Noether’s Theorems and Energy in General Relativity

Sebastian De Haro

Institute for Logic, Language and Computation, University of Amsterdam
Institute of Physics, University of Amsterdam
<EMAIL_ADDRESS>

Abstract

This paper has three main aims: first, to give a pedagogical introduction to
Noether’s two theorems and their implications for energy conservation in
general relativity, which was a central point of discussion between Hilbert,
Klein, Noether and Einstein. Second, it introduces and compares two proposals
for gravitational energy and momentum, one of which is very influential in
physics: and, so far as I know, neither of the two has been discussed in the
philosophical literature. Third, it assesses these proposals in connection
with recent philosophical discussions of energy and momentum in general
relativity.

After briefly reviewing the debates about energy conservation between Hilbert,
Klein, Noether and Einstein, I give Noether’s two theorems. I show that
Einstein’s gravitational energy-momentum pseudo-tensor, including its
superpotential, is fixed, through Noether’s theorem, by the boundary terms in
the action. That is, the freedom to add an arbitrary superpotential to the
gravitational pseudo-tensor corresponds to the freedom to add boundary terms
to the action without changing the equations of motion. This freedom is fixed
in the same way for both problems. I also review two proposals for energy and
momentum in GR, of which one is a quasi-local alternative to the local
expressions, and the other builds on Einstein’s local pseudo-tensor approach.
I discuss the recent philosophical literature on the conservation of energy
and momentum in general relativity, and I assess and compare the two proposals
in the light of this literature: especially, in light of questions about
diffeomorphism invariance and background-independence.

Forthcoming in: J. Read, N. Teh and B. Roberts (Eds.), The Philosophy and
Physics of Noether’s Theorems, Cambridge University Press, 2021.

###### Contents

  1. 1 Introduction
    1. 1.1 Noether’s theorem and general relativity
    2. 1.2 Illustrating Noether’s theorems in Maxwell theory
    3. 1.3 Einstein’s proposal, and five problems for energy and momentum in GR
  2. 2 Noether’s Theorems and the Energy-Momentum Tensor
    1. 2.1 Noether’s first theorem
    2. 2.2 Noether’s second theorem and the boundary theorem
    3. 2.3 Two consequences of Noether’s second theorem
  3. 3 Noether’s Theorems in General Relativity
  4. 4 Quasi-local Stress-Energy Tensors in GR
    1. 4.1 The Brown-York quasi-local stress-energy tensor
    2. 4.2 The holographic stress-energy tensor
    3. 4.3 Diffeomorphisms and redundancy of the description
  5. 5 Gravitational Energy and Momentum, Revisited
    1. 5.1 Superpotentials are boundary terms in the Hamiltonian of GR
    2. 5.2 Recent philosophical discussions of energy conservation in GR and pseudo-tensors
  6. 6 A Comparison of Two Proposals
    1. 6.1 What else is required for us to accept pseudo-tensors?
    2. 6.2 Quasi-local expressions: superpotentials vs. the Brown-York stress-energy tensor
    3. 6.3 Interpreting quasi-local quantitites
  7. 7 Summary and Conclusion
  8. Acknowledgements
  9. References

##  1 Introduction

This paper has three main aims: first, to give a pedagogical introduction to
Noether’s two theorems: and in particular to the role that her second theorem
played in important discussions between Hilbert, Klein, Noether and Einstein
about the conservation of energy and momentum in general relativity (GR).
These discussions have been the subject of recent scholarship, and are still
valuable today.111See for example Kastrup (1987: pp. 116-125), Brading (2005:
pp. 125-127), Brading and Brown (2003: pp. 91, 06), Kosmann-Schwarzbach (2011:
pp. 37-46), Rowe (1999: pp. 209-221; 2019: pp. 7-11, 29-34, 2019a).

Second, I introduce for a philosophical audience two proposals for the
gravitational energy-momentum tensor: namely, Einstein’s pseudo-tensor (and
the quasi-local expressions derived from it222Especially in recent work by
Chang et al. (1999) and Chen et al. (2017, 2018).) and the quasi-local
definition by Brown and York, especially as modified in the recent literature
on the holographic principle. I argue that arguments from gauge-gravity
duality support the quasi-local quantities as correctly expressing the energy
and momentum of general relativity.

Third, I discuss and critically assess the various proposals in light of a
recent philosophical discussion of energy and momentum in general relativity,
by Hoefer (2000), Pitts (2010), Lam (2011), Duerr (2019a), and Read (2020).

General relativity suffers from a number of problems regarding its local
conservation laws for energy and momentum. This was the subject of a crucial
discussion between Hilbert, Klein, Noether and Einstein between 1915 and 1918.
What is perhaps less well-known is that the problem of the definition and
conservation of energy333For brevity, I will sometimes follow the literature
in using ‘energy’ or ‘energy-momentum’ for the longer ‘energy-momentum and
angular momentum’, especially in expressions like ‘conservation of energy-
momentum and angular momentum’. played a crucial role in both: the genesis of
general relativity and Noether’s reasons for proving her two theorems.

To define gravitational energy and momentum, Einstein introduced a
gravitational pseudo-tensor, i.e. a quantity that is non-covariant, and is in
some ways analogous to the Christoffel symbols. However, unlike the
Christoffel symbols, pseudo-tensors are not geometric objects. The question of
energy and momentum, and the pseudo-tensor proposal, are discussed in
virtually all of Einstein’s papers on the subject.444See for example Janssen
and Renn (2007: p. 844). Part of the reason it played such a prominent role,
including in his November (1915, 1915a, 1915b) papers, is because the pseudo-
tensor is intimately linked to Einstein’s field equations and to the
contracted Bianchi identities.

Noether’s own work was directly motivated by the question of energy-momentum
conservation, and prior to publishing her two theorems she had worked on
comparing Hilbert’s and Einstein’s approach to it. According to Hilbert,
Klein, and Noether, the conservation law in general relativity is improper,
because it is given in terms of a superpotential whose conservation law is an
identity. On Klein’s interpretation, the conservation law does not make direct
use of the equations of motion, and so it has no physical content.

Einstein’s pseudo-tensor, and thus the energy, are known to have a number of
other problems: the pseudo-tensor is non-covariant, even non-geometric, and
there appears to be no unique expression for it. For many different
expressions have appeared in the literature. This non-uniqueness stems from
the ability to add to the pseudo-tensor the derivative of an arbitrary
‘superpotential’. The superpotential is such that its divergence is
identically zero, and so an arbitrary superpotential can be added into the
conservation law without modifying it. Thus, the very definition of the
pseudo-tensor appears to be ambiguous.

The philosophical interest in energy and momentum in general relativity has so
far been moderate.555For a detailed analysis of Hilbert’s (1916, 1917) two
papers on the ‘Foundations of Physics’, see Renn and Stachel (2007: pp.
878-902, 914-957). Hilbert’s first paper, including the changes that he
introduced into the proofs, is discussed, and compared with Einstein’s four
November 1915 papers, in Sauer (1999: pp. 539-557), Renn and Stachel (2007:
pp. 872-933), and Corry et al. (1997: pp. 1271-1273). A brief account of the
interactions between Einstein and Hilbert in 1915 is in Corry (1999: pp.
177-179). For example, Hoefer (2000), Lam (2011), and Duerr (2019, 2019a) have
criticised, and Read (2020) has defended, local definitions of energy and
momentum, in particular those involving the energy pseudo-tensor. I will give
considerable attention to this recent philosophical discussion in Sections 5.2
and 6.

There are two aspects that these recent discussions have not focussed on, and
which I will address in this paper: namely, the alleged non-uniqueness of the
superpotential, and the quasi-local (as opposed to local) definitions of
energy and momentum.

I will show in Section 2 that, once the Lagrangian and its variation are
fixed, Noether’s second theorem fixes the superpotential. This holds not only
for general relativity, but for any theory with a local symmetry. In general
relativity, the gravitational pseudo-tensor, which is the Noether current
corresponding to the local translation symmetry, is thus fixed, through
Noether’s theorem, by the choice of the action. In particular, the freedom to
add an arbitrary superpotential corresponds to the freedom to add boundary
terms to the action without changing the equations of motion.666In a different
context, Chang et al. (1999: pp. 3-4) and Chen et al. (2018: p. 4) have also
made a related connection between the superpotential and boundary conditions
in the Hamiltonian, although they do not mention Noether’s theorem in that
connection. Thus the superpotential is not arbitrary, but is fixed, through
Noether’s theorem, by the choice of boundary terms in the action. Although
this is known to the experts, I have not seen these consequences of Noether’s
theorem discussed in the literature: in fact, the “problem of non-uniqueness”
is still frequently cited without a discussion of the implications of the
choice of boundary conditions and boundary terms. For example, even Brading
and Brown (2003: pp 101-102), whose formulation of Noether’s second theorem I
largely follow, have the superpotential implicit in their formulas, but do not
seem to recognise it as such, and they do not comment on the definition and
alleged non-uniqueness of the pseudo-tensor in connection with Noether’s
theorems.777For a discussion of the literature on this point, see Section 2.3.

The detailed consideration of Noether’s second theorem also allows one to
address Klein’s worry about the alleged ‘lack of physical content’ of the
local energy conservation law of general relativity. Indeed, I will argue that
the distinction between ‘proper’ and ‘improper’ conservation laws, in the
context of comparing general relativity with classical mechanics, is a red
herring: and that the claim that local energy conservation laws lack physical
content is misleading.

I will also introduce two different quasi-local expressions: one, due to Chang
et al. (1999), Nester (2004), and Chen et al. (2017, 2018, 2018a), directly
obtained from the total energy-momentum (i.e. the energy-momentum tensor plus
the gravitational pseudo-tensor) suitably integrated over a volume. These
authors show that, under assumptions that I will discuss in Sections 5 and 6,
this expression can be obtained from the covariant Hamiltonian formalism of
general relativity. Indeed, part of their aim appears to be to justify the use
of the pseudo-tensor from the Hamiltonian formalism.

The second expression is the Brown-York (1993) quasi-local stress-energy
tensor, which is directly defined on a two-dimensional surface. This stress-
energy tensor has been studied extensively in the physics literature,
especially in the presence of a cosmological constant (positive or negative),
and has been shown to have many of the desired properties of the quasi-local
stress-energy, as I will also review.

I also compare, in Section 6, the two quasi-local proposals. While quasi-local
expressions have sometimes been easily dismissed (or avoided altogether) in
the philosophical literature, I will argue that neither dismissal nor
avoidance are justified. Also, we will see that the analysis of the Brown-York
stress-energy tensor in the gauge-gravity literature gives crucial insights
into the correct definition and properties of this tensor.

One obstruction to the construction of normal conservation laws for energy and
momentum in curved spacetimes is the presence of a covariant derivative in the
conservation law of the energy-momentum tensor.888Notice that this fact is not
specific to general relativity: for any translation-invariant classical field
theory on a curved spacetime has a covariantly-conserved, two-index, energy-
momentum tensor. But GR offers a specific interpretation of the source or sink
terms in terms of a dynamical gravitational field. One can integrate such a
law to get an integral form of the conservation law that is properly covariant
only if the metric admits a timelike Killing vector field, i.e. if the metric
has a timelike isometry.

An important property of the quasi-local energy-momentum tensors is that they
can lead to covariant integral conservation laws under weaker conditions than
the energy-momentum tensor does. For, while the latter requires a timelike
Killing vector field in the whole spacetime, the quasi-local expression from
Section 4 only requires a (conformal) Killing vector field at the quasi-local
surface, i.e. usually at asymptotic spatial or null infinity. I show this in
Section 4.2.

In physics, there are well-known global expressions (called the Bondi and ADM
masses) for the total energy at infinity, depending on whether they are
integrated over spheres at spatial infinity (in the case of asymptotically
flat and AdS cases) or null infinity (in the case of the Bondi mass). The two
kinds of quasi-local quantities discussed in this paper reproduce these
expressions. But these expressions, including the quasi-local quantities, do
not necessarily have well-defined local densities.999This is often further
explicated by an appeal to the principle of equivalence, that a local observer
cannot detect a gravitational field at a single point. This often stated,
infinitesimal or strong, form of the principle of equivalence is in Pauli
(1958: p. 145); see also Brown (2005: pp. 169-170). For its role in Einstein’s
own thinking about general relativity, and its relation with the Einstein
equivalence principle, see Lehmkuhl (2019: pp. 2-7, 19-22) and Janssen (2014:
pp. 174-182). The strong equivalence principle is criticised by Norton (1985:
pp. 207-208, 239-242, 245) and Curiel (2019: p. 91). Chen et al. (1999: p. 2)
discuss a quasi-local formula that can be localised: its localisation gives
rise to the gravitational pseudo-tensor, which depends on the choice of
coordinates. They view the interplay between the local and quasi-local
quantities as follows: ‘Appreciating the fundamental nonlocality of the
gravitational interaction yet believing in the basically local nature of
physical interactions led to the idea of quasi-local quantities: quantities
that take on values associated with a compact orientable spatial 2-surface’.

The plan of the paper is as follows. In the rest of this Introduction, I
summarise the discussion of energy and momentum between Hilbert, Klein,
Noether and Einstein (Section 1.1), illustrate Noether’s theorems for Maxwell
theory as my running example (Section 1.2), and summarise Einstein’s basic
proposal of pseudo-tensors as candidates for energy and momentum and its
problems (Section 1.3). Section 1.3 lays out five problems that organise the
subsequent discussion. Section 1.1 has more of a historical character; the
rest of the paper has a more philosophical and foundational character. Section
2 then gives Noether’s two theorems and discusses two consequences of it, one
of which I have not seen discussed in the literature (Section 2.3). Section 3
applies Noether’s theorems to general relativity. Section 4 discusses the
quasi-local Brown-York stress-energy tensor. Sections 4.1 and 4.2 are slightly
more technical than the rest of the paper, and can be skipped by readers only
interested in the philosophical discussion. Section 4.3 discusses whether
diffeomorphisms are redundancies of the description. Section 5 reviews
proposals by Chang et al. (1999) and Chen et al. (2017, 2018, 2018a) and
Nester (2004) that link the gravitational pseudo-tensor with a quasi-local
expression that can be reproduced from the covariant Hamiltonian of general
relativity, under assumptions that I will point out. Section 6 then compares
the two proposals for quasi-local expressions and discusses the status of
energy-momentum in general, and the pseudo-tensors in particular, in the
philosophical literature. Sections 4 to 6 also discuss two themes that run
through the philosophical literature on energy and momentum in general
relativity: namely, covariance and background-independence.

###  1.1 Noether’s theorem and general relativity

Emmy Noether’s (1918) two theorems, on the correspondence between symmetries
of the action under infinitesimal transformations of a continuous group, and
local conservation laws, are better known for their applications in classical
mechanics and gauge theories than in general relativity. Yet Noether’s
motivation to study this problem came chiefly out of discussions between David
Hilbert, Felix Klein, Albert Einstein, and herself about the interpretation of
general relativity, between 1915 and 1918: specifically, about the nature of
the conservation equations in GR.101010For some of the scientific context of
Noether’s theorems, including her own biography, see Kosmann-Schwarzbach
(2020: pp. 1-9). A clear exposition of Noether’s theorems in their original
form is in Rowe (1999: pp. 223-226). Norton (1993) is a useful reference about
the history of some issues in general relativity that are complementary to the
ones discussed in this Section: in particular, about general covariance.
Noether’s theorem still gives the most general—and, as I will argue in Section
2—best approach to the gravitational pseudo-tensor of general relativity and
its associated superpotential: which was at the heart of the debate.

Since at least 1912, Einstein had been searching for an expression of the laws
of gravitation, invariant under general coordinate transformations.111111See
e.g. Sauer (2005: pp. 806-808) and Kosmann-Schwarzbach (2011: p. 37). Already
in his (1913: p. 318) Entwurf paper with Marcel Grossmann, and in his (1914:
p. 99) ‘Formal Foundation’ paper, which amongst other things gave a detailed
development of tensor calculus and worked out the geodesic equation, he
discussed at length (especially in his 1914) a covariant balance equation for
the energy-momentum and stress tensor density, subject to external forces.
When the external forces were set to zero, he got an analogue of the local
conservation law of energy and momentum (pp. 100, 120):

$\displaystyle\partial_{\mu}(\mathfrak{T}^{\mu}{}_{\nu}+t^{\mu}{}_{\nu})=0~{}.$
(1)

Here, $\mathfrak{T}$ is the energy-momentum tensor density of
matter,121212Compare this equation with Eq. (21). and $t$ is the ‘energy
tensor of the gravitational field’ (later called the gravitational pseudo-
tensor), in analogy with the Christoffel symbols, which are the components of
the ‘field strength of the gravitational field’. He gave an explicit
expression for $t^{\mu}{}_{\nu}$ in terms of the Christoffel symbols (p. 120)
and noted that it is not a tensor, but it is only invariant under linear (i.e.
affine) transformations. At this point, Einstein wrote his tentative field
equations explicitly in terms of the pseudo-tensor. Thus from very early on,
the gravitational energy-momentum pseudo-tensor was, in Einstein’s mind, an
important ingredient of the theory that he was seeking.

In the spring of 1915, Hilbert and Klein invited Noether to Göttingen to work
with them on the implications of the general theory of relativity (Kosmann-
Schwarzbach, 2011: p. 45; Corry, 1999: p. 177).131313For the prehistory of
this invitation, see Sauer (1999: p. 539-540). The prehistory of Hilbert’s
acquaintance with Einstein is on pp. 527-539. Although Noether was probably
not an expert on variational methods (however, Hilbert was an expert on
that),141414See Sauer (1999: pp. 535, 558, 569). she was well-acquainted with
the related methods for using formal differential operators to generate
algebraic and differential invariants (Rowe 2019: p. 6). Also, shortly after
her arrival in 1915, Einstein gave a series of lectures on general
relativity—he was staying at Hilbert’s house during his visit. Thus by
mid-1915, Hilbert himself was working intensely to understand Einstein’s
papers. Early on, Hilbert saw a connection between two topics: (i) the theory
of invariants and geometry, and (ii) the problem of extending the special
theory of relativity to general coordinate systems. This was of course a seed
for Noether’s theorems (and Hilbert himself would soon go on to give a special
case of Noether’s second theorem). Likewise, Klein would regard general
relativity from the point of view of his Erlangen programme, which he had laid
out with Sophus Lie, of geometry as the search for invariants of a
group.151515See Rowe (1999: p. 215; 2019a: p. 72) and Kosmann-Schwarzbach
(2011: pp. 26, 40, 64).

In two of his papers from November 1915, Einstein gave a very general
expression (1915: p. 784) for the gravitational pseudo-tensor in terms of the
Lagrangian density161616See Eq. (47), which is only a minor generalisation of
this equation through the addition of a total-derivative term. However, at
this point Einstein’s Lagrangian was incomplete, since its variation gave only
(part of) the the Ricci tensor, rather than the Einstein tensor. A problem in
this paper, to which he returned in (1915a: pp. 799-800), was the
incompatibility between his restriction to coordinate systems where
$\sqrt{-g}=1$ and the requirement that the energy-momentum tensor has non-zero
trace. The full, covariant equations without any restrictions on the trace of
the energy-momentum tensor appreared in Einstein (1915b: p. 845). and found
the correct gravitational field equations (1915b: p. 845).

As is well-known, on 20 November 1915, i.e. five days before Einstein
submitted the paper that contained the final form of his field equations
(1915b),171717For the now accepted resolution of the priority issue, see Corry
et al. (1997: pp. 1271-1273), Sauer (1999: pp. 539-557), and Renn and Stachel
(2007: pp. 872-933). Hilbert submitted his ‘Die Grundlagen der Physik’, which
is an attempt to axiomatise general relativity (and make a connection with
Gustav Mie’s unification programme).181818Hilbert’s reference to Mie in the
introduction is as follows: ‘I would like to formulate—in the sense of the
axiomatic method—a new system of basic equations of physics, out of two simple
axioms, which are of ideal beauty, and which, I believe, contain the solution
of the problems by both Einstein and Mie’ (1915: p. 395; see also pp. 396,
407). For more on Mie’s influence on this paper by Hilbert, see Sauer (1999:
pp. 538, 552-555), Rowe (2019: pp. 5-7), Renn and Stachel (2007: pp. 857-867,
871-874, 899-900), Corry (1999: p. 176). Hilbert reproduced Einstein’s field
equations from his action and, among other results, he derived the four
contracted Bianchi identities from the invariance of the action under
arbitrary transformations of coordinates. Thus in this paper Hilbert derived a
special case of Noether’s second theorem. At this point, however, the nature
of these relations was unclear. Even less clear was the meaning of the ‘energy
equation’, based on an ‘energy vector’, that he derived.191919Hilbert would
later rewrite his two ‘Grundlagen der Physik’ papers, (1916) and (1917),
publishing them in sequence in Hilbert (1924).

Noether also began to work on this problem. In 1916, she wrote an unpublished
manuscript in which she studied Hilbert’s energy vector, and used Hilbert’s
work to analyse Einstein’s conservation law in general relativity. Although
the manuscript is not extant, its content is partly known from two sources:
namely, the exchange of letters between Klein and Hilbert published in the
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen,
Mathematisch-Physikalische Klasse, in 1917, and a partial transcription by R.
J. Humm, who studied at Berlin and Göttingen at the time (Rowe 2019, 2019a).
In this manuscript, she got the contracted Bianchi identities, and in fact she
anticipated the main points of Klein’s influential lecture of 22 January 1918
for the Göttingen Gesellschaft.

Einstein (1916a: p. 1111) endorsed Hilbert’s action in another paper,
published in November 1916, and solved the conservation equation, Eq. (1), in
terms of what is now called a ‘superpotential’,202020See Eq. (22) below. i.e.
a quantity whose divergence is identically zero.212121Such an expression
already appeared in Einstein (1914: p. 1077), but this was only for his old
theory, in which $\sqrt{-g}=1$.

On 22 January 1918, Klein delivered his famous lecture at the Mathematische
Gesellschaft zu Göttingen (i.e. the Göttingen Mathematical Society), which
elicited a reaction from Hilbert one week later (Rowe 1999: p. 212; 2019: p.
8). Klein’s lecture would be published in the Nachrichten von der Gesellschaft
der Wissenschaften zu Göttingen, in the unusual form of an exchange of letters
with Hilbert (also, Klein and Hilbert lived a short distance from one another
in the Wilhelm Weber Strasse!).222222Klein (1917). For a reason why they chose
this unusual form of publication, see Rowe (1999: p. 212). In it, he
explicitly acknowledged Noether’s help and her earlier contribution (Kosmann-
Schwarzbach, 2011: p. 66):

> You know that Miss Noether advises me continually regarding my work, and
> that in fact it is only thanks to her that I have understood these
> questions. When I was speaking recently to Miss Noether about my result
> concerning your energy vector, she was able to inform me that she had
> derived the same result on the basis of developments of your note… more than
> a year ago, and that she had then put all of that in a manuscript (which I
> was subsequently able to read). She simply did not set it out as forcefully
> as I recently did at the Mathematical Society.

Like Noether had done before, Klein compares Hilbert’s energy equation and
Einstein’s conservation law, Eq. (1). Klein’s main point was the disanalogy
between Hilbert’s energy equation (and Einstein’s conservation law) and the
conservation laws in classical mechanics. While, in classical mechanics,
energy conservation follows as a consequence of the equations of motion, in
general relativity Eq. (1) is an identity, i.e. valid without requiring the
equations of motion to be satisfied. He explained it thus:

> Thus the above statement… cannot be regarded analogous to the conservation
> law of energy in ordinary mechanics. For when we write the latter as
> $d(T+U)/dt=0$, this differential equation is not satisfied identically [as
> the Bianchi identity is], but only as a consequence of the differential
> equations of mechanics.

He concluded the note to Hilbert with a sceptical remark: ‘After all this, I
can hardly believe that it is appropriate to characterise the very arbitrarily
built quantities $t^{\mu}{}_{\nu}$ as ‘energy components of the gravitational
field”. This agreed with Noether’s own view, which she had explained to
Hilbert more than a year before.

In his reply,232323Klein (1917: p. 477). Hilbert completely agreed with
Klein’s comments about the lack of an analogy between the energy equation and
conservation of energy in classical mechanics. He took this one step further,
by expressing his belief that no true law of energy conservation exists in
general relativity, and claiming that it is possible to prove a theorem to
this effect—to which Klein responds that he ‘would be very interested in
seeing the mathematical proof to which you alluded to in your reply’ (p. 482).
Such a proof would not be provided by Hilbert, but by Noether’s second
theorem, in the following year.

Meanwhile, in 1917-1918, Einstein and Klein corresponded about Klein’s (1917)
note, which Einstein (1918: p. 673) praised for its clarity.242424For an
account of the correspondence, see Rowe (1999: pp. 209-213). Einstein
disagreed that his conservation law for general relativity was an identity,
since the equations of motion are used. To this, Klein (1918a: p. 504) replied
by saying that the divergence in Eq. (1) consists of two terms:252525This can
be seen by expressing the pseudo-tensor $t$ in terms of two contributions: the
Einstein tensor and the superpotential: see Eq. (24) below.

(1) the divergence of the Einstein field equations;

(2) the divergence of the superpotential.

Klein’s point was that the vanishing of (2) is a mathematical identity, while
(1) contains the divergence of the equations of motion, rather than the
equations of motion themselves—and so, that this term is also zero, even
before one takes its divergence in the pseudo-tensor. (This differs from
classical mechanics, where the equations of motion, rather than their
divergence, appear in the conservation equation). Einstein (1918a: p. 513)
disagreed: for the equations of motion were after all used, and the
conservation equation used at least part of the field equations (i.e. their
contraction with the divergence).262626The above summary of the argument in
Klein (1918a: p. 504) and Einstein (1918, 1918a: p. 513) follows Brading
(2005: p. 126), which indeed seems to me a correct interpretation both of
Klein’s letter and of Einstein’s (1918a: p. 513) reply.

In the winter-spring of 1918, Noether understood the reasons for the
difficulties in the interpretation of the conservation laws of general
relativity. What she set out to investigate was indeed Klein and Hilbert’s
central problem of the analogy between the energy equations of classical
mechanics and general relativity. On 23 July 1918, she presented her work at
the Göttingen Mathematical Society. In short, her second theorem could account
for the four identities that Hilbert had pointed out, so that the conservation
laws were ‘improper’, and thus disanalogous to classical mechanics.

Noether had a deeper reason for the difference between general relativity and
other theories: her theorems allowed her to ‘deduce… the proof of an assertion
of Hilbert concerning the relationship between the lack of a proper law of
energy and “general relativity”’ (Kosmann-Schwarzbach, 2011: p. 19). Namely,
given an action invariant under the group of translations, the energy
relations are improper iff the action is invariant under an infinite group
which contains the group of translations as a subgroup.272727Noether’s use of
the word ‘subgroup’ here appears to be stronger than the modern use. Her
statement assumes not just the group of translations being a subgroup of an
infinite group, but also its being embedded in a specific way: see Section
2.3. I thank Bartlomiej Czech for pushing me on this point.

Thus Noether interpreted her theorems along Hilbertian lines: ‘As Hilbert
expresses his assertion, the lack of a proper energy law constitutes a
characteristic of the general theory of relativity’.

The paper by Klein (1918) clarifies the relationship between Einstein’s and
Hilbert’s theories, and in particular concerning their conservation laws. In
this paper, he derives what Brading and Brown (2003) call the ‘boundary
theorem’, from which it follows that the current is proportional to the field
equations (and to the superpotential), so that its divergence is identically
zero—as I already mentioned above. In doing so, he made essential use of
Noether’s work (Kosmann-Schwarzbach, 2011, p. 68):282828Klein (1918: p. 189).

> I would be remiss if I did not thank Miss Noether once again for her active
> participation in my new work. She has by herself completely set out in
> proper form the mathematical ideas that I use in connection with the physics
> problems related to [the action], which will be presented in a note to
> appear shortly in these Nachrichten.

Thus by 1918, Noether’s theorem was, in the eyes of Hilbert, Klein, and
Noether, the proof that there is no proper conservation law in general
relativity: which Klein interpreted—with Hilbert’s and Noether’s approval, one
should add—as the energy equations’ lacking physical content.

I will return to Einstein’s proposal for a conservation law in general
relativity, and its problems, in more detail in Section 1.3. In the next
Section, I first illustrate Noether’s two theorems in a well-known example.

###  1.2 Illustrating Noether’s theorems in Maxwell theory

In this paper, I will use the Maxwell theory of electromagnetism as a running
example to illustrate various aspects of Noether’s theorems, since this theory
already exemplifies some of the issues that appear in general
relativity.292929For an introduction to Noether’s theorem for a physics
audience, see e.g. Bañados et al. (2016). For a mathematical treatment, see
Olver (1986: pp. 272-283, 334-336). Bessel-Hagen (1921: pp. 44-51) was the
first to apply Noether’s first theorem to electrodynamics, using the
generalised theorem involving quasi-symmetries. He credits Klein (p. 33) with
suggesting the problem: ‘Klein expressed a desire that Miss Emmy Noether’s
theorems on invariant variational problems, proved approximately two years
ago, should be applied to the Maxwell equations’. I thank Harvey Brown for
recommending the work by Bessel-Hagen.

Thus let me recall how the theorems work for elementary electromagnetism
without sources: take the following action, defined on Minkowski spacetime:

$\displaystyle S[A]=\int\mbox{d}^{4}x\,{\cal L}(A;x)$ $\displaystyle=$
$\displaystyle-{1\over 4}\int\mbox{d}^{4}x\,F_{\mu\nu}F^{\mu\nu}$
$\displaystyle F_{\mu\nu}$ $\displaystyle:=$
$\displaystyle\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu}~{}.$ (2)

The field equations, derived by setting the variation of the action to zero,
are of course:

$\displaystyle\partial_{\nu}F^{\mu\nu}=0~{}.$ (3)

I will give Noether’s theorems in general form in Section 2. I here give the
specific cases that apply to Maxwell theory.

Noether’s first theorem asserts that the conservation of the energy-momentum
tensor density $T_{\mu\nu}$ follows from the global translational symmetry of
the action in Minkowski spacetime (and vice versa), i.e. under a constant
translation $\delta x^{\mu}=\epsilon^{\mu}$.303030Under such a translation, we
have a new field $A^{\prime}_{\mu}(x^{\prime})$, where
$x^{\prime}=x+\epsilon$, and $A_{\mu}^{\prime}(x^{\prime})$ is related to
$A_{\mu}(x)$ by the usual tensor transformation rule for the components of
one-forms, linearised in $\epsilon$. In the variation, we evaluate
$A_{\mu}^{\prime}$ at $x$. Defining the variation

$\displaystyle\delta
A_{\mu}:=A_{\mu}^{\prime}(x)-A_{\mu}(x)=-\epsilon^{\nu}\,\partial_{\nu}A_{\mu}~{},$
(4)

Noether’s first theorem then gives:

$\displaystyle T^{\mu}{}_{\nu}$ $\displaystyle:=$
$\displaystyle\delta^{\mu}_{\nu}\,{\cal L}-{\partial{\cal
L}\over\partial(\partial_{\mu}A_{\lambda})}\,\partial_{\nu}A_{\lambda}=F^{\mu\lambda}\,\partial_{\nu}A_{\lambda}-{1\over
4}\,\delta^{\mu}_{\nu}\,F^{\lambda\sigma}F_{\lambda\sigma}~{}.$ (5)

One readily finds the following conservation law:

$\displaystyle\partial_{\mu}T^{\mu}{}_{\nu}=\left(\partial_{\mu}\,F^{\mu\lambda}\right)\partial_{\nu}\,A_{\lambda}\overset{{\mbox{\scriptsize
Eq.}}\,(\ref{Mxeom})}{=}0~{}.$ (6)

The term between brackets vanishes only if the equations of motion, Eq. (3),
are satisfied: and thus the energy-momentum is conserved if and only if (a
linear combination of) the equations of motion are satisfied, i.e. it is a
weak conservation law.

Notice that the first term in the energy-momentum tensor Eq. (5) is not gauge
invariant (it is also not symmetric in its two indices upon lowering one of
them).313131Goldberg (1958: pp. 315, 319) discusses the desirability of
symmetry for the discussion of angular momentum. See also Barbashov and
Nesterenko (1983: p. 559-560). This lack of gauge invariance is unsurprising,
given that we used a variation of the Lagrangian, Eq. (4), that is not gauge
invariant. One can get a gauge-invariant energy-momentum tensor by
constructing a gauge-invariant variation of the gauge potential, $A$: namely,
by allowing this variation to contain a gauge transformation, i.e. $\delta
A_{\mu}=-\epsilon^{\nu}\,\partial_{\nu}A_{\mu}+\partial_{\mu}\lambda$, where
$\lambda$ is the gauge parameter. There is a unique gauge-invariant variation
of this form, viz. $\delta A_{\mu}=F_{\mu\nu}\,\epsilon^{\nu}$, where
$\lambda=\epsilon^{\nu}A_{\nu}$. In effect, this adds an ‘improvement term’ to
the energy-momentum tensor, i.e. a term that is a total derivative in an
antisymmetric tensor $U$, without modifying the equations of motion, such that
the energy-momentum tensor is still conserved:323232Improvement terms go back
to Belinfante (1938: p. 888). See also Anderson (1967: p. 273).

$\displaystyle
T^{\prime}{}^{\mu}{}_{\nu}:=T^{\mu}{}_{\nu}+\partial_{\lambda}\,U^{[\mu\lambda]}{}_{\nu}~{}.$
(7)

The above variation gives $U^{[\mu\lambda]}{}_{\nu}=-F^{\mu\lambda}\,A_{\nu}$,
which brings the energy-momentum tensor to the standard form, and is
automatically symmetric in its indices:

$\displaystyle
T^{\prime}{}^{\mu}{}_{\nu}=F^{\mu\lambda}\,F_{\nu\lambda}-{1\over
4}\,\delta^{\mu}_{\nu}\,F^{\lambda\sigma}\,F_{\lambda\sigma}~{}.$ (8)

This is of course the energy-momentum tensor that also appears on the right-
hand side of the Einstein field equations. (Thus general relativity uniquely
detects the energy-momentum tensor of its sources: it fixes the improvement
term.)

Noether’s second theorem is also exemplified by the Maxwell theory,333333For a
discussion of Noether’s theorems for Maxwell theory coupled to a complex
scalar field, see Barbashov and Nesterenko (1983: pp. 550-552) and Sus (2017:
pp. 273-274). which has a local gauge symmetry:

$\displaystyle A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\lambda\,,$ (9)

for an arbitrary smooth function
$\lambda:\mathbb{R}^{4}\rightarrow\mathbb{R}$. Indeed, since $F_{\mu\nu}$ is
itself invariant under this symmetry, the Maxwell action is trivially
invariant.

It is instructive to study the gauge invariance when we couple Maxwell’s
action to a gauge-invariant source $J_{\mu}$:

$\displaystyle S[A,J]=\int\mbox{d}^{4}x\left(-{1\over
4}\,F_{\mu\nu}\,F^{\mu\nu}+J^{\mu}A_{\mu}\right),$ (10)

and require that the action is invariant up to a boundary term that does not
modify the equations of motion, so that the transformed action gives the same
equations of motion as the action Eq. (10).

The variation of this action under the gauge symmetry, Eq. (9), gives:

$\displaystyle\delta_{\lambda}S[A,J]=\int\mbox{d}^{4}x~{}J^{\mu}\partial_{\mu}\lambda=\int\mbox{d}^{4}x~{}\partial_{\mu}(J^{\mu}\lambda)-\int\mbox{d}^{4}x~{}(\partial_{\mu}J^{\mu})\,\lambda~{},$
(11)

where in the last step I used partial integration. The first integral on the
right contains a total derivative, which integrates to a boundary term.
Although physics texts often require that $\lambda$ has compact support so
that it vanishes at infinity and this boundary term is zero,343434See e.g.
Maggiore (2005: p. 69). this is not necessary. For, as we will see in Section
2, Noether’s theorem can be generalised to admit a variation of the action
with a non-zero boundary term, because such a term does not modify the
equations of motion. Thus this term does not need to be zero and we simply
keep it. The second term must be zero if the action is to be invariant up to a
boundary term, which requires that the source is conserved:

$\displaystyle\partial_{\mu}J^{\mu}=0~{}.$ (12)

Writing this out in components gives the equation for the local conservation
of charge:

$\displaystyle{\partial\rho\over\partial t}+\nabla\cdot{\bf J}=0~{}.$ (13)

Integrating over the volume $V$, bounded by the two-surface $S=\partial V$
with outward pointing unit vector $\hat{\bf n}$, gives the time derivative of
the total charge $Q_{V}$ enclosed by the volume:

$\displaystyle{\mbox{d}Q_{V}\over\mbox{d}t}$ $\displaystyle=$
$\displaystyle{\mbox{d}\over\mbox{d}t}\int_{V}\mbox{d}^{3}x~{}\rho\left(t,{\bf
x}\right)$ (14) $\displaystyle\overset{{\mbox{\scriptsize
Eq.}}\,(\ref{localC})}{=}$
$\displaystyle-\int_{V}\mbox{d}^{3}x~{}\nabla\cdot{\bf
J}=-\int_{S}\mbox{d}^{2}x~{}\hat{\bf n}\cdot{\bf J}=-I_{S}~{},$

where in the second line I used Gauss’ theorem. $I_{S}$ is the current flowing
out of the boundary $S$, so that this equation expresses the conservation of
charge in the volume: if the charge in $V$ decreases, it is carried away
through the boundary by a current $I_{S}$, and vice versa.

Thus, although gauge invariance is trivially satisfied by the purely
electromagnetic part of the action, it implies a physical constraint on the
gauge-invariant source $J$ to which it is coupled; the Maxwell theory can only
be consistently coupled to a conserved current (I will return to this at the
end of Section 2.2).

I will now make a point that will recur in general relativity, and which was
important in the discussions between Noether, Hilbert, Klein and Einstein,
reviewed in Section 1.1. It is sometimes said that local gauge symmetry is a
“mere redundancy”, a consequence of our reformulation of the electomagnetic
fields ${\bf E}$ and ${\bf B}$ in terms of the gauge potential $A_{\mu}$ with
no physical significance. However, it is not true that gauge symmetry has no
physical significance. If it did not, it would indeed be very surprising that
it could entail the local charge conservation law Eq. (12).

The point is that the local gauge symmetry reduces the number of physical
degrees of freedom of the gauge potential $A_{\mu}$ from three to two, thus
leaving a field that only has transverse polarisations. Physically, the
freedom to do a gauge transformation secures that the longitudinal
polarisation of the photon does not contribute to any of the quantities that
are usually deemed physical. This is of course crucial to get the correct
phenomenology of the photon. Thus, despite the fact that a choice of gauge
does not fix anything physical, gauge invariance does have physical content:
it secures the correct number of degrees of freedom for the photon.

Recent discussions have also concluded that the alleged lack of physical
significance of the conservation laws derived from the second theorem, as
claimed by Klein (see Section 1.1), is misleading. For example, Brading and
Brown (2003: p. 99) have emphasised that symmetry transformations, such as
local gauge symmetries, do have an empirical significance, because they
restrict the kinds of terms that can appear in the action and in the equations
of motion.353535More precisely, they admit that local gauge symmetries can
have indirect, but not direct, empirical significance, i.e. as in Galileo’s
ship-type scenarios. While, in the philosophical literature, it is widely
recognised that global gauge (i.e. internal) symmetries applied to subsystems
can have direct empirical significance thus understood, there is a debate
about whether local gauge symmetries can have such significance. Early work by
Kosso (2000: pp. 95, 97), Brading and Brown (2004: pp. 656-657), and Healey
(2009) stated that local gauge (internal) symmetries have no direct empirical
significance, but only indirect empirical significance (e.g. through Noether’s
theorems). Their reasons for this are diverse—e.g. for Brading and Brown, it
is only ‘relative system transformations’ (i.e. subsystem transformations),
but not local gauge symmetries, that have direct empirical significance. More
recently, Greaves and Wallace (2014), and Teh (2016) have argued that local
gauge symmetries, applied to subsystems, can have a direct empirical
significance, through Galileo’s ship-type scenarios. I will return to the
discussion of the empirical significance of local symmetries, in the context
of general relativity, in Section 4.3. For a characterisation of gauge
symmetries as transformations that only alter the description, but not the
physical system itself, see e.g. Horský and Novotný (1969: p. 420). For a more
general discussion of symmetries in terms of ‘redundant structure’, see e.g.
Ismael and van Fraassen (2003: pp. 378-380), Caulton (2015: p. 156-158), and
Belot (2013: p. 2). Indeed, as the above already shows, imposing local gauge
symmetry implies, through Noether’s theorem, that Maxwell theory can only be
coupled to a conserved current.

Before we proceed, let us recall how one normally gets an expression analogous
to the charge conservation law, Eq. (14), for energy and momentum conservation
in general relativity. Such an expression can be derived in the presence of
spacetime symmetries, in particular in spacetimes with timelike Killing
vectors $K$:363636I will discuss the general case in Section 3. Also, this
equation can be relaxed to a conformal Killing equation, as in Eq. (63) below.

$\displaystyle\nabla_{\mu}K_{\nu}+\nabla_{\nu}K_{\mu}=0~{}.$ (15)

The idea here is to derive an analogue of the charge conservation equation Eq.
(14) for energy and momentum, starting from the covariant analogue of the
energy and momentum conservation equation:

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0~{}.$ (16)

If $K$ is a timelike Killing vector satisfying Eq. (15), then we can define a
vector current $P^{\mu}:=T^{\mu\nu}\,K_{\nu}$, such that:

$\displaystyle\nabla_{\mu}P^{\mu}=\nabla_{\mu}(T^{\mu\nu}\,K_{\nu})=(\nabla_{\mu}T^{\mu\nu})\,K_{\nu}+T^{\mu\nu}\,(\nabla_{\mu}K_{\nu})\overset{{\mbox{\scriptsize
Eqs.}}\,(\ref{Killxi}),(\ref{GRcon})}{=}0~{},$ (17)

where in the last equality I used the fact that $T^{\mu\nu}$ is symmetric, and
then used Eq. (15) to set this term to zero. We can now readily apply to $P$
the divergence theorem for a $d$-dimensional region of space(time) $\Sigma$
with boundary $\partial\Sigma$:

$\displaystyle
0=\int_{\Sigma}\mbox{d}^{d}x~{}\sqrt{-g}~{}\nabla_{\mu}P^{\mu}=\int_{\Sigma}\mbox{d}^{d}x~{}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu}K_{\mu})=\int_{\partial\Sigma}\mbox{d}^{d-1}x~{}\sqrt{-h}~{}n_{\mu}\,P^{\mu}~{},$
(18)

where $g$ is the metric on $\Sigma$, $h$ is the induced metric on
$\partial\Sigma$, and $n$ is the unit vector normal to
$\partial\Sigma$.373737After the second equality sign, I used:
$\sqrt{g}\,\nabla_{\mu}P^{\mu}=\partial_{\mu}(\sqrt{g}\,P^{\mu})$. For an
overview of integration on manifolds, see Wald (1984: pp. 430-434) and
Landsman (2020: pp. 133-135).

Here, the Killing vector is important because it allows us to construct the
vector quantity $P^{\mu}$ from the tensor $T^{\mu}{}_{\nu}$. For, while Eq.
(16) does not give rise to a partial derivative that can be integrated
(because it has one free index), the vanishing covariant divergence of $P$ in
Eq. (17), which has no free indices, does allow for this, in Eq. (18). The
result is a conservation law that is completely analogous to charge
conservation in the Maxwell theory, as I will now show.

Take $\Sigma$ to be a three-dimensional spacelike volume $V$ with two-boundary
$S$, and write out the index $\mu=(0,i)$ in terms of timelike and spatial
components, with coordinates $x^{i}$ that cover the whole boundary surface
$S$, but are otherwise arbitrary:

$\displaystyle\int_{V}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu}K_{\mu})={1\over
c}{\mbox{d}\over\mbox{d}t}\int_{V}\mbox{d}^{3}x~{}\sqrt{-g}\,\,T^{0\nu}K_{\nu}+\int_{S}\mbox{d}^{2}x~{}\sqrt{h}\,Nn_{i}\,T^{i\nu}K_{\nu}=0~{}.$
(19)

Here, $N$ is the lapse function, and $h$ is the induced metric on $S$. In the
last term, I used the divergence theorem Eq. (18) applied to $V$ with boundary
$S$, to give the integral over the surface $S$ of the momentum density, ${\cal
P}_{S}=\sqrt{h}\,Nn_{i}\,T^{i\nu}K_{\nu}$, through the boundary $S$. The first
term in the middle expression is the time derivative of the energy associated
with the region $V$, with energy density: ${\cal
E}_{V}=\sqrt{-g}~{}T^{0\nu}K_{\nu}$. Thus the expression can be rewritten as:

$\displaystyle{1\over c}\,{\mbox{d}E_{V}\over\mbox{d}t}=-P_{S}~{},$ (20)

where $E_{V}$ is the total energy in $V$ and $P_{S}$ is the integrated
momentum flux through $S$. This is analogous to the charge conservation law,
Eq. (14). Notice that this interpretation is possible thanks to the presence
of a timelike Killing vector $K$ in the spacetime, which defines the energy.
As we will see in Section 1.3, in the absence of Killing vectors, defining an
analogous expression is problematic.

###  1.3 Einstein’s proposal, and five problems for energy and momentum in GR

Having illustrated Noether’s theorems in the example of Maxwell theory, I
return, in this Section, to Einstein’s idea of energy and momentum
conservation laws in general relativity.383838Janssen and Renn (2007: pp. 842,
844) note that Einstein (1916a: p. 1115) is an instantiation of Noether’s
second theorem, predating the theorem itself. I will begin by listing five
problems that are prima facie associated with this notion.393939See for
example the discussion in Horký and Novotný (1969: pp. 426-430).

The first problem is the one that motivated Noether’s research into invariance
and symmetries: namely, Klein’s, Hilbert’s and Noether’s view that:

(i) Energy and momentum conservation laws in GR are improper, i.e. identities
with no physical content. This is a special property of the notion of energy
in general relativity, and prompts the question whether this is the correct
interpretation of Noether’s theorem. It also bears on the conceptual relation
between energy in general relativity and in classical mechanics. (I will
return to the resolution of this problem at the end of Section 2, after I
discuss Noether’s theorem).

In the literature, two further problems about the physical consequences of
Noether’s second theorem for general relativity have been discussed. Both
concern the interpretation of the energy-momentum tensor:

(ii) When general relativity is coupled to matter, the Bianchi identity
implies the analogue of the conservation law, Eq. (16). But $T^{\mu}{}_{\nu}$
is the energy-momentum tensor of matter, rather than of the gravitational
field. So, the question is what is the definition of the energy and momentum
of the gravitational field itself?404040Even though the energy-momentum tensor
will have this property for any field theory in a curved spacetime, there is a
specific problem to GR here: namely, how to define the energy and momentum of
the gravitational field from its dynamical principles.

(iii) The conservation law, Eq. (16), is not a conservation law in the usual
sense. For the energy-momentum tensor cannot be straightforwardly integrated
over space to give a conserved vector current (though this can be done if
there is a Killing vector, as we saw in Eqs. (17)-(18)).414141Note that this
is not an exclusive property of general relativity, since it holds for the
covariant divergence of the energy-momentum tensor in any field theory on a
curved spacetime. However, in general relativity, because the gravitational
field is dynamical, Eq. (16) can be reinterpreted in terms of the
gravitational energy-momentum pseudo-tensor $t^{\mu}{}_{\nu}$, as we will
discuss. Namely, this equation contains two additional terms that depend on
the connection, i.e. it is not a pure derivative like the expression Eq. (12)
for charge conservation in Minkowski space.424242In particular, in Eq. (18),
it is possible to replace the covariant derivative in the first expression by
the partial derivative in the middle expression because we have the covariant
divergence of a vector quantity, $P^{\mu}$. If we had simply integrated the
covariant divergence of the tensor $T^{\mu}{}_{\nu}$, this would pick up
additional terms involving the connection and it would not give a boundary
term. This is also the origin of the extra term $t^{\mu}{}_{\nu}$, in Eq. (21)
below. This suggests the question of the correct interpretation of the
putative conservation law Eq. (16).

Einstein’s (1913, 1914, 1915, 1915a, 1916a) idea to solve these problems was
to use the particular form of the Christoffel symbols that appear in the
covariant analogue of the conservation law for matter, Eq. (16), and to
reinterpret the extra terms as a gravitational contribution to the energy and
momentum. So, one can try to construct an energy-momentum tensor for
gravitation from the Christoffel symbols—viz. introduce a pseudotensor
$t^{\mu}{}_{\nu}$ such that the conservation law contains an ordinary
derivative. This can be done by defining $t^{\mu}{}_{\nu}$ so that Eq. (16)
takes the form of a total derivative:

$\displaystyle\partial_{\mu}(\sqrt{g}\,T^{\mu}{}_{\nu}+t^{\mu}{}_{\nu})=0~{}.$
(21)

As I already discussed in Section 1.1, Einstein interpreted $t^{\mu}{}_{\nu}$
as a ‘gravitational energy-momentum tensor’ density. One can associate with
this tensor a superpotential, ${\cal U}^{\mu\lambda}{}_{\nu}$, such that:

$\displaystyle\sqrt{g}\,T^{\mu}{}_{\nu}+t^{\mu}{}_{\nu}=\partial_{\lambda}\,{\cal
U}^{\mu\lambda}{}_{\nu}\,,~{}~{}~{}~{}\partial_{\mu}\,\partial_{\lambda}\,{\cal
U}^{\mu\lambda}{}_{\nu}=0~{}.$ (22)

Einstein (1916a: p. 244) obtained the superpotential from an action that
differs from the Einstein-Hilbert action by a boundary term.

Freud’s (1939) expression for the superpotential is different from Einstein’s,
namely:

$\displaystyle{\cal
U}^{\mu\lambda}{}_{\nu}=-\sqrt{g}\,g^{\beta\sigma}\,\Gamma^{\alpha}_{\beta\gamma}\,\delta^{\mu\lambda\gamma}_{\alpha\sigma\nu}~{},$
(23)

which automatically satisfies Eq. (22), because the symbol
$\delta^{\mu\lambda\gamma}_{\alpha\sigma\nu}$ is totally antisymmetric in its
upper (and lower) indices. Another well-known expression is e.g. Landau and
Lifschitz’s (1971: p. 306)434343For a critique of the Landau-Lifshitz pseudo-
tensor, see Goldberg (1958: pp. 316, 320) and Ohanian (2010: p. 10). pseudo-
tensor (see also Bergmann, 1958: p. 289).

However, even if Einstein’s strategy appears to solve the two previous
problems, (ii)-(iii), of finding an expression for the energy and momentum of
the gravitational field, and of recasting Eq. (16) with an ordinary
derivative, so that we can integrate to get a conserved charge, it does so at
the expense of introducing two new challenges:444444These concerns can be
found e.g. in Hoefer (2000: pp. 191-195). My (iv) corresponds to Duerr’s
(2019a: Section 3.2) label (H2), and my (v) to his (H1). I will return to
Duerr’s objections in Section 6.2.

(iv) Non-uniqueness: the ability to add an arbitrary superpotential ${\cal U}$
to the pseudotensor means that the pseudotensor is not unique (I have so far
mentioned three expressions: Einstein’s, Freud’s, and Landau and
Lifschitz’s).454545Horký and Novotný (1969: pp. 426-430) is an enlightening
review of various approaches. See also Komar’s (1958: pp. 935-936)
expressions. In fact, there are an infinite number of pseudo-tensors
compatible with Eq. (21). For define a gravitational (i.e. matter-independent)
pseudo-tensor with an arbitrary superpotential:

$\displaystyle t^{\mu}{}_{\nu}$ $\displaystyle:=$
$\displaystyle-{1\over\kappa}\,\sqrt{g}\,G^{\mu}{}_{\nu}+\partial_{\lambda}\,{\cal
U}^{\mu\lambda}{}_{\nu}~{},$ (24)

where $\kappa=8\pi G_{\mbox{\tiny N}}/c^{4}$, and $G^{\mu}{}_{\nu}$ is the
Einstein tensor. Provided ${\cal U}$ is antisymmetric in its upper indices,

$\displaystyle{\cal U}^{\mu\lambda}{}_{\nu}={\cal U}^{[\mu\lambda]}{}_{\nu}$

(see e.g. the Freud expression, Eq. (23)), the gravitational pseudo-tensor
thus defined automatically satisfies the conservation law Eq. (21) when the
Einstein field equations are satisfied:464646The expression that Einstein
obtained was not antisymmetric: for a discussion, see Chen et al. (2017: p.
12).

Thus the definition of the pseudo-tensor, Eq. (24), is ambiguous. Nester
(2004: p. 264) summarises the problem thus: ‘there is no physical meaning
here, only formal math—unless one has a good criterion for choosing the
superpotential’. In my opinion, this is a serious problem that defenders of
pseudo-tensors need to solve.

(v) Lack of covariance, and lack of appropriate transformation properties: the
pseudotensor is not a tensor, as can be seen from its definition in Eq. (21).
For example, it can be made to vanish at any given point in appropriately
chosen coordinates.474747This argument was put forward to Einstein by
Schrödinger and Bauer: for details, see Horký and Novotný (1969: p. 430). See
the responses by Pitts (2010: p. 613) and Duerr (2019: p. 8). It is only
invariant under affine transformations.

Thus Klein (1917) concluded: ‘After all this, I can barely believe that it is
useful to designate [Einstein’s] very arbitrarily built quantities
$t^{\mu}{}_{\nu}$ as the components of the energy of the gravitational field.’
Schrödinger (1950: p. 104) called the gravitational pseudo-tensor a ‘sham
tensor’. Twenty-three years later, Misner, Thorne, Wheeler (1973, p. 467)
wrote:

> Anyone who looks for a magic formula for “local gravitational energy-
> momentum” is looking for the right answer to the wrong question. Unhappily,
> enormous time and effort were devoted in the past to trying to “answer this
> question” before investigators realized the futility of the enterprise.

So, we have a strong warning from these authors that we should avoid looking
for a general local expression for the energy and momentum of the
gravitational field.

As I will mention in Section 5.2, beyond the lack of covariance, the problem
is also that the pseudo-tensor does not have well-defined transformation
properties: for example, it is not a geometric object, in the sense of
Anderson (1967: pp. 14-16, 424), Kucharzewski and Kuczma (1964: pp. 15-18),
and Trautman (1965: p. 84-87):484848See also Nijenhuis (1952: pp. 23-30),
Schouten (1954: p. 67-68), and Bergmann (1958: p. 287). i.e. roughly, such
that its components in a given coordinate system, together with suitable
transformation properties, are well-defined.494949As Trautman (1965: p. 86)
mentions, spinors are not geometric objects either. In Section 6.1 I will
explain what appears to be the real worry. See the discussion in Pitts (2010:
pp. 603-604, 606) and Duerr (2019a: Section 3.3).

In Section 3, I will propose a solution of the non-uniqueness problem (iv)
based on an extension of Noether’s theorem. Although the facts that I will use
are known to the experts, I have not found this solution discussed in any
detail in the literature on Noether’s theorems that I studied—not as a
solution of the alleged problem of non-uniqueness. In Section 5.1, I will
discuss a recent related proposal. In Section 5.2, I will assess how this
approach deals with problem (iii) (in particular, the problem of the need for
Killing vectors or similar structures) and with problem (v), i.e. covariance.

Penrose (1982) made a new proposal to solve (ii) that does not require a local
expression for the energy and momentum, thereby circumventing problems
(iii)-(v), because he did not use the gravitational energy-momentum pseudo-
tensor or other local expressions. He argued that one can still define quasi-
local energy-momentum and angular momentum, i.e. energy and momentum that are
associated with a closed spacelike surface. It refers only to the geometry of
the two-surface and the extrinsic curvature quantities for its embedding in
the spacetime (Penrose, 1988: p. 1). The idea has been widely accepted, and
developed in various ways, in the literature on general relativity.505050For
some of the properties that the quasi-local energy should (presumably) have,
and its relation to other definitions of energy such as the asymptotic ADM
mass (for asymptotically flat spacetime regions) and Bondi mass (for
asymptotically null regions), see for example Jaramillo and Gourgoulhon (2011:
pp. 97-102, 105-107) and Christodoulou and Yau (1988: pp. 9-10). I will return
to it in Section 4, when I discuss the Brown-York quasi-local stress-energy
tensor.

##  2 Noether’s Theorems and the Energy-Momentum Tensor

In this Section, I give Noether’s first theorem (Section 2.1), her second
theorem (Section 2.2), and I discuss two consequences (Section 2.3) that, in
my opinion, have not been sufficiently clarified in the literature.

###  2.1 Noether’s first theorem

Noether’s first theorem is a statement about the invariance of an action of
the form:515151For the connection between symmetries and conservation laws
without the appeal of Lagrangian methods, see Brown (2020: p. 3).

$\displaystyle S[\phi;x]=\int\mbox{d}^{4}x~{}{\cal L}(\phi;x)~{},$ (25)

where $\phi$ is the set of fields present in the problem (labelled by $a$, so
one writes $\phi_{a}$ if one wishes to indicate this) under an infinitesimal
transformation of the dependent and-or independent variables, up to a boundary
term.525252This boundary term was not included in Noether’s paper, and so this
is a slight generalisation of her two theorems, first proven by Bessel-Hagen
(1921: p. 36), who attributes it directly to Noether: ‘I owe this
generalization to an oral communication from Emmy Noether’. See also Kosmann-
Schwarzbach (2020: p. 11) and Brown (2020: pp. 6, 8). This generalisation is
standardly used in e.g. Brading and Brown (2003: p. 93) and Bañados and Reyes
(2016: p. 17). The physical significance of this invariance of the action up
to boundary term is that the equations of motion derived from the new action,
expressed in the new variables, will take the same form as in the old ones.
Following Brading and Brown (2003: pp. 91-97), on which the exposition in this
Section is largely based,535353For other treatments of Noether’s theorems,
closely related to the one given here, see Trautman (1962: pp. 176-179),
Horský and Novotný (1969: pp. 420-424), Barbashov and Nesterenko (1983: pp.
541-544), Kastrup (1987: pp. 116-117), and Brading and Brown (2000: pp. 4-8).
I take the invariance to mean that, when the equations of motion are
satisfied, the action is form-invariant up to a boundary term that depends on
an arbitrary vector quantity $\Lambda^{\mu}$:

$\displaystyle\delta
S=S[\phi^{\prime};x^{\prime}]-S[\phi;x]=\int\mbox{d}^{4}x^{\prime}\,{\cal
L}(\phi^{\prime},x^{\prime})-\int\mbox{d}^{4}x\,{\cal
L}(\phi,x)=\int\mbox{d}^{4}x\,\partial_{\mu}\Lambda^{\mu}~{}.$ (26)

Such a transformation, which leaves the action invariant up to a boundary
term, is called a ‘quasi-symmetry’.545454Echoing Bessel-Hagen (1921: p. 36),
Olver (1986: p. 278) calls them ‘divergence symmetries’. See also Kosmann-
Schwarzbach (2020: p. 11).

Noether’s first theorem states that, if the action Eq. (25) is invariant with
respect to a continuous group of transformations $G_{n}$ that is generated by
$n$ constant parameters $\xi^{k}$ (where $k=1,\ldots,n$), in the sense just
indicated, then there are $n$ linearly independent combinations of the field
equations that are equal to the total divergence of a current, as follows:

$\displaystyle\sum_{a}\left({\partial{\cal
L}\over\partial\phi_{a}}-\partial_{\mu}\,{\partial{\cal
L}\over\partial(\partial_{\mu}\phi_{a})}\right)\eta_{ak}$ $\displaystyle=$
$\displaystyle\partial_{\mu}J^{\mu}_{k}~{}.$ (27)

Since the index $k$ labels the $n$ generators of $G_{n}$, there is one such
equation for each generator. The index $a$ labels the various fields present
in the problem, and $\eta_{ak}$ is a field-dependent expression, defined by
the variation of the fields:

$\displaystyle\delta_{\xi}\,\phi_{a}=\eta_{ak}\,\xi^{k}~{}.$ (28)

The current is given by the following expression:

$\displaystyle J^{\mu}_{k}=-\sum_{a}{\partial{\cal
L}\over\partial(\partial_{\mu}\,\phi_{a})}\,{\partial\delta\phi_{a}\over\partial\xi^{k}}-{\cal
L}\,{\partial\delta
x^{\mu}\over\partial\xi^{k}}+{\partial\Lambda^{\mu}\over\partial\xi^{k}}~{}.$
(29)

The converse is also true: if the field equations satisfy Eq. (27), then this
implies the invariance of the action under the corresponding group $G_{n}$.
The theorem remains valid in the limiting case of an infinite number of
parameters, $n\rightarrow\infty$.

The above current is weakly conserved, since the left-hand side of Eq. (27) is
a linear combination of the Euler-Lagrange field equations for $\phi$.

Example: Maxwell theory. Let us illustrate the above for Maxwell theory,
familiar from Section 1.2.555555Noether’s theorems for Yang-Mills theory are
discussed in Barbashov and Nesterenko (1983: pp. 552-556). Thus we take the
group generated by $\xi^{k}$ to be the group of constant spacetime
translations. Since $A$ is a vector field, the index $a$ labelling the
different fields can be taken to be the spacetime index $\mu$ of the vector
field components $A_{\mu}$. Since $\xi^{k}$ is the generator of translations,
$\xi$ is also a vector field in Minkowski space, and the index $k=1,\ldots,n$,
labelling the generators, is also a spacetime index, $\nu=1,\ldots,4$. Thus
the group $G_{4}$ here under consideration is the group of constant spacetime
translations generated by $\delta x^{\mu}=\epsilon^{\mu}$.

To see how $\eta_{ak}$ depends on the vector field $A$, we temporarily write
it as $\eta_{\mu\nu}$ (not to be confused with the Minkowski metric!), and
compare the general variation Eq. (28) with the variation of the gauge field
Eq. (4) (which is obtained by expanding the field components $A_{\mu}$ in
terms of the constant coefficients, $\delta x^{\mu}=\epsilon^{\mu}$). The
comparison gives $\eta_{\mu\nu}=-\partial_{\nu}A_{\mu}$.

Let us check that the Noether current Eq. (29) reproduces the energy-momentum
tensor of electromagnetism, Eq. (5). Since $\delta x^{\mu}=\epsilon^{\mu}$, we
have ${\partial\delta x^{\mu}\over\partial\epsilon^{\nu}}=\delta^{\mu}_{\nu}$.
The last term in Eq. (29) is zero, and the first term is $-{\partial{\cal
L}\over\partial(\partial_{\mu}A_{\lambda})}\,\eta_{\lambda\nu}={\partial{\cal
L}\over\partial(\partial_{\mu}A_{\lambda})}\,\partial_{\nu}A_{\lambda}=-F^{\mu\lambda}\,\partial_{\nu}A_{\lambda}$.
Thus we get $J^{\mu}_{\nu}=-F^{\mu\lambda}\,\partial_{\nu}A_{\lambda}-{\cal
L}\,\delta^{\mu}_{\nu}=-T^{\mu}{}_{\nu}$, i.e. the energy-momentum tensor Eq.
(5). The Noether current conservation equation Eq. (27) of course reproduces
Eq. (6).

We can repeat this for the gauge-invariant variation $\delta
A_{\mu}=F_{\mu\nu}\,\epsilon^{\nu}$ leading to the energy-momentum tensor with
the improvement term, i.e. Eq. (7). Comparing this variation with Eq. (28), we
find that $\eta_{\mu\nu}=F_{\mu\nu}$. Evaluating the current Eq. (29), we
indeed get $J^{\mu}_{\nu}=-T^{\prime}{}^{\mu}{}_{\nu}$, i.e. we reproduce the
standard energy-momentum tensor, Eq. (8).

###  2.2 Noether’s second theorem and the boundary theorem

In this Section, I will give Noether’s second theorem and the identities that
it implies. I will follow the terminology in Brading and Brown (2003: pp.
100-104), who distinguish a set of formulas that they call the ‘boundary
theorem’, due to Klein (1918), from Noether’s second theorem. (I will
occasionally refer to both sets of formulas, collectively, as ‘Noether’s
second theorem’, since both are entailed by Noether’s general expression for
the invariance of the action under a continuous group).

The boundary theorem and Noether’s second theorem, in generalised form, assume
that the action Eq. (25) is invariant (up to a boundary term) with respect to
a continuous group of transformations $G_{\infty n}$ that is generated by $n$
arbitrary functions $\xi^{k}(x)$ ($k=1,\ldots,n$).

Before giving the theorems, it will be useful to introduce three important
pieces of notation. I use the Einstein summation convention for repeated
indices $a$ and, below, also for repeated $k$ indices:

$\displaystyle E^{a}$ $\displaystyle:=$ $\displaystyle{\partial{\cal
L}\over\partial\phi_{a}}-\partial_{\mu}\,{\partial{\cal
L}\over\partial(\partial_{\mu}\phi_{a})}$ (30) $\displaystyle J^{\mu}_{k}$
$\displaystyle:=$ $\displaystyle-{\partial{\cal
L}\over\partial(\partial_{\mu}\phi_{a})}\,{\partial\delta\phi_{a}\over\partial\xi^{k}}-{\cal
L}\,{\partial\delta
x^{\mu}\over\partial\xi^{k}}+{\partial\Lambda^{\mu}\over\partial\xi^{k}}$ (31)
$\displaystyle U^{\mu\nu}_{k}$ $\displaystyle:=$ $\displaystyle{\partial{\cal
L}\over\partial(\partial_{\nu}\phi_{a})}\,\gamma^{\mu}_{ak}+{\partial\Lambda^{\mu}\over\partial(\partial_{\nu}\xi^{k})}~{}.$
(32)

The quantities here defined are, respectively, the Euler-Lagrange equations of
motion, the Noether current, and the superpotential (more on this later).
$\gamma^{\nu}_{ak}$ is a (field-dependent) variation of the field, to be
defined below. $\Lambda^{\mu}$ is a boundary term that may appear in the
variation of the action, Eq. (26). Notice that we can also apply Noether’s
theorem to a single term in (i.e. to a part of) the action, to require the
invariance of that particular term. In that case, for example, $E^{a}$ is
simply the functional variation of that term in the action with respect to the
fields $\phi_{a}$, and likewise for the other two quantities.

Using this notation, Noether’s first theorem takes a particularly simple form:

$\displaystyle E^{a}\,\eta_{ak}=\partial_{\mu}J^{\mu}_{k}~{}.$ (33)

The expression $\gamma^{\nu}_{ak}$ in Eq. (32), together with $\eta_{ak}$, are
the (field-dependent) variations of the field with respect to $\xi$ and its
first derivative (repeated $k$ indices are again summed over) relevant to
Noether’s second theorem and her boundary theorem:

$\displaystyle\delta\phi_{a}=\eta_{ak}(\phi;x)\,\xi^{k}(x)+\gamma^{\mu}_{ak}(\phi;x)\,\partial_{\mu}\xi^{k}(x).$
(34)

This variation is analogous to Eq. (28), where the second term is non-zero
because in Noether’s second theorem $\xi$ is a function of $x$.

The boundary theorem, with this notation, takes the following simple form:

$\displaystyle\partial_{\mu}(E^{a}\gamma^{\mu}_{ak})$ $\displaystyle=$
$\displaystyle\partial_{\mu}J^{\mu}_{k}$ (35) $\displaystyle
E^{a}\gamma^{\mu}_{ak}$ $\displaystyle=$ $\displaystyle
J^{\mu}_{k}-\partial_{\nu}U^{\mu\nu}_{k}$ (36) $\displaystyle U^{\mu\nu}_{k}$
$\displaystyle=$ $\displaystyle-U^{\nu\mu}_{k}~{}.$ (37)

Again, the index $k=1,\ldots,n$ labels the functions $\xi^{k}(x)$ that
generate $G_{\infty n}$, and so there is one such equation for each generator.
Note that the first equation here follows from the second and third.565656My
notation, using the Einstein convention for the three different types of
indices, also corrects errors in Brading and Brown (2003: pp. 92, 95, 97, 101)
here and in their equation corresponding to my Eq. (29), where summation signs
over $a$ (i.e. their $i$) are incorrectly extended to terms that contain no
index $a$.

Noether’s second theorem reads:

$\displaystyle E^{a}\,\eta_{ak}=\partial_{\mu}(E^{a}\gamma^{\mu}_{ak})$ (38)

Again, the full statement is that, if the action Eq. (25) is invariant under
the group $G_{\infty n}$ of variations as indicated, then the Euler-Lagrange
equations satisfy the identities Eq. (38). The converse statement is here also
valid.

Notice that both the boundary theorem and Noether’s second theorem, Eqs.
(35)-(38), take forms that differ from Noether’s first theorem Eq. (33).
However, it is possible to derive Noether’s first theorem, which is the
special case that the continuous group of transformations $G_{n}$ is a
subgroup of $G_{\infty n}$, i.e. $G_{n}\subset G_{\infty n}$. In this case,
the generators $\xi^{k}$ of $G_{n}$ are constant, and the corresponding second
term in Eq. (34) vanishes. Combining the first of Eq. (35) with Eq. (38), we
recover Noether’s first theorem, Eq. (33), with the $k$ generators thus
identified.

Example: Maxwell theory. It will be instructive to compare with our previous
notation and reproduce the conservation law of Maxwell’s action coupled to a
gauge-invariant source, Eq. (10). The gauge symmetry modifies the field $A$
as: $\delta A_{\mu}=\partial_{\mu}\lambda$. Since $\lambda$ is a function of
$x$, Noether’s second theorem is relevant here. Comparing with Eq. (34), since
there is a single generator, namely $\lambda$, we see that $k=1$ and
$\xi=\lambda$ (and thus, we can drop the index $k$). As before, the index $a$
is a spacetime index, $a=\nu$. Further, $\delta A_{\mu}$ only depends on
$\lambda$’s derivative and not on $\lambda$ itself, so $\eta_{\mu}=0$ and
$\gamma^{\mu}_{\nu}=\delta^{\mu}_{\nu}$. In particular, this means that there
is no non-trivial subgroup of transformations with constant parameters in this
case, and so no Noether first theorem for the gauge transformations. Also,
recall, from Eq. (11), that the variation of the action was zero up to a
boundary term, i.e. a term with a total derivative
$\int\mbox{d}^{4}x~{}\partial_{\mu}\Lambda^{\mu}$. We find that $\Lambda$ is
given by:

$\displaystyle\Lambda^{\mu}=J^{\mu}\,\lambda~{}.$ (39)

This confirms my claim in Section 1.2 that, with this generalised form of
Noether’s theorem, the gauge parameter $\lambda$ does not need to be zero at
the boundary. It is straightforward to check the boundary theorem and
Noether’s second theorem, Eqs. (35)-(38). For the quantities defined in Eqs.
(30) and (32), we get the following expressions:

$\displaystyle E^{\mu}$ $\displaystyle=$ $\displaystyle
J^{\mu}-\partial_{\nu}F^{\mu\nu}$ $\displaystyle U^{\mu\nu}$ $\displaystyle=$
$\displaystyle F^{\mu\nu}~{}.$ (40)

The first equation is of course the expression for the Euler-Lagrange equation
derived from the action Eq. (10).

With these expressions in hand, it is straightforward to verify that the
boundary theorem, i.e. Eqs. (35)-(37), is trivially satisfied, given the anti-
symmetry of the field tensor. Interestingly, the left-hand side of the
expression from Noether’s second theorem, Eq. (38), is zero (because $\eta$ is
zero), and so Noether’s second theorem takes the form
$\partial_{\mu}E^{\mu}=0$. Given the anti-symmetry of the field tensor, this
is identically satisfied iff $J$ is conserved, independent of the satisfaction
of the equations of motion of Maxwell theory. Thus imposing gauge symmetry
implies, through Noether’s theorem, that Maxwell theory can only be coupled to
a conserved current $J$.

###  2.3 Two consequences of Noether’s second theorem

I will first make a comment that I have not found in the literature on
Noether’s theorems, although it may well be known to the experts.575757For
example, Barbashov and Nesterenko (1993: p. 562) mention, for the specific
energy-momentum (pseudo-)tensors considered by Einstein and by Lorentz, that
they are related by a total derivative, and so lead to the same equations of
motion. But, a few paragraphs later, they categorise the non-uniqueness of the
pseudo-tensor as a “difficulty”, without mentioning the relevance of their
previous remark, nor noting that the pseudo-tensor is unique after fixing the
boundary terms of the action. Likewise, Ohanian (2010: p. 10) has a formula
for the pseudo-tensor in terms of the gravitational action, but he drops a
total derivative term from it, and he does not comment on the effect of the
dropped boundary term on the pseudo-tensor. Similar comments apply to Goldberg
(1958: p. 318), Bergmann (1958: p. 288), Komar (1958: pp. 934-935), Trautman
(1962: pp. 171, 174, 190-191), and Horský-Novotný (1969: pp. 425-427), all of
whom explicitly survey and compare different expressions for the pseudo-
tensor, mention the lack of uniqueness and sometimes propose a criterion for
fixing a unique expression, but fail to mention the most obvious criterion:
namely, that of fixing the boundary terms. That the superpotential is fixed,
through Noether’s theorem, by the choice boundary terms of the action, seems
implicit in Wald (1993: p. 3429), but he does not discuss or mention the
gravitational pseudo-tensor or the superpotential at all. In Sections 3 and
4.2, I will further comment on the physical significance of fixing the
boundary terms. These remarks are made visible by the simplified notation
above, and they will bear on the question of energy and momentum conservation
discussed in the next Section. Then I will answer Klein’s critique of the
local energy-momentum conservation law in GR.

The superpotential is fixed, through Noether’s theorem, by the boundary terms.
In studies of Noether’s theorem in connection with general relativity,
emphasis is often laid on the integration of Eq. (35) (see Brading and Brown,
2003: p. 102), as follows:

$\displaystyle J^{\mu}_{k}=E^{a}\gamma^{\mu}_{ak}+\partial_{\nu}{\cal
U}^{\mu\nu}_{k}~{},$ (41)

where ${\cal U}^{\mu\nu}_{k}$ is an arbitrarily chosen superpotential,
analogous to the one in Eq. (24), whose divergence is automatically zero:

$\displaystyle\partial_{\mu}\,\partial_{\nu}\,{\cal U}^{\mu\nu}_{k}=0~{}.$
(42)

Indeed, this is how Einstein himself arrived at his result, Eq. (24), for the
gravitational pseudo-tensor.

However, note that Noether’s theorem gives us, in Eq. (36), an expression for
the current that is not arbitrary but fixed:

$\displaystyle
J^{\mu}_{k}=E^{a}\gamma^{\mu}_{ak}+\partial_{\nu}U^{\mu\nu}_{k}~{},$ (43)

where $U$ is given by Eq. (32). Since $U$ is antisymmetric in its upper
indices, it indeed satisfies Eq. (42).585858For diffeomorphism invariant
theories, the superpotential corresponds, in Wald’s (1993: p. 3429)
formulation, to the Noether charge ${\bf Q}$, which in a 4-dimensional
spacetime is a 2-form.

The compatibility of Eqs. (41) and (43) requires that we must have:595959This
holds up to an antisymmetric, total derivative, term of the form
$\partial^{[\mu}w^{\nu]}$, i.e. the superpotential obtained by integrating Eq.
(35) must be compatible with Eq. (36). I thank Henrique Gomes for pointing
this out. Note that this ambiguity only appears if one integrates Eq. (35)
regardless of the other equations. Most important, this ambiguity does not
affect the superpotential $U$ of Noether’s theorem, which is completely fixed
by Noether’s theorem, through Eq. (36). ${\cal
U}^{\mu\nu}_{k}=U^{\mu\nu}_{k}$. Since $U$ was defined in Eq. (32), and the
boundary theorem depends on its having this exact form, this means that the
superpotential is not arbitrary, but is fixed through Eq. (32).

How exactly is the superpotential fixed by Noether’s theorem? The first term
in Eq. (32) is fixed by the choice of the Lagrangian density and of the
variation of the field. The second term (depending on $\Lambda^{\mu}$) is
fixed by the boundary term that appears in the variation of the action. This
means that Noether’s theorem determines the superpotential, $U$, once the
action and its first variation are fixed.

In the next Section, I will return to the physical interpretation of this
result, and to how it solves the problem (iv) of non-uniqueness in general
relativity from Section 1.1.

A response to the Hilbert-Klein-Noether critique. The above also casts light
on the Klein-Hilbert worry, problem (i) from Section 1.3, that energy and
momentum conservation laws in GR are identities that do not require using the
Einstein field equations, and hence they lack physical content. We can see
that this follows from Eq. (43). Since the Noether current $J$, rather than
its derivative, is linear in the equations of motion, the satisfaction of the
equations of motion does not play a role in securing current conservation,
i.e. in the vanishing of the current’s derivative. In other words, by the time
one wants to prove the conservation of the current, the equations of motion
have “disappeared” from it. They play a role in the current, but not in the
conservation. This is unlike Noether’s first theorem, Eq. (27), where the
divergence of the current, rather than the current itself, is linear in the
equations of motion.

Noether herself took her theorems to confirm Hilbert’s view that there is no
proper energy conservation law in general relativity. She wrote (see Kosmann-
Schwarzbach, 2011: p. 20):

> Given [an action] $S$ invariant under the group of translations, then the
> energy relations [i.e. the conservation laws corresponding to translations]
> are improper [i.e. the divergences vanish identically] iff $S$ is invariant
> under an infinite group which contains the group of translations as a
> subgroup…606060The group of translations being a subgroup of an infinite
> group is not sufficient to get improper energy relations. For example, in
> the Maxwell theory in Minkowski space, the group of translations, $G_{4}$,
> is a sugroup of the total group containing both translations and local gauge
> transformations, i.e. $G_{4}\times G_{\infty 1}$, where $G_{\infty 1}$ is
> the group of local gauge transformations: and yet, the energy relations of
> the Maxwell theory are proper. Thus, for improper energy relations, it is
> also necessary (and this is obviously what Noether had in mind) that the
> four constant generators of translations, $\epsilon^{\mu}$, are obtained by
> setting four corresponding generators of the infinite group, i.e. the four
> functions $\xi^{\mu}(x)$, to be constant. The latter condition is not
> satisfied in the Maxwell theory, but it is satisfied in general relativity.
> As Hilbert expresses his assertion, the lack of a proper law of energy
> constitutes a characteristic of the “general theory of relativity”.

It is of course true that the details of the conservation are different in the
two cases, because they satisfy different theorems. Noether’s second theorem
imposes more constraints.

However, as we also saw above, the boundary theorem, combined with Noether’s
second theorem, does imply an identity that looks exactly like Noether’s first
theorem, Eq. (27). The two theorems differ in that, in the case of the
boundary theorem, the equations of motion appear in the form of the current
itself, while they do not appear in Noether’s first theorem. Thus the
disappearance of the equations of motion from the divergence of the Noether
current in the boundary theorem is not a fact that is additional to its
disappearance from the current itself (the two are connected through Noether’s
second theorem, Eq. (38)).

Let me make this reply to Klein (in particular) more precise. Recall his
points (1) and (2), from Section 1.1, that the divergence of the total energy-
momentum tensor density, i.e.
$\tau^{\mu}{}_{\nu}:=\sqrt{g}\,T^{\mu}{}_{\nu}+t^{\mu}{}_{\nu}$, has no
physical meaning because it is an improper conservation law, i.e. it consists
of two terms:616161Here, and in the rest of the paper, the notation $\sqrt{g}$
means $\sqrt{|\det(g)|}$. (1) the divergence of the Einstein field equations,
and (2) the divergence of the superpotential, which is identically zero. This
follows from the boundary theorem, Eq. (36), which gives:
$\partial_{\mu}J^{\mu}_{k}=\partial_{\mu}(E^{a}\gamma^{\mu}_{ak})+\partial_{\mu}\partial_{\nu}U^{\mu\nu}_{k}$,
where: (1) $E^{a}$ appears here under the divergence, (2) the second term
vanishes identically. By (1), the equations of motion play no direct role in
the vanishing of the divergence (as they do in e.g. classical
mechanics).626262Klein’s (1918a: p. 504) argument is not quite as explicit as
this. This reconstruction of the argument follows Brading (2005: p. 126): see
footnote 26. Thus the equations of motion play no direct role in the vanishing
of the divergence (as they do in e.g. classical mechanics).

But using Noether’s second theorem, i.e. Eq. (38), for non-zero
$\gamma^{\mu}_{ak}$, we rewrite this equation as:
$\partial_{\mu}J^{\mu}_{k}=E^{a}\eta_{ak}+\partial_{\mu}\partial_{\nu}U^{\mu\nu}_{k}$.636363This
is different from the case of the local gauge symmetry of the Maxwell theory,
because $\gamma^{\mu}_{ak}$ there is indeed zero, while it is non-zero in
general relativity. Since the second term is identically zero, we see that the
Noether current $J$ is conserved if and only if (a linear combination of) the
equations of motion are satisfied, in complete analogy with Noether’s first
theorem.646464The (indirect) physical significance of Noether’s second theorem
has long been recognised by philosophers. Recently, a number of authors have
also emphasised the (direct) physical significance of improper conservation
laws for local gauge symmetries: see Teh (2016: p. 99) and Sus (2017: pp.
274-275).

Slightly anticipating the results of the next Section, we can show this
explicitly in the case of general relativity. Rewrite:
$\tau^{\mu}{}_{\nu}=2E^{\mu}{}_{\nu}+\partial_{\lambda}U^{\mu\lambda}_{\nu}$,
where $E^{\mu}{}_{\nu}={1\over 2\kappa}\sqrt{g}(G^{\mu}{}_{\nu}-\kappa
T^{\mu}{}_{\nu})$ is proportional to the Einstein field equations. Using the
contracted Bianchi identities, we find:
$\partial_{\mu}\tau^{\mu}{}_{\nu}=E^{\alpha\beta}\partial_{\nu}g_{\alpha\beta}+\partial_{\mu}\partial_{\lambda}U^{\mu\lambda}_{\nu}$.
Since the second term vanishes identically, the total energy-momentum
(pseudo-) tensor density is conserved if and only if a certain linear
combination of Einstein’s equations are satisfied. This seems to be what
Einstein (1918: p. 513) had in mind in his reply that part of the field
equations are used,656565Also in classical mechanics, it is only part, i.e. a
linear combination, of the equations of motion that appear on the right-hand
side. Consider, as an elementary example, the Lagrangian $L={1\over
2}\,m\dot{x}_{i}^{2}+V(x)$, where $i=1,2,3$. The time derivative of the
Hamiltonian gives: $\mbox{d}H/\mbox{d}t=E_{i}\dot{x}_{i}$, where
$E_{i}=m\ddot{x}_{i}+\partial_{i}V$, i.e. the conservation law is proportional
to a linear combination of the equations of motion. and he was indeed
right.666666Klein’s argument can go through if $E^{a}\gamma^{\mu}_{ak}=0$
without use of the equations of motion, e.g. if $\gamma^{\mu}_{ak}=0$. In that
case, it is true that the first term vanishes before its derivative is taken.
However, this is not the case in general relativity.

##  3 Noether’s Theorems in General Relativity

In this Section, I will apply Noether’s theorem to general relativity coupled
to matter.676767For a discussion of Noether’s theorem and Kretschmann’s
objection to Einstein’s treatment of general covariance, see Brown and Brading
(2002: pp. 22-25). This paper also contains a treatment of Noether’s theorem
for electrodynamics with an arbitrary background metric. I will first focus on
the case of pure gravity, and then discuss adding matter.

Take the action Eq. (25) with Lagrangian density given by ${\cal L}={\cal
L}_{\mbox{\tiny E}}+{\cal L}_{\mbox{\tiny m}}$, where ${\cal L}_{\mbox{\tiny
E}}$ is the Einstein Lagrangian, and ${\cal L}_{\mbox{\tiny m}}$ is the matter
Lagrangian. The Einstein Lagrangian is given by:

$\displaystyle 2\kappa\,{\cal L}_{\mbox{\tiny
E}}(g_{\mu\nu},\partial_{\lambda}g_{\mu\nu})$ $\displaystyle:=$
$\displaystyle\sqrt{g}~{}g^{\mu\nu}(\Gamma^{\lambda}_{\mu\nu}\Gamma^{\lambda}_{\lambda\sigma}-\Gamma^{\sigma}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma})$
(44) $\displaystyle=$
$\displaystyle\sqrt{g}~{}R-\partial_{\mu}(\sqrt{g}\,(g^{\alpha\beta}\Gamma^{\mu}_{\alpha\beta}-g^{\mu\nu}\Gamma^{\lambda}_{\nu\lambda}))~{},$

and so the Einstein action differs from the usual Einstein-Hilbert action by a
total derivative term that does not affect the equations of motion.686868There
are various boundary terms that one can add, depending on the choice of
boundary conditions, and these lead to different expressions for the pseudo-
tensor. For a treatment of Noether’s second theorem for the Einstein-Hilbert
action, see for example Horký and Novotný (1969: p. 424). In this Section, I
follow some of the historical literature in using the Einstein action. I will
return to the issue of boundary terms and boundary conditions in Sections 4
and 5.

Let us first write out the notation for Noether’s theorems applied to the
Einstein term in the action. For, although we have a coupled gravity-matter
system, we may of course require the diffeomorphism invariance of a single
term.

The field $\phi_{a}$ is then the metric tensor, and the index $a$ is a
symmetric pair of spacetime indices, $a=\alpha\beta$, ranging from 1 to $4$,
i.e. the dimension of spacetime. Under a local translation $\delta
x^{\mu}=\xi^{\mu}(x)$, the metric transforms as follows:

$\displaystyle\delta
g_{\alpha\beta}=-\nabla_{\alpha}\xi_{\beta}-\nabla_{\beta}\xi_{\alpha}=-\xi^{\mu}\,\partial_{\mu}\,g_{\alpha\beta}-2g_{\nu(\alpha}\,\partial_{\beta)}\xi^{\nu}~{}.$
(45)

The index $k$ is also a spacetime index $\mu$. Comparing with Eq. (34), we see
that $\eta_{ak}$ is equal to $-\partial_{\mu}g_{\alpha\beta}$, and
$\gamma^{\mu}_{ak}$ is equal to $-2\delta^{\mu}_{(\alpha}\,g_{\beta)\nu}$.

Noether’s second theorem, Eq. (38), applied to the gravitational part of the
action, gives the contracted Bianchi identity for the Einstein
tensor:696969This formulation follows Noether’s (1918: p. 241) original
methods, reviewed in Section 2. However, there is an important and useful
reformulation of Noether’s theorem, for diffeomorphism-invariant Lagrangians,
in terms of differential forms: see Wald (1993: pp. 3428-3429) and Wald and
Zoupas (2000: pp. 2-3).

$\displaystyle\partial_{\mu}(\sqrt{g}\,G^{\mu}{}_{\nu})={1\over
2}\,\sqrt{g}\,G^{\alpha\beta}\,\partial_{\nu}g_{\alpha\beta}~{}~{}\Leftrightarrow~{}~{}\nabla_{\mu}G^{\mu}{}_{\nu}=0~{},$
(46)

Noether’s second theorem, applied to the matter part of the action, gives the
covariant analogue of the energy and momentum conservation equation, i.e. Eq.
(16). That equation is, as expected, obtained as an identity, without the use
of the equations of motion.

Let us write down the Noether current, Eq. (31),
$J^{\mu}_{\nu}=:-t^{\mu}{}_{\nu}$, for the gravitational part of the action:

$\displaystyle t^{\mu}{}_{\nu}=-{\partial{\cal L}_{\mbox{\tiny
E}}\over\partial(\partial_{\mu}g_{\alpha\beta})}\,\partial_{\nu}\,g_{\alpha\beta}+{\cal
L}_{\mbox{\tiny
E}}\,\delta^{\mu}_{\nu}-{\partial\Lambda^{\mu}\over\partial\xi^{\nu}}~{}.$
(47)

This gives a gravitational analogue of the matter energy-momentum tensor. In
fact, as I shall discuss in a moment, it is not a tensor: it is the
gravitational energy-momentum pseudo-tensor.

The boundary theorem gives the two following results. The first equation of
the boundary theorem, Eq. (35), applied to the Einstein action, gives an
expression for the divergence of the gravitational pseudo-tensor:

$\displaystyle\partial_{\mu}t^{\mu}{}_{\nu}=-{1\over
2\kappa}\,\sqrt{g}\,G^{\alpha\beta}\,\partial_{\nu}g_{\alpha\beta}=-{1\over\kappa}\,\partial_{\mu}(\sqrt{g}\,G^{\mu}{}_{\nu})~{},$
(48)

where in the last step I used the Bianchi identity, Eq. (46). (The same result
can be obtained by taking the divergence of Eq. (47) directly). Using
Einstein’s equations, $G_{\mu\nu}=\kappa\,T_{\mu\nu}$, we find Einstein’s form
of the conservation law, Eq. (21). The second equation of the boundary
theorem, Eq. (36), reproduces the expression Eq. (24) for the pseudo-tensor in
terms of the superpotential.

I will now argue that this form of the boundary theorem shows that the alleged
non-uniqueness of the pseudo-tensor, i.e. problem (iv) in Section 1.3, only
arises if one treats Eq. (21) independently of Noether’s theorems. For if one
takes Eq. (21) as the definition of the pseudo-tensor and integrates it, then
the pseudo-tensor indeed admits an arbitrary superpotential, as in Eq.
(24).707070For example, Lam (2011: pp. 1017-1018) defines the energy-momentum
pseudo-tensor in terms of an expression that automatically satisfies the
conservation equation Eq. (21), as in Eq. (24), and thus contains an arbitrary
superpotential.

But if one, correctly, applies Noether’s theorem to the gravitational part of
the action, Eq. (44), then both the pseudo-tensor and the superpotential are
fixed by their Noether expressions in Eqs. (31) and (32), in terms of:

(1) the Lagrangian density,

(2) the boundary term $\Lambda^{\mu}$ that appears in the Noether variation of
the action, Eq. (26).

Noether’s expression Eq. (32) for the superpotential indeed shows that the
superpotential is not arbitrary, but is fixed when (1) and (2), including the
boundary terms of the action, are fixed. One may add boundary terms to the
action without changing the equations of motion, and in this sense there is
freedom to change the superpotential by changing the boundary terms. However,
boundary terms often do change the boundary conditions, and so the boundary
terms must be such that they give the desired boundary conditions. (Typically,
the boundary conditions are not sufficient to uniquely determine the boundary
terms; the finiteness of the on-shell action is also required to fix them: see
Section 4).

This means that the freedom to choose the superpotential has a physical
meaning: it is the freedom to change the action, without affecting the
equations of motion, by boundary terms. The latter have a physical meaning,
because they determine:

(a) the boundary conditions in the variation of the metric that gives the
equation of motion;

(b) the on-shell value of the (Euclidean) action, which e.g. in thermodynamic
approaches to gravity, especially in the context of black holes, is
interpreted as a gravitational potential;717171See e.g. Hawking (1979: pp.
772, 778-779). Wald (1993: p. 3431) and Hawking and Horowitz (1996: pp.
1494-1495) contain reformulations of the first and second laws of black hole
mechanics in terms of the Lagrangian and its variation. For some examples, see
Wald and Zoupas (2000: p. 8-14), Jacobson et al. (1994: pp. 6592-6593).

(c) the boundary terms of the Hamiltonian are also determined by the boundary
terms of the Lagrangian.727272For the relation between the Lagrangian and
Hamiltonian approaches to GR, see for example Wald (1984: pp. 464-465) Thus
the significance of the boundary terms does not hinge on interpreting the
Lagrangian physically.

Chang et al. (1999) notice that, in a Hamiltonian approach, the freedom to
change the superpotential is related to the freedom to change the boundary
conditions, i.e. analogous to my points (a) and (c) above. However, so far as
I can see, they do not notice that it is in general Noether’s theorem that
completely fixes the superpotential from the variation of the action, in
particular through Eq. (32).

In the absence of Killing vectors that allow us to construct the energy by
integrating a scalar expression like Eq. (19), some authors use the pseudo-
tensor to convert the covariant derivative in Eq. (16) into a partial
derivative and integrate. Thus these authors define a total momentum vector
$P_{\mu}$ as an integral over a spacelike volume $V$:737373See Weinberg (1972:
p. 167), Landau and Lifshitz (1971: p. 306), Chen et al. (2018: p. 3, 2018a:
p. 4). For the philosophical literature on this point, see Read (2020: p. 217)
and Hoefer (2000: p. 194).

$\displaystyle
P_{\mu}:=\int_{V}\mbox{d}\Sigma_{\nu}\left(T^{\nu}{}_{\mu}+{1\over\sqrt{g}}~{}t^{\nu}{}_{\mu}\right).$
(49)

By choosing special coordinates, where the normal vector to the boundary $S$
of the volume $V$ points along the time direction, these authors derive:

$\displaystyle{1\over
c}{\mbox{d}P_{\mu}\over\mbox{d}t}=-\int_{S}\mbox{d}^{2}x\,Nn_{i}\,(\sqrt{h}~{}T^{i}{}_{\mu}+t^{i}{}_{\mu})~{},$
(50)

where, as before, $n_{i}$ is the outward-pointing unit vector normal to $S$,
$N$ is the lapse function, and $h$ is the metric on $S$ induced from $g$ in
these coordinates. This expression can be shown to reproduce the well-known
ADM (for Arnowitt, Deser, and Misner) formulas for asymptotically flat
spacetimes: see Arnowitt et al. (1962: p. 16), Wang (2015: p. 3), Chen et al.
(2017: p. 200), Chen et al. (2018: p. 3), Chen et al. (2018a: p.
158).747474For a discussion of the history of the ADM mass formula, and
various versions and derivations of it, see Ohanian (2010: p. 9). This
expression is regarded as an analogue of the charge conservation law Eq. (14).

However, as I will discuss in Section 5, the above definition is
unsatisfactory: for it transforms covariantly only under affine
transformations, and the analogy with charge conservation depends on a very
special embedding of the volume $V$ in the spacetime and on the associated
choice of coordinates.

##  4 Quasi-local Stress-Energy Tensors in GR

It is time to discuss other proposals for the energy-momentum and angular
momentum of the gravitational field. This Section introduces two such. Section
4.1 introduces the Brown-York quasi-local stress-energy tensor, which in
Section 4.2 is then modified and gets a holographic interpretation that, I
will argue, casts light on the nature of energy and momentum in GR. In Section
4.3, I will discuss the transformations of the Brown-York stress-energy tensor
under diffeomorphisms.

###  4.1 The Brown-York quasi-local stress-energy tensor

At the end of Section 1.3, where I discussed five problems with the
gravitational pseudo-tensor, I mentioned Penrose’s idea of giving up local
expressions for energy and momentum in GR, and to consider quasi-local notions
instead. This Section presents one such expression in some detail.

Brown and York’s (1993) proposal is to define the quasi-local stress-energy
tensor, associated with a spatially bounded region $M$, as the conjugate of
the classical action with respect to an appropriately defined metric on the
boundary of that region.

The action to consider contains six terms. There is the usual Einstein-Hilbert
action, coupled to matter, and possibly including a cosmological constant
term. Furthermore, since the spacetime region $M$ is bounded, appropriate
boundary terms are required in order to make the variational problem well-
defined. Since $\partial M$ consists of two spacelike hypersurfaces
$\Sigma_{\pm}$ to the past and future of $M$, and the timelike three-surface
$N$ that spatially bounds $M$, there are three such boundary terms (where the
first two are abbreviated into a single term, see Eq. (51) below).757575Brown
and York’s (1993: p. 1410) coordinates are such that the hypersurfaces
$\Sigma_{\pm}$ and $N$ are orthonormal to each other on the two-surface where
they intersect. Furthermore, no additional term in the action for this two-
surface is required. To make the variational problem of the Einstein-Hilbert
action well-defined, a choice of boundary conditions for the metric on the
boundary is required: Brown and York consider Dirichlet boundary conditions,
i.e. keeping the metric on the boundary fixed.767676These boundary conditions
also render the gravitational part of the action well-defined in case a
cosmological constant term is present. The appropriate boundary term for
Dirichlet boundary conditions is the extrinsic curvature, denoted $K$ on the
hypersurfaces $\Sigma_{\pm}$, and $\Theta$ on the three-surface $N$. Finally,
a reference term is required that renders the value of the action and the
stress-energy tensor well-defined (I will have more to say about this term
later). Thus the action is:

$\displaystyle S={1\over
2\kappa}\int_{M}\mbox{d}^{4}x\,\sqrt{-g}~{}R\\!\\!\\!\\!\underset{\mbox{\tiny
extrinsic curvature of hypersurfaces
$\Sigma_{\pm}$}}{\underbrace{+{1\over\kappa}\int_{\Sigma_{\pm}}\mbox{d}^{3}x\,\sqrt{h}~{}K}}~{}~{}\underset{\mbox{\tiny
extrinsic curvature of
$N$}}{\underbrace{-{1\over\kappa}\int_{N}\,\sqrt{-\gamma}~{}\Theta}}+S^{\mbox{\tiny
ref}}+S^{\mbox{\tiny matter}}$ (51)

Here, $h$ is the metric on the hypersurfaces $\Sigma_{\pm}$ embedded in $M$,
and $\gamma$ is the metric on the three-boundary $N$ embedded in $M$. $K$ and
$\Theta$ are their corresponding extrinsic curvatures.

Varying this action with respect to the metric, we get:

$\displaystyle\delta S=-\int_{M}\mbox{d}^{4}x~{}E^{\mu\nu}\delta
g_{\mu\nu}+{1\over
2}\,\int_{\Sigma_{\pm}}\mbox{d}^{3}x~{}\sqrt{h}~{}P^{ij}\,\delta
h_{ij}+{1\over
2}\,\int_{N}\mbox{d}^{3}x~{}\sqrt{\gamma}~{}\tau^{ij}_{\mbox{\tiny
BY}}\,\delta\gamma_{ij}~{}.$ (52)

The first term contains the Einstein field equations including matter, and
thus vanishes if the field equations are satisfied. The second and third terms
define the momenta conjugate to the three-metrics $h$ on $\Sigma_{\pm}$ and
$\gamma$ on $N$, respectively (with Dirichlet boundary conditions, as
announced). $\tau^{ij}_{\mbox{\tiny BY}}$ is a genuine (quasi-local) energy-
momentum tensor, because it contains both spacelike and timelike components.
It also contains contributions from both the gravitational and the matter part
of the action. The subscript ‘BY’ emphasises that this is the energy-momentum
tensor, as defined in the proposal by Brown and York.

Restricting for the moment to the gravitational contribution to this last
term, we get the following result for the Brown-York stress-energy tensor
density:

$\displaystyle\tau^{ij}_{\mbox{\tiny
BY}}=-{1\over\kappa\sqrt{\gamma}}\left(\Theta\,\gamma^{ij}-\Theta^{ij}\right)+\mbox{\scriptsize(subtracted
terms)}~{}.$ (53)

The subtracted terms subtract a potentially divergent piece, in cases where
one is evaluating this tensor at infinity. These subtracted terms are obtained
by subtracting to the action Eq. (51) the same action, evaluated on an
appropriate reference spacetime. For example, if the solutions of the Einstein
field equations that one considers are asymptotically flat, one usually
subtracts from Eq. (51) the same action, but evaluated on a Minkowski
solution. In this way, the ‘on-shell’ value of the action (i.e. the action
evaluated on a particular solution) is finite, and also the Brown-York stress-
energy tensor is finite. I will return to this procedure in the next Section,
where I will mention an alternative way to resolve this problem that does not
appeal to the reference spacetime.

The boundary stress-energy tensor (density) satisfies one of Einstein’s
equations:

$\displaystyle\nabla_{i}\tau^{ij}_{\mbox{\tiny BY}}=-T^{nj}\,,$ (54)

where $T^{nj}$ is the matter energy-momentum tensor with one index projected
normally, and the other tangentially, to $N$. This is a covariant balance
equation for the quasi-local stress-energy tensor, where the contribution on
the right is from the matter that passes through the boundary.

If there is no matter passing through the boundary, and if the boundary three-
metric $\gamma_{ij}$ possesses a Killing vector field $K_{i}$,777777In the
context of spaces with a cosmological constant, this condition can be weakened
for $K_{i}$ to be only a conformal Killing vector, i.e. one that satisfies Eq.
(63). then $\tau^{ij}$ defines a conserved charge, i.e. we have the
conservation law $\nabla_{i}(\tau^{ij}_{\mbox{\tiny BY}}K_{j})=0$, which is
analogous to Eq. (17). Thus by the same manipulations as in Section 1.2 we
derive an analogue of Eqs. (19)-(20) on the boundary.

Note that, in the Brown-York proposal, the spacetime region $M$ can be finite
and embedded in an ambient spacetime: and so, the three-surface $N$ that
spatially bounds $M$ need not be at infinity. In the next Section, I will
discuss the case $\Lambda<0$, and $N$ will be taken to be the asymptotic
boundary.787878For studies of the Brown-York stress-energy tensor at surfaces
that are not asymptotic boundaries, see e.g. De Haro et al. (2001a: pp. 3174).
For gauge-gravity dualities (both AdS-CFT and dS-CFT) with finite boundaries,
see Harlow and Stanford (2011: pp. 4-5) and Heemskerk and Polchinski (2011:
pp. 2-6).

As any quasi-local energy-momentum must, the Brown-York expression agrees with
the ADM energy at infinity (see Section 3) for the special case of
asymptotically flat spacetimes (Brown and York, 1993: p. 1408).

###  4.2 The holographic stress-energy tensor

Consider the case of pure gravity with a negative cosmological constant,
$\Lambda<0$, , i.e. no matter. (The Brown-York stress-energy tensor can be
equally well be defined in the case $\Lambda>0$).797979See Balasubramanian et
al. (2001: p. 3) and Anninos et al. (2011: pp. 4-5). Quite independent of
quantum gravity or holographic ideas, the Brown-York energy-momentum tensor
has been used to define notions of mass and momentum in spaces with a positive
cosmological constant: see Balasubramanian et al. (2001: p. 4). The asymptotic
diffeomorphism group at future null infinity of asymptotically de Sitter
spaces, and the conservation equations for the associated charges, are
considered in Anninos et al. (2011: pp. 3-6) and Strominger (2001: pp. 4-5).
According to AdS-CFT, the gravitational theory in the spacetime region $M$
with boundary $\partial M$ is dual to a quantum field theory on the boundary,
$\partial M$.808080For a general introduction to gauge-gravity dualities, see
Ammon and Erdmenger (2015). For a conceptual introduction, see De Haro et al.
(2016). For a sample of the recent philosophical literature on dualities, see
for example: Rickles (2011), Dieks et al. (2015), Read (2016), Huggett and
Wüthrich (2020), De Haro (2020), Read and Møller-Nielsen (2020), Butterfield
(2020). As we will see, one can use gauge-gravity duality to reinterpret the
Brown-York stress-energy tensor, evaluated on this asymptotic boundary, in
terms of the dual CFT.

Since we are interested in the Brown-York stress-energy tensor that is claimed
to represent the energy and stress of the spacetime $M$, it will be useful to
briefly mention how it fits within gauge-gravity duality.818181In what
follows, I will restrict to the case of a spacetime with a Euclidean
signature, so that $M$ has a single boundary, $N$, with the topology of a
three-sphere. For the case with Lorentzian signature, see Skenderis and van
Rees (2008: p. 2; 2009: p. 6).

Following the recent literature,828282I here follow De Haro and Butterfield
(2018: pp. 335-339), De Haro (2020: pp. 264-266), and Butterfield (2020: p.
12). a duality is, roughly, an isomorphism between two theory formulations. A
theory can be construed as a triple of states, quantities, and dynamics. The
statement of duality says that the sets of states, quantities, and the
dynamics of the two theories are isomorphic.

On the gravity side of the duality, a distinguished quantity should be the
stress-energy tensor of the spacetime region $M$. The isomorphism should
relate this to an appropriate quantity in the dual CFT. The question is: which
stress-energy tensor has a dual in the CFT? If the Brown-York stress-energy
tensor is to be a candidate for the energy and stress in the spacetime, then
it ought to be related by the duality to a quantity in the CFT that has those
properties. In fact, gauge-gravity duality gives us the following
correspondence:

The Brown-York stress-energy tensor, defined by the variation of the action
(evaluated on the equations of motion) with respect to a boundary three-metric
on $N$, and with appropriate boundary terms that render the action finite, is
dual to the expectation value of the stress-energy tensor of the CFT defined
on the background of that metric.838383I here omit technical details, such as
that the boundary metric is defined only up to a conformal transformation.
This point will resurface in Section 4.3.

This correspondence follows from the identification of the gravitational
action Eq. (51) with a given set of matter fields, evaluated on a given
solution of the Einstein field equations, with the generating functional of
(connected) correlation functions in the dual CFT, usually called
$W_{\mbox{\tiny CFT}}$:

$\displaystyle S[\gamma,\cdots]|_{\mbox{\tiny EFE}}\equiv W_{\mbox{\tiny
CFT}}[\gamma,\cdots]~{}.$ (55)

The left-hand side of this equation is the action Eq. (51), evaluated for a
metric and set of matter fields that satisfy Einstein’s field equations (hence
the subscript ‘EFE’). (Physicists call this an ‘on-shell’ action). $\gamma$ is
the metric on $\partial M$, and the dots indicate that there is also a
dependence on the boundary values of the matter fields. The right-hand side of
this equation is the generating functional of connected correlation functions,
which is related to the CFT’s partition function by: $Z_{\mbox{\tiny
CFT}}=\exp(-W_{\mbox{\tiny CFT}})$. Again, this generating functional depends
on the metric $\gamma$, which is now viewed as a curved background on which
the CFT is defined, and other sources that the theory is defined with (these
are dual to the boundary values of the matter fields on the left).

On the right-hand side, the expectation value of the stress-energy tensor of
the CFT is defined by the functional derivative with respect to the metric:

$\displaystyle\langle T_{ij}(x)\rangle_{\mbox{\tiny
CFT}}={2\over\sqrt{\gamma}}{\delta W_{\mbox{\tiny
CFT}}\over\delta\gamma^{ij}(x)}~{}.$ (56)

But substituting Eq. (55) into Eq. (56), and comparing with Eq. (52), we see
that we indeed have the following correspondence:

$\displaystyle\tau_{{\mbox{\tiny BY}}}^{\prime}{}^{ij}\equiv\langle
T^{ij}\rangle_{\mbox{\tiny CFT}}~{}.$ (57)

The dash on the Brown-York stress-energy tensor indicates that this tensor is
modified, by replacing the ‘subtracted terms’ in Eq. (53) by local
counterterms added to the action, according to a standard method in
holographic renormalisation.848484See Henningson and Skenderis (1998: p. 5),
Balasubramanian and Kraus (1999: p. 415), De Haro et al. (2001: p. 603). For a
review of holographic renormalisation, see Skenderis (2002). In what follows,
I will call $\langle T^{ij}\rangle_{\mbox{\tiny CFT}}$ ‘holographic stress-
energy tensor’.

The holographic stress-energy tensor can be calculated from Eq. (53) for any
given solution of Einstein’s equations.858585In general this is tedious, but
there is a systematic procedure to do this calculation. See De Haro et al.
(2001: pp. 602-607, 609-612), Skenderis (2002: pp. 5855-5861). For an
alternative method, see Papadimitriou and Skenderis (2004: pp. 14-18).

The gravitational equation Eq. (54) can be reproduced from the CFT
perspective. For example, consider a two-dimensional CFT with a single
(scalar) operator ${\cal O}$ which has a non-zero expectation value
$\langle{\cal O}\rangle$, coupling to a source $J$, the invariance of the CFT
partition function under infinitesimal changes of coordinates,

$\displaystyle\delta\gamma_{ij}=\nabla_{i}\xi_{j}+\nabla_{j}\xi_{i}~{},$ (58)

yields the following identity:

$\displaystyle\nabla^{j}\langle T_{ij}\rangle=\langle{\cal
O}\rangle\,\partial_{i}J~{}.$ (59)

One can check that this result is reproduced, on the gravity side with a
three-dimensional spacetime, by the identity of Brown and York, Eq. (54),
since, given the form of the normal energy-momentum tensor $T^{nj}$ for the
dual scalar field, the above indeed reproduces Eq. (54).868686See De Haro et
al. (2001: p. 615).

Noether’s theorem in connection with asymptotic symmetries is discussed in
Papadimitriou and Skenderis (2005: pp. 15-19), who show, for various types of
fields, that the holographic charges are the Noether charges.

Implications of gauge-gravity duality for the gravitational energy. Let us
briefly discuss how these results bear on the problem of energy conservation.
Gauge-gravity dualities give independent support to the claim that the
modified Brown-York stress-energy tensor represents the quasi-local energy. We
will consider the two main features of the holographic stress-energy tensor of
the dual quantum field theory that bear on the status of the quasi-local
energy in GR:

(1) The duality map, Eq. (57), relates the Brown-York stress-energy tensor to
the holographic stress-energy tensor. This gives further support to the
interpretation of the Brown-York stress-energy tensor as representing quasi-
local stress and energy. For recall that a duality is an isomorphism of models
of a single common core theory.878787This ‘common core’ view is discussed and-
or endorsed by e.g. Rickles (2011: p. 66), Dieks et al. (2015: p. 209), De
Haro (2017: p. 116), Huggett (2017: p. 84), De Haro and Butterfield (2018: p.
321), Le Bihan and Read (2018: p. 3). The interpretation of two dual models
need not be the same, but if it is the same,888888That it is the same depends
on our developing an internal interpretation of the dual models: for a
discussion, and conditions under which this is possible, see Dieks et al.
(2015: p. 209), De Haro (2020: pp. 267-271), Read and Møller-Nielsen (2020: p.
284-288). then the two dual quantities on either side of Eq. (57) receive the
same interpretation. Thus the right-hand side of this equation, i.e. the
holographic stress-energy tensor, can give us guidance about the
interpretation of the side under discussion, i.e. the gravitational, left-
hand, side. And the interpretation of the left-hand side is indeed as energy,
momentum, and stress. This is a further argument to the effect that the Brown-
York stress-energy tensor represents energy, momentum, and stress.898989Note
that the internal interpretation of the common core theory, as currently
worked out, does not support the interpretation of this energy as
gravitational energy—even though this can be done on an external
interpretation (see footnote 91). For understanding the nature of the common
degrees of freedom depends on having a good development of this common core
theory, which we do not currently have. For a more detailed discussion, see De
Haro (2020: pp. 275-279).

Thus, in AdS-CFT, the modified Brown-York stress-energy tensor in spacetime
with $\Lambda<0$ is precisely dual to the quantity that, in the CFT, is
interpreted as describing energy and momentum: namely, the expectation value
of the stress-energy tensor. The match is a consequence of the general AdS-CFT
correspondence: namely, the identification of the gravitational action,
evaluated on solutions, with the generating functional of the CFT, Eq. (55),
and of the conformal metric on the boundary, $\gamma_{ij}$, with the metric of
the dual quantum field theory. This precise correspondence, which has been
illustrated in many examples, gives additional support to the statement that
the Brown-York stress-energy tensor is indeed to be thought of as energy and
momentum, since it plays that role in the boundary theory.

(2) The boundary conservation equation, Eq. (54), has a similar dual
interpretation, and is not analogus to the usual (disputed) conservation laws
of GR.909090There are three important disanalogies between the boundary
balance equation, Eq. (54), and the (disputed) conservation law, Eqs.
(16)-(21), which are relevant to points (i)-(iv) from Section 1.3. First, Eq.
(54) distinguishes between the quasi-local gravitational energy,
$\tau_{\mbox{\tiny BY}}$, and matter energy-momentum, $T^{nj}$. Second, the
covariant derivative is calculated using the boundary, metric-compatible,
connection, which is non-dynamical, and fixed by the boundary conditions.
Third, the right-hand side is in general non-zero, so that this is a balance
equation. I will address points (iii) and (v), from Section 1.3, in Section
6.2 This boundary conservation equation is dual to Eq. (59): and so, it can be
interpreted as a balance equation for the energy and stress of the common core
theory, and this again strengthens that interpretation in GR.919191An internal
interpretation (as in ‘Eq. (54) is the energy balance equation’) is usually
compatible with a judiciously chosen external interpretation (as in ‘Eq. (54)
is the balance equation for the gravitational energy’), through the process of
abstraction outlined in De Haro (2021: Section 4.4). This is why, if the
internal interpretation is correct, it bears on the interpretation on GR.

###  4.3 Diffeomorphisms and redundancy of the description

In our discussion, in Section 5, of the two main proposals for defining energy
and momentum in general relativity, diffeomorphism invariance will play a
crucial role. Therefore, we should understand the transformation properties of
the Brown-York stress-energy under various diffeomorphisms: namely, those that
are descriptive redundancies and those that are not.929292My discussion here
essentially summarises De Haro, Teh, and Butterfield (2015, 2017) and De Haro
(2017). For physics references on this topic, see Imbimbo et al. (2000: pp.
1130-1132), Skenderis (2001: pp. 742-747).

The asymptotic symmetries of gravity have been a central foundational topic in
general relativity since at least the work of Arnowitt et al. (1959, 1962),
Bondi et al. (1962), Penrose (1963, 1964), Newman et al. (1966), Geroch
(1972), Ashtekar et al. (1978), among others. For a philosophical discussion
of some aspects of asymptotic symmetries, see Belot (2018).

A central question is which asymptotic diffeomorphisms are true symmetries of
a spacetime, and which asymptotic diffeomorphisms act non-trivially on the
physical degrees of freedom of general relativity: and how these different
kinds of diffeomorphisms are to be characterised.

In spaces with a negative cosmological constant, Brown and Henneaux (1986)
discovered that the asymptotic diffeomorphism of anti-de Sitter space is the
conformal group of the asymptotic boundary.939393The analysis by Brown and
Henneaux has been extended to the case $\Lambda>0$, including in higher
dimensions: see Strominger (2001: pp. 4-5) and Balasubramanian et al. (2001:
pp. 7-8). For a more detailed analysis of diffeomorphisms in the case
$\Lambda>0$, see Anninos et al. (2011: pp. 3-4). As we will see in this
Section, these results can be generalised to any space with $\Lambda<0$ that
is a solution of Einstein’s equations, with arbitrary metric at infinity. As
we will see in this Section, this asymptotic symmetry group acts non-trivially
on the quasi-local energy-momentum tensor.

Thus the question of which subgroup of the diffeomorphism group is a symmetry
of a spacetime is important from the point of view of quasi-local energy and
momentum: for the quasi-local quantities will be either invariant or else have
a well-defined transformation law under the diffeomorphisms that are true
asymptotic symmetries of a spacetime, and they transform non-trivially under
diffeomorphisms that are not true symmetries.

What do we mean by the ‘true asymptotic symmetries’ of a spacetime? What we
are looking for here is a criterion that distinguishes the symmetries
(diffeomorphisms) that are true redundancies of a description, from those that
entail a physical change in the solution. We can define redundancies as
follows (De Haro et al., 2017: p. 69):949494There is an ongoing debate,
following Greaves and Wallace (2015)—which itself argues against earlier work:
see footnote 35—whether local gauge symmetries express mere redundancies of
the description, or whether they can be empirically significant, as in
Galileo’s ship scenarios: see, for example, Friederich (2015: p. 537),
Murgueitio (2020: p. 1), Murgueitio and Teh (2020: p. 3), Gomes (2020: p. 11),
Gomes and Riello (2020: pp. 3-5, 21). The approach I adopt here is, in effect,
to construct a class of diffeomorphisms (the ‘visible’ diffeomorphisms below)
that are GR-analogues of the class of gauge symmetries that can have a direct
empirical equivalence in gauge theories. The latter are called ‘non-interior,
boundary-preserving symmetries’ by Greaves and Wallace (2014: p. 71), and
their definition is further refined by Teh (2016: p. 116).

> (Redundant): If a physical theory’s formulation is redundant (i.e. roughly:
> it uses more variables than the number of degrees of freedom of the system
> being described), one can often think of this in terms of an equivalence
> relation, ‘physical equivalence’, on its states; so that gauge-invariant
> quantities are constant on an equivalence class, and gauge-symmetries are
> maps leaving each class (called a ‘gauge-orbit’) invariant. Leibniz’s
> criticism of Newtonian mechanics provides a putative example: he believed
> that shifting the entire material contents of the universe by one meter must
> be regarded as changing only its description, and not is physical state.

I will introduce the notion of the ‘invisibility’ of diffeomorphisms to make
precise the idea of the asymptotic symmetries that are (Redundant). I will
distinguish three relevant classes of diffeomorphisms that qualify differently
under redundancy: ‘invisible’ diffeomorphisms are always (Redundant), ‘large’
diffeomorphisms are never (Redundant), and the intermediate category of
‘visible’ diffeomorphisms can sometimes be (Redundant) and sometimes not,
depending on the interpretation that the theory is given.959595Belot (2018:
pp. 964-967) also distinguishes local translations that are redundant, from
those that are physical: and he also connects this to the question of boundary
conditions—although he does not appear to consider my intermediate category of
‘visible diffeomorphisms’. The three classes are distinguished by the
different boundary geometric structures that they preserve.969696Class (1) is
the class of ‘QFT-invisible’ diffeomorphisms in De Haro (2017a: pp. 1468,
1484), also called ‘(simple) gravity-invisible’. Class (2) is the class of
‘QFT-visible’ diffeomorphisms, also called ‘weakly gravity-invisible’
diffeomorphisms’. For the reasons why these classes are cases of (Redundant)
or non-(Redundant), see De Haro, Teh, and Butterfield (2015, 2017) and De Haro
(2017). They are as follows:

(1) Invisible diffeomorphisms: these are diffeomorphisms under which the bulk
action, Eq. (55), evaluated on a given solution, is invariant. Also the other
quantities derived from the action, like the holographic stress-energy tensor
Eq. (57), are invariant. These diffeomorphisms fix the boundary manifold and
the conformal metric.979797It is sometimes said that the diffeomorphisms that
are redundancies of the theory are those that “go to unity” at the boundary
(this being the class by which the allowed diffeomorphisms have to be
quotiented in order to obtain the asymptotic symmetry group). But this is
imprecise, because a diffeormorphism can go to unity at the boundary while
still inducing a non-trivial transformation of the boundary metric
$\gamma_{ij}$, through its dependence on the parametrisation of the ‘radial
direction” $r$, which results in a rescaling of the boundary metric, and hence
a diffeomorphism of type (2) below. Thus, going to unity at the boundary is
not sufficient for (Redundant). Instead, the definition of invisible
diffeomorphisms makes precise the sense in which diffeomorphisms are
(Redundant).

These diffeomorphisms count as (Redundant) symmetries, because all of the
physical quantities, for a given solution of the equations of motion, are
invariant under them. In particular, these diffeomorphisms allow us to set up
an analogue of the hole argument in a space with a negative cosmological
constant, which in turn leads to the conclusion that the diffeomorphisms in
question are (Redundant).989898See De Haro et al. (2017: p. 79).

(2) Visible diffeomorphisms: these are diffeomorphisms that generate conformal
transformations of the boundary metric, i.e.
$\gamma_{ij}\mapsto\tilde{\gamma}_{ij}:=e^{-2\xi(x)}\,\gamma_{ij}$. The
stress-energy tensor transforms by a conformal factor under them, and in the
case of even boundary dimensions the stress-energy tensor (and the action)
pick up an additional, anomalous term under them:

$\displaystyle\langle
T_{ij}[\tilde{\gamma}](x)\rangle=e^{(d-2)\,\xi(x)}\left(\langle
T_{ij}[\gamma](x)\rangle+{1\over\sqrt{\gamma}}{\delta{\cal
A}[\gamma]\over\delta\gamma^{ij}(x)}\right),$ (60)

where $\gamma_{ij}$ is the metric induced on the boundary, $d$ is the
dimension of the boundary, and ${\cal A}[\gamma]$ is the anomalous term, which
can be calculated from the action.999999To simplify the discussion, I do not
distinguish between conformal transformations and Weyl rescalings of the
metric. For details, see De Haro (2017a: p. 1479). It is a local term written
in terms of curvature invariants (for the explicit expressions, see Deser and
Schwimmer (1993), Henningson and Skenderis (1998: Section 3), De Haro et al.
(2001: Section 4.2.1)). The anomaly vanishes when $d=3$, i.e. for a four-
dimensional spacetime.100100100The anomaly can be understood as coming from
the divergence of the gravitational action, Eq. (51), at large distances. To
cancel this divergence, the action is regularised and boundary counterterms
are introduced, but this renormalisation procedure breaks covariance, albeit
in a controlled way, i.e. through the additional term Eq. (60). Since the
anomaly term vanishes in the case of a four-dimensional spacetime, which is
the case of most interest, I will not discuss it further.101101101For
spacetimes of odd dimension, the anomaly is proportional to powers of the
boundary curvature that depend on the dimension, and so it is also zero if the
boundary metric is flat.

Visible diffeomorphisms are potentially physical symmetries, rather than
redundancies of the description: namely, they act as conformal transformations
of the boundary quantities. Whether they are physical depends on one’s
interpretation of the spacetime within one’s theory.102102102In De Haro et al.
(2017: p. 79), it was argued that, on an ‘internal’ interpretation (typically,
in the context of a theory of the whole universe), two states related by
visible diffeomorphisms are physically equivalent, so that the transformations
count as (Redundant) gauge symmetries. On the other hand, on an ‘external
interpretation’ (as e.g. in a Galileo’s ship scenario, where the inside of the
ship is not the whole world, but can be related to other parts of the world,
e.g. the shore) these diffeomorphisms are not (Redundant), but express ‘the
motion of the ship relative to the quay’.

(3) Large diffeomorphisms: these are diffeomorphisms that do not preserve
sufficient boundary structure, and so they map solutions of Einstein’s
equations to physically inequivalent solutions of the equations (e.g.
equations with a different total mass or angular momentum). Such
diffeomorphisms are therefore not (Redundant), but physical transformations
that relate different states. In particular, they will change the value of the
holographic stress-energy tensor.103103103For example, Grumiller et al. (2016:
Eq. (3.6)) report a metric that is claimed to be physically inequivalent to
the corresponding metric in normal form.

An analysis of the diffeomorphisms as detailed as the preceding one has, so
far as I am aware, not been done for $\Lambda>0$, and so one should proceed
with some caution. However, much work has been done on this case as well: see
Anninos et al. (2011: pp. 3-6), Strominger (2001: pp. 4-5), Balasubramanian et
al. (2001: pp. 7-8).

There are also a few general things one can say. As noted in Skenderis (2002:
Appendix A) and De Haro (2017a: pp. 1490-1491), the geometric techniques that
enabled the analysis summarised in Section 4.3 and just above, apply in the
case $\Lambda>0$ as well, due to the fact that the differential form of
Einstein’s equations in the radial variable was turned into a set of coupled
algebraic relations. The results depend only on the asymptotic solutions, and
can be continued from one case to the other, where the radial coordinate $r$
goes into a timelike coordinate $t$ (the anti-de Sitter boundary is thereby
mapped to the timelike future infinity in de Sitter space).

To summarise, in case (1) we have a hole-type argument that leads us to
conclude that invisible diffeomorphisms correspond to redundancies of the
theory. In case (3), such an argument is not available, because the
diffeomorphisms in question change the physical properties of the spacetime,
and we should not characterise spacetimes related by a large diffeomorphism as
physically equivalent. Case (2) is an intermediate case, where the
diffeomorphisms act as conformal transformations on the boundary quantities,
and their nature as (Redundant) or as non-(Redundant) depends on one’s
interpretation (including further use) of the solutions, say as being
cosmological models of the whole universe, or as being merely parts of the
universe.

Let me say a bit more about this last case, (2), of visible diffeomorphisms,
and how they give rise to transformations of the conformal boundary: namely,
conformal transformations. Choose an open neighbourhood of the boundary, such
that the boundary of $M$ is the surface $r=0$. A theorem of Fefferman and
Graham (1985, 2012) guarantees that in such an open neighbourhood of the
boundary, we can write the metric in Poincaré normal form:

$\displaystyle\mbox{d}s^{2}={\ell^{2}\over