11institutetext: Berkeley Center for Theoretical Physics and Department of
Physics,
University of California, Berkeley, CA 94720, USA22institutetext: Stanford
Institute for Theoretical Physics,
Stanford University, Stanford, CA 94305, USA

# Quantum Singularities

Raphael Bousso 2 and Arvin Shahbazi-Moghaddam<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Two spatial regions $B$ and $R$ are _hyperentangled_ if the generalized
entropy satisfies $S_{\rm gen}^{B\cup R}<S_{\rm gen}^{R}$. If in addition all
future (or all past) directed inward null shape deformations of $B$ decrease
$S_{\rm gen}^{B\cup R}$, then we show that the causal development of $B$, with
$R$ held fixed, must be incomplete. This result eliminates the Null Energy
Condition from the assumptions of a recently proven singularity theorem.
Instead, we assume a quantum version of the Bousso bound.

Taking $R$ to contain the Hawking radiation after the Page time, our theorem
predicts a singularity in the _past_ causal development of the black hole
interior. This is surprising because the classical spacetime is nonsingular in
the past. However, one finds that Cauchy slices that are required to contain
$R$ do not remain in the semiclassical regime. The _quantum singularities_
predicted by our theorem are an obstruction to further semiclassical
evolution, generalizing the singularities of classical general relativity.

##  1 Introduction

A spacetime $M$ is singular if it contains an incomplete timelike or null
geodesic wald2010general  (an inextendible geodesic of finite affine length).
Physically relevant examples include the past singularity in certain
cosmological solution—the “big bang”—and the future singularity that
terminates time evolution inside a Schwarzschild black hole.

Singularities are generic in classical general relativity. A theorem by
Penrose Penrose:1964wq  guarantees that at least one of the null geodesics
orthogonal to a trapped surface is incomplete. A surface is trapped if both
sets of future-directed orthogonal null geodesics have negative expansion.

Penrose’s theorem requires two crucial assumptions about the spacetime: $M$
must admit a noncompact Cauchy surface; and $M$ must satisfy the Null
Curvature Condition, $R_{ab}k^{a}k^{b}\geq 0$, where $R_{ab}$ is the Ricci
tensor and $k^{a}$ is any null vector.

A recent result Bousso:2022cun  has established a connection between
singularities and quantum information: the noncompactness assumption can be
eliminated from Penrose’s theorem, if instead the spacetime is assumed to
satisfy the Bousso bound Bousso:1999xy  on the entropy of matter.

Figure 1: The quantum Bousso bound (Conjecture 2.22): if the quantum expansion
at $\partial B$ in the direction of $B^{\prime}$ is nonpositive then $S_{\rm
gen}^{B^{\prime}}\leq S_{\rm gen}^{B}$.

The Null Curvature Condition can also be eliminated. This is important,
because it it is known not to hold in Nature. By Einstein’s equation, it is
equivalent to the Null Energy Condition, that $T_{ab}k^{a}k^{b}\geq 0$, where
$T_{ab}$ is the stress tensor. Any relativistic quantum field theory, such as
the Standard Model, contains states in which the expectation value of the
stress tensor, $\braket{T_{ab}}$, violates this condition Epstein:1965zza .
Wall Wall:2010jtc  eliminated the Null Curvature Condition from Penrose’s
theorem, by assuming instead that the Generalized Second Law (GSL) holds in
$M$. The GSL is the statement that the generalized entropy—the sum of horizon
area and von Neumann entropy of the matter fields outside a causal
horizon—cannot decrease. A causal horizon is the boundary of the past of a
timelike or null curve of infinite affine length; examples include black hole,
Rindler, and de Sitter horizons. Unlike for the NCC, there is no known
counterexample to the GSL. There is considerable evidence for its validity,
and it has been proven to hold on Killing horizons Wall:2011hj .

In this paper, we combine the advances of Refs. Bousso:2022cun ; Wall:2010jtc
, using a single assumption, a quantum refinement of the Bousso bound
Bousso:2015mna . This bound says that if the generalized entropy outside a
Cauchy-splitting null hypersurface $L$ is decreasing towards the future (resp.
past) at some moment of time, then it must be lower at all future (past)
times. See Fig. 1; and see Conj. 2.22 below for a more precise statement. The
quantum Bousso bound implies the GSL as a special case.

Figure 2: Quantum singularity theorem for a hyperentangled region $B$. (a)
Collapsing star entangled with a distant reference system. At late times the
entanglement entropy exceeds the surface area of the star and the theorem
predicts a future singularity. This is the (ordinary) singularity of the black
hole. (b) Evaporating black hole after the Page time. The black hole interior
$B$ and distant Hawking radiation $R$ are hyperentangled, and the theorem
predicts both a future and a past singularity. The latter is a quantum
singularity: it arises from the breakdown of semiclassical evolution in the
red region when $R$ is held fixed.

We will prove that the quantum Bousso bound implies a singularity theorem for
certain hyperentangled regions, Theorem 3.1 below. We call a spatial region
$B$ hyperentangled if

$S_{\rm gen}^{B\cup R}<S_{\rm gen}^{R}$ (1)

for some spacelike-separated region or external system $R$, where $S_{\rm
gen}$ is the generalized entropy. Suppose that, in addition, $B\cup R$ has
negative inward quantum expansion at $\partial B$; that is, $S_{\rm
gen}^{B\cup R}$ decreases under shape deformations of $B$ along a past-
directed (or a future-directed) ingoing null congruence. See Fig. 2.

Under these assumptions, we prove that at least one null geodesic in the
congruence is incomplete, _in any spacetime obtained from semiclassical
evolution on Cauchy surfaces that all contain $R$._

We shall see through the study of examples that the quantum Bousso bound, and
indeed the GSL, are surprisingly restrictive when Cauchy evolution is limited
to slices containing $R$. As a result, spacetimes that satisfy this bound
admit a novel, $R$-dependent notion of singularity. Our theorem captures such
singularities. We believe that the notion of $R$-dependent singularities in
semiclassical gravity has not been discussed in the literature, so we will do
so now.

Figure 3: $M_{\not{R}}$ (green) is a spacetime covered by nice slices that
contain $R$. In general this semiclassical region is a proper subset of the
region $M-I(R)$ (red+green) covered classically by Cauchy slices that contain
$R$. Generators of $H^{\pm}(B)$ may be incomplete in $M_{\not{R}}$ even if
they would be complete in $M-I(R)$. This reflects a real obstruction to
further semiclassical evolution which we call a _quantum singularity_.

In classical General Relativity, the inclusion of $R$ in all Cauchy slices
would be a trivial restriction for the purposes of our theorem. As shown in
Fig. 3, slices that contain $R$ foliate $M-I(R)$, where $I(R)$ is the union of
the chronological past and future of $R$. The Cauchy horizon $H^{\pm}(B)$ is a
subset of $M-I(R)$, so if it contains an incomplete geodesic, then so does
$M$.

However, in semiclassical gravity, Cauchy slices must be “nice.” That is, the
data on each slice must be compatible the validity of an effective field
theory description with a cutoff below the Planck scale.111We are not aware of
a first-principles derivation of the precise inequalities that “niceness”
implies for scalar quantities extracted from the initial data on a slice. A
plausible set of criteria was outlined in Ref. Lowe:1995ac ; Polchinski:1995ta
. To understand the quantum singularities predicted by our theorem in the
examples we study here, we need only require a rather weak niceness condition
on the trace of the extrinsic curvature; see Sec. 4. The boundary of $M-I(R)$
is null; therefore, it contains distinct points with zero spatial distance,
and nice slices that contain $R$ cannot approach it arbitrarily closely; see
Fig. 3.

Thus, only a subset $M_{\not{R}}\subset M-I(R)$ can be obtained by
semiclassical evolution along Cauchy surfaces that all contain $R$. Hence it
is possible for $H^{\pm}(B)$ to contain a geodesic that is incomplete in any
semiclassically allowed spacetime $M_{\not{R}}$, even though it would be
complete in the (larger but semiclassically unattainable) $M-I(R)$. We call a
singularity that arises in this manner a _quantum singularity_.

#### Outline

In Sec. 2, we define a semiclassical spacetime as a globally hyperbolic
manifold $M$ with metric $g_{ab}$ and quantum state $\rho$ whose stress tensor
expectation value satisfies the Einstein equation, $G_{ab}=8\pi
G\braket{T_{ab}}$. In addition, $M$ must admit a “nice slicing.” That is, time
evolution must be consistent with the validity of an effective field theory
description, with an ultraviolet cutoff $\Lambda\ll M_{P}$, where $M_{P}$ is
the Planck mass. We formulate the main assumption of our theorem, the quantum
Bousso bound, and we clarify that it applies to nice slices only.222We also
comment on Ref. Rolph:2022csa , which asserts a different regime of validity
of the quantum Bousso bound (and, implicitly, of the GSL), and which arrives
at conclusions different from ours.

After proving our theorem in Sec. 3, we discuss two nontrivial applications.
In Sec. 4.1, we consider an evaporating black hole formed from regular initial
conditions. We apply our theorem to the black hole interior $B$ after the Page
time; $R$ is the Hawking radiation emitted so far. In this case a singularity
is predicted along the black hole horizon to the _past_ of $B$.

This is a quantum singularity. It appears because we are holding $R$ fixed,
thus excluding the region $R$ from participating in the semiclassical
evolution. On nice slices that contain the Hawking radiation, the horizon
cannot be evolved far into the past. Hence the horizon generators are
incomplete in $M_{\not{R}}$. The semiclassically allowed spacetime
$M_{\not{R}}$ is substantially smaller than $M-I(R)$, which contains the
entire black hole horizon.

In Sec. 4.2, we consider the Kerr-Newman solution in thermal equilibrium with
a bath. We again consider the black hole interior $B$ at a sufficiently late
time, when it is entangled with distant radiation $R$. Our theorem predicts a
singularity before the inner horizon. In the classical Kerr-Newman solution,
the region between the inner and outer horizons is regular, so this conclusion
is of some interest. We discuss its potential relevance to the strong cosmic
censorship hypothesis.

##  2 Semiclassical Gravity

###  2.1 Causal Structure

######  Convention 2.1.

Everywhere below, $M$ will denote a time-orientable globally hyperbolic
spacetime. ($M$ may be extendible.) We use an overdot to represent the
boundary of a subset of $M$.

######  Convention 2.2.

Everywhere below, $B$ will denote a closed subset of a Cauchy slice $N$ of
$M$, such that $\partial B$ is a compact codimension 2 submanifold of $M$ and
$B-\partial B\neq\varnothing$. Here $\partial B$ denotes the boundary of $B$
in the induced topology of $N$.

######  Definition 2.3.

The _chronological_ and _causal future_ and _past_ , $I^{\pm}(K)$ and
$J^{\pm}(K)$, of any set $K\subset M$ are defined as in Wald wald2010general .
For $K=\set{p}$, we drop the set brackets. Key consequences of these
definitions include: $p\notin I^{+}(p)$ but $p\in J^{+}(p)$, and $I^{+}(K)$ is
open.

######  Definition 2.4.

For any set $K\subset M$, we define its _domain of influence_ as the union of
$K$ and all points that can be reached by a timelike curve from $K$:
$I(K)\equiv I^{+}(K)\cup I^{-}(K)\cup K$.

######  Definition 2.5.

For any closed achronal set $K\subset M$, the _future domain of dependence_ ,
$D^{+}(K)$, is the set of points $p$ such that every past-inextendible causal
curve through $p$ must intersect $K$. The _past domain of dependence_ ,
$D^{-}(K)$, is defined analogously. The _domain of dependence_ is $D(K)\equiv
D^{+}(K)\cup D^{-}(K)$. The _future Cauchy horizon_ is
$H^{+}(K)\equiv\overline{D^{+}(K)}-I^{-}[D^{+}(K)]$.

######  Definition 2.6.

Let $M$ be a spacetime with Cauchy surface $N$. ($N$ or $M$ may be
extendible.) Let $B\subset\Sigma$. We say that $B$ is _future singular in $M$_
if its Cauchy horizon $H^{+}(B)$ contains an incomplete geodesic; _i.e._ , a
geodesic that is future-inextendible in $M$ but of finite affine length.
Otherwise, we call $B$ _future complete in $M$_.

###  2.2 Kinematics

######  Definition 2.7.

A _nice global slice_ is an inextendible Cauchy surface $\Sigma$ whose
intrinsic and extrinsic geometry and quantum state $\rho(\Sigma,\Lambda)$ can
be fully described using a cutoff scale $\Lambda\ll M_{P}$. In particular,
curvature scalars and energy densities that can be constructed from the normal
vector to $\Sigma$ must be sub-Planckian.

######  Definition 2.8.

Let $\Sigma$ be a nice global slice of $M$, with associated cutoff scale
$\Lambda\ll M_{P}$. Let $B$ be defined as in Convention 2.2. If the intrinsic
and extrinsic geometry of $\partial B$ is well resolved at the cutoff
$\Lambda$ (that is, when $\partial B$ is put on a lattice with characteristic
scale $\Lambda^{-1}$), then we call $B$ a _nice partial slice_ , and the
quantum state on $B$ is defined as

$\rho(B,\Lambda)=\operatorname{tr}_{\Sigma-B}\rho(\Sigma,\Lambda)~{},$ (2)

######  Definition 2.9.

We call $N$ a _nice slice_ if $N$ is a nice global slice or a nice partial
slice.

######  Definition 2.10.

Let $N$ be a nice slice with associated cutoff scale $\Lambda$. The
_generalized entropy_ of $N$ is

$S_{\rm gen}^{N}=\frac{\operatorname{Area}(\partial
N)}{4G(\Lambda)\hbar}+\ldots+S(N,\Lambda)~{},$ (3)

where $G(\Lambda)$ is the effective Newton constant at the scale $\Lambda$,
and

$S(N,\Lambda)=-\operatorname{tr}_{N}\rho(N,\Lambda)\log\rho(N,\Lambda)$ (4)

is the von Neumann entropy of the quantum fields on $N$ at the scale
$\Lambda$. The area term is the leading gravitational counterterm. The
subleading gravitational counterterms are indicated by “$\ldots$”; see Ref.
Dong:2013qoa  for details.

######  Remark 2.11.

Niceness of $N$ is required in the above definition since otherwise
$G(\Lambda)$ is not operationally defined, for example as the effective
gravitational coupling in a scattering process. The boundary of a nice global
slice ($N=\Sigma$) vanishes. The boundary of a nice partial slice ($N=B$) is
understood to be defined in a completion $\Sigma\supset N$, where $\Sigma$ is
a nice global slice. Neither the boundary area nor the generalized entropy of
$N$ will depend on the choice of completion.

######  Conjecture 2.12.

The generalized entropy is cutoff-independent, in the following sense. Suppose
that the slice $N$ is nice with respect to two different scales $\Lambda$ and
$\Lambda^{\prime}$. Under $\Lambda\to\Lambda^{\prime}$, both terms in Eq. (3)
will change, but their sum will not. For references supporting this claim, see
the Appendix of Ref. Bousso:2015mna .

###  2.3 Dynamics

######  Definition 2.13.

A _slicing_ of the spacetime $(M,g)$ is a continuous map from an open interval
to achronal subsets of $M$, $t\to N(t)$, such that every point in $M$ is
contained in at least one $N(t)$, and $N(t^{\prime})\subset J^{+}[N(t)]$ for
$t^{\prime}>t$. [Thus, a slicing is not a foliation. Along a timelike curve
$\gamma$, the proper time of $\gamma\cap N(t)$ increases monotonically with
$t$, but not strictly so.]

######  Definition 2.14.

A _Cauchy slicing_ of $(M,g)$ is a slicing such that each $N(t)$ is a Cauchy
surface of $M$.

######  Definition 2.15.

A _nice Cauchy slicing_ of $(M,g)$ is a Cauchy slicing such that each $N(t)$
is a nice slice with the same associated cutoff $\Lambda\ll M_{P}$. A
collection of nice Cauchy slicings with cutoff $\Lambda$ will be denoted
${\cal S}_{\Lambda}$.

######  Definition 2.16.

A _semiclassical spacetime_ is a quadruplet $(M,g,{\cal S}_{\Lambda},\rho$).
Here $M$ is a globally hyperbolic manifold with metric $g$. ${\cal
S}_{\Lambda}$ is a nonempty set of nice Cauchy slicings. For each slicing,
$\rho(N(t),\Lambda)$ solves the Schrödinger equation of the quantum fields.
The expectation values of local operators do not depend on the slicing.
Finally,

$G_{ab}=8\pi G(\Lambda)\braket{T_{ab}}+\ldots~{},$ (5)

where $G_{ab}$ is the Einstein tensor computed from $g$, $T_{ab}$ is the
stress tensor (viewed as an operator), and “$\ldots$” stands for higher-
curvature corrections.

######  Remark 2.17.

The above definition ignores gravitons; this can be justified by taking the
number of matter fields to be large. The Cauchy slices of $M$ may be partial
and hence extendible; and in any case $M$ may be extendible.

######  Remark 2.18.

Given a nice slice $N(0)$, one can solve the quantum field theory and
Einstein’s equation iteratively in $G\hbar$, in some open neighborhood of
$N(0)$, and thus generate a semiclassical spacetime.

######  Definition 2.19.

Let $N$ be a nice Cauchy slice in a semiclassical spacetime $M$, and let
$R\subset N$, $R\neq N$. A semiclassical spacetime with nice slicing
$N_{\not{R}}(t)$ will be called a _reduction of $M$ by $R$_ and will be
denoted $M_{\not{R}}$, if for every $t$, $N_{\not{R}}(t)\cup R$ is a nice
slice of $M$. (See Fig. 3 for an example.)

######  Definition 2.20.

Let $M$ be a semiclassical spacetime, and let $B\subset M$ be a nice partial
slice. The future-directed ingoing _quantum expansion_ of $B$ at $y\in\partial
B$ is the rate of change of the generalized entropy under a shape deformation
of $B$ along the ingoing future-directed null congruence orthogonal to
$\partial B$:

$\Theta_{+}^{B}(y)=\frac{4G\hbar}{\sqrt{h(y)}}\left.\frac{\delta S_{\rm
gen}[V]}{\delta V(y)}\right|_{\partial B}~{}.$ (6)

Here $h$ is the area element of the induced metric on $\partial B$. The
functional derivative is taken with respect to the affine parameter $V(y)$
along the congruence that specifies the location of cuts of the congruence
such as $\partial B$.

The past-directed ingoing quantum expansion is defined analogously. Outgoing
quantum expansions are related to the ingoing ones by a change of sign and
exchange of past and future.

######  Remark 2.21.

The functional derivative in Eq. (6) is an idealization that suppresses the
cutoff $\Lambda$. The quantum expansion is well-defined only if one of the
nice Cauchy slicings of $M$ contains slices that contain $B$ and its shape
deformation. In particular, this excludes deformations whose transverse
support near $y$ is localized to better than $\Lambda^{-1}$
Leichenauer:2017bmc .

######  Conjecture 2.22 (Quantum Bousso Bound).

Let $M$ be a semiclassical spacetime, and let $N$ and $N^{\prime}$ be slices
in one of the nice Cauchy slicings of $M$. Let $B\subset N$ be a nice partial
slice, and let $\partial B^{+}$ ($\partial B^{-}$) be the subset of $\partial
B$ with positive future (past)-directed inward quantum expansion. Let
$B^{\prime}=D(B)\cap N^{\prime}$. If $N^{\prime}\cap I^{+}(\partial
B^{+})=N^{\prime}\cap I^{-}(\partial B^{-})=\varnothing$, then

$S_{\rm gen}^{B^{\prime}}\leq S_{\rm gen}^{B}~{}.$ (7)

######  Remark 2.23.

The above conjecture was originally obtained as a consequence of the Quantum
Focusing Conjecture Bousso:2015mna . However, its derivation was somewhat
heuristic and omitted a careful regularization of points where null generators
leave $\dot{I}(C)$. Here we will assume Conjecture 2.22 directly. The original
Bousso bound Bousso:1999xy  follows in the limit where $\Theta$ is well
approximated by the classical expansion and $M$ satisfies the Null Curvature
Condition.

######  Remark 2.24.

(Note added.) Compared to our Conj. 2.22, Sec. 2 of Rolph:2022csa  imposes the
additional requirement that $B^{\prime}\neq\varnothing$, and more strongly
that $N^{\prime}$ must intersect every connected component of $D(B)$. We
believe this restriction is too strong and also unnecessary. An important
manifestation of the GSL is the fact that the generalized entropy outside the
horizon of a black hole is larger than the the (ordinary) entropy before the
black hole has formed. This key feature follows from the quantum Bousso bound
only if $B^{\prime}=\varnothing$ is allowed. On the other hand, Ref.
Rolph:2022csa  does not restrict the application of the bound and of the GSL
to nice slices. We believe that this is too permissive, even when combined
with the restriction to nonempty $B^{\prime}$ advocated in Ref. Rolph:2022csa
. As we discuss at the end of Sec. 4.1, the generalized entropy becomes
negative and stops making sense on slices allowed by this set of
criteria.333With Rolph’s conditions on the GSL and the quantum Bousso bound,
the proof of the “island finder” theorem Bousso:2021sji  would indeed have a
loophole as claimed in Ref. Rolph:2022csa . With ours, it does not. In the
case of concern to Ref. Rolph:2022csa , Conj. 2.22 would be violated, so a
(possibly quantum) singularity must form. (In the maximin formalism
Wall:2012uf , it is necessary to assume that the maximin slice is repelled by
singularities. We propose that this feature extends to quantum singularities.)

##  3 Singularity Theorem

######  Theorem 3.1 (Singularity Theorem for Hyperentangled Regions).

Let $M$ be a semiclassical spacetime with nice Cauchy slice $N$. Let the
disjoint union $B\cup R\subset N$ be a nice slice, with $\partial B$ compact.
Suppose that the future-directed inward quantum expansion of $B\cup R$ is
negative everywhere on $\partial B$:

$\Theta_{+}^{B\cup R}(y)<0~{}~{}\mbox{for~{}all}~{}~{}y\in\partial B~{}.$ (8)

Suppose moreover that $B$ is hyperentangled with $R$, that is:

$S_{\rm gen}^{BR}<S_{\rm gen}^{R}~{}.$ (9)

Let $M_{\not{R}}$ be a reduction of $M$ by $R$ (see Def. 2.19). Assuming
Conjecture 2.22 (Quantum Bousso Bound), $B$ is future singular in
$M_{\not{R}}$, _i.e._ , $H^{+}(B)\cap M_{\not{R}}$ contains an incomplete null
geodesic.

###### Proof.

The Cauchy horizon $H^{+}(B)$ is topologically the direct product of $\partial
B$ with the future-inward directed null geodesics orthogonal to $\partial B$,
up to possible identifications of their endpoints on $H^{+}(B)$. By Eq. (8),
no null geodesic can remain on $H^{+}(B)$ for infinite affine time Wall:2011hj
.444For if such a geodesic $\gamma$ did exist, then $\dot{I}^{-}(\gamma)$
would be a causal horizon, and by the Generalized Second Law,
$\Theta_{+}^{(I^{-}(\gamma)\cap N)\cup R}\geq 0$. By construction,
$I^{-}(\gamma)\subset B$ and $\partial B$ touches $\dot{I}^{-}(\gamma)$ at
$p=\gamma\cap\partial B$. By Theorem 3 of Ref. Wall:2010jtc ,
$\Theta_{+}^{B\cup R}(p)\geq\Theta_{+}^{(I^{-}(\gamma)\cap N)\cup R}(p)\geq
0$, which contradicts Eq. (8). Assuming for contradiction that $B$ is future
complete in $M_{\not{R}}$, it follows that $H^{+}(B)$ contains the endpoints
of all of its generators. Compactness of $\partial B$ then implies that
$H^{+}(B)$ is compact.

Let $N_{\not{R}}(0)=N-R$, and assume for contradiction that
$N_{\not{R}}(t)\cap H^{+}(B)\neq\emptyset$ for all $t\geq 0$. Let $t_{n}$ be a
monotonically increasing sequence that converges to the upper bound of the
time interval for which the slicing $N_{\not{R}}(t)$ is defined (or diverges
to $\infty$ if there is no upper bound), and let $x_{n}\in
N_{\not{R}}(t_{n})\cap H^{+}(B)$.555This step invokes the axiom of choice;
perhaps this can be eliminated. By compactness of $H^{+}(B)$, the sequence
$x_{n}$ has an accumulation point $p\in H^{+}(B)$. Let $q\in I^{+}(p)$ and let
$N_{\not{R}}(t_{q})$ be a slice that contains $q$. Because a slicing moves
forward in time monotonically by Def. 2.13, there exists a small neighborhood
$O(p)$ that no slice with $t\geq t_{q}$ can intersect. This contradicts the
fact that $p$ is an accumulation point.

Therefore $M_{\not{R}}$ admits a nice slice such that $N_{\not{R}}(t_{\rm
above})\cap H^{+}(B)=\emptyset$, $t_{\rm above}>0$, and by Def. 2.19, $M$
admits a nice slice

$N^{\prime}\equiv N_{\not{R}}(t_{\rm above})\cup R$ (10)

that contains $R$ and fails to intersect $D(B)$. $N^{\prime}$ satisfies the
assumptions of the quantum Bousso bound as applied to $B\cup R\subset N$. (In
Conjecture 2.22, substitute $B\to B\cup R$.) Hence

$S_{\rm gen}^{R}\leq S_{\rm gen}^{BR}~{},$ (11)

which contradicts Eq. (9). Hence $B$ must be future singular in $M_{\not{R}}$.
∎

######  Remark 3.2.

Note that the assumption (9) cannot be satisfied if $R=\varnothing$, so any
nontrivial application of the theorem requires a nonempty choice of $R$.
However, $R$ can be arbitrarily far from $B$. After a straightforward
adaptation of the relevant definitions, $R$ can even be treated as a
nongravitating quantum system that is external to the spacetime. In that case,
$M_{\not{R}}$ can be an inextendible spacetime.

######  Remark 3.3.

The singularity theorem for hyperentropic regions Bousso:2022cun  emerges in
the limit as $\hbar\to 0$. In this limit, $\Theta\to\theta$, so the quantum
expansion is well approximated by the classical expansion. Moreover, nice
slices will cover all of $M-I(R)$ in this limit. If the entropy in $B$ is not
purified by some disjoint region $R$ then an appropriate external purification
can be added.

######  Remark 3.4.

By the previous remark, Theorem 3.1 applies to all of the examples discussed
in Ref. Bousso:2022cun , which include several settings where Penrose’s
theorem would not apply. In all cases we must first introduce an external
purification $R$ of the matter entropy in $B_{i}$. In the following section,
we study examples of singularities predicted by Theorem 3.1 that have no
classical analogue.

##  4 Hyperentangled Black Holes

The conditions of Theorem 3.1 can be satisfied by choosing $B$ to be a region
in an evaporating black hole after the Page time, with $R$ a region containing
the Hawking radiation. One can arrange that both quantum expansions are
negative, $\Theta^{B\cup R}_{\pm}|_{B}<0$, so the Theorem predicts a
singularity both along $H^{+}(B)$ and along $H^{-}(B)$ when $R$ is held fixed.

Let us discuss this in more detail. A slice of the black hole interior after
the Page time is by definition hyperentangled with the Hawking radiation
emitted so far; let $R$ be the region containing this radiation. By picking
$B$ to be the interior of a sphere which is slightly outside of the horizon
one can arrange $\Theta^{B\cup R}_{\pm}|_{B}<0$.

Alternatively, one can obtain a region $B$ with these properties by deforming
the island $I$ Penington:2019npb ; Almheiri:2019psf  associated to $R$. By
quantum maximin Akers:2019lzs , islands generically satisfy
$\partial_{\ell}\Theta_{k}=\partial_{k}\Theta_{\ell}<0$, where $k$ and $\ell$
are null future-directed orthogonal vectors fields on $\partial I$ outward and
inward to $I$ respectively. Therefore, by slightly deforming the island in the
future-outward and past-outward null directions, one obtains a hyperentangled
region which satisfies the conditions of Theorem 3.1 both in the future and
past directions. One can also use this method in an eternal black hole coupled
to a bath Almheiri:2019yqk  to find a region $B$ with these properties.

Naively, both the future and the past applications of Theorem 3.1 to such a
region are quite puzzling. Schwarzschild black holes have a singularity along
$H^{+}(B)$; but for a Schwarzschild black hole formed from regular initial
conditions, $H^{-}(B)$ is complete by construction. At the classical level,
even $H^{+}(B)$ is complete when charge or angular momentum is present. Small
classical perturbations are believed to produce a spacelike singularity before
the inner horizon, but the conditions of our theorem are satisfied in the
unperturbed Kerr-Newman solution (see Fig. 4).

However, the spacetime $M_{\not{R}}$ covered by _nice slices that all contain_
$R$ is smaller than $M-I(R)$. We will now argue that this implies that the
null generators of both $H^{+}(B)$ and $H^{-}(B)$ are incomplete in the
semiclassically allowed spacetime $M_{\not{R}}$, as predicted by Theorem 3.1.
In particular, we demonstrate that _any_ Cauchy slicing of the spacetime $M$
with slices that contain $R$, the slices that probe the region beyond the
endpoints of $H^{\pm}(B)$ necessarily have exponentially large extrinsic
curvature.

Figure 4: Reissner–Nordstrom black hole. The inner and outer horizons are
shown as dashed lines. In the Hartle-Hawking state, the conditions of the
singularity theorem can be satisfied if $R$ is a large enough region near null
infinity containing Hawking radiation and $B$ the interior of the black hole
containing the purification of the radiation. This is puzzling since all null
generators of both Cauchy horizons $H^{\pm}(B)$ contain their endpoints in the
classical spacetime. However, semiclassically, $H^{-}$ encounters a quantum
singularity when $R$ is fixed; $H^{+}$ does too, or else quantum corrections
significantly alter the geometry near the inner horizon.

In this section, we speculate on why the semiclassical spacetime $M_{\not{R}}$
might be smaller than $M-I(R)$ leading to the incompleteness of the generators
of both $H^{+}(B)$ and $H^{-}(B)$, upholding Theorem 3.1. We demonstrate that
_any_ Cauchy slicing of the spacetime $M$ with slices that contain $R$, the
slices that probe the region beyond the endpoints of $H^{\pm}(B)$ necessarily
have exponentially large extrinsic curvature.

###  4.1 Quantum Singularity On a Classically Regular Horizon

For concreteness, consider a maximally extended Schwarzschild black hole of
radius $r_{S}$ in the Hartle-Hawking state, in 3+1 spacetime dimensions.
Advanced and retarded time are defined by $u=t-r^{*}$ and $v=t+r^{*}$, where
$r^{*}=r+r_{S}\log|(r/r_{S})-1|$. The near horizon zone is the region
$r_{S}<r<3r_{S}/2$; its outer boundary will be denoted $Z$. Below, we will
also use Kruskal coordinates, $U=-2r_{S}e^{1-u/2r_{S}}$ and
$V=2r_{S}e^{1+v/2r_{S}}$, which cover the entire spacetime. (These are
slightly nonstandard to match the standard Rindler coordinates.) We define
$T=(U+V)/2$ and $X=(U-V)/2$.

Let $R$ be the union of a right asymptotic bulk region and its left mirror
image; see Fig. 5. On the right, $R$ is given by the portion $U<-U_{0}$ of a
constant $t$ slice, with $t$ chosen large enough for $R$ to be far from the
black hole. We choose $U_{0}$ past the Page time, that is, $U_{0}\lesssim
r_{S}e^{-\gamma_{1}S}$ where $S$ is the Bekenstein-Hawking entropy of the
black hole and $\gamma_{1}\sim O(1)$. The boundary of the past of $R$
intersects the boundary of the near-horizon zone at $v_{\dot{I}(R)\cap Z}$.
Choosing $B$ to be the black hole interior at the same (or slightly earlier)
value of $v$, Theorem 3.1 predicts a singularity along $H^{-}(B)$.

Since the spacetime is classically regular in the past of $B$, this must be a
quantum singularity. We will now verify this prediction.

Figure 5: Schwarzschild black hole. In the Hartle-Hawking state, $B$ and $R$
can be chosen such that the conditions of the theorem can be satisfied towards
the past of $B$. This is a quantum singularity: Cauchy slices containing $R$
which intersect the horizon around a Page time in the past of $B$ (red lines)
have exponentially large extrinsic curvature.

For $v<v_{\dot{I}(R)\cap Z}$, $\dot{I}^{-}(R)$ lies within the Rindler region
where the metric is well-approximated by

$\displaystyle ds^{2}=-dT^{2}+dX^{2}+r_{S}^{2}d\Omega^{2}+O(T^{2},X^{2})$ (12)

Intuitively, this leaves little room between $\dot{I}(R)$ for spacelike slices
that contain $R$ and enter the horizon very early. We will now argue that
there exist no nice slices containing $R$ that intersect the horizon at or
earlier than

$v_{0}=v_{\dot{I}(R)\cap Z}-\gamma_{2}\,t_{\rm Page}~{},$ (13)

where $\gamma_{2}\sim O(1)$.

One of the necessary conditions for niceness is that the trace of the
extrinsic curvature, $K$, is not too large. The precise condition is not clear
to us. The early literature on nice slices Lowe:1995ac ; Polchinski:1995ta
suggests $|K|\ll 1/l_{P}$, but this may be too stringent. (In four or more
spacetime dimensions, it would exclude slices that contain the Hawking
radiation $R$ and its entanglement island.) We shall use the more lenient
necessary condition

$|K|\ll\frac{1}{l_{P}}\left(\frac{\ell}{l_{P}}\right)^{n}$ (14)

where $n>0$ is an unknown fixed constant, and $\ell$ is a characteristic scale
of the geometry (here, $\ell\sim r_{S}$). We will see that even this rather
weak niceness condition cannot be satisfied for any $n$, because $|K|$ becomes
exponentially large at early times.

Let $\Sigma$ be a left-right symmetric smooth Cauchy slice. (Thus, we assume
that niceness cannot be rescued by using a slicing that spontaneously breaks
the left-right symmetry.) $\Sigma$ is fully determined by a function
$T=T_{\Sigma}(X)$. $\dot{I}(R)$ is given by $T=|X|-U_{0}$. For $\Sigma$ to
contain $R$ we must have $T_{\Sigma}>|X|-U_{0}$. In the Rindler region, the
extrinsic curvature is well approximated by

$\displaystyle K_{\Sigma}=\frac{T^{\prime\prime}}{(1-T^{\prime 2})^{3/2}}~{}.$
(15)

So long as $\Sigma$ is spacelike, this quantity is real.

Let $\Sigma$ intersect the horizon at $(U=0,V_{h})$. The timelike proper
distance between the intersection and $(-U_{0},-U_{0})$ is given by
$\sqrt{(V_{h}+U_{0})U_{0}}$. The smaller this distance, the larger
$|K_{\Sigma}|$ needs to be if $\Sigma$ is not to become spacelike separated
from $R$. To see this, first consider the special case where $T_{\Sigma}(0)$
is small enough to be in the Rindler region, with $T^{\prime}(0)$ arbitrary.
Given an upper bound $K_{\text{max}}$ on the magnitude of the extrinsic
curvature, one finds for $|X|\gg 1/K_{\text{max}}$:

$\displaystyle
T_{\Sigma}(X)<T_{\Sigma}(0)-\frac{1}{\sqrt{1-T^{\prime}(0)^{2}}}\frac{1}{K_{\text{max}}}[1-T^{\prime}(0)\mbox{sgn}(X)]+|X|~{}.$
(16)

Now consider the situation of interest: a slice $\Sigma$ which intersects the
horizon at $(U=0,V_{h})$ with $U_{0}\lesssim r_{S}e^{-\gamma_{1}S}$ and
$V_{h}\lesssim\frac{r_{S}}{U_{0}}e^{-\gamma_{2}S}$. (This corresponds to
$u_{0}$ past the Page time and $v_{h}\leq v_{0}$ with $v_{0}$ given by Eq.
(13).) In the Rindler region, the sphere $(U=0,V_{h})$ is related by a boost
to the sphere $(X=0,T(0))$ with $T(0)=-U_{0}+\sqrt{(V_{h}+U_{0})U_{0}}$. By
Eq. (16), there exists no solution for $\Sigma$ with subexponential extrinsic
curvature.

The only alternative to the presence of a quantum singularity is that the
theorem fails, which means that one of its assumptions must fail. Indeed, if
there were no restriction on the extrinsic curvature on a semiclassical slice,
then the example in this subsection could be viewed as a violation of the
quantum Bousso bound. Moreover, in the special case where $B$ is precisely the
black hole interior, the example would furnish a violation of the Generalized
Second Law of thermodynamics. Our viewpoint is that these assumptions are
valid in the semiclassical regime, and that the theorem has simply uncovered a
(possibly surprising) limitation of the semiclassical regime.

In simple models models where the entropy is approximated by a two dimensional
CFT (see e.g. Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019yqk ;
Hartman:2020swn ), it is easy to show that the quantum focusing conjecture is
satisfied along $H^{-}(B)$, even if we ignore the restriction to nice slices.
In particular, the quantum expansion formally exists and remains negative
along $H^{-}(B)$. However, the quantum Bousso bound is still violated by the
(non-nice) slice that stays below the black hole. It is important to emphasize
that starting with $S_{\rm gen}$ on a nice slice, and then integrating the
quantum expansion, is a valid method for computing the generalized entropy of
any other _nice_ slice. In particular one is permitted to continue past
caustics and self-intersections. Such features are generic, so this is a
crucial ingredient in important semiclassical generalization of classical
theorems. The problem in the present example is different: the slices do not
stay nice.

If one ignores this limitation, integration of $\Theta^{B\cup R}_{-}$ formally
yields negative values of $S_{\rm gen}$ well before the tip of the event
horizon is reached, followed by a discontinuity when the slices no longer
intersect $H^{-}(B)$. (To see this for an evaporating black hole, note that
the quantum expansion along the horizon is to a good approximation independent
of whether slices end at spatial infinity or at null infinity, and hence, so
is the integrated change in $S_{\rm gen}$. But the latter can be much greater
than the generalized entropy of the complement of $B\cup R$ when $R$ is the
Hawking radiation sufficiently far past the Page time.) $S_{\rm gen}<0$ has no
interpretation as a von Neumann entropy in a fundamental theory, so this would
be a nonsensical conclusion.

One might be tempted to “save” the GSL and the quantum Bousso bound for non-
nice slices, by observing that the exact von Neumann entropy of $R$ receives
non-perturbative corrections, which cause it to be bounded above by the
Bekenstein-Hawking entropy of the black hole. This is not correct.

First, the quantum Bousso bound is a semiclassical bound and is expected to
apply to the semiclassical state, not to the nonperturbatively correct state.
The same is true for the generalized second law. Calculating the generalized
entropy outside of an evaporating black hole using the exact von Neumann
entropy of radiation results in the violation of the generalized second law
after the Page time, but in the semiclassical state the entropy of radiation
increases throughout the process of evaporation, upholding the generalized
second law.

Secondly, the conditions of our theorem can be satisfied even when there is no
difference between the semiclassical and exact von Neumann entropy of $R$. For
example, take $U_{0}$ to correspond to a few scrambling times, rather than the
Page time. Then there exists a nonminimal quantum extremal surface associated
to $R$. Now, consider moving this surface in the outward past null direction
towards $\dot{I}(R)$. In simple 1+1 models with CFT matter, one finds that the
generalized entropy of the enclosed region union $R$ decreases without bound.
Therefore, at some point along the deformation the regions become
hyperentangled. Furthermore, one can check that the quantum expansion also has
the correct sign needed for the singularity theorem. However, again any Cauchy
slice containing $R$ and dipping below the past tip of the event horizon
necessarily has exponential extrinsic curvature. We view this as additional
evidence for our nice slice criterion. (In fact, the above construction fails
to yield a region that satisfies the assumptions of our theorem on a nice
slice.)

###  4.2 Classical vs. Quantum Singularity in Kerr-Newman Black Holes

Here we will discuss the singularity theorem applied to the future of $B$. In
the Schwarzschild solution, the generators of $H^{+}(B)$ are obviously
incomplete due to the curvature singularity at $r=0$, validating the
prediction of our theorem. In charged or rotating black holes, however, the
generators of $H^{+}(B)$ contain their endpoints, apparently violating Theorem
3.1. For simplicity, we will discuss this in detail for the Hartle-Hawking
state of the maximally extended Reissner-Nordstrom black holes of nonzero
charge, though we expect the main lessons to generalize to Kerr and Kerr-
Newman black holes. The metric is given by:

$\displaystyle ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega_{d-1}^{2}~{},$
(17)

where

$\displaystyle
f(r)=\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)~{}.$ (18)

We pick $R$ to be the union of the asymptotic region similar to subsection 4.1
and $B$ a late time slice of the interior such that $B$ is hyperentangled with
$R$. Furthermore, the quantum expansion of $B\cup R$ along $H^{+}(B)$ can be
easily arranged to be negative since the area variation towards the interior
is large and negative. Therefore, it is easy to satisfy the conditions of our
theorem towards $H^{+}(B)$. See Fig. 4.

In the classical Reissner-Nordstrom background, the generators of $H^{+}(B)$
contain their endpoints (see Fig. 6). These lie on a sphere $\mu$ near the
inner horizon bifurcation surface, of radius $r_{-}+\delta r$. The region $B$
is located around the Page time at the earliest, which implies

$\displaystyle\delta r\lesssim L\exp{\left(-\frac{\alpha
r_{+}^{4}}{r_{-}^{2}G}\right)}~{},$ (19)

where $L$ is some function of $r_{+}$ and $r_{-}$, and $\alpha$ is an order
one coefficient. Therefore, $\mu$ is exponentially close to the inner horizon
bifurcation surface.

Figure 6: In the Reissner–Nordstrom geometry, the region which is
exponentially close to the inner horizon in area radius (shown in red) has the
property that any Cauchy slice intersecting it necessarily has exponentially
large extrinsic curvature somewhere. In the classical geometry, the generators
of $H^{+}(B)$ come to an end on a sphere $\mu$ inside the red region.
Therefore, a Cauchy slice (shown in black) which is nowhere to the past of
$H^{+}(B)$ has exponentially large extrinsic curvature.

We do not expect that any nice slice containing $R$ reaches the future of
$\mu$. As evidence for this, consider a constant $r$ slice which crosses $\mu$
or its future. Its extrinsic curvature satisfies

$\displaystyle
K\gtrsim\left[\frac{\left(\frac{r_{+}}{r_{-}}-1\right)}{L~{}r_{-}}\right]^{\frac{1}{2}}\exp{\left(\frac{\alpha
r_{+}^{4}}{2r_{-}^{2}G}\right)}~{}.$ (20)

There is a second, seemingly independent reason why the semiclassical geometry
may not contain $\mu$, and hence will satisfy the prediction of a singularity
by Theorem 3.1. Quantum corrections to the matter stress tensor are known to
become important near the inner horizon for a generic set of black hole
parameters Birrell:1978th ; Sela:2018xko ; Hollands:2019whz ;
Zilberman:2019buh ; Zilberman:2022aum .666For rotating BTZ, the stress tensor
at leading order can be regular at the inner horizon Dias:2019ery . However,
it has been argued that subleading corrections lead to a divergence
Emparan:2020rnp . For a simple toy model in which this can be shown, consider
conformal matter in a 1+1 dimensional Reissner-Nordstrom background. The
metric is

$\displaystyle ds^{2}=-f(r)\,du\,dv~{},$ (21)

with $u=t-r^{*}$ and $v=t+r^{*}$ where $dr^{*}=dr/f(r)$. Setting the infalling
flux to zero, the trace anomaly and conservation of the stress tensor imply:

$\displaystyle\langle T_{\mu\nu}k^{\mu}k^{\nu}\rangle\sim
c\frac{\kappa_{-}^{2}-\kappa_{+}^{2}}{r^{2}}~{},$ (22)

where $k^{\mu}=\partial_{v}$, $\kappa_{+}$ and $\kappa_{-}$ denote the outer
and inner horizon surface gravities and $c$ denotes the central charge of the
CFT. Here we have added powers of $r$ by dimensional analysis to turn (22)
into an equation for 3+1 dimensions. A detailed derivation of the stress
tensor can be found in Zilberman:2019buh  where it is shown that the
coefficients in Eq. (22) are more complicated. We assume here that Eq. (22) is
valid qualitatively for all $d$, though the detailed dependence on $r_{\pm}$
may be more complicated.

The stress tensor given in Eq. (22) seems regular because in $(u,v)$
coordinates the inner horizon is at infinity. In coordinates which are regular
in a neighborhood of the inner horizon, the stress tensor can be shown to
diverge. Here it is important to show that Eq. (22) will cause a large
deviation of the metric from the classical geometry in a neighborhood of the
inner horizon. The location $r$ at which the geometry gets $O(1)$ corrections
from this stress tensor can be estimated by inspecting Raychaudhuri’s equation
for a spherically symmetric congruence near the inner horizon:

$\displaystyle\partial_{v}\theta_{v}=\kappa_{v}\theta_{v}-\frac{\theta_{v}^{2}}{d-1}-8\pi
G\langle T_{vv}\rangle$ (23)

where $\kappa_{v}$ is the inaffinity. The quantum stress tensor becomes
comparable to the other terms for $r-r_{-}\lesssim r_{\text{max}}$ where

$\displaystyle r_{\text{max}}-r_{-}\sim\sqrt{G}$ (24)

Therefore, it is clear that the generators of $H^{+}(B)$ exit the region which
is well-approximated by the classical solution (17). To further understand the
nature of the incompleteness of $H^{+}(B)$ would require knowledge of the
correct geometry which is beyond the scope of this work. A natural guess would
be that the geometry terminates at a spacelike singularity, directly upholding
theorem 3.1. See Fig. 7.

Figure 7: (a) Generically the expectation value of the quantum stress tensor
diverges near the inner horizon, causing large backreaction (grey). The
generators of $H^{+}(B)$ encounter this region, and our singularity theorem
predicts that they are incomplete. The quantum-corrected geometry is not
known, but the singularity theorem predicts that it either fails to admit nice
slices or has a curvature singularity as shown in (b).

Our analysis has shown that quantum singularities blurs the line between an
ordinary singularities and a Cauchy horizon. Even if quantum backreaction does
not create an ordinary singularity, a quantum singularity forms before a
Cauchy horizon can be reached. This may have some bearing on the strong cosmic
censorship hypothesis, that physically reasonable spacetimes are globally
hyperbolic. Strong cosmic censorship appears to be violated in the final
stages of black hole evaporation and of the Gregory-Laflamme instability.
These violations are in some sense small Emparan:2020vyf  and should perhaps
be ignored. The classical Kerr-Newman solution is of greater concern. The fact
that our theorem treats quantum singularities on the same footing as classical
singularities encourages us to think of the evolution near the inner horizon
as becoming singular, since no nice slices are available. We should treat this
quantum singularity no differently than a classical one. Hence we need not
rely on arguments that the backreaction from quantum effects would invalidate
the classical Kerr-Newman solution near the inner horizon. Let $N$ be the past
neighborhood of the inner horizon that cannot be reached by nice global Cauchy
slices. We should treat $N$ the same as the small past neighborhood $N$ of the
Schwarzschild singularity in which scalar curvature invariants approach or
exceed the Planck scale: $N$ should not be part of the physical spacetime, the
semiclassical geometric description terminates at the past boundary of $N$,
and any geometric extensions of $M-N$ that we could consider are physically
meaningless.

## Acknowledgements

We are grateful to Aidan Chatwin-Davies, Chitraang Murdia, Andrew Rolph, and
Edgar Shaghoulian for their comments on Ref. Bousso:2021sji , which encouraged
the present study. We also thank Ven Chandrasekaran, Tom Hartman, Adam Levine,
Raghu Mahajan, Leonard Susskind, Marija Tomasevic, Aron Wall, Zhenbin Yang,
and especially Juan Maldacena for useful discussions. We further thank
Alexander Frenkel, Andrew Rolph, and Michelle Xu for comments on the first
version of this paper. This work was supported in part by the Berkeley Center
for Theoretical Physics; by the Department of Energy, Office of Science,
Office of High Energy Physics under QuantISED Award DE-SC0019380 and under
contract DE-AC02-05CH11231; and by the National Science Foundation under Award
Numbers 2112880 (RB) and 2014215 (ASM).

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#  Jack the Rippler: Arbitrage on the Decentralized Exchange of the XRP Ledger

Gaspard Peduzzi  APWine
Lausanne, Switzerland
<EMAIL_ADDRESS>Jason James  UCL
London, UK
<EMAIL_ADDRESS>Jiahua Xu  UCL Centre for Blockchain Technologies
London, UK
<EMAIL_ADDRESS>
###### Abstract

The XRP Ledger (XRPL) is a peer-to-peer cryptographic ledger. It features a
decentralized exchange (DEX) where network participants can issue and trade
user-defined digital assets and currencies. We present Jack the Rippler, a bot
that identifies and exploits arbitrage opportunities on the XRPL DEX. We
describe Jack’s arbitrage process and discuss risks associated with using
arbitrage bots.

###### Index Terms:

Ripple, XRP, Arbitrage, Decentralized Exchange, Blockchain, Distributed Ledger
Technology

††publicationid: pubid: 978-1-6654-3924-4/21/$31.00 ©2021 IEEE

##  I Background

###  I-A Cross-currency arbitrage

Arbitrage is the simultaneous purchase and sale of an asset to profit from an
imbalance in price. It is a trade that profits by exploiting the price
differences of identical or similar financial instruments on different markets
or in different forms. In the conventional currency market, arbitrage is
difficult to execute due to various market frictions such as excessive order
book spread, high transaction fees, capital controls, etc. On the XRP Ledger
(XRPL)’s decentralized exchange (DEX), users are able to conduct cross-
currency payments with little friction thanks to its high speed and low
transaction fees.

In this study, we illustrate the existing arbitrage behavior on the DEX of
XRPL, and present Jack the Rippler, a bot that takes arbitrage opportunities
on XRPL.

###  I-B Key XRPL components

####  I-B1 Transaction types

Multiple types of transactions can be conducted on XRPL:

##### OfferCreate

an order for a given currency pair.

##### OfferCancel

cancellation of an existing order.

##### Payment

a peer-to-peer payment or a fulfillment of an existing order.

##### TrustSet

setting a trustline between two accounts.

####  I-B2 Flags

XRPL transactions apply different flags that dictate a transaction’s behavior:

##### tfPartialPayment

a “best flow effort" flag that allows partial payment and prevents the
transaction being cancelled in case of insufficient liquidity on the path or
insufficient balance in the recipient trustline.

##### tfNoDirectRipple

specifying the exact path (ordered list of offers) to be used when consuming
the transaction. This is intended to force the transaction to take the
arbitrage opportunity.

####  I-B3 Trustlines

The success of a Payment transaction necessitates the existence of a trustline
established by the asset receiver towards the asset issuer. A trustline can
either be a direct linkage between the receiver and the issuer, or, can be an
indirect path if the flag tfNoDirectRipple is not activated. Absent a
trustline, a payment transaction fails with tecPATH_DRY error. XRP is the only
asset on XRPL that does not require a trustline for successful transactions.

####  I-B4 Assets

With the exception of the native currency XRP, a digital asset on XRPL exists
in the form of an IOU (“I owe you”) [3], and is primarily characterized by
(i) a currency code (i.e. ticker), e.g. BTC, USD, CNY; and  (ii) the
introduction of further criteria or constraints.

Anyone can act as a gateway on the XRP ledger and issue their own IOU tokens.
An XRPL account must create a trustline to a gateway through a TrustSet
transaction to indicate willingness to hold a specific currency issued by the
gateway. Ripple provides a list of featured issuers, based on their business
practices, volume, and other measures. Yet there is no formal process for
ascertaining whether or not a gateway can be trusted, and whether their issued
IOU tokens are ultimately redeemable.

##  II Arbitrage on XRPL

Figure 1: Stylized arbitrage on XRPL

XRPL does not allow a cross-currency Payment cycle from XRP to XRP. Therefore,
two transactions must be performed to complete an XRP to XRP arbitrage cycle.
Fig. 1 illustrates a stylized XRP to XRP arbitrage scheme on XRPL, where
accounts $\mathcal{A}$ and $\mathcal{B}$ are controlled by the same entity.
Specifically, $\mathcal{A}$ first performs an OfferCreate transaction with the
intention to buy XRP with self-issued currency CUR that only $\mathcal{B}$ has
the necessary trustline to accept. $\mathcal{A}$ then sends a cross-currency
Payment to $\mathcal{B}$ where $\mathcal{A}$ pays XRP and $\mathcal{B}$
receives CUR, triggering the fulfillment of $\mathcal{A}$’s outstanding order.
Fig. 1 shows that the aggregate gain of $\mathcal{A}$ and $\mathcal{B}$ equals
$x^{\prime}-x-e$. Naturally, the process is only profitable if this number is
positive. Historically, this scheme is found to be heavily used by
rhAenHTxnAAxpMsrWorCcgfdzjyg6zsXRP and rsdMbYxHmYswHCg1V6vBsnxmHuCjpn6SC4.

###  II-A Arbitrage process with Jack the Rippler

Fig. 2 illustrates the arbitrage process employed by Jack. Note that one round
of arbitrage needs to be completed within one ledger interval, around 3.5
seconds.

Figure 2: Jack’s arbitrage process.

####  II-A1 Building the order book

We use the XRPL websocket APIs to listen to new blocks. At the launch of Jack,
the initial state of the order book is pulled with the book_offers request,
which returns the current order book state of an issuer, and after the initial
state is constructed, it is updated continually with every new block broadcast
to the network.

####  II-A2 Building the graph

We use a modified Bellman-Ford algorithm [1] to build the graph. We first sort
orders upon their retrieval. We construct two lists of orders for each trading
pair (in both directions). We then construct a graph with the “best offer”
(highest bid, or lowest ask) for each pair as the only edge between the two
currencies. While new orders are added to the order book at each new block, we
keep track of whether or not we modified the position of a previous “best
offer" in our sorted lists. If not, the graph remains the same with no need to
re-run the cycle detection algorithm. If at least one edge has been modified
due to the new block, we replace all the modified edges in the smaller graph
and run cycle detection again. Thus, the algorithm (Algorithm 1) progressively
relaxes all edges on the graph, updating the shortest distance to each edge.

Result: distance[], predecessor[]

Input: list $vertices$, list $edges$, vertex $source$

1

2// Initialization

3 foreach  _vertex $v\in vertices$ _ do

4 distance[$v$] := inf

5 predecessor[$v$] := null

6distance[$source$] := 0

7// Relax the edges repeatedly

8 for  _i from 1 to size(vertices)-1_ do

9 foreach  _$w$ -weighted edge $(u,v)\in edges$ _ do

10 if  _distance[ $u$] + $w$ < distance[$v$]_ then

11 distance[$v$] := distance[$u$] + $w$

12 predecessor[$v$] := $u$

13

14

15

16// Look for negative cycles

17 foreach  _$w$ -weighted edge $(u,v)\in edges$ _ do

18 if  _distance[ $u$] + $w$ < distance[v]_ then

19 output "The graph contains a negative cycle"

20

21

22return distance[], predecessor[]

Algorithm 1 Bellman-Ford cycle detection

Running this algorithm $|v|-1$ times, we are able to obtain the shortest
distance to each edge. Running one extra iteration on this graph, we can
determine if there are negative cycles [2] in the graph, depending on whether
or not the minimum distances were updated after this iteration. A negative
cycle represents a path with a negative cost, i.e. a positive gain.

####  II-A3 Taking an opportunity

Upon detection of a valid negative cycle, our bot Jack is able to build
transaction paths according to the cycle. We then select the corresponding
trustline to issue our payment and sign the transaction before publishing it
through our validator.

###  II-B Caveats

Risks associated with the usage of arbitrage bots still exist.

####  II-B1 Incomplete cycle

We may identify opportunities of arbitrage that become invalid in the next
ledger, due to pending transactions being fulfilled by other actors prior to
our bot. Such unsuccessful attempts still incur transaction fees, resulting in
a pure loss.

####  II-B2 Trusted actors

XRP is the only native asset in the ledger. Other currencies are issued in the
form of IOU debt by issuing gateways who are responsible for honoring the debt
[4]. The solvency of those gateways cannot be guaranteed and there always
exists a certain degree of irredeemability of IOU tokens. We have chosen for
our experiment to open trustlines only with issuers featured on the list
provided by Ripple to mitigate the counterparty risk.

##  III Conclusion

We present the existing arbitrage behavior on XRPL DEX and introduce Jack the
Rippler, a bot that can automatically detect and take arbitrage opportunities.
We discuss the limitations of our algorithm and argue that there still exist
risks associated with using arbitrage bots.

## Acknowledgment

This paper is based on SpringBlock Labs’ entry in the 2019 Block-Sprint
Hackathon. We thank teammates Hugo Roussel, Henry Declety and Matthieu Baud
for their contribution to the early development of the
software.111https://github.com/GaspardPeduzzi/spring_block

## References

  * [1] R. Bellman, “On a routing problem,” _Quarterly of Applied Mathematics_ , vol. 16, no. 1, pp. 87–90, 1958.
  * [2] B. V. Cherkassky and A. V. Goldberg, “Negative-cycle detection algorithms,” _Mathematical Programming_ , vol. 85, no. 2, pp. 277–311, 6 1999. [Online]. Available: http://link.springer.com/10.1007/s101070050058
  * [3] D. Perez, J. Xu, and B. Livshits, “Revisiting Transactional Statistics of High-scalability Blockchains,” in _The ACM Internet Measurement Conference_ , 10 2020, pp. 535–550. [Online]. Available: https://dl.acm.org/doi/10.1145/3419394.3423628
  * [4] XRPL, “Become an XRP Ledger Gateway - XRP Ledger Dev Portal,” 2020. [Online]. Available: https://xrpl.org/become-an-xrp-ledger-gateway.html