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

Abstract

In a previous paper we investigate a Lagrangian field theory for the gravitational field, which is there represented by a section {gα} of the coframe bundle over Minkowski spacetime (M≃R4,g̊,D̊,τg̊,↑). Such theory, under appropriate conditions, has been proved to be equivalent to a Lorentzian spacetime structure (M≃R4,g,D,τg,↑) where the metric tensor g satisfies the Einstein field equation. Here, we first recall that according to quantum field theory ideas gravitation is described by a Lagrangian theory of a possible massive graviton field (generated by matter fields and coupling also to itself) living in Minkowski spacetime. The massive graviton field is moreover supposed to be represented by a symmetric tensor field h carrying the representations of spin two and zero of the Lorentz group. Such a field, then (as it is well known) must necessarily satisfy the gauge condition given by Eq.(10) below. Next, we introduce an ansatz relating h with the 1‐form fields {gα}. Then, using the Clifford bundle formalism we derive from our Lagrangian theory the exact wave equation for the graviton and investigate the role of the gauge condition given by Eq.(10) by asking the question: does Eq.(10) fix any gauge condition for the field g of the effective Lorentzian spacetime structure (M≃R4,g,D,τg,↑) that represents the field h in our theory? We show that no gauge condition is fixed a priory, as it is the case in General Relativity. Moreover we prove that if we use Logunov gauge condition, i.e., D̊γ(−detggγκ = 0 then only a restricted class of coordinate systems (including harmonic ones) are allowed by the theory.