Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond

TL;DR

The paper tackles the gauge-dependence problem in relativistic perturbation theory beyond linear order by building a rigorous geometrical framework for higher-order gauge transformations. It introduces knight diffeomorphisms as a practical representation of arbitrary one-parameter diffeomorphisms and derives a generating formula that yields explicit transformation rules up to third order. The authors define gauge invariance to order n and provide concrete second- and third-order transformation rules, applying them to a flat Robertson–Walker cosmology to relate synchronous and Poisson gauges. This approach unifies perturbation theory at higher orders and offers a systematic method applicable to cosmology and other spacetime theories, with clear guidance on constructing gauge-invariant quantities and transforming between gauge choices.

Abstract

We consider in detail the problem of gauge dependence that exists in relativistic perturbation theory, going beyond the linear approximation and treating second and higher order perturbations. We first derive some mathematical results concerning the Taylor expansion of tensor fields under the action of one-parameter families (not necessarily groups) of diffeomorphisms. Second, we define gauge invariance to an arbitrary order $n$. Finally, we give a generating formula for the gauge transformation to an arbitrary order and explicit rules to second and third order. This formalism can be used in any field of applied general relativity, such as cosmological and black hole perturbations, as well as in other spacetime theories. As a specific example, we consider here second order perturbations in cosmology, assuming a flat Robertson-Walker background, giving explicit second order transformations between the synchronous and the Poisson (generalized longitudinal) gauges.

Figures (4)

• Figure 1: The action of a knight diffeomorphism $\Psi_\lambda$ generated by $\xi_{(1)}$ and $\xi_{(2)}$. Solid lines: integral curves of $\xi_{(1)}$. Dashed lines: integral curves of $\xi_{(2)}$. The parameter lapse between $p$ and $\phi^{(1)}_\lambda(p)$ is $\lambda$, and that from $\phi^{(1)}_\lambda(p)$ to $\phi^{(2)}_{\lambda^2/2}\circ \phi^{(1)}_\lambda(p)$ is $\lambda^2/2$.
• Figure 2: The action of a knight diffeomorphism of rank two, represented in a chart to second order.
• Figure 3: The action of a gauge transformation $\Phi_\lambda$, represented on the background spacetime ${\cal M}_0$ by its second order approximation, generated by the two vector fields $\xi_{(1)}$ and $\xi_{(2)}$.
• Figure 4: If $X$ and $Y$ do not commute, $\Phi$ is not a flow on ${\cal M}_0$.