TITLE:
Cauchy, Cosserat, Clausius, Einstein, Maxwell, Weyl Equations Revisited
AUTHORS:
J.-F. Pommaret
KEYWORDS:
Janet Sequence, Spencer Sequence, Poincaré Sequence, Gauge Sequence, Lie Group of Transformations, Lie Pseudogroup of Transformations, Conformal Group of Transformations, Adjoint Representation
JOURNAL NAME:
Journal of Modern Physics,
Vol.15 No.13,
December
23,
2024
ABSTRACT: The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are
n,n
(
n−1
)/2
,1,n
with a total of
N=(
n+1
)
(
n+2
)/2
, that is 15 when
n=4
for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations,
n
(
n−1
)/2
rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary
n≥1
because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when
n=4
, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through
Nn
(
n−1
)/2
potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.