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Geodesics on H-type quaternion groups with sub-Lorentzian metric and their physical interpretation

We study the existence and cardinality of normal geodesics of different causal types on H(eisenberg)-type quaternion group equipped with the sub-Lorentzian metric. We present explicit formulas for geodesics and describe reachable sets by geodesics of different causal character. We compare results with the sub-Riemannian quaternion group and with the sub-Lorentzian Heisenberg group, showing that there are similarities and distinctions. We show that the geodesics on H-type quaternion groups with the sub-Lorentzian metric satisfy the equations describing the motion of a relativistic particle in a constant homogeneous electromagnetic field.


Introduction
The term sub-Riemannian manifold means the triple (M, H, d), where M is an n-dimensional manifold, H is a smoothly varying k-dimensional distribution inside the tangent bundle T (M ) of the manifold M with k < n, and d is a Riemannian metric defined on H, i. e., a positively definite quadratic form. Recently the study of geometric structures, where the Riemannian metric d on H is substituted by a semi-Riemannian metric g, that is a non-degenerate indefinite metric, started e. g., in [2,3,4,5,6,8,11,12]. There is no special attribution so far for such kind of manifolds (M, H, g), thus we propose to call them sub-semi-Riemannian manifolds or shortly ssr-manifolds. In the particular case, when the metric g has index 1, an ssr-manifold receives the name sub-Lorentzian manifold by the analogy to Lorentzian manifold.
In the present article we study an example of H-type group furnished with the sub-Lorentzian metric. This is an interesting example not only as an almost unique known example of sub-Lorentzian manifold but also because it has a precise physical meaning. In the article we reveal the connection between sub-Lorentzian geometry and physics of relativistic electrodynamics basing on the example of H-type quaternion group equipped with the Lorentzian metric. We also compare characterising features of sub-Riemannian and sub-Lorentzian geometries. The notion of H-type groups was introduced in [9]. It is known that Riemannian manifolds have applications in classical mechanics. Sub-Riemannian manifolds of step 2 (Heisenberg manifolds) play important role in quantum mechanics. Sub-Riemannian geodesics even localy behave very differently from the ones in Riemannian geometry, where the energy minimising motion is described by a unique geodesic. A sub-atomic particle behaves in a way similar to an electron which moves only along a given set of directions. There can be infinitely many geodesics with different length joining two points. In its turn, the sub-Lorentzian structure underlise the motion in an electro-magnetic field. Just like space and time emerge in special relativity, the electric and magnetic fields can not be considered separately. The sub-Lorentzian structure absorbes both phenomena, the presence of the electro-magnetic field and the the space-time geometry. That is why the H-type quaternion group with a sub-Lorentzian metric is an interesting example to work with.
In the introduction in order to explain the main idea we would like to mention the Heisenberg group as the simplest noncommutative example of H-type (Heisenberg type) groups and its numerous connections with physics. The Heisenberg group is the manifold R 3 with the noncommutative group law (x, y, z) • (x ′ , y ′ , z ′ ) = x + x ′ , y + y ′ , z + z ′ + 1 2 (xy ′ − yx ′ ) .
Left-invariant vector fields X = ∂ x − 1 2 y∂ z , Y = ∂ y + 1 2 x∂ z are obtained from the group law and span a 2-dimensional distribution H which is called horizontal. The horizontal distribution can be also defined as the kernel of the contact one-form ω = dz − 1 2 (xdy − ydx) in R 3 . The differential of ω is the 2-form Ω = dω = −dx ∧ dy that satisfies the Maxwell's equation dΩ = 0 for the magnetic field Ω in R 3 . Let us define a Riemannian metric ds 2 R = dx 2 + dy 2 on H. Then the sub-Riemannian manifold H 1 R = (R 3 , H, ds 2 R ) is also called the Heisenberg group. It turned out that the geodesic equation for geodesics γ(t) satisfying the non-holonomic constraintsγ(t) ∈ H(γ(t)) coincides up to a constant with the Lorentz equation of motion of the charged particle in the magnetic field Ω. If we change Riemannian metric on H on the Lorentzian one ds 2 L = −dx 2 + dy 2 we come to the notion of sub-Lorentzian Heisenberg group H 1 L = (R 3 , H, ds 2 L ). In this case the geodesic equation for non-holonomic geodesics coincides with analogue of the Lorentz equation for the motion of a charged particle in the electromagnetic field defined by Ω and the Lorentzian metric tensor. The geodesics, metrics properties, and other related questions on H 1 L were studied in [2,3,4,11]. The lack of the dimension of the horizontal distribution on the Heisenberg group does not allow to reveal the peculiarity of the applications in the case of the magnetic and electromagnetic fields. Therefore, we chose the analogue of the Heisenberg group admitting a 4-dimensional distribution, that we called quaternionic H-type group. This example allows also to show similarities and differences between the sub-Riemannian and sub-Lorentzian geometries.
The article is organized in the following way. In Section 2 we introduce the H-type quaternion groups endowed with different metrics: Riemannian and Lorentzian. We also present the differential equations for the geodesics in both cases. Section 3 is the collection of the definitions related to the motion of charged particles in electro-magnetic fields. We give the explanation of the geometrical results from the physical point of view. Section 4 is devoted to the solution of geodesic equations, where we fined the explicit formulas for the horizontal and vertical parts of geodesics. Section 5 is dedicated to study of reachable sets by geodesics of different causal types and estimation of the cardinality of geodesics, connecting two different points. Section 6 shows a brief overview of reachable sets for H 1 L for the sake of comparison with obtained results for the quaternion H-type group.
2. H-type quaternion groups Q R and Q L We remind that quaternions form a noncommutative division algebra that extends the system of complex numbers. It is convenient to define any quaternion q in the algebraic form The scalar a ∈ R is called the real part Re q = a and a vector (b, c, d) ∈ R 3 recieved the name pure imaginary quaternion and is denoted by Pu q. Thus q = (Re q, Pu q). With this notations it is easy to introduce the noncommutative multiplication * between two quaternions q 1 and q 2 using the usual inner product · and vector product × in R 3 . Namely, Notice that this structure suggests itself the analogy with a Lorentzian geometry where R 4 consists of a time part t ∈ R and space part ( are the basis of quaternion numbers in the representation given by real (4 × 4)-matrices.
We introduce an H-type group whose noncommutative multiplication law makes use of quaternion multiplication rule. Let us take the background manifold M as R 7 and define the noncommutative law by for (x, z) and (x ′ , z ′ ) from R 4 × R 3 . Here Pu(x * x ′ ) is the imaginary part of the productx * x ′ defined in (2.1) of the conjugate quaternionx to x by another quaternion x ′ . The introduced multiplication law (2.3) makes M = (R 7 , •) into a noncommutative Lie group with the unity (0, 0) and the inverse element (−x, −z) to (x, z). The group law defines the left translation . . , ∂ ∂z 3 be a standard basis of the tangent space T p M to M at p ∈ M . The basic left-invariant vector fields can be obtained by the action of the tangent map dL (x,z) of L (x,z) to the standard basis as dL (x,z) . Then the vector fields is a copy of R 4 . The commutation relations are as follows Therefore, {X 0 , . . . , X 3 } and their commutators span the entire tangent space T M . This property of the distribution Q is called bracket-generating of step 2. The Lie algebra with the basis {X 0 , . . . , X 3 , Z 1 , Z 2 , Z 3 } is nilpotent of step 2.
The horizontal distribution Q can be defined by making use of one-forms. Namely, the one-forms annihilate the distribution Q. Here x = (x 0 , . . . , x 3 ), dx = (dx 0 , . . . , dx 3 ), and dx T is the transposed vector to dx. Thus, Q is the common kernel of forms ω k , k = 1, 2, 3. Let us consider the external differential of the linear combination ω = 3 k=1 ϑ k ω k . We get the twoform that is defined in 4-dimensional space Any vector v ∈ Q p , p ∈ M , is called horizontal, a vector field X tangent to Q p at each point p is also called horizontal. An absolutely continuous curve γ : [0, 1] → M that has its velocity vectorγ(t) tangent to Q γ(t) almost everywhere is called horizontal curve.
2.1. Sub-Riemannian manifold Q R . Let us define the Riemannian metric (·, ·) R on the distribution Q in such a way that (X i , X j ) = δ ij , where δ ij is the Kronecker symbol. With this we get the sub-Riemannian manifold Q R = R 7 , Q, (·, ·) R with the sub-Riemannian structure Q, (·, ·) R .
Geodesics (or normal extremals) in the sub-Riemannian geometry are defined as a projection of solutions of the Hamiltonian equations on the underlying manifold. Let us present the corresponding Hamiltonian system. We denote by T p Q R , T * p Q R the tangent and cotangent space at p ∈ Q R respectively and by T Q R , T * Q R the corresponding tangent and cotangent bundle. Thus, if (p, λ) ∈ T * Q R then the restriction of λ on the subspace Q(p) of T p Q R is well defined and making use of the inner product we define a Hamiltonian function on T * p Q R by where by ·, · we denoted the pairing between vector spaces T p Q R and T * p Q R . This definition coincides with the definition of the norm of the linear functional λ over the vector space Q(p). If we write p = (x 0 , . . . , x 3 , z 1 , z 2 , z 3 ) and λ = (ξ 0 , . . . , ξ 3 , θ 1 , θ 2 , θ 3 ) then the Hamiltonian function can be rewritten in the following form We get the corresponding Hamiltonian system After the simplification, we obtain that θ k are constant and The solution of these equations and detailed calculations can be found in [7].

2.2.
Sub-Lorentzian manifold Q L . Let us change the positively definite metric (·, ·) R on Q on the Lorentz metric (that is nondegenerate metric of index 1) (·, ·) L such that We call the triple Q L = (R 7 , Q, (·, ·) L ) the sub-Lorentzian manifold or the sub-Lorentzian Htype group and the pair (Q, (·, ·) L ) is named by the sub-Lorentzian structure on R 7 . We define the casual character on A horizontal curve is called timelike if its tangent vector is timelike at each point. Spacelike, null and nonspacelike curves are defined similarly. The choice of the sub-Lorentzian metric 2.9 implies that the horizontal vector field X 0 is timelike and other horizontal vector fields X i , i = 1, 2, 3, are spacelike. We call X 0 the time orientation on Q L . Then a nonspacelike vector v ∈ Q p is called future directed if (v, X 0 (p)) L < 0, and it is called past directed if (v, X 0 (p)) L > 0. Throughout this paper f.d. stands for "future directed", t. for "timelike", and nspc. for "nonspacelike".
We would like to start the description of Q L with finding geodesics, that is by definition, the projections of a solution of the associated Hamiltonian system on Q L .
We construct a Hamiltonian system with respect to sub-Lorentzian metric. Locally the Hamiltonian function associated with the Lorentzian metric can be defined in the following way: If we use the coordinates for (p, λ) as in the previous subsection, then the Hamiltonian becomes Here Ω(θ) is the matrix given by (2.5) and η is the matrix of the Minkowskii metric tensor.
The corresponding Hamiltonian system takes the forṁ where the paricipating matrix A = ηΩ is a constant matrix of the parameters and Here by symbol |θ| is denoted the expression θ 2 1 + θ 2 2 + θ 2 3 . Notice that A T = −Ωη. After not intricate calculations two first equations of system (2.11) roll up to the following linear system of ordinary differential equations (2.13)ẍ = Aẋ, (2.14)ż k =ẋ T I k x, k = 1, 2, 3, that gives the equations for geodesics on Q L . Here the conditions (2.14) are derived from the second line of the system (2.11) by substituting ξ from the first line of this system. The exact formulae for obtained geodesics see in [11].

Electromagnetic fields
In this section we briefly introduce the notion of an electromagnetic field in order to explain the relation between the motion of the charged particle in an electromagnetic field and a sub-Lorentzian geodesic.
Consider the Minkowski spacetime M with the Lorentzian metric η in it. Let (m, α) be a charged particle in M of a unite charge and a constant mass m with a trajectory α. Charged particles create an electromagnetic field and also respond on the fields created by other particles. An electromagnetic field in M can be described by using two 3-dimensional vectors − → E and − → B that express electric and magnetic components respectively. Electromagnetic fileds in the space free of charge satisfies to four Maxwell's equations where x 0 stands for the time coordinate in M and permittivity and permeability are supposed to be constant and equal to 1. If we use the covariant formulation then Maxwell's equations can be written in the nice symmetric form where F is a 2-form field in 4-dimensional spacetime corresponding to antisymmetric electromagnetic tensor field The operator d is the exterior derivative, a coordinate and metric independent differential operator, and * is the Hodge star operator that is linear transformation from the space of 2 form into the space of two-forms defined by the metric in Minkowski space, see for instance [10]. While Maxwell equations describes how electrically charged particles and objects give rise to electric and magnetic fields, the Lorentz force law completes that picture by describing the force acting on a moving charged particle in the presence of electromagnetic fields, see, for instance [14]. This effect is described by Lorentz equation where U is the particle's world velocity, P = mU its world momentum, t is the proper time of the particle, and F is an electromagnetic tenzor field. Let (e 0 , e 1 , e 2 , e 3 ) be any admissible basis in M ; that is orthonormal, e 0 responds for the time coordinate and (e 1 , e 2 , e 3 ) for space coordinates. As it was mentioned at each point of M the linear transformation F can be defined in terms of the classical electric and magnetic The transformation A which influences on the charged particle is often called Lorentz force. In order to find eigenspaces of A that are invariant subspaces of this linear transformation we come to the characteristic equation.
are the same in all admissible frames and are called Lorentz invariants. If both of them are equal to zero (i.e., − → E and − → B are perpendicular and have the same magnitudes): Every regular skew-symmetric with respect to the Lorentz metric linear transformation There exist a basis, that is called the canonical basis, relative to which the matrix of regular skew-symmetric linear tranformation A has the form . Now the eigenvalues of A are easy to calculate since the characteristic equation becomes λ 4 + (δ 2 − ε 2 )λ 2 − δ 2 ε 2 = 0, i. e., (λ 2 − ε 2 )(λ 2 + δ 2 ) = 0, which has the following solutions: λ 1,2 = ±ε and λ 3,4 = ±iδ.
Observe that T is symmetric with respect to the Lorentzian inner product and is trace-free, i. e. tr T = 0. The term −T 1 is called the energy density in the given frame of reference for the electromagnetic field F with the form (3.3). The 3-vector is called the Poynting 3-vector and describes the energy flux of the field. Finally, the 3 × 3 matrix (T i j ) i,j=1,2,3 is known as Maxwell stress tensor of the field in the given frame. Thus, the content of the matrix of T determines the energy content of the field F in the corresponding basis.

Sub-Lorentzian geodesics and the trajectories of the particles
The noncommutative multiplication law (2.3) of quaternion H-type group (R 7 , •) defines the non-integrable distribution Q = span{X 0 , . . . , X 3 } or in the covariant language the nonholonomic constraints ω 1 = ω 2 = ω 3 = 0. The curvature of the distribution gives rise to the skew symmetric transformation Ω in 4-dimensional space (2.5). Independently whether this space has Euclidean structure or it is the Minkowskii space the antisymmetric 2-form Ω defines the electromagnetic field, since it trivially satisfies the Maxwell equations (3.2). We emphasize that the geometry of the nonholonomic manifold (R 7 , •) is related to the geometry of 4-dimensional space where a constant electromagnetic field acts. Given a positively definite metric (·, ·) R (Riemannian metric) on the nonholonomic distribution Q we obtain a sub-Riemannian manifold Q R . The Hamiltonian function H R in this case is reduced to the Lorentz equation in the Euclidean space R 4 given by (2.7). Equation (2.7) describes a motion of charged particle of unit charge in magnetic field Ω in 4-dimentional Euclidean space. Replacement of a positively definite metric (·, ·) R on a nondegenerate indefinite metric, the Lorentzian metric (·, ·) L , leads to the relativistic Lorentz equation (2.13) in the Minkowskii space. It has more connections to physics since it is related to the motion of the charged particle in electromagnetic field in the Minkowskii space that is closely connected to the general relativity.
Our aim is to find geodesics that are projections on R 7 of the solutions to the Hamiltonian system for H L defined on Q. The Hamiltonian system is reduced to the equations (2.13) and (2.14). Equations (2.13) are ordinary differential equations in 4-dimensional space with A that is skew-symmetric with respect to the Lorentzian metric η. This makes us to endow the 4-dimensional space with the Lorentzian metric, producing the Minkowskii space M = (R 4 , η). Therefore, at each point of M we can elect a model of electromagnetic field which corresponds to a linear skew-symmetric transformation A : M → M that assigns to the world velocityẋ = U of a charged particle passing through that point the change in world momentum dP dt that the particle should expect due to the field. Since we can assume that the charge and the mass of the particle equals 1 we get P = U and the Lorentz force law becomes dU dt = AU that is exactly equations (2.13). If we set then we conclude that equation (2.13) describes the motion of a particle of unit charge in the constant electromagnetic field with In this case one of the Lorentz invariant is zero: The matrix A 2 given by (2.12) has zero trace. Therefore, the energy momentum transformation is given up to a constant multiplied by symmetric matrix A 2 . The energy density is equal to −2|θ| 2 . The Poynting 3-vector in our case is zero vector. The Maxwell stress tensor is given by  Let us find a canonical basis for the matrix A. First we find a plane in R 3 in which both of the vectors − → E and − → B are lying, then we rotate it in such a way that it will coincide with a plane x 3 = 0 in R 3 . That is, let us choose a right-handed orthonormal basis { e 1 , e 2 , e 3 } of the space R 3 = span(e 1 , e 2 , e 3 ) in which E 3 = B 3 = 0. There are infinitely many planes containing both vectors − → E and − − → E . Fix one of these planes: for example, the one passing through the axis x 2 = (0, 1, 0): With the help of the rotation , −E 2 , 0) and matrix A can be defined in the following way: subspace for A and there exist real numbers ε 0 and δ 0 such that [14] A 2 u = ε 2 u for all u = (u 1 , u 2 ) ∈ U and Take any future directed unit timelike vectorẽ 0 in U , for example (1, 0, 0, 0) = v 1 +v 2 2|θ| , where v i are eigenvectors of matrix A (see [11]). Then spacelike unit vectorẽ 1 in U and the real value ε 0 can be found from the conditions Aẽ 0 = εẽ 1 and Aẽ 1 = εẽ 0 . We getẽ 1 = (0, − θ 1 |θ| , θ 3 |θ| , θ 2 |θ| ) and ε = |θ|. Now, letẽ 2 be an arbitrary unit spacelike vector in U ⊥ , for example, select . Then constructẽ 3 satisfying Aẽ 2 = δẽ 3 and Aẽ 3 = −δẽ 2 . We obtainẽ 3 = (0, − θ 2 √ Electric and magnetic fields corresponding to this transformation are − → E = εẽ 3 = |θ|ẽ 3 and − → B = δẽ 3 = |θ|ẽ 3 , so that an observer in this frame will measure them in the same direction (of x 3 -axis) and of magnitude |θ|. Matrix A is a block-type matrix, where (2 × 2)-block in the left upper corner coincides with a matrix in sub-Lorentzian Heisenberg case and the right lower (2 × 2)-block coincides with a matrix in usual sub-Riemannian Heisenberg case (see [11]).
Let us denote by P the matrix which columns are orthnormal basis vectorsẽ 0 , . . . ,ẽ 3 . Then in new basisÃ = P −1 AP and the vector x ∈ M is of the form x = P x old , where x old is a vector in old coordinates. Therefore, x old = P −1 x.
The Lorentz equation (2.13) in canonical coordinates takes the form 

Reachable sets by geodesics
We wish to describe the set of points in Q L that can be reached from the origin (x, z) = (0, 0) by a geodesic: timelike, lightlike or spacelike. We can fix the starting point at the origin O = (0, 0), since the solutions of the Hamiltonian equations are invariant under the left translation. We start from the simple lemma, that is related to the case of null transformation A.
The description of the reachable set by causal curves is very complicated in general, therefore, we present here some particular cases. We mostly reduce our considerations to the sets that can be reached by geodesics, since the geodesics can not change its causal character. (0, 0) and (x 1 , z 1 ), where |x 1 | 2 L = 0. We need to solve the equationsẍ = Aẋ,ż k =ẋ T J k x, k = 1, 2, 3 with the boundary conditions
Notice that the equations of the horizontal part (5.5) and (5.6) of geodesics in Q L coincides up to reparametrization with the equations of horizontal part for geodesics in H 1 L provided the initial velocityẋ 0 (0) = ±ẋ 1 (0) (see [11]).
The results of cases 1 and 2 may be united in the following statement Then there is a unique lightlike geodesic joining the origin and A which is a straight line. 3.1.1) Let us suppose thatẋ 0 (0) =ẋ 1 (0) = 0 andẋ 2 (0),ẋ 3 (0) are arbitrary. The main result is expressed in the following Theorem 2. Given a point A = (0, z 1 ) there are uncountably many spacelike geodesics connecting the origin with A. The geodesics are given by equations The geodesics have the lengths 2πn|z 1 |, n ∈ N \ {0}.
Proof. We start from the equations for x-coordinates of geodesics. Since x(t) = Wẋ 0 we get by (4.6). The projection into (x 2 , x 3 )-plane is (5.9) x 2 2 (t) + x 2 3 (t) = 4v 2 sin 2 (πnt)), which by lemma (2) can be rewritten as the canonical equation of the circle on the (x 2 , x 3 )-plane with the center at −ẋ 3 (0) 2πn ,ẋ 2 (0) 2πn of the radius √ v 2 2πn : The number n ∈ N \ {0} reflects the number of turns along the circle for any fixed initial velocity v 2 and the radius of the circle. To exclude v 2 from the equations we use (4.10) and findz Set t = 1 in the expression forz 1 (t) and get v 2 = −2πnz 1 (1). Since z(t) = Pz(t) with P given by (4.9) we get (5.11) It implies that −z 1 (1) = |z 1 | and v 2 = 2πn|z 1 |. Fixing n ∈ N \ {0}, we fix the speed v 2 of a geodesic, but we still have the choice in the directions of (ẋ 2 (0),ẋ 3 (0)) that are parametrized by the unit circle. It gives uncountably many geodesics. We get the equations for z-coordinates from z(t) = Pz(t) taking into account that the values of θ k related to the values of z 1 by (5.11).
Since the geodesics are spacelike the length l(γ) of a geodesic γ can be calculated from the formula Remark 1. Notice that in Lorentzian Heisenberg group H L there is no geodesic of any causal type joining (0, 0, 0) and (0, 0, z) with z = 0 (see [11]).

3.2)
Consider the case v 1 > 0, i. e., the initial velocityẋ 0 is spacelike. It suggests that the ratio − v 1 v 2 is negative, which contradicts to (5.7). Hence, in this case there are no spacelike geodesics with the property (5.1).
3.3) Now suppose v 1 < 0. It implies that the ratio − v 1 v 2 is positive and here we have 3 more subcases: In this case the initial velocity appears to be timelike. There are no points where this can happen because the function ν(|θ|) = sin 2 |θ| 2 sinh 2 |θ| 2 is wave decaying with exponential speed on the right halfplane and bounded with 0 from below and with 1 from above. Thus, there are no timelike geodesics with the considered timelike initial velocity.
This lemma allows us to prove the following theorem.
Theorem 8. Let us define the following sets Then there exists a unique geodesic connecting the point O = (0, 0, 0) with a point P that belongs to one of the sets R t , R sp or R l . Particularly, if P ∈ R t , then the geodesic is timelike, if P ∈ R sp , then the geodesic is spaselike, and if P ∈ R l , then the geodesic is lightlike.
The connectivity on H 1 L is dependent on the solutions |θ| of the equation 4z −x 2 + y 2 = µ(θ), where the function µ(τ ) = τ /2 sinh 2 (τ /2) − coth(τ /2) is strictly decreasing on the interval (−∞, ∞) from −1 to 1. It means that if the point A = (x, y, z) is such that 4z −x 2 +y 2 1, i. e. if A ∈ (R t ∪ R sp ∪ R l ) c , then there are no geodesics of any causal type joining the origin with A. In particular, there are no Hamiltonian geodesics joining the origin with the points of the surfaces { 4|z| x 2 −y 2 = 1, −x 2 + y 2 < 1} and { 4|z| −x 2 +y 2 = 1, −x 2 + y 2 > 0}. We observe that in [6] it is shown that it is possible to find nonHamiltonian lightlike geodesics joining 0 with the points of these surfaces.
Thus, in contrast with Q L , the Heisenberg Lorentzian group has the property of uniqueness of geodesics starting from the origin with given tangent vector. It happens due to lower dimension of the "spacelike"part of H 1 L .