Modeling inertia through the interaction with quantum fluctuations

Jaume Giné and Giuseppe Gaetano Luciano, Modeling inertia through the interaction with quantum fluctuations, Results in Physics 28, article 104543, 2021. Reproduced under the Creative Commons Attribution 4.0 International licence stated in the article; the published version is at doi.org/10.1016/j.rinp.2021.104543. The equations do not survive extraction from the PDF, so each is described in words and identified by the number it carries in the source.

(On this site, the 1994 paper this one builds from is Haisch, Rueda and Puthoff at /library/stm-0f2b09effd, with the authors’ own overview at /library/stm-7be6973b35 and Rueda’s extension to weight at /library/stm-dfc45c66c7. The critique the authors cite is Levin’s, at /library/stm-acd09d2067. Mike McCulloch’s quantised inertia — the theory revisited and corrected in the second half of this paper — is at /library/stm-830d5d4620, /library/stm-76d39277ec and /library/stm-5f5929fc9c. The entropic-gravity route the authors set aside is Verlinde’s, at /library/stm-44f7206c3a and /library/stm-601c8315b2. The companion paper in this series, arguing that the carrier of inertia is paired vacuum photons, is at /library/stm-b284bdebd0.)

Abstract

The origin of inertia of macroscopic bodies has never been thoroughly elucidated. In this paper we provide a new explanation based on the following assumptions: (i) we can think of any body as being composed by resonant parts of Planck size, (ii) inertia arises from the interaction among these elementary constituents and quantum fluctuations. In compliance with such prescription, we propose two frameworks within which inertia can be modeled. The first one relies on the direct application of Heisenberg Uncertainty Principle to the fluctuations nearby a body, the other involves the asymmetric (Casimir-like) damping of the radiation experienced by an accelerated object due to the appearance of a Rindler horizon. Consistency between the two approaches is then discussed.

Introduction

Inertia is the tendency of physical objects to resist any change in their state of motion. More practically, we can say that it is the property that lets objects stay still if they are still, or keeps them moving if they are moving. From Aristotle’s general considerations on natural motion to Galileo’s experiments on falling objects and inclined planes, the understanding of the very nature of inertia has always attracted extensive attention, culminated with the formulation of Newton’s laws of dynamics. Nevertheless, in spite of formalizing the definition of inertia, such laws did not tackle the issue of its physical origin.

Two hundred years after the development of Newton’s theory, it was Mach who seriously approached the matter. Criticizing Newton’s concepts of absolute space and time, he proposed that the inertia of bodies was holistically caused by their interactions with the rest of the Universe. Although Mach’s principle (as later loosely termed by Einstein) was soon set aside due to inconsistencies with the then emerging relativity theory, its impact was so much influential that a number of physicists tried to investigate the origin of inertia in greater detail. Amongst them, we mention the proposal by Sciama and Dicke, who ascribed inertia to a field contact inductive effect of distant matter, and the attempt by Moon and Spencer, later improved by Brown, who introduced the concept of retarded action-at-a-distance of cosmic matter on objects in the laboratory.

From classical shores, in recent years the debate on the origin of inertia has landed on more quantum grounds. For instance, Haisch and colleagues suggested a model for inertia that uses the electromagnetic part of Unruh radiation, which is the radiation perceived by a uniformly accelerated probe in the inertial vacuum. The idea is that, due to the interaction with the zero-point field, oscillating partons within an accelerated object feel a magnetic Lorentz force which opposes the acceleration, behaving just like inertia. In a similar vein, inspired by previous works by Milgrom, McCulloch conjectured a modification of the inertial mass resulting from Unruh radiation imbalance between the cosmic and Rindler horizons (Quantized Inertia). However, both these two approaches have been partly criticized, thus leaving the problem of the origin of inertia still open.

Starting from the outlined picture, in this work we propose a new model for inertia based on the following assumptions. First, we suppose that any body can be conceived as a collection of resonant parts of Planck size; this is in line with the hypothesis that the Planck scale has a fundamental rôle in the quantization of space-time, and we assume that the same happens for the energy and the mass of macroscopic bodies and their interactions in a quantification at a deeper level. Second, inertia is the result of the interaction of these components with vacuum fluctuations, since at the Planck level these fluctuations become relevant and, in the same way as for the Casimir force, they produce macroscopic effects. The first assumption generalizes the corpuscular picture of black holes to any macroscopic body. The second requirement is the core of our analysis, as it provides us with a recipe to trace the origin of a macroscopic property like inertia back to a more fundamental microscopic mechanism.

Within this framework, we present two derivations of inertia which turn out to be two sides of the same coin. The first one is investigated in the section on inertia from the Heisenberg Uncertainty Principle and involves the application of that principle and the particle event horizon description of macroscopic bodies. The section on inertia from the Rindler-scale Casimir effect revisits McCulloch’s theory of quantized inertia, showing that inertia can be modeled by the interaction of accelerated bodies with the asymmetrically damped radiation they perceive due to the appearance of a Rindler horizon. Consistency between the two approaches is then discussed.

Throughout the paper the Planck length is the square root of the reduced Planck constant times the gravitational constant divided by the speed of light cubed, and the Planck mass is the square root of the reduced Planck constant times the speed of light divided by the gravitational constant.

Inertia from the Heisenberg Uncertainty Principle

We assume that the inertial mass of bodies is given by quantum fluctuations spontaneously popping out around it. It is well-known that these fluctuations can be regarded as the temporary appearance of virtual particle-antiparticle pairs, which annihilate shortly after their creation in a time interval delta-t. Denoting by delta-E the energy of each fluctuation, equation 1 in the source follows from the uncertainty principle: the product of the uncertainty in position and the uncertainty in momentum is of order half the reduced Planck constant, so that the energy of the fluctuation is about the reduced Planck constant times the speed of light divided by twice the distance over which it is allowed to propagate. Here the momentum uncertainty has been taken as the energy divided by the speed of light, and the distance as the speed of light times the time interval. Henceforth we denote this distance by r.

Clearly, the higher the energy, the shorter the distance traveled by the fluctuation, and vice-versa. This means that the only virtual particles which manage to reach the body are those originating in its neighborhood or of very low energy. To account for this intrinsic cutoff, we introduce an effective radius, representing the threshold beyond which quantum fluctuations can be safely neglected.

Let us now depict a generic macroscopic body as a collection of elementary resonant parts of Planck size, which acts as minimal length scale in our quantum picture of the gravitational interaction. The specific features of the interaction of quantum fluctuations with these resonant elements are still object of investigation. Here we just give a simple model in a completely inelastic collision that must be confirmed by further works in this direction. The most relevant result is that the two approaches to inertia we consider lead to very similar outcomes.

Equation 2 in the source gives the force exerted by a vacuum fluctuation on each resonant part as minus the derivative of the fluctuation energy with respect to distance, which is about the reduced Planck constant times the speed of light divided by twice the squared distance. Equation 3 in the source integrates that force from the Planck length out to the effective radius, giving the energy transferred to each resonant part as the reduced Planck constant times the speed of light, times the difference between the reciprocal of the Planck length and the reciprocal of the effective radius, divided by two — which is approximately the reduced Planck constant times the speed of light divided by twice the Planck length. In the second step we have exploited the fact that the Planck length is expected to be much smaller than the effective radius, and that the closer the fluctuation, the higher it contributes to the energy.

At this stage, it is worth introducing the so-called particle event horizon description of elementary particles. According to this scheme, since elementary particles represent field singularities, they can be interpreted as black holes. Then, by the scale invariance of general relativity, it is possible to formulate an Einstein-like equation which allows us to derive all their fundamental properties. Consequently, one finds that any elementary particle is confined within a region whose binding surface is four pi times the squared Schwarzschild radius, that radius being twice the gravitational constant times the mass divided by the speed of light squared — the singularity of the exact solution of Einstein’s equation for the gravitational field outside of a non-rotating, spherically symmetric body. This model can be extended to macroscopic bodies as well. Naively speaking, in this case one can associate to any object an event horizon with a radius given by the Schwarzschild radius of the equivalent black hole having the same mass.

With the above scheme in mind, let us consider a macroscopic body of mass M. It is a simple matter to understand that the number of its resonant parts of Planck size is then given by the ratio of the associated Schwarzschild radius to the Planck length: equation 4 in the source states that this number is twice the mass divided by the Planck mass, and equation 5 in the source inverts it, so that the mass is that number times half the Planck mass.

Notice that, in the framework of Corpuscular Gravity theory, black holes are described as Bose-Einstein condensates of N gravitons stuck at the critical point. In light of the above correspondence between generic macroscopic bodies and black holes, we can then rephrase also the concept of resonant parts in the language of that theory: equation 6 in the source identifies the number of resonant parts with the square root of the number of gravitons.

It is now easy to prove that the mass relation coincides with the expression of inertia arising from the vacuum energy which arrives on the body for each elementary component. Indeed, by assuming that the macroscopic body has that number of resonant parts and that it owes its inertia to the interaction with vacuum fluctuations, equation 7 in the source gives the mass equivalent to the energy that can interact with the environment: the number of parts times the transferred energy, divided by the speed of light squared, which is the number of parts times half the Planck mass — that is, exactly M.

Therefore, in agreement with the corpuscular-gravity picture, the inertial mass of a body can be conceived as the result of the interaction among the gravitational fluctuations popping out around the body and its resonant parts of Planck size, or gravitons in a Bose-Einstein condensate. In a broader sense, this is a realization of the old Mach’s principle, already sought after by Einstein, which stated that the inertia of a body could be somehow influenced by the rest of the Universe — more precisely, Mach referred to the background of distant stars that allow to fix the inertial reference systems. This conjecture did not make sense anymore, and was abandoned by Einstein himself, after the development of relativity theory. However, its statement is now realized but with the quantum fluctuations that actually give the fundamental contribution to the energy of the Universe. Notice that a similar attempt to establish a connection between inertia and vacuum fluctuations has been performed in Quantum Field Theory by revisiting the relationship between the mass of charged particles and zero-point electromagnetic fields.

Before proceeding with the next modeling of inertia, let us come back to the concept of resonant parts of Planck size. For a given body, one may wonder how to relate this microscopic property to more familiar macroscopic features. In this regard, the calculation of the entropy comes to our aid. In statistical mechanics, it is well-known that entropy is a measure of the number of possible microstates corresponding to the system’s macrostate. For the particular case of corpuscular-gravity black holes, equation 8 in the source gives the entropy as the logarithm of the number of possible states per graviton raised to the number of gravitons, which is approximately the number of gravitons itself, that is the squared number of resonant parts; Boltzmann’s constant has been set to one. Similarly, for a generic macroscopic body of the same mass, equation 9 in the source writes the entropy as the number of gravitons times the logarithm of the number of states available to each.

The problem is how to quantify this number, since we know only that it is much larger than its black-hole value. To infer an estimate, let us follow this reasoning: suppose we consider two systems, the one made of a single atom of a heavy element, for example lead with atomic number 82, the other made of atoms of a lighter element, for example 82 atoms of hydrogen with atomic number 1. Although the two systems have the same number of excitable, that is resonant, states, namely 82, due to the Pauli exclusion principle the total number of states in which the electrons can be arranged is clearly higher for the system made of hydrogen atoms. In other terms, we expect that the lower the density, the higher the total number of accessible states for the system and vice versa. Let us then assume that the number of states per graviton is a suitable constant divided by the density of the body. Equation 10 in the source substitutes that into the entropy, giving the entropy as the number of gravitons times the logarithm of that constant over the density.

The constant can be fixed by requiring that this relation reduces to the black-hole expression when the density equals the black-hole density, which makes the constant the black-hole density times the black-hole number of states. In this way equation 11 in the source gives the entropy as the squared number of resonant parts, times one plus the logarithm of the ratio of the black-hole density to the body’s density. This provides us with the relation we were looking for, since it links the number of resonant parts of a body conceived as a condensate of gravitons to its entropy and density.

Inertia from the Rindler-scale Casimir effect

Let us consider a body of mass M moving rightwards along the x-axis with uniform proper acceleration. As a result of this motion, it is well-known that a dynamic Rindler horizon appears at a distance of the speed of light squared divided by the acceleration on the left side of the body, since information coming from farther away can never catch up with the body. Hence, the impact of vacuum fluctuations turns out to be weaker from the left than the right side, where no shielding occurs, giving rise to a net force which pushes the object back against its acceleration. As a footnote to this, the authors remark that, due to the accelerated expansion of the Universe, any body has an acceleration at least given by the cosmic acceleration.

The accompanying figure in the source shows the geometry: a body moving rightwards with acceleration a experiences a Rindler horizon far away to its left, at a distance of the speed of light squared over the acceleration. The net contribution to the inertia is given by those fluctuations which originate at a distance greater than that to its right. The radiation pressure imbalance will produce a force against the direction of acceleration. The angle theta is the angle of integration in the x-y plane, and phi the azimuthal angle.

Notice that in the original framework of Quantized Inertia, McCulloch ascribes inertia to the damping of fluctuations of Unruh radiation at the temperature given by equation 12 in the source — the reduced Planck constant times the acceleration, divided by two pi times the speed of light — due to both Rindler and cosmic horizons, the latter appearing at a distance far greater than the former and being relevant only on the right side of the body. In that case, the author obtains a modified expression for the inertial mass, with a correction scaling as twice the speed of light squared divided by the acceleration times the Hubble diameter, that diameter being about ten to the twenty-sixth metres. Nevertheless, since in the present study we are interested in explaining the origin of pure inertia, we shall consider accelerations large enough to neglect this Hubble-scale Casimir effect and the ensuing correction to inertia. This amounts to saying that the Hubble diameter is much larger than any other characteristic length scale in our analysis. The authors note in a footnote that twice the speed of light squared divided by the Hubble diameter is about ten to the minus tenth metres per second squared, which means that for accelerations of the order of Earth’s gravity we are by far in the regime where corrections to the inertia can be neglected.

Let us show how this asymmetric Rindler-scale Casimir effect can model inertia intuitively. In this regard, we recall that, in the case of an isotropic radiation, equation 13 in the source gives the pressure exerted on each resonant part of Planck size of the body as the radiation energy density times the intercepted surface area, divided by three.

However, in our case we need to estimate the net difference between the forces acting on the body from the left and the right, respectively. This can be done by considering a virtual line through the particle forming an arbitrary angle theta with the x-axis. Equation 14 in the source gives the infinitesimal contribution to the net force as the difference between the left and right energy densities times the area, divided by three, with the component along the x-axis carrying an additional factor of the cosine of theta. Clearly, the relative sign arises from the fact that while on the left side the radiation pressure pushes the body from the rear, in the other region it opposes the acceleration.

Now, since the radiation coming from the right does not experience any damping, its energy density is simply the undamped value. On the other side, the presence of the Rindler horizon reduces the radiation that impacts onto the body through the mechanism explained above. Equation 15 in the source quantifies that reduction: the left-hand energy density is the undamped density multiplied by one minus the peak Unruh wavelength times the acceleration times the cosine of theta, divided by eight times the speed of light squared. It has been argued elsewhere that the above formula is incorrect for accelerations around ten to the minus ninth metres per second squared, shown by deriving Planck’s law in a cavity and numerically computing the ratio of the discrete to continuous blackbody radiances. However, for accelerations far below and far above that value — the regime under investigation in this work — that result has approximately the same behavior as the relation used here, which justifies its use in our calculations.

Substituting, equation 16 in the source gives the infinitesimal force along the x-axis as minus the energy density times the peak Unruh wavelength times the area times the squared cosine of theta times the acceleration, divided by twenty-four times the speed of light squared, where the minus sign indicates that the net force always opposes the acceleration.

In order to derive the total force, we have to add up the contributions from all angles. This first requires an integration over the azimuthal angle from minus theta to theta, which trivially gives a factor of twice theta. The result must then be integrated over theta from zero to pi over two and doubled in order to span all the x-y plane. Equation 17 in the source is the outcome: the total force is minus the quantity pi squared minus four, over ninety-six, times the energy density, times the peak Unruh wavelength, times the area, times the acceleration, divided by the speed of light squared. We remark that the above calculation differs from that in the earlier quantised-inertia papers, where the integration over the azimuthal angle is performed over the range from zero to pi, resulting in an incorrect numerical factor.

Let us now focus on the estimation of the energy density. The classical energy density is defined as the energy stored in a given region of space per unit volume. However this definition must be modified in its quantum version, since the volume under consideration cannot be taken arbitrarily small. This is because the energy comes from quantum fluctuations and these cannot occur at any point in a continuous space. If this were the case, the contribution of any volume would then be an infinity of the same order — it is the same as the number of real numbers in any interval of the real line. To resolve the contradiction, it must be required that there exists a minimum length and therefore a minimum volume.

In turn, the space-time discretization poses another problem, since it usually breaks Lorentz invariance. However, Causal Set theory discretizes space-time without breaking Lorentz symmetry. This is due to the fact that Causal Set theory discretizes the causal structure of space-time using the ideas of Hawking, Malament and Sorkin. In this framework, each element of the causal set has a Planck volume. Therefore, it emerges that space-time should exhibit a continuous-to-discrete transition at very fine scales, the effective threshold being represented by the Planck scale. This is also pointed out by most candidate theories of quantum gravity. As a result, we have that the correct evaluation of the energy density in the quantum realm must involve the computation of the energy in a Planck volume, but inside this minimum volume only a single quantum fluctuation can happen.

Now we analyze what is the wavelength of each particle of this fluctuation. As stated above, the fluctuations which are actually responsible for the radiation imbalance are those appearing beyond the hypothetical Rindler horizon in the right region. Clearly, the uncertainty in the position of a photon from this region is at least pi times the speed of light squared divided by the acceleration, assuming a half-sphere form for the Rindler horizon. Then, very straightforward considerations allow one to derive the condition of equation 18 in the source: the temperature of these fluctuations, identified with the fluctuation energy, is at most the Unruh temperature, that temperature corresponding to the fluctuations coming from the Rindler horizon.

Accordingly, denoting the energy of each photon of any virtual pair as half the fluctuation energy, that photon energy is at most the energy of an Unruh photon, which is half the Unruh temperature. But since photon energy is Planck’s constant times the speed of light divided by wavelength, it follows that the photon wavelength is at least the Unruh wavelength, and likewise that the peak wavelength of the fluctuation spectrum in all elementary Planck volumes is at least the peak Unruh wavelength. So we can assume that the former is a factor k times the latter, with k greater than or of order one.

According to the previous discussion, equation 19 in the source writes the energy density as the energy divided by the volume, which is Planck’s constant times the speed of light divided by k, the peak wavelength and the volume — the volume of the elementary block of discrete space being assumed spherical, that is four thirds pi times the Planck length cubed.

In this way equation 20 in the source becomes the force along the x-axis as approximately minus the quantity pi squared minus four, times pi, times the Planck mass, divided by thirty-two times k, times the acceleration — that is, about minus six tenths of the Planck mass over k, times the acceleration. Here we have approximated the generic resonant part of the body to a sphere of radius equal to the Planck length, so that the area intercepted by the radiation imbalance is simply twice pi times the squared Planck length; it should be remembered that the net radiation only impinges on the right side of the body. From the above equation, it follows that the mass of each resonant part due to the radiation imbalance is nothing but six tenths of the Planck mass divided by k, leading to the total inertial mass in equation 21 in the source: the number of resonant parts times six tenths of the Planck mass over k.

It is worth emphasizing that this expression coincides with the earlier mass relation, up to the numerical factor k. We can pick its exact value by requiring consistency between the two, obtaining k of about 1.2, which is indeed greater than one. In turn, this implies that the wavelength of the fluctuation spectrum which mainly contributes to the energy density is about 1.2 times the peak Unruh wavelength, consistently with our previous considerations.

Concerning the comparison between the two models of inertia, we also remark that in the first approach we only take into account all the quantum fluctuations around the body that can interact simultaneously with it, the restriction being given by the number of resonant parts. We can perform this analysis by considering either the body at rest or moving with a certain acceleration. In the first case, the forces in any two opposed directions are equal to each other and the body is kept at rest. On the other hand, if the body is initially accelerated, the situation would be similar to that depicted in the second model: there would be an unbalance in the radiation produced by the quantum fluctuations that oppose the movement. This happens because the radiation coming from the direction opposed to the movement has to travel longer distances and consequently is less energetic. By contrast, in the direction of the movement there are high-energy fluctuations that could not reach the body at rest, but now they can. Hence, a force opposed to the movement would appear, similarly to what was discussed for the asymmetric Casimir effect.

Some comments are in order here. First, we notice that the computation of the energy density carried out in the earlier quantised-inertia papers is not properly justified, since the volume that appears in the energy-density relation is not the volume of the particle, as instead assumed there, but rather the volume of the space with respect to which the energy density is estimated. On the other hand, the criticism raised elsewhere does not take into account the crucial feature of space-time discretization, thus leading to the conclusion that the peak wavelength contribution tends to zero for large accelerations.

Second, we stress that in our model inertia is ascribed to the asymmetric pressure of fluctuations appearing beyond the Rindler horizon, rather than Unruh radiation on its own. The latter is indeed isotropic and extremely faint, as it only consists of those fluctuations originating very close to the horizon. By contrast, the radiation imbalance involves fluctuations coming anisotropically from an infinite volume. Thus, just like the Casimir effect, it is perfectly eligible to be at the root of a macroscopic phenomenon like inertia.

The authors add a footnote to that point. One might wonder why the standard Casimir effect between two plates is typically so weak, while the Rindler-scale Casimir effect gives rise to an easily observable phenomenon like inertia. In our model, this can be explained by noticing that in the first case the resonant parts of Planck size of the plates do not play any rôle in determining the difference between the inward and outward radiation pressure responsible for the Casimir attraction; for instance the mass or the density of the plates does not matter. In other terms, each plate acts as a unique resonant part of a certain surface pushed by the boiling vacuum energy. On the other hand, in the Rindler-scale Casimir effect we have seen that each resonant part composing a given body contributes to its inertia through the interaction with fluctuations from any point of a Planck volume of the quantized space. This entails that the ensuing effect is visible at macroscopic scale, being the sum of the interactions over the large number of resonant parts of the body.

In this context, we would like to emphasize a meaningful comparison between the quantum radiation imbalance and the classical inertial forces. Concerning these forces, it is well-known that, in spite of being experienced only by accelerated observers, their effects can be somehow deduced by external, that is inertial, observers as well. In fact, it is an everyday experience that, when a car accelerates, the driver is pushed back into the seat due to the appearance of an inertial force. Even though an observer at rest outside the car does not feel this action directly, he becomes aware of its consequences by looking at the driver tossed toward the seat. In a sense, what we have found here is that the same happens for the radiation imbalance: it is indeed true that this imbalance can only be experienced by accelerated bodies through the emergence of a force opposite to the acceleration. Nevertheless, its existence can be inferred by any other inertial observer, the universal macroscopic manifestation being exactly what we call inertia.

Finally, we remark that a derivation of inertia involving the Unruh effect has been proposed on the basis of Verlinde’s theory of the entropic origin of gravity and inertia. However, this theory has been shown to possess some possible inconsistencies and its current version is not complete.

Conclusions and outlook

Despite many attempts, the origin of inertia has not been adequately explained yet. In this paper, we have introduced a new model that traces the origin of this macroscopic property back to a more fundamental microscopic mechanism, that is, the interaction among the resonant parts of Planck size of any body and the vacuum fluctuations around it. In this sense, our model is conceptually similar to the Higgs mechanism, where mass loses its status as a primary quality, becoming the result of elementary massless particles interacting with the Higgs field. Here, however, we stress that our considerations are applicable to elementary, as well as composite objects.

By employing the above prescription, we have presented two frameworks in which inertia can be easily modeled. The first one is based on the use of the Heisenberg Uncertainty Principle and the computation of the energy transferred by fluctuations to each resonant part of the body. The second approach is inspired by McCulloch’s theory of quantized inertia and ascribes inertia to the radiation imbalance due to the appearance of the Rindler horizon, the Rindler-scale Casimir effect. In spite of the underlying differences, the two frameworks are consistent as regards the resulting expressions for inertia.

We emphasize that we have focused on the simplest analysis of a non-inertial motion with linear acceleration. Clearly, the case of a circular orbit is expected to be much more complicated, if only for the concerns on the existence of a rotational analogue of the Rindler horizon. It would also be interesting to address how our model of inertia is intertwined with the equivalence principle.

Some aspects remain to be addressed. For instance, in the context of a quantum description of gravity, several models predict a modification of the standard uncertainty principle to a Generalized Uncertainty Principle which accounts for the emergence of the minimal length at the Planck scale. The question thus arises as to how this deformation is related to our result. In turn, a possible GUP-modified expression of inertia could allow us to put some bound on the GUP parameter through current limits on equivalence principle violations. On the other hand, possible non-thermal behaviors of the radiation perceived by accelerated observers have been pinpointed in various scenarios. In light of the established connection between such phenomenon and inertia, it would be worth studying whether the validity of our considerations is somehow affected by this deviation from thermality, and, if so, how it might be constrained by our analysis. More work is inevitably required along these and other directions.

Acknowledgments

The first author is partially supported by a MINECO/FEDER, Spain grant number PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya), Spain grant number 2017SGR-1276.