Functional Gravitational Energy

Functional Gravitational Energy Abstract Does the gravitational field described in general relativity possess genuine stress-energy? We answer this question in the affirmative, in (i) a weak sense applicable in a certain class of models of the theory, and (ii) arguably also in a strong sense, applicable in all models of the theory. In addition, we argue that one can be a realist about gravitational stress-energy in general relativity even if one is a relationist about spacetime ontology. In each case, our reasoning rests upon a functionalist approach to the definition of physical quantities. 1 Introduction 2 Dramatis Personae   2.1 General relativity   2.2 Differential and integral conservation laws   2.3 Conservation equations in special and general relativity   2.4 Killing vector fields and spacetime isometries   2.5 The gravitational stress-energy pseudotensor 3 Interpreting Conservation Laws for Total Stress-Energy   3.1 The differential conservation law for total stress-energy   3.2 The integral conservation law for total stress-energy   3.3 Gravitational stress-energy 4 Gravitational Stress-Energy and Spacetime Ontology   4.1 Relationism and gravitational stress-energy   4.2 The cosmological constant 5 Conclusion 1 Introduction In the study of general relativity (GR), a range of claims have been advanced regarding the nature of stress-energy in the theory, and its status as a conserved quantity. For example, the intuition that stress-energy must be conserved in GR sometimes leads to the expression ∇aTab=0 being treated as the conservation law for the stress-energy of matter fields (see, for example, Misner et al. [1973], p. 152); yet, on reflection, one may question whether this is a genuine conservation law at all.1 On the other hand, it is sometimes claimed that gravitational stress-energy exists in GR, but is non-local (Hartle [2003]; Misner et al. [1973]; Schutz [2009]). In this article, we argue that gravitational stress-energy does exist in the theory, in both (i) a weak sense applicable in a certain class of models of the theory, and (ii) arguably also in a strong sense, applicable in all models of the theory. In each case, our reasoning rests upon a functionalist approach to the definition of physical quantities. In Section 2, we provide an interpretationally neutral presentation of the various stress-energy ‘conservation laws’ in GR. In Section 3, we argue for the existence of the above weak notion of gravitational stress-energy in certain models of GR, and also raise the possibility of a strong notion of gravitational stress-energy in the theory. In Section 4, we consider ways in which the related debate between substantivalism and relationism on the ontology of spacetime connects with the debate on gravitational stress-energy in GR. 2 Dramatis Personae In this section, we introduce all mathematical concepts essential to foundational discussions of gravitational stress-energy in GR. In Section 2.1, we introduce the kinematically and dynamically possible models (DPMs) of GR; we then discuss in Section 2.2 differential and integral conservation laws in physical theories; before in Section 2.3 presenting specific stress-energy conservation laws that arise in special and general relativity. In Section 2.4, we introduce the mathematical apparatus of isometries and Killing vector fields, before finally discussing in Section 2.5 a conservation law for total (that is, matter-plus-gravitational) stress-energy in GR. 2.1 General relativity Kinematically possible models (KPMs) of GR are picked out by triples ⟨M,gab,Φ⟩, where M is a four-dimensional differentiable manifold; gab is a Lorentzian metric field on M; and Φ is a placeholder for matter fields in the theory.2 Associated to ⟨M,gab⟩ there exists a unique derivative operator ∇a, which is: (i) torsion free—in the sense that the associated torsion tensor Tbca, defined through TbcaXbYc=Xb∇bYa−Yb∇bXa−[X,Y]a, vanishes—and (ii) metric compatible—in the sense that ∇agbc=0. Since this derivative operator ∇a is uniquely determined by ⟨M,gab⟩, it is not included as an independent variable in the KPMs of GR. DPMs of GR are those KPMs the geometrical objects of which satisfy the Einstein field equations Gab=8πTab (1) —the dynamical equations of the theory, which relate gab to the stress-energy tensor Tab of the Φ—in addition to the dynamical equations of the Φ.3,4 2.2 Differential and integral conservation laws 2.2.1 Case 1: Flat connection For a given derivative operator ∇a and tensor field Ta1…an, suppose: ∇a1Ta1…an=0. (2) In a coordinate basis, Equation (2) reads: ∂μ1Tμ1…μn+Γσμ1μ1Tσμ2…μn+…+Γσμ1μnTμ1…μn−1σ=:∂μ1Tμ1…μn+Δμ2…μn=0, (3) where the Γνσμ are the connection components associated to the derivative operator ∇a in this coordinate basis, and we have introduced the notation Δμ2…μn to represent the terms featuring connection components in the above expression.5 In GR, at any p∈M we can find Riemann normal coordinates such that connection components vanish, and Equation (3) becomes: ∂μ1Tμ1…μn(p)=0. (4) This result holds only at p. If we wish Equation (4) to obtain at some other q∈M in the neighbourhood of p, we require that derivatives of connection components vanish—and a fortiori that the derivative operator ∇a be flat.6 In this case, we may write: ∂μ1Tμ1…μn=0. (\rm{ {Par {\hbox{-}}T) Given the local validity of Equation (Par-T), we may integrate over a volume V⊂M and apply Gauss's theorem, to obtain (na is a unit normal vector to S=∂V; na=gabnb): ∫S=∂Vnμ1Tμ1…μndS=0. (\rm{ {Int {\hbox{-}}T)  Note that Equation (Par-T) is invariant only under affine transformations—that is, only retains the same form in coordinate systems related by such transformations.7 This means that an equation of the form Int-T also holds only in such coordinate systems. If one applies a non-affine transformation xμ→xμ′=fμ′(xμ) to Equation (Par-T), one will generically pick up extra terms (which we may schematically represent by Θμ2…μn), so that this equation reads instead8,9: ∂μ′1Tμ′1…μ′n+Θμ′2…μ′n=0. (5) Integrating over the volume V as before, we have by analogy with Int-T: ∫S=∂Vnμ′1Tμ′1…μ′ndS=−∫VΘμ′2…μ′ndV. (6) 2.2.2 Case 2: Non-flat connection Consider now the case in which ∇a is not flat. In that scenario, Equation (Par-T) does not hold in the neighbourhood of p, and one may not integrate to construct a result of the form Int-T.10 Instead, one has only the more general Equation (3); integrating this expression and applying Gauss's theorem, one obtains: ∫Snμ1Tμ1…μndS=−∫VΔμ2…μndV. (7) In a new coordinate basis xμ′=fμ′(xμ), Equation (3) transforms to the same expression written in primed indices; integrating and applying Gauss's theorem, one obtains Equation (7) written in primed indices.11 Since the right-hand side of Equation (7) is coordinate-dependent (in the sense that it consists of non-tensorial geometrical objects), in general the value of the integral on the left-hand side of Equation (7) will differ when this expression is written in unprimed versus primed coordinates. From these results, one concludes the following. If the derivative operator ∇a is flat, then an equation of the form Equation (Par-T) holds in a class of frames related by affine transformations; given this, an equation of the form Int-T also holds in such frames.12 In a generic coordinate system at p∈M, however, such results do not hold (even when ∇a is flat)—one has, rather, Equation (5) and Equation (6). If the derivative operator in question is not flat, then in general equations of the form Equation (Par-T) or Equation (Int) do not hold; one has instead equations of the form Equation (3) and Equation (7). Since the right-hand side of Equation (7) is coordinate-dependent (in the sense of being composed of non-tensorial quantities), so too is the value of the integral on the left-hand side. 2.2.3 Conservation laws In physics, one often speaks both of differential and integral conservation laws—the former of form Equation (Par-T); the latter of form Int-T. To illustrate why these are referred to as ‘conservation laws’, substitute the second rank tensor (TΦ1μν+TΦ2μν) for Tμν1…νi—with TΦiμν here interpreted as the stress-energy tensor of the ith matter field (in the coordinate basis under consideration). In this case, the above two laws become ∂μ(TΦ1μν+TΦ2μν)=0 and ∫Snμ(TΦ1μν+TΦ2μν)dS=0, respectively. Trivially, moving the terms for Φ2 to the right-hand side, we obtain for Equation (Par-T) ∂μTΦ1μν=−∂μTΦ2μν, (8) and for Int-T ∫SnμTΦ1μνdS=−∫SnμTΦ2μνdS. (9)Equation (8) tells us that, at any point, any change in the stress-energy of the Φ1 field must be balanced by an equal and opposite change in the stress-energy of the Φ2 field; stress-energy is therefore conserved between Φ1 and Φ2 at that point. Analogously, Equation (9) tells us that any change in the stress-energy of Φ1 across a boundary of a spacetime region S⊂M must be balanced by an equal and opposite change in the stress-energy of Φ2 across that boundary; stress-energy is therefore conserved in the volume enclosed by S. Prima facie, both Equations (Par-T) and (Int-T) are legitimate conservation laws. In a sense though, the differential conservation law in Equation (Par-T) is stronger than the integral conservation law Int-T, as while the former implies the latter, the reverse is not true. Though we return to this point in Section 3.2, we emphasize at this stage that there is no reason to think that Equations (Par-T) and (Int-T) are not both legitimate conservation laws, albeit different in scope. 2.3 Conservation equations in special and general relativity 2.3.1 Special relativity The above in hand, consider the role of stress-energy conservation laws in the context of relativity theory. In special relativity (SR), the ‘differential conservation law’ for the stress-energy tensor of matter fields Tab reads ∇aTab=0, (\rm{ {Cov {\hbox{-}}T) where ∇a is the unique torsion-free derivative operator compatible with the fixed Minkowski metric field ηab of SR.13 Crucially, this derivative operator is flat—meaning that the discussion of Section 2.2 applies, and in SR we have a differential conservation law for the stress-energy tensor of matter fields in the form of Equation (Par-T) in the neighbourhood of a given p∈M, when Equation (Cov-T) is written in normal coordinates at p, for all frames related to such coordinates by affine transformations. By integrating, we obtain a well-defined ‘integral conservation law’ of the form Int-T—in the same class of frames in which the associated ‘differential conservation law’ holds. When the above conditions are not satisfied, Equation (Cov-T) instead takes the form of Equation (5). 2.3.2 General relativity In GR, the fixed metric field ηab of SR is replaced with a generic Lorentzian metric field gab, and the derivative operator ∇a is now the unique torsion-free such operator compatible with gab.14 In this case, ∇a can no longer be assumed to be flat; moreover, the operator is rendered dynamical, due to the coupling of curvature and matter degrees of freedom via Equation (1). Given this, in GR, Equation (Cov-T) yields ‘conservation laws’ of the form Equation (3) and Equation (7). 2.3.3 Discussion Though many textbooks claim that Equation (Cov-T)—subject to its GR interpretation—is the differential conservation law for matter stress-energy in GR (see, for example, Wald [1984]; Misner et al. [1973]; Schutz [2009]), foundational authors often claim to the contrary that this is not a legitimate conservation law (see, for example, Hoefer [2000]; Baker [2005]; Lam [2011]; Ohanian [unpublished]). To assess such claims, first consider Equation (3), to which Equation (Cov-T), subject to its GR interpretation, reduces in a coordinate basis. Since Equation (3) does not take the form of Equation (Par-T), it cannot prima facie be understood as a differential conservation law in the sense of Section 2.2.3 (Hoefer [2000]; Baker [2005]; Lam [2011]). In fact, a common interpretation of Equation (Cov-T), subject to its GR interpretation and written in a coordinate basis, is that locally no rest mass or momentum is created, except by the interaction between the gravitational field and matter fields. Indeed, for this reason Ohanian states ‘[Equation (3)] is really a non-conservation law!’ (Ohanian [unpublished], p. 3). While it is true that Equation (3) does not take the canonical form Par-T, there are two immediate problems with this interpretation: To claim that Δμ1…μn terms in equations such as Equation (3) represent interactions between the gravitational and matter fields is to overlook a debate in the history and philosophy of GR as to which mathematical object should be identified with the gravitational field. Specifically, while some argue that the metric field gab represents the gravitational field (Lehmkuhl [2008], Section 4.3), others (arguably including Einstein ([1950])) argue that the connection coefficients Γλνμ (in any given coordinate basis) play this role, and still others argue that the gravitational field is represented by the Riemann tensor Rbcda (Synge [1960], p. 8).15,16 Though ultimately one may be able to argue that the gravitational field is best represented by the connection coefficients Γλνμ (a point to which we return shortly), to simply assert this is to overlook an important debate in the foundations of GR. To claim that Δμ1…μn terms in Equation (3) denote interactions between the gravitational and matter fields is to conflate the Φ in a given KPM of GR with the associated stress-energy tensor Tab. These objects are not the same, so saying that such terms represent the interaction of matter with the gravitational field must be accompanied with a precise explication of the sense in which this is so. We return to the first of these points in the forthcoming sections. For now, it suffices to note that the fact that Equation (3) contains extra terms that result in it not taking the form Equation (Par-T) results in difficulties in interpreting this as a standard differential conservation law. While the problems with treating Equation (Cov-T) subject to its GR interpretation as a conservation law are widely discussed in the literature, the analogous case of Equation (Cov-T) subject to its special relativistic interpretation also merits consideration. In that case, in an arbitrary frame of reference in the neighbourhood of a point p∈M, the very same points as outlined above apply, for in such a case the differential and integral ‘conservation laws’ for the theory become, respectively, Equations (5) and (6). Thus, one can say that it is only in particular frames of reference—those related to a choice of normal coordinates at p by affine transformations—that one can construct uncontroversial differential and integral conservation laws (respectively Equation (Par-T) and Int-T) in SR. In the following section we introduce some further mathematical apparatus, which will enable us to shed light upon this result. 2.4 Killing vector fields and spacetime isometries A diffeomorphism φ:M→M is said to be a ‘symmetry transformation’ of a tensor field T just in case φ*T=T, where φ*T is the push-forward of T. A symmetry transformation of the metric field is called an ‘isometry’. Such isometries can be characterized by their generators—that is, their associated Killing vector fields—defined through Killing’s equation,17 ∇(aKb)=0. (10) To every such Killing vector field there corresponds an integral stress-energy conservation law. To see this, note that if a given ⟨M,gab⟩ possesses a Killing vector field Ka, then one can build the quantity TabKb with vanishing divergence: ∇a(TabKb)=∇aTabKb+Tab∇aKb=Tab∇(aKb)=0. (11) In the penultimate step here, we have used the symmetry of Tab; in the final step we have used Equation (10). To see how this leads to a well-defined integral conservation law even in the case in which ∇a is not flat, now recall the covariant divergence theorem, ∫Mdnx|g|∇aXa=∫∂Mdn−1x|h|naXa, (12) which holds whenever ∂M is time-like or space-like; Xa is a vector field on M; ∇a is the unique torsion-free derivative operator compatible with gab; g denotes the determinant of gab; and h denotes the determinant of the metric on ∂M induced by pulling back the metric on M.18 Using Equation (12), we may integrate Equation (11) to obtain ∫V∇aTabKbdV=∫S=∂VnaTabKbdS=0, (13) where na is the unit normal to S ( na=gabnb), and dS and dV are respectively surface and volume elements. This integral formulation unequivocally means (non-gravitational) stress-energy conservation, with respect to the integral curves of the Killing vector field Ka. Equation (13) shows that we can have (non-gravitational) stress-energy integral conservation laws in curved spacetime in cases where this latter instantiates certain global symmetries—namely, when spacetime structure remains stationary along the integral curves of a Killing vector field.19,20 How does this relate to the work undertaken above? To see the connection, first recall that a special relativistic ‘spacetime’ ⟨M,ηab⟩ has ten independent Killing vector fields; these correspond to the generators of Poincaré transformations.21 Thus, if one projects ∇aTab onto such a Killing vector Ka as in Equation (11) at every point in the neighbourhood of some p∈M, one projects onto the integral curves of a vector field that generates Poincaré transformations—and so along which the conservation laws Equation (Par-T) and Int-T hold (see Section 2.2). By using the covariant divergence theorem, one is then able to construct an integral conservation law Equation (13) that holds in any arbitrary frame. 2.5 The gravitational stress-energy pseudotensor We have seen in Section 2.2.1 that in SR, a conservation law of the form Equation (Par-T) holds in a class of frames related by Poincaré transformations (in fact, all affine transformations); this can be expressed in covariant language through contraction with the Killing vector fields Ka associated to the isometries of ⟨M,ηab⟩, as per Equation (13). In GR, by contrast, no such move is possible in general (see Section 2.2.2)—as can be seen through the fact that in general a ‘spacetime’ ⟨M,gab⟩ satisfying Equation (1) possesses no non-trivial Killing vector fields. This notwithstanding, however, one might seek to construct an alternative stress-energy conservation principle in GR. One way to reconcile one’s intuition that stress-energy must be conserved in GR with the observation that Equation (Cov-T)—subject to its GR interpretation—is not a conservation law of the form Equation (Par-T) is to argue that there instead exists a conservation law of the form Equation (Par-T) in models of the theory for material plus ‘gravitational’ stress-energy (represented in a given coordinate basis by tμν)22: ∂μTμν=∂μ(Tμν+tμν)=0. (\rm{ {Par {\hbox{-}}Tt) The idea is that, though generically Tμν alone is not conserved (in every frame, at some p∈M) in the sense of Par-T in GR, perhaps a quantity representing the stress-energy of the gravitational field exists such that the sum of the two is a conserved quantity (that is, such that Equation (Par-Tt) holds in every frame, at a given p∈M). The vanishing of ∂μTμν can be encoded in terms of an antisymmetric ‘superpotential’ Uμλν=Uμ[λν], by writing (see, for example, Trautmann [1962]; Lam [2011]): Tμν+tμν=∂λUμλν. (14) Bearing in mind Equations (1) and (14), we may choose to define tμν (in a given frame) via tμν:=∂λUμλν−18πGμν. (15) In fact, there is a freedom of choice of the superpotential, since Equation (Par-Tt) does not specify this object uniquely. This leads to distinct, non-equivalent expressions for tμν, including (among others) the so-called ‘Einstein pseudotensor’ and ‘Landau-Lifshitz pseudotensor’.23 For our purposes, it is crucial to note that tμν does not transform as a tensor under a coordinate change; it is for this reason that it is often referred to as the gravitational stress-energy ‘pseudotensor’. In particular, at any point p∈M there is a coordinate system in which tμν vanishes. It is worth emphasizing that what we are referring to as ‘the’ gravitational stress-energy pseudotensor is doubly ambiguous, in the following sense: There are many distinct but non-equivalent choices for this pseudotensor, based upon one’s choice of superpotential. Hence, when we refer to ‘the’ gravitational stress-energy pseudotensor, we are implicitly supposing that a choice has been made from the family of possible candidates. Once one such definition of this pseudotensor is chosen, the resulting object is still a frame-dependent (that is, non-tensorial) quantity. With these points regarding the gravitational stress-energy pseudotensor in hand, we consider in Section 3 possible interpretations of Equation (Par-Tt). Before doing so, however, one further observation is in order: the requirement that there be a conservation principle of the form Equation (Par-Tt) in GR that holds in every frame is prima facie a very strong condition—for typically one does not consider such a conservation principle to hold even in SR (see Section 2.3.1). That said, she who requires Equation (Par-Tt) to hold in every frame in GR could (in principle) define an analogous principle to hold in every frame in SR. This issue is discussed further in Section 3. 3 Interpreting Conservation Laws for Total Stress-Energy In this section, we assess whether putative conservation laws for total (that is, matter-plus-gravitational) stress-energy in GR such as Equation (Par-Tt) can indeed be regarded as legitimate conservation principles, and whether there is any physical significance to the notion of ‘gravitational stress-energy’ in models of GR. To this end, in Section 3.1 we consider possible interpretations of Equation (Par-Tt), finding that one’s verdict on whether this equation counts as a conservation law for total stress-energy in GR depends upon one’s view on whether non-tensorial objects such as tμν may represent physical quantities. In Section 3.2, we use Equation (Par-Tt) to construct an integral conservation law, initially drawing similar conclusions. However, we then show that in certain physical circumstances, this integral version of Equation (Par-Tt) yields a notion of gravitational stress-energy at least as robust as that in SR. In Section 3.3, we reflect on the correct attitude that one should take towards this quantity, and therefore on whether one should be a realist about gravitational stress-energy in GR in this sense. 3.1 The differential conservation law for total stress-energy Does Equation (Par-Tt) qualify as a conservation law for total stress-energy in GR? One’s answer to this question hinges upon one’s understanding of which mathematical objects in a theory should be taken to represent physical entities. In modern works on GR, something like the following is often asserted: ‘Since different coordinate representations are just different mathematical descriptions, relevant physical entities are usually taken to correspond to coordinate-independent entities’ (Lam [2011], p. 1018). On this understanding, the coordinate-dependence (that is, non-tensorial nature) of tμν shows that this object is unphysical, and there can be no local notion of gravitational stress-energy. Accordingly, on this view Equation (Par-Tt) cannot express a conservation principle relating physical quantities in GR. We dub this position ‘antirealism’ about non-tensorial objects, and in particular tμν. On the other hand, the historical Einstein did not endorse this position. Instead, Einstein maintained that the stress-energy pseudotensor could represent a physical quantity, writing ‘I do not see why only those quantities with the transformation properties of the components of a tensor should have physical meaning’ (Einstein [2002a], p. 167). On this second understanding, one views tμν as a physical but frame-dependent quantity; we dub this ‘realism’ about tμν. This is in accord with Einstein’s view that the presence of a gravitational field is intimately tied to the non-vanishing of connection coefficients (Einstein [1950], [1996]): since the connection coefficients are frame-dependent, it is prima facie plausible that gravitational stress-energy also be frame-dependent.24,25 For the realist, Equation (Par-Tt) is a legitimate but frame-dependent conservation principle.26 Let us reflect further on what follows if one is a realist about pseudotensorial quantities, such as tμν. There are several questions that deserve consideration here, for example: (i) If we assert that coordinate-dependent objects such as pseudotensors are candidates for representing real physical quantities, which coordinate system is the ‘right’ one for accurately so representing such quantities? (ii) Are coordinate systems supposed to be associated with observers in some way? (iii) If so, are pseudotensors relative quantities, like relative velocity? One plausible line of reasoning that may be advanced in response to these questions proceeds as follows. In every coordinate system, the gravitational stress-energy pseudotensor tμν will take some value (possibly zero). Just as Einstein understood the connection coefficients Γνσμ to represent the value of the gravitational field in a given frame of reference, so too may the realist about tμν understand this to represent the magnitude of gravitational stress-energy in a given frame of reference. Accordingly, on this view there is no ‘right’ frame for accurately representing the quantity of gravitational stress-energy. Rather, on this position gravitational stress-energy is always defined with respect to a given frame of reference—so the above-suggested analogy with relative velocity is indeed apt. 3.2 The integral conservation law for total stress-energy We return to whether one should be a realist about pseudotensorial quantities such as tμν in Section 3.3.3. In this section, however, we consider the possibility of the construction of integral conservation laws for total stress-energy in GR. Note first that Equation (Par-Tt) can be used to construct an integral conservation law describing the interchange of stress-energy between gravitational and matter stress-energy: ∂ν(Tμν+tμν)=0 ⇒∫VdV∂ν(Tμν+tμν)=∫S=∂VdS(Tμν+tμν)nν=0. (16) This tells us that any change in matter stress-energy (from Tμν) in a region must be balanced by an opposite change in gravitational stress-energy (from tμν); it thereby encodes total stress-energy exchange within a volume S⊂M. However, as Hoefer ([2000], p. 194) notes, there exist here conceptual difficulties: the pseudotensorial nature of tμν results in Equation (16) being ill defined and coordinate-dependent in general. The sense in which this is so is clear: moving the part of the integral in Equation (16) involving tμν to the right-hand side, we obtain ∫S=∂VTμνnνdS=−∫S=∂VtμνnνdS (\rm{ {Int {\hbox{-}}Tt) Since tμν is coordinate-dependent in general, the same is true of the right-hand side of Equation (Int-Tt); hence the left-hand side—that is, the surface integral of the matter stress-energy tensor—is also coordinate-dependent. In other words, although the integral of the sum of matter and gravitational stress-energy in Equation (16) evaluates to zero, the (equal) magnitudes of these quantities are in general not well defined, and hence this integral ‘conservation principle’ is still frame-dependent. This being said, it is important to note that there do exist physical circumstances in which one can obtain well-defined results for the left- and right-hand sides of Equation (Int-Tt). One set of sufficient conditions is the following (see Hoefer [2000], p. 194): Integrals must be taken in the limit r→∞. Asymptotic flatness of the spacetime is assumed: gab→ηab as r→∞. The coordinate system must be Lorentzian asymptotically, but can vary arbitrarily in the interior. As Nerlich ([2013], p. 159) states, these conditions ‘impose time translation symmetry in a cryptic form’. In effect, imposing condition (ii)—asymptotic flatness—allows one to treat the bulk spacetime and its content as a physical system on a Minkowski background; in this way one recovers the isometries of Minkowski space and their associated Killing vector fields, and thereby the associated conserved quantities when constructing integral conservation laws (in the r→∞ limit: condition (i)) with respect to the integral curves of these Killing vector fields (condition (iii)), à laEquation (13). Hence, by the work of Section 2.4, the amount of matter stress-energy in the volume V must be well-defined (with respect to the integral curves of these Killing vector fields). Then, by Equation (16), the amount of gravitational stress-energy in the volume V must also be well defined. In fact, this is easy to see mathematically, by contracting Equation (Int-Tt) with the components of a Killing vector field Ka in this coordinate basis ( Ka=gabKb): ∫S=∂VtμνKμnνdS=−∫S=∂VTμνKμnνdS=0. (\rm{ {Int {\hbox{-}}TtK) Here, we have rearranged Equation (Int-Tt) and used Equation (13) and the symmetry of the stress-energy tensor. Hence, in this case we find that just as matter stress-energy is conserved with respect to the integral curves of the Killing vector field Ka, the same is true of gravitational stress-energy. Thus the amounts of both matter and gravitational stress-energy in the volume are well defined, so the splitting in Equation (16) is well defined. This is why conditions (i)–(iii) yield a well-defined (that is, frame-independent) conservation principle for total stress-energy. When conditions (i)–(iii) hold, Equation (16) can be applied in order to calculate (for example) stress-energy loss by a system due to gravitational wave transportation; such calculations agree with observations on binary star/pulsar pairs (Hoefer [2000], p. 194). Hence, in such cases there appears to exist a well-defined quantity (with respect to a class of frames) that balances any change of matter stress-energy of the system, in exactly the same manner as in SR. Regardless of whether one is a realist or antirealist about tμν in the sense above, this new quantity is prima facie a candidate for a well-defined notion of gravitational stress-energy in GR.27 Finally, with the above in hand, we are in a position to understand why some authors (for example, Hoefer [2000]; Baker [2005]; Lam [2011]) have claimed that only integral conservations laws should be considered conservation principles ‘properly speaking’, contra Section 2.2.3. It is likely that such claims are made post hoc, in light of the frame-dependence of expressions such as Equation (Par-Tt). However, there exist at least two issues with this view. First, such a position is highly revisionary: it is more in line with physical practice to state that both differential and integral conservation laws are a priori legitimate; it is only if these contain non-tensorial quantities and one is an antirealist about these quantities that one can claim that such conservation laws are not genuine. Second, this position faces the obvious objection that some integral conservation laws such as Equation (16) are also generically ill defined.28 3.3 Gravitational stress-energy The logic of the previous two subsections was as follows: Though Equation (Par-Tt) has the form of a differential conservation law in Equation (Par-T), one of its relata is a frame-dependent quantity. Whether one views Equation (Par-Tt) as a legitimate conservation law (that is, as a conservation law relating physical entities) will therefore depend upon whether one thinks that frame-dependent mathematical objects such as tμν can represent physical quantities. Whatever one’s take on this though, it is also true that in some physical circumstances, an integral version of Equation (Par-Tt)—that is, Equation (Int-Tt)—appears to relate frame-independent physical quantities, one of which corresponds to a notion of gravitational stress-energy.29 The question to be pursued now is whether such a quantity does indeed represent gravitational stress-energy as a physical magnitude. In this subsection, we evaluate two positions in the literature on this point. According to the former (advocated by Hoefer [2000]) there exists no genuine gravitational stress-energy in GR in any sense. According to the latter (advocated by Lam [2011]), there does exist frame-independent gravitational stress-energy in GR, in the sense that the conservation law in Equation (Int-Tt) holds in some models of the theory. Though both Hoefer and Lam take an antirealist attitude towards tμν (in the sense of Section 2.5), the position of Lam is compatible with a realist understanding of tμν. 3.3.1 Against weak gravitational stress-energy? Based upon the results of the previous section, Hoefer argues that in GR: (a) the stress-energy of the gravitational field is ill defined both locally (that is, at a point) and globally (that is, in the sense of an integral conservation law); and (b) there is no general principle of total stress-energy conservation in GR. He claims: ‘we should abandon this effort to gloss over the facts. Let the textbooks admit openly that gravitational field stress-energy is not well-defined or fundamental, and that neither it nor ordinary stress-energy is conserved’ (Hoefer [2000], p. 195). Hoefer adopts an antirealist line regarding pseudo-tensorial quantities, and thereby rejects both Equation (Par-Tt) and Equation (16) as conservation principles. (While Hoefer also argues that equations such as Equation (Par-Tt) should be rejected as conservation principles on the grounds that they are not integral conservation laws, we have seen above that such reasoning is misguided.) With this in mind, the most interesting of Hoefer’s claims is the assertion that genuine gravitational stress-energy does not exist in GR even when a frame-independent quantity playing this functional role exists in the theory, as with Equation (Int-Tt). Hoefer argues that the stringent limitations on the applicability of Equation (Int-Tt) imposed by (i)–(iii) make this no genuine conservation principle in these circumstances either, for two reasons: The actual world is not asymptotically Minkowski, so ‘[Equation (Int-Tt)] does not apply to gravity in the actual world’ (Hoefer [2000], p. 194). Holding Equation (Int-Tt) as an important physical result ‘goes against the most important and philosophically progressive approach to spacetime physics: that of downplaying coordinate-dependent notions and effects, and stressing the intrinsic, covariant and coordinate-independent as what is important’ (Hoefer [2000], pp. 194–5). What should we make of these two claims? Beginning with (i), one might object to this on various grounds. First, the first statement of (i) is undeniable in the sense that the entire universe is not asymptotically Minkowski. Nevertheless, this does not preclude us from applying Equation (Int-Tt) when modelling certain physical systems in the actual world (for example, binary star systems). Hence, it appears that the second claim of (i) does not follow from the first, at least when physical systems within the world are considered in isolation.30 A second, related reason to object to (i) is the following: every theory of physics is an idealization and does not ‘apply to the actual world’ in this strong sense. So, Hoefer’s objection levied at Equation (Int-Tt) seems at the same time to apply to an unacceptably broad class of physical laws and theories. In addition, one might object to (i) on the grounds that our concern is not with the specific DPM of GR (see Section 2.1) that is taken to model the (cosmology of) the actual world, but rather with the entire space of DPMs of GR. If a frame-independent physical quantity corresponding to gravitational stress-energy can be defined in a certain subclass of those DPMs, then that is sufficient to conclude that a frame-independent notion of gravitational stress-energy does exist in GR. On this way of understanding the dialectic, consideration of the actual world is broadly irrelevant to the question of whether a frame-independent notion of gravitational stress-energy exists in GR. Once this point is recognized, one is also capable of responding to any objection to the statement that a frame-independent notion of gravitational stress-energy exists in GR on the grounds that the models of the theory in which this notion may be defined are ‘rare’, or ‘unstable’—in the sense that perturbing the model slightly yields a new model of GR in which such a notion of frame-independent gravitational stress-energy cannot be applied.31 The nature of this response is straightforward: such issues are (once again) irrelevant to the question of whether a frame-independent notion of gravitational stress-energy can be defined in GR simpliciter. On (ii), this claim is again objectionable. First, it is clear that the statement is a mixture of an appeal to a majority view (if indeed this is a majority view, as Hoefer asserts) and a re-assertion of the antirealist position presented in Section 3.1, with no concrete argument presented for this position. Indeed, even if Hoefer can argue for this antirealist position, it is not clear it applies in the case under consideration, that is, Equation (Int-Tt). This is because here we have a frame-independent notion of gravitational stress-energy, which seems to match the desiderata laid out in (ii) anyway! Perhaps Hoefer has in mind the following worry: although in such cases we appear to have a frame-independent notion of gravitational stress-energy, this is only after projecting onto Killing vector fields. In fact, as we saw in Section 3.2, stress-energy in GR (sans such projection) is only a well-defined quantity in a restricted class of frames. While this is true, commitment to the view that we do not have total stress-energy conservation law in GR due to the fact that the relevant conservation principles do not hold in every frame leads to potentially undesirable consequences. Most notably, such a claim would also commit one to the statement that there exists no genuine stress-energy conservation law in SR—a theory in which the conservation of total stress-energy typically is taken to be uncontroversial. While Hoefer is free to adopt such a position, he does not appear to do so (see, for example, Hoefer [2000], p. 189). A further worry regarding such general antirealism about gravitational stress-energy in GR is the following. As Baker ([2005], p. 1305) notes, the advocate of a Hoefer-type view is apparently committed to the denial of the claim that gravitational waves and other forms of purely gravitational radiation are energetic. For example, for the case of a binary star system where conditions (i)–(iii) hold and Equation (Int-Tt) is well defined, textbook accounts state that the matter stress-energy of the system decreases as some stress-energy is carried away in gravitational radiation (see, for example, Misner et al. [1973]; Hartle [2003]; Schutz [2009]). For Hoefer, such a story cannot be told. Instead, he will have to assert that the matter stress-energy of the system just decreases, and stress-energy is not conserved. Though it is likely that Hoefer will bite the bullet on this point, it is certainly a revisionary view. 3.3.2 Gravitational stress-energy relative to background structure The above in mind, it does not appear that strong reasons have been given to support the view that no genuine gravitational stress-energy exists in GR. Let us now lay out an alternative perspective on gravitational stress-energy, presented by Lam, who makes the weaker claim that ‘the very notion of energy—gravitational or not—is well defined in the theory only with respect to some background structure’ (Lam [2011], p. 1012). What is meant by ‘background structure’ here? Suppose that for a given ⟨M,gab⟩ there exists a Killing vector field Ka. Then there exists an isometry φ*gab=gab generated by Ka, where φ is a diffeomorphism along the integral curves of Ka. In this sense, the Killing vector field is associated with spacetime structure ‘stationary’ under φ, and can thereby be taken to indicate non-dynamical ‘background structure’ with respect to which integral stress-energy conservation can be demonstrated. By contrast, a fully dynamical metric will in general lack the above stationarity and so preclude the existence of integral conservation laws. With the above characterization of ‘background structure’ in hand, Lam’s position can be presented as follows: (a) conserved quantities such as stress-energy are only well defined in the presence of a Killing vector field (the ‘background structure’); (b) conditions (i)–(iii) provide a specific situation in which Killing vector fields can be constructed in GR, with associated integral conservation laws; (c) in such cases, the stress-energy (including gravitational stress-energy) associated with those Killing vector fields, constructed via the integral conservation law in Equation (Int-Tt), is a frame-independent quantity and therefore (Lam claims) genuine. In light of this, Lam ([2011], pp. 1022–3) concludes ‘in the cases in which total energy-momentum is well defined (and conserved), it is a global notion’. The advocate of this view will therefore maintain that there appear to be contexts in which sufficient ‘background structure’ exists that conditions such as (i)–(iii) hold for the system under consideration; here a frame-independent quantity representing gravitational stress-energy is well defined, and in such contexts (including real-world contexts, such as binary star systems) it does make sense to speak of the gravitational stress-energy of the system, and of Equation (16) as being a legitimate conservation law.32 Though we concur with Lam on this point, we diverge in our interpretation of pseudotensorial quantities such as tμν—see below. 3.3.3 Functionalism about gravitational stress-energy The above two positions in mind, we must ask two questions: (a) is it correct to call the quantity appearing in Equation (Int-Tt) associated with tμν ‘gravitational stress-energy’, and (b) does such ‘gravitational stress-energy’ really exist in GR?33 Begin with (a). As Lam notes, ‘energy and mass might not be fundamental properties of the world, in the sense that they make sense only in some particular (but very useful) settings; this does not lessen the fact that the notions of energy and mass constitute extremely powerful tools for many concrete and practical cases’ (Lam [2011], p. 1023).34 This point is important: gravitational stress-energy à laEquation (Int-Tt) is not a fundamental concept in GR, insofar as it is only applicable in a limited range of DPMs of the theory—namely, those in which certain Killing vector fields may be defined. (That is, the term ‘gravitational energy’ is associated with structures—namely terms such as that on the left-hand side of Equation (Int-Tt)—which are not used to construct the space of DPMs of GR, but rather which are only well defined in a certain subset of those DPMs.) Nevertheless, in such instances it is extremely useful to make use of this term, within that subclass of DPMs. Hence, at a practical level, it is legitimate to call such a quantity gravitational stress-energy. Clearly though, this does not settle the putative ontological issue (b), concerning whether gravitational stress-energy ‘really’ exists in GR. In our view, it is plausible to maintain that in situations such as those in which Equation (Int-Tt) holds, there exists a quantity in GR that fulfils the functional role of gravitational stress-energy. The reasons for this are the following. First, for a structure in a certain model of a theory to play the ‘functional role of gravitational stress-energy’, it must (i) fulfil a function analogous to that of gravitational energy as traditionally conceived—namely, as a quantity (gravitational potential energy) in Newtonian gravitation, which balances the matter (in Newtonian mechanics: kinetic) energy of the system in question such that their sum is conserved; and (ii) bear some relation to the ‘gravitational’ degrees of freedom in the theory in question. Second, arguably terms such as that on the left-hand side of Equation (Int-Tt) do satisfy (i) and (ii)—the former holds in virtue of a comparison of the form of Equation (Int-Tt) with Newtonian energy conservation equations; the latter in virtue of the connections between tμν and, for example, the connection components—themselves (at least, on views such as those indicated by Einstein mentioned above) understood to be associated with ‘gravitational’ degrees of freedom, as elaborated in Section 2.3.3 and 3.1.35 A functionalist may, therefore, speak of the existence of gravitational stress-energy in such situations.36 On the assumption that (i) and (ii) are satisfied, the alternative to functionalism is to say that ‘the structure of certain DPMs of GR is such that it appears that there exists gravitational stress-energy in those models, but really there is no such stress-energy there’; the payoff to be gained from making such a claim is unclear. Still, doubts may linger. In particular, one might argue as follows: ‘Surely there is a much more plausible alternative that disputes that gravitational energy “really” exists, which says that we can describe everything that is going on in terms of solutions to Equation (1) (and its consequences, including Equation (Cov-T)), without any need to help ourselves to talk involving tμν’.37 With this point, we are in broad agreement: one could indeed explain all general relativistic phenomena, in any model of the theory, simply using the apparatus used to pick out the DPMs of the theory. Nevertheless, we would respond that there may exist other structures only in certain models of the theory, which play certain functional roles. In our view, there is nothing illegitimate in regarding such structures as also existing in (the worlds represented by) those models of the theory—and, moreover, doing so may open up more perspicuous avenues for the explanation of phenomena within those models (recall from Section 3.3.2 the case of gravitational radiation from binary star systems). Thus, while we concur with the above argument, in our view this does not constitute an objection to a notion of functional gravitational energy in GR, since advocates of such a concept simply have more explanatory apparatus available to them. Thus, a functionalist may assert that gravitational stress-energy in the sense of Equation (Int-Tt) does exist in GR, but only in a restricted class of situations; this aligns with the case of SR. Since we have already found Hoefer’s objections to the existence of gravitational stress-energy in GR wanting, and since such functionalist principles are practical and simple, we conclude that in such situations (corresponding to certain models of GR) it is best to state that a quantity that plays the functional role of gravitational stress-energy does exist in the theory, and hence should be labelled as genuine. We conclude that frame-independent gravitational stress-energy does exist in GR, in the limited sense above. Whether one also maintains that a frame-dependent notion of gravitational stress-energy exists in GR will, for the reasons discussed, depend on whether one is a realist or antirealist about pseudotensorial quantities, in particular tμν. Note, however, that realism about such quantities is arguably compatible with the above functionalist principles: in each given frame of reference, one may define a quantity, represented by tμν, such that total stress-energy is conserved. Again, this plays the functional role of gravitational stress-energy in that frame, for (i) and (ii) as delineated above are both satisfied. Thus, it is also the case that speaking of frame-dependent gravitational stress-energy may be justified on functionalist grounds. On this latter point, there exist connections with other, ongoing debates in the foundations of spacetime theories—in particular, over the primacy of coordinate-dependent versus -independent explanations. According to advocates of the latter, such as Friedman ([1983]) and Maudlin ([2012]), explanations of physical phenomena within models of a given spacetime theory should proceed by appeal only to coordinate-independent structures. By contrast, according to advocates of the former, such as Brown ([2005]; Brown and Read [forthcoming]) and Wallace ([forthcoming]), presentations of spacetime theories need not proceed in a coordinate-independent manner; rather, spacetime theories may be defined in terms of equations written in a coordinate basis and their transformation properties (this is what Brown ([2005], p. 9) and Wallace ([forthcoming], p. 5) refer to as the ‘Kleinian conception of geometry’), and explanations may be given by appeal to those laws, written in a coordinate basis.38 For advocates of the coordinate-independent perspective in the context of, for example, SR, explanations of phenomena proceeding by appeal to coordinate-dependent effects (for example, the twin paradox differential in terms of the relativity of simultaneity, assuming some clock synchrony convention) are to be rejected, as the associated physical effects not considered ‘real’; for advocates of the coordinate-dependent perspective, there is nothing wrong with issuing such explanations, and with viewing such effects as physical. Those who buy into the latter programme may view the notion of gravitational energy in question as frame-dependent, but no less real for all that. 4 Gravitational Stress-Energy and Spacetime Ontology 4.1 Relationism and gravitational stress-energy In the above, we endorsed a position according to which gravitational stress-energy does exist in GR, in at least (i) a weak sense applicable in a certain class of models of the theory, and arguably also (ii) a strong sense, applicable in all models of the theory. Nevertheless, there exists a residual worry, related to the debate between advocates of substantivalism versus relationism about spacetime ontology. In this article, we take substantivalists to claim that spacetime exists as an entity in its own right, and relationists to deny this—that is, to claim that all talk of spacetime is reducible to talk of (relational properties of) matter fields. This in mind, the worry regarding gravitational stress-energy runs as follows: if one is a relationist, how can one maintain that there indeed exists gravitational stress-energy in the world? The thought that relationism implies the nonexistence of gravitational stress-energy (or, equivalently, that genuine gravitational stress-energy implies substantivalism) is intuitive. In this section, however, we argue that it is not correct. There are two different pictures of the substantivalism/relationism debate that are relevant here. Let ⟨M,gab,Φ⟩ be a model of GR. Then, according to ‘manifold substantivalism’, spacetime is identified with M; to be a relationist is to maintain that manifold points do not have an ontological status independent of the fields defined upon them. On the other hand, according to ‘metric substantivalism’, a spacetime is identified with ⟨M,gab⟩—that is, with both the manifold and the metric field. To be a relationist is then to maintain that neither manifold points nor the metric field have a distinct existence over and above the matter fields Φ. Clearly, it is harder to be a relationist in the second sense than the first.39 For our purposes, it is more relevant to consider relationism in the second sense above, since we are concerned with the ontological status of quantities associated with gab. In this case, our question becomes: how can those who do not believe that the metric field is fundamental (insofar as they think it reducible to properties of matter fields) maintain the existence of genuine gravitational stress-energy? On the face of it, such a claim is implausible, and indeed many (for example, Hoefer [2000]; Baker [2005]) maintain that the answer is a simple negative: they cannot. We answer to the contrary. Our positive story runs as follows. Whether one is a relationist or a substantivalist, it is a fact that there are some situations in which results such as Equation (Int-Tt) are well defined, and there appears to be a well-defined quantity in the theory that plays the functional role of frame-independent gravitational stress-energy. Of course, the account given by the relationist of this quantity will differ from that given by the substantivalist: the relationist will assert that this quantity is associated (in some way to be made precise) with the metric field, which is in turn a codification of properties of the fields; the substantivalist will appeal to the metric field simpliciter. In either case though, this quantity exists in the theory: the two sides are not debating its existence, but rather the fields to which it is ultimately attributable. 4.2 The cosmological constant Finally, it is worth commenting on Baker’s claims regarding the bearing of the possibility of a non-zero cosmological constant Λ on both the substantivalism/relationism debate, and the existence of gravitational stress-energy (Baker [2005], Section 4). Including a cosmological constant term, Equation (1) reads: Gab+Λgab=8πTab (17) We focus on the following two claims made by Baker: Λ≠0 commits us to (manifold-plus-metric) substantivalism, because this quantity is associated with spacetime yet cannot be reduced to relations amongst the matter fields (Baker [2005], Section 4.1). Λ≠0 results in a non-zero vacuum energy density, for which a relationist cannot account (Baker [2005], Section 4.2). (a) questions whether relationism is compatible with a non-zero cosmological constant. (b) assumes that such is the case, but claims that relationism nevertheless cannot account for the vacuum energy density arising from non-zero Λ. Let us first consider (b). In fact, there are several reasons to be suspicious of this claim. First, Baker ([2005], p. 1301) justifies that Λ is ‘associated with spacetime’ as follows: [Λ’s] role in the field equations [Equation (17)] is to influence, by itself or in combination with other terms, the metric structure of spacetime, and thereby to affect the physical behaviour of matter. This is exactly the sort of influence that accounts for gravitational forces in GR, the only difference being that Λ does not depend on matter as its source. This claim is suspect, because Baker has not ruled out the possibility that the Λ term in Equation (17) can appear on the right-hand side of this equation, with Λ being treated as another matter field; such an interpretation of Equation (17) is also prima facie possible, yet the above assertion does nothing to rule it out. Therefore, Baker needs to do more to make convincing any claim that Λ is ‘associated with spacetime’. Indeed, there is an ongoing debate in cosmology over whether to consider the cosmological constant term in Equation (17) as being attributable to spacetime (that is, as sitting on the left-hand side), or as another type of matter-like field (that is, as sitting on the right-hand side); Baker cannot simply presuppose an answer to this question. Second, the claim that Λ cannot be reduced to relations among the matter fields is also too quick. On this, Baker ([2005], p. 1306) states: I do not doubt that a persistent relationist could describe Λ’s effects as mere relational properties, but the price will be high. Considering [the] example of distant objects moving apart under the influence of Λ, the relationist would have to posit a brute fact that material objects possess a tendency to accelerate away from one another at a rate proportional only to the distance between them. In fact, this seems to be roughly in line with the relationist-type approach to the metric field in relativity theory in the context of SR outlined in (Brown [2005]). More generally, Baker cannot infer from the fact that such a programme appears to be undesirable to him that it cannot be done (and indeed, he openly admits that it can be done), or even that such a task might not be desirable or acceptable in some relationist research programmes. On (b), Baker ([2005], p. 1310) claims that a relationist cannot account for the vacuum energy density ρΛ=Λ/8πG implied by a non-zero Λ: ‘I can see no easy way for the relationist to explain the energy density of empty space’.40 In fact though, this is not so: the relationist can account for the energy density of empty spacetime. To see this, suppose that the cosmological constant can be reduced to properties of matter fields. Then, as with gravitational stress-energy in the previous subsection, the equations of the theory will still state that there exists a quantity that plays the functional role of a vacuum energy density. Once again, the only difference will be the story that is told to account for this quantity. While the substantivalist will appeal directly to gab and Λ, the relationist will resort to an elliptic story about how this quantity arises from properties amongst the matter fields themselves. But on either account, a functional vacuum energy density exists according to the theory. 5 Conclusion In this article, we have reconsidered the existence of gravitational stress-energy in GR; adopting a functionalist attitude to the definition of physical quantities, we have argued that gravitational stress-energy can be considered to exist in GR, in both (i) a weak sense applicable in a certain class of models of the theory (namely, models that instantiate certain symmetries, and therefore that possess Killing vector fields), and (ii) arguably also in a stronger sense (as represented by the gravitational stress-energy pseudotensor in a given frame), applicable in all models of the theory. This latter approach runs against contemporary orthodoxy, but is in line with the thinking of the historical Einstein ([2002a], p. 167). In addition, we have adopted a revisionary line regarding whether gravitational stress-energy is compatible with relationism, arguing that regardless of whether one thinks that the metric field gab is reducible to properties of matter fields, gravitational stress-energy still exists in the theory, if one again embraces functionalism about physical quantities. Accordingly, one’s position on the ontology of spacetime does not affect one’s commitment to gravitational stress-energy in GR; this point also applies to the claim that a non-zero vacuum energy density is incompatible with relationism. Acknowledgements First and foremost, I am grateful to Patrick Dürr for many rewarding discussions on gravitational energy. Many thanks also to Harvey Brown, Eleanor Knox, Dennis Lehmkuhl, Tushar Menon, Brian Pitts, and Simon Saunders for valuable comments on earlier drafts of this article. I am supported by an Arts and Humanities Research Council scholarship at the University of Oxford, and am also indebted to Hertford College, Oxford for a graduate senior scholarship. Footnotes 1 Throughout, abstract (that is, coordinate-independent) indices are written in Latin script; indices in a coordinate basis are written in Greek script; and the Einstein summation convention is used. Round brackets around indices denote symmetrization over those indices; square brackets around indices denote antisymmetrization. We set GN=c=1. 2 One should avoid, at this stage, asserting M to be the ‘spacetime manifold’, for to do so is to conflate the mathematical model under consideration with the possible world to which that model is ultimately interpreted as corresponding. Indeed, in light of the debate over the hole argument (Earman and Norton [1987]), it is not necessarily correct to interpret M as representing substantial spacetime—see Section 4.1. 3 Until Section 4, we restrict to the sector of GR with vanishing cosmological constant Λ. For Λ≠0, Equation (1) reads Gab+Λgab=8πTab. 4 Strictly, independence of these equations from Equation (1) depends on the case—see (Brown [2005], Section 9.3; Misner et al. [1973], Section 20.6). 5 In a coordinate basis {eμ}, the connection components are defined through ∇ρeν=Γμνρeμ. 6 Since in a coordinate basis, the (unique) Riemann tensor Rbcda associated to ∇a—defined through Rbcdaξb=−2∇[c∇d]ξa for all smooth fields ξa (Malament [2012], p. 68)—reads Rμνρσ=∂ρΓνσμ−∂σΓνρμ+ΓνστΓτρμ−ΓνρτΓτσμ. 7 Consider an affine coordinate transformation xμ′=Mμ′μxμ+aμ′. If an (r, s) tensor Fμ1…μrν1…νs transforms under this coordinate change as Fμ1…μrν1…νs→Mμ1μ1′…Mμrμr′Mν1′ν1…Mνs′νsFμ1′…μr′ν1′…νs′, then we say that it is ‘covariant’ with this coordinate transformation. If a dynamical equation retains the same form in either of the two coordinate systems under consideration, then we say that it is ‘invariant’ under the coordinate change. 8 The notation xμ′=fμ′(xμ) signifies that in this case xμ′ may be defined in terms of arbitrary contractions with xμ, provided that there is one free primed index. 9 To take an explicit example (relevant to our discussions of stress-energy below), consider the expression ∂μTμν=0. Transforming to a new coordinate basis xμ→xμ′=fμ′(xμ), one obtains ∂μ′Tμ′ν′+∂xμ′∂xμ∂2xμ∂xμ′xλ′Tλ′ν′+∂xν′∂xν∂2xν∂xλ′∂xσ′Tλ′σ′=:∂μ′Tμ′ν′+Θν′=0. 10 Rather, Equation (Par-T) holds only at p. 11 This result is straightforward: Equation (3) follows from Equation (2), which involves only tensorial quantities. 12 Defined through specifying normal coordinates at p∈M. 13 As discussed by Pooley ([2017], p. 115), and unlike the case of the metric field gab of GR, one may understand ηab as being fixed identically in all KPMs of SR. Otherwise, KPMs of SR—as with GR—are again denoted by triples, this time of the form ⟨M,ηab,Φ⟩. 14 Performing these two replacements in any dynamical equations of SR, to obtain general relativistic dynamical equations, is sometimes dubbed ‘minimal coupling’. For philosophical discussion, see (Brown and Read [2016]; Read et al. [forthcoming]). 15 To claim that Einstein identified the gravitational field with the connection coefficients may be to oversimplify his position. In fact, Einstein saw GR as unifying gravity and inertia, in the same way that SR had unified electricity and magnetism. If one takes this view, then perhaps one need never speak of the gravitational field in GR. See (Lehmkuhl [2014]; Brown and Read [2016]). 16 For further discussion of these matters, see (Lehmkuhl [2008]). 17 The ‘Lie derivative’ LXT of a tensor field T represents how that tensor field changes as one acts with a diffeomorphism along the integral curves of a vector field Xa; the condition φ*T=T imposes that LXT=0. Since the Lie derivative of the metric field reads (LXg)ab=2∇(aXb) (assuming that the unique metric-compatible, torsion free derivative operator ∇a is used), this condition yields ∇(aXb)=0, which is Killing’s equation. See (Wald [1984], pp. 437–44). Note also that Ka=gabKb. 18 See Wald ([1984], pp. 433–4). 19 This does not preclude the possibility of other unspecified ways of obtaining genuine (non-gravitational) stress-energy conservation in curved spacetime: it specifies sufficient but not necessary conditions. 20 The covariant divergence theorem can only be applied to vector fields, but not to higher rank tensor fields: this is why we obtained in Equation (13) an integral conservation law from ∇a(TabKb)=0, but we cannot obtain an analogous integral conservation law from ∇aTab=0. 21 Consider a Killing vector field Ka associated to ⟨M,gab⟩, which by definition satisfies Equation (10) (the derivative operator ∇a in this equation is the unique torsion-free such operator compatible with gab, as usual). From this, one can derive straightforwardly that Ka must also satisfy ∇a∇bKc=RbadcKd. Restricting to the special relativistic case ⟨M,ηab⟩, one has Rbadc=0, in which case ∇a∇bKc=0. Then, restricting to normal coordinates at some p∈M, one has that ∂μ∂νKρ=0—and integrating this equation, one finds that Kμ=aμνxν+bμ where bμ is a constant one-form, and the coefficients of aμν are also constant. Therefore, the components Kμ are linear functions of the inertial frame coordinates. Now, substituting this result into Equation (10) reveals that aμν must be antisymmetric—that is, has six independent components. Since bμ has four independent components, in total we find that there are ten independent Killing vector fields associated to the ‘spacetime’ ⟨M,ηab⟩; these are the isometries of the Minkowski metric ηab. Since the Poincaré group is the group of coordinate transformations that leave the Minkowski metric invariant, we see that this group must have ten independent generators, each corresponding to a Killing vector field. 22 Three notes on tμν are in order: (i) This term and the notation used to denote it were originally introduced by Einstein (Einstein [1996]; Einstein and Grossmann [1996]). Einstein went on to state in 1918 that ‘nearly all my colleagues raise objections to my definition of the momentum-energy theorem’ (Einstein [2002b], p. 448); here he had in mind particularly Levi-Civita ([1917]), Schrödinger ([1918]), and Bauer ([1918]), with whom he had corresponded heavily on this topic in the preceding four years. (ii) Some, such as Hoefer ([2000], p. 193) and Nerlich ([2013], p. 162), have taken tμν to be defined implicitly through Equation (3). Unfortunately, this does not work out smoothly, as there is more than one candidate for tμν (as discussed below), so Equation (3) is not sufficient to fix tμν uniquely. (iii) For simplicity in this article we use the symbol tμν to refer both to the object representing gravitational stress-energy, and to its components in a given coordinate basis. 23 There exist interesting questions regarding which of these non-equivalent versions of the gravitational stress-energy pseudotensor best describes gravitational stress-energy. See (Trautmann [1962], pp. 190–1). 24 Though see Footnote 15. 25 This position is also advanced at (Lehmkuhl [2008], p.94); see in addition (Bergmann [1976], p. 197; Lehmkuhl [2014], Section 5). 26 For a recent attempt to make sense of Equation (Par-Tt) as a legitimate conservation principle, see (Pitts [2010]). In a sense, Pitts’ view is a halfway house between realism and antirealism: though he argues that pseudotensors are ‘physically meaningful’ (Pitts [2010], p. 15) with ‘no vicious coordinate dependence’ (Pitts [2010], p. 14), he does this by demonstrating that they can be unified into an infinite-component geometric object. For further discussion of Pitts’ position, see (Lam [2011], Section 5; Dürr [unpublished]). 27 Localizability is a stronger condition than satisfying an integral conservation principle, because it is possible to have the latter in the absence of the former, as is the case in Equation (16) when satisfying (i)–(iii). 28 This may be evaded straightforwardly if one claims that being an integral conservation law is a necessary but not sufficient condition to be a conservation principle ‘properly speaking’. 29 Recall from Section 2.4 that, while conserved quantities such as energy strictly only exist in frames related by the appropriate coordinate transformations, by projecting onto the Killing vector fields associated with such transformations, we can obtain results that hold in any frame. It is this that we mean by a ‘frame-independent’ notion of gravitational stress-energy—even though strictly such stress-energy (sans projection) is conserved only in a restricted class of frames. 30 In other words, the notion of gravitational stress-energy may still be applicable to (subsystems of) the actual world at an approximate, functional level—see Section 3.3.3 below. 31 For example, in a model of GR in which conditions (i)–(iii) are satisfied, it suffices to introduce a small perturbation in the metric field such that it is not asymptotically Minkowski at infinity for the above notion of frame-independent gravitational stress-energy to no longer be applicable. 32 At least at an approximate level—see Footnote 30. 33 Note that (b) is not the same as asking whether gravitational stress-energy really exists in the (possibly general relativistic) actual world, for the reasons delineated in Section 3.3.1. 34 Here, Lam is referring to both matter and gravitational (stress-)energy. Note also that Lam says ‘properties of the world’, rather than ‘properties in GR’. For our purposes, it is legitimate (and preferable) to read him as making the latter, more general statement (see Footnote 33). 35 Though one should recall the caveats of Section 2.3.3 regarding the question of which object in GR should be associated with the ‘gravitational field’, strictly speaking. 36 Such a line accords with general functionalist attitudes in science. Wallace ([2012], p. 58) summarizes this as follows: ‘Science is interested with interesting structural properties of physical systems, and does not hesitate at all in studying those properties just because they are instantiated “in the wrong way”’. 37 We are grateful to an anonymous referee for putting the point in this way. 38 We set aside here the question of the extent to which this ‘Kleinian conception’ is faithful to the definition of geometry in Klein’s ‘Erlangen’ programme (Klein [1892]). 39 Two points are in order here. First, work such as (Lehmkuhl [2011]) demonstrates that the stress-energy tensor Tab of the matter fields Φ of GR in fact presupposes metric structure in its definition. It is perhaps then misguided to attempt to reduce gab to Tab. That said, one should of course recall that one should not conflate matter fields Φ with their associated stress-energy tensor Tab. Thus, even if it is misguided to attempt to reduce gab to Tab, perhaps it is still possible to reduce gab to Φ. Our second point is related: suppose one does seek to reduce gab to Φ. To do so is to endorse a version of ‘Mach’s principle’ (Lehmkuhl [2014], p. 455). There exist problems with attempting to achieve this in GR—for example, one faces the problem that a priori the metric field has ten independent components, whereas the electromagnetic field tensor (for example) has only six—so there is a question of how the former can be reduced to the latter. Another problem in this vicinity lies in the existence of vacuum solutions in GR. As a result, it is questionable whether the form of relationism considered here is ultimately defensible in GR—so the work of this section is best viewed as (a) a response to (Hoefer [2000]; Baker [2005]), where it is claimed that such views are incompatible with the existence of gravitational energy; and (b) an exploration of the consequences of the functionalism about gravitational stress-energy developed above. We are grateful to an anonymous referee for pushing us on this point. 40 As Baker ([2005], p. 1309) states, this does not mean that Λ arises from the non-zero ρΛ; rather, its role in the field equations is equivalent to an energy density of empty space. References Baker D. [ 2005 ]: ‘Spacetime Substantivalism and Einstein’s Cosmological Constant’ , Philosophy of Science , 72 , pp. 1299 – 311 . Google Scholar CrossRef Search ADS Bauer H. [ 1918 ]: ‘Über die Energiekomponenten des Gravitationsfeldes’ , Physikalische Zeitschrift , 19 , pp. 163 – 6 . Bergmann P. G. [ 1976 ]: Introduction to the Theory of Relativity , New York : Dover . Brown H. R. [ 2005 ]: Physical Relativity: Spacetime Structure from a Dynamical Perspective , Oxford : Oxford University Press . Google Scholar CrossRef Search ADS Brown H. R. , Read J. [ 2016 ]: ‘Clarifying Possible Misconceptions in the Foundations of General Relativity’ , American Journal of Physics , 84 , pp. 327 – 34 . Google Scholar CrossRef Search ADS Brown H. R. , Read J. [forthcoming]: ‘The Dynamical Approach to Spacetime Theories’, in E. Knox and A. Wilson (eds), The Routledge Companion to Philosophy of Physics, London: Routledge. Dürr P. [unpublished]: ‘Fantastic Beasts and Where (Not) to Find Them, Part I: Local Gravitational Energy and Energy Conservation in General Relativity’. Earman J. , Norton J. [ 1987 ]: ‘What Price Spacetime Substantivalism? The Hole Story’ , British Journal for the Philosophy of Science , 38 , pp. 515 – 25 . Google Scholar CrossRef Search ADS Einstein A. [ 1950 ]: Letter to Max von Laue, 12 September 1950, Einstein Archive, Boston, MA (EA 16-148). Einstein A. [ 1996 ]: ‘Die formale Grundlage der allgemeinen Relativitätstheorie’, in Klein M. J. , Kox A. J. , Schulman R. (eds), The Collected Papers of Albert Einstein , Volume 6, Princeton, NJ: Princeton University Press, pp. 72ff . Einstein A. [ 2002a ]: ‘Über Gravitationswellen’, in Janssen M. , Schulmann R. , Illy J. , Lehner C. , Buchwald D. K. (eds), The Collected Papers of Albert Einstein , Volume 7, Princeton, NJ: Princeton University Press, pp. 11ff . Einstein A. [ 2002b ]: ‘Der Energiesatz in der allgemeinen Relativitätstheorie and Nachtrag zur Korrektur’, in Janssen M. , Schulmann R. , Illy J. , Lehner C. , Buchwald D. K. (eds), The Collected Papers of Albert Einstein , Volume 7, Princeton, NJ: Princeton University Press, pp. 63ff . Einstein A. , Grossmann M. [ 1996 ]: ‘Kovarianzeigenschaften der Feldgleichungen der auf die verallgemeinerte Relativitätstheorie geggründeten Gravitationstheorie’, in Klein M. J. , Kox A. J. , Schulman R. (eds), The Collected Papers of Albert Einstein , Volume 6, Princeton, NJ: Princeton University Press, pp. 6ff . Friedman M. [ 1983 ]: Foundations of Space-Time Theories , Princeton, NJ : Princeton University Press . Hartle J. [ 2003 ]: Gravity: An Introduction to Einstein’s General Relativity , San Francisco, CA : Addison Wesley . Hoefer C. [ 2000 ]: ‘Energy Conservation in GTR’ , Studies in the History and Philosophy of Modern Physics , 31 , pp. 187 – 99 . Google Scholar CrossRef Search ADS Klein F. [ 1892 ]: ‘A Comparative Review of Recent Researches in Geometry’ , Bulletin of the New York Mathematical Society , 2 , pp. 215 – 49 . Google Scholar CrossRef Search ADS Lam V. [ 2011 ]: ‘Gravitational and Nongravitational Energy: The Need for Background Structures’ , Philosophy of Science , 78 , pp. 1012 – 24 . Google Scholar CrossRef Search ADS Lehmkuhl D. [ 2008 ]: ‘Is Spacetime a Gravitational Field?’, in Dieks D. (ed.), The Ontology of Spacetime II , Amsterdam : Elsevier , pp. 83 – 110 . Google Scholar CrossRef Search ADS Lehmkuhl D. [ 2011 ]: ‘Mass–Energy–Momentum: Only There Because of Spacetime?’ , British Journal for the Philosophy of Science , 62 , pp. 453 – 88 . Google Scholar CrossRef Search ADS Lehmkuhl D. [ 2014 ]: ‘Why Einstein Did Not Believe That General Relativity Geometrizes Gravity’ , Studies in the History and Philosophy of Modern Physics , 46 , pp. 316 – 26 . Google Scholar CrossRef Search ADS Levi-Civita T. [ 1917 ]: ‘Sulla espressione analitica spettante al tensore gravitazionale nella teoria di Einstein’ , Rendiconti Accademia dei Lincei , 26 , pp. 381 – 91 . Malament D. [ 2012 ]: Topics in the Foundations of General Relativity and Newtonian Gravitation Theory , Chicago, IL : University of Chicago Press . Google Scholar CrossRef Search ADS Maudlin T. [ 2012 ]: Philosophy of Physics: Space and Time , Princeton, NJ : Princeton University Press . Misner C. , Thorne T. , Wheeler J. [ 1973 ]: Gravitation , San Francisco, CA : Freeman . Nerlich G. [ 2013 ]: Einstein’s Genie: Spacetime out of the Bottle , Montreal : Minkowski Institute Press . Ohanian H. [unpublished]: ‘The Energy–Momentum Tensor in General Relativity and in Alternative Theories of Gravitation, and the Gravitational vs. Inertial Mass’, available at <arxiv.org/abs/1010.5557>. Pitts J. B. [ 2010 ]: ‘Gauge-Invariant Localization of Infinitely Many Gravitational Energies from All Possible Auxiliary Structures’ , General Relativity and Gravitation , 42 , pp. 601 – 22 . Google Scholar CrossRef Search ADS Pooley O. [ 2017 ]: ‘Background Independence, Diffeomorphism Invariance, and the Meaning of Coordinates’, in Lehmkuhl D. , Schiemann G. , Scholz E. (eds), Towards a Theory of Spacetime Theories , New York : Birkhäuser , pp. 105 – 44 . Google Scholar CrossRef Search ADS Read J. , Brown H. R. , Lehmkuhl D. [forthcoming]: ‘Two Miracles of Relativity’, Studies in History and Philosophy of Modern Physics. Schrödinger E. [ 1918 ]: ‘Die Energiekomponenten des Gravitationsfeldes’ , Physikalische Zeitschrift , 19 , pp. 4 – 7 . Schutz B. [ 2009 ]: A First Course in General Relativity , Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Synge J. [ 1960 ]: Relativity: The General Theory , Amsterdam : North-Holland . Trautmann A. [ 1962 ]: ‘Conservation Laws in General Relativity’, in Witten L. (ed.), Gravitation: An Introduction to Current Research , New York : John Wiley and Sons , pp. 169 – 98 . Wald R. [ 1984 ]: General Relativity , Chicago, IL : University of Chicago Press . Google Scholar CrossRef Search ADS Wallace D. [ 2012 ]: The Emergent Multiverse: Quantum Theory According to the Everett Interpretation , Oxford : Oxford University Press . Google Scholar CrossRef Search ADS Wallace D. [forthcoming]: ‘Who’s Afraid of Coordinate Systems? An Essay on Representation of Spacetime Structure’, Studies in the History and Philosophy of Modern Physics, available at <philsci-archive.pitt.edu/11988/>. © The Author(s) 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The British Journal for the Philosophy of Science Oxford University Press

Functional Gravitational Energy

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Abstract

Abstract Does the gravitational field described in general relativity possess genuine stress-energy? We answer this question in the affirmative, in (i) a weak sense applicable in a certain class of models of the theory, and (ii) arguably also in a strong sense, applicable in all models of the theory. In addition, we argue that one can be a realist about gravitational stress-energy in general relativity even if one is a relationist about spacetime ontology. In each case, our reasoning rests upon a functionalist approach to the definition of physical quantities. 1 Introduction 2 Dramatis Personae   2.1 General relativity   2.2 Differential and integral conservation laws   2.3 Conservation equations in special and general relativity   2.4 Killing vector fields and spacetime isometries   2.5 The gravitational stress-energy pseudotensor 3 Interpreting Conservation Laws for Total Stress-Energy   3.1 The differential conservation law for total stress-energy   3.2 The integral conservation law for total stress-energy   3.3 Gravitational stress-energy 4 Gravitational Stress-Energy and Spacetime Ontology   4.1 Relationism and gravitational stress-energy   4.2 The cosmological constant 5 Conclusion 1 Introduction In the study of general relativity (GR), a range of claims have been advanced regarding the nature of stress-energy in the theory, and its status as a conserved quantity. For example, the intuition that stress-energy must be conserved in GR sometimes leads to the expression ∇aTab=0 being treated as the conservation law for the stress-energy of matter fields (see, for example, Misner et al. [1973], p. 152); yet, on reflection, one may question whether this is a genuine conservation law at all.1 On the other hand, it is sometimes claimed that gravitational stress-energy exists in GR, but is non-local (Hartle [2003]; Misner et al. [1973]; Schutz [2009]). In this article, we argue that gravitational stress-energy does exist in the theory, in both (i) a weak sense applicable in a certain class of models of the theory, and (ii) arguably also in a strong sense, applicable in all models of the theory. In each case, our reasoning rests upon a functionalist approach to the definition of physical quantities. In Section 2, we provide an interpretationally neutral presentation of the various stress-energy ‘conservation laws’ in GR. In Section 3, we argue for the existence of the above weak notion of gravitational stress-energy in certain models of GR, and also raise the possibility of a strong notion of gravitational stress-energy in the theory. In Section 4, we consider ways in which the related debate between substantivalism and relationism on the ontology of spacetime connects with the debate on gravitational stress-energy in GR. 2 Dramatis Personae In this section, we introduce all mathematical concepts essential to foundational discussions of gravitational stress-energy in GR. In Section 2.1, we introduce the kinematically and dynamically possible models (DPMs) of GR; we then discuss in Section 2.2 differential and integral conservation laws in physical theories; before in Section 2.3 presenting specific stress-energy conservation laws that arise in special and general relativity. In Section 2.4, we introduce the mathematical apparatus of isometries and Killing vector fields, before finally discussing in Section 2.5 a conservation law for total (that is, matter-plus-gravitational) stress-energy in GR. 2.1 General relativity Kinematically possible models (KPMs) of GR are picked out by triples ⟨M,gab,Φ⟩, where M is a four-dimensional differentiable manifold; gab is a Lorentzian metric field on M; and Φ is a placeholder for matter fields in the theory.2 Associated to ⟨M,gab⟩ there exists a unique derivative operator ∇a, which is: (i) torsion free—in the sense that the associated torsion tensor Tbca, defined through TbcaXbYc=Xb∇bYa−Yb∇bXa−[X,Y]a, vanishes—and (ii) metric compatible—in the sense that ∇agbc=0. Since this derivative operator ∇a is uniquely determined by ⟨M,gab⟩, it is not included as an independent variable in the KPMs of GR. DPMs of GR are those KPMs the geometrical objects of which satisfy the Einstein field equations Gab=8πTab (1) —the dynamical equations of the theory, which relate gab to the stress-energy tensor Tab of the Φ—in addition to the dynamical equations of the Φ.3,4 2.2 Differential and integral conservation laws 2.2.1 Case 1: Flat connection For a given derivative operator ∇a and tensor field Ta1…an, suppose: ∇a1Ta1…an=0. (2) In a coordinate basis, Equation (2) reads: ∂μ1Tμ1…μn+Γσμ1μ1Tσμ2…μn+…+Γσμ1μnTμ1…μn−1σ=:∂μ1Tμ1…μn+Δμ2…μn=0, (3) where the Γνσμ are the connection components associated to the derivative operator ∇a in this coordinate basis, and we have introduced the notation Δμ2…μn to represent the terms featuring connection components in the above expression.5 In GR, at any p∈M we can find Riemann normal coordinates such that connection components vanish, and Equation (3) becomes: ∂μ1Tμ1…μn(p)=0. (4) This result holds only at p. If we wish Equation (4) to obtain at some other q∈M in the neighbourhood of p, we require that derivatives of connection components vanish—and a fortiori that the derivative operator ∇a be flat.6 In this case, we may write: ∂μ1Tμ1…μn=0. (\rm{ {Par {\hbox{-}}T) Given the local validity of Equation (Par-T), we may integrate over a volume V⊂M and apply Gauss's theorem, to obtain (na is a unit normal vector to S=∂V; na=gabnb): ∫S=∂Vnμ1Tμ1…μndS=0. (\rm{ {Int {\hbox{-}}T)  Note that Equation (Par-T) is invariant only under affine transformations—that is, only retains the same form in coordinate systems related by such transformations.7 This means that an equation of the form Int-T also holds only in such coordinate systems. If one applies a non-affine transformation xμ→xμ′=fμ′(xμ) to Equation (Par-T), one will generically pick up extra terms (which we may schematically represent by Θμ2…μn), so that this equation reads instead8,9: ∂μ′1Tμ′1…μ′n+Θμ′2…μ′n=0. (5) Integrating over the volume V as before, we have by analogy with Int-T: ∫S=∂Vnμ′1Tμ′1…μ′ndS=−∫VΘμ′2…μ′ndV. (6) 2.2.2 Case 2: Non-flat connection Consider now the case in which ∇a is not flat. In that scenario, Equation (Par-T) does not hold in the neighbourhood of p, and one may not integrate to construct a result of the form Int-T.10 Instead, one has only the more general Equation (3); integrating this expression and applying Gauss's theorem, one obtains: ∫Snμ1Tμ1…μndS=−∫VΔμ2…μndV. (7) In a new coordinate basis xμ′=fμ′(xμ), Equation (3) transforms to the same expression written in primed indices; integrating and applying Gauss's theorem, one obtains Equation (7) written in primed indices.11 Since the right-hand side of Equation (7) is coordinate-dependent (in the sense that it consists of non-tensorial geometrical objects), in general the value of the integral on the left-hand side of Equation (7) will differ when this expression is written in unprimed versus primed coordinates. From these results, one concludes the following. If the derivative operator ∇a is flat, then an equation of the form Equation (Par-T) holds in a class of frames related by affine transformations; given this, an equation of the form Int-T also holds in such frames.12 In a generic coordinate system at p∈M, however, such results do not hold (even when ∇a is flat)—one has, rather, Equation (5) and Equation (6). If the derivative operator in question is not flat, then in general equations of the form Equation (Par-T) or Equation (Int) do not hold; one has instead equations of the form Equation (3) and Equation (7). Since the right-hand side of Equation (7) is coordinate-dependent (in the sense of being composed of non-tensorial quantities), so too is the value of the integral on the left-hand side. 2.2.3 Conservation laws In physics, one often speaks both of differential and integral conservation laws—the former of form Equation (Par-T); the latter of form Int-T. To illustrate why these are referred to as ‘conservation laws’, substitute the second rank tensor (TΦ1μν+TΦ2μν) for Tμν1…νi—with TΦiμν here interpreted as the stress-energy tensor of the ith matter field (in the coordinate basis under consideration). In this case, the above two laws become ∂μ(TΦ1μν+TΦ2μν)=0 and ∫Snμ(TΦ1μν+TΦ2μν)dS=0, respectively. Trivially, moving the terms for Φ2 to the right-hand side, we obtain for Equation (Par-T) ∂μTΦ1μν=−∂μTΦ2μν, (8) and for Int-T ∫SnμTΦ1μνdS=−∫SnμTΦ2μνdS. (9)Equation (8) tells us that, at any point, any change in the stress-energy of the Φ1 field must be balanced by an equal and opposite change in the stress-energy of the Φ2 field; stress-energy is therefore conserved between Φ1 and Φ2 at that point. Analogously, Equation (9) tells us that any change in the stress-energy of Φ1 across a boundary of a spacetime region S⊂M must be balanced by an equal and opposite change in the stress-energy of Φ2 across that boundary; stress-energy is therefore conserved in the volume enclosed by S. Prima facie, both Equations (Par-T) and (Int-T) are legitimate conservation laws. In a sense though, the differential conservation law in Equation (Par-T) is stronger than the integral conservation law Int-T, as while the former implies the latter, the reverse is not true. Though we return to this point in Section 3.2, we emphasize at this stage that there is no reason to think that Equations (Par-T) and (Int-T) are not both legitimate conservation laws, albeit different in scope. 2.3 Conservation equations in special and general relativity 2.3.1 Special relativity The above in hand, consider the role of stress-energy conservation laws in the context of relativity theory. In special relativity (SR), the ‘differential conservation law’ for the stress-energy tensor of matter fields Tab reads ∇aTab=0, (\rm{ {Cov {\hbox{-}}T) where ∇a is the unique torsion-free derivative operator compatible with the fixed Minkowski metric field ηab of SR.13 Crucially, this derivative operator is flat—meaning that the discussion of Section 2.2 applies, and in SR we have a differential conservation law for the stress-energy tensor of matter fields in the form of Equation (Par-T) in the neighbourhood of a given p∈M, when Equation (Cov-T) is written in normal coordinates at p, for all frames related to such coordinates by affine transformations. By integrating, we obtain a well-defined ‘integral conservation law’ of the form Int-T—in the same class of frames in which the associated ‘differential conservation law’ holds. When the above conditions are not satisfied, Equation (Cov-T) instead takes the form of Equation (5). 2.3.2 General relativity In GR, the fixed metric field ηab of SR is replaced with a generic Lorentzian metric field gab, and the derivative operator ∇a is now the unique torsion-free such operator compatible with gab.14 In this case, ∇a can no longer be assumed to be flat; moreover, the operator is rendered dynamical, due to the coupling of curvature and matter degrees of freedom via Equation (1). Given this, in GR, Equation (Cov-T) yields ‘conservation laws’ of the form Equation (3) and Equation (7). 2.3.3 Discussion Though many textbooks claim that Equation (Cov-T)—subject to its GR interpretation—is the differential conservation law for matter stress-energy in GR (see, for example, Wald [1984]; Misner et al. [1973]; Schutz [2009]), foundational authors often claim to the contrary that this is not a legitimate conservation law (see, for example, Hoefer [2000]; Baker [2005]; Lam [2011]; Ohanian [unpublished]). To assess such claims, first consider Equation (3), to which Equation (Cov-T), subject to its GR interpretation, reduces in a coordinate basis. Since Equation (3) does not take the form of Equation (Par-T), it cannot prima facie be understood as a differential conservation law in the sense of Section 2.2.3 (Hoefer [2000]; Baker [2005]; Lam [2011]). In fact, a common interpretation of Equation (Cov-T), subject to its GR interpretation and written in a coordinate basis, is that locally no rest mass or momentum is created, except by the interaction between the gravitational field and matter fields. Indeed, for this reason Ohanian states ‘[Equation (3)] is really a non-conservation law!’ (Ohanian [unpublished], p. 3). While it is true that Equation (3) does not take the canonical form Par-T, there are two immediate problems with this interpretation: To claim that Δμ1…μn terms in equations such as Equation (3) represent interactions between the gravitational and matter fields is to overlook a debate in the history and philosophy of GR as to which mathematical object should be identified with the gravitational field. Specifically, while some argue that the metric field gab represents the gravitational field (Lehmkuhl [2008], Section 4.3), others (arguably including Einstein ([1950])) argue that the connection coefficients Γλνμ (in any given coordinate basis) play this role, and still others argue that the gravitational field is represented by the Riemann tensor Rbcda (Synge [1960], p. 8).15,16 Though ultimately one may be able to argue that the gravitational field is best represented by the connection coefficients Γλνμ (a point to which we return shortly), to simply assert this is to overlook an important debate in the foundations of GR. To claim that Δμ1…μn terms in Equation (3) denote interactions between the gravitational and matter fields is to conflate the Φ in a given KPM of GR with the associated stress-energy tensor Tab. These objects are not the same, so saying that such terms represent the interaction of matter with the gravitational field must be accompanied with a precise explication of the sense in which this is so. We return to the first of these points in the forthcoming sections. For now, it suffices to note that the fact that Equation (3) contains extra terms that result in it not taking the form Equation (Par-T) results in difficulties in interpreting this as a standard differential conservation law. While the problems with treating Equation (Cov-T) subject to its GR interpretation as a conservation law are widely discussed in the literature, the analogous case of Equation (Cov-T) subject to its special relativistic interpretation also merits consideration. In that case, in an arbitrary frame of reference in the neighbourhood of a point p∈M, the very same points as outlined above apply, for in such a case the differential and integral ‘conservation laws’ for the theory become, respectively, Equations (5) and (6). Thus, one can say that it is only in particular frames of reference—those related to a choice of normal coordinates at p by affine transformations—that one can construct uncontroversial differential and integral conservation laws (respectively Equation (Par-T) and Int-T) in SR. In the following section we introduce some further mathematical apparatus, which will enable us to shed light upon this result. 2.4 Killing vector fields and spacetime isometries A diffeomorphism φ:M→M is said to be a ‘symmetry transformation’ of a tensor field T just in case φ*T=T, where φ*T is the push-forward of T. A symmetry transformation of the metric field is called an ‘isometry’. Such isometries can be characterized by their generators—that is, their associated Killing vector fields—defined through Killing’s equation,17 ∇(aKb)=0. (10) To every such Killing vector field there corresponds an integral stress-energy conservation law. To see this, note that if a given ⟨M,gab⟩ possesses a Killing vector field Ka, then one can build the quantity TabKb with vanishing divergence: ∇a(TabKb)=∇aTabKb+Tab∇aKb=Tab∇(aKb)=0. (11) In the penultimate step here, we have used the symmetry of Tab; in the final step we have used Equation (10). To see how this leads to a well-defined integral conservation law even in the case in which ∇a is not flat, now recall the covariant divergence theorem, ∫Mdnx|g|∇aXa=∫∂Mdn−1x|h|naXa, (12) which holds whenever ∂M is time-like or space-like; Xa is a vector field on M; ∇a is the unique torsion-free derivative operator compatible with gab; g denotes the determinant of gab; and h denotes the determinant of the metric on ∂M induced by pulling back the metric on M.18 Using Equation (12), we may integrate Equation (11) to obtain ∫V∇aTabKbdV=∫S=∂VnaTabKbdS=0, (13) where na is the unit normal to S ( na=gabnb), and dS and dV are respectively surface and volume elements. This integral formulation unequivocally means (non-gravitational) stress-energy conservation, with respect to the integral curves of the Killing vector field Ka. Equation (13) shows that we can have (non-gravitational) stress-energy integral conservation laws in curved spacetime in cases where this latter instantiates certain global symmetries—namely, when spacetime structure remains stationary along the integral curves of a Killing vector field.19,20 How does this relate to the work undertaken above? To see the connection, first recall that a special relativistic ‘spacetime’ ⟨M,ηab⟩ has ten independent Killing vector fields; these correspond to the generators of Poincaré transformations.21 Thus, if one projects ∇aTab onto such a Killing vector Ka as in Equation (11) at every point in the neighbourhood of some p∈M, one projects onto the integral curves of a vector field that generates Poincaré transformations—and so along which the conservation laws Equation (Par-T) and Int-T hold (see Section 2.2). By using the covariant divergence theorem, one is then able to construct an integral conservation law Equation (13) that holds in any arbitrary frame. 2.5 The gravitational stress-energy pseudotensor We have seen in Section 2.2.1 that in SR, a conservation law of the form Equation (Par-T) holds in a class of frames related by Poincaré transformations (in fact, all affine transformations); this can be expressed in covariant language through contraction with the Killing vector fields Ka associated to the isometries of ⟨M,ηab⟩, as per Equation (13). In GR, by contrast, no such move is possible in general (see Section 2.2.2)—as can be seen through the fact that in general a ‘spacetime’ ⟨M,gab⟩ satisfying Equation (1) possesses no non-trivial Killing vector fields. This notwithstanding, however, one might seek to construct an alternative stress-energy conservation principle in GR. One way to reconcile one’s intuition that stress-energy must be conserved in GR with the observation that Equation (Cov-T)—subject to its GR interpretation—is not a conservation law of the form Equation (Par-T) is to argue that there instead exists a conservation law of the form Equation (Par-T) in models of the theory for material plus ‘gravitational’ stress-energy (represented in a given coordinate basis by tμν)22: ∂μTμν=∂μ(Tμν+tμν)=0. (\rm{ {Par {\hbox{-}}Tt) The idea is that, though generically Tμν alone is not conserved (in every frame, at some p∈M) in the sense of Par-T in GR, perhaps a quantity representing the stress-energy of the gravitational field exists such that the sum of the two is a conserved quantity (that is, such that Equation (Par-Tt) holds in every frame, at a given p∈M). The vanishing of ∂μTμν can be encoded in terms of an antisymmetric ‘superpotential’ Uμλν=Uμ[λν], by writing (see, for example, Trautmann [1962]; Lam [2011]): Tμν+tμν=∂λUμλν. (14) Bearing in mind Equations (1) and (14), we may choose to define tμν (in a given frame) via tμν:=∂λUμλν−18πGμν. (15) In fact, there is a freedom of choice of the superpotential, since Equation (Par-Tt) does not specify this object uniquely. This leads to distinct, non-equivalent expressions for tμν, including (among others) the so-called ‘Einstein pseudotensor’ and ‘Landau-Lifshitz pseudotensor’.23 For our purposes, it is crucial to note that tμν does not transform as a tensor under a coordinate change; it is for this reason that it is often referred to as the gravitational stress-energy ‘pseudotensor’. In particular, at any point p∈M there is a coordinate system in which tμν vanishes. It is worth emphasizing that what we are referring to as ‘the’ gravitational stress-energy pseudotensor is doubly ambiguous, in the following sense: There are many distinct but non-equivalent choices for this pseudotensor, based upon one’s choice of superpotential. Hence, when we refer to ‘the’ gravitational stress-energy pseudotensor, we are implicitly supposing that a choice has been made from the family of possible candidates. Once one such definition of this pseudotensor is chosen, the resulting object is still a frame-dependent (that is, non-tensorial) quantity. With these points regarding the gravitational stress-energy pseudotensor in hand, we consider in Section 3 possible interpretations of Equation (Par-Tt). Before doing so, however, one further observation is in order: the requirement that there be a conservation principle of the form Equation (Par-Tt) in GR that holds in every frame is prima facie a very strong condition—for typically one does not consider such a conservation principle to hold even in SR (see Section 2.3.1). That said, she who requires Equation (Par-Tt) to hold in every frame in GR could (in principle) define an analogous principle to hold in every frame in SR. This issue is discussed further in Section 3. 3 Interpreting Conservation Laws for Total Stress-Energy In this section, we assess whether putative conservation laws for total (that is, matter-plus-gravitational) stress-energy in GR such as Equation (Par-Tt) can indeed be regarded as legitimate conservation principles, and whether there is any physical significance to the notion of ‘gravitational stress-energy’ in models of GR. To this end, in Section 3.1 we consider possible interpretations of Equation (Par-Tt), finding that one’s verdict on whether this equation counts as a conservation law for total stress-energy in GR depends upon one’s view on whether non-tensorial objects such as tμν may represent physical quantities. In Section 3.2, we use Equation (Par-Tt) to construct an integral conservation law, initially drawing similar conclusions. However, we then show that in certain physical circumstances, this integral version of Equation (Par-Tt) yields a notion of gravitational stress-energy at least as robust as that in SR. In Section 3.3, we reflect on the correct attitude that one should take towards this quantity, and therefore on whether one should be a realist about gravitational stress-energy in GR in this sense. 3.1 The differential conservation law for total stress-energy Does Equation (Par-Tt) qualify as a conservation law for total stress-energy in GR? One’s answer to this question hinges upon one’s understanding of which mathematical objects in a theory should be taken to represent physical entities. In modern works on GR, something like the following is often asserted: ‘Since different coordinate representations are just different mathematical descriptions, relevant physical entities are usually taken to correspond to coordinate-independent entities’ (Lam [2011], p. 1018). On this understanding, the coordinate-dependence (that is, non-tensorial nature) of tμν shows that this object is unphysical, and there can be no local notion of gravitational stress-energy. Accordingly, on this view Equation (Par-Tt) cannot express a conservation principle relating physical quantities in GR. We dub this position ‘antirealism’ about non-tensorial objects, and in particular tμν. On the other hand, the historical Einstein did not endorse this position. Instead, Einstein maintained that the stress-energy pseudotensor could represent a physical quantity, writing ‘I do not see why only those quantities with the transformation properties of the components of a tensor should have physical meaning’ (Einstein [2002a], p. 167). On this second understanding, one views tμν as a physical but frame-dependent quantity; we dub this ‘realism’ about tμν. This is in accord with Einstein’s view that the presence of a gravitational field is intimately tied to the non-vanishing of connection coefficients (Einstein [1950], [1996]): since the connection coefficients are frame-dependent, it is prima facie plausible that gravitational stress-energy also be frame-dependent.24,25 For the realist, Equation (Par-Tt) is a legitimate but frame-dependent conservation principle.26 Let us reflect further on what follows if one is a realist about pseudotensorial quantities, such as tμν. There are several questions that deserve consideration here, for example: (i) If we assert that coordinate-dependent objects such as pseudotensors are candidates for representing real physical quantities, which coordinate system is the ‘right’ one for accurately so representing such quantities? (ii) Are coordinate systems supposed to be associated with observers in some way? (iii) If so, are pseudotensors relative quantities, like relative velocity? One plausible line of reasoning that may be advanced in response to these questions proceeds as follows. In every coordinate system, the gravitational stress-energy pseudotensor tμν will take some value (possibly zero). Just as Einstein understood the connection coefficients Γνσμ to represent the value of the gravitational field in a given frame of reference, so too may the realist about tμν understand this to represent the magnitude of gravitational stress-energy in a given frame of reference. Accordingly, on this view there is no ‘right’ frame for accurately representing the quantity of gravitational stress-energy. Rather, on this position gravitational stress-energy is always defined with respect to a given frame of reference—so the above-suggested analogy with relative velocity is indeed apt. 3.2 The integral conservation law for total stress-energy We return to whether one should be a realist about pseudotensorial quantities such as tμν in Section 3.3.3. In this section, however, we consider the possibility of the construction of integral conservation laws for total stress-energy in GR. Note first that Equation (Par-Tt) can be used to construct an integral conservation law describing the interchange of stress-energy between gravitational and matter stress-energy: ∂ν(Tμν+tμν)=0 ⇒∫VdV∂ν(Tμν+tμν)=∫S=∂VdS(Tμν+tμν)nν=0. (16) This tells us that any change in matter stress-energy (from Tμν) in a region must be balanced by an opposite change in gravitational stress-energy (from tμν); it thereby encodes total stress-energy exchange within a volume S⊂M. However, as Hoefer ([2000], p. 194) notes, there exist here conceptual difficulties: the pseudotensorial nature of tμν results in Equation (16) being ill defined and coordinate-dependent in general. The sense in which this is so is clear: moving the part of the integral in Equation (16) involving tμν to the right-hand side, we obtain ∫S=∂VTμνnνdS=−∫S=∂VtμνnνdS (\rm{ {Int {\hbox{-}}Tt) Since tμν is coordinate-dependent in general, the same is true of the right-hand side of Equation (Int-Tt); hence the left-hand side—that is, the surface integral of the matter stress-energy tensor—is also coordinate-dependent. In other words, although the integral of the sum of matter and gravitational stress-energy in Equation (16) evaluates to zero, the (equal) magnitudes of these quantities are in general not well defined, and hence this integral ‘conservation principle’ is still frame-dependent. This being said, it is important to note that there do exist physical circumstances in which one can obtain well-defined results for the left- and right-hand sides of Equation (Int-Tt). One set of sufficient conditions is the following (see Hoefer [2000], p. 194): Integrals must be taken in the limit r→∞. Asymptotic flatness of the spacetime is assumed: gab→ηab as r→∞. The coordinate system must be Lorentzian asymptotically, but can vary arbitrarily in the interior. As Nerlich ([2013], p. 159) states, these conditions ‘impose time translation symmetry in a cryptic form’. In effect, imposing condition (ii)—asymptotic flatness—allows one to treat the bulk spacetime and its content as a physical system on a Minkowski background; in this way one recovers the isometries of Minkowski space and their associated Killing vector fields, and thereby the associated conserved quantities when constructing integral conservation laws (in the r→∞ limit: condition (i)) with respect to the integral curves of these Killing vector fields (condition (iii)), à laEquation (13). Hence, by the work of Section 2.4, the amount of matter stress-energy in the volume V must be well-defined (with respect to the integral curves of these Killing vector fields). Then, by Equation (16), the amount of gravitational stress-energy in the volume V must also be well defined. In fact, this is easy to see mathematically, by contracting Equation (Int-Tt) with the components of a Killing vector field Ka in this coordinate basis ( Ka=gabKb): ∫S=∂VtμνKμnνdS=−∫S=∂VTμνKμnνdS=0. (\rm{ {Int {\hbox{-}}TtK) Here, we have rearranged Equation (Int-Tt) and used Equation (13) and the symmetry of the stress-energy tensor. Hence, in this case we find that just as matter stress-energy is conserved with respect to the integral curves of the Killing vector field Ka, the same is true of gravitational stress-energy. Thus the amounts of both matter and gravitational stress-energy in the volume are well defined, so the splitting in Equation (16) is well defined. This is why conditions (i)–(iii) yield a well-defined (that is, frame-independent) conservation principle for total stress-energy. When conditions (i)–(iii) hold, Equation (16) can be applied in order to calculate (for example) stress-energy loss by a system due to gravitational wave transportation; such calculations agree with observations on binary star/pulsar pairs (Hoefer [2000], p. 194). Hence, in such cases there appears to exist a well-defined quantity (with respect to a class of frames) that balances any change of matter stress-energy of the system, in exactly the same manner as in SR. Regardless of whether one is a realist or antirealist about tμν in the sense above, this new quantity is prima facie a candidate for a well-defined notion of gravitational stress-energy in GR.27 Finally, with the above in hand, we are in a position to understand why some authors (for example, Hoefer [2000]; Baker [2005]; Lam [2011]) have claimed that only integral conservations laws should be considered conservation principles ‘properly speaking’, contra Section 2.2.3. It is likely that such claims are made post hoc, in light of the frame-dependence of expressions such as Equation (Par-Tt). However, there exist at least two issues with this view. First, such a position is highly revisionary: it is more in line with physical practice to state that both differential and integral conservation laws are a priori legitimate; it is only if these contain non-tensorial quantities and one is an antirealist about these quantities that one can claim that such conservation laws are not genuine. Second, this position faces the obvious objection that some integral conservation laws such as Equation (16) are also generically ill defined.28 3.3 Gravitational stress-energy The logic of the previous two subsections was as follows: Though Equation (Par-Tt) has the form of a differential conservation law in Equation (Par-T), one of its relata is a frame-dependent quantity. Whether one views Equation (Par-Tt) as a legitimate conservation law (that is, as a conservation law relating physical entities) will therefore depend upon whether one thinks that frame-dependent mathematical objects such as tμν can represent physical quantities. Whatever one’s take on this though, it is also true that in some physical circumstances, an integral version of Equation (Par-Tt)—that is, Equation (Int-Tt)—appears to relate frame-independent physical quantities, one of which corresponds to a notion of gravitational stress-energy.29 The question to be pursued now is whether such a quantity does indeed represent gravitational stress-energy as a physical magnitude. In this subsection, we evaluate two positions in the literature on this point. According to the former (advocated by Hoefer [2000]) there exists no genuine gravitational stress-energy in GR in any sense. According to the latter (advocated by Lam [2011]), there does exist frame-independent gravitational stress-energy in GR, in the sense that the conservation law in Equation (Int-Tt) holds in some models of the theory. Though both Hoefer and Lam take an antirealist attitude towards tμν (in the sense of Section 2.5), the position of Lam is compatible with a realist understanding of tμν. 3.3.1 Against weak gravitational stress-energy? Based upon the results of the previous section, Hoefer argues that in GR: (a) the stress-energy of the gravitational field is ill defined both locally (that is, at a point) and globally (that is, in the sense of an integral conservation law); and (b) there is no general principle of total stress-energy conservation in GR. He claims: ‘we should abandon this effort to gloss over the facts. Let the textbooks admit openly that gravitational field stress-energy is not well-defined or fundamental, and that neither it nor ordinary stress-energy is conserved’ (Hoefer [2000], p. 195). Hoefer adopts an antirealist line regarding pseudo-tensorial quantities, and thereby rejects both Equation (Par-Tt) and Equation (16) as conservation principles. (While Hoefer also argues that equations such as Equation (Par-Tt) should be rejected as conservation principles on the grounds that they are not integral conservation laws, we have seen above that such reasoning is misguided.) With this in mind, the most interesting of Hoefer’s claims is the assertion that genuine gravitational stress-energy does not exist in GR even when a frame-independent quantity playing this functional role exists in the theory, as with Equation (Int-Tt). Hoefer argues that the stringent limitations on the applicability of Equation (Int-Tt) imposed by (i)–(iii) make this no genuine conservation principle in these circumstances either, for two reasons: The actual world is not asymptotically Minkowski, so ‘[Equation (Int-Tt)] does not apply to gravity in the actual world’ (Hoefer [2000], p. 194). Holding Equation (Int-Tt) as an important physical result ‘goes against the most important and philosophically progressive approach to spacetime physics: that of downplaying coordinate-dependent notions and effects, and stressing the intrinsic, covariant and coordinate-independent as what is important’ (Hoefer [2000], pp. 194–5). What should we make of these two claims? Beginning with (i), one might object to this on various grounds. First, the first statement of (i) is undeniable in the sense that the entire universe is not asymptotically Minkowski. Nevertheless, this does not preclude us from applying Equation (Int-Tt) when modelling certain physical systems in the actual world (for example, binary star systems). Hence, it appears that the second claim of (i) does not follow from the first, at least when physical systems within the world are considered in isolation.30 A second, related reason to object to (i) is the following: every theory of physics is an idealization and does not ‘apply to the actual world’ in this strong sense. So, Hoefer’s objection levied at Equation (Int-Tt) seems at the same time to apply to an unacceptably broad class of physical laws and theories. In addition, one might object to (i) on the grounds that our concern is not with the specific DPM of GR (see Section 2.1) that is taken to model the (cosmology of) the actual world, but rather with the entire space of DPMs of GR. If a frame-independent physical quantity corresponding to gravitational stress-energy can be defined in a certain subclass of those DPMs, then that is sufficient to conclude that a frame-independent notion of gravitational stress-energy does exist in GR. On this way of understanding the dialectic, consideration of the actual world is broadly irrelevant to the question of whether a frame-independent notion of gravitational stress-energy exists in GR. Once this point is recognized, one is also capable of responding to any objection to the statement that a frame-independent notion of gravitational stress-energy exists in GR on the grounds that the models of the theory in which this notion may be defined are ‘rare’, or ‘unstable’—in the sense that perturbing the model slightly yields a new model of GR in which such a notion of frame-independent gravitational stress-energy cannot be applied.31 The nature of this response is straightforward: such issues are (once again) irrelevant to the question of whether a frame-independent notion of gravitational stress-energy can be defined in GR simpliciter. On (ii), this claim is again objectionable. First, it is clear that the statement is a mixture of an appeal to a majority view (if indeed this is a majority view, as Hoefer asserts) and a re-assertion of the antirealist position presented in Section 3.1, with no concrete argument presented for this position. Indeed, even if Hoefer can argue for this antirealist position, it is not clear it applies in the case under consideration, that is, Equation (Int-Tt). This is because here we have a frame-independent notion of gravitational stress-energy, which seems to match the desiderata laid out in (ii) anyway! Perhaps Hoefer has in mind the following worry: although in such cases we appear to have a frame-independent notion of gravitational stress-energy, this is only after projecting onto Killing vector fields. In fact, as we saw in Section 3.2, stress-energy in GR (sans such projection) is only a well-defined quantity in a restricted class of frames. While this is true, commitment to the view that we do not have total stress-energy conservation law in GR due to the fact that the relevant conservation principles do not hold in every frame leads to potentially undesirable consequences. Most notably, such a claim would also commit one to the statement that there exists no genuine stress-energy conservation law in SR—a theory in which the conservation of total stress-energy typically is taken to be uncontroversial. While Hoefer is free to adopt such a position, he does not appear to do so (see, for example, Hoefer [2000], p. 189). A further worry regarding such general antirealism about gravitational stress-energy in GR is the following. As Baker ([2005], p. 1305) notes, the advocate of a Hoefer-type view is apparently committed to the denial of the claim that gravitational waves and other forms of purely gravitational radiation are energetic. For example, for the case of a binary star system where conditions (i)–(iii) hold and Equation (Int-Tt) is well defined, textbook accounts state that the matter stress-energy of the system decreases as some stress-energy is carried away in gravitational radiation (see, for example, Misner et al. [1973]; Hartle [2003]; Schutz [2009]). For Hoefer, such a story cannot be told. Instead, he will have to assert that the matter stress-energy of the system just decreases, and stress-energy is not conserved. Though it is likely that Hoefer will bite the bullet on this point, it is certainly a revisionary view. 3.3.2 Gravitational stress-energy relative to background structure The above in mind, it does not appear that strong reasons have been given to support the view that no genuine gravitational stress-energy exists in GR. Let us now lay out an alternative perspective on gravitational stress-energy, presented by Lam, who makes the weaker claim that ‘the very notion of energy—gravitational or not—is well defined in the theory only with respect to some background structure’ (Lam [2011], p. 1012). What is meant by ‘background structure’ here? Suppose that for a given ⟨M,gab⟩ there exists a Killing vector field Ka. Then there exists an isometry φ*gab=gab generated by Ka, where φ is a diffeomorphism along the integral curves of Ka. In this sense, the Killing vector field is associated with spacetime structure ‘stationary’ under φ, and can thereby be taken to indicate non-dynamical ‘background structure’ with respect to which integral stress-energy conservation can be demonstrated. By contrast, a fully dynamical metric will in general lack the above stationarity and so preclude the existence of integral conservation laws. With the above characterization of ‘background structure’ in hand, Lam’s position can be presented as follows: (a) conserved quantities such as stress-energy are only well defined in the presence of a Killing vector field (the ‘background structure’); (b) conditions (i)–(iii) provide a specific situation in which Killing vector fields can be constructed in GR, with associated integral conservation laws; (c) in such cases, the stress-energy (including gravitational stress-energy) associated with those Killing vector fields, constructed via the integral conservation law in Equation (Int-Tt), is a frame-independent quantity and therefore (Lam claims) genuine. In light of this, Lam ([2011], pp. 1022–3) concludes ‘in the cases in which total energy-momentum is well defined (and conserved), it is a global notion’. The advocate of this view will therefore maintain that there appear to be contexts in which sufficient ‘background structure’ exists that conditions such as (i)–(iii) hold for the system under consideration; here a frame-independent quantity representing gravitational stress-energy is well defined, and in such contexts (including real-world contexts, such as binary star systems) it does make sense to speak of the gravitational stress-energy of the system, and of Equation (16) as being a legitimate conservation law.32 Though we concur with Lam on this point, we diverge in our interpretation of pseudotensorial quantities such as tμν—see below. 3.3.3 Functionalism about gravitational stress-energy The above two positions in mind, we must ask two questions: (a) is it correct to call the quantity appearing in Equation (Int-Tt) associated with tμν ‘gravitational stress-energy’, and (b) does such ‘gravitational stress-energy’ really exist in GR?33 Begin with (a). As Lam notes, ‘energy and mass might not be fundamental properties of the world, in the sense that they make sense only in some particular (but very useful) settings; this does not lessen the fact that the notions of energy and mass constitute extremely powerful tools for many concrete and practical cases’ (Lam [2011], p. 1023).34 This point is important: gravitational stress-energy à laEquation (Int-Tt) is not a fundamental concept in GR, insofar as it is only applicable in a limited range of DPMs of the theory—namely, those in which certain Killing vector fields may be defined. (That is, the term ‘gravitational energy’ is associated with structures—namely terms such as that on the left-hand side of Equation (Int-Tt)—which are not used to construct the space of DPMs of GR, but rather which are only well defined in a certain subset of those DPMs.) Nevertheless, in such instances it is extremely useful to make use of this term, within that subclass of DPMs. Hence, at a practical level, it is legitimate to call such a quantity gravitational stress-energy. Clearly though, this does not settle the putative ontological issue (b), concerning whether gravitational stress-energy ‘really’ exists in GR. In our view, it is plausible to maintain that in situations such as those in which Equation (Int-Tt) holds, there exists a quantity in GR that fulfils the functional role of gravitational stress-energy. The reasons for this are the following. First, for a structure in a certain model of a theory to play the ‘functional role of gravitational stress-energy’, it must (i) fulfil a function analogous to that of gravitational energy as traditionally conceived—namely, as a quantity (gravitational potential energy) in Newtonian gravitation, which balances the matter (in Newtonian mechanics: kinetic) energy of the system in question such that their sum is conserved; and (ii) bear some relation to the ‘gravitational’ degrees of freedom in the theory in question. Second, arguably terms such as that on the left-hand side of Equation (Int-Tt) do satisfy (i) and (ii)—the former holds in virtue of a comparison of the form of Equation (Int-Tt) with Newtonian energy conservation equations; the latter in virtue of the connections between tμν and, for example, the connection components—themselves (at least, on views such as those indicated by Einstein mentioned above) understood to be associated with ‘gravitational’ degrees of freedom, as elaborated in Section 2.3.3 and 3.1.35 A functionalist may, therefore, speak of the existence of gravitational stress-energy in such situations.36 On the assumption that (i) and (ii) are satisfied, the alternative to functionalism is to say that ‘the structure of certain DPMs of GR is such that it appears that there exists gravitational stress-energy in those models, but really there is no such stress-energy there’; the payoff to be gained from making such a claim is unclear. Still, doubts may linger. In particular, one might argue as follows: ‘Surely there is a much more plausible alternative that disputes that gravitational energy “really” exists, which says that we can describe everything that is going on in terms of solutions to Equation (1) (and its consequences, including Equation (Cov-T)), without any need to help ourselves to talk involving tμν’.37 With this point, we are in broad agreement: one could indeed explain all general relativistic phenomena, in any model of the theory, simply using the apparatus used to pick out the DPMs of the theory. Nevertheless, we would respond that there may exist other structures only in certain models of the theory, which play certain functional roles. In our view, there is nothing illegitimate in regarding such structures as also existing in (the worlds represented by) those models of the theory—and, moreover, doing so may open up more perspicuous avenues for the explanation of phenomena within those models (recall from Section 3.3.2 the case of gravitational radiation from binary star systems). Thus, while we concur with the above argument, in our view this does not constitute an objection to a notion of functional gravitational energy in GR, since advocates of such a concept simply have more explanatory apparatus available to them. Thus, a functionalist may assert that gravitational stress-energy in the sense of Equation (Int-Tt) does exist in GR, but only in a restricted class of situations; this aligns with the case of SR. Since we have already found Hoefer’s objections to the existence of gravitational stress-energy in GR wanting, and since such functionalist principles are practical and simple, we conclude that in such situations (corresponding to certain models of GR) it is best to state that a quantity that plays the functional role of gravitational stress-energy does exist in the theory, and hence should be labelled as genuine. We conclude that frame-independent gravitational stress-energy does exist in GR, in the limited sense above. Whether one also maintains that a frame-dependent notion of gravitational stress-energy exists in GR will, for the reasons discussed, depend on whether one is a realist or antirealist about pseudotensorial quantities, in particular tμν. Note, however, that realism about such quantities is arguably compatible with the above functionalist principles: in each given frame of reference, one may define a quantity, represented by tμν, such that total stress-energy is conserved. Again, this plays the functional role of gravitational stress-energy in that frame, for (i) and (ii) as delineated above are both satisfied. Thus, it is also the case that speaking of frame-dependent gravitational stress-energy may be justified on functionalist grounds. On this latter point, there exist connections with other, ongoing debates in the foundations of spacetime theories—in particular, over the primacy of coordinate-dependent versus -independent explanations. According to advocates of the latter, such as Friedman ([1983]) and Maudlin ([2012]), explanations of physical phenomena within models of a given spacetime theory should proceed by appeal only to coordinate-independent structures. By contrast, according to advocates of the former, such as Brown ([2005]; Brown and Read [forthcoming]) and Wallace ([forthcoming]), presentations of spacetime theories need not proceed in a coordinate-independent manner; rather, spacetime theories may be defined in terms of equations written in a coordinate basis and their transformation properties (this is what Brown ([2005], p. 9) and Wallace ([forthcoming], p. 5) refer to as the ‘Kleinian conception of geometry’), and explanations may be given by appeal to those laws, written in a coordinate basis.38 For advocates of the coordinate-independent perspective in the context of, for example, SR, explanations of phenomena proceeding by appeal to coordinate-dependent effects (for example, the twin paradox differential in terms of the relativity of simultaneity, assuming some clock synchrony convention) are to be rejected, as the associated physical effects not considered ‘real’; for advocates of the coordinate-dependent perspective, there is nothing wrong with issuing such explanations, and with viewing such effects as physical. Those who buy into the latter programme may view the notion of gravitational energy in question as frame-dependent, but no less real for all that. 4 Gravitational Stress-Energy and Spacetime Ontology 4.1 Relationism and gravitational stress-energy In the above, we endorsed a position according to which gravitational stress-energy does exist in GR, in at least (i) a weak sense applicable in a certain class of models of the theory, and arguably also (ii) a strong sense, applicable in all models of the theory. Nevertheless, there exists a residual worry, related to the debate between advocates of substantivalism versus relationism about spacetime ontology. In this article, we take substantivalists to claim that spacetime exists as an entity in its own right, and relationists to deny this—that is, to claim that all talk of spacetime is reducible to talk of (relational properties of) matter fields. This in mind, the worry regarding gravitational stress-energy runs as follows: if one is a relationist, how can one maintain that there indeed exists gravitational stress-energy in the world? The thought that relationism implies the nonexistence of gravitational stress-energy (or, equivalently, that genuine gravitational stress-energy implies substantivalism) is intuitive. In this section, however, we argue that it is not correct. There are two different pictures of the substantivalism/relationism debate that are relevant here. Let ⟨M,gab,Φ⟩ be a model of GR. Then, according to ‘manifold substantivalism’, spacetime is identified with M; to be a relationist is to maintain that manifold points do not have an ontological status independent of the fields defined upon them. On the other hand, according to ‘metric substantivalism’, a spacetime is identified with ⟨M,gab⟩—that is, with both the manifold and the metric field. To be a relationist is then to maintain that neither manifold points nor the metric field have a distinct existence over and above the matter fields Φ. Clearly, it is harder to be a relationist in the second sense than the first.39 For our purposes, it is more relevant to consider relationism in the second sense above, since we are concerned with the ontological status of quantities associated with gab. In this case, our question becomes: how can those who do not believe that the metric field is fundamental (insofar as they think it reducible to properties of matter fields) maintain the existence of genuine gravitational stress-energy? On the face of it, such a claim is implausible, and indeed many (for example, Hoefer [2000]; Baker [2005]) maintain that the answer is a simple negative: they cannot. We answer to the contrary. Our positive story runs as follows. Whether one is a relationist or a substantivalist, it is a fact that there are some situations in which results such as Equation (Int-Tt) are well defined, and there appears to be a well-defined quantity in the theory that plays the functional role of frame-independent gravitational stress-energy. Of course, the account given by the relationist of this quantity will differ from that given by the substantivalist: the relationist will assert that this quantity is associated (in some way to be made precise) with the metric field, which is in turn a codification of properties of the fields; the substantivalist will appeal to the metric field simpliciter. In either case though, this quantity exists in the theory: the two sides are not debating its existence, but rather the fields to which it is ultimately attributable. 4.2 The cosmological constant Finally, it is worth commenting on Baker’s claims regarding the bearing of the possibility of a non-zero cosmological constant Λ on both the substantivalism/relationism debate, and the existence of gravitational stress-energy (Baker [2005], Section 4). Including a cosmological constant term, Equation (1) reads: Gab+Λgab=8πTab (17) We focus on the following two claims made by Baker: Λ≠0 commits us to (manifold-plus-metric) substantivalism, because this quantity is associated with spacetime yet cannot be reduced to relations amongst the matter fields (Baker [2005], Section 4.1). Λ≠0 results in a non-zero vacuum energy density, for which a relationist cannot account (Baker [2005], Section 4.2). (a) questions whether relationism is compatible with a non-zero cosmological constant. (b) assumes that such is the case, but claims that relationism nevertheless cannot account for the vacuum energy density arising from non-zero Λ. Let us first consider (b). In fact, there are several reasons to be suspicious of this claim. First, Baker ([2005], p. 1301) justifies that Λ is ‘associated with spacetime’ as follows: [Λ’s] role in the field equations [Equation (17)] is to influence, by itself or in combination with other terms, the metric structure of spacetime, and thereby to affect the physical behaviour of matter. This is exactly the sort of influence that accounts for gravitational forces in GR, the only difference being that Λ does not depend on matter as its source. This claim is suspect, because Baker has not ruled out the possibility that the Λ term in Equation (17) can appear on the right-hand side of this equation, with Λ being treated as another matter field; such an interpretation of Equation (17) is also prima facie possible, yet the above assertion does nothing to rule it out. Therefore, Baker needs to do more to make convincing any claim that Λ is ‘associated with spacetime’. Indeed, there is an ongoing debate in cosmology over whether to consider the cosmological constant term in Equation (17) as being attributable to spacetime (that is, as sitting on the left-hand side), or as another type of matter-like field (that is, as sitting on the right-hand side); Baker cannot simply presuppose an answer to this question. Second, the claim that Λ cannot be reduced to relations among the matter fields is also too quick. On this, Baker ([2005], p. 1306) states: I do not doubt that a persistent relationist could describe Λ’s effects as mere relational properties, but the price will be high. Considering [the] example of distant objects moving apart under the influence of Λ, the relationist would have to posit a brute fact that material objects possess a tendency to accelerate away from one another at a rate proportional only to the distance between them. In fact, this seems to be roughly in line with the relationist-type approach to the metric field in relativity theory in the context of SR outlined in (Brown [2005]). More generally, Baker cannot infer from the fact that such a programme appears to be undesirable to him that it cannot be done (and indeed, he openly admits that it can be done), or even that such a task might not be desirable or acceptable in some relationist research programmes. On (b), Baker ([2005], p. 1310) claims that a relationist cannot account for the vacuum energy density ρΛ=Λ/8πG implied by a non-zero Λ: ‘I can see no easy way for the relationist to explain the energy density of empty space’.40 In fact though, this is not so: the relationist can account for the energy density of empty spacetime. To see this, suppose that the cosmological constant can be reduced to properties of matter fields. Then, as with gravitational stress-energy in the previous subsection, the equations of the theory will still state that there exists a quantity that plays the functional role of a vacuum energy density. Once again, the only difference will be the story that is told to account for this quantity. While the substantivalist will appeal directly to gab and Λ, the relationist will resort to an elliptic story about how this quantity arises from properties amongst the matter fields themselves. But on either account, a functional vacuum energy density exists according to the theory. 5 Conclusion In this article, we have reconsidered the existence of gravitational stress-energy in GR; adopting a functionalist attitude to the definition of physical quantities, we have argued that gravitational stress-energy can be considered to exist in GR, in both (i) a weak sense applicable in a certain class of models of the theory (namely, models that instantiate certain symmetries, and therefore that possess Killing vector fields), and (ii) arguably also in a stronger sense (as represented by the gravitational stress-energy pseudotensor in a given frame), applicable in all models of the theory. This latter approach runs against contemporary orthodoxy, but is in line with the thinking of the historical Einstein ([2002a], p. 167). In addition, we have adopted a revisionary line regarding whether gravitational stress-energy is compatible with relationism, arguing that regardless of whether one thinks that the metric field gab is reducible to properties of matter fields, gravitational stress-energy still exists in the theory, if one again embraces functionalism about physical quantities. Accordingly, one’s position on the ontology of spacetime does not affect one’s commitment to gravitational stress-energy in GR; this point also applies to the claim that a non-zero vacuum energy density is incompatible with relationism. Acknowledgements First and foremost, I am grateful to Patrick Dürr for many rewarding discussions on gravitational energy. Many thanks also to Harvey Brown, Eleanor Knox, Dennis Lehmkuhl, Tushar Menon, Brian Pitts, and Simon Saunders for valuable comments on earlier drafts of this article. I am supported by an Arts and Humanities Research Council scholarship at the University of Oxford, and am also indebted to Hertford College, Oxford for a graduate senior scholarship. Footnotes 1 Throughout, abstract (that is, coordinate-independent) indices are written in Latin script; indices in a coordinate basis are written in Greek script; and the Einstein summation convention is used. Round brackets around indices denote symmetrization over those indices; square brackets around indices denote antisymmetrization. We set GN=c=1. 2 One should avoid, at this stage, asserting M to be the ‘spacetime manifold’, for to do so is to conflate the mathematical model under consideration with the possible world to which that model is ultimately interpreted as corresponding. Indeed, in light of the debate over the hole argument (Earman and Norton [1987]), it is not necessarily correct to interpret M as representing substantial spacetime—see Section 4.1. 3 Until Section 4, we restrict to the sector of GR with vanishing cosmological constant Λ. For Λ≠0, Equation (1) reads Gab+Λgab=8πTab. 4 Strictly, independence of these equations from Equation (1) depends on the case—see (Brown [2005], Section 9.3; Misner et al. [1973], Section 20.6). 5 In a coordinate basis {eμ}, the connection components are defined through ∇ρeν=Γμνρeμ. 6 Since in a coordinate basis, the (unique) Riemann tensor Rbcda associated to ∇a—defined through Rbcdaξb=−2∇[c∇d]ξa for all smooth fields ξa (Malament [2012], p. 68)—reads Rμνρσ=∂ρΓνσμ−∂σΓνρμ+ΓνστΓτρμ−ΓνρτΓτσμ. 7 Consider an affine coordinate transformation xμ′=Mμ′μxμ+aμ′. If an (r, s) tensor Fμ1…μrν1…νs transforms under this coordinate change as Fμ1…μrν1…νs→Mμ1μ1′…Mμrμr′Mν1′ν1…Mνs′νsFμ1′…μr′ν1′…νs′, then we say that it is ‘covariant’ with this coordinate transformation. If a dynamical equation retains the same form in either of the two coordinate systems under consideration, then we say that it is ‘invariant’ under the coordinate change. 8 The notation xμ′=fμ′(xμ) signifies that in this case xμ′ may be defined in terms of arbitrary contractions with xμ, provided that there is one free primed index. 9 To take an explicit example (relevant to our discussions of stress-energy below), consider the expression ∂μTμν=0. Transforming to a new coordinate basis xμ→xμ′=fμ′(xμ), one obtains ∂μ′Tμ′ν′+∂xμ′∂xμ∂2xμ∂xμ′xλ′Tλ′ν′+∂xν′∂xν∂2xν∂xλ′∂xσ′Tλ′σ′=:∂μ′Tμ′ν′+Θν′=0. 10 Rather, Equation (Par-T) holds only at p. 11 This result is straightforward: Equation (3) follows from Equation (2), which involves only tensorial quantities. 12 Defined through specifying normal coordinates at p∈M. 13 As discussed by Pooley ([2017], p. 115), and unlike the case of the metric field gab of GR, one may understand ηab as being fixed identically in all KPMs of SR. Otherwise, KPMs of SR—as with GR—are again denoted by triples, this time of the form ⟨M,ηab,Φ⟩. 14 Performing these two replacements in any dynamical equations of SR, to obtain general relativistic dynamical equations, is sometimes dubbed ‘minimal coupling’. For philosophical discussion, see (Brown and Read [2016]; Read et al. [forthcoming]). 15 To claim that Einstein identified the gravitational field with the connection coefficients may be to oversimplify his position. In fact, Einstein saw GR as unifying gravity and inertia, in the same way that SR had unified electricity and magnetism. If one takes this view, then perhaps one need never speak of the gravitational field in GR. See (Lehmkuhl [2014]; Brown and Read [2016]). 16 For further discussion of these matters, see (Lehmkuhl [2008]). 17 The ‘Lie derivative’ LXT of a tensor field T represents how that tensor field changes as one acts with a diffeomorphism along the integral curves of a vector field Xa; the condition φ*T=T imposes that LXT=0. Since the Lie derivative of the metric field reads (LXg)ab=2∇(aXb) (assuming that the unique metric-compatible, torsion free derivative operator ∇a is used), this condition yields ∇(aXb)=0, which is Killing’s equation. See (Wald [1984], pp. 437–44). Note also that Ka=gabKb. 18 See Wald ([1984], pp. 433–4). 19 This does not preclude the possibility of other unspecified ways of obtaining genuine (non-gravitational) stress-energy conservation in curved spacetime: it specifies sufficient but not necessary conditions. 20 The covariant divergence theorem can only be applied to vector fields, but not to higher rank tensor fields: this is why we obtained in Equation (13) an integral conservation law from ∇a(TabKb)=0, but we cannot obtain an analogous integral conservation law from ∇aTab=0. 21 Consider a Killing vector field Ka associated to ⟨M,gab⟩, which by definition satisfies Equation (10) (the derivative operator ∇a in this equation is the unique torsion-free such operator compatible with gab, as usual). From this, one can derive straightforwardly that Ka must also satisfy ∇a∇bKc=RbadcKd. Restricting to the special relativistic case ⟨M,ηab⟩, one has Rbadc=0, in which case ∇a∇bKc=0. Then, restricting to normal coordinates at some p∈M, one has that ∂μ∂νKρ=0—and integrating this equation, one finds that Kμ=aμνxν+bμ where bμ is a constant one-form, and the coefficients of aμν are also constant. Therefore, the components Kμ are linear functions of the inertial frame coordinates. Now, substituting this result into Equation (10) reveals that aμν must be antisymmetric—that is, has six independent components. Since bμ has four independent components, in total we find that there are ten independent Killing vector fields associated to the ‘spacetime’ ⟨M,ηab⟩; these are the isometries of the Minkowski metric ηab. Since the Poincaré group is the group of coordinate transformations that leave the Minkowski metric invariant, we see that this group must have ten independent generators, each corresponding to a Killing vector field. 22 Three notes on tμν are in order: (i) This term and the notation used to denote it were originally introduced by Einstein (Einstein [1996]; Einstein and Grossmann [1996]). Einstein went on to state in 1918 that ‘nearly all my colleagues raise objections to my definition of the momentum-energy theorem’ (Einstein [2002b], p. 448); here he had in mind particularly Levi-Civita ([1917]), Schrödinger ([1918]), and Bauer ([1918]), with whom he had corresponded heavily on this topic in the preceding four years. (ii) Some, such as Hoefer ([2000], p. 193) and Nerlich ([2013], p. 162), have taken tμν to be defined implicitly through Equation (3). Unfortunately, this does not work out smoothly, as there is more than one candidate for tμν (as discussed below), so Equation (3) is not sufficient to fix tμν uniquely. (iii) For simplicity in this article we use the symbol tμν to refer both to the object representing gravitational stress-energy, and to its components in a given coordinate basis. 23 There exist interesting questions regarding which of these non-equivalent versions of the gravitational stress-energy pseudotensor best describes gravitational stress-energy. See (Trautmann [1962], pp. 190–1). 24 Though see Footnote 15. 25 This position is also advanced at (Lehmkuhl [2008], p.94); see in addition (Bergmann [1976], p. 197; Lehmkuhl [2014], Section 5). 26 For a recent attempt to make sense of Equation (Par-Tt) as a legitimate conservation principle, see (Pitts [2010]). In a sense, Pitts’ view is a halfway house between realism and antirealism: though he argues that pseudotensors are ‘physically meaningful’ (Pitts [2010], p. 15) with ‘no vicious coordinate dependence’ (Pitts [2010], p. 14), he does this by demonstrating that they can be unified into an infinite-component geometric object. For further discussion of Pitts’ position, see (Lam [2011], Section 5; Dürr [unpublished]). 27 Localizability is a stronger condition than satisfying an integral conservation principle, because it is possible to have the latter in the absence of the former, as is the case in Equation (16) when satisfying (i)–(iii). 28 This may be evaded straightforwardly if one claims that being an integral conservation law is a necessary but not sufficient condition to be a conservation principle ‘properly speaking’. 29 Recall from Section 2.4 that, while conserved quantities such as energy strictly only exist in frames related by the appropriate coordinate transformations, by projecting onto the Killing vector fields associated with such transformations, we can obtain results that hold in any frame. It is this that we mean by a ‘frame-independent’ notion of gravitational stress-energy—even though strictly such stress-energy (sans projection) is conserved only in a restricted class of frames. 30 In other words, the notion of gravitational stress-energy may still be applicable to (subsystems of) the actual world at an approximate, functional level—see Section 3.3.3 below. 31 For example, in a model of GR in which conditions (i)–(iii) are satisfied, it suffices to introduce a small perturbation in the metric field such that it is not asymptotically Minkowski at infinity for the above notion of frame-independent gravitational stress-energy to no longer be applicable. 32 At least at an approximate level—see Footnote 30. 33 Note that (b) is not the same as asking whether gravitational stress-energy really exists in the (possibly general relativistic) actual world, for the reasons delineated in Section 3.3.1. 34 Here, Lam is referring to both matter and gravitational (stress-)energy. Note also that Lam says ‘properties of the world’, rather than ‘properties in GR’. For our purposes, it is legitimate (and preferable) to read him as making the latter, more general statement (see Footnote 33). 35 Though one should recall the caveats of Section 2.3.3 regarding the question of which object in GR should be associated with the ‘gravitational field’, strictly speaking. 36 Such a line accords with general functionalist attitudes in science. Wallace ([2012], p. 58) summarizes this as follows: ‘Science is interested with interesting structural properties of physical systems, and does not hesitate at all in studying those properties just because they are instantiated “in the wrong way”’. 37 We are grateful to an anonymous referee for putting the point in this way. 38 We set aside here the question of the extent to which this ‘Kleinian conception’ is faithful to the definition of geometry in Klein’s ‘Erlangen’ programme (Klein [1892]). 39 Two points are in order here. First, work such as (Lehmkuhl [2011]) demonstrates that the stress-energy tensor Tab of the matter fields Φ of GR in fact presupposes metric structure in its definition. It is perhaps then misguided to attempt to reduce gab to Tab. That said, one should of course recall that one should not conflate matter fields Φ with their associated stress-energy tensor Tab. Thus, even if it is misguided to attempt to reduce gab to Tab, perhaps it is still possible to reduce gab to Φ. Our second point is related: suppose one does seek to reduce gab to Φ. To do so is to endorse a version of ‘Mach’s principle’ (Lehmkuhl [2014], p. 455). There exist problems with attempting to achieve this in GR—for example, one faces the problem that a priori the metric field has ten independent components, whereas the electromagnetic field tensor (for example) has only six—so there is a question of how the former can be reduced to the latter. Another problem in this vicinity lies in the existence of vacuum solutions in GR. As a result, it is questionable whether the form of relationism considered here is ultimately defensible in GR—so the work of this section is best viewed as (a) a response to (Hoefer [2000]; Baker [2005]), where it is claimed that such views are incompatible with the existence of gravitational energy; and (b) an exploration of the consequences of the functionalism about gravitational stress-energy developed above. We are grateful to an anonymous referee for pushing us on this point. 40 As Baker ([2005], p. 1309) states, this does not mean that Λ arises from the non-zero ρΛ; rather, its role in the field equations is equivalent to an energy density of empty space. References Baker D. [ 2005 ]: ‘Spacetime Substantivalism and Einstein’s Cosmological Constant’ , Philosophy of Science , 72 , pp. 1299 – 311 . Google Scholar CrossRef Search ADS Bauer H. 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