Gravitational radiation at infinity with negative cosmological constant and AdS$_4$ holography

TL;DR

This work extends the tidal, Bel–Robinson framework for characterizing gravitational radiation to AdS$_4$ spacetimes with negative cosmological constant by using a timelike conformal boundary $\mathscr{J}$. Radiation at infinity is diagnosed through the transverse flux of asymptotic superenergy, equivalently encoded by a linear dependence between the boundary Cotton–York tensor $C_{\alpha\beta}$ and the holographic stress tensor $D_{\alpha\beta}$ of the rescaled Weyl tensor at $\mathscr{J}$; this yields a covariant, observer-independent criterion and a Weyl-scalar formulation that encompasses all Petrov types. The paper also derives restrictions on the principal null directions at $\mathscr{J}$, discusses initial-data implications for the IBVP, and provides explicit new boundary-condition families controlling incoming versus outgoing radiation in terms of Weyl scalars. Together, these results connect boundary holographic data with the presence or absence of gravitational radiation, offering a robust framework for AdS/CFT and holographic investigations of bulk dynamics. The approach generalizes previous Λ≥0 results and integrates Fefferman–Graham data, boundary geometry, and matter decay conditions into a coherent criterion for radiation at timelike infinity.

Abstract

The covariant characterization of the existence of gravitational radiation traversing infinity $\mathscr{J}$ in the presence of a negative cosmological constant is presented. It is coherent and consistent with the previous characterizations put forward for the cases of non-negative cosmological constant, relying on the properties of the asymptotic super-Poynting vector; or in more transparent terms, based on the intrinsic properties of the flux of tidal energy at infinity. The proposed characterization is fully satisfactory, it can be covariantly typified in terms of boundary data at infinity, and it can also be categorized according to the geometric properties of the rescaled Weyl tensor at $\mathscr{J}$. The cases with no incoming radiation entering from (or no outgoing radiation escaping at) $\mathscr{J}$ can similarly be determined in terms of the boundary data or geometric properties of the rescaled Weyl tensor. In particular, we identify the most general boundary conditions that, in an initial-boundary value problem, ensure absence of gravitational radiation traversing $\mathscr{J}$, namely (functional) proportionality between the Cotton-York tensor field and the holographic stress tensor field at $\mathscr{J}$. We also present novel conditions ensuring the absence of just incoming (outgoing) radiation at $\mathscr{J}$. These are given in a covariant way and also in terms of standard rescaled Weyl tensor scalars. The results are compatible with any matter content of the physical spacetime.

Figures (2)

• Figure 1: Schematic representation of the first set of tetrads and their transformations used to derive \ref{['eq:formula270']}. The null directions, labelled with $i=1,2,3,4$, are aligned with the four PNDs of the rescaled Weyl tensor $(k_1,k_2,k_3,k_4)$. The starting basis is $\left({k}{_{1}^\alpha},{k}{_{2}^\alpha}, \tensor{{m}}{^\alpha}\right)$, and the rest of the tetrads are obtained by doing a null rotation along $k_1^\alpha$ (with complex coefficients $\hat{c}$ and $c^\prime$) or $k_2^\alpha$ (with complex coefficientes $\tilde{c}$ and $\breve{c}$). The different decorations indicate relative normalisations (e.g., $k^\prime_3$ and $\tilde{k}_{3}$ are proportional and aligned with the PND $k_3$). See \ref{['sec:appendix']} for full expressions.
• Figure 2: Similarly to \ref{['fig:set1']}, this represents a second set of null tetrads. In this case, the starting point is $\left({k}{_{3}^\alpha},{k}{_{4}^\alpha}, \underset{\bar{}}{m}^\alpha\right)$ and $f^\prime$, $\hat{f}$, $\tilde{f}$, $\breve{f}$ are the complex coefficients of the different null rotations. The bar appearing below some of the complex vectors in the tetrad is used just to distinguish them from those of the first set in \ref{['fig:set1']} (that is, to distinguish $\underset{\bar{}}{m}^\alpha$ from $\tensor{{m}}{^\alpha}$).

Theorems & Definitions (19)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 1: No gravitational radiation condition with $\Lambda<0$
• proof
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4