Gravitational charges and radiation in asymptotically locally de Sitter spacetimes

We present a comprehensive discussion of gravitational charges and radiation in four-dimensional asymptotically locally de Sitter (AldS) spacetimes. Such spacetimes have compact spatial sections and possess spacelike past and future infinities, $\mathscr{I}^\pm$. We show that the variational problem is well-posed if one specifies a conformal class up to three-dimensional diffeomorpshims at $\mathscr{I}^\pm$, provided one adds at $\mathscr{I}^\pm$ suitable local terms. These are analogues of the AdS covariant counterterms of holographic renormalisation. Radiation is possible only when the conformal class at infinity is non-trivial. Bulk diffeomorphisms that asymptote to conformal Killing vectors at $\mathscr{I}^\pm$ lead to gravitational charges that are conserved under radial translations, while transformations that asymptote to a general three-dimensional vector imply that the gravitational charges satisfy flux-balance laws. We also present quantities that are invariant under temporal translations, if gravitational flux is absent, but otherwise satisfy flux-balance laws. We derive these results using first principles application of Noether’s method as well as covariant phase space methods. We apply our formalism to several exact solutions, including the Robinson-Trautman-dS class, which we use to demonstrate the existence of conserved charges even in the absence of asymptotic conformal Killing vectors and the existence of monotonic charges, including the Bondi mass.

💡 Research Summary

This paper provides a comprehensive treatment of gravitational charges and radiation in four‑dimensional asymptotically locally de Sitter (AldS) spacetimes. Unlike asymptotically flat or asymptotically locally anti‑de Sitter (AlAdS) geometries, AldS spacetimes have compact spatial sections and spacelike past and future infinities, 𝓘⁻ and 𝓘⁺. Because there is no spatial infinity, the usual definitions of conserved quantities must be reformulated in terms of data on these spacelike boundaries.

The authors first specify the boundary conditions that render the variational problem well‑posed. They fix a conformal class of the three‑dimensional boundary metric up to Weyl rescalings σ(x) and diffeomorphisms ζⁱ(x). To cancel the divergences that arise from the infinite volume of de Sitter space, they add local counterterms on 𝓘⁻ and 𝓘⁺ that are the direct analogues of the holographic counterterms used in AlAdS holographic renormalisation. With these terms the on‑shell action is finite and its variation depends only on the prescribed boundary data.

Using a Fefferman‑Graham‑type expansion near 𝓘⁺/𝓘⁻, the bulk metric takes the form
ds² = –dτ² + e^{2τ} g^{(0)}{ij}dxⁱdxʲ + e^{–2τ} g^{(2)}{ij} + … .
Einstein’s equations imply that the coefficient g^{(3)}{ij} is transverse and traceless with respect to g^{(0)}{ij}. The authors introduce the tensor
T_{ij} ≡ –(3ℓ/16πG) g^{(3)}_{ij},
which plays the role of the stress‑energy tensor of a putative three‑dimensional dual theory.

The paper then derives gravitational charges in two complementary ways.

1. Noether’s first theorem (global symmetries) is applied to the renormalised action. For a bulk diffeomorphism generated by a vector ξ that asymptotes to a boundary vector ξ^{(0)}ⁱ, the associated charge on a timelike hypersurface C intersecting 𝓘⁺/𝓘⁻ is
   Q^{±}_ξ