Static Vacuum Einstein Equations Research Articles

5-dimensional space-periodic solutions of the static vacuum Einstein equations

An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S1 × S2 and sphere S3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.

A complete classification of S1-symmetric static vacuum black holes

In a seminal paper of 1917, Weyl presented a remarkable reduction of the static axisymmetric vacuum Einstein equations, serving as a relatively straightforward technique to generate and explore new solutions. Weyl’s reduction was used by Myers in 1987, and independently by Korotkin–Nicolai in 1994, to construct a new family of static and axisymmetric solutions with compact non-empty horizon, however with non-trivial topology and asymptotically Kasner. This family, together with the Schwarzschild and the Boost families, remained until now as the only known -symmetric static black hole solutions, namely, (metrically complete) -symmetric static vacuum solutions with compact and non-empty horizon. In this article we prove that, indeed, these three families exhaust all the examples of -symmetric static vacuum black holes.

Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature, Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of $(\bar B, g)$. We prove the validity of the second statement, i.e.~such mass-minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true in general; there is a rather large class of bodies $(\bar B, g)$ for which a minimal mass extension does not exist.

A classification theorem for static vacuum black holes, Part II: the study of the asymptotic

This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumption on the topology or the asymptotic. Specifically, it is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In this Part II we show that the only end of a static black hole data set is either asymptotically flat or asymptotically Kasner. This proves the third and last step required in the proof of the classification theorem, as was explained in Part I. The analysis requires a thorough study of static data sets with a free $S^{1}$-symmetry and a delicate investigation of the geometry of static black hole ends with sub-cubic volume growth. Many of the conclusions of this article are hitherto unknown and have their own interest.

On the Bartnik conjecture for the static vacuum Einstein equations

We prove that given any smooth metric and smooth positive function H on there is a constant depending on and an asymptotically flat solution of the static vacuum Einstein equations on such that the induced metric and mean curvature of at are given by This gives a partial resolution of the conjecture of Bartnik.

On static Poincaré-Einstein metrics

The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian n-manifold (M, g) and a positive function N, called the lapse. We study this problem on Asymptotically Poincare-Einstein n-manifolds, n ≥ 3, when the conformal boundary-at-infinity is either a round sphere, a flat torus or smooth quotient thereof, or a compact hyperbolic manifold. Such manifolds have well-defined Wang mass, and are time-symmetric slices of static, vacuum, asymptotically anti-de Sitter spacetimes. By integrating a mildly generalized form of an identity used by Lindblom, Shen, Wang, and others, we give a mass formula for such manifolds. There are no solutions with positive mass. In consequence, we observe that either the lapse is trivial and (M, g) is Poincare-Einstein or the Wang mass is negative, as in the case of time symmetric slices of the AdS soliton. As an application, we use the mass formula to compute the renormalized volume of the warped product (X, γ) ≃ (M 3 , g) × N 2 (S 1 , dt 2). We also give a mass formula for the case of a metric that is static in the region exterior to a horizon on which the lapse function is zero. Then the manifold (X, γ) is said to have a “bolt” where the S 1 factor shrinks to zero length. The renormalized volume of (X, γ) is expected on physical grounds to have the form of the free energy per unit temperature for a black hole in equilibrium with a radiation bath at fixed temperature. When M is 3-dimensional and admits a horizon, we apply this mass formula to compute the renormalized volume of (X, γ) and show that it indeed has the expected thermodynamically motivated form. We also discuss several open questions concerning static vacuum asymptotically Poincare-Einstein manifolds.

Local existence and uniqueness for exterior static vacuum Einstein metrics

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean space, there exists a unique AF solution to the static vacuum equations realizing the boundary data, which is close to the standard flat solution.

Static Vacuum Einstein Metrics on Bounded Domains

We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.

On the Bartnik extension problem for the static vacuum Einstein equations

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations on with geometric boundary conditions on ∂M ≃ S2. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.

Static axisymmetric vacuum solutions and non-uniform black strings

We describe new numerical methods to solve the static axisymmetric vacuum Einstein equations in more than four dimensions. As an illustration, we study the compactified non-uniform black string phase connected to the uniform strings at the Gregory–Laflamme critical point. We compute solutions with a ratio of maximum to minimum horizon radius of up to nine. For a fixed compactification radius, the mass of these solutions is larger than the mass of the classically unstable uniform strings. Thus they cannot be the end state of the instability.

Scalar curvature, metric degenerations, and the static vacuum Einstein equations on 3-manifolds, II

Scalar curvature, metric degenerations, and the static vacuum Einstein equations on 3-manifolds, II

On the Structure of Solutions to the Static Vacuum Einstein Equations

A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the time-symmetric space-like hypersurface has only finitely many ends, each of which is either asymptotically flat (AF) or parabolic, as in the (static) Kasner metric. Examples are given with both types of behavior, together with an extensive discussion and new characterization of Weyl metrics. The asymptotics result allows one in most circumstances to drop the AF assumption from the static black hole uniqueness theorems and replace it with just a completeness assumption.

Scalar Curvature, Metric Degenerations and the Static Vacuum Einstein Equations on 3-manifolds, I

((Without Abstract)).

A classification theorem for static vacuum black holes, Part I: the study of the lapse

The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we extend this result by proving a classification theorem for all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without making any further assumption on the topology or the asymptotic. It is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In a broad sense, the theorem classifies all the static vacuum black holes in 3+1-dimensions. In this Part I we use introduce techniques in conformal geometry and comparison geometry a la Bakry-Emery to prove that vacuum static black holes have only one end, and, furthermore, that the lapse is bounded away from zero at infinity. The techniques have interest in themselves and could be applied in other contexts as well, for instance to study higher-dimensional static black holes.