formulas become "generalized Lorentz transformations" by inserting
\Gamma(3)=X', \Gamma(4)=Y', \Gamma(2)=Z', \Gamma(1)=ic(T'-\tau)
thus
X-x=ic(T'-\tau)c1+Z'c2+X'c3+Y'c4
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
ak{X}=x+\Delta(2)c2+\Delta(3)c3+\Delta(4)c4
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:
Hyperbolic motion | Uniform circular motion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1914b | 1914a | 1921 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle
=0,&&
=0,&&
\sinhu,&&
=\cosh
=0,&&
=0,&&
\coshu,&&
=-\sinh
=1,&&
=0,&&
=0,&&
=0,&&
=1,&&
=0,&&
=0, \end{matrix}\\ \boldsymbol{\downarrow}\\ X=x+\Delta(2)c2+\Delta(3)c3+\Delta(4)c4\\ \boldsymbol{\downarrow}\\ \begin{align} X&=x0+ak{X}'\\ Y&=y0+ak{Y}'\\ Z&=\left(b+ak{Z}'\right)\coshak{u}\\ cT&=\left(b+ak{Z}'\right)\sinhak{u} \end{align}\\ \left(\Delta(2)=ak{X}', \Delta(3)=ak{Y}', \Delta(4)=ak{Z}'\right)\\ \boldsymbol{\downarrow}\\ \begin{align} ak{X}'&=X0-x0+qxT\\ ak{Y}'&=Y0-y0+qyT\\ b+ak{Z}'&=\sqrt{\left(Z0+qxT\right)2-c2T2 | \scriptstyle
| \scriptstyle\begin{matrix}\begin{align} X&=(a+x')\cos\omegat-
|
In (1916a) Kottler gave the general metric for acceleration-relative motions based on the three curvatures
{\begin{align}dS2=&d\xi\prime2+dη\prime2+d\zeta\prime2-2c d\tau'd\xi' ⋅ η'i/R2+2c d\tau'dη' ⋅ \left(\xi'i/R2-\zeta'i/R3\right)+c d\tau'd\zeta' ⋅ η'i/R3\\ &-c2d\tau\prime2\left[\left(1-\xi'/R1\right)2+η\prime2
2 | |
/R | |
2 |
+η\prime
2 | |
/R | |
3 |
+\left(\xi'/R2-\zeta'/R3\right)2\right] \end{align} }
In (1916b) he gave it the form:
{ds2=dx2+dy2+dz2+2g14dx dit+2g24dy dit+2g34dz dit+g44(dit)2
where
{g14g24g34g44
t
\partialgi4 | + | |
\partialxk |
\partialgk4 | |
\partialxi |
=0
\partialgi4 | - | |
\partialxk |
\partialgk4 | |
\partialxi |
=const.
\sqrt{g}
xyz
Møller (1952)[5] defined the following transport equation
dei | = | |
d\tau |
| |||||||||||
c2 |
in agreement with Fermi–Walker transport by (without rotation). The Lorentz transformation into a momentary inertial frame was given by him as
xi=fi
\prime | |
(\tau)+x | |
k |
\alphaki(\tau)
in agreement with . By setting
xi
\prime | |
=x | |
l |
\prime | |
x | |
4 |
=0
t=\tau
Xi=fi(t)+x\prime\kappa\alpha\kappa(\tau)
in agreement with the Fermi coordinates, and the metric
ds2=dx2+dy2+dz2-c2dt2\left[1+
g\kappax\kappa | |
c2 |
\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):
Hyperbolic motion | Uniform circular motion | |||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1943 | 1952 | 1952 | ||||||||||||||||||||||||||||||||||||||||||||||||
{\scriptstyle\begin{matrix}\begin{align}x&=
\left\{\sqrt{(1+gX)2-g2T2 | \scriptstyle\begin{matrix}\alphaik=\left(\begin{matrix}U4/ic&0&0&iU1/c\\ 0&1&0&0\\ 0&0&1&0\\ U1/ic&0&0&U4/ic \end{matrix}\right)\\ Ui=\left(c\sinh
\right)\\ \boldsymbol{\downarrow}\\ Xi=fi(t)+x\prime\kappa\alpha\kappa(\tau)\\ \boldsymbol{\downarrow}\\ \begin{align} X&=
\\ Y&=y\\ Z&=z\\ T&=
\end{align}\\ \boldsymbol{\downarrow}\\ ds2=dx2+dy2+dz2-c2dt2\left(1+gx/c2\right)2\\ \\ \end{matrix} | \scriptstyle\begin{matrix}\alphaik=\left(\begin{matrix}\cos\alpha\cos\beta+\gamma\sin\alpha\sin\beta&\sin\alpha\cos\beta-\gamma\cos\alpha\sin\beta&0&-i
\sin\beta\\ \cos\alpha\sin\beta-\gamma\sin\alpha\cos\beta&\sin\alpha\sin\beta+\gamma\cos\alpha\cos\beta&0&i
\cos\beta\\ 0&0&1&0\\ i
\sin\alpha&-i
\cos\alpha&0&\gamma \end{matrix}\right)\\ {\alpha=\omega\gamma\tau}, {\beta=\gamma\alpha=\omega\gamma2\tau}. \end{matrix} |
General case | Uniform rotation | Catenary | Semicubical parabola | Hyperbolic motion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Herglotz (1909) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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loxodromic | elliptic | hyperbolic | parabolic | hyperbolic \scriptstyle(\alpha=0) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle\begin{matrix}D\ne0\\ p21=-p12=1\\ p34=-p43=i\\ qi=[0,0,0,0] \end{matrix} | \scriptstyle\begin{matrix}D=0, \Delta>0\\ p21=-p12=1\\ \\ qi=[0,0,0,\deltai] \end{matrix} | \scriptstyle\begin{matrix}D=0, \Delta<0\\ p34=-p43=i\\ \\ qi=[\alpha,0,0,0] \end{matrix} | \scriptstyle\begin{matrix}D=0, \Delta=0\\ p31=-p13=1\\ p41=-p14=i\\ qi=[0,\beta,0,\deltai] \end{matrix} | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Lorentz-Transformations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle\begin{align}x+iy&=(x'+iy')eiλ\vartheta\\ x-iy&=(x'-iy')e-iλ\vartheta\\ t-z&=(t'-z')e\vartheta\\ t+z&=(t'+z')e-\vartheta\end{align} | \scriptstyle\begin{align}x+iy&=(x'+iy')ei\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')+
\delta\vartheta2\\ y&=y'+\beta\vartheta\\ z&=z'+\varthetax'+
\vartheta2(t'-z')+
\delta\vartheta3\\ 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} | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Trajectories (time) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle\begin{align}x+iy&=(x0+iy0)eiλ\\ x-iy&=(x0-iy0)e-iλ\\ z&
+t2 | \scriptstyle\begin{align}x+iy&=(x0+iy0
\\ x-iy&=(x0-iy0
\\ z&=z0\end{align} | \scriptstyle\begin{align}x&=x0+\alphalg
0 | \scriptstyle\begin{align}x&=x0+
\delta\vartheta2\\ y&=y0+\beta\vartheta\\ z&=z0+x0\vartheta+
\delta\vartheta3\\ t-z&=\delta\vartheta \end{align} | \scriptstyle\begin{align}x&=x0\\ y&=y0\\ z&
+t2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Kottler (1912, 1914) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
hyperspherical curve | uniform rotation | catenary | cubic curve | hyperbolic motion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Curvatures | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle\begin{align}\left(
\right)2&=
\right)2&=-
\right)2&=-
\end{align} | \scriptstyle\begin{align}\left(
\right)2&=
\right)2&=-
\right)2&=0 \end{align} | \scriptstyle\begin{align}\left(
\right)2&=
\right)2&=-
\right)2&=0 \end{align} | \scriptstyle\begin{align}\left(
\right)2&=
\right)2&=-
\right)2\\ \left(
\right)2&=0 \end{align} | \scriptstyle\begin{align}\left(
\right)2&=
\right)2&=0\\ \left(
\right)2&=0 \end{align} | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Trajectory of \scriptstyleS4 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle\begin{align}x(1)&=a\cosλ\left(u-u0\right)\\ x(2)&=a\sinλ\left(u-u0\right)\\ x(3)&=b\cosiu\\ x(4)&=b\siniu \end{align} | \scriptstyle\begin{align}x(1)&=a\cosλ\left(u-u0\right)\\ x(2)&=a\sinλ\left(u-u0\right)\\ x(3)&
\\ x(4)&=iu \end{align} | \scriptstyle\begin{align}x(1)&
+\alphau\\ x(2)&
\\ x(3)&=b\cosiu\\ x(4)&=b\siniu \end{align} | \scriptstyle\begin{align}x(1)&
\alphau2\\ x(2)&
\\ x(3)&
\alphau3\\ x(4)&
\alphau2\right)+i\alphau \end{align} | \scriptstyle\begin{align}x(1)&
\\ x(2)&
\\ x(3)&=b\cosiu\\ x(4)&=b\siniu \end{align} | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Trajectory (time) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
\scriptstyle\begin{align}x&=a\cosλ\left(u-u0\right)\\ y&=a\sinλ\left(u-u0\right)\\ z&=\sqrt{b2+c2t2 | \scriptstyle\begin{align}x&=a\cos\omegaz\left(t-t0\right)\\ y&=a\sin\omegaz\left(t-t0\right)\\ z&=z0\end{align} | \scriptstyle\begin{align}x&=x0+\alphaln
| \scriptstyle\begin{align}x&=x0+
\alphau2\\ y&=y0\\ z&=z0+x0u+
\alphau3\\ ct&=z0+x0u+
\alphau3+\alphau\\ &=z+\alphau \end{align} | \scriptstyle\begin{align}x&=x0\\ y&=y0\\ z&=\sqrt{b2+c2t2 |