On the local well-posedness of Lovelock and Horndeski theories
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Publication Date
2017-08-15Journal Title
Physical Review D - Particles, Fields, Gravitation and Cosmology
ISSN
2470-0010
Publisher
American Physical Society
Volume
96
Number
044019
Language
English
Type
Article
This Version
AM
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Papallo, G., & Reall, H. (2017). On the local well-posedness of Lovelock and Horndeski theories. Physical Review D - Particles, Fields, Gravitation and Cosmology, 96 (044019) https://doi.org/10.1103/PhysRevD.96.044019
Abstract
We investigate local well-posedness of the initial value problem for Lovelock and Horndeski theories of gravity. A necessary condition for local well-posedness is strong hyperbolicity of the equations of motion. Even weak hyperbolicity can fail for strong fields so we restrict to weak fields. The Einstein equation is known to be strongly hyperbolic in harmonic gauge so we study Lovelock theories in harmonic gauge. We show that the equation of motion is always weakly hyperbolic for weak fields but, in a generic weak-field background, it is not strongly hyperbolic. For Horndeski theories, we prove that, for weak fields, the equation of motion is always weakly hyperbolic in any generalized harmonic gauge. For some Horndeski theories there exists a generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a weak-field background. This includes “k-essence” like theories. However, for more general Horndeski theories, there is no generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a generic weak-field background. Our results show that the standard method used to establish local well-posedness of the Einstein equation does not extend to Lovelock or general Horndeski theories. This raises the possibility that these theories may not admit a well-posed initial value problem even for weak fields.
Sponsorship
Science and Technology Facilities Council (ST/P000673/1)
Science and Technology Facilities Council (ST/L000636/1)
Identifiers
External DOI: https://doi.org/10.1103/PhysRevD.96.044019
This record's URL: https://www.repository.cam.ac.uk/handle/1810/265748
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