# Substantive general covariance and the Einstein-Klein dispute: A Noetherian
approach

Laurent Freidel1 Nicholas Teh2

(1Perimeter Institute for Theoretical Physics
<EMAIL_ADDRESS>
2Department of Philosophy, University of Notre Dame
<EMAIL_ADDRESS>
)

###### Abstract

Famously, Klein and Einstein were embroiled in an epistolary dispute over
whether General Relativity has any physically meaningful conserved quantities.
In this paper, we explore the consequences of Noether’s second theorem for
this debate, and connect it to Einstein’s search for a ‘substantive’ version
of general covariance as well as his quest to extend the Principle of
Relativity. We will argue that Noether’s second theorem provides a clear way
to distinguish between theories in which gauge or diffeomorphism symmetry is
doing real work in defining charges, as opposed to cases in which this
symmetry stems from Kretchmannization. Finally, we comment on the relationship
between this Noetherian form of substantive general covariance and the notion
of ‘background independence’.

###### Contents

  1. 1 Introduction
  2. 2 What work should ‘general covariance’ do?
    1. 2.1 Basic General Covariance
    2. 2.2 Extending the Relativity Principle
    3. 2.3 Non-trivial corner charges
    4. 2.4 The Einstein test for ‘substantive’
  3. 3 General covariance through a Noetherian lens
    1. 3.1 The covariant phase space approach to Noether
    2. 3.2 Test cases
      1. 3.2.1 BGC with trivial superpotential
      2. 3.2.2 BGC with non-trivial superpotential
    3. 3.3 Substantive general covariance
    4. 3.4 Revisiting the case of General Relativity
    5. 3.5 Background independence
  4. 4 Conclusion

##  1 Introduction

In this note, we will explore the enduring impact of Noether’s legacy on some
of the deepest issues in the philosophical foundations of field theory. The
import of Noether’s two theorems is certainly undisputed in both the physics
and the philosophy literature, but an articulation of the why and wherefore of
her second theorem’s import is rather more difficult to come by. Indeed, most
textbooks on field theory only stress Noether’s first theorem, and even in
those places where Noether’s second theorem is mentioned, little more physical
insight is offered than the remark that the Noether identities relate the
Euler-Lagrange quantities and thus express the ‘redundancies’ that are
characteristic of a gauge theory.

The short shrift given to Noether’s second theorem goes hand in hand with a
glib narrative about a famous epistolary dispute between Einstein and Felix
Klein (letters 480 and 492 in [9]). Contra Einstein, Klein contended that
general relativity does not have any physically contentful conserved
quantities; the glib narrative takes Noether’s second theorem to settle the
dispute decisively in favor of Klein. As we will see shortly, the truth is
somewhat more complex and to a large extent vindicates Einstein’s strategy for
responding to Klein.

A good reason to get clear on the Einstein-Klein dispute is that it holds the
key to understanding just how fundamentally Noether’s second theorem (albeit
not as a piece of mathematics in itself, but as taken up and deployed by
physicists) has transformed our understanding of Lagrangian field theory.
Nonetheless, our reason for considering this controversy is not primarily
historical; we wish instead to bear witness to the continuing vitality of
Noether’s theorems—as a living force that energizes the physical imagination
and leads us to make profound new discoveries. Our witness consists in using
Noether’s second theorem to provide a novel and incisive analysis of what
substantive general covariance really amounts to: the vision that Einstein so
desired to contemplate but could only see in a mirror dimly. While there has
of course been much discussion of substantive general covariance—and the
related notion of background independence—in the philosophy and physics
literature,111See [19, 21] and the references therein. our discussion is (to
the best of our knowledge) the first to fully exploit the power of Noether’s
second theorem in this setting.222[8, 4] also discuss Noether’s second theorem
in relation to general covariance, but our use of the theorem goes far beyond
these discussions. Furthermore, we conjecture in Section 3.5 that our
Noetherian approach to substantive general covariance provides a royal road to
understanding the core phenomenon underlying the elusive notion of ‘background
independence’.

We begin our reflections in Section 2 in a historico-philosophical fashion: by
surveying two questions with which Einstein wrestled long and hard, and which
have been re-articulated in various guises in the contemporary philosophy of
physics literature. First, in Section 2.1, we remind the reader of the
received view of general covariance (which we call Basic General Covariance)
in the context of Lagrangian field theory. Section 2.2 then discusses the
question of whether and how general covariance plays a role in extending the
Principle of Relativity, and Section 2.3 discusses subject of the Einstein-
Klein dispute, viz. whether generally covariant theories have non-trivial
conserved charges. On the basis of the previous two subsections, Section 2.4
formulates what we will call the ‘Einstein test’ for whether a theory has
substantive general covariance.

Section 3 develops our novel contribution to the literature: we argue that the
machinery of Noether’s second theorem provides a powerful technique for
determining whether theories have substantive general covariance. Crucially
(and unlike many other treatements), we do not just make use of the Noether
identities, but also of the entire mathematical nexus that relates the Noether
identities, the Noether current, and the corner Noether charge which is fixed
by a specific choice of Lagrangian. Section 3.1 summarizes in an elementary
and conceptually transparent way the reformulation of Noether’s theorems in
the covariant phase space framework (much of which was already anticipated in
Noether’s original work [15]). Section 3.2 then applies this machinery to run
the Einstein test on several toy theories around which much of the
philosophical debate has been centered. On the basis of the results of this
test, we articulate in Section 3.3 a proposal for what substantive general
covariance à la Einstein amounts to: we give a less demanding form (which only
requires a non-trivial corner charge), and a more demanding form (which
requires an extension of the Principle of Relativity). Section 3.4 revisits
the case of General Relativity and the Einstein-Klein dispute from this
perspective, and Section 3.5 comments briefly on the relationship between our
notion of substantive general covariance and the idea of ‘background
independence’: while [19] attempts to make the case that the connection
between some substantive form of general covariance is ‘messier, and less
illuminating’ than the quantum gravity literature might suggest, our analysis
suggests that such pessimism is largely a function of considering
insufficiently robust versions of substantive general covariance.

##  2 What work should ‘general covariance’ do?

Although the discussion that follows will apply to Lagrangian field theories
quite generally (mutatis mutandis), we remind the reader that the themes of
interest to us were first explicitly problematized in the context of
Einstein’s General Relativity. Thus, it is here that our meditation will
originate.

The central concept that Einstein struggled so long and hard to formulate, to
interpret, and to employ in modeling certain material aspects of the world is
that of ‘general covariance’. Today, the descendant of this notion that is
commonly invoked by physicists is more plainly referred to as the
‘diffeomorphism invariance’ of General Relativity, but it is important to
appreciate that in the early days of the subject, Einstein’s conception of
general covariance was very much in flux, both in terms of its mathematical
definition, as well as its physical interpretation.

Here, then, are two key questions that Einstein wrestled with, which are
supposed to provide desiderata for a substantive—or physically
contentful—notion of general covariance:333Einstein’s struggles with general
covariance are of course famously various: for instance, we leave aside in
this essay the question of the relationship between general covariance and
determinism, which has been discussed in the ‘Hole Problem’ literature.

  1. 1.

Does general covariance play a role in extending the Principle of Relativity?

  2. 2.

Do generally covariant theories have non-trivial conserved charges?

As regards the second question, we emphasize that it is clear from Einstein’s
correspondence with Klein (see letter 480 in [9] and our discussion in Section
3.4) that the charges Einstein had in mind were corner ones, where by ‘corner’
we mean the boundary of a Cauchy surface —thus a ‘corner’ from the perspective
of the entire spacetime manifold.444Of course, Einstein did not use the
terminology of ‘corners’. What he says is that the construction of the charges
is analogous to the is analogous to the use of Gauss’ Law in electrostatics,
and that the ‘…possibility of this physically important interpretation is
based on the fact that the same quantities […] which enter in the conservation
law […] can, by virtue of the field equations […] be written as a
“divergence”,’[9] where by ‘quantities’ he means what we call the Noether
charge $J$ below. This means that, returning to standard modern terminology,
the on-shell charge integral will descend to a corner by Stokes’ theorem.

Whatever substantive general covariance amounted to for Einstein, it was
supposed to earn its spurs by means of a ‘yes’ answer to the above two
questions. We will proceed by laying out in Section 2.1 a ‘basic’ or minimal
notion of general covariance that physicists would typically invoke. Sections
2.2 and 2.3 will explain Einstein’s desiderata for the work that the notion of
substantive general covariance needs to do. Section 2.4 then uses these
respective desiderata to formulate a two-part ‘Einstein test’ for substantive
general covariance.

###  2.1 Basic General Covariance

According to standard accounts of the history of General Relativity, Einstein
initially intended to use general covariance as an adequacy constraint that
would single out General Relativity.555For the history, see the discussion in
[16] and the references therein. To achieve this goal, he initially proposed
that a generally covariant theory is one that can be formulated in such a way
that the theory is expressible in any coordinate system (where the different
coordinate systems/frames are related by diffeomorphisms).

The problem with this notion of general covariance, as Kretschmann [14]
pointedly remarked, is that it does not achieve Einstein’s goal of singling
out a particular theory—just about any theory is generally covariant according
to this definition, including Newtonian mechanics, as Cartan showed in his
invariant, curved geometric, formulation of Newtonian gravitation (often
called Newton-Cartan theory). In light of Kretschmann’s response to Einstein,
it is now commonly accepted (see [17] and references therein for discussion)
that Einstein’s early formulation of general covariance was a misstep.
Instead, one should try to provide a ‘physically contentful’ or substantive
notion of general covariance that really does pick out some proper subset of
physical theories, and perhaps General Relativity in particular.

Within the philosophical literature, one finds various attempts to articulate
a version of substantive general covariance that distinguishes it from the
mere freedom to express a theory’s fields, equations of motion, and solutions
in arbitrary frames (see [20] for a review). For the purposes of this piece,
however, it will be helpful to build up to substantive general covariance by
starting with the following notion that most physicists take as standard
(albeit typically under the head of ‘diffeomorphism invariance’):666We will
see in Section 3.2 below that only some cases of basic general covariance are
inequivalent to the freedom to express a theory in an arbitrary frame.

> (BGC) First, distinguish between the background (non-dynamical) and the
> dynamical fields of the theory. We then say that a Lagrangian of a theory is
> basically generally covariant (BGC) just in case diffeomorphisms of its
> dynamical fields are variational symmetries777In other words,
> diffeomorphisms leave the Lagrangian invariant up to exact terms; we remind
> the reader that, as a consequence, such diffeomorphisms also take solutions
> of the theory’s equations of motion (EOM) to solutions. of the
> Lagrangian.888For ease of comparison, we note that our BGC is the
> variational version of what Pooley calls ‘GC5’ in [20] and what he calls
> ‘Diffeomorphism Invariance (final version)’ in [19]. Pooley also identifies
> the variational form with what he calls ‘Background Independence (version
> 2)’ in [19].

By extension, we will also use BGC below to refer to the analogous case in
which diffeomorphisms are replaced by internal gauge symmetries; which case we
mean will be obvious from the context. The broad contrast here is between
theories whose actions have only rigid symmetries (and thus do not satisfy
BGC) and theories with BGC, i.e. whose actions have symmetries that are
functions of spacetime (such as diffeomorphism symmetry and internal gauge
symmetry).

It should be clear that BGC is just what physicists typically mean when they
say that a Lagrangian theory is diffeomorphism invariant (mutatis mutandis for
internal gauge symmetry, where the term ‘gauge-invariant’ is used), and it
should be equally clear that physicists generally take this feature to be
informative about the physical content of the system being modeled.999As
Norton [17] says, the feature is taken to physically contentful given an
interpretation of the formalism, including which fields in the Lagrangian
count as dynamical and which as background. For reasons that will become clear
in Section 3 (and which are amply discussed in [10, 11]) we note that we do
not assume that a theory has a unique action: thus, the Einstein-Hilbert
Lagrangian should be understood as just one out of a possible class of
standardly stipulated Lagrangians for General Relativity.

Having put on the table the standard notion of BGC, let us now try to
understand what work Einstein would have general covariance do in order to
qualify as substantive.

###  2.2 Extending the Relativity Principle

We now turn to our first question, which is about the relationship between
general covariance and the Principle of Relativity (PR). At one point in the
development of General Relativity, Einstein thought that general covariance
was simply an extension of the PR in Special Relativity. Later, as part of his
response to Kretschmann’s critique [14], he retreated to the view that general
covariance is necessary but insufficient for extending PR (see [19] for a
discussion of Einstein’s retreat). For our purposes, it will be more fruitful
to frame the question thus: does one’s preferred notion of general covariance
play a distinctive role in extending the Special Relativistic PR (or more
generally, analogs of PR for internal gauge symmetry)? If ‘yes’, then we can
consider it to be substantive in virtue of playing this role.

Here again one finds the familiar story of Einstein making a bold and
potentially physically insightful conjecture, but with murky conceptual
foundations, and which was thus amenable to being problematized by others.
Correspondingly, there has been much controversy about whether any notion of
general covariance can play a role in extending PR. At least within the
philosophical literature, one finds various authors (e.g. [2] and [16]) who
answer ‘no’ because they claim that the desired extension of the PR to General
Relativity must be trivial, in the sense that the resulting ‘relativity’ group
is just the identity. In brief, these authors argue for their claim by
appealing to the following controversial assumption about PR, which we will
call (Geometric PR): in such an extension, the empirically significant group
of symmetries invoked in the Principle of Relativity should be identified with
the stabilizer group of the spacetime metric in General Relativity (just as it
is typically identified with the stabilizer group of the Minkowski metric in
Special Relativity).

If one accepts Geometric PR, then since a generic metric—which is now a
dynamical field—in General Relativity has no non-trivial automorphisms, it
immediately follows that the PR is trivial in General Relativity. Thus, no
form of general covariance (including BGC) can play a role in enabling such an
extension. We stress that this conclusion is wholly driven by the assumption
of Geometric PR: if the argument works, it works regardless of what one takes
substantive general covariance to be.

The reason Geometric PR is controversial is that it is far from clear that
this is the most physically perspicuous and relevant interpretation of the PR.
Indeed, in [5], Brown and Sypel articulate a version of the PR which they take
to be both the more physically perspicuous interpretation, as well as the one
that more faithfully reflects Einstein’s own understanding of PR. According to
Brown and Sypel, the symmetry group of PR is to be understood as the symmetry
group relating inertial frames, which are in turn to be fleshed out as—in
idealization—ways of encoding the equivalence of outcomes of experiments set
up with the same initial conditions in an isolated subsystem. Thus, when in
Special Relativity one goes on to define a geometric object (the Minkowski
spacetime metric) whose stabilizer group is precisely the symmetry group of
the PR, one is merely codifying ‘…aspects of the comparative behaviour of
different systems of physical rods and clocks in relative motion’, where the
behavior of the target subsystem (whose inertial frames are related by these
symmetries) is being measured with respect to an environment frame, from which
the subsystem is dynamically isolated. Of course, once one understands this
point, one can abstract away from a concrete material system and even its
idealization in terms of inertial frames, and arrive at a kind of ‘iconic’ or
‘generic’ image that we call ‘Minkowski spacetime’, which provides an
efficacious vehicle for abstract reasoning. This generic ‘type’ can then be
made to descend once again into a representation of a concrete empirical
scenario when it is filled in with the description of the relevant subsystem-
environment decomposition, initial and boundary conditions including the
relevant conditions for ‘isolation’, and the practical knowledge of the
experimentalist.

Let us call the understanding of PR that we have just articulated Embodied PR,
because of its emphasis on how the Principle of Relativity (and its
symmetries) needs to be understood not just formally, but as an embodiment of
concrete empirical scenarios and their idealizations (such as boundary
conditions). It is this embodied understanding of PR (and of symmetries more
generally) that we will adopt below, because we are interested in the physical
and empirical content of symmetries, which stems from an understanding of how
the formal apparatus is used to represent particular subsystem-environment
relationships and the related boundary conditions.

Notice that, according to Embodied PR, one needs to be able to understand how
a theory models an ‘isolated subsystem’ before one can even articulate PR
within that theory. Thus, advocates of Embodied PR will not accept the above
argument for Geometric PR, because it is based on a definition of PR that
ignores the question of how ‘dynamical isolation’ is to be modeled in the
extended scenario.101010Ironically, Brown [3] has expressed sympathy for
Geometric PR on the grounds that the diffeomorphism symmetry of GR ‘…does not
have an active interpretation in terms of isolated subsystems of the
universe.’ The debate about whether and in what sense local symmetry (of which
the diffeomorphism symmetry of GR is an instance) can have such an
interpretation has by now played out quite fully (see [28] and references
therein), but our interest here is in highlighting that, on an embodied view
(and again: in contrast to Geometric PR), one’s conception of substantive
general covariance has to do real work in adjudicating whether (Non-extension)
is true. Furthermore, a moment’s thought will show that ‘dynamical isolation’
is a much more sophisticated notion in General Relativity than it is in
Special Relativity: we typically model it as the asymptotic flatness of a
subsystem spacetime.

To recapitulate, we have arrived at the following way of sharpening our second
question: is there a notion of general covariance that has a distinctive role
to play in extending PR—in the sense of Embodied PR—from Special Relativity to
General Relativity? If we frame things from Einstein’s perspective, then the
question becomes: can we articulate a suitably generalized version of PR if we
are concerned with asymptotically flat subsystems? But it is important to note
that the question can be generalized so that we consider isolated finite
subsystems as well, and not just asymptotic infinity.111111Furthermore
(although it lies beyond the scope of this paper), it is also physically
relevant to consider non-isolated subsystems in which charges are not
conserved because of symmetry-breaking.

In Section 3, we will see that Noether’s theorems strongly suggest that the
answer to the above question is ‘yes’, but before that we will need to turn to
a second related facet of Einstein’s struggle with general covariance, and the
one that led Noether to write her 1918 paper.

###  2.3 Non-trivial corner charges

The second question of this section is: is there a notion of general
covariance that yields physically meaningful corner charges (such as energy)?
For instance, General Relativity is a theory whose Lagrangian has the
diffeomorphism symmetries of BGC: does it in virtue of this have physically
meaningful charges? (We will see in Section 3 that, from the perspective of
the framework developed by Noether, it is clear that this question is
intimately related to our first question.)

Historically speaking, this question found its genesis in Hilbert’s
variational treatment of General Relativity (see [25, 26] for a masterful
treatment of the history, which we follow here). In addition to deriving the
field equations in his treatment, Hilbert’s main contribution (which he called
‘the most important goal of his theory’ ) was the formulation of an invariant
‘energy vector’ and a proof of its conservation. In response, Einstein
provided his own somewhat different variational approach to General
Relativity; in this note he derived an energy current which can be written in
modern notation as $J_{X}=C_{X}+dU_{X}$, where $C_{X}$ is the constraint
associated with the gauge transformation $X$. $C_{X}$ vanishes on-shell.
$U_{X}$ is referred to as the charge aspect or superpotential (these formulae
are generic for a gauge theory, but we will provide explicit details for
electromagnetism in the next section).121212We direct the reader to [6],
forthcoming in this volume, for an explication of Einstein’s notation
$J=\mathfrak{T}+\mathfrak{t}$ and how he thought about the objects
$\mathfrak{T}$ and $\mathfrak{t}$. Einstein also stressed the importance of
the conservation law $dJ_{X}=0$ for the on-shell current.

It was this series of developments that set the stage for Emmy Noether’s
contribution to the mathematical foundations of General Relativity. First, as
Felix Klein’s assistant, Noether worked out the exact relationship between
Hilbert’s energy vector and Einstein’s energy current. This analysis, in
conjunction with Klein’s own work on the topic, led to a open letter from
Klein to Hilbert in which Klein explained that the energy current $J_{X}$
could always be decomposed as the sum of two parts, the first of which
vanishes on-shell, and the second of whose divergence is identically zero
(where we emphasize that the latter is an off-shell statement). In
consequence, Klein asserted, the statement that $dJ_{X}=0$ is merely a
mathematical identity and does not have any physical content.

Upon reading this letter, Einstein wrote Klein to express admiration for his
insights, but also to protest that ‘I regard what you remark about my
formulation of the conservation laws as incorrect.’ According to Einstein, the
conservation of the current was not itself a mathematical identity, but was
instead the consequence of a mathematical identity and the equations of
motion. Einstein further argued that the physical interpretation of such a
conservation statement was analogous to the integral form of Gauss’s law in
the following sense: since the current $J_{X}$ can be shown to be on-shell
exact, i.e. $J_{X}\approx dU_{X}$, one can use Stokes’ theorem to write the
on-shell charge (the integrated current) as a corner (relative to the
spacetime manifold) quantity, i.e.
$\int_{\Sigma}dU_{X}=\int_{\partial\Sigma}U_{X}$. He then proceeded to sketch
a particular subsystem for whose boundary conditions (essentially a primitive
version of asymptotic flatness) such a computation of charge made sense.

As regards Einstein and Klein, their further exchange only led to a stalemate,
with Einstein insisting that his conception of energy conservation in General
Relativity was physically contentful, and Klein denying this claim on the
grounds that ‘mathematical identities’ cannot be physically contentful
statements. But against the background of this impasse concerning physical
interpretation, mathematical progress was made: Noether published her 1918
‘Invariante Variationsprobleme’ [15]. Herein, she proved two theorems
(referred to in the physics community as Noether’s First Theorem and Noether’s
Second Theorem respectively) and derived a corollary (the so-called
‘Hilbertian assertion’) that bore directly on the Einstein-Klein debate. In
our terminology, the corollary states that Lagrangian field theories with BGC
only have (in virtue of that BGC) trivial currents, where we remind the reader
that a ‘trivial current’ is one that can be written as the sum of an exact
form and a term that vanishes on-shell (i.e. our $C_{X}$ and $dU_{X}$ above
respectively). Thus, the on-shell current is identically conserved and is said
to have an ‘improper conservation law’.

It would be a mistake to read Noether here as weighing in on the question of
physical interpretation; rather, one should understand her as offering a
mathematician’s conceptual analysis: given that you think the concepts of BGC
and trivial currents are physically relevant, you should be aware of the
following mathematical relationship between them. Read this way, Noether’s
corollary does not answer our second question, but it does prompt the
following specific version of it: ‘Does a theory with BGC have—on that
basis—physically meaningful conserved charges?’ where we should now understand
the latter as ‘physically meaningful charges that stem from trivial currents’,
since (as Noether showed) non-trivial currents cannot be derived from the
diffeomorphism (or gauge) symmetry of BGC.

###  2.4 The Einstein test for ‘substantive’

To sum up this section, reflection on Einstein’s struggles with ‘extending the
principle of relativity’ and ‘defining conserved charges’ within the context
of GR leads us to what we might call the ‘Einstein test’ for a substantive
notion of general covariance:

  1. 1.

The ‘corner charge’ part of the Einstein test (corresponding to Section 2.3):
Does the notion of substantive general covariance yield non-trivial corner
charges?131313We note that, based on the history, one might reasonably take
the corner charge part of the test to include the conservation of non-trivial
corner charges. However, for the purposes of analytical clarity, it will be
convenient for us to make the minimum criterion the existence of non-trivial
corner charges, with conservation left as a further criterion.

  2. 2.

The ‘Extending PR’ part of the Einstein test (corresponding to Section 2.2):
Does the notion of substantive general covariance allow us to extend the
principle of relativity (in the sense of Embodied PR that we discussed above)?

Given the status of BGC as the ‘received’ version of general covariance (when
formulated variationally) and Einstein, Klein and Noether’s own recognition of
the relevance of BGC to the question of conserved charges, it is natural for
us to first try to run the Einstein test on BGC. In the next section, we will
see how the raw mathematical materials of Noether’s theorems can taken up—in
the art of the physicist—to give a particularly trenchant formulation of this
test: we will find that BGC does not pass the test, but also that the
machinery of Noether’s second theorem suggests a version of substantive
general covariance that does.

##  3 General covariance through a Noetherian lens

Since Kasia Rejzner’s essay [22] in this volume expounds on Noether’s two
theorems from a more technical perspective, namely the BV complex, we shall
here take a more down-to-earth approach in which we work in ghost degree zero.
However, we stress that the fundamental observations that we make can be
reconstructed in terms of the cohomology of the BV complex—a beautiful
connection with another part of Noether’s career since she was one of the
developers of algebraic homology groups, albeit not in the context of her two
theorems. To set the stage, we note that at a highly schematic level, the
chief moral of Noether’s two theorems for Lagrangian field theory is that the
space of such schematic theories can be partitioned according to whether or
not the theories possess two features, which in turn correspond to the
symmetry-structure of the theories’ Lagrangians. These two features are:

  * •

Noether currents. A non-trivial Noether current is one that cannot be written
as the sum of an exact form and a term that vanishes on-shell. A trivial
Noether current is one that can be so written.

  * •

Noether identities, viz. relations between the Euler-Lagrange quantities that
take the form $N^{i}[\frac{\delta L}{\delta\phi^{i}}]=0$; we emphasize that
since these are mathematical identities, they hold off-shell. A nontrivial
Noether identity is one whose coefficients $N^{i}$ do not vanish on-shell. A
trivial Noether identity is one whose coefficents vanish on-shell.

Furthermore, the correspondence to the symmetry-structure is given by
Noether’s two theorems (and their modern interpretations), as
follows:141414Here our presentation of the ‘iff’ statement is heuristic.
Making this an honest theorem requires considerable care about how to define
equivalence classes of symmetries and currents, as demonstrated by Olver in
[18].

  * •

Rigid symmetries (symmetries with a finite set of parameters) are in 1-1
correspondence with non-trivial Noether currents. Furthermore, these currents
are conserved when we assume the on-shell property in computing the divergence
of the current.

  * •

Gauge symmetries (symmetries with an infinite set of parameters, i.e.
symmetries in the sense of BGC) are in 1-1 correspondence with non-trivial
Noether identities. Furthermore, they yield trivial currents whose on-shell
expression is conserved as a mathematical identity, i.e. without using the on-
shell property in computing the divergence.

Let us now descend from this schematic discussion to a more realistic physical
context in which we consider boundary conditions of subsystems. In particular,
we will use the covariant phase space framework to understand how Noether’s
theorems get applied in such a context.

###  3.1 The covariant phase space approach to Noether

First, we sketch the spacetime geometry of the subsystems that we will
consider. It will suffice to consider the elementary case of a (simply
connected) spacetime $M$ with timelike boundary $\partial M$, and into which
we insert an initial Cauchy surface $\Sigma$; the corner $S=\partial\Sigma$
lies in $\partial M$. Introducing such a boundary $\partial M$ is crucial for
modeling the boundary conditions of a subsystem; furthermore, one can think of
such a finite boundary model as a regularized version of a subsystem with
asymptotic boundary conditions.

Consider the equivalence class $[L]$ of Lagrangians on $M$, where the
equivalence relation is given by $L\sim L+d\ell$, i.e. two representatives of
the class differ by a boundary Lagrangian $\ell$ (on $\partial M$). Such an
equivalence class corresponds to a set of bulk equations of motion. In the
literature, there are two ways of interpreting the representatives of this
class, corresponding to two ways of interpreting the fundamental variational
formula, a version of which appeared in Noether’s 1918 paper, i.e.

$\delta L=-\mathcal{E}\delta\phi+d\theta,$ (1)

where we remind the reader that $\mathcal{E}=E_{\phi}\delta\phi$ is a field
space $1$-form whose coefficient $E_{\phi}$ is the Euler-Lagrange quantities,
and $\theta$ is a field space $1$-form called the (pre-)symplectic potential
current.

According to the first approach, $\theta$ is defined by the fundamental
variational formula and the representatives correspond to an ambiguity in the
definition of $\theta$. According to the second approach (for which see [11,
6] and references therein), the variations are only well-defined when we know
the full bulk and boundary data—corresponding to a particular representative
$L$ (including a possible boundary Lagrangian piece)—and this fully specified
data picks out a unique $\theta$ by means of Anderson’s homotopy operators
[1]; this unique $\theta$ is then the one that appears in the fundamental
variational formula.151515We also assume that we have fixed a particular set
of coordinates in field space. Here we will follow the second approach,
because the incorporation of the boundary data is crucial to resolving the
questions we have raised in Section 2.

We remind the reader that given the unique $\theta$ corresponding to some $L$,
we can construct the presymplectic form $\Omega:=\int_{\Sigma}\delta\theta$.
If the Lagrangian has a symmetry $\xi$, we can also construct a corresponding
Noether current $J_{\xi}:=I_{\xi}\theta$ and Noether charge
$Q:=\int_{\Sigma}J_{\xi}$, $\xi$ is the symmetry vector field on field space
and $I$ is the field space interior product.161616We note that, below, we will
adopt a standard abuse of notation in which the notation for a vector field on
field space $\xi$ is also used to represent the gauge parameter that appears
in the co-efficient of that vector field. The statement of Noether’s first and
second theorem can then be understood in terms of these quantities.

From this perspective, the basic statement of Noether’s first theorem arises
when $\xi$ is a variational symmetry of $L$, i.e. $L_{\xi}L=dl_{\xi}$ (where
$L_{\xi}$ is the field space Lie derivative). We can then use Cartan’s magic
formula and (1) to obtain the conservation statement
$dJ_{\xi}=I_{\xi}\mathcal{E}$, where the Noether current $J_{\xi}$ is defined
as $J_{\xi}:=I_{\xi}\theta-l_{\xi}$. Furthermore, when $\xi$ is a rigid
symmetry then $J_{\xi}$ is a non-trivial Noether current, in the sense
discussed above, and so the current is conserved only on-shell.

On the other hand, when the symmetry $\xi$ is a non-constant function of
spacetime and thus has an infinite number of parameters, we find that we can
write $I_{\xi}\mathcal{E}=dC_{\xi}+\mathcal{N}_{\xi}$ where $C_{\xi}$ is a
constraint term that vanishes on-shell, and $\mathcal{N}_{\xi}$ is the sum of
terms linear in $\xi$, each of which contains a differential operator acting
on the Euler-Lagrange quantities. We can then use the locality of $\xi$ to
argue that $\mathcal{N}$ must vanish as an identity—this yields the non-
trivial Noether identity (often also called a Bianchi identity) corresponding
to the gauge symmetry $\xi$. By combining this observation with
$L_{\xi}L=dl_{\xi}$ and (1), we can conclude that $d(J_{\xi}-C_{\xi})=0$ and
so the Noether current takes the form $J_{\xi}=C_{\xi}+dU_{\xi}$, where the
‘superpotential’ or ‘charge aspect’ $U_{\xi}$ depends on the gauge parameter
$\xi$ in a linear but non-trivial manner. In other words, we have just shown
that gauge symmetry corresponds to a trivial Noether current, in the above
sense. There are two extremely important points to emphasize about the
superpotential $U_{\xi}$ at this stage:

  * •

The superpotential $U_{\xi}$ is uniquely determined by our choice of
Lagrangian in the equivalence class $[L]$ and the symmetry generator
$\xi$.171717See [11] for discussion and see [6] for an explicit formula for
$U_{\xi}$.

  * •

For gauge theories, i.e. theories with BGC, the Noether charge
$Q^{\Sigma}_{\xi}:=\int_{\Sigma}J_{\xi}=\int_{\partial\Sigma}U_{\xi}$ is
always a corner quantity, because the Noether current is on-shell exact, and
determined by the superpotential $U_{\xi}$.

With these observations in hand, we can see that it is the ‘corner charge’
part of the Einstein test which is more fundamental than the ‘Extending PR’
part of the test: if the corner charges for a gauge theory are trivial (i.e.
vanishing), then there is no sense in which the PR can be extended (from a
rigid precursor theory) to that gauge theory. If, on the other hand, the
corner charges are non-trivial, then subject to understanding what kinds of
boundary conditions model an ‘isolated system’, it may be possible to obtain a
generalized version of PR in that gauge theory.

###  3.2 Test cases

With the above Noetherian machinery in place, we are now in a position to run
the ‘corner charge’ part of the Einstein test on BGC, i.e. to pose the
question: does a theory with BGC thereby have non-trivial corner charges? To
that end, it will be helpful to make a connection here with some of the toy
models that Pooley [20, 19] and others use to test the validity of different
definitions of general covariance, although we warn the reader that for
reasons of simplicity, we have sometimes opted for a different labeling
convention from that of Pooley [19].

Consider first the following theory of a free scalar field on Minkowski space
that we will call SR1:181818Pooley [19] calls the non-variational formulation
SR1.

$L(\phi,\eta)=-\frac{1}{2}\eta^{ab}\partial_{a}\phi\partial_{b}\phi,$ (2)

where the scalar field $\phi$ is the dynamical variable and the Minkowski
metric $\eta$ is a non-dynamical (or background) structure. Our variational
formula for this theory is $\delta L=E_{\phi}\delta\phi+\nabla_{a}\theta^{a}$,
where the Euler-Lagrange expression is $E_{\phi}=\square\phi$ and the pre-
symplectic potential current is $\theta^{a}=-\nabla^{a}\phi\delta\phi$. We
note that the conservation of the stress-energy tensor $T^{ab}$ follows from
applying Noether’s first theorem to the rigid spacetime symmetries of this
theory.

Clearly, this theory does not satisfy BGC, because diffeomorphisms are not a
variational symmetry of $\phi$. In our next example, we will modify SR1
slightly to show that one can produce an example of a theory that satisfies
BGC, but which fails the ‘corner charge’ part of the Einstein test, because it
only has a trivial superpotential, and thus the corner charge vanishes.

####  3.2.1 BGC with trivial superpotential

Consider now a theory that we will call SR2 which is given by,

$L^{\text{{SR2}}}(\phi,Y;\eta)=L(\phi,Y^{*}\eta),$ (3)

where $Y:\mathbb{R}^{d}\rightarrow M$ is a parametrization field that captures
our freedom to choose coordinates on the spacetime $M$.191919Pooley [19] calls
this SR5, and his example of SR4 is similar. Due to the latter interpretation
of $Y$, it is appropriate to call SR2 a ‘Kretchmannized’ version of
SR1.202020We note in passing that Yang-Mills theory and General Relativity can
be analogously Kretchmannized, and this leads to the construction in [7]. Our
dynamical variables in this theory will be $\phi$ and $Y$, whereas the
Minkowski metric $\eta$ still retains its status as a non-dynamical background
structure.

We can again compute the variational formula (1) for this theory, thus
obtaining $\delta
L^{\text{{SR2}}}=E_{\phi}\delta\phi+E_{a}\chi^{a}+\nabla_{a}\theta^{a}$, where
$\chi:=\delta Y\circ Y^{-1}$, $E_{a}=-2\nabla_{b}T^{ba}$, and
$\theta^{a}=2T^{ab}\chi_{b}-\partial^{a}\phi\delta\phi$. Notice that in
addition to the previous equation of motion $E_{\phi}=0$, we now have a new
equation of motion $E_{a}=0$, i.e. the conservation equation for the stress-
energy tensor. Furthermore, by inspecting $\theta$, we can see that in
addition to the original symplectic pair $\phi$ and its momentum
$\nabla^{a}\phi$, we now have a new symplectic pair $Y^{a}$ and its momentum
$T^{ab}$.

We now draw the reader’s attention to an interesting point: clearly,
$L^{\text{{SR2}}}$ is covariant in the sense of BGC (this is true even though
$\eta$ remains a non-dynamical field and is not varied) and thus this theory
must have a non-trivial Noether identity. By turning the crank of the Noether
machine, we find that this identity is
$E_{\phi}\partial_{a}\phi-2\nabla_{b}T^{ba}=0$ (that this is an identity can
be straightforwardly checked from the definition of $T_{ab}$). In other words,
the content of the Noether identity SR2 is actually the content of Noether’s
first theorem for SR1—one might call this the transmutation of Noether’s first
theorem into Noether’s second theorem via Kretchmannization!

It is also interesting to compute the (trivial) Noether current, i.e
$J^{a}_{\xi}=I_{\xi}\theta-\iota_{\xi}L^{\text{{SR2}}}=0$. In other words, the
current vanishes identically off-shell, and we can deduce that the
superpotential $U_{\xi}$—and thus the corner charge algebra—is trivial.

To sum up, the theory SR2 demonstrates why BGC fails the ‘corner charge’ part
of the Einstein test for substantive general covariance (and in consequence
fails the ‘extending PR’ part of the test): even though this theory clearly
has BGC, it nonetheless fails to deliver non-trivial corner charges.

####  3.2.2 BGC with non-trivial superpotential

Next, we turn to a familiar example in which BGC does yield a non-trivial
corner charge (and thus passes the ‘corner charge’ part of the Einstein test).
Consider the Maxwell gauge theory Lagrangian

$L=-\frac{1}{4}F\wedge\star F,$ (4)

where we have chosen a representative $L\in[L]$ with vanishing boundary
Lagrangian, i.e. $\ell=0$. The variational formula is

$\delta L=-E~{}\delta A+d\theta,$ (5)

where $\delta$ is the exterior derivative on the space of fields, $E=d\star F$
is the Euler-Lagrange expression, and $\theta=\star F~{}\delta A$ is the
presymplectic potential current.

In this case, the gauge symmetry vector field on field space has the form
$\hat{X}=dX(x,t)\delta/\delta A$, and $L_{\hat{X}}$ is the field space Lie
derivative with respect to this vector field, although we will now abuse
notation by using $X$ to denote both the field space vector field and the
local gauge parameter. Since $L_{X}L=0$, we can repeat the general analysis
summarized in Section 3.1 to obtain

$0=L_{X}L=d(I_{X}\theta)-I_{X}(E\delta A)$ (6)

and then by using integration by parts and the definition of the current
$J_{X}=I_{X}\theta$, we have

$d(J_{X}-EX)=-XdE.$ (7)

Since $X$ is an arbitrary gauge symmetry, we can integrate both sides over a
domain with boundary, apply Stokes’ theorem to the LHS, and—by assuming that
$X$ and its derivatives vanish on the boundary—deduce that $dE=0$ as an
identity. This is a simple example of a non-trivial Noether identity (here the
Noether operator $N$ is just the exterior derivative $d$, which does not
vanish on-shell). We note that $dE=dd*F=0$ is clearly a mathematical identity
(the Bianchi identity) since it follows from $d^{2}=0$.

By applying the Noether identity $dE=0$ and the Poincare lemma, we see from
(7) that

$J_{X}=EX+dU_{X}\approx dU_{X},$ (8)

where ‘$\approx$’ denotes ‘on-shell equality’.212121This is nothing other than
the $U(1)$ gauge theory version of Einstein’s formula
$J=\mathfrak{T}+\mathfrak{t}$, of which the reader can find an analysis in
[6]. It explains why energy pseudotensors in GR are often said to be defined
only up to a superpotential. In other words, we see that the current $J_{X}$
that is associated with the gauge symmetry is trivial, although—as we are
about to see—this of course does not imply that the Noether charge is trivial!

Given that we have a current $J$, we can define a charge by integrating $J$
over a spacelike Cauchy surface $\Sigma$ to obtain:

$Q^{\Sigma}=\int_{\Sigma}J$ (9)

We now remind the reader that a unique possibility arises for trivial currents
(but not non-trivial currents), which is a direct consequence of the non-
trivial Noether identity: by Stokes’ theorem, a trivial on-shell current
$J_{X}\approx dU$ can be converted into a purely boundary Noether charge

$Q^{\Sigma}_{X}=\int_{\Sigma}dU_{X}\approx\int_{\partial\Sigma}U_{X}=\int_{\partial\Sigma}X(\star
F),$ (10)

where in the last equality we used the explicit form of the superpotential
$U_{X}$ (fixed by $X$ and our choice of $L$), which—unlike the case of SR2—is
non-trivial and leads to a non-trivial corner charge algebra.

To go further and say more about the conservation of this corner charge, one
needs to impose boundary conditions on the subsystem that rule out the leakage
of ‘symplectic flux’ from the system [13, 11].222222In general, one also needs
to make sure that the chosen boundary configuration has stabilizers, although
this condition is guaranteed in our example, since all $U(1)$ gauge field
configurations have stabilizers (the constant gauge transformations). In the
context of the present example, consider the case in which the subsystem has
Dirichlet boundary conditions, i.e. $\delta A|_{\partial M}=0$. Notice that
this sets the symplectic potential to zero at the boundary, which prevents
symplectic flux leakage. Then, the symmetries of the subsystem need to
preserve these boundary conditions, and so $dX|_{\partial M}=0$. Evidently,
when $X|_{\partial M}=0$, the symmetry is in the kernel of $\Omega$ and so
does not represent a physical symmetry generated by a charge, leaving non-zero
constant transformations at the boundary $X$ as the only remaining possibility
(which are precisely the non-trivial stabilizers of the boundary condition).

In the above example, we see that the Noether charges are also the canonical
conserved charges associated with our boundary condition of interest, namely
the Dirichlet boundary condition. The latter are also called ‘Hamiltonian
charges’ because the charges are Hamiltonians for the corresponding symmetry
transformations in phase space. We note that the Dirichlet boundary conditions
represent an isolated subsystem, for which one expects the relevant
Noether/Hamiltonian charge to be conserved, unlike an open system which is
continually experiencing an exchange of charge with its environment.

However, as we will soon see, there can be more complex scenarios in which a
Hamiltonian charge associated with a particular boundary condition is the
Noether charge of $L+d\ell$, but not of $L$. In Section 3.4, we will discuss
how one can resolve this tension between Noether and Hamiltonian charges,
which is relevant to the ‘extending PR’ part of the Einstein test of
substantive general covariance.

###  3.3 Substantive general covariance

We have just seen that, due to examples such as SR2, BGC does not on the whole
pass the ‘corner charge’ part of the Einstein test for substantive general
covariance. On the other hand, we have also seen that central examples of
theories with BGC (electromagnetism, and by a simple extension: Yang-Mills
theory and General Relativity) do yield non-trivial corner charges. The
general moral is that while BGC is insufficient for the existence of non-
trivial corner charges, it is nonetheless necessary, because one needs a
trivial Noether current in order to obtain a corner charge. What then needs to
be added to BGC to obtain a non-trivial corner charge?

As it turns out, this is one of the cases where the analysis of the difficulty
(by means of the Noether machine) also makes the remedy obvious: what needs to
be added is precisely the requirement that the theory’s Lagrangian $L$ be such
that its superpotential is non-trivial (we again remind the reader that a
choice of $L$ also fixes the superpotential). We thus propose the following
definition of substantive general covariance in response to the ‘corner
charge’ part of the Einstein test:

> (Corner-SGC) A theory has Corner-SGC when, in addition to BGC, its
> Lagrangian $L$ has a non-trivial superpotential $U_{\xi}$.232323NB: The
> corresponding Noether charge will be conserved if we impose boundary
> conditions that rule out flux leakage, and the Lagrangian $L$ is consistent
> with those boundary conditions.

It is worth pausing to emphasize that the potency of Noether’s second theorem
is not that of a purely mathematical result. However, just as the character of
a line or a stroke can begin to suggest the animation or dynamism of a figure
in painting, so too can mathematics be taken up into the mobilization of
physical thought. What Noether’s second theorem does here is offer physicists
a heightened awareness of what the mathematical medium of field theory
(understood through the lens of the variational calculus and Noether’s
theorems) is capable of when it is pressed into the service of physical
representation: our discussion thus far is precisely an illustration of how
the second theorem can be so recruited! Through that discussion, we come to
see that the substantiveness of general covariance is not just about the
symmetry of a Lagrangian, but also about how—through that symmetry—some kinds
of Lagrangians determine a non-trivial corner charge algebra.

We now turn to the ‘Extending PR’ part of the Einstein test, for which
(Corner-SGC) is a necessary but insufficient condition, because of the
aforementioned potential tension between the Noether charge and the
Hamiltonian charge relative to some boundary condition. Note that for a gauge
theory (where we mean to include both General Relativity and Yang-Mills
theory), whatever the symmetries of generalized PR are, they will need to be
generated by the non-trivial corner charges from (Corner-SGC). However, there
is an added complication present in generalizing Embodied PR: we are now
considering the symmetries of dynamically isolated subsystems, which are
generated by the Hamiltonian charges for the boundary conditions that model
dynamical isolation—thus, we will only succeed in extending PR if the corner
charges coming from the trivial Noether current of some Lagrangian are also
Hamiltonian charges. In Section 3.2.2, we saw an example (Maxwell gauge theory
with Dirichlet boundary conditions) where this coincidence clearly obtains: we
can think of it as a regularized (finite) version of a case in which the
asymptotically rigid $U(1)$ symmetries of some subsystem provide a generalized
version of PR relative to an environment subsystem (where the environment has
been idealized away into the boundary conditions). However, as we are about to
see in Section 3.4, the corner charges from some particular Lagrangian can
also fail to be Hamiltonian charges (relative to some choice of dynamically
isolated boundary conditions). We thus propose the following more demanding
notion of substantive general covariance in response to the ‘extending PR’
part of the Einstein test:

> (PR-SGC) Fix boundary conditions that represent ‘dynamical isolation’ in the
> context of some theory. The theory’s Lagrangian $L$ has PR-SGC just in case,
> in addition to satisfying Corner-SGC, the Noether charge coming from $L$ is
> also a Hamiltonian charge relative to the ‘dynamically isolated’ boundary
> conditions.

We now turn to a brief consideration of General Relativity, which illustrates
the subtleties of PR-SGC.

###  3.4 Revisiting the case of General Relativity

In [11], the general way in which Noether charges can fail to be Hamiltonian
charges is considered, and this framework is then applied to the case of
General Relativity (Section 3.4 of [11]). Here we only summarize the relevant
features for us to revisit the Einstein-Klein dispute.

First, consider the Einstein-Hilbert Lagrangian $L_{EH}$. We can follow
through the covariant phase space recipe sketched in Section 3.1 and compute
the superpotential corresponding to $L_{EH}$, which turns out to yield what is
known as the Komar charge (see (2.38) of [10] for a definition). This shows
that we have satisfied (Corner-SGC).

Next, let us consider whether this Lagrangian satisfies (PR-SGC). To do so,
suppose that we wish to use Dirichlet boundary conditions to model an isolated
system (or a finite boundary regularization thereof): that is to say, we wish
to set the induced metric $\bar{g}_{ab}$ on the timelike boundary to a fixed
value, thereby also setting $\delta\bar{g}_{ab}\stackrel{{\scriptstyle\partial
M}}{{=}}0$. More specifically, we typically take $\bar{g}_{ab}$ to be the
Minkowski metric, thus leading to an ‘asymptotically flat’ solution in the
limit where we take the boundary to infinity. It is well-known that given
these Dirichlet boundary conditions, one can construct a Hamiltonian charge;
however, this is not the Komar charge, but a different quasi-local charge
known as the Brown-York charge (see (2.31) of [10] for a definition).

Thus we have the complication that we discussed earlier: if one chooses
$L_{EH}$ as one’s Lagrangian and the Dirichlet boundary condition as the
‘isolated boundary condition’ in the ‘Extending PR’ part of the Einstein test,
then one ends up with a mismatch between the Noether charge (the Komar charge)
and the the Hamiltonian charge (the Brown-York charge). The resolution of this
complication, discussed in [13, 11], is to choose a different Lagrangian—which
is associated with a different boundary condition—in order to satisfy (PR-
SGC). In particular, if we choose the Lagrangian $L=L_{EH}+d\ell_{GHY}$, where
$\ell_{GHY}$ is the celebrated Gibbons-Hawking-York boundary term, then the
corresponding Noether charge is precisely the Brown-York charge, so we have
now brought our Lagrangian into alignment with the boundary condition—and
corner charge—that we are after. Furthermore, as [13] shows, if we assume that
the metric tends towards the Minkowski metric in the asymptotic limit, then we
recover precisely the ADM charges for an asymptotically flat spacetime, which
is precisely what one expects of an isolated subsystem in General Relativity.
In particular, we note that these charges generate the ‘asymptotic Poincare
symmetries’, which are the generalized analogs of the rigid Poincare
symmetries (from the standard PR) in this context.242424Here we are only
discussing spatial infinity; once one takes into account null infinity, one is
forced to consider the infinite-dimensional BMS group if one wants a rich set
of solutions. We thus see that $L=L_{EH}+d\ell_{GHY}$ satisfies (PR-SGC), but
$L_{EH}$ does not.

With these clarifications in hand, we can also return to an assessment of the
dispute between Einstein and Klein that we mentioned in our Introduction (we
refer the reader to [6] for a complementary analysis of this dispute, with an
emphasis on holographic renormalization). Is it really true that Einstein was
terribly confused about the status of conserved charges in General Relativity,
and that Noether’s Second Theorem served to highlight his confusion by
securing Klein’s claim that General Relativity does not have any physically
meaninful conservation laws? To answer this question, let us consider several
key points from Einstein’s March 1918 response to Klein (letter 480 in [9]):

  * •

Einstein says the importance of the quantity
$J_{\xi}=\mathfrak{T}+\mathfrak{t}$ (the on-shell exact Noether current, cf.
(8) in Section 3.1) is its physical interpretation as the energy of a point
mass when we go very far away from the subsystem; in other words, when we
treat the subsystem as dynamically isolated from an environment.

  * •

He also says that the possibility of this physical interpretation is
underwritten by the fact that $J_{\xi}$ (on-shell) is an exact term,
presumably because he knows that via Stokes’ theorem, a bulk integral of an
exact term can be transformed into a boundary integral (which is why we call
such charges ‘quasi-local’).

  * •

Einstein emphasizes that the equations of motion are necessary in order to
show that $J_{\xi}$ is exact, and thus $J_{\xi}$ should be regarded as
physically meaningful.

And then, a litle over a week later, in letter 492 to Klein [9], Einstein goes
on to place a heuristic ‘isolated’ boundary condition on the subsystem that
allows for the construction of a conserved charge!

We can now ask ourselves how Einstein’s more physically-oriented (and more
mathematically imprecise) strategy looks with the clarity of hindsight. It
would seem that Einstein already understood much of the intuitive physical
content that can be expressed by means of Noether’s Second Theorem: certainly,
he understood that the distinctive qualitative feature of a gauge theory
(implied by Noether’s Second Theorem) is the on-shell exactness of the Noether
current, and that this feature leads to a boundary charge; and he was
undoubtedly sensitive to the need to choose boundary conditions in order to
construct such a charge. In all this, we believe that Einstein was vindicated
in resisting Klein’s criticisms. On the other hand, the place where Einstein’s
understanding was most lacking was in his grasp of (i) the relationship
between a choice of $L$ and a choice of boundary conditions, (ii) the
potential mismatch between the Noether charge corresponding to some $L$ and
the Hamiltonian charge corresponding to some choice of boundary conditions,
and (iii) the potential appearance of infinite-dimensional symmetries (such as
the BMS group at null infinity) in the asymptotic limit. As we discussed in
Section 3.2, this lack is closely related to his struggle to find an
appropriate extension of the Relativity Principle to the context of General
Relativity.

###  3.5 Background independence

While this is not the place for us to offer a fully-developed account of
‘background independence’, it seems appropriate to make some brief remarks
about how this notion is related to our preferred conceptualization of
substantive general covariance, especially since Pooley [19] has attempted to
pour cold water on attempts by physicists [23, 27, 24] to find a connection
between some form of general covariance and background
independence.252525According to Pooley, Smolin [27] seems to identify
background independence with BGC, whereas Rovelli [23] seems to want a
slightly more full-blooded form of general covariance.

Roughly speaking, Pooley’s narrative runs as follows. There is a standard line
of thought in the quantum gravity literature according to which a theory’s
‘general covariance’ is tightly linked to its background independence,
conceived of as a theory’s spacetime structure being dynamical, as opposed to
playing the role of a ‘fixed background’. As evidence, Pooley cites Rovelli’s
claim [24] that the background independence of classical GR is ‘…realized by
the gauge-invariance of the action under active diffeomorphisms.’ But, Pooley
goes on to argue, this and similar claims seem to run counter to the moral of
Kretschmann’s point that any theory can be formulated in a generally covariant
manner—and it is so much the worse for these claims! Pooley goes on to
prosecute his case in various ways, but the part of his argument that will be
of interest to us here is his attempt to show that an identification of
background independence with (what we call) BGC misclassifies certain theories
which are intuitively background dependent as background independent.

There are two cases of relevance for us. First, Pooley says that the theory
(that we call) SR2 clearly satisfies BGC, but seems to be the same theory as
SR1, which is a paradigm case of background dependence. Second, Pooley
introduces a theory that we shall call SR3, defined by the
Lagrangian262626This theory was introduced by Rosen and rediscovered by Sorkin
(see [19] for the history and references). Its most careful analysis so far
seems to have been by [12].

$L^{\text{{SR3}}}=\sqrt{-g}(-\frac{1}{2}\nabla_{a}\phi\nabla^{a}\phi+\frac{1}{4}\lambda^{abcd}R_{abcd}).$
(11)

Pooley again notes that this theory satisfies BGC, but seems to be background
dependent, and so the identification of BGC with background independence seems
to misclassify the example. On this basis, Pooley concludes that ‘…one should
recognise that we are now far past the point where one might hope to
articulate a simple and illuminating connection between diffeomorphism
invariance and background independence.’

We of course agree with Pooley that SR2 and SR3 are clear cases of Lagrangians
that satisfy BGC. But as we have just argued above, BGC is not plausibly a
substantive notion of general covariance (certainly, it would not have been to
Einstein in light of his desiderata), and the Noetherian analysis that reveals
its implausibility also suggests a minimal notion of substantive general
covariance, viz. what we called Corner-SGC in Section 3.3; thus, it would be
premature to judge on the basis of BGC that there is no illuminating
connection between some form of general covariance and background
independence. With this background in mind, we now offer a conjecture for a
simple and illuminating connection between background independence and Corner-
SGC:

> Conjecture: A theory is background independent only if its Lagrangian
> satisfies Corner-SGC with respect to spacetime diffeomorphisms.

The intuition driving this conjecture is that a non-trivial corner charge
associated with diffeomorphisms (and the Lagrangian) is required for spacetime
structure to have dynamical degrees of freedom, and thus for the theory to be
background independent.

How does Pooley’s misclassification argument against BGC—using SR2 and SR3 as
test cases—fare against our conjecture? Let us take the test cases in turn.
The case of SR2 is especially simple: as we discussed in Section 3.2.1, this
theory does not satisfy (Corner-SGC) because its Noether current is
identically zero and its corner charge is trivial. Thus, according to our
conjecture, this theory counts as background dependent, and we see that
Pooley’s argument is no threat to the conjecture.

The case of SR3 is somewhat more interesting. When we put this theory through
the Noether machine, we find that the equations of motion are
$g_{ab}\nabla^{a}\nabla^{b}\phi=0$, $R_{abcd}(g)=0$, and
$\nabla_{a}\nabla_{d}(\lambda^{abcd}+\lambda^{acbd})=T^{bc}$. We also obtain a
set of non-trivial Noether identities and a pre-symplectic potential current

$\theta=-\partial^{a}\phi\delta\phi+\frac{1}{2}(\nabla_{d}\lambda^{abcd}\delta
g_{bc}-\lambda^{abcd}\nabla_{d}\delta g_{bc})$ (12)

from which we can compute the Noether current rCl J_ξ^a &= - T^a_b ξ^b + 12
∇_b (∇_d λ^a (bc)d ξ_c) \- 12 ∇_d ( λ^a(bc)d ∇_b ξ_c )
\- λ^abcd ∇_d ∇_b ξ_c +14( ∇_d λ^abcd ∇_c ξ_b - λ^abcd ∇_d ∇_c ξ_b). From this
we can compute the superpotential $2$-form
$U_{\xi}=\epsilon_{ab}\lambda^{abcd}\nabla_{c}\xi_{d}$, where $\epsilon_{ab}$
is an area $2$-form obtained by contracting $\partial_{a}$ and $\partial_{b}$
with the volume form. In other words, we see that this theory has a non-
trivial corner Noether charge, and thus our conjecture classifies it as
potentially background independent.

What then are we to make of Pooley’s ‘intuition’ that SR3 is background
dependent? Note that the basis for this intuition seems to have two parts. The
first is the uncontroversial observation that SR3’s equations of motion
$R^{abcd}(g)=0$ can be replicated in a non-Lagrangian theory, viz. the theory
specified just by the equations of motion $\nabla_{a}\nabla^{a}\phi=0$ and
$R^{a}_{bcd}=0$. The second is Pooley’s claim that if the latter theory is
regarded as background dependent, then the former theory should also be
regarded as background dependent.

It will help to introduce an analogy to put the observation in context and to
expose the defects of Pooley’s line of thought here. Consider an example in
which we couple some matter field degrees of freedom to a $U(1)$ BF theory,
which in 4 dimensions has the Lagrangian $B\wedge dA$, where the dynamical
fields are a $2$-form $B$ and a gauge field $A$ with curvature $F=dA$; the
equations of motion of the pure BF theory are $F=0$ and $dB=0$. We will take
this example to be our first theory, to be compared to a second theory in
which the same matter field degrees of freedom are set against the fixed
background of a flat $U(1)$ connection, which is of course described by the
equation $F=0$; in the former case, we can think of the flatness condition as
being enforced by a Lagrange multiplier field, whereas it is imposed by fiat
in the latter case. If we applied Pooley’s reasoning to this analogous
scenario, then we would say that since the latter theory is background
dependent (now in the gauge field, and not the spacetime, sense), we should
also regard the former theory as background dependent. But now it should be
evident that this line of thought cannot be right, because a BF theory is a
topological field theory, which is supposed to be one of the paradigm cases of
background independence! The mistake here is to conflate local dynamical
degrees of freedom—which both theories lack—with global dynamical degrees of
freedom, which only BF theory possesses, and in virtue of which it is called
‘topological’. For instance, in the case of BF (or Chern-Simons) theories of
gravity, what we would standardly say is that the spacetime is dynamical in
the sense that it has these global degrees of freedom.

With the analogy in hand, we are now in a position to better understand the
physics of SR3. It is precisely a BF-like theory, and as noted by Giulini
[12], the spacetime part has global—but not local—dynamical degrees of freedom
which make the solution space for SR3 inequivalent to the theory specified
just by the equations of motion $\nabla_{a}\nabla^{a}\phi=0$ and
$R^{a}_{bcd}=0$. The equation of motion for $\lambda$, in which this dynamical
parameter now appears as a potential for the matter field’s stress-energy, and
the non-trivial corner charge from $U_{\xi}$, serve to describe how the global
spacetime dynamics is coupled to the dynamics of the matter field.

To sum up: we again see that in the case of SR3, Pooley’s misclassification
argument poses no threat to our conjecture; on the contrary, a careful
Noetherian analysis of SR3 indicates that Pooley has misinterpreted this
theory.

##  4 Conclusion

Noether’s theorems, as we said early on in the Introduction, do not in
themselves resolve any questions about the interpretation of General
Relativity, or lend clarity to the physical (not merely mathematical) dispute
between Einstein and Klein. Their true glory lies in their being a uniquely
subtle medium for physical representation—a set of mathematical materials that
physicists can bend to the ends of a certain kind of representational
activity.

We pay tribute to the author of these theorems by closing with a broader
philosophical reflection of the kind of activity into which her theorems are
taken up. When describing the productive activity of house-building, Aristotle
introduces the idea of dunamis, i.e. the potency or capacity of the building
materials, which the builder actualizes in pursuing his goal of building a
house. This kind of activity Aristotle calls a kinesis, because it aims at a
further product outside itself, namely the completed house. And he goes on to
contrast kinesis with what he calls energeia, i.e. the kind of activity which
aims at nothing beyond itself and is its own end (two of his examples are
‘seeing’ and ‘dancing’).

Evidently, Noether’s theorems are a dunamis in Aristotle’s sense: they have an
incredible capacity to be actualized in representation, which Einstein merely
intuited, and which we have tried to flesh out more fully in Section 3,
although we still only managed to scratch the surface. A further interesting
question is whether this activity into which Noether’s theorems are taken
up—the activity of using them to represent the material world—is more akin to
kinesis or energeia. And here we would like to make the suggestion that
energeia is the better description, for the kind of insight achieved by
absorbing physical subject matter into the framework of Noether’s theorems
(e.g. in the reconceptualization of Section 3.2) leads to ever new insights
that energize and propel the physicist’s theorizing further forward—the end
here is internal to the activity of doing theoretical physics, and not a
further product that is external to the activity. And perhaps this is close to
what Noether had in mind when she said that ‘My methods are really methods of
working and thinking; this is why they have crept in everywhere anonymously.’

## Acknowledgements

We thanks James Read for helpful comments on this draft. NT’s research is
partially supported by JTF grant 61521 and NSF grant 1947155. LF’s research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Industry Canada and by the Province
of Ontario through the Ministry of Colleges and Universities.

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