Gauge symmetry of linearised Nordstr\"om gravity and the dual spin two field theory

TL;DR

This paper develops a gauge-invariant formulation of linearised Nordstr"om gravity using a third-rank hook tensor, revealing its relation to a dual spin-two field theory in any dimension.

Contribution

It introduces a new gauge symmetry framework for the linearised Nordstr"om equation and derives the corresponding spin-two field equations from topological and gauge considerations.

Findings

The gauge transformations form a reducible sequence starting with a hook tensor parameter.

The derived field equations describe an irreducible spin-two massless representation.

The degree of freedom count confirms the theory describes a spin-two field.

Abstract

The field equations are proposed for the third rank tensor field with the hook Young diagram. The equations describe the irreducible spin two massless representation in any $d\geq 3$. The starting point of the construction is the linearised system of Einstein equations which includes the Nordstr\"om equation. This equation, being considered irrespectively to the rest of the Einstein system, corresponds to the topological field theory. The general solution is a pure gauge, modulo topological modes which we neglect in this article. We find the sequence of the reducible gauge transformations for the linearised Nordstr\"om equation, with the hook tensor being the initial gauge symmetry parameter. By substituting the general solution of the Nordstr\"om equation into the rest of the Einstein's system, we arrive at the field equations for the hook tensor. The degree of freedom number count confirms, it is the spin two theory.