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.
In the frame of special relativity theory, the history of an inertial particle is described by a geodesic straight line in the four dimensional Minkowski space, endowed with the
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 [
Let us underline that this formalism can be used both in Special and in General Relativity.
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
Let’s start with the group of
Its Lie algebra
This manifold is obviously isomorphic to the euclidean space
tangent vectors
The left translation on the group by
equal to its differential
Derivating the relation
Application to kinematics of rotation of a rigid body around a fixed point.
Keeping in mind later comparisons, we shall apply the results of the previous section to the study of the motion of a rigid body. We want to show the interest of looking at the action of
Let us denote by
composing by the left translation
This relation expresses the derivation rule of the movement of a point X in the moving coordinate system
An analogous process starting from the right translation
There is another interesting application of the identification
is a diffeomorphism of the open ball
Such a formula has an obvious geometrical meaning:
with the initial condition
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
The columns of such a matrix have a clear physical and geometrical interpre- tation: 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 shall denote by
This group
Let
From the relation
As a conclusion to this subsection, the Lie algebra
The skew-symmetric tensor associated to the Lie bracket
The exponential mapping from the Lie algebra to the group
defines a diffeomorphism from
Every matrix belonging to the Lie algebra
where
Note the relation
We then have the following proposition about the reduced forms of the matrices: Given any matrix
where
If
Proof
We shall use following notations:
n1) Let
n2) To every 3-vector
Any 4-vector
n3) With the aim of more elegant computations we shall write C the cross product
n4) With the aim of studying the action of
Let
To obtain the reduced form of
where
Note by the way the formulas linking the roots of the polynomial (11)):
The first columns of the matrices
The relation (12) means that the columns of the matrix
The eigenspace IIα associated to the eigenvalue α2:
Writing
shows that
Apart from the relation
which means that
Writing now
The eigenspace IIω associated to the eigenvalue −ω2:
Writing
Apart from the relation
The plane
These two relations (19), with the former relations (18) linking
The situation where A and B are orthogonal: In the case
1) Assume
and
As above, normalizing the four vectors and writing them
2) Assume
The plane
As above, normalizing the four vectors (the first one being timelike) and writing them
3) Assume
Noting
We thus obtain the third reduced form in (8).
Corollary:
The Lie group of special and orthochronous Lorentz matrices has four one- parameter subgroups which can be obtained by integrating the linear differential equation
where
The solution of this equation is
Let O and M be two inertial particles in the Minkowski space
Let
with t he relations:
where
In order to define the Lorentz-Poincar transform we may apply the orthonormalization Gram-Schmidt process to the basis
In this result,
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
where A is an orthogonal matrix of size 3. A left and a right multiplication correspond to a change of basis in the rest space of O and of M respectively.
2. The writing of the boost (20) can be simplified by choosing an appropriate basis of
We can define an h-orthonormal basis
We also know that the two dimensional orthogonal complement is L- invariant. This can be seen by noting that the two 4-vectors
These two vectors are eigenvectors of L associated to the double eigenvalue 1. We thus obtain a new h-orthonormal basis
Noting
the above expression can also be written
To summarize: there is a basis
With respect to
Let us now consider the case where O is an inertial particle and where M is not. Then, the wordline
matrix, its associated matrix in the Lie algebra
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
Equation (23) shows that the first column of
Now, let
Changing
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 lines. The term
The matrix
In the referential frame
where:
The mere knowledge of
Let us then calculate its associated matrix
Using (25) in computing
where
Its components
Calculating
and the 4-acceleration is uniform in the rest frame
Let us now consider two nearby particles N and M, N being at rest with respect to M and their coordinates in
Knowing that X does not depend on
The components of
where
the proper time of N is not the same as the one of M. In fact, the norm
This shows that in the case of a non inertial motion of M, it is impossible to synchronize the clocks in the rest frame of M.
Let us add that N has not the same acceleration as M. In fact, knowing that
In the referential frame of O, the parametric equations of the worldline
Noting
The tangent boost (20) is:
and its associated matrix in the Lie algebra
To summarize: using notations (6) we see that
• the 3-vector A is the acceleration of M in its rest frame
• the 3-vector B gives the instantaneous Thomas rotation by its components in
The rotating basis
1. Using the definition of
and applying the derivation rule to the tangent boost
where
We thus obtain
where
2. Using (29) and (30) which give the 3-vectors A and B from
Using
and Equation (30) gives the Thomas rotation
To conclude: from the
These calculations show that we must clearly distinguish between the instantaneous rotation of
In order to get a better insight on Thomas rotation, let us consider the infinitesimal Lorentz matrix relating
A left-multiplication by
At first order,
We will see later that the Thomas rotation is a rotation of the rest frame of M with respect to the referential frame
With respect to the frame
where
Noting that the Lorentz factor
where
Let us recall that the
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
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 [
A gyroscopic torque along a worldline
These relations permit to calculate k. Noting
The proportionality condition implies that the 4-vector
In the inertial referential frame, Equation (32) thus becomes:
Using the covariant derivative in cylindrical coordinates and noting derivatives with respect to
Identifying this result with
Taking initial conditions
say when M goes a 180 degree turn. It shows that in that case, the gyroscope indicates a half turn plus a rotation which corresponds to the Thomas rotation (in the clockwise direction). The gyroscope rotation in the plane
It is also possible to highlight the Thomas rotation by applying the derivation rule (23) to
and using (23) in that moving frame:
we get
Using then
we obtain:
The left hand side of this equation is the Fermi-Walker derivative of
Consequently, the gyroscope rotates with respect to
It can be noted that the solution of (33) (with the initial condition
The main results of every dynamical system are contained in the tangent boost L (which gives its 4-velocity
The age of the electron with respect to the atom nucleus is then obtained by integrating
The gyroscope rotation
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
Langlois, M., Meyer, M. and Vigoureux, J.-M. (2017) Using Tangent Boost along a Worldline and Its Associated Matrix in the Lie Algebra of the Lorentz Group. Journal of Modern Physics, 8, 1190-1212. https://doi.org/10.4236/jmp.2017.88079