The positive mass theorem with angular momentum and charge for manifolds with boundary

Edward, Bryden,; Khuri, Marcus; Sokolowsky, Benjamin

doi:10.1063/1.5070080

Cited by 7 publications

(5 citation statements)

References 27 publications

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“…We mention this result does not contradict the results found in [13,14] as in addition to the weak energy condition, the authors again assume maximality. Also, a lower bound for the ADM energy, independent of any energy conditions, was recently obtained in [15].…”

Section: The Positive Mass Theoremsupporting

confidence: 74%

Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition

Jaracz¹

2022

Class. Quantum Grav.

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Of the various energy conditions which can be assumed when studying mathematical general relativity, intuitively the simplest is the weak energy condition μ≥0 which simply states that the observed mass-energy density must be non-negative. This energy condition has not received as much attention as the so-called dominant energy condition. When the natural question of the Penrose inequality in the context of the weak energy condition arose, we could not find any results in the literature, and it was not immediately clear whether the inequality would hold, even in spherical symmetry. This led us to constructing a spherically symmetric asymptotically flat initial data set satisfying the weak energy condition which violates the "usual" formulations of the Penrose conjecture. This does not contradict [18], as there the data is assumed to be maximal. Our result is unexpected and interesting because the heuristic argument behind the Penrose inequality relies on the black hole area law which holds when the weak energy condition is satisfied. Our methods also lead to a counter example to the positive mass theorem assuming the weak energy condition.

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Section: The Positive Mass Theoremsupporting

confidence: 74%

Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition

Jaracz¹

2022

Class. Quantum Grav.

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“…It should be pointed out that the hypotheses used in Theorem 2.3 is strong, but it is necessary to address the rigidity cases. In general, we remove simple connectivity and completeness for non-smooth initial data sets in three and four dimensions and prove strict inequalities which are generalization of [9]. (b) If M 4 is spin, π 1 pΣ min q " 0, and H 2 pM 4 zW q " 0, where BW " BM n Y Σ min , then the strict inequality in Theorem 2.3-(b) holds.. (c) If M 4 is spin, E " B " 0, π 1 pΣ min q " Z, H 2 pM 4 zW q " Z and J 1 ě J 2 , then the strict inequality in Theorem 2.3-(c) holds.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…This shows that the simple connectivity and two ends assumptions in above inequalities are only technical restriction and not physical features of black holes. Therefore, recently, Khuri, Bryden, and Sokolowsky [9] showed that the strict inequality of (1.2) is true for initial data sets with minimal axially symmetric boundary BM 3 without simple connectivity assumption.…”

Section: Introductionmentioning

confidence: 99%

Positive mass theorem for initial data sets with corners along a hypersurface

Alaee¹,

Yau²

2019

Preprint

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We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory and 5-dimensional minimal supergravity theory which metrics fail to be C 1 and second fundamental forms and electromagnetic fields fail to be C 0 across an axially symmetric hypersurface Σ. Furthermore, we remove the completeness and simple connectivity assumptions in this result and prove it for manifold with boundary such that the mean curvature of the boundary is non-positive.

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“…Proof. The following argument is a generalization of that in [8] for outermost minimal surfaces in axisymmetry. Suppose that the outermost apparent horizon Σ does not admit the stated symmetry.…”

Section: Manifolds With Spherical Symmetrymentioning

confidence: 97%

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

Edward¹,

Khuri²,

Sormani³

2019

Preprint

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The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in L p , 1 ≤ p < 2. In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.

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The positive mass theorem with angular momentum and charge for manifolds with boundary

Cited by 7 publications

References 27 publications

Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition

Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition

Positive mass theorem for initial data sets with corners along a hypersurface

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

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