Stationary trajectories in Minkowski spacetimes Open Access

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. Letaw¹ 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 Frenet² and Serret³ 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 Jordan⁴ 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 effect⁵ 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 independent⁹ 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 ds² = −dt² + dx² and use the standard set of Minkowski coordinates (t, x, y, z, …).

Stationary trajectories are the timelike solutions to the Frenet–Serret equations with proper-time-independent curvature invariants. Letaw¹ 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.

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 $a∈Rn,1$⁠. The restricted Lorentz group is the subgroup of Lorentz transformations preserving orientation and time orientation. The Poincaré group acts on $Rn,1$ by $gxμ=Λμνxν+aν$⁠. The Poincaré group is equipped with the group multiplication law $g̃⋅g=(Λ̃,ã)⋅(Λ,a)=(Λ̃Λ,Λ̃a+ã)$ and inverse elements are given by g⁻¹ = (Λ⁻¹, −Λ⁻¹a). 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, $ISO+(n,1)=Rn,1⋊SO+(n,1)$⁠. This structure as a semidirect product of Lie groups is inherited at the level of Lie algebras, $iso(n,1)=Rn,1⋊so(n,1)$⁠.

Given a Lie group G and associated Lie algebra $g$⁠, G acts naturally on $g$ by conjugation, $G×g→g$⁠, (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 $ϕ(x)$ leads to $ϕ(x)↦ϕ(x)−(aμ∂μ+12ωμν(xν∂μ−xμ∂ν))ϕ(x)$⁠, where ω^μν is antisymmetric. This is the standard vector field representation and may be written as $ϕ(x)↦(1−aμPμ−12ωμνMμν)ϕ(x)$⁠, where P_μ = ∂_μ is the generator of spacetime translations and $Mμν=(xν∂μ−xμ∂ν)$ is the generator of spacetime rotations.

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.

To classify the conjugacy classes of SO⁺(n, 1), we first consider SO⁺(3, 1), which is isomorphic to the Möbius group $PSL(2,C)=SL(2,C)/{1,−1}$ 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 $xμ$ to $i(x01+x⋅σ)$⁠, where σ are the Pauli matrices. The determinant of the resulting matrix is the Minkowski squared distance from the origin, $xμxμ$⁠. 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 $SL(2,C)→SO+(3,1)$⁠. The kernel of this map is ${1,−1}$⁠. Therefore, by the first isomorphism theorem, $PSL(2,C)=SL(2,C)/{1,−1}≅SO+(3,1)$⁠.

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 R₁₂, the parabolic conjugacy class by NR₀₁₂, the hyperbolic conjugacy class by B₀₁, and the loxodromic conjugacy class by B₀₁ ⊕ R₂₃.

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 $ξ0⨁η⨁1$⁠, where ξ₀ ∈ 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, $η=⨁i=1m/2ηi$ and for m odd, $η=⨁i=1⌊m/2⌋ηi⨁1$⁠, where η_i ∈ SO(2).

We consider now the elliptic $ξEk$ (6a) and parabolic $ξPk$ (6b) Killing generators. Neither of these can be timelike anywhere. However, both $T0⨁ξEk$ and $T0⨁ξPk$ can be timelike somewhere. Since conjugation does not change the timelike/null/spacelike nature of a Killing vector, we conclude that $T0⨁ξE(P)k≁ξE(P)k$⁠.

In the case m = 1, this translation is parallel to the boost. By choosing $aβ=αηβρ(Λ−1)0ρ$⁠, one can make the vector contribution (12) vanish. In the case 1 < m ≤ 2l − 1, this translation is parallel to an axis of rotation and can be conjugated away. For m even, $aβ=−α/(b(m+2)/2)ηβρ(Λ−1)m+1ρ$ and for m odd, $aβ=α/(b(m+1)/2)ηβρ(Λ−1)m−1ρ$ will make (12) vanish. However, for 2l − 1 < m ≤ n with $l<⌈n2⌉$⁠, one is unable to conjugate away S_m. If $l=⌈n2⌉$⁠, then the result depends on the parity of n. If n is odd, then all available spatial dimensions are filled by the boost along x¹ 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, $Sm⨁ξLl∼ξLl$ for 1 ≤ m ≤ 2l − 1 and $1≤l<⌈n2⌉$⁠. Whereas, for 2l − 1 < m ≤ n and $1≤l≤⌈n2⌉$⁠, $Sm⨁ξLl$ forms a new conjugacy class.

The analyses for $Sm⨁ξEk$ and $Sm⨁ξPk$ are characteristically and computationally similar to the loxodromic case. We summarise the results now. For 1 ≤ m ≤ 2k and $1≤k≤⌊n2⌋$⁠, one may choose a translation a suitably to conjugate away the translation S_m. However, for 2k < m ≤ n with $1≤k≤⌊n2⌋$⁠, there is a free axis, parallel to which one may perform a spatial translation, producing two more sets of conjugacy classes, $Sm⨁ξE(P)k$ for 2k < 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.

We consider now the case of spatial and temporal translations, T₀ ⊕ ξ ⊕_i S_i. If the translations T₀ 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 $ξ=ξLl$ (6c). The timelike translation T₀ can be conjugated away so we have $ξLl⊕iSi$⁠. 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 $ξLl⊕i≥2lSi$⁠. Finally, one can perform rotations in the hyperplanes containing the S_i to align them along one axis, leaving $ξLl⨁S2l$⁠.

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 $T0⨁ξE(P)k⨁S2k+1$⁠. In this case, we consider the relative magnitudes of the translations T₀ and S_2k+1. Let T₀ ⊕ S_2k+1 = α∂_t − β∂_2k+1, then: if |α| > |β|, this is conjugate to T₀; 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 $T0⨁1⊕iSi$⁠. Depending on whether T₀ ⊕ _iS_i is timelike, spacelike, or null, this is conjugate to T₀, S₁, or NT₀₁.

Considering now the case n = 4, $TKV(4)={ξ0,ξLM1,ξLM2,ξdL1,ξdL2,ξSP1,ξSP2,ξCM1,ξCM2}$ with #TKV(4) = 9. We will exhibit and classify these trajectories explicitly in the following Section.

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 Letaw¹ to a vielbein formulation, applicable to Minkowski spacetime of dimension n + 1 with n ≥ 1. We present explicitly the stationary trajectories in $R4,1$⁠.

We will consider only the case where χ_a are constant in τ and when $V0μ$ 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.

The dots in (28c) represent successive insertions of terms of the form $∑pk=2q−2kpk−1−2χpk2$⁠. For example, $b8a=∑p1=6a−2χp12∑p2=4p1−2χp22∑p3=2p2−2χp32∑i,j=0p3−2ηijχiχj$⁠. One may prove (27) using strong induction and using the relation $b2qm+χm−12b2(q−1)m−1=b2qm+1$⁠, which one may derive by expanding (28).

A priori, one is able to express solutions to (30) in terms of radicals only for n ≤ 8.²² We demonstrate this extended formalism in the following Section to explicitly calculate and classify the stationary trajectories in 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.

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 x¹–x² plane with circular motion in the x³–x⁴ plane (B14). Case IVb: the class of motions with semicubical parabolic spatial projection in the x¹–x² plane with circular motion in the x³–x⁴ plane (B16). Case IVc: the class of circular motions in the x¹–x² plane with circular motion in the x³–x⁴ 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.

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 Letaw¹ 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.

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.

The author has no conflicts to disclose.

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

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