CASIMIR ENERGY FOR A 3-PIECE RELATIVISTIC STRING

The Casimir energy for the transverse oscillations of a piecewise unif orm closed string is calculated. The string consists of three pieces I , II, III of equal length, endowed with different tensions and mass de nsities, but adjusted in such a way that the velocity of sound always equals the velocity of Light. In this sense the string forms a relativ istic mechanical system. In the present paper the string is subjected to the following analysis: the dispersion function is derived and the zero-point energy is regularized using (i) a contour integration techn ique, being most convenient for the generalization of the theory to th e case of finite temperatures, and (ii) the Hurwitz zeta-function tech nique, being usually the most compact method when the purpose is to ca lculate the Casimir energy numerically at T=0. The energy, being alway s nonpositive, is shown graphically in some cases as functions of the tension ratios I/II and II/III. The generalization to finite temperatu re theory is also given.