Hu, Yichao; (2024) Covariant Radiative Transfer in Dynamical Spacetime: a 5 Dimensional Formulation. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

We propose a novel approach for constructing a covariant formulation of radiative transfer in dynamical spacetimes, which overcomes limitations of previous methods when they are applied in the strong-field regime, by promoting the 3+1 numerical relativity (NR) decomposition via an embedding of a 4 dimensional spacetime into a 5 dimensional non-flat pseudo-Riemannian manifold. This new formulation uses a 4+1 approach: one is able to calculate, in a physically-consistent way, the null geodesics emitted from gravitational wave (GW) sources, e.g., from black hole and neutron star coalescence. Chapters 1–3 introduce the fundamental knowledge for this work and review previous studies of the general relativistic radiative transfer formulation in stationary spacetimes (e.g., Kerr). Chapter 4 introduces the level set method, which is applied to evolving the 4 dimensional spacetime (and null geodesics) in higher dimensional manifolds. Chapter 5 discusses the causal structure of a generic spacetime and studies the embedding of a 4 dimensional spacetime in a 5 dimensional flat and a non-flat manifold. We recover the Lorentz structure by choosing a specific isometric embedding and by defining an appropriately-chosen form of the 5 dimensional metric (e.g., the Schwarzschild and Oppenheimer-Snyder metrics). Chapter 6 discusses the embedding of the 3+1 numerical representation of the Kerr black hole and a 4 dimensional Brill-Lindquist spacetime. In Chapter 7, we present the proof that the isometric embedding of a 4 dimensional Lorentzian manifold in a 5 dimensional manifold with chosen metric, where the Lorentz structure is enforced, exists and is non-unique. Chapter 8 looks at the construction of a covariant radiative transfer formulation for a binary black hole system. We apply the embedding method for an equal mass non-spinning black hole merger using 3+1 numerical relativity and find the evolution equation of the geometric flow (spacetime flow). A summary of the work presented in this thesis, together with discussions and additional remarks, is presented in Chapter 9. Finally, directions for future work are presented in Chapter 10.

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