scholarly journals Bartnik mass via vacuum extensions

2019 ◽  
Vol 30 (13) ◽  
pp. 1940006
Author(s):  
Pengzi Miao ◽  
Naqing Xie
Keyword(s):  
Initial Data ◽  
Scalar Curvature ◽  
Apparent Horizon ◽  
Gauss Curvature ◽  
Positive Mass Theorem ◽  
Data Sets ◽  
Penrose Inequality ◽  
Asymptotically Flat ◽  
Apparent Horizons ◽  
Positive Gauss Curvature

We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

scholarly journals Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

2017 ◽  
Vol 2017 (727) ◽  
Author(s):  
Lan-Hsuan Huang ◽  
Dan A. Lee ◽  
Christina Sormani
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Positive Mass Theorem ◽  
Flat Manifold ◽  
Positive Mass ◽  
Asymptotically Flat ◽  
Intrinsic Flat

AbstractThe rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and

scholarly journals Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary

2020 ◽  
Author(s):  
Sérgio Almaraz ◽  
Levi Lopes de Lima ◽  
Luciano Mari
Keyword(s):  
Initial Data ◽  
Energy Conditions ◽  
Data Sets ◽  
Positive Mass ◽  
Momentum Vector ◽  
Asymptotically Flat ◽  
The Common ◽  
Underlying Manifold ◽  
Asymptotically Hyperbolic ◽  
Compact Boundary

Abstract In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is light-like. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

scholarly journals The Conformal Flow of Metrics and the General Penrose Inequality

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Qing Han ◽  
Marcus Khuri
Keyword(s):  
Lower Bound ◽  
Initial Data ◽  
Total Mass ◽  
Einstein Equations ◽  
Data Sets ◽  
Penrose Inequality ◽  
General Initial Data ◽  
Time Symmetry ◽  
Conformal Flow ◽  
Special Case

The conformal flow of metrics has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the conformal flow of metrics, so that it may be applied to the Penrose inequality for general initial data sets of the Einstein equations. The Penrose conjecture without the assumption of time symmetry is then reduced to solving a system of PDE with desirable properties.

scholarly journals Initial Data Rigidity Results

2021 ◽  
Author(s):  
Michael Eichmair ◽  
Gregory J. Galloway ◽  
Abraão Mendes
Keyword(s):  
Lower Bound ◽  
Initial Data ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Positive Mass Theorem ◽  
Positive Mass ◽  
Rigidity Result ◽  
Trapped Surfaces ◽  
Rigidity Results ◽  
Special Case

AbstractWe prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.

scholarly journals FORMATION OF TRAPPED SURFACES FOR THE SPHERICALLY SYMMETRIC EINSTEIN–VLASOV SYSTEM

2010 ◽  
Vol 07 (04) ◽  
pp. 707-731 ◽  
Author(s):  
HÅKAN ANDRÉASSON ◽  
GERHARD REIN
Keyword(s):  
Initial Data ◽  
Event Horizon ◽  
Apparent Horizon ◽  
Cosmic Censorship ◽  
Einstein Equations ◽  
Spherically Symmetric ◽  
Asymptotically Flat ◽  
Trapped Surfaces ◽  
Matter Model ◽  
Trapped Surface

We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is analogous to what is found in certain Vaidya spacetimes. The analysis is carried out in Eddington–Finkelstein coordinates.

scholarly journals Apparent horizons in vacuum Robinson-Trautman spacetimes

1999 ◽  
Vol 41 (2) ◽  
pp. 217-230 ◽  
Author(s):  
E. W. M. Chow ◽  
A. W.-C. Lun
Keyword(s):  
Numerical Simulation ◽  
Surface Area ◽  
Gravitational Radiation ◽  
Apparent Horizon ◽  
Retarded Time ◽  
Asymptotically Flat ◽  
The Past ◽  
Apparent Horizons

AbstractVacuum asymptotically flat Robinson-Trautman spacetimes are a well known class of spacetimes exhibiting outgoing gravitational radiation. In this paper we describe a method of locating the past apparent horizon in these spacetimes and discuss the properties of the horizon. We show that the past apparent horizon is non-timelike and that its surface area is a decreasing function of the retarded time. A numerical simulation of the apparent horizon is also discussed.

scholarly journals Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

2021 ◽  
Vol 74 (4) ◽  
pp. 865-905
Author(s):  
Otis Chodosh ◽  
Michael Eichmair ◽  
Yuguang Shi ◽  
Haobin Yu
Keyword(s):  
Scalar Curvature ◽  
Asymptotically Flat
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