Space-Time as a Minkowski Fluid

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 non-zero 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 non-zero 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
before the transformation, then the same must be true after the transformation. This zero set has been called the definition of a signal, but V. Fock has shown that the basic idea is that it represents a condition (the Eikonal condition) when Maxwell's equations can admit a propagating amplitude discontinuity.

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 Minkowski-Lorentz 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 Rayliegh-Taylor and the Kelvin-Helmoltz instability patterns. See
Wakes as topological limit sets

For a discussion of the Cartan Method (using matrix rather than tensor methods), see projfram.pdf
For a number of examples for various Lorentz maps, and a Maple program to do the (horrible) algebra for you,
see lorentz.pdf


Copyright © 1995-2009, CSDC Inc. All rights reserved.
Last update 01/23/2009
to HomePage