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 ${\displaystyle \mathbf {e} _{(\eta )}}$ (also called vielbein, moving frame, frame field, local frame, repère mobile in arbitrary dimensions):^{[16][17][18][19]}

${\displaystyle {\begin{aligned}{\frac {d\mathbf {e} _{(0)}}{d\tau }}&=\kappa _{1}\mathbf {e} _{(1)},&{\frac {d\mathbf {e} _{(1)}}{d\tau }}&=\kappa _{1}\mathbf {e} _{(0)}+\kappa _{2}\mathbf {e} _{(2)},\\{\frac {d\mathbf {e} _{(2)}}{d\tau }}&=-\kappa _{2}\mathbf {e} _{(1)}+\kappa _{3}\mathbf {e} _{(3)},\quad &{\frac {d\mathbf {e} _{(3)}}{d\tau }}&=-\kappa _{3}\mathbf {e} _{(2)},\end{aligned}}}$

(1)

Here, ${\displaystyle \tau }$ is the proper time along the worldline, the timelike field ${\displaystyle \mathbf {e} _{(0)}}$ is called the tangent that corresponds to the four-velocity, the three spacelike fields are orthogonal to ${\displaystyle \mathbf {e} _{(0)}}$ and are called the principal normal ${\displaystyle \mathbf {e} _{(1)}}$, the binormal ${\displaystyle \mathbf {e} _{(2)}}$ and the trinormal ${\displaystyle \mathbf {e} _{(3)}}$. The first curvature ${\displaystyle \kappa _{1}}$ corresponds to the magnitude of four-acceleration (i.e., proper acceleration), the other curvatures ${\displaystyle \kappa _{2}}$ and ${\displaystyle \kappa _{3}}$ 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^[20] or generalized Fermi transport^[21] of tetrad ${\displaystyle \mathbf {e} _{(\eta )}}$, namely^{[10][12][22][21][20][23]}

${\displaystyle {\frac {d\mathbf {e} _{(\eta )}}{d\tau }}=-{\boldsymbol {\vartheta }}\mathbf {e} _{(\eta )}}$

(2)

where

${\displaystyle \vartheta ^{\mu \nu }={\underset {\text{Fermi–Walker}}{\underbrace {A^{\mu }U^{\nu }-A^{\nu }U^{\mu }} }}+{\underset {\mathrm {\text{spatial rotation}} }{\underbrace {U_{\alpha }\omega _{\beta }\epsilon ^{\alpha \beta \mu \nu }} }}}$

or together in simplified form:

${\displaystyle {\frac {d\mathbf {e} _{(\eta )}}{d\tau }}=-\left[(\mathbf {U} \wedge \mathbf {A} )\mathbf {e} _{(\eta )}+\mathbf {R} \cdot \mathbf {e} _{(\eta )}\right]}$

with ${\displaystyle \mathbf {U} }$ as four-velocity and ${\displaystyle \mathbf {A} }$ as four-acceleration, and "${\displaystyle \cdot }$" indicates the dot product and "${\displaystyle \wedge }$" the wedge product. The first part ${\displaystyle (\mathbf {U} \wedge \mathbf {A} )\mathbf {e} _{(\eta )}=\mathbf {A} \left(\mathbf {U} \cdot \mathbf {e} _{(\eta )}\right)-\mathbf {U} \left(\mathbf {A} \cdot \mathbf {e} _{(\eta )}\right)}$ represents Fermi–Walker transport,^[13] 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 ${\displaystyle \mathbf {R} }$ consists of an antisymmetric second rank tensor with ${\displaystyle \omega }$ as the angular velocity four-vector and ${\displaystyle \epsilon }$ 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 ${\displaystyle \mathbf {e} _{(i)}}$ of a rotating tetrad (such as a Frenet–Serret tetrad) with respect to the non-rotating spacelike fields ${\displaystyle \mathbf {f} _{(i)}}$ of a Fermi–Walker tetrad along the same world line.

Deriving Fermi–Walker tetrads from Frenet–Serret tetrads

Since ${\displaystyle \mathbf {f} _{(i)}}$ and ${\displaystyle \mathbf {e} _{(i)}}$ on the same worldline are connected by a rotation matrix, it is possible to construct non-rotating Fermi–Walker tetrads using rotating Frenet–Serret tetrads,^[24][25] which not only works in flat spacetime but for arbitrary spacetimes as well, even though the practical realization can be hard to achieve.^[26] For instance, the angular velocity vector between the respective spacelike tetrad fields ${\displaystyle \mathbf {f} _{(i)}}$ and ${\displaystyle \mathbf {e} _{(i)}}$ can be given in terms of torsions ${\displaystyle \kappa _{2}}$ and ${\displaystyle \kappa _{3}}$:^{[12][13][27][28]}

${\displaystyle {\boldsymbol {\omega }}=\kappa _{3}\mathbf {e} _{(1)}+\kappa _{2}\mathbf {e} _{(3)}}$ and ${\displaystyle \left|{\boldsymbol {\omega }}\right|={\sqrt {\kappa _{2}^{2}+\kappa _{3}^{2}}}}$

(3a)

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 ${\displaystyle \mathbf {e} _{(1)}-\mathbf {e} _{(3)}}$ plane by constant counter-clockweise rotation, then the resulting intermediary spatial frame ${\displaystyle \mathbf {h} _{(i)}}$ is constantly rotated around the ${\displaystyle \mathbf {h} _{(3)}}$ axis by the angle ${\displaystyle \Theta =\left|{\boldsymbol {\omega }}\right|\tau }$, which finally gives the spatial Fermi–Walker frame ${\displaystyle \mathbf {f} _{(i)}}$ (note that the timelike field remains the same):^[25]

${\displaystyle {\begin{array}{c|c|c}{\begin{aligned}\mathbf {h} _{(1)}&={\frac {\kappa _{2}\mathbf {e} _{(1)}-\kappa _{3}\mathbf {e} _{(3)}}{\left|{\boldsymbol {\omega }}\right|}}\\\mathbf {h} _{(2)}&=\mathbf {e} _{(2)}\\\mathbf {h} _{(3)}&={\frac {\boldsymbol {\omega }}{\left|{\boldsymbol {\omega }}\right|}}\end{aligned}}&{\begin{aligned}\mathbf {f} _{(1)}&=\mathbf {h} _{(1)}\cos \Theta -h_{(2)}\sin \Theta \\\mathbf {f} _{(2)}&=\mathbf {h} _{(1)}\sin \Theta +h_{(2)}\cos \Theta \\\mathbf {f} _{(3)}&=\mathbf {h} _{(3)}\end{aligned}}&\mathbf {e} _{(0)}=\mathbf {h} _{(0)}=\mathbf {f} _{(0)}\end{array}}}$

(3b)

For the special case ${\displaystyle \kappa _{3}=0}$ and ${\displaystyle \mathbf {e} _{(3)}=[0,0,0,1]}$, it follows ${\displaystyle {\boldsymbol {\omega }}=\left[0,0,0,\ \kappa _{2}\right]}$ and ${\displaystyle \Theta =\left|{\boldsymbol {\omega }}\right|\tau =\kappa _{2}\tau }$ and ${\displaystyle \mathbf {h} _{(i)}=\mathbf {e} _{(i)}}$, therefore (3b) is reduced to a single constant rotation around the ${\displaystyle \mathbf {e} _{(3)}}$-axis:^{[29][30][31][24]}

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {f} _{(1)}&=\mathbf {e} _{(1)}\cos \Theta -\mathbf {e} _{(2)}\sin \Theta \\\mathbf {f} _{(2)}&=\mathbf {e} _{(1)}\sin \Theta +\mathbf {e} _{(2)}\cos \Theta \\\mathbf {f} _{(3)}&=\mathbf {e} _{(3)}\end{aligned}}&\mathbf {e} _{(0)}=\mathbf {f} _{(0)}\end{array}}}$

(3c)

Hyperbolic motion

The curvatures ${\displaystyle \kappa _{1}=\alpha ,}$ ${\displaystyle \kappa _{2}=\kappa _{3}=0}$, where ${\displaystyle \alpha }$ is the constant proper acceleration in the direction of motion, produce hyperbolic motion because the worldline in the Minkowski diagram is a hyperbola:^{[55][56][57][58][59][60]}

${\displaystyle \mathbf {X} =\left[{\frac {1}{\alpha }}\sinh(\alpha \tau ),\quad {\frac {1}{\alpha }}\left(\cosh(\alpha \tau )-1\right),\quad 0,\quad 0\right]}$

(5a)

The corresponding orthonormal tetrad is identical to an inverted Lorentz transformation matrix with hyperbolic functions ${\displaystyle \gamma =\cosh \eta }$ as Lorentz factor and ${\displaystyle v\gamma =\sinh \eta }$ as proper velocity and ${\displaystyle \eta =\operatorname {artanh} v=\alpha \tau }$ as rapidity (since the torsions ${\displaystyle \kappa _{2}}$ and ${\displaystyle \kappa _{3}}$ are zero, the Frenet–Serret formulas and Fermi–Walker formulas produce the same tetrad):^{[56][61][62][63][64][65][66]}

${\displaystyle {\begin{aligned}\mathbf {e} _{(0)}&=(\cosh(\alpha \tau ),\ \sinh(\alpha \tau ),\ 0,\ 0)\\\mathbf {e} _{(1)}&=(\sinh(\alpha \tau ),\ \cosh(\alpha \tau ),\ 0,\ 0)\\\mathbf {e} _{(2)}&=(0,\ 0,\ 1,\ 0)\\\mathbf {e} _{(3)}&=(0,\ 0,\ 0,\ 1)\end{aligned}}}$

(5b)

Inserted into the transformations (4b) and using the worldline (5a) for ${\displaystyle \mathbf {q} }$, the accelerated observer is always located at the origin, so the Kottler-Møller coordinates follow^{[67][68][62][69][70]}

${\displaystyle {\begin{array}{c|c}{\begin{aligned}T&=\left(x+{\frac {1}{\alpha }}\right)\sinh(\alpha \tau )\\X&=\left(x+{\frac {1}{\alpha }}\right)\cosh(\alpha \tau )-{\frac {1}{\alpha }}\\Y&=y\\Z&=z\end{aligned}}&{\begin{aligned}\tau &={\frac {1}{\alpha }}\operatorname {artanh} \left({\frac {T}{X+{\frac {1}{\alpha }}}}\right)\\x&={\sqrt {\left(X+{\frac {1}{\alpha }}\right)^{2}-T^{2}}}-{\frac {1}{\alpha }}\\y&=Y\\z&=Z\end{aligned}}\end{array}}}$

which are valid within ${\displaystyle -1/\alpha <X<\infty }$, with the metric

${\displaystyle ds^{2}=-(1+\alpha x){}^{2}d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}}$.

Alternatively, by setting ${\displaystyle \mathbf {\mathbf {q} } =0}$ the accelerated observer is located at ${\displaystyle X=1/\alpha }$ at time ${\displaystyle \tau =T=0}$, thus the Rindler coordinates follow from (4b) and (5a, 5b):^[71][72][73]

${\displaystyle {\begin{array}{c|c}{\begin{aligned}T&=x\sinh(\alpha \tau )\\X&=x\cosh(\alpha \tau )\\Y&=y\\Z&=z\end{aligned}}&{\begin{aligned}\tau &={\frac {1}{\alpha }}\operatorname {artanh} {\frac {T}{X}}\\x&={\sqrt {X^{2}-T^{2}}}\\y&=Y\\z&=Z\end{aligned}}\end{array}}}$

which are valid within ${\displaystyle 0<X<\infty }$, with the metric

${\displaystyle ds^{2}=-\alpha ^{2}x^{2}d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}}$

Uniform circular motion

The curvatures ${\displaystyle \kappa _{2}^{2}-\kappa _{1}^{2}>0}$, ${\displaystyle \kappa _{3}=0}$ produce uniform circular motion, with the worldline^{[74][75][76][77][78][79][80]}

${\displaystyle X=\left[\gamma \tau ,\ {\frac {n}{p}}\cos(p\tau ),\ {\frac {n}{p}}\sin(p\tau ),\ 0\right]}$

(6a)

where

${\displaystyle {\begin{array}{c|c|c}{\begin{aligned}\kappa _{1}&=-\gamma ^{2}hp_{0}^{2}\\\kappa _{2}&=\gamma ^{2}p_{0}\end{aligned}}&{\begin{aligned}p&={\sqrt {\kappa _{2}^{2}-\kappa _{1}^{2}}}={\frac {\kappa _{2}}{\gamma }}=\gamma p_{0}\\p_{0}&={\frac {\kappa _{2}^{2}-\kappa _{1}^{2}}{\kappa _{2}}}={\frac {\kappa _{2}}{\gamma ^{2}}}={\frac {p}{\gamma }}\\\theta &=p\tau =p_{0}t=\gamma p_{0}\tau \end{aligned}}&{\begin{aligned}n&={\frac {\kappa _{1}}{p}}=v\gamma ={\sqrt {\gamma ^{2}-1}}\\h&={\frac {\kappa _{1}}{\kappa _{2}^{2}-\kappa _{1}^{2}}}={\frac {n}{p}}\\v&={\frac {\kappa _{1}}{\kappa _{2}}}=hp_{0}={\frac {n}{\gamma }}\\\gamma &={\frac {\kappa _{2}}{p}}={\frac {1}{\sqrt {1-v^{2}}}}={\sqrt {n^{2}+1}}\end{aligned}}\end{array}}}$

(6b)

with ${\displaystyle h}$ as orbital radius, ${\displaystyle p_{0}}$ as coordinate angular velocity, ${\displaystyle p}$ as proper angular velocity, ${\displaystyle v}$ as tangential velocity, ${\displaystyle n}$ as proper velocity, ${\displaystyle \gamma }$ as Lorentz factor, and ${\displaystyle \theta }$ as angle of rotation. The tetrad can be derived from the Frenet–Serret equations (1),^{[74][76][77][80]} or more simply be obtained by a Lorentz transformation of the tetrad ${\displaystyle d_{(\nu )}}$ of ordinary rotating coordinates:^[81][82]

${\displaystyle {\begin{array}{c|c}{\begin{aligned}d_{(0)}&=(1,\ 0,\ 0,\ 0)\\d_{(1)}&=(0,\ \cos \theta ,\ \sin \theta ,\ 0)\\d_{(2)}&=(0,\ -\sin \theta ,\ \cos \theta ,\ 0)\\d_{(3)}&=(0,\ 0,\ 0,\ 1)\end{aligned}}&{\begin{alignedat}{1}\mathbf {e} _{(0)}&\ =\gamma \left(d_{(0)}+vd_{(2)}\right)&\ =\gamma (1,\ -v\sin \theta ,\ v\cos \theta ,\ 0)\\\mathbf {e} _{(1)}&\ =d_{(1)}&\ =(0,\ \cos \theta ,\ \sin \theta ,\ 0)\\\mathbf {e} _{(2)}&\ =\gamma \left(d_{(2)}+vd_{(0)}\right)&\ =\gamma \left(v,\ -\sin \theta ,\ \cos \theta ,\ 0\right)\\\mathbf {e} _{(3)}&\ =d_{(3)}&\ =(0,\ 0,\ 0,\ 1)\end{alignedat}}\end{array}}}$

(6c)

The corresponding non-rotating Fermi–Walker tetrad ${\displaystyle \mathbf {f} _{(\eta )}}$ on the same worldline can be obtained by solving the Fermi–Walker part of equation (2).^[83][84] Alternatively, one can use (6b) together with (3a), which gives

${\displaystyle {\boldsymbol {\omega }}=\left[0,0,0,\gamma ^{2}p\right],\quad \left|{\boldsymbol {\omega }}\right|=\gamma ^{2}p,\quad \Theta =\left|{\boldsymbol {\omega }}\right|\tau =\gamma ^{2}p_{0}\tau =\gamma p\tau =\gamma \theta }$

The resulting angle of rotation ${\displaystyle \Theta }$ together with (6c) can now be inserted into (3c), by which the Fermi–Walker tetrad follows^[31][24]

${\displaystyle {\begin{alignedat}{1}\mathbf {f} _{(0)}&\ =\mathbf {e} _{(0)}&\ =\gamma (1,\ -v\sin \theta ,\ v\cos \theta ,\ 0)\\\mathbf {f} _{(1)}&\ =\mathbf {e} _{(1)}\cos \Theta -\mathbf {e} _{(2)}\sin \Theta &\ =\left(-\gamma v\sin \Theta ,\ \cos \theta \cos \Theta +\gamma \sin \theta \sin \Theta ,\ \sin \theta \cos \Theta -\gamma \cos \theta \sin \Theta ,\ 0\right)\\\mathbf {f} _{(2)}&\ =\mathbf {e} _{(1)}\sin \Theta +\mathbf {e} _{(2)}\cos \Theta &\ =\left(\gamma v\cos \Theta ,\ \cos \theta \sin \Theta -\gamma \sin \theta \cos \Theta ,\ \sin \theta \sin \Theta +\gamma \cos \theta \cos \Theta ,\ 0\right)\\\mathbf {f} _{(3)}&\ =\mathbf {e} _{(3)}&\ =(0,\ 0,\ 0,\ 1)\end{alignedat}}}$

In the following, the Frenet–Serret tetrad is used to formulate the transformation. Inserting (6c) into the transformations (4b) and using the worldline (6a) for ${\displaystyle \mathbf {q} }$ gives the coordinates^{[74][76][85][86][87][38]}

${\displaystyle {\begin{array}{c|c}{\begin{aligned}T&=\gamma \left(\tau +\gamma yv\right)\\X&=(x+h)\cos \theta -y\gamma \sin \theta \\Y&=(x+h)\sin \theta +y\gamma \cos \theta \\Z&=z\end{aligned}}&{\begin{aligned}\tau &=\gamma ^{-1}\left(T-\gamma yv\right)\\x&=X\cos \theta +Y\sin \theta -h\\y&=\gamma ^{-1}\left(-X\sin \theta +Y\cos \theta \right)\\z&=Z\end{aligned}}\end{array}}}$

(6d)

which are valid within ${\displaystyle (X+h)^{2}+(\gamma Y)^{2}\leqq 1/p_{0}^{2}}$, with the metric

${\displaystyle ds^{2}=-\gamma ^{2}\left[1-(x+h)^{2}p_{0}^{2}-\gamma ^{2}p_{0}^{2}y^{2}\right]d\tau ^{2}+2\gamma ^{2}p_{0}(x\ dy-y\ dx)d\tau +dx^{2}+dy^{2}+dz^{2}}$

If an observer resting in the center of the rotating frame is chosen with ${\displaystyle h=0}$, the equations reduce to the ordinary rotational transformation^[88][89][90]

${\displaystyle {\begin{array}{c|c|c}{\begin{aligned}T&=t\\X&=x\cos \theta -y\sin \theta \\Y&=x\sin \theta +y\cos \theta \\Z&=z\end{aligned}}&{\begin{aligned}t&=T\\x&=X\cos \theta +Y\sin \theta \\y&=-X\sin \theta +Y\cos \theta \\z&=Z\end{aligned}}&{\text{or}}\quad {\begin{aligned}T&=t\\X+iY&=(x+iy)e^{i\theta }\\Z&=z\end{aligned}}\end{array}}}$

(6e)

which are valid within ${\displaystyle 0<{\sqrt {X^{2}+Y^{2}}}<1/p_{0}}$, and the metric

${\displaystyle ds^{2}=-\left[1-p_{0}^{2}\left(x^{2}+y^{2}\right)\right]dt^{2}+2p_{0}(-y\ dx+x\ dy)dt+dx^{2}+dy^{2}+dz^{2}}$.

The last equations can also be written in rotating cylindrical coordinates (Born coordinates):^{[91][92][93][94][95]}

${\displaystyle {\begin{array}{c|c|c|c}{\begin{aligned}T&=t\\X&=r\cos(\phi +\theta )\\Y&=r\sin(\phi +\theta )\\Z&=z\end{aligned}}&{\begin{aligned}t&=T\\x&=r\cos(\Phi -\theta )\\y&=r\sin(\Phi -\theta )\\z&=Z\end{aligned}}\rightarrow &{\begin{aligned}T&=t\\R&=r\\\Phi &=\phi +\theta \\Z&=z\end{aligned}}&{\begin{aligned}t&=T\\r&=R\\\phi &=\Phi -\theta \\z&=Z\end{aligned}}\end{array}}}$

(6f)

which are valid within ${\displaystyle 0<r<1/p_{0}}$, and the metric

${\displaystyle ds^{2}=-\left(1-p_{0}^{2}r^{2}\right)dt^{2}+2p_{0}r^{2}dt\ d\phi +dr^{2}+r^{2}d\phi ^{2}+dz^{2}}$

Frames (6d, 6e, 6f) can be used to describe the geometry of rotating platforms, including the Ehrenfest paradox and the Sagnac effect.

Catenary

The curvatures ${\displaystyle \kappa _{1}^{2}-\kappa _{2}^{2}>0}$, ${\displaystyle \kappa _{3}=0}$ produce a catenary, i.e., hyperbolic motion combined with a spacelike translation^{[96][97][98][99][100][101][102]}

${\displaystyle X=\left[{\frac {\gamma }{a}}\sinh(a\tau ),\quad {\frac {\gamma }{a}}\cosh(a\tau ),\quad n\tau ,\quad 0\right]}$

(7a)

where

${\displaystyle {\begin{array}{c|c|c}{\begin{aligned}\kappa _{1}&=\gamma a\\\kappa _{2}&=na\end{aligned}}&{\begin{aligned}a&={\sqrt {\kappa _{1}^{2}-\kappa _{2}^{2}}}\\n&={\frac {\kappa _{2}}{a}}=v\gamma ={\sqrt {\gamma ^{2}-1}}\\\eta &=a\tau \end{aligned}}&{\begin{aligned}v&={\frac {\kappa _{2}}{\kappa _{1}}}={\frac {n}{\gamma }}\\\gamma &={\frac {\kappa _{1}}{a}}={\frac {1}{\sqrt {1-v^{2}}}}={\sqrt {n^{2}+1}}\end{aligned}}\end{array}}}$

(7b)

where ${\displaystyle v}$ is the velocity, ${\displaystyle n}$ the proper velocity, ${\displaystyle \eta }$ as rapidity, ${\displaystyle \gamma }$ is the Lorentz factor. The corresponding Frenet–Serret tetrad is:^[97][99]

${\displaystyle {\begin{aligned}\mathbf {e} _{(0)}&=\left(\gamma \cosh \eta ,\ \gamma \sinh \eta ,\ n,\ 0\right)\\\mathbf {e} _{(1)}&=\left(\sinh \eta ,\ \cosh \eta ,\ 0,\ 0\right)\\\mathbf {e} _{(2)}&=\left(-n\cosh \eta ,\ -n\sinh \eta ,\ -\gamma ,\ 0\right)\\\mathbf {e} _{(3)}&=\left(0,\ 0,\ 0,\ 1\right)\end{aligned}}}$

The corresponding non-rotating Fermi–Walker tetrad ${\displaystyle \mathbf {f} _{(\eta )}}$ on the same worldline can be obtained by solving the Fermi–Walker part of equation (2).^[102] The same result follows from (3a), which gives

${\displaystyle {\boldsymbol {\omega }}=\left[0,0,0,na\right],\quad \left|{\boldsymbol {\omega }}\right|=na,\quad \Theta =\left|{\boldsymbol {\omega }}\right|\tau =na\tau }$

which together with (7a) can now be inserted into (3c), resulting in the Fermi–Walker tetrad

${\displaystyle {\begin{alignedat}{1}\mathbf {f} _{(0)}&\ =\mathbf {e} _{(0)}&\ =\left(\gamma \cosh \eta ,\ \gamma \sinh \eta ,\ n,\ 0\right)\\\mathbf {f} _{(1)}&\ =\mathbf {e} _{(1)}\cos \Theta -\mathbf {e} _{(2)}\sin \Theta &\ =\left(\sinh \eta \cos \Theta +n\cosh \eta \sin \Theta ,\ \cosh \eta \cos \Theta +n\sinh \eta \sin \Theta ,\ \gamma \sin \Theta ,\ 0\right)\\\mathbf {f} _{(2)}&\ =\mathbf {e} _{(1)}\sin \Theta +\mathbf {e} _{(2)}\cos \Theta &\ =\left(\sinh \eta \sin \Theta -n\cosh \eta \cos \Theta ,\ \cosh \eta \sin \Theta -n\sinh \eta \cos \Theta ,\ -\gamma \cos \Theta \ 0\right)\\\mathbf {f} _{(3)}&\ =\mathbf {e} _{(3)}&\ =\left(0,\ 0,\ 0,\ 1\right)\end{alignedat}}}$

The proper coordinates or Fermi coordinates follow by inserting ${\displaystyle \mathbf {e} _{(\eta )}}$ or ${\displaystyle \mathbf {f} _{(\eta )}}$ into (4b).

Semicubical parabola

The curvatures ${\displaystyle \kappa _{1}^{2}-\kappa _{2}^{2}=0}$, ${\displaystyle \kappa _{3}=0}$ produce a semicubical parabola or cusped motion^{[103][104][105][106][107][108][109]}

${\displaystyle X=\left[\tau +{\frac {1}{6}}a^{2}\tau ^{3},\ {\frac {1}{2}}a\tau ^{2},\ {\frac {1}{6}}a^{2}\tau ^{3},\ 0\right]}$ with ${\displaystyle a=\kappa _{1}=\kappa _{2}}$

(8)

The corresponding Frenet–Serret tetrad with ${\displaystyle \theta =a\tau }$ is:^[104][106]

${\displaystyle {\begin{aligned}\mathbf {e} _{(0)}&=\left(1+{\frac {1}{2}}\theta ^{2},\ \theta ,\ {\frac {1}{2}}\theta ^{2},\ 0\right)\\\mathbf {e} _{(1)}&=\left(\theta ,\ 1,\ \theta ,\ 0\right)\\\mathbf {e} _{(2)}&=\left(-{\frac {1}{2}}\theta ^{2},\ -\theta ,\ 1-{\frac {1}{2}}\theta ^{2},\ 0\right)\\\mathbf {e} _{(3)}&=\left(0,\ 0,\ 0,\ 1\right)\end{aligned}}}$

The corresponding non-rotating Fermi–Walker tetrad ${\displaystyle \mathbf {f} _{(\eta )}}$ on the same worldline can be obtained by solving the Fermi–Walker part of equation (2).^[109] The same result follows from (3a), which gives

${\displaystyle {\boldsymbol {\omega }}=\left[0,0,0,a\right],\quad \left|{\boldsymbol {\omega }}\right|=a,\quad \Theta =\left|{\boldsymbol {\omega }}\right|\tau =a\tau =\theta }$

which together with (8) can now be inserted into (3c), resulting in the Fermi–Walker tetrad (note that ${\displaystyle \Theta =\theta }$ in this case):

${\displaystyle {\begin{alignedat}{1}\mathbf {f} _{(0)}&\ =\mathbf {e} _{(0)}&\ =\left(1+{\frac {1}{2}}\theta ^{2},\ \theta ,\ {\frac {1}{2}}\theta ^{2},\ 0\right)\\\mathbf {f} _{(1)}&\ =\mathbf {e} _{(1)}\cos \Theta -\mathbf {e} _{(2)}\sin \Theta &\ =\left(\theta \cos \theta +{\frac {1}{2}}\theta ^{2}\sin \theta ,\ \cos \theta +\theta \sin \theta ,\ \theta \cos \theta +\left({\frac {1}{2}}\theta ^{2}-1\right)\sin \theta ,\ 0\right)\\\mathbf {f} _{(2)}&\ =\mathbf {e} _{(1)}\sin \Theta +\mathbf {e} _{(2)}\cos \Theta &\ =\left(\theta \sin \theta -{\frac {1}{2}}\theta ^{2}\cos \theta ,\ \sin \theta -\theta \cos \theta ,\ \theta \sin \theta -\left({\frac {1}{2}}\theta ^{2}-1\right)\cos \theta ,\ 0\right)\\\mathbf {f} _{(3)}&\ =\mathbf {e} _{(3)}&\ =\left(0,\ 0,\ 0,\ 1\right)\end{alignedat}}}$

The proper coordinates or Fermi coordinates follow by inserting ${\displaystyle \mathbf {e} _{(\eta )}}$ or ${\displaystyle \mathbf {f} _{(\eta )}}$ into (4b).

General case

The curvatures ${\displaystyle \kappa _{1}\neq 0}$, ${\displaystyle \kappa _{2}\neq 0}$, ${\displaystyle \kappa _{3}\neq 0}$ produce hyperbolic motion combined with uniform circular motion. The worldline is given by^{[110][111][112][113][114][115][116]}

${\displaystyle X=\left[{\frac {\gamma }{a}}\sinh(a\tau ),\quad {\frac {\gamma }{a}}\cosh(a\tau ),\quad {\frac {n}{p}}\sin(p\tau ),\quad {\frac {n}{p}}\cos(p\tau )\right]}$

(9a)

where

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\kappa _{1}&={\sqrt {n^{2}p^{2}+\gamma ^{2}a^{2}}}\\\kappa _{2}&={\frac {1}{\kappa _{1}}}\left(a^{2}+p^{2}\right)\gamma n\\\kappa _{3}&={\frac {1}{\kappa _{1}}}ap\end{aligned}}&{\begin{aligned}a&={\sqrt {{\frac {1}{2}}\left(\kappa _{1}^{2}-\kappa _{2}^{2}-\kappa _{3}^{2}+r\right)}}\\p&={\sqrt {{\frac {1}{2}}\left(-\kappa _{1}^{2}+\kappa _{2}^{2}+\kappa _{3}^{2}+r\right)}}=\gamma p_{0}\\n&={\sqrt {{\frac {1}{2}}\left({\frac {1}{r}}\left[\kappa _{1}^{2}+\kappa _{2}^{2}+\kappa _{3}^{2}\right]-1\right)}}={\sqrt {\frac {\kappa _{1}^{2}-a^{2}}{p^{2}+a^{2}}}}=v\gamma ={\sqrt {\gamma ^{2}-1}}\\\gamma &={\sqrt {{\frac {1}{2}}\left({\frac {1}{r}}\left[\kappa _{1}^{2}+\kappa _{2}^{2}+\kappa _{3}^{2}\right]+1\right)}}={\sqrt {\frac {\kappa _{1}^{2}+p^{2}}{p^{2}+a^{2}}}}={\frac {1}{\sqrt {1-v^{2}}}}={\sqrt {n^{2}+1}}\\r&={\sqrt {\left(\kappa _{1}^{2}-\kappa _{2}^{2}-\kappa _{3}^{2}\right)^{2}+4\kappa _{1}^{2}\kappa _{3}^{2}}}\\&p_{0}={\frac {p}{\gamma }},\quad v=hp_{0}={\frac {n}{\gamma }},\quad h={\frac {n}{p}},\quad \eta =a\tau ,\quad \theta =p\tau =p_{0}t=\gamma p_{0}\tau \end{aligned}}\end{array}}}$

(9b)

with ${\displaystyle v}$ as tangential velocity, ${\displaystyle n}$ as proper tangential velocity, ${\displaystyle \eta }$ as rapidity, ${\displaystyle h}$ as orbital radius, ${\displaystyle p_{0}}$ as coordinate angular velocity, ${\displaystyle p}$ as proper angular velocity, ${\displaystyle \theta }$ as angle of rotation, ${\displaystyle \gamma }$ is the Lorentz factor. The Frenet–Serret tetrad is^[111][113]

${\displaystyle {\begin{aligned}\mathbf {e} _{(0)}&=\left(\gamma \cosh \eta ,\ \gamma \sinh \eta ,\ -n\sin \theta ,\ -n\cos \theta \right)\\\mathbf {e} _{(1)}&={\frac {1}{\kappa _{1}}}\left(\gamma a\sinh \eta ,\ \gamma a\cosh \eta ,\ -np\cos \theta ,\ -np\sin \theta \right)\\\mathbf {e} _{(2)}&=\left(-n\cosh \eta ,\ -n\sinh \eta ,\ \gamma \sin \theta ,\ -\gamma \cos \theta \right)\\\mathbf {e} _{(3)}&={\frac {1}{\kappa _{1}}}\left(np\sinh \eta ,\ np\cosh \eta ,\ \gamma a\cos \theta ,\ \gamma a\sin \theta \right)\end{aligned}}}$

The corresponding non-rotating Fermi–Walker tetrad ${\displaystyle \mathbf {f} _{(\eta )}}$ on the same worldline is as follows: First inserting (9b) into (3a) gives the angular velocity, which together with (9a) can now be inserted into (3b, left), and finally inserted into (3b, right) produces the Fermi–Walker tetrad. The proper coordinates or Fermi coordinates follow by inserting ${\displaystyle \mathbf {e} _{(\eta )}}$ or ${\displaystyle \mathbf {f} _{(\eta )}}$ into (4b) (the resulting expressions are not indicated here because of their length).

Herglotz

Herglotz (1909)^{[H 5]} argued that the metric

${\displaystyle ds^{2}=d\sigma ^{2}+{\frac {1}{A_{44}}}(d\nu )^{2}}$