Abstract
In a work by Visser, Bassett and Liberati (VBL) (Nucl Phys B Proc Suppl 88:267, 2000) a relation was suggested between a null energy condition and the censorship of superluminal behaviour. Their result was soon challenged by Gao and Wald (Class Quantum Grav 17:4999, 2000) who argued that this relation is gauge dependent and therefore not appropriate to find such connections. In this paper, we clear up this controversy by showing that both papers are correct but need to be interpreted in distinct paradigms. In this context, we introduce a new paradigm to interpret gravitational phenomena, which we call the Harmonic Background Paradigm. This harmonic background paradigm starts from the idea that there exists a more fundamental background causality provided by a flat spacetime geometry. One of the consequences of this paradigm is that the VBL relation can provide an explanation of why gravity is attractive in all standard weak-field situations.
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1 Introduction
Since at least the works of Rosen [1, 2], it is known that General Relativity (GR) can be interpreted from the point of view of two distinct paradigms. One is the so-called Geometric Paradigm (GP), which is the most standard manner in which General Relativity is presented and understood. The other can be called Field-Theoretic Paradigm (FTP) and corresponds to understanding GR as a non-linear, relativistic theory of a tensorial field \(h_{\mu \nu }\) over a flat background. Just as in electromagnetism, the field is considered to be massless and subject to a gauge symmetry, which reduces the physical degrees of freedom, leaving less of them than what one might naively expected. In this case, the symmetry leaves only the 2 helicities of a massless spin-2 particle as physical degrees of freedom. The FTP puts gravity in parallel with the description of all the other fundamental interactions found in Nature (for a description of these two paradigms, see e.g. [3]). We call them “paradigms”, instead of just formulations, because we think they affect one’s view on gravitational phenomena and how to transcend the GR regime. Nonetheless, the literature shows no clear consensus on whether these paradigms as applied to local GR phenomena are similar or on the contrary crucially distinct. The reader can find interesting discussions e.g. in [4,5,6] and a historical overview of different positions in [7]. Here we want to mention an interesting and sharp but not very well-known result about the non-equivalence of these two paradigms, presented by Penrose [8]. In this paper he showed that the conformal (or causal) asymptotic structures of Minkowski spacetime and simple physical geometries like stellar geometries in GR do not match together.
The first central theme of this work is to describe and discuss what we argue to be a potential third paradigm in which to frame the gravitational interaction. We shall call this third paradigm the Harmonic Background Paradigm (HBP) for reasons we will explain later. We qualify it as “potential” because in this work we only provide a linear version of it, as we do not yet know its full non-linear completion. This HBP seems close to the FTP, but significantly differs from it: essentially in that it considers from the start that the metrical configuration space is not that of generic \(h_{\mu \nu }\) fields, but a constrained subspace. In standard GR language, this looks like a gauge fixing condition, but in our proposal it is very different in spirit.
Similar ideas have been contemplated from time to time, starting from Rosen himself [1], through the Relativistic Theory of Gravitation by Logunov and collaborators [9] up to the Special Relativistic Approach by Pitts and Schieve [7]. What these approaches have in common is to ascribe a crucial role to a background geometric structure. However, as these names might evoke entire sets of results, many of them different from what we are concretely proposing here, we have opted to use a different name.
For the purpose of this introduction, let it suffice to say that in weak-field gravitational phenomena—i.e. local phenomena like relativistic stars or solar systems tests—, these three paradigms are physically identical and can be considered as just different interpretations of a single theory: General Relativity. Beyond that, they might be distinguished by some future observations. For example, strictly speaking, a flat background is not compatible with a \(S^3\) topology for the universe; a comprehensive discussion of this issue can be found in [10]. However, in our view, the relevance of having different interpretations or paradigms becomes more pertinent by suggesting rather different extensions beyond the current standard theoryFootnote 1
The other central theme of this work is a subtle connection between superluminality and energy condition violations put forward by Visser–Bassett–Liberati (VBL) in [11]. This theme might at first seem unrelated to the first central theme stated previously. But, as the reader will see in due time, they are on the contrary tightly related. Let us briefly describe this second theme.
In GR, by construction, there is nothing that can travel faster than the local speed of light. This is because GR precisely prescribes at each spacetime point a causal-cone structure; the metric is nothing more than this causal-cone or conformal structure multiplied by a conformal or volume factor. This observation does however not completely eliminate the possibility of considering faster-than-light configurations. Such behaviours typically involve some non-local definition of superluminal behaviour. For example, in cosmology we can have geometries with cosmological horizons, in which space is expanding so rapidly (exponentially) that light rays cannot connect points which are initially sufficiently far apart. Another paradigmatic example are warp drive configurations [12].
All examples we know of superluminal behaviour within standard GR are based on configurations in which the causal cones stretch more open in some regions of spacetime with respect to some other referential causal cones in other regions.Footnote 2 In general, it turns out that generating such configurations requires introducing some exotic matter content: The matter stress-energy tensor has to violate some of the energy conditions (EC) of GR [15]. Armed with this intuition, and starting with [16], several proposals have been made to prove general theorems which make an explicit connection between EC violations and superluminal behaviour. Among these results there is the one due to VBL that links EC satisfaction with the impossibility of travelling faster than a referential speed of light [11]. However, this result was soon challenged by Gao and Wald [17], who criticised the VBL approach. Gao and Wald agree that there should exist connections between superluminality and EC violations, but they convey that such connections should be built using a different strategy, and in particular an explicitly gauge-invariant strategy. For instance, they construct two theorems establishing links between superluminality and EC violations.
Both the VBL paper and the Gao-Wald response have (separately) received a fair amount of attention. In the VBL case, these range from warp drives [18] and wormholes [19] over extensions of the energy conditions [20] to more general considerations of null cones [21] and causality [22]. The Gao-Wald is also cited in the context of causality [23], its relation to energy conditions [24] and their impact on IR/UV physics [25], but has furthermore found its way to discussions on holography [26] and the (A)dS-CFT correspondence [27, 28]. Curiously, the actual controversy between the VBL and Gao-Wald arguments has, to our knowledge, never been analyzed in detail.
The second central point of the present paper is thus to clarify this controversy. As we will argue, both arguments are in fact correct but they require the assumption of different conceptual setups or paradigms. It is precisely in making the VBL approach rigorously correct that we will find one motivation for introducing the HBP. We will argue that this not only clears up the VBL/Gao-Wald debate, but leads to many other interesting consequences. For instance, in this paper we contend that “inverting” the VBL interpretation of their own result offers a potential explanation of why gravity is attractive, at least under gravitational-weak-field conditions. By “attractive”, in this paper we simply mean that masses/energies are only of a positive nature, as encoded in a specific EC.Footnote 3
The structure of the paper is the following. In Sects. 2 and 3 we review the VBL result and the Gao-Wald criticism, respectively. Then, we present the Harmonic Background Paradigm as a setup for discussing gravitational phenomena, in Sect. 4. The discussion in this section also serves, among other things, to shed light onto the previous controversy. After that, we turn in Sect. 5 to expose one of the central results of this work: we explain why gravity is attractive in standard weak-field situations. Section 6 is devoted to a discussion of a potential obstruction to our proposal identified by Penrose in [8]. Finally, we summarize our proposal and make further remarks in Sect. 7.
2 The Visser–Bassett–Liberati argument
Let us suppose at this stage that VBL assume a standard geometric paradigm in their discussion. We assume this because in their paper there is no explicit discussion of a need for any paradigm shift. Presented in steps, the VBL result is essentially the following.
-
1.
Whenever we have a metric \(g_{\mu \nu }\) which is close to being flat we can write \(g_{\mu \nu } \simeq \eta _{\mu \nu } + h_{\mu \nu }\), where \(\eta _{\mu \nu } \) is the Minkowski metric.
-
2.
From the Fierz-Pauli Lagrangian for linearized gravity [29] one obtains the following equation for \(h_{\mu \nu }\):
$$\begin{aligned} \square h_{\mu \nu } - 2\eta _{\beta (\mu } \nabla _{\nu )} \nabla _{\alpha } h^{\alpha \beta } +\left( \nabla _{\mu } \nabla _{\nu } h + \eta _{\mu \nu } \nabla _{\alpha } \nabla _{\beta } h^{\alpha \beta } \right) -\eta _{\mu \nu } \square h = -16\pi T_{\mu \nu }, \nonumber \\ \end{aligned}$$(1)where \(T_{\mu \nu }\) represents the Stress-Energy Tensor (SET). Now, in the de Donder or harmonic gauge [30]Footnote 4
$$\begin{aligned} \nabla _\mu \left( h^{\mu \nu }-\frac{1}{2}\eta ^{\mu \nu }h \right) =0, \end{aligned}$$(2)the Einstein equations read
$$\begin{aligned} \square h_{\mu \nu } = -16\pi \bar{T}_{\mu \nu }, \end{aligned}$$(3)where \(\bar{T}_{\mu \nu }\) is related to the SET as
$$\begin{aligned} \bar{T}_{\mu \nu } = T_{\mu \nu } -{1 \over 2}\eta _{\mu \nu } T. \end{aligned}$$(4)In all the previous expressions, \(\nabla \) represents the covariant derivative with respect to the background metric \(\eta \), and we have used \(\nabla \) instead of \(\partial \) (as in the VBL paper) to stress the covariance of the construction under changes of coordinates.
-
3.
In the VBL paper, the next condition imposed is that the system does not contain any incoming gravitational waves, or equivalently, that there are no free gravitational waves superposed to the inhomogenous solutions of the D’Alembertian equation for \(h_{\mu \nu }\). Once this condition is imposed, they assert that one can write down the further evolution of the perturbation \(h_{\mu \nu }\) as
$$\begin{aligned} h_{\mu \nu }({\varvec{x}},t)= 4\int d {\varvec{x}}' \frac{\bar{T}_{\mu \nu }({\varvec{x}}',t')}{|{{\varvec{x}}-{\varvec{x}}'}|}. \end{aligned}$$(5)The \(\bar{T}_{\mu \nu }\) in the above equation depends on \({\varvec{x}}'\) and the retarded time \(t'=t-|{{\varvec{x}}-{\varvec{x}}'|}\) (we use \(c=1\) and \(G=1\) units).
-
4.
Finally, they assume that matter satisfies a Background Null Energy Condition or BNEC, namely that \(T_{\mu \nu } l^\mu l^\nu \ge 0\) for any null vector \(l^\mu \) with respect to \(\eta \), i.e. any \(l^\mu \) such that \(\eta _{\mu \nu }l^\mu l^\nu =0\). Then, under the previous conditions they show that the causal cones of the g metric will always lie inside the cones of the \(\eta \) metric. The argument is simple: contracting (5) with \(l^\mu l^\nu \) one sees that
$$\begin{aligned} h_{\mu \nu } l^\mu l^\nu \ge 0 \end{aligned}$$(6)But if \(l^\mu \) is null with respect to \(\eta \), then (6) is also the norm of \(l^\mu \) with respect to the full metric g. Hence, if \(l^\mu \) was null with respect to \(\eta \), it can never be timelike, and in fact, unless \(T_{\mu \nu }\equiv 0\), it will necessarily be spacelike, with respect to g. In other words, the causal cones of the g metric always lie inside the cones of the \(\eta \) metric. In the following we will write symbolically \([g]<[\eta ]\). Note that the BNEC is not the standard NEC condition, since the latter would involve null vectors with respect to g rather than \(\eta \).
Thus, the essence of the VBL argument is the observation that, if the BNEC is satisfied, the g-lightcones will necessarily become squeezed inside the \(\eta \)-lightcones, and can never lie outside them (the proof they provide is only at the linear regime). That is why they called their result a “superluminal censorship”. Note that the use of the word superluminal here is different from that in e.g. [16]. In the context of the VBL analysis it refers to the possibility of surpassing a fixed background causality.
2.1 Some further subtleties
Before continuing with the Gao-Wald counterargument, let us point out some subtleties of the VBL argument that were not explicitly discussed in the VBL paper. Before discussing the linear gravitational case, it is interesting to recall a very standard way of calculating the electromagnetic field associated, for instance, with an arbitrarily moving point-like charge. In the standard treatment of electromagnetism as a gauge theory, we know that the physical electric and magnetic fields do not depend on the gauge used to calculate them in terms of four-potentials. Nevertheless, it is typically useful to adopt some particular gauge to perform the calculations. One standard path is to first assume the Lorenz gauge \(\nabla _\mu A^\mu =0\) .Footnote 5 The four-potential then has to satisfy the D’Alembertian equation with the charge current as a source: \(\square A_\mu =4\pi j_\mu \). The solution is easily found in any textbook, e.g. [33], namely
where \(G^\textrm{R}(x,x')\) is the retarded Green function
where \(x=\{t, {\varvec{x}}\}\), \(x'=\{t', {\varvec{x}}'\}\). This specific Green function is selected by imposing specific boundary conditions or a concrete integration contour if one is working in the Fourier space [34]. Here, to give a completely explicit expression, we have used Cartesian coordinates in the Minkowski spacetime in which the theory is defined. Then, for the example of a single point-like charge \(j^\mu =q \gamma ^{-1} u^\mu \delta ({\varvec{x}}-{\varvec{x}}_\textrm{T}(t))\), one obtains the Lienard-Wiechert retarded potentials
In this expression, the quantities on the right-hand side are evaluated at the retarded position of the point charge which is following the trajectory \(x_\textrm{T}(t)\), \(u^\mu =\gamma \{1,\varvec{\beta } \}\) is its normalized four velocity, \(\gamma \) the Lorentz factor, and \(\kappa =1-\widehat{{\varvec{R}}} \cdot \varvec{\beta }\) with \(\widehat{{\varvec{R}}}\) a unit vector with direction between the point \(\textbf{x}\) and the retarded position of the charge. Then, the total potential is
Having selected a unique Green function, the non-uniqueness of the solution to the D’Alembertian equation \(\square A_\mu =0\) is encoded in the presence of the term \(A^\textrm{in}_\mu \) which represents any free solution of the equation.
Now, to calculate the field produced just by the moving charge it is customary to take \(A^\textrm{in}_\mu =0\). Here we want to emphasize that setting \(A^\textrm{in}_\mu =0\) selects a unique four-potential to represent the physical situation, that is, completely fixes the initial gauge freedom for such situations.
In a more detailed manner, we could have written \(A^\textrm{in}_\mu \) as consisting of two parts:
The first part embodies the presence of physically real waves incoming from infinity (i.e. waves with non-trivial electric and magnetic fields). The second part, the “residual gauge”, is characterized by changes of \(\delta A_\mu = \partial _\mu \phi \) such that \(\square \phi =0\), and hence represents pure gauge waves. Imposing boundary conditions so that both terms are zero not only eliminates the physical incoming waves, but also fixes completely the gauge used in the calculation, including the residual freedom left by the choice of the Lorenz gauge. This second issue is typically not made explicit, probably because it is not important for the calculation at hand. Indeed, any consistent choice of \(A^\mathrm{in-gauge}_\mu \) results in a zero contribution to the relevant electric and magnetic fields anyway. Finally, let us emphasize that completely fixing the gauge cannot be attained using only local conditions on the 4-potential. We have in addition needed to impose some asymptotic boundary conditions.
Let us now look at the linear gravity case. As we will see, it is parallel but a bit more subtle than the situation in electromagnetism that we have just discussed. If one sticks to the geometric paradigm, and thus adheres to the idea that there is just one real, physical metric tensor \(g_{\mu \nu }\), then an immediate question that arises is how to make the separation \(g_{\mu \nu }=\eta _{\mu \nu }+ h_{\mu \nu }\). This separation is not unique; in fact a priori there are infinitely many possible such separations. A first condition in the VBL analysis that restricts this set of partitions consists in imposing the harmonic condition (2) on \(h_{\mu \nu }\). This condition allows one to write down the general solution for \(h_{\mu \nu }\) as
where \(h^\textrm{in}_{\mu \nu }\) are free solutions of the d’Alembertian equation \(\square h_{\mu \nu }=0\).
The second term on the right-hand side results from the integration in the \(t'\) variable with the retarded Green function (8). For clarity we have used the same expression for the retarded Green function for linear gravity as for electromagnetism: we are again using Cartesian coordinates in the Minkowski spacetime. The first term on the right-hand side is
Now, for the VBL result to hold, the entire \(h^\textrm{in}_{\mu \nu }\) must be zero. Among the free solutions of the D’Alembertian equation (3) there are some \(h^\mathrm{in-phys}_{\mu \nu }\) that represent real gravitational waves (for instance, the \(\eta _{\mu \nu } + h^\mathrm{in-phys}_{\mu \nu }\) metric leads through the Levi-Civita connection to non-zero curvature) while other solutions represent just pure gauge waves associated to the residual gauge symmetry, which in a GR language would correspond to wavy changes of coordinates. Indeed, the term \(h^\mathrm{in-gauge}_{\mu \nu }\) represents the residual gauge freedom, namely all transformations that preserve the de Donder gauge condition (2) on \(h_{\mu \nu }\). It is easy to check that these are transformations of the form:
where the generator \(\xi ^\mu \) fulfills the condition
Therefore, when the VBL paper states that they have “tacitly assumed that there is no incoming gravitational radiation”, this is in fact more subtle than just eliminating physically real incoming waves; one has to eliminate also any “gauge wave”. Otherwise, non-trivial \(h^\mathrm{in-gauge}_{\mu \nu }\) terms could cause the causal cones of \(g_{\mu \nu }\) to lie outside the selected Minkowskian reference cones (the cones of \(\eta _{\mu \nu }\)), even if all matter satisfies the BNEC condition.
In standard GR language we know that any \(h^\mathrm{in-gauge}_{\mu \nu }\) can be absorbed into the arbitrariness of the partition \(\eta _{\mu \nu } + h_{\mu \nu }\), since \(\eta _{\mu \nu } + h^\mathrm{in-gauge}_{\mu \nu }\) is in general equivalent to another Minkowskian \(\tilde{\eta }_{\mu \nu }\) metric written in suitable coordinates. Therefore, to fix the term \(h^\mathrm{in-gauge}_{\mu \nu }\) to zero is tantamount to restricting completely and uniquely what one takes to be \(\eta _{\mu \nu }\).
Let us advance at this point that, although not mentioned anywhere in the VBL paper itself, the VBL result actually finds a better accommodation within a non-geometric paradigm. We will discuss this issue later on.
3 The Gao-Wald counterargument
In [17], Gao and Wald argue that, given a metric \(g_{\mu \nu }\) and a decomposition \(g_{\mu \nu }= \eta _{\mu \nu }+h_{\mu \nu }\), whether the light cones of g lie inside or outside those of \(\eta \) is a highly gauge dependent issue and they do not follow further this pathway. As a first observation, it is clear that the Gao-Wald analysis assumes the GP explicitly (or equivalently the FTP): They maintain themselves firmly based in general relativity as a gauge theory of a single universal metric.
The example provided by Gao-Wald is based on applying a linear gauge transformation \(\delta h_{\mu \nu }=2\nabla _{(\mu } \xi _{\nu )}\). They present a transformation of the form
defined in polar coordinates in a Minkowski spacetime. Here \(r^\mu \) is the radial four-vector and f(r) is a function such that that \(f(r)= r^2\) for \(0 \le r \le 1/2\), \(f(r)= r\) for \(r \ge 1\), and in between any smooth interpolation with \(f(r) \ge 0 \) and \(0 \le f'(r)< 2\). For instance, they show that this gauge transformation opens up everywhere the lightcones associated with g with respect to those associated with \(\eta = g-h \). They do not discuss whether this transformation could maintain a harmonic condition or not. In fact, it is clear that \(\nabla ^\mu \xi _\mu \ne 0\); they just recall a result [35] showing that the harmonic condition is ill-defined at null infinity; a point to which we will come back later. But even if one finds a gauge transformation that preserves the harmonic gauge, the problem of the pure residual gauge additions described above will still persist.
3.1 Some further discussion
Let us discuss the Gao-Wald position in some more detail. When standard GR is written as if it depended on two metrics \(\eta \) and g, the theory acquires additional gauge symmetries besides the standard diffeomorphism invariance. This is easily seen by looking at the following simple example with just two particles (no fields). Consider a Lagrangian of the form
This Lagrangian is invariant under the transformation \(\delta x_1 = \alpha (t)\) and \(\delta x_2 = - \alpha (t)\) with \(\alpha (t)\) an arbitrary function of t. This system therefore has one gauge symmetry.
For instance, even though the system might appear as having 2 degrees of freedom, \(x_1\) and \(x_2\), it actually has only one degree of freedom. This can be verified trivially by defining \(q=x_1+x_2\) and \(r=x_1-x_2\), which allows one to rewrite the Lagrangian as
The elimination of a redundant coordinate, r, has automatically made the gauge symmetry \(\alpha (t)\) disappear.
Notice the difference with a Lagrangian of the form
This is invariant only under \(\delta x_1 = \alpha \) and \(\delta x_2 = - \alpha \) with \(\alpha \) constant, not an arbitrary function \(\alpha (t)\). Thus, this symmetry is physical, not gauge.
Within standard GR, the decomposition of the metric g into a background \(\eta \) and a perturbation h follows the same pattern just explained. Apart from the usual gauge symmetry (diffeomorphism invariance in the standard jargon) of the theory written with just one g, there is an additional gauge symmetry associated with the (fictitious) partition into \(\eta \) and h, which thus could be called a “background gauge symmetry” to distinguish it from the former.
Gao-Wald argue that, given a metric \(g_{\mu \nu }\) and a decomposition \(g_{\mu \nu }= \eta _{\mu \nu }+h_{\mu \nu }\), whether the causal cones of g lie inside or outside those of \(\eta \) is just a gauge issue and thus not physical. We can easily see their point by taking the Schwarzschild metric external to a relativistically non-compact star of radius \(R\gg 2m\) (this is to ensure that we are within a perturbative regime with respect to a flat metric). For the simplicity of the argument, let us concentrate just on the Schwarzschild external metric. Written in standard Kerr-Schild coordinates \(\{t,r,\theta ,\varphi \}\), the Schwarzschild metric \(g_S\) reads
thus with \(h_{\mu \nu } = l_\mu l_ \nu \) and with the \(\eta \) metric written in polar coordinates
This setup satisfies the VBL idea that the lightcones of \(g_S\) lie inside those of \(\eta \): \([g_s] \le [\eta ]\).
Let us add and subtract from the previous expression
with
where C is a constant such that \(C^2 > 2m/R\). One can easily check that the metric
is again a flat metric,Footnote 6 but we have managed to close inwards (in the radial direction) its lightcones with respect to the former flat metric, \([\tilde{\eta }] \le [\eta ]\), as can easily be seen by checking that \(\tilde{\eta }_{\mu \nu } l^\mu l^\nu \ge 0\) for any radial null vector with respect to \(\eta \).
Of course, \(g_S\) has not changed under the operation (22). The assignment (24) is compensated by the new \(h_{\mu \nu }\) piece, that we will denote by \(\tilde{h}_{\mu \nu }\), which reads
This piece contributes to the opening of the cones of \(g_S\) with respect to \(\tilde{\eta }_{\mu \nu }\). The constant C has been set so that now \( [g_S] \ge [\tilde{\eta }]\), and thus, with respect to the new flat referential metric \(\tilde{\eta }\), the metric \(g_S\) has its causal cones outside. Note that a similar trick could be applied to any internal stellar metric.
Which part of the VBL theorem has precisely been violated in this manipulation? Strictly speaking the former \(h_{\mu \nu }\) is already non-harmonic, but the really crucial element is that in the change from \(h_{\mu \nu }\) to \(\tilde{h}_{\mu \nu }\), we have introduced a non-trivial \(h^\mathrm{in-gauge}_{\mu \nu }\).
Let us explain a subtle point in performing this gauge transformation. Notice that at the non-linear level the gauge symmetry of GR becomes identified with diffeomorphisms or coordinate transformations. If we apply a non-infinitesimal coordinate transformation to \(\eta _{\mu \nu } + h_{\mu \nu } \) we have
with
The matrix transformation applies to both \(\eta _{\mu \nu }\) and \(h_{\mu \nu }\). At the infinitesimal level, the transformation of coordinates can be written as \(x'^\mu =x^\mu + \xi ^\mu (x)\), and we have
It is well known that
Gao-Wald decided to use the presence of the additional background gauge symmetry to apply the entire transformation just to \(h_{\mu \nu }\). To be even more clear, we can perform first a standard (gauge) coordinate transformation
and then a background gauge transformation
In this way we have the following total transformations
which at the linear level become
This is precisely the transformation scheme used by Gao-Wald. If the initial \(h_{\mu \nu }\) satisfies the condition
then the new \(h_{\mu \nu }\) will satisfy
which in general will not be zero.
To summarize, the central message of Gao-Wald is that to be or not to be inside the causal cones of a flat reference metric is a highly gauge-dependent notion and thus not appropriate (within standard GR) to construct physical connections, for instance, between superluminal behaviour (which are rather to be extracted just from the characteristics of the g metric) and EC violations.
4 The harmonic background paradigm
The Gao-Wald argument is absolutely correct within the geometric and field-theoretic paradigms. If one maintains any of these views, any attempt to establish a hierarchy between the causal cones of a general relativistic metric g and a Minkowski metric \(\eta \) extracted from g is doomed to fail. But is there a way to make the VBL vision consistent? We think there is. The key point, in our opinion, lies in introducing a background geometric structure and taking its concept and implications further than in the standard FTP. Before introducing our specific paradigm proposal, let us briefly discuss some additional motivations for such a proposal.
4.1 Notions absent in standard GR
In a FTP one assumes that there exists a flat metric on top of which a tensor field \(h_{\mu \nu }\) is evolving. The dynamical equations of the tensor field contain non-linearities. More importantly, they also exhibit a gauge symmetry. This gauge symmetry is precisely
Note that, from the point of view of other gauge symmetries in field theories, such as Yang-Mills fields, it can be argued that this gauge symmetry is a bit odd as it depends on the very h field itself. But the key point is that in this paradigm there is still no way to attribute a physical reality to the relation between \(\eta \)-cones and \((\eta +h)\)-cones, precisely because it is a gauge-dependent notion.
There is another important notion that is not gauge-invariant, and therefore lacks physical meaning: to associate to a curve in the \((\eta +h)\) space a single curve as seen from the point of view of \(\eta \). Indeed, take for instance a geodesic of \((\eta +h)\). Changing \(h_{\mu \nu }\) by a gauge transformation changes the form of the geodesic equation
Therefore, given a specific geodesic trajectory of \((\eta +h)\), it cannot be associated with a unique curve from the point of view of hypothetical observers who only perceive the \(\eta \) metric (more on these observers below). The interpretation within the GP is that the trajectory of a celestial body \(x(\lambda )\) makes sense only in its relation with other trajectories, but not “in itself”. In other words, there is no unique way to construct an (external) Minkowskian reference system. In the context of perturbation theory in GR it is well known that the selection of a gauge is tantamount to the identification of the points of two metric manifolds, the physical metric and the background metric [36].
A related question is that of the stress-energy tensor (SET) for the gravitational field h. Already Rosen [1] realized that adding an extra flat structure allows one to construct a proper SET for h as opposed to a pseudo-tensor. However, he did not mention in this paper that this SET is not gauge-invariant, not even at the linear level. This was pointed out as early as [37] and discussed in further detail e.g. in [38, 39]. In GR this is easily interpreted as the absence of a local notion of gravitational energy. Thus, this is another instance of notions that one would like to have when building a field-theoretic view, but these notions fail to acquire physical meaning because they are not gauge invariant.Footnote 7 This can be taken as part of the more general problem of absence of local observables in GR: all the physical (i.e. gauge-invariant) quantities are highly non-local [42] or relational.Footnote 8 This relational interpretation is in stark contrast with the way in which many problems in GR are analyzed in practice, such as the geodesics of bodies around a black hole: to understand these problems one typically fixes a particular set of coordinates.
For the reasons described, the GP and FTP appear to us closer to each other than one could expect at first glance. From a conceptual point of view there is not much advantage in using FTP. In fact, in our view the GP provides a simpler interpretative framework to understand such conundrums as the cone hierarchy, relational trajectories and non-local energy.
However, one could solve all these quandaries by taking the FTP one step further. One direct manner to give complete physical reality to the \(\eta \) background metric is to introduce a physical or privileged gauge for \(h_{\mu \nu }\). Or, from a different perspective, to assume that we do not have a gauge theory of gravitation but a theory of a tensor field \(h_{\mu \nu }\) subject to certain restrictions (the necessity to impose such restrictions has been identified previously e.g. by Pitts and Schieve in [21]; however, the algebraic restrictions they impose differ from ours). Before describing our specific proposal, which the reader can foresee based on our discussion so far, let us make two additional digressions, one on the notion of diffeomorphism invariance in the context of bi-metrical theories, and the other on Fock’s harmonic coordinates.
4.2 Diffeomorphisms and bi-metricity
Let us recall a few basic notions. In the mathematical literature, a diffeomorphism between two differentiable manifolds is a differentiable one-to-one, onto map between the manifolds whose inverse is also differentiable. When such a map exists, we say that both manifolds are diffeomorphic. A diffeomorphism induces a change of coordinates, but the contrary need not be the case. The diffeomorphism notion is thus in principle not connected with any field one can define on the manifolds. Let us use the term t-diffeomorphism to denote this strict diffeomprphism notion. Now, a pseudo-Riemannian manifold is a manifold endowed with a metric tensor of Lorentzian signature. With a certain abuse of language we can say that two pseudo-Riemannian manifolds, \(\{ \mathcal{M}, g \}\) and \(\{ \mathcal{M}',g' \}\), are diffeomorphic
if \(\mathcal{M}\) and \(\mathcal{M}'\) are t-diffeomorphic and there exist coordinates x in \(\mathcal{M}\) and \(x'\) in \(\mathcal{M}'\), which might be defined only patch by patch, and which are related by the diffeomorphism (one is the pull-back of the other), such that g(x) has the same functional form as \(g'(x')\). Note that we say “diffeomorphic”: \(\{ \mathcal{M}, g \} \sim \{ \mathcal{M}', g' \}\), not “equal”: \(g=g'\).
In mathematical terms, two different metrics in a single manifold represent in principle two different structures, even if they were diffeomorphic when considered in two manifolds. In the mathematics of manifolds, one is giving an ontological reality to the points that constitute the manifold. Then, a different value for a field in the same point signals a different field. However, to this initial mathematical structure we can add additional features motivated by the physics one wants to represent. For instance, if one considers that the physics does not offer any operational way to distinguish two metric manifolds with diffeomorphic metrics, for example assuming a diffeomorphism-invariant dynamical theory for the metric, then one can take one step further and consider that the real physical objects are not the metrics themselves but equivalence classes of metrics related by diffeomorphisms:
where the double brackets represent the idea of an equivalence class. Then, we can say that instead of a metrical theory we have a geometrical theory. This is equivalent to saying that the points in the manifold do not have a physical meaning. This is the usual physical interpretation of GR, which of course goes back to Einstein himself: only diffeomorphism invariant properties have physical reality. A consequence of this point of view, as mentioned above, is that it only allows to define relational quantities. This can be taken as a fundamental characteristic, but also as a serious drawback. For instance, as mentioned before, a concept such as a local gravitational energy is in principle ill-defined in GR, since it cannot be described in any straightforward way as a tensor.
Consider now a manifold with two pseudo-Riemannian metrics \(\{ \mathcal{M},g_1,g_2 \}\). We can then follow the same steps as in the 1-metric case, up to a certain point. We can define a notion of diffeomorphic equivalence between \(\{ \mathcal{M}, g_1, g_2 \}\) and another structure \(\{ \mathcal{M}',g'_1,g'_2 \}\):
under the adequate conditions, which so far are a relatively straightforward extension from the 1-metric case. We can consider that physical theories are not able to discern diffeomorphisms. We can then use the freedom in the diffeomorphism group to consider, for example, \(g_1\) as a geometry instead of as a metric, thus worrying only about equivalence classes of \(g_1\):
However, at this point, it is not straightforward at all to consider that one should also necessarily deal with geometric equivalence classes \([[g_2]]\). In fact, it is more natural that \(g_2\) will continue to operate physically as a metric. Only in very specific cases, for example when the dynamical theory has a second, independent group of symmetries, formally equivalent to another diffeomorphism invariance, might one speak of a genuinely “bi-geometric” theory. In generic dynamical theories with two metrics and differmorphism invariance, once one assumes that \(g_1\) is a geometry, the second metric \(g_2\) in principle still maintains more physical degrees of freedom than a geometry.
To illustrate the previous point, let us consider a couple of simple examples of bi-metric constructions. As a first example take the simple action
There is a gauge symmetry of this action that could be directly taken as representing standard diffeomorphisms. The symmetry is under changes of the form \(\delta g_{1\mu \nu }=2 \nabla ^{g_1}_{(\mu } \xi _{\nu )}\), \(\delta g_{2\mu \nu }=2 \nabla ^{g_2}_{(\mu } \xi _{\nu )}\) with a single \(\xi _\mu (x)\) field (we use the superindex over \(\nabla \) to specify that it is the covariant derivative associated with that particular metric). Any hierarchical relation between causal cones of the form \([g_1] > [g_2]\), for example, is maintained under these symmetries that amount to changes of coordinates. However, the problem is that the theory is also invariant under two independent sets of transformations. The transformation \(\delta g_{1\mu \nu }=0\), \(\delta g_{2\mu \nu }=2 \nabla ^{g_2}_{(\mu } \xi _{\nu )}\), for instance, is also a gauge symmetry of the system. This transformation cannot be associated directly to a strict diffeomorphism (unless one interprets that the theory is that of two independent metric manifolds). Realizing that the system has two sets of independent symmetries, \(\delta g_{1\mu \nu }=2 \nabla ^{g_1}_{(\mu } \xi ^{(1)}_{\nu )}\), \(\delta g_{2\mu \nu }=2 \nabla ^{g_2}_{(\mu } \xi ^{(2)}_{\nu )}\), one can realize that actually the physical degrees of freedom can be associated with two equivalence classes of metrics \([[g_1]], [[g_2]]\). But then, there is no natural way to define a causal cone hierarchy at the level of equivalence classes, as the relation would not apply to all members in the classes.
As a second example take an action \(S=S_1+S_2+S_I\) with
The action \(S_2\) alone is invariant under transformations of the form
(which without further scrutiny might be confused as standard diffeomorphisms) Because of the interaction term, the total action is however not invariant under these transformations. Nonetheless, it is possible to interpret for example \(g_{\mu \nu }\) as a tensor field living on a geometrical Manifold represented by an equivalence class of \(\eta \) metrics. The total action is indeed invariant under
These transformations (actual diffeomorphisms) preserve any hierarchy present between \(\eta _{\mu \nu }\) and \(g_{\mu \nu }\) but at the cost that \(g_{\mu \nu }\) cannot be considered anymore as describing just a geometry. The interaction term \(S_I\) makes \(g_{\mu \nu }\) contain more degrees of freedom than those corresponding to a geometry.
Let us take the perspective that there is a high-energy regime in which there is a geometry \(\eta \) and a further tensorial object h with, in principle, up to 10 physical degrees of freedom. For h sufficiently small, \(g=\eta +h\) would seem to constitute a second metric (but it will not describe a second geometry). But then, we know that in Nature there is a low-energy regime well approximated by the GR equations where no additional (i.e.: metrical) degrees of freedom appear. We also know that these GR equations find an excellent accommodation in both the geometric and the field-theoretic paradigms. It thus seems obvious to require that h should, at some point, stop being a completely arbitrary tensor so that \(\eta +h\) itself can be taken as a geometry. This can be obtained by having an emergent gauge symmetry for h, but also by imposing restrictions on h, which mimic the elimination of the degrees of freedom.Footnote 9 This requirement is by no means sufficient to obtain GR, nor is it compulsory. For example, one could have a strict bi-geometrical theory in which at some point one metric becomes operationally invisible. However, we think it is interesting to appreciate its naturalness.
4.3 Fock harmonic condition
Our investigation into these topics started by looking at the VBL/Gao-Wald controversy. However, in revising the scientific literature we encountered several ideas which resonated with our own proposal. One of them deserves special mention because of its relevance and strong connection with our own proposal. This is the notion of harmonic coordinate system put forward by Fock [45]. This section is presented from the point of view of the geometric or field-theoretical paradigms.
Consider a Minkowski spacetime and some Cartesian coordinates \(x^\mu \). Let us solve the D’Alembertian equation \(\square _\eta X^{(\mu )}(x)=0\) for 4 scalar functions \(X^{(\mu )}=X^{(\mu )}(x)\). The general solution of this equation is
with \(C^{(\mu )},\Lambda ^{(\mu )}_{~\nu },A_{\textbf{k}}^{(\mu )},B_{\textbf{k}}^{(\mu )}\) different constants. If we remove any wavy solutions and set the trivial constant term to zero (corresponding to a shift in the origin of coordinates) we are left with some special solutions
which are linear combinations of the Cartesian coordinates themselves. In more detail, Fock proved the following:
Given a global Cartesian system of coordinates, any other harmonic set of coordinates, i.e. \(\square _\eta X^{(\mu )}(x)=0\), which
-
(i)
is asymptotically Cartesian, i.e. any departure from Cartesianity dies off as \(f(x) \sim 1/r\), where f(x) represents any of the coordinate functions,
-
(ii)
is bounded, i.e. any departure from Cartesianity obeys \(f(x)<C\), and
-
(iii)
contains at most outgoing waves, i.e.
then this new set of coordinates is related to the original set of Cartesian coordinates by a global Lorentz transformation. For this reason, we can say that Cartesian coordinates are precisely the unique non-wavy solutions of the harmonic equation, or that Cartesian coordinates are harmonic for the flat geometry.
From a dual perspective, instead of imposing restrictions on the coordinates, one could impose restrictions on the form in which the metric \(\eta \) is expressed. If we write \(\eta \) in (in principle: arbitrary) coordinates \(X^{(\mu )}\) and impose the restriction that these coordinates are harmonic:
then, this restriction amounts to the condition \(\eta ^{\mu \nu } \Gamma ^{\sigma }_{\mu \nu }=0\) or equivalently \(\partial _\mu \sqrt{-\eta }\eta ^{\mu \sigma }=0\). Thus, the harmonic condition restricts the forms that the flat metric can acquire. To single out a unique form for the metric, one has to add additional restrictions on \(\eta \) besides the harmonic condition: essentially that it does not contain wavy components.
Fock then argues that the same uniqueness should hold true at the non-linear level for asymptotically flat and regular \(\mathbb {R}^4\) spacetimes under equivalent conditions. The equation \(\square _g X^\mu (x)=0\) can be written also as
where 
and \(C^\alpha _{\mu \nu }\) is a tensor which has the same structure as the Levi-Civita symbols but with \(\partial \)’s substituted by \(\nabla \)’s. Fock imposed the non-linear harmonic condition
which at linear order in \(h_{\mu \nu }\) corresponds precisely to the harmonic gauge condition (2). Then, one is left with a pure second-order equation for the coordinates: 
. Under the previous conditions the unique solution will again be Cartesian coordinates. The condition “containing at most outgoing waves” can equivalently be imposed by removing from the set of solutions of \(h_{\mu \nu }\) anything containing incoming waves at infinity.
So Fock’s assertion is that to describe gravitational phenomena, within an asymptotically flat and regular \(\mathbb {R}^4\) spacetime, one can always select a unique and privileged set of coordinates, which Fock called the harmonic coordinate system. He gave to this fact an important significance. But one can ask oneself, how can a selection of coordinates have any physical significance? To answer this question and explain why we think Fock’s result (or our elaboration on it) is really significant, we need to make a brief digression.
First of all let us mention that Einstein himself gave a strong significance, not to selection of coordinates itself, but to the fact that: given a regular neighbourhood of a point (i.e. a neighbourhood with a well defined metric), there is always a small-enough sub-neighbourhood in which the gravitational field can be absorbed into a change of coordinates. Thus, in many situations and from this local point of view, gravity can be seen as just an illusion of the coordinates chosen to analyze an experiment. In fact, it appears that for Einstein this was the most significant contribution of GR, as it unified gravity with inertia. This is consistent with his explicit lack of enthusiasm for the idea defended by many others that the most relevant contribution of GR is its reduction of gravity to geometry; see e.g. the discussion in [46] on both issues. This does not put into question the well-established idea that one should be able to describe physics in any system of coordinates. In general, one can always generate a parametrized version of any theory such that the construction appears independent of the coordinates used in the description. This is done by introducing non-dynamical fields that transform in prescribed ways under changes of coordinates. One can for example parametrize classical dynamics (see e.g. discussions such as [47, 48]). Another example, already relativistic, is that of a scalar field theory over a fixed background metric. When one writes
one typically considers that \(\eta _{\mu \nu }\) is an object that transforms under changes of coordinates as a covariant 2-tensor. At this stage the invariance of the construction under changes of coordinates should not be considered a proper gauge symmetry, essentially because the \(\eta \) field is not a dynamical field of the theory. However, in GR one goes one step further and requires that the objects that absorb the coordinate transformations are themselves dynamical fields. The minimal version of this idea is to make \(\eta \) a dynamical field. As is standard in GR, one can write
so that the previous changes of coordinates are promoted to a (dynamical) gauge symmetry. When interpreted from this dynamical point of view, this is often qualified as a diffeomorphism gauge symmetry.
There is a subtle distinction between diffeomorphism and changes of coordinates. For example, a change from Cartesian to polar coordinates is not per se a diffeomorphism. Instead, a diffeomorphism can be associated with a global and one-to-one relabelling of the points of the manifold; at the infinitesimal level they correspond to infinitesimal local translation of a labelling system. In any case, the previous dynamical gauge symmetry maintains a strong tie with coordinate-choice invariance, and from there its name (diffeomorphism gauge symmetry). Thus, in spite of their differences, we will use diffeomorphism symmetry and symmetry under arbitrary changes of coordinates as synonyms.
The previous heuristics proved crucial for Einstein in finding his equations. The success of this heuristics, together with the strong association in GR between gauge invariance and coordinate change invariance is, in our view, one of the main reasons for the reluctance of most physicists to even consider a non-gauge invariant theory of gravitation.
Now, the crucial point that we want to make here is that we should not consider a putative dynamical gauge symmetry in a gravitational theory as necessarily equivalent to a diffeomorphisms gauge symmetry. In linear gravity (i.e., Fierz-Pauli theory), there is a clear distinction between coordinate changes and the gauge symmetry of the system. This gauge symmetry has no relation with diffeomorphisms or coordinate changes. However, when going to full non-linear gravity, the two notions of coordinate selection and choice of gauge marvelously merge into one single notion. When thinking of GR as an effective low-energy description, however, this coincidence could in fact be misleading.
For example, one can consider that there is one fundamental metric plus some other tensorial objects defined in a single manifold. In this way one has by construction invariance under changes of coordinates; in other words, there is one geometry in which the rest of the fields live. This geometry could itself be dynamical in such a way that the theory exhibits a dynamical diffeormorphism invariance. But the geometry could also be almost rigid at the relevant low-energy scale. Independently of whether it is dynamical or not at a low-energy regime, among those extra objects defined in the manifold, one could have tensorial objects (e.g. a \(h_{\mu \nu }\) field) which in certain circumstances deform the causality perceived by the fields with respect to that provided by the geometry itself. The theory which describes these additional fields could be gauge or not; one would have to construct the theory that best describes the phenomenology. But the important message is that the arguments used to construct the dynamical theory for the h field need not be related to invariance under diffeomorphism, which is already warranted by the fundamental geometry. Imagine that the best-fit theory was one in which the graviton had a significant mass. This theory will not impose any additional gauge symmetry or restriction on the h field, but invariance under changes of coordinates (aka diffeomorphism) is still warranted. Now, the success of GR at low energies already tells us that gravity appears to excite significantly less degrees of freedom than those that a tensor h could in principle accommodate. This idea can be incorporated in the theory by enforcing a further gauge theory over the h field, as is done in the heuristics of the FTP, or alternatively by constraining the h field in some way, which is the option that we will argue for in our proposal. Our central message here is that there is no physically compelling argument to opt for additional extra gauge symmetries, because these extra symmetries need not have anything to do with changes of coordinates. Indeed, from the perspective of the FTP, it is usual to think that adding a mass for the graviton would break gauge invariance. But, in the line of our argument, such a gauge invariance breaking does not pose any physical problem, and has nothing to do with a breaking of diffeormorphism invariance (which would be a problem).
After this digression, let us summarize the main point that we want to make with respect to our reconstruction of Fock’s results. This lies not so much in the importance of choosing a specific coordinate system, but in the importance of adopting one concrete definition of gravitational field based on physical considerations (or using a more standard language, in the importance of selecting one specific physical gauge). As we will argue below, this would allow for example a well-defined local notion of gravitational energy. Let us emphasize that, once this “physical gauge” has been selected, one can still use any coordinate system.
As a final twist, once one opts for describing the gravitational field by a unique h field (rather than by an equivalence class of h fields), the further possibility of selecting Cartesian coordinates, under Fock’s conditions, allows, as an additional bonus, to give a precise and consistent definition of what it means to be in a linear gravitational regime: namely when the components of the unique \(h_{\mu \nu }\) in those coordinates are small with respect to the unit value of the \(\eta \) metric in those same coordinates (i.e, simply \(\{-1,+1,+1,+1\}\)). (More precisely, it is also necessary that the first and second derivatives of \(h_{\mu \nu }\) are small, to avoid large curvatures by some rapid wiggling of the spacetime). In this way, selection of coordinates can acquire a role after all.
4.4 The harmonic background paradigm
We are now ready to present what we call the “Harmonic Background Paradigm” (HBP). The paradigm is constituted by a series of hypotheses, or fundamental assumptions. The idea would then be to develop the consequences of these hypotheses and to check their predictive power.
1. Background causality: The first assumption is that there exists something playing the role of a flat background geometry, represented by a metric \(\eta _{\mu \nu }\) written in whatever system of coordinates. With this assumption, we are not committing to an ontological reality of this spacetime, only that such a structure exists pragmatically as a way of describing the fundamental causality in the system. We think of \(\eta \) as a fixed background geometry, not in an absolutely rigid sense, but in the sense of an effective decoupling because of the large “inertia” of this background with respect to all the other fields. The background metric \(\eta \) influences the other fields (“geometry tells effective fields how to move”), but these fields barely affect the background metric (“field content tells the geometry how to curve” is no longer valid for the background metric \(\eta \)). Notice that situations of this sort are very standard in the physical description of phenomena. Physically speaking, we can think of this background flat geometry as a constant extrapolation to the entire manifold of the geometry in regions very far away from any source and in moments in which there are no gravitational waves passing by. Once we have this background geometry installed, we will impose that no effective causality can surpass its propagation speed limit.
2. Gravity as a causality deformation field: Our second assumption is that gravity is represented by a tensor object \(h_{\mu \nu }\) living on this geometrically flat manifold. In many circumstances in which \(h_{\mu \nu }\) is sufficiently small compared to \(\eta _{\mu \nu }\), we have that \(g_{\mu \nu }:=\eta _{\mu \nu } + h_{\mu \nu }\) can be considered formally as a second pseudo-Riemannian metric. If \(h_{\mu \nu }\) is too big, \(g_{\mu \nu }\) might not even be a Lorentzian metric. Note that this regime need not coincide with the linear regime. Now, for the reasons explained above, we do not want to consider this second metric as a second geometry but something disguised as such. Instead of working with a completely general tensor \(h_{\mu \nu }\), we will therefore from the very start impose several restrictions on \(h_{\mu \nu }\). On the one hand, we impose four differential conditions. At the linear level, these conditions are precisely the de Donder conditions (2). We leave their extension to the full non-linear regime for future work. This extension need not be in the standard non-linear harmonic condition. Note however that in fact we do not need the non-linear regime to find one of the principal results of this paper: our hypothesis as to why gravity is attractive. On the other hand, as an additional restriction we consider that the physical solutions of \(h_{\mu \nu }\) do not contain waves of any kind coming in from infinity, neither physical nor gauge waves. Note that this additional condition on the allowed forms of \(h_{\mu \nu }\) cannot be imposed through a local differential condition but requires a boundary condition. These restrictions on h plus the presence of a background are precisely what inspire the name of the paradigm.
3. Causality hierarchy: Finally, our third assumption is that gravity exhibits a cone hierarchy such that one always has \([g]<[\eta ]\). This assumption would make gravity to be a special relativistic field theory (again with the caveat of the extension to the non-linear regime).
Therefore, the paradigm we propose here is essentially bi-metric; it assigns an “ontological” priority to the \(\eta \)-geometry with respect to the gravitational g metric; it considers that gravity is not described by a gauge theory but instead assumes that there is a unique \(h_{\mu \nu }\) representing each gravitational situation. Thus, although Fock’s philosophical view on these matters is not completely clear to us, it is probably safe to say that we go further than Fock himself did: our proposal assumes a specific definition of what is the gravitational field (in the GP and FTP this would correspond to the selection of a specific gauge).Footnote 10 The important point is that one can not impose any gauge one likes. In fact, in our view such an assertion makes little sense: we take as hypothesis that the restrictions on h are physical, not originating from any gauge fixing. For example, if the unique h becomes singular, then this singularity should be considered physically relevant; it would make no sense to verify “whether the singularity disappears in a different gauge” because the theory is not a gauge theory.
Before ending this subsection let us warn again that the HBP as specified here is incomplete, as we have not prescribed yet how the linear harmonic constraint generalizes to the non-linear regime (in this context, note that there are already proposals to go beyond the standard non-linear harmonic condition [52, 53]). We leave this issue for a future analysis.
4.5 Benefits of the harmonic background paradigm
Our proposal can probably be better understood if framed within the following philosophical position. We suppose that GR is an effective description that dissolves at increasing energies. On the one hand, one could start perceiving additional degrees of freedom in h, e.g. by experiencing a small graviton mass. One could also start realizing that not everything follows the effective causality provided by \(\eta +h\). We foresee situations in which the h field disappears altogether, in the same sense that an order parameter in condensed matter can disappear through a phase transition. However, the more fundamental \(\eta \) causality could still survive in those regimes, and could play a central role in the causality perceived by high-energy observers. For these high-energy observers, it would be natural to describe any low-energy phenomena using their high-energy language. For example, they would describe the trajectories of bodies in their fundamental Minkowski spacetime. Low-energy observers, on the other hand, who have access only to the GR degrees of freedom, are perfectly justified to use a GP or a FTP to describe the phenomenology which they observe, and to reject as unphysical any description in terms of a background metric which they do not perceive.
Note that in our proposal, apart from the two causal structures to which we focus most of our attention in this paper, we are assuming the existence of two, in principle independent, volume elements. We are assuming that the two affine structures are however those compatible with their respective metrics and have no torsion (at this stage we don’t have compelling reasons to further complicate the model). This implies that low-energy observers will have a different notion of length than high-energy observers. All low-energy physical clocks and rulers will follow the effective metric. The presence of a gravitational field however would not affect the perception of time and space by high-energy observers, which would always follow the Minkowski standard.Footnote 11
Again, whenever the phenomenology remains in a weak-field and local GR regime, the three paradigms we have been discussing (geometric, field-theoretic, and harmonic background) can be considered as just three interchangeable and equivalent approaches, with some calculations easier in one or another paradigm. So, from the perspective of the HBP, most calculations performed within a gauge perspective make perfectly good sense; only the interpretation would be different. However, as we will see in the next section, even in standard situations there can be important differences of interpretation, in particular the HBP can be understood as pointing towards an explanation of why gravity is attractive. Nonetheless, as said, the more crucial distinction between these paradigms really crystallizes when one tries to go beyond the GR regime, for example, when trying to solve the singularity problem. The HBP gives an important role to the background geometry and thus suggests that the first departures from GR might appear precisely because of this prominence of the background geometry (for instance, in terms of effective models and situations not describable in terms of just one single \(g=\eta +h\)). For example, it is well known that trying to incorporate even a tiny mass for the graviton forces one to introduce an additional metric besides the GR metric [55]. In a bi-metric paradigm, GR can be seen as a small-mass limit of contemporary massive-gravity dynamical theories. From this point of view, a bi-metrical formulation of GR appears as more robust.
Now, by this point, it should be obvious that, by construction, the VBL result is absolutely and rigorously valid in the HBP: the VBL conditions are in fact equivalent to those of the paradigm. Moreover, this paradigm not only allows one to establish a hierarchy of causal cones whenever the Background NEC is satisfied (at least at the linear level). It also allows one to define a unique SET for all the physical gravity tensors \(h_{\mu \nu }\). Furthermore, it permits one to associate a single trajectory in the flat background to the geodesic movement of any celestial body. All these highly desirable characteristics are completely absent in the other two paradigms in which gauge symmetries play a crucial role.
Let us also stress that the HBP is compatible with situations in which there is emission of gravitational waves (these waves are outgoing, not incoming). The full conceptual consistency of the paradigm occurs when considering bounded and isolated closed systems. To accommodate in the formalism a gravitational wave coming from far away in the universe, one only needs to incorporate this emitting system as part of the isolated closed system. If one wants to analyze in simple mathematical terms what happens when a gravitational wave generated far away interacts with some local system, one could alternatively relax the formalism and allow the presence of physical waves incoming from infinity. Here we just want to remark that this calculational convenience should not be confused with a fundamental property of the physics.
To end this section and re-connect with the VBL argument, let us recall that the VBL paper was originally formulated within the GP, and correctly criticized within that same paradigm by Gao-Wald. Note that the VBL argument can also be understood within the FTP that we mentioned earlier. But even in such a field-theoretic view, it does not acquire a convincing power, since it would still be gauge-dependent. The alternative paradigm that we have discussed here stresses the importance of the flat background geometry and the specifically restricted h. It is thus within this paradigm that the VBL idea acquires a renewed rigour and interest.
5 Why is gravity attractive?
We started this paper with the interesting debate between VBL and Gao-Wald, explaining why both are correct. The clash appeared because of their different initial positions. While Gao-Wald maintained themselves clearly and completely within the GP, or equivalently the FTP (but in any case a gauge-independent formalism), the VBL paper can be considered as a cautious but insightful first heuristic step in the exploration of extensions of GR beyond these paradigms. We have shown that the VBL proposal can be made totally rigorous in a harmonic background setup. In this framework gravity appears as a modification of a more rigid background causality described by a flat background Minkowski metric.
In the present section, we will discuss a result that immediately follows from the HBP simply by inverting the logical arrow of the VBL paper. VBL emphasized that the satisfaction of an energy condition forbids superluminal propagation (with superluminal understood as surpassing a fixed and background velocity). Here we explain, vice versa, how the existence of a fundamental background causality can be interpreted as automatically imposing a null energy condition. To be clear, in this work by “attractive” gravity we just mean that matter satisfies a specific null energy condition. (Note that in the standard literature on energy conditions (see e.g. [10]) attractive gravity is more tightly associated with the satisfaction of the strong energy condition than with the standard null energy condition, although the latter can still be loosely associated with attractive gravity.) Having an additional metric structure to compare with, we could go even further and say that by attractive gravity we mean that matter can only delay signalling with respect to the background metric. This would immediately hold by definition in the HBP. However, this does not guarantee that null rays are always convergent as a NEC does. In any case, it is interesting to realize that the presence of two metrical structures opens up ways to new energy conditions and probably to novel interpretations.
GR is a peculiar theory. It establishes a connection between matter/energy content and spacetime curvature, but it says nothing about what kind of matter can exist in Nature. In fact, the distinctive attractive character of gravity is not explained by GR, nor by Newtonian gravity: it is an additional postulate about the positive nature of matter/energy. Curiously, though, the proof of many of the most important results in GR requires this additional postulate, i.e. the satisfaction of some energy condition.
Given the crucial role we are assigning to the flat background geometry, it was most reasonable to impose as a defining characteristic of the HBP the restriction that the effective causality is always subsumed into a more fundamental and insurmountable background causality, in other words:
always. When h is small, we are in fact probing the possible first deviations from a flat causality. It is an empirical fact that, in all usual low-gravity situations, mass and energy are always positive. The VBL result tells us that the BNEC, or background NEC, can be interpreted (at the linear level) as a consistency condition for the absence of superluminality, i.e. \([g] \le [\eta ]\). We have discussed in detail that the VBL result automatically follows from the HBP. We therefore propose to interpret \([g] \le [\eta ]\), i.e. the presence of a more fundamental background causality, as the real reason why gravity is attractive, at least in normal weak-gravity situations. The idea is that the obstruction to make the g-cones exit the \(\eta \)-cones restricts the stress-energy characteristics that any matter content may have.
In Eq. (3) the implication “BNEC satisfaction” \(\rightarrow \) “cone hierarchy \([g] \le [\eta ]\)” is clear. Obviously, the straight inverse implication does not apply in general. However, in situations in which the causality is almost flat, \(h_{\mu \nu } \rightarrow 0\), any appreciable violation of BNEC could make \(h_{\mu \nu } l^\mu l^\nu \) become negative. In this very limit is where we argue that a condition \(h_{\mu \nu } l^\mu l^\nu \ge 0\) implies a non-violation of BNEC, and the one-way logical arrow thus becomes an equivalence.
Indeed, this energy condition need not necessarily be maintained in regions where the cones of g are already well inside those of \(\eta \), in other words in the non-linear regime for h. First, from a technical point of view, it appears that neither the VBL result nor the harmonic condition can be extended in any straightforward way to the full non-linear case. But beyond this technical issue, there is a more profound reason why energy conditions might be violated in high-curvature regimes. There is nothing in the HBP which indicates that energy conditions should always be imposed. Within this paradigm, it is only when the effective cones approach the Minkowski cones that one should start appreciating an obstruction or constraint to the behaviour of matter. That is, precisely in the weak-field regime to which our analyses are restricted.
Beyond this regime, although not straightforward to prove, we suspect that imposing suitable energy conditions under all circumstances would indeed force the g-cones to become narrower and narrower. In such a picture, the formation of a singularity seems inevitable in situations such as a sufficiently massive standard gravitational collapse. Our framework, on the contrary, suggests to appeal to the background metric, and to proper bi-metrical generalizations of the GR regime, whenever the effective field \(h_{\mu \nu }\) approaches a singular situation. Such ideas and how they could help to avoid the formation of singularities in gravitational collapse have been explored in [51, 56, 57]. There, we advocate for the presence of a flat background geometry and the characterization of the strong gravity regime as a switching off of the effective causality, leading for example to black-hole to white-hole bounces.
To summarize this section, let it suffice to repeat that the HBP provides an explanation, which in our opinion is very natural, for the attractive character of gravity, as an immediate consequence of the fundamental causality of the background structure.
6 About the Penrose property
We cannot finish this paper without mentioning and briefly discussing a potential problem with the HBP and in particular the assumption that it contains a cone hierarchy. The problem comes from Penrose’s argument that GR as a geometric theory cannot be reduced simply to a Lorentz-covariant field theory [8]. The essence of the problem has to do with the possibility or impossibility of interpreting gravity as a dynamical theory of cones always inside some fixed Minkowski structure, \([g]\le [\eta ]\), a point with an obvious relevance with respect to our previous discussion. We will explain in this section why Penrose’s idea does not pose an essential problem for our proposal.
Let us start by reviewing the logic of Penrose’s argument, see also its review and extension in [58]. Penrose implicitly works with a single manifold with two metric structures, \(\{ \mathcal{M},\eta , g \}\), so that changes of coordinates in the manifold do not change the potential hierarchy between the cones of the two metrics. So far, this view is completely in tune with our HBP presented above, see Sec. 4, and not so much with a field-theoretic view, precisely because gauge transformations can affect the cone hierarchy.
Second, he defines what the authors of [58] call the Penrose property, which is obeyed by a spacetime if and only if given two arbitrary timelike curves there is always a third timelike curve that intersects them both. Third, Penrose proves that global Minkowski spacetime does not satisfy the Penrose property, contrarily to spacetimes representing stellar structures with a positive mass and a Schwarzschild exterior. Finally, based on the previous point, he proves that it is impossible to define a conformally compactified manifold and two compactified metrics on the manifold \(\{ \hat{\mathcal{M}}, \hat{\eta }, \hat{g} \}\), related respectively to a Minkowski and a positive-mass asymptotically Schwarzschild spacetime, such that \([\hat{g}] \le [\hat{\eta }]\).
The authors of [58] argue that Penrose’s proof also implies the impossibility of imposing a condition \([g] \le [\eta ]\). However, as we will now explain, we do not agree with this assertion. The crucial observation of Penrose is that any spacetime that approaches Minkowski spacetime at infinity in a 1/r Schwarzschild manner makes any radial null ray acquire an infinite delay in reaching infinity with respect to what would have happened when traversing instead a purely Minkowskian region. This can be seen by looking at the radial null geodesics outgoing from any positive central matter:
Consider any null particle (a photon) escaping away from the central mass towards infinity and already very far from the mass. Compare this with a second photon, equally far from the central mass but coming in from an antipodal position. Then, the continuously accumulated delay of the first photon means that the second photon will be able to catch up with the first one by following a trajectory that avoids the center of the configuration. The infinite delay accumulated with respect to pure Minkowski is therefore a characteristic of any asymptotically Schwarzschild geometry. However, the infinite character of the involved spacetimes is essential for this to hold. To be more explicit, take a conformally compactified Minkowski spacetime, which amounts to a 4D diamond with boundary. Then, prescribe a second metric \(\hat{g}\) on this compactified manifold. In order for \(\hat{g}\) to incorporate delays with respect to \(\hat{\eta }\), assume \([\hat{g}] \le [\hat{\eta }]\). But null rays moving according to \(\hat{g}\) will never accumulate an infinite delay with respect to the Minkowski rays in a finite amount of space. Such infinite delay could be obtained if cones of the initial Minkowski compactified representative \(\hat{\eta }\) completely open up at the conformal boundary. But then this conformal Minkowski metric no longer possesses the appropriate null infinities. Therefore, it is true (as Penrose demonstrates) that \(\{ \hat{\mathcal{M}}, \hat{\eta }, \hat{g} \}\) with \([\hat{g}] \le [\hat{\eta }]\) is impossible. However, and contrarily to the claim made in [58], \(\{ \mathcal{M}, \eta , g \}\) with \([g] \le [\eta ]\) is possible.
Penrose’s correct analysis should not be taken as proof that there is an inconsistency in treating GR from the point of view of a Lorentz covariant construction. More precisely: Penrose does not prove it impossible to construct a non-compact manifold in which standard physical metrics, such as those associated with regular stars, are inside a Minkowski metric (in fact, his paper starts by showing precisely an example in which this happens). What is not possible is to compare the causal asymptotic structures of Minkowski and of physical, Schwarzschild-type metrics.
But how relevant is this for actual physical situations? Our viewpoint is that asymptotic structures in GR allow a better mathematical and conceptual control over the theory. The same happens in Quantum Field Theory: a mathematical control of scattering theory requires the definition of asymptotically free states. But care should be taken when physical conclusions depend crucially on these asymptotic properties. In particle collider experiments, the asymptotically free states obviously do not exist physically. The fact that such theories work can be considered precisely as an illustration of how the existence of a strict asymptotic region does not play any role. In fact, we do not know of any example of observed physical phenomena whose explanation depends crucially on the existence of an asymptotic structure. In the gravitational case, let us stress that the Penrose property strongly depends on the 1/r behaviour of the Newtonian potential in (3+1) dimensions (e.g. in [58] it is shown that stellar configurations in higher dimensions do not exhibit the Penrose property). It is also worth recalling that 1/r interactions (in other words, massless fields) always have subtleties in scattering theory (see e.g. [59]). Thinking of GR as the small-mass limit of a massive gravity theory, for example, could immediately solve this problem.
Let us take the opportunity offered by this discussion to clarify that with our proposal we are not advocating for the reality of a strict Minkowski background (i.e. spatially infinite). We take this simple background as a proxy for a large but spatially-compact setting, again with the idea of not making a crucial usage of asymptotic structures, beyond their mathematical and conceptual convenience. For instance, we leave for a future work to analyze if and how the HBP described here could be generalized to spatially compact backgrounds such as an Einstein static universe.
To end this section let us also briefly comment on another potential challenge for the HBP, as briefly mentioned above in Sect. 3. In [35] the authors discuss that the evolution equation for the perturbations over conformally compactified asymptotically simple spacetimes appears ill defined if one uses the harmonic gauge for the perturbations. Here, let us just say that this does not imply that the harmonic condition applied to perturbations of the physical metric is ill-defined.
7 Conclusions and further remarks
Our investigation started by trying to understand a result by Visser–Bassett–Liberati [11] and a subsequent criticism by Gao-Wald [17]. VBL found that under certain conditions the satisfaction of an energy condition forbids propagation faster than light. Gao-Wald counter-argued that this result is highly gauge dependent and therefore has no physical significance within standard GR. Here, we have cleared up this debate and reached the conclusion that both were right but that a proper defense and a correct interpretation of the VBL result requires a paradigm shift, for which we have proposed the Harmonic Background Paradigm.
The relativistic revolution of the 20th century has made an anathema of thinking about any kind of fundamental background, anything that acts but cannot be acted upon [60].Footnote 12 Even bi-metrical theories are usually interpreted as being better motivated if both metrics are dynamical [62]. However, if one agrees that our current knowledge of spacetime and gravitation is purely effective, then it might be necessary to take one step back when thinking about a more fundamental theory (at the Planck scale, say), let alone any hypothetical “final theory” (whatever that may mean). We believe it only very reasonable to hypothesize that such a more fundamental structure might be sufficiently rigid with respect to our effective physics that it can be treated as a background structure.
Concretely, in this work we propose an alternative way to understand the gravitational phenomena. We propose that the gravitational field (at least at the linear level) is a tensor field \(h_{\mu \nu }\) living in a flat background geometry. In the GR regime this tensor is sourced by the stress-energy tensor of matter fields and acts as an effective and universal deformation of the causal structure of the background geometry: all matter fields couple to \(\eta +h\). The tensor \(h_{\mu \nu }\) solves the Einstein equations and in addition is subject to three restrictions: i) it satisfies the harmonic condition \(\nabla ^\mu h_{\mu \nu }- (1/2) \nabla _\nu h=0\); ii) it contains no incoming waves from asymptotic infinity, neither physical nor “gauge waves”; and iii) it satisfies the consistency condition that the cones of this effective causality always lie inside the fundamental Minkowski cones, i.e. \([g] \le [\eta ]\); we have discussed that Penrose’s property does not pose a fundamental obstruction to such a construction. Concentrations of matter thus deform the causality in their surroundings (through g) and produce delays (never advances) with respect to the more fundamental causality \(\eta \). This complies with the original special relativity idea that nothing can travel faster than the speed of light.
In standard weak-gravity situations this gravitational theory is phenomenologically indistinguishable from GR and can be thought of as just an alternative interpretation. This is true because from the perspective of the geometric or the field-theoretical paradigms, the harmonic background paradigm just amounts to the selection of a particular gauge in which to perform the calculations. However, by granting these conditions a physical reality (not merely based on a gauge choice), we obtain several very interesting bonuses. The first one is a possible explanation of the observational fact that matter is universally attractive in all standard (weak field) situations. By inverting the logical arrow of the VBL reasoning, we have argued that this fundamental causality hypothesis automatically implies, for consistency, that matter fields in weak field configurations must satisfy a background null energy condition, in other words: that in all weak-gravity situations, matter must universally make the null rays in its surroundings converge.
Our proposal has other advantages with respect to the geometric and the field-theoretical paradigms. We highlight specifically three points: i) As already observed by Rosen [1], introducing a background metric \(\eta \) allows one to construct a SET for the gravitational field, rather than just a pseudo-tensorial quantity. Moreover, having a unique prescription for the gravitational field \(h_{\mu \nu }\), it does not suffer from the interpretation problem it has in the FTP, where the gravitational SET is not gauge-invariant. ii) The total fixing of \(h_{\mu \nu }\) amounts to an unambiguous identification of the effective metric and the background geometry. Then, a curve defined over g has a unique correspondence with respect to \(\eta \). Thus, the geodesic trajectory, for example, of a planet around the Sun has a perfectly defined meaning for an observer who only perceives the background metric. iii) The bi-metrical theory easily accommodates a non-zero graviton mass. Since experimentally, it will never be possible to establish that the graviton mass must be exactly zero (only to put upper bounds on its mass), it makes sense to expect this to be reflected in “theory space” also, and thus that the limit for zero graviton mass should be continuous with respect to theories with a massive graviton. Beyond these concrete points, in more general terms we expect this paradigm might simplify many conceptual issues that appear within the standard paradigms.
In any case, the most important asset of the HBP lies in the possible extensions from GR that it can accommodate, from Rosen-like modifications [63] to contemporary massive gravity theories [55, 64], to name just the most obvious. In this context, the present authors proposed several years ago [56] that a possible way of solving gravitational collapse singularities is by considering that gravity “switches off” when matter reaches Planckian densities (see also [51, 57]). This idea was originally motivated by condensed-matter models of gravity, in which hierarchies between effective and more fundamental levels of description emerge naturally.Footnote 13 We then argued that, if “real” gravity shows a similar emergent hierarchy, there might exist some energy scale at which the effective metric disappears as an order parameter of the system, leaving only the background causality to take control of the physics. From the point of view of such emergent-gravity scenarios, let us also mention that in a generic linear Lorentz-invariant theory of a tensorial field \(h_{\mu \nu }\), without imposing any gauge symmetries, one recovers precisely the De Donder or harmonic condition (2) as a natural restriction on the space of solutions [44]. Thus, starting from a theory without gauge symmetries, one deduces a dynamical theory for gravity equivalent to GR when written in a particular physical gauge, precisely corresponding to the harmonic background argued for in this paper.
In our opinion, the conflation between selection of coordinates and selection of gauge might be an obstacle to our understanding of gravitational phenomena and utterly to progressing beyond GR. In our proposal, there is a clear coexistence of subjective gravitational fields, based on selection of coordinates, with objective gravitational fields, the unique h in our Harmonic Background Paradigm. Given that there is an objective gravitational field one might try to quantize it, but this need not imply that one is quantizing the entire causality in the system; the background causality could still be maintained as a reference causality avoiding some of the problems associated with background-independent quantizations of GR.
There are many subtleties still to be analyzed within the HBP, mostly in its extension to the full non-linear regime. We hope this paper serves to clear up the VBL/Gao-Wald controversy and to stimulate further analyses and alternative ways of thinking about gravity.
Notes
Having different interpretations also gives one insight into, and so a deeper understanding of, the internal conceptual structure of the theory itself, as relationships among (e.g.) some concepts and principles may become more perspicuous, or may show themselves in a new light, in one interpretation rather than another. This comment, which we totally subscribe, has been borrowed from an anonymous referee with his/her permission.
In the FTP, one can prove that equal-sign masses interacting through a massless spin-2 field attract each other [4], just the contrary than in the case of a spin-1 field. In this paper we do not discuss why the Newton constant is positive in the first place.
None of De Donder’s works on relativity have, as far as we are aware, ever been translated in English. For a rare work of De Donder in English, see his lecture notes [31].
The Lorenz gauge is often mistakenly attributed to Lorentz, see Jackson and Okun [32] for an extensive discussion. Jackson’s own classic textbook [33] in fact commits exactly this confusion up to the second edition. The third edition corrects the mistake and contains a note of explanation, but ironically still commits the misspelling in a few places.
Strictly speaking, this metric has a cone singularity in \(r=0\), but this is irrelevant for the illustrative purpose of our example.
It appears that one cannot even solve the classification problem of observables within standard analysis [43].
It is interesting to note that both possibilities appear intertwined in the mechanism of emergent gauge symmetries proposed in [44]
In this regard, it is also interesting to recall that Weyl transverse gravity, a theory almost equivalent to GR [54] is formulated by already using a fixed volume element, which can be considered to be Minkowskian. The necessity of having two or only one volume factors is related to the final selection of non-linear restrictions. From this perspective, a bi-metrical construction like ours could have two causal structures but only one volume element. In fact, one could think of Weyl transverse gravity as a reformulation of GR that partially recalls the presence of a background structure. Therefore, apart from the fundamental Minkowski volume form, the existence of an additional low-energy volume form is related to the way in which one formulates the low-energy theory. We leave a further analysis of this issue for future work.
Note that, contrary to common lore, this was not a position that Einstein himself strongly defended, see for example our discussion in [61].
It might be interesting to note that a bi-metric effective gravity with a flat background very similar to Logunov’s RTG is spontaneously realized in \(^3\)He-A [65], see the discussion in [66]. Other scenarios of emergent gravity are also possible, which do not necessarily depend on a flat background, see e.g. [67, 68].
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Acknowledgements
The authors would like to thank Gerardo García-Moreno, Stefano Liberati and Matt Visser for enlightening discussions on the subject, as well as Germain Rousseaux and Grigory Volovik for useful comments on the manuscript. Financial support was provided by the Spanish Government through projects PID2020-118159GB-C43 and PID2020-118159GB-C44 (with FEDER contribution), and by the Junta de Andalucía through project FQM219. C.B. acknowledges financial support from the grant CEX2021-001131-S funded by MCIN/AEI/ 10.13039/501100011033.
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Barceló, C., Jannes, G. The harmonic background paradigm, or why gravity is attractive. Gen Relativ Gravit 56, 116 (2024). https://doi.org/10.1007/s10714-024-03306-9
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DOI: https://doi.org/10.1007/s10714-024-03306-9