Scientific programme

In this talk, our object of study is the Einstein-Vlasov system with a massless Vlasov matter field. Our main contribution is the provision of a novel technique for controlling the Vlasov matter in a double null gauge, relying purely on vector field commutation and bypassing the need for introducing Jacobi fields, which has so far been the only existing technique outside of symmetry in this gauge. To obtain it, we have to overcome stringent regularity issues that are existent along this path. Because it relies solely on vector field commutation, our technique is fundamentally simple and flexible enough to be applicable in any data regime. We thus anticipate this to be a helpful tool for many subsequent problems regarding the Einstein-Vlasov system, including, for instance, simplified approaches to the proof of nonlinear stability of Minkowski spacetime with massless Vlasov matter, scattering problems for data close to black holes etc. We further introduce and work in an optimal regularity framework, using this new technique, to obtain a large data semi-global existence theorem and a dynamical trapped surface formation statement for our system. Joint work with Puskar Mondal and Shing-Tung Yau.

Robinson-Trautman waves are a class of exact solutions to Einstein's equation that describe a system that settles down to boosted Schwarzschild black holes through the emission of gravitational radiation. By analyzing them through the lens of physics on null infinity, we are able to explicitly work out their displacement and non-linear memory as well as the thermodynamics of boosted Schwarzschild black holes, which correspond to the equilibrium manifold of this dissipative system.

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In this talk, we will discuss the initial boundary value problem in general relativity from a mathematical point of view. We will first make the analogy to the classical Cauchy problem for the Einstein equations, first resolved by Yvonne Choquet-Bruhat in 1952, and highlight some desired properties that a potential treatment of timelike boundaries should have. Then, we will go over the various technical difficulties that arise in such an endeavor. The main result that we will present in the end concerns the local well-posedness of the problem for homogeneous Neumann conditions, in geometric terms, when the boundary is umbilic. This is joint work with Jacques Smulevici.

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Null boundaries provide a natural setting for exploring soft degrees of freedom and infinite-dimensional symmetries. I will discuss the Hamiltonian structure of electromagnetism in three and four spacetime dimensions, considering two representative types of null boundaries: null infinity in flat spacetime and black hole or cosmological horizons. Besides the standard charges associated with gauge transformations, the null foliation reveals additional edge observables generated by an intrinsic symmetry of the boundary. Together, these charges form infinite-dimensional algebras with nontrivial central extensions, whose detailed structure depends on the spacetime dimension and the geometry of the boundary. These results suggest that additional soft sectors are a general feature of field theories formulated on null hypersurfaces.

In this talk I will present two protocols---one for asymptotically AdS spacetimes and one for asymptotically flat spacetimes---for how to detect small excitations in the bulk of the spacetime, from measurements performed in a small time-band at the AdS boundary in the asymptotically AdS case and  from measurements in a small cut at the past of the future null infinity, in the asymptotically flat case, respectively. The protocols are possible due to the failure of the split property for quantum gravitational theories and hint to the holographic nature of quantum gravity. Moreover, I will emphasise the various assumptions and limitations and I will connect these results with recent works on the type of the algebras of observables in the two cases.

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In this talk, I will present how techniques from quantum information theory and quantum thermodynamics can be used to derive the second law of thermodynamics for quantum fields. I will begin with the case of a finite quantum system undergoing a Markovian evolution driven by an infinite bath, whose dynamics are governed by the Lindblad equation and whose thermodynamic properties are well understood. Building on this framework, I will show how a natural generalization allows us to extend these methods to the thermodynamics of quantum fields defined on causal horizons. Unlike finite quantum systems, quantum field theory admits many unitarily inequivalent Hilbert spaces, each constructed from a distinct choice of vacuum state. I will argue that these different vacua can be interpreted as corresponding to different reservoirs driving the dynamics. In particular, I will demonstrate how Wall’s proof of the generalized second law fits naturally within this framework, and how analogous thermodynamic laws describing black hole thermodynamics emerge for asymptotic observers. These laws correspond to different thermodynamic potentials associated with distinct vacuum states, such as the Boulware, Hartle–Hawking, and Unruh vacua.

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The fine asymptotic properties of dispersive gauge theories, the archetypal examples of which are the Maxwell-Klein-Gordon and the Yang-Mills-Higgs equations, depend crucially on the choice of gauge as well as the scale and regularity of the initial data. In physics, these asymptotic properties are expected to play a role as some of the conditions for a good definition of a unitary S-matrix and an analytic understanding of infrared divergences. Motivated by these questions, in this talk I will discuss the analytic properties of the Cauchy and Goursat problems for the Maxwell-Klein-Gordon equations, the appearance of null and weak null structures in the equations depending on gauge, and discuss recent work (with J.-P. Nicolas) in which we prove that certain dynamical charged scalar fields possess scattering states on null infinity. Time permitting, I will finish by discussing ongoing work which develops a novel time-asymmetric ``radiation gauge’’ adapted to the Goursat problem from null infinity, and touch on the question of time-asymmetric peeling going back to Penrose, Damour, and Christodoulou.

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