de Sitter Asymptotic Symmetries and Charges

An asymptotic charge is defined only after boundary conditions select an asymptotic symmetry. At four-dimensional spacelike de Sitter infinity, the minimal conformal conditions allow an arbitrary Riemannian boundary metric, so their universal symmetry group is essentially $\operatorname{Diff}(\mathscr I)$ and contains no preferred translations. A finite de Sitter-like group and associated surface charges emerge only after additional restrictions. The most common restriction—conformal flatness of the boundary metric—also suppresses radiative data and makes the corresponding gravitational flux vanish, so the resulting charges are conditional rather than universal.

Required background. We use The Einstein Equation with $\Lambda\gt 0$ to obtain the boundary Weyl constraint and its matter source.

Helpful background. Pseudo-Riemannian Metrics and Signature explains why every tangent to spacelike infinity is spatial, Affine and Variational Geodesics supplies boundary completeness, Causality and Global Structure supplies asymptotic equivalence, Exact Solutions of Einstein’s Equations supplies charge checks, and Formulations of General Relativity supplies constrained phase-space language.

The generic symmetry group at spacelike infinity

Fix a smooth four-dimensional conformal completion with $\Lambda\gt 0$, boundary topology $\mathscr I$, complete induced metric $q_{ab}$, and matter falloff $\Omega^{-1}\widehat T_{ab}$ smooth. An asymptotic symmetry is represented at $\mathscr I$ by a diffeomorphism induced by a bulk diffeomorphism that preserves the chosen class of boundary conditions, modulo transformations approaching the identity.

For asymptotically flat null infinity, the degenerate metric and preferred null ruling constrain this group to BMS transformations. Spacelike $\mathscr I$ has neither feature. Its normal is transverse, and a general Riemannian three-manifold has no unique conformal structure. If the phase space permits arbitrary $q_{ab}$ in the stated topology, every diffeomorphism maps one admissible boundary metric to another. The resulting universal group is

$$\mathcal G_{\rm weak}=\operatorname{Diff}(\mathscr I).$$

This statement does not mean that every solution has infinitely many Killing fields. A Killing field preserves one metric; an asymptotic symmetry preserves the entire class of allowed boundary structures. Conversely, fixing one particular conformal class $[q]$ as an extra boundary condition leaves only its conformal isometries, which may be few or absent for a generic $q$. The derivation of $\operatorname{Diff}(\mathscr I)$ from the minimal asymptotically de Sitter structure is given in Ashtekar, Bonga, and Kesavan 2015, §IV.B, pp. 19–20.

The group $\operatorname{Diff}(\mathscr I)$ has no canonically selected four-dimensional translation subgroup. Therefore the minimal structure supplies no preferred global energy–momentum or angular momentum analogous to the Bondi quantities. Exact de Sitter’s ten isometries do not cure the problem for the full phase space: nearby radiative solutions can induce non-round, non-conformally-flat metrics on $\mathscr I$.

Finite groups require conformal flatness and topology

Now impose the ABK condition

$$q_{ab}\text{ is conformally flat} \quad\Longleftrightarrow\quad \mathcal B_{ab}=0,$$

where $\mathcal B_{ab}$ is the magnetic part of the leading rescaled Weyl tensor. The surviving asymptotic transformations preserve the conformally flat class. Completeness and topology then decide which conformal transformations are globally admissible:

• for $\mathscr I\cong S^3$, the group is the ten-dimensional de Sitter group $SO(1,4)$;
• for $\mathscr I\cong\mathbb R^3$, completeness removes the three special conformal transformations, leaving three translations, three rotations, and a dilation;
• for $\mathscr I\cong S^2\times\mathbb R$, the punctures remove the translations as well, leaving three rotations and a distinguished dilation interpreted as the stationary time translation in Schwarzschild–de Sitter.

The seven- and four-dimensional groups are proper subgroups of $SO(1,4)$. A locally available conformal Killing field is not an asymptotic symmetry if its finite flow makes the chosen noncompact boundary metric incomplete or moves a removed point into the manifold. The topology-sensitive reductions are proved in Ashtekar, Bonga, and Kesavan 2015, §IV.C, pp. 20–25.

Calling these groups “the de Sitter asymptotic symmetry group” without giving the topology and conformal-flatness condition is incomplete. The $S^3$, $\mathbb R^3$, and $S^2\times\mathbb R$ cases answer different physical questions.

Conditional surface charges

Continue in four-dimensional vacuum, or with the ABK matter falloff and corresponding matter term. Let

$$\mathcal E_{ab} =q_a{}^c q_b{}^dK_{cedf}\nu^e\nu^f, \qquad K_{abcd}=\lim_{\Omega\to0}\Omega^{-1}C_{abcd},$$

be the electric part of the leading Weyl curvature. For a conformal Killing field $\xi^a$ of $q_{ab}$ and an oriented compact two-surface $S\subset\mathscr I$ with unit normal $r^a$ within $\mathscr I$, the vacuum charge is

$$Q_\xi[S] =-\frac{\ell}{8\pi G} \int_S \mathcal E_{ab}\xi^a r^b\,d^2V, \qquad \ell=\sqrt{\frac{3}{\Lambda}}.$$

Under $q'_{ab}=\omega^2q_{ab}$ and $\mathcal E'_{ab}=\omega^{-1}\mathcal E_{ab}$, the transformations of $r^a$ and $d^2V$ cancel the remaining powers, so $Q_\xi$ is conformally invariant. With the minimum allowed matter decay, the integrand acquires the term $(8\pi G/3)\mathcal T q_{ab}$, where $\mathcal T$ is the finite trace of the appropriately rescaled mixed stress tensor; its inclusion is required for conformal covariance. The vacuum expression, matter correction, and balance identity are derived in Ashtekar, Bonga, and Kesavan 2015, §V.C.1–2, pp. 29–31.

The formula passes stationary checks. For the standard $S^2\times\mathbb R$ completion of Schwarzschild–de Sitter, the charge of the preferred time translation is the physical mass $M$ when the metric is written with $1-2GM/r-r^2/\ell^2$, while rotational charges vanish. For Kerr–de Sitter, the normalization of the time generator matters and differs across the family. These calculations are given in Ashtekar, Bonga, and Kesavan 2015, §V.C.3, pp. 32–33. No unit-norm normalization can be imposed at $\mathscr I$ because the physical norm of a boundary-tangent “time translation” does not approach a finite negative constant.

Why no universal positive global energy follows

The boundary $\mathscr I$ is spacelike. Every nonzero tangent vector is spacelike in the unphysical metric, and a de Sitter Killing field is spacelike near every boundary point where its boundary value is nonzero. At its boundary zeros this argument gives no sign; for example, a static generator stays timelike along its central observer while tending to zero at the observer’s ideal endpoint. No Killing field is timelike everywhere on global de Sitter. Even when matter satisfies standard energy conditions, flux associated with a spacelike generator need not have a definite sign. The four generators often called de Sitter translations also fail to commute like Poincaré translations. Consequently their charges do not form a globally preferred future-causal four-momentum whose time component is automatically positive.

There is nevertheless a useful patch notion. The static Killing field is timelike inside one observer’s diamond and null on its cosmological horizons. With a specified orientation and energy condition, it can define positive flux for test matter through those horizons. That energy is tied to the chosen static patch, not to a universally timelike generator at global $\mathscr I$. Background-based and Hamiltonian constructions can define other charges after selecting falloffs, a reference geometry, or a spatial end; their assumptions must be compared rather than merged.

The conformal-flatness price

The charge formula above is not a generic radiation formula. Conformal flatness sets $\mathcal B_{ab}=0$ and removes half of the freely radiative boundary data. It can leave a nonzero electric TT tensor—Schwarzschild–de Sitter already demonstrates that—but the ABK balance law has no gravitational-wave flux term. In vacuum, $Q_\xi[S]$ is the same on homologous cuts with consistent orientation; only the allowed matter current changes it across the intervening region. Homology is essential: the divergence theorem compares surfaces that bound a region in $\mathscr I$.

For $S^3$ or $\mathbb R^3$ topology, a smooth cut can be shrunk to a point and the source-free charges vanish. For $S^2\times\mathbb R$, a sphere surrounding a puncture cannot be contracted within the boundary, so mass and angular momentum charges can be nonzero. The vanishing radiative flux and the statement that $\mathcal B_{ab}=0$ removes half—not necessarily all—of the radiative modes are established in Ashtekar, Bonga, and Kesavan 2015, §V.C.4, pp. 33–36.

Thus there is a genuine tradeoff. Keeping generic $q_{ab}$ retains radiative freedom but offers no canonical finite translation group. Imposing conformal flatness selects tractable symmetries and charges but excludes generic radiation and makes the chosen gravitational flux vanish. A satisfactory charge framework for broader positive-$\Lambda$ radiation must add structure without concealing this restriction.

Exercises

Assume vacuum, conformal flatness, and a conformal Killing field satisfying

$$D_{(a}\xi_{b)}=\alpha q_{ab}.$$

Use $D^a\mathcal E_{ab}=0$ and $\mathcal E^a{}_a=0$ to show that $Q_\xi[S]$ is independent of the cut. Explain why this forces zero charge on smooth $S^3$ data but permits nonzero charge on $S^2\times\mathbb R$.

Solution

The boundary current $j^a=\mathcal E^{ab}\xi_b$ has divergence

$$D_aj^a =(D_a\mathcal E^{ab})\xi_b +\mathcal E^{ab}D_a\xi_b.$$

The first term vanishes by the vacuum constraint. Only the symmetric part of $D_a\xi_b$ contributes to the second, so

$$\mathcal E^{ab}D_a\xi_b =\alpha\mathcal E^{ab}q_{ab}=0.$$

The divergence theorem therefore equates the flux of $j^a$ through any two homologous cuts, which is precisely cut independence of $Q_\xi$. On $S^3$, every embedded two-sphere bounds a compact ball; shrinking through smooth source-free data gives zero. The same applies to contractible spheres in $\mathbb R^3$. In $S^2\times\mathbb R$, a cross-sectional $S^2$ represents a nontrivial homology class associated with the punctured ends and cannot be shrunk to a regular point. A nonzero Schwarzschild–de Sitter mass charge is therefore compatible with the local divergence equation.

References

• Ashtekar, Abhay, Béatrice Bonga, and Aruna Kesavan. “Asymptotics with a Positive Cosmological Constant: I. Basic Framework.” Classical and Quantum Gravity 32, 025004 (2015). arXiv.