Program

1. Thomas Mädler, Jeffrey Winicour: Bondi-Sachs Formalism. arXiv link
2. Robert Bartnik: Einstein equations in the null quasi-spherical gauge. arXiv link.

Einstein equation in the null quasi-spherical gauge (I)

In this lecture, I will discuss the null quasi-spherical gauge, which is a coordinate gauge to study Einstein's equation based on expanding null hypersurfaces foliated by metric 2-spheres. Introduced by Robert Bartnik, this framework provides an effective way to study Einstein's equation in which the equation takes a simple form.

Memory effect in gravitational radiation (II)

In this lecture, I will discuss the memory effect in gravitational radiation at null infinity of spacetime. This memory effect is introduced by Christodoulou based upon his earlier work on the stability of the Minkowski space with Klainerman. In particular, it demonstrates the non-linear nature of Einstein’s equation.

Wave equation in black hole spacetimes (I, II)

The study of wave equation plays a central role in general relativity and especially in understanding the dynamics of Einstein equations. In this lecture we will discuss linear wave equations in Schwarzschild and slowly rotating Kerr spacetimes and its application to semilinear wave equations under the null condition.

The BMS group: past and future development (I, II)

In general relativity, the gravitational fields of isolated systems are modeled by asymptotically flat spacetimes. Such systems emit gravitational waves which travel at the speed of light and reach future null infinity of the spacetime. In clarifying the nature of this gravitational radiation, Bondi, Metzner, van der Burg, and Sachs discovered a novel group of asymptotic symmetries at future null infinity, the eponymous Bondi-Metzner-Sachs (BMS) group, extending the expected Poincare group via supertranslations. There has been a recent resurgence of interest in the BMS group, both in proposed extensions of the group to include superrotations and in the associated charge algebra following from the Wald-Zoupas procedure. We discuss these developments, along with possible application of the Chen-Wang-Yau definition of quasi-local conserved quantities to this setting.

On the formation of trapped surfaces and the weak cosmic censorship (I, II)

On the formation of trapped surfaces and the weak cosmic censorship Abstract: The study of the formation of trapped surfaces and the large data problem of the Einstein equations without symmetries was pioneered by Christodoulou in 2008. I will discuss this work and its generalizations. Furthermore, I will discuss its relation to the weak cosmic censorship conjecture, which is one of fundamental open problems in general relativity.

Null geometry and the Penrose inequality (I, II)

In the early seventies, Roger Penrose conjectured the inequality:

where |Σ_i| is the area of an ‘initial’ black hole boundary, and M is the total mass of the system. In the appropriate setting, this provides not only a strengthened version of the famous Positive Mass Theorem of Schoen-Yau, and Witten, but also insight regarding the mathematical validity of the weak cosmic censorship hypothesis that Penrose employed in the formulation of his conjecture:

According to the famous Hawking-Penrose singularity theorems, a variety of ‘physically reasonable’ initial data for an isolated system (a cluster of stars, a galaxy, etc...) support solutions of the Einstein field equations with singularities. Under cosmic censorship, a spherical boundary forms prior to the singularity ‘wrapping it up’, hiding any chaotic physics likely to ensue. This boundary traces out the event horizon, a semi-permeable barrier trapping even light from escaping to the outside (i.e a black hole). As matter continues to fall through the horizon, the Hawking area theorem describes that the area of the boundary expands to the future. Outside the horizon, the system approaches equilibrium via the dissipation of gravitational radiation, measured by the loss in Trautman-Bondi mass along null infinity. It is expected that the spacetime consequently settles towards a stationary vacuum solution of the field equations, namely a member of the Kerr family of rotating black holes. Here, all cross-sections of the event horizon are isometric with (‘final’) area |Σ_f|. Moreover,

|Σ_f| ≤ 16π(m^f_TB)²

for ^mf_TB the irreducible Trautman-Bondi mass. Therefore, Penrose concludes:

The existence and exact location of the event horizon assumes knowledge of the complete future evolution of the spacetime and its structure. Consequently, cosmic censorship posits a fundamental mathematical structure to solutions of Einstein’s field equations. The conjecture is therefore not only an interesting bound on a global geometric invariant (the mass), it hints at the validity of cosmic censorship.

In these talks we consider a formulation of the Penrose conjecture for null hypersurfaces in spacetime. A hypersurface being null for inheriting a degenerate metric from the ambient spacetime. The bulk of our time will be spent developing the fundamentals of null geometry and showing why it represents the geometry of light propagation in general relativity. We will also construct and motivate old and new quasi-local mass quantities in order to formulate the notion of total mass. We will conclude with previous and recent progress on the, so called Null Penrose Conjecture, including some interesting open problems.

The structure of null infinity of spacetime (I, II)

Null infinity is where the detection of gravitational waves takes place. It also determines the global structure of spacetime, in particular the location of the black hole region. I shall discuss topics such as the Bondi-Trautman mass, the mass loss formula, and the BMS (Bondi-MetzlerSachs) group.

Conformal Infinity (I, II)

The conformal framework was introduced by Penrose in 1963 to study the gravitational field at large distance. The aim is to provide a coordinate-dependent description to null infinity of an asymptotically flat spacetime. We will follow Geroch’s lecture note ”Asymptotic Structure of Space-Time” to study this subject and compare it with the other coordinate based approaches introduced in the summer course.