U.U.D.M. Project Report 2020:27 Spacetime Penrose Inequality For Asymptotically Hyperbolic Spherical Symmetric Initial Data Mingyi Hou Examensarbete i matematik, 30 hp Handledare: Anna Sakovich Examinator: Julian Külshammer Augusti 2020 Department of Mathematics Uppsala University Spacetime Penrose Inequality For Asymptotically Hyperbolic Spherically Symmetric Initial Data Mingyi Hou Abstract In 1973, R. Penrose conjectured that the total mass of a space-time containing black holes cannot be less than a certain function of the sum of the areas of the event horizons. In the language of differential geometry, this is a statement about an initial data set for the Einstein equations that contains apparent horizons. Roughly speaking, an initial data set for the Einstein equations is a mathematical object modelling a slice of a space-time and an apparent horizon is a certain generalization of a minimal surface. Two major breakthroughs concerning this conjecture were made in 2001 by Huisken and Ilmanen respectively Bray who proved the conjecture in the so-called asymptotically flat Riemannian case, that is when the slice of a space-time has no extrinsic curvature and its intrinsic geometry resembles that of Euclidean space. Ten years later, Bray and Khuri proposed an approach using the so-called generalized Jang equation which could potentially be employed to deal with the general asymptotically flat case by reducing it to the Riemannian case. Bray and Khuri have successfully implemented this strategy under the assumption that the initial data is spherically symmetric. In this thesis, we study a suitable modification of Bray and Khuri's approach in the case when the initial data is asymptotically hyperbolic (i.e. modelled on a hy- perboloidal slice of Minkowski spacetime) and spherically symmetric. In particular, we show that the Penrose conjecture for such initial data holds provided that the generalized Jang equation has a solution with certain asymptotic behaviour near infinity. Furthermore, we prove a result concerning the existence of such a solution. Contents 1 Introduction1 2 Preliminaries4 2.1 Apparent Horizons............................4 2.2 Asymptotically Flat Initial Data.....................5 2.3 Asymptotically Hyperbolic Initial Data.................5 2.4 Spherically Symmetric Asymptotically Hyperbolic Initial Data..........................6 3 The Generalized Jang Equation8 3.1 The Generalized Jang Equation Approach...............8 3.2 Spherical Symmetry...........................9 4 Motivation for Asymptotics and the Warping Factor 12 4.1 Behavior at Infinity............................ 12 4.2 Proof of the Inequality.......................... 15 5 Existence and Asymptotics 18 5.1 Existence................................. 18 5.2 Asymptotics................................ 19 Chapter 1 Introduction In the 1970s, physicist R. Penrose conjectured that the total mass of a spacetime containing black holes with event horizons of total area A should be at least pA=16π. In his argument, cosmic censorship, a famous conjecture saying that singularities arising in the solutions of the Einstein equations are typically hidden within event horizons, is one of the fundamental ingredients. In fact, a proof of a suitable version of the Penrose inequality would give indirect support to cosmic censorship and a counter example of the inequality would very likely involve a spacetime for which cosmic censorship fails to hold [Mar09]. Thus, finding a proof for the inequality has become an active area of research. An important case of the Penrose inequality is a statement about asymptotically flat initial data. Recall that a spacetime is a Lorentzian manifold (N ; gN ) satisfying the Einstein equations, 1 Ric − R g = 8πT (1.1) N 2 N N where T is the so-called stress-energy tensor, RicN is the Ricci curvature tensor and RN is the scalar curvature. We call a triple (M; g; K) an initial data set (or Cauchy data) if (M; g) is a Riemannian manifold and K is a symmetric 2-tensor, such that the so-called constraint equations ( 1 µ = (R − jKj2 + tr (K)2) 2 g g g (1.2) J = divg(K − trg(K)g) are satisfied. Here Rg is the scalar curvature and µ and J are physically energy and momentum density respectively. In the case when (M; g) is a space-like slice of a spacetime (N ; gN ), with unit normal n, we have µ = T (n; n) and J = T (n; ·). In what follows, we will assume that any given initial data satisfies the so-called dominant energy condition µ > jJj (1.3) which is known to be satisfied in all physically reasonable spacetimes. Note that if (1.2) is satisfied then (M; g; K) is indeed the initial data required to pose an initial value problem for the Einstein equations in the sense that the time evolution of the initial data is a spacetime (N ; gN ) satisfying (1.1) and (M; g) is a space-like hypersurface in the spacetime with second fundamental form K [BI04]. 1 Roughly speaking, an initial data (M; g; K) is asymptotically flat if near infinity g approaches the Euclidean metric δ and K approaches 0. For an asymptotically flat initial data set there is a well known notion of mass [ADM08] that in a certain sense measures how fast the initial data set becomes “flat” at infinity. The argument which can be used to motivate the Penrose inequality roughly goes as follows [Mar09]. Assume a spacetime (N ; gN ) which is asymptotically flat, in the sense that it looks like the Minkowski spacetime [ONe83] at infinity. Intuitively, the event horizon H, is the boundary of the region around a black hole which not even the light can escape. Note that it is a null hypersurface which is at least Lipschitz continuous [Wal84]. Assume now, that the spacetime admits an asymptotically flat slice with total ADM mass MADM and which intersects H on a cut S. If H is a smooth hypersurface, then this cut is a smooth embedded surface which has a well-defined area jSj. Consider any other cut S1 to the future of S along the event horizon. The black hole area law [Haw71; Haw72] states jS1j > jSj, provided that the null energy condition, which is a certain strengthening of (1.3), holds. From physical considerations, the spacetime is expected to settle down to some equilibrium configuration. Assuming that all matter fields are swallowed by the black hole in the process, uniqueness theorems for stationary black holes (see e.g. [Heu96]) imply that the spacetime must approach the Kerr black hole spacetime, where the area of the section S of the event horizon is independent of the cut and takes the value p 2 2 2 2 AKerr = 8πM(M + M − L =M ) 6 16πM , where M is the total mass and L is the total angular momentum of the spacetime. In particular, M should be the asymptotic value of the Bondi mass along the future null infinity. Since gravitational waves carry positive energy, the Bondi mass cannot increase to the future [BVM62; SB62]. The Penrose inequality p MADM > jSj=16π follows provided that the Bondi mass approaches the ADM mass MADM which is known to be true under certain assumptions (see e.g. [Hay03; Kro03; AM79]). After a period of heuristic proofs and partial results, important breakthroughs have been made in the past decades. First, Huisken and Ilmanen [HI01] were able to prove the Penrose inequality in the so called time symmetric case, i.e. when the initial data has K = 0, using a geometric evolution equation known as the inverse mean curvature flow. Shortly afterwards, another proof using the so-called conformal flow was found by Bray. Roughly speaking, the result of [HI01] is for a single black hole, while that of [Bra01] allows for multiple ones. In summary, we have the following result in the time symmetric case: Theorem 1 (Riemann Penrose Inequality). Let (M; g) be a complete, smooth, asymptotically flat 3-manifold with nonnegative scalar curvature, total mass M, and with the boundary (possibly disconnected) which is an outermost minimal surface of total area A. Then r A M (1.4) > 16π with equality if and only if (M; g) is isometric to a Schwarzschild manifold 3 (R n f0g; gSchw). 2 The Penrose inequality is a strengthening of the famous positive mass theorem, saying that an asymptotically flat initial data (M; g; K) satisfying dominant energy condition has MADM > 0, with equality if and only if it is a slice of the Minkowski spacetime. This result was proven in [SY82] by reducing it to the time symmetric case K = 0 that was earlier settled in [SY79]. The proof uses a certain quasi-linear elliptic partial differential equation, the so-called Jang equation. It was suggested by Bray and Khuri (see [BK11; BK10]) that a generalization of the Jang equation could be a way to reduce the general Penrose inequality to the case of time symmetry. This idea has been successfully implemented in the asymptotically flat spherically symmetric case [BK10]. In this thesis, we provide some further support to the generalized Jang equation approach of Bray and Khuri by implementing it in the case when the initial data set is asymptotically hyperbolic spherically symmetric. This setting is of great interest for both geometrical and physical reasons, in particular, it is important for the study of gravitational radiation [Fra04]. The thesis is organized as follows: In Chapter 2, we introduce terminology re- quired in this thesis. In Chapter 3, we briefly review the generalized Jang equation and rewrite it as an ordinary differential equation under the assumption of spherical symmetry. In Chapter 4, we prove a theorem stating that the Penrose inequality for asymptotically hyperbolic spherically symmetric initial data holds provided that a solution to the generalized Jang equation exists and has certain asymptotic behavior at infinity. Finally, in Chapter 5, we prove the existence of a solution to the gener- alized Jang equation and obtain a (non-optimal) result regarding its asymptotics.