* SO(3,1): Lorentz Transformations -*- outline -*-
* _SO(3,1): Lorentz Transformations_ -*- outline -*-
The _SO31.plt_ package provides procedures for manipulating elements
of the _group_ SO(3,1) (the group of Lorentz transformations in four
You can obtain access to the functions exported by this package by
(require (planet "SO31.ss" ("wmfarr" "SO31.plt" 1 1)))
The package comes with _SchemeUnit_ _unit tests_, which you can run by
(require (planet "run-tests.ss" ("wmfarr" "SO31.plt" 1 1)))
Currently the package will only run on macosx-ppc, but it should be
easy to port to any system with the _BLAS_ and _LAPACK_.
This package was developed and is maintained by Will M. Farr
<email@example.com>; please direct bug reports and feature requests to that
address. SO31.plt is released under the GPL---see the License section
of this document for more information.
** SO(3,1): The Lorentz Group
SO(3,1) is a six-dimensional group. Its canonical representation is
the set of four by four matrices which preserve the _Minkowski_
M^T * eta * M = eta,
where eta = diag(-1,1,1,1). These matrices implement _boosts_ and
_rotations_ on four-vectors. In this package, we parameterize the
group by the map from three boost parameters, vx, vy, and vz and three
rotation parameters x, y, z to the matrix M, representing a group
M = Bx(vx)*By(vy)*Bz(vz)*Rx(x)*Ry(y)*Rz(z),
where the Bi are "boost matrices", for example
cosh(vx) -sinh(vx) 0 0
-sinh(vx) cosh(vx) 0 0
0 0 1 0
0 0 0 1
and the Ri are rotation matrices, for example
1 0 0 0
0 1 0 0
0 0 cos(x) -sin(x)
0 0 sin(x) cos(x)
Note that, under this definition, the v parameters are the _rapidity_
in each dimension, and the x,y,z parameters are angles of _rotation_
about the corresponding axis. For any single non-zero parameter, the
parameterization of SO(3,1) is additive: Bx(vx2)*Bx(vx1) =
Bx(vx1+vx2), etc. If more than one parameter is non-zero, then the
situation is more complicated due to the _non-abelian_ nature of
The code doesn't make any effort to ensure that this representation is
single-valued. It works for Lorentz transforms "close enough" to the
identity, but will fail (spectacularly, since SO(3) isn't
simply-connected) for matrices which depart from the identity too
In code, the parameters of an element of SO(3,1) will be represented
by a 6-element _f64vector_, and an element will be represented by a 4
by 4 _matrix_ from the _PLT-Linalg_ library:
> (SO31-params? obj) : #f unless obj is a 6-element f64vector.
> (SO31-matrix? obj) : #f unless obj is a 4 by 4 matrix (but not
necessarily one which preserves eta---this predicate is not
** Boost and Rotation Matrices
The code provides functions which consume a number and produce a boost
or rotation matrix:
> (Rx x)
> (Ry y)
> (Rz z)
> (Bx vx)
> (By vy)
> (Bz vz)
** Matrix<->Parameter Conversion
> (matrix->params M) : Takes an element of SO(3,1) and returns the
parameters for that element.
> (params->matrix p) : Takes parameters for an element of SO(3,1) and
returns the matrix representing that element.
** Group Operations
> (params->inverse-params p) : Returns the parameters for the group
inverse of the element described by p.
> (params-compose p1 p2) : Returns the params for the composition of
the elements described by p1 and p2.
The analog of these two functions on matrices are already provided by
the _plt-linalg.plt_ package: _matrix-inverse_ and _matrix-mul_.
** Derivatives of SO(3,1) Elements
The _derivative_ of an SO(3,1) element is a length-6 scheme vector
containing the matrices which are the derivative of the matrix
representation of that element with respect to #(vx, vy, vz, x, y, z),
> (params->dM p) : Derivative of the SO(3,1) element specified by p
wrt vx, vy, vz, x, y, z.
> (matrix->dM M) : Derivative of the matrix M wrt vx, vy, vz, x, y, z.
As an example, the following code produces a linearized approximation
to the matrix at incremented parametrs,
(params->matrix (f64vector-add p0 dps)):
(let ((dMs (params->dM p0)))
(fold-ec (params->matrix p0) (:parallel (:vector dM dMs)
(:vector dp dps))
(matrix-scale dM dp)
(lambda (M-inc M-acc)
(matrix-sum M-inc M-acc))))
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