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Ryan Unger

Publications and source records attributed to Ryan Unger.

12 recordsLinked to original sources
arXiv2026

The moduli space of dynamical spherically symmetric black hole spacetimes and the extremal threshold

In this paper, we give a complete description of the black hole threshold, locally near the Reissner-Nordstr\"om family, in the infinite-dimensional moduli space $\mathfrak M$ of dynamical spherically symmetric solutions to the Einstein-Maxwell-neutral scalar field system. In a neighborhood of the full Reissner-Nordstr\"om family in $\mathfrak M$, we prove the following: (i) Any solution that forms a black hole eventually decays to a Reissner-Nordstr\"om black hole. (ii) Any solution that fails to collapse into a black hole eventually becomes superextremal along null infinity and exists globally in the domain of dependence of the bifurcate characteristic initial data. (iii) The subset of this neighborhood consisting of black hole solutions admits a $C^1$ foliation by hypersurfaces of constant final charge-to-mass ratio, up to and including extremality. (iv) The mutual boundary between the set of black hole solutions and noncollapsing solutions, i.e., the black hole threshold, is the extremal leaf of the foliation. Black holes which are not on the threshold are asymptotically subextremal. Our quantitative control of near-threshold solutions allows us to prove "universal" scaling laws for the location of the event horizon and its final area and temperature (surface gravity), with critical exponent $\frac 12$. Moreover, we show that the celebrated Aretakis instability is activated for an open and dense set of threshold solutions and that generic near-threshold subextremal black holes experience a transient horizon instability on the timescale of their inverse final temperature.

gr-qc↗
arXiv2025

Semilinear wave equations on extremal Reissner-Nordstr\"om black holes revisited

We revisit global existence and decay for small-data solutions of semilinear wave equations on extremal Reissner-Nordstr\"om black hole backgrounds satisfying the classical null condition, a problem which was previously addressed by the first author in joint work with Aretakis and Gajic (Ann. of PDE, 2020). In this paper, we develop a new approach based on propagating a significantly weaker set of estimates, which allows for a simpler and more streamlined proof. Our proof does not require tracking sharp estimates for the solution in the near-horizon region, which means that it is compatible with, but does not imply, the non-decay and growth hierarchy of derivatives of the solution along the event horizon expected from the Aretakis instability. In particular, this approach is in principle compatible with other settings where stronger horizon instabilities are expected, such as nonlinear charged scalar fields on extremal Reissner-Nordstr\"om, or nonlinear waves on extremal Kerr. We also sketch how our proof applies to semilinear problems on spacetimes settling down to extremal Reissner-Nordstr\"om, such as those constructed in our joint work with Kehle (arXiv:2410.16234, 2024).

math.AP↗
arXiv2024

Nonlinear stability of extremal Reissner-Nordstr\"om black holes in spherical symmetry

In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordstr\"om family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" $\mathfrak M_\mathrm{stab}$ of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordstr\"om family, such that any data in $\mathfrak M_\mathrm{stab}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordstr\"om family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordstr\"om family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordstr\"om black holes in the full nonlinear Einstein-Maxwell-scalar field model.

gr-qc↗
arXiv2024

Gravitational collapse to extremal black holes and the third law of black hole thermodynamics

We construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry which are exactly isometric to extremal Reissner-Nordström after a finite advanced time along the event horizon. Moreover, in each of these examples the apparent horizon of the black hole coincides with that of a Schwarzschild solution at earlier advanced times. In particular, our result can be viewed as a definitive disproof of the "third law of black hole thermodynamics." The main step in the construction is a novel $C^k$ characteristic gluing procedure, which interpolates between a light cone in Minkowski space and a Reissner-Nordström event horizon with specified charge to mass ratio $e/M$. Our setup is inspired by the recent work of Aretakis-Czimek-Rodnianski on perturbative characteristic gluing for the Einstein vacuum equations. However, our construction is fundamentally nonperturbative and is based on a finite collection of scalar field pulses which are modulated by the Borsuk-Ulam theorem.

gr-qc↗
arXiv2024

Extremal black hole formation as a critical phenomenon

In this paper, we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein-Maxwell-Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically, these solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter, we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion, or undergoes gravitational collapse. At the critical value of the parameter, an extremal Reissner-Nordström black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.

gr-qc↗
arXiv2023

Event horizon gluing and black hole formation in vacuum: the very slowly rotating case

In this paper, we initiate the study of characteristic event horizon gluing in vacuum. More precisely, we prove that Minkowski space can be glued along a null hypersurface to any round symmetry sphere in a Schwarzschild black hole spacetime as a $C^2$ solution of the Einstein vacuum equations. The method of proof is fundamentally nonperturbative and is closely related to our previous work in spherical symmetry [KU22] and Christodoulou's short pulse method [Chr09]. We also make essential use of the perturbative characteristic gluing results of Aretakis-Czimek-Rodnianski [ACR21a; CR22]. As an immediate corollary of our methods, we obtain characteristic gluing of Minkowski space to the event horizon of very slowly rotating Kerr with prescribed mass $M$ and specific angular momentum $a$. Using our characteristic gluing results, we construct examples of vacuum gravitational collapse to very slowly rotating Kerr black holes in finite advanced time with prescribed $M$ and $0\le |a|\ll M$. Our construction also yields the first example of a spacelike singularity arising from one-ended, asymptotically flat gravitational collapse in vacuum.

gr-qc↗
arXiv2022

Density and positive mass theorems for incomplete manifolds

For manifolds with a distinguished asymptotically flat end, we prove a density theorem which produces harmonic asymptotics on the distinguished end, while allowing for points of incompleteness (or negative scalar curvature) away from this end. We use this to improve the "quantitative" version of the positive mass theorem (in dimensions $3\leq n\leq 7$), obtained by the last two named authors with S.-T. Yau [LUY21], where stronger decay was assumed on the distinguished end. We also give an alternative proof of this theorem based on a relationship between MOTS and $μ$-bubbles and our recent work on the spacetime positive mass theorem with boundary [LLU21].

math.DG↗
arXiv2022

Noncompact Fill-Ins of Bartnik Data

We generalize Y. Shi and L.-F.\ Tam's \cite{ShiTam} nonnegativity result for the Brown-York mass, by considering nonnegative scalar curvature (NNSC) fill-ins that need only be complete rather than compact. Moreover, the NNSC fill-ins need not even be complete as long the incompleteness is ``shielded'' by a region with positive scalar curvature and occurs occurs sufficiently far away. We accomplish this by generalizing P.~Miao's~\cite{Miao02} positive mass theorem with corners to asymptotically flat manifolds that may have other complete ends, or possibly incomplete ends that are appropriately shielded. We can similarly extend other results on the compact NNSC fill-in problem to allow for complete (or shielded) NNSC fill-ins. In particular, we prove the following generalization of a theorem of Miao~\cite{Miao20}: Given any metric $γ$ on a closed manifold $Σ^{n-1}$, there exists a constant $λ$ such that for any complete (or shielded) NNSC fill-in $(Ω^n, g)$ of $(Σ^{n-1},γ)$, we have $\min_ΣH \le λ$, where $H$ is the mean curvature of $Σ$ with respect to $g$.

math.DG↗
arXiv2022

Density and positive mass theorems for initial data sets with boundary

We prove a harmonic asymptotics density theorem for asymptotically flat initial data sets with compact boundary that satisfy the dominant energy condition. We use this to settle the spacetime positive mass theorem, with rigidity, for initial data sets with apparent horizon boundary in dimensions less than $8$ without a spin assumption.

math.DG↗
arXiv2021

The Positive Mass Theorem with Arbitrary Ends

We prove a Riemannian positive mass theorem for manifolds with a single asymptotically flat end, but otherwise arbitrary other ends, which can be incomplete and contain negative scalar curvature. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau.

math.DG↗
arXiv2020

Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem

Using minimal hypersurfaces, we obtain topological obstructions to admitting complete metrics with positive scalar curvature on a given class of non-compact n-manifolds with n less than 8. We show that the Liouville theorem for a locally conformally flat n-manifold of non-negative scalar curvature follows from the impossibility of there being a positive scalar curvature metric on its connect sum with the n-torus. With the recent work of Chodosh-Li, the Liouville theorem is now proved in all remaining cases. Finally, using MOTS instead of minimal hypersurfaces, we show an Initial Data Set version of these results with the Dominant Energy Scalar appearing instead of positive scalar curvature.

math.DG↗