Black hole gluing in de Sitter space

Abstract

We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild–de Sitter or Kerr–de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions on the black hole parameters. We give a self-contained treatment of solving the Einstein equation directly for the metric, given the scattering data we encounter at the future conformal boundary. The main step in the construction is the solution of a linear divergence equation for trace-free symmetric 2-tensors; this is closely related to Friedrich’s analysis of scattering problems for the Einstein equation on asymptotically simple spacetimes.

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 83C05
• 83C57
• Secondary 35B40
• 35C20
• 35L05

KEYWORDS AND PHRASES:

• Many-black-hole spacetime
• asymptotically de Sitter space
• gluing

Acknowledgments

Part of this research was conducted during the period I served as a Clay Research Fellow. I would like to thank Sara Kališnik and Maciej Zworski for their enthusiasm and support, and Richard Melrose and András Vasy for discussions on a related project. I am also grateful to Piotr Chruściel for helpful suggestions, and to two referees for helpful comments.

Notes

1 This means that the right hand side of (1.1(1.1) $$\text{Ric}(g)-\Lambda g=0.$$(1.1) ) is no longer 0, but related to the energy-momentum tensor of an electromagnetic field satisfying Maxwell’s equation.

2 A manifold M with boundary, and a metric g on $M^{°}$ satisfying (1.1(1.1) $$\text{Ric}(g)-\Lambda g=0.$$(1.1) ) such that, for a boundary defining function τ, the ‘unphysical metric’ ${\tau }^{2}g$ is a smooth Lorentzian metric on M, with $\partial M$ spacelike when $\Lambda >0.$

3 Just this one time, we also include the past conformal boundary.

4 This region, $r>\sqrt{3/\Lambda },$ is the interior of the complement of the static region $r<\sqrt{3/\Lambda }$ (i.e. $\tilde{R}<\tilde{\tau }$). In the cosmological region, r is a time function whereas t has spacelike differential; by contrast, the static region is foliated by the spacelike level sets of t (which the Killing vector field ${\partial }_t$ is orthogonal to).

5 One has ${\text{tr}}_g°{\mathsf{G}}_g=-{\text{tr}}_g$ only for $n+1=4.$

6 The reader familiar with b-analysis [53] will recognize this as the Taylor expansion of A into dilation-invariant (with respect to τ) b-differential operators on ${[0,ϵ)}_{\tau }\times \partial M.$

7 On M, thus this does encode uniformity down to compact subsets of $\partial M.$

8 We write $A\sim B$ to mean the existence of a constant C > 1, independent of u, so that $C^{-1}B\le A\le CB.$

9 Here, h in the (3, 1) component of ${\delta }_g^{*}$ simply multiplies the scalar λ it acts on by h, producing the tangential-tangential 2-tensor λh; similarly for other occurrences of h.

10 Indeed, if (X, h) is a Riemannian manifold and V is a conformal Killing vector fields, so ${\mathcal{L}}_{V}h=fh$ for some $f\in {\mathcal{C}}^{\infty }\left(X\right),$ then ${\mathcal{L}}_V\left(e^{2\phi }h\right)=e^{2\phi }(f+2V\phi )h$ for any $\phi \in {\mathcal{C}}^{\infty }\left(X\right).$

11 Note that if Ω has several connected components, the space of such V is larger than the space of conformal Killing vector fields on X.

12 In particular, $g-g_{00}\in {\tau }^m{\mathcal{C}}^{\infty }.$

13 A systematic and more precise way of accomplishing this is to use geometric microlocal techniques [54]. For a single SdS black hole centered at $p\in {\mathbb{S}}^3,$ one starts with the total space $[0,{\lambda }_0)\times M$ and blows up $[0,{\lambda }_0)\times \left\{p\right\}$ and then $\left\{0\right\}\times L,\hspace{0.5em}L=(0,\pi /2]\times \left\{p\right\}.$ The first blow-up resolves the singular nature—due to its r-dependence—of the SdS metric near p, and the second blow-up resolves the event horizon, whose r-coordinate goes to 0 roughly linearly with λ. The family of SdS metrics with mass $\lambda \mathfrak{m}$ can then be defined as a smooth section of the pullback of $S^2{ }^0{T}^{*}M$ to this resolved space, and, crucially, in such a manner that it equals the de Sitter metric on the lift of λ = 0.

14 For consistency with §3, the roles of $t_0,r_0,\dots$ and $t,r,\dots$ are reversed compared to the reference.

15 The definition of r² implies that $r_0^2{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2{\theta }_0\le r^2,$ hence θ is well-defined.

16 This can also be checked directly. Indeed, the equality of ${\text{tr}}_{h_s}{\left({\gamma }_3\right)}_{TT}=\frac{9}{{\Lambda }^2}{\left({\gamma }_3\right)}_{tt}+\frac{3}{\Lambda }{\hspace{0.17em}\text{sin}\hspace{0.17em}}^{-2}\theta {\left({\gamma }_3\right)}_{\varphi \varphi }$ and ${\left({\gamma }_3\right)}_{NN}$ is equivalent to ${\Delta }_{\theta }^2+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^{-2}\theta \frac{\Lambda }{3}{\mathbf{a}}^2{\hspace{0.17em}\text{sin}\hspace{0.17em}}^4{\theta }_0={\Delta }_{\theta }{\Delta }_0$ and thus to ${\Delta }_{\theta }=\frac{{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2{\theta }_0}{{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\theta };$ this is easily verified by plugging in ${\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\theta =1-\frac{r_0^2}{r^2}{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2{\theta }_0=1-\frac{{\Delta }_0}{{\Delta }_{\theta }}{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2{\theta }_0,$ which holds at ${\tau }_s=0.$

Additional information

Funding

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while I was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2019 semester. During the final revisions, I was supported by a Sloan Research Fellowship and the NSF under Grant No. DMS-1955614 and Alfred P. Sloan Foundation; Clay Mathematics Institute.