1. Introduction

Since it encodes both the inertio-gravitational effects and the space-time structure, the gravitational field has a very peculiar nature within the general theory of relativity (GTR). An important aspect of the peculiarity of the gravitational field lies in the nature of its energy (and momentum). Intuitively, and from a nonrelativistic point of view, it does not seem very surprising that the gravitational field possesses some energy—we experience the transfer of gravitational energy into kinetic energy all the time in everyday life, every time we drop something on the floor, for instance. Within GTR, because of the dual role of the gravitational field, it seems that gravitational energy could be understood equally as the energy of the space-time structure itself. As will be discussed in this article, such understanding of gravitational energy may shed some interesting light on the difficulties to ascribe the usual (related) features of localization and conservation to gravitational energy.

I will review some well-known aspects of these difficulties (secs. 2–4) and will discuss them in relation to fundamental issues such as conservation laws, uniqueness (sec. 5), background independence (sec. 6), and the nature and status of energy in general within GTR (sec. 7). The upshot of the discussion is to highlight the fact that the fundamental property of background independence, which is an aspect of the dual nature of the gravitational field, might constitute the real crux of the difficulties with energy localization; moreover, strictly speaking, the introduction of background structures is actually mandatory for the very definition of energy within GTR, then understood as a global notion. We first start to look at the way standard, nongravitational energy is defined according to the theory.

2. Energy-Momentum Tensor

Within the special and general theories of relativity, nongravitational (mass-)energy and momentum densities (as well as stress) of any physical system (matter fields indeed) are described by the (stress-)energy-momentum tensor field. The vacuum Einstein field equations can be derived by means of a variational principle from the Einstein-Hilbert action S _EH. Now if the matter field equations can also be derived from a Lagrangian density (like in many physically interesting cases such as electrodynamics), then the energy-momentum tensor can be defined as the variation of with respect to the metric

so that the variation of the total gravity-matter action with respect to the metric leads to the full Einstein field equations

This is a very general definition of the energy-momentum tensor associated with any given matter fields, whose dynamics is derived from a Lagrangian density .Footnote ¹ An important aspect of this definition is that it entails, assuming the matter field equations (the Euler-Lagrange equations for ) and the diffeomorphism invariance of , that the covariant divergence of the energy-momentum tensor vanishes:

By analogy to special relativity, this identity is often and loosely called energy-momentum ‘conservation law’. Indeed, within special relativity, the vanishing of the ordinary divergence of the energy-momentum tensor

can be straightforwardly interpreted as a genuine conservation law in the sense that for any space-time region the total energy-momentum flux is zero (in rough terms, ‘all that enters the space-time region goes out’).Footnote ² But such interpretation of the vanishing of the covariant divergence (3) of the energy-momentum tensor is in general not available within GTR. There are several interrelated ways to see this. From an intuitive point of view, a mere look at the rewriting of the covariant divergence (3) in terms of ordinary divergence shows that there are additional terms due to the gravitational field:

These terms, containing the metric connection coefficients (or Christoffel symbols) , can be understood as resulting from the interaction between the gravitational field and matter fields; there is a strong analogy between the Christoffel symbols and the gauge potential in non-Abelian Yang-Mills theories (both are coefficients of a connection on the respective relevant principal fiber bundle), where the Yang-Mills covariant derivative , resulting from the interaction between matter fields and a gauge field, is defined in complete analogy with the GTR covariant derivative .

So, the vanishing of the covariant divergence (3) does not constitute a genuine conservation law since it contains interaction terms with the gravitational field but does not account for any gravitational energy-momentum. In the light of the dual nature of the gravitational field, it might therefore be helpful to discuss this failure of (nongravitational) energy-momentum conservation from the point of view of the space-time structure.

3. Conservation and Symmetries

Indeed, energy and momentum conservation are fundamentally linked to invariant properties of the space-time structure. This can be seen within the Lagrangian and Hamiltonian formalisms as an instance of Noether's first theorem, which states (without entering into the details) that if the action is invariant under a continuous group of (global) transformations that depend on p constant parameters, then there are p associated vanishing divergences of the so-called Noether currents , (assuming that all Euler-Lagrange equations are satisfied); these p Noether currents naturally lead to p conserved quantities (‘charges’) by suitable integration using Gauss's theorem (see below). For instance, within special relativity, for an action that is invariant under the Poincaré group, which has 10 parameters, there are then 10 conserved quantities, among which are energy and linear and angular momentum. Energy and linear momentum conservation is specifically associated with the invariance under temporal and spatial translations.Footnote ³ So energy(-momentum) conservation does not only depend on properties of the matter fields but is also deeply related to symmetries of the space-time structure. Such symmetries (isometries indeed) can be characterized by their generators, that is, by their associated Killing vector fields.Footnote ⁴ The existence of a genuine (i.e., integral) energy(-momentum) conservation law is then bound to the existence of a timelike Killing vector field since this latter is the generator of an isometry that can be understood in terms of temporal translation invariance: the metric ‘does not change’ along the timelike integral curves of the Killing vector field (such space-times are called stationary). More precisely, if a space-time (M, ) possesses a timelike Killing vector field K ^μ, then one can build the quantity , which represents the energy-momentum current density relative to such that its divergence vanishes:

And this allows us to write for the three-dimensional boundary S of any four-dimensional volume V, using Gauss's theorem,

where N _μ is the unit normal to S (with standard orientation) and dS and dV are three- and four-volume elements. Now the integral formulation (6) genuinely means (nongravitational) energy-momentum conservation (with respect to the integral curves of the timelike vector field K ^μ): for any four-dimensional space-time volume V, all energy-momentum that enters V is balanced by what goes out (change in time equals the flux), which precisely means that energy-momentum is conserved. Another way to consider this is to see that the quantity of nongravitational energy in a spatial (three-dimensional) region R contained in a spacelike hypersurface Σ, defined by

is conserved (it does not depend on the choice of Σ).Footnote ⁵

Such conservation law seems to be fundamental for the very notions of mass, energy, and momentum (and it plays indeed a fundamental role in pregeneral relativistic physics): conservation is what makes these notions so fundamental, at least to matter (e.g., according to the Newtonian characterization of matter) since it grounds a notion of persistence.Footnote ⁶ Now (6) shows that we can have genuine (nongravitational) energy-momentum conservation in the presence of a gravitational field, that is, in curved space-time, but only in the cases where this latter instantiates certain global symmetries, namely, when the space-time structure remains unchanging (stationary) along the integral curves of the timelike Killing field. This is a very peculiar feature indeed, which can be very useful in relevant approximations of many physical situations (from everyday laboratory life to Kerr-Newman black holes) but which is in general (and globally) not instantiated in our evolving universe.

One can analyze this failure of integral nongravitational energy-momentum conservation from two interrelated points of view. First, the timelike Killing vector field associated with a stationary space-time structure can be understood in a certain sense as a nondynamical background structure with respect to which integral nongravitational energy-momentum conservation can be obtained; indeed, a timelike Killing vector field can be understood as defining a global inertial frame, which can represent a global family of inertial observers all at rest with each other. A fully dynamical metric field would prevent the existence of such global symmetries. So, in other terms, certain fundamental aspects of the space-time structure itself prevent nongravitational energy-momentum conservation. Second, as already mentioned above, from the point of view of the universally interacting gravitational field, the failure of integral nongravitational energy-momentum conservation is an obvious consequence of not taking into account gravitational energy: strictly speaking, there cannot be nongravitational energy-momentum conservation since any material system interacts with the gravitational field and its energy can transform into gravitational energy and vice versa. Because of the fundamental dual nature of the gravitational-metric field, these two points of view are indeed equivalent. In this sense, the nature of gravitational energy seems to be linked to the failure of certain global symmetries and, most importantly, to the lack of nondynamical background structures, that is, to background independence.

4. Pseudotensors and Real Ambiguities

So, it seems natural to ask whether the gravitational field possesses some energy(-momentum) that can also be described by an energy-momentum tensor, written , such that the total (matter and gravitational) energy-momentum is conserved; in differential form, this means that the ordinary divergence of the total energy-momentum vanishes:

so that integral conservation can be obtained. Thus, can be defined using this conservation requirement and the Einstein field equations. Indeed, the vanishing of the ordinary divergence of the total energy-momentum tensor (eq. [8]) can be encoded in terms of an antisymmetric (so-called) ‘superpotential’ , so that one can write

where (8) is guaranteed by the antisymmetry of the superpotential. Using Einstein field equations, can then be defined in the following way:Footnote ⁷

where is Einstein's tensor (2). There is a freedom in the choice of the superpotential (the conservation law (8) does not define in an unambiguous way), which leads to distinct nonequivalent expressions for , such as the ones of Einstein and Landau-Lifschitz among others. Another way to understand the formal definition (10) is to consider in terms of the nonlinear corrections to the linearization of Einstein's tensor: from the point of view of the linearized theory, gravitational energy-momentum is encoded in the gravitational (nonlinear) corrections.Footnote ⁸

Now, the important point to highlight is that the definition (10) of the geometric object depends on a local coordinate system (partial derivatives are defined with respect to a coordinate system): does not represent an invariant entity under coordinate transformations, so that it is not a tensor (it is called a pseudotensor).Footnote ⁹ This means that does not describe the amount of local gravitational energy-momentum in an unambiguous and unique way: it depends on the choice of a coordinate system.Footnote ¹⁰ In particular, at any space-time point or along any world line, there is a coordinate system in which vanishes. Since different coordinate representations are just different mathematical descriptions, relevant physical entities are usually taken to correspond to coordinate-independent entities (which can be represented in different coordinate systems, then providing different descriptions of the same physical entity—clearly assuming some uniqueness about the latter; see sec. 5). So, according to this understanding, the coordinate (or background or gauge) dependence of shows that there is indeed no local (unique) gravitational energy-momentum, in the sense that such a quantity cannot be, in general, unambiguously defined at any space-time point.

It should be intuitively clear that this difficulty of localizing gravitational energy actually has an impact on all forms of energy since one form of energy can transform into another. As a consequence of the above discussion, the amount of gravitational energy-momentum present in any given space-time region (and not only at space-time points) cannot be unambiguously defined in the general case. This ambiguity is valid for total energy-momentum as well since the integral conservation law that can be derived from (8) for any space-time region is obviously coordinate dependent too. In particular, it is important to emphasize that nongravitational energy (and mass) as well cannot be unambiguously defined in the general case: in the absence of the relevant symmetries, the definition of matter energy (7) in the spatial region R is not unique and not conserved (it now depends on the choice of Σ).

5. Nongenuine or Nonunique?

If one considers that the satisfaction of a proper conservation principle is an essential aspect of the definition of energy (see n. 6), then it is clear from the discussion above that gravitational energy is not a genuine form of energy. Hoefer (Reference Hoefer2000) seems to defend such a view. He explicitly claims that, due to the lack of a tensorial definition of gravitational energy as well as the lack of proper energy-momentum conservation in GTR, gravitational energy is not fundamental. As a consequence, he argues that the standard argument in favor of space-time substantivalism based on the attribution of some form of energy to space-time itself is not sound. In the light of the difficulties to define in any space-time region gravitational and nongravitational energy-momentum, this objection is actually rather weak: we are entitled to believe (or not) that the gravitational field or space-time itself possesses some energy to the same extent that we believe (or not) that fundamental material entities such as matter fields possess some energy (some mass). So if one wants to hold that material entities do possess some well-defined energy, one needs to account for gravitational energy and the lack of proper energy conservation law.

Pitts (Reference Pitts2010) recently discusses such an account based on the possible nonuniqueness of gravitational energy-momentum. The idea is that the difficulties linked with the definition of (local) gravitational energy-momentum and the lack of proper energy conservation can be removed if one drops the assumption of uniqueness for gravitational energy-momentum: we simply have to accept that there are infinitely many (local) gravitational energies, which can be represented by a mathematically precisely defined infinite-component (geometric) object. As discussed above, this move explicitly recognizes that any definition of (local) gravitational energy needs additional background structures such as a (flat) background metric or a coordinate system, for instance: gauge invariance is then restored by collecting all the background-dependent expressions for a given background structure into one (infinite-component) object. For instance, such an object can be constituted by the expressions for Einstein's pseudotensor in all coordinate systems; contrary to the different sets of components of a tensor in different coordinate systems, the different sets of components of this object in different coordinate systems do not constitute different descriptions of the same entity but distinct entities, namely, distinct but conserved energies (since [8] holds in the corresponding coordinate system).

Since the conceptual importance within GTR of coordinate-free methods has been widely recognized, it might seem a bit awkward to rely so heavily on background-dependent (coordinate-dependent) entities. However, as such, there does not seem to be any contradiction in such a move. Indeed, it has the virtue of emphasizing the importance of additional background structure to make sense of gravitational energy (as well as of the very notion of energy itself; see sec. 7). What remains however obscure is the physical improvement brought by and the physical meaning of such an infinite-component object. In particular, the definition of (matter or gravitational) energy contained in a given spatial region R remains ambiguous in the sense of nonunique (trivially, since the very assumption of uniqueness has been dropped): one cannot say how much energy there is in a given spatial region R (in the general case) since each component may provide a distinct answer. It might be the case that looking for such energy quantity is fundamentally hopeless since, as a matter of principle, local gravitational energy cannot be unambiguously defined.

6. From Equivalence Principle to Background Independence

Indeed, the ambiguity about local gravitational energy-momentum is in general taken in the literature (at least in most GTR textbooks) as a consequence of some ‘infinitesimal equivalence principle’. According to this principle, in any infinitesimal space-time region, the gravitational field, and therefore gravitational energy-momentum, can always be transformed away by a suitable choice of coordinate system. The idea is that in considering infinitesimal space-time regions, any arbitrary gravitational field can be considered homogenous and transformed away in virtue of the equivalence of inertial reference frames and uniformly accelerated ones. Despite its heuristic force, one should be a bit wary here of the significance of this argument. The notion of an infinitesimal region and the sense in which the gravitational field and, above all, its energy-momentum can be transformed away are not clear.Footnote ¹¹ Obviously, there is in general no coordinate system in which the components of the curvature tensor and the second derivatives of the metric components can be made to vanish. In other words, tidal (or ‘geodesic deviation’) effects, which are often invoked in examples aiming to show that the gravitational field possesses some substantial energy, cannot be made to vanish. Simply ignoring second and higher derivatives of the metric components by invoking sufficiently small (infinitesimal) regions (i.e., no proper neighborhoods) is indeed problematic since such restriction does not allow one to distinguish between sufficiently smooth curves and geodesics any more.

I suggest considering, rather, the fundamental GTR property (call it ‘general equivalence principle’ if you like) according to which the metric or gravitational field accounts for both inertial and gravitational effects. More precisely, the fact that the metric-gravitational field cannot be decomposed uniquely into an inertial (nondynamical) part plus a gravitational (dynamical) part is called ‘background independence’.Footnote ¹² This property is taken by many physicists and philosophers of physics to be the distinctive feature of GTR and one of the main difficulties toward a coherent view between GTR and quantum field theory, which relies on a nondynamical space-time background. Background independence (and diffeomorphism invariance in particular) forces us to modify our common notion of localization with respect to a nondynamical background, in particular background space-time points (represented by manifold points) within field theories. In the framework of GTR, physical entities can only be meaningfully localized with respect to the dynamical gravitational field, and more generally (Dirac) observables of the theory might only be meaningfully defined in terms of relations between dynamical variables. From this point of view, the trouble with gravitational energy-momentum localization might well be linked to the fundamental GTR feature of background independence: there is no (nondynamical) background with respect to which gravitational (and nongravitational indeed) energy-momentum can be defined and localized. For practical purposes, the introduction of some background structure seems mandatory for defining and localizing gravitational (and nongravitational) energy-momentum. The case is actually similar to the possibility of defining meaningful time evolution and initial value formulation within GTR: some ‘background gauge fixing’ in the form of a decomposition (together with the choice of some coordinate system) is required in order to implement any useful calculation. But this does not mean that such decomposition or such background structure is fundamental.

7. Asymptotic Symmetries and Quasi Localization: The Need for Background Structures

In the general case, one cannot define unambiguously the amount of total (gravitational and nongravitational) energy(-momentum) there is in any space-time region; obviously, there is no question of conservation for the (ill-defined) total energy ‘contained in’ a given spacelike three-region. Now, if one takes this conclusion seriously, that is, if one takes GTR ‘seriously’, it may well have important metaphysical consequences: according to some standard conception, conservation is what makes the very notions of energy and mass (and momentum) so fundamental (recall the quotation in n. 6). In the light of the difficulties with energy conservation as well as with the very notion of energy itself, this standard conception leads us to the conclusion that energy and mass (and momentum) may not be fundamental properties (within GTR).

To be more precise, strictly speaking, these notions make sense only once appropriate background structures are introduced: especially useful for practical purposes is, for instance, the notion of asymptotic flatness, which encodes the notion of an isolated system within GTR where one therefore assumes that space-time becomes flat as one ‘approaches’ (spatial or null) infinity. The relevant asymptotic symmetries with the corresponding conserved quantities at (spatial or null) infinity can then be defined, namely, the Arnowitt-Deser-Misner (ADM) and Bondi energies. The ADM energy associated with some (asymptotically flat) spacelike hypersurface (with respect to some foliation) represents the total energy ‘contained’ in (whole) and is conserved under the ‘time’ evolution . These notions of energy depend crucially on the background structures inserted in the form of asymptotic symmetries and are fundamentally global in the sense that they are associated with the whole (asymptotically flat) space-time.

The ‘quasi-local’ research program has hoped for several decades to remedy this last point with its various attempts to define energy-momentum for extended but finite space-time regions, such as, typically, a compact spacelike (three-dimensional) region (or its closed two-dimensional boundary). There are mainly two different ways to define quasi-local energy: first, using the variational method within the Lagrangian or Hamiltonian setting and, second, trying to ‘quasi-localize’ global expressions such as the ADM energy. Unfortunately, these different methods lead to different expressions (Brown-York energy, Bartnik mass, etc.), the links between which are not always clear; as acknowledged in a recent review on the topic, there is for the time being no consensus on how to build a ‘quasi-local’ notion of energy, that is, how to define the energy or mass associated with some finite space-time region (Szabados Reference Szabados2009). More importantly, all the quasi-local expressions depend on particular background structures (or gauge choices), such as the dependence on some particular embedding or on some particular boundary conditions, for instance.Footnote ¹³

As a conclusion, I would like to highlight two points whose metaphysical implications need to be further investigated: first, in the cases in which total energy-momentum is well defined (and conserved), it is a global notion. Second, it seems that within GTR all meaningful notions of (gravitational and nongravitational) energy-momentum (and mass) require the introduction of some background structures,Footnote ¹⁴ which correspond to some particular ways of describing the world (gauge choices) but which might not be intrinsic to it. That is, energy and mass might not be fundamental properties of the world, in the sense that they make only sense in some particular (but very useful) settings; this does not lessen the fact that the notions of (local, quasi-local, or global) energy and mass constitute extremely powerful tools for many concrete and practical cases.