1.1 Motivations, Constraints, and Desiderata

The motivations for QG are the reasons for seeking a theory that satisfies the Primary Motivation: answers to the question ‘why do we want this theory in the first place?’ The project that this Element aims to promote starts as a descriptive one, considering any motivations cited by physicists, but also looking for other potential motivations. The next stage is an evaluative and prescriptive one. The motivations are framed as principles, and their utility, and implications, assessed in terms of their potential roles in theory-development in various approaches to QG. Both of these stages are merely sketched in this Element, with the aim of inspiring the reader to engage with this project in their own research. Ultimately, the goal is to better understand the theory being sought, and determine whether it is actually what we want or require of QG.

A principle may be used in different ways in different research programmes, and at different stages of theory-development within a given programme (see Crowther (Reference Crowther2021) for a discussion of these, with examples from QG). Since I am concerned with motivations here, it is useful to distinguish them from those non-motivational principles and features that share some of the same roles.

Broadly, motivations can be translated into either:

1. 1. Heuristics or guiding principles: these are desirable or useful features that can aid in the discovery of the theory by leading to new insights, but which may or may not actually be retained in the resulting theory; they are non-necessary desiderata: features that it would be nice if the theory possessed, and which would make us more inclined to accept the theory; or,

2. 2. (Strong) constraints or criteria of theory acceptance: the new theory should not be accepted if it is incompatible with the principle (unless there is strong evidence in favour of the theory, and/or the principle is shown to be violated under the relevant conditions); when a motivation is adopted as a constraint, it typically is taken to form part of the definition of the new theory.

Not all of the constraints on QG come from motivations, however, and nor do all desiderata motivate the need for a new theory. Typically, the non-motivational constraints are not taken to define the theory, as the motivational ones are. Some examples of constraints on QG that are not motivations include those principles coming from current theories within their domains of applicability (e.g., QG predictions should not violate Lorentz invariance in the domains where we know this symmetry holds). There is also the requirement that the new theory explain the success of the theories it replaces, in the domains where we know those theories work well. This constraint is related to the Generalised Correspondence Principle, introduced next (§1.2), which also plays a significant role in ensuring the new theory satisfies certain empirical constraints, namely those observations that are explained by current theories. Some of the non-motivational constraints, in spite of being non-motivational, can nevertheless still also be heuristically useful. Some possible examples of desiderata that might not be motivational include the ‘theoretical virtues’ (if these are viewed as guides in developing a new theory rather than motivations to seek a new theory), as well as unexpected explanation of (theoretical or empirical) problems that are not hitherto thought to be quantum-gravitational.

The relationship between motivations, constraints (necessary features), and desiderata (non-necessary features), is illustrated in Figure 1.

Figure 1 Motivations, constraints, and desiderata: Motivations may be necessary (constraints), or non-necessary (desiderata).

Motivations, constraints, and desiderata can all play important roles as means of confirmation or as indicators of pursuit-worthiness of a program: compatibility with the principle (or empirical result) serving to increase credence in the theory, or as being suggestive of the theory’s potential future success, if it were to be developed (‘pursued’). I believe this holds regardless of whether the principle has been used in the development of the theory, or whether the theory’s possession of the feature in question is unexpected (‘prediction’).Footnote ³ Keep in mind, too, that principles are not always purely theoretical, but can be based on empirical results. Empirical results can serve in various roles, though different results may feature in different roles, as discussed further (§1.3).

So, how are the motivations listed earlier used, and how are they related to the Primary Motivation? These questions will be explored in the rest of this Element, though my conclusions here are certainly more provisional than conclusive, and I invite further engagement. The tentative list below is illustrated in Figure 2.

• Incompatibilities between GR and QFT motivate the Primary Motivation, representing the constraint of consistency (which itself requires philosophical assessment in this context), but also frustrate attempts at satisfying it;

• Unification motivates the Primary Motivation, but also represents different (i–ii) ways of satisfying the Primary Motivation; it is not a constraint, but a desideratum;

• Putative inconsistency of semiclassical and other ‘hybrid’ approaches to QG arise from attempts to satisfy the Primary Motivation, underpinned by the constraint of consistency; they are not motivations for the Primary Motivation, but are motivations for particular ways of satisfying it (through unification, typically taken to indicate the need for quantization), which I argue are not necessary ways;

• Singularity resolution or explanation motivates the Primary Motivation, through the constraint of consistency; can serve as a guiding principle and desideratum, but not all singularities in GR and QFT require resolution by QG;

• Black hole thermodynamics stems from attempts to bring together GR and QFT (plus thermodynamics), it motivates the Primary Motivation; leads to various desiderata and potential constraints;

• Complete cosmology: these are problems related to empirical observations which QG is expected to solve, and so they motivate the Primary Motivation; arguably, they represent constraints on QG;

• Hierarchy problem: stems from empirical concerns; motivates the Primary Motivation, but is (arguably) a desideratum rather than a constraint;

• The measurement problem of QM: as above;

• The problem of becoming in GR: as above.

Figure 2 Various motivations as related to the Primary Motivation. Arrows represent ‘motivates’; unification and quantization are possible routes towards fulfilling the Primary Motivation, so can also be thought of as motivated by the Primary Motivation (as well as independently motivated).

1.2 Relative Fundamentality

The general motivations for QG listed earlier may be seen as reasons for not treating GR and QFT as fundamental. There are different notions of, or requirements for, fundamentality in physics,Footnote ⁴ but the key notion here concerns completeness (or maximality) of description. In this Element, I take it that QG need not be a fundamental theory, in the sense of being a final theory or theory of everything. All that is required is fulfilment of the Primary Motivation. As such, we should not require as strong constraints on the QG all those features that are expected of a final theory.

However, QG is expected to be more fundamental than GR and QFT. The notion of relative fundamentality here can be understood simply as an asymmetric relation of non-causal dependence (Crowther, Reference Crowther2018). QG is more fundamental than GR and QFT by virtue of its having a broader domain of applicability, one that subsumes the domains of success of the previous theories – QG describes ‘more stuff’ than either of these frameworks. GR and QFT are thus expected to be effective theories (or frameworks): valid as approximations to QG in the restricted domains where they are known to be successful.

Asymmetric dependence between an older, ‘predecessor’ theory $O$ , and a newer, ‘successor’ theory, $N$ , may be demonstrated through the establishment of particular correspondence relations between them, such that $O$ is taken to be derivable from $N$ . There are various types of inter-theory correspondence relations,Footnote ⁵ though the most familiar are characteristic limiting relations (a standard example is the transition from relativistic to classical mechanics in the Newtonian limit of small velocities compared to the speed of light, that is, $v/c\to 0$ ). The idea is essentially what Nickles (Reference Nickles1973) calls ‘reduction₂’, where $N$ is said to ‘reduce to’ or recover $O$ in the domains where $O$ is known to be successful. This is standardly taken as a constraint on any new theory of physics that is more fundamental than its predecessor. This is also a way of satisfying the Generalised Correspondence Principle, which forms an important constraint on QG, and states that any two theories whose domains of applicability overlap share approximately the same results within these domains.Footnote ⁶

Given that QG is taken to be more fundamental than GR and QFT, we expect that GR and QFT be derivable from QG in this way. (In spite of this, the recovery of QFT has not been appealed to as a constraint on QG to the same extent as has the recovery of GR or spacetime, except within the context of string theory.) On the other hand, QG is not standardly expected to be more fundamental than the framework of quantum theory: most approaches to QG attempt to satisfy the Primary Motivation by simply providing a quantum description of spacetime (or, more precisely, the quantum physics underlying classical spacetime). According to this ‘standard perspective’, QG is itself a quantum theory, rather than a more fundamental framework supposed to explain quantum mechanics (§3). This said, however, it is a significant, and underexplored, question as to whether the explanation of quantum mechanics should be understood as part of the motivation for QG – for instance, if we expect QG to explain the measurement problem of QM. There are some approaches that do take this as a goal, most notably the work by Penrose (Reference Penrose2002, Reference Penrose2014). If standard quantum theory is modified in the construction of QG, or QG were to represent a new framework, of which quantum mechanics were a special case, then QG would be considered more fundamental than quantum theory.

1.3 Empirical Constraints

Although QG is primarily motivated by non-empirical concerns, the need to be consistent with empirical observations is still a principle that is used extensively in the search for QG. Three different types are employed: empirical observations that are not explained by current theories; empirical observations that are explained by current theories; future empirical observations. I briefly describe each in turn.

1.4 Approaches to QG

There are various different research programs, or approaches, towards QG, with different motivations, assumptions, formalism, and of varying degrees of completeness. Each of these have their own problems, which, from their own perspective could be called ‘the problem of QG’. This Element, however, is written from a general perspective, discussing issues relevant to any possible theory of QG: indeed, it aims, specifically, at better understanding the common features by virtue of which we can even conceptualize these different approaches to QG as being ‘approaches to QG’. The idea, as stated, is to clarify, and begin to assess, the most basic motivations for seeking QG in the first place. The different approaches to QG each assign varying levels of importance to the various principles and motivations, and utilize them in different ways. As I go through the motivations and principles, I will mention the ways they have been used in some approaches to QG as illustration. But, given the limited space of this Element, I will not introduce or explain what these approaches are in any detail.

We can broadly classify the approaches of three types, owing to their historical development as stemming from the different methods of attempting to quantize gravity (see, e.g., Carlip, Reference Carlip2001; Rovelli, Reference Rovelli, Gurzadyan, Jantzen and Ruffini2000). As described in §2.3, one natural approach to quantizing GR is to use familiar techniques from QFT to treat gravity, which broadly captures the ‘particle physics perspective’ (or ‘high-energy physics perspective’) towards QG. An early attempt utilizes quantum perturbation theory, which is known as the perturbative covariant quantization approach to QG. This approach to quantization splits the GR metric into a background part plus small perturbations; the perturbations are quantized, while the background remains classical and non-dynamical. [Splitting apart the dual role played by the metric field in GR, and restoring spacetime as a fixed, static, ‘background’, goes against the principle of background independence in QG (§2.1), and was thus objectionable from the ‘GR perspective’ on QG]. This approach was shown to suffer from gravity being non-renormalizable (see §2.3). In response, the years 1980–2000 saw the development of research into non-perturbative QG, and the majority of research since has focused on this (Loll et al., Reference Loll, Fabiano, Frattulillo and Wagner2022). The main approach to QG that historically stems from the covariant quantization methods is string theory (see, e.g., Zwiebach, Reference Zwiebach2009).

Asymptotic safety is an approach based on the Wilsonian idea of renormalization, and the conjecture that quantized gravity is non-perturbatively renormalizable (discussed in §2.3).

Closely related to the covariant approach is Feynman quantization, which applies Feynman’s path integral (sum over histories) quantization techniques to the GR metric. Some important current examples of approaches that aim to implement these methods include causal dynamical triangulations (Loll, Reference Loll2019), and causal set theory (Surya, Reference Surya2019).

Alternatively, the canonical or Hamiltonian quantization of GR aims to construct QG as a theory of the fluctuations of the metric as a whole, so no fixed metric is involved. This approach represents the ‘GR perspective’ on QG. It involves first casting GR into canonical form, and then using canonical quantization methods. This requires splitting spacetime into space and time, with a fixed topological structure. The metric information is represented, in full, by operators on a Hilbert space. The goal is then to find the eigenvalues of the operators along with their transition probabilities. While the canonical QG approaches aim at preserving the principle of background independence, they suffer more manifestly from the problem of time, §2.2 (though problems of time also arise in other approaches to QG). The main approach to QG stemming from the canonical quantization procedure is loop quantum gravity (although now loop quantum gravity also is being developed using a covariant formalism, see Rovelli (Reference Rovelli2004); Rovelli & Vidotto (Reference Rovelli and Vidotto2014)).

Two other classes of approaches mentioned in this Element are hybrid theories and emergent gravity approaches (§3.3,5). Note, however, that this general classification by no means exhausts the range of approaches pursued in QG.

2.1 Background Independence

Background independence, and the principle of relationalism are used widely in the search for QG. Several authors view the history of physics in terms of the increasing relationalism of its theories, and thus motivate the quest for QG along these lines. Smolin (Reference Smolin, Huggett, Matsubara and Wüthrich2020), for instance, adopts relationalism as a methodological imperative to ‘make our theories more relational by eliminating background structure’.Footnote ¹⁹ One might wonder, however, whether it actually represents a motivation for QG, rather than just being, for example, a constraint or desideratum upon a new theory of QG. On the one hand, we might argue that the goal of having a background independent theory is not a reason to seek QG, since we already have a theory of gravity that is background independent, namely GR. But, while background independence can certainly be viewed as part of the motivation that drove the development of GR, it is not straightforward to identify what exactly is meant by background independence in GR as I discuss below. On the other hand, though, QM and QFT are straightforwardly not background independent theories, so a drive for increasing relationalism through the elimination of background structure could promote bringing these theories in closer connection to GR, and thus to motivate the Primary Motivation.

There are four general, but distinct, principles associated with background independence:

1. a. That the theory not feature an absolute, fixed, ‘background’ structure (typically: spacetime);

2. b. That the theory be compatible with relationalism;

3. c. That the theory be generally covariant;

4. d. That the theory be non-perturbative.

The first of these (a.) is most properly called ‘background independence’: basically, a theory that is background independent does not posit the existence of a fixed background object, or require the existence of such an object in order to define the properties of other objects in the theory. A well-known example of a background dependent theory is Newtonian mechanics, which features absolute space and time as fixed background structures used to define position and motion. For Newton, space was not only absolute and immutable (meaning that it is fixed, existing unchanged throughout all time, totally unaffected by matter, but also responsible for affecting matter, being used to define the properties of – or itself conferring properties upon – matter) but also substantival (meaning that space itself is a ‘substance’, as real as matter, and which can exist independently of matter).

For QG, background independence standardly means that the theory not feature a fixed (background) spacetime, but there are several meanings, corresponding to different notions of ‘fixity’ (Butterfield & Isham, Reference Butterfield, Isham and Butterfield1999, §4.1), as well as the different (layers of) structures that could be identified as ‘spacetime’ (metric, manifold, affine structure, etc.). There are (at least) four proposals that attempt to formally capture the idea of background independence in the sense of (a.), earlier. Note that each is a separate proposal:

1. i. A theory is background independent if it features no absolute objects (i.e., structures that are the same in all models of the theory);

2. ii. A theory is background independent if it has no formulation which features fixed fields (i.e., there is no formulation of the theory that is not diffeomorphism invariant);

3. iii. A theory is background independent if its solution space is determined by a generally-covariant action (all of whose dependent variables are subject to Hamilton’s principle, and represent physical fields);

4. iv. A theory is background independent if all of its physical degrees of freedom correspond to geometrical degrees of freedom.

For details on (i–iii), Pooley (Reference Pooley, Lehmkuhl, Schiemann and Scholz2017), and for (iv), Belot (Reference Belot2011) (a comprehensive discussion of background independence in classical and quantum gravity can be found in Read (Reference Read2023)). Each of these attempted characterizations has its own difficulties, however – and, arguably, none are adequate to do the job.Footnote ²⁰

The idea of (b.) relationalism in regard to background independence comes from the thought that a theory that does not rely on a fixed background structure will have to define the properties of its objects purely in terms of the relations between these objects, rather than relying on the background structure. In this sense, relationalism is opposed to absolutism (that there be a structure that affects objects without itself being affected), as well as substantivalism (that space itself has an independent existence, rather than just being a means of representing the relations between objects). Also, an absolute object is typically understood as being fundamental, in the sense that it not ontologically dependent on any further, more basic, underlying objects. So, when it is claimed that a background independent theory is relational, it means that the properties in the theory are defined only with respect to one another, and that these relationships are not fixed (in the sense of being non-dynamical), but evolve according to the equations of motion of the theory. While we might for this reason identify ‘relationalism’ and ‘background independence’ as synonymous,Footnote ²¹ Rickles (Reference Rickles and Deiks2008) argues against the identification on the grounds that a substantival view of spacetime is also compatible with background independence.

A major driving force for seeking a background independent theory is the desire to preserve, or continue, what is seen as one of the most important insights of GR. Yet, while GR is taken as the prime exemplar of a relational or background independent theory, it is difficult to precisely identify what it is that ‘makes GR special’ in this regard. The intuitive notion we have is that GR is special because – owing to the theory’s diffeomorphism invariance or general covariance – we cannot define the physical observables of the theory without solving the dynamics, or, in other words, that ‘there is no kinematics independent of dynamics’ (Stachel, Reference Stachel, Rickles and French2006). However, it is not easy to define explicitly the idea of a dynamical object along the lines of ‘being solved for’. The main problem, following Kretschmann (Reference Kretschmann1917), is that we can turn any background field into a dynamical object by making it satisfy some equations of motion, however physically vacuous they might happen to be. Even the metric in special relativity is able to be made dynamical in the sense that it satisfies some equations of motion – hence special relativity can be made background independent, which then conflicts with our basic intuitions of what background independence is supposed to be (Rickles, Reference Rickles and Deiks2008).Footnote ²²

It is not impossible that general covariance (c.) be used in defining ‘what is special’ about GR,Footnote ²³ and thus feature in a principle of theory selection, though general covariance can not by itself be the principle of background independence. However, most of the recent literature has instead moved on in to principle (a.) in their attempts to salvage the general intuition that GR is special by virtue of possessing some particular conception of background independence. In this context, the proposal (i) earlier fails for GR (and any theory with massive fields),Footnote ²⁴ and while proposal (ii) arguably works for GR, it is difficult to implement as a guiding principle in the search for a new theory (Read, Reference Read2023). Nevertheless, the success of GR as a theory of spacetime, together with the significant heuristic role of the principle in its discovery, mean, however, that background independence (in some sense, which may, arguably, be any of (ii–iv)) continues to be viewed as a desirable principle in QG, in all roles.

Background independence is not standardly considered a motivation for QG: it is not a reason for seeking a new theory in the first place. One way of framing it as a motivation would be to argue (1) QM/QFT are not fundamental because they are not background independent, and (2) that fundamental physics, as it progresses through theory-change, trends towards increasingly relational, or less-background-dependent theories. Such an argument by induction is not very strong, however, without additional reasons for expecting a fundamental theory should be background independent, or an argument as to why background independent theories are better or more successful than background dependent ones. The main argument is that background independent theories are more explanatory than background dependent ones because they do not require structures to be fixed in advance, that is, put in ‘by hand’, but instead have their structures follow from the dynamical laws and principles specified by the theory itself. In this case, if background independence is a motivation for QG, then it would mean that QG is motivated by the need to explain the background elements of QM/QFT (and potentially also those in GR), and itself to not feature these elements. If we were to reject the principle as a motivation for QG, then, arguably, the principle need not be treated as a constraint on QG, unless QG were conceived of as a final theory – if the theory required background structures, not themselves explained by the theory, then we would be motivated to search for a more fundamental theory that did explain these structures.

Finally, there is the distinct notion, (d.), that the theory be non-perturbative. Perturbative treatments of gravity involve fixing a background metric (e.g., Minkowski metric) and studying gravitons propagating on this background. The desire that a theory be non-perturbative is a natural one, given that perturbation theory is a set of techniques used to construct approximations when the exact theory is unknown or unsolvable. These techniques are of limited applicability, and can easily lead to problems when misapplied. Some problems with perturbative gravity are considered in §2.3.

I close this section with a brief mention of background independence in string theory and some other approaches to QG. From early on, string theory has been intensely criticized for being a background dependent approach – on account of its being based on perturbative methods, and describing strings moving on a background spacetime – it apparently fails on all (a.–d.).Footnote ²⁵ In recent years, however, the criticism has died down, as string theorists came to appreciate the significance of background independence, and the theory itself shows hints of being background independent in various ways. (For this reason, the philosophical debate concerning the importance of background independence in QG is no longer so active, having achieved its goal, in a sense.) As Read (Reference Read2023, §5.2) demonstrates (building on a claim made by Huggett & Vistarini (Reference Huggett and Vistarini2015)), there are in fact several senses in which string theory (considered both at the level of spacetime fields, and at the level of worldsheet fields) may be considered background independent, on some of the definitions (ii–iv) earlier, given that the background fields in the theory are required to be dynamically coupled together in the same dynamical equations of GR (the Einstein field equations, plus higher order corrections), and thus are not ‘fixed’ after all, but possess background independence in ways similar to GR.

Additionally, there are attempts to construct a non-perturbative version of string theory; the most celebrated development being the AdS/CFT duality (also known as the Maldacena conjecture, after Maldacena (Reference Maldacena1998)). This is a relationship – a conjectured exact equivalence – between a string theory featuring gravity, describing closed strings propagating on a spacetime (anti-de Sitter space (AdS)), known as the ‘bulk’, and a gauge theory without gravity (a conformal field theory (CFT)), defined on the boundary that contains the bulk spacetime. In the regime where the string theory is strongly coupled, and computations difficult – the CFT is weakly coupled, meaning that perturbative techniques can be used to make calculations. Since the theories are supposed to be equivalent (describe the same physics), the weak coupling regime of the CFT can be used to shed light on the strong coupling regime of the string theory.Footnote ²⁶ This is closely related to the holographic principle, §7.4. While string theorists hope to ultimately arrive at a theory that is background independent, the approach began as a background dependent one (perturbative string theory). Credence in the approach increases by its being shown to be background independent, it may also feature as a criterion of theory-acceptance.

These comments also apply, broadly, to the program of asymptotic safety, which is an approach to QG that treats gravity in the framework of QFT, but avoids the problems associated with perturbative non-renormalizablity (see §2.3). While the approach is premised on non-perturbative QFT techniques, it still relies on splitting the full metric $g_{\mu \nu }$ into a fixed, but arbitrary, background metric ${\bar{g}}_{\mu \nu }$ plus a dynamical fluctuation field $h_{\mu \nu }$ . The principles of diffeomorphism invariance and background dependence (in the sense of (a.)) pose challenges – both technical and conceptual – for the theory, associated with its use of the functional renormalization group techniques and the background field method (Bonanno et al., Reference Bonanno, Eichhorn and Gies2020).

Other approaches to QG have placed heavy emphasis on background independence from their inception, including LQG (Rovelli & Vidotto, Reference Rovelli and Vidotto2014), causal set theory (Surya, Reference Surya2019), causal dynamical triangulations (Ambjørn et al., Reference Ambjørn, Jurkiewicz, Loll and Oriti2009), group field theory (Oriti, Reference Oriti and Oriti2009), and other ‘pre-geometric’ approaches (Caravelli & Markopoulou, Reference Caravelli and Markopoulou2011). These approaches strive for a fully background independent theory, and sell themselves as upholding what they see as a central insight of GR (and of any acceptable theory). These approaches argue that the key to finding a successful theory of QG is in complete (as much as possible) elimination of any background structure or geometry. Adopting Smolin’s (Reference Smolin, Rickles, French and Saatsi2006) slogan, the idea is ‘the more relational the better’. These approaches use the principle of background independence in all roles.

The principle of background independence is now widely accepted as an important feature for QG to possess: it is presented as the main candidate for satisfying the condition that QG ‘take into account’ the lessons of GR in the Primary Motivation (§3.3). Nevertheless, it is an open question whether background independence is a sufficient means for satisfying this aspect of the Primary Motivation: there may be other principles from GR that could serve this role. Background independence is furthermore desirable since it functions as means of principle correspondence, connecting QG with GR, and thus helping, in part, to satisfy the Generalised Correspondence Principle and establish a connection between QG and known empirical results.

2.2 Problem of Time

There is an incompatibility between the treatment of time quantum theory and in GR. According to quantum theory, time is external to the system being studied: it is fixed background in the sense that it is specified from the outset and is the same in all models of the theory. In GR, however, time forms part of what is being described by the theory. Time in GR is subject to dynamical evolution, and it is not ‘given once and for all’ in the sense that it is the same across all models (Butterfield & Isham, Reference Butterfield, Isham and Butterfield1999). This clash between the external time in quantum theory and the dynamical time in GR leads to problems in combining the theories, and is often taken as a suggestion – along with other heuristics, such as the suggestion of ‘quantum foam’, or the Planck length as a minimal length – that time ‘disappears’, or ‘does not exist’ in QG.

The problem of time is a cluster of problems, usually framed along the lines just suggested, but specifically as manifesting in the context of canonical QG. This is a class of approaches to QG that attempts to quantize the canonical (or Hamilitonian) formulation of GR, via familiar canonical quantization techniques. The oldest and most straightforward of such approaches is known as quantum geometrodynamics, which proceeds in a spirit similar to Schrödinger’s original heuristic approach to quantization (via the Hamiltonian-Jacobi formalism). In this context, one finds the primary form of the ‘problem of time’: the supposed ‘dynamical’ equation of the resulting quantum theory (the Wheeler-de Witt equation) does not feature a time variable, and so presents us with a ‘frozen formalism’.Footnote ²⁷ Yet, as Thébault (Reference Thébault, Knox and Wilson2021) emphasizes, the problem as I have just set it up, is misleading in the sense that it does not in fact arise from ‘from forcing a background independent theory of spacetime onto the Procrustean bed of quantization with respect to a background time’. Instead, Thébault (Reference Thébault, Knox and Wilson2021) argues, ‘the problem arises generically from the manipulation of a particular class of classical theories, that includes general relativity, according to the standard formal steps that are preparatory for quantization. That is, the problem of time becomes apparent in the process of preparing any reparametrization invariant theory for quantization’ (p. 390).

Thus, the problem of time (or, at least, the global problem of time that Thébault (Reference Thébault, Knox and Wilson2021) focuses on) may not be specific to GR, and may arise even before quantization. This does not ameliorate the fact, however, that there are numerous problems related to the treatment of time in QG. The relevant question for this Element is whether these problems represent motivations for seeking the new theory, or rather challenges that crop up once we embark on this quest. As mentioned in §1, one motivation for seeking QG (which is not discussed in this Element) is the ‘problem of becoming’: if we are dissatisfied with the lack of ‘real change’, or temporal passage, in the standard ‘block universe’ picture of spacetime as presented by SR and GR, a possible option is to see if we can find a more hospitable notion of time in the more-fundamental theory beyond GR. This type of motivation is based on a perceived inconsistency between time in SR/GR, and our own human experience of time (i.e., of time flowing, and the experience of events occurring ‘Now’). One prima facie difficulty with this strategy is, of course, analogous to that faced by attempts to overcome the determinism of GR (i.e., in the hopes of finding ‘free will’) by looking to more fundamental theory: given that relativity does describe our macroscopic world, and that any more-fundamental theory needs to be compatible with this in the regimes where GR is successful, we may question what relevance any concept of time in QG has to our everyday experiences of ‘change’ and ‘becoming’. Plausibly, though, the connection with becoming could be relevant in the context of QG cosmology, where the theory is expected to replace the Standard Model of cosmology (based on GR), including the idea of the ‘big bang’, in describing the initial conditions of the universe.

More generally, we might take the disparate treatments of time in GR and QM as motivation for seeking a new theory that has a single, coherent account of time. The exploration of time in QG has spawned a huge amount of discussion, with all sorts of different solutions being proposed: from those that feature space without time (e.g., Barbour (Reference Barbour2000); Gomes (Reference Gomes, Huggett, Matsubara and Wüthrich2020)), time without space (e.g., Markopoulou (Reference Markopoulou2009b); Smolin (Reference Smolin2013), (Reference Smolin, Huggett, Matsubara and Wüthrich2020), neither time nor space (e.g., Rovelli, Reference Rovelli2004), and disagreements over the implications of QG for the notion of ‘becoming’ (e.g., Dowker, Reference Dowker, Huggett, Matsubara and Wüthrich2020; Rovelli, Reference Rovelli, Huggett, Matsubara and Wüthrich2020). So, is ‘the problem of QG’ about finding a fundamental description of time, or of timeless fundamental physics that can nevertheless explain the description of spacetime in GR, or of providing an account of change and/or becoming? Arguably, none of these are necessitated by the Primary Motivation. In order to translate these into positive constraints or heuristics for the new theory, we need to evaluate the arguments (including metaphysical arguments) for one solution over another – assuming, of course, we follow Huggett & Wüthrich (Reference Huggett and Wüthrich2013), in not outright rejecting such theories as unphysical or ‘empirically incoherent’ on account of not featuring spacetime fundamentally.

2.3 Perturbative Non-renormalizability of Quantum GR

A natural way of approaching QG is to utilize familiar techniques from QFT – for example, using quantum perturbation theory to treat gravity. The framework of QFT within which the Standard Model of particle physics is formulated relies upon perturbative techniques in order to make predictions in most cases. A perturbative calculation of any particular physical process involves a summation over all possible intermediate states, and this is done at all orders of perturbation theory (though, in practice, often only the first few terms are taken, and it is hoped that the higher-orders decay rapidly). The dynamics of QFT is local, utilizing point sources and interactions – it features arbitrarily small distances and, thus, arbitrarily large momenta. This, as well as the integration over all the momentum-energy states, implies that there are an infinite number of intermediate states. Consequently, perturbative calculations within the theory lead to divergent integrals (if the terms are not sufficiently suppressed). These infinities are dealt with in particular QFTs by the process of renormalization, which first imposes a finite-valued cutoff in momenta (so that the integral is not taken over all possible high values of momenta), and then the parameters of the theory are re-defined in terms of ‘physical’ coupling constants that absorb the divergences, and whose values are then fixed by experiment. A theory is renormalizable if only a finite number of counterterms are required to absorb the divergences, whereas a non-renormalizable theory requires an infinite number of such terms. Because an infinite number of terms cannot be determined experimentally, a non-renormalizable theory is not predictive.Footnote ²⁸

The strong and electroweak interactions are renormalizable, and thus the Standard Model of particle physics is a renormalizable theory. Gravity, however, is non-renormalizable: the Einstein-Hilbert action (1), treated in the perturbative framework, has divergences appear in loop diagrams at first order (in the matter case, or second order in the matter-free case), and there is an expectation that infinitely more infinities appear at higher orders.Footnote ²⁹ The divergences occur at high energy scales, near the Planck scale, and we are prevented from calculating anything in this regime.

In standard QFT, however, even non-renormalizable theories can be predictive (and impressively so), if they are treated as effective field theories. This is thanks to the development of renormalization group (RG) techniques, which led to a framework for studying the way in which the physical coupling constants change depending on the energy scale at which the theory is being applied (the couplings are said to ‘run’ or ‘flow’ with the RG).Footnote ³⁰ Effective field theories are valid at low energies compared to some high energy valued cutoff scale: they are not formally predictive to arbitrarily high energies; that is, they are not thought to be UV-complete (§6.1).

Perturbative quantum GR can be treated as an effective field theory which makes predictions for low energies compared to the Planck scale. It has proven to be an extremely good approximation for all the present experimental tests of gravity (Burgess, Reference Burgess2004; Wallace, Reference Wallace2022). However, the low-energy nature of its predictions is taken to imply that quantum effects are important where gravitational fields are very strong, such as inside black holes or cosmological singularities. In other words, its failure here is taken to indicate the domains of necessity of QG. It also means that this approach towards finding QG fails to satisfy the Primary Motivation, and thus motivates the search for a new theory.

However, it is possible that the proliferation of infinities in the ultraviolet (UV) regime does not signal the failure of GR treated in the framework of QFT, but instead signals the limitations of the perturbative approach in this regime. The asymptotic safety scenario claims that the physical couplings, in the non-perturbative RG flow, do not actually diverge, but instead flow to a finite value: a ‘fixed point’ in the UV.Footnote ³¹ This is similar to QCD, where the couplings flow to a fixed value of 0 in the UV, and the theory is said to be asymptotically free (the theory is non-interacting in this regime). In the case of gravity, however, the fixed point is supposed to be non-zero (the theory is interacting in at least one of the couplings), and the theory is said to be asymptotically safe, since the physical quantities are ‘safe’ from divergences as the cutoff is removed (taken to infinity). If there is a fixed point, then following the RG trajectory (almost) to it, one can in principle extract unambiguous answers for physical quantities on all energy scales (Niedermaier & Reuter Reference Niedermaier and Reuter2006). At the fixed point, the dependence on the UV cutoff is lost, and the theory is scale invariant: it does not change as smaller length scales are probed. While this approach may satisfy the Primary Motivation for QG, it does suffer other problems, including a conflict with the principle of unitarity. This problem of unitarity, means – roughly–that probabilities will not be conserved in time.Footnote ³²

2.4 UV Completeness

The failure of perturbative quantum GR to provide a predictive theory at the Planck scale is motivation for the Primary Motivation. The problem, from this perspective, is that the theory fails in the regime where we expect quantum effects to be significant for gravity. Nevertheless, the problem is often misdiagnosed as the failure of the theory to be UV-complete, and as indicating the need for a UV-completion of this theory. Thus, UV-completeness is often taken as a criterion of theory acceptance for QG. Here, following Crowther & Linnemann (Reference Crowther and Linnemann2019), I argue that it need not be.

A theory is UV-complete if it is formally predictive to all possible high energy scales (or, equivalently, all short distance scales). In other words, the theory can be used to generate predictions at all possible short distance scales. Being UV-complete does not guarantee that all of these predictions are correct, however (Newtonian mechanics is UV complete, for instance, since it does not formally ‘break down’ at any short distance scale, and yet we know that it is not the correct theory to use at all short distance scales). There are several ways of obtaining a UV-complete theory, including having a QFT that is non-perturbatively renormalizable, such as quantum chromodynamics.Footnote ³³ Yet, the idea of UV-completion is not restricted to QFT, and there are other ways in which a theory may be UV-complete, apart from being renormalizable.Footnote ³⁴

In the heyday of QFT, UV-completion was usefully employed as a criterion of theory success in QFT: that is, a means of selecting theories that were not only viable, but physical. The development of the RG and effective field theory techniques (and associated philosophy) led to a change in this perspective, where we recognize that theories can be physical without being fundamental – effective field theories correctly describe the world within their limited domains, and are consistent when applied within these domains. Nevertheless, it seems there remains a sort of hangover, where the general perception of UV-incomplete theories as being faulty, or mathematically inconsistent, still lingers.

Although approaches such as asymptotic safety are guided by the principle of UV-completion, QG need not be a UV-complete theory, unless QG is conceived of as a fundamental (final) theory. If we conceive of QG in a more minimal sense, then all that is required to satisfy the Primary Motivation is a ‘UV-better’ theory, predictive at the Planck scale (Crowther & Linnemann, Reference Crowther and Linnemann2019). Following Figure 3, we may thus distinguish between two options: either the ‘amalgamation’ of QG (as not ToE)Footnote ³⁵ plus the Standard Model (i.e., a non-unified set of theories) are together fundamental (i), or they are not. If they are not, then (some of) these theories are emergent, and they can either emerge from a ToE (ii), or they emerge from another amalgam (iii). And so on. Ultimately, there are three possible scenarios: (1) some non-unified set of theories is fundamental, (2) there is a ToE, or (3) there is no fundamental level, but a never-ending tower of theories.

Figure 3 Options on the road towards higher-energy physics: Suppose we have a non-unified set of theories ‘amalgam’ of QG and the Standard Model. It is possible that there is no deeper theory beyond, or that this amalgam emerges from a unified ‘final theory’, or that it emerges from another amalgam. This could happen at each ‘level’ of inquiry as we seek increasingly fundamental theories.

It is only in scenario (i) in which QG is necessarily UV-complete. This scenario is distasteful, however, in that it represents a disunified amalgamation of several theories as standing in place of a final ToE. The disunified nature of the amalgam would prevent us from believing it fundamental,Footnote ³⁶ and we would push on in the search for a more unified theory beyond. This would be the case in spite of all the theories in the amalgam being UV-complete. The lesson from this is that UV-completion should only be used as a constraint on QG if you accept that, at the fundamental level, there is no final theory, but rather a disunified collection of several theories.Footnote ³⁷

String theory is supposed to be a ToE, rather than just QG – as such, it must be UV-complete. The extendedness of the theory’s basic entities is very likely to ensure that it is UV-complete, although no proof has yet been found that it is (see Hagar (Reference Hagar2014, §7.2), Dawid (Reference Dawid2013, Ch. 1)). Textbooks on string theory often present an analogy between the perturbative non-renormalizability of quantum GR and that of 4-Fermi theory, which is a non-renormalizable theory that was revealed to be the low-energy limit of the renormalizable theory of electroweak interactions. Similarly, proponents of string theory claim that string theory is the renormalizable theory underlying perturbative quantum GR. Thus, the alleged UV-completeness of string theory is treated as a means of confirmation or indicator of pursuit-worthiness.

Other approaches to QG use UV-completion in different roles, particularly as a guiding principle, constraint, and means of confirmation. In some cases, including in LQG, UV-completion is obtained automatically in virtue of the theory naturally describing a minimal length scale (Rovelli, Reference Rovelli2004). Here, without taking UV-completeness as a constraint on QG, it may make sense to treat the principle as a means of confirmation, especially given the consilience of the idea of minimal length with other motivations for QG, such as singularity resolution, §6.

3.1 ‘Domains’

Smeenk (Reference Smeenk and Batterman2013) describes the ‘overlapping domain’ argument for QG, arguing that the early universe falls within the domains of applicability of both QFT and GR, and thus requires a theory of QG for its description. As stated earlier, however, both GR and quantum theory are supposed to be universal theories, so technically their domains of applicability overlap completely! Thus, QG should also be a universal theory, not restricted to particular scales or parts of the world. Properly, the motivation for QG is to describe domains where QG effects cannot be neglected – where the predictions of QG are thought to differ from those of GR and QFT. In other words, it should describe the overlapping domains of necessity – rather than applicability – of GR and quantum theory, as viewed from the current paradigm.Footnote ³⁸

Wallace (Reference Wallace2022), as well as the physicists interested in QG phenomenology, emphasizes that not all the domains of QG are exotic and inaccessible. Also, in fact, we have a working theory that satisfies the second condition of the Primary Motivation as a combination of GR and quantum theory: this is just QFT applied to GR,Footnote ³⁹ and which describes QG phenomena and observations – that is, observations of QG phenomena that we in fact already have access to – and the theory is relatively well-confirmed (depending on the different domains where it is applied, its support ranges from strong, to moderate, to weak). This is a low energy theory of QG (LEQG); it is an effective field theory, meaning that it is valid in a particular, restricted domain. This domain includes QG phenomena – being self-gravitating objects that also require QM to describe – in the semiclassical regime, both non-relativistic (describing planets and white dwarfs), and relativistic (describing stars, neutron stars, supernovas, and cosmology), and also the perturbative regime (having some support from observations of the solar system and cosmic microwave background).

The domain of LEQG, however, is low-energyFootnote ⁴⁰ – the theory is not thought to describe the Planck scale (explained next), the interior of black holes, or very early universe cosmology. It is these domains for which we need a full theory of QG, and which properly serve to characterize the Primary Motivation for QG. Thus, ‘the problem of QG’ from the perspective of high-energy physics, is that LEQG does not describe the high-energy (UV) regime. (The theory also faces particular theoretical and conceptual problems, but these are not what prevents it from satisfying the Primary Motivation.)

The Planck scale is characterized by combining the fundamental constants from our current theories: the speed of light, $c$ , the gravitational constant (Newton’s constant), $G$ , and the quantum of action (the reduced Planck’s constant), $\hslash$ . It was shown by Planck in 1899 that there is a unique way (apart from numerical factors) to do so. We arrive at the Planck length, $l_{\text{P}}$ , the Planck time, $t_{\text{P}}$ and the Planck mass (equivalently, the Planck energy), $m_{\text{P}}$ , respectively:Footnote ⁴¹

$\begin{array}{cc}{l}_{\text{P}}& =\sqrt{\frac{\hslash G}{c^3}}\approx 1.62\times {10}^{-35}\text{m}\end{array}$(6)

$\begin{array}{cc}{t}_{\text{P}}& =\sqrt{\frac{\hslash G}{c^5}}\approx 5.40\times {10}^{-44}\text{s}\end{array}$(7)

$\begin{array}{cc}{m}_{\text{P}}& =\sqrt{\frac{\hslash c}{G}}\approx 1.22\times {10}^{19}\text{GeV}\end{array}$(8)

In cosmology, we also refer to the Planck temperature $T_{\text{P}}$ , and Planck density, ${\rho }_{\text{P}}$ ,

$\begin{array}{cc}{T}_{\text{P}}& =\frac{m_{\text{P}}{c}^2}{k_{\text{B}}}\approx 1.41\times {10}^{-32}\text{K}\end{array}$(9)

$\begin{array}{cc}{\rho }_{\text{P}}& =\frac{m_{\text{P}}}{{l^3}_{\text{P}}}\approx 5\times {10}^{93}{\text{g/cm}}^3\end{array}$(10)

where $k_{\text{B}}$ is Boltzmann’s constant.

These are obtained by dimensional analysis, a familiar and useful technique in physics. Theories are typically characterized by their own fundamental constants, and because this dimensional analysis combines both the characteristic constants of GR and QM in a unique way to form a new set of characteristic constants, it is standardly thought that these characterize the regime where QG is necessary.

In spite of the popularity of the Planck scale as characterizing the domains of QG, the dimensional analysis that gives us the Planck scale is heuristic: it does not establish that this is the characteristic scale of QG. The relevance of the Planck scale can be criticized as a naive general estimate. Nevertheless, it is possible to make the case that the Planck scale is the relevant scale at which to expect new physics, based on arguments from both particle- and gravitational-physics which may be taken to suggest that our current picture of familiar ‘large scale’ (i.e., energies lower than the Planck energy) physics is complete (Held, Reference Held2019). Such a conclusion, however, neglects the problems of dark matter and dark energy (the cosmological constant Lambda), which may be considered as part of QG phenomenology.

Perhaps stronger than the heuristic arguments for the relevance of the Planck scale is the argument that the Planck scale is where LEQG – as standardly conceived, perturbatively – breaks down, so necessarily a new theory of QG is required here (§2.3).

The other domains that characterize QG are the interior of black holes and the very early universe. This is motivated by the idea that GR ‘breaks down’ in the vicinity of curvature singularities because of the neglect of expected quantum effects in these domains (§6). Another motivation comes from the theoretical predictions of black hole thermodynamics (§7).

3.2 ‘Describes’

Ideally, for a theory to describe particular domains, the theory gives predictions in these domains that are tested and not falsified. This is a challenge given the domains in question, however. More minimally, then, we might require that the theory give predictions that are physically reasonable when applied in these domains – although, what ‘physically reasonable’ means is a non-trivial question in this context! More work needs to be done to explore this, particularly in connection to discussions in general philosophy of science.

Of course, a theory being formally predictive, and that the predictions are taken as ‘physically reasonable’, in these domains does not guarantee that what it says is correct when applied here, but we can usefully appeal to the empirical constraints described in §1.3, including correspondence relations with GR and QFT to provide at least some guidance.

3.3 ‘Takes into Account Both QM and GR’

Finally, consider the second main condition of the Primary Motivation: that the theory take into account both QM and GR. This condition is motivated by the success of GR and QM, and also the fact that both are expected to be necessary in the particular domains indicated earlier – and yet, each are thought to be inaccurate here. There are several ways of understanding this condition, representing different viewpoints on the problem of QG. Ashtekar expresses some of these,

This quote demonstrates two ways in which QG could take into account GR and quantum theory: by recovering both theories in their respective domains of applicability (through correspondence relations, including limiting relations), and combining the principles of both theories. These two conditions represent what we might call the ‘new framework perspective’ of the problem of QG. It is also described in Ashtekar & Geroch (Reference Ashtekar and Geroch1974), in a passage where the authors state that GR and QM each can be understood as a distinct ‘body of universal rules’, which means that the quantization of gravity is essentially different from quantizing the electromagnetic field: ‘From this viewpoint, then, the problem is to obtain a new body of rules which suitably encompasses the essential features of those of quantum mechanics and of general relativity. (p. 1214).

This ‘new framework perspective’ on the problem of QG, however, is not the standard one adopted by physicists working on QG.Footnote ⁴³ Rather than seeking a new theory that replaces GR and QM, and recovers both theories as approximations in their respective domains, most physicists tend to treat QM as fundamental and GR as not.Footnote ⁴⁴ For instance, Loll et al. (Reference Loll, Fabiano, Frattulillo and Wagner2022, p. 16) expresses the sentiment that the ‘universality of quantum theory’ is one of the fundamental principles of modern physics: the physical world at the fundamental level is governed by quantum laws, and that the classical picture is only an approximation, valid at sufficiently low energies and on sufficiently large scales.

Thus, we find at least two different ways of satisfying the condition that QG ‘take into account’ the lessons of GR and QM. What I have called the New Framework Perspective holds (1) we need a new body of rules that express insights of both GR and QM, and (2) we need to recover both GR and QM as approximations in their respective domains. The Standard Perspective, however, holds (1) we need a quantum theory, and (2) from this we can recover GR as an approximation in the domains where we know GR is successful (on such a view, the recovery of QM is not required). While I will not here argue for one view or the other, I would at least like to suggest that the fundamentality of QM is an assumption that has not been questioned seriously enough in the search for QG, and that the problem of QG may well be that we require a whole new framework that could ‘explain’ the success of quantum theory just as it would explain the success of GR.

There are more-specific ways of satisfying the second main condition of the Primary Motivation. Butterfield & Isham (Reference Butterfield, Isham, Callender and Huggett2001), §3.1.3. describes four different types of approach to QG: (1) quantize GR, (2) GR as the low-energy limit of a quantization of a different classical theory, (3) GR as the low-energy limit of a theory that is not a quantization of a classical theory, (4) start ab initio with a radically new theory. Approaches 1 and 2 represent the Standard Perspective, approach 4 represents the New Framework Perspective, while approach 3 could represent either perspective, depending on whether or not the theory is formulated in the framework of quantum theory as we currently understand it. Any of these approaches could produce a theory that accords with the condition of ‘taking into account’ both GR and QM, if it satisfies the two necessary aspects described earlier. Nevertheless, the by-far dominant approach is (1), where not only is QG supposed to be a quantum theory, but one in which gravity is quantized.

It is possible that QG take into account QM without itself being a (purely) quantum theory. The most familiar way of thinking about this is along the lines of semiclassical gravity (§2) or other ‘hybrid’ approaches (§5), which attempt to couple the classical and the quantum as a sort of ‘amalgamation’ (as opposed to a unification) or ‘mongrel gravity’ theory (a term borrowed from Mattingly (Reference Mattingly2009); cf. Tilloy (Reference Tilloy2018) for discussion of some other hybrid theories). The New Framework Perspective countenances the possibility that the framework of QM – as it stands – is not applicable in QG, and gets modified somehow (in which case QG would be more fundamental than both GR and QM, §1.2, but need not be a full unification of GR and QM §4).

As stated earlier, Penrose believes that the measurement problem of QM points to the need for QG, and that the unitarity of QM will be modified, so that QM will become a non-linear theory at the Planck length. More recently, the unitarity of QM has been called into question in the context of quantum cosmology, where Cotler & Strominger (Reference Cotler and Strominger2022) replace linear unitary time evolution with linear isometric time evolution (note: this is very different from the quite radical proposal of Penrose). Other recent work in the New Framework perspective is the hybrid, ‘post-quantum classical gravity’ theory which modifies both GR and QM Oppenheim (Reference Oppenheim2023). This approach is motivated by problems with semiclassical gravity as well as perceived inadequacies to mainstream responses to the black hole information loss paradox.

QG can ‘take into account’ the lessons of GR and QG by means of principle correspondence: the new theory features one or more principles from each of the older theories (Hartmann, Reference Hartmann2002). As part of taking into account GR, it is popular to utilize background independence as a principle-correspondence, featuring in both GR and QG (§2.1). For principle correspondence in QM, the favourite principle to use is quantization (§5), which accords with the Standard Perspective. These principles are naturally, and conservatively, motivated by the universality of both GR and QM.

3.4 Three Pillars Perspective

An increasingly significant perspective upon the problem of QG is that it involves bringing together not just GR and QM, but also the third pillar of modern physics: thermodynamics (also including statistical mechanics). There are several approaches to QG that tie gravity to the thermodynamic concept of entropy. For instance, Jacobson (Reference Jacobson1995) argues that the Einstein Field Equations can be derived from the proportionality of entropy and black hole horizon area together with the first law of thermodynamics. Padmanabhan (Reference Padmanabhan2004), (Reference Padmanabhan2010) and Verlinde (Reference Verlinde2011), (Reference Verlinde2017) also present arguments for gravity being an emergent phenomenon of entropic origin. According to the ‘emergent gravity’ approaches, discussed in §5, spacetime is an effective thermodynamic entity; as such, a quantization of gravity (the metric field) would not lead us to the ‘micro’ degrees of freedom which we seek to describe by a more fundamental theory of QG. (These micro degrees of freedom, however, could themselves be described by a quantum theory, and so these approaches need not represent the New Framework perspective.) The link between GR, QFT and thermodynamics is motivated, and further revealed, through the theoretical results of black hole thermodynamics, discussed in §7.

3.5 Delimitations

The Primary Motivation is supposed to represent the minimal definition of QG: any approach that does not (attempt to) satisfy it would not be accepted, by the current mainstream consensus, as an approach to QG. The definition thus serves to delimit what would ‘count’ as an acceptable theory, and necessarily excludes alternative possibilities. We should bear in mind that there may be other solutions to the problems discussed here, which are overshadowed if we take them exclusively as problems that motivate QG (as a theory that satisfies the Primary Motivation). Examining the particular motivations in more detail can help us also to explore the other possibilities for their resolution, or reconsider the need to resolve them at all.

This is nicely illustrated by the example of the differing attitudes one can take towards spacetime singularities (§6): not all cases of singularities motivate the need for a new theory, and in the cases where the (particular) singularities do motivate a new theory, it is not certain that these point to the need for QG rather than a different theory formulated ‘at the level of current theory’ (i.e., a new classical theory of gravity such as a modification of GR). There is the possibility that other motivations for QG may also be addressed by reconsidering our current best theories, including their principles and formulation, rather than moving to a theory of QG.

5.1 Hybrid Theories and Emergent Gravity Approaches

If gravity is not quantized, one possibility is that it remains fundamentally classical, as in semiclassical gravity or other hybrid approaches. Recall that LEQG (§3) is an effective field theory for QG (it is taken to reproduce the results of QG at low energies), and comprises both a semiclassical treatment of gravity and a description of the quantum fluctuations of the LEQG field. The dominant attitude towards semiclassical gravity is that it is an approximation to LEQG in the situations where quantum fluctuations of the spacetime geometry can be ignored; that is, semiclassical gravity is a mean field description in LEQG. This attitude holds that gravity is fundamentally quantized (and will be described as such by the more fundamental, unknown, theory of QG), and is only effectively treated as classical (i.e., as an approximation in the ‘low-energy’ domains where we know GR is successful).

As Huggett et al. (Reference Huggett, Linnemann and Schneider2023) points out, though, there are two alternative attitudes towards semiclassical gravity. One of these holds that it is not the mean field description of LEQG, but some other theory. If the low-energy approximation of the more fundamental theory of QG were not LEQG, this would allow that QG is not necessarily a theory of quantized gravity. This view fits the various examples of ‘emergent gravity’ approaches mentioned next, as well as proposals such as that of Oppenheim (Reference Oppenheim2023). Huggett et al. (Reference Huggett, Linnemann and Schneider2023) take this view as an ‘epistemically careful’ one, where we accept that semiclassical gravity approximates some theory X, which in turn approximates QG, but we take no stance on whether X is LEQG, or some other alternative (one in which gravity may even be classical – or not). It allows us to remain neutral on the exact relationship between semiclassical gravity and QG. Alternatively, there is the possibility that semiclassical gravity (§2) is the fundamental theory of QG. This attitude is not a popular one, being plagued with both conceptual and mathematical difficulties.Footnote ⁵⁰

The label ‘emergent gravity’ refers to a broad and eclectic class of approaches to QG which treat gravity as a low-energy effective phenomenon that emerges from the collective behaviour of more fundamental microscopic degrees of freedom. The microphysics itself is not based on a discretization or quantization of GR (Linnemann & Visser, Reference Linnemann and Visser2018). Simple examples of this approach are analogue models of spacetime from Bose-Einstein condensates (BECs).Footnote ⁵¹ Others include Sakarov’s induced gravity (Sakharov, Reference Sakharov1967, Reference Sakharov2000), Jacobson’s gravitational thermodynamics (Jacobson, Reference Jacobson1995), Padmanabhan’s gravitational thermodynamics (Padmanabhan, Reference Padmanabhan2010), and Verlinde’s entropic gravity (Verlinde, Reference Verlinde2011).

Typically, these approaches share the conviction that gravity is analogous to hydrodynamics (or thermodynamics), and the Einstein equations are akin to the wave equation for sound in a medium (i.e., they are higher-level equations of state). Accordingly, they hold that quantization of GR does not give us a theory of the fundamental micro degrees of freedom – by analogy, we would arrive at a theory of phonons rather than a description of the underlying ‘atoms’ of spacetime. If approaches that quantize the metric tensor produce theories of particles analogous to phonons, then it is unsurprising that they should break down at high-energy. Their breakdown can motivate the search for a high-energy theory beyond (quantum) GR, but we cannot say that the degrees of freedom of the high-energy theory would themselves need to be quantized in order to produce a theory of QG.

5.2 Tabletop Experiments

In spite of the issues discussed earlier, there is widespread agreement that the quantum nature of gravity cannot be established (or ruled out) by conceptual niceties, theoretical arguments or mathematical deduction: it must be confirmed empirically. Even Rosenfeld maintained that the Bohr & Rosenfeld (Reference Bohr and Rosenfeld1933) argument against semiclassical gravity by analogy with ‘semiclassical electromagnetism’ (described in §2) did not demonstrate the inconsistency of semiclassical approaches, because no formal arguments can establish the necessity of quantization. Rather, according to Rosenfeld (Reference Rosenfeld1963), only experiment can answer such questions.

And, indeed, since then, various experiments have been proposed and carried out. The Colella-Overhauser-Werner neutron interferometry experiment, touted as ‘observation of gravitationally induced quantum interference’ demonstrated that the gravitational field affects the behaviour of quantum systems (Overhauser & Colella, Reference Overhauser and Colella1974; Colella et al., Reference Colella, Overhauser and Werner1975). The much-discussed Page and Geilker quantum Cavendish experiment (Page & Geilker, Reference Page and Geilker1981) claimed to provide evidence supporting the hypothesis that a consistent theory of gravity coupled to quantized matter should also have the gravitational field quantized; however, their results were shown to be dependent on the particular interpretation of QM adopted.Footnote ⁵² These experiments, respectively, show that quantum systems both feel the effect of gravity, and that quantum systems can be sources of gravitational fields – in accordance with predictions of LEQG (Adlam, Reference Adlam2022). However, neither of these experiments probe states in which gravity remains in a coherent quantum superposition, and as such, we have no empirical observations in the regime of LEQG where superpositions of spacetimes are important. More recently, a new class of ‘tabletop QG experiments’ have been proposed which claim to do exactly this (Bose et al., Reference Bose, Mazumdar and Morley2017; Marletto & Vedral, Reference Marletto and Vedral2017). Huggett et al. (Reference Huggett, Linnemann and Schneider2023) argue that how we assess the empirical scope of such experiments, though, is ‘a much more intricate affair than may be at first thought, and crucially depends on matters of physical interpretation that are not settled by rational argument’ (p. 74). Note that these experiments are – at this time – proposals, likely years away from being carried out.

6.1 UV Divergences in QFT

In §2.3, I described the problem of divergences associated with the perturbative methods in ‘conventional QFT’ (CQFT), and which lie at the heart of the formalism itself. Historically, these infinities were removed in particular theories, such as quantum electrodynamics (QED), via the procedure of renormalization which rendered the theory finite and (impressively) predictive. This procedure, however, was physically suspicious, and the perturbative approach to QFT itself remained intrinsically approximate and conceptually problematic.Footnote ⁵⁶

One response was the development of axiomatic QFT (AQFT): instead of introducing informal renormalization techniques to treat interactions, this approach attempts to put QFT on a firm, non-perturbative footing, by specifying a mathematically precise set of axioms at the outset. Then, models of the axioms are constructed (constructive QFT). Importantly, this approach is not an attempt at QG, but simply a new formulation of QFT at the level of QFT – that is, as a combination of QM and special relativity without any singularities in the theory.Footnote ⁵⁷ Unlike CQFT, it is not an intrinsically approximate theory, since it is supposed to be directly defined on Minkowski spacetime, and so remain well-defined at arbitrarily small and arbitrarily large length scales. Although there are various simplified toy models satisfying the axioms of AQFT, there have been no realistic models constructed. In particular, it has not been demonstrated that QED or any other successful theories in high-energy physics admit formulations that satisfy the axioms of AQFT.

Meanwhile, in mainstream high-energy physics, the development of RG techniques led to a framework for studying CQFT systems at different energy scales, and ultimately to the discovery of the Standard Model. The RG analysis demonstrates that the means by which the cutoff is implemented has no bearing on the low-energy physics; the only effects that are significant at these scales are changes in the coefficients of finitely many interaction terms (the renormalizable interactions). In other words, RG theory tells us that large-scale phenomenology gives us almost no information about the nature of the short-distance cutoff, if there is one.

The dominant philosophical interpretation of this CQFT picture is that the UV divergences are not a real physical problem, but rather indications of the limitations of the perturbative framework of QFT. The framework itself is taken to be inherently approximate,Footnote ⁵⁸ and its models are supposed to be effective theories: not to be valid to arbitrarily high energy scales. ‘This, in essence, is how modern particle physics deals with the renormalization problem:Footnote ⁵⁹ it is taken to presage an ultimate failure of quantum field theory at some short length scale, and once the bare existence of that failure is appreciated, the whole of renormalization becomes unproblematic, and indeed predictively powerful in its own right’ (Wallace, Reference Wallace2011, p. 120). The idea is that, whatever the unknown physics of QG turns out to be, the success of the CQFT models at known energies is explained, thanks to the RG.

This interpretation of CQFT as effective means that the theory is not supposed to be reliable at short length scales. In particular, the need to employ a short-distance cutoff is not taken to indicate anything regarding the physics beyond. This is in contrast with the case in condensed matter physics, where the RG is also employed because the system is described by a theory which diverges in the UV, but in which case the divergences, and the need to employ a short length scale cutoff is consistent with the existence of something physical – we know that we cannot treat matter as continuous at arbitrarily short length scales, because matter has a discrete structure at the atomic scale. In CQFT, however, there is no empirical evidence for the existence of a real physical cutoff (e.g., a discrete structure for spacetime at extremely high energies). It is possible that the UV divergences in CQFT simply reflect limitations of the theory, rather than any new physics.

6.2 Landau Poles

A motivation for treating particular QFTs (rather than the framework of QFT itself) as effective field theories is the existence of Landau poles. This type of infinity is thought more concerning than the UV-divergences because it is not taken to be merely due to the limitations of perturbative analysis.

The most familiar example of Landau pole is in QED. This theory is renormalizable, so in principle it should be able to be extended to arbitrarily high energies. Yet, the renormalized coupling grows with energy scale, and becomes infinite at a finite (though extremely large) energy scale, estimated as ${10}^{286}$ eV (the original result comes from Landau et al. (Reference Landau, Abrikosov and Khalatnikov1954)). The existence of this ‘pole’ could mean the theory is mathematically inconsistent. This is avoided if the renormalized charge is set to zero, that is, if the theory has no interactions. There is indication that in QED, the renormalized charge goes to zero as the cutoff is taken to infinity (a physical interpretation of this is that the charge is completely screened by vacuum polarization). This is a case of quantum triviality, where quantum corrections completely suppress the interactions in the absence of a cutoff. Since the theory is supposed to represent physical interactions, the coupling constant should be non-zero, and so the Landau pole and the associated triviality might be interpreted as a symptom of the theory being effective, or incomplete (i.e., that it fails to take into account other fundamental interactions relevant at high energy scales).

QED and ${\varphi }^4$ theory are thought to be trivial in the continuum limit in this way. In other words, RG analysis of QED and ${\varphi }^4$ does not indicate that these theories possess a stable UV fixed point (Lüsher & Weisz, Reference Lüsher and Weisz1987; Gies & Jaeckel, Reference Gies and Jaeckel2004). This means that the Standard Model of QFT is thought to suffer Landau poles both for the electron charge, and the Higgs.Footnote ⁶⁰ Although the Landau pole in QED is problematic for the theory, it is usually ignored because it concerns an energy scale where QED is not thought to be valid anyway, given that electroweak unification occurs at an energy scale lower than this. The Higgs triviality is thus more interesting for the Standard Model, as potentially indicative of new physics at high energy.Footnote ⁶¹ However, it also concerns an energy scale where QFT itself is not thought to be valid, based on the external arguments for the Planck length.

In summary, the Landau pole divergences in CQFT are typically interpreted as part of the formal (mathematical) grounds – internal to the theory – for treating particular QFTs as effective. But there are also external grounds (not from the theory itself) for treating particular QFTs – and perhaps the framework of QFT itself – as effective, which come from the motivations for QG. These external motivations hold regardless of whether or not there are divergences inherent to QFT, but they are reasons why one might not be concerned with finding a singularity-free theory of QFT in order to describe the world at arbitrarily small length scales.

6.3 Responses to QFT Singularities

There are three different types of divergences associated with QFT that may bear on the problem of QG: non-renormalizability of gravity (§2.3), UV-divergences in QFT, and Landau poles. The perturbative non-renormalizability of quantum GR (LEQG, §2.3) does indicate the need for a theory of QG valid at the Planck scale, though this theory may be non-perturbative quantum GR (i.e., the asymptotic safety scenario). The UV divergences of QFT may be artefacts of perturbation theory, and there is a danger of misinterpreting the need for a high energy cutoff as indicating a physical or operational minimal length. They do not point directly to the need for QG. Landau poles in the Standard Model of particle physics do (likely) represent an incompleteness of the theory and motivate the need for a new theory – though this new theory need not be QG. Landau poles, however, are typically ignored by physicists who already expect QG to replace QFT at the Planck scale for external reasons (§3.1, also Held (Reference Held2019, §1).

From the preceding discussion, we can identify four different possibilities in response to these singularities (following Crowther & De Haro (Reference Crowther, De Haro and Vassallo2022)):

1. i. AQFT view: Singularities in CQFT motivate a different QFT framework, one whose theories are singularity-free, but which does not include gravity (as in AQFT);

2. ii. New physics: Singularities in CQFT motivate a new, more fundamental theory at high-energy, and motivate treating our current theories as effective (applicable only at low energy scales), consistent with external motivations for QG;

3. iii. Effective theory view: Ignore the Landau poles in the Standard Model and perturbative non-renormalizability of gravity (we shouldn’t worry about interpreting them), since we have external reasons for thinking of these as non-fundamental effective theories, to be replaced by QG at high-energy scales (i.e., we appeal only to the external motivations for new physics, and the singularities in current theories do not count as motivations for new physics);

4. iv. Asymptotic Safety view: UV divergences due to perturbative non-renormalizability and Landau poles do not motivate new physics or a new theory according to the asymptotic safety scenario for gravity and the Standard Model; these singularities do not appear in the full (non-perturbative) theory.

There is one more response to singularities in CQFT, which has been expressed prominently by Jackiw (Reference Jackiw and Cao1999, Reference Jackiw, Fokas, Grigoryan, Kibble and Zegarlinski2000) and Batterman (Reference Batterman2011).

1. v. Emergent physics view: Singularities in CQFT are of physical significance, but not motivation for new, more fundamental physics. The singularities are important for facilitating and understanding the emergent, low-energy physics.

Following Crowther & De Haro (Reference Crowther, De Haro and Vassallo2022), we can classify these positions under four different ‘attitudes’ towards singularities . Attitude 1 means that the singularities are resolved ‘at the level of QFT’, without pointing to a more-fundamental theory or framework. Here, (i) and (iv) fall under Attitude 1. Attitude 2 means that the singularities are to be resolved by a more fundamental theory of QG, and here (ii) is an example. Attitude 3 is a positive attitude towards singularities, which means not resolving them at any level, because we have reason to keep them (or reasons why they cannot be removed). Example (v) is a case of Attitude 3, where the singularities are seen as importantly explanatory, or ‘essential’. Finally, Attitude 4 means that we do not care about the singularities in question; they are of no significance. (iii) is an example of this attitude. Importantly, one can (and should) take a different attitude towards different types of singularities. This classification also holds when considering spacetime singularities: taking them as pointing to QG is only one possible option (depending on which singularities are being considered).

The QFT divergences, in contrast to the spacetime singularities to be discussed, are not typically appealed to as motivation for QG (though this argument occasionally appears); more commonly the resolution of these is taken as a means of confirmation, or indication of pursuit-worthiness of approaches to QG that resolve these divergences (though we still classify this falling under Attitude 2). The adherence to quantization as the means of solving the problem of QG is consilient with the resolution of QFT divergences: if we quantize gravity, the UV divergences are usually expected to be resolved ‘for free’ (via a minimal length), and this is taken to support approaches to QG that quantize GR. As stated in §2.3, the (alleged) resolution of the UV-divergence of quantum GR by string theory is presented as one of the selling points of string theory.

Some believe that the singularities in GR and QFT are to be cured by the existence of a minimal length, even without quantization of gravity (note, too, that the minimal length need not represent an actual discretization of spacetime, but may be an operational minimal length, e.g., due to an extended probe).Footnote ⁶² Henson (Reference Henson and Oriti2009) expresses this view, particularly in regard to the need for QG to resolve the divergences associated with the non-renormalizability of GR, and uses it as motivation for the discreteness postulated by an approach to QG known as causal set theory.

6.4 Geodesic Incompleteness

The definition of spacetime singularities in terms of geodesic incompleteness states that a spacetime is singular if and only if it contains an incomplete, inextendible timelike geodesic. Such a geodesic is the worldline of a freely falling test object; the property that makes it singular is that the worldline ends within finite proper time and cannot be further extended. While this definition forms the basis of the Penrose and Hawking singularity theorems, it is not without problems (see Curiel, Reference Curiel and Zalta2023, §1.1). Geodesic incompleteness appears to be a genuine physical worry, because it means that ‘particles could pop in and out of existence right in the middle of a singular spacetime, and spacetime itself could simply come to an end, though no fundamental physical mechanism or process is known that could produce such effects’ (Curiel, Reference Curiel1999, p. S140). Geodesic incompleteness leads to a lack of predictability and determinism, and so could indicate that the theory is incomplete (Earman, Reference Earman1995, §2.6).Footnote ⁶³ If the breakdown of determinism were visible to external observers, ‘then those observers would be sprayed by unpredictable influences emerging from the singularities’ (Earman, Reference Earman1992, p. 171). This would represent a nasty form of inconsistency – as Earman puts it, the laws would ‘perversely undermine themselves’.

The breakdown of determinism inside black holes occurs beyond the Cauchy horizon (inside the event horizon): beyond this surface, the Einstein equations no longer give a unique solution. In response, Penrose (Reference Penrose, Hawking and Israel1979) proposed strong cosmic censorship (SCC), which postulates that the appearance of the Cauchy horizon in Schwarzschild black holes is non-generic, and that the interior region of these black holes is in some way unstable (under small perturbation of initial data) in the vicinity of the Cauchy horizon. Any passing gravitational waves would prevent the formation of Cauchy horizons, meaning that instead, spacetime would terminate at a ‘spacelike singularity’, across which the metric is inextendable.

SCC ensures that no violations of predictability are detectable even by local observers (i.e., an astronaut on a geodesically incomplete worldline would detect nothing up until, and presumably after, her disappearance), and so, the truth of this conjecture would render any singularities (incomplete geodesics) harmless in regard to determinism. As Dafermos & Luk (Reference Dafermos and Luk2017, p. 5) states, ‘The singular behaviour of Schwarzschild, though fatal for reckless observers entering the black hole, can be thought of as epistemologically preferable for general relativity as a theory, since this ensures that the future, however bleak, is indeed determined’. Thus, SCC may be able to save GR from the charge of incompleteness.

There are two issues that have been raised in the philosophy literature that bear on the question of indeterminism and incompleteness in GR, however. The first regards the nature of the spacetime singularities themselves: (essential) singularities are not located at a point of spacetime (if a singularity is located at a point in spacetime, then this is indicative of having a non-essential singularity, that is, one that can be removed, in the case of incomplete geodesics, by extending the geodesics beyond that point). As Curiel (Reference Curiel1999, Reference Curiel and Zalta2023) emphasizes, a singular spacetime does not have any ‘missing points’ of spacetime. (For this reason, Curiel and others have stressed the global, rather than local nature of singularities.) Earman (Reference Earman1995, Reference Earman1996) also uses this argument against the claim that spacetime singularities indicate that GR (alternatively abbreviated as GTR) is incomplete,

Smeenk (Reference Smeenk and Batterman2013) makes the same argument in regard to the big bang singularity: that the laws of GR apply throughout the entire spacetime, and there is no obvious incompleteness. If GR is the correct final theory, ‘then there is nothing more to be said in regard to singularities’ (p. 634). While Smeenk does not view spacetime singularities as motivation for seeking QG, he states that there are other reasons for doubting that GR is the correct final theory, and good reasons to expect that the successor to GR will have novel implications for singularities – that is, he recognizes external motivations for QG, without counting the GR singularities as internal motivation for seeking a new theory.

Although Earman generally advocates a ‘tolerance for spacetime singularities’ – arguing that we can treat them as predictions, rather than pathologies of GR – he nevertheless believes there is one way in which the charge of incompleteness may be justified. This is the idea, described earlier, that if SCC does not hold, then the determinism of GR is undermined (Earman, Reference Earman1995, Reference Earman1996). Earman ties the determinism of GR to spacetime models that are globally hyperbolic and thus admit of a globally well-posed initial value formulation. The problem with this, however, is a ‘dirty open secret’, that determinism in GR fails without help by fiat (i.e., the imposition of ad hoc constraints that simply rule out those spacetimes that do not admit a globally well-posed initial value formulation).

This leads to the second issue discussed in the philosophy literature – whether GR is a deterministic theory, and whether all models of the theory represent physically possible spacetimes. These questions are explored in Smeenk & Wüthrich (Reference Smeenk and Wüthrich2021), which also highlights a tension between the ‘philosopher’s conception’ of determinism and the conditions needed for having a well-posed initial value formulation in GR physics. Doboszewski (Reference Doboszewski2019, Reference Doboszewski2020) also discusses the problems of defining determinism in GR, with the former paper arguing for a pluralistic conception. These issues with determinism in GR may take the bite out of the worry that geodesic incompleteness without SCC is a problem motivating QG (though, of course, this depends on one’s convictions regarding the need for a deterministic theory).

6.5 Curvature Singularities

According to the definition of curvature singularities, a spacetime is singular if the curvature, especially the scalar quantities constructed by contracting powers of the Riemann tensor, grows without bound in some region of the spacetime.Footnote ⁶⁵ This gives rise to various problems, such as unbounded tidal forces and the lack of consistency of the semiclassical approximation. The semiclassical approximation being referred to here treats GR as an effective field theory, with the Einstein-Hilbert action supplemented with additional higher-curvature terms that represent the quantum corrections to the theory.Footnote ⁶⁶ If we take a model, M, of GR with a curvature singularity, and then check whether it is an approximate solution of the equations with the quantum corrections, we find that in general, M may be a good solution far away from the singularity, but as we approach the singularity, the higher-order (high curvature) terms will start to dominate over the lowest order term (Einstein tensor). Because of this, M will not be a model of the quantum-corrected theory. This shows how the curvature singularities can be used to predict that GR ‘breaks down’, since it reveals an inconsistency between GR and (expected) QG effects that manifest at high curvature in the region close to the curvature singularity.

This is a standard argument found in the physics literature, and physicists tend to find the curvature singularities more concerning than geodesic incompleteness, which has been the focus in the philosophy literature. Notice, however, that here it is not the singularity itself that is the problem (depending on the outcome of the charge of indeterminism described earlier), but the expected quantum effects in regimes of high curvature – it is this that motivates the need for a new theory.

6.6 Responses to Spacetime Singularities

Earlier, I argued that geodesic incompleteness may not represent an incompleteness of GR, depending on the status of the SCC and/or the assessment of (in)determinism of GR. Curvature singularities, meanwhile, are arguably problematic insofar as they represent domains of extreme curvature where GR is no longer thought to be reliable, and thus do motivate the need for a new theory to replace GR. If we take it that GR is unreliable due to its neglect of quantum effects in these domains (as is the standard interpretation), then this new theory required is QG.

The dominant attitude towards spacetime singularities – especially the big bang singularity and black hole singularities – is as motivation for QG. This attitude treats GR as a classical approximation, an ‘effective theory’, and the singularities in GR signal its limitations as such. Accordingly, singularities should be resolved by a more fundamental theory of QG.Footnote ⁶⁷ This attitude is inspired by the history of physics, in particular the suggestion that singularities in classical theories signal the breakdown of these theories and the need for quantization. An oft-cited example in this context is the resolution of the instability of the hydrogen atom in classical electrodynamics, by quantum mechanics. Electrodynamics predicts that the electron radiates energy as it orbits the nucleus and eventually falls into the nucleus, at which point the force becomes infinite. QM solves this by, first, rejecting the classical picture of an electron ‘orbiting’ around the nucleus. Second, by confining the electron to discrete energy levels, and allowing it to emit energy only in discrete packets of energy (photons). And finally, the lowest energy level allowed by the theory is where the electron is, on average, located at a finite distance from the nucleus (the ‘Bohr radius’), so that it can never fall in. In regard to the spacetime singularities, the idea is that, analogously, gravity should be quantized and that QG solutions have a discrete spectrum, similar to the discrete spectrum of QM – and that, accordingly, singularities can be avoided.

While this argument by analogy appears often in the literature, it may be criticized on various grounds. Earman (Reference Earman1996) argues against the applicability of this analogy on two points (1) the instability of the atom in this picture comes not from a single, well-confirmed theory, but by combining the Rutherford model of the atom with classical electrodynamics, (2) this combination leads to a prediction that is empirically ‘falsified by the fact that the material world exists’ (p. 629). By contrast, Earman states that there is no such definitive conflict with observation in the case of GR. The predictions of GR regarding the formation of black holes in gravitational collapse are meeting with increasing empirical success (though there is more difficulty in assessing the empirical success of Standard Model of cosmology which gives us the big bang singularity). Another line of criticism is expressed by Tilloy (Reference Tilloy2018): just because quantization might solve the problem of singularities in GR does not mean it is the only solution, and promoting it as such is, in a way, avoiding the problem by saying it will be dealt with by an unknown, more-fundamental theory. Indeed, by looking to QG rather than seeking to understand singularities in GR ‘at the level of GR’, we may be missing key physical insights of the theory. This attitude is advocated by Earman (Reference Earman1996) (following Misner (Reference Misner1969)), who argues that singularities in GR should be treated as predictions of the theory.

In Crowther & De Haro (Reference Crowther, De Haro and Vassallo2022), we find there are four different attitudes one may take towards singularities in GR, the same as those mentioned earlier for the divergences in QFT. Attitude 1 for spacetime singularities means that they are to be resolved classically, or ‘at the level of GR’, rather than pointing to QG. Attitude 2 means they are to be resolved by QG; this is the attitude that dominates the physics literature, and which I have focused mostly on here. Attitude 3 is a tolerance for spacetime singularities, which means not resolving them at any level, because we have reason to keep them. This is exemplified by Earman’s attitude expressed earlier, where they are treated as predictions of GR, not pointing to QG. Attitude 4 is an indifference to spacetime singularities: they are not of any significance. We find examples of each of these attitudes in the literature.

The choice of which attitude to take depends on the singularity being considered. We can – and should – adopt different attitudes towards different singularities. The choice also depends, amongst other things, on one’s disposition towards the internal versus external motivations for QG, and one’s position in regard to scientific realism. For instance, if the spacetime singularities (geodesic incompleteness) are not necessarily problematic for GR, then these would not count as internal motivations for QG. By contrast, in QFT the Landau poles are more readily interpreted as signalling incompleteness of the theories; these are, however, typically dismissed because they occur in regimes where the theory is not thought to be applicable anyway for external reasons (e.g., the existence of the Planck scale as a minimal length). The tension between the internal and external reasons for treating a theory as effective is reflected in differing possible attitudes towards singularities in current theories: whether they are to be resolved by a more fundamental theory of QG, whether we should instead look to resolve them ‘at the level of current theories’ (i.e., through developing a new framework for QFT, or modifying GR, etc.). Those who put weight on the external motivations for QG may tend to disregard alternative, internal, possibilities for singularity resolution.

Even if singularity resolution is not, in all cases, a motivation necessitating QG, it can be used as a guiding principle inspiring new work. And, if a new theory of QG naturally resolves the problematic spacetime singularities (i.e., if QG describes the ‘domains’ where GR is thought to ‘break down’, or be problematically in deterministic, as well as explains why these occur in GR, and the theory itself not feature any problematic singularities) then, just as in the case of the QFT divergences, this may be promoted as a means of confirmation, or an indication of pursuit-worthiness. Making the case for a stronger role, where singularity resolution is taken as a criterion of acceptance for any theory of QG, however, requires further argument.

7.1 Hawking Radiation

Hawking (Reference Hawking1974, Reference Hawking1975) showed that the QFT vacuum state becomes a state with (real) particles present if there is a black hole – that is, that black holes radiate. More precisely, Hawking considered the vacuum state of a scalar field in the presence of an event horizon and found that it evolves into a thermal state of temperature $T$ . Because the thermal spectrum of the radiation encodes a Planckian temperature that is proportional to the surface gravity, and the surface gravity plays the role of temperature in the zeroth and first laws of BHT, the temperature is associated with the black hole. Whereas, classically, a black hole is thought to absorb everything and emit nothing (including light), Hawking’s result shows that black holes emit thermal radiation with the Planckian power spectrum characteristic of a perfect blackbody at a fixed temperature. It results in energy being radiated away, leading to the eventual ‘evaporation’ of the black hole. A primordial black hole of mass ${10}^{12}$ kg, produced in the early universe, would evaporate around now. The Hawking temperature of a solar mass black hole is ${10}^{-7}$ K and its lifetime is ${10}^{64}$ years. Figures 5 and 6, from Almheiri et al. (Reference Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini2021), depict this process.

Figure 5 The Hawking process of black hole evaporation: (a) Stellar collapse: the interior geometry continues to elongate in one direction, while pinching towards zero size in the angular directions; (b) Hawking process creates entangled pairs of particles, with one partner trapped behind the horizon and the other escaping to infinity where it is observed as blackbody radiation. The black hole shrinks as its mass is slowly carried away; (c) Eventually, angular directions shrink to zero, as does the event horizon; (d) Final stage: a smooth spacetime containing thermal radiation but no black hole.

Figure 6 Penrose diagram for the formation and evaporation of a black hole. Spatial slices (a)–(d) correspond to the slices drawn in Figure 5.

(Almheiri et al., Reference Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini2021).

A non-technical heuristic explanation attributes the Hawking effect to vacuum fluctuations leading to the separation of positive and negative modes through the black hole’s event horizon. The QFT vacuum state swarms with pairs of particles that are being constantly created and annihilated. If such a pair forms near the event horizon, one of the entangled partners may fall into the black hole, while the other escapes to infinity – the separation of the pair precludes their otherwise typical recombination, and results in the exterior region having an effective excess energy (real particles). The escape of the excess particles to the asymptotic region is then what gives rise to outgoing radiation.Footnote ⁷²

Hawking’s original calculation was done using the framework of QFT in curved spacetime.Footnote ⁷³ This is a non-trivial, but consistent, theory of quantum fields on the background of a classical relativistic spacetime, which differs from semiclassical gravity in its neglect of the backreaction of matter on the curved spacetime. This neglect is justified in general if the relevant length scales are large and curvature effects small. Thus, we expect this framework to naturally apply in the cases where the curvature of spacetime is well above the Planck length, and where we can assume (i.e., we have some theoretical grounds for expecting) that we can ignore any quantum properties of spacetime itself. Nevertheless, the empirical support for QFT in curved spacetime is both very indirect and highly nonspecific, since it comes not from indirect measurements concerning the predictions of particular QFTs in curved spacetime, but rather from claims that particular QFTs in curved spacetime show the right behaviour in various limits.Footnote ⁷⁴ An example is the geometric optics limit of electrodynamics in curved spacetime, which is a limit of quantum electrodynamics in curved spacetime, describes light rays as tracing out null geodesics given sufficiently high frequency relative to the curvature scale. That light indeed moves on null geodesics in curved spacetime has been observationally verified, for example, in gravitational lensing effects (see Dyson et al. (Reference Dyson, Eddington and Davidson1920)). The prediction of Hawking radiation is one of only two main specific predictions that could serve as a test for this framework.Footnote ⁷⁵

Another problem with Hawking’s original derivation is that it involves field modes whose wavelengths become shorter than the Planck length near the black hole horizon – this ‘trans-Planckian’ regime is beyond the domain of applicability of QFT in curved spacetime, and so we should not expect to trust its results there (instead, requiring a theory of QG to describe this regime). The problem arises due to an exponential gravitational redshift assumed to occur at the horizon, which means that the black hole radiation detected at late times (i.e., the outgoing particles) corresponds to extremely high (divergent) frequency when traced back to the horizon.

Yet, there are at least four other derivations of the Hawking effect in addition to Hawking’s original method (Wallace, Reference Wallace2018a). Together, Wallace claims, these strongly suggest that Hawking radiation is a real consequence of QFT in curved spacetime. Curiel (Reference Curiel and Zalta2023), on the other hand, argues that not all these derivations are consilient,Footnote ⁷⁶ given that some appear to rely on assumptions that contradict assumptions made by others: leaving the physical content of the prediction of Hawking radiation obscure at the foundational level. Curiel argues against identifying Hawking radiation with blackbody radiation given that, in the semiclassical picture, it is not generated by micro-degrees of freedom on the event horizon – emphasizing, instead, that we need further assumptions to bridge the derivation of the statistical mechanical entropy in LEQG (which does, at least naively, appear to treat the quantum microstates as located in a thin skin around the event horizon) with the derivation of Hawking radiation (which describes quantum fields external to the horizon).Footnote ⁷⁷

Similarly, while Wallace argues that we can understand QFT on curved spacetime as an application of the general machinery of modern QFT, Curiel (Reference Curiel and Zalta2023, §6.1) highlights not only a profound difference between the two but also calls attention to the fact that QFT itself (even in the flat spacetime of special relativity) is fraught with difficulties.Footnote ⁷⁸

While the detection of Hawking radiation may not reveal anything of its fundamental nature, it is nevertheless fair to say that an empirical test of Hawking radiation is required in order to entrench BHT. Yet, the Hawking radiation of typical astrophysical black holes has an extremely weak empirical signature, making its detection a near impossibility. As a black hole’s temperature $T$ is proportional to its surface gravity $\kappa$  – and, thereby, inversely proportional to its mass $M$  – the temperature of the radiation becomes smaller the bigger the black hole considered. Consequently, black holes detectable in astronomical settings would emit radiation of an order several million times smaller than the temperature of the cosmic microwave background (Thébault, Reference Thébault, Dardashti, Dawid and Thébault2019).

7.2 Black Hole Information Paradox

If black holes radiate – if they are ordinary statistical mechanical systems obeying standard QM – then we would expect that their radiation and eventual evaporation be described as a fully unitary process, in accordance with QM. Yet, the calculations of the Hawking effect in QFT in curved spacetime describe a process that appears strongly non-unitary. This leads to what is known as the ‘information loss paradox’. In this context, there are two different problems that have been discussed: the ‘final state problem’ (or ‘evaporation time problem’) and the ‘Page time problem’; I briefly introduce each in turn.

Quantum systems are subject to unitary evolution: the initial conditions together with the Schrödinger equation fix the future state of the system. This means that unitary evolution ‘preserves information’: if we start with a quantum system in a known state, then quantum theory ensures that the system will evolve in such a ways that we can infer the state of the system at some later time, and vice versa – the quantum evolution takes a pure state to a pure state, and this process is reversible. This is challenged by black hole evaporation. If we drop a cup of coffee into the black hole, information about its state still exists (though inaccessible) beyond the horizon, but once the black hole completely evaporates, all that is left is Hawking radiation (as shown in Figure 5 (d)). Hawking radiation is perfectly thermal, meaning that the ‘information’ about what was dropped in cannot be recovered: we cannot infer the previous state of the system given the final state of the Hawking radiation. Another way of seeing the problem is to imagine forming a black hole through the collapse of a pure state (such as a large amplitude gravity wave), and then noting that at the final state of evaporation is a mixed state: to an outside observer, the process of black hole formation and evaporation is a pure-to-mixed, irreversible transition. In other words: the evolution is non-unitary.

To see why we end up with thermal (mixed state) radiation despite starting from an initial pure state, recall that the effect arises from considering the QFT vacuum in the presence of a horizon. In doing so, we are essentially splitting the original vacuum state into two parts: one interior and one exterior. While the QFT vacuum state is a pure state overall, its degrees of freedom are entangled at short distances, which means that when we consider only half of the space, we find a mixed state on that half. In the heuristic picture described earlier (§7.1) involving pair creation, the two particles are entangled with one another, forming a pure state, but when we consider only the outgoing radiation, we find it in a mixed state looking like a thermal state at the Hawking temperature. [The presentation of the information loss paradox in terms of ‘information’ thus risks being a bit misleading – the basic problem has really to do with the entanglement, or correlations, between matter within the black hole and matter far outside the black hole.]

One difficulty with this story, however, is that the final stages of black hole evaporation occur when the black hole’s curvature is of the Planck scale, and so the semiclassical description that gives rise to the ‘paradox’ is not itself valid here. It cannot tell us what actually happens in the final stages of evaporation. There are various proposals that attempt to exploit this ‘loophole’ in order to save unitarity: for instance by having the ‘information’ ejected in the final bursts of energy, or by having it encoded in a non-radiating Planck scale ‘remnant’ black hole, or something even more exotic (Hawking & Stewart, Reference Hawking and Stewart1993). None of these proposals appear satisfactory (Marolf, Reference Marolf2017; Unruh & Wald, Reference Unruh and Wald2017).

For GR physicists, and most philosophers, this black hole ‘final state problem’ is not really seen as a paradox – or, indeed, a problem.Footnote ⁷⁹ On this view, ‘the problem’ is simply an argument to the conclusion that the black hole evaporation is a non-unitary process (this is often supported, for instance, by claiming that the black hole evaporating spacetime is not globally hyperbolic, and we do not expect unitary evolution in such spacetimes). Early on, Hawking himself (Hawking et al., Reference Hawking, King and McCarthy1976) stated that the full theory of QG will be one that violates the standard dynamical principle of quantum evolution. Hawking later changed his view, persuaded by arguments in favour of the holographic principle, that ‘information’ is stored on the event horizon, §7.3. The opposing side in this debate holds that the evolution of an initial pure state to a final mixed state is in conflict with QM. Arguments to this effect are mainly found from within the ‘particle physics perspective’ on QG (typically, string theorists), which treats unitary evolution as an inviolable principle. Why exactly this need be so, however, is debatable.

The other main form of the ‘information loss paradox’ is the ‘Page time problem’. Here, the problem occurs not in the final stages of evaporation where we expect the semiclassical theory to be invalid anyway, but rather in a regime where the black hole is still big, and where we have no reason to expect the semiclassical description to be incorrect. To understand this paradox, we need to first distinguish between two notions of entropy that we ordinarily use in physics.Footnote ⁸⁰ The first is the von Neumann entropy, $S_{vN}$ , also called the fine-grained entropy, or ‘entanglement entropy’. Given the density matrix, $\rho$ , describing the quantum state of the system, it is defined as,

$S_{vN}=-\text{Tr}\left[\rho \text{log}\rho \right]$(15)

This equation quantifies our ignorance about the precise quantum state of the system. It vanishes for a pure state, indicating our complete knowledge of the quantum state. Importantly, it is invariant under unitary time evolution, $\rho \to U\rho U^{-1}$ .

The second notion of entropy is the coarse-grained entropy, $S_T$ , also known as the thermodynamic entropy. In this case, we have the density matrix $\rho$ , but instead of measuring all observables, we measure only a subset of simple, or coarse-grained, observables, $A_i$ . We then consider all possible density matrices $\tilde{\rho }$ that give the same result as our system for the observables we are tracking, and then compute the von Neumann entropy, $S\left(\tilde{\rho }\right)$ . Finally, we maximize this over all choices of $\tilde{\rho }$ . A simple example is the ordinary entropy used in thermodynamics: here, $A_i$ are often chosen to be a few observables, for example, the approximate energy and the volume, and then the thermodynamic entropy is obtained by maximizing the von Neumann entropy among all states with that approximate energy and volume. The coarse-grained entropy obeys the second law of thermodynamics: it tends to increase under unitary time evolution. The generalized entropy, $S_{tot}$ in eq. 14, comprising the entropy of the black hole plus the entropy outside – if it is to be either of these two entropies – would be the coarse grained, thermodynamic entropy (Almheiri et al., Reference Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini2021).

The von Neumann entropy cannot be larger than the thermodynamic entropy: to see this, realize that we can always consider $\rho$ a candidate $\tilde{\rho }$ . Or, in other words, because the thermodynamic entropy provides a measure of the total number of degrees of freedom available to the system, it sets an upper bound on how much the system can be entangled with something else.

$S_{vN}\ge S_T$(16)

Now, consider the black hole as an ordinary thermodynamic system. The von Neumann entropy of the matter that formed the black hole will be extremely small compared to the black hole’s initial thermodynamic entropy.Footnote ⁸¹ The system cools through the emission of thermal radiation – that is, through the emission of quanta of radiation in highly mixed states. In the early stages of black hole evaporation, the von Neumann entropy will be almost exactly thermal because the radiation is entangled with the system. [If the original state of the black hole, considered as a thermodynamic system, is pure, and its dynamics are unitary, then the total state of the system plus radiation is pure. But if the radiation is thermal, this means each emitted particle is in a mixed state, and so must be entangled with some other system (in this case, the black hole) in order for the total state to be pure (no two emitted quanta can be entangled in thermal radiation).] As the system cools, the von Neumann entropy of the radiation rises: the entanglement of the emitted particles increases. However, as the black hole radiates, its area shrinks, and at some point it will not be possible for the emitted radiation to be thermal anymore. This is because the number of degrees of freedom of the black hole is given by its thermodynamic entropy – that is, the area of its horizon. When the area is sufficiently small, it will no longer be possible for the outgoing radiation to be entangled with the system, and at this point, the von Neumann entropy of the radiation will decrease along with the thermodynamic entropy of the black hole.

In summary: if, as in accordance with BHT, black holes are ordinary thermodynamic systems that can be described, by an external observer, as a quantum system with $e^{A/4G}$ microstates, that evolves unitarily, then – from the argument earlier – the von Neumann entropy of the Hawking radiation needs to decrease at some point in order that it not exceed the thermodynamic entropy of the black hole. This argument was made by Page (Reference Page1993), who suggested that the entropy of the radiation would need to follow the curve (the Page curve) indicated in Figure 7, rather than the Hawking curve. The entropy of the radiation at early stages increases until it equals the decreasing thermodynamic entropy of the black hole, and then the two remain equal for the rest of the evaporation process. The time at which the von Neumann entropy of the radiation is maximal (the turnover point) is known as the Page time. The Page time for a Schwarzschild black hole is approximately half the total evaporation time: at this point, it is not possible for the black hole’s radiation to be thermal, and instead it should be maximally entangled with the early-time radiation.

Figure 7 The Page curve: Schematic behaviour of the entropy of the outgoing radiation. In Hawking’s calculation remove, the entropy monotonically increases until t_end, when the black hole completely evaporates. If the process is unitary, the entropy of radiation must be less than the thermodynamic entropy of the black hole remove, and follows the Page curve (‘Expected from unitarity’) it saturates this maximum at t_page.

Now, the problem is that this result is in clear conflict with the predictions regarding Hawking radiation in QFT in curved spacetime. In these calculations, the Hawking radiation is exactly thermal, with no entanglement between the early-time and late-time emitted particles. Indeed, there is no possibility of such entanglement on this picture, given that each mode of the Hawking radiation is maximally entangled with a partner mode inside the event horizon (a principle known as the monogamy of entanglement ensures that these modes cannot be entangled with others), which has no means of escaping.

The Page time problem is indeed a paradox: a clash between the predictions of QFT in curved spacetime and the predictions of black hole statistical mechanics. As stated, it occurs when the black hole is still macroscopic in scale, a regime where both these frameworks are expected to be valid, and so closes the ‘loophole’ left open by the evaporation time problem (where we could attempt to appeal to unknown, exotic QG effects in the Planckian regime). Prima facie, there are two ways to resolve the paradox:

1. 1. Accept that QFT in curved spacetime fails as a description of the entire spacetime of an evaporating black hole; retain the statistical-mechanical underpinnings of black hole thermodynamics; try to understand why and when the QFT description breaks down.

2. 2. Retain QFT in curved spacetime, reject black hole statistical mechanics, and find some non-statistical-mechanical understanding of BHT.

Wallace (Reference Wallace, Huggett, Matsubara and Wüthrich2020) presents these two options as each packaged with their own explanatory burdens. Option 1, according to Wallace, necessitates explaining when and why the QFT description breaks down, given that the breakdown occurs in regimes ‘well within the applicability domain of the theory’. Option 2 necessitates explaining not only why we have such a striking theoretical analogy between BHT and ordinary thermodynamic systems, but also why we have a managed to reproduce the thermodynamic entropy of black holes in statistical-mechanical calculations in various different contexts. Failure to address the compatibility of this vast network of calculations, he says, would leave these results ‘miraculous’.

Again, though, we might follow Curiel in questioning the necessity of taking on these additional explanatory demands in order to reject one or the other framework, given the tenuous theoretical status of each. In the case of option 1, we might need only to appeal to the fact that ‘semiclassical gravity is an effective field theory with no empirical support – deal with it’. Or, more seriously, it may be philosophically fruitful to question whether, or why, we should be compelled to believe that results developed in the framework of semiclassical gravity are revelatory of genuine insights into a more fundamental theory – given not only the problems associated with semiclassical gravity, but also due to the very nature of effective field theories in general (this is not to imply, of course, that Wallace does not provide sound arguments for his trust in LEQG in various non-Planckian regimes, see Wallace (Reference Wallace2022)). In the case of option 2, we might again wish to look more closely at the ‘consilience’ of derivations and explore the inter-theory relations between the various frameworks, and/or bemoan the underavailability of physical assumptions bridging the frameworks generating the statistical-mechanical results with the BHT ones.

Rather than fuel debate here, my goal is to encourage more philosophical interest in exploring these issues. Whichever of these routes one chooses to pursue, it is likely that interesting philosophical insights will be gained – whether concerning QG, BHT, semiclassical gravity, QFT, effective field theory, or more general issues in the philosophy of science such as inter-theory relations, scientific methodology, and scientific progress. In regard to the specific questions being considered in this Element, we can ask how the information loss paradox should be used in motivating and constraining the search for QG. It is a theoretical puzzle that arises by taking seriously our best current theories, developing natural ways of combining them, and then pushing these until breaking point. Surely the paradox serves as an extremely stimulating heuristic and guiding principle, and any approach towards QG that indicates a solution could take this as an indication of pursuit-worthiness, or means of confirmation (given that it is an indication of consistency of the approach with low-energy theoretical results). At this stage, however, the resolution of the paradox via new fundamental physics should not be taken as a constraint (criterion of acceptance) on a theory of QG: a dissolution is not ruled out, neither a solution ‘at the level of current theories’.

On that note, I close this section by gesturing to the wide variety of different solutions to the paradox being explored in physics. Particularly noteworthy are recent results involving replica wormholes, which show not only that entanglement entropy follows the Page curve,Footnote ⁸² but also suggest a possible mechanism that allows the ‘information’ to escape via wormholesFootnote ⁸³ (perhaps as a proxy for nonlocal effects.Footnote ⁸⁴) Exactly what this could mean physically, and what it could imply for QG, is still very much open to speculation.

7.3 Entropy Bounds

Various theoretical results in BHT suggest that there is an upper bound on how much entropy can be contained in a given region of spacetime. There are several different proposals for the entropy bound, with varying degrees of generality. These results are striking, and in conflict with the picture of the world presented by current theories – the explanation of the entropy bound is motivation for a new, more fundamental, theory.

The idea of an entropy bound originally came from considering the Geroch process, which is a mechanism via which the GSL can apparently be violated.Footnote ⁸⁵ Imagine we prepare a massless box full of energetic radiation with high entropy, far away from the black hole. The mass of the radiation will be attracted by the black hole’s gravitational force. We could then use this weight to drive an engine to produce energy while slowly lowering the box towards the event horizon of the black hole – this process extracts energy (work), but not entropy, from the radiation in the box. If we keep slowly lowering the box towards the event horizon, we can arrange for all the mass-energy of the radiation to have been exhausted when the box reaches the event horizon. Dropping the box into the black hole, then, will not increase the size of the event horizon (because the mass-energy of the black hole does not increase), but will mean that the thermodynamic entropy outside the black hole has decreased.

In response, Bekenstein proposed that there must be a limit to how much entropy can be contained in a given region of spacetime (Bekenstein, Reference Bekenstein1981). The Bekenstein bound states that maximum entropy contained in a spherical system of radius R and total gravitating energy E is $S\le \frac{2\pi ER}{\hslash }$ . For a weakly gravitating system, the Bekenstein bound reduces to $S\le \frac{A}{4\hslash G}$ , where A is the area of a spherical surface fully containing the system.

If we assume, however, that the Bekenstein bound holds also for strongly gravitating systems, we can arrive at a more general conjecture known as the spherical entropy bound, using the GSL and considering the Susskind process, where, instead of lowering a thermodynamic system into a black hole, we convert the system into a black hole (Susskind, Reference Susskind1995). This bound states that the maximum entropy contained in a region of space is $S\le \frac{A}{4\hslash G}$ , where A is the area of any surface bounding the region. To see why this is, imagine that we find a region with entropy greater than $\frac{A}{4\hslash G}$ , then we could add energy to it adiabatically without changing its entropy, and thereby transform it into a black hole of area A. Such a black hole would then have entropy greater than the Bekenstein-Hawking entropy, (eq. 13). Thus, if the Bekenstein-Hawking entropy is correct, the entropy of a black hole of surface area A must be the maximum possible entropy of any region of space with surface area A. By definition, then, the entropy of a single Schwarszchild black hole precisely saturates the spherical entropy bound: a black hole is the most entropic object we can put inside a given spherical surface (’t Hooft, Reference ’t Hooft1993). This form of the entropy bound is associated with the holographic principle, discussed next.

The heuristic derivation of the spherical entropy bound rests on a large number of fairly strong assumptions (e.g., suitable asymptotic conditions, spherical surface, and gravitational stability of the region). Additionally, the bound suffers several counter examples, so does not have general validity. Bousso (Reference Bousso1999) proposed a successful generalization of this bound, known as the covariant entropy bound. Heuristically, we can imagine a spherical star with nonzero entropy, $S_0$ that burns out and undergoes catastrophic gravitational collapse. For an outside observer, the star will form a black hole whose surface area will be at least $4S_0$ in accordance with the GSL. But we can imagine following the star as it falls through its own horizon, and watch it shrink to zero radius as it crushes to a singularity: $A\to 0$ . But, the entropy of the star must still be at least $S_0$ . This is a case where the spherical entropy bound does not apply, being a regime of dominant gravity. But we can imagine taking a snapshot of the star just before it crunches to a point: light does not have time to cover the entire area of the star before it completely crunches, and so the entropy of the area traversed by the light will always be less than the area of the star’s surface. The covariant entropy bound is formulated in terms of light sheets, where a light sheet is a way of taking a ‘snapshot’ of a matter system in light-cone time (so the entropy on a light sheet is given by the entropy of the matter system). More technically, as illustrated in Figure 8, a light sheet associated with a surface is a set of null geodesics leaving the surface orthogonally such that the expansion of the set in the direction going away from the surface is zero or negative; that is, the geodesics are remaining parallel or coming closer together as they get further from the surface. The light sheet continues up until the geodesics intersect or encounter a singularity of spacetime.

Figure 8 Bousso’s light sheet construction, depicting four null hypersurfaces orthogonal to a spherical surface, B (also, the four null geodesics orthogonal to a point x on B, to make the illustration clearer). The two cones $F_1$ , $F_3$ have negative expansion and hence are light sheets. The covariant entropy bound states that the entropy of each light sheet will not exceed the area of B. The other two families of light rays, $F_2$ , $F_4$ give the ‘skirts’; these have increasing cross-sectional area, and so are not light sheets and their entropy does not relate to the area of B (Bousso, Reference Bousso2002).

The covariant entropy bound simply states that the entropy on a light sheet associated with a surface B will not exceed the area of B, that is, $S\le A/4G$ . This bound appears to hold universally in nature, at least at the semiclassical level – it holds for arbitrary physically realistic matter systems and arbitrary surfaces in any spacetime that is a solution of Einstein’s equations. This bound reduces to the spherical entropy bound in special cases where the latter is known to hold. In addition, Bousso (Reference Bousso2002) argues that the covariant entropy bound implies the GSL, that is, that the GSL can be regarded as a consequence of the covariant entropy bound.

Yet, as Bousso (Reference Bousso2002) emphasizes, the origin of the bound remains mysterious: aside from its success, little motivation for a light-like bound can be offered. Although the discovery of the covariant entropy bound was initially inspired by BHT (via the need for a generalization of the Bekenstein bound, which itself was proposed in order to save the GSL), the covariant entropy bound is no longer connected, via derivation, to these discoveries, and stands on its own. Essentially, the principle has outstripped its original motivations. [Here, it is interesting to recall, as well, that BHT has strange motivations itself. The analogy of the area theorem with the second law of thermodynamics gives perhaps unreasonable weight to the second law itself, which is known not to be fundamental, but open to statistical violations: its appearance is due to perhaps contingent features of our universe.] We cannot look to its mode of discovery in order to justify the covariant entropy bound: instead, its justification is that it holds true in all calculations.

That there could be a maximal bound on entropy inside a region of spacetime is itself a striking result that conflicts with our current picture of the world: according to QFT, the number of degrees of freedom in any given region of spacetime is infinite, so this result calls into question the fundamental status of field theory. But the entropy bound is more incredible given its form: it implies that the number of degrees of freedom grows in proportion to the area surrounding the region, rather than the volume of the region. This clashes with the standard dynamics of all other known types of physical systems, where entropy increases as the volume of any spatial region. Notably, even if QG describes a minimal length (e.g., through a discretization of spacetime), which could serve as a short distance cutoff for QFT, the information content of a given region would still be expected to grow with the volume.

As with the other results coming from BHT, however, it is precisely the inexplicability of the entropy bound that excites researchers: its origin is expected to be found in a new theory. Because the bound relates aspects of spacetime geometry to the number of quantum states of matter (it involves the fundamental constants of both G and h), the suggestion is that any theory that explains the bound will be a theory of QG: a theory that ‘combines’ GR and QFT. The theory will also need to explain why it is that field theory (QFT) gives the wrong answer. Still, however, it is possible that established physics is correct, and that the entropy bound result is an artefact of the semiclassical approximation. While the entropy bound is promoted as holding true in all calculations, we have to keep in mind that these calculations are still remote from empirical tests.

We can ask how the entropy bound can, or should, be used in the search for QG. Bousso (Reference Bousso2002) suggests that the bound may be useful as a guiding principle in the context of string theory. This is because the bound holds in situations where string theory suffers limitations, for instance, in highly dynamical geometries, as well as in the non-perturbative regimes of string theory: the hope is that the bound can help aid the development of string theory in these areas. (The AdS/CFT duality is an example of a non-perturbative formulation of string theory where the entropy bound, as well as the holographic principle is satisfied; the idea is to find others, which could physically represent our universe.) It seems possible that the covariant entropy bound (or the holographic principle) be used as a postulate for a candidate theory of QG, similar to how approaches such as causal set theory take discreteness as a postulate, for instance. The explanation of the bound can, and has, been appealed to as a means of confirmation for QG. Calculations in causal set theory, for instance, give evidence for the existence of an entropy bound which leads to Susskind’s spherical entropy bound in the continuum (Rideout & Zohren, Reference Rideout and Zohren2006). Finally, at this stage, it seems difficult that the explanation of the entropy bound be taken as a criterion of acceptance for QG, given its status in semiclassical gravity. The use of the entropy bound in various roles may also be influenced by the attitude we take towards it: Adlam (Reference Adlam2023) discusses arguments for and against the epistemic and ontological interpretations of the entropy bound, highlighting the different commitments, challenges, and implications of each.

7.4 Holographic Principle

The most common understanding of the holographic principle is based on the idea that the description of a volume – ‘bulk’ – of space can be equivalently defined on a surface bounding the volume of space. In other words, a physical theory defined only on the (N-1)-dimensional boundary of the region it encloses completely describes the N-dimensional physics of the bulk. The principle is then the claim that the full theory of QG can be reformulated as a theory all of whose degrees of freedom are defined on the boundary – a concrete example is the AdS/CFT duality, mentioned earlier (p. 2.1). The idea, originally from ’t Hooft (Reference ’t Hooft1993), was popularized, and given a precise interpretation in string theory, by Susskind (Reference Susskind1995). The picture is that ‘The three-dimensional world of ordinary experience – the universe filled with galaxies, stars, planets, houses, boulders, and people – is a hologram, an image of reality coded on a distant two-dimensional surface’ (Susskind, Reference Susskind2008, p. 410). Susskind’s presentation, following t’Hooft, is based on a response to the information loss paradox, where the suggestion is that the ‘information’ is stored on the black hole horizon.

There are slightly different statements of the holographic principle, and its relation to entropy bounds. Bousso (Reference Bousso2002), for instance, takes the holographic principle as simply the claim that the origin of the entropy bound is to be found in a new theory of QG. Smolin (Reference Smolin2001), states that, while entropy bounds provide a limit on the number of degrees of freedom in a region of space, the holographic principle is supposed to go further in telling us about the nature of those degrees of freedom, by postulating a form of dynamics in which the quantum evolution of the spacetime and matter fields is described in terms of observables measurable on the screens (surfaces). ‘The holographic principle is not a relationship between two independent sets of concepts: bulk theories and measures of geometry vs. boundary theories and measures of information. Instead, it is the assertion that in a fundamental theory the first set of concepts must be completely reduced to the second’ (Smolin, Reference Smolin2001). These claims suggest, however, that the more fundamental principle is the entropy bound itself, rather than the holographic duality, given that the latter is supposed to explain the former.

Jaksland & Linnemann (Reference Jaksland and Linnemann2020) usefully distinguish between the holographic duality and entropy bounds, in that the former is inter-representational, while the latter are intra-representational. The duality is inter-representational because it describes a relationship between different equivalent representations of the same physics; that is, it relates two separate descriptions (different theories): one on the ‘bulk’ and one on the ‘boundary’. In the case of the AdS/CFT duality, for instance, the duality relates a string theory in N-dimensional spacetime ‘bulk’, to a gauge theory (CFT) formulated on the (N-1)-dimensional ‘boundary’. On the other hand, the entropy bounds apply only in a single theory, and are thus intra-representational. For this reason (though as should already be evident from the characterization of the holographic principle, earlier), the existence of an entropy bound does not imply the holographic principle. Nevertheless, the holographic principle does suggest an entropy bound in the bulk theory, and its aim is to motivate or explain the existence of the bound. The main question is, then, how the holographic duality can be used to develop a theory of QG.

One suggestion is made by De Haro (Reference De Haro2019), which utilizes an account of dualities as isomorphic representations of a single ‘common-core’ theory.Footnote ⁸⁶ We can then distinguish between different functions of dualities in theory-construction. The theoretical function of dualities is their use in developing the common-core theory that is represented on both sides of the duality: the idea is to utilize the dual theories, and exploit the relationship between them, to extract the content of this common physics that is being represented differently by each. The heuristic function, rather than discovering and describing the ‘equivalent physics’ (the existing physics of the underlying common-core theory), aims at discovering new physics beyond the common-core theory: constructing a new theory that supersedes the common-core theory. While the theoretical function relies on the exactness of the duality, and it being a non-perturbative description of equivalent physics (in order to identify the physics of the common-core), the heuristic function requires breaking the (exactness of the) duality, and showing that it is recovered only approximately (e.g., in some appropriate limit, or perturbatively) from a more fundamental theory beyond the common-core theory. The heuristic function thus treats the duality as an ‘intermediate step’ in discovering new physics.

Jaksland & Linnemann (Reference Jaksland and Linnemann2020) argue that, given that holography is inter-representational, it cannot, as it stands, tell us anything ‘outside’ of itself (it only tells us about the relationship between features on the two different sides of the duality). In order for the duality to be useful for discovering new physics, we need to render it intra-representational – so that we are working within a single description, which, they argue, need not be the common-core theory. The heuristic use of the duality, for Jaksland & Linnemann (Reference Jaksland and Linnemann2020), is to get at the intra-representational description. On this account, it does not make sense to recover the duality, whether approximately or not. As a simple example of the heuristic impotency of duality, take the Ryu-Takayanagi formula (Ryu & Takayanagi, Reference Ryu and Takayanagi2006). If we consider a spatial slice, $\Sigma$ of an AdS spacetime on whose boundary we define the dual CFT, the formula is a conjecture relating the entanglement entropy $S$ of the CFT (boundary theory) in some spatial subregion, with the area $A$ of a ‘Ryu-Takayanagi surface’ in the bulk theory, as $S=A/\left(4G\right)$ . This formula is clearly inter-representational. As such, Jaksland & Linnemann (Reference Jaksland and Linnemann2020) argue that it cannot testify to the origin of the area in terms of entropy or vice versa: in the representation where the area exists, the entropy is not there, and where the entropy exists, the area is not there.

So, the holographic duality can be used as a guiding principle in the search for QG, but this involves either the inexactness of the duality, following De Haro (Reference De Haro2019), or its transformation into a non-dual description, according to Jaksland & Linnemann (Reference Jaksland and Linnemann2020). The holographic principle can also be used as a postulate, as for instance Smolin (Reference Smolin, Bagla and Engineer2017), who presents it as one of four principles of QG, in the attempt to build QG as a principle theory, in Einstein’s sense. The holographic principle could potentially function as a constitutive principle – indeed, it seems the most likely candidate for a coordinating principle defining a new physical framework (Crowther, Reference Crowther2021). For instance, Bousso (Reference Bousso2002, p. 861) suggests that the holographic principle may not only aid the search for a non-perturbative version of string theory, but ‘could also contribute to a background-independent formulation that would illuminate the conceptual foundation of string theory’. Smolin (Reference Smolin2001) also presents his weak holographic principle as playing something like a coordinating role. The holographic principle can also be appealed to as a means of confirmation: for instance the AdS/CFT duality as embodying the holographic principle is presented as an argument in favour of string theory, especially in its background independent, non-perturbative form. Additionally, if we follow De Haro (Reference De Haro2019), the approximate recovery of the duality from a more-fundamental theory of QG (a theory of new physics, beyond the common-core theory embodied by the duality), could serve as a means of confirmation (though Jaksland & Linnemann (Reference Jaksland and Linnemann2020) argue against this). Finally, considering taking the holographic principle as a criterion of acceptance for QG requires further argument. This may be prompted, perhaps, if the entropy bound is taken as a fundamental result motivating QG – the holographic principle would then be required in order to explain the origin of the entropy bound.