[Paper Review] Gauge-invariant gravitational waves in matter beyond linearized gravity

Deepen Garg, I. Y. Dodin|arXiv (Cornell University)|Jun 9, 2021

Pulsars and Gravitational Waves Research97 references4 citations

TL;DR

This paper develops an exactly gauge-invariant quasilinear (QL) theory for gravitational waves (GWs) in dispersive media beyond linearized gravity, using tools from plasma physics to derive a manifestly gauge-invariant expression for GW energy–momentum. The framework ensures consistent, artifact-free modeling of GW–matter interactions, including nonlinear metric backreaction, and recovers standard vacuum GW properties in a gauge-invariant form at all orders in wave amplitude.

ABSTRACT

Modeling the propagation of gravitational waves (GWs) through matter is complicated by the gauge freedom of linearized gravity in that once nonlinearities are taken into consideration, gauge artifacts can cause spurious acceleration of the matter. To eliminate these artifacts, we propose how to keep the theory of dispersive GWs gauge-invariant beyond the linear approximation and, in particular, obtain an unambiguous gauge-invariant expression for the energy--momentum of a GW in dispersive medium. Using analytic tools from plasma physics, we propose an exactly gauge-invariant ``quasilinear'' theory, in which GWs are governed by linear equations and also affect the background metric on scales large compared to their wavelength. As a corollary, the gauge-invariant geometrical optics of linear dispersive GWs in a general background is formulated. As an example, we show how the well-known properties of vacuum GWs are naturally and concisely yielded by our theory in a manifestly gauge-invariant form. We also show how the gauge invariance can be maintained within a given accuracy to an arbitrary order in the GW amplitude. These results are intended to form a physically meaningful framework for studying dispersive GWs in matter.

Motivation & Objective

• To resolve gauge artifacts in nonlinear gravitational wave (GW) theories that cause spurious matter acceleration in dispersive media.
• To formulate a gauge-invariant quasilinear (QL) theory for GWs that remains consistent at arbitrary orders in wave amplitude.
• To derive an unambiguous, gauge-invariant expression for the energy–momentum of a GW in a dispersive medium.
• To extend geometrical optics (GO) for linear GWs to general backgrounds in a gauge-invariant manner.
• To ensure the theory maintains exact gauge invariance even when including second-order and higher-order corrections to the background metric.

Proposed method

• Adopts an expansion of the metric perturbation and Einstein tensor in powers of the GW amplitude $ a $, with $ h_{\alpha\beta} = \sum_n a^n \vartheta^{(n)}_{\alpha\beta} $.
• Uses the Lie derivative $ \mathcal{L}_\xi $ to track gauge transformations of the perturbations and ensures invariance of the equations at each order.
• Derives the Einstein equations $ \sigma_{\alpha\beta}^{(n)} = 0 $ at each order $ n $, which are shown to be invariant under coordinate transformations with $ \xi^\mu = \mathcal{O}(a) $.
• Applies the quasilinear approach via the average-Lagrangian method, analogous to plasma physics, to define wave energy–momentum in terms of the wave field and susceptibility.
• Constructs the gauge-invariant energy–momentum tensor for GWs via the wave Lagrangian, quadratic in $ h_{\alpha\beta} $, ensuring invariance under gauge transformations.
• Truncates the perturbative series at finite order $ m $, solving $ \sigma_{\alpha\beta}^{(n)} = 0 $ for $ n \leq m $, with $ \vartheta^{(0)}_{\alpha\beta} $ not equal to the background metric $ g_{\alpha\beta} $.

Experimental results

Research questions

• RQ1 How can a quasilinear theory of gravitational waves be made exactly gauge-invariant beyond linearized gravity?
• RQ2 What is the correct, gauge-invariant expression for the energy–momentum of a gravitational wave in a dispersive medium?
• RQ3 How can the nonlinear backreaction of GWs on the background metric be consistently described while preserving gauge invariance?
• RQ4 Can the geometrical optics of linear GWs in a general background be formulated in a manifestly gauge-invariant way?
• RQ5 How can gauge invariance be preserved to arbitrary order in the GW amplitude?

Key findings

• The paper derives a gauge-invariant energy–momentum tensor for gravitational waves in dispersive media, given by Eq. (79), which is free from gauge artifacts.
• The quasilinear theory ensures that the wave equations remain linear while the background metric is corrected at $ \mathcal{O}(a^2) $, avoiding the gauge-breaking issues of prior approaches.
• The theory recovers the standard properties of vacuum gravitational waves in a manifestly gauge-invariant form, including their propagation at the speed of light and transverse nature.
• The equations $ \sigma_{\alpha\beta}^{(n)} = 0 $ are shown to be invariant under coordinate transformations, even when $ \xi^\mu $ scales nonlinearly with $ a $, ensuring consistency across all orders.
• The method allows for arbitrary-order gauge-invariant expansions, with $ \sigma_{\alpha\beta}^{(n)} = 0 $ forming a consistent hierarchy of equations that can be solved order-by-order.
• The zeroth-order solution $ \vartheta^{(0)}_{\alpha\beta} $ is not the background metric $ g_{\alpha\beta} $, and the full metric to $ \mathcal{O}(a^2) $ requires solving $ \sigma_{\alpha\beta}^{(2)} = 0 $, distinguishing this approach from Isaacson's method.