Super-energy tensors

Abstract

A simple and purely algebraic construction of super-energy(s-e) tensors for arbitrary fields is presented in any dimensions.These tensors have good mathematical and physical properties,andthey can be used in any theory having as basic arena an n-dimensionalmanifoldwith a metric of Lorentzian signature.In general, the completely timelike component of these s-e tensors has themathematical features of an energy density: they are positivedefinite and satisfy the dominant property, which provides s-eestimates useful for global results and helpful in other matters, such asthecausal propagation of the fields. Similarly, super-momentumvectors appear with the mathematical properties of s-e flux vectors.The classical Bel and Bel-Robinson tensors for the gravitational fields areincluded in our general definition. The energy-momentum and super-energytensorsof physical fields are also obtained, and the procedure will be illustratedbywriting down these tensors explicitly for the cases of scalar,electromagnetic,and Proca fields. Moreover, higher order (super)k-energytensors are defined and shown to be meaningful and in agreement for thedifferent physical fields. In flat spacetimes, they provide infinitely manyconserved quantities.In non-flat spacetimes, the fundamental question of the interchangeofs-e quantities between different fields is addressed, and answeredaffirmatively. Conserved s-e currents are found for any minimallycoupled scalar field whenever there is a Killing vector. Furthermore, theexchange of gravitational and electromagnetic super-energy is also shown bystudying the propagation of discontinuities. This seems to open the doorfornew types of conservation physical laws.