Using Tangent Boost along a Worldline and Its Associated Matrix in the Lie Algebra of the Lorentz Group

In order to generalize the relativistic notion of boost to the case of non inertial particles and to general relativity, we look closer into the definition of the Lie group of Lorentz matrices and its Lie algebra and we study how this group acts on the Minskowski space. We thus define the notion of tangent boost along a worldline. This very general notion gives a useful tool both in special relativity (for non inertial particles or/and for non rectilinear coordinates) and in general relativity. We also introduce a matrix of the Lie algebra which, together with the tangent boost, gives the whole dynamical description of the considered system (acceleration and Thomas rotation). After studying the properties of Lie algebra matrices and their reduced forms, we show that the Lie group of special Lorentz matrices has four one-parameter subgroups. These tools lead us to introduce the Thomas rotation in a quite general way. At the end of the paper, we present some examples using these tools and we consider the case of an electron rotating on a circular orbit around an atom nucleus. We then discuss the twin paradox and we show that when the one who made a journey into space in a high-speed rocket returns home, he is not only younger than the twin who stayed on Earth but he is also disorientated because his gyroscope has turned with respect to earth referential frame.


Introduction
In the frame of special relativity theory, the history of an inertial particle is − metric. This geodesic is a timelike straight line and its orthogonal complement is the physical space of the particle formed by all its simultaneous events. The passage from one inertial particle to another one is done through a special Lorentz matrix, which is called the boost, and this process is the Lorentz-Poincar?? transformation. But for a non inertial particle, all this is lost since its worldline is no more a straight line and there is no Lorentz transformation and boost associated to it. In order to fill this gap we suggest a deeper insight into the action of the Lie group of Lorentz matrices (and its Lie algebra) on the Minkowski space. This leads us to a new definition of a tangent boost along a worldline. This notion may be used in both situations of special or general relativity theories. Therefore we introduce a matrix belonging to the Lie algebra, which, together with the tangent boost, describes completely the dynamical system: acceleration and instantaneous Thomas rotation.
In a first part, we present properties of Lie matrices and of their reduced forms and we show that the Lie group of special and orthochronous Lorentz matrices has four one-parameter subgroups. These tools permit to introduce the Thomas rotation in a quite general way. Then, we give some applications of these tools: we first consider the case of an uniformly accelerated system and the one of an electron rotating on a circular orbit around the atom nucleus. We then present the case of the so-called "Langevin's twins" and we show that, when the twin who made a journey into space returns home, he is not only younger than the twin who stayed on Earth but he is also disorientated with respect to the terrestial frame because his gyroscope has turned with respect to the earth referential frame [1].
Let us underline that this formalism can be used both in Special and in General Relativity.

The Lie Algebra of a Lie Group
A Lie group is a smooth manifold with a compatible group structure, which means that the product and inverse operations are smooth. The Lie algebra of this Lie group can be seen as the tangent space e T to the manifold at the unit element e of the group multiplication. This tangent space is a vector space endowed with the Lie bracket of two tangent vectors.

SO 3 and Its Lie Algebra
Let's start with the group of 3 3 × -matrices having 1 + determinant. As a smooth manifold, it can be regarded as a 3-dimensional submanifold of 9  defined by the 6 equations resulting from the orthogonal matrix definition: Let's denote it, as usual, by ( ) 3 SO .
Its Lie algebra  is the 3-dimensional vector space of skew-symmetric matrices endowed with the bracket [ ] , Ω Ω = Ω ⋅ Ω − Ω ⋅ Ω This manifold is obviously isomorphic to the euclidean space 3  endowed with the cross product. The vectors of our Lie algebra should be regarded as The left translation on the group by Derivating the relation T AA I = gives T 0 Ω + Ω = , which means that Ω is skew-symmetric. Of course it would be possible to obtain the same I T through right translation by Application to kinematics of rotation of a rigid body around a fixed point.  into an open subset of ( ) 3

SO
, and we thus obtain an interesting parametrization of ( ) writing ϖ the norm of ω and ( ) Such a formula has an obvious geometrical meaning: exp Ω is a rotation through the angle ϖ about the axis having the direction of the vector ω . Note that, Ω being independant of t, the function defines a one-parameter subgroup of ( ) 3

SO
, and that the matrix product is the solution of the linear differential equation

The Lie Group of Lorentz Matrices Application to
Special Relativity

Preliminaries
In special relativity the motion of an inertial particle with respect to an inertial observer is described by a Lorentz-Poincar transformation. This transformation is associated to a 4 4 × -matrix belonging to the subgroup of orthochronous Lorentz matrices of 1 + determinant: they map the subset of timelike future oriented vectors into itself. They transform a η-orthonormal basis (associated to the inertial observer) into another η-orthonormal basis (associated to the particle).
The columns of such a matrix have a clear physical and geometrical interpretation: the first column is the 4-velocity of the particle (a unitary timelike 4-vector tangent to the worldline), and the three other columns define an orthonormal basis of the physical space of the particle. We turn now to the more general situation of a non inertial particle: the relative motion between two non inertial particles, or between a non inertial particle and another (inertial or non inertial) observer will be described by a time-dependent function with values in the group of Lorentz transformations. We thus naturally come to the notion of tangent boost along a worldline, we shall now study its main properties.

The Lie Group of Lorentz Matrices and Its Associated Lie Algebra
The shall denote by  the subgroup of the Lie group of Lorentz matrices consisting of all orthochronous (Lorentz) matrices with 1 + determinant. It is a 6-dimensional submanifold of 16  as defined by the 10 equations involving the 16 coordinates ij s of the matrix S, obtained from the relation T S S η η = .
This group  acts as a group of isometries on the Minkowski space ( ) We shall now be interested in the tangent space taken at the identity matrix I. η Ω =  is skew-symmetric (this is obtained by derivating the relation T S S η ), and so the element Λ of the Lie algebra can be rewritten: As a conclusion to this subsection, the Lie algebra  is the linear space of η Ω matrices, where Ω is a skew-symmetric covariant tensor of type ( ) 0, 2 .
The skew-symmetric tensor associated to the Lie bracket [ ] The exponential mapping from the Lie algebra to the group

Properties of the Lie Algebra Matrices
Every matrix belonging to the Lie algebra  can be written ( ) with respect to which Λ can be written: where α and ω are two real numbers and where 0 E is a timelike 4-vector, the three other 4-vectors i E are spacelike.
We shall use following notations: , , can be associated, which obviously is space like. Note that this process leeds us to the definition of a linear mapping q.
can be written 0 0 V v e qV = + . The inner product of two 4-vectors U and V shall be written U, V , and we have the formula: With the aim of more elegant computations we shall write C the cross product B A × , and , , a b c the euclidean norms of the 3-vectors , A B and C respectively. The following formulas will be useful: With the aim of studying the action of Λ on 4-vectors, we write be any 4-vector, we get the following formulas for the matrix products: To obtain the reduced form of Λ lets start with the study of 2 Λ . Indeed its where 2 α and 2 ω − are the two zeros of ( ) m P X . We wrote I and O for the identity and the zero 4 4 × -matrices. Note by the way the formulas linking the roots of the polynomial (11)): The first columns of the matrices ω ω The relation (12) means that the columns of the matrix ( ) , W W is an η-orthogonal basis of α Π . Indeed, on the one hand: shows that 2 W belongs to α Π and on the other hand, using (n3) to compute 2 W we get: Apart from the relation we also have: which means that 1 W is timelike and that 2 W is spacelike. Also note the relation 2 Writing now 0 , E E defines an orthonormal basis of the space like plane α Π with: The eigenspace II ω associated to the eigenvalue −ω 2 : the vector defined by the first column of ( ) Here is an outline of the computations: we also have: The plane ω Π is obviously spacelike, and writing: , , E E E are spacelike; they define an orthonormal basis of As above, normalizing the four vectors and writing them ( ) respectively, the reduced form of the matrix Λ in this new basis shall be the first matrix of (8) Π is timelike since it is generated by the two vectors belonging to the kernel of Λ : As above, normalizing the four vectors (the first one being timelike) and writing them ( ) , , , E E E E , the reduced form of Λ in this new basis shall be the second matrix of (8) form an η-orthonormal basis and ( ) generate an eigenspace 0 Π with the relations: We thus obtain the third reduced form in (8). Corollary: The Lie group of special and orthochronous Lorentz matrices has four oneparameter subgroups which can be obtained by integrating the linear differential where Λ is one of the four reduced forms obtained above.

Inertial Particles in Special Relativity
Let O and M be two inertial particles in the Minkowski space ( ) ( ) is the Lorentz factor. All these quantities are constants.
In order to define the Lorentz-Poincar transform we may apply the orthonormalization Gram-Schmidt process to the basis ( ) between the two inertial particles: In this result, V is the column matrix of its components and 3 I is the unit matrix of size 3.
L being a constant matrix, its associated matrix in the Lie algebra is the zero matrix. All this corresponds to the classical case of Special Relativity and can be summarized as follows: Any constant matrix L ∈  defines a Lorentz transform relating two inertial particles. Remarks: 1. The relation between O and M can be characterized by an infinity of Lorentz matrices. Each of them can be deduced from L by a left or a right multiplication of L with a pure rotation (a Lorentz matrix) R To summarize: there is a basis e′ deduced from e through a space rotation of e for which the boost L can be written in the following canonical form: With respect to e′ , ( ) , , M E E is the plane of the Lorentz transformation and ( ) 2 3 , , M E E is the invariant plane of that transformation.

Non Inertial Particles in Special Relativity. Tangent Boost along a Worldline
Let us now consider the case where O is an inertial particle and where M is not.
Then, the wordline M  of M is no more a straight line and its 4-velocity V is a vector field along M  . This leeds us to define the tangent boost along M  as being the boost of the inertial particle M ′ which coincides with M and the worldine of which is the tangent straight line at M. We thus obtain a field give some examples of using this latter.

Derivation Rule of a Vector X Defined by Its Components in the Referential Frame of M
Let us consider the two basis where the subscripts e and E correspond to the basis e and E respectively. The above relation gives the derivative rule by its E-components that is the intrinsic vectorial relation: Let us now apply that law to the 4-velocity of M the components of which are ( ) Note that there is a minor abuse of notation in the last line: B and W must be understood here as 3-vectors and no more as components in E as in previous shows that B is an instantaneous rotation in the (physical) space of 3-vectors. It corresponds to Thomas rotation.
The matrix Λ thus contains the 4-acceleration of M and the Thomas rotation. It therefore undoubtedly constitutes a valuable tool to describe the motion of any physical system.

Example of an Uniformly Accelerated Particle
In The mere knowledge of V permits to calcule the tangent boost L. Inserting into Equation (20) we get: Let us then calculate its associated matrix Λ in the Lie algebra > , a is the norm of the 4-acceleration). Using the derivation rule, we obtain the components E A of the 4-acceleration in ( )

Tangent Boost of a Worldline and Its Associated Matrix in the Lie Algebra in Special Relativity
In and its associated matrix in the Lie algebra 1 d d x y y x z y y z y    In order to get a better insight on Thomas rotation, let us consider the We will see later that the Thomas rotation is a rotation of the rest frame of M with respect to the referential frame ( ) , M E which is defined by the tangent boost.

Application to a Particle in Circular Motion at Constant Velocity
With respect to the frame ( ) Noting that the Lorentz factor Γ is constant, the 4-acceleration is: where D V is the covariant derivative in the direction of V expressed in cylindrical coordinate. In order to calculate the tangent boost we have to express V in the η-orthonormal system ( ) . The tangent boost is then defined by using (20): Let us recall that the B -columns give the referential frame ( ) ( ) The matrix of the Lie algebra It directly gives the 3-acceleration and the instantaneous Thomas rotation (which both are in the physical space of M). Let us note that it is also possible to obtain the 3-vectors A and B of ( )

Discussion
In order to understand the meaning of Thomas rotation, let us consider a gyroscope and let us recall the definition of a gyroscopic torque along a worldline as given in [2] and in [3] [4]: A gyroscopic torque along a worldline  the 4-velocity and the proper time of which are V and τ respectively is a 4-vector G defined along  , orthogonal to V and such that its derivative with respect to τ is proportional to V, that is to say: These relations permit to calculate k. Noting A the 4-acceleration d dτ V , we in fact get: The proportionality condition implies that the 4-vector G (which belongs to the physical space of M along the worldline  ) rotates in that space. In fact, let us write the differential Equation (32)     We thus see that in the case of Langevin's twins, (here, in the case of a uniform circular motion), when the twin who made a journey into space returns home he is not only younger than the twin who stayed on Earth but he is also disorientated with respect to the terrestrial frame because his gyroscope has turned with respect to earth referential frame. This effect is illustrated in Figure 3 in the case of an electron rotating on a circular orbit around the atom nucleus.