Consider space time to be an algebraic variety {x,y,z,t} upon which can be
constructed a matrix of functions that will serve as a basis for vectors at any
point p. This matrix of sixteen functions will be defined as a Frame Field, or
Repere Mobile in the language of E. Cartan.
First restrict the class of class of admissable functions to those matrices that
have nonzero determinant. Such a distinction creates what are called
projective transformations. As such, the large class of projective
transformations may be viewed as a mapping that takes differential elements
{dx,dy,dz,dt} into new differential elements. The nonzero determinantal
condition gives a global perspective to the domain of support for the Repere
Mobile.
The projective matrix functions can can be grouped into subsets of equivalence
classes by applying constraints or conditions that the matrices must satisfy.
For example, the class of transformations deemed interesting (or admissable)
might be constrained to be elements of some Lie Group, say SU2, etc. The study
of such arbitrary constraints are quite popular in the current physical
literature.
Herein, the interest will focus on those projective transformations that belong
to the equivalence class of Lorentz transformations. The Lorentz
transformations are defined as those maps such that if dx^2+dy^2+dz^2(d{Ct})^2
= 0 Note that C is not necessarily a global constant. In fact, the cases where C = C(x,y,z,t) are those interesting cases that define the Minkowski Fluid.
Recall, the projective transformations at a point consist of 16 functions of
(x,y,z,t), but the MinkowskiLorentz Eikonal constraint limits the functional
format of these functions to form the subset, or Lorentz equivalence class, of
admissable matrices.
It is possible to construct a Lorentz basis frame from six 4 x 4 matrix
generators, that are built on 7 independent parameters. The parameters
represent an expansion, three Lorentz rotations, and three Lorentz translations.
The Cartan methods can then be applied to this equivalence class of restrictions
on the 4D variety to compute the induced torsion and curvature coefficients on
the 3D {x,y,z} subspace.
Such is the study of a Minkowski Fluid. What is extraordinary is that the
Lorentz Translation has coeffients for the inverse Frame Matrix that correspond
to the generators of the RaylieghTaylor and the KelvinHelmoltz instability
patterns. See
For a discussion of the Cartan Method (using matrix rather than tensor methods),
see projfram.pdf
