Gauge Fixing in the Maxwell Like Gravitational Theory in Minkowski Spacetime and in the Equivalent Lorentzian Spacetime

摘要

In a previous paper we investigate a Lagrangian field theory for thegravitational field (which is there represented by a section g^a of theorthonormal coframe bundle over Minkowski spacetime. Such theory, underappropriate conditions, has been proved to be equivalent to a Lorentzianspacetime structure, where the metric tensor satisfies Einstein fieldequations. Here, we first recall that according to quantum field theory ideasgravitation is described by a Lagrangian theory of a possible massive gravitonfield (generated by matter fields and coupling also to itself) living inMinkowski spacetime. The graviton field is moreover supposed to be representedby a symmetric tensor field h carrying the representations of spin two and zeroof the Lorentz group. Such a field, then (as it is well known), mustnecessarily satisfy the gauge condition given by Eq.(3) below. Next, weintroduce an ansatz relating h to the 1-form fields g^a. Then, using theClifford bundle formalism we derive, from our Lagrangian theory, the exact waveequation for the graviton and investigate the role of the gauge condition givenby Eq.(3) in obtaining a reliable conservation law for the energy-momentumtensor of the gravitational plus the matter fields in Minkowski spacetime.Finally we ask the question: does Eq.(3) fix any gauge condition for the fieldg of the effective Lorentzian spacetime structure that represents the field hin our theory? We show that no gauge condition is fixed a priory, as is thecase in General Relativity. Moreover we investigate under which conditions wemay fix Logunov gauge condition.