Marcus Khuri

On the Penrose inequality for charged black holes

Bray and Khuri (2011 Asian J. Math. 15 557–610; 2010 Discrete Continuous Dyn. Syst. A 27 741766) outlined an approach to prove the Penrose inequality for general initial data sets of the Einstein equations. In this paper we extend this approach so that it may be applied to a charged version of the Penrose inequality. Moreover, assuming that the initial data are time-symmetric, we prove the rigidity statement in the case of equality for the charged Penrose inequality, a result which seems to be absent from the literature. A new quasi-local mass, tailored to charged initial data sets is also introduced, and used in the proof.

Existence and Blow-Up Behavior for Solutions of the Generalized Jang Equation

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique.

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 Bartniks definition of quasi-local mass.

Rigidity in the positive mass theorem with charge

In this paper, we show how a natural coupling of the Dirac equation with the generalized Jang equation leads to a proof of the rigidity statement in the positive mass theorem with charge, without the maximal slicing condition, provided a solution to the coupled system exists.

Existence of black holes due to concentration of angular momentum

A bstractWe present a general sufficient condition for the formation of black holes due to concentration of angular momentum. This is expressed in the form of a universal inequality, relating the size and angular momentum of bodies, and is proven in the context of axisymmetric initial data sets for the Einstein equations which satisfy an appropriate energy condition. A brief comparison is also made with more traditional black hole existence criteria based on concentration of mass.

Deformations of Axially Symmetric Initial Data and the Mass-Angular Momentum Inequality

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.

Hoop conjecture in spherically symmetric spacetimes

We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is, they do not require any auxiliary assumptions such as maximality, time symmetry, or special extrinsic foliations, and most importantly they can easily be generalized to the nonspherical case once an existence theory for a modified version of the Jang equation is developed. Moreover, our methods also yield positivity and monotonicity properties of the Misner-Sharp energy.

The positive mass theorem for multiple rotating charged black holes

In this paper a lower bound for the ADM mass is given in terms of the angular momenta and charges of black holes present in axisymmetric initial data sets for the Einstein–Maxwell equations. This generalizes the mass-angular momentum-charge inequality obtained by Chrusciel and Costa to the case of multiple black holes. We also weaken the hypotheses used in the proof of this result for single black holes, and establish the associated rigidity statement.

Deformations of Charged Axially Symmetric Initial Data and the Mass–Angular Momentum–Charge Inequality

We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi:10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case.

A note on the nonexistence of generalized apparent horizons in Minkowski space

We establish a positive mass theorem for initial data sets of the Einstein equations having generalized trapped surface boundary. In particular, we answer a question posed by R Wald concerning the existence of generalized apparent horizons in Minkowski space.