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Non-extremal Black-Hole Solutions of \(\mathcal{N }=2,\;d=4,\;5\) Supergravity

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Black Objects in Supergravity

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 144))

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Abstract

Black holes have been intensely studied in the framework of string theory for the last 20 years. They are described by classical solutions of the supergravity theories that describe effectively the low-energy dynamics of different string compactifications. Being solutions of theories with local supersymmetry one can distinguish among them the particular class of those that preserve some unbroken supersymmetries (called supersymmetric or, less precisely, BPS).

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Notes

  1. 1.

    See also [10] for a review.

  2. 2.

    By this we mean that the metric of the spatial part of its worldvolume is Euclidean. As we are going to see, the metric of the full worldvolume is not flat.

  3. 3.

    This metric has also been derived from the equations of motion in Refs. [27].

  4. 4.

    In higher dimensions \(\omega \) is not a length, hence the change in notation.

  5. 5.

    Not to be mistaken for the non-extremality parameter \(\omega \).

  6. 6.

    This is true for \(\omega \ne 0\). The near-horizon behavior for \(\omega =0\) is given by Eq. (4.46).

  7. 7.

    The same result can be obtained by reducing first the action Eq. (4.1) to \((d-p)=(\tilde{p}+4)\) dimensions in such a way that the action only contains the Einstein-Hilbert term, scalars and 1-forms and then by using the FGK formalism of Ref. [20] for \(d\)-dimensional black holes (\(p=0\)). See the Appendix in Ref. [19].

  8. 8.

    Only the supersymmetric attractors, that is, r the values of the scalars of supersymmetric black-brane solutions, are guaranteed to depend only on the charges. The general situation for extremal non-supersymmetric black branes is that the scalars on the horizon keep some dependence on their values at spatial infinity. This situation is sometimes referred to as the existence of a moduli space of attractors parametrized by some numbers whose physical meaning (i.e. their expressions in terms of the physical constants) is seldom given in the literature. These parameters are functions of the moduli, as shown explicitly in the \(\overline{\mathbb{CP }}^{n}\) model studied in [19].

    Of course, there are some moduli-independent non-supersymmetric attractors but it is important to realize that this is what happens in general.

  9. 9.

    In that reference the coordinate used was \(\tau =-\rho \).

  10. 10.

    It would be stressed that this a transformation that relates solutions, and not a coordinate transformation.

  11. 11.

    In the construction of the solution this is achieved by requiring the positivity of certain constants that appear in it, such as the mass or the entropy.

  12. 12.

    A related result valid for horizons of arbitrary topology has been recently found in [37].

  13. 13.

    We have chosen, for convenience, the normalization \(\alpha ^{2}=3/32\).

  14. 14.

    For more information on these theories see, for instance, Ref. [42], the review [43], and the original works [44, 45].

  15. 15.

    These results have been extended to theories with hypermultiplets and tensor multiplets in Refs. [40, 48, 49] but these only include regular black-hole solutions when the additional fields vanish and, therefore, we will not consider them here.

  16. 16.

    The change \(\rho =r^{-2}\) brings the metric to the standard form.

  17. 17.

    The change \(\rho =r^{-1}\) brings the metric to the standard form.

  18. 18.

    These are the supersymmetric solutions such that the vector constructed as a bilinear from its Killing spinor is timelike. In particular, it is a timelike Killing vector. The other possible class is the null class. The supersymmetric extremal black-hole solutions belong to the timelike class.

  19. 19.

    In the gauged theories there are asymptotically-\(AdS\) black holes [55] and also asymptotically-flat, regular black holes with non-Abelian hair [14–16], but here we are not going to consider these cases.

  20. 20.

    The change \(\rho =r^{-1}\) brings the metric to the standard form.

  21. 21.

    Apart from the examples studied in Refs. [19, 20], the assumption is true in all the supersymmetric solutions of \(\mathcal N =2,\;d=4,5\) theories, for all matter couplings and gaugings.

  22. 22.

    A similar, more general, formalism that reduces to the H-FGK one for single, static, spherically-symmetric black holes of \(\mathcal N =2,\;d=4,5\) has been given in Refs. [17, 18, 21]. The \(\mathcal{N }=2,\;d=5\) string case has not been treated with this method.

  23. 23.

    This relation can be derived from the identities in Ref. [57].

  24. 24.

    Observe that we do not really need it, since it does not appear in the original FGK action anyway.

  25. 25.

    It has also been studied via the analogous method mentioned before in Ref. [59].

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Acknowledgments

The author would like to thank his collaborators in the work reviewed here: P. Galli, P. Meessen, J. Perz and C. S. Shahbazi, and the organizers of the BOSS2011 school and workshop. This work has been supported in part by the Spanish Ministry of Science and Education grant FPA2009-07692, the Comunidad de Madrid grant HEPHACOS S2009ESP-1473 and the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042. The author wishes to thank M. M. Fernández for her permanent support.

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  1. Instituto de Física Teórica UAM/CSIC C/ Nicolás Cabrera, 13–15, C.U. Cantoblanco, 28049 , Madrid, Spain

    Tomás Ortín

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    Stefano Bellucci

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Ortín, T. (2013). Non-extremal Black-Hole Solutions of \(\mathcal{N }=2,\;d=4,\;5\) Supergravity. In: Bellucci, S. (eds) Black Objects in Supergravity. Springer Proceedings in Physics, vol 144. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00215-6_4

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