Energy-momentum tensor and duality symmetry of linearized gravity in the Fierz formalism

TL;DR

This paper formulates linearized gravity using the Fierz tensor, deriving an energy-momentum tensor with electromagnetic-like properties and revealing a duality symmetry analogous to electromagnetism, along with associated conserved currents.

Contribution

It introduces a Fierz tensor-based formulation of linearized gravity, constructs a new energy-momentum tensor, and demonstrates duality symmetry and conserved currents similar to electromagnetism.

Findings

Derived a first-order PDE form of linearized Einstein equations.

Found an electromagnetic-like energy-momentum tensor for linearized gravity.

Established duality symmetry and conserved currents in the absence of matter.

Abstract

A formulation of linearized gravity in flat background, based on the Fierz tensor as a counterpart of the electromagnetic field strength, is discussed in detail and used to study fundamental properties of the linearized gravitational field. In particular, the linearized Einstein equations are written as first order partial differential equations in terms of the Fierz tensor, in analogy with the first order Maxwell equations. An energy-momentum tensor ($T_{\mathrm{lg}}^{ab}$) with favourable properties and exhibiting remarkable similarity to the standard energy-momentum tensor of the electromagnetic field is found for the linearized gravitational field. $T_{\mathrm{lg}}^{ab}$ is quadratic in the Fierz tensor (which is constructed from the first derivatives of the linearized metric), traceless, and satisfies the dominant energy condition in a gauge that contains the transverse traceless gauge. It is further shown that in suitable gauges, including the transverse traceless gauge, linearized gravity in the absence of matter has a duality symmetry that maps the Fierz tensor, which is antisymmetric in its first two indices, into its dual. Conserved currents associated with the gauge and duality symmetries of linearized gravity are also determined. These currents show good analogy with the corresponding currents in electrodynamics.