Thickening the string. I. The string perfect dust

Abstract

The classical theory of the geometrical string is developed as the theory of a simple, surface-forming timelike bivector field in an arbitrary background space-time. The stress-energy tensor for a perfect dust o f such strings is written down, and the conservation laws for such a dust, as well as the equations of motion of the string, are derived from the vanishing of the divergence of the stress-energy tensor. (The boundary conditions for the open string are also derived from the junction conditions for the stress-energy tensor in Appendix A.) The generalization of this model to null strings, and to a perfect fluid of strings, are discussed, and will form the subject o f the second and third papers in this series. The problem of a fully general-relativistic string theory, and an alternate approach to the string, based upon defining an acceleration tensor for two-(and higher) dimensional subspaces, are also discussed. I . INTRODUCTIONIn recent y e a r s there has been extensive discussion of the theory of the geometrical string, primarily because of i t s application a s a possible interpretation of the dual resonance model of elementary particles.' However, the theory has also been discussed on the classical level2; and, indeed, f r o m this point of view it provides a r emarkably natural generalization of the relativistic theory of a structureless point particle. This generalization could have been made a t any time after 1905, and it is r a t h e r surprising that such a natural structure should not have been investigated much earlier. This paper will be exclusively concerned with the classical theory of the string; the hope of generalizing special-relativistic string theory into a fully general-relativistic one provides i t s ultimate m~t i v a t i o n .~The geometrical string will b e treated a s the theory of a surface-forming simple bivector field, subject to field equations which determine the s u rface. This bivector field i s usually treated mathematically by introducing a parametrization of the surface, and a pair of linearly independent vectors, derived f r o m the parametrization, which span the surface. While t h e r e i s nothing wrong with this procedure, it may tend to make one lose sight of the fact that the resulting theory must b e invariant under the family of possible reparametrizations of the surface which leave the bivector unaltered. However, it is equally simple to t r e a t the bivector field intrinsically, a s we shall demons t r a t e , and perhaps more natural t o do so, in the sense that it is m o r e natural to discuss any geometrical figure intrinsically r a t h e r than via i t s components with respect to some basis.The next section will review some well-known mathematical results on simple surface-forming bivector fields that will be needed in the following In Sec. 111, we shall review the "thickening" of a point particle provided by the incoherent dust model of matter, and r e c a l l how the equations of motion and conservation law for the dust may b e derived f r o m the van...

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