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The multiple realizability of general relativity in quantum gravity

Synthese volume 199, pages 441–467 (2021)Cite this article

Abstract

Must a theory of quantum gravity have some truth to it if it can recover general relativity in some limit of the theory? This paper answers this question in the negative by indicating that general relativity is multiply realizable in quantum gravity. The argument is inspired by spacetime functionalism—multiple realizability being a central tenet of functionalism—and proceeds via three case studies: induced gravity, thermodynamic gravity, and entanglement gravity. In these, general relativity in the form of the Einstein field equations can be recovered from elements that are either manifestly multiply realizable or at least of the generic nature that is suggestive of functions. If general relativity, as argued here, can inherit this multiple realizability, then a theory of quantum gravity can recover general relativity while being completely wrong about the posited microstructure. As a consequence, the recovery of general relativity cannot serve as the ultimate arbiter that decides which theory of quantum gravity that is worthy of pursuit, even though it is of course not irrelevant either qua quantum gravity. Thus, the recovery of general relativity in string theory, for instance, does not guarantee that the stringy account of the world is on the right track; despite sentiments to the contrary among string theorists.

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Notes

  1. The reader is referred to Dawid (2006, 2013a, b) for a more thorough analysis of non-empirical confirmation in the context of string theory. Consult Dawid (2009) for general epistemic assessment of the status of string theory.

  2. See also Greene (1999, p. 145) and Polchinski (1998, p. 5).

  3. Indeed, ‘on the right track’-talk is a recurring theme among string theorists as documented by Camilleri and Ritson (2015).

  4. Thank you to an anonymous reviewer of Synthese for pressing me on this point.

  5. It is charitable to observe that string theory has other features to recommend it such as indications toward the recovery of the Standard Model via D-branes (Antoniadis et al. 2003), providing the expected entropy for black holes (Strominger and Vafa 1996), and renormalizability. Still, in most accounts the recovery of general relativity is presented as the prominent success of string theory.

  6. The functionalism assumed here is an ontological functionalism in the terminology of Bealer (1997).

  7. Exactly what qualifies a theory as a prospective microtheory—and thus a potential candidate of a theory that can reconcile general relativity and quantum mechanics—shall not concern us here. The important point is simply that the class of theories of quantum gravity includes not only the actual theory of quantum gravity, but also microtheories of other possible worlds with varying similarity to our own.

  8. Whether general relativity is multiply realizable within any one theory of quantum gravity is not at issue here. See Huggett (2017) for a discussion relating to this other theme in the context of string theory.

  9. Having three different, independent approaches leading to the same conclusion establishes its “robustness” beyond what is achieved by each individual approach (Wimsatt 1981). However, as observed by Weisberg (2006): “Robustness analysis helps to identify robust theorems, but it does not confirm them” (742).

  10. I would like to thank an anonymous reviewer of Synthese for pressing me on this issue.

  11. To question the significance of the recovery of general relativity in string theory is not in itself novel and has already been argued to be part of a larger debate where “string theorists and their critics typically adopt different attitudes to the heuristic significance of solved, partially solved, and unsolved problems” (Camilleri and Ritson 2015, p. 54). However, when it comes to the recovery of general relativity in string theory, the dispute over its heuristic significance has only ensued as the debate whether this problem is solved or only partially solved by string theory (see footnote 16). In contrast, the present argument directly informs the heuristic significance of recovering general relativity in a theory of quantum gravity and contends that it should be moderated, if general relativity, as argued, is multiply realizable in quantum gravity.

  12. See the contributions to Dardashti et al. (2019) for some recent instalments in the debate over the role and relevance of such factors in theory assessment in fundamental physics.

  13. See Cabrera (2018) for a discussion of the difference between the context of pursuit and the context of justification in relation to string theory.

  14. Strictly speaking, the massless modes of the bosonic string also includes the Kalb–Ramond field, and this derivation of the EFEs can therefore be regarded as one where the Kalb–Ramond field is assumed to vanish. This, however, makes no difference for the present purposes.

  15. See Callan et al. (1987) for higher order corrections.

  16. Critics argue that this derivation of the EFEs is still short of recovering general relativity, because it—in perturbing around a fixed background—is not properly background independent (e.g. Smolin 2006, pp. 185–186; Rovelli 2013). Some string theorists have simply denied this proposing a weaker notion of background independence, while others argue that this is merely an artefact of the background dependent perturbative string theory used for the derivation, but maintain that there are indications that the (unknown) non-perturbative formulation of string theory is background independent (Camilleri and Ritson 2015). We shall set this issue aside, but observe that the same worry applies to the three routes to the EFEs considered in Sect. 5. They may therefore also be background dependent and thus short of full derivations of general relativity.

  17. See Dardashti et al. (2015) and Crowther et al. (2019) for a discussion of the significance of such analogue black holes. See Barceló et al. (2011) for a general review of analogue gravity.

  18. Though we will assume this throughout, more is, as observed by Crowther and Linnemann (2017), to be said about the relation between theories of quantum gravity, fundamentality, and the final theory of everything.

  19. Depending on one’s view on reductionism in the context of philosophy of mind, the talk of levels of description might be regarded with suspicion. However, in the present context it seems uncontroversial that general relativity belongs to one length scale and quantum gravity to smaller length scale; at least if one does not promote this to a claim about ontological levels (see Le Bihan (2018) for a discussion).

  20. For identity theorists, this entails a identification between the mental state and its realizer, but this semantic component of the functionalism/identity theory debate shall not concern us here.

  21. For some recent instalments of this debate, see Lam and Esfeld (2013), Oriti (2014), Carlip (2014) and Huggett (2017).

  22. See also Padmanabhan (2010b) and Verlinde (2011).

  23. See for instance Ryu and Takayanagi (2006), Van Raamsdonk (2010), Maldacena and Susskind (2013), Chirco et al. (2014), Lashkari et al. (2014), Jacobson (2016), Han and Hung (2017), Baytaş et al. (2018), Cao and Carroll (2018) and Chirco et al. (2018).

  24. It could not be since none of the three approaches are theories of quantum gravity, but merely intermediate structures from which the EFEs can be derived.

  25. For a survey of emergent gravity, see Linnemann and Visser (2018).

  26. In the reconstruction by Visser (2002), the metric, \(g_{ab}\), is expanded as the sum of a background, \(g^{0}_{ab}\), and a perturbation, \(h_{ab}(x)\) (nothing is assumed about the size of the perturbation). For the purposes of a Feynman diagram picture, the Fourier transform of the metric perturbation, \(h_{ab}(k)\), takes the form of external gravitons. Sakharov’s result is then that one-loop diagrams with an arbitrary number of external gravitons give rise to terms in the effective action of the form:

    $$\begin{aligned} \int d^4 x \sqrt{-g} \left[ c_1 + c_2 R(g) \right] \end{aligned}$$

    where R(g) is the Ricci tensor, and \(c_1,c_2\) are dimensionless constants that depend on the particular particle species. Compare this to the Einstein–Hilbert action:

    $$\begin{aligned} \int d^4 x \sqrt{-g} \left[ - \Lambda - \frac{R(g)}{16 \pi G_N} \right] \end{aligned}$$

    where \( \Lambda \) is the cosmological constant. See Visser (2002, p. 5) for further details on the connection between these two actions.

  27. See Visser (2002) for a survey of different routes to such a limit.

  28. Indeed, the Weinberg–Witten (1980) theorem suggests that induced gravity scenarios faces serious difficulties. See Sindoni (2012) for a discussion.

  29. There are many approaches going by the name of thermodynamic gravity all building to some extent on Jacobson’s insight. See Padmanabhan (2010b) for a review.

  30. This is an equilibrium thermodynamic relation.

  31. The most well known example of a causal horizon is the event horizon of a black hole. The type of causal horizons considered here are those observed by a constantly accelerating observer as known from the Unruh (1976) effect.

  32. Again this is well known from black hole thermodynamics (Bekenstein 1973).

  33. The rigorous foundation of the relation between the macroscopic thermodynamic level of description and the microstructure on which it supervenes remains elusive (Sklar 1993; Ridderbos 2002). Here it suffices to observe that thermodynamics somehow does work to make reliable predictions.

  34. This and many other aspects in the foundations of equilibrium thermodynamics are disputed. These will not be discussed further here. For a review, see Frigg (2008).

  35. See Sindoni (2013) for further details on how such an equilibration might obtain.

  36. See Padmanabhan (2010b, section 4) for further details. In essence, Jacobson’s thermodynamic gravity derives from the thermodynamic nature of causal horizons. While horizon thermodynamics can be derived from the assumption that the EFEs are equations of state (e.g. Hansen et al. 2017), horizon thermodynamics can also be defended on independent grounds and then shown to recover general relativity (Padmanabhan 2005).

  37. This argument by analogy to the proportionality between entropy and horizon area is at risk of being circular since the derivation of black hole thermodynamics assumes gravity. We shall here ignore this concern for two reasons: First, since there are routes to thermodynamic gravity that do not assume a proportionality between entropy and horizon area (e.g. Padmanabhan (2010a)) and second, that the proportionality can be defended on independent grounds if the entropy is identified as entanglement entropy (more on this below). Observe that a similar concern arises if the proportionality is defended with reference to the holographic principle as formulated by ’t Hooft (1994), Susskind (1995) and Bousso (2002) since this principle also ultimately relies on black hole thermodynamics.

  38. This view is disputed by Dougherty and Callender (2017).

  39. It is disputed whether entanglement entropy is genuine thermodynamic entropy and the approaches to the EFEs based on entanglement entropy might as a consequence not be examples of thermodynamic gravity. There are indications that entanglement entropy is indeed connected to thermodynamic entropy (Kaufman et al. 2016) and regardless, the entanglement approaches to EFEs considered below at most assume an entanglement thermodynamics—genuine or not—analogous to ordinary thermodynamics (see Alishahiha et al. (2013) for more details on entanglement thermodynamics).

  40. Formally, the entanglement entropy, \(S_B\), of a quantum subsystem living in the manifold subregion B is defined as \(S_B = - \rho _B \log (\rho _B)\). Here \(\rho _B\) is the reduced density matrix for the quantum subsystem.

  41. Anti-de Sitter spacetimes are vacuum solutions to the EFEs with a negative cosmological constant, i.e. solutions whose curvature is the other way around as compared to the current best cosmological models. It appears, therefore, that the AdS/CFT correspondence does not obtain in the actual world.

  42. The correspondence has its origin in string theory and conjectures more precisely that certain closed string theories in asymptotically AdS are dual to certain CFTs defined on a fixed background identical to the asymptotic boundary of the dual AdS spacetime. See Butterfield et al. (2016) for a philosophical introduction to the AdS/CFT correspondence.

  43. See for instance de Haro (2017) for more details about the notion of duality in the context of the AdS/CFT correspondence. All metaphysical questions, for instance whether the AdS side and the CFT side represent metaphysically distinct possible worlds or not will be set aside since both worlds—by definition of the notion of duality—would be empirically indistinguishable (see Read (2016) and Le Bihan and Read (2018) for an overview of these metaphysical questions).

  44. Formally we have \(dS_B = \delta E^{hyp}_B\) if

    $$\begin{aligned} \delta E^{hyp}_B = \pi \int _B d^{d-1} x \frac{R^2-(\vec {x}-\vec {x_0})^2}{R} \delta \langle {T^{tt}(x)}\rangle \end{aligned}$$

    where R is the radius of the ball shaped region and \(\delta \langle {T^{tt}(x)}\rangle \) is the variation of the energy density.

  45. In mathematical terms, the Ryu–Takayanagi formula takes the form: \(S_B = \text {Area}( {\tilde{B}})/4G\) where \(S_B\) is the entanglement entropy of a subregion of the manifold, B, on the CFT side and \({\tilde{B}}\) is the area on the AdS side related to the region B on the CFT side. For further details about the Ruy–Takayanagi formula including the evidence for it within the AdS/CFT correspondence, see Rangamani and Takayanagi (2017).

  46. There are indications, though conjectural, that the Ryu–Takayanagi formula generalizes to other prospective theories of quantum gravity: causal set theory (Sorkin and Yazdi 2018), loop quantum gravity (Smolin 2016; Han and Hung 2017), and the more general framework of generic group field theories employing a relation to tensor networks with promises of even further generalization (Chirco et al. 2018). But the worries raised in Sect. 4 might be raised anew, if too much emphasis is put on our current field of prospective quantum gravity theories.

  47. Jacobsen explicitly refers to Lashkari et al. (2014) and Faulkner et al. (2014) as central sources of inspiration (2016, p. 1).

  48. For the sake of brevity, we shall here limit the attention to Jacobson (2016), but there are several other approaches to the EFEs from entanglement that are inspired by AdS/CFT without stipulating it, e.g. Verlinde (2017) and Cao and Carroll (2018).

  49. For the purposes of Jacobson’s argument, it is even sufficient that the UV part of the entanglement entropy is proportional with area, an assumption that looks even less contestable.

  50. Jacobson also uses the (contracted) Bianchi identity which assumes that the connection is metric compatible. Though this is a very common assumption in general relativity, reality might prove to be otherwise, but we will not pursue this further here. Thank you to one of the anonymous reviewers of Synthese for raising this issue.

  51. Crowther and Linnemann (2017) argue that a theory of quantum gravity does not have to be fundamental and that UV-completeness cannot be assumed for quantum gravity. We will disregard this subtlety here.

  52. Some care must be taken when it comes to the holographic principle since its derivation assumes parts of general relativity. Using it to argue that general relativity is multiply realizable might therefore be circular. This, however, it not the place to discuss this issue further. See (Bousso 2002, section ii) for a discussion of the various routes to the holographic principle.

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Acknowledgements

I would like to express my gratitude to Niels Linnemann, Kian Salimkhani, Astrid Rasch, Richard Dawid, Sorin Bangu, and two anonymous reviewers of Synthese for valuable feedback on and helpful discussion of earlier drafts of this paper. I also send my thanks for constructing comments to the participants at the Spacetime Functionalism Workshop (University of Geneva) and 1st Scandinavian Workshop on the Metaphysics of Science (NTNU) where earlier versions of this paper was presented.

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    Rasmus Jaksland

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Jaksland, R. The multiple realizability of general relativity in quantum gravity. Synthese 199 (Suppl 2), 441–467 (2021). https://doi.org/10.1007/s11229-019-02382-8

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