BEGIN:VCALENDAR VERSION:2.0 PRODID:-//linuxsoftware.nz//NONSGML Joyous v1.4//EN BEGIN:VEVENT SUMMARY:Characteristic formulations of general relativity and applications DTSTART:20220926T160000Z DTEND:20220926T180000Z DTSTAMP:20260810T220800Z UID:bc9f9983-197a-4c5e-9294-60c1632bb193 SEQUENCE:2 CREATED:20220923T214651Z DESCRIPTION:Abstract:General relativity can describe various gravitational systems of astrophysical relevance\, like black holes and neutron stars\, or even strongly coupled systems through the holographic duality. In addi tion\, more formal aspects of the theory like the stability of spacetimes and the formation of singularities are still topics of active research. In several cases\, solutions in closed analytic form are not known\, and per turbative methods are inadequate\, leading to the employment of numerical techniques. The characteristic initial (boundary) value problem has numero us applications in general relativity involving numerical studies and is o ften formulated using Bondi-like coordinates. Well-posedness of the result ing systems of partial differential equations\, however\, remains an open question. The answer to this question affects the accuracy\, and potential ly the reliability of conclusions drawn from numerical studies based on su ch formulations. A numerical approximation can converge to the continuum l imit only for well-posed systems. The notion of well-posedness is tightly related to that of hyperbolicity and includes the specification of a norm. In the first part of this thesis\, we expand our understanding of the hyp erbolicity and well-posedness of Bondi-like free evolution systems. We sho w that several prototype Bondi-like formulations are only weakly hyperboli c and examine the root cause of this result. In a linear analysis we ident ify the gauge\, constraint and physical blocks in the principal part of th e Einstein field equations in such a gauge\, and we show that the subsyste m related to the gauge variables is only weakly hyperbolic. Weak hyperboli city of the full system follows as a consequence in many cases. We demonst rate this explicitly in specific examples\, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite gen eral conditions. Consequently\, the characteristic initial (boundary) valu e problem of general relativity in these gauges is rendered ill-posed in t he simplest norms one would like to employ. We discuss the implications of this result in accurate gravitational waveform modeling methods and work towards the construction of alternative norms that might be more appropria te. We also present numerical tests that demonstrate weak hyperbolicity in practice and highlight important features to perform them effectively. In the second part\, we turn our attention to applications of these formulat ions to strongly coupled systems via holography. We expect these studies t o shed more light on the qualitative behavior of strongly coupled plasmas\ , but due to weak hyperbolicity\, we cannot perform rigorous error estimat es to our satisfaction. We present Jecco\, a newly developed characteristi c code that allows us to simulate the dynamics of strongly coupled plasmas . Representative examples of the simulations that can be achieved with thi s code are provided\, namely the out-of-equilibrium dynamics of said plasm as that undergo phase transitions. This is a putative scenario of the earl y universe and such simulations might provide insights into questions of f undamental nature. LAST-MODIFIED:20220923T214815Z LOCATION:Online URL:http://df.vps.tecnico.ulisboa.pt/pt/eventos/characteristic-formulation s-of-general-relativity-and-applications/ X-ALT-DESC;FMTTYPE=text/html:

Abstract:

General relativity can describe various gravitational systems of astrophysical relevance\, like black holes and neutron stars\, or even strongly coupled systems through the holographic duality. In addit ion\, more formal aspects of the theory like the stability of spacetimes a nd the formation of singularities are still topics of active research. In several cases\, solutions in closed analytic form are not known\, and pert urbative methods are inadequate\, leading to the employment of numerical t echniques.

The characteristic initial (boundary) value problem h as numerous applications in general relativity involving numerical studies and is often formulated using Bondi-like coordinates. Well-posedness of t he resulting systems of partial differential equations\, however\, remains an open question. The answer to this question affects the accuracy\, and potentially the reliability of conclusions drawn from numerical studies ba sed on such formulations.

A numerical approximation can converge to the continuum limit only for well-posed systems. The notion of well-po sedness is tightly related to that of hyperbolicity and includes the speci fication of a norm. In the first part of this thesis\, we expand our under standing of the hyperbolicity and well-posedness of Bondi-like free evolut ion systems. We show that several prototype Bondi-like formulations are on ly weakly hyperbolic and examine the root cause of this result. In a linea r analysis we identify the gauge\, constraint and physical blocks in the p rincipal part of the Einstein field equations in such a gauge\, and we sho w that the subsystem related to the gauge variables is only weakly hyperbo lic. Weak hyperbolicity of the full system follows as a consequence in man y cases.

We demonstrate this explicitly in specific examples\, a nd thus argue that Bondi-like gauges result in weakly hyperbolic free evol ution systems under quite general conditions. Consequently\, the character istic initial (boundary) value problem of general relativity in these gaug es is rendered ill-posed in the simplest norms one would like to employ. W e discuss the implications of this result in accurate gravitational wavefo rm modeling methods and work towards the construction of alternative norms that might be more appropriate.

We also present numerical tests that demonstrate weak hyperbolicity in practice and highlight important f eatures to perform them effectively. In the second part\, we turn our atte ntion to applications of these formulations to strongly coupled systems vi a holography. We expect these studies to shed more light on the qualitativ e behavior of strongly coupled plasmas\, but due to weak hyperbolicity\, w e cannot perform rigorous error estimates to our satisfaction. We present Jecco\, a newly developed characteristic code that allows us to simulate t he dynamics of strongly coupled plasmas.

Representative examples of the simulations that can be achieved with this code are provided\, nam ely the out-of-equilibrium dynamics of said plasmas that undergo phase tra nsitions. This is a putative scenario of the early universe and such simul ations might provide insights into questions of fundamental nature.

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