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, ), il est facile de vérifier que l'action Einstein-Maxwell-dilaton-Katz "on shell" satisfait encore la relation de Gibbs, présentée au chapitre précédent, bien que ce calcul ne fasse pas l
,
), supposé régulier. Dans le cas d'un trou noir, qui est singulier en r = 0, l'intégration du Lagrangien d'Einstein (où de celui d'Einstein-Katz, qui en est la généralisation covariante), ne peut s'étendre que de l ,
énergie libre" de Gibbs, suggérant là encore la thermodynamique comme outil privilégié pour interpréter la "skeletonisation ,
de sorte que son entropie S reste elle aussi constante. Le fait que son mouvement soit entièrement décrit par les deux constantes q A et µ A trouve donc une explication naturelle dans le cadre de la thermodynamique : si q A s'identifie à sa charge U(1) conservée, la constante d'intégration notée µ A dans le chapitre 4, s'avérait coïncider avec la masse irréductible du trou noir, µ A = ? A + /16? (avec A + l'aire de son l'horizon) pour la bonne raison que son entropie ,
, Elle permet en effet d'interpréter, et peutêtre même de guider, la façon dont la trajectoire d'un trou noir peut être réduite à une ligne d'univers. Elle montre en particulier que les échanges de masse sous forme d'ondes gravitationnelles et de charges sont ignorés. De plus, il est clair que, dans les derniers stades précédant la fusion d'un système binaire, où la période orbitale devient comparable au temps de relaxation d'un trou noir, il n'est plus possible de le considérer comme étant à l'équilibre, aussi bien en théories EMD qu'en relativité générale d'ailleurs : son comportement ne satisfait plus la première loi de la thermodynamique, L'article qui suit ouvre donc une interface entre la thermodynamique des trous noirs d'une part
, ) doit donc, probablement, être étendue pour pouvoir décrire une particule sensible aussi aux gradients {? t , ? i } des champs, ce qui permettra d'encoder, cette fois-ci, la réponse du trou noir à un environnement non plus quasi-statique, mais dynamique, cf
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, La dégénérescence avec la relativité générale étant ainsi levée, le formalisme EOB permettra de contraster les prédictions des théories EMD à celles de la relativité générale, même lorsque le dipôle est nul, dans le régime de champ fort, Conclusion en calculant le flux d'ondes gravitationnelles à l'ordre 1PK (l'approximation du dipôle nul suffisant)
, Cette information s'avérera utile dans le régime de champ fort, et pourra faire l'objet de travaux futurs afin de l'incorporer au sein du formalisme EOB. De la même façon, il serait utile de calculer le lagrangien EMD à l'ordre 2PK
, Afin de rendre compte de l'augmentation de l'entropie des trous noirs à laquelle on s'attend près de leur coalescence, une façon de procéder est de faire dépendre leurs sensibilités des gradients des champs, par exemple, du champ scalaire : comme l'ont montré T. Damour et G. Esposito-Farèse dans, comme nous l'avons suggéré dans le chapitre 7 par l'exemple des trous noirs EMD
, c 2 ) 2 . Comme de plus ? = O(v 2 /c 2 ), on trouve que ce terme en gradients est accompagné d'un préfacteur 1/c 6 . C'est donc un terme d'ordre 3PK, contrairement à 5PN pour celui décrivant les effets de taille finie en relativité générale, faisant intervenir par covariance le tenseur de Weyl au carré, La fonction n A (?) a la dimension d'une masse fois une longueur au carré, ce qui permet de l'estimer
, Une autre extension possible de cette thèse concerne l'étude de la relaxation du trou noir final issu de la fusion d'un système binaire en théories EMD. En effet
, Un projet futur (en collaboration avec E. Berti) est donc d'étudier la théories des perturbations des trous noirs EMD, afin d'en déterminer les modes quasi-normaux (notons d'ailleurs que des travaux préliminaires ont été effectués dans cette direction par R, Mais si nous voulons pousser son évolution au-delà de l'ISCO, il faut
, Enfin, les méthodes développées ici pourront être étendues à d'autres gravités modifiées, telles que les théories scalaire-tenseur couplées à l'invariant de Gauss-Bonnet, dans lesquelles les trous noirs manifestent des phénomènes de scalarisation spontanée similaires à ceux des théories scalaire-tenseur, vol.118
, Une autre possibilité à explorer, dans la continuité des théories EMD, serait celle où le couplage du champ scalaire ? au "graviphoton" A µ devient une fonction générique
EOBisée" de corps compacts (y compris de trous noirs) en théories de gravité modifiée, et qui, vu la nouveauté du sujet, s'est arrétée aux premières étapes ,
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