Completing the Bousso–Hawking analysis of nucleated primordial black holes in de Sitter space: the explicit Lorentzian-conjugate slicing for the constrained instanton

In their 1999 paper 'Primordial Black Holes: Pair Creation, Lorentzian Condition, and Evaporation', Bousso and Hawking computed the semiclassical pair creation of black holes in de Sitter space using constrained instantons selected by a Lorentzian condition on the wave function in the conjugate representation, and analyzed the quantum evolution of the nucleated black holes in a two-dimensional s-wave large-N dilaton model, finding that nearly maximal Schwarzschild–de Sitter black holes antievaporate under displacement perturbations while a second perturbative mode, always excited by the no-boundary condition, leads to evaporation. One element that the authors explicitly left open remains without a solution in the later literature: the analytic continuation of the conical constrained instanton to a Lorentzian pair of black holes requires a slicing of Schwarzschild–de Sitter spacetime — a family of spacelike slices all passing through the conical singularity, with the proper time between points on different slices going to zero at least quadratically as a function of proper distance from the black hole horizon — and no such slicing has ever been constructed explicitly.

Background

In the very early universe, quantum fluctuations can in principle produce black holes — primordial black holes — and during inflation the ambient geometry is close to de Sitter space, the maximally symmetric solution of Einstein's equations with positive cosmological constant $\Lambda$. Static, spherically symmetric black holes in such a background are described by the Schwarzschild–de Sitter solution, which has two horizons: a black hole horizon and a cosmological horizon. The black hole mass is bounded above by the value $M_N=1/(3\sqrt{\Lambda})$ (in units with Newton's constant set to 1) at which the two horizons coincide; the degenerate limiting geometry, called the Nariai spacetime, is the smooth product of two-dimensional de Sitter space with a two-sphere, and its spatial sections have topology $S^1\times S^2$, describable as a pair of black holes.

Semiclassically, the creation of pairs of black holes in de Sitter space is treated as a tunneling event. In the Euclidean path integral approach, the leading contribution to the tunneling rate $\Gamma$ comes from an instanton, a compact solution of the Euclidean field equations. For a sub-maximal Schwarzschild–de Sitter black hole no smooth compact instanton exists: the Euclidean time coordinate can be made periodic so as to remove the conical singularity — a delta-function curvature defect analogous to the tip of a cone, produced when the Euclidean time period does not match the inverse surface gravity of a horizon — at one horizon, but a conical singularity then remains at the other. Bousso and Hawking nevertheless assigned such configurations a role as constrained instantons: saddles of a path integral in which an additional parameter (the black hole mass) is held fixed, a construction long known from ordinary quantum tunneling problems. Their framework uses the Hartle–Hawking no-boundary wave function of the universe, normally a functional $\Psi[h_{ij}]$ of the three-metric $h_{ij}$ induced on a spacelike boundary surface $\Sigma$ (the metric representation), defined by a path integral over compact geometries with no boundary other than $\Sigma$. Bousso and Hawking instead worked partly in the conjugate (momentum) representation, in which the wave function depends on a quantity related to the second fundamental form $K_{ij}$ of $\Sigma$ — the extrinsic curvature describing how the induced metric changes along the normal to $\Sigma$ — and the requirement that this argument be purely imaginary guarantees that $\Sigma$ is part of a universe with Lorentzian signature; they called this the Lorentzian condition. In this formalism they obtained a nonzero semiclassical pair creation rate for sub-maximal black holes, which had been thought to vanish. For the prescription to be consistent, the surface of measurement $\Sigma$ must contain the conical singularity (so the metric is not varied there), and the analytic continuation to a real Lorentzian pair of black holes requires a slicing of Schwarzschild–de Sitter — a family of spacelike slices all passing through the conical singularity, since the metric must be specified there for the conical Euclidean solution to be a saddlepoint — with a specific regularity property: the proper time between points on different slices must go to zero at least quadratically as a function of proper distance from the black hole horizon, ensuring that all Lorentzian spacelike slices are regular on the black hole horizon. In the 1999 paper the authors state that they 'shall not give any such slicing explicitly', leaving the geometric realization of the prescription unresolved.

For the quantum evolution of the nucleated black holes, the same authors reduced the spherically symmetric (s-wave, that is, angular-momentum-zero) sector of four-dimensional gravity coupled to $N$ minimally coupled scalar fields to a two-dimensional dilaton gravity theory, in which the dilaton field $\phi$ measures the area of the symmetry two-spheres and the scalars couple to the dilaton. Quantum corrections were included at one loop in the large-$N$ approximation ($N\to\infty$, with $\kappa\equiv 2N/3$ large, so that quantum fluctuations of the $N$ scalars dominate those of the metric). The trace anomaly — the anomalous nonzero trace of the energy-momentum tensor of conformally invariant matter, encoded in the nonlocal Polyakov term and made local by introducing an auxiliary field — then drives Hawking evaporation, and the analysis was perturbative in the small parameter $b\equiv\kappa\Lambda\ll 1$ and about the degenerate Nariai solution. The main dynamical results were: (i) a perturbation with nonzero initial displacement from the quantum-corrected Nariai solution but vanishing time derivative grows at early times, so the black hole 'antievaporates' ('stable at least initially', in the authors' words); (ii) a perturbation with vanishing displacement and nonvanishing time derivative evaporates; and (iii) complex constrained instantons implementing the no-boundary condition select a linear combination of the two mode types containing the evaporating one, so that pair-created maximal cosmological black holes evaporate rather than antievaporate. In deriving the field equations the authors dropped the nonlocal effective-action term $(\nabla\phi)^2(1/\Box)R$ on the grounds that it enters only at second order in the small expansion parameters.

The 1999 analysis is thus built on a chain of approximations: the s-wave truncation, the two-dimensional reduction, the large-$N$ limit, first order in $b$, and an unconstructed geometric ingredient of the pair-creation prescription. The dynamical results are early-time claims within this truncation; the geometric ingredient — the explicit slicing that validates the Lorentzian-conjugate prescription — was left unconstructed and is the open element addressed here.

Problem Statement

The problem is the element that Bousso and Hawking explicitly left open in their 1999 semiclassical analysis of nucleated primordial black holes in de Sitter space: the explicit geometric realization of the Lorentzian-conjugate constrained-instanton prescription for sub-maximal black hole pair creation.

Construct an explicit slicing of Schwarzschild–de Sitter spacetime — one family of spacelike slices — that analytically continues the conical constrained instanton (cosmological horizon chosen regular in the Euclidean sector, conical singularity at the black hole horizon) to a Lorentzian pair of black holes, satisfying the conditions imposed by the source: every slice passes through the conical singularity, since the metric must be specified there in order for the conical Euclidean solution to be a saddlepoint of the constrained path integral (equivalently, the surface of measurement $\Sigma$ contains the conical singularity); the proper time between points on different slices goes to zero at least quadratically as a function of proper distance from the black hole horizon, ensuring that all Lorentzian spacelike slices are regular on the black hole horizon; and the slicing extends, as schematically indicated in the source's Carter–Penrose diagram (Fig. 2 of hep-th/9807148), through the regions between the black hole and cosmological horizons, the asymptotic de Sitter region, and the black hole interior.

The scientific objective is to place the semiclassical prediction of a nonzero pair-creation rate for sub-maximal Schwarzschild–de Sitter black holes in the conjugate representation on a complete geometric footing, by exhibiting the slicing that the Lorentzian condition requires and whose construction the source explicitly declined to provide.

References

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