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Summary: arXiv:gr-qc/000509923May2000
HERMITIAN EXTENSION OF THE FOUR-DIMENSIONAL
HOOKE'S LAW
S. ANTOCI
Abstract. It has been shown recently that the classical law of elas-
ticity, expressed in terms of the displacement three-vector and of the
symmetric deformation three-tensor, can be extended to the four di-
mensions of special and of general relativity with a physically meaning-
ful outcome. In fact, the resulting stress-momentum-energy tensor can
provide a unified account of both the elastic and the inertial properties
of uncharged matter. The extension of the displacement vector to the
four dimensions of spacetime allows a further possibility. If the real dis-
placement four-vector i is complemented with an imaginary part i,
the resulting complex "displacement" four-vector allows for a complex,
Hermitian generalisation of the four-dimensional Hooke's law.
Let the complex, Hermitian "stress-momentum-energy" tensor den-
sity Tik
built in this way be subjected to the "conservation condition"
Tik
;k = 0. It turns out that, while the real part of the latter equation
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