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Spin-two fields and general covariance Heiderich, Karen Rachel


It has long been presumed that any consistent nonlinear theory of a spin-two field must be generally covariant. Using Wald's consistency criteria, we exhibit classes of nonlinear theories of a spin-two field that do not have general covariance. We consider four alternative formulations of the spin-two equations. As a first example, we consider a conformally invariant theory of a spin-two field coupled to a scalar field. In the next two cases, the usual symmetric rank-two tensor field, γab, is chosen as the potential. In the fourth case, a traceless symmetric rank-two tensor field is used as the potential. We find that consistent nonlinear generalization of these different formulations leads to theories of a spin-two field that are not generally covariant. In particular, we find types of theories which, when interpreted in terms of a metric, are invariant under the infinitesimal gauge transformation γab→γab + ∇ (a∇[symbol omitted]K[symbol omitted]), where Kab is an arbitrary two-form field. In addition, we find classes of theories that are conformally invariant. As a related problem, we compare the types of theories obtained from the nonlinear extension of a divergence- and curl-free vector field when it is described in terms of two of its equivalent formulations. We find that nonlinear extension of the theory is quite different in each case. Moreover, the resulting types of nonlinear theories may not necessarily be equivalent. A similar analysis is carried out for three-dimensional electromagnetism.

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