In this paper, we combine and generalize to higher dimensions the approaches to proving the uniqueness of connected (3+1)-dimensional static vacuum asymptotically flat black hole spacetimes by Müller zum Hagen--Robinson--Seifert and by Robinson. Applying these techniques, we prove and/or reprove geometric inequalities for connected (n+1)-dimensional static vacuum asymptotically flat spacetimes with either black hole or equipotential photon surface or specifically photon sphere inner boundary. In particular, assuming a natural upper bound on the total scalar curvature of the boundary, we recover and extend the well-known uniqueness results for such black hole and equipotential photon surface spacetimes. We also relate our results and proofs to existing results, in particular to those by Agostiniani--Mazzieri and by Nozawa--Shiromizu--Izumi--Yamada.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

It is a well-known fact that the Schwarzschild spacetime admits a maximal spacetime extension in null coordinates which extends the exterior Schwarzschild region past the Killing horizon, called the Kruskal-Szekeres extension. This method of extending the Schwarzschild spacetime was later generalized by Brill-Hayward to a class of spacetimes of "profile h" across non-degenerate Killing horizons. Circumventing analytical subtleties in their approach, we reconfirm this fact by reformulating the problem as an ODE, and showing that the ODE admits a solution if and only if the naturally arising Killing horizon is non-degenerate. Notably, this approach lends itself to discussing regularity across the horizon for non-smooth metrics.

We will discuss applications to the study of photon surfaces, extending results by Cederbaum-Galloway and Cederbaum-Jahns-Vičánek-Martínez beyond the Killing horizon. In particular, our analysis asserts that photon surfaces approaching the Killing horizon must necessarily cross it.

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41Geometrically defined asymptotic coordinates in general relativity

We present a new proof of the Willmore inequality for an arbitrary bounded domain Ω⊂ℝ^n with smooth boundary. Our proof is based on a parametric geometric inequality involving the electrostatic potential for the domain Ω; this geometric inequality is derived from a geometric differential inequality in divergence form. Our parametric geometric inequality also allows us to give new proofs of the quantitative Willmore-type and the weighted Minkowski inequalities by Agostiniani and Mazzieri.

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41Geometrically defined asymptotic coordinates in general relativity

We present different proofs of the uniqueness of 4-dimensional static vacuum asymptotically flat spacetimes containing a connected equipotential photon surface or in particular a connected photon sphere. We do not assume that the equipotential photon surface is outward directed or non-degenerate and hence cover not only the positive but also the negative and the zero mass case which has not yet been treated in the literature. Our results partially reproduce and extend beyond results by Cederbaum and by Cederbaum and Galloway. In the positive and negative mass cases, we give three proofs which are based on the approaches to proving black hole uniqueness by Israel, Robinson, and Agostiniani--Mazzieri, respectively. In the zero mass case, we give four proofs. One is based on the positive mass theorem, the second one is inspired by Israel's approach and in particular leads to a new proof of the Willmore inequality in (ℝ^3,δ), under a technical assumption. The remaining two proofs are inspired by proofs of the Willmore inequality by Cederbaum and Miehe and by Agostiniani and Mazzieri, respectively. In particular, this suggests to view the Willmore inequality and its rigidity case as a zero mass version of equipotential photon surface uniqueness.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We study four-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani–Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal, and super-extremal Reissner–Nordström spacetimes. We also obtain new results for connected photon spheres and equipotential photon surfaces in the extremal case. Finally, we provide, up to a restrictionon the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) equipotential photon surfaces in the super-extremal case, not yet treated in the literature.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We conduct a review of the basic definitions and the principal results in the study of wavelike spacetimes, that is spacetimes whose metric models massless radiation moving at the speed of light, focusing in particular on those geometries with parallel rays. In particular, we motivate and connect their various definitions, outline their coordinate descriptions and present some classical results in their study in a language more accessible to modern readers, including the existence of "null coordinates" and the construction of Penrose limits. We also present a thorough summary of recent work on causality in pp-waves, and describe progress in addressing an open question in the field - the Ehlers-Kundt conjecture.

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41Geometrically defined asymptotic coordinates in general relativity

We study timelike, totally umbilic hypersurfaces -- called photon surfaces -- in n+1-dimensional static, asymptotically flat spacetimes, for n+1≥4. First, we give a complete characterization of photon surfaces in a class of spherically symmetric spacetimes containing the (exterior) subextremal Reissner--Nordström spacetimes, and hence in particular the (exterior) positive mass Schwarzschild spacetimes. Next, we give new insights into the spacetime geometry near equipotential photon surfaces and provide a new characterization of photon spheres (not appealing to any field equations).

We furthermore show that any asymptotically flat electrostatic electro-vacuum spacetime with inner boundary consisting of equipotential, (quasi-locally) subextremal photon surfaces and/or non-degenerate black hole horizons must be isometric to a suitable piece of the necessarily subextremal Reissner--Norström spacetime of the same mass and charge. Our uniqueness result applies work by Jahns and extends and complements several existing uniqueness theorems. Its proof fundamentally relies on the lower regularity rigidity case of the Riemannian Positive Mass Theorem.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

The coordinate freedom of General Relativity makes it challenging to find mathematically rigorous and physically sound definitions for physical quantities such as the center of mass of an isolated gravitating system. We will argue that a similar phenomenon occurs in Newtonian Gravity once one ahistorically drops the restriction that one should only work in Cartesian coordinates when studying Newtonian Gravity. This will also shed light on the nature of the challenge of defining the center of mass in General Relativity. Relatedly, we will give explicit examples of asymptotically Euclidean relativistic initial data sets which do not satisfy the Regge--Teitelboim parity conditions often used to achieve a satisfactory definition of center of mass. These originate in our joint work with Jan Metzger. This will require appealing to Bartnik's asymptotic harmonic coordinates.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

In 2015, Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be ``far away'' from being round. The resulting manifolds, called extensions, are geometrically not ``close'' to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality. Their construction was later adapted to n+1 dimensions by Cabrera Pacheco and Miao, suggesting instability of the higher dimensional Riemannian Penrose Inequality. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum and by Cabrera Pacheco, Cederbaum, and McCormick, a similar construction was performed for asymptotically Euclidean, electrically charged initial data sets and for asymptotically hyperbolic Riemannian manifolds, respectively, obtaining 3-dimensional extensions that suggest instability of the Riemannian Penrose Inequality with electric charge and of the conjectured asymptotically hyperbolic Riemannian Penrose Inequality in 3 dimensions. This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in n+1 dimensions for n greater or equal to 2.

Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

Related project(s):
40Construction of Riemannian manifolds with scalar curvature constraints and applications to general relativity41Geometrically defined asymptotic coordinates in general relativity