Asymptotic Data at de Sitter Infinity

Because de Sitter conformal infinity is spacelike, its geometry can be treated as asymptotic initial data rather than as data on a null radiative boundary. In four-dimensional vacuum gravity, the freely specifiable information consists of a conformal Riemannian three-metric and a conformally weighted symmetric, trace-free, divergence-free tensor represented by the electric part of the leading Weyl curvature. Topology, completeness, differentiability, conformal gauge, and matter limits remain part of the problem. Data at future and past infinity are not two independent lists for one global solution.

Required background. We use The Einstein Equation to obtain the conformal constraints at a smooth spacelike boundary for $\Lambda\gt 0$.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes the induced Riemannian metric and unit normal, Affine and Variational Geodesics clarifies boundary completeness, Causality and Global Structure supplies the conformal completion, Exact Solutions of Einstein’s Equations supplies de Sitter and Schwarzschild–de Sitter checks, and Formulations of General Relativity supplies constrained evolution and gauge language.

Geometry in a boundary-adapted conformal gauge

Let $(\widehat M,\widehat g_{ab})$ be a smooth four-dimensional vacuum solution,

$$\widehat R_{ab}=\Lambda\widehat g_{ab}, \qquad \Lambda\gt 0, \qquad \ell^2=\frac{3}{\Lambda},$$

with a smooth completion $g_{ab}=\Omega^2\widehat g_{ab}$. Set $n_a=\nabla_a\Omega$. At $\mathscr I$,

$$n^a n_a=-\ell^{-2}.$$

Choose the unit normal $\nu^a=\pm\ell n^a$ with the desired time orientation and define

$$q_{ab}=g_{ab}+\nu_a\nu_b.$$

The pullback of $q_{ab}$ to $\mathscr I$ is positive definite. The freedom $\Omega\mapsto\omega\Omega$ can be used locally to choose a divergence-free frame, $\nabla_a n^a=0$ at $\mathscr I$. In that frame the field equations give $\nabla_a n_b=0$ at the boundary, so its second fundamental form vanishes. A nearby form of the unphysical metric can then be chosen schematically as

$$g=-\ell^2d\Omega^2+q_{ij}(\Omega,x)dx^i dx^j,$$

with no term linear in $\Omega$ in the tangential metric expansion. This “totally geodesic” statement is a conformal gauge choice, not a claim that physical expansion vanishes. The boundary gauge and its residual conformal freedom are derived in Ashtekar, Bonga, and Kesavan 2015, §II.B, pp. 7–9.

The topology must be fixed before data are prescribed. The complete Riemannian manifold can be $S^3$ for global type, $\mathbb R^3$ for a Poincaré-patch completion, or $S^2\times\mathbb R$ for an isolated-system end. On noncompact choices, completeness and falloff toward their punctures are extra requirements; local conformal constraints do not supply them.

Leading Weyl data

The unphysical Weyl tensor vanishes on a regular four-dimensional $\mathscr I$, but its first normal coefficient need not. Define the leading rescaled Weyl tensor

$$K_{abcd}=\lim_{\Omega\to0}\Omega^{-1}C_{abcd},$$

assuming the stated smoothness. Its electric part is

$$\mathcal E_{ab} =q_a{}^c q_b{}^d K_{cedf}\nu^e\nu^f.$$

In vacuum, $\mathcal E_{ab}$ is tangential, symmetric, and trace free. The conformal constraint is

$$D^a\mathcal E_{ab}=0,$$

where $D$ is the Levi-Civita derivative of $q_{ab}$. Thus $\mathcal E_{ab}$ is a transverse-tracefree, or TT, tensor on the boundary. In a formal expansion

$$q_{ij}(\Omega,x) =q_{(0)ij}(x) +\Omega^2q_{(2)ij}(x) +\Omega^3q_{(3)ij}(x)+\cdots,$$

$q_{(2)}$ is fixed locally by the curvature of $q_{(0)}$, while the TT part $q_{(3)}$ represents the same free information as $\mathcal E_{ij}$, up to convention-dependent factors and signs. The expansion is quoted only schematically because changing the defining function changes its coefficients. In four bulk dimensions the boundary has odd dimension three, so the smooth vacuum formal expansion has no even-boundary-dimensional obstruction term; higher-dimensional generalizations require their own regularity analysis.

Under a residual conformal change, representatives transform as

$$q'_{ab}=\omega^2q_{ab}, \qquad \mathcal E'_{ab}=\omega^{-1}\mathcal E_{ab}.$$

The weight of $\mathcal E_{ab}$ follows from its definition: the all-lowered $C_{abcd}$ contributes $\omega^2$, $\Omega^{-1}$ contributes $\omega^{-1}$, and the two unit normals contribute $\omega^{-2}$. The TT constraint is compatible with the same transformation. In three boundary dimensions, for any trace-free symmetric tensor,

$$D'{}^a(\omega^{-1}\mathcal E_{ab}) =\omega^{-3}D^a\mathcal E_{ab}.$$

Thus a vacuum TT datum remains TT after rescaling. The physical datum is an equivalence class $[(q_{ab},\mathcal E_{ab})]$, not a preferred pair. Coordinates on $\mathscr I$ and diffeomorphically related representatives are also redundant. Friedrich’s four-dimensional asymptotic initial-value result identifies a freely specifiable Riemannian metric and a symmetric trace-free tensor satisfying a divergence equation, modulo conformal rescaling, and gives a unique semiglobal past-asymptotically-simple vacuum solution for admissible data Friedrich 1986, pp. 101–117.

Magnetic data and conformal flatness

The magnetic part of $K_{abcd}$ is

$$\mathcal B_{ab} =q_a{}^c q_b{}^d({}^\star K)_{cedf}\nu^e\nu^f.$$

It is not an additional independent TT tensor once the conformal constraints are imposed. In three dimensions the Cotton tensor of $q_{ab}$ measures failure of conformal flatness, and the boundary equations identify its dual with $\mathcal B_{ab}$. Consequently,

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

Imposing this condition restricts the freely specifiable conformal geometry. It removes the magnetic half of generic radiative boundary data, while leaving possible nonzero electric TT data. It is therefore incorrect both to call conformal flatness harmless and to say that it forces every curvature datum to vanish. The equivalence and its radiative severity are established in Ashtekar, Bonga, and Kesavan 2015, §II.B, pp. 8–9 and §V.C, pp. 29–36.

This distinction is crucial for later charges. A conformally flat boundary admits a finite-dimensional conformal group in the standard topologies, but the same restriction makes the associated gravitational momentum flux vanish in the ABK framework. A generic radiative treatment must retain arbitrary $q_{ab}$ and accept that no canonical finite-dimensional asymptotic symmetry group follows from the leading structure alone.

Matter data and constraints

For the chapter’s nonvacuum definition, $\Omega^{-1}\widehat T_{ab}$ must extend smoothly. Einstein’s equation and the Bianchi identities then force the limiting tangential-tangential and mixed components to vanish; only a normal-normal component can survive at that order. At the next order, boundary matter currents source the divergence of $\mathcal E_{ab}$. The vacuum TT equation must therefore be replaced by the appropriate sourced constraint, together with boundary data satisfying the matter field equations. The component restrictions are derived in Ashtekar, Bonga, and Kesavan 2015, §II.B, pp. 8–9.

No universal matter prescription follows from positive $\Lambda$. Maxwell and null-fluid stress–energy can decay faster than the minimum condition; dust motivates allowing the weaker normal-normal limit. Massive scalar fields are conformally regular only for special equations or masses. Every existence or scattering statement must name the matter model and its conformal regularity rather than silently using the vacuum constraints.

What one boundary determines

Since $\mathscr I$ is spacelike, admissible conformal data on one component can be evolved inward by a regular conformal formulation. The result is local or semiglobal until a theorem proves more. Caustics of a chosen gauge, loss of regularity, singularity formation, or failure to reach the opposite temporal end are dynamical questions.

If a solution has both $\mathscr I^-$ and $\mathscr I^+$, their data are related by the bulk evolution. Prescribing arbitrary independent pairs at both ends generally overdetermines the hyperbolic problem. Where a classical scattering map exists, it maps an allowed class of data at one spacelike boundary to data induced at the other. It is not the assertion that every two boundary pairs are connected.

The free tensor is sometimes called the “normalizable” coefficient by analogy with other asymptotic expansions. Here it is simply classical gravitational data. Nothing in the conformal constraints assigns it a quantum state, a dual operator, or a probability measure. Those interpretations lie outside this volume.

Exercises

On $\mathscr I\cong S^2\times\mathbb R$, take

$$q_{ab}dx^a dx^b=dt^2+\ell^2d\omega_2^2,$$

and let $u_a=D_at$, so $u^au_a=1$ and $D_au_b=0$. Show that

$$\mathcal E_{ab} =-\frac{3GM}{\ell^3} \left(u_au_b-\frac13q_{ab}\right)$$

is TT. What does this example demonstrate about conformal flatness and mass data?

Solution

The trace is

$$q^{ab}\mathcal E_{ab} =-\frac{3GM}{\ell^3} \left(u^au_a-\frac13q^{ab}q_{ab}\right) =-\frac{3GM}{\ell^3}(1-1)=0.$$

Metric compatibility and $D_au_b=0$ give

$$D^a\mathcal E_{ab}=0.$$

Thus the tensor satisfies the vacuum conformal constraint. It is the leading electric Weyl datum of Schwarzschild–de Sitter in the standard conformal frame Ashtekar, Bonga, and Kesavan 2015, §V.C.3, pp. 32–33. The product metric is conformally flat, so its magnetic datum vanishes, but $\mathcal E_{ab}$ and the mass parameter $M$ need not. Conformal flatness removes magnetic boundary data; it does not force the electric TT datum or every charge to be zero on the punctured topology.

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.
• Friedrich, Helmut. “Existence and Structure of Past Asymptotically Simple Solutions of Einstein’s Field Equations with Positive Cosmological Constant.” Journal of Geometry and Physics 3, 101–117 (1986). DOI.