Abstract
General relativity cannot be formulated as a perturbatively renormalizable quantum field theory. An argument relying on the validity of the Bekenstein–Hawking entropy formula aims at dismissing gravity as non-renormalizable per se, against hopes (underlying programs such as Asymptotic Safety) that d-dimensional GR could turn out to have a non-perturbatively renormalizable d–dimensional quantum field theoretic formulation. In this note we discuss various forms of highly problematic semi-classical extrapolations assumed by both sides of the debate concerning what we call The Entropy Argument, and show that a large class of dimensional reduction scenarios leads to the blow-up of Bekenstein–Hawking entropy.
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Notes
The perturbative non-renormalizability of gravity is typically only heuristically established in textbooks, that is by means of the power counting criterion of renormalizability (as in Zee [35]; see diagram 10.2 on p. 318 in Peskin and Schroeder [24] for examples of how easily the power counting criterion fails). More accurately, one argues for gravity’s perturbative non-renormalizability by pointing at the incurable infinities in loop diagrams of first order (in the matter case; second order in the matter-free case) while expressing an expectation of infinitely more infinities at higher orders (cf. [16]).
Some authors, such as Doughty [14], can even be interpreted as suggesting that there are two “unsatisfactory” features of classical general relativity: non-renormalizability and singularities. Whereas singularities have been thoroughly discussed in philosophy of science, non-renormalizability of GR has so far received at best passing attention.
There is much more to be said about the status of the alleged inadequacy of the framework, choice of features of QFT which are to be preserved (cf. for instance [25]), and the importance of non-renormalizability for the theory choice and theory assessment; these philosophical considerations are crucial when it comes to the evaluation of exotic options we are seemingly faced with. But this is not our focus here.
A standard example is due to Parisi [23], who has shown that a perturbatively non-renormalizable four-fermion model is nevertheless describable up to high energies by a finite amount of parameters when expanding it in the number of field copies rather than in a coupling constant as usually done.
In standard terminology, the Bekenstein–Hawking formula is relation between entropy and area, not energy.
This formula only applies to spacetimes with a cosmological constant \(\varLambda \le 0\). Unfortunately, for the most interesting case of positive \(\varLambda \)—it is believed that we live in a universe with positive cosmological constant, after all—there is no straightforward dimension-dependent formula available. Provided that talk about a de Sitter/CFT holographic scenario is physically sensible (for a critical take see for instance Strominger [29]), the Cardy–Verlinde entropy formula could be used (see [32]).
Needless to say, black hole states are only expected to arise for sufficiently high spatial concentration of energy, as for instance upon probing spacetime at sufficiently high energies with electromagnetic waves—provided that the usual Planck–Einstein relation \(E\propto G/R\) (where R is the size of the region to be probed) holds.
The concept of black hole dominance requires a notion of density of states, that is, it presupposes that different spacetime-matter-settings can be binned into the probing energy E for which they occur.
This can be derived as follows. Assuming that (1) energy and entropy are extensive and that (2) temperature sets the dimensions in an otherwise scale-invariant theory, it follows that \(S\propto R^{d-1} T^{d-1}\), and \(E\propto R^{d-1} T^d\). Combining these two expressions, one finds that the entropy density \(S/R^{d-1}\) scales with the energy density \((E/R^{d-1})^{\nu }\) where \(\nu =\frac{d-1}{d}\). Precisely speaking, this gives a relation between entropy density and energy density. Taking R to be independent of energy (\(R(E)=R\)), will render it as a relation between entropy and energy. This point will form an essential part of the criticism of the Entropy Argument in Sect. 3.3.2.
What counts as minimal departure from the QFT framework is of course far from being unambiguous—simply observe how completely different frameworks for quantum gravity are each being advertised as being close to standard QFT.
A generic microcanonical ensemble of particle configurations is characterized by a fixed total energy E, fixed particle number N and a fixed volume V with in which the particles move (NVE-ensemble). The entropy for such an ensemble is then defined as a function of the density of states which consequently is a function of N, V and E. For a generic black hole, the microcanonical ensemble of states realizing the black hole is analogously characterized by a fixed energy, charge and angular momentum of the black hole. In particular, the entropy of a Schwarzschild black hole thereby becomes a function of energy.
However, there may be ways to defuse this worry: for instance, Rovelli and Vidotto [26] argue—within the context of covariant loop quantum gravity—that the microcanonical entropy on the one hand and entanglement entropy on the other hand can be understood as two sides of the same coin and are thus mutually compatible after all.
Additionally, in certain situations the canonical or microcanonical ensemble description may fail to exist; an instructive example is discussed in Hawking and Page [17]. Even though (at least at the thermodynamical limit) these two ensembles can and in practice are used interchangeably (see [33, 36]), one should, in general, make sure that in particular physical situations under consideration the necessary conditions for the existence of the canonical and/or microcanonical ensemble are satisfied.
Which is an asymptotic safety term (see [22]) for any kind of curvature-corrected variant of GR where the standard Einstein–Hilbert action is extended by further terms depending on the curvature of spacetime, with the quantization of gravitational Hamiltonian which can include metric terms different from the Einstein–Hilbert action term and is the result of RG flow on the Einstein–Hilbert action from IR towards the UV.
Basu and Mattingly conclude instead: “Hence the standard minimum length argument fails to necessarily be true in this case.” This is not uncontroversial (at least from an operational point of view) since in a certain sense the asymptotic safety program always comes with a minimal length ([18], p. 39):
It is essentially a tautology that an asymptotically-safe theory comes with this upper bound when measured in appropriate units.
The discussion, then, seems to boil down to whether asymptotic safety can provide a workaround for the operational minimal length argument or not.
In case of stationary and perfectly symmetric matter, the hoop conjecture has been proven by Bizon et al. [6]; but whether the proof generalizes if these symmetry assumptions are dropped is an open question.
Alternatively, one might say that black holes in lower dimensions have no well-defined entropy at all.
Carlip and Grumiller [9] use 2-dimensional dilation gravity to study an (Euclidean) Schwarzschild black hole in arbitrary dimensions.
Why this talk about an uphill battle? Now, the Bekenstein–Hawking entropy formula is not only a trusted result from black hole thermodynamics but it also provides the basis for one of the most dearly held principles—at least within some communities—towards a theory of quantum gravity, namely the holographic principle. Cf. Bousso [7].
Another issue is that the derived CFT formula is a formula for the entropy density, not the entropy itself. See Sect. 3.3.2.
Note that (1) \(S \propto R^{d-2}\) and (2) \(E \propto R^{d-3}\). From (2), we get (3) \(R \propto E^{\frac{1}{d-3}}\). Now, \(\frac{S}{R^{d-1}} \propto \frac{E^{\frac{d-2}{d-3}}}{R^{d-1}} \propto R^{d-1}\). As \(\frac{E}{R^{d-1}} \propto R^{-2}\), the Bekenstein–Hawking entropy density \(s_{BH}\propto \epsilon ^{1/2}\) follows.
It is the dimensionless coupling constant \(g=G k^2\) with respect to which asymptotic safety is said to have a non-trivial fixed point \(g*:=\lim _{k \rightarrow \infty } g(k) \ne 0\) where k is the energy scale.
It would of course be interesting to know whether the match between entropy density formulae in the asymptotic safety improvement-scenario holds for d dimensions (rather than just for 4 dimensions)—the correction from dimensional reduction would always work provided that the reduction is to \(d=2\). Unfortunately we are not aware of any asymptotic safety-improved Bekenstein–Hawking entropy density formula for higher dimensions.
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Acknowledgements
J. D.’s work was partly performed under a collaborative agreement between the University of Illinois at Chicago and the University of Geneva and made possible by Grant Number 56314 from the John Templeton Foundation and its contents are solely the responsibility of the authors and do not necessarily represent the official views of the John Templeton Foundation. N. L. would like to thank the Swiss National Science Foundation for financial support (105212_165702), and the Lorentz Center in Leiden for providing the platform for an enriching conference on Quantum Spacetime and the Renormalization Group. N. L. would also like to thank the the participants of this conference for valuable feedback. Both authors would like to thank Claus Beisbart, Vincent Lam, Max Niedermaier, Carina Prunkl, Christian Wüthrich and an anonymous referee for valuable feedback.
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This work was partly performed under a collaborative agreement between the University of Illinois at Chicago and the University of Geneva and made possible by Grant Number 56314 from the John Templeton Foundation and its contents are solely the responsibility of the authors and do not necessarily represent the official views of the John Templeton Foundation. This work was also partly funded by a grant from the Swiss National Science Foundation (Project Number: 105212_165702).
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Doboszewski, J., Linnemann, N. How Not to Establish the Non-renormalizability of Gravity. Found Phys 48, 237–252 (2018). https://doi.org/10.1007/s10701-017-0136-x
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DOI: https://doi.org/10.1007/s10701-017-0136-x