SUB-LORENTZIAN METRIC AND THEIR PHYSICAL INTERPRETATION
ANNA KOROLKO, IRINA MARKINA
Abstract. 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.
1. 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 ap-plications 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 underlies the motion in an electro-magnetic field. Just like space and time emerge in special relativity, the
2000 Mathematics Subject Classification. 53C50, 53B30 53C17.
Key words and phrases. Quaternion H-type group, sub-Lorentzian metric, electromagnetic field, special relativity.
The authors are partially supported by the grant of the Norwegian Research Council # 177355/V30, by the grant of the European Science Foundation Networking Programme HCAA, and by the NordForsk Research Network Programme # 080151.
electric and magnetic fields can not be considered separately. The sub-Lorentzian structure unites both phenomena, the presence of an electro-magnetic field and the space-time geome-try. That is why an H-type quaternion group with a sub-Lorentzian metric is an interesting example to work with.
In order to explain the main idea we would like to mention the Heisenberg group in the introduction as the simplest noncommutative example of H-type (Heisenberg type) groups and its numerous connections with physics. The Heisenberg group is the manifold R3 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− 12y∂z, Y = ∂y+ 12x∂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 a kernel of the contact one-form ω = dz − 12(xdy − ydx) in R3. 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 R3. Let us define a Riemannian metric ds2R = dx2+ dy2 on H. Then the sub-Riemannian manifold H1R = (R3, H, ds2R) is also called the Heisenberg
group. It turns out that the geodesic equation for geodesics γ(t) satisfying the non-holonomic constraints ˙γ(t) ∈ H(γ(t)) coincides up to a constant with the Lorentz equation of motion of a charged particle in the magnetic field Ω. If we change the Riemannian metric on H to the Lorentzian one ds2
L = −dx2 + dy2, we come to the notion of sub-Lorentzian Heisenberg
group H1
L = (R3, H, ds2L). In this case the geodesic equation for non-holonomic geodesics
coincides with an 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, metric properties, and other related questions for H1L were studied in [2, 3, 4, 11]. The lack of dimension of the horizontal distribution on the Heisenberg group does not allow to reveal the peculiarity of applications in the case of magnetic and electromagnetic fields. Therefore, we choose an analogue of the Heisenberg group admitting a 4-dimensional distribution, that we call a 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 H-type quater-nion groups endowed with different metrics: Riemannian and Lorentzian. We also present differential equations for the geodesics in both cases. Section 3 is the collection of definitions related to the motion of charged particles in electro-magnetic fields. We give an explanation of geometrical results from the physical point of view. Section 4 is devoted to the solution of geodesic equations, where we find 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 H1Lfor the sake of comparison with obtained results for the quaternion H-type group.
2. H-type quaternion groups QR and QL
We remind that quaternions form a noncommutative division algebra that extends the sys-tem of complex numbers. It is convenient to define any quaternion q in the algebraic form q = a + i1b + i2c + i3d, where i21 = i22 = i23 = i1i2i3 = −1. The scalar a ∈ R is called the real
part Re q = a and a vector (b, c, d) ∈ R3 recieved the name of the 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 q1 and q2 using the usual inner
product · and the vector product × in R3. Namely,
(2.1) q1∗ q2 = (Req1Req1− Pu q1· Pu q2, Req1Pu q2+ Req2Pu q1+ Pu q1× Pu q2).
Notice that this structure suggests an analogy with the Lorentzian geometry where R4consists
of the time part t ∈ R and the space part (x1, x2, x3) ∈ R3. The conjugate quaternion q to q
is q = a − i1b − i2c − i3d. It is known that a quaternion q = a + i1b + i2c + i3d can also be
represented in the 4 × 4-matrix form
q = a b −d −c −b a −c d d c a b c −d −b a = aI + bI1+ cI2+ dI3,
where the matrices
I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , I1 = 0 1 0 0 −1 0 0 0 0 0 0 1 0 0 −1 0 , I2= 0 0 0 −1 0 0 −1 0 0 1 0 0 1 0 0 0 , I3= 0 0 −1 0 0 0 0 1 1 0 0 0 0 −1 0 0 (2.2)
are the basis of quaternions 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 R7 and define the
noncommutative law by
(2.3) (x, z) ◦ (x′, z′) = x + x′, z + z′+1
2Pu(¯x ∗ x
′)
for (x, z) and (x′, z′) from R4 × R3. Here Pu(¯x ∗ x′) is the imaginary part of the product ¯
x ∗ x′ defined in (2.1) of the conjugate quaternion ¯x to x by another quaternion x′. The
introduced multiplication law (2.3) makes M = (R7, ◦) 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 L(x,z)(x′, z′) = (x, z) ◦ (x′, z′). Let ∂ ∂x0, . . . , ∂ ∂x3, ∂ ∂z1, . . . , ∂ ∂z3 be a standard basis of
the tangent space TpM 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) ∂x∂
i = Xi(x, z), dL(x,z) ∂z∂ k
= Zk(x, z). Then the vector fields
X0 = ∂ ∂x0 +1 2 +x1 ∂ ∂z1 − x3 ∂ ∂z2 − x2 ∂ ∂z3 , X1 = ∂ ∂x1 +1 2 −x0 ∂ ∂z1 − x2 ∂ ∂z2 + x3 ∂ ∂z3 , X2 = ∂ ∂x2 +1 2 +x3 ∂ ∂z1 + x1 ∂ ∂z2 + x0 ∂ ∂z3 , X3 = ∂ ∂x3 +1 2 −x2 ∂ ∂z1 + x0 ∂ ∂z2 − x1 ∂ ∂z3 ,
span a 4-dimensional distribution Q, which we call horizontal. The left-invariant vector fields Zβ = ∂z∂β, β = 1, 2, 3 form a basis of the complement V to Q in T M . At each point (x, z) ∈
R4× R3 the distribution Q(x, z) is a copy of R4. The commutation relations are as follows
[X0, X1] = −Z1, [X0, X2] = Z3, [X0, X3] = Z2,
[X1, X2] = Z2, [X1, X3] = −Z3, [X2, X3] = −Z1.
Therefore, {X0, . . . , X3} 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 {X0, . . . , X3, Z1, Z2, Z3} is nilpotent of step 2.
The horizontal distribution Q can be defined by making use of one-forms. Namely, the one-forms ω1 =dz1− 1 2(+x 1dx0 − x0dx1+ x3dx2− x2dx3) = dz1−1 2dx TI 1x, ω2 =dz2− 1 2(−x 3dx0− x2dx1+ x1dx2+ x0dx3) = dz2−1 2dx TI 2x, ω3 =dz3− 1 2(−x 2dx0+ x3dx1+ x0dx2− x1dx3) = dz3−1 2dx TI 3x, (2.4)
annihilate the distribution Q. Here x = (x0, . . . , x3), dx = (dx0, . . . , dx3), and dxT is the transposed vector to dx. Thus, Q is the common kernel of forms ωk, k = 1, 2, 3. Let us
consider the external differential of the linear combination ω =P3k=1ϑkωk. We get the
two-form that is defined in 4-dimensional space
Ω = ϑ1(dx0∧ dx1+ dx2∧ dx3) + ϑ2(−dx0∧ dx3− dx1∧ dx2) + ϑ3(−dx0∧ dx2+ dx1∧ dx3) = 1 2 X ij Ωijdxi∧ dxj, where (2.5) Ω(ϑ) = Ωij(ϑ) = 0 ϑ1 −ϑ3 −ϑ2 −ϑ1 0 −ϑ2 ϑ3 ϑ3 ϑ2 0 ϑ1 ϑ2 −ϑ3 −ϑ1 0 .
Any vector v ∈ Qp, p ∈ M, is called horizontal, a vector field X tangent to Qp at each point
p is also called horizontal. An absolutely continuous curve γ : [0, 1] → M with the velocity vector ˙γ(t) tangent to Qγ(t) almost everywhere is called horizontal curve.
2.1. Sub-Riemannian manifold QR. Let us define the Riemannian metric (·, ·)R on the
distribution Q in such a way that (Xi, Xj) = δij, where δij is the Kronecker symbol. With
this, we get the sub-Riemannian manifold QR = R7, Q, (·, ·)R with the sub-Riemannian
structure Q, (·, ·)R
.
Geodesics (or normal extremals) in the sub-Riemannian geometry are defined as the projec-tion of soluprojec-tions of the Hamiltonian equaprojec-tions onto the underlying manifold. Let us present the corresponding Hamiltonian system. We denote by TpQR, Tp∗QRthe tangent and cotangent
space at p ∈ QR respectively and by T QR, T∗QR the corresponding tangent and cotangent
bundle. Thus, if (p, λ) ∈ T∗QR, then the restriction of λ onto the subspace Q(p) of TpQR is
well defined, and making use of the inner product we define a Hamiltonian function on Tp∗QR by HR(p, λ) = 1 2 3 X i=0 hλ, Xii2,
where by h·, ·i we denoted the pairing between vector spaces TpQRand Tp∗QR. This definition
coincides with the definition of the norm of the linear functional λ over the vector space Q(p). If we write p = (x0, . . . , x3, z1, z2, z3) and λ = (ξ0, . . . , ξ3, θ1, θ2, θ3), then the Hamiltonian
function can be rewritten in the following form HR(p, λ) = 1 2|ξ| 2+1 8|x| 2 |θ|2+ ξTΩx. We get the corresponding Hamiltonian system
˙x = ∂HR ∂ξ = ξ +12Ωx ˙zk = ∂H∂θkR = 14|x|2θk+12ξTIkx, k = 1, 2, 3 ˙ξ = −∂HR ∂x = − 1 4|θ|2x + 1 2Ωξ ˙θ = −∂HR ∂z = 0. (2.6) Here |θ| = (P3k=1θk2)1/2, |x| = ( P3
i=0x2i)1/2. After simplification, we obtain that θk are
constant and
(2.7) x = Ω ˙x,¨ x ∈ R4,
(2.8) ˙zk = ˙xTIkx, k = 1, 2, 3.
The solution of these equations and detailed calculations can be found in [7].
2.2. Sub-Lorentzian manifold QL. Let us change the positively definite metric (·, ·)R on Q
on the Lorentz metric (that is nondegenerate metric of index 1) (·, ·)L such that
(2.9) (Xi, Xj)L= δij, (X0, X0)L= −1, (Xi, Xi)L= 1, i = 1, 2, 3.
We call the triple QL = (R7, Q, (·, ·)L) the sub-Lorentzian manifold or the sub-Lorentzian
H-type group and the pair (Q, (·, ·)L) is named the sub-Lorentzian structure on R7.
We define the casual character on QL. Fix a point p ∈ R7. A horizontal vector v ∈ Qp is
called timelike if (v, v)L < 0, spacelike if (v, v)L > 0 or v = 0, null if (v, v)L = 0 and v 6= 0,
nonspacelike if (v, v)L60. 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 X0 is timelike and other
horizontal vector fields Xi, i = 1, 2, 3, are spacelike. We call X0 the time orientation on QL.
Then a nonspacelike vector v ∈ Qp is called future directed if (v, X0(p))L < 0, and it is called
past directed if (v, X0(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 QLwith finding geodesics, which are by definition,
the projections of a solution of the associated Hamiltonian system onto QL.
We construct a Hamiltonian system with respect to the sub-Lorentzian metric. Locally the Hamiltonian function associated with the Lorentzian metric can be defined in the following way HL(p, λ) = − 1 2hλ, X0i 2+1 2 3 X i=1 hλ, Xii2.
If we use the coordinates for (p, λ) as in the previous subsection, then the Hamiltonian becomes HL(ξ, θ, x, z) = 1
2ξ
Tηξ + 1
8(Ωx)
Here Ω(θ) is the matrix given by (2.5) and η is the matrix of the Minkowskii metric tensor. (2.10) η = −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 .
The corresponding Hamiltonian system takes the form ˙x = ∂HL ∂ξ = ηξ + 1 2Ax, ˙zk= ∂HL ∂θk = 1 4(Ikx) TηΩx +1 2ξ TηI kx, k = 1, 2, 3, ˙ξ = −∂HL ∂x = 1 4ηA 2x − 1 2A Tξ, ˙θ = −∂HL ∂z = 0, (2.11)
where the paricipating matrix A = ηΩ is a constant matrix of the parameters and
A2 = |θ|2 0 0 0 0 θ12− θ22− θ32 −2θ1θ3 −2θ1θ2 0 −2θ1θ3 −θ21− θ22+ θ32 2θ2θ3 0 −2θ1θ2 2θ2θ3 −θ12+ θ22− θ23 . (2.12)
Here by symbol |θ| we denote the expressionpθ21+ θ22+ θ23. Notice that AT = −Ωη.
After non intricate calculations, two first equations of system (2.11) roll up to the following linear system of ordinary differential equations
(2.13) ¨x = A ˙x,
(2.14) ˙zk = ˙xTIkx, k = 1, 2, 3,
that gives the equations for geodesics on QL. 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 formulas for obtained geodesics see in [11].
3. Electromagnetic fields
In this section we briefly introduce the notion of an electromagnetic field in order to explain the relation between the motion of a charged particle in an electromagnetic field and sub-Lorentzian geodesics.
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 par-ticles. 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 satisfy four Maxwell’s equations
(3.1) ∇ · − → B = 0, ∇ ·−→E = 0, ∇ ×−→E + ∂∂x−→B 0 = 0, ∇ × − → B −∂x∂−→E0 = 0,
where x0 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
(3.2) dF = 0, d ∗ F = 0,
where F is a 2-form field in 4-dimensional spacetime corresponding to antisymmetric electro-magnetic tensor field
(3.3) F = Fαβ =
0 −E3 −E2 −E1
E3 0 B1 −B2 E2 −B1 0 B3 E1 B2 −B3 0 .
The operator d is the exterior derivative, a coordinate and a metric independent differential operator, and ∗ is the Hodge star operator that is linear transformation from the space of two-form into the space of two-two-forms defined by the metric in Minkowski space, see for instance [10]. While Maxwell equations describe how electrically charged particles and objects give rise to electric and magnetic fields, the Lorentz force law completes this 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 the Lorentz equation
dP
dt = eηF U = F
µνU β,
where U is particle’s world velocity, P = mU is its world momentum, t is the proper time of the particle, and F is an electromagnetic tenzor field.
Let (e0, e1, e2, e3) be any admissible basis in M ; that is orthonormal, e0 responds to the time
coordinate and (e1, e2, e3) to space coordinates. As it was mentioned the linear transformation
F can be defined in terms of the classical electric and magnetic 3-vectors −→E = E1e1+ E2e2+
E3e3 and −→B = B1e1+ B2e2+ B3e3 at each point of M . Set
(3.4) A = ηF = 0 E3 E2 E1 E3 0 B1 −B2 E2 −B1 0 B3 E1 B2 −B3 0 .
The transformation A that influences on a charged particle is often called the Lorentz force. In order to find eigenspaces of A, that are invariant subspaces of this linear transformation, we come to the characteristic equation.
det(A − λI) = λ4+ (|−→B |2− |−→E |2)λ2− (−→E ·−→B )2 = 0,
where |−→E |2 = (E1)2 + (E2)2 + (E3)2 and −→E ·−→B = E1B1 + E2B2 + E3B3. The algebraic
combinations |−→B |2 − |−→E |2 and −→E ·−→B 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 magnitude): |−→B |2 − |−→E |2 = −→E ·−→B = 0, then A is called null transformation, otherwise A is said to be regular.
Every regular linear transformation A : M → M that is skew-symmetric with respect to the Lorentz metric has a 2-dimensional invariant subspace V , such that V TV⊥ = {0}. There
skew-symmetric linear tranformation A has the form (3.5) A = 0 ε 0 0 ε 0 0 0 0 0 0 −δ 0 0 δ 0 ,
where δ and ε are nonnegative real values, such that |−→B |2− |−→E |2 = δ2− ε2,−→E ·−→B = δε. Now
the eigenvalues of A are easy to calculate since the characteristic equation becomes λ4+ (δ2− ε2)λ2− δ2ε2 = 0, i. e., (λ2− ε2)(λ2+ δ2) = 0. It has the following solutions: λ1,2 = ±ε and
λ3,4 = ±iδ.
Definition 1. The linear transformation T : M → M defined by T = 1 4π 1 4tr(A 2)I − A2 ,
where A2= A ◦ A, I is the identity transformation I(x) = x for every x ∈ M and tr(A2) is the
trace of A2, is called the energy-momentum transformation associated with A.
Observe that T is symmetric with respect to the Lorentzian inner product and is traceless, i. e., tr T = 0. The term −T11 = T e0· e0 = 8π1 (|−→E |2 + |−→B |2) is called the energy density in
the given frame of reference for the electromagnetic field F of the form (3.3). The 3-vector
1 4π
− →
E ×−→B = (E2B3− E3B2)e1+ (E3B1− E1B3)e2+ (E1B2− E2B1)e3 is called the Poynting
3-vector and describes the energy flux of the field. Finally, the 3 × 3 matrix (Ti
j)i,j=1,2,3 is
known as the 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.
4. Sub-Lorentzian geodesics and the trajectories of the particles
The non-commutative multiplication law (2.3) of quaternion H-type group (R7, ◦) defines the non-integrable distribution Q = span{X0, . . . , X3}, or in the covariant language, the
non-holonomic constraints ω1 = ω2 = ω3 = 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 (R7, ◦) is related to the
geom-etry 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 QR. The Hamiltonian function HR in this case is reduced to the
Lorentz equation in the Euclidean space R4 given by (2.7). Equation (2.7) describes the mo-tion of a charged particle of unit charge in the magnetic field Ω in the 4-dimenmo-tional Euclidean space. Replacement of a positively definite metric (·, ·)R by 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 a charged par-ticle in an electromagnetic field in the Minkowskii space that is closely connected to general relativity.
Our aim is to find geodesics that are projections onto R7of the solutions to the Hamiltonian system for HLdefined on Q. The Hamiltonian system is reduced to equations (2.13) and (2.14).
Equations (2.13) are ordinary differential equations in the 4-dimensional space with A which 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 = (R4, η).
Therefore, at each point of M we can select a model of electromagnetic field which corresponds to a linear skew-symmetric transformation A : M → M that assigns to the world velocity ˙x = U of a charged particle passing through that point the change in world momentum dPdt 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 dUdt = AU , that is exactly equations (2.13). If we set E1 = −B1 = θ2, E2 = −B2 = θ3, E3 = −B3 = θ1,
then we conclude that equation (2.13) describes the motion of a particle of unit charge in the constant electromagnetic field with−→E = −−→B . In this case one of the Lorentz invariants is zero: |−→B |2− |−→E |2 = 0, and the other−→E ·−→B is equal to −|θ|2, where |θ|2= θ2
1+ θ22+ θ23. The matrix
A2 given by (2.12) is traceless. Therefore, the energy-momentum transformation is given up to a constant multiplier by symmetric matrix A2. 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 θ12− θ22− θ32 −2θ1θ3 −2θ1θ2 −2θ1θ3 −θ12− θ22+ θ23 2θ2θ3 −2θ1θ2 2θ2θ3 −θ21+ θ22− θ32 . (4.1)
Let us find a canonical basis for the matrix A. First we find a plane in R3 in which both of the vectors −→E and −→B are lying, then we rotate it in such a way that it will coincide with the plane x3 = 0 in R3. That is, let us choose a right-handed orthonormal basis {be1, be2, be3} of the
space R3 = span(e1, e2, e3) in which bE3 = bB3= 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 x2 = (0, 1, 0):
(4.2) −x1E3+ x3E1= 0.
With the help of the rotation
(4.3) R = cos α 0 sin α 0 1 0 − sin α 0 cos α ,
we turn it to the angle α = arccos√ E1
E2 1+E
2 3
between the plane (4.2) and x3 = 0 about the axis
x2, so that the third coordinate of −→E equals zero. Consider now the Lorentz transformation
R1=
1 0 0 R
of the basis {e0, e1, e2, e3}, where R is given by (4.3). It yields a new admissible coordinate
system {be0, be1, be2, be3}, in which
− → b E = (pE2 1 + E32, E2, 0) and − → b B = (−pE2 1 + E32, −E2, 0), and
the matrix bA can be defined as
b A = 0 0 Eb2 Eb1 0 0 − bE1 Eb2 b E2 Eb1 0 0 b E1 − bE2 0 0 , where bE1 = p E2 1 + E32 = − bB1 = p θ2 1+ θ22, bE2 = E2 = − bB2 = θ3, and bE3= − bB3= 0.
Since A : M → M is a regular skew-symmetric transformation, then it has a 2-dimensional invariant subspace U such that U ∩ U⊥ = {0}. Then U⊥ is also a 2-dimensional invariant
subspace for A and there exist real numbers ε > 0 and δ > 0 such that [14] A2u = ε2u for all u = (u1, u2) ∈ U and
A2v = −δ2v for all v = (v1, v2) ∈ U⊥.
Take any future directed unit timelike vector ˜e0 in U , for example (1, 0, 0, 0) = v12|θ|+v2, where
vi are eigenvectors of the matrix A (see [11]). Then the spacelike unit vector ˜e1 in U and
the real value ε > 0 can be found from the conditions A˜e0 = ε˜e1 and A˜e1 = ε˜e0. We get
˜
e1 = (0, −|θ|θ1,|θ|θ3,|θ|θ2) and ε = |θ|. Now, let ˜e2 be an arbitrary unit spacelike vector in U⊥,
for example, select ˜e2 = v3+v4 2|θ|√θ2 1+θ 2 2 = 1 |θ|√θ2 1+θ 2 2 (0, θ1θ3, θ21 + θ22, −θ2θ3). Then construct ˜e3
satisfying A˜e2 = δ˜e3 and A˜e3 = −δ˜e2. We obtain ˜e3 = (0, −√θ2 θ2 1+θ 2 2, 0, − θ1 √ θ2 1+θ 2 2) and δ = |θ|.
Thus, {˜e0, ˜e1, ˜e2, ˜e3} is an orthnormal basis for M, which is called canonical and
(4.4) A =˜ 0 |θ| 0 0 |θ| 0 0 0 0 0 0 −|θ| 0 0 |θ| 0 .
Electric and magnetic fields corresponding to this transformation are −→E = ε˜e3 = |θ|˜e3 and
− →B = δ˜e
3 = |θ|˜e3, so that an observer in this frame will measure them in the same direction (of
x3-axis) and of magnitude |θ|. Matrix eA is a block-type matrix, where (2 × 2)-block in the left
upper corner coincides with the matrix in sub-Lorentzian Heisenberg case and the right lower (2 × 2)-block coincides with the matrix in usual sub-Riemannian Heisenberg case (see [11]).
Let us denote by P the matrix which columns are orthnormal basis vectors ˜e0, . . . , ˜e3. Then
˜
A = P−1AP in new basis and the vector x ∈ M is of the form x = P xold, where xoldis a vector
in the old coordinates. Therefore, xold= P−1x.
It is clear that the real eigenvalues of A are λ = ±|θ| from the canonical form (4.4) . Therefore, eigenspaces are span{˜e3+ ˜e4} and span{˜e3− ˜e4} respectively. Directions ˜e3± ˜e4 are
called principal null directions of A.
The Lorentz equation (2.13) in canonical coordinates takes the form d ˙x0 dt d ˙x1 dt d ˙x2 dt d ˙x3 dt = 0 |θ| 0 0 |θ| 0 0 0 0 0 0 −|θ| 0 0 |θ| 0 ˙x0 ˙x1 ˙x2 ˙x3 = |θ| ˙x1 |θ| ˙x0 − |θ| ˙x3 |θ| ˙x2 .
It splits into 2 independent systems ( d ˙x0 dt = |θ| ˙x1, d ˙x1 dt = |θ| ˙x0, ( d ˙x2 dt = −|θ| ˙x3, d ˙x3 dt = |θ| ˙x2.
We wish to solve this system under the initial conditions x(0) = 0, ˙x(0) = ˙x0. The solution
for ˙x(t) is ˙x0(t) = ˙x0(0) cosh(|θ|t) + ˙x1(0) sinh(|θ|t), ˙x1(t) = ˙x0(0) sinh(|θ|t) + ˙x1(0) cosh(|θ|t), ˙x2(t) = ˙x2(0) cos(|θ|t) − ˙x3(0) sin(|θ|t), ˙x3(t) = ˙x2(0) sin(|θ|t) + ˙x3(0) cos(|θ|t). (4.5)
Integrating these expressions, we get the solution x(t), that we write in the matrix form as x(t) = W ˙x0 with (4.6) W = 1 |θ| sinh(|θ|t) cosh(|θ|t) − 1 0 0 cosh(|θ|t) − 1 sinh(|θ|t) 0 0 0 0 sin(|θ|t) cos(|θ|t) − 1 0 0 1 − cos(|θ|t) sin(|θ|t). .
Now, let us introduce the following notation: v1= − ˙x20(0) + ˙x21(0), v2 = ˙x22(0) + ˙x23(0).
Lemma 1. The projection of the geodesic onto the (x0, x1)-plane is a brunch of the hyperbola
with the canonical equation (4.7) x0+ ˙x1(0) |θ| 2 − x1+ ˙x0(0) |θ| 2 = v1 |θ|2. Proof. Since x0(t) = 1 |θ| ˙x0(0) sinh(|θ|t) + ˙x1(0)(cosh(|θ|t) − 1) , x1(t) = 1 |θ| ˙x0(0)(cosh |θ|t − 1) + ˙x1(0) sinh(|θ|t) , we calculate −x2 0(t) + x21(t) = 2(− ˙x 2 0(0)+ ˙x 2 1(0)) |θ|2 · sinh2 |θ|t
2 . This expression can be rewritten as
stated in (4.7)
Lemma 2. The projection of the geodesic onto the (x2, x3)-plane is a circle with the center at
−˙x 0 3 |θ|, ˙x0 2 |θ| of radius √v2 |θ| . Proof. Since x2(t) = 1 |θ| ˙x2(0) sin |θ|t + ˙x3(0)(cos |θ|t − 1) , x3(t) = 1 |θ| ˙x2(0)(1 − cos |θ|t) + ˙x3(0) sin(|θ|t) , we get x22(t) + x23(t) = 4v2
|θ|2 · sinh2|θ|t2 . This expression leads to
(4.8) x2+ ˙x3(0) |θ| 2 + x3− ˙x2(0) |θ| 2 = v2 |θ|2. The horizontality conditions (2.8) in the canonical basis have the form
˙zk = (Ikxold, ˙xold) = (IkP−1x, P−1˙x) = (P IkP−1x, ˙x),
where the matrices
J1 = P I1P−1 = − 1 |θ|Ω(θ), J2 = P I2P−1= p 1 θ2 1+ θ22 0 θ2 0 θ1 −θ2 0 θ1 0 0 −θ1 0 θ2 −θ1 0 −θ2 0 ,
and J3 = P I3P−1= 1 |θ|pθ21+ θ22 0 −θ1θ3 −(θ12+ θ22) θ2θ3 θ1θ3 0 θ2θ3 θ21+ θ22 θ21+ θ22 −θ2θ3 0 −θ1θ3 −θ2θ3 −(θ12+ θ22) θ1θ3 0
are skew-symmetric with respect to the usual euclidean metric. More explicitely, ˙z = P ˙˜z, where ˙z are coordinates in the canonical basis, ˙˜z = ( ˙˜z1, ˙˜z2, ˙˜z3) are auxiliary expressions ˙˜zk = (Ikx, ˙x),
˙x is given by (4.5), and (4.9) P = −θ1 |θ| −θ 2 |θ| −θ 3 |θ| θ2 √ θ2 1+θ 2 2 −√θ1 θ2 1+θ 2 2 0 − θ1θ3 |θ|√θ2 1+θ 2 2 − θ2θ3 |θ|√θ2 1+θ 2 2 √ θ2 1+θ 2 2 |θ| . .
Notice that P is an orthogonal transformation in R3 while the matrix P represents the orthog-onal transformation R4. It is more convenient for us to work with the expressions ˜z
1, ˜z2, ˜z3.
Then z1, z2, z3 can be obtained by the orthogonal transformation P of ˜z1, ˜z2, ˜z3. Taking into
account that x = W ˙x0 and (4.5), we calculate
˙˜z1(t) = 1 |θ| v1(cosh(|θ|t) − 1) + v2(cos(|θ|t) − 1) , ˙˜z2(t) = 1 |θ| h
2 cosh |θ|t cos |θ|t ˙x0(0) ˙x2(0) + 2 sinh |θ|t cos |θ|t ˙x1(0) ˙x2(0)
− 2 sinh |θ|t sin |θ|t ˙x1(0) ˙x3(0) − 2 cosh |θ|t sin |θ|t ˙x0(0) ˙x3(0)
+ cosh(|θ|t) ˙x1(0) ˙x3(0) − ˙x0(0) ˙x2(0) + sinh(|θ|t) ˙x0(0) ˙x3(0) − ˙x1(0) ˙x2(0) − cos(|θ|t) ˙x0(0) ˙x2(0) + ˙x1(0) ˙x3(0) + sin(|θ|t) ˙x0(0) ˙x3(0) − ˙x1(0) ˙x2(0) i , ˙˜z3(t) = 1 |θ| h
− 2 cosh |θ|t cos |θ|t ˙x0(0) ˙x3(0) − 2 sinh |θ|t cos |θ|t ˙x1(0) ˙x3(0)
− 2 sinh |θ|t sin |θ|t ˙x1(0) ˙x2(0) − 2 cosh |θ|t sin |θ|t ˙x0(0) ˙x2(0)
+ cosh(|θ|t) ˙x0(0) ˙x3(0) + ˙x1(0) ˙x2(0) + sinh(|θ|t) ˙x0(0) ˙x2(0) + ˙x1(0) ˙x3(0) + cos(|θ|t) ˙x0(0) ˙x3(0) − ˙x1(0) ˙x2(0) + sin(|θ|t) ˙x0(0) ˙x2(0) + ˙x1(0) ˙x3(0) i . Let us use the following notation for the constants
a1= ˙x0(0) ˙x3(0) + ˙x1(0) ˙x2(0), a3 = ˙x0(0) ˙x2(0) + ˙x1(0) ˙x3(0), a2= ˙x0(0) ˙x3(0) − ˙x1(0) ˙x2(0), a4 = ˙x0(0) ˙x2(0) − ˙x1(0) ˙x3(0). Thus, we integrate ˜ z1(t) = 1 |θ|2 v1 sinh(|θ|t) − |θ|t+ v2 sin(|θ|t) − |θ|t,
˜ z2(t) = 1 |θ|2 a1 cos(|θ|t) cosh(|θ|t) − 1
− a2 sin(|θ|t) sinh(|θ|t + cos(|θ|t) − cosh(|θ|t))
+ a3 cos(|θ|t) sinh(|θ|t) − sin(|θ|t)
+ a4 sin(|θ|t) cosh(|θ|t) − sinh(|θ|t)
, ˜ z3(t) = 1 |θ|2 a4 cos(|θ|t) cosh(|θ|t) − 1
− a3 sin(|θ|t) sinh(|θ|t + cos(|θ|t) − cosh(|θ|t))
− a2 cos(|θ|t) sinh(|θ|t) − sin(|θ|t)
− a1 sin(|θ|t) cosh(|θ|t) − sinh(|θ|t)
, (4.10) Observe that a21+ a24= a22+ a32 = ˙x20(0) + ˙x21(0)v2, a1a2+ a3a4 = −v1v2, a1a3− a2a4 = 2 ˙x0(0) ˙x1(0)v2.
Then the direct calculations yield ˜
z22(t) + ˜z23(t) = 4v2
|θ|4 cosh(|θ|t) − cos(|θ|t) − sin(|θ|t) sinh(|θ|t)
, ( ˙x20(0) + ˙x21(0)) cosh(|θ|t) + 2 ˙x0(0) ˙x1(0) sinh(|θ|t) − v1.
(4.11)
5. Reachable sets by geodesics
We wish to describe the set of points in QLthat 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 a simple lemma, that is related to the case of a transformation A.
Lemma 3. If |θ| = 0, then the system (2.11) with initial data x(0) = z(0) = 0, ξ(0) = ξ0, η(0) = η0 has the solution x(t) = ξ0t, z(t) = 0, ξ(t) = ξ0, η(t) = η0. The projections onto
(x, z)-space are straight lines that are timelike if |ξ0|2
L< 0, lightlike if |ξ0|2L = 0, and spacelike
if |ξ0|2 L> 0.
Proof. The condition |θ| = 0 immediately implies that ξ(t) and η(t) are constant and x(t) = ξ0t.
Then ˙˜zk= 2t(ξ0)TIkξ0= 0 since the matrix Ik is skew symmetric.
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 their causal character. 5.1. Connectivity by geodesics between (0, 0) and (x1, z1), where |x1|2
L = 0. We need
to solve the equations
¨
x = eA ˙x, ˙zk= ˙xTJkx, k = 1, 2, 3,
with the boundary conditions
We find the relation between the initial velocity and the value x(1). Substituting t = 1 in x(t) = W ˙x0, we obtain (5.2) |x1|2L= 4 |θ|2 − ˙x20(0) + ˙x21(0) sinh2 |θ| 2 + ˙x 2 2(0) + ˙x23(0) sin2|θ| 2 . We also have ˙x(0) = ˙x0= W−1(t)x(t), where
(5.3) W−1(t) = |θ| 2 sinh(|θ|t) cosh(|θ|t)−1 −1 0 0 −1 cosh(|θ|t)−1sinh(|θ|t) 0 0 0 0 1−cos(|θ|t)sin(|θ|t) −1 0 0 1 sin(|θ|t) 1−cos(|θ|t) .
from (4.6). Putting t = 1, we calculate | ˙x0|2L= (η ˙x0, ˙x0) =|θ| 2 4 − x20(1) + x21(1) 1 sinh2|θ|+ x 2 2(1) + x23(1) 1 sin2|θ| . The expression (5.2) can be written in the form
(5.4) 0 = |x1|2L= 4 |θ|2 v1sinh 2 |θ| 2 + v2sin 2|θ| 2 . Let us describe all possible initial velocities vanishing the norm |x1|2L.
Case 1. |θ| = 0. In this case Lemma 3 implies |x1|2L= | ˙x0|2Land we conclude that if (x1, z1)
is such that |x1|2
L = 0 and z1 = 0, then there is a unique geodesic, that is lightlike straight
line, connecting (x1, z1) with (0, 0).
Case 2. v2 = 0, |θ| 6= 0. Therefore, v1 = 0. In this case | ˙x0|2L = 0 and the possible
geodesic is lightlike that remains lightlike for all t ∈ [0, ±∞]. Let us write the equations for these geodesics taking into account the condition on the initial velocity ˙x0(0) = ± ˙x1(0),
˙x2(0) = ˙x3(0) = 0. (5.5) If ˙x0(0) = ˙x1(0), then x0(t) = x1(t) = ˙x0(0) |θ| (sinh |θ|t + cosh |θ|t − 1), x2(t) = x3(t) = 0, (5.6) if ˙x0(0) = − ˙x1(0), then x0(t) = −x1(t) = ˙x0(0) |θ| (sinh |θ|t − cosh |θ|t + 1), x2(t) = x3(t) = 0. This is a non obvious parametrization of straight lines x(s) = ( ˙x0(0)s, ˙x1(0)s, 0, 0), ˙x20(0) =
˙x2
1(0). In both cases ˙˜zk(t) = ˙xT(t)Ikx(t) = 0, k = 1, 2, 3, that implies ˜zk(t) = zk(t) ≡ 0. We
conclude that the origin can be joined with a point (x0(1), x1(1), 0, 0), where x20(1) = x21(1) by
lightlike geodesic that is a straight line. Thus, case 2 is a particular situation of the case 1. Notice that the equations of the horizontal part (5.5) and (5.6) of geodesics in QLcoincides
up to reparametrization with the equations of the horizontal part for geodesics in H1Lprovided the initial velocity ˙x0(0) = ± ˙x1(0) (see [11]).
The results of cases 1 and 2 may be united in the following statement Theorem 1. Let A = (x1, z1) be a point such that |x1|2
L = 0, zk = 0, k = 1, 2, 3, and
x2
0(1) + x21(1) 6= 0. Then there is a unique lightlike geodesic joining the origin and A which is
Case 3. v2 6= 0, |θ| 6= 0. We write the expression (5.4) in the equivalent form (5.7) sin 2 |θ| 2 sinh2 |θ|2 = − v1 v2
and consider the following subcases.
3.1) Let v1 = 0 then | ˙x0|2 = v2 > 0 and in this case all geodesics are spacelike. Then (5.7)
implies that |θ| = 2πn, n ∈ N \ {0}.
3.1.1) Let us suppose that ˙x0(0) = ˙x1(0) = 0 and ˙x2(0), ˙x3(0) are arbitrary. The main result
is expressed in the following
Theorem 2. Given a point A = (0, z1) there are uncountably many spacelike geodesics con-necting the origin with A. The geodesics are given by equations
x0(t) = x1(t) ≡ 0, x22(t) + x23(t) = 2|z 1| πn sin 2(πnt)) z(t) = |z1|sin(2πnt) 2πn − t z1(1), z2(1), z3(1).
The geodesics have the lengths p2πn|z1|, n ∈ N \ {0}.
Proof. We start from the equations for x-coordinates of geodesics. Since x(t) = W ˙x0 we get
(5.8) x0(t) x1(t) x2(t) x3(t) = 1 2πn 0 0 ˙x2(0) sin(2πnt) + ˙x3(0)(cos(2πnt) − 1) ˙x2(0)(1 − cos(2πnt)) + ˙x3(0) sin(2πnt)
by (4.6). The projection onto (x2, x3)-plane is
(5.9) x22(t) + x23(t) = 4v2sin2(πnt)),
which by lemma (2), can be rewritten as the canonical equation of a circle on the (x2, x3)-plane
with the center at −˙x3(0)
2πn, ˙x2(0) 2πn of radius √v2 2πn: (5.10) x2+ ˙x3(0) 2πn 2 + x3− ˙x2(0) 2πn 2 = v2 (2πn)2.
The number n ∈ N \ {0} reflects the number of turns along the circle for any fixed initial velocity v2 and the radius of the circle. To exclude v2 from the equations we use (4.10) and
find
˜ z1(t) =
v2
4π2n2(sin(2πnt) − 2πnt), z˜2(t) ≡ 0, z˜3(t) ≡ 0.
Set t = 1 in the expression for ˜z1(t) and get v2= −2πn˜z1(1). Since z(t) = P ˜z(t) with P given
by (4.9), we get (5.11) z1= ˜z1(1)(−θ1 |θ| , θ2 p θ21+ θ22, − θ1θ3 |θ|pθ12+ θ22), |θ| = 2πn.
It implies that −˜z1(1) = |z1| and v2 = 2πn|z1|. Fixing n ∈ N \ {0}, we fix the speed v2 of a
geodesic, but we still have the choice in the directions of ( ˙x2(0), ˙x3(0)) that are parametrized
by the unit circle. It gives uncountably many geodesics.
We get the equations for z-coordinates from z(t) = P ˜z(t) taking into account that the values of θk are related to the values of z1 by (5.11).
Since the geodesics are spacelike the length l(γ) of a geodesic γ can be calculated from the formula l(γ) = Z 1 0 √v 2dt = p 2πn|z1|. Remark 1. Notice that in the Lorentzian Heisenberg group HL there is no geodesic of any
causal type joining (0, 0, 0) and (0, 0, z) with z 6= 0 (see [11]).
3.1.2) We suppose now that ˙x0(0) = ˙x1(0) 6= 0. We present first auxiliary calculations and
then formulate the main statement. We have a1 = a3= ˙x0(0)( ˙x2(0) + ˙x3(0)
, −a2 = a4= ˙x0(0)( ˙x2(0) − ˙x3(0)
. Then the equations of geodesics take the form
(5.12) x0(t) x1(t) x2(t) x3(t) = 1 2πn ˙x0(0)(e2πnt− 1) ˙x0(0)(e2πnt− 1) ˙x2(0) sin(2πnt) + ˙x3(0)(cos(2πnt) − 1) ˙x2(0)(1 − cos(2πnt)) + ˙x3(0) sin(2πnt)
The relations (4.10) lead to ˜ z1(t) = v2 (2πn)2(sin(2πnt) − 2πnt), (5.13) ˜ z2(t) = ˙x0(0) (2πn)2 h ˙x2(0) cos(2πnt) − 1 e2πnt+ 1+ sin(2πnt) e2πnt− 1) + ˙x03 cos(2πnt) + 1 e2πnt− 1− sin(2πnt) e2πnt+ 1)i, ˜ z3(t) = ˙x0(0) (2πn)2 h ˙x2(0) cos(2πnt) + 1 e2πnt− 1− sin(2πnt) e2πnt+ 1) − ˙x3(0)
cos(2πnt) − 1 e2πnt+ 1− sin(2πnt) e2πnt− 1)i, where n ∈ N \ {0}. Setting t = 1 in (5.12) we get
(5.14) ˙x0(0) = x0(1)
2πn e2πn− 1.
The equations (5.13) imply
(5.15) v2 = −˜z1(1)2πn, ˙x0(0) ˙x2(0) (2πn)2 = ˜ z3(1) 2(e2πn− 1), ˙x0(0) ˙x3(0) (2πn)2 = ˜ z2(1) 2(e2πn− 1).
Theorem 3. Given a point A = (x1, z1), |x1|
L = 0, x0(1) = x1(1) 6= 0, |z1| 6= 0, there are
uncountably many spacelike geodesics connecting the origin O = (0, 0) with A. The equations of geodesics are given by expressions
(5.16) x0(t) = x1(t) = x0(1) e2πnt− 1 e2πn− 1, x 2 2(t) + x23(t) = − 2˜z1(1) πn sin 2(πnt))
and z(t) = P ˜z(t), where ˜ z1(t) = ˜z1(1)(t − sin(2πnt) 2πn ), ˜ z2(t) = 1 2(e2πn− 1) ˜
z3(1) (cos(2πnt) − 1)(e2πnt+ 1) + sin(2πnt)(e2πnt− 1)
+˜z2(1) (cos(2πnt) + 1)(e2πnt− 1) − sin(2πnt)(e2πnt+ 1)
, ˜ z3(t) = 1 2(e2πn− 1) ˜
z3(1) (cos(2πnt) + 1)(e2πnt− 1) − sin(2πnt)(e2πnt+ 1)
−˜z2(1) (cos(2πnt) − 1)(e2πnt+ 1) + sin(2πnt)(e2πnt− 1),
˜
z1= P−1z1, and n ∈ N \ {0}.
Proof. Fix n ∈ N and choose any θ1, θ2, θ3, such thatqP2k=1|θ|k= 2πn. We have uncountably
many triples that are parameterized by the 3-d sphere. This choice defines the orthogonal transformation P given by (4.9). Given z1 we find the auxiliarry parameters ˜z1by ˜z1 = P−1z1. Setting the values of ˙x0(0), v2 and ˙x2(0), ˙x3(0) from (5.14) and (5.15) into the general
solutions (5.12) and (5.13) and applying the orthogonal transformation P to ˜z(t) we finish the
proof.
We observe that projections of the geodesics onto (x0, x1)-plane are straight lines x0(s) =
x1(s) = s. The projections onto (x2, x3)-plane are circles from Lemma 2.
3.1.3) For the case ˙x0(0) = − ˙x1(0) 6= 0, in the same way as above, we obtain the following.
Theorem 4. Given a point A = (x1, z1), |x1|
L= 0, x0(1) = −x1(1) 6= 0, |z1| 6= 0, there are
uncountably many spacelike geodesics connecting the origin O = (0, 0) with A. The equations of geodesics are given by expressions
x0(t) = −x1(t) = x0(1) · e−2πnt− 1 e−2πn− 1, x22(t) + x23(t) = −2˜z1(1) πn sin 2(πnt), and z(t) = P ˜z(t), where ˜ z1(t) = ˜z1(1)(t −sin(2πnt) 2πn ), ˜ z2(t) = 1 2(e−2πn+ 1)
−˜z3(1) (cos(2πnt) − 1)(e−2πnt− 1) − sin(2πnt)(e−2πnt+ 1)
+˜z2(1) (cos(2πnt) + 1)(e−2πnt+ 1) + sin(2πnt)(e−2πnt− 1)
, ˜ z3(t) = 1 2(e−2πn+ 1) ˜
z3(1) (cos(2πnt) + 1)(e−2πnt+ 1) + sin(2πnt)(e−2πnt− 1)
−˜z2(1) (1 − cos(2πnt))(e−2πnt− 1) + sin(2πnt)(e−2πnt+ 1).
˜
z1= P−1z1, and n ∈ N \ {0}.
3.2) Consider the case v1 > 0, i. e., the initial velocity ˙x0 is spacelike. It suggests that
the ratio −v1
v2 is negative, which contradicts (5.7). Hence, in this case there are no spacelike
geodesics with the property (5.1).
3.3) Now suppose v1 < 0. It implies that the ratio −vv12 is positive and here we have 3 more
3.3.1) −v1
v2 > 1. 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
sinh2|θ| 2
is a wave decaying exponentially in 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.
3.3.2) −v1
v2 = 1. Then sin
2 |θ|
2 = sinh 2 |θ|
2 , which is possible if and only if |θ| = 0. This case
corresponds to Case 1), when the initial velocity is lightlike. 3.3.3) 0 < −v1
v2 < 1. Let us investigate the quantity of Hamiltonian geodesics in this case.
Using (4.11) and the fact that ˙x0= W−1(1)x1 we calculate that ˜ z22(1) + ˜z23(1) = (−x 2 0(1) + x21(1))x20(1) 2|θ|2sinh2 |θ| 2 sin2 |θ|2
cosh |θ| − cos |θ| − sin |θ| sinh |θ|, (5.17) ˜ z1(1) = −x 2 0(1) + x21(1) 4|θ|2sinh2 |θ| 2 sin2 |θ|2 sin2|θ| 2 (sinh |θ| − |θ|) − sinh 2|θ| 2 (sin |θ| − |θ|)
and notice that the ratio z˜
2 2(1)+˜z
2 3(1)
−2˜z1(1)x20(1) depends only on |θ| and
(5.18) z˜
2
2(1) + ˜z32(1)
−2˜z1(1)x20(1)
= cosh |θ| − cos |θ| − sin |θ| sinh |θ|
sin2 |θ|2 · (sinh |θ| − |θ|) − sinh2|θ|2 · (sin |θ| − |θ|). Lemma 4. The function
(5.19) µ(|θ|) = cosh |θ| − cos |θ| − sin |θ| sinh |θ| sin2 |θ|
2 · (sinh |θ| − |θ|) − sinh2 |θ|2 · (sin |θ| − |θ|)
is nonnegative, has countably many points |θ|k where µ(|θ|k) vanishes. The function µ has a
unique critical point mkon each interval |θ|k, |θ|k+1
, k = 0, 1, 2, . . . where it strictly increases from 0 to µ(mk), and then, strictly decreases from µ(mk) to 0.
Proof. Let us denote the numerator cosh |θ| − cos |θ| − sin |θ| sinh |θ| by f(|θ|) and study this function. Its derivative f′(|θ|) = 4 sin|θ|
2 sinh|θ|2 cos|θ|2 cosh|θ|2 (tan|θ|2 − tanh|θ|2 ) turns into 0 if
|θ| = 2πn, n = 0, 1, 2, . . . , or
(5.20) tan|θ|
2 = tanh |θ|
2 .
The latter equation has infinitely many solutions |θ|k, k = 0, 1, 2, . . ., where |θ|0 = 0 and all
other roots |θ|k∈ ((2k − 1)π, (2k + 1)π) and |θ|k>2kπ, k = 1, 2, . . .. Since
f (|θ|k) = 2 cosh2 |θ|k 2 − 2 cos 2 |θ|k 2 − 4 cos 2|θ|k 2 cosh 2|θ|k 2 tan |θ|k 2 tanh |θ|k 2 = 2 cosh2|θ|k 2 − 2 cos 2|θ|k 2 − 2 sin 2 |θ|k 2 cosh 2 |θ|k 2 − 2 sinh 2|θ|k 2 cos 2|θ|k 2 = 0
and f (2πn) = 2 sinh2(πn), n = 1, 2, . . ., we conclude that the function f has local minimums at |θ|k, local maximums at points 2πn, and it is non-negative for |θ| > 0, see Figure 1.
The denominator g(|θ|) = sin2 |θ|2 (sinh |θ| − |θ|) − sinh2 |θ|2 (sin |θ| − |θ|) is an increasing
func-tion, passing through 0 only at |θ| = 0, since the derivative g′(|θ|) = |θ|2(sinh |θ| − sin |θ|) +
4 sinh2|θ|2 sin2 |θ|2 is always positive and equals to 0 only at |θ| = 0. Hence, g is positive for |θ| > 0.
q 0 2 4 6 8 10 12 14 f 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000
Figure 1. The graph of function f (θ)
Since g 6= 0 for |θ| 6= 0 and since f(|θ|k) = 0, k = 1, 2, . . . the function µ = fg equals 0 only
at points |θ|k, k = 1, 2, . . . and is positive at all other points |θ| 6= |θ|k and |θ| 6= 0. The point
|θ|0 = 0 needs additional consideration. Using the Taylor expansion of the function µ near
|θ|0 we get that µ(|θ|) = 15|θ| + o(|θ|2), which goes to 0 if |θ| → 0. Therefore, µ(0) = 0. Let
us consider the derivative µ′(|θ|) = f′(|θ|)g(|θ|)−f(|θ|)g′(|θ|)
g2
(|θ|) . We already know that |θ|k are the
solutions of µ′(|θ|) = 0, since f(|θ|k) = f′(|θ|k) = 0, therefore, they are local extremal points
of function µ. Since µ(|θ|) is nonnegative for |θ| > 0 and µ(|θ|k) = f (|θ||θ|kk) = 0, we conclude
that the points |θ|kare local minimums of function µ.
Moreover, since µ is a smooth function on (0, ∞) it reaches its maximums on each interval |θ|k, |θ|k+1
, k = 0, 1, 2, . . ., which we denote by mk.
Let us make one more observation. Since g is a monotonically increasing function it has its local minimums at points 2πk on each interval [2πk, 2π(k + 1)], k = 1, 2, . . .. Then, taking into account that f attains local maximums at points 2πn, n = 1, 2, . . ., we can estimate the function µ on each of these intervals from above by the value f (2π(k+1))g(2πk) = 2 sinh2(π(k+1))
πk·2 sinh2
(πk),
k = 1, 2, . . .. From here we obtain that mk 6 2 sinh
2
(π(k+1)) πk·2 sinh2
(πk) → 0 with k → ∞. Therefore, the
function µ(|θ|) decays to 0 when |θ| goes to infinity. Note, that we can estimate m0from above
by f (2π)g(π) = 2 sinh π+π cosh π−3π4 sinh2π .
Figure 2 shows the graph of function µ.
This lemma allows us to prove the following theorem.
Theorem 5. Let A = (x1, z1) be a point such that |x1|L2 = 0, x21(1) < x20(1) and m0 is a global
maximum of µ. Suppose that z
2 2(1)+z 2 3(1) −2z1(1)x20(1)
< m0 and that |θ| is a solution of the equation
(5.21) z2 2(1) + z23(1) −2z1(1)x20(1) = µ(|θ|).
q 0 10 20 30 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 2. The graph of function µ(θ)
The equations of geodesics are given by −x20(t) + x21(t) = −x 2 0(1) + x21(1) sinh2|θ|2 sinh 2 |θ|t 2 , (5.22) x22(t) + x23(t) = x 2 2(1) + x23(1) sin2 |θ| 2 sin2 |θ|t 2 and z1(t) = −x 2 0(1) + x21(1) 4 sinh2 |θ|2 sinh(|θ|t) − |θ|t +x 2 2(1) + x23(1) 4 sin2 |θ|2 sin(|θ|t) − |θ|t , z22(t) + z32(t) = x 2 2(1) + x23(1)
8 sin2 |θ|2 sinh2 |θ|2 cosh(|θ|t) − cos(|θ|t) − sinh(|θ|t) sin(|θ|t) × (x20(1) + x21(1)) cosh(|θ|t − |θ|) + 2x0(1)x1(1) sinh(|θ|t − |θ|) − 2x20(1) + 2x21(1) , (5.23) If z22(1)+z 2 3(1) −2z1(1)x20(1)
> m0, then there are no geodesics of any causal type joining 0 and A.
Proof. Given coordinates (x1, z1) of the end point we find |θ| as a solution to the equa-tion (5.21). For any of these soluequa-tions the expression W−1(t)x(t) = ˙x(0) for t = 1 leads to ˙x0(0) = |θ| 2 x0(1) + x1(1) e|θ|− 1 + −x0(1) + x1(1) e−|θ|− 1 , ˙x1(0) = |θ| 2 x0(1) + x1(1) e|θ|− 1 − −x0(1) + x1(1) e−|θ|− 1 , (5.24) ˙x2(0) = |θ| 2 x2(1) − ix3(1) e−i|θ|− 1 + x2(1) + ix3(1) ei|θ|− 1 , ˙x3(0) = −i|θ| 2 x2(1) − ix3(1) e−i|θ|− 1 − x2(1) + ix3(1) ei|θ|− 1 .
From here we calculate v1= − ˙x20+ ˙x21 = −|θ| 2 − x2 0(1) + x21(1) 4 sinh2 |θ|2 , (5.25) v2 = ˙x22+ ˙x23 = |θ| 2 x2 2(1) + x23(1) 4 sin2 |θ|2 , ˙x20+ ˙x21 = |θ| 2 16 sinh2 |θ|2 e−|θ|(x0(1) + x1(1))2+ e|θ|(−x0(1) + x1(1))2 , (5.26) ˙x0˙x1 = |θ| 2 16 sinh2 |θ|2 e−|θ|(x0(1) + x1(1))2− e|θ|(−x0(1) + x1(1))2 . (5.27)
We substitute the values of v1
|θ| and v2
|θ| by (5.25) in general equations and get (5.22).
Next, for each solution to (5.21) we find an orthogonal transformation (4.9) of z-space that fixes z1-coordinate, and therefore, leaves the expression z22(t) + z32(t) = ˜z22(t) + ˜z32(t) invariant.
Thus, using (4.11), we substitute there the necessary combinations by (5.25) and (5.26), and get (5.23).
We observe that the projections of geodesics onto the (x2, x3)-plane are circles centered
−˙x3(0) |θ| , ˙x0 2 |θ| of radius √v2
|θ| according to Lemma 2. The projections onto (x0, x1)-plane are
suitable brunches of hyperbolas passing through (0, 0), see Lemma 1. The parameters of the circles and hyperbolas can be rewritten in terms of the final point of a geodesic by making use of (5.1).
Notice that if z22(1) + z32(1) → 0, then quantity of solutions to (5.21) is growing and the values of solutions tend to 2πk. In this case, (5.17) shows that the part of the velocity v1 tends
to 0 and we come to a particular situation of Theorem 2.
Remark 2. Theorem 5 gives an answer about the existence of spacelike geodesics and allows us to estimate their cardinality. Unfortunately, the estimation is not complete since we use only the orthogonal transformations in z-space leaving invariant z1-coordinate. The difficulty
is purely technical, since it is complicated to find the relation between the coordinates of finite point and the values of |θ|. In the case of orthogonal transformations in z-space leaving invariant z1-coordinate the relation is expressed by equation (5.21).
Remark 3. Given a point A = (x1, z1) on the surface |x1|2L= 0, z1 6= 0, there are no timelike geodesics joining 0 to A.
5.2. Connectivity between (0, z1) and (x1, z1), where z1 = const. Let us use the hyper-spherical coordinates (r, ϕ, ξ1, ξ2): x0+ ix1 = reiξ1cos ϕ 2, x2+ ix3 = re iξ2 sinϕ 2 Then the horizontality conditions (2.8) become
˙z1 = − 1 2r 2 ˙ξ 1cos2 ϕ 2 + ˙ξ2sin 2 ϕ 2 , ˙z2 = 1 4r 2 ϕ sin(ξ˙ 1+ ξ2) + sin ϕ cos(ξ1+ ξ2)( ˙x2− ˙x1), ˙z3 = 1 4r 2 ϕ cos(ξ˙ 1+ ξ2) + sin ϕ sin(ξ1+ ξ2)( ˙x1− ˙x2).
Then, ˙z12+ ˙z22+ ˙z32= 1 4r 2 ϕ˙2 4 + ˙ξ 2 1cos2 ϕ 2 + ˙ξ 2 2sin2 ϕ 2 .
Theorem 6. A smooth curve c(t) is horizontal with constant z-coordinates if and only if c(t) is a straight line in a 4-dimensional affine subspace: c(t) = (α0t, . . . , α3t, z11, z12, z31) with
α0, . . . α3 ∈ R4 and α20+ . . . α23 6= 0.
Proof. Let c(t) be a horizontal curve with constant vertical components. Then the equation | ˙z|2= 14r2 ϕ˙42 + ˙ξ12cos2 ϕ2 + ˙ξ22sin2 ϕ2= 0 implies
˙ ϕ = 0, ˙ξ1cos ϕ 2 = 0, ˙ξ2sin ϕ 2 = 0.
The first equation says that ϕ = ϕ0 is constant. In the case ϕ0 6= π + 2πk and ϕ0 6= 2πn,
k, n ∈ Z, we see that ξ1 and ξ2 are constants ξ10 and ξ20. Therefore, the horizontal components
of the curve c(t) are of the form x0= t cos ξ10cos ϕ0 2 , x1= t cos ξ 0 2sin ϕ0 2 , x2 = t sin ξ 0 1cos ϕ0 2 , x3 = t sin ξ 0 2sin ϕ0 2 . If ϕ0 = π + 2πk, k ∈ Z, then x0= 0, x1 = ±t cos ξ20, x2 = 0, x3 = ±t sin ξ02, and if ϕ0 = 2πn, n ∈ Z, then x0= ±t cos ξ01, x1= 0, x2 = ±t sin ξ10, x3 = 0.
Conversely, let us assume that c(t) = (α0t, . . . , α3t, z11, z21, z31), where z11, . . . , z31 are some
con-stants. Set αt = (α0t, . . . , α3t). Observe, that (I1α, α) = (I2α, α) = (I3α, α) = 0 for any vector
α = (α0, . . . , α3). Then, ˙˜z1 = 0 = 1 2(I1(αt), ˙(αt)) = t 2(I1α, α), ˙˜z2 = 0 = 1 2(I2(αt), ˙(αt)) = t 2(I2α, α), ˙˜z3= 0 = 1 2(I3(αt), ˙(αt)) = t 2(I3α, α).
This implies that z-coordinates are constants.
6. Reachable sets by geodeics on Lorentzian Heisenberg group
In this section we would like to compare the results obtained for the H-type Quaternion group with Lorentzian metric with the 3-dimensional Lorentzian Heisenberg group H1L.
We remind that H1
L is a triple (R3, H, g), where R3 is equipped with the non-commutative
multiplication law (x0, x1, z) ◦ (x′0, x′1, z′) = x0+ x′0, x1+ x′1, z + z′+ 1 2(x1x ′ 0− x0x′1) , the subbundle H is a span of two left invariant vector fields X0 = ∂x∂0 +
1
2x1∂z∂ , X1 = ∂x∂0 −
1
2x0∂z∂, for which [X0, X1] = Z = ∂z∂, and g is a Lorentzian metric on H defined by
g(X0, X0) = −1, g(X1, X1) = 1, g(X0, X1) = 0.
In [11] authors investigated the connectivity by geodesics in H1
L. In particular, they obtained
Theorem 7. Let A = (x, y, z) be a point such that x > 0 (x < 0), −x2+ y2 < 0, 4|z| x2
−y2 < 1.
Then there is a unique future directed (past directed) geodesic, joining O = (0, 0, 0) with the point A. Let θ be a solution of the equation
4z −x2+ y2 =
|θ|/2
sinh2(|θ|/2) − coth(|θ|/2). Then the equations of timelike future directed geodesic γ : [0, 1] → H1L are
x(t) = sinh2(|θ| 2 t)(x(coth( |θ| 2 t) coth( |θ| 2 ) − 1) + y(coth( |θ| 2 t) − coth( |θ| 2 ))), y(t) = sinh2(|θ| 2 t)(y(coth( |θ| 2 t) coth( |θ| 2 ) − 1) + x(coth( |θ| 2 t) − coth( |θ| 2 ))), z(t) = z|θ|t − sinh(|θ|t) |θ| − sinh(|θ|) .
Moreover, the authors obtained the following result about the reachability by causal Hamil-tonian geodesics.
Theorem 8. Let us define the following sets
Rt= {−x2+ y2 < 0, 4|z|
x2− y2 < 1},
Rsp= {−x2+ y2> 0, 4|z|
−x2+ y21 < 1},
Rl= {−x2+ y2= 0, z = 0}.
Then there exists a unique geodesic connecting the point O = (0, 0, 0) with a point P that belongs to one of the sets Rt, Rsp or Rl. Particularly, if P ∈ Rt, then the geodesic is timelike,
if P ∈ Rsp, then the geodesic is spacelike, and if P ∈ Rl, then the geodesic is lightlike.
The connectivity on H1L depends in the solutions |θ| of the equation 4z
−x2+ y2 = µ(θ),
where the function µ(τ ) = sinhτ /22
(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 −x4z2+y2
> 1, i. e. if A ∈ (Rt∪ Rsp∪ Rl)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 {x24|z|
−y2 = 1, −x2+ y2 < 1} and { 4|z|
−x2
+y2 = 1, −x2 + y2 > 0}. We observe that in [6]
it was shown that it is possible to find non-Hamiltonian lightlike geodesics joining 0 with the points of these surfaces.
Thus, in contrast to QL, 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 H1L.
7. Acknowledgment
The final version of this paper was written while authors visited the National Center for Theoretical Sciences and National Tsing Hua University during December of 2009. They would like to express their profound gratitude to Professors Wen-Ching Li and Der Cheng Chang for their invitation and for the warm hospitality extended to them during their stay Taiwan. We
also thank the Norwegian Research Council for the financial support of Taiwan - Norway Joint Workshop on Geometric Analysis and Mathematical Physics that took place in the National Center for Theoretical Sciences, December 15-16, 2009 and that was very useful and fruitful for us.
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Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Nor-way
E-mail address: [email protected]
Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Nor-way
