Glimmers of a post-geometric perspective

Federico Piazza

Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

<EMAIL_ADDRESS>
###### Abstract

Quantum gravitational effects could become important at low energy if the
wavefunction of the metric field fails to be peaked around a classical
configuration. We try to understand such deviations from classicality within
canonical quantum gravity by introducing a “fluid of observers” in the low
energy theory and defining a distance operator “at equal time” among them. We
find that, even in the presence of relevant fluctuations in the metric field,
a “locally flat” limit is recovered in the neighbourhood of each observer.
Deviations from classicality have no local consequences. However, at larger
separations the expectation value of the distance operator behaves differently
than a standard Riemannian distance. In particular, it is non-additive and
thus cannot be obtained by the integral of a differential line element. This
emerging “beyond Riemannian” geometry is a metric space similar to embedded
Riemannian manifolds equipped with _chord_ distances that “cut through” the
ambient space. We study deviations from flat space by looking at triangles in
the limit where their sizes go to zero. Beyond-Riemannian deviations with
respect to flat space are of the same order as standard Riemannian ones, but
qualitatively different. Some implications on the bulk picture of holographic
theories and black hole evaporation are discussed.

###### Contents

  1. 1 Introduction
    1. 1.1 Why are you doing this?
    2. 1.2 Constraints and range of applicability
  2. 2 Main ideas and plan of the paper
    1. 2.1 Observers
    2. 2.2 Time
    3. 2.3 Classical basis
    4. 2.4 Average and beyond-Riemannian distances
  3. 3 Setup
    1. 3.1 Fluid of observers
    2. 3.2 Time-field
    3. 3.3 Distance operator
  4. 4 Properties of distances on quantum states
    1. 4.1 Unitary gauge
    2. 4.2 Normal coordinates
    3. 4.3 Immediate neighborhood: local flatness
    4. 4.4 A little further away: average distances are non-additive
  5. 5 Sketches of beyond-Riemannian geometry
    1. 5.1 The triangle diagnostic
    2. 5.2 Beyond-Riemannian discrepancy
  6. 6 Discussion
    1. 6.1 Islands, fuzzballs and (the breakdown of) classicality
    2. 6.2 Beyond-Lorentzian?
  7. A The AdS boundary and the rest

##  1 Introduction

Spacetime is portrayed in general relativity as a (pseudo-) Riemannian
manifold. What are the limits of such a classical description? Quantum
gravitational effects must become important at least when curvature invariants
are of order Planck as, for example, in the regions approaching a singularity.
However, aspects of quantum gravity could be at play already in the low energy
theory and at macroscopic scales, if the wavefunction of the metric field
fails to be peaked around any classical configuration.

The classicality of the wavefunction is a known issue in quantum cosmology
(see e.g. [1]), where one usually assumes a strong correlation among the
canonical variables so that a classical metric can emerge as a coherent state
in the appropriate limit. However, such a correlation is not guaranteed to
exist nor to be preserved [2, 3, 4] during time evolution. Typical examples of
highly quantum states are those of a system undergoing tunneling. The use of
instanton methods for bubble nucleation [5] tends to conceal that between the
two classical configurations that the instanton interpolates spacetime is not
defined in any classical sense. In more general situations one could imagine a
tunneling that does not even reach a classical configuration eventually.
Depending on the characteristic time scales, rather than a pathology this
could simply represent a peculiar quantum behavior that cannot be captured by
classical evolution.

We speculate that such a non-classical behavior of the gravitational field is
a general feature of certain time dependent situations (see Sec. 1.2 below)
whose complete and accurate description should thus include deviations from
the classical geometrical picture of general relativity. In order to explore
and better understand these states, in the broad framework of canonical
quantum gravity we introduce a distance operator between observers and show
that its expectation value displays indeed “ _beyond-Riemannian_ ” properties.

In the rest of this section we provide motivations for this research. We
dedicate Sec. 2 to summarizing the main ideas and the plan of the rest of the
paper.

###  1.1 Why are you doing this?

Quantum gravity is best understood and under control in asymptotically anti-de
Sitter space, where one can benefit of the insights of the AdS/CFT
correspondence [6]. Classical spacetime emerges in this setup as the saddle
point approximation for quantities defined on the boundary. Notably, fine-
grained Von Neumann entropies can be calculated with Euclidean path integrals
already at the level of the low-energy effective theory [7, 8]. By using these
tools a number of breakthroughs on the black hole information paradox have
been obtained and the Page curve of Hawking radiation correctly reproduced
(see e.g. [9, 10, 11, 12, 13] and references therein). Different Euclidean
spacetimes [14] and different topologies in the bulk [15, 16] contribute to
these calculations. These sparkling Euclidean constructions, however, leave
the Lorentzian side of the picture relatively in the dark. How to reconstruct
the actual Lorentzian spacetime, say, as experienced by an observer falling
into the black hole? More generally, Euclidean saddles seem far more telling
and efficient computational tools than Lorentzian ones. How does the
information about some measurement happening inside the black hole eventually
evade to the outside without violating causality? We argue that some of these
struggles, along with other puzzles recently discussed in [17], are rooted in
insisting to stage black hole evaporation in a classical Lorentzian spacetime.

Another inspiration for the present work comes from a promising approach to
bulk reconstruction, that of integral geometry [18]. The idea is to recover a
geometry from the lengths of the geodesics living therein. In asymptotically
AdS spaces of arbitrary dimension one should consider the set of minimal
codimension two surfaces anchoring to the AdS boundary. The authors of [18]
concentrate on AdS3, where such surfaces are in fact geodesics. One can assign
a length to these curves by calculating the Von Neuman entropy of the
subtended spatial regions of the CFT and applying the Ryu Takayanagi formula
[19, 20]. Without all subtleties related to quantum gravity, and for
arbitrarily “highly quantum” states, the entropy of a boundary region is
always a well defined QFT quantity. The corresponding bulk lengths can thus be
taken as the fundamental building blocks to reconstruct the metric. But what
if such lengths are not supported by any Riemannian geometry? Clearly, a
metric tensor can be used to calculate all geodesic distances between points.
But randomly chosen “distances” cannot always be obtained as the integral of
some differential line element (see the rest of the paper for examples).

One last motivation for venturing beyond classical geometry is provided by
cosmology. After years of successful confirmations, data of increasing
precision have stopped converging so enthusiastically towards a single
$\Lambda$CDM model. The disagreement between the direct and indirect
measurements of the Hubble constant [21, 22] looks difficult to fix with just
minor adjustments and might be calling for a deeper reassessment of our basic
geometrical description of the universe. Among other (milder) inconsistencies
[23], one is particularly smelling of “geometrical anomaly”, namely the fact
that a positive spatial curvature seems preferred by some observables at
around $3\sigma$ and excluded by most others [24, 25]. But this is not the end
of the story, if the reader can tolerate any more philosophy at this point.
Accelerating expansion must be invoked  _twice_ within the standard geometric
framework, at disparate epochs and disparate energy scales. The whole
evolution of the universe is sandwiched in between two accelerations,
inflation and dark energy. Such a dissociated picture is by now very familiar
and accepted among professional cosmologists. Nonetheless, it remains a
curious circumstance, that some alternative geometric framework could perhaps
unify away.

###  1.2 Constraints and range of applicability

In this paper we suggest that a beyond-Riemannian geometry could indeed emerge
in quantum gravity by studying the expectation value of a _spatial_ distance
operator, defined on suitable “equal time surfaces”. Our approach is
kinematical because we consider from the onset states with relevant metric
fluctuations. The _dynamical_ question to address next would be: in which low
energy regimes (if any) could quantum gravity display such relevant deviations
from classicality? Here follow some qualitative/phenomenological
considerations.

Static or quasi static backgrounds provide the most severe experimental
constraints to any beyond-Riemannian ambition. The satellite-based global
positioning system (GPS) on earth together with the many earth-moon and solar
system tests include direct measurements of lengths and time intervals which
show exquisite agreement with classical general relativity. If macroscopic
quantum gravity effects are at play in this world, they better be related with
time evolution and die off in quasi-static situations. Successful
gravitational wave experiments—and our full understanding of the underlying
dynamics—extend this no-go zone to those time-dependent environments where on-
shell gravitons—or Weyl curvature invariants—account for most of the dynamics.
Time-varying Ricci curvatures are far less tested. One example is the interior
of a collapsing star and the spacetime around an evaporating black hole on
which the matter content of Hawking radiation back-reacts. The main other
notable example is cosmology. The expanding universe entails important time
variations of the Ricci curvature over Hubble scales. As argued before, this
is a realm where macroscopic quantum gravity effects, while not completely
excluded, may actually be longed for.

##  2 Main ideas and plan of the paper

One notorious difficulty in quantum gravity is that of defining gauge
invariant quantities. In practice, as we aim to define a distance operator, we
should first understand _between what and what_ we want to measure distances.
The problem can be posed on firmer grounds by introducing a set of dynamical
objects that can serve as a reference frame (see e.g. [26, 27]). A very
natural type of reference is a certain number of observers/test-particles
following their worldlines. In the next sections of the paper we will opt for
an improved and mathematically smoother version of this, namely a continuum
“fluid of observers” (Sec. 3.1). Here however we stick with the more intuitive
test-particles picture and convey the main ideas in this simplified language.

###  2.1 Observers

The test particles that we include in our system are actually real observers,
recognizable and distinguishable from each other. Instead of calling them e.g.
Charlie, Mick, Keith etc. we will name them by capital latin letters for
notational convenience. They look like points, or worldlines, just because we
deal with them at lowest order in derivatives in a low energy approximation
[28],

$S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}R\ -\ m_{A}\\!\int d\tau_{A}\ -\
m_{B}\\!\int d\tau_{B}\ +\ \dots\,,$ (2.1)

where ellipses stand for other observers, finite size effects on the
worldlines and matter fields not further specified. The canonical ADM
variables of this system are the spatial metric $h_{ij}(x^{k})$ and the
spatial positions111They are contained inside the worldline actions because
$d\tau_{A}=dt\sqrt{g_{\mu\nu}\frac{dx^{\mu}_{A}}{dt}\frac{dx^{\nu}_{A}}{dt}}\,,\quad{\rm
etc.}$ of the observers $x^{i}_{A}$, $x^{i}_{B}$, etc. on some given time
foliation, together with their conjugate momenta.

We aim for a low energy—and still non perturbative—description of this system.
We thus consider states that are functionals of the metric field and functions
of the positions,

$\Psi=\Psi\left[h_{ij}(x^{k}),\ x^{i}_{A},\ x^{i}_{B},\ \dots\right]\,.$ (2.2)

Now we know _between what and what_ we want to measure the distance—between
$A$ and $B$! But the problem of gauge invariance is not completely solved yet.
We still have to deal with time evolution. Gauge invariant distances imply the
full four-dimensional extremisation of the length of a curve between two
points/events. Even when $\Psi$ is peaked on a classical configuration, it
displays the system on a time slice that is chosen arbitrarily. We cannot use
such a spatial configuration to calculate any meaningful gauge invariant
distance.

###  2.2 Time

There is a trick for fixing the time slice in a gauge invariant way. Let us
introduce a dynamical scalar field $T(x^{k})$ (see Sec. 3.2 for more details)
and extract from the full state $\Psi$ a _conditional probability amplitude_
$\tilde{\Psi}$ for all other fields. The condition to satisfy here is that
$T(x^{k})$ be equal to some constant value $T_{0}$,

$\tilde{\Psi}\left[h_{ij}(x^{k}),\,\ x^{i}_{A},\ \dots;\ T_{0}\right]\ \equiv\
{\cal N}\Psi\left[h_{ij}(x^{k}),\,\ x^{i}_{A},\ \dots\,,\
T(x^{k})=T_{0}\right],\ $ (2.3)

The “state” $\tilde{\Psi}$ is a fair description of the system “at a given
time”. Thanks to the scalar field $T$ the surface at constant time is gauge
invariantly defined and we can measure distances on it. Other subtleties
related with the momentum and Hamiltonian constraints are dealt with in Sec.
3.3 and will not be discussed further in this summary.

###  2.3 Classical basis

The next problem is how to define a distance in the presence of relevant
departures from classicality. To this aim we can expand $\tilde{\Psi}$ on a
basis of “classical states” $|h_{ij}(x^{\kappa}),x^{i}_{A},\dots\rangle$ i.e.
on a basis of common eigenvectors of the commuting operators $\hat{h}_{ij}$,
$\hat{x}^{i}_{A}$, etc.222Of course, the wavefuntional (2.3) _is_ already the
scalar product of the state $|\tilde{\Psi}\rangle$ with the generic element of
this basis, i.e.  $\tilde{\Psi}[h_{ij}(x^{k}),\ x^{i}_{A},\ \dots;T_{0}]\
\equiv\ \langle
h_{ij}(x^{\kappa}),x^{i}_{A},\dots|\tilde{\Psi};T_{0}\rangle\,.$ On each
element of such a basis the distance between $A$ and $B$ is defined with
infinite accuracy. It depends on the metric configuration $h_{ij}(x^{k})$, on
the positions $x_{A}$ and $x_{B}$ and nothing else. One has to find the
geodesic between the points $x_{A}$ and $x_{B}$ in that classical geometry and
calculate its length. Let us call $d({A},{B})_{\\{h,x_{A},x_{B}\\}}$ the
result of this computation. We can define a distance operator between $A$ and
$B$ by its action on the classical basis,

$\hat{d}({A},{B})\
|h_{ij}(x^{\kappa}),x^{i}_{A},\dots\rangle=d({A},{B})_{\\{h,x_{A},x_{B}\\}}\
|h_{ij}(x^{\kappa}),x^{i}_{A},\dots\rangle\,.$ (2.4)

In other words, we define the distance operator by declaring that the elements
of the classical basis are its eigenvectors, of eigenvalues
$d({A},{B})_{\\{h,x_{A},x_{B}\\}}$.

###  2.4 Average and beyond-Riemannian distances

By linear superposition we can compute the expectation value of the distance
operator between $A$ and $B$ on any state $\tilde{\Psi}$,

$\langle\hat{d}({A},{B})\rangle=\int{\cal D}h\ d^{3}x_{A}\ d^{3}x_{B}\
d^{3}x_{C}\dots\ \left|\langle
h_{ij}(x^{\kappa}),x^{i}_{A},\dots|\tilde{\Psi};T_{0}\rangle\
\right|^{2}d({A},{B})_{\\{h,x_{A},x_{B}\\}}\,.$ (2.5)

For reasons of analyticity, however, the quantity that should replace the
classical notion of distance is probably a different expression, obtained by
taking the square root of the expectation value of the _square_ of the
distance operator.

$\overline{d(A,B)}\ \equiv\ \sqrt{\left\langle\hat{d}(A,B)^{2}\right\rangle},$
(2.6)

where, clearly,

$\left\langle\hat{d}({A},{B})^{2}\right\rangle=\int{\cal D}h\ d^{3}x_{A}\
d^{3}x_{B}\ d^{3}x_{C}\dots\ \left|\langle
h_{ij}(x^{\kappa}),x^{i}_{A},\dots|\tilde{\Psi};T_{0}\rangle\
\right|^{2}\left(d({A},{B})_{\\{h,x_{A},x_{B}\\}}\right)^{2}\,.$ (2.7)

As we need a short name for $\overline{d(A,B)}$ we will call it _beyond-
Riemannian distance_ , or just _BR-distance_ , not to be confused the average
of the distance operator defined in (2.5). One main message of this paper is
that BR-distances are not additive and thus cannot be expressed as integrals
of some “average differential line element”—although they can in the limit of
small separation. It is quite straightforward to see that averaging over
different geometries can produce this effect, and the two examples discussed
at the end of Sec. 4 might satisfy the immediate curiosity of the reader.

We argue in Sec. 4.3, on the other hand, that the BR-distance respects a
_locally flat_ limit in the vicinity of each observer and bears similarity
with the chord distance of embedded manifolds. In Sec. 5 we attempt to
characterize intrinsically the behavior of such distances by considering
triangles of small sizes in the standard Riemannian case first and then in the
beyond-Riemannian one.

Finally, in App. A we will make a last use of the discrete set of observers
introduced in this section by setting them in the bulk of an asymptotically
AdS space.

##  3 Setup

The purpose of this section is to introduce a distance operator $\hat{d}$
between spacetime points. The broad framework here is that of canonical
quantum gravity333No introduction nor self-contained review of canonical
quantum gravity and ADM formalism will be attempted here. The unbeaten
reference on the subject is still probably [29] where even identifying a point
in space is notoriously subtle. So the idea is to introduce a bunch of
observers. Our worldlines of observers (2.1) help the intuition but are
problematic in several respects. Among others, their back reaction on the
gravitational field is singular and unavoidable.

###  3.1 Fluid of observers

We can smooth out this difficulty by considering a _continuum_ or “a fluid” of
free falling observers, each one labeled by the three numbers, or “internal
coordinates” $X^{I}$, $I=1,2,3$. This suggests the use of the effective field
theory description of a non-relativistic fluid [30, 31]. In this formalism the
quantities $X^{I}$ that label the fluid particles/observers are promoted to
fields. To lowest order in derivatives we can write a Lagrangian for this
system in terms of the spacetime scalar

$B^{IJ}=g^{\mu\nu}\partial_{\mu}X^{I}\partial_{\nu}X^{J}\,.$ (3.1)

The system is thus described by an action of the type

$S\ =\ \frac{1}{16\pi G}\int d^{4}x\sqrt{-g}R\ -\ \mu^{4}\int
d^{4}x\sqrt{-g}\sqrt{\det B^{IJ}}\ +\ S_{m}[\Phi]\ +\ \dots\,,$ (3.2)

where $\mu$ is a coupling with dimensions of mass (we adopt conventions where
the fields $X^{I}$ have dimensions of length) and $\Phi$ collectively denotes
all matter fields. The energy momentum tensor for the fields $X^{I}$ is that
of a perfect fluid with zero pressure, with the fluid trajectory given by the
condition $X^{I}(x^{\mu})=const.$ [30, 31]. It follows that in the
corresponding classical system $X^{I}=const.$ are geodesics, as expected from
the test-particle version (2.1).444This “fluid of observers” formalism
naturally contains the test-particles one of Sec. 2. Without considering
fluctuations of the metric, and with a little help from the _classical basis_
of eigenstates to be introduced momentarily, the state functional in the fluid
formalism is $\Psi[X^{I}(x^{k})]\equiv\langle X^{I}(x^{k})|\Psi\rangle$. Upon
inversion, the basis of states $|X^{I}(x^{k})\rangle$ admits the equivalent
notation $\simeq|x^{k}(X^{I})\rangle$. Inside the fluid we can focus on a
number of observers of interest, e.g.
$A=\\{X^{1}\\!=1,X^{2}\\!=0,X^{3}\\!=0\\},B=\dots$ etc. Then e.g.
$\psi(x^{k}_{A})\equiv\langle x^{k}(\\{1,0,0\\})|\Psi\rangle$ is the
wavefunction of the observer $A$.

The canonical field operators of (3.2), denoted by a hat symbol, are the
spatial metric $\hat{h}_{ij}(x^{k})$, the matter fields $\hat{\Phi}(x^{k})$
and the observers’ fields $\hat{X}^{I}(x^{k})$. The wavefunction for this
system, before imposing any constraint, is thus a functional of these fields,

$\Psi=\Psi[h_{ij}(x^{k}),\,\Phi(x^{k}),\,X^{I}(x^{k})]\,.$ (3.3)

The purpose of this section is that of introducing a “distance along the same
time slice” operator between our observers. In order to make the argument as
sharp as possible, we need to discuss two main assumptions.

  * •

Assumption 1 In order for the fields $X^{I}$ to label observers, we need their
gradients to be space-like vectors. So we will introduce the three scalar
operators $g^{\mu\nu}\partial_{\mu}X^{I}\partial_{\nu}X^{I}$ (no summation
over $I$ here) and make the very strong assumption that _all possible values
that these operators can assume when evaluated on $\Psi$ be strictly
positive_—which in our mostly-positive convention for the metric means that
the gradients are spacelike. Such gradients should also never become parallel
to each other. To this purpose, we can introduce a “vector product”
operator555This quantity is parallel to the fluid velocity.,

$J^{u}=\epsilon^{\mu\alpha\beta\gamma}\epsilon_{IJK}\partial_{\alpha}X^{I}\partial_{\beta}X^{J}\partial_{\gamma}X^{K},$
(3.4)

and ask that it be strictly non zero when evaluated on $\Psi$.

###  3.2 Time-field

Next, we would like to limit our discussion to states that represent “equal
time slices”, in order to avoid the further level of complication related with
time evolution in quantum gravity. We thus assume that there is a very weakly
coupled scalar field $T$ among the matter fields $\Phi$ and we make another
strong assumption, namely that

  * •

Assumption 2 the possible values that the operator666In the canonical
formalism and for a canonical scalar field $T$, up to operator ordering
issues,
$g^{\mu\nu}\partial_{\mu}T\partial_{\nu}T=N\left(-\frac{\hat{\pi}_{T}^{2}}{\sqrt{\hat{h}}}+\hat{h}^{ij}\partial_{i}\hat{T}\partial_{j}\hat{T}\right),$
where $\hat{\pi}_{T}$ is the conjugate momentum to $\hat{T}$ and $N$ is the
lapse function. $g^{\mu\nu}\partial_{\mu}T\partial_{\nu}T$ takes on the state
$\Psi$ are _strictly negative_.

In other words, the gradient of $T$ is timelike and $T$ can be effectively
used as a “clock”. This assumption is strong but is nevertheless probably
implicit in all models of eternal inflation, although somewhat hidden by the
semiclassical treatement. Moreover, assumptions 1 and 2 can be weakened by
restricting them to some slice of interest within the configuration space, as
we discuss in what follows.

For systems without boundaries there is no time evolution, only one state
$\Psi$ satisfying the Hamiltonian constraint of the theory (or Wheeler-DeWitt
equation [29]). However, the presence of the field $T$ can allow us to
interpret the correlations between $T$ and the other fields as a time
evolution. Let us denote all matter fields _except_ $T$ with the lowercase
greek letter $\phi$. We can evaluate the state $\Psi$ on a section of the
configuration space characterized by some constant value of $T$,
$T(x^{k})=T_{0}$.

$\tilde{\Psi}[h_{ij}(x^{k}),\,\phi(x^{k}),\,X^{I}(x^{k});T_{0}]\ \equiv\ {\cal
N}\Psi[h_{ij}(x^{k}),\,\phi(x^{k}),\,X^{I}(x^{k}),\,T(x^{k})=T_{0}],\ $ (3.5)

where ${\cal N}$ is a normalization constant. This defines a “conditional
probability amplitude” to find the remaining fields in some given
configuration given that $T(x^{k})=T_{0}$. Equivalently, this is the relative
state _à la_ Everett [32] for the rest of the system when the subsystem $T$
finds itself in the “state $T_{0}$”. If there is sufficient correlation, or
entanglement, between $T$ and the rest of the system, a smooth change of the
value of $T_{0}$ will produce a continuum of different relative states
$\tilde{\Psi}$, thus describing a “time evolution” for the remaining fields.
The relative state $\tilde{\Psi}$ is probably the best we can do to picture
the system “at a given time” and thus to make the notion of “spatial distance
on a given time slice” gauge invariant.

This trick of conditional probability can also be used to weaken some
statements, such as the assumptions 1 and 2 above, and restrict them to some
“time slice”. Basically, these statements can be valid at the level of
conditional probability—given that $T(x^{k})=T_{0}$—without requiring that
they be valid within the entire state $\Psi$.

###  3.3 Distance operator

We can now attempt a formal definition of the distance operator. The canonical
fields $\hat{h}_{ij}(x^{k}),\hat{\phi}(x^{k})$ and $\hat{X}^{I}(x^{k})$
commute with each other and thus share a basis of common eigenvectors, that we
denote concisely by $|h,\phi,X^{I}\rangle$,

$\displaystyle\hat{h}_{ij}(x^{k})|h,\phi,X^{I}\rangle=h_{ij}(x^{k})|h,\phi,X^{I}\rangle,\quad\hat{X}^{I}(x^{k})|h,\phi,X^{I}\rangle=X^{I}(x^{k})|h,\phi,X^{I}\rangle\,,\
\ {\rm etc.}$ (3.6)

On the LHS of these equations the ket is acted upon by the quantum operator
while on the RHS it is simply multiplied by the corresponding eigenvalue. The
meaning of the states $|h,\phi,X_{I}\rangle$ is quite straightforward. We call
these vectors a _classical basis_ because they represent classical _spatial_
geometries with perfectly defined field configurations and localized
observers. Note however that these states have infinite uncertainties on the
conjugate momenta, and are thus very bad approximations of classical space
_times_.

As said, a triplet of values for the fields $X^{I}$ identifies a given
observer. However, on the eigenbasis $|h,\phi,X^{I}\rangle$, they also
entirely define a point in space. Let us indicate such triplets of values by
$\vec{X}$, $\vec{Y}$ etc. The quantum operator $\hat{d}(\vec{X},\vec{Y})$
“distance between observer $\vec{X}$ and observer $\vec{Y}$” (what would
correspond in the discrete model (2.1) to a distance operator between, say,
observer $B$ and observer $C$) is straightforwardly defined on this basis,
although in the usual implicit manner. One has to find the spatial geodesic
between the corresponding points for the geometry $h_{ij}(x^{k})$ and then
calculate its length by integrating the differential line element. The outcome
of this operation has no uncertainty on the classical basis and thus
$|h,\phi,X^{I}\rangle$ must be an eigenvector of $\hat{d}(\vec{X},\vec{Y})$,

$\hat{d}(\vec{X},\vec{Y})|h,\phi,X^{I}\rangle=d(\vec{X},\vec{Y})_{\\{h,X^{I}\\}}|h,\phi,X^{I}\rangle\,.$
(3.7)

The notation above wants to emphasize that the eigenvalue of the distance
operator depends only on the geometry and on the position of the observers,
not on the other fields.

Before discussing the distance operator any further we should attempt to
clarify the relation between the basis $|h,\phi,X^{I}\rangle$ and the actual
Hilbert space of the theory. The point is the possible redundancy of this
basis due to the constraints that the physical vectors should satisfy.

Let us first address the redundancy of $|h,\phi,X^{I}\rangle$ due to the
_momentum constraint_. Because of spatial gauge invariance, infinitely many
classical field configurations correspond to the same physical state. For each
class of gauge-equivalent metrics we should thus pick a representative, that
we denote by $\tilde{h}$ and restrict our basis accordingly,

$|h,\phi,X_{I}\rangle\ \longrightarrow\ |\tilde{h},\phi,X_{I}\rangle.$ (3.8)

In the case of a closed universe, $\tilde{h}$ really represents some three-
dimensional geometry, rather than a tensor field [29]. When taking quantum
averages, we should sum over $\tilde{h}$ and not over $h$. Apart from
(possibly insurmountable) mathematical difficulties, the physical picture is
rather clear here. The state $|\tilde{h},\phi,X^{I}\rangle$ still
represents—this time non-redundantly—some definite three-dimensional geometry,
with field configurations and observer positions perfectly specified in it.
And thus it is still an eigenvector of $\hat{d}(\vec{X},\vec{Y})$.

The Hamiltonian constraint is more subtle because it deals with possibly
different time foliations. The point is that some given three-dimensional
classical configuration most of the time will fail to satisfy the Hamiltonian
constraint. The typical example of an unphysical state is that of a
spherically symmetric Newtonian potential that is not centered around the mass
that is supposed to source it. Of course we could restrict our considerations
to the _kinematic_ Hilbert space (the one before imposing some of the
constraints) hoping that they could still be of some interest. The very
optimistic attitude suggested here is that a solution to this problem is
already hinted by the time-field $T$ that we have introduced before. The
fields denoted by $\phi$ are already a subset of all matter fields and we
assume that their configuration can be chosen arbitrarily within the Hilbert
space of the theory. This means that there will be other auxiliary fields
beyond $T$. The Hamiltonian constraint can then be used to solve for these
remaining fields: $T$ and whatever other degrees of freedom we need to
introduce in order to complete this operation.

With this picture in mind, the basis that we denoted with
$|\tilde{h},\phi,X^{I}\rangle$ does not need to be further restricted. We can
thus optimistically write a resolution of the identity with it,

$\mathbb{1}\ =\ \sum_{I}\int{\cal D}\tilde{h}\,{\cal D}\phi\,{\cal D}X^{I}\,\
|\tilde{h},\phi,X^{I}\rangle\langle\tilde{h},\phi,X^{I}|\,.$ (3.9)

At the same time, the constraints that we have just discussed should apply to
the physical states of the theory,

$\tilde{\Psi}=\tilde{\Psi}[\tilde{h}_{ij}(x^{k}),\,\phi(x^{k})\,X^{I}(x^{k});T_{0}]\,.$
(3.10)

We can formally express the expectation value of the any power of the distance
operator on the state $\tilde{\Psi}$ as

$\left\langle\hat{d}(\vec{X},\vec{Y})^{n}\right\rangle=\sum_{I}\int{\cal
D}\tilde{h}\,{\cal D}\phi\,{\cal
D}X^{I}\left|\langle\tilde{\Psi}|\tilde{h},\phi,X^{I}\rangle\right|^{2}\
\left(d(\vec{X},\vec{Y})_{\\{\tilde{h},X^{I}\\}}\right)^{n}\,.$ (3.11)

One could simply consider the average of the distance operator
$\langle\hat{d}\rangle$ and find that it has interesting non-additive
properties. However this quantity does not have a very smooth local limit. The
expression that best generalizes standard classical distances, we argue, is
what we call in short _beyond-Riemannian distance_ (_BR-distance_), defined as

$\overline{d(\vec{X},\vec{Y})}\equiv\sqrt{\left\langle\hat{d}(\vec{X},\vec{Y})^{2}\right\rangle}\,.$
(3.12)

##  4 Properties of distances on quantum states

In this section we explore the properties of the expectation value defined
above and show that it can be different than the corresponding classical
quantity in a standard three-dimensional manifold. In order to perceive
deviations from the classical behavior we need to consider “highly quantum
states” $\tilde{\Psi}$ on which the fluctuations of the distance operator are
relevant.

The state (3.10) clearly contains possible uncertainties in both the metric
field and the observers’ positions. One naive but legitimate question is then
the following. Can we distinguish between $(i)$ _states where the metric is
highly fluctuating_ from $(ii)$ _states where the positions of the observers
are highly fluctuating_? As we are going to see, there is no clear separation
between these two situations, which reiterates the known statement that
observables are relational in quantum gravity. However, one of the two
limiting situations $(ii)$ is clearly well defined. Imagine that
$\tilde{\Psi}$ is extremely peaked on some geometry $\tilde{h}$ but highly
uncertain about the observers’ configuration $X^{I}(x^{k})$. This would
clearly represent situation $(ii)$, where the indeterminacy of the distance
operator is entirely due to the spreading of the observers’ positions. The
opposite situation, on the other hand, does not appear to be well defined. In
other words, there seems to be no sense in which the metric fluctuates but the
observers are kept “fixed”.

Let us look more deeply into these issues with the aid of specific coordinate
choices.

###  4.1 Unitary gauge

In this gauge we choose the spatial coordinates $x^{i}$ in such a way that
they coincide with the values of the fields $X^{I}$,

$X^{I}=x^{i}\ ;\qquad\frac{\partial X^{I}}{\partial
x^{i}}=\delta^{I}_{i};\qquad B^{IJ}=g^{ij}=h^{ij}-\frac{N^{i}N^{j}}{N^{2}},$
(4.1)

where $g^{ij}$ are the spatial components of the inverse metric $g^{\mu\nu}$,
$h^{ij}$ is the inverse of the spatial metric $h_{ij}$ and $N^{i}$ and $N$ are
the standard shift and lapse functions of the ADM formalism. The first and
last equations contain a slight abuse of notation, as the indexes $I$ and $i$
are identified. Note that by using this gauge directly inside the action (3.2)
the invariance under spatial diffeomorphisms is lost. There are terms in the
action where the metric appears without derivatives, giving rise to one of the
many variants of massive gravity. The degrees of freedom previously contained
in the fields $X^{I}$ have migrated into the metric. It is customary to call
_unitary gauge_ such gauge choices in which a certain number of matter degrees
of freedom are eaten by the gauge fields—the metric in this case (see e.g.
[33] and references therein).

Within this gauge the state of the system does not depend any longer on the
configuration of the fields $X^{I}$,

$\tilde{\Psi}_{\rm U}=\tilde{\Psi}_{\rm
U}[h_{IJ}(X^{K}),\,\phi(X^{K});T_{0}]\,,$ (4.2)

where the index U stands for “unitary gauge”. These coordinates have physical
meaning because they are attached to physical observers. The momentum
constraint is automatically satisfied in this gauge without the need to select
a representative of the metric, $\tilde{h}$. The reason is that two metric
fields $h_{IJ}(X^{I})$ and $h^{\prime}_{IJ}(X^{I})$ which differ by a
coordinate transformation still represent the same geometry but with the
observers distributed differently. Therefore, they represent two different
states. Equivalently, the two corresponding configurations can give distinct
probability amplitudes when evaluated within $\tilde{\Psi}$.

In this gauge the position of the observers does not seem to fluctuate.
However, the situation $(ii)$ depicted at the beginning of this section—i.e. “
_only the observers fluctuate_ ”—is easily recognizable. This will be some
$\tilde{\Psi}$ whose entire support is on a set of metrics $h_{ij}$ that are
all equivalent to one single metric up to gauge transformations.

Let’s make an example also for later use. On a two-dimensional sphere we
smoothly rotate the two poles with respect to the equator of an angle
$\epsilon$. This is achieved by a coordinate transformation

$\Theta\rightarrow\Theta\,,\qquad\varPhi\rightarrow\varPhi+\epsilon
f(\Theta)\,,$ (4.3)

Figure 1: On the same geometry (a sphere), red dashed and thick blue lines
represent $\varPhi=const.$ observers for $\epsilon=0$ and $\epsilon=0.4$
respectively. The choice of function here is
$f(\Theta)=1-e^{-\tan^{2}(\Theta-\pi/2)}$

where $\Theta\in[0,\pi]$ and $\varPhi\in[0,2\pi]$ and the smooth function
$f(\Theta)$ is such that $f(0)=f(\pi)=1$ and has a minimum at $f(\pi/2)=0$.
For instance, $f(\Theta)=1-e^{-\tan^{2}(\Theta-\pi/2)}$ will do the job (see
Fig. 1). The metric in the new coordinates reads

$h_{IJ}^{(\epsilon)}\ =\
\begin{pmatrix}1+\epsilon^{2}{f^{\prime}(\Theta)}^{2}\sin^{2}\Theta&-\epsilon
f^{\prime}(\Theta)\sin^{2}\Theta\\\ -\epsilon
f^{\prime}(\Theta)\sin^{2}\Theta&\sin^{2}\Theta\,\end{pmatrix}\,.$ (4.4)

We stress again that $\Theta$ and $\varPhi$ are the labels, the “names”, of
our distinguishable observers and thus different values of the parameter
$\epsilon$ represent different states. Say that
$|h^{(\epsilon)}(\Theta,\varPhi)\rangle_{\rm U}$ is the corresponding
eigenvector in the presence of some fixed matter fields’ configuration777We
will pay no attention to the remaining fields $\phi$ from now on and thus we
will omit them from the notation. The state

$|\tilde{\Psi}\rangle=\int d\epsilon\
\psi(\epsilon)|h^{(\epsilon)}(\Theta,\varPhi)\rangle_{\rm U},$ (4.5)

where $\psi(\epsilon)$ is some suitable probability amplitude, is entirely
supported on metrics that represent the same geometry. We can thus say that
the uncertainty of the corresponding distances in this state is due to the
uncertainty in the (relative) positions of the observers.

As already mentioned, however, there is no such thing as situation $(i)$ in
this framework, where “ _only the metric fluctuates_ ”. In this gauge one
could try to build a state $\tilde{\Psi}$ that has support on a continuous
class of metrics that are all “transverse” to one another, so that they are
not reachable one from another by means of a gauge transformation. However,
this automatically implies also a relational displacement among our observers.
The point is that even the value of the metric tensor in a point is meaningful
in this gauge, because it tells the distances between the observers that are
infinitesimally close to that point. In order to see this in more detail, let
us discuss another choice of coordinates.

###  4.2 Normal coordinates

Let us focus on the observer who sits at the origin of the coordinate system
in unitary gauge, $X^{I}=0$. Let us decompose the state $\tilde{\Psi}$ into
the eigenvectors of the metric field $|h_{IJ}(X^{K})\rangle_{\rm U}$. Every
such state represents a definite classical metric, which we can express in
Riemann normal coordinates around our chosen observer,

$h_{ij}(y^{k})=\delta_{ij}-\frac{1}{3}R_{ikjl}\ y^{k}y^{l}+\dots\,,$ (4.6)

where we have set $y^{k}=0$ to coincide with $X^{I}=0$. For any metric
$h_{IJ}(X^{I})$ on which $\tilde{\Psi}_{\rm U}$ has support, there is a
transformation law connecting Riemann normal coordinates and unitary gauge
coordinates,

$y^{k}=y^{k}(X^{I})\simeq A^{k}_{\ I}\,X^{I}+B^{k}_{\
IJ}\,X^{I}X^{J}+\dots\,.$ (4.7)

The coefficients $A$, $B$ etc. of this expansion depend on the functional form
of the metric in a neighborhood of the origin. In particular, $A^{k}_{\ I}$
depends just on the actual value $h_{IJ}(0)$, $A^{k}_{\ I}A^{k}_{\
J}=h_{IJ}(0)$. This transformation is defined up to a global rotation $R$
(e.g. $A^{k}_{\ I}\rightarrow R^{kl}A^{l}_{\ I}$ etc.) that leaves the leading
flat term in the metric (4.6) invariant. Before moving further it is
convenient to get rid of this ambiguity and fix a convention for such a
rotation. At linear order in the transformation (4.7) we can ask the
coordinate $y^{1}$ to be parallel to $X^{1}$. We are then left to fix the
rotation around such a common axis, for instance by insisting that the axis
$y^{2}$ belongs to the $X^{1}\\!-\\!X^{2}$ plane. All this corresponds to
choosing

$A^{1}_{\ 2}=A^{1}_{\ 3}=A^{2}_{\ 3}=0\,.$ (4.8)

By inverting (4.6) we can then express $X^{I}$ in normal coordinates. Our
“classical basis” of vectors which in unitary gauge are specified by the
metric configuration $h_{IJ}(X^{K})$, are now defined by the curvatures in the
origin and by the positions of the observers around the origin,

$|h_{IJ}(X^{I})\rangle_{\rm U}\ \simeq\
|X^{I}(y^{k}),R_{ikjl}(0),\dots\rangle_{\rm NC}\,,$ (4.9)

where the ellipsis stands for the tower of higher order curvature tensors
needed to expand the metric to an arbitrary order. By superposition, a generic
state is expressed in this gauge as a functional of the observers’ positions
and a function of the curvatures in the origin,

$\tilde{\Psi}_{\rm NC}=\tilde{\Psi}_{\rm
NC}[X^{I}(y^{k}),R_{ikjl}(0),\dots;T_{0}]\,.$ (4.10)

The normal coordinate (NC) gauge conveniently conveys the observer’s
viewpoint. We want to discuss in turn the dependence of this wavefunction on
the curvatures in the origin and on the observers positions.

###  4.3 Immediate neighborhood: local flatness

If we hope to make sense of the low energy theory that we are using it looks
reasonable to make a “low energy hypothesis” on the state $\tilde{\Psi}$, that
we formulate as follows.

  * •

Assumption 3 Scalar curvature invariant operators (both 3- and 4-dimensional)
will acquire a whole range of values within $\tilde{\Psi}$, depending on the
position at which they are evaluated and on the quantum fluctuations present
in $\tilde{\Psi}$. We require that, well below the cutoff of the theory, there
is a _maximal curvature scale_ for such values, above which the corresponding
probability rapidly drops to zero888Of course, quantum fluctuations are
expected to be present up to the UV but with an exponentially suppressed
distribution. The standard vacuum wavefunction for a scalar $\phi$ goes in
Fourier space as
$\Psi[\phi_{\vec{k}}]\sim\exp\left(-\frac{1}{2}\sum_{\vec{k}}k\
\phi_{\vec{k}}\phi_{-\vec{k}}\right).$ In order for effective field theory to
make sense, the dynamics should not affect such a “UV tail” and it should be
possible to characterize a generic low-energy state by the range of modes that
are populated _on top of_ this universal behavior..

If this hypothesis applies, the NC-gauge perspective suggests that _there is
always a neighborhood around each observer that looks locally flat._ This
“quantum version” of the equivalence principle is suggested by the fact that
being locally flat around one point is nothing more than a gauge condition.
The NC gauge is not necessarily convenient to quantize the system [34] but it
explicitly shows this point. Even if the gravitational wavefunction is
supported on a range of widely different metrics, all such metrics look flat
around each observer in this gauge, if we are granted an upper bound on the
possible values of the _curvature_. The one metric in the quantum ensemble
with the highest curvature will set the length within which the equivalence
principle applies.

Within the locally flat neighborhood of each observer we can still have a
large uncertainty in the positions of the nearby observers. Let us see this in
more detail.

On each element of the classical basis (4.9) the distance from the origin is
simply the Euclidean distance expressed in the $y^{k}$ coordinates. This means
that, on such basis,

$\displaystyle\hat{d}(0,\vec{Y})^{2}|X^{I}(y^{k}),\dots\rangle_{\rm NC}\ $
$\displaystyle=\ y^{k}(\vec{Y})\ y^{k}(\vec{Y})\
\,|X^{I}(y^{k}),\dots\rangle_{\rm NC}$ (4.11) $\displaystyle=\ A^{k}_{\
I}A^{k}_{\ J}Y^{I}Y^{J}\ |X^{I}(y^{k}),\dots\rangle_{\rm NC}+{\cal
O}(X)^{3}\,,$ (4.12)

where in the last line the matrix $A^{k}_{\ I}$ is the linear piece of the
coordinate transformation (4.7) relative to the configuration $X^{I}(y^{k})$
of that basis vector. In other words, $X^{I}(y^{k})$ can be considered as a
proxy for the matrices $A^{k}_{\ I}$, $B^{k}_{\ IJ}$ etc. that perform the
transformation $y^{k}\rightarrow X^{K}$. This observation justifies the change
of variable in the following functional integral, where we express the average
of the square distance of a nearby observer,

$\displaystyle\left\langle\hat{d}(0,\vec{Y})^{2}\right\rangle\ $
$\displaystyle\simeq\ \int{\cal D}[X^{I}]\left|\tilde{\Psi}_{\rm
NC}[X^{I}(y^{k})]\right|^{2}A^{k}_{\ I}A^{k}_{\ J}Y^{I}Y^{J}$ (4.13)
$\displaystyle\simeq\ \int{\cal D}[(A^{-1})^{\ I}_{k}]\left|\tilde{\Psi}_{\rm
NC}[(A^{-1})^{\ I}_{k}]\right|^{2}A^{k}_{\ I}A^{k}_{\ J}Y^{I}Y^{J}$ (4.14)
$\displaystyle=\ {\cal G}_{IJ}\ Y^{I}Y^{J}\,.$ (4.15)

So in the vicinity of each point the BR-distance of the nearby observers can
be smoothly expressed by an “average metric” ${\cal G}_{IJ}$,

$\overline{d(0,\vec{Y})}\ =\ \sqrt{{\cal G}_{IJ}\ Y^{I}Y^{J}}\ +\ {\cal
O}(Y)^{3}\,.$ (4.16)

This is the claimed local limit of BR-distances, which show no particular
anomaly at this point.

The uncertainty in the unitary gauge metric $h_{IJ}(0)$ thus translates, in
NC-gauge, into the uncertainty in the distances of the nearby observers. This
uncertainty grows with the distance itself and, in any reasonable scenario, we
shold assume that it is extremely small. So that when calculated, say, at the
typical curvature scale, it represents a small fraction of the curvature scale
itself.

If the fluctuations of the metric vanish at one point, then the uncertainty of
the nearby observers’ positions also vanishes at linear order. This is what
happens, for dynamical reasons, at the boundary of asymptotically AdS. In this
context a nice analogue NC gauge are Fefferman-Graham coordinates. See the
Appendix for more details.

###  4.4 A little further away: average distances are non-additive

As previously discussed, the average distance between two observers is an
implicit but rather well-defined calculation in this framework. Let us
summarize here the procedure by specializing the previous formulae to the
unitary gauge and by omitting the matter fields for notational convenience.

Our classical basis of vectors in unitary gauge is $|h_{ij}(X^{I})\rangle_{\rm
U}$. They represent classical field configurations with metric and observers
entirely defined. They are therefore eigenvectors of the distance operator,

$\hat{d}(\vec{X},\vec{Y})|h_{ij}(X^{I})\rangle_{\rm
U}=d(\vec{X},\vec{Y})_{\\{h\\}}|h_{ij}(X^{I})\rangle_{\rm U}.$ (4.17)

The eigenvalue $d(\vec{X},\vec{Y})_{\\{h\\}}$ depends on the metric because
given some metric one can find the geodesic connecting $\vec{X}$ and $\vec{Y}$
and then calculate its length. When the average of some power of the distance
operator is calculated on some generic vector,

$\left\langle\hat{d}(\vec{X},\vec{Y})^{n}\right\rangle=\int{\cal
D}h\,\left|\langle\tilde{\Psi}|h_{ij}(X^{I})\rangle_{\rm U}\right|^{2}\
\left(d(\vec{X},\vec{Y})_{\\{h\\}}\right)^{n}\,,$ (4.18)

this procedure should be done for every element of the basis on which
$|\tilde{\Psi}\rangle$ decomposes.

Something interesting happens if geodesics calculated for (slightly) different
basis vectors $|h_{ij}(X^{I})\rangle_{\rm U}$ inside the ensemble take (very)
different paths. Let us consider the following one-dimensional example.

#### Example 1

The state $\tilde{\Psi}$ is a superposition of only two vectors belonging to
the “classical basis”,

$|\tilde{\Psi}\rangle=\frac{1}{\sqrt{2}}\left(|\tilde{\Psi}_{1}\rangle+|\tilde{\Psi}_{2}\rangle\right)$
(4.19)

where the classical states $|\tilde{\Psi}_{1}\rangle$ and
$|\tilde{\Psi}_{2}\rangle$ are depicted in the figure below. Such a
superposition gives the following expectation values for the distances:

$\displaystyle\ |\tilde{\Psi}_{1}\rangle\ \
\qquad\quad\qquad\qquad\qquad\qquad\qquad|\tilde{\Psi}_{2}\rangle$

$\displaystyle\langle\hat{d}(A,B)\rangle$
$\displaystyle=\langle\hat{d}(B,C)\rangle=\langle\hat{d}(A,D)\rangle=\langle\hat{d}(C,D)\rangle=\ell\,,$
(4.20) $\displaystyle\langle\hat{d}(B,D)\rangle$ $\displaystyle=2\ell\,,$
(4.21) $\displaystyle\langle\hat{d}(A,C)\rangle$
$\displaystyle=2(\ell-\epsilon)\,,$ (4.22)

where $\ell=r\pi/2$, $r$ being the radius of the circle. To first order in
$\epsilon$ these are also the BR-distances between the observers
$\overline{d(A,B)}\simeq\langle\hat{d}(A,B)\rangle$ etc. No Riemannian one-
dimensional manifold can reproduce the above set of distances. The point is
simply that _quantum averaged_ Riemannian distances behave differently than
Riemannian distances. This anomaly is not due to the topology of the problem
(the fact that the circle is not simply connected) as our second example
shows.

#### Example 2

Let us go back to the example of Sec. 4.1 and chose the state as follows

$|\tilde{\Psi}\rangle=\frac{1}{\sqrt{2}}\left(|h^{(\epsilon)}(\Theta,\varPhi)\rangle_{\rm
U}+|h^{(-\epsilon)}(\Theta,\varPhi)\rangle_{\rm U}\right)\,$ (4.23)

(any smoother superposition would also do the job). We can evaluate the
eigenvalue of the distance operator on each classical state
$|h^{(\epsilon)}(\Theta,\varPhi)\rangle_{\rm U}$ by using the “haversine
formula” for distances on a sphere of unit radius. In standard coordinates it
reads

$d\left(\\{\theta_{1},\varphi_{1}\\},\\{\theta_{2},\varphi_{2}\\}\right)_{\\{\epsilon=0\\}}=2\arcsin\sqrt{\sin^{2}\frac{\theta_{1}-\theta_{2}}{2}\
+\
\sin(\theta_{1})\sin(\theta_{2})\sin^{2}\frac{\varphi_{1}-\varphi_{2}}{2}}.$
(4.24)

Then we should perform the coordinate transformation (4.3),

$\displaystyle\Theta$ $\displaystyle=\theta$ (4.25) $\displaystyle\varPhi$
$\displaystyle=\varphi+\epsilon f(\theta)$ (4.26)
$\displaystyle=\varphi+\epsilon\left(1-e^{-\tan^{2}(\theta-\pi/2)}\right)\,,$
(4.27)

where a specific choice for $f$ has been made, in order to obtain
$d(\\{\Theta_{1},\varPhi_{1}\\},\\{\Theta_{2},\varPhi_{2}\\})_{\\{\epsilon\\}}$.999Notice
that in this two-dimensional example we are pedantically indicating the
observer $\vec{X}$ with a doublet of components $\\{\Theta,\varPhi\\}$. The
outcome is a rather complicate expression which however can be simplified in
some cases.

Figure 2: A Lorentzian manifold embedded into a higher dimensional one. The
intrinsic lightcone structure is represented by dashed blue lines. Null
geodesics in the ambient space (the red line) can connect events that are at
spacelike separation.

  1. 1.

The distance between two points at the same longitude, say
$\varPhi_{1}=\varPhi_{2}=0$ and of opposite latitude (i.e. one at distance
$\lambda$ from the north pole and the other at the same distance $\lambda$
from the south pole), $\Theta_{1}=\lambda,\Theta_{2}=\pi-\lambda$, is simply

$d(\\{\lambda,0\\},\\{\pi-\lambda,0\\})_{\\{\epsilon\\}}=\pi-2\lambda\,.$
(4.28)

The result is independent of $\epsilon$ because of the parity of the function
$f$ around the equator. Geodesics between points at the same longitude simply
follow a meridian.

  2. 2.

Let us now consider instead the distance between one of the two points above,
$\\{\lambda,0\\}$, and a point on the equator at some longitude $\varPhi$,
$\\{\pi/2,\varPhi\\}$. Such a distance should now depend on $\epsilon$ because
the point near the pole “has rotated” while the point on the equator has not.
If the first point is very close to the pole we can assume that the function
$f$ is very close to one and the relative rotation between the pole and the
equator is $\epsilon$. In this limit we have

$d(\\{\lambda,0\\},\\{\pi/2,\varPhi\\})_{\\{\epsilon\\}}=\frac{\pi}{2}-\lambda\cos(\varPhi+\epsilon)+{\cal
O}(\lambda^{2})\,.$ (4.29)

We can now focus on 5 specific observers, $A,B,C,D,F$ as defined and displayed
in Fig. 2. We have all the instruments to calculate the average squared
distances among them. For instance,

$\left\langle\hat{d}(A,B)^{2}\right\rangle=\frac{1}{2}\left(d(A,B)^{2}_{\\{+\epsilon\\}}+d(A,B)^{2}_{\\{-\epsilon\\}}\right)\quad{\rm
etc.}\,,$ (4.30)

as follows from the general formula (4.18). Let us now compute BR-distances
$\overline{d(A,B)}=\sqrt{\langle\hat{d}(A,B)^{2}\rangle}$ etc. By using (4.28)
and (4.29) we obtain

$\displaystyle\overline{d(A,B)}$
$\displaystyle=\overline{d(A,C)}=\frac{\pi}{2}-\lambda\cos\epsilon+{\cal
O}(\lambda^{2})\,,$ (4.31) $\displaystyle\overline{d(A,D)}$
$\displaystyle=\overline{d(A,F)}=\varPhi\,,\quad\overline{d(D,F)}=2\Phi$
(4.32) $\displaystyle\overline{d(D,C)}$
$\displaystyle=\overline{d(C,F)}=\overline{d(D,B)}=\overline{d(B,F)}=\frac{\pi}{2}-\lambda\cos\varPhi\,\cos\epsilon+{\cal
O}(\lambda^{2})\,,$ (4.33) $\displaystyle\overline{d(B,C)}$
$\displaystyle=\pi-2\lambda\,.$ (4.34)

Again, the direct average distances $\langle\hat{d}(A,B)\rangle$ would give
the same results at this order in $\lambda$. The most striking aspect of this
set of distances is their non-additivity along the meridian,

$\overline{d(A,B)}+\overline{d(A,C)}>\overline{d(B,C)}$ (4.35)

for $\epsilon\neq 0$. The point is that the geodesics between $B$ and $C$ do
not pass through $A$ for any of the two states in the classical basis, they
cut through a shorter path. The observers $D$ and $F$ are “control observers”,
symmetrical with respect to the meridian. Distances along the equator are not
anomalous in this example.

In summary, the observers in this example describe a two dimensional manifold
that we can label with the coordinates $\Theta$ and $\varPhi$. There is a
distance naturally defined on this manifold, $\overline{d}$. Equipped with
such a distance, the manifold is a metric space. Such a distance resembles a
standard Riemannian distance at short separations. However it behaves
anomalously at larger separations. In particular it looses the additivity
property that Riemannian distances enjoy along any geodesic. It is a _beyond-
Riemannian_ distance.

One could object that these examples are too stupid. After all they deal with
superpositions of identical geometries on top of which we are just moving the
observers around. As already emphasized, however, the basis states that we are
superimposing are two honestly different states of the theory. Moreover,
fluctuations in the geometry would always imply also fluctuations in the
relative positions of the observers as we have seen in Sec. 4.2. At any rate,
it would be easy to slightly perturb the two states that we are superimposing
in such a way that they would also differ in their geometries, and still
maintain the results above.

##  5 Sketches of beyond-Riemannian geometry

We now attempt to sketch the general geometrical properties of BR-distances. A
more throughout study of this geometry will be given elsewhere [35]. The
geometrical description that we are after has to recover some notion of “local
flatness” around each point. A natural generalization of Riemannian distances
with this feature is suggested by _chord_ distances in embedded manifolds.
Consider a two dimensional sphere embedded in the standard three dimensional
euclidean space as in Fig. 3 and say that we define distances on the sphere by
“cutting through” the ambient space. This distance is not additive and thus
cannot be obtained as an integral of a line element. Nevertheless, such a
metric space has the desired locally flat limit to satisfy the equivalence
principle—eq. (4.16), more precisely. Corrections to flat space geometry are
of the same order as the standard Riemannian ones (${\cal O}(d^{2}/r^{2})$
where $d$ is the size of the region considered and $r$ the radius of the
sphere), but qualitatively different. Most notably, three aligned points
(along the same geodesic) fail to saturate the triangle inequality because
chord distances are not additive.

Figure 3: 3 Chord distances (in blue), as opposed to arc distances (in black),
are non-additive. Although $A$, $B$ and $C$ are aligned on the sphere,
$d(A,B)+d(B,C)\neq d(A,C)$. 3 What is the length of $AB$ given $L$ and the
angle $\alpha$? With this triangle diagnostic we measure discrepancies from
flat space induced by Riemannian and beyond Riemannian geometry. In both cases
the corrections are of order $L^{2}$ with respect to flat space. However, the
dependence on the angle $\alpha$ is different in the two cases [eq. (5.1) vs.
eq. (5.3)]

###  5.1 The triangle diagnostic

Of course the choice of distances on a manifold can be highly arbitrary.
Nevertheless their local behavior depends on a finite number of parameters. In
order to better understand this we can use triangles as a diagnostic while
taking the limit for the size of the triangle that goes to zero. From some
point $O$ we send two geodesics of equal length $L$ to the two points $A$ and
$B$ as in Fig. 3. The two curves span an angle $\alpha$ between them. In a
flat Euclidean space the distance between $A$ and $B$ is $d_{\rm
Euclidean}=L\sqrt{2(1-\cos\alpha)}$.

####  5.1.1 Standard-Riemannian discrepancies

But in the presence of curvature the distance gets corrected by a fractional
amount

$\delta_{R}=-\frac{R\,L^{2}}{24}(1+\cos\alpha)+{\cal O}(L^{4})\,,$ (5.1)

where $R$ is the Ricci scalar of the two dimensional surface subtended by the
two vectors $OA$ and $OB$ and we have defined

$\delta_{R}=\frac{d_{\rm Riemannian}-d_{\rm Euclidean}}{d_{\rm Euclidean}}\,.$
(5.2)

As noted, the Riemannian discrepancies vanish at $\alpha=\pi$, when the three
points $A$, $O$ and $B$ are aligned. This is when the problem becomes one-
dimensional, and one-dimensional manifolds have no intrinsic curvature.
Whenever we reiterate that “Riemannian distances are additive” in this paper
we mean, more precisely, that $\delta_{R}=0$ when $\alpha=\pi$.

Another feature of standard Riemannian discrepancies is isotropy on any given
plane. Once the plane subtended by the two vectors $OA$ and $OB$ is assigned
within a manifold of arbitrary dimension, the result in (5.1) only depends on
the angle $\alpha$ between them and not on their overall orientation.

###  5.2 Beyond-Riemannian discrepancy

Let us now look at what type of discrepancies we can expect in a beyond-
Riemannian scenario. The triangle construction can still be made rigorously in
this extended framework but the details will be given elsewhere [35]. For now
let us look at the concrete example of embedded manifolds for inspiration and
discuss the corrections induced by chord distances. Locally, they can be
expressed by means of the extrinsic curvature of the embedding, which
potentially introduces a lot of new degrees of freedom. Such corrections are
generally anisotropic, in the sense that they depend on the orientation of
$AB$ with respect to the principal axes of the extrinsic curvature. For
example, if we embed our manifold “like a cylinder”, the correction vanishes
when $AB$ is parallel to the axis of the cylinder and extremizes when it is
orthogonal. Among all such possibilities it looks natural to restrict to the
isotropic case and assign to beyond-Riemannian corrections the same number
degrees of freedom as the Riemannian ones. This corresponds to a “sphere-like”
embedding where the extrinsic curvature tensor is proportional to the
identity, i.e. $K^{i}_{j}=\delta^{i}_{j}/r$ for some embedding radius $r$
which possibly depends on the plane to which the two vectors $OA$ and $OB$
belong. It is not difficult to see that this prescription leads to a beyond
Riemannian discrepancy with respect to flat space of the form

$\delta_{BR}=-\frac{L^{2}}{12r^{2}}(1-\cos\alpha)+{\cal O}(L^{4})\,.$ (5.3)

The above expression is somewhat complementary to (5.1) in that it is
_maximized_ for aligned points ($\alpha=\pi$). Notice also that the embedding
intuition is consistent with $\delta_{BR}$ being negative-definite because
“cutting through” some ambient space always represents a shortcut. The
negative sign in (5.3) is a necessary condition for the new distance to still
define a metric space. We argue, without proving, that the average metrics
defined in the last section also define a metric space, in the sense that they
satisfy the triangle inequality.

Notice that spatial curvature, when measured with the triangle diagnostic,
becomes an ambiguous quantity in this beyond-Riemannian framework, because the
correction (5.3) is of the same order as the Riemannian one (5.1). So for
instance there is a choice of the parameter $1/r^{2}$ such that the total
discrepancy $\delta_{R}+\delta_{BR}$ becomes independent of the angle
$\alpha$. When we measure the spatial curvature of the universe we effectively
look for some $\alpha$ dependence in our geometric probes. It is intriguing to
imagine that the _curvature problem_ , traditionally addressed by primordial
inflation, could “just not be there” in some extended geometric framework.

##  6 Discussion

In this paper we have considered averaged spacelike distances in states with
relevant quantum fluctuations and found that they do not add up at large
separation. In what follows we sketch some possible implications of these
reasonings on holography and black hole evaporation. Then we turn to timelike
and lightlike distances and discuss what it would mean, for them, to be non
additive. The emerging “beyond-Lorentzian” picture looks too preposterous to
be omitted.

###  6.1 Islands, fuzzballs and (the breakdown of) classicality

A standard prescription to make local operators gauge invariant is that of
“dressing” them [36, 37]. In holographic AdS this can be achieved by defining
a non-local object that contains the metric operator evaluated all the way
from the boundary to the point that we want to reach. For example, a Wilson
line implicitly contains the geometrical prescription of how to reach some
given point of the bulk from the boundary (e.g. “Leave the boundary at point
$x^{\mu}$ in the orthogonal direction and proceed straight along a geodesic
for 120 meters”).

It is clear that in the presence of relevant fluctuations in the metric field
the Wilson line procedure would miss its target, and define instead some non
local field operator spread about an _extended region of space_. This is a
vague statement which our observers can make more precise, as detailed in App.
A. We are saying that a Wilson line operator will hit different observers with
different calculable probabilities, rather than picking up just one of them.

We argue that something similar might be happening inside real Lorentzian
spacetimes when Euclidean calculations start being dominated by replica
wormholes [10]. The “island” that at some point forms inside an evaporating
black hole [10] is disconnected from the boundary, separated by a region
which, we argue, could be characterized by relevant quantum fluctuations in
the metric field. As a consequence, the local operators inside the island
cannot be dressed by any simple Wilson line procedure and require a much more
complicated superposition of boundary operators.

Once the island forms, reaching any given observer with a simple “classical
prescription” becomes impossible. As a consequence, the picture of the hole
from the boundary looks blurry, like a “fuzzball”. However, and contrary to
what the fuzzball paradigm [38] seems to suggest, the picture looks extremely
smooth as we get close to the hole. As argued in Sec. 4.3, the effects of
“highly quantum” gravitational states are locally invisible. So nothing
dramatic would happen at the horizon of the black hole.

One main takeaway of the present paper is that the breakdown of classicality
has no local consequences. The consequences are relegated to the infrared, in
the sense that beyond-Riemannian effects build up at large mutual separations
and cannot be detected with local experiments. The anomalous _chord_ -like
distance emerging from our picture—with its property of being “locally flat”
around each point—is an example of this phenomenon. Our fluid of observers
could safely populate the region very close to the black hole101010Such a
region should be suitably defined, for instance by trying to generalize
quantum mechanically the concept of closed trapped surface. and inside it, and
such regions will look smooth and locally flat for each one of them.

It could be that these novel points of view could help addressing e.g. the
difficulties recently raised in [17]. A more in depth analysis of these issues
is left for future work.

###  6.2 Beyond-Lorentzian?

Thanks to the time-field $T(x^{\kappa})$ introduced in Sec. 3.2 we have
managed to deal with spacelike distances “along some given time-slice” and
keep away from the subtleties related with time evolution in quantum gravity.
We see no reason why the non-additivity property of spacelike distances should
disappear once time evolution will be properly taken into account. If
anything, searching for spacelike geodesics not restricted to the $T=const.$
surface should make the picture even richer. Moreover, we generally expect
this non-additivity to leak also in the timelike and lightlike directions. At
least this is the extremely naive picture that we get if we add an orthogonal
straight time direction to an embedded Riemannian manifold to make it an
embedded _Lorentzian_ manifold (Fig. 4).

Figure 4: Time moves along the vertical axis and space is horizontal in this
portrait of an embedded Lorentzian manifold. Between emission and absorption
coherent photons states connect events that are at null separation with
respect to the beyond-Lorentzian distance. They “jump” inside the ambient
space. The classical lightcone structure of general relativity emerges in the
limit in which photons are continuously absorbed and reemitted in the same
direction.

Timelike distances represent the proper time of some observer—how could they
possibly be non-additive? After all, time intervals just add up to one another
as much as the minute hand of a clock continuously probes the circumference of
the clock itself. In this sense, time intervals are additive by construction,
at least if the clock is macroscopic and we keep watching it. But suppose
instead that we keep track of proper time by (quantum-) measuring a system
that evolves coherently without external interactions until the measurement
takes place. For instance we could consider the relative phase of the
wavefunction of a massive particle beam-splitted into two slightly different
paths. Between the instant when the particle is beam splitted and the one when
the relative phase is measured there is no such thing as “keeping watching the
clock”.

So a way to make sense of non-additive time intervals seems to be that of
associating them with subsystems that maintain quantum coherence and do not
entangle with anything else. And to argue that, on the opposite, classical
spacetime is the domain of decoherence and dynamical entanglement. A
macroscopic clock measures proper time as given by the metric line element
integrated along the geodesic. The “quantum-coherent” clock described above
measures instead the beyond-Lorentzian time interval between the events
“preparation of the system” and “measurement”.

Even more strikingly, the Lorentzian distances inherited from the ambient
space can change the causal structure. Null geodesics of the ambient manifold
can connect events that are spacelike separated for the embedded one (Fig. 4).
Again, these “jumps” could characterise the behavior of a photon that
maintains quantum coherence between its production and its absorption. On the
opposite, the lightcone structure of general relativity is defined by photons
that continuously decohere with the environment.

The locally flat limit of chord distances is respected also in Lorentzian
signature so these “superluminal” effects would never be observable locally
and would become important only at some large separation set, say, by a
curvature scale (e.g. a CMB photon produced at recombination and collected by
the Planck satellite). The healthy physicist should immediately become
suspicious when it comes to superluminality. On the other hand, the leak of
information from an evaporating black hole should also come to terms with
slight violations of causality in a way or another [39, 11] and a beyond-
Lorentzian geometry could perhaps stage this process.

## Acknowledgements

I am particularly grateful to Alberto Nicolis for endless discussions about
many of the topics touched upon in this paper. I also acknowledge useful
exchanges with Raphael Bousso, Sergio Cacciatori, Tom Hartman, Federico Re,
Alberto Verga and Kenza Zeghari.

##  Appendix A The AdS boundary and the rest

In asymptotically Anti de Sitter (AdS) space the boundary can efficiently
serve as an “observer”: a most reliable one, because the fluctuations of the
metric dynamically vanish on it. When discussing geometric constructions in
AdS that anchor to its boundary it might be useful to specify also _what we
anchor to the other side_. To this purpose we populate the AdS bulk with
observers. Let us consider the observers “test particles” of Sec. 2: $A$, $B$,
$C$ etc.

With minor adjustments we can extend the definition of distance operator to
include the boundary, so that quantities like e.g. $\hat{d}(A,{\rm Boundary})$
can be evaluated on some state. As for the distance operator among pointlike
observers, we define $\hat{d}(A,{\rm Boundary})$ by its action on a classical
basis like in eq. (2.4). On each element of this basis a straightforward
geometrical computation produces a definite eigenvalue for the distance
operator. While doing this, we have to pay attention to consistently
renormalizing the (otherwise, infinite) distance, which we leave implicit in
what follows. Moreover, when we minimize the length of all possible curves to
obtain the geodesic, one should also vary over all possible boundary points.

In this setup several questions can be asked. What is the distance from $A$ to
the boundary? This is a quantum mechanical quantity with an average (expressed
by the appropriate generalization of (2.5)) an uncertainty etc.

Alternatively, we can consider a typical “Wilson line” geometrical
prescription to define a point in the bulk. For instance

> _Prescription_ : Leave the boundary at point $y^{\mu}$ in the orthogonal
> direction and proceed along a geodesic for 120 meters.

And ask: will such a prescription define a point in the bulk? The problem with
this question is that if we are dealing with superpositions of geometries it
is not even clear what it means “a point in the bulk”. Our observers, however,
can help setting this question on a slightly firmer basis.

To each element of the classical basis the _Prescription_ assigns a coordinate
point unambiguously,

$|h_{ij}(x^{\kappa}),x^{i}_{A},\dots\rangle\quad\longrightarrow\quad
X^{i}[h],$ (A.1)

which is a functional of the corresponding metric and does not depend on the
observers’ positions. The probability amplitude that $(i)$ the metric is in
some configuration $h_{ij}(x^{k})$, $(ii)$ the observers $B$, $C$ etc. are at
positions $x_{B}$, $x_{C}$ etc. _and_ $(iii)$ that the _Prescription_ has
reached observer $A$ is given by the wavefunction $\tilde{\Psi}$ with
$X^{i}[h]$ instead of the variable $x_{A}^{i}$.

$\tilde{\Psi}\left[h_{ij}(x^{k}),\,\ X^{i}[h],\ x_{B}^{i},\ \dots\right]\,.$
(A.2)

Notice that this quantity depends on the metric “twice”. We can now calculate
the probability that the _Prescription_ has reached observer $A$ by tracing
the corresponding density matrix elements over all variables except $x_{A}$,

$P_{A}\ =\ \int{\cal D}\tilde{h}\ d^{3}x_{B}\
d^{3}x_{C}\dots\left|\tilde{\Psi}\left[\tilde{h}_{ij}(x^{k}),\,\ X^{i}[h],\
x_{B}^{i},\ \dots\right]\right|^{2}\,.$ (A.3)

In general, several observers will be hit by the _Prescription_ with non
vanishing probability. This means that, in the presence of relevant
fluctuations of the metric, _Prescription_ will not define anything “local”,
in the operational sense of “being attached to some given observer”.

Making the same operation in different coordinate systems could also be
interesting. In unitary gauge, by definition, there is no uncertainty in the
positions of the observers: $\tilde{\Psi}$ is delta-like in the variables
$x_{A}$, $x_{B}$ etc. This means that all the uncertainty about _Prescription_
reaching or not some observer is contained in the metric fluctuations, and
builds up at large separations from the boundary. On the opposite, we can go
to Fefferman-Graham (FG) coordinates where the line element reads

$ds^{2}=\frac{dz^{2}+\gamma_{\mu\nu}(z,y)\ dy^{\mu}dy^{\nu}}{z^{2}}.$ (A.4)

These coordinates are built in such a way that _Prescription_ always produces
the same result,

$X^{i}_{FG}=\left(z,y^{i}\right),$ (A.5)

independently of the metric $\gamma_{ij}$. In particular, $z$ is always the
exponential of the distance from the boundary in these coordinates. In FG-
gauge, when we calculate the probability of reaching some observer like in
(A.3), the fluctuations of the metric become irrelevant. The uncertainty is
now directly encoded in the wave _function_ of the observer’s position.

FG coordinates clearly bear similarities with the normal coordinates of Sec.
4.2—in normal coordinates the distance from _the origin_ trivially becomes the
Euclidean one. The boundary of AdS, as opposed to our generic observer, has
the further advantage that the metric fluctuations vanish in its vicinity. So
the uncertainty in observers’ positions of the FP-wavefunction should also
vanish for the observers close the boundary. In other words, we do not expect
for those observers any uncertainty of quantum gravitational origin, i.e.
other than the usual quantum mechanical indeterminacy of the position, which
is negligible for macroscopic objects.

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