Spontaneously Generated Tensor Field Gravity

TL;DR

This paper proposes that a symmetric tensor field in Minkowski spacetime, constrained by a spontaneous Lorentz invariance violation, naturally leads to a theory equivalent to linearized general relativity with emergent diffeomorphism invariance and new Lorentz-violating couplings that are physically unobservable.

Contribution

It demonstrates that a symmetric tensor field theory with a length-fixing constraint can reproduce linearized general relativity and explains the emergence of diffeomorphism invariance from spontaneous Lorentz violation.

Findings

The theory reproduces linearized general relativity in the weak field limit.

Emergent diffeomorphism invariance arises as a necessary condition.

New Lorentz and CPT violating couplings do not produce observable violations in the extended theory.

Abstract

An arbitrary local theory of a symmetric two-tensor field $H_{\mu \nu}$ in Minkowski spacetime is considered, in which the equations of motion are required to be compatible with a nonlinear length-fixing constraint $H_{\mu \nu}^{2}=\pm M^{2}$ leading to spontaneous Lorentz invariance violation, SLIV ($M$ is the proposed scale for SLIV). Allowing the parameters in the Lagrangian to be adjusted so as to be consistent with this constraint, the theory turns out to correspond to linearized general relativity in the weak field approximation, while some of the massless tensor Goldstone modes appearing through SLIV are naturally collected in the physical graviton. In essence the underlying diffeomophism invariance emerges as a necessary condition for the tensor field $H_{\mu \nu}$ not to be superfluously restricted in degrees of freedom, apart from the constraint due to which the true vacuum in the theory is chosen by SLIV. The emergent theory appears essentially nonlinear, when expressed in terms of the pure Goldstone tensor modes and contains a plethora of new Lorentz and $CPT$ violating couplings. However, these couplings do not lead to physical Lorentz violation once this tensor field gravity is properly extended to conventional general relativity.