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

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.

References in corpus (8)

• Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models
• A Joint Analysis of BICEP2/Keck Array and Planck Data
• Stable and unstable cosmological models in bimetric massive gravity
• Inflation in (Super-)renormalizable Gravity
• Gravitational waves in bigravity cosmology
• Introductory lectures on the Effective One Body formalism
• Surfing gravitational waves: can bigravity survive growing tensor modes?
• Instabilities in tensorial nonlocal gravity

Cited by in corpus (2)

• Nonlinear stability in nonlocal gravity
• The transverse-traceless gauge and the gauge problem of second order gravitational waves