r^{2}}\left(\mbox{d}r^{2}+\gamma_{ij}(r,x)\,\mbox{d}x^{i}\,\mbox{d}x^{j}\right)~{},$
(61)

where $x^{i}$ are coordinates tangent to the boundary directions, and
$\gamma_{ij}(r,x)$ is a metric along this direction, that in general depends
on both $r$ and $x$. We obtain the earlier-used $\gamma_{ij}$ by setting
$r=0$, that is: $\gamma_{ij}(x):=\gamma_{ij}(0,x)~{}.$ The diffeomorphisms we
are interested in are those that “fix the location of the boundary $r=0$”.
They take the following form:

$\displaystyle\delta x^{i}=\xi^{i}(x)~{},~{}~{}\delta r=-r\,\xi(x)~{}.$ (62)

Requiring that these diffeomorphisms satisfy the definition of ‘being
visible’, i.e. fix: (i) the Poincaré normal form of the metric, i.e. Eq. (61),
and (ii) the boundary metric, implies the following condition on the
diffeomorphism generated by the boundary vector field $\xi$:104104104See Brown
and Henneaux (1986), De Haro (2017a: p. 1477).

$\displaystyle\nabla_{i}\,\xi_{j}(x)+\nabla_{j}\,\xi_{i}(x)$ $\displaystyle=$
$\displaystyle{2\over d}\,\gamma_{ij}(x)\,\nabla^{k}\,\xi_{k}(x)~{},$ (63)

where the covariant derivatives are with respect to the boundary metric
$\gamma_{ij}(x)$. This equation is the conformal Killing equation, which is
precisely the condition for the vector field $\xi$ to be a conformal
transformation, rather than an isometry, of the boundary metric
$\gamma_{ij}(x)$.105105105The conformal Killing equation reproduces the
ordinary Killing equation, Eq. (15), in the special case
$\nabla^{k}\xi_{k}=0$. This gives us a very general relation between the
symmetries of the metric in $M$, in Poincaré normal form Eq. (61), and the
conformal group on $\partial M$:

The asymptotic symmetry group of the Poincaré normal form of the metric, Eq.
(61), on the manifold $M$, is the conformal group on the asymptotic boundary,
$\partial M$, with boundary metric $\gamma$.

This generalises the result by Brown and Henneaux (1986), for any solution of
Einstein’s equations with $\Lambda<0$.

These diffeomorphisms give us a conformal Killing vector $\xi$ of the boundary
manifold. As I noticed after Eq. (54), especially in footnote 77, using the
properties of the holographic stress-energy tensor, Eq. (56), this can be used
to derive a boundary analogue of the energy conservation equation Eq. (20).

##  5 Gravitational Energy and Momentum, Revisited

In this Section, I first review two recent discussions of pseudo-tensors that
I aim to justify: one is from physics (Section 5.1) and the other is
philosophical (Section 5.2). I will agree with much of these defences of the
pseudo-tensor approach—but I will nevertheless, in Section 6, argue for the
superiority, in the present state of knowledge, of the Brown-York approach.

###  5.1 Superpotentials are boundary terms in the Hamiltonian of GR

In recent work, Chang, Chen, Nester, and collaborators (1999, 2017, 2018,
2018a) have defended a rehabilitation of pseudo-tensors that thereby casts
light on the physical nature of pseudo-tensors. Their work can be seen as
arguing for a resolution of problem (iv) in Section 1.3. Namely, they argue
that pseudo-tensors:

(a) Do provide a description of energy and momentum conservation. They do
this, in particular, by establishing a relationship between pseudo-tensors and
the Hamiltonian of general relativity. The superpotential,
$U^{\mu\nu}_{\lambda}$, can be identified with a total divergence in the
Hamiltonian of general relativity, and so different choices of the
superpotential correspond to different boundary terms/conditions. Thus problem
number (iv) is thereby resolved, because fixing the boundary terms determines
the pseudotensor uniquely.

(b) Give, for apropriate choices of boundary conditions, the expected values
of the total energy and momentum as measured at infinity, i.e. the ADM and
Bondi formulas,106106106See for example Jaramillo and Gourgoulhon (2011: pp.
97-102, 105-107) and Christodoulou and Yau (1988: pp. 9-10). thus further
strengthening the contention, in the resolution of problem (iv), that pseudo-
tensors are physical.

(c) Have well-defined values in each system of coordinates. In particular, the
pseudo-tensors give quasi-local quantities, which are naturally identified
with the two-surfaces that bound regions of space, and give correct results at
infinity, as in (b).

Here I will focus on (a), which is the most distinctive part of the approach
by Chang, Chen, Nester, and collaborators. Then I will more briefly comment on
(b) and (c). Let me state, in words, the main statement that goes into arguing
for (a):

Pseudotensor-Hamiltonian correspondence: The quasi-local energy-momentum over
a hypersurface $\Sigma$ with constant vector $N$, associated with the pseudo-
tensor $t^{\mu}{}_{\nu}$ and superpotential $U^{\nu\lambda}_{\mu}$,
corresponds to a Hamiltonian $H(\Sigma,N)$ whose boundary term is given by the
expression $B(N)$ (and vice versa). The choice of the superpotential
$U^{\nu\lambda}_{\mu}$ in the pseudo-tensor determines a boundary term in the
Hamiltonian (and vice versa).

Working this out in the forward direction, i.e. showing that the quasi-local
energy-momentum gives rise to the Hamiltonian in the way stated, goes in three
main steps:

(1) Follow Einstein’s derivation of the pseudo-tensor $t^{\mu}{}_{\nu}$, in
Section 1.3, and express it in terms of the superpotential
$U^{\mu\lambda}_{\nu}$, in Eq. (24).

(2) “Integrate Einstein’s derivation” to get a pseudo-tensor, analogously to
the steps in Eqs. (16)-(18). Here, Chen, Nester et al. construct an energy-
momentum quantity $P(N)$ by integrating the total local energy-momentum
density (including its matter and gravitational contributions) over a three-
dimensional spacelike hypersurface $\Sigma$. Upon the use of the Einstein
field equations, the result is a boundary term, i.e. an integral over
$\partial\Sigma$. Thus in this step they associate to each pseudo-tensor, and
solution of Einstein’s field equations, a quasi-local quantity—and this is
their preferred formulation of the problem.

The details are as follows: they define energy and momentum by integrating Eq.
(21) with an arbitrary vector $N$:

$\displaystyle P(\Sigma,N)$ $\displaystyle:=$
$\displaystyle\int_{\Sigma}\mbox{d}\Sigma_{\nu}\,(T^{\nu}{}_{\mu}+t^{\nu}{}_{\mu})N^{\mu}$
(64) $\displaystyle=$
$\displaystyle{1\over\kappa}\int_{\Sigma}\mbox{d}\Sigma_{\nu}\,[(-G^{\nu}{}_{\mu}+\kappa\,T^{\nu}{}_{\mu})N^{\mu}+{1\over
2}\,\partial_{\lambda}(N^{\mu}\,U^{\nu\lambda}_{\mu})]$ $\displaystyle=$
$\displaystyle\int_{\Sigma}{\cal
H}_{\mu}\,N^{\mu}+\oint_{\partial\Sigma}B(N)~{},$

where I have introduced the notation ${\cal
H}_{\mu}=\mbox{d}\Sigma_{\nu}({1\over\kappa}\,G_{\mu}{}^{\nu}-T_{\mu}{}^{\nu})$
and $B(N)=-{1\over
4\kappa}\,\mbox{d}S_{\nu\lambda}\,N^{\mu}\,U^{\nu\lambda}_{\mu}$ for the first
and second terms, respectively, in the last line. Going from the first to the
second line uses Eq. (24), and going from the second to the third line uses
Gauss’ law to rewrite the total derivative, integrated over the hypersurface
$\Sigma$, as an integral over the boundary $\partial\Sigma$.

In this derivation, $N$ is taken to be an arbitrary vector that is constant
with respect to the given system of coordinates. Such a vector always exists,
although it will be different for each system of coordinates. This will be an
important point of discussion in Section 6, in connection with point (c)
above.107107107Nester (2020, personal communication) argues that, in Eq. (64),
‘the volume integrand is proportional to the field equations, and so vanishes
on-shell. The value of the integral comes only from the 2-surface boundary
term. This means that we can extend our formula, modifying the vector field
$N$ within the region, without changing the value of the integral, as long as
one keeps $N$ constant on the boundary. Thus it is only necessary for the
vector field to match a translational Killing field of the Minkowski
coordinate reference geometry on the boundary’. Although this is of course
true, the independence of the value of the vector field $N$ in the region
inside the volume $V$ is only valid in the last two lines of Eq. (64), i.e.
for the Hamiltonian expression. However, the step from the first to the second
line of Eq. (64) does not hold if $N$ is not constant, because the second term
is no longer a total derivative. In other words, the above direct relation
between the integrated total energy-momentum pseudo-tensor (i.e. energy-
momentum plus gravitational pseudo-tensor) and the Hamiltonian expression only
holds if $N$ is everywhere constant in a system of coordinates adapted to the
boundary. (Nester’s main interest is of course the Hamiltonian formalism, and
less so the pseudo-tensor. But, in so far as these papers aim to justify the
use of pseudo-tensors through the Hamiltonian approach, my point is that this
only works for constant vector fields $N$ and special systems of coordinates,
as just argued). For the moment, I will set this aside.

As announced above, and as we see from the middle expression in Eq. (64), the
volume term in the Hamiltonian vanishes when Einstein’s field equations are
satisfied, and we are left only with the boundary term $B(N)$, integrated over
a 2-surface. Thus we have succeeded in constructing, from the local energy and
momentum, a quasi-local quantity.108108108This is true at least formally. See
Section 6 for a discussion.

(3) Recognise that the expression Eq. (64), which is already written in
suggestive notation, takes exactly the form of the Hamiltonian of general
relativity, associated with a finite region $\Sigma$ and constant vector $N$:

$\displaystyle P(\Sigma,N)=-H(\Sigma,N)~{}.$ (65)

The Hamiltonian is the generator of spacetime displacements of the
hypersurface $\Sigma$ along a vector field $N^{\mu}$, and ${\cal H}_{\mu}$
indeed has the form in Eq. (64), and vanishes if the field equations are
satisfied.

$B(N)$ is the boundary term of the Hamiltonian, which gives the quasi-local
energy and momentum, i.e. it is an expression for the energy and momentum on
the 2-surface $\partial\Sigma$. This boundary term, as it appears in the above
calculation starting from the pseudo-tensor, contains the superpotential.
Given this correspondence, between superpotentials and boundary terms, the
Hamiltonian approach casts light on the physical significance of the various
choices of superpotentials. In particular, the variation of the boundary term,
$\delta B(N)$, must be set to zero in order to get the Einstein field
equations in their Hamiltonian form. Filling in various the choices of
superpotential $U^{\nu\lambda}_{\mu}$ in this variation, one finds that each
choice of superpotential leads to a specific class of boundary conditions. For
example, the Freud expression, Eq. (23), corresponds to keeping
$\sqrt{g}\,g^{\mu\nu}$ fixed at the boundary,109109109The condition is
$\delta(\sqrt{g}\,g^{\mu\nu})=0$, which gives $\delta g^{\mu\nu}={1\over
2}\,g^{\mu\nu}g_{\alpha\beta}\,\delta g^{\alpha\beta}$. This allows, but does
not require, the Dirichlet boundary condition $\delta g^{\mu\nu}=0$, and thus
it is slightly more general than a Dirichlet boundary condition. while keeping
the vector field $N$ constant. Thus the Freud superpotential allows Dirichet
boundary conditions.

Other choices of superpotential lead to other boundary conditions, e.g.
Neumann boundary conditions where the momentum conjugate to the metric is held
fixed. For each choice of superpotential there is a corresponding boundary
condition, and a quasi-local energy expression.

To show that the pseudotensor-Hamiltonian correspondence also works in the
opposite direction, one reverses the above steps: begin with the Hamiltonian,
and show how its boundary terms correspond to various superpotentials in the
energy-momentum pseudo-tensor.

The expression Eq. (64) for the Hamiltonian, and in particular the various
choices of its boundary term, can be motivated further by using the
Hamiltonian formalism. To do this, Chen, Nester et al. construct the
Hamiltonian of general relativity in form notation. In this formalism, the
various boundary terms $B(N)$, that naturally enforce Dirichlet or Neumann
boundary conditions in the variation, take simple forms in terms of three
geometrical quantities and their corresponding conjugate momenta: namely, the
metric, and the orthonormal frame and coframe fields. These relatively simple
forms of the boundary terms reproduce, when worked out in tensor components,
the various expressions for the superpotential found in the literature. Thus
the form notation for the Hamiltonian explains the relatively long tensor
formulas for the superpotential through simple boundary terms.

###  5.2 Recent philosophical discussions of energy conservation in GR and
pseudo-tensors

The philosophical discussion of energy conservation in general relativity
revolves around two main problems, mainly to do with problems (v) (lack of
covariance, and lack of appropriate transformation properties) and (iii) (non-
integrability) in Section 1.3. (Surprisingly, Lam (2011: p. 1017) and Read
(2020: p. 215) mention the problem (iv) of non-uniqueness, which in my view is
pressing, only in passing; other papers do not appear to mention
it.)110110110Duerr (2019: pp. 2, 6, 9-10) does stress the importance of this
ambiguity, which he calls ‘baneful’ (p. 9). The first is about coordinate-
dependence; the second is about background-dependence. I give them new labels,
(A) and (B), to cover other aspects of recent discussions:

(A) Lack of covariance, and lack of appropriate transformation properties. The
known expressions for the local energy and momentum in general relativity are
not tensorial. In particular, the pseudo-tensors Eq. (24) are coordinate-
dependent, and so they do not have good local (or global) definitions. (As I
mentioned in Section 1.3, pseudo-tensors are only invariant under affine
transformations).

For example, Hoefer (2000: pp. 194-195) has stressed that the expressions for
energy and momentum, even in their integrated form Eq. (50), are coordinate-
dependent, and so are ill-defined—as I already mentioned at the end of Section
3.

One immediate comment to make about (A) is that it is diffeomorphism
invariance, rather than covariance, that is most important here: and that, as
we saw in Section 4.3, the various kinds of diffeomorphisms require a
different treatments. Second, although the lack of covariance can be
technically and physically inconvenient, the main worry is the pseudo-tensor’s
lack of appropriate transformation properties.

(B) “Background-dependence”. The conservation of energy and momentum in
general relativity requires the introduction of “background structures”: such
as a flat metric (usually, at infinity) or timelike Killing vectors in the
spacetime, see Eq. (15) and the discussion following.111111111Absolute objects
are defined e.g. in Anderson (1967: pp. 83-84; 1971: p. 166). However, as we
will discuss below, these background structures are not absolute objects in
this sense. Without such additional structures, the existing expressions for
the energy-momentum tensors do not express genuine conservation laws, and do
not give unique answers for the physical energy and momentum. While both
Hoefer (2000) and Lam (2011) admit that expressions like Eq. (50) do yield
unambiguous results when such structures are added, they ultimately reject
them, because they amount to introducing undesired “background
structures”.112112112Background-independence is obviously a large topic that I
cannot enter in detail into here, nor will that be required. For, as I
mentioned before, the “background structures” discussed in this context are
not the same as the ones discussed in the literature on background-
independence. Thus my comments will be restricted to the specific discussion
of the role of background structures in Noether’s second theorem. For more
general treatments of background-independence: see, for example, Pooley (2017:
pp. 24-29) and Read (2016a). For my own views about background-independence,
see De Haro (2017).

For example, Hoefer (2000: p. 194) criticises the calculations of the energy-
momentum conservation equation Eq. (50) that evaluate integrals at infinity to
get the total energy and mass of the spacetime, because they assume that the
spacetime is asymptotically flat and that the coordinate system at infinity is
Minkowskian. This does not give a genuine conservation principle for general
relativity, the most important reason being that fixing the asymptotics in
this way ‘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’. Thus points (A) and (B) are, for Hoefer,
really part of a single worry: namely, coordinate-
dependence.113113113Furthermore, Curiel (2019: p. 91) proves a theorem that
there exists no local, divergence-free, covariant two-tensor other than a
multiple of the Einstein tensor, and thus no such local candidates for a
gravitational energy-momentum tensor.

Lam (2011: p. 1010) is more tolerant about coordinate-dependence: he concedes
that ‘there does not seem to be any contradiction’ in relying on ‘background-
dependent (coordinate-dependent) entities’. However, he worries that there is
no unique definition of pseudo-tensors, because the possibility to add a
superpotential to them means that ‘one cannot say how much energy there is in
a given spatial region’, i.e. problem (iv), non-uniqueness.114114114Pitts
(2010: pp. 605-606) proposes to restore the covariance of the gravitational
energy-momentum expressions by embracing their multiplicity: namely, by
constructing a single geometric object whose infinitely many components are
the components of a given pseudotensor with respect to all possible coordinate
systems, so that they transform into each other under coordinate
transformations. In the text quoted, Lam is commenting on Pitts’ proposal, but
similar remarks could apply to the pseudo-tensor itself.

But his main worry is about the introduction of “background structures”. For
example, although the standard integral formulation of the conservation
equation with a Killing vector, Eq. (19), is a genuine energy-momentum
conservation law for the matter fields, it requires the introduction of the
Killing vector $K$, which Lam considers is an undesired “non-dynamical
background”. And likewise for the introduction of asymptotically flat metrics
at infinity in order to reproduce the ADM and Bondi formulas: these ‘depend
crucially on the background structures inserted in the form of asymptotic
symmetries’ (p. 1022).

Read (2020) has recently responded (at least partially) to the critiques (A)
and (B), by arguing that a genuine stress-energy tensor does exist in general
relativity, in both: (i) a weak sense applicable in a certain class of models
of the theory (namely, those that admit appropriate Killing vectors), (ii) a
strong sense, applicable in all models.

The above distinction between cases (i) and (ii) is underpinned by the
existence of a timelike Killing vector in the spacetime. Namely, in response
to (B), Read (2020: p. 222) argues that the Killing vector $K$ that gives the
well-defined energy conservation law Eq. (20), should not be seen as
“background structure”. For the existence of a Killing vector is merely a
property of the metric: if the metric has an isometry, then a Killing vector
exists. This simply means that the models with a Killing vector are a subset
of all the dynamically possible models of the theory: no additional structures
are introduced into the theory.

Read’s advocacy of energy-momentum conservation appeals to functionalism (i.e.
briefly, the idea that there exists in GR a quantity that fulfils the
functional role of gravitational stress-energy: cf. p. 224) and to the
possibility of being a realist about pseudo-tensors. The possibility of
realism is motivated by Einstein’s (1918b: pp. 449-451) own position that
quantities can be coordinate-dependent, but still well-defined. Just as the
connection coefficients are frame-dependent and are intimately tied to the
gravitational field115115115See Einstein (1916: p. 802), and also the
discussion in Norton (1985: p. 233). (recall that the Newtonian gravitational
potential arises from the Christoffel symbols), it is prima facie plausible to
interpret the gravitational pseudo-tensor as representing gravitational
energy, momentum, and stress in a given frame of reference. Thus Read proposes
that gravitational stress-energy is always defined with respect to a given
frame of reference, analogously to how the relative velocity of a body is
evaluated in the rest frame of another body (but see the objections in the
next Section).

He argues that in case (i), i.e. in those models admitting timelike Killing
vectors, in which the conservation law Eq. (19) exists, the quantity
$t^{\mu}{}_{\nu}$ does fulfil the role of a gravitational energy-momentum
tensor, i.e. it fulfils a role analogous to that of gravitational energy as
traditionally conceived, and bears an appropriate relation to the
gravitational degrees of freedom. He proposes, based on (a) the pseudo-
tensor’s fulfilling this functional role, on (b) the larger explanatory
apparatus that the advocates of gravitational energy have available to them,
and on (c) the simple and practical character of functionalism involved: that,
for this subset of models of the theory, the gravitational stress-energy does
exist, and should be labelled as genuine energy, momentum, and stress. Thus a
frame-independent gravitational stress-energy tensor does, in this sense,
exist in general relativity.

In case (ii), although a frame-independent stress-energy tensor is not
available, one can still be a realist about the pseudo-tensor. Namely, realism
about the pseudo-tensor is compatible with functionalism, since the pseudo-
tensor may be defined, in each frame, such that it is conserved. In
particular, he argues from the broader idea that it is acceptable to give
explanations of phenomena by appealing to coordinate-dependent effects. If one
accepts this realist line, then the gravitational stress-enegy exists in the
stronger sense, which is applicable to all the models of the theory.

While I will in part agree with Read’s two arguments in response to (A) and
(B), I will discuss, in the next Section, whether these arguments conclusively
establish the existence of local energy and momentum in general
relativity—since indeed they do not offer a satisfactory response to (A).

##  6 A Comparison of Two Proposals

In this Section, I will compare the two proposals for energy and momentum from
the previous two Sections. In Section 6.1, I discuss three worries about the
gravitational pseudo-tensor that, in my opinion, should be addressed before
pseudo-tensors are taken to be part of the ontology of general relativity. In
Section 6.2, I will defend the holographic stress-energy tensor from two
general worries, articulated in the philosophical literature on energy and
momentum in general relativity, and already mentioned in the previous Section.
In Section 6.3, I will discuss further questions that one might have about the
physical interpretation of quasi-local quantities.

###  6.1 What else is required for us to accept pseudo-tensors?

In this Section I make three qualifications about Read’s (2020) rebuttal of
(A) and (B).116116116A fourth point is that pseudo-tensors are non-unique
because one can add an arbitrary superpotential to them, and so it is not a
priori clear how to interpret them. However, as I discussed in Section 3, this
is clarified by Noether’s second theorem, which establishes a clear relation
between the superpotential and the boundary term in the first variation of the
action, see Eq. (32). My second qualification is in the neighbourhood of a
more technical objection by Duerr (2019a: Section 3.3) (I will return to the
resolution of a number of Duerr’s other objections at the end of Section 6.2).
My third qualification leads in to Section 6.2.

The first qualification is that, while Read (2020) frames his defence of
pseudo-tensors in terms of scientific realism, this is not necessary. For also
the empiricist, e.g. van Fraassen (1980), could argue along Read’s
functionalist lines, and conclude that pseudo-tensors are physical quantities,
which describe properties of some entities in the world—in this case, an
unobservable entity, viz. gravitational energy and momentum. Agreed: this
depends upon the empiricism under consideration. But the point is that the
significance of pseudo-tensors goes beyond the question of whether there is a
justified realism about them: rather, it is a matter of whether the semantics
of GR must invoke such an entity.

Thus while I agree that there is no a priori reason why coordinate-dependent
quantities could not be physical, and also that, in general, some coordinate-
dependent quantities are physical, I think that more work is required to admit
these specific coordinate-dependent quantities—pseudo-tensors—into one’s
theory. This leads in to my next two points, about further conditions that are
required to ascribe to pseudo-tensors a semantics that is physically
acceptable.

The second qualification is that, to accept pseudo-tensors as good physical
quantities, one requires that they have well-defined transformation properties
under (different kinds of) diffeomorphisms. (Of course, pseudo-tensors
transform covariantly only under affine transformations). Even if one is in
principle sympathetic to the idea of coordinate-dependent quantities, surely
some relationship between the quantities in the different coordinates, at
least for special classes of coordinates, must be required before such
quantities can be accepted in the theory.117117117For example, when the number
of spacetime dimensions is odd, under visible diffeomorphisms the quasi-local
stress-energy tensor picks up an anomalous term that is in general non-zero,
i.e. Eq. (60). Although this breaks covariance, it is a well-defined
transformation law, and it is admissible. As the analogy below should make
clear, the real physical worry is that different coordinate systems would be,
quite literally, incommensurable. This worry is, of course, in the
neighbourhood of a closely related one: namely, that pseudo-tensors are not
geometric objects118118118However, as Trautman (1965: p. 86) points out,
spinors are not geometric objects either. (see Anderson (1967: p. 424) and
Duerr (2019a: Section 3.3)). Therefore, it is not clear that the gravitational
pseudo-tensor gives well-defined values for the energy in the various
coordinate systems.119119119In this context, Nester argues that the
Hamiltonian is a fully covariant object, and that its significance is
independent of the coordinates. But this is not entirely satisfying, for two
reasons: (i) As I already mentioned in footnote 107, the direct relation Eq.
(64) between the Hamiltonian and the pseudo-tensor only holds for everywhere
constant vector fields $N$: and so, a putative covariance of the Hamiltonian
does not automatically apply to the pseudo-tensor (which is expressly known to
not be non-covariant). (ii) As I mentioned in my reply to (A) in Section 5.2,
covariance is not what matters the most, because it can be easily achieved by
introducing additional quantities. Specifically, the Hamiltonian depends on a
choice of vector field $N$, and the Hamiltonian approach is only covariant
relative to a given such choice. But since spacetimes do not come naturally
equipped with vector fields, the Hamiltonian approach is dependent on such a
choice, which makes it liable to either: (a) introducing background
structures, or (b) treating different models differently. This is something
that Nester’s work, in my opinion, does not clarify.

Thus Hoefer’s (2000) argument that the quantities are not well-defined still
seems to have some force, because a satisfactory definition of energy and
stress would involve either giving their transformation rules or else giving a
physical interpretation of the non-covariance (e.g. as in the case with
boundaries, see Section 6.3). Lacking these, the realist story about pseudo-
tensors seems less convincing.

Consider the analogy with special relativity, where the 4-momentum vector $p$
is a physical quantity that transforms covariantly under Lorentz
transformations, but its components are not invariant. Choose a coordinate
system where we write $(p^{\mu})=(E/c,{\bf p})$, where $E$ is the energy and
${\bf p}$ the spatial momentum as measured in those coordinates. Then we know
that $E^{2}/c^{4}$ and ${\bf p}^{2}$ are not invariant quantities, whereas
their difference, $-E^{2}/c^{4}+{\bf p}^{2}$, is invariant (and likewise for
covariant quantities). The energy and the momentum transform into each other
under Lorentz transformations, and so their definition depends on the
coordinates chosen. Despite this coordinate-dependence, $E$ and ${\bf p}$ are
physical quantities, and we can:

(1) define energy and momentum for each choice of coordinates (these
definitions being incompatible in the sense that what is defined as energy in
one system is not the energy in another system), and furthermore these
definitions can be made operational with respect to the corresponding frame,
e.g. a preferred measuring apparatus;

(2) give well-defined transformation laws for $E$ and ${\bf p}$ from one
coordinate system to another: namely, the Lorentz transformation law.

Property (2) seems to fail in the case of pseudo-tensors, and thus there is a
disanalogy with other coordinate-dependent quantities that we are familiar
with. The point is not that pseudo-tensors fail to be covariant, but rather
that there is no well-defined transformation law between them.120120120As I
showed in Section 4.3, in spaces with boundaries, different kinds of
diffeomorphisms need to be distinguished, and for some diffeomorphism one may
obtain a modified transformation law, involving the anomalous term in Eq.
(60). Even if it were still possible to define energy and momentum in every
coordinate system, as in (1), these different notions would be completely
unrelated to each other—different coordinate systems would be, quite
literally, incommensurable.

The point is that it is not quite clear that one can simply drop the
requirement of well-defined transformation properties under diffeomorphisms
and put nothing in its place: a coherent alternative for it seems required. At
any rate, this is an objection that the programme needs to
address.121121121Pitts (2010) is one such proposal. Duerr (2019a: Section 3.3)
notices, that, while ‘nothing is inherently baneful about coordinate-dependent
objects… [f]or pseudotensors… the coordinate-dependence is “vicious” (Pitts):
in general, the preferred coordinate transformations do not pick out the
characteristic invariants of the spacetime. The spacetime symmetries do not
align with the pseudotensor’s symmetry group’.

A third qualification is that, even if Read’s arguments indeed allow taking
pseudo-tensors as physical quantities, they do not require this. Pseudo-
tensors might be the correct definition of energy and momentum in general
relativity—or they might not: for there are alternative proposals—in
particular, the quasi-local proposals. Thus the different proposals need to be
compared. I do this in the following Section.

###  6.2 Quasi-local expressions: superpotentials vs. the Brown-York stress-
energy tensor

In this Section, I will defend the ‘modified’ version of the Brown-York quasi-
local stress-energy tensor from Section 4 as a preferred quasi-local
definition of energy, momentum, and stress in general relativity. I will
partly do this by comparison with the other main proposal, from Section 5:
namely, the pseudo-tensor, and the quasi-local Hamiltonian expression obtained
from it. I will argue that the former is so far better developed, and thus
better able to answer various queries. Hence my favouring, on balance, the
Brown-York quasi-local stress-energy tensor. But since some of the details
about the pseudo-tensor approach have not been fully worked out, the latter
approach, with all its open questions, should still be considered.

By the ‘modified’ version of the Brown-York quasi-local stress-energy tensor’,
or simply the ‘Brown-York stress-energy tensor’, I will mean the quasi-local
Brown-York stress-energy tensor from Section 4, modified, where this technique
is available (i.e. in the case $\Lambda\not=0$, regardless of its
sign),122122122The modified Brown-York stress-energy tensor, in the case
$\Lambda>0$, has been worked out in the references in footnote 79. by the
inclusion of counterterms: as opposed to the older reference field
configuration subtraction (see (B)-(b) below). This is all, in principle,
regardless of its possible holographic interpretation.

The quasi-local perspective has been almost entirely missed in recent
philosophical discussions of energy conservation. Lam (2011: p. 1022) mentions
the existence of quasi-local stress-energy tensors, but then dismisses them,
on two grounds:

(1) A ‘divide and conquer’ argument to the effect that there are different
expressions in the literature for such tensors, and that ‘the links betwen
[them] are not always clear… there is for the time being no consensus on how
to build a ‘quasi-local’ notion of energy’.

(2) The quasi-local expressions allegedly also depend on particular background
structures, ‘such as the dependence on some particular embedding or on some
particular boundary conditions’.

I will respond to both arguments below. Let me here say, about (1), that it is
not a strong argument: for the mere existence of more than one expression,
without further analysis or elaboration, is not a good reason to reject them
collectively, but rather to further investigate which one is correct, or
whether they are related. Furthermore, at least in cases with a non-zero
cosmological constant, and pace Szabados (2009) which Lam cites, it is not
really true that there is no consensus in—a large portion of—the physics
community about which is the preferred expression123123123The literature here
is too large to survey. Some examples are e.g. the textbook Ammon and
Erdmenger (2015: p. 210), Anninos et al. (2011: p. 5), Balasubramanian and
Kraus (1999: p. 413), and Skenderis (2001, 2002). (by itself, this need of
course not be a sufficient reason to join the consensus, but neither should
the reasons for the consensus be ignored).

My comparison here will be along two main lines, which I will label (A) and
(B), that parallel the philosophical discussion in Section 5.2 (and go back to
points (iii) and (v) in Section 1.3). The first is a modification of point (A)
in Section 5.2, namely: diffeomorphism-invariance; the second is, like (B) in
Section 5.2, about “background-dependence”, or the requirement that the theory
should not contain any “background structures”. Then, at the end of the
Section, I will ask how the two approaches, the pseudo-tensor/Hamiltonian and
the Brown-York approaches, are related.

(A) The issue of diffeomorphism invariance. I begin with the Brown-York
approach. As I discussed in some detail in Section 4.3 for the case of
spacetimes with a negative cosmological constant, the properties of the
modified Brown-York stress-energy tensor under diffeomorphisms of different
kinds are well-understood, and can be summarised in three points:

(1) The modified Brown-York stress-energy tensor is invariant under
diffeomorphisms that express true redundancies (in the sense of (Redundant)),
i.e. invisible diffeomorphisms.

(2) It is covariant under visible diffeomorphisms, i.e. those that generate a
conformal transformation on the boundary (see Eq. (63)).124124124As I said
before, I am ignoring the anomaly term in Eq. (60), which vanishes identically
in four-dimensional spacetimes.

(3) It is not invariant under those diffeomorphisms that express physical
transformations and are not (Redundant), i.e. those that give physically
inequivalent solutions of Einstein’s field equations. This is as one would
expect (see also the discussion in the next Section).

Thus the modified Brown-York stress-energy tensor has correct transformation
properties under these different kinds of diffeomorphisms.

Also, up to a possible anomaly of the diffeomorphism group of the type
discussed in Section 4.3, it is clear from the Brown-York construction that
their stress-energy tensor transforms covariantly with respect to arbitrary
reparametrisations, $\xi_{i}(x)$, of the boundary manifold (i.e. a visible
diffeomorphism) for any value of $\Lambda$. This shows that the modified
Brown-York stress-energy tensor is not liable to the allegation (A) from
Section 5.2.

By contrast, in the case of pseudo-tensors and the quasi-local expressions
derived from them in Section 5, an analysis along these lines has, so far as I
know, not been carried out. There are, however, some remarks in the
literature. For example, Chang et al. (1999: p. 3) write:

> [T]he energy-momentum defined by such a pseudotensor does not really depend
> on the local value of the reference frame, it is actually quasi-local—it
> depends (through the superpotenial) on the values of the reference frame
> (and the fields) only on the boundary of a region.

The quote is perhaps somewhat optimistic. First, as I noticed in Section 5.1,
there are assumptions that go into deriving the quasi-local expression Eq.
(64) from the pseudo-tensor, which surely do not involve just the boundary
itself, but at least an open neighbourhood of it. Furthermore, the ‘dependence
on the values of the reference frame’ of the quasi-local energy that Chang et
al. mention seems incompatible with its being covariant, since the reference
frame dependence comes in through the superpotential, which is not a tensor,
also not when evaluated at the boundary. In other words, we appear to be in
the situation that I discussed in Section 5.2, where there is a strong
dependence on the frame of reference chosen at the boundary, and apparently no
rule to compare one system to another.125125125One might think that the
Hamiltonian formalism comes to the rescue here. However, two problems need to
be addressed: first, the Hamiltonian depends on a choice of vector field $N$,
and there is no natural way to choose this vector field in a general manifold;
second, relating the pseudo-tensor to the Hamiltonian requires a constant
vector field $N$, as already discussed in footnote 107. At any rate, these
authors do not clarify how the coordinate-dependence disappears. This
contrasts with the Brown-York stress-energy tensor, which as I discussed in
(2) does have satisfactory properties under changes of frame on the boundary.
Also, Chang et al. (1999) do not make a distinction between the
diffeomorphisms (1)-(3), whereas this is key to understand the transformation
properties of the quasi-local energy.

In Chen et al. (1999: Section 2) and Nester (2004: pp. 272-275), a ‘covariant
Hamiltonian formalism’ is worked out, in which as I already mentioned the
various superpotentials are reproduced from relatively simple geometric
expressions. Apart from the physical significance that one may ascribe to such
covariance (see footnote 119), it is not clear how the quasi-local energy and
momentum transform under (non-affine) changes of frame on the boundary, i.e.
whether there is any way to compare quantities between different boundary
frames.

Either way, although the Hamiltonian approach to pseudo-tensors seems
promising, it needs more work before a full judgment about diffeomorphism
invariance can be made.

(B) “Background-dependence”. Here, I begin with the pseudo-tensor/Hamiltonian
approach. I already endorsed, in Section 5.2 (B), Read’s response to claims
that pseudo-tensors require the introduction of background structures. Here I
will defend the Brown-York approach, as regards background-independence. This
will be three points, (a), (b), (c), of which (a) is a brief response to
certain criticisms of AdS-CFT:126126126For a fuller treatment and references,
see De Haro (2017: pp. 113-116); cf. footnote 112.

(a) It has occasionally, and informally, been claimed that the methods in AdS-
CFT are “background-dependent”, because they allegedly require the use of
anti-de Sitter boundary conditions. This is part of a more general criticism,
in Section 5.2-(B), that defining asymptotic expressions for energy and
momentum requires fixing the structure at infinity (typically, a flat metric).
Here, there are two points to be made.

First, the AdS-CFT correspondence, and in particular the calculation of the
modified Brown-York stress-energy tensor from Section 4.2, works for an
arbitrary, smooth boundary metric $\gamma$: and so, the claim that AdS-CFT
requires ‘AdS boundary conditions’ is incorrect. This means that the space is
not even asymptotically AdS: it is asymptotically locally AdS, i.e. there
exists, in an open neighbourhood of the boundary, always a choice of
coordinates where the space looks like AdS—locally! But the space is in
general not AdS: it is only required that the boundary metric have conformal
Killing vectors, as in Eq. (63), if the CFT is to have a non-trivial conformal
symmetry group. Thus the boundary metric is taken to have conformal
symmetries, but it does not need to have any pre-determimed form, like e.g.
being flat or being a round sphere. Examples in AdS-CFT have been worked out
for boundary manifolds of many different kinds.

Second, the above objection perhaps originates in a confusion about the role
of boundary conditions in AdS-CFT. As I discussed in Section 4.1, the original
Brown-York definition of the stress-energy tensor did have a dependence on a
given, non-dynamical, background spacetime, through the introduction of the
reference term in the action, Eq. (51). But as I also noted after Eq. (53),
this is not how the tensor is rendered finite in AdS-CFT. Rather, one adds
covariant counterterms that are intrinsic to the boundary. The renormalisation
method, from Section 4.2, of adding boundary counterterms in the action, has
the advantage of removing another kind of background: namely, the reference
spacetime. Thus there is no reference spacetime that one needs to compare
with.

(b) The role of boundary conditions in the approach that I am advocating here
thus differs from the idea of a reference field configuration or background
subtraction. For the boundary conditions required are merely those that are
dictated by the fact that Einstein’s equation is a second-order differential
equation. For example, in globally hyperbolic spacetimes, one standardly
specifies data on an initial Cauchy surface in order to solve Einstein’s
equations (think of the analogy with the hole argument!).127127127There are of
course also disanalogies: see Stachel (2014: pp. 14-15, 50). But this is not
in any sense a dependence on a background: rather, the specification of the
initial data is the specification of a solution among the set of all possible
solutions. If the spacetime had a boundary somewhere at a finite distance, in
order to specify a solution one would, in addition, need to specify boundary
conditions. This is what one does when one solves differential equations in
spaces with boundaries: it has nothing to do with imposing backgrounds, but
rather with choosing one’s solution of interest out of the total set.

Thus, the background-independence of the modified Brown-York stress-energy
tensor is “as good as it gets” in general relativity.128128128Janssen (2014:
pp. 198-202) discusses Einstein’s struggle, in his debate with de Sitter, with
the need to impose boundary conditions at infinity, and in connection his
(failed) attempts to incorporate Mach’s principle into general relativity. See
also Norton (1993: pp. 806-809). In De Haro (2017: p. 115), I argue that there
are two distinct senses of ‘background-independence’: only one of which is
implemented in general relativity—hence the ‘as good as it gets’.

The situation is quite different for the pseudo-tensor and its quasi-local
energy and momentum. First of all, these methods have only been worked out
using reference spacetimes, both in the cases $\Lambda=0$ and $\Lambda\not=0$.
Furthermore, to my knowledge, the asymptotics considered is only the simplest
one: namely, Minkowski space and (A)dS space. Thus only cases with a flat
boundary metric $\gamma$ have been considered. This may be only a matter of
technical development: but it is of course yet to be seen whether the pseudo-
tensor gives sensible (e.g. finite!) results in other cases.129129129It also
seems a sign of lack of communication between, at least some parts of, the
communities that the intrinsic renormalisation methods have not been applied,
or even noticed, by those working on pseudo-tensors. For example, the review
by Szabados (2009), written 12 years after gauge-gravity dualities were
discovered by Maldacena, makes a single passing mention of it, and takes no
notice of how holographic renormalisation improves on the existing methods, by
not appealing to a reference spacetime (see the discussion after Eq. (53)).

Nester (2004: p. 178) describes the problem as follows: ‘The pseudotensor
coordinate reference frame ambiguity now reappears in the guise of the
reference field configration, which must be specified on the boundary’.

Despite their differences, both approaches emphasise the physical importance
of choosing boundary terms to define energy and momentum. In the
superpotential approach, boundary conditions/terms resolve the ambiguity in
the specification of the superpotential. Also in the Brown-York approach are
boundary terms central, since the stress-energy tensor is defined directly in
terms of boundary quantities.

This dependence of the energy and momentum on boundary conditions is natural:
for example, in classical mechanics, the momentum of a bullet of course
depends on the initial velocity with which it is shot.

(c) Nor should the requirement that the boundary spacetime has conformal
Killing vectors be seen as a dependence on a background, in the usual way. As
Read (2020: p. 222) mentions, choosing a Killing vector field is making a
choice of a model among all the possible dynamical models.

There is also a question of whether the pseudo-tensor and the Brown-York
approach are directly related. It is unclear that they are, and a direct
calculation of the Freud superpotential for the case $\Lambda<0$, which
includes Dirichlet boundary conditions, does not exhibit a clear or direct
relationship to the Brown-York stress-energy tensor. Nester (2004: p. 178)
claims that the Hamiltonian formalism establishes a relation between the two:
‘Quasilocally, with certain choices for the reference fields and displacement,
[our quasi-local expressions] reduce to the famous expressions of Brown and
York’. However, the exact relationship remains obscure.

As already discussed in Section 4.2, AdS-CFT gives, through its holographic
correspondence with the expectation value of the stress-energy tensor of the
dual quantum field theory, Eq. (57), independent support to the modified
Brown-York stress-energy tensor as being the gravitational energy and
momentum.

The advantage of the quasi-local approach is perhaps best illustrated by
mentioning how it solves several problems that Duerr (2019a: Section 3.3)
discusses in connection with the pseudo-tensor (and, as I said earlier, I
agree with much of Duerr’s critique).

The first has already been discussed: it is the lack of covariance, and the
pseudo-tensor’s lacking appropriate transformation properties, i.e. my (v)
from Section 1.3, which is Duerr’s label (H1). The quasi-local stress-energy
is both a geometric object and a (quasi-local) tensor under the relevant
diffeomorphisms (see Section 4.3), and so this problem does not appear.

Second, asymptotic flatness (Duerr 2019a: Section 3.3) is not a requirement of
this approach: since, as I have stressed, the quasi-local stress-energy tensor
is well-defined for both positive and negative values of $\Lambda$, and for
arbitrary boundary conditions on the metric, i.e. not just the purely de
Sitter or purely anti-de Sitter boundary conditions. In particular, this can
be done for de Sitter-type solutions, which, in cosmological settings, are
more realistic than the asymptotically flat ones. Furthermore, the divergences
can be correctly renormalised by the method of holographic renormalisation.
This is one of Duerr’s main points of critique, and so this is a clear
advantage of the quasi-local approach.

Third, as a general point (not specific to quasi-local energy): one of Duerr’s
(2019a: Section 3.2) main criticisms to Read (2020) is that the latter
assumes, rather than argues, that the ambiguity of the pseudo-tensor can be
resolved. But we have seen that Read’s “assumption” is in fact vindicated by
the details of Noether’s second theorem. For the ambiguity is only apparent:
it is fixed by a choice of boundary conditions, and of corresponding boundary
terms (and this important point is common to all three approaches I have
discussed).

Gauge-gravity dualities also cast further light on the interpretation of these
boundary conditions. Each choice of boundary conditions corresponds to a
different CFT (and also a different common core theory): namely, a CFT where a
different kind of source is held fixed, with a different Lagrangian, and with
a different path integral. The choices between such CFT’s are different
physical choices: which explains why their energies are different.

###  6.3 Interpreting quasi-local quantitites

Although, in previous Sections, I have addressed various objections to the
quasi-local energy-momentum expressions, one can still ask about the physical
meaning of such quasi-local expressions. After all, the traditional view in
general relativity is that physical quantities are closely associated with
geodesics and, in particular, with point coincidences, that are used to
delimit these quantities. Indeed, the view goes back to Einstein
himself.130130130See Einstein (1916: p. 776) and Kretschmann (1917: pp.
576-577, 579). However, elsewhere Einstein (1915c: p. 228) considers
physically real events as consisting of more general spatio-temporal
coincidences. For a discussion of Einstein’s (1916) paper and its prehistory,
see Sauer (2005). The point-coincidence argument, and how Einstein sees in it
an implicit objection to the hole argument, are discussed on p. 816. In more
substantivalist terms,131131131For a review of substantivalist and anti-
substantivalist arguments, see Pooley (2012: pp. 47-63). it is points that are
fundamental in general relativity, and other meaningful geometric quantities
ultimately reduce to quantities defined on points. But the pseudo-quantities,
especially the ones that—like the Brown-York stress-energy tensor—are directly
defined quasi-locally, without a local definition, would seem to give some
physical significance to spacelike two-surfaces, possibly located at infinity:
and it is not immediately clear what this means, both in terms of the theory’s
ontology (should we include surfaces in the theory’s ontology? And, if so,
which ones?) and in empirical terms (what is the empirical content of such
terms?).

I will here offer a number of arguments to the effect that surfaces and
boundaries should be counted as part of general relativity’s ontology, and
that they do have an empirical meaning.

First, notice that the fact that points are regarded, by some substantivalists
at least, as fundamental, does not exclude the reality of finite surfaces,
which, just like the manifold itself, are made of points. Also boundaries can
be understood in terms of neighbourhoods of the manifold,132132132The boundary
of a manifold $M$ is the set of points $p$ such that $p$ has no neighbourhood
homeomorphic with an open set of $\mathbb{R}^{n}$, where $n$ is the dimension
of the manifold.  and so as such what we call ‘the existence of a boundary’ is
a topological property of the manifold, which just is part of the ontology of
the theory. Thus, once we have a semantics for points and for neighbourhoods
in general relativity (and the latter are required for both topological and
geometric reasons), this very semantics gives a meaning to surfaces and
boundaries. This point can also be argued from two philosophical directions:

(a) An ontology that accepts points as objects should surely accept sets,
classes, or mereological fusions, of points. For example, this is standard in
“free lunch” supervenience metaphysics: see Lewis (1986: pp. 211-212),
Armstrong (1997: pp. 12-13), Sider (2001: pp. xvi, 7, 59).

(b) As Butterfield (2011) argues, in his campaign against pointillisme, not
all classical physical quantities are defined at a point (where ‘defined at a
point’ is here taken in the sense of Lewis’ (1986a: pp. ix-xvi; 1994) Humean
supervenience). Classical continuum mechanics gives us many examples of
‘spatial extrinsicality: the way that an ascription “reaches across space”’.
In particular, boundaries and quasi-local energy and momentum are as
admissible in general relativity as they are in classical mechanics:
‘continuum mechanics embraces spatial extrinsicality due to surface forces,
acting at a point, with respect to a surface through that point (see
Butterfield, 2011: pp. 354, 356; my emphasis).

Second, there is no contradiction with any fact about general relativity if it
turns out that energy and momentum can only be defined quasi-locally but not
everywhere locally. There is no a priori reason why energy and momentum should
exist everywhere locally on the manifold (see e.g. Lam, 2011: p. 1023). So, if
some quasi-local notion of energy and momentum is well-defined, rather than
being regarded with suspicion, it should be a welcome possibility.

Third, there are several physical reasons to consider boundaries, and stress-
energy tensors as physical quantities defined on them. One reason is that one
is interested in considering open systems, i.e. finite regions of the
universe. In such cases, the non-vanishing of the right-hand side of Eq. (54)
is interpreted as matter entering or leaving the region through the boundary.
These open systems are very important for several reasons: as modelling a real
system (e.g. a black hole, and the gravitational waves that it emits), or as a
way to generate new solutions by gluing a given solution to other solutions
through the boundary (for example, solutions with different values of the
cosmological constant, with so-called ‘domain walls’ in between: see e.g.
Freedman et al. (1999: pp. 376-377), Ammon and Erdmenger (2015, p. 302-304);
for gluing of spacetimes in a cosmological context, see Carrera and Giulini
(2010: pp. 170, 181)). One then needs to match the conditions of the two
solutions at the boundary, and the boundary metric and its conjugate variable,
viz. the stress-energy tensor, are the two canonical variables to control at
the boundary.133133133 If the Brown-York stress-energy tensor is evaluated not
at conformal infinity, but at some finite distance, then this can have several
other applications in string physics. For example, the universe is modelled in
some models as the four-dimensional boundary of a five-dimensional space,
placed at finite distance from the centre of the space (so that we ‘live on a
D-brane’: see e.g. Randall and Sundrum (1999: p. 4690)). In that case, the
Brown-York stress-energy tensor of the five-dimensional space acts as an extra
source for the matter energy-momentum appearing in the four-dimensional
Einstein field equations on the D-brane (De Haro et al. 2001a: p. 3171).

More generally, in semi-classical descriptions of quantum gravity theories,
where the solutions of general relativity coupled to matter contain objects
such as strings and D-branes, higher-dimensional surfaces such as the ones
considered here “stand ready” for an interpretation in terms of extended
material membranes, i.e. as a natural generalisation of the notion of a
geodesic. In other words, such surfaces must be considered, if general
relativity is to be the semi-classical limit of these theories of quantum
gravity.

##  7 Summary and Conclusion

In Section 1.3, I identified five putative, inter-dependent, problems for
local definitions of energy and momentum in general relativity:

(i) the Hilbert-Klein-Noether worry that the conservation laws of GR are
identities, with no physical content;

(ii) the lack of a gravitational energy-momentum tensor, analogous to the
matter energy-momentum tensor;

(iii) the lack of integral conservation laws, because of the lack of a
timelike Killing vector field in a general spacetime;

(iv) the non-uniqueness of the gravitational energy-momentum pseudo-tensor;
and

(v) its lacking covariance, and its lacking appropriate geometric properties.

The Hilbert-Klein-Noether worry, (i), was addressed in Sections 2.2 and 2.3,
where I argued that the conservation of energy and momentum in GR does have
physical content.

The lack of a gravitational energy-momentum tensor, (ii), can be resolved in
two ways: either by introducing the gravitational pseudo-tensor, or by
defining quasi-local quantities. I then analysed two quasi-local proposals,
and specialised questions (iii) to (v) to these proposals.

About the lack of a timelike Killing vector field, (iii), in a general
spacetime: while I accept that this is prima facie a problem for local
expressions, I noted that the modified Brown-York stress-energy tensor only
requires the existence of a conformal Killing vector field at the quasi-local
surface (usually: the asymptotic boundary).

I also argued that the putative non-uniqueness, (iv), of the gravitational
energy-momentum pseudotensor, i.e. the ability to add an “arbitrary”
superpotential, is fixed by Noether’s theorem directly in terms of the
variation of the action, as one sees from Eqs. (32)-(26). Specifically, it is
fixed by a choice of boundary terms. So far as I know, this possibility has
not been used in the literature before, to solve the problem of non-
uniqueness.134134134See footnote 57 for a brief discussion of the literature
on this point. In more detail: one can understand the ability to add an
abitrary term to the superpotential as the ability to add an arbitrary
boundary term to the action without changing the equations of motion; (this
then agrees with results by Chang et al. (1999: p. 1898) in the Hamiltonian
formalism).

An important point is that standard arguments in gauge-gravity dualities,
which interpret the duality in terms of a common core shared between two
isomorphic models, further support the statement that the Brown-York stress-
energy tensor correctly represents the quasi-local energy and stress of
general relativity, in the corresponding spacetimes.

Finally, point (v) about covariance, and the lack of appropriate
transformation properties, was re-phrased in terms of two different notions:
(A) diffeomorphism invariance, and (B) background-independence. This was done
in Section 5.2 onwards.

About (A): I summarised, in Section 4.3, the properties of the modified Brown-
York stress-energy tensor under three relevant kinds of diffeomorphisms. (1)
The modified Brown-York stress-energy tensor is correctly invariant under
‘invisible’, i.e. (Redundant), diffeomorphisms. (2) It is correctly non-
invariant under ‘large’, i.e. always non-redundant, diffeomorphisms, because
they relate different models. (3) ‘Visible’ diffeomorphisms correctly give
conformal transformations at infinity. These diffeomorphisms thus correspond
to the conformal transformations of the conformal field theory. (The results
were obtained for $\Lambda<0$, but I argued that they can be extended to
$\Lambda>0$).

About (B): I agreed with Read’s (2020) defence of the background-independence
of expressions involving Killing vectors: although, as mentioned under (iii),
the Killing vectors required for the modified Brown-York stress-energy tensor
to give an integral conservation law satisfy a weaker condition. Also, I noted
that the Brown-York stress-energy tensor can be defined for any smooth
boundary metric $\gamma$, and I clarified the role of the boundary conditions.

In sum, the modified Brown-York stress-energy tensor gives the clearest
analysis and resolution, so far, of the above problems. As for admitting the
gravitational pseudo-tensor itself into the ontology of general relativity: I
argued that there are valid points in Read’s (2020) defence of pseudo-tensors,
although important questions remain open (especially, about its transformation
properties, and about which proposal is correct). The Chang et al. (1999) and
Chen et al. (2017, 2018) quasi-local proposal, and its relation to the
Hamiltonian of GR, also seems promising, and it clarifies the differences
between the different pseudo-tensors used in the literature: but it also faces
additional questions (especially, about the status of the vector field $N$).

## Acknowledgements

I thank Bryan Roberts and Nic Teh for organising the conference ‘The
Philosophy and Physics of Noether’s Theorems’ at the University of Notre Dame,
London Global Gateway, and James Read, Bryan Roberts, and Nic Teh, for their
invitation to contribute to this volume. I thank James Nester for detailed
correspondence and for his patience answering my questions, and James Read,
Henrique Gomes, Harvey Brown, Jeremy Butterfield, and Jeroen van Dongen for
comments. I also thank audiences at the University of Notre Dame, London
Global Gateway, the Hong Kong University of Science and Technology, and the
Tsinghua Sanya International Mathematics Forum in Hainan, China.

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