#  On the Hole Argument and the physical interpretation of General Relativity

Jaume de Haro111E-mail<EMAIL_ADDRESS>Departament de Matemàtiques,
Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

###### Abstract

Einstein presented the Hole Argument against General Covariance, understood as
invariance with respect to a change of coordinates, as a consequence of his
initial failure to obtain covariant equations that, in the weak static limit,
contain Newton’s law. Fortunately, about two years later, Einstein returned to
General Covariance and found these famous equations of gravity. However, the
rejection of his Hole Argument carries a totally different vision of space-
time. Its substantivalism notion, which is an essential ingredient in
Newtonian theory and also in his special theory of relativity, has to be
replaced, following Descartes and Leibniz’s relationalism, by a set of ”point-
coincidences.”

Hole Argument, Diffeomorphisms, Gauge Covariance, ADM formalism

##  I Introduction

The Einstein Hole Argument is a notable and intriguing challenge to General
Covariance. This argument, presented by Einstein in 1913, supports his
rejection of a general covariant (covariant with respect change of co-
ordinates) theory of gravitation, because in his initial attempts well before
1915, he was unable to construct a general covariant theory that, for static
weak gravitational fields, would yield the Newtonian theory.

At its core, the Hole Argument challenges the conventional substantivalism
doctrine, which entails understanding space-time as an objective entity with
intrinsic properties independent of its content. This raises questions about
the individuality of space-time points and whether one can attribute distinct
physical characteristics to them. Einstein’s original argument, adopting a
substantivalism point of view, explores the tension between the co-ordinate
freedom inherent in general covariant theories and the uniqueness of their
solutions.

In this context, the Hole Argument emerges as a thought experiment that probes
the consequences of the mathematical structure of the theory on our
conceptualization of physical reality. By scrutinizing the implications of
General Covariance, the Hole Argument prompts us to reconsider our intuitions
about the nature of space-time and grapple with the profound interplay between
mathematical formalism and physical interpretation. Its resolution has far-
reaching implications for our understanding of the ontology of space-time and
continues to stimulate philosophical discussions within the realm of General
Relativity Stachel ; Earman-Norton ; Pooley ; Pooley-Read ; Weatherall ;
Norton ; Landsman ; Norton1 , although some of these discussions seem to me to
take on a metaphysical character, which can lead to losing the physical
content of the problem. In the same sense, sometimes the abuse and misuse of
complicated mathematical structures obscure the reasoning and make it
difficult to understand their real physical meaning.

In fact, as we will discuss throughout the work, there exists a disconnect
between Einstein’s Hole Argument against General Covariance and the modern
iteration used by some authors, where the covariance or lack thereof of the
stress-energy tensor does not play a substantial role. On the contrary, the
new version constitutes a discussion, albeit sometimes concealed and poorly
understood, about the diffeomorphism invariance of the field equations. The
true essence of some of these papers lies in the implicit assumption of Gauge
Covariance, leading to the trivial observation that the field equations
possess infinitely many solutions. Consequently, the Hole Argument emerges as
a simple corollary devoid of any essential significance.

Once the trivial link between the assumed Gauge Covariance and the lack of
uniqueness of solutions is established, albeit through unnecessarily
convoluted mathematical tools, a philosophical debate ensues. This debate,
contrary to physical intuition, revolves around the mathematical
viewpoint—especially the geometric aspect—of the field equations and spacetime
structure, sometimes resulting in erroneous physical interpretations of
General Relativity.

From a physical approach to the problem, the main goal of the present essay
is, following Rovelli’s reasoning in Rovelli , to explain the most agreed-upon
point of view in the physics community regarding the resolution of the Hole
Argument. It can be summarized as follows: To preserve General Covariance, one
has to reject substantivalism of space-time, meaning the rejection of absolute
space-time, and accept the relationalistic doctrine. In this perspective,
events in space-time can be defined ”à la Einstein” as coincidence points of
world-lines. Adopting this viewpoint, the Hole Argument only shows that a
deterministic covariant theory demands the abandonment of absolute space-time,
where events have an intrinsic physical meaning independently of what
surrounds them.

However, one has to be cautious with the fact that sometimes physicists argue
that in General Relativity, ”the space-time is dynamic” in the sense that it
interacts with the matter contained within it, providing the space-time with a
metric. This could lead to the temptation to attribute some individuality to
the events of space-time, as in Newtonian absolute space. However, in General
Relativity, one cannot dissociate space-time from matter fields and other
fields involved in the theory; that is, points are related to fields.
Following this line of thinking, one can argue that positions are defined with
respect to fields via the coincidence of world-lines or geodesics whose nature
is intrinsically related to the metric, and thus with the fields that provide
it.

The work is organized as follows: In Section II, we provide a review of
pseudo-Riemann geometry, demonstrating that diffeomorphisms preserve the
intrinsic curvatures of a manifold. The presentation of Einstein’s equation in
this intrinsic form is given in Section III, where we also discuss the Hole
Argument as a challenge to tensor field equations. In that section, we also
introduce the concept of Gauge Covariance (invariance by diffeomorphisms),
eliminating the need for a Hole to present arguments for or against
substantivalism. Section IV is dedicated to the presentation of the original
Einstein Hole Argument against general covariance, i.e., invariance with
respect to the change of coordinates or, in modern language, invariance with
respect to passive diffeomorphisms. This section includes Hilbert’s
elaboration of the Hole Argument and its most popular interpretation in the
physics community. In Section V, I attempt to shed light on works with a
profound philosophical essence and a highly complex mathematical foundation.
Finally, in the last Section, I present my conclusions.

##  II Brief review of pseudo-Riemann manifolds

The objective of this section is to provide a foundational understanding of
differential geometry, establishing the necessary groundwork to navigate
tensor equations within a manifold.

Let me start with an $n+m$-dimensional pseudo-Riemann manifold represented by
a pair $(\mathcal{M},\mathfrak{g})$, where it is possible to select a frame
field ${E_{1},\ldots,E_{n},E_{n+1},\ldots,E_{n+m}}$ in
$\mathfrak{X}(\mathcal{M})$ (the set of vector fields in $\mathcal{M}$ or in a
more technical language the sections of the the tangent bundle $T\mathcal{M}$)
that satisfies:

$\displaystyle\mathfrak{g}(E_{1},E_{1})=\ldots=\mathfrak{g}(E_{n},E_{n})=1,\quad\mathfrak{g}(E_{n+1},E_{n+1})=\ldots=\mathfrak{g}(E_{n+m},E_{n+m})=-1,$
(1)

and $\mathfrak{g}(E_{i},E_{i})=0$ when $i\neq j$.

Consider an active diffeomorphism (a diffeomorphism different from the
identity), denoted as $\phi$, from $\mathcal{M}$ to itself. This induces an
isometry defined by

$\displaystyle\Phi:(\mathcal{M},\widetilde{\mathfrak{g}})\longrightarrow(\mathcal{M},\mathfrak{g}),$
(2)

where the metric $\widetilde{\mathfrak{g}}$ is the pull-back of
$\mathfrak{g}$, expressed as $\widetilde{\mathfrak{g}}=\phi^{*}\mathfrak{g}$.
Effectively, for two vector fields $X$ and $Y$ in $\mathfrak{X}(\mathcal{M})$,
we have

$\displaystyle\widetilde{\mathfrak{g}}(X,Y)=\mathfrak{g}(\phi_{*}X,\phi_{*}Y),$
(3)

where $\phi_{*}Z$ represents the push-forward of a vector $Z$. This push-
forward is defined in the usual manner, as
$\phi_{*}Z(f)=Z(\phi^{*}f)=Z(f\circ\phi)$.

Since isometries preserve angles and distances, one can argue mathematically
that the pseudo-Riemann manifolds
$(\mathcal{M},\widetilde{\mathfrak{g}}=\phi^{*}\mathfrak{g})$ and
$(\mathcal{M},\mathfrak{g})$ are identical. More precisely, any measure (for
example, the scalar product of two vector fields or the calculation of the
scalar curvature), denoted as $\widetilde{E}$, in
$(\mathcal{M},\widetilde{\mathfrak{g}})$ at the point $p\in\mathcal{M}$, when
mapped by $\Phi$ to the measure $E$ in $(\mathcal{M},\mathfrak{g})$ at the
point $\phi(p)\in\mathcal{M}$, predicts the same result as $E$.

To express this fact mathematically, following Iftime-Stachel , let us to
introduce the group of active diffeomorphisms and the class or orbit of a
tensor as follows:

$\displaystyle\mathcal{M}_{diff}=\\{\phi:\mathcal{M}\longrightarrow\mathcal{M},\quad\phi\mbox{
diffeomorphism}\\},$ (4)

and given a $2$-covariant tensor
$\mathfrak{H}:\mathfrak{X}(\mathcal{M})\times\mathfrak{X}(\mathcal{M})\longrightarrow\mathcal{C}^{\infty}(\mathcal{M})$,
we define its class or orbit by

$\displaystyle[\mathfrak{H}]=\\{\phi^{*}\mathfrak{H},\quad\forall\phi\in\mathcal{M}_{diff}\\}.$
(5)

These orbits are the elements of the topological quotient space
$\mathfrak{Q}_{2}(\mathcal{M})=\mathfrak{TEN}_{2}/\mathcal{M}_{diff}$, where
$\mathfrak{TEN}_{2}$ is the vector space of all symmetric $2$-covariant
tensors. So, given a metric $\mathfrak{g}^{0}$ in $\mathcal{M}$ we can say
that its orbit $[\mathfrak{g}^{0}]\in\mathfrak{Q}_{2}(\mathcal{M})$ contains
all the metrics mathematically equivalent to $\mathfrak{g}^{0}$ in
$\mathcal{M}$.

Next, we consider the Levi-Civita connection of the metric $\mathfrak{g}$,
namely $\nabla^{\mathfrak{g}}$, which is obtained imposing the symmetry
condition or torsion free:

$\displaystyle\nabla^{\mathfrak{g}}_{X}Y-\nabla^{\mathfrak{g}}_{Y}X=[X,Y],$
(6)

where $[X,Y]=XY-YX$ is the Lie bracket, and the compatibility with
$\mathfrak{g}$, i.e.,

$\displaystyle\nabla^{\mathfrak{g}}\mathfrak{g}=0,$ (7)

where the covariant derivative of a $2$-covariant tensor is defined as

$\displaystyle\nabla^{\mathfrak{g}}_{Z}\mathfrak{H}(X,Y)=Z\mathfrak{H}(X,Y)-\mathfrak{H}(\nabla^{\mathfrak{g}}_{Z}X,Y)-\mathfrak{H}(X,\nabla^{\mathfrak{g}}_{Z}Y).$
(8)

The fundamental property of the Levi-Civita connection, which comes from the
uniqueness of this connection, is that (see the Proposition 5.6 of Lee )

$\displaystyle\phi^{*}\nabla^{\mathfrak{g}}=\nabla^{\widetilde{\mathfrak{g}}}\Longrightarrow\phi_{*}(\nabla^{\widetilde{\mathfrak{g}}}_{X}Y)=\nabla^{\mathfrak{g}}_{\phi_{*}X}\phi_{*}Y.$
(9)

Next we introduce the Riemann tensor. First of all we introduce the Riemann
curvature endomorphism
$\mathfrak{R}_{\mathfrak{g}}:\mathfrak{X}(\mathcal{M})\times\mathfrak{X}(\mathcal{M})\times\mathfrak{X}(\mathcal{M})\rightarrow\mathfrak{X}(\mathcal{M})$
defined by

$\displaystyle\mathfrak{R}_{\mathfrak{g}}(X,Y)Z=\nabla^{\mathfrak{g}}_{X}\nabla^{\mathfrak{g}}_{Y}Z-\nabla^{\mathfrak{g}}_{Y}\nabla^{\mathfrak{g}}_{X}Z-\nabla^{\mathfrak{g}}_{[X,Y]}Z.$
(10)

With this endomophism we can define the Riemann curvature tensor for the
metric $\mathfrak{g}$, namely $\mathfrak{Rm}_{\mathfrak{g}}$, as the
$4$-covariant tensor satisfying

$\displaystyle\mathfrak{Rm}_{\mathfrak{g}}(X,Y,Z,W)=\mathfrak{g}(\mathfrak{R}_{\mathfrak{g}}(X,Y)Z,W).$
(11)

Since $\Phi$ is an isometry, because
$\widetilde{\mathfrak{g}}=\phi^{*}\mathfrak{g}$, one has the fundamental
property (see the Lemma 7.2 of Lee )

$\displaystyle\phi^{*}\mathfrak{Rm}_{\mathfrak{g}}={\mathfrak{Rm}}_{\widetilde{\mathfrak{g}}},$
(12)

which comes from (9) and tells us that the Riemann tensor is invariant for
diffeomorphisms.

Once we have defined the curvature we can define the $2$-covariant Ricci
tensor as the trace of the Riemann curvature endomorphism, as follows:

$\displaystyle\mathfrak{Rc}_{\mathfrak{g}}(X,Y)=Tr(Z\rightarrow\mathfrak{R}_{\mathfrak{g}}(Z,X)Y).$
(13)

To calculate this trace we define the flat operator
$\flat:\mathfrak{X}(\mathcal{M})\rightarrow\mathfrak{X}^{*}(\mathcal{M})$,
defined as follows:

$\displaystyle\flat:X\rightarrow X^{\flat},\quad\mbox{where}\quad
X^{\flat}(Y)=\mathfrak{g}(X,Y),$ (14)

and its inverse $\sharp=\flat^{-1}$ named sharp operator.

Let now $E^{i}$ be the dual of the vector $E_{i}$, that is
$E^{i}(E_{j})=\delta_{ij}$, being $\delta_{ij}$ the Kronecker’s delta. We want
to calculate ${E^{i}}^{\sharp}$. Since the sharp is the inverse of the flat,
we will have:

$\displaystyle\mathfrak{g}({E^{i}}^{\sharp},E_{j})={{E^{i}}^{\sharp}}^{\flat}(E_{j})=E^{i}(E_{j})=\delta_{j}^{i},$
(15)

meaning that

$\displaystyle{E^{i}}^{\sharp}=\mathfrak{g}^{-1}(E^{i},E^{k})E_{k},$ (16)

where we have used the Einstein’s summation convention and we have introduced
the $2$-contravariant tensor
$\mathfrak{g}^{-1}:\mathfrak{X}^{*}(\mathcal{M})\times\mathfrak{X}^{*}(\mathcal{M})\longrightarrow\mathcal{C}^{\infty}(\mathcal{M})$
as the inverse of $\mathfrak{g}$, i.e.,
$\mathfrak{g}^{-1}(E^{i},E^{k})\mathfrak{g}(E_{k},E_{j})=\delta_{j}^{i}$.
Then, the Ricci tensor can be calculated as follows:

$\displaystyle\mathfrak{Rc}_{\mathfrak{g}}(X,Y)=\mathfrak{Rm}_{\mathfrak{g}}(E_{i},X,Y,{E^{i}}^{\sharp}),$
(17)

and, from the property (12) we have

$\displaystyle\phi^{*}\mathfrak{Rc}_{\mathfrak{g}}={\mathfrak{Rc}}_{\widetilde{\mathfrak{g}}}.$
(18)

To define the scalar curvature (referred to as the Ricci scalar in physics),
denoted as $R$, as the trace of the Ricci tensor, we first need to introduce
the concept of the trace of a $2$-covariant tensor
$\mathfrak{H}:\mathfrak{X}(\mathcal{M})\times\mathfrak{X}(\mathcal{M})\rightarrow\mathcal{C}^{\infty}(\mathcal{M})$,
which we can be defined as follows

$\displaystyle
Tr_{\mathfrak{g}}\mathfrak{H}=\mathfrak{H}(E_{i},{E^{i}}^{\sharp}),$ (19)

independently of the chosen basis.

Then, applying this definition to the Ricci tensor we will have

$\displaystyle
R_{\mathfrak{g}}=\mathfrak{Rc}_{\mathfrak{g}}(E_{i},{E^{i}}^{\sharp}).$ (20)

In summary, the lesson we have to learn here is that the pull-back preserves
the intrinsic curvatures of the manifold, in the sense that

$\displaystyle\phi^{*}:(\mathfrak{Rm}_{\mathfrak{g}},\mathfrak{Rc}_{\mathfrak{g}},R_{\mathfrak{g}})\rightarrow({\mathfrak{Rm}}_{\widetilde{\mathfrak{g}}},{\mathfrak{Rc}}_{\widetilde{\mathfrak{g}}},{R}_{\widetilde{\mathfrak{g}}}).$
(21)

##  III Einstein equation

With all this machinery, one can introduce the $2$-covariant Einstein’s
equation in $\mathfrak{TEN}_{2}$, in its intrinsic form:

$\displaystyle\mathfrak{G}_{\mathfrak{g}}\equiv\mathfrak{Rc}_{\mathfrak{g}}-\frac{1}{2}R_{\mathfrak{g}}\mathfrak{g}=\kappa\mathfrak{T}_{\mathfrak{g}},$
(22)

where $\mathfrak{G}_{\mathfrak{g}}$ is the so-called Einstein tensor and
$\mathfrak{T}_{\mathfrak{g}}$ is the $2$-covariant stress-energy tensor, for
example, the one corresponding to a perfect fluid

$\displaystyle\mathfrak{T}_{\mathfrak{g}}=\left(\rho+\frac{p}{c^{2}}\right)\mathfrak{u}\otimes\mathfrak{u}+p\mathfrak{g},$
(23)

where $\rho$ is the energy density of the fluid, $p$ is its pressure and
$\mathfrak{u}$ is the vector field of $4$-velocities.

In this context, $\kappa=\frac{8\pi G}{c^{4}}$, where $G$ represents Newton’s
constant, and $c$ signifies the velocity of light. It is crucial to grasp that
the metric $\mathfrak{g}$ is the unknown in this equation. According to
Einstein’s ”Mach principle,” the mass of bodies dictates geodesic motion. This
concept serves as a criterion for selecting solutions to Einstein’s equation
Raine . In this sense, the stress-energy tensor $\mathfrak{T}_{\mathfrak{g}}$
determines the metric $\mathfrak{g}^{0}$, the sole ”physically meaningful”
solution to the Einstein equation (22), thereby establishing the pseudo-
Riemann manifold $(\mathcal{M},\mathfrak{g}^{0})$.

###### Remark III.1

In a $4$-dimensional Lorentz manifold ($\mathfrak{g}(E_{4},E_{4})=-1$ and
$\mathfrak{g}(E_{j},E_{j})=1$ for $j=1,2,3$), one has
$Tr_{\mathfrak{g}}(\mathfrak{g})=g(E_{i},{E^{i}}^{\sharp})=4$, and thus,
taking the trace in both sides of Einstein’s equation one has
$R_{\mathfrak{g}}=-\kappa T_{\mathfrak{g}}$ where
$T_{\mathfrak{g}}=Tr_{\mathfrak{g}}(\mathfrak{T}_{\mathfrak{g}})$. Therefore,
the Einstein equation can also be written as

$\displaystyle\mathfrak{Rc}_{\mathfrak{g}}=\kappa\mathfrak{T}-\frac{\kappa}{2}T_{\mathfrak{g}}\mathfrak{g}.$
(24)

Next we consider an active diffeomorphism
$\phi:\mathcal{M}\rightarrow\mathcal{M}$ which pulls-back the stress-tensor
$\mathfrak{T}_{\mathfrak{g}}$ to $\phi^{*}\mathfrak{T}_{\mathfrak{g}}$. We
have
$\phi^{*}\mathfrak{G}_{\mathfrak{g}}={\mathfrak{G}}_{\phi^{*}\mathfrak{g}}$,
but in general
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}\not={\mathfrak{T}}_{\phi^{*}\mathfrak{g}}$,
and the question is to compare the Einstein equations

$\displaystyle\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}\qquad\mbox{and}\qquad{\mathfrak{G}}_{{\phi^{*}\mathfrak{g}}}=\kappa\phi^{*}\mathfrak{T}_{\mathfrak{g}}.$
(25)

It is clear that if $\mathfrak{g}^{0}$ is the solution (we will assume
uniqueness) of
$\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}$, then
$\mathfrak{\widetilde{g}}^{0}=\phi^{*}\mathfrak{g}^{0}$ is only a solution of
$\mathfrak{G}_{\mathfrak{g}}=\kappa\phi^{*}\mathfrak{T}_{\mathfrak{g}}$,
provided that
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}={\mathfrak{T}}_{\phi^{*}\mathfrak{g}}$.

Therefore, it is important to stress that when
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}\not={\mathfrak{T}}_{\phi^{*}\mathfrak{g}}$,
the diffeomorphism $\phi$ does not induce another solution of the Einstein
equation.

However, although in general
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}\not={\mathfrak{T}}_{\phi^{*}\mathfrak{g}}$,
there is a class of active diffeomorphisms, which we will name hole-
diffeomorphisms, that pulled-back the stress-energy tensor to itself, i.e.,
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}=\mathfrak{T}_{\phi^{*}\mathfrak{g}}$

To define this kind of diffeomorphism we consider an open subset $\mathcal{H}$
(the hole) of $\mathcal{M}$ where the stress-energy tensor vanishes. Then, we
consider diffeomorphisms $\phi_{h}$ from $\mathcal{M}$ to itself, satisfying:

$\displaystyle\phi_{h}=Id.\quad\mbox{in}\quad\mathcal{M}\setminus\mathcal{H};\qquad\phi_{h}\not=Id.\quad\mbox{in}\quad\mathcal{H},$
(26)

where $Id.$ denotes the identity. From this definition, we can see that
$\phi_{h}^{*}\mathfrak{T}_{\mathfrak{g}}={\mathfrak{T}}_{\phi_{h}^{*}\mathfrak{g}}$,
and thus from a solution $\mathfrak{g}^{0}$ of the Einstein equation
$\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}$ we can
construct another different one
$\mathfrak{\widetilde{g}}^{0}=\phi_{h}^{*}\mathfrak{g}^{0}$.

Note that outside the hole, both solutions coincide, but inside it, they
differ. This is the foundation of the ”hole argument” against tensor field
equations. One way to express it is as follows:

Assuming tensor field equations in the form of a geometric tensor constructed
by intrinsic curvatures of the manifold being proportional to the stress-
energy tensor, the lack of uniqueness in solutions introduces indeterminism.

Now, there are two distinct positions we can take to address this assertion:

  1. 1.

Accepting that the field equation doesn’t necessarily have to be a tensor
equation.

  2. 2.

Abandoning substantivalism and embracing a relationalistic approach to define
events on the manifold. This allows us to interpret different solutions
provided by a hole-diffeomorphism as representing the same physical reality.

In the following sections, we will explore why the latter option is the
correct one for formulating a theory of Gravity, including General
Covariance.”

###  III.1 Gauge Covariance

We can extend our exploration by imposing Gauge Covariance, signifying
invariance under diffeomorphism (for a more schematic definition, refer to
Landsman ). The formulation can be expressed as follows:

Let’s assume the presence of fields $\mathfrak{A}_{1}$, …, $\mathfrak{A}_{k}$
within a physical system, adhering to tensor equations of the following form:

$\displaystyle\mathfrak{B}_{1}(\mathfrak{g},\mathfrak{A}_{1},..,\mathfrak{A}_{k})=0,...,\mathfrak{B}_{m}(\mathfrak{g},\mathfrak{A}_{1},..,\mathfrak{A}_{k})=0.$
(27)

The complete dynamical system includes the Einstein equation and the equations
satisfied by the fields, i.e.,

$\displaystyle\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}(\mathfrak{A}_{1},...,\mathfrak{A}_{k}),\quad\mathfrak{B}_{1}(\mathfrak{g},\mathfrak{A}_{1},..,\mathfrak{A}_{k})=0,...,\mathfrak{B}_{m}(\mathfrak{g},\mathfrak{A}_{1},..,\mathfrak{A}_{k})=0.$
(28)

Then, given a diffeomorphism $\phi$, a Gauge Covariant theory transform the
equations (28) via the pull-back $\phi^{*}$ as follows:

$\displaystyle\mathfrak{G}_{\mathfrak{\phi^{*}g}}=\kappa\mathfrak{T}_{\phi^{*}\mathfrak{g}}(\phi^{*}\mathfrak{A}_{1},...,\phi^{*}\mathfrak{A}_{k}),\quad\mathfrak{B}_{1}(\phi^{*}\mathfrak{g},\phi^{*}\mathfrak{A}_{1},..,\phi^{*}\mathfrak{A}_{k})=0,...,\mathfrak{B}_{m}(\phi^{*}\mathfrak{g},\phi^{*}\mathfrak{A}_{1},..,\phi^{*}\mathfrak{A}_{k})=0.$
(29)

Therefore, let $\mathfrak{R}(\mathcal{M})$ the space of solutions of (28).
Given a solution
$\\{\mathfrak{g}^{0},\mathfrak{A}_{1}^{0},..,\mathfrak{A}_{k}^{0}\\}$ of (28)
, for each diffeomorphism we obtain another solution
$\\{\phi^{*}\mathfrak{g}^{0},\phi^{*}\mathfrak{A}_{1}^{0},..,\phi^{*}\mathfrak{A}_{k}^{0}\\}$.
That is, we obtain a class of solutions
$[\mathfrak{g}^{0},\mathfrak{A}_{1}^{0},..,\mathfrak{A}_{k}^{0}]$ belonging to
$\mathfrak{R}(\mathcal{M})/\mathcal{M}_{diff}$.

In a Gauge Covariant theory, one that remains invariant under diffeomorphism,
the resolution to eliminate indeterminacy requires acknowledging that all
elements within a class of solutions represent the same physical state (refer
to Chapter 2 of Rovelli ).

This acknowledgment entails abandoning substantivalism and embracing the
Earman-Norton ’Leibniz Equivalence,’ as articulated in Davis  where it is
stated, ’Diffeomorphic models represent the same physical situation.’ This
shift involves adopting a relationalistic approach to define events on the
manifold.

A final point worth noting:

###### Remark III.2

Embracing Gauge Covariance obviates the need for a Hole to argue against
covariance. Given a solution to the equations, all diffeomorphisms applied to
this solution lead to solutions belonging to the same class, resulting in
indeterminism without the need for the Hole. This indeterminacy, naturally,
dissipates when rejecting substantivalism in favor of Einstein’s relational
viewpoint.

##  IV Einstein’s hole argument

The history of the Hole Argument comes from its initial failure to obtain
General Covariant field equations that in the static weak limit contain the
Newtonian equations. I direct to the reader to papers about the foundation of
General Relativity Norton ; Norton2 ; Norton3 ; Straumann ; Weinstein ; Sauer
. In this section we will present the original Hole Argument against General
Covariance (covariance with respect change of co-ordinates or in more modern
lenguage with respect to passive diffeomorphisms) applied to the Einstein
equation, but it also holds for other tensor equations relating some intrinsic
$2$-curvature tensors with the stress-energy tensor.

Let us consider a $4$-dimensional Lorentz manifold
$(\mathcal{M},\mathfrak{g})$, and to simplify the reasoning, we assume that
there is only a chart $(\mathcal{M},\psi)$ mapping $\mathcal{M}$ to
$\mathbb{R}^{4}$ ”in $\\{x\\}$ coordinates”, that is, given an event
$p\in\mathcal{{M}}$, $\psi$ assigns the point $p$ to $x\in\mathbb{R}^{4}$ with
coordinates $(x^{1},x^{2},x^{3},x^{4})$ in a basis
$\\{e_{1},e_{2},e_{3},e_{4}\\}$ defined as follow:

$\displaystyle e_{j}f=\frac{\partial f}{\partial x^{j}},\quad\forall
f:\mathbb{R}^{4}\rightarrow\mathbb{R},$ (30)

which can be pushed-forward to $\mathcal{M}$ by $\psi^{-1}$, obtaining a basis
in $\mathfrak{X}(\mathcal{M})$

$\displaystyle E_{j}=\psi^{-1}_{*}e_{j}.$ (31)

The Einstein equation
$\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}$, where I
assume a simplified form without intricate fields for the stress-energy tensor
(recalling that originally, Einstein dealt with static matter to derive, in
the weak approximation, the Newtonian law), undergoes a pull-back operation by
${\psi^{-1}}^{*}$, resulting in:

$\displaystyle{\psi^{-1}}^{*}\mathfrak{G}_{\mathfrak{g}}(e_{i},e_{j})(x)=\kappa{\psi^{-1}}^{*}\mathfrak{T}_{\mathfrak{g}}(e_{i},e_{j})(x).$
(32)

Then, giving a $2$-covariant tensor $\mathfrak{H}$ and denoting
${\psi^{-1}}^{*}\mathfrak{H}(e_{i},e_{j})(x)=\mathfrak{H}(E_{i},E_{j})(p)$ by
$H_{ij}({\psi^{-1}}^{*}\mathfrak{g},x)$, the Einstein equations in
$\\{x\\}$-coordinates, becomes

$\displaystyle
R_{ij}({g},x)-\frac{1}{2}R({g},x)g_{ij}(x)=T_{ij}(g,x)\Longleftrightarrow
G_{ij}({g},x)=T_{ij}(g,x),$ (33)

where we have used the notation ${\psi^{-1}}^{*}\mathfrak{g}=g$ and

$\displaystyle
R_{ij}({g},x)={\psi^{-1}}^{*}\mathfrak{Rc}(e_{i},e_{j})(x)=\mathfrak{Rc}(E_{i},E_{j})(p),$
$\displaystyle R({g},x)=R(p),\quad
g_{ij}(x)=\mathfrak{g}(E_{i},E_{j})(p),\quad
T_{ij}(g,x)=\mathfrak{T}_{\mathfrak{g}}(E_{i},E_{j})(p).$ (34)

The expression for the entries of the Ricci tensor in ${x}$-coordinates in
terms of the Christoffel symbols

$\displaystyle\Gamma_{ij}^{k}(x)=\frac{1}{2}g^{kl}(x)\left(\frac{\partial
g_{li}}{\partial x^{j}}(x)+\frac{\partial g_{lj}}{\partial
x^{i}}(x)-\frac{\partial g_{ij}}{\partial x^{l}}(x)\right),$ (35)

are

$\displaystyle R_{ij}({g},x)=\frac{\partial\Gamma_{ij}^{k}}{\partial
x^{k}}(x)-\frac{\partial\Gamma_{ik}^{k}}{\partial
x^{j}}(x)+\Gamma_{ij}^{k}(x)\Gamma_{kl}^{l}(x)-\Gamma_{il}^{k}(x)\Gamma_{jk}^{l}(x).$
(36)

Next, let’s consider another coordinate system, denoted as $\\{x^{\prime}\\}$,
where now a homeomorphism $\psi^{\prime}$ maps an event $p$ to $x^{\prime}$
with coordinates $(x^{\prime 1},x^{\prime 2},x^{\prime 3},x^{\prime 4})$ in
the basis $\\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime},e_{4}^{\prime}\\}$

$\displaystyle e^{\prime}_{j}f=\frac{\partial f}{\partial x^{\prime j}}.$ (37)

This basis can be pushed-forward by $\psi^{\prime-1}$, obtaining in
$\mathfrak{X}(\mathcal{M})$, the basis

$\displaystyle E^{\prime}_{j}=\psi^{\prime-1}_{*}e^{\prime}_{j}.$ (38)

In these coordinates, the entries of the Ricci tensor are denoted as
$R_{ij}(g^{\prime},x^{\prime})$, where
$g^{\prime}={\psi^{\prime-1}}^{*}\mathfrak{g}$, and the Christoffel symbols
${\Gamma^{\prime}}_{ij}^{k}$ are obtained from (35) by replacing the
$\\{x\\}$-coordinates with the $\\{x^{\prime}\\}$ ones and $g_{ij}$ with
$g^{\prime}_{ij}$. However, the entries of the stress-energy tensor take the
form $T^{\prime}_{ij}(g^{\prime},x^{\prime})$, which, in general, differs from
$T_{ij}(g^{\prime},x^{\prime})$.

Now we define the passive diffeomorphism (a change of coordinates)

$\displaystyle F:\mathbb{R}^{4}\longrightarrow\mathbb{R}^{4},\quad
F(x)=(F^{1}(x),F^{2}(x),F^{3}(x),F^{4}(x))=(\psi^{\prime}\circ\psi^{-1})(x),$
(39)

which sends $x=\psi(p)$ to $x^{\prime}=\psi^{\prime}(p)$.

Clearly, $F^{*}$ pulls back the solution of the Einstein equations in
$\\{x\\}$-coordinates to the solution in $\\{x^{\prime}\\}$-coordinates. In
other words, $F^{*}$ transforms the solution ${g^{\prime}}^{0}$ of
$G_{ij}(g^{\prime},x^{\prime})=\kappa T_{ij}^{\prime}(g^{\prime},x^{\prime})$
to the solution $g^{0}\equiv F^{*}{g^{\prime}}^{0}$ of $G_{ij}(g,x)=\kappa
T_{ij}(g,x)$.

This embodies Einstein’s perspective on ’general covariance’: the field
equations in ${x}$-coordinates, $G_{ij}(g,x)=\kappa T_{ij}(g,x)$, must be
transformed into the equations $G_{ij}(g^{\prime},x^{\prime})=\kappa
T_{ij}^{\prime}(g^{\prime},x^{\prime})$ in the ${x^{\prime}}$-coordinates,
where

$\displaystyle T=F^{*}T^{\prime}\Longleftrightarrow T_{ij}(g,x)=\frac{\partial
F^{k}}{\partial x^{i}}(x)\frac{\partial F^{l}}{\partial
x^{j}}(x)T^{\prime}_{kl}(g^{\prime},F(x)).$ (40)

To raise the Einstein’s Hole Argument, we can also define a passive hole-
diffeomorphism as follows: consider a subset $\mathcal{H}$ of $\mathcal{M}$
where the stress-energy tensor vanishes. Consider two different co-ordinate
systems, namely $\psi_{h}$ and $\psi^{\prime}_{h}$, satisfying

$\displaystyle\psi_{h}=\psi^{\prime}_{h}\quad\mbox{in}\quad\mathcal{M}\setminus\mathcal{H}\quad\mbox{and}\quad\psi_{h}\not=\psi^{\prime}_{h}\quad\mbox{in}\quad\mathcal{H},$
(41)

define $F_{h}=\psi^{\prime}_{h}\circ\psi_{h}^{-1}$, and let
$H=\psi(\mathcal{H})=\psi^{\prime}(\mathcal{H})$ the hole in $\mathbb{R}^{4}$.
Then,

$\displaystyle F_{h}=Id.\quad\mbox{in}\quad\mathbb{R}^{4}\setminus
H,\qquad\mbox{and}\qquad F_{h}\not=Id.\quad\mbox{in}\quad H.$ (42)

The crucial observation lies in examining the stress-energy tensor in the
coordinate systems $\psi_{h}$ and $\psi^{\prime}_{h}$. This leads to the
relationship:

$\displaystyle{\psi_{h}^{-1}}^{*}\mathfrak{T}_{\mathfrak{g}}={{\psi^{\prime}}_{h}^{-1}}^{*}\mathfrak{T}_{\mathfrak{g}}\Longrightarrow
T_{ij}^{\prime}(g^{\prime},x^{\prime})=T_{ij}(g,x)=T_{ij}(g^{\prime},x^{\prime}),$
(43)

which implies that the Einstein equations in different coordinate systems are
$G_{ij}(g,x)=\kappa T_{ij}(g,x)$ and $G_{ij}(g^{\prime},x^{\prime})=\kappa
T_{ij}(g^{\prime},x^{\prime})$.

Now, let ${g^{\prime}}^{0}(x^{\prime})$ be a solution in
$\\{x^{\prime}\\}$-coordinates, and
$g^{0}(x)=F^{*}{g^{\prime}}^{0}(x^{\prime})$ the corresponding solution in
$\\{x\\}$-coordinates. We can find another solution in $\\{x\\}$-coordinates
simply by replacing $x^{\prime}$ in ${g^{\prime}}^{0}$ with $x$, essentially
relabeling the variables. In $\\{x\\}$-coordinates, we then have two distinct
solutions, $g^{0}(x)$ and ${g^{\prime}}^{0}(x)$.

Effectively, the metric tensor satisfies the following coordinate
transformation

$\displaystyle g^{0}_{ij}(x)=\frac{\partial F_{h}^{k}}{\partial
x^{i}}(x)\frac{\partial F_{h}^{l}}{\partial
x^{j}}(x){g^{\prime}}^{0}_{kl}(x^{\prime}).$ (44)

Thus, outside the hole $H$, we will have
$g^{0}_{ij}(x)={g^{\prime}}^{0}_{ij}(x^{\prime})={g^{\prime}}^{0}_{ij}(F_{h}(x))={g^{\prime}}^{0}_{ij}(x)$.
However, inside the hole,
$g^{0}_{ij}(x)\neq{g^{\prime}}^{0}_{ij}(x^{\prime})$, and by relabeling the
right-hand side of this last equation, we find that inside the hole
$g^{0}_{ij}(x)\neq{g^{\prime}}^{0}_{ij}(x)$.

In conclusion, within a given system of coordinates, one can find different
solutions—all induced by a passive hole-diffeomorphism—of the Einstein
equations. When transported to $\mathcal{M}$ by any $\psi^{*}_{h}$, these
solutions result in different metrics. For instance, a solution
${g^{\prime}}^{0}$ and its induction by $F_{h}^{*}$, denoted as
${g}^{0}=F^{*}_{h}{g^{\prime}}^{0}$, are transported in ${\mathcal{M}}$ to
$\mathfrak{g^{\prime}}^{0}=\psi_{h}^{*}{g^{\prime}}^{0}$ and
$\mathfrak{g}^{0}=\psi_{h}^{*}g^{0}$. Once again, this forms the basis of
Einstein’s hole argument.

Next, we will see that a passive diffeomorphism
$F=\psi^{\prime}\circ\psi^{-1}$ induces active diffeomorphisms. Effectively,
define

$\displaystyle\phi:\mathcal{M}\longrightarrow\mathcal{M};\quad\phi=\psi^{-1}\circ
F\circ\psi=\psi^{-1}\circ\psi^{\prime},$ (45)

where clearly $\phi\not=Id$, is satisfied.

Applied to a passive hole-diffeomorphism
$F_{h}=\psi_{h}^{\prime}\circ\psi_{h}^{-1}$, this leads to the active hole-
diffeomorphism $\phi_{h}=\psi_{h}^{-1}\circ\psi_{h}^{\prime}$. Note that
$\phi_{h}$ pulls-back $\mathfrak{g}^{\prime}=\psi_{h}^{*}g^{\prime}$ to
$\psi_{h}^{*}g=\mathfrak{g}$, because $\phi_{h}^{*}=\psi_{n}^{*}\circ
F_{h}^{*}\circ{\psi^{-1}_{h}}^{*}$, and thus,

$\displaystyle\phi_{h}^{*}\mathfrak{g}^{\prime}=(\psi_{h}^{*}\circ
F_{h}^{*}\circ{\psi^{-1}_{h}}^{*})\psi_{h}^{*}g^{\prime}=\psi_{h}^{*}\circ
F_{h}^{*}g^{\prime}=\psi_{h}^{*}g=\mathfrak{g}.$ (46)

Therefore, $\phi_{h}$ induces the isometry

$\displaystyle\Phi_{h}:(\mathcal{M},{\mathfrak{{g}}})\longrightarrow(\mathcal{M},\mathfrak{g}^{\prime}),$
(47)

what demonstrates that the ’trick’ of relabeling $x^{\prime}$ by $x$ can be
understood as the isomorphism $\Phi_{h}$.

Finally, at the tensor level, we can see that the hole-diffeomorphism
$\phi_{h}$ satisfies
$\phi_{h}^{*}\mathfrak{T}_{\mathfrak{g}}=\mathfrak{T}_{\phi_{h}^{*}\mathfrak{g}}$.
Consequently, the Einstein equation
$\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}$ transforms via
$\phi_{h}^{*}$ to
$\mathfrak{G}_{\phi_{h}^{*}\mathfrak{g}}=\kappa\mathfrak{T}_{\phi_{h}^{*}\mathfrak{g}}$,
indicating that if $\mathfrak{g}^{0}$ is solution of
$\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}$, then
$\phi_{h}^{*}\mathfrak{g}^{0}$ is also solution of the same equation.

###  IV.1 Hilbert’s view of the Hole Argument

As previously discussed, Einstein’s interpretation of the hole argument
revolves around a boundary problem that can be reformulated as an initial
value problem.

In concise terms, this presents a version of Hilbert’s hole argument as a
Cauchy problem: in $\\{x\\}$-coordinates, consider an initial value spacelike
surface defined by $x^{4}=\text{const.}$ with specific data (refer to ADM  for
insights into the Hamiltonian approximation to General Relativity and Eric ;
Isenberg  for its application to the Cauchy problem in General Relativity).
Assume that the evolution of the stress-energy tensor is predetermined,
meaning its form is known throughout the Lorentz manifold. Subsequently, we
seek to determine the metric as an evolution problem in time $x^{4}$. Through
a coordinate transformation that relocates points in the future of the initial
value surface $x^{4}=\text{const.}$—while leaving the initial surface and the
complementary of an open subset in the future, where the stress-energy tensor
vanishes, unaffected—we find that the Einstein equations fail to uniquely
determine the future. This is because the newly transformed metric in these
coordinates is equally valid as a solution to the same field equations in the
original coordinate system. Consequently, we can infer that the initial value
problem lacks a unique solution in General Relativity.

###  IV.2 Physical interpretation of the Hole Argument

To overcome the Hole Argument, Einstein approached it in a relational manner,
asserting that the events of a manifold must be defined as coincidence points.
Specifically, let $\widetilde{\gamma}_{1}$ and $\widetilde{\gamma}_{2}$ be two
geodesics in $(\mathcal{M},\widetilde{\mathfrak{g}})$ intersecting at the
point $p$ (a coincidence). Both geodesics are transformed by a diffeomorphism
$\phi$ into two other ones ${\gamma}_{1}$ and ${\gamma}_{2}$ in
$(\mathcal{M},{\mathfrak{g}}={\phi^{-1}}^{*}\widetilde{\mathfrak{g}})$,
intersecting at the point $\phi(p)$.

The key question is not the value of a measure at a point $p$ of the pseudo-
Riemann manifold $(\mathcal{M},\widetilde{\mathfrak{g}})$ or at the same point
of $(\mathcal{M},{\mathfrak{g}})$. Instead, the meaningful question is about
the value of a measure at the coincidence points. These coincidence points in
both pseudo-Riemann manifolds are respectively $p$ and $\phi(p)$, and clearly,
the same measure at both points leads to the same result. I refer the reader
to Rovelli’s book Rovelli , specifically to Section 2.2.5 where Figure 2.4
vividly illustrates the meaning of coincidence points.

In conclusion, given the Einstein equation
$\mathfrak{G}_{\mathfrak{g}}=\kappa\mathfrak{T}_{\mathfrak{g}}$, a hole-
diffeomorphism transforms this equation into
$\mathfrak{G}_{\phi_{h}^{*}\mathfrak{g}}=\kappa\mathfrak{T}_{\phi_{h}^{\mathfrak{g}}}$.
Thus, following the point-coincidence interpretation, the solutions
$\mathfrak{g}^{0}$ and $\phi_{h}^{*}\mathfrak{g}^{0}$ represent the same
physical reality.

In the context of Gauge Covariance, i.e., assuming
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}=\mathfrak{T}_{\phi^{*}\mathfrak{g}}$, one
can employ a rather complicated mathematical apparatus to present the
relationalistic resolution of the Hole Argument.

In fact, one must contemplate solutions to the Einstein equation within the
space $\mathfrak{Q}_{2}(\mathcal{M})$, with the field equation taking the
form:

$\displaystyle\mathfrak{G}_{[\mathfrak{g}]}=\kappa\mathfrak{T}_{[\mathfrak{g}]}.$
(48)

Solving this equation is equivalent to solving, utilizing a Numerical
Relativity scheme Eric , the equation:

$\displaystyle\mathfrak{G}_{\mathfrak{g}}=\kappa{\mathfrak{T}}_{\mathfrak{g}},$
(49)

where, one might assume that $\mathcal{M}$ is globally hyperbolic, meaning it
contains a Cauchy hyper-surface ${\Sigma}$.

More precisely, choose a normal field $N\in\mathfrak{X}(\mathcal{M})$ on
$\Sigma$, that is, satisfying:

$\displaystyle\mathfrak{g}(N,N)=-1,\quad\mathfrak{g}(N,X)=0\quad\forall
X\in\mathfrak{X}(\Sigma).$ (50)

Next, define the extrinsic curvature tensor in $\Sigma$, as the $2$-covariant
tensor
$\mathfrak{K}:\mathfrak{X}(\Sigma)\times\mathfrak{X}(\Sigma)\rightarrow\mathcal{C}^{\infty}(\Sigma)$
satisfying:

$\displaystyle\mathfrak{K}(X,Y)=-\mathfrak{g}(X,\nabla_{Y}N),\quad\forall
X,Y\in\mathfrak{X}(\Sigma),$ (51)

and consider an orthonormal basis $\\{E_{j}\\}_{j=1,2,3}$ on
$\mathfrak{X}(\Sigma)$, that is, $\mathfrak{g}(E_{i},E_{j})=\delta_{ij}$.

Then, Einstein’s equation leads to four constraints in $\Sigma$:

$\displaystyle\mathfrak{G}_{\mathfrak{g}}(N,E_{i})=\kappa{\mathfrak{T}}_{\mathfrak{g}}(N,E_{i}),\quad\mathfrak{G}_{\mathfrak{g}}(N,N)=\kappa{\mathfrak{T}}_{\mathfrak{g}}(N,N),$
(52)

which using the Gauss-Codazzi equations

$\displaystyle
R+2\mathfrak{Rc}(N,N)=R_{\mathfrak{h}}+(Tr_{\mathfrak{h}}\mathfrak{K})^{2}-\mathfrak{K}(E_{i},E_{j})\mathfrak{K}({E^{i}}^{\sharp},{E^{j}}^{\sharp}),$
$\displaystyle\mathfrak{Rc}(N,E_{i})=(D^{\mathfrak{h}}_{E_{i}}\mathfrak{K})(E_{j},{E^{j}}^{\sharp})-(D^{\mathfrak{h}}_{E_{j}}\mathfrak{K})(E_{i},{E^{j}}^{\sharp}),$
(53)

where $\mathfrak{h}=\mathfrak{g}_{|_{\Sigma}}$ is the restriction of the
metric $\mathfrak{g}$ to $\Sigma$, $R_{\mathfrak{h}}$ is the Ricci scalar on
$\Sigma$ and $D^{\mathfrak{h}}$ is the Levi-Civita connection in $\Sigma$, the
constrains become:

$\displaystyle
R_{\mathfrak{h}}+(Tr_{\mathfrak{h}}\mathfrak{K})^{2}-\mathfrak{K}(E_{i},E_{j})\mathfrak{K}({E^{i}}^{\sharp},{E^{j}}^{\sharp})=2\kappa{\mathfrak{T}}_{\mathfrak{g}}(N,N)\quad\mbox{and}\quad(D^{\mathfrak{h}}_{E_{i}}\mathfrak{K})(E_{j},{E^{j}}^{\sharp})-(D^{\mathfrak{h}}_{E_{j}}\mathfrak{K})(E_{i},{E^{j}}^{\sharp})=\kappa{\mathfrak{T}}_{\mathfrak{g}}(E_{i},N).$
(54)

What is crucial to realize here is that the extrinsic curvature tensor and,
when fixing the values of ${\mathfrak{T}}_{\mathfrak{g}}(N,N)$ and
${\mathfrak{T}}_{\mathfrak{g}}(E_{i},N)$ on $\Sigma$, the constraint equations
(54) depend solely on the metric $\mathfrak{h}$.

With these considerations, the Einstein equation becomes an evolution problem
with initial data $(\Sigma,\mathfrak{h},\mathfrak{K})$ and fixed values of the
stress-energy tensor on $\Sigma$, satisfying the constraints (54). If the
initial value problem is well-posed (Ivonne Choquet-Bruhat proved this for the
vacuum case Choquet , but as far as I know, such a result does not exist in
the general case), it would lead, up to diffeomorphisms, to a unique metric
${\mathfrak{g}}^{0}$. Assuming Gauge Covariance, the unique solution of (48)
is $[{\mathfrak{g}}^{0}]$.

####  IV.2.1 Meaning of the coordinates

In pre-relativistic mechanics, the $4$-dimensional manifold takes the form
$\mathcal{M}=\mathcal{M}_{3}\times\mathbb{R}$, where the space-manifold
$\mathcal{M}_{3}$ is equipped with a fixed metric $\mathfrak{g}_{3}$ that
imparts meaning to any coordinate systems employed. This approach holds true
in the special theory of relativity as well, where spacetime is represented by
the Minkowski manifold. In both cases, the metric structure of the manifold
plays a crucial role in defining the physical framework and interpreting the
coordinates within it. Therefore, the adoption of a fixed metric provides a
consistent and meaningful foundation for understanding the geometry and
physics associated with the respective pre-relativistic theories.

However, as we have already discussed, in General Relativity, a metric
determined as the solution of the Einstein equation can be transformed by an
active diffeomorphism into an equivalent one, placing them within the same
orbit.

Then, to elucidate the significance of a coordinate system within this metric
framework, we turn to a poignant statement by Misner, as quoted in Macdonald :

The metric [after a Gauge choice] defines not only the gravitational field
that is assumed, but also the coordinate system in which it is presented.
There is no other source of information about the coordinates apart from the
expression for the metric. It is also not possible to define the coordinate
system in any way that does not require a unique expression for the metric. In
most cases where the coordinates are chosen for computational convenience, the
expression for the metric is the most efficient way to communicate clearly the
choice of coordinates that is being made.

This is the key to comprehending Einstein’s original Hole Argument. As
elucidated earlier, we consider two distinct charts, denoted as
$(\mathcal{M},\psi_{h})$ and $(\mathcal{M},\psi^{\prime}h)$, where $\psi_{h}$
and $\psi^{\prime}_{h}$ coincide outside the hole but exhibit differences
within it. Employing the technique of relabeling the $\\{x^{\prime}\\}$
coordinates to $\\{x\\}$, Einstein discovers two distinct solutions,
$g_{ij}(x)$ and $g^{\prime}_{ij}(x)$, to the equations in
$\\{x\\}$-coordinates. However, as we have observed, this relabeling is
essentially a means of effecting an active hole-diffeomorphism, denoted as
$\phi_{h}=\psi_{h}^{-1}\circ\psi^{\prime}_{h}$, allowing the expression of the
metrics $\mathfrak{g}^{\prime}$ and
$\mathfrak{g}=\phi_{h}^{*}\mathfrak{g}^{\prime}$ in terms of
$\\{x\\}$-coordinates. Thus, in accordance with Misner’s perspective, the
physical interpretation of the $\\{x\\}$-coordinates differs when describing
$(\mathcal{M},\mathfrak{g})$ and $(\mathcal{M},\mathfrak{g}^{\prime})$.

Following Davis’s line of thinking depicted in Davis1 , in General Relativity,
the metric is only available to define a Lorentz manifold after the Einstein
equations are solved. However, the field equations have to be solved using
coordinates of unknown physical meaning. This endorses a privileged role to
each metric solution, consisting of determining the physical meaning of the
${x}$-coordinates in terms of which it is written.

Well understood, different metric solutions to the field equations may give
different physical meanings to the same set of local coordinates. The fact
that the ${x}$-coordinates used to find a metric $g_{ij}(x)$ are the same
quadruple of real numbers to find another solution ${g}^{\prime}_{ij}(x)$ does
not mean that the ${x}$-coordinates have the same physical meaning in both
solutions. Each metric solution brings its own assignment of physical or
geometrical meaning to the local coordinates used to write it.

In a few words, as stated by Davis: Coordinates have no independent meaning,
independent of the metric solution. Each of the multiple metric solutions
carries its own physical interpretation of its own local coordinates. The
example of the Schwarzschild metric studied in Macdonald  is very illuminating
and helps to clearly understand what was mentioned earlier.

##  V Some remarks on the philosophical viewpoint of the Hole Argument

First of all, and this is my opinion, there are trees (mathematical
structures) that, unfortunately, does not always lead some authors to see all
the forest (its physical implications). Even worse, some researchers do not
look at the trees with the correct glasses, often getting entangled in the
intricate branches and details without grasping the broader implications for
the physical reality they seek to understand. Let me elaborate on this
observation:

In delving into the intricacies of mathematical formalism, it is crucial to
maintain a clear perspective on the overarching physical principles and
concepts that these mathematical structures are meant to capture. It is akin
to navigating through a dense forest of mathematical symbols and
equations—while the details of individual trees (a concrete mathematical
apparatus) are undoubtedly important, it is equally essential to step back and
appreciate the collective landscape they form (physical implications). Failure
to do so may result in a disconnect between the elegance of mathematical
formulations and their true significance in describing the underlying physics.

In this context, my aim is to analyze certain works associated with the Hole
Argument. For instance, in Earman-Norton , the authors characterize a model,
denoted as $T$ in a given theory, as an $n+1$ tuple
$(\mathcal{M},O_{1},\ldots,O_{n})$, where $O_{j}$ represents geometric objects
defined on $\mathcal{M}$ that adhere to specific tensor equations, namely
$O_{k}=0,\ldots,O_{n}=0$, where $k$ is less than $n$.

The authors articulate the Gauge Theorem in the following manner: ”If
$(\mathcal{M},O_{1},\ldots,O_{n})$ represents a model of a theory and $\phi$
is a diffeomorphism of $\mathcal{M}$ onto itself, then
$(\mathcal{M},\phi^{*}O_{1},\ldots,\phi^{*}O_{n})$ also constitutes a model of
the theory.” Subsequently, the authors proceed to establish this theorem in an
unconventional manner, employing coordinates.

Essentially, their objective is to demonstrate that the vanishing equations
$O_{k}=0,\ldots,O_{n}=0$ are ”preserved by diffeomorphisms.” However, this
proof appears trivial, as (assuming, for simplicity, that the equations are
given by $2$-covariant tensors), for any $j=k,\ldots,n$, one has

$\displaystyle\phi^{*}O_{j}(X,Y)=O_{j}(\phi_{*}X,\phi_{*}Y)=0,$ (55)

because $O_{j}$ vanishes for all pairs of vector fields.

Here, the crucial point is that this does not imply that the ”carry along” of
the tuple $(\mathcal{M},\phi^{*}O_{1},\ldots,\phi^{*}O_{n})$ is inherently a
model of the theory. I have previously addressed this matter in Section III.1
following the introduction of the concept of Gauge Covariance. A clear
understanding of Gauge Covariance implies, particularly in terms of a simple
stress-energy tensor $\mathfrak{T}_{\mathfrak{g}}$ (with the general
definition provided in III.1),

$\displaystyle\phi^{*}\mathfrak{T}_{\mathfrak{g}}=\mathfrak{T}_{\phi^{*}\mathfrak{g}}\Longleftrightarrow(\phi^{*}\mathfrak{T}_{\mathfrak{g}})(X,Y)=\mbox{(by
definition)}=\mathfrak{T}_{\mathfrak{g}}(\phi^{*}X,\phi^{*}Y)=\mathfrak{T}_{\phi^{*}\mathfrak{g}}(X,Y)\quad\forall
X,Y\in\mathfrak{X}(\mathcal{M}).$ (56)

In my view, the scenario in Earman-Norton  is characterized by non-vanishing
fields $\mathfrak{A}_{1},\ldots,\mathfrak{A}_{k}$, while the vanishing ones
$O_{j}=0$ represent equations such as
$\mathfrak{B}_{j}(\mathfrak{g},\mathfrak{A}_{1},\ldots,\mathfrak{A}_{k})=0$.
Consequently, Gauge Covariance simply signifies:

$\displaystyle\phi^{*}\mathfrak{B}_{j}(\mathfrak{g},\mathfrak{A}_{1},...,\mathfrak{A}_{k})=\mathfrak{B}_{j}(\phi^{*}\mathfrak{g},\phi^{*}\mathfrak{A}_{1},...,\phi^{*}\mathfrak{A}_{k}),$
(57)

and therefore, $(\mathcal{M},\phi^{*}O_{1},...,\phi^{*}O_{n})$ also qualifies
as a model of the theory if the equations governing the theory remain
invariant under diffeomorphisms.

This fact is crucial because the statement provided in Earman-Norton , which
asserts that given a model $(\mathcal{M},O_{1},\ldots,O_{n})$ and $\phi$ a
diffeomorphism, then $(\mathcal{M},\phi^{*}O_{1},...,\phi^{*}O_{n})$ is also a
model, is fundamentally incorrect. This assertion holds only if the equations
$O_{j}=0$ are Gauge Covariant in the sense of (57). If it is not, then the
equations represent a different physical situation, and thus, one cannot argue
that different solutions represent the same reality.

Einstein clearly understood this when formulating the Hole Argument, as he did
not consider Gauge Covariance. For this reason, he introduced hole-
diffeomorphisms, which are Gauge Covariant with respect to the stress-energy
tensor. However, this crucial fact is not addressed at all in Earman-Norton .

Consequently, if one assumes Gauge Covariance, the Hole Argument loses its
meaning, as one can see in Earman-Norton , where the Hole Argument is
presented as a corollary, stated as follows: ”Let $T$ be a model with a
manifold $\mathcal{M}$ and $\mathcal{H}$ (the hole) be any neighborhood of
$\mathcal{M}$. Then there exist arbitrarily many distinct models of the theory
on $\mathcal{M}$ which differ from one another only within $\mathcal{H}$.”

To substantiate this claim, the authors employ hole-diffeomorphisms—those that
deviate from the identity only within the hole. They argue that there are
arbitrarily many such pulled-back models that fulfill the specified
requirement. This highlights, as I have mentioned in Remark III.2, that in a
Gauge Covariant theory, the Hole serves as a mere tool to illustrate that
choosing diffeomorphisms different from the identity yields another distinct
model of the theory $T$. That is, the key point is that under Gauge
Covariance, all the diffeomorphisms pull back models to models, and one does
not need any kind of special diffeomorphisms. This can be clearly seen in the
proof of the Hole corollary presented on page 523 of Earman-Norton , where, in
order to assume that the hole-diffeomorphisms pull back models to models, the
authors use their ”Gauge Theorem”, which, if well understood, not as the
authors do, is only the assumption of Gauge Covariance.

Exactly, the crux of Einstein’s Hole Argument lies in this fact—the invariance
under certain diffeomorphisms (specifically, hole-diffeomorphisms in the
absence of a Gauge Invariant theory). As I have previously detailed, with a
hole diffeomorphism, one observes that
$\phi_{h}^{*}\mathfrak{T}_{\mathfrak{g}}=\mathfrak{T}_{\phi_{h}^{*}\mathfrak{g}}$,
implying that the Einstein equation remains unchanged. Consequently, for a
given solution $\mathfrak{g}^{0}$, both $\mathfrak{g}^{0}$ and
$\phi_{h}^{*}\mathfrak{g}^{0}$ stand as solutions to the same Einstein
equation.

Furthermore, the authors of Earman-Norton  state: ”The name of this corollary
stems from Einstein’s original discovery of it in a specialized form. He
considered a matter-free hole in a source mass distribution and showed that
the gauge freedom of any generally covariant gravitational field equation in
general relativity allowed multiple metric fields within the hole.” This, of
course, encapsulates the true essence of the Hole Argument. However, in their
formulation, the authors do not delve into the core of the argument, which
lies in the vanishing of the stress-energy tensor—distinct from an intrinsic
curvature tensor—in the hole. They directly assume Gauge Covariance, rendering
the Hole inconsequential in their reasoning.

To wrap up, once one embraces the relationalist viewpoint through Einstein’s
coincidence points, all hole-diffeomorphisms define the same model.
Essentially, the hole serves to bolster a substantivalist perspective against
General Covariance. Expanding on this, if one assumes Gauge Covariance, then
all diffeomorphisms define the same model. From a substantivalist standpoint,
one can argue that the theory becomes indeterministic.

Having clarified the Hole Argument, it appears to me that some authors
continues employing an inaccurate version of it. For instance, in Gomes , the
authors define General Covariance, in terms of models, as follows:

If $(\mathcal{M},\mathfrak{g},\mathfrak{T}_{\mathfrak{g}})$ is a model, and
$\phi:\mathcal{M}\rightarrow\mathcal{M}$ is a diffeomorphism. Then,
$(\mathcal{M},\phi^{*}\mathfrak{g},\phi^{*}\mathfrak{T}_{\mathfrak{g}})$ is
also considered a model.

Clearly, this statement holds true only if one accepts Gauge Covariance, i.e.,
$\phi^{*}\mathfrak{T}_{\mathfrak{g}}=\mathfrak{T}_{\phi^{*}\mathfrak{g}}$.
Subsequently, the authors embark on a ”mathematico-philosophical” discussion,
employing mathematical concepts such as category theory to distinguish
different models related via diffeomorphisms and arguing that they represent
different possibilities. The issue here is that, in all their arguments,
everything is intertwined with mathematics—algebraic structures, Lie
derivatives, Noether’s second theorem, considering $4$ space-times as leaves
of a $5$ dimensional manifold, and more.

However, the crux of the matter is that there is a lack of physics in these
discussions. While there is a reference to ”Quantum reference frames,” it is
evidently unrelated to the Hole Argument. What I would emphasize is that to
genuinely address the Hole Argument, one must incorporate physics,
particularly Einstein’s point-coincidences, which appears to be absent in
Gomes .

Another example is Curiel , where the author presents the Hole Argument as an
evolution problem, as follows: ”Fix an space-time model
$(\mathcal{M},g_{\mu\nu})$. For ease of exposition, we stipulate that the
space-time be globally hyperbolic, and so possesses a global Cauchy surface
$\Sigma$. Say that we know the metric tensor on $\Sigma$ and on the entire
region of space-time to its causal past. It is known that this forms a well
set Cauchy problem, and so there is a solution to the Einstein field equation
that uniquely extend $g_{\mu\nu}$ on the causal pass of $\Sigma$ to metric
tensor on all of $\mathcal{M}$, yielding the original space-time we fixed. In
particular, the solution to the Cauchy problem fixes the metric on the region
to the causal future of $\Sigma$.”

Next, the author considers a diffeomorphism $\phi$ that leaves the causal past
of $\Sigma$ invariant. Subsequently, the author performs an unusual
”mathematical” operation—applying $\phi^{*}$ to the metric but not to the
manifold itself. This can be seen as a less developed version of the Einstein
trick, where one typically changes $g^{\prime}_{\mu\nu}(x^{\prime})$ to
$g^{\prime}_{\mu\nu}(x)$. Then, the author assesses:

This yields a seemingly different metric—a different “physical state of the
gravitational field”- on the causal future of $\Sigma$, in the sense that the
same points of the causal future of $\Sigma$ now carry (in general) a
different value for the metric. This is the crux of the issue, that the
diffeomorphism applied to the metric has yielded a different tensor field in
the sense that the same points of the space-time manifold now carry a
different metric tensor than before.

Certainly, this appears to be a misinterpretation of the initial value
problem. Firstly, the author specifies the values of the metric (and
presumably those of the extrinsic curvature) on $\Sigma$ based on the metric
across the entire manifold, without any mention of the stress-tensor. This
suggests, to me, an assumption that they are dealing with the vacuum case.
Utilizing the Choquet-Bruhat theorem Choquet , one can establish that, for the
vacuum Einstein equation, there exists only one extension of the metric, up to
diffeomorphisms (the mathematical ones), to the entire manifold $\mathcal{M}$.

This contradicts the argument presented in Curiel , as their
”operation”—applying $\phi^{*}$ solely to the metric—yields another metric
that is distinct and not related via diffeomorphisms to the original one. This
contradicts the uniqueness guaranteed by the Choquet-Bruhat theorem Choquet
(see also Landsman ; Isenberg ). In other words, the newly constructed metric
in this manner is not a solution to Einstein’s vacuum equation with the fixed
initial conditions on $\Sigma$ provided by the original metric.

Once again, it seems that the author is dealing with two different problems,
and consequently, the distinct solutions correspond to two different physical
situations, avoiding indeterminism. (See also Arledge  for a critical review
of Curiel ).

Another paper that approaches the Hole Argument in an unconventional manner is
Weatherall  (see Pooley-Read  for a critical review of it). In this paper, the
author explicitly utilizes the Earman-Norton trivial definition of the Hole
Argument, implicitly assuming, although not realizing it, Gauge Covariance.
The author then posits that isometric pseudo-Riemann manifolds are empirically
equivalent, meaning that active diffeomorphisms (in the mathematical sense,
not in the sense assumed in Weatherall ) provide the same physical reality.
However, this argument is presented without employing a relationalism
viewpoint to support it, indicating to me that the author may lack some depth
in understanding the subject, particularly in relation to Einstein’s
resolution of the problem.

However, the author goes beyond and employs the same unusual ”mathematical”
reasoning, as explained earlier: applying $\phi^{*}$ to the metric but not to
the manifold. The author compares $(\mathcal{M},\mathfrak{g})$ with
$(\mathcal{M},\phi^{*}\mathfrak{g})$ at the same point $p$, which the author
considers an ”active” diffeomorphism in contrast to the mathematical
definition (the author also uses the term ”push-forward” to describe the pull-
back operation and employs various incorrect mathematical concepts).

Once again, the use of the Hole seems to be a mere instrument to express that
the diffeomorphism $\phi$ is not the identity, devoid of any real connection
to the actual problem concerning General Covariance.

Regardless, the author concludes that in this case,
$(\mathcal{M},\mathfrak{g})$ and $(\mathcal{M},\phi^{*}\mathfrak{g})$ are not
equivalent. This conclusion, however, seems trivial, as using this approach,
the Lorentz manifolds $(\mathcal{M},\mathfrak{g})$ and
$(\mathcal{M},\phi^{*}\mathfrak{g})$ stem from different initial values on a
Cauchy surface, as I have already explained in the context of Curiel .
Therefore, this viewpoint appears to be disconnected from the real Hole
Argument, which necessitates a genuine active diffeomorphism to determine
whether one adopts a substantival viewpoint and accepts indeterminism or
rejects indeterminism by choosing the relationalist option, as Einstein did.

Finally, in my opinion, more metaphysical arguments like the ones presented in
Fletcher , discussed critically in Arledge , seem to obscure the real debate
about the Hole Argument. They do not contribute to visualizing its importance
in the context of the ongoing debate about substantivalism vs. relationalism
in the physical interpretation of the world.

##  VI Conclusions

In the course of this essay, I have observed the significance of the
Hole—defined as a set where the stress-energy tensor vanishes—particularly in
the context of General Covariance, which entails invariance with respect to
changes in coordinates without assuming Gauge Covariance. It is noteworthy
that, under the assumption of Gauge Covariance, the Hole ceases to play a
role, as the equations maintain invariance under diffeomorphisms.

Additionally, I have delved into Einstein’s original Hole Argument, revealing
its connection to coordinates as a manifestation of Hole Argument on the
manifold through hole-diffeomorphisms. From a physical standpoint, there is a
compelling argument for resolving the Hole Argument ”à la Einstein,” which
involves abandoning substantivalism and embracing a relationalistic
perspective where events are defined by coincidences. Einstein’s rejection of
the substantivalism vision of space-time is succinctly expressed in his words:
The requirement of General Covariance takes away from space and time the last
remnant of physical objectivity. Rovelli echoes this sentiment, stating: Not
anymore fields on space-time, just fields on fields.

Throughout the essay, I have emphasized the unconventional treatment of the
Hole Argument by some philosophers. It appears to me that they fail to
recognize the importance of the stress-energy tensor, as opposed to the
intrinsic curvatures of the manifold, which may not be preserved by certain
pull-backs. This is precisely why Einstein introduces passive hole
diffeomorphisms—to ensure that exactly the same equations are preserved.
Consequently, this results in different metrics satisfying the same field
equations. Initially, this led Einstein to reject General Covariance. However,
he later realized, by removing substantivalism, that these different metrics
define the same physical reality, allowing for the preservation of General
Covariance in the theory.

Hence, when this physical perspective is overlooked, the Hole Argument becomes
a mere artifact, used to implicitly support hidden Gauge Covariance, even if
it is not always consciously recognized. For instance, in Earman-Norton ,
authors use this argument to trivially demonstrate that the Gauge Covariant
field equations yield different solutions. Subsequently, they delve into a
philosophical-mathematical debate that, from a physical standpoint, seems
irrelevant.

I have endeavored to elaborate on this concept in Section V, expressing my
reservations about an overreliance on intricate geometric structures such as
fiber bundles, morphisms, categories, homotopy theory, spin networks,
$C^{*}$-algebras, rings, or functors within the realm of physics (see, for
instance, Ladyman ; Halvorson ; Rickle ; Yaghmaie ; Bain  and the references
therein). From my perspective as a cosmologist and mathematician, such an
approach risks losing sight of the fundamental physical essence of the
problem. To be more precise, I advocate for a return to the core of the issue:
”Less math, just physics.”

## Acknowledgments

This work is supported by the Spanish grant PID2021-123903NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

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