2024/05/19 by Jaroslaw S. Jaracz, Jaracz, Jaroslaw S.
Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #Classical mechanics #Conformal map #Energy (signal processing) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geophysics and Gravity Measurements #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Poisson distribution #Poisson's equation #Quantum mechanics #Spacetime #Statistics
paper · pdf · doi:10.48550/arxiv.2405.12249
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2024/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We give an alternate proof of one of the results given in [16] showing that initial data sets with boundary for the Einstein equations (M, g, k) satisfying the dominant energy condition can be conformally deformed to the strict dominant energy condition, while preserving the character of the boundary (minimal, future trapped, or past trapped) while changing the area of the boundary and ADM energy of the initial data set by an arbitrarily small amount. The proof relies on solving an equation that looks like the equation for spacetime harmonic functions studied in [7], but with a Neumann boundary condition and non-zero right hand side, which we refer to as a spacetime Poisson equation. One advantage of this method of proof is that the conformal deformation is explicitly constructed as a solution to a PDE, as opposed to only knowing the solution exists via an application of the implicit function theorem as in [16]. We restrict ourselves to the physically relevant case of a 3-manifold M, though the proof can be generalized to higher dimensions.