We determine the conjugacy classes of the Poincaré group ISO^{+}(*n*, 1) and apply this to classify the stationary trajectories of Minkowski spacetimes in terms of timelike Killing vectors. Stationary trajectories are the orbits of timelike Killing vectors and, equivalently, the solutions to Frenet–Serret equations with constant curvature coefficients. We extend the 3 + 1 Minkowski spacetime Frenet–Serret equations due to Letaw to Minkowski spacetimes of arbitrary dimension. We present the explicit families of stationary trajectories in 4 + 1 Minkowski spacetime.

## I. INTRODUCTION

Stationary trajectories in Minkowski spacetime can be defined as the timelike solutions to the Frenet-Serret equations, whose curvature invariants are constant in proper time. Letaw^{1} showed in the case of 3 + 1 Minkowski spacetime that these trajectories equivalently correspond to the orbits of timelike Killing vectors.

The study of curves defined in terms of their curvature invariants began with the work of Frenet^{2} and Serret^{3} in three-dimensional, flat, Euclidean-signature space with the standard metric. Frenet and Serret found a coupled set of differential equations for the tangent vector, normal vector, and binormal vector, which together form an orthonormal basis in $R3$. These differential equations are now known as the Frenet–Serret equations and were later generalised by Jordan^{4} to flat, Euclidean spaces of arbitrary dimension.

We are interested in the stationary trajectories of Minkowski spacetimes. One motivation in physics where stationary trajectories are extensively used is in the study of the Unruh effect^{5} and Unruh-like effects.^{6–10} They have the exploitable property that, in quantum field theory, the two-point function with respect to the Minkowski vacuum when pulled back to a stationary worldline is only a function of the total proper time.^{6} As such, the response of a particle detector^{5,11} is time independent^{9} and hence there is no time dependence in the detector’s associated spectrum.^{1}

The purpose of this paper is to classify the stationary trajectories of Minkowski spacetimes of any dimension. Because stationary trajectories can be defined in terms of curves with constant curvature invariants and alternatively in terms of timelike Killing vectors, we develop the classification in both of these formalisms. First, we determine the conjugacy classes of the connected component of the Poincaré group, referred to as the restricted Poincaré group, and second, we extend the Frenet–Serret equations in 3 + 1 Minkowski spacetime to *n* + 1 Minkowki spacetime. Related previous work includes the classification of the adjoint and co-adjoint orbits of the Poincaré group.^{12,13} A classification of the stationary trajectories in terms of the eigenvalues of a matrix constructed from the Killing vector was given in Ref. 14, with explicit results up to six spacetime dimensions.

We begin in Sec. II with the first method by reviewing the isometries of Minkowski spacetime in terms of the restricted Poincaré group ISO^{+}(*n*, 1) where *n* + 1 is the dimension of Minkowki spacetime. We then classify the conjugacy classes of the restricted Lorentz group SO^{+}(*n*, 1) in Sec. II B by generalising the known results for SO^{+}(3, 1). In Sec. II C, we extend the classification to fully determine the conjugacy classes of the Poincaré group.

Specialising to the conjugacy classes whose associated Killing vector is timelike somewhere, we classify the stationary trajectories in *n* + 1 Minkowski spacetime in Sec. II D, presenting the set of all timelike Killing vectors (18) and giving a formula for the number of classes of timelike trajectories (19).

In Sec. III, we consider the second method and extend the Frenet–Serret equations of 3 + 1 Minkowski spacetime to *n* + 1 Minkowski spacetime. We present the ordinary differential equation satisfied by the four-velocities of the stationary trajectories. Finally, in Sec. III B, we use this formalism to explicitly present the stationary trajectories of 4 + 1 Minkowski spacetime, showing that these trajectories fall into nine distinct families.

We use units in which the speed of light is set to unity. Sans serif letters $(x)$ denote spacetime points and boldface Italic letters (** x**) denote spatial vectors. We adopt the mostly plus convention for the metric of Minkowski spacetime d

*s*

^{2}= −d

*t*

^{2}+ d

*x*^{2}and use the standard set of Minkowski coordinates (

*t*,

*x*,

*y*,

*z*, …).

## II. KILLING VECTORS IN MINKOWSKI SPACETIMES

Stationary trajectories are the timelike solutions to the Frenet–Serret equations with proper-time-independent curvature invariants. Letaw^{1} demonstrated that the stationary trajectories of 3 + 1 Minkowski spacetime can be alternatively defined as the orbits of timelike Killing vectors. Each solution to the Frenet–Serret equations is determined only up to a Poincaré transformation of the worldline, leading to equivalence classes of trajectories. In terms of Killing vectors, each is determined up to conjugation of the generator associated to the Killing vector. In this Section, we determine the conjugacy classes of the Poincaré group, then restrict to the classes whose associated Killing vector is timelike somewhere, thereby classifying the stationary trajectories of Minkowski spacetime of dimension *n* + 1, where *n* ≥ 1.

### A. Isometries of Minkowski spacetime

The isometry group of Minkowski spacetime $Rn,1$ is the Poincaré group. We consider only the connected component of the Poincaré group, the restricted Poincaré group ISO^{+}(*n*, 1), consisting of the connected component of the Lorentz group and translations. A general element of ISO^{+}(*n*, 1) is a pair *g* = (Λ, *a*), where Λ is an element of the restricted Lorentz group SO^{+}(*n*, 1) and $a\u2208Rn,1$. The restricted Lorentz group is the subgroup of Lorentz transformations preserving orientation and time orientation. The Poincaré group acts on $Rn,1$ by $gx\mu =\Lambda \mu \nu x\nu +a\nu $. The Poincaré group is equipped with the group multiplication law $g\u0303\u22c5g=(\Lambda \u0303,a\u0303)\u22c5(\Lambda ,a)=(\Lambda \u0303\Lambda ,\Lambda \u0303a+a\u0303)$ and inverse elements are given by *g*^{−1} = (Λ^{−1}, −Λ^{−1}*a*). The restricted Lorentz group is a subgroup of the restricted Poincaré group with elements (Λ, 0). Pure spacetime translations $h=(1,a)$ form a normal subgroup of the Poincaré group, which may be verified by explicitly computing *g* · *h* · *g*^{−1}. As such, this decomposes ISO^{+}(*n*, 1) as a semidirect product, $ISO+(n,1)=Rn,1\u22caSO+(n,1)$. This structure as a semidirect product of Lie groups is inherited at the level of Lie algebras, $iso(n,1)=Rn,1\u22caso(n,1)$.

Given a Lie group *G* and associated Lie algebra $g$, *G* acts naturally on $g$ by conjugation, $G\xd7g\u2192g$, (*g*, *X*) ↦ *gXg*^{−1}. We use a matrix group notation, anticipating its use in Sec. II C. We define a conjugacy class in the Lie algebra in the following sense *Y* ∼ *X* ⇔ *∃g* ∈ *G* such that *Y* = *gXg*^{−1}.

When considering the Poincaré group acting on Minkowski spacetime, the generators of the Lie algebra $iso(n,1)$ are the Killing vector fields. A Killing vector field is the velocity vector field of a one-parameter isometry group at the identity. It is natural then to consider representations of these generators. The infinitesimal Poincaré transformation of a scalar field $\varphi (x)$ leads to $\varphi (x)\u21a6\varphi (x)\u2212(a\mu \u2202\mu +12\omega \mu \nu (x\nu \u2202\mu \u2212x\mu \u2202\nu ))\varphi (x)$, where *ω*^{μν} is antisymmetric. This is the standard vector field representation and may be written as $\varphi (x)\u21a6(1\u2212a\mu P\mu \u221212\omega \mu \nu M\mu \nu )\varphi (x)$, where *P*_{μ} = *∂*_{μ} is the generator of spacetime translations and $M\mu \nu =(x\nu \u2202\mu \u2212x\mu \u2202\nu )$ is the generator of spacetime rotations.

*ξ*=

*ξ*

^{μ}

*∂*

_{μ}of a spacetime $M$ are defined by,

^{15}

*i*<

*j*:

_{ij}is the Killing vector associated with a rotation in the

*x*

^{i}–

*x*

^{j}plane, NR

_{0ij}is the Killing vector associated with a null rotation consisting of a boost along the

*x*

^{i}–axis and a rotation in the

*x*

^{i}–

*x*

^{j}plane, B

_{0i}is the Killing vector associated with a boost along the

*x*

^{i}–axis, T

_{0}is the Killing vector associated with a timelike translation, S

_{i}is the Killing vector associated with a spacelike translation parallel to the

*x*

^{i}-axis, and NT

_{0i}is the Killing vector associated with a null translation with spatial translation parallel to the

*x*

^{i}-axis. We use ⊕ as a shorthand for Killing vectors with scalars suppressed. For example,

*ξ*= T

_{0}⊕ R

_{12}is shorthand for

*ξ*=

*a∂*

_{t}+

*b*(

*x*

_{2}

*∂*

_{1}−

*x*

_{1}

*∂*

_{2}) with

*a*,

*b*both nonzero.

### B. Conjugacy classes of the restricted Lorentz group

ISO^{+}(*n*, 1) is a semidirect product of the restricted Lorentz group and the group of spacetime translations. Owing to this decomposition, one may methodically approach determining the conjugacy classes of the Poincaré group by first beginning with the restricted Lorentz group and then considering the effect of spacetime translations.

#### 1. Conjugacy classes of the Möbius group

To classify the conjugacy classes of SO^{+}(*n*, 1), we first consider SO^{+}(3, 1), which is isomorphic to the Möbius group $PSL(2,C)=SL(2,C)/{1,\u22121}$ where $1$ is the identity matrix. To see this, one notes that there is a homomorphism between $R3,1$ and anti-hermitian matrices by sending $x\mu $ to $i(x01+x\u22c5\sigma )$, where ** σ** are the Pauli matrices. The determinant of the resulting matrix is the Minkowski squared distance from the origin, $x\mu x\mu $. The special linear group $SL(2,C)$ acts naturally on the set of anti-hermitian matrices by conjugation, which preserves the determinant and hence preserves the Minkowski squared distance. This implies a (surjective) homomorphism $SL(2,C)\u2192SO+(3,1)$. The kernel of this map is ${1,\u22121}$. Therefore, by the first isomorphism theorem, $PSL(2,C)=SL(2,C)/{1,\u22121}\u2245SO+(3,1)$.

The Möbius group is well studied with well-known conjugacy classes.^{16} There are five conjugacy classes: identity, elliptic, parabolic, hyperbolic and loxodromic. In the context of the restricted Lorentz group, these correspond to the identity, spatial rotations, null rotations, boosts, and boosts combined with rotations. Without loss of generality, the elliptic conjugacy class is generated by R_{12}, the parabolic conjugacy class by NR_{012}, the hyperbolic conjugacy class by B_{01}, and the loxodromic conjugacy class by B_{01} ⊕ R_{23}.

#### 2. Conjugacy classes of the restricted Lorentz group

We are now in a position to classify the conjugacy classes of SO^{+}(*n*, 1). First, we note the conjugacy classes of SO^{+}(*n*, 1) for *n* < 3: SO^{+}(1, 1) contains only the identity and hyperbolic classes, whereas SO^{+}(2, 1) contains the identity, elliptic, parabolic, and hyperbolic conjugacy classes. Both follow from reducing the available dimensions in the SO^{+}(3, 1) case.

For *n* > 3, it has been demonstrated that elements of SO^{+}(*n*, 1) are conjugate to one of three canonical forms depending on the eigenvalues of the element.^{17,18} In particular, any element of SO^{+}(*n*, 1) can be reduced to the form $\xi 0\u2a01\eta \u2a011$, where *ξ*_{0} ∈ SO^{+}(3, 1) or SO^{+}(2, 1), *η* ∈ SO(*m*), $1$ is the (*n* − *m* − 3) or (*n* − *m* − 2)-dimensional identity matrix respectively, and ⊕ is the direct sum of matrices. Furthermore, elements of SO(*m*) can be reduced to a canonical form:^{18} for *m* even, $\eta =\u2a01i=1m/2\eta i$ and for *m* odd, $\eta =\u2a01i=1\u230am/2\u230b\eta i\u2a011$, where *η*_{i} ∈ SO(2).

#### 3. Summary

*n*≥ 3 [cf., classification of the orthochronous components of O(

*n*, 1)

^{19,20}],

*n*+ 1 is the spacetime dimension and ⌊·⌋ and ⌈·⌉ are the floor and ceiling functions respectively. We will refer to $\xi Ek$ as the elliptic conjugacy class, $\xi Pk$ as the parabolic conjugacy class, and $\xi Ll$ as the loxodromic conjugacy class. It is important to note that the boosts and rotations appearing in (Sec. II B 3) are considered to have non-zero rotation angles, that is to say the scalar coefficients are non-zero. The set of non-identity conjugacy classes of SO

^{+}(

*n*, 1) are then given by ${\xi Ek,\xi Pk,\xi Ll:1\u2264k\u2264\u230an2\u230b,1\u2264l\u2264\u2308n2\u2309}$ for

*n*≥ 2. We remark that, using the terminology of the conjugacy classes of the Möbius group, $\xi L1$ is the hyperbolic conjugacy class. As a consistency check, we note that this set recovers what we reported earlier for

*n*= 3. We denote the identity conjugacy class by $1$.

### C. Conjugacy classes of the restricted Poincaré group

^{+}(

*n*, 1) as

^{+}(

*n*, 1), $1\u2208R$, and $a,0\u2208Rn,1$ and are viewed as column vectors. Then, ISO

^{+}(

*n*, 1) acts on $Rn,1$ by

*A*,

*B*= 0, 1, …,

*n*+ 1. In this notation, the spacetime rotation generators are given in matrix form by R

_{ij}=

*M*

_{ij}, B

_{0i}=

*M*

_{0i}, NR

_{0ij}=

*M*

_{0i}−

*M*

_{ij}.

*N*and a vector (translation) component

*K*. Under conjugation by

*g*= (Λ,

*a*), we have

#### 1. Temporal translation

_{0}=

*P*

_{0}. One may add a timelike translation to the identity conjugacy class, resulting in the class of inertial trajectories T

_{0}. Adding a timelike translation to the loxodromic conjugacy class (6c) results in $T0\u2a01\xi Ll=\alpha \u2202t+(t\u22021+x1\u2202t)+\u2211i=2lbi(x2i\u22121\u22022i\u22122\u2212x2i\u22122\u22022i\u22121)$. This linear combination as a matrix results in a matrix part $N\mu \nu =(\delta 0\mu \eta 1\nu \u2212\delta 1\mu \eta 0\nu )+\u2211i=2lbi(\delta 2i\u22122\mu \eta 2i\u22121\nu \u2212\delta 2i\u22121\mu \eta 2i\u22122\nu )$ and a vector part $K\mu =\alpha \delta 0\mu $. The loxodromic conjugacy class $\xi Ll$ contains the same matrix contribution and no vector contribution. We can force the vector part of $T0\u2a01\xi Ll$, (−Λ

*N*Λ

^{−1}

*a*+ Λ

*K*), in the conjugation (10) to vanish by choosing $a\mu =\alpha \eta \mu \beta (\Lambda \u22121)1\beta $. Therefore, with this choice of

*a*,

We consider now the elliptic $\xi Ek$ (6a) and parabolic $\xi Pk$ (6b) Killing generators. Neither of these can be timelike anywhere. However, both $T0\u2a01\xi Ek$ and $T0\u2a01\xi Pk$ can be timelike somewhere. Since conjugation does not change the timelike/null/spacelike nature of a Killing vector, we conclude that $T0\u2a01\xi E(P)k\u2241\xi E(P)k$.

#### 2. Spatial translation

_{m}=

*P*

_{m}with 1 ≤

*m*≤

*n*fixed. A spacelike translation added to the identity conjugacy class results in a spatial curve. Consider now the loxodromic conjugacy class, $Sm\u2a01\xi Ll$. This can be written in terms of Killing vectors as $Sm\u2a01\xi Ll=\alpha \u2202m+(x1\u2202t+t\u2202t)+\u2211i=2lbi(x2i\u22121\u22022i\u22122\u2212x2i\u22122\u22022i\u22121)$. The vector part of the conjugation $(\Lambda ,a)\u22c5(Sm\u2a01\xi Ll)\u22c5(\Lambda ,a)\u22121$ reads

In the case *m* = 1, this translation is parallel to the boost. By choosing $a\beta =\alpha \eta \beta \rho (\Lambda \u22121)0\rho $, one can make the vector contribution (12) vanish. In the case 1 < *m* ≤ 2*l* − 1, this translation is parallel to an axis of rotation and can be conjugated away. For *m* even, $a\beta =\u2212\alpha /(b(m+2)/2)\eta \beta \rho (\Lambda \u22121)m+1\rho $ and for *m* odd, $a\beta =\alpha /(b(m+1)/2)\eta \beta \rho (\Lambda \u22121)m\u22121\rho $ will make (12) vanish. However, for 2*l* − 1 < *m* ≤ *n* with $l<\u2308n2\u2309$, one is unable to conjugate away S_{m}. If $l=\u2308n2\u2309$, then the result depends on the parity of *n*. If *n* is odd, then all available spatial dimensions are filled by the boost along *x*^{1} and rotations in the remaining (*n* − 1)/2 independent planes. By contrast, if *n* is even, there is then one free axis, parallel to which one may perform a spatial translation.

To summarise, $Sm\u2a01\xi Ll\u223c\xi Ll$ for 1 ≤ *m* ≤ 2*l* − 1 and $1\u2264l<\u2308n2\u2309$. Whereas, for 2*l* − 1 < *m* ≤ *n* and $1\u2264l\u2264\u2308n2\u2309$, $Sm\u2a01\xi Ll$ forms a new conjugacy class.

The analyses for $Sm\u2a01\xi Ek$ and $Sm\u2a01\xi Pk$ are characteristically and computationally similar to the loxodromic case. We summarise the results now. For 1 ≤ *m* ≤ 2*k* and $1\u2264k\u2264\u230an2\u230b$, one may choose a translation *a* suitably to conjugate away the translation S_{m}. However, for 2*k* < *m* ≤ *n* with $1\u2264k\u2264\u230an2\u230b$, there is a free axis, parallel to which one may perform a spatial translation, producing two more sets of conjugacy classes, $Sm\u2a01\xi E(P)k$ for 2*k* < *m* ≤ *n*.

We remark that in all cases where one may add a spatial translation, it is parallel to an axis, along which and parallel to which there are no other motions. As such, if one were to add multiple spatial translations, each along axes independent of the other motions, one could choose a Λ to align all translations along one axis. This is possible since in each conjugacy class, Λ was hitherto arbitrary. Hence, we need only consider one spatial translation.

#### 3. Temporal and spatial translation

We consider now the case of spatial and temporal translations, T_{0} ⊕ *ξ* ⊕_{i} S_{i}. If the translations T_{0} or S_{i} are parallel to a plane of boost or plane of rotation, they can be conjugated away as seen in Secs. II C 1 and II C 2. We first consider the loxodromic conjugacy class $\xi =\xi Ll$ (6c). The timelike translation T_{0} can be conjugated away so we have $\xi Ll\u2295iSi$. Any S_{i} parallel to the rotations or the boost can be conjugated away. This will either conjugate away all S_{i} or we are left with $\xi Ll\u2295i\u22652lSi$. Finally, one can perform rotations in the hyperplanes containing the S_{i} to align them along one axis, leaving $\xi Ll\u2a01S2l$.

We consider now the parabolic and elliptic conjugacy classes (6a) and (6b). Once again, we can conjugate away any translations parallel to a plane of rotation and then rotate in the planes containing the remaining spatial translations, leaving $T0\u2a01\xi E(P)k\u2a01S2k+1$. In this case, we consider the relative magnitudes of the translations T_{0} and S_{2k+1}. Let T_{0} ⊕ S_{2k+1} = *α∂*_{t} − *β∂*_{2k+1}, then: if |*α*| > |*β*|, this is conjugate to T_{0}; if |*α*| < |*β*|, this is conjugate to S_{2k+1}; and if |*α*| = |*β*|, this is a null translation NT_{0,2k+1}.

Finally, we consider the identity conjugacy class $1$. We can perform temporal and spatial translations $T0\u2a011\u2295iSi$. Depending on whether T_{0} ⊕ _{i}S_{i} is timelike, spacelike, or null, this is conjugate to T_{0}, S_{1}, or NT_{01}.

#### 4. Summary of conjugacy classes

^{+}(

*n*, 1) for small

*n*, we note that ISO

^{+}(0, 1) contains only $1$ and T

_{0}, and ISO

^{+}(1, 1) contains $1$, T

_{0}, B

_{01}, S

_{1}and NT

_{01}. We can now list the conjugacy classes of the Poincaré group, ISO

^{+}(

*n*, 1). We first have the conjugacy classes of SO

^{+}(

*n*, 1),

*x*

^{n}-axis without loss of generality,

*n*is even and $k=\u230an2\u230b$; and $\xi Lk\u2a01Sn\u223c\xi Lk$ if

*n*is odd and $l=\u2308n2\u2309$. Finally, we have the conjugacy classes with a null translation, which again may be confined to the $x0$–$xn$ plane without loss of generality,

*n*is even and $k=\u230an2\u230b$.

### D. Stationary trajectories in Minkowski spacetimes

^{+}(

*n*, 1) which correspond to a stationary trajectory in $Rn,1$ are those whose associated Killing vector is timelike somewhere. For

*n*≥ 3, we list them:

^{1,21}We remark two special cases: $\xi LM1$ is accelerated (Rindler) motion parallel to the

*x*

^{1}-axis and $\xi dL1$ is drifted Rindler motion.

^{8}The semicubical parabolic motions have the spatial projection of a semicubical parabola in the

*x*

^{1}–

*x*

^{2}plane with circular motions in the remaining independent planes. The circular motions exhibit circular motion in each independent plane.

*n*) denote the set of conjugacy classes of timelike Killing vectors of $Rn,1$,

Considering now the case *n* = 4, $TKV(4)={\xi 0,\xi LM1,\xi LM2,\xi dL1,\xi dL2,\xi SP1,\xi SP2,\xi CM1,\xi CM2}$ with #TKV(4) = 9. We will exhibit and classify these trajectories explicitly in the following Section.

## III. VIELBEIN FORMULATION

Stationary trajectories can also be defined as the timelike solutions to the Frenet–Serret equations with proper-time-independent curvature invariants. In this Section, we extend the vierbein formalism of Letaw^{1} to a vielbein formulation, applicable to Minkowski spacetime of dimension *n* + 1 with *n* ≥ 1. We present explicitly the stationary trajectories in $R4,1$.

### A. Frenet–Serret equations in Minkowski spacetimes

*n*+ 1 Minkowski spacetime, where

*τ*is the proper time. These are constructed out of derivatives of the worldline with respect to proper time. We assume that the first

*n*+ 1 derivatives are linearly independent and none of the first

*n*− 1 derivatives are vanishing or null i.e., $x\mu (k)(\tau )x\mu (k)(\tau )\u22600$ for

*k*= 1, …,

*n*− 1. Orthonormality is imposed by the relation

*K*

_{ab}is antisymmetric. Furthermore, since each $Vb\mu $ is constructed out of the first

*b*+ 1 derivatives of the worldline, whereas $V\u0307a\mu $ is constructed out of the first

*a*+ 2 derivatives, we have the $Kab$ vanishes for

*b*>

*a*+ 1. This tells us that the only non-vanishing components are the off-diagonal components and one may write this matrix as

*δ*

_{ab}is the Kronecker delta.

We will consider only the case where *χ*_{a} are constant in *τ* and when $V0\mu $ is future-directed. These *χ*_{a} are then referred to as the curvature invariants. Combining (24) and (25) then yields a set of equations referred to as the Frenet–Serret equations.

*χ*

_{a}= 0 renders the Frenet–Serret equations ill defined. We discuss this further in Appendix A. The Frenet–Serret equations (26) enable one to write each vielbein as an ordinary differential equation for $V0\mu $. One sees that these differential equations can be written down explicitly in the general case,

The dots in (28c) represent successive insertions of terms of the form $\u2211pk=2q\u22122kpk\u22121\u22122\chi pk2$. For example, $b8a=\u2211p1=6a\u22122\chi p12\u2211p2=4p1\u22122\chi p22\u2211p3=2p2\u22122\chi p32\u2211i,j=0p3\u22122\eta ij\chi i\chi j$. One may prove (27) using strong induction and using the relation $b2qm+\chi m\u221212b2(q\u22121)m\u22121=b2qm+1$, which one may derive by expanding (28).

*n*+ 1 Minkowski spacetime are therefore given by

*a*= 3, …,

*n*+ 1. Using terminology from differential equations, the characteristic equation of (29d) is then

*τ*= 0 (31) determines which orthonormal basis one uses. This is the geometric equivalent of the conjugacy class of a Killing vector, as found in Sec. II D.

### B. Example: 4 + 1 Minkowski spacetime

In this Section, we use the formalism of Sec. III A to present the stationary trajectories in 4 + 1 Minkowski spacetime. We demonstrate the equivalence between solutions to the Frenet–Serret equations with constant curvature invariants and the integral curves of timelike Killing vectors in 4 + 1 dimensions. We classify the resulting trajectories into nine equivalence classes.

*m*= 0 and $m2=a\xb1a2+b$, which we write as $m2=a2+b+a$ or $m2=\u2212(a2+b\u2212a)$. Hence

*m*= 0, ±

*R*

_{+}, ±i

*R*

_{−}, where $R\xb12=a2+b\xb1a$. The general solution for $V0\mu $ is then

We explicitly calculate the stationary trajectories in 4 + 1 Minkowski spacetime in Appendix B. We report the results case-by-case. For *m* ≤ *n*, the stationary trajectories of $Rm,1$ are embedded in $Rn,1$. Hence to find all stationary trajectories in $Rn,1$, one solves the Frenet–Serret equations (29) for each *m* = 0, 1, …, *n*.

The stationary trajectories of 4 + 1 Minkowski spacetime are as follows: Case 0: the class of inertial trajectories (B2). Case I: the class of Rindler trajectories (B3). Case IIa: the class of drifted Rindler motions (B4). Case IIb: the class of motions with semicubical parabolic spatial projection (B5). Case IIc: the class of circular motions (B6). Case III: the class of loxodromic motions (B9). Case IVa: the class of drifted loxodromic motions in the *x*^{1}–*x*^{2} plane with circular motion in the *x*^{3}–*x*^{4} plane (B14). Case IVb: the class of motions with semicubical parabolic spatial projection in the *x*^{1}–*x*^{2} plane with circular motion in the *x*^{3}–*x*^{4} plane (B16). Case IVc: the class of circular motions in the *x*^{1}–*x*^{2} plane with circular motion in the *x*^{3}–*x*^{4} plane (B18).

As expected from the previous calculation of #TKV(4) in Sec. II D, there are nine classes of stationary trajectory. In agreement with Refs. 1 and 21, we recover the six classes of stationary trajectory in 3 + 1 Minkowski spacetime.

## IV. CONCLUSIONS

In this paper, we determined the conjugacy classes of the restricted Poincaré group ISO^{+}(*n*, 1) and used this to classify the stationary trajectories in Minkowski spacetimes. We found there were five classes of trajectories, which we name: the inertial motions, the loxodromic motions, the drifted loxodromic motions, the semicubical parabolic motions, and the circular motions. Each type of trajectory has conjugacy classes with qualitatively similar motion. The Rindler and drifted Rindler motions are special cases of the loxodromic and drifted loxodromic motions. We then generalised the work of Frenet,^{2} Serret,^{3} Jordan,^{4} and Letaw^{1} to provide a framework for the computation of stationary trajectories in terms of their curvature invariants in Minkowski spacetime. In doing so, we have provided the ordinary differential equation satisfied by the four-velocity of the stationary worldline. We finally utilised this framework to present explicitly the stationary trajectories in 4 + 1 Minkowski spacetime.

Minkowski spacetime is a space of zero curvature. A natural extension of this work would be to the spacetimes of constant positive or negative curvature, de Sitter and anti-de Sitter spacetimes respectively.^{23} Previous work in this direction includes the study of the Frenet–Serret equations in general curved spacetimes.^{24} The isometry group of *n* + 1 de Sitter spacetime is the de Sitter group, whose connected component is isomorphic to SO^{+}(*n* + 1, 1). The classification of the stationary vacuum states in de Sitter,^{25} as well as the classification of the conjugacy classes of the restricted Lorentz group in Sec. II B would therefore be relevant and adaptable to this classification. However, the connected component of the isometry group of *n* + 1 anti-de Sitter spacetime, the anti-de Sitter group, is isomorphic to SO^{+}(*n*, 2). This change in signature in comparison to the Minkowski or de Sitter isometry groups means that new techniques will be required in classifying the conjugacy classes of anti de Sitter spacetime.

## ACKNOWLEDGMENTS

C.R.D.B. is indebted to Leo Parry and Jorma Louko for invaluable discussions. I thank Carlos Peón-Nieto and Marc Mars for bringing the work in Refs. 12, 13, 19, and 20 to my attention, Prasant Samantray for bringing the work in Refs. 25 and 26 to my attention, and Hari K for bringing the work in Ref. 24 to my attention. I thank the anonymous referees for helpful comments and for bringing the work in Ref. 14 to my attention. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Cameron R D Bunney**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX A: GENERALISED FRENET–SERRET EQUATIONS

**(a)**, we changed summation variable

*q*↦

*q*+ 1 in the second summation. In equality

**(b)**, we isolated the

*q*= 0 term and used that for odd integers $\u230an2\u230b=\u230an+12\u230b$. For odd integers, consider the effect of replacing $\u230an2\u230b$ by $\u230an+12\u230b$. Let

*n*= 2

*m*− 1, an odd integer. Then $\u230an+12\u230b=m$. Hence, the coefficient of this term would be $b2m2m\u22121$. Calculating this using (28c), one finds that the first summation is over $\u22112m\u221222m\u22123$, which identically vanishes. Hence, one may replace $\u230an2\u230b$ by $\u230an+12\u230b$ in the first sum. In equality

**(c)**, we used the relation $b2qm+\chi m\u221212b2(q\u22121)m\u22121=b2qm+1$ and then combined the two terms under one summation and finally multiplied by

*χ*

_{n}/

*χ*

_{n}. In equality

**(d)**, we recognised that the resulting summation was (27) in the case

*a*=

*n*+ 1.

Therefore, one may find the ordinary differential equation for $V0\mu $ (and hence the four-velocity) by forcing $Vn+1\mu $ to vanish. This fully determines the Frenet–Serret equations.

### APPENDIX B: STATIONARY TRAJECTORIES IN 4 + 1 MINKOWSKI SPACETIME

In this Section, we present the stationary trajectories in 4 + 1 Minkowski spacetime. One first solves the ordinary differential equation for $V0\mu $, then calculates the constants of integration using (31) and finally brings the motion into a more familiar form by a suitable Lorentz transformation.

*χ*

_{a}= 0, then the Frenet–Serret equations (29) are no longer well defined. The geometric effect of setting

*χ*

_{a}= 0 is to confine the motion to $Ra+1,1$. Note, however, that the stationary trajectories of $Rm,1$ are also present in $Rn,1$ for

*m*≤

*n*. Therefore, to calculate the stationary trajectories present in $Rn,1$, one solves the Frenet–Serret equations (29) for each

*m*≤

*n*. The trajectories are then written via the inclusion map,

*n*−

*m*)–many zeros (in the case

*m*=

*n*, then there are no zeros present).

We present the stationary trajectory in cases. **Case** ** m** gives the solution(s) to the Frenet–Serret equations (29) in $Rm,1$.

**Case 0**— The class of inertial trajectories,(B2)$V0\mu =x\u0307\mu =(1,0,0,0,0).$**Case I**—*χ*_{0}> 0. Rindler motion,(B3)$V0\mu =(cosh(\chi 0\tau ),sinh(\chi 0\tau ),0,0,0).$**Case II**— The solutions to the Frenet–Serret equations in 2 + 1 have two free parameters, the curvature invariants*χ*_{0}and*χ*_{1}. The classification of the stationary trajectory depends on their relation.**Case IIa**—*χ*_{0}> |*χ*_{1}| > 0. After a suitable Lorentz transformation, this is drifted Rindler motion,^{8}(B4)$V0\mu =1\chi 02\u2212\chi 12\chi 0cosh(\chi 02\u2212\chi 12\tau ),\chi 0sinh(\chi 02\u2212\chi 12\tau ),\chi 1,0,0.$**Case IIb**— |*χ*_{0}| = |*χ*_{1}| ≠ 0,whose spatial profile is that of the semicubical parabola $y2=29\chi 0x3$.(B5)$V0\mu =1+12\chi 02\tau 2,\chi 0\tau ,12\chi 02\tau 2,0,0,$**Case IIc**—*χ*_{1}> |*χ*_{0}| > 0. After a suitable Lorentz transformation, this is circular motion in the*x*^{1}–*x*^{2}plane.In the following cases, we give the Lorentz transformation explicitly owing to their more involved calculations.(B6)$V0\mu =1\chi 12\u2212\chi 02\chi 1,\u2212\chi 0sin(\chi 12\u2212\chi 02\tau ),\chi 0cos(\chi 12\u2212\chi 02\tau ),0,0.$**Case III**— The general solution to (29d) is(B7a)$V0\mu =B\mu cosh(R+\tau )+C\mu sinh(R+\tau )+D\mu cos(R\u2212\tau )+E\mu sin(R\u2212\tau ),$(B7b)$B\mu =1R2R\u22122+\chi 02,0,\chi 0\chi 1,0,0,$(B7c)$C\mu =1R20,\chi 0R+(\chi 02\u2212\chi 12+R\u22122),0,\chi 0\chi 1\chi 2R+,0,$(B7d)$D\mu =1R2R+2\u2212\chi 02,0,\u2212\chi 0\chi 1,0,0,$(B7e)$E\mu =1R20,\u2212\chi 0R\u2212(\chi 02\u2212\chi 12\u2212R+2),0,\u2212\chi 0\chi 1\chi 2R\u2212,0.$This is loxodromic motion, which may more clearly be seen by the following Lorentz transformation,(B8a)$\Lambda \mu \nu =\alpha 0\beta 000\gamma 0\delta 00C0D0A0B0000001,$(B8b)$\alpha =\Delta R,\beta =\Delta (R+2\u2212\chi 02)\chi 0\chi 1R,\gamma =\Delta R+\chi 0R,\delta =\u2212\Delta R+(\chi 02\u2212\chi 12\u2212R+2)\chi 0\chi 1\chi 2R,A=\chi 0\chi 1\Delta R,B=\u2212\Delta R,C=\u2212\chi 1R\u2212\Delta R,D=R\u2212(\chi 02\u2212\chi 12+R\u22122)\chi 2\Delta R,\Delta 2=R\u22122+\chi 02,$(B9)$\Lambda \mu \nu V0\nu =1R\Delta \u2061cosh(R+\tau ),\Delta \u2061sinh(R+\tau ),\u2212\chi 0\chi 1\Delta sin(R\u2212\tau ),\chi 0\chi 1\Delta cos(R\u2212\tau ),0.$**Case IV**— The classification of the solutions to the Frenet–Serret equations in 4 + 1 Minkowski spacetime depends on the relationship between the four curvature invariants. In particular, on the sign of*b*(32).**Case IVa**— $b=\chi 22\chi 02+\chi 32(\chi 02\u2212\chi 12)>0$. The four-velocity is given bywhere(B10)$V0\mu =A\mu +B\mu cosh(R+\tau )+C\mu sinh(R+\tau )+D\mu cos(R\u2212\tau )+E\mu sin(R\u2212\tau ),$(B11a)$A\mu =1\u2212\chi 02b(\chi 23+\chi 32),0,\u2212\chi 0\chi 1\chi 32b,0,\u2212\chi 0\chi 1\chi 2\chi 3b,$(B11b)$B\mu =1R2\chi 02R+2(\chi 02\u2212\chi 12+R\u22122),0,\chi 0\chi 1R+2(\chi 02\u2212\chi 12\u2212\chi 22+R\u22122),0,\chi 0\chi 1\chi 2\chi 3R+2,$(B11c)$C\mu =1R20,\chi 0R+(\chi 02\u2212\chi 12+R\u22122),0,\chi 0\chi 1\chi 2R+,0,$(B11d)$D\mu =1R2\chi 02R\u22122(\chi 02\u2212\chi 12\u2212R+2),0,\chi 0\chi 1R\u22122(\chi 02\u2212\chi 12\u2212\chi 22\u2212R+2),0,\chi 0\chi 1\chi 2\chi 3R\u22122,$(B11e)$E\mu =1R20,\u2212\chi 0R\u2212(\chi 02\u2212\chi 12\u2212R+2),0,\u2212\chi 0\chi 1\chi 2R\u2212,0.$Given the classifications in Sec. II D, one may hope to identify this motion as a boost, combined with a drift and circular motion. We make an ansatz of the desired form of the four-velocity and find the appropriate Lorentz transformation,By imposing that Λ is a Lorentz transformation, one may identify the coefficients(B12)$V0\mu =\Lambda \mu \nu V\u03030\nu =B0\alpha 0A0\beta 0D0\gamma 0C1\alpha 0\u2212E1\gamma 0B2\alpha 0A2\beta 0D2\gamma 0C3\alpha 0\u2212E3\gamma 0B4\alpha 0A4\beta 0D4\gamma \alpha cosh(R+\tau )\alpha sinh(R+\tau )\beta \u2212\gamma sin(R\u2212\tau )\gamma cos(R\u2212\tau ).$*α*,*β*, and*γ*as(B13a)$\alpha =\u2212B\mu B\mu =C\mu C\mu ,$(B13b)$\beta =A\mu A\mu ,$(B13c)$\gamma =D\mu D\mu =E\mu E\mu .$Intermediate steps include the verification that*A*^{μ}*D*_{μ}=*C*^{μ}*E*_{μ}=*A*^{μ}*B*_{μ}=*B*^{μ}*D*_{μ}= 0. This brings the four-velocity into the more recognisable formcorresponding to a boost along the(B14)$V0\mu =\u2212B\mu B\mu cosh(R+\tau ),\u2212B\mu B\mu sinh(R+\tau ),A\mu A\mu ,\u2212D\mu D\mu sin(R\u2212\tau ),D\mu D\mu cos(R\u2212\tau ),$*x*^{1}-axis, a drift in the*x*^{2}-axis and circular motion in the*x*^{3}–*x*^{4}plane.**Case IVb**— $b=0\u27fa\chi 02=\chi 1\chi 3\chi 22+\chi 32$. In this case, one finds $2a=\chi 02\u2212\chi 12\u2212\chi 22\u2212\chi 32=\u2212(\chi 22+\chi 32)2+\chi 12\chi 22\chi 22+\chi 32<0$, leading to $R+2=0$ and $R\u22122=\u22122a>0$. The four-velocity reads(B15a)$V0\mu =A\u0303\mu +12B\u0303\mu \tau 2+C\u0303\mu \tau +D\u0303\mu cos(R\u2212\tau )+E\u0303\mu sin(R\u2212\tau ),$(B15b)$A\u0303\mu =1\u2212\chi 02R\u22124(\chi 02\u2212\chi 12),0,\u2212\chi 0\chi 1R\u22124(\chi 02\u2212\chi 12\u2212\chi 22),0,\u2212\chi 0\chi 1\chi 2\chi 3R\u22124,$(B15c)$B\u0303\mu =\chi 02R\u22122(\chi 02\u2212\chi 12+R\u22122),0,\chi 0\chi 1R\u22122(\chi 02\u2212\chi 12\u2212\chi 22+R\u22122),0,\chi 0\chi 1\chi 2\chi 3R\u22122,$(B15d)$C\u0303\mu =0,\chi 0R\u22122(\chi 02\u2212\chi 12+R\u22122),0,\chi 0\chi 1\chi 2R\u22122,0,$(B15e)$D\u0303\mu =\chi 02R\u22124(\chi 02\u2212\chi 12),0,\chi 0\chi 1R\u22124(\chi 02\u2212\chi 12\u2212\chi 22),0,\chi 0\chi 1\chi 2\chi 3R\u22124,$(B15f)$E\u0303\mu =0,\u2212\chi 0R\u22123(\chi 02\u2212\chi 12),0,\u2212\chi 0\chi 1\chi 2R\u22123,0.$We proceed as in (B12) to findwhose spatial profile is the semicubical parabola $y2=29(A\u0303\mu B\u0303\mu )2(\u2212A\u0303\nu A\u0303\nu )(C\u0303\rho C\u0303\rho )3x3$ in the(B16)$V0\mu =\u2212A\u0303\mu A\u0303\mu \u221212A\u0303\mu B\u0303\mu \u2212A\u0303\mu A\u0303\mu \tau 2,C\u0303\mu C\u0303\mu \tau ,\u221212A\u0303\mu B\u0303\mu \u2212A\u0303\mu A\u0303\mu \tau 2,\u2212D\u0303\mu D\u0303\mu sin(R\u2212\tau ),D\u0303\mu D\u0303\mu cos(R\u2212\tau ),$*x*–*y*plane, combined with circular motion in the*x*^{3}–*x*^{4}plane. We use (*x*,*y*) in place of (*x*^{1},*x*^{2}) for clarity.**Case IVc**— $b=\chi 22\chi 02+\chi 32(\chi 02\u2212\chi 12)<0$. In this case, we have both*a*< 0 and*b*< 0. Hence, $R+2=a2+b+a<0$, yet $R\u22122=a2+b\u2212a>0$. We then write*a*= −*α*,*b*= −*β*such that $R+2=\u2212(\alpha \u2212\alpha 2\u2212\beta )=\u2212\rho \u22122$. The four-velocity reads(B17a)$V0\mu =A\mu +B\mu cos(\rho \u2212\tau )+C\mu sin(\rho \u2212\tau )+D\mu cos(R\u2212\tau )+E\mu sin(R\u2212\tau ),$(B17b)(B17c)$B\mu =1R2\u2212\chi 02\rho \u22122(\chi 02\u2212\chi 12+R\u22122),0,\u2212\chi 0\chi 1\rho \u22122(\chi 02\u2212\chi 12\u2212\chi 22+R\u22122),0,\u2212\chi 0\chi 1\chi 2\chi 3\rho \u22122,$(B17d)$C\mu =1R20,\chi 0\rho \u2212(\chi 02\u2212\chi 12+R\u22122),0,\chi 0\chi 1\chi 2\rho \u2212,0,$(B17e)$D\mu =1R2\chi 02R\u22122(\chi 02\u2212\chi 12+\rho \u22122),0,\chi 0\chi 1R\u22122(\chi 02\u2212\chi 12\u2212\chi 22+\rho \u22122),0,\chi 0\chi 1\chi 2\chi 3R\u22122,$where $R2=R\u22122\u2212\rho \u22122$. Proceeding once more as in (B12), one may rewrite this four-velocity as(B17f)$E\mu =1R20,\u2212\chi 0R\u2212(\chi 02\u2212\chi 12+\rho \u22122),0,\u2212\chi 0\chi 1\chi 2R\u2212,0,$identifying the trajectory as independent circular motions in the(B18)$V0\mu =\u2212A\mu A\mu ,\u2212B\mu B\mu sin(\rho \u2212\tau ),B\mu B\mu cos(\rho \u2212\tau ),\u2212D\mu D\mu sin(R\u2212\tau ),D\mu D\mu cos(R\u2212\tau ),$*x*^{1}–*x*^{2}and*x*^{3}–*x*^{4}planes.

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*General Relativity: An Einstein Centenary Survey*

*Spacetime and Geometry: An Introduction to General Relativity*

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