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

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

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

(2)

where

\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:

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

with $\mathbf {U}$ as four-velocity and $\mathbf {A}$ as four-acceleration, and " $\cdot$ " indicates the dot product and " $\wedge$ " the wedge product. The first part $(\mathbf {U} \wedge \mathbf {A} )\mathbf {e} _{(\eta )}=\mathbf {A} \left(\mathbf {U} \cdot \mathbf {e} _{(\eta )}\right)-\mathbf {U} \left(\mathbf {A} \cdot \mathbf {e} _{(\eta )}\right)$ represents Fermi–Walker transport,^[13] which is physically realized when the three spacelike tetrad fields do not change their orientation with respect to the motion of a system of three gyroscopes. Thus Fermi–Walker transport can be seen as a standard of non-rotation. The second part $\mathbf {R}$ consists of an antisymmetric second rank tensor with $\omega$ as the angular velocity four-vector and $\epsilon$ 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 $\mathbf {e} _{(i)}$ of a rotating tetrad (such as a Frenet–Serret tetrad) with respect to the non-rotating spacelike fields $\mathbf {f} _{(i)}$ of a Fermi–Walker tetrad along the same world line.

Deriving Fermi–Walker tetrads from Frenet–Serret tetrads

Since $\mathbf {f} _{(i)}$ and $\mathbf {e} _{(i)}$ 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 $\mathbf {f} _{(i)}$ and $\mathbf {e} _{(i)}$ can be given in terms of torsions $\kappa _{2}$ and $\kappa _{3}$ :^[12]^[13]^[27]^[28]

{\boldsymbol {\omega }}=\kappa _{3}\mathbf {e} _{(1)}+\kappa _{2}\mathbf {e} _{(3)}

and

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

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

{\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)