114 research outputs found
The Lanczos potential for Weyl-candidate tensors exists only in four dimensions
We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd
does not generally exist for dimensions higher than four. The technique is
simply to assume the existence of such a potential in dimension n, and then
check the integrability conditions for the assumed system of differential
equations; if the integrability conditions yield another non-trivial
differential system for L_abc and W_abcd, then this system's integrability
conditions should be checked; and so on. When we find a non-trivial condition
involving only W_abcd and its derivatives, then clearly Weyl candidate tensors
failing to satisfy that condition cannot be written in terms of a Lanczos
potential L_abc.Comment: 11 pages, LaTeX, Heavily revised April 200
On Effective Constraints for the Riemann-Lanczos System of Equations
There have been conflicting points of view concerning the Riemann--Lanczos
problem in 3 and 4 dimensions. Using direct differentiation on the defining
partial differential equations, Massa and Pagani (in 4 dimensions) and Edgar
(in dimensions n > 2) have argued that there are effective constraints so that
not all Riemann tensors can have Lanczos potentials; using Cartan's criteria of
integrability of ideals of differential forms Bampi and Caviglia have argued
that there are no such constraints in dimensions n < 5, and that, in these
dimensions, all Riemann tensors can have Lanczos potentials. In this paper we
give a simple direct derivation of a constraint equation, confirm explicitly
that known exact solutions of the Riemann-Lanczos problem satisfy it, and argue
that the Bampi and Caviglia conclusion must therefore be flawed. In support of
this, we refer to the recent work of Dolan and Gerber on the three dimensional
problem; by a method closely related to that of Bampi and Caviglia, they have
found an 'internal identity' which we demonstrate is precisely the three
dimensional version of the effective constraint originally found by Massa and
Pagani, and Edgar.Comment: 9pages, Te
A local potential for the Weyl tensor in all dimensions
In all dimensions and arbitrary signature, we demonstrate the existence of a
new local potential -- a double (2,3)-form -- for the Weyl curvature tensor,
and more generally for all tensors with the symmetry properties of the Weyl
curvature tensor. The classical four-dimensional Lanczos potential for a Weyl
tensor -- a double (2,1)-form -- is proven to be a particular case of the new
potential: its double dual.Comment: 7 pages; Late
Generalized hydrodynamics and extended irreversible thermodynamics
The thermodynamic implications of the first deviations with respect to the classical hydrodynamic behavior in high-frequency, short-wavelength phenomena are examined. The constitutive equations arising from an extended irreversible-thermodynamic formalism taking into account spatial inhomogeneities in the space of state variables are compared with those used in generalized hydrodynamics. The so-called exponential model for the memory function of the transverse-velocity correlation function is derived under the assumptions of extended irreversible thermodynamics only. Furthermore, it is also shown how more complicated memory functions can be derived. The results are carefully analyzed and compared with some microscopic derivations
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