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:
KEYWORDS AND PHRASES:
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 (Equation1.1(1.1)
(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 satisfying (Equation1.1
(1.1)
(1.1) ) such that, for a boundary defining function τ, the ‘unphysical metric’
is a smooth Lorentzian metric on M, with
spacelike when
3 Just this one time, we also include the past conformal boundary.
4 This region, is the interior of the complement of the static region
(i.e.
). 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
is orthogonal to).
5 One has only for
6 The reader familiar with b-analysis [Citation53] will recognize this as the Taylor expansion of A into dilation-invariant (with respect to τ) b-differential operators on
7 On M, thus this does encode uniformity down to compact subsets of
8 We write to mean the existence of a constant C > 1, independent of u, so that
9 Here, h in the (3, 1) component of 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 for some
then
for any
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,
13 A systematic and more precise way of accomplishing this is to use geometric microlocal techniques [Citation54]. For a single SdS black hole centered at one starts with the total space
and blows up
and then
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
can then be defined as a smooth section of the pullback of
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 and
are reversed compared to the reference.
15 The definition of r2 implies that hence θ is well-defined.
16 This can also be checked directly. Indeed, the equality of and
is equivalent to
and thus to
this is easily verified by plugging in
which holds at