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From Lovelock to Horndeski’s Generalized Scalar Tensor Theory

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Modifications of Einstein's Theory of Gravity at Large Distances

Part of the book series: Lecture Notes in Physics ((LNP,volume 892))

Abstract

We review and discuss some recent progress in Lovelock and Horndeski theories modifying Einstein’s General Relativity. Using as our guide the uniqueness properties of these modified gravity theories we then discuss how Kaluza-Klein reduction of Lovelock theory can lead to effective scalar-tensor actions including several important terms of Horndeski theory. We show how this can be put to practical use by mapping analytic black hole solutions of one theory to the other. We then elaborate on the subset of Horndeski theory that has self-tuning properties and review a generic method giving scalar-tensor black hole solutions.

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Notes

  1. 1.

    Local distance scales range up to 30 odd astronomical units, size of the solar system, but also size of typical binary pulsar systems. The astronomical unit is a rough earth to sun distance.

  2. 2.

    For a interesting proposal tackling the cosmological constant problem including the crucial radiative corrections see, [8].

  3. 3.

    The result depicted here is easily extended to manifolds with boundaries [12].

  4. 4.

    When a surface has a boundary an analogous result holds.

  5. 5.

    In the action one can always add terms that can be written as a total divergence. Therefore the term “unique action” refers to the unique class of equivalence which is in turn defined modulo total divergence terms. In other words two actions are equal if and only if they are in the same class of equivalence or they differ only by a totally divergent term.

  6. 6.

    The special relation between the coupling parameters corresponds to the strong coupling limit of EGB-literally the case where the Gauss-Bonnet term is of maximal relative strength to the Einstein-Hilbert term and gives a very special theory with enhanced symmetries, usually referred to as Chern–Simons theory (see the nice review [27]).

  7. 7.

    We will see that when the horizon sections carry non-zero curvature there is a global change in the topology of the solution related to the presence of a solid angle deficit. This will end up having important consequences that we will discuss in detail later with the solution at hand.

  8. 8.

    In higher order Lovelock theory there are more according to the order of the highest order Lovelock term [12].

  9. 9.

    Clearly, had we been seeking a self-tuning solution in the presence of an arbitrary cosmological constant this linear anzatz would not do. We know rather that there must be at large distance a t 2 dependence on the scalar field. This unfortunately renders the field equations t-dependent and the system cannot admit a non zero mass solution. In other words a self-tuning black hole would have to be part of a radiating space-time. Again this is an open problem.

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Acknowledgements

I am very happy to acknowledge numerous interesting discussions with my colleagues during the course of the Aegean summer school. I am also indebted to E Papantonopoulos for the very effective and smooth organisation of the school. I am very happy to thank Eugeny Babichev for numerous comments as well as Stanley Deser for remarks on the first version of this paper. I also thank the CERN theory group for hosting me during the final stages of this work.

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  1. Laboratoire de Physique Théorique (LPT), Univ. Paris-Sud, CNRS UMR 8627, 91405, Orsay, France

    Christos Charmousis

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Correspondence to Christos Charmousis .

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  1. Department of Physics, National Technical University of Athens, Athens, Greece

    Eleftherios Papantonopoulos

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Charmousis, C. (2015). From Lovelock to Horndeski’s Generalized Scalar Tensor Theory. In: Papantonopoulos, E. (eds) Modifications of Einstein's Theory of Gravity at Large Distances. Lecture Notes in Physics, vol 892. Springer, Cham. https://doi.org/10.1007/978-3-319-10070-8_2

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