# Do Large GPT Models Discover Moral Dimensions in Language Representations? A
Topological Study Of Sentence Embeddings.

Stephen Fitz
<EMAIL_ADDRESS>

###### Abstract

Since the release of ChatGPT, and subsequently GPT-4, there has been an
increased interest in AI safety and alignment research. As Large Language
Models are deployed within Artificial Intelligence systems, that are
increasingly integrated with human society, it becomes more important than
ever to study their internal structures. Higher level abilities of LLMs such
as GPT-3.5 emerge in large part due to informative language representations
they induce from raw text data during pre-training on trillions of words.
These embeddings exist in vector spaces of several thousand dimensions, and
their processing involves mapping between multiple vector spaces, with total
number of parameters on the order of trillions. Furthermore, these language
representations are induced by gradient optimization, resulting in a black box
system that is hard to interpret. In this paper, we take a look at the
topological structure of neuronal activity in the “brain” of Chat-GPT’s
foundation language model, and analyze it with respect to a metric
representing the notion of _fairness_. We develop a novel approach to
visualize GPT’s moral dimensions. We first compute a fairness metric, inspired
by social psychology literature, to identify factors that typically influence
fairness assessments in humans, such as legitimacy, need, and responsibility.
Subsequently, we summarize the manifold’s shape using a lower-dimensional
simplicial complex, whose topology is derived from this metric. We color it
with a heat map associated with this fairness metric, producing human-readable
visualizations of the high-dimensional sentence manifold. Our results show
that sentence embeddings based on GPT-3.5 can be decomposed into two
submanifolds corresponding to fair and unfair moral judgments. This indicates
that GPT-based language models develop a moral dimension within their
representation spaces and induce an understanding of fairness during their
training process.

## INTRODUCTION

Figure 1: Large transformer language models develop their understanding of
sentences by inducing informative vector space representation of linguistic
units within their neural activations. The above figure summarizes this idea.
A corpus of text is transformed into a point cloud of vectors by the base
language model. This induces an implicit embedding of any text into a
submanifold of a high dimensional ambient vector space, whose metric is
learned in self-supervised fashion by backpropagation gradient descent on
large amounts of text extracted from the internet. It is precisely the
topology and geometry of that vector space that allow AI assistants such as
Chat-GPT to perform their cognitive functions. (The figure above shows the
point cloud in 3D for visualization purposes, while the manifold we analyze in
this study is embedded within a much larger space $\cong\mathbb{R}^{1536}$.)

Large Language Models and Natural Language Processing systems based on them
are currently at the forefront of research and applications of Artificial
Intelligence. Progress in this subfield of AI was made possible through
increasingly more advanced representation methods of natural language inputs.
Initially shallow pre-training of early model layers became standard in NLP
research through methods such as word2vec [1]. Subsequent progress followed
trends similar to those in Computer Vision, which naturally led to pre-
training of multiple layers of abstraction. These advancements resulted in
progressively deeper hierarchical language representations, such as those
derived using self-attention mechanisms in transformer-based architectures
[2]. Currently SOTA NLP systems use representations derived from pre-training
of entire language models on large quantities of raw text, and often involve
billions of parameters. Those informative distributed representations of
language in high dimensional vector spaces induced from large quantities of
raw text are at the core of emergent abilities of Large Language Models.

Vector space representations of linguistic units within language models
reflect the social propensities determined by the psychology literature, as
language reflects the social values of its speakers [3] [4]. Therefore,
sentences describing fair acts will be more closely associated with sentences
describing responsibility, benefit, joy, and reward than their antithesis
terms of irresponsibility, harm, sadness, and punishment. This pro-social bias
can be leveraged to construct a moral compass within LLM embedding manifolds.
To this end linear spans of specially selected vectors can be used [5] to
narrow the implicit ontological associations. Word embeddings implicitly
reflect ontological knowledge [6] [7] [8] [9], such as grammatical ontologies
due to the co-occurrence of specific grammatical knowledge in the co-
occurrence of words [10]. The term "fairness," being a collection of several
social ontologies, may be represented using linear combinations of hand
crafted basis vectors.

The seminal work of [1], [11], and [12] was foundational in providing
machinery for progress in this area. Later studies have demonstrated that
language models hold implicit representations of moral values as seen in [13]
[14] [15]. The authors of [16] applied word embeddings to analyze cultural
meaning and [17] used them for a study on social justice. Moral Foundation
Theory developed in [18] was used to analyze texts, and previous studies
labeled datasets with categories and applied machine learning algorithms to
learn the distinctions between each category. Similarly, the authors of [19]
[20] [21] [22] [23] worked on predefined measures of moral language. Prior
exploration of psychological factors influencing fairness assessments includes
Dictator Game and its variations, including [24] [25] [26] [27] [28]. These
studies emphasize the role of legitimacy and need in pro-social behavior. The
principal factor of responsibility is discussed with reference to several
studies, such as [29] [30] [31] [32]. They highlight the influence of
responsibility on pro-social behavior and its dependency on cultural climate.
Additionally, contingent factors that influence responsibility perception were
explored in studies by [33] [34] [35] [36] for the benefit-harm gained; [37]
[38] [39] for the consideration of wider public benefit and harm; [40] [41]
[42] [43] for the emotional salience of the context; [44] [45] [46] [47] [48]
[42] [49] for the possible consequences of rewards and punishments. Several
studies aimed to remove or diminish the role of responsibility, including [50]
[51] [44]. These studies show that when responsibility is diminished, pro-
social behavior decreases. By integrating these observations, the authors of
[15] developed a fairness metric for natural language text, that approximates
human perceptions of fairness.

Figure 2: Vector space representations induced by LLMs such as GPT-3.5 are the
result of multiple nonlinear mappings through a sequence of high dimensional
ambient vector spaces. Simple dimensionality reduction techniques such as
linear projections can be very misleading. We would like to visualize the
distribution of sentence embeddings with respect to the fairness dimension (as
defined here) while preserving some notion of shape of the original space. In
order to do so, instead of simply projecting sentence embeddings onto that
direction, we cluster the original data by the mapping technique described in
this paper. This results in a low dimensional simplicial complex, whose shape
can be thought of as a coarse grained summary of the topological features of
the original high dimensional representation manifold. Here vertices
correspond to clusters of sentence embeddings, while edges correspond to
intersections of clusters in the projection onto the fairness direction.
Higher dimensional simplices (triangles, tetrahedra, …) would correspond to
multiple clusters intersecting. In our study we study this representation only
up to the graph dimension, which gives a good enough sketch of the manifold’s
shape to conclude separation into two submanifolds corresponding to fair
versus unfair moral judgments.

In this paper we probe the topological structure of vector space
representations of sentences induced by GPT-3.5 - the foundation language
model behind Chat-GPT in order to gauge its understanding of moral values. For
this purpose we introduce novel approaches from computational algebraic
topology into the study of LLMs. We compute a metric of fairness based on
prior studies in computational social sciences, and use it to perform a
topological summary of 1536-dimensional sentence embedding manifold with a
one-dimensional simplicial complex (i.e. a graph) whose vertices correspond to
clusters of sentence representations under a projection onto a "fairness"
dimension, and edges to pairwise cluster intersections. This procedure
produces a sketch of the original high-dimensional manifold (which is
impossible to visualize directly) with a topological object that can be
visualized in human readable form. Furthermore, the shape of the resulting
visualization gives clues about the general shape of the representation space
of the language model. We also produce a heat map with colors corresponding to
the degree of fairness. This coloring allows us to inspect the manifold for
separation into submanifolds corresponding to fair vs unfair moral judgments.
We observe a clear pattern of separation, which suggests the notion of
fairness is encoded in the shape of the high dimensional representation
manifold induced by the language model.

## FAIRNESS METRIC

Figure 3: Topological summary of the 1536-dimensional vector space
representation within GPT-3.5 of the _fairness corpus_ [15] composed of equal
number of fair and unfair sentences (according to human judgement). Each node
is composed of multiple sentences. The size of the node corresponds to the
cardinality, and the color to fairness metric (dark purple corresponding to
unfair, and bright yellow corresponding to fair directions within the
embedding manifold). The heat map on the left corresponds to the fairness
measure based on the criterion described here (magnitude of the projection
onto the fairness subspace), while the one on the right to human judgments (-1
for unfair, 1 for fair).

The authors of [15] propose to harness implicit social biases in language as a
metric for an explainable assessment of sentences related to fairness. They
delve into the psychology literature to identify factors that influence
humans’ fairness assessments. They discuss the Dictator Game and its
variations, which have been used to isolate factors that predict pro-social
behavior, such as legitimacy and need. They also highlight the principal
factor of responsibility, which has been shown to play a critical role in pro-
social behavior and fairness assessments. We compute a fairness metric based
on those ideas with respect to sentence representations induced by the
foundation language model behind Chat-GPT3.5. We subsequently use this metric
to analyze the shape of high dimensional point clouds of sentence embeddings
induced by the language model.

Similarly to [15] we define the fairness dimension as a linear combination of
5 factors representing concepts of responsibility, pleasure, benefit, reward,
and harm. In our computation, every factor is derived from pairs of sentences
representing two polarities of each factor. The vector representations are
obtained from stimulating neurons of the transformer encoder blocks within
GPT3.5 by a set of several hundred examples of sentences representing fair vs
unfair moral situations according to human feedback.

In particular the fairness dimension is defined to be the linear span of the
vector defined by:

$\vec{\textbf{resV}}-\vec{\textbf{irresV}}+\vec{\textbf{joyV}}-\vec{\textbf{painV}}+\vec{\textbf{libV}}-\vec{\textbf{priV}}+\vec{\textbf{benV}}-\vec{\textbf{harV}}+\vec{\textbf{appV}}-\vec{\textbf{inappV}}$

, where resV, irresV, joyV, painV, libV, priV, benV, harV, appV, inappV are
GPT-3.5 derived vector space representations of respective concepts: "it was
very respondible", "it was very irresponsible", "it was joyous", "it was sad",
"it was beneficial to society", "it was not beneficial to society", "was free
to and rewarded", "was sent to prison and punished", "it was beneficial", "it
was harmful".

Because large GPT models are trained in autoregressive fashion on trillions of
tokens of text, the metric they induce in the sentence representations encodes
various features of natural language sentences, including semantics and moral
judgments. This translates to the inner product within those spaces expressing
similarity across multiple aspects of natural language. Convex combinations of
sentence embeddings can be used to express a semantic gradient ranging between
the chosen concepts expressed by these sentences. Very large language models
like GPT-3.5 induce surprisingly powerful sentence representations, leading to
the observed emergent properties of these models. If we consider vector space
representations of sentences "it was beneficial" and "it was harmful", and
compare them to a representation of a test sentence, such as "the guard helped
the man" by computing cosine similarity, the result will be a score in the
interval $[-1,1]$. The more associated the sentence is with benefit, the
closer to 1 will this normalized inner product tend to be. Whereas sentences
that are more associated with harmfulness will provide an outcome closer to
-1.

In figure 3 we summarize this by displaying side by side comparison between
human judgment and the distribution of the fairness metric within GPT-3.5’s
text representation manifold.

## METHOD OF ANALYSIS

Figure 4: Inducing topological structure from a point cloud representing noisy
samples from a neighborhood of a 1-dimensional submanifold of
$\mathbb{R}^{2}$. The resulting simplicial complex is a graph with four
vertices corresponding to clusters of the projection on the left, and four
edges corresponding to their intersections in the original embedding space.
Note that the simplicial complex obtained is homotopy equivalent to the circle
$\mathbb{S}^{1}$ (a 1-dimensional topological manifold) and gives a reasonable
summary of the shape of this original 2D point cloud of vectors.

The technique we use to study GPT-3.5’s sentence embeddings can be thought of
as a topological dimensionality reduction method, where the goal is to
summarize the shape of our representation space with a rough sketch in form of
a low dimensional topological manifold. This reduced representation can be
thought of as a map approximating the shape of our embedding space. Such
description can be visually inspected by a human, while remaining more
topologically informative than a naive projection. Instead of studying those
high dimensional vectors directly, we can map them to a different space first,
define an open cover, and then cluster the original points within the preimage
of each cover set. This produces a summary of the topological features in the
embedding with a simplicial complex of a chosen dimension [52]. In particular
we generated 1-dimensional simplicial complexes (i.e. graphs) from GPT-3.5
based sentence representations by projecting them onto the fairness dimension
(from the last section) and then analyzing clusters of original high
dimensional sentence embeddings with respect to an open cover of that
projection. We also applied a coloring according to a heat map corresponding
to the magnitude and sign of that projection. The resulting data structure
allows for a visual exploration of the shapes of these high dimensional
embedding manifolds in order to identify distribution of fairness in the
representation space. Figure 4 shows a visualization of this process for a
point cloud sampled from the circle ($\mathbb{S}^{2}$).

The general procedure can be summarized as follows.

Given data points $\mathbb{X}=\\{x_{1},\dots,x_{n}\\},x_{i}\in\mathbb{R}^{d}$,
a function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{m},m<d$, and a cover
$\mathcal{U}=\bigcup_{i\in\mathcal{I}}U_{i}$ of the image $f(\mathbb{X})$
(where $\mathcal{I}$ is some index set) we construct a simplicial complex as
follows: 1. For each $U_{i}\in\mathcal{U}$, cluster $f^{-1}(U_{i})$ into
$k_{U_{i}}$ clusters $C_{U_{i,1}},\dots,C_{U_{i},k_{U_{i}}}$ 2.
$\underset{U_{i}\in\mathcal{U}}{\bigsqcup}\\{C_{U_{i,1}},\dots,C_{U_{i},k_{U_{i}}}\\}$
now define a cover of $\mathbb{X}$; calculate the nerve of this cover Nerve is
defined in the following way. Given a cover
$\mathcal{U}=\bigcup_{i\in\mathcal{I}}U_{i}$, the nerve of $\mathcal{U}$ is
the simplicial complex $\mathcal{C}(\mathcal{U})$ where the 0-skeleton is
formed by the sets in the cover (each $U_{i}$ is a vertex) and
$\sigma=[U_{j_{0}},\dots,U_{j_{k}}]$ is a k-simplex
$\iff\bigcap\limits_{l=0}^{k}U_{l_{k}}\neq 0$

## DISCUSSION OF RESULTS

Figure 5: The fairness subspace clearly subdivides the 1536-dimensional
representation space within the GPT-3.5 language model, into two submanifolds.
When the language model is prompted with fair sentences, different patterns of
neuronal activation are manifested within the transformer layers than when the
model is prompted with morally unfair examples. This leads to the sentence
representation manifold being separated into submanifolds corresponding to
these two extremes. Each part is itself a topologically complex space, but the
two parts are linked at a single cluster of morally ambiguous sentences. The
sentences represented in the green part of the manifold are all judged to be
fair, while all the sentences in the blue part represent unfair situations.

Visualization in figure 5 shows the results of applying our method to
1536-dimensional vector space representations of sentences obtained from
GPT-3.5 based embedding model. This is the foundation language model used by
ChatGPT, and we believe its emergent abilities to judge fairness of situations
are partially enabled by the topological structure of this embedding manifold.
Each node corresponds to a cluster of sentences obtained by first projecting
their embeddings onto the fairness subspace defined earlier in this paper, and
then grouping the original sentences based on preimages of an open cover of
this projection. This procedure gives a topological sketch of the general
shape of this high dimensional sentence representation manifold. Node sizes
are defined in proportion to the number of sentences in each cluster. Their
colors are based on the heat map shown to the right, which is a gradient
corresponding to the sign and magnitude of the projection. High values on that
scale correspond to fair, while low values to unfair situations (as described
by the sentences). The edges come from cluster intersections in the original
high dimensional space, and thus give visual cues to how the clusters are
distributed within the embedding manifold.

The two boxes contain fair and unfair sentences. We see that each box bounds a
topologically complex manifold (due to the nontrivial pattern of edges), but
there are no edges connecting nodes from different boxes. This means the
entire manifold is composed of two submanifolds corresponding to fair and
unfair situations. This separation of colors suggests that there are
subregions in this high dimensional representation space corresponding to
activation patterns within the encoder neural network that occur only when
fair sentences are processed, and separate disconnected regions corresponding
to neural activations caused by unfair sentences. Before the model is trained,
this separation could not occur (other than with infinitesimal probability
from random initialization), and the fact that GPT-3.5 arrives at this
topology in its neural representations of language, implies it (and hence
models based on it, such as ChatGPT) possess the ability to represent moral
dimensions of fairness.

The behavior of these models is determined by the vector space representations
of inputs they induce from text during training. Therefore, the existence of
clear patterns in the shape of those representation manifolds, such as
separation into regions corresponding to moral aspects of language, can serve
as a tool to intrinsically examine their abilities. This is in contrast to
currently used mainstream approaches, which are extrinsic and behavioral in
nature. In these contemporary approaches the abilities of the model are
examined with techniques based on prompting and analysing the outputs, or by
proxy of performance on downstream tasks. We believe the approach presented
here is novel in the field of AI, and NLP in particular. It is more analogous
to performing studies on the brain with something like fMRI or a Neuralink
device, and analyzing topological patterns in neural communication in order to
study the subject, instead of making judgment based on the subject’s apparent
behavior, or reports from introspection. We believe this alternative way of
looking at language models could benefit alignment research and AI safety, and
we hope this paper serves as an inspiration for the community to develop more
tools of this kind.

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††institutetext: Department of Physics,
University of California,
Berkeley, CA 94720, USA

# The Holographic Entropy Cone in AdS-Vaidya Spacetimes

Reginald J. Caginalp<EMAIL_ADDRESS>
###### Abstract

We examine the five-region holographic entropy cone inequalities for the
special case of the AdS3-Vaidya metric for a variety of boundary
configurations. This is done by numerically solving the geodesic equation in
the bulk for various boundary configurations. In all the cases we examine, we
find that all the inequalities are satisfied when the bulk satisfies the null
energy condition, while the inequalities are all violated when the bulk
spacetime violates the null energy condition. A proof of the five-region
holographic entropy cone inequalities for the dynamical bulk case remains an
open problem–our results provide evidence that these inequalities hold for
dynamical bulk spacetimes.

##  1 Introduction

Recent work has unveiled deep connections between gravity and entanglement.
The AdS/CFT correspondence Maldacena ; Gubser ; Witten  states that any theory
of quantum gravity in $(d+1)$-dimensional anti-de Sitter space (AdSd+1) is
equivalent to a conformal field theory (CFT) in $d$ dimensions. The Ryu-
Takayanagi (RT) RT  formula and its covariant generalization, the Hubeny-
Rangamani-Takayanagi (HRT) formula HRT , posit that the entanglement entropies
of holographic CFTs are given in terms of minimal or extremal areas. These
have been derived from the basic AdS/CFT dictionary RT_deriv ; HRT_deriv . In
general, entanglement entropies of quantum field theories are difficult to
compute. It is of great interest to try to determine which types of states are
dual to semi-classical AdS bulks. The fact that holographic entanglement
entropies are given by minimal areas should therefore enable us to constrain
the entanglement structure of holographic states.

It is simple to show that if we have three spatial CFT regions, $A,B,C$, then
the RT formula implies strong subadditivity RT_SSA ; MMI_SSA2 :

$S(ABC)+S(B)\leq S(AB)+S(BC).$

The above inequality is, of course, true for all quantum states Lieb1 ; Lieb2
, though the general proof is technically complicated. In addition,
holographic entropies obeying the RT formula obey the constraint of monogamy
of mutual information RT_MMI ; MMI_SSA2 :

$I(A:BC)\geq I(A:B)+I(A:C).$

Unlike strong subadditivity, this constraint is not obeyed by all quantum
systems. In addition, recent work Cone  has shown that holographic
entanglement entropies for $n$ regions obey a set of inequalities known as the
holographic entropy cone, assuming the RT formula holds. Recently, the exact
holographic entropy cone for five regions has been obtained Cuenca . However,
it is not known in general if these inequalities are valid for the covariant
HRT formula.

Using the maximin formalism of Wall Wall , it is possible to show that
(assuming the null-energy condition holds in the bulk), strong subadditivity
and monogamy of mutual information hold for the HRT formula. However, the
validity of the inequalities for the entropy cone for the HRT formula for five
or more regions remains unknown. Indeed, Rota  showed that the set of five-
region inequalities provable with the maximin formalism is less strong than
the entropy cone inequalities.

Understanding the validity of the entropy cone inequalities in the dynamical,
HRT case is thus an important step towards understanding the structure of
holographic states. In this paper, we will numerically calculate the
entanglement entropies for an AdS3-Vaidya spacetime, and examine the validity
of the five-region entropy cone inequalities, using the HRT formula. This is a
very simple setting to test these inequalities, since the HRT surfaces will be
geodesics (not higher-dimensional surfaces), and the AdS3-Vaidya solution is a
very simple dynamical spacetime.

We find that the inequalities are all valid, in the cases we examined, as long
as the bulk obeys the null energy condition. If the bulk violates the null
energy condition, then all the inequalities are violated. This is analogous to
the situation for strong subadditivity, which requires the NEC to hold in the
bulk. We believe that this provides strong evidence for the validity of the
five-region inequalities when the bulk is dynamical. Moreover, the shape of
the curves resemble those of the strong subadditivity curves. This may hint
that there is a reformulation of the HRT prescription for which both strong
subadditivity and the five-region inequalities are valid. Indeed, this has
already been done in certain limits for the positive-energy spacetime we
considered here BaoMezei .

Understanding the validity of these inequalities in general, as well as
further study of the entanglement of holographic states, will be very
important in furthering our understanding of quantum gravity.

Our results build on previous work on the validity of the five-region
inequalities for dynamical bulks. Flory  numerically verified the inequalities
for a holographic model of two 1+1 dimensional heat baths joined at $t=0$. It
has also been shown that they are valid for large, late-time CFT regions in
collapsing black hole spacetimes BaoMezei . Our work is closely analogous to
Headrick ; Allais , which numerically studied the validity of strong
subadditivity and monogamy of mutual information for AdS3-Vaidya spacetimes.

##  2 Setup

Consider a holographic CFT with a Cauchy slice $\Sigma$ of a static bulk, at a
moment of time reflection symmetry. Let $A$ be a boundary subregion. The Ryu-
Takayanagi formula posits that

$S(A)=\frac{\min_{m}\text{Area}}{4G_{N}},$

where $m$ is a codimension-2 surface in the bulk (with $\partial m=\partial
A$) homologous to $A$. That is, there is a bulk region $\chi$ such that
$\partial\chi=A\cup m$. The Hubeny-Rangamani-Takayanagi formula is the
covariant generalization of this equation. If $A$ is some spacelike CFT
subregion, then the HRT formula says that

$S(A)=\frac{\min_{\text{extremal }m}\text{Area}}{4G_{N}},$

where extremal $m$ means that $m$ is a co-dimension 2 spacelike surface that
extremizes the area and has $\partial m=\partial A$ and is homologous to $A$.

We will consider the planar AdS3-Vaidya spacetime, with metric

$ds^{2}=-(r^{2}-m(v))dv^{2}+2drdv+r^{2}dx^{2}.$

We wish to see when the null-energy condition (i.e.,
$T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$ for all lightlike $k$) is satisfied for this
metric. The Einstein tensor of this metric is given by

$G_{vr}=G_{rv}=1,G_{xx}=r^{2},G_{vv}=-r^{2}+m(v)+\frac{1}{2r}\frac{dm}{dv},$

with all other components vanishing. We need to find the null vectors
$k^{\mu}$ in the $r$-$v$ plane. Suppose $k^{\mu}=(C,D,0)$ so that we require

$-f(r,v)C^{2}+2CD=0,$

where we have introduced the function $f(r,v)\equiv r^{2}-m(v)$. There are two
solutions to this:

$C=0\text{ or }D=\frac{f}{2}C,$

so there are two such linearly-independent null vectors

$n^{\mu}=(0,1,0),\ell^{\mu}=(1,\frac{f}{2},0).$

Einstein’s equations are

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G_{N}T_{\mu\nu}.$

These imply

$T_{\mu\nu}n^{\mu}n^{\nu}=G_{\mu\nu}n^{\mu}n^{\nu}=G_{rr}=0,$
$T_{\mu\nu}\ell^{\mu}\ell^{\nu}=G_{\mu\nu}\ell^{\mu}\ell^{\nu}=G_{vv}+fG_{vr}=-f+f+\frac{1}{2r}\frac{dm}{dv}=\frac{1}{2r}\frac{dm}{dv}.$

Therefore, we see that the null energy condition is satisfied if and only if
$dm/dv$ is positive.

We can re-write this metric in more standard coordinates, $t$ and $r$, with

$v=t+g(r),g^{\prime}(r)=\frac{1}{f}$ $\implies
dv=dt+g^{\prime}(r)dr=dt+\frac{dr}{f},$

which means

$ds^{2}=-f(dt^{2}+\frac{dr^{2}}{f^{2}}+2\frac{dtdr}{f})+2drdt+2\frac{dr^{2}}{f}+r^{2}dx^{2}=-fdt^{2}+\frac{dr^{2}}{f}+r^{2}dx^{2}.$

We will consider a thin-shell limit,

$m(v)=\pm m\Theta(v),$

which represents a shell of infalling null matter. The plus sign satisfies the
null energy condition and corresponds to positive energy matter–inside the
shell, the metric is pure AdS3, outside the shell, the metric corresponds to a
black hole, i.e., the BTZ metric. The minus sign violates the null energy
condition; this choice represents a shell of negative-energy null matter so
that outside the matter, the metric is pure AdS, while inside it is BTZ. We
will consider both cases, starting with the positive-energy metric. For
simplicity, we will set $m=1.$

For AdS, we have $f=r^{2}$ so that

$g^{\prime}=-\frac{1}{r}\implies v=t-\frac{1}{r}.$

Meanwhile for BTZ, we have $g=r^{2}-1$, which gives

$g^{\prime}=-\frac{1}{r^{2}-1}\implies g=-\tanh^{-1}\frac{1}{r}\implies
v=t-\tanh^{-1}\frac{1}{r}.$

Our discussion of geodesic kinematics largely follows that of Headrick .

##  3 Positive Energy Vaidya Metric

We wish to obtain enanglement entropies in the CFT dual to the Vaidya metric.
The HRT prescription tells us that we need to calculate the areas of the
extremal codimension 2 surfaces that are anchored at the boundary of the CFT
subregion. In our case, this corresponds to spacelike geodesics. The geodesic
equation for $v$ is given by

$\ddot{v}+\frac{1}{2}\partial_{r}f\dot{v}^{2}-r\dot{x}^{2}=0,$

where an overdot denotes a derivative with respect to the affine parameter
$\tau$. In addition, the tangent vector $dx^{\mu}/d\tau$ has unit norm, so
that

$-f(r,v)\dot{v}^{2}+2\dot{v}\dot{r}+r^{2}\dot{x}^{2}=1.$

Because the metric has no explicit $x$-dependence, the quantity $p_{x}\equiv
g_{xx}\dot{x}=r^{2}\dot{x}$ is conserved along the geodesic. When we express
the metric in terms of $t$ and $r$ (instead of $v$ and $r$), we found that the
metric takes the form

$ds^{2}=-h(r)dt^{2}+\frac{1}{h(r)}dr^{2}+r^{2}dx^{2},$

where $h(r)=r^{2}$ in AdS, and $h(r)=r^{2}-m$ in BTZ. For simplicity, we will
set $m=1$ throughout. Other than at the shell, there is no explicit dependence
in the metric on $t$, so there is a quantity that is conserved except at the
shell. It is given by

$E\equiv g_{tt}\dot{t}=h(r)\dot{t}.$

In these coordinates, the normalization condition for the vector
$dx^{\mu}/d\tau$ becomes

$-h(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{h(r)}+r^{2}\dot{x}^{2}=1$
$\dot{r}^{2}=h+E^{2}-\frac{hp_{x}^{2}}{r^{2}}$

Alternatively, we can write this in terms of $r^{\prime}$, where the prime
denotes a derivative with respect to $x$. Then we obtain

$r^{\prime 2}=\frac{r^{4}h}{p_{x}^{2}}+\frac{r^{4}E^{2}}{p_{x}^{2}}-hr^{2}.$

We will first solve these equations for constant time intervals, and then
solve them for general covariant regions. For both of these, we will verify
strong subadditivity and the 5-body entropy cone inequalities.

###  3.1 Constant Time Intervals

We begin by considering a region $A$ on the boundary that is constant in time,
and is a single interval in $x$. That is,

$A=\\{(x,t)\in\text{CFT}|x\in[0,\ell_{x}],t=const=t_{b}\\}.$

To calculate the entanglement entropy $S(A)$ of this region in the CFT, the
HRT prescription tells us that we need to find the extremal boundary-anchored
curve (i.e., spacelike geodesic) $\chi_{A}$ such that
$\partial\chi_{A}=\partial A$ and $\chi_{A}$ is homologous to $A$.

Figure 1: A Penrose diagram of the Vaidya spacetime (the red line represents
the shell of null matter), showing the three cases for the spacelike
geodesics. (i) Entirely in the AdS bulk, (ii) entirely in the BTZ bulk, and
(iii) partially in the BTZ bulk, and partially in the AdS bulk.

There are three cases (i) the geodesic is entirely in the AdS bulk, (ii) the
geodesic is entirely in the BTZ bulk, or (iii) the geodesic is in both the BTZ
bulk and the AdS bulk. See Figure 1. We consider each of these cases in turn.

####  3.1.1 Geodesics Entirely in the AdS Bulk

We are considering a constant time geodesic in the AdS bulk, so $E=0$.
Therefore, we have

$\frac{dr}{d\tau}=\sqrt{r^{2}-p_{x}^{2}},$

which has solution

$r(\tau)=\frac{1}{2}(p_{x}^{2}e^{-\tau}+e^{\tau}).$

Now, $x$ obeys the equation

$\dot{x}=\frac{p_{x}}{r^{2}},$

which has solution

$x(\tau)=Const-\frac{2p_{x}}{p_{x}^{2}+e^{2\tau}}.$

This means that

$\ell_{x}=x(\tau=\infty)-x(\tau=-\infty)=\frac{2}{p_{x}}.$

We have normalized our affine parameter so that $\tau$ measures the length of
the curve. For $\tau$ approaching $\pm\infty,$ $r$ approaches $\infty$. For
large $R$, there are two roots of $\tau$., one large and positive, the other
large and negative. They are

$\tau_{b-}=-\log(2R)+2\log(p_{x})$ $\tau_{b+}=\log(2R),$

so the total length of the curve is

$L=\tau_{b+}-\tau_{b-}=2\log(2R)-2\log(p_{x}).$

To get to the boundary, of course, we need to send $R\rightarrow\infty$, and
the length diverges. Thus, to obtain a regularized, finite length, we need to
subtract the UV-divergent term. Thus, we obtain:

$L_{reg}=2\log\frac{\ell_{x}}{2}.$

This is a concave function, so it satisfies strong subadditivity.

####  3.1.2 Geodesics Entirely in the BTZ bulk

In this case we have

$\dot{r}=\sqrt{r^{2}-1+E^{2}-p_{x}^{2}+\frac{p_{x}^{2}}{r^{2}}}.$

There are two solutions to this equation:

$r_{1}(\tau)=\frac{1}{2}\sqrt{-2E^{2}+2p_{x}^{2}+2+e^{2\tau}-E^{4}e^{-2\tau}+2E^{2}(p_{x}^{2}+1)e^{-2\tau}-(p^{2}-1)^{2}e^{-2\tau}},$
$r_{2}(\tau)=\frac{1}{2}\sqrt{(-E^{2}+2E^{2}p_{x}^{2}+2E^{2}-p_{x}^{4}+2p_{x}^{2}-1)e^{2\tau}-e^{-2\tau}-2E^{2}+2p_{x}^{2}+2}.$

With a few lines of algebra, we can cast these in the following form:

$r_{1}(\tau)^{2}=\frac{1}{4}(e^{\tau}+B_{+}e^{-\tau})(e^{\tau}+B_{-}e^{-\tau}),$
$r_{2}(\tau)^{2}=-\frac{1}{4}(B_{+}e^{\tau}-e^{-\tau})(B_{-}e^{\tau}-e^{-\tau}),$

where we have defined the quantities

$B_{\pm}=(p_{x}\pm 1)^{2}-E^{2}.$

We are, of course, looking for geodesics that are boundary anchored. As $\tau$
goes to minus infinity, $r_{2}^{2}$ goes to $-1/4$. Thus, the solution $r_{2}$
can never describe the geodesics we are interested in. Therefore, we restrict
our attention to the solution $r_{1}$. We first obtain expressions for $x$ and
$t$. We find

$t(\tau)=const+\frac{1}{2}\log\left(\frac{A_{-}+e^{2\tau}}{A_{+}+e^{2\tau}}\right),$
$x(\tau)=const-\frac{1}{2}\log\left(\frac{B_{-}+e^{2\tau}}{B_{+}+e^{2\tau}}\right),$

where we have defined

$A_{\pm}=p_{x}^{2}-(1\pm E)^{2}.$

An analysis exactly the same as above gives us that the regularized length is
(subtracting off the UV-divergent term $2\log(2R)$)

$L_{reg}=-\frac{1}{2}\log(B_{+}B_{-}).$

Meanwhile,

$\Delta t=\frac{1}{2}\log(A_{-}/A_{+}),\text{
}\ell_{x}=-\frac{1}{2}\log(B_{-}/B_{+}).$

In particular, if we have a constant-time interval, then $E=0$ and

$L_{reg}=-\frac{1}{2}\log(p_{x}^{2}-1)^{2},\ell_{x}=\frac{1}{2}\log\left(\frac{p_{x}+1}{p_{x}-1}\right).$

Finally, we turn to geodesics that are partially in AdS and partially in BTZ.

###  3.2 Geodesics in both AdS and BTZ

Because the shell is located at $v=0$, if the interval is at boundary time
$t_{b}<0$, it will be in pure AdS. If $t_{b}>0,$ then we calculate
$v(\tau)=t(\tau)-\tanh^{-1}(1/r(\tau))$ for the pure BTZ geodesic, and see if
it dips below $0$. If it does, then it will have a component that is in the
AdS bulk. If not, it will be contained entirely in the BTZ bulk.

We begin by considering what happens at the junction of AdS and BTZ. Because
we do not want a delta function in $v^{\prime\prime}$, then $v^{\prime}$ needs
to be continuous. This means:

$v^{\prime}=2(r_{A}^{\prime}-r_{B}^{\prime}).$

In AdS space, we have that

$v^{\prime}=t^{\prime}+\frac{r_{A}^{\prime}}{r_{c}^{2}}=\frac{E_{A}}{p_{x}}+\frac{r_{A}^{\prime}}{r_{c}^{2}},$

where $r_{c}$ is the value of $r$ when the geodesic crosses the shell, and
$E_{A}$ is the value of $E$ in the AdS region. We combine these two to get

$r_{B}^{\prime}=-\frac{E_{A}}{2p_{x}}+\left(1-\frac{1}{2r_{c}^{2}}\right)r_{A}^{\prime}$

From the BTZ side we know that

$v^{\prime}=t^{\prime}+\frac{r_{B}^{\prime}}{r_{c}^{2}-1}=\frac{r_{c}^{2}}{r_{c}^{2}-1}\frac{E_{B}}{p_{x}}+\frac{r_{B}^{\prime}}{r_{c}^{2}}=\frac{r_{c}^{2}}{r_{c}^{2}-1}\frac{E_{B}}{p_{x}}+\frac{r_{B}^{\prime}}{r_{c}^{2}-1}.$

This gives us:

$(r_{c}^{2}-1)\frac{E_{A}}{p_{x}}+r_{A}^{\prime}-\frac{r_{A}^{\prime}}{r_{c}^{2}}=r_{c}^{2}\frac{E_{B}}{p_{x}}+r_{B}^{\prime}$
$(r_{c}^{2}-\frac{1}{2})\frac{E_{A}}{p_{x}}-\frac{r_{A}^{\prime}}{2r_{c}^{2}}=r_{c}^{2}\frac{E_{B}}{p_{x}}$
$E_{B}=(1-\frac{1}{2r_{c}^{2}})E_{A}-\frac{p_{x}r_{A}^{\prime}}{2r_{c}^{4}}.$

In addition, we know that

$r_{A}^{\prime}=\sqrt{\frac{r_{c}^{6}}{p_{x}^{2}}+\frac{r_{c}^{4}E_{A}^{2}}{p_{x}^{2}}-r_{c}^{4}}.$

The value of the affine paramter in the BTZ geodesic when $r=r_{c}$ is given
by

$\alpha_{c}\equiv\exp(2\tau_{c})=\frac{1}{2}\left[-(B_{+}+B_{-})+4r_{c}^{2}+\sqrt{-4B_{+}B_{-}+(B_{+}+B_{-}-4r_{c}^{2})^{2}}\right].$

There are two BTZ components. For second component, from $\tau_{c}$ to
$\tau=\infty$, we have

$\Delta
x_{B}=x(\tau=\infty)-x(\tau_{c})=-\frac{1}{2}\log\left(\frac{B_{-}+\alpha_{c}}{B_{+}+\alpha_{c}}\right),$
$\Delta
t_{B}=\frac{1}{2}\log\left(\frac{A_{-}+\alpha_{c}}{A_{+}+\alpha_{c}}\right).$

At the shell, we have that $t_{c}=\tanh^{-1}(1/r_{c})$, so the boundary time
is

$t_{b}=\tanh^{-1}(1/r_{c})+\Delta t_{B}.$

Meanwhile, the length of the curve is

$L_{B}=\log 2R-\frac{1}{2}\log\alpha_{c}.$

By symmetry, the portion of the curve in AdS has $E=0$, and the two BTZ
components have the same $\Delta x$ and the same length. The curve in AdS
obeys

$r(\tau)=\frac{1}{2}(p_{x}^{2}e^{-\tau}+e^{\tau})$

This satisfies $r=r_{c}$ at two values of the affine parameter

$\tau_{\pm}=\log(r_{c}\pm\sqrt{r_{c}^{2}-p_{x}^{2}}).$

Figure 2: The regularized geodesic length $L_{reg}$ as a function of the
boundary interval length, $\ell_{x}$ at boundary time $t_{b}=0.8$.

Using our expression for $x(\tau),$ we find that

$\Delta
x_{A}=x(\tau_{+})-x(\tau_{-})=\frac{2}{r_{c}p_{x}}\sqrt{r_{c}^{2}-p_{x}^{2}}$

and that

$L_{A}=\tau_{+}-\tau_{-}=\log\left(\frac{r_{c}+\sqrt{r_{c}^{2}-p_{x}^{2}}}{r_{c}-\sqrt{r_{c}^{2}-p_{x}^{2}}}\right).$

We know that $L=L_{A}+2L_{B}$ and $\ell_{x}=2\Delta x_{B}+\Delta x_{A}$. We
then find

$\ell_{x}=\frac{2}{r_{c}p_{x}}\sqrt{r_{c}^{2}-p_{x}^{2}}+\frac{1}{2}\log\left(\frac{B_{+}+\alpha_{c}}{B_{-}+\alpha_{c}}\right),$
$L_{reg}=\log\left(\frac{r_{c}+\sqrt{r_{c}^{2}-p_{x}^{2}}}{r_{c}-\sqrt{r_{c}^{2}-p_{x}^{2}}}\right)-\log\alpha_{c}.$

For given values of $t_{b}$ and $\ell_{x}$, we can solve numerically to find
the corresponding values of $r_{c}$ and $p_{x}$. We do this numerically for
$t_{b}=0.8$, and show the result in Figure 2.

(a) The quantity $T$ plotted vs $t_{b}$.

(b) The quantity $M$ plotted vs $t_{b}$.

Figure 3: The quantities $T$ and $M$ plotted vs $t_{b}$. We see that $T>0$ and
$M>0$ so strong subadditivity and monogamy of mutual information are always
satisfied. The jumps at late times are due to numerical errors.

We begin by testing strong subadditivity. We consider adjacent three regions
$A,B,C$, all at constant time $t_{b}$. We choose
$\ell_{A}=2,\ell_{B}=4,\ell_{C}=2$. We then consider the quantity

$T(A,B,C)\equiv 4G_{N}[S(AB)+S(BC)-S(ABC)-S(B)]$

as a function of the boundary time $t_{b}$. Strong subadditivity will be
satisfied if and only if $T\geq 0$. $S(AB)$ is given by the length of a
geodesic with $\ell_{x}=6$ over $4G_{N}$. $S(BC)$ is identical. $ABC$ is an
interval of length 8, while $B$ is an interval of length 4. Using this
information, we can plot $T$ as a function of $t_{b}$. This is done in Figure
3(a). We see that $T>0$, so strong subadditivity is always satisfied.

We do the same calculation for monogamy of mutual information. Define

$M\equiv 4G_{N}[I(A:BC)-I(A:B)-I(A:C)]\\\
=4G_{N}[S(A)+S(BC)-S(ABC)-S(A)-S(B)+S(AB)-S(A)-S(C)+S(AC)]\\\
=4G_{N}[-S(A)+S(BC)-S(ABC)-S(B)+S(AB)-S(C)+S(AC)].$ (1)

We plot this quantity as a function of $t_{b}$ in Figure 3(b). Again, since
$M>0$, we see that monogamy of mutual information is always satisfied.

Figure 4: The quantity $S_{0}$ plotted vs $t_{b}$. We see that $S_{0}>0$ so
the corresponding inequality is always satisfied. The jump at late times is
due to numerical errors.

Next, we consider five regions. There are several inequalities that are valid
for the RT formula for five regions. These bound the holographic entropy cone.
For example, we have that

$S(A|BC)+S(B|CD)+S(C|DE)+S(D|EA)+S(E|AB)\geq S(ABCDE)$

(see Cone ). Here $S(X|Y)\equiv S(XY)-S(Y)$ is the conditional entropy. To
test this in the non-static case, consider 5 regions $A,B,C,D,E$, all of which
are constant time intervals on the boundary. Take
$\ell_{A}=\ell_{C}=\ell_{E}=2$, $\ell_{B}=\ell_{D}=4$. Then define the
quantity

$S_{0}\equiv 4G_{N}[S(A|BC)+S(B|CD)+S(C|DE)+S(D|EA)+S(E|AB)-S(ABCDE)].$

We plot this as a function of $t_{b}$. See Figure 4. We see that $S_{0}>0$ so
this inequality is always satisfied in this non-static case.

There are several more inequalities for the holographic five-region case Cone
. For example,

$2S(ABC)+S(ABD)+S(ABE)+S(ACD)+S(ADE)+S(BCE)+S(BDE)\geq\\\
S(AB)+S(ABCD)+S(ABCE)+S(ABDE)+S(AC)+S(AD)+S(BC)+S(BE)+S(DE),$ (2)

$S(ABE)+S(ABC)+S(ABD)+S(ACD)+S(ACE)+S(ADE)+S(BCE)+S(BDE)+S(CDE)\geq\\\
S(AB)+S(ABCE)+S(ABDE)+S(AC)+S(ACDE)+S(AD)+S(BCD)+S(BE)+S(CE)+S(DE),$ (3)

$S(ABC)+S(ABD)+S(ABE)+S(ACD)+S(ACE)+S(BC)+S(DE)\geq\\\
S(AB)+S(ABCD)+S(ABCE)+S(AC)+S(ADE)+S(B)+S(C)+S(D)+S(E),$ (4)

$3S(ABC)+3S(ABD)+3S(ACE)+S(ABE)+S(ACD)+S(ADE)+S(BCD)+S(BCE)\\\
+S(BDE)+S(CDE)\geq 2S(AB)+2S(ABCD)+2S(ABCE)+2S(AC)+2S(BD)+2S(CE)+\\\
S(ABDE)+S(ACDE)+S(AD)+S(AE)+S(BC)+S(DE).$ (5)

(a) The quantity $S_{1}$ plotted vs $t_{b}$.

(b) The quantity $S_{2}$ plotted vs $t_{b}$.

(c) The quantity $S_{3}$ plotted vs $t_{b}$.

(d) The quantity $S_{4}$ plotted vs $t_{b}$.

Figure 5: The quantities $S_{i}$ ($i=1,2,3,4$) plotted vs $t_{b}$. We see that
$S_{i}>0$ for all $i$ so the corresponding inequalities given in the main text
are always satisfied. The jumps at late times are due to numerical errors.

For each of these inequalities, we define the quantities $S_{i}$ to be
$4G_{N}$ times the left-hand side minus $4G_{N}$ times the right-hand side for
$i=1,2,3,4$. Inequality $i$ will be satisfied if and only if $S_{i}$ is
positive. We plot each of these quantities as functions of $t_{b}.$ See Figure
5. We see that for each $i$, $S_{i}$ is positive so that the five-region
inequalities are all satisfied in this case, even though the spacetime is not
static.

We now consider the case where the interval is not constant-time.

###  3.3 Spacelike Intervals with Nonzero $\Delta t$

Figure 6: A Penrose diagram of the Vaidya spacetime (the red line represents
the shell of null matter), showing the four cases for the general spacelike
geodesics. (i) Entirely in the AdS bulk, (ii) entirely in the BTZ bulk, (iii)
starts in AdS, crosses into BTZ, and (iv) starts in BTZ, crosses into AdS, and
crosses back into BTZ.

In this situation, there are four cases: (i) entirely in AdS, (ii) entirely in
BTZ, (iii) starts in AdS, crosses into BTZ, (iv) starts in BTZ, crosses into
AdS, crosses back into BTZ. See Figure 6. Without loss of generality, suppose
$\Delta t\geq 0$. Then the interval is characterized by three parameters,
$\Delta x,\Delta t,$ and the starting boundary time of the interval $t_{b}.$
Again, because the shell is located at $v=0$, if $t_{b}<0$, the curve is
either (i) or (iii). If $t_{b}\geq 0$, the curve is either (ii) or (iv).

####  3.3.1 Geodesics entirely in AdS or BTZ

We begin with the AdS case. The solution is:

$r(\tau)=\frac{1}{2}((p_{x}^{2}-E^{2})e^{-\tau}+e^{\tau}),$
$t(\tau)=Const-\frac{2E}{p_{x}^{2}-E^{2}+e^{2\tau}},$
$x(\tau)=Const-\frac{2p_{x}}{p_{x}^{2}-E^{2}+e^{2\tau}}.$

This is very similar to the case considered above with $E=0$. We calculate

$\Delta x=\frac{2p_{x}}{p_{x}^{2}-E^{2}},\Delta
t=\frac{2E}{p_{x}^{2}-E^{2}},L_{reg}=-\log(p_{x}^{2}-E^{2}).$

We calculated the solution to the BTZ case with $E\neq 0$ above. The solution
is

$r(\tau)^{2}=\frac{1}{4}(e^{\tau}+B_{+}e^{-\tau})(e^{\tau}+B_{-}e^{-\tau}),$
$t(\tau)=Const+\frac{1}{2}\log\left(\frac{A_{-}+e^{2\tau}}{A_{+}+e^{2\tau}}\right),$
$x(\tau)=Const-\frac{1}{2}\log\left(\frac{B_{-}+e^{2\tau}}{B_{+}+e^{2\tau}}\right),$

where we have defined the quantities

$B_{\pm}=(p_{x}\pm 1)^{2}-E^{2},A_{\pm}=p_{x}^{2}-(1\pm E)^{2}.$

This tells us that

$\Delta x=\frac{1}{2}\log\left(\frac{A_{-}}{A_{+}}\right),\Delta
t=-\frac{1}{2}\log\left(\frac{B_{-}}{B_{+}}\right),L_{reg}=-\frac{1}{2}\log(B_{+}B_{-}).$

We now turn our attention to the geodesics that cross the shell.

####  3.3.2 Geodesics that start in AdS, end in BTZ

In this case, the geodesic intersects the shell once, say at coordinate
$r_{c}$. The affine parameter (in the AdS portion) at which the crossing
occurs is

$\tau_{c}=\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})$

Because this occurs at the shell, we need $t_{c}=t(\tau_{c})=1/r_{c},$ which
fixes the constant in the equation for $t(\tau)$. The length of the geodesic
in AdS is given by

$L_{A}=\log
2R+\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})-\log(p_{x}^{2}-E_{A}^{2}).$

Meanwhile,

$\Delta
x_{A}=\frac{2p_{x}}{p_{x}^{2}-E_{A}^{2}}-\frac{p_{x}}{r_{c}(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})},\Delta
t_{A}=\frac{E_{A}}{p_{x}}\Delta x_{A}.$

As we calculated above, the value of $E$ in the BTZ portion is given by

$E_{B}=(1-\frac{1}{2r_{c}^{2}})E_{A}-\frac{p_{x}r_{A}^{\prime}}{2r_{c}^{4}},\text{
}r_{A}^{\prime}=\sqrt{\frac{r_{c}^{6}}{p_{x}^{2}}+\frac{r_{c}^{4}E_{A}^{2}}{p_{x}^{2}}-r_{c}^{4}}.$

The value of the affine parameter in the BTZ portion of the crossing is

$\alpha_{B}\equiv\exp(2\tau_{B})=\frac{1}{2}\left[-(B_{+}+B_{-})+4r_{c}^{2}+\sqrt{-4B_{+}B_{-}+(B_{+}+B_{-}-4r_{c}^{2})^{2}}\right],$

where $B_{\pm}$ is as defined above, using the energy $E_{B}$. Furthermore, we
have

$\Delta
x_{B}=x(\tau=\infty)-x(\tau_{B})=-\frac{1}{2}\log\left(\frac{B_{-}+\alpha_{B}}{B_{+}+\alpha_{B}}\right),$
$\Delta
t_{B}=\frac{1}{2}\log\left(\frac{A_{-}+\alpha_{B}}{A_{+}+\alpha_{B}}\right),$
$L_{B}=\log 2R-\frac{1}{2}\log\alpha_{B}.$

In the AdS region, the time of the boundary crossing is given by
$t_{c}=1/r_{c}$. Therefore, the starting time of the interval is

$t_{b}=\frac{1}{r_{c}}-\Delta t_{A}.$

The total (regularized) length of the curve is

$L_{reg}=-\frac{1}{2}\log\alpha_{c}=\log
2R+\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})-\log(p_{x}^{2}-E_{A}^{2}),$

while

$\Delta x=\Delta x_{A}+\Delta x_{B},\Delta t=\Delta t_{A}+\Delta t_{B}.$

####  3.3.3 Geodesics that start in BTZ, cross into AdS, end in BTZ

Finally we consider the geodesics that start in the BTZ bulk (so that
$t_{b}=0$), cross over into the AdS bulk, and then cross back to the BTZ bulk.
These geodesics cross the shell twice, say at $r_{1}$ and $r_{2}$, with
$r_{1}>r_{2}$. If the part of the geodesic in AdS has $E_{A}$, then the length
of the AdS portion is given by

$L_{A}=\tau_{1}-\tau_{2}=\log(r_{1}+\sqrt{r_{1}^{2}+E_{A}^{2}-p_{x}^{2}})-\log(r_{2}+\sqrt{r_{2}^{2}+E_{A}^{2}-p_{x}^{2}}).$

Also,

$\Delta t_{A}=\frac{1}{r_{2}}-\frac{1}{r_{1}},\Delta
x_{A}=\frac{p_{x}}{E_{A}}\Delta t_{A}.$

We now consider the BTZ portions of the geodesics. Consider the upper BTZ arc
of the geodesic. The shell is at $v=0$, so since $r_{1}>r_{2}$,
$r_{1}^{\prime}<0$. Thus, we obtain

$r_{A}^{1\prime}=-\sqrt{\frac{r_{1}^{6}}{p_{x}^{2}}+\frac{r_{1}^{4}E_{A}^{2}}{p_{x}^{2}}-r_{1}^{4}},$

and

$E_{B1}=(1-\frac{1}{2r_{1}^{2}})E_{A}-\frac{p_{x}r_{A}^{1\prime}}{2r_{1}^{4}}.$

We know $r$ $x$ and $t$ as functions of $\tau$ for the BTZ curve for these
values of the conserved momenta. We can numerically solve for $\tau_{B}^{1}$
when $r_{B1}$ is equal to $r_{1}$. Because $t_{B1}(\tau_{B}^{1})$ has to be
equal to $\tanh^{-1}(1/r_{1})$, this fixes the constant. We then compute

$\Delta x_{B1}=x_{B1}(\tau_{\infty})-x_{B1}(\tau_{B}^{1}),\Delta
x_{B1}=x_{B1}(\infty)-x_{B1}(\tau_{B}^{1}),L_{B1}=\tau_{\infty}-\tau_{B}^{1}.$

We repeat this procedure for the bottom BTZ arc. For $r_{2}$, however,
$r_{A}^{2\prime}>0$, so that

$r_{A}^{1\prime}=\sqrt{\frac{r_{2}^{6}}{p_{x}^{2}}+\frac{r_{2}^{4}E_{A}^{2}}{p_{x}^{2}}-r_{2}^{4}},$

which means

$E_{B2}=(1-\frac{1}{2r_{2}^{2}})E_{A}-\frac{p_{x}r_{A}^{1\prime}}{2r_{2}^{4}}.$

We then follow the same procedure to compute $\Delta x_{B2},\Delta t_{B2}$ and
$L_{B2}$. The totals are, of course,

$\Delta x=\Delta x_{A}+\Delta x_{B1}+\Delta x_{B2},\Delta t=\Delta
t_{A}+\Delta t_{B1}+\Delta t_{B2}.$

To obtain the regularized geodesic length, we have to subtract off the usual
UV-divergent term:

$L_{reg}=L_{A}+L_{B1}+L_{B2}-2\log(2R).$

The boundary time of the start point of the interval is given by

$t_{b}=\frac{1}{r_{1}}-\Delta t_{B1}.$

If we are given $r_{1}$ and $E_{A}$, we can calculate $r_{2}$ as follows. In
the AdS region, we can solve for the value of $\tau$ at which $r$ is equal t
o$r_{1}$. We know that $t$ evaluated at this value is $1/r_{1}$, which fixes
the value of the integration constant in the $t(\tau)$ function. We then find
the other value of $\tau$ for which the function
$v(\tau)=r(\tau)-\frac{1}{t(\tau)}$ vanishes. Evaluating the function
$r(\tau)$ at this value then gives us $r_{2}$. Therefore, the geodesic is
specified by three parameters: $E_{A}$, $p_{x}$, and $r_{1}$. From these we
can calculate the starting time $t_{b}$ and the values of $\Delta x$ and
$\Delta t$. For values of $t_{b}$, $\Delta x$ and $\Delta t$, we can
numerically find the corresponding values of $E_{A}$, $p_{x},$ and $r_{1}$,
and then use these to calculate the geodesic lengths.

###  3.4 Testing Entropy Inequalities

Figure 7: The covariant "trapezoid" and "zigzag" configurations. In both
cases, each of the components ($A,B$, and $C$) have $\Delta x=1.$

(a) $S(AB)+S(BC)-S(B)-S(ABC)$ versus $t_{b}$ plotted for the trapezoid
configuration. It is always positive so strong subadditivity holds..

(b) $S(AB)+S(BC)-S(B)-S(ABC)$ versus $t_{b}$ plotted for the zigzag
configuration. It is always positive so strong subadditivity holds.

Figure 8: Strong subadditivity is verified for the trapezoid and zigzag
configurations.

To find the geodesic length for a given set of parameters, we proceed as
follows. If $t_{b}<0$, we solve for the AdS geodesic. We then calculate
$v(\tau)=t(\tau)-\frac{1}{r(\tau)}$. If $v$ never crosses 0, the geodesic is
entirely in the AdS bulk. If it crosses $0$, then the geodesic has a portion
in the BTZ spacetime. We then numerically find the values of
$r_{c},p_{x},E_{A}$ that correspond to the given values of $\Delta x,\Delta
t,t_{b}$. We then substitute these results back into our formula for geodesic
length. Similarly, if $t_{b}\geq 0$, we calculate the pure BTZ solution, and
see if $v(\tau)=t(\tau)-\tanh^{-1}(\frac{1}{r(\tau)})$ is ever negative, there
is a component in the AdS bulk. We numerically find the values of
$r_{1},E_{A},p_{x}$ that correspond to the values of $\Delta x,\Delta t,t_{b}$
and use these to find the geodesic length.

Finally, we are ready to test the entropy inequalities for regions that are
not purely spacelike. We begin by testing strong sub-additivity. We test two
cases, the "trapezoidal" case, and the "zigzag" case; both of these are shown
in Figure 7. We plot the quantity

$S(AB)+S(BC)-S(B)-S(ABC)$

(times $4G_{N}$) for these regions as a function of the boundary start time of
the region $A$, for a variety of values of $\Delta t/\Delta x$, fixing $\Delta
x=1.$ We show the results in Figures 8(a) and 8(b). These curves show that
strong sub-additivity is obeyed for these regions.

Figure 9: The three configurations we consider. Each of the components has its
$\Delta x$ fixed to be 1.

(a) Inequality 0 plotted vs $t_{b}$.

(b) Inequality 1 plotted vs $t_{b}$.

(c) Inequality 2 plotted vs $t_{b}$.

(d) Inequality 3 plotted vs $t_{b}$.

(e) Inequality 4 plotted vs $t_{b}$.

Figure 10: The inequalities plotted vs $t_{b}$ for the zigzag region for a
variety of values of $\Delta t/\Delta x$. We see that the inequalities are all
satisfied.

(a) Inequality 0 plotted vs $t_{b}$.

(b) Inequality 1 plotted vs $t_{b}$.

(c) Inequality 2 plotted vs $t_{b}$.

(d) Inequality 3 plotted vs $t_{b}$.

(e) Inequality 4 plotted vs $t_{b}$.

Figure 11: The inequalities plotted vs $t_{b}$ for the Region 2 (with 2 flat
components) for a variety of values of $\Delta t/\Delta x$. We see that the
inequalities are all satisfied.

(a) Inequality 0 plotted vs $t_{b}$.

(b) Inequality 1 plotted vs $t_{b}$.

(c) Inequality 2 plotted vs $t_{b}$.

(d) Inequality 3 plotted vs $t_{b}$.

(e) Inequality 4 plotted vs $t_{b}$.

Figure 12: The inequalities plotted vs $t_{b}$ for the Region 3 (with 1 flat
component) for a variety of values of $\Delta t/\Delta x$. We see that the
inequalities are all satisfied.

We now test the five-region inequalities. We use the same labeling scheme for
the inequalities as used above, in the constant-time case. We consider the
three configurations shown in Figure 9. We consider a variety of values of
$\Delta t/\Delta x$, again fixing the value of $\Delta x$ for each component
to be 1. We plot these curves as functions of the boundary start time $t_{b}$.
We do this for the zigzag configuration in Figure 10, the configuration with 2
flat components in Figure 11, and the configuration with 1 flat component in
Figure 12 We see that the inequalities are all satisfied, and that the shapes
of the curves strongly resemble those of the strong subadditivity.

##  4 Negative Energy Vaidya Metric

We now consider the negative-energy Vaidya metric. As discussed above, this
violates the null energy condition. We will see that strong subadditivity is
violated, as well as the five-body inequalities.

###  4.1 Geodesic Kinematics

We consider constant-time intervals. For boundary time $t_{b}<0$ the geodesics
will be entirely in the BTZ bulk, while for large enough $t_{b}$ it will be
entirely in the AdS bulk. These cases were treated above; we will now consider
the case where the geodesic is partially in the AdS region, and partially in
the BTZ region. We first consider the BTZ part. By the symmetry of the
problem, $E=0$ in the BTZ arc. Suppose that the geodesic crosses the shell at
$r_{c}$. The value of the affine parameter at this value of $r$ is

$\tau_{c}=\log(\sqrt{\absolutevalue{r_{c}^{2}-1}}+\sqrt{\absolutevalue{r_{c}^{2}-p_{x}^{2}}}).$

From the equation for $\dot{r}$, it is clear that $r=p_{x}$ is the turning
point. Therefore, by symmetry, the length of the BTZ part of the geodesic is

$L_{B}=2(\tau_{c}-\tau_{p_{x}})=2\log\left(\frac{\sqrt{\absolutevalue{r_{c}^{2}-1}}+\sqrt{\absolutevalue{r_{c}^{2}-p_{x}^{2}}}}{\sqrt{\absolutevalue{p_{x}^{2}-1}}}\right).$

Similarly, the change in $x$ is given by

$\Delta
x_{B}=-\log\left(\frac{(p_{x}-1)^{2}+(\sqrt{\absolutevalue{r_{c}^{2}-1}}+\sqrt{\absolutevalue{r_{c}^{2}-p_{x}^{2}}})^{2}}{(p_{x}+1)^{2}+(\sqrt{\absolutevalue{r_{c}^{2}-1}}+\sqrt{\absolutevalue{r_{c}^{2}-p_{x}^{2}}})^{2}}\right)+\log\left(\frac{(p_{x}-1)^{2}+(\sqrt{\absolutevalue{p_{x}^{2}-1}})^{2}}{(p_{x}+1)^{2}+(\sqrt{\absolutevalue{p_{x}^{2}-1}})^{2}}\right).$

We now turn to the AdS components. Similar to the positive-energy case, we
require that at the shell $v=0$ we must have

$v^{\prime}=2(r_{A}^{\prime}-r_{B}^{\prime}).$

$E_{B}=0$ so

$v^{\prime}=\frac{r_{B}^{\prime}}{r_{c}^{2}-1},$

which means that

$r_{A}^{\prime}=r_{B}^{\prime}+\frac{v^{\prime}}{2}=r_{B}^{\prime}+\frac{r_{B}^{\prime}}{2(r_{c}^{2}-1)}=\frac{(2r_{c}^{2}-1)r_{B}^{\prime}}{2(r_{c}^{2}-1)}.$

Note that this means that $r_{A}^{\prime}$ becomes negative when
$r_{c}<\frac{1}{2}$. In AdS, we know that

$r_{A}^{\prime
2}=\frac{r_{c}^{6}}{p_{x}^{2}}+\frac{r_{c}^{4}E_{A}^{2}}{p_{x}^{2}}-r_{c}^{4},$
$r_{A}^{\prime 2}p_{x}^{2}=r_{c}^{6}+r_{c}^{4}E_{A}^{2}-r_{c}^{4}p_{x}^{2},$
$\frac{r_{A}^{\prime 2}p_{x}^{2}}{r_{c}^{4}}+p_{x}^{2}-r_{c}^{2}=E_{A}^{2},$
$E_{A}^{2}=\frac{(2r_{c}^{2}-1)^{2}r_{B}^{\prime
2}p_{x}^{2}}{4r_{c}^{4}(r_{c}^{2}-1)^{2}}+p_{x}^{2}-r_{c}^{2}.$

Also, we know that

$r_{B}^{\prime 2}=(r_{c}^{2}-1)r_{c}^{2}(\frac{r_{c}^{2}}{p_{x}^{2}}-1),$

which means

$E_{A}^{2}=\frac{(2r_{c}^{2}-1)^{2}(r_{c}^{2}-p_{x}^{2})}{4r_{c}^{2}(r_{c}^{2}-1)}+p_{x}^{2}-r_{c}^{2}=\frac{((2r_{c}^{2}-1)^{2}-4r_{c}^{4}+4r_{c}^{2})(r_{c}^{2}-p_{x}^{2})}{4r_{c}^{2}(r_{c}^{2}-1)}=\frac{(r_{c}^{2}-p_{x}^{2})}{4r_{c}^{2}(r_{c}^{2}-1)}.$

If $r_{c}>\frac{1}{2}$ then $r^{\prime}>0$ and $\dot{r}>0$, and the solution
is

$r(\tau)=\frac{1}{2}(e^{\tau}+(p_{x}^{2}-E^{2})e^{-\tau}).$

We have that

$\tau_{c}=\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}}).$

For large $R$, the corresponding (large, positive) value of $\tau$ is $\log
2R$ so the length of the AdS arc is

$L_{A}=\log 2R-\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}}).$

The change in $x$ is given by

$\Delta
x_{A}=\frac{p_{x}}{r_{c}(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})}.$

We find the starting point of the interval in the usual way:

$t_{b}=t_{c}-\Delta
t_{A}=\frac{1}{r_{c}}-\frac{E_{A}}{r_{c}(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})},$

since $\Delta t_{A}=\Delta x_{A}\frac{E_{A}}{p_{x}}$. The total arc length and
displacement are

$L=L_{B}+2L_{A},\Delta x=2\Delta x_{A}+\Delta x_{B}.$

On the other hand, if $r_{c}<\frac{1}{2},$ $\dot{r}<0$ and the solution is
given by

$r(\tau)=\frac{1}{2}(e^{-\tau}+(p_{x}^{2}-E^{2})e^{\tau}),$

and we have that

$\tau_{c}=-\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}}).$

The positive affine parameter for large $R$ is given by

$\tau_{\infty}=\log(2R)-\log(p_{x}^{2}-E_{A}^{2}),$

which means that the total length of the AdS arc is given by

$L_{A}=\tau_{\infty}-\tau_{c}=\log(2R)-\log(p_{x}^{2}-E_{A}^{2})+\log(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}}).$

Meanwhile,

$\Delta
x_{A}=x(\tau_{\infty})-x(\tau_{c})=\frac{p_{x}}{r_{c}(r_{c}+\sqrt{r_{c}^{2}+E_{A}^{2}-p_{x}^{2}})}-\frac{2p_{x}}{-E_{A}^{2}+p_{x}^{2}+(p_{x}^{2}-E_{A}^{2})^{2}}$

and

$t_{b}=\frac{1}{r_{c}}-\frac{E_{A}}{p_{x}}\Delta x_{A}.$

Once again, the total geodesic length and displacement are

$L=L_{B}+2L_{A},\Delta x=2\Delta x_{A}+\Delta x_{B}.$

Figure 13: The regularized geodesic length $L_{reg}$ as a function of the
boundary interval length, $\Delta x$ for various values of the boundary time
$t_{b}$ in the negative-energy Vaidya spacetime. We see that the curves are
not convex, meaning there will be violations of strong subadditivity

###  4.2 Tests of Entropy Inequalities

To find the geodesic for the negative-energy metric for a given interval, we
proceed as follows. First, if $t_{b}\leq 0$, the geodesic is of course
entirely in BTZ. If $t_{b}>0$, we find the trajectory of the geodesic in AdS,
and calculate $v(\tau)$. If at any point it dips below 0, then there will be a
portion of the geodesic that is in the BTZ bulk. We then use a numerical
algorithm to find the values of $r_{c}$ and $p_{x}$ that correspond to the
desired $t_{b}$ and $\Delta x$. We show a plot of the (regularized) geodesic
length as a function of the displacement for various values of the boundary
time $t_{b}$ in Figure 13. We see the non-convex behavior of some of these
curves, which means that strong subadditivity will be violated.

Figure 14: Strong subadditivity versus boundary time $t_{b}$ for the negative-
energy Vaidya spacetime. We see that strong subadditivity is violated. Here,
we consider strong subadditivity for regions $A,B,$ and $C,$ as well as for
$AB,C,$ and $D.$

We consider five adjacent constant-time intervals, $A,B,C,D,E$. $A,C,$ and $E$
have width 2, while $B$ and $D$ have width 4. To start with, we plot strong
subadditivity for a couple collections of regions in Figure 14. We see that
strong subadditivity is violated, which is expected since our metric violates
the null energy condition.

(a) Inequality 0 plotted vs $t_{b}$.

(b) Inequality 1 plotted vs $t_{b}$.

(c) Inequality 2 plotted vs $t_{b}$.

(d) Inequality 3 plotted vs $t_{b}$.

(e) Inequality 4 plotted vs $t_{b}$.

Figure 15: The inequalities plotted vs $t_{b}$. We see that the inequalities
are all violated for this spacetime that violates the null energy condition.

Next, we check the five-region inequalities. We use the same numbering scheme
as before (with labels 0 through 4), and we plot $4G_{N}$ times the left hand
side minus $4G_{N}$ times the right hand side of each of the inequalities. We
show the results in Figure 15. We see that all of the inequalities are
violated for this spacetime, roughly in the places where strong subadditivity
is violated. Furthermore, we once again see that the curves for the five-
region inequalities resemble the strong subadditivity curves.

###### Acknowledgements.

I am very grateful to Matt Headrick for suggesting this problem and fruitful
discussions, as well as useful comments on the manuscript. I would also like
to thank Ning Bao, Alan Guth, Daniel Harlow, Mark Kon, Sam Leutheusser, and
Hong Liu for useful conversations.

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