# On the occurrence of gauge-dependent secularities in nonlinear gravitational waves

## TL;DR

This paper investigates gauge-dependent secular instabilities in nonlinear gravitational waves, demonstrating that such instabilities are gauge artifacts and can be eliminated by choosing appropriate coordinates, highlighting the importance of gauge choice in gravitational wave analysis.

## Contribution

The study shows that nonlinear gravitational wave instabilities in harmonic gauge are gauge artifacts and can be removed by a suitable coordinate transformation, providing exact solutions at second order.

## Key findings

- Instability in harmonic gauge is gauge-dependent and non-physical.
- Exact second-order solutions can be obtained in a specific gauge.
- The instability disappears in a gauge used by Belinski and Zakharov.

## Abstract

We study the plane (not necessarily monochromatic) gravitational waves at nonlinear quadratic order on a flat background in vacuum. We show that, in the harmonic gauge, the nonlinear waves are unstable. We argue that, at this order, this instability can not be eliminated by means of a multiscale approach, i.e. introducing suitable long variables, as it is often the case when secularities appear in a perturbative scheme. However, this is a non-physical and gauge-dependent effect that disappears in a suitable system of coordinates. In facts, we show that in a specific gauge such instability does not occur, and that it is possible to solve exactly the second order nonlinear equations of gravitational waves. Incidentally, we note that this gauge coincides with the one used by Belinski and Zakharov to find exact solitonic solutions of Einstein's equations, that is to an exactly integrable case, and this fact makes our second order nonlinear solutions less interesting. However, the important warning is that one must be aware of the existence of the instability reported in this paper, when studying nonlinear gravitational waves in the harmonic gauge.