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17002 results

Page 195 of 851

  • 13 - Hyperbolic reductions

  • from Part III - Methods of the theory of partial differential equations
  • Juan A. Valiente Kroon, Queen Mary University of London
  • Book:
    Conformal Methods in General Relativity
    Published online:
    05 July 2016
    Print publication:
    21 July 2016, pp 331-389

  • Summary

    This chapter discusses several methods for the construction of symmetric hyperbolic evolution systems out of the conformal Einstein field equations. Once suitable evolution systems have been obtained, the methods of Chapter 12 allow, in turn, one to make statements about the existence of solutions to the equations. Direct inspection of the conformal field equations reveals that these are overdetermined – there are more equations than unknowns, even if the symmetries of the various tensorial and spinorial fields are taken into account. Thus, the process of hyperbolic reduction for the conformal field equations necessarily requires discarding some of the equations. The discarded equations are then treated as constraints. It is a remarkable structural property of the conformal field equations that these constraints satisfy a system of evolution equations – a so-called subsidiary evolution system – from where it can be concluded that the constraint equations will be satisfied if they hold at some initial hypersurface and the evolution equations are imposed. This construction is called the propagation of the constraints. The solution of the evolution system together with the propagation of the constraints yields the required solution of the conformal Einstein field equations.

    In this chapter, two different procedures for the hyperbolic reduction of the conformal Einstein field equations are considered. The first method, based on the notion of gauge source functions, exploits the fact that certain derivatives of the conformal fields are not directly determined by the equations and, thus, can be freely specified. In the spinorial formulation of the equations, once the required gauge source functions have been specified, the irreducible decomposition of the various zero quantities leads to the required evolution equations. The equations obtained by this procedure include the conformal factor as an unknown.

    The second hyperbolic reduction procedure presented in this chapter exploits the properties of congruences of conformal geodesics to construct conformal Gaussian gauge systems. As discussed in Chapter 5, the connection coefficients and components of the Schouten tensor with respect to a frame which is Weyl propagated along the congruence satisfy certain relations which lead to a particularly simple system of equations in which the evolution of all the geometric unknowns, save for the components of the rescaled Weyl spinor, are either fixed by the gauge or given by transport equations along the congruence.

Conformal Methods in General Relativity
  • Juan A. Valiente Kroon
  • Published online:
    05 July 2016
    Print publication:
    21 July 2016
  • This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this the perfect reference companion on the topic.

Page 195 of 851