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Lost horizon? – modeling black holes in string theory

  • Paper in Philosophy of the Natural Sciences
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Abstract

The modeling of black holes is an important desideratum for any quantum theory of gravity. Not only is a classical black hole metric sought, but also agreement with the laws of black hole thermodynamics. In this paper, we describe how these goals are achieved in string theory. We review black hole thermodynamics, and then explicate the general stringy derivation of classical spacetimes, the construction of a simple black hole solution, and the derivation of its entropy. With that in hand, we address some important philosophical and conceptual questions: the confirmatory value of the derivation, the bearing of the model on recent discussions of the so-called ‘information paradox’, and the implications of the model for the nature of space.

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Notes

  1. For instance, see Dardashti et al. (2017), and Crowther et al. (2019).

  2. That black holes have entropy was originally claimed in Bekenstein (1973). After Hawking (1975) showed that black holes radiate, BHT was taken much more seriously. Philosophical work on BHT include Belot et al. (1999), Wallace (2018, 2019, 2020), and Wüthrich (2017). Reviews by physicists include Susskind and Lindesay (2005), Mathur (2009), Harlow (2016), and Polchinski (2017).

  3. See Susskind et al. (1993).

  4. See Belot et al. (1999) and van Dongen and de Haro (2004) for philosophical discussions.

  5. Or in the sources from which it is drawn, e.g. Polchinski (1998). See Vistarini (2019) or Huggett and Wüthrich (forthcoming) for longer philosophical analyses.

  6. A bosonic string field theory, with a 3-point interaction exists (e.g., Taylor, 2009), but is not viewed as a candidate fundamental string theory.

  7. Green et al. (1987, §3.4.1) give the following demonstration of their equivalence: a coherent state of strings, each in a massless spin-2 state, introduces a term γμν in the path integral (2) which adds to the Minkowski metric to produce ημν → gμν = ημν + γμν. Since the path integral determines all physical quantities in a quantum theory, we have fully equivalent theories whether we introduce the curved metric as a classical field or as a graviton state.

  8. It is worth stressing that the expansion is in \(\alpha ^{\prime }\), so that the approximation is prima facie valid when the radius of spacetime curvature is small compared to the string length: say, compared to the Planck length – far beyond the regime of linear gravity. We will, however, see that it seems to break down in a ‘fuzzball’, even for moderate curvature.

  9. True, the full metric contains Minkowski and stringy parts: gμν = ημν + γμν. But the conclusion that ημν is a non-stringy classical background can be resisted: Witten (1996), Matsubara (2013), Huggett (2015a), and Luboš (2012). See Read (2019) for more on fungibility.

  10. Hence it is popular in pedagogical presentations, e.g. Das and Mathur (2000) and Zwiebach(2004, chapter 22.) The origin of this type of construction to show that the entropy agrees with the Bekenstein-Hawking formula (1) is Strominger and Vafa (1996), but the specific approach discussed was proposed in Callan and Maldacena (1996). It is studied in historical and conceptual depth in De Haro et al. (2020): inter alia, this paper describes the state of string theory before and after 1996, the technical and conceptual details of the calculation, and its subtle logic (drawing on a number of approximations and correspondences) and limitations. We strongly recommend it for a full treatment beyond the sketch given for our purposes (see also van Dongen et al., 2020).

  11. Three other comments: (1) The calculation does not appeal to perturbative string theory laid out earlier. (2) For details we refer the reader to our other references, and especially to De Haro et al. (2020) for the role of BPS states (§3.1), and the complex relations between the free and interacting pictures (§3.3) justifying the result. (3) We thank a referee for patiently clarifying the significance of the following point.

  12. An earlier program due to Susskind, on which he reflects in (2006), approached the same problem by adiabaticity; that slowly lowering the string interaction to zero would not change the state counting. This method is more general, allowing the Boltzmann entropy to be calculated for a range of realistic, non-extremal black holes, but is less reliable because it doesn’t have the BPS guarantee that the density of states is constant.

  13. See Wallace (2018, 2019); van Dongen et al. (2020) and De Haro et al. (2020) for more detailed discussions.

  14. See De Haro et al. (2020, §3.3) for a detailed discussion of the role of this assumption in the entropy calculation.

  15. Huggett and Wüthrich (2013) explores spacetime emergence.

  16. In addition, see Wallace (2020) for a convincing demonstration that there are forms of the paradox that resist Maudlin’s analysis.

  17. See Susskind (2012a, 2012b, 2012c) for further discussions.

  18. Rosaler (2013) is a significant exception, and it is explored further in Huggett and Wüthrich (forthcoming).

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Acknowledgements

Some of this work was performed under a collaborative agreement between the University of Illinois at Chicago and the University of Geneva and made possible by grant numbers 56314 and 61387 from the John Templeton Foundation. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the John Templeton Foundation. We also thank John Dougherty for considerable assistance at the start of this work, and an anonymous referee.

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John Templeton Foundation, 56314, 61387.

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  1. Department of Philosophy, University of Illinois at Chicago, 601 South Morgan Street, Chicago, IL, 60607, USA

    Nick Huggett

  2. Department of Philosophy, Uppsala University, Box 627, SE-751 26, Uppsala, Sweden

    Keizo Matsubara

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Correspondence to Nick Huggett.

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Huggett, N., Matsubara, K. Lost horizon? – modeling black holes in string theory. Euro Jnl Phil Sci 11, 70 (2021). https://doi.org/10.1007/s13194-021-00376-3

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