# TranS: Transition-based Knowledge Graph Embedding
with Synthetic Relation Representation

Xuanyu Zhang Qing Yang Dongliang Xu
DXM-DI-AI

###### Abstract

Knowledge graph embedding (KGE) aims to learn continuous vectors of relations
and entities in knowledge graph. Recently, transition-based KGE methods have
achieved promising performance, where the single relation vector learns to
translate head entity to tail entity. However, this scoring pattern is not
suitable for complex scenarios where the same entity pair has different
relations. Previous models usually focus on the improvement of entity
representation for 1-to-N, N-to-1 and N-to-N relations, but ignore the single
relation vector. In this paper, we propose a novel transition-based method,
TranS, for knowledge graph embedding. The single relation vector in
traditional scoring patterns is replaced with synthetic relation
representation, which can solve these issues effectively and efficiently.
Experiments on a large knowledge graph dataset, ogbl-wikikg2, show that our
model achieves state-of-the-art results.

##  1 Introduction

Knowledge graphs (KGs), such as Freebase Bollacker et al. (2008), Yago Rebele
et al. (2016) and Wikidata Vrandečić and Krötzsch (2014), play an important
role in many applications, including question answering, information retrieval
and so on. KG is usually represented as the form of triplets (head entity,
relation, tail entity)(denoted as (h, r, t)), where relation indicates the
relationship between the two entities. With the rapid development of deep
learning, many representation learning methods Bordes et al. (2013); Wang et
al. (2014); Fan et al. (2014); Lin et al. (2015); Ji et al. (2015, 2016); Xie
et al. (2017); Qian et al. (2018); Chao et al. (2020); Yu et al. (2021); Chen
et al. (2021); Wang et al. (2022) are proposed to obtain low-dimensional
embedding vectors of entities and relations in KG.

Figure 1: Examples from ogbl-wikikg2. It is difficult for single relation
vector to represent different relations between the same entity pairs.

Generally speaking, knowledge graph embedding (KGE) methods can be roughly
divided into the following directions: translational distance Bordes et al.
(2013); Wang et al. (2014); Fan et al. (2014); Lin et al. (2015); Ji et al.
(2015), semantic matching Nickel et al. (2011); García-Durán et al. (2014);
Yang et al. (2014); Nickel et al. (2016); Trouillon et al. (2016) and neural
networks Bordes et al. (2014); Liu et al. (2016); Li et al. (2016); Gilmer et
al. (2017); Xu et al. (2019). Because transition-based KGE method like TransE
Bordes et al. (2013) is simple but effective, this series of models are
becoming more and more popular in both academia and industry. Specifically,
TransE makes the difference between two entity vectors ($\bf{h}$ and $\bf{t}$)
approximate to the relation vector ($\bf{r}$), i.e., $\bf{t}-h\approx r$. That
is to say, the relation r is characterized by the translating vector $\bf{r}$.
However, TransE is not suitable to deal with complex relations like one-to-
many/many-to-one/many-to-many. Especially, the same entity pair usually has
many different relations. For example, in Figure 1, after graduating from
Erasmus University Rotterdam, the professor became a professor of the same
university. The composer, producer, screenwriter, editor and director of the
film, Indramalati, can be the same person, Jyoti Prasad Agarwala. As shown in
Table 1, although previous models like TransH/R/D Wang et al. (2014); Lin et
al. (2015); Ji et al. (2015) solved the problem of 1-to-N, N-to-1 and N-to-N,
they still continued the TransE pattern, $\bf{R_{t}}-\bf{R_{h}}\approx\bf{r}$,
where $\bf{R_{t}}$ and $\bf{R_{h}}$ are the deformation of $\bf{t}$ and
$\bf{h}$ respectively. Even if the entity vector is represented by hyperplane
or multiple embedding spaces, single relation vector $\bf{r}$ still cannot
represent different relationships when facing the same entity pair.

To solve these issues, we propose a novel transition-based knowledge graph
embedding model, TranS, which replaces tradition scoring pattern with
synthetic relation pattern, i.e.,
$\bf{R_{t}}-\bf{R_{h}}\approx\bar{r}+r+\hat{r}$. The final relation
representation is the sum of multiple relation vectors. It can handle not only
complex relations in knowledge graph, but also the situation in Figure 1,
where orange solid lines denote $\bf{r}$, and blue dotted lines denote
$\bf{\bar{r}},\bf{\hat{r}}$. Experiments on a large knowledge graph dataset,
ogbl-wikikg2, show that our proposed model achieves state-of-the-art results.

Model | Embedding | Scoring Function | Scoring Pattern
---|---|---|---
TransE |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$, $\mathbf{r}\in\mathbb{R}^{d}$ | $-\|\mathbf{h}+\mathbf{r}-\mathbf{t}\|_{1/2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TransR |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$ , $\mathbf{r}\in\mathbb{R}^{k},\mathbf{M}_{r}\in\mathbb{R}^{k\times d}$ | $-\left\|\mathbf{M}_{r}\mathbf{h}+\mathbf{r}-\mathbf{M}_{r}\mathbf{t}\right\|_{2}^{2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TransH |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$, $\mathbf{r},\mathbf{w}_{r}\in\mathbb{R}^{d}$ | $-\left\|\left(\mathbf{h}-\mathbf{w}_{r}^{\top}\mathbf{h}\mathbf{w}_{r}\right)+\mathbf{r}-\left(\mathbf{t}-\mathbf{w}_{r}^{\top}\mathbf{t}\mathbf{w}_{r}\right)\right\|_{2}^{2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
ITransF |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$, $\mathbf{r}\in\mathbb{R}^{d}$ | $\left\|\bm{\alpha}_{r}^{H}\cdot\mathbf{D}\cdot\mathbf{h}+\mathbf{r}-\bm{\alpha}_{r}^{T}\cdot\mathbf{D}\cdot\mathbf{t}\right\|_{\ell}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TransAt |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$, $\mathbf{r}\in\mathbb{R}^{d}$ | $P_{r}\left(\sigma\left(\mathbf{r}_{h}\right)\mathbf{h}\right)+\mathbf{r}-P_{r}\left(\sigma\left(\mathbf{r}_{t}\right)\mathbf{t}\right)$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TransD |  $\mathbf{h},\mathbf{t},\mathbf{w}_{h}\mathbf{w}_{t}\in\mathbb{R}^{d}$, $\mathbf{r},\mathbf{w}_{r}\in\mathbb{R}^{k}$ | $-\left\|\left(\mathbf{w}_{r}\mathbf{w}_{h}^{\top}+\mathbf{I}\right)\mathbf{h}+\mathbf{r}-\left(\mathbf{w}_{r}\mathbf{w}_{t}^{\top}+\mathbf{I}\right)\mathbf{t}\right\|_{2}^{2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TransM |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$, $\mathbf{r}\in\mathbb{R}^{d}$ | $-\theta_{r}\|\mathbf{h}+\mathbf{r}-\mathbf{t}\|_{1/2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TranSparse |  $\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$, $\mathbf{r}\in\mathbb{R}^{k},\mathbf{M}_{r}\left(\theta_{r}\right)\in\mathbb{R}^{k\times d}$ | $-\left\|\mathbf{M}_{r}\left(\theta_{r}\right)\mathbf{h}+\mathbf{r}-\mathbf{M}_{r}\left(\theta_{r}\right)\mathbf{t}\right\|_{1/2}^{2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
$\mathbf{h},\mathbf{t}\in\mathbb{R}^{d}$,$\mathbf{M}_{r}^{1}\left(\theta_{r}^{1}\right),\mathbf{M}_{r}^{2}\left(\theta_{r}^{2}\right)\in\mathbb{R}^{k\times d}$ | $-\left\|\mathbf{M}_{r}^{1}\left(\theta_{r}^{1}\right)\mathbf{h}+\mathbf{r}-\mathbf{M}_{r}^{2}\left(\theta_{r}^{2}\right)\mathbf{t}\right\|_{1/2}^{2}$ | $||\bf{R_{h}}-\bf{R_{t}}+\bf{r}||$
TranS | $\mathbf{h},\mathbf{t},\mathbf{\tilde{h}},\mathbf{\tilde{t}}\in\mathbb{R}^{d}$, $\mathbf{r},\bf{\bar{r}},\bf{\hat{r}}\in\mathbb{R}^{d}$ | $-||\bf{h}\circ\tilde{t}-\bf{t}\circ\tilde{h}+\uwave{\bf{\bar{r}}\circ\bf{h}+\bf{r}+\bf{\hat{r}}\circ\bf{t}}||$ | $||\bf{R_{h}}-\bf{R_{t}}+$
$\bf{\bar{r}}+r+\hat{r}||$
Table 1: Transition-based knowledge graph embedding models.

##  2 Methodology

In this section, we will first introduce our proposed TranS model. And we then
introduce the loss function during training and differences among different
KGE methods.

###  2.1 TranS

Our proposed TranS model first breaks the traditional scoring patterns
$\bf{R_{t}}-\bf{R_{h}}\approx r$ in previous models Bordes et al. (2013); Wang
et al. (2014); Fan et al. (2014); Lin et al. (2015); Chao et al. (2020); Yu et
al. (2021); Wang et al. (2022). It replaces single relation vector $\bf{r}$
with synthetic relation vectors $\bf{\bar{r}}+r+\bf{\hat{r}}$, where
$\bf{\bar{r}},\bf{\hat{r}}$ are related to head and tail entities. That is to
say, we want $\bf{R_{t}}-\bf{R_{h}}\approx\bar{r}+r+\hat{r}$. The illustration
of TranS is shown in Figure 2 (f). Two entity and three relation
representations together make up our proposed scoring function $f_{r}(h,t)$.
Especially, the synthetic relation representation consists of the sum of three
different relation vectors. To make full use of context information, we use
combined vectors with Hadamard product to represent $\bf{h}$, $\bf{t}$,
$\bf{\bar{r}}$ and $\bf{\hat{r}}$ separately:

$\displaystyle f_{r}(h,t)$
$\displaystyle=-||\bf{R_{h}}-\bf{R_{t}}+\bf{R_{r}}||$ (1)
$\displaystyle\bf{R_{h}}$ $\displaystyle=\bf{h}\circ\tilde{t}$
$\displaystyle\bf{R_{t}}$ $\displaystyle=\bf{t}\circ\tilde{h}$
$\displaystyle\bf{R_{r}}$
$\displaystyle=\bf{\bar{r}}\circ\bf{h}+\bf{r}+\bf{\hat{r}}\circ\bf{t}$

where $\mathbf{h}$, $\mathbf{t}$ and $\mathbf{r}$ denote main vectors similar
to traditional scoring patterns. $\mathbf{\tilde{h}}$ is auxiliary head entity
vector and $\mathbf{\tilde{t}}$ is auxiliary tail entity vector. And
$\mathbf{\bar{r}}$ is auxiliary relation vector related to head entity and
$\mathbf{\hat{r}}$ is auxiliary relation vector related to tail entity.
$\bf{R_{h}}$ is the representation of the head that combines information of
the tail, and $\bf{R_{t}}$ is the representation of the tail integrating
information of the head. $\bf{\bar{r}}\circ\bf{h}$ is the representation of
the relation that combines information of the head, and
$\bf{\hat{r}}\circ\bf{t}$ is the representation of another relation
integrating information of the tail. Thus the final equations can be
represented as follows:

$\displaystyle-||\bf{h}\circ\tilde{t}-\bf{t}\circ\tilde{h}+\uwave{\bf{\bar{r}}\circ\bf{h}+\bf{r}+\bf{\hat{r}}\circ\bf{t}}||$
(2)

Following previous works Yu et al. (2021); Wang et al. (2022), we add an unit
vector $\bf{e}$ to $\bf{R_{h}}$ and $\bf{R_{t}}$, i.e.,
$\bf{h}\circ\tilde{t}\rightarrow\bf{h}\circ(\tilde{t}+e)$,
$\bf{t}\circ\tilde{h}\rightarrow\bf{t}\circ(\tilde{h}+e)$. And considering the
OOV problem, we also use the NodePiece Galkin et al. (2021) to learn a fixed-
size entity vocabulary.

Figure 2: Comparison of different transition-based KGE models.

###  2.2 Training

In the training process, we use the self-adversarial negative sampling loss
Sun et al. (2019) as the loss function. The loss function is as follows:

$\displaystyle\mathcal{L}=$ $\displaystyle-log\ \sigma(\gamma-
d_{r}(\mathbf{h},\mathbf{t}))$ (3)
$\displaystyle-\sum_{i=1}^{n}\frac{1}{k}log\
\sigma(d_{r}(\mathbf{h_{i}^{\prime}},\mathbf{t_{i}^{\prime}})-\gamma)$

where $\gamma$ is a fixed margin, $\sigma$ is the sigmoid function, and
$(\mathbf{h_{i}^{\prime}},\mathbf{r},\mathbf{t_{i}^{\prime}})$ is the $i$-th
negative triplet.

###  2.3 Comparison

As shown in Figure 2, the main difference between our model (f) and previous
transition-based KGE methods (a,b,c,d,e) is the synthetic relation
representation. That is to say, it changes single relation representation
$\bf{r}$ in tradition scoring pattern $\bf{R_{t}}-\bf{R_{h}}\approx r$ to
synthetic relation representation $\bf{\bar{r}}+r+\hat{r}$ in our proposed new
pattern $\bf{R_{t}}-\bf{R_{h}}\approx\bar{r}+r+\hat{r}$. Specifically,
different from InterHT Wang et al. (2022), the relation part of our scoring
function is the sum of multiple relation vectors
$\bf{R_{r}}=\bf{\bar{r}}\circ\bf{h}+\bf{r}+\bf{\hat{r}}\circ\bf{t}$ rather
than single vector $\bf{r}$. Comparing with TripleRE Yu et al. (2021), where
three relations are applied into three parts ($\bf{R_{h}}=\bf{h}\circ r^{h}$,
$\bf{R_{t}}=\bf{t}\circ r^{t}$ , $\bf{R_{r}}=\bf{r^{m}}$) of traditional
scoring patterns with addition and subtraction operations, our proposed TranS
only applies synthetic relation vectors into the relation part
$\bf{R_{r}}=\bf{\bar{r}}\circ\bf{h}+\bf{r}+\bf{\hat{r}}\circ\bf{t}$ of scoring
functions with vector addition operation.

##  3 Experiments

In this section, we will first introduce the dataset and metric. Then we will
introduce implementation details and experimental results further.

Model | #Params | #Dims | Test MRR | Valid MRR
---|---|---|---|---
TransE | 1251M | 500 | 0.4256 $\pm$ 0.0030 | 0.4272 $\pm$ 0.0030
RotatE | 1250M | 250 | 0.4332 $\pm$ 0.0025 | 0.4353 $\pm$ 0.0028
PairRE | 500M | 200 | 0.5208 $\pm$ 0.0027 | 0.5423 $\pm$ 0.0020
AutoSF | 500M | - | 0.5458 $\pm$ 0.0052 | 0.5510 $\pm$ 0.0063
ComplEx | 1251M | 250 | 0.5027 $\pm$ 0.0027 | 0.3759 $\pm$ 0.0016
TripleRE | 501M | 200 | 0.5794 $\pm$ 0.0020 | 0.6045 $\pm$ 0.0024
ComplEx-RP | 250M | 50 | 0.6392 $\pm$ 0.0045 | 0.6561 $\pm$ 0.0070
AutoSF + NodePiece | 6.9M | - | 0.5703 $\pm$ 0.0035 | 0.5806 $\pm$ 0.0047
TripleREv2 + NodePiece | 7.3M | 200 | 0.6582 $\pm$ 0.0020 | 0.6616 $\pm$ 0.0018
InterHT + NodePiece | 19.2M | 200 | 0.6779 $\pm$ 0.0018 | 0.6893 $\pm$ 0.0015
TripleREv3 + NodePiece | 36.4M | 200 | 0.6866 $\pm$ 0.0014 | 0.6955 $\pm$ 0.0008
TranS + NodePiece | 19.2M | 200 | 0.6882 $\pm$ 0.0019 | 0.6988 $\pm$ 0.0006

Table 2: Results on the ogbl-wikikg2 dataset.

###  3.1 Dataset and Metric

The ogbl-wikikg2 Hu et al. (2020) is a large KG dataset extracted from
Wikidata Vrandečić and Krötzsch (2014). It contains a set of triplet edges,
capturing the different types of relations between entities in the world. The
statistics of the dataset are show in Table 3. It contains 2,500,604 entities,
535 relation types and 17,137,181 edges. Following official guidelines, we
evaluate the KGE performance by predicting new triplet edges according to the
training edges. The evaluation metric follows the standard filtered metric
widely used in KG. Specifically, each test triplet edges are corrupted by
replacing its head or tail with randomly-sampled 1,000 negative entities,
while ensuring the resulting triplets do not appear in KG. The goal is to rank
the true head (or tail) entities higher than the negative entities, which is
measured by Mean Reciprocal Rank (MRR).

We use original dataset splitting configuration. It splits the triplets
according to time, simulating a realistic KG completion scenario that aims to
fill in missing triplets that are not present at a certain timestamp.
Specifically, the training set contains 16,109,182 triplets, the validation
set contains 429,456 triplets, and the test set contains 598,543 triplets.

Data | #Number
---|---
Nodes | 2,500,604
Relations | 535
Edges | 17,137,181
Train | 16,109,182
Validation | 429,456
Test | 598,543
Table 3: Statistics of the ogbl-wikikg2 dataset.

###  3.2 Implementation Details

In our experiments, the Adam Kingma and Ba (2014) is used as our optimizer
with 0.0005 learning rate. The batch size of the model is set to 512. To
prevent overfitting, we use dropout technique, and set it to 0.1. The negative
sampling size is set to 128. And the dimension of each embedding vector in Eq.
2 is set to 200. The maximum number of training steps is 800 thousand. We
validate the model every 20 thousand steps. The number of anchors for
NodePiece are 20 thousand. And $\gamma$ in the loss function is set to 6.
These hyper-parameters are selected according to the performance on the
validation set.

###  3.3 Results

The results are shown in Table 2. Our model achieves 0.6988 (validation set)
and 0.6882 (test set) on MRR, which outperforms previous best model,
TripleREv3, on the ogbl-wikikg2 dataset. Especially, the parameters of our
model (19.2M) are about half of TripleREv3 (36.4M). The experimental results
show that our proposed method can improve the model performance effectively
with fewer parameters. Besides, we also construct the 38.4M TranS (large)
model, the best score of which can reach 0.7101 (validation set) and 0.6992
(test set) on MRR.

##  4 Conclusion

In this paper, we propose a novel transition-based knowledge graph embedding
model, TranS, to solve the issues of complex relations, especially the
situation of same entity pair with different relations. TranS replaces
tradition scoring pattern with synthetic relation pattern. And the final
relation representation is the sum of different relation vectors. Experiments
on the ogbl-wikikg2 dataset show that our model achieves state-of-the-art
results.

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# Implication of the swampland distance conjecture on the Cohen-Kaplan-Nelson
bound in de Sitter space

Min-Seok Seo<EMAIL_ADDRESS>Department of Physics Education, Korea
National University of Education,
Cheongju 28173, Republic of Korea

###### Abstract

The Cohen-Kaplan-Nelson (CKN) bound formulates the condition that black hole
is not produced by the low energy effective field theory dynamics. In de
Sitter space it also constrains the maximal size of the matter distribution to
be smaller than the cosmological horizon determined by black hole. On the
other hand, the swampland distance conjecture (SDC) predicts that de Sitter
space becomes unstable by the descent of the low energy degrees of freedom
from UV. This results in the rapid increase in the energy inside the
cosmological horizon, the distribution of which can be constrained by the CKN
bound. We study the CKN bound in de Sitter space in detail and point out that
when compared with the slow-roll in the inflation, the bound on the slow-roll
parameter which forbids the eternal inflation is obtained.

##  I Introduction

The swampland program Vafa:2005ui  (for reviews, see Brennan:2017rbf ;
Palti:2019pca ) aims at finding out bounds on the low energy effective field
theory (EFT) parameter space consistent with quantum gravity. The swampland
distance conjecture (SDC) Ooguri:2006in  provides the string-inspired
explanation to these bounds by assuming that all parameters are determined by
the dynamical stabilization of the moduli. It claims that as we move away from
any point in the allowed parameter region to the boundary, some modulus
traverses trans-Planckian geodesic distance and an infinite tower of states
becomes exponentially light. Then the ‘species cutoff’ Dvali:2007hz ;
Dvali:2007wp , above which gravity is no longer weakly coupled to matter,
falls off drastically, which signals the breakdown of the EFT.

Combined with the covariant entropy bound Bousso:1999xy , the SDC is used to
support the de Sitter (dS) swampland conjecture (dSC) Obied:2018sgi  (see also
Andriot:2018mav ; Garg:2018reu ; Ooguri:2018wrx  for a refined version) which
claims the quantum instability of dS space. The increasing number of
excitations of light degrees of freedom along the trajectory of the inflaton,
the modulus responsible for the vacuum energy, results in the rapid increase
in the entropy. Once this new entropy contribution exceeds the covariant
entropy bound given by the horizon area, the EFT defined below the almost
constant cosmological horizon scale $H$ is no longer valid Ooguri:2018wrx
(for relevant discussions, see, e.g., Seo:2018abc ; Seo:2019mfk ). From this
we can estimate the duration of the quasi-dS phase Seo:2019wsh ; Cai:2019dzj ,
which is similar to that suggested by the trans-Planckian censorship
conjecture Bedroya:2019snp  (see also Brahma:2020zpk ) but enhanced by the
slow-roll parameter.

The argument above illustrates how the excitations of a large number of states
deform the cosmological horizon in thermodynamic language. Meanwhile, such a
rapid production of matter also enhances the energy inside the cosmological
horizon and furthermore, may result in the formation of black hole, in which
case quantum gravity is no longer irrelevant to the particle interactions. The
similar issue was considered by Cohen, Kaplan, and Nelson (CKN) Cohen:1998zx .
They pointed out that for the EFT defined in the region of the size $L$ to be
valid, the UV cutoff should be low enough not to produce black hole through
the EFT dynamics. It also reads that matter should not be concentrated within
the radius of black hole having the same amount of energy. In flat space, this
condition is written as

$\begin{split}L^{3}\Lambda_{\rm UV}^{4}\lesssim LM_{\rm Pl}^{2},\quad{\rm
or}\quad\Lambda_{\rm UV}\lesssim\Big{(}\frac{M_{\rm
Pl}}{L}\Big{)}^{1/2}.\end{split}$ (1)

Comparing with the Bekenstein entropy bound Bekenstein:1980jp ,

$\begin{split}L^{3}\Lambda_{\rm UV}^{3}\lesssim S_{\rm BH}=\pi L^{2}M_{\rm
Pl}^{2},\quad{\rm or}\quad\Lambda_{\rm UV}\lesssim\Big{(}\frac{M_{\rm
Pl}^{2}}{L}\Big{)}^{1/3},\end{split}$ (2)

the CKN bound provides the more stringent bound on $\Lambda_{\rm UV}$.

We can also impose that black hole should not be produced in the dS background
for the validity of the EFT description. This dS CKN bound was formulated in
Banks:2019arz , which also constrains the maximal size of the matter
distribution to be smaller than the cosmological horizon determined by the
backreaction of black hole. We expect from the SDC that as the inflaton moves
in quasi-dS space, a large amount of matter is produced inside the
cosmological horizon. Then the maximal size of the matter distribution is
identified with the deformed cosmological horizon. The dS CKN bound in
Banks:2019arz  requires that this deformed horizon is smaller than the
cosmological horizon determined by black hole of the same energy. In this
article, we investigate the physical implications of this SDC-assisted dS CKN
bound.

The plan of this article is as follows. After summarizing essential features
of the SDC, we formulate the dS CKN bound in Banks:2019arz  more concretely.
Then we focus on the specific limit $GMH\ll 1$, where $M$ is interpreted as a
total energy, to describe the contraction of the cosmological horizon at the
early stage of the matter production analytically. By combining the SDC with
the dS CKN bound, we provide the EFT validity condition and compared this with
the slow-roll during the inflation. As we will see, our condition forbids the
eternal inflation.

##  II Swampland distance conjecture

Before considering the CKN bound, we briefly review aspects of the SDC
relevant to our discussion. The universality of the gravitational interaction
implies that given the UV cutoff scale $\Lambda_{\rm UV}$ and ${\cal N}$
particle species of the EFT, the strength of the gravitational interaction is
given by $\alpha_{\rm grav}={\cal N}(\Lambda_{\rm UV}/M_{\rm Pl})^{2}$. Thus,
if $\alpha_{\rm grav}$ is ${\cal O}(1)$, the EFT weakly coupled to gravity is
no longer valid, which sets the species cutoff $\Lambda_{\rm sp}=M_{\rm
Pl}/\sqrt{{\cal N}}$.

On the other hand, the SDC assumes the existence of a tower of states, the
mass gap of which depends on some modulus $\varphi$ as $\Delta m=M_{\rm
Pl}{\rm exp}[-\lambda\Delta\varphi]$, where $\Delta\varphi$ is the geodesic
distance traversed by $\varphi$ in Planck unit and $\lambda$ is a model
dependent ${\cal O}(1)$ constant. Taking an ansatz ${\cal N}=(M_{\rm
Pl}/\Delta m)^{\alpha}$ for the number of particle species below $\Lambda_{\rm
sp}$, the requirement that the heaviest particle mass in EFT is smaller than
$\Lambda_{\rm sp}$,

$\begin{split}{\cal N}\Delta m\lesssim\Lambda_{\rm sp}=\frac{M_{\rm
Pl}}{\sqrt{\cal N}}\end{split}$ (3)

gives $\alpha=2/3$, or Grimm:2018ohb  111See also Hebecker:2018vxz  for the
earlier derivation, which also discussed the ‘loophole’ of the entropy
argument in Ooguri:2018wrx .

$\begin{split}\Lambda_{\rm sp}\sim M_{\rm
Pl}e^{-\frac{1}{3}\lambda\Delta\varphi},\quad\quad{\cal N}\sim
e^{\frac{2}{3}\lambda\Delta\varphi}.\end{split}$ (4)

The semi-classical EFT on the (quasi-)dS background has a natural UV cutoff
scale $H$ Cheung:2007st ; Weinberg:2008hq  (see also Prokopec:2010be ;
Gong:2016qpq ). For this EFT to be valid against the SDC,
$H\lesssim\Lambda_{\rm sp}$ must be satisfied, which reads

$\begin{split}\Delta\varphi\lesssim\frac{3}{\lambda}\log\Big{(}\frac{M_{\rm
Pl}}{H}\Big{)}.\end{split}$ (5)

This bound indeed is consistent with the entropy argument of Ooguri:2018wrx
which supports the dSC. Meanwhile, the low energy excitations can be seen as
in thermal equilibrium with the (quasi-)dS background, in which case the
temperature is given by Gibbons-Hawking temperature $T=H/(2\pi)$
Gibbons:1977mu . The condition that their contribution to the entropy inside
the horizon given by $S\sim{\cal N}T^{3}H^{-3}\sim{\cal N}$ cannot exceed the
covariant entropy bound $\pi M_{\rm Pl}^{2}/H^{2}$ gives the same bound on
$\Delta\varphi$ as (5).

We note that in the entropy argument above, we used ${\cal N}T^{3}\sim{\cal
N}H^{3}$, rather than ${\cal N}\Lambda_{\rm sp}^{3}$ for the entropy density.
For this estimation to be valid, the number density of states having the
energy $M$ in the range from $H$ to $\Lambda_{\rm sp}$ needs to be Boltzmann-
suppressed such that ${\cal N}{\rm exp}[-M/T]<1$. At the early stage of the
inflaton excursion, i.e., $\Delta\varphi\ll 1$, this is trivially satisfied.
As $\Delta\varphi$ gets close to the bound in (5), ${\cal N}$ becomes
exponentially large enough to overcome the Boltzmann-suppression factor, but
then $\Lambda_{\rm sp}$ is close to $H$. Thus, in any case, we can employ
${\cal N}H^{3}$ for the entropy density. For example, the states having the
energy $M\simeq\Lambda_{\rm sp}$ are not negligible if

$\begin{split}\log{\cal N}-\frac{\Lambda_{\rm
sp}}{T}&=\frac{2}{3}\lambda\Delta\varphi-\frac{2\pi M_{\rm
Pl}}{H}e^{-\frac{1}{3}\lambda\Delta\varphi}\end{split}$ (6)

becomes positive. For the observational bound $H<10^{13-14}$GeV imposed by the
recent Planck observation Akrami:2018odb , this happens after
$\Delta\varphi\simeq(0.9)[(3/\lambda)]\log(M_{\rm Pl}/H)$, at which
$\Lambda_{\rm sp}$ is very close to $H$. For the same reason, we use
$\sim{\cal N}H^{4}$ for the energy density throughout this article.

##  III dS CKN bound

The CKN bound Cohen:1998zx  stems from the requirement that black hole cannot
be the low energy excitation, or equivalently, the total energy inside the
given volume is sufficiently low that black hole is not produced. In this
section, we apply this argument to the dS background.

An uncharged, nonrotating black hole in the dS background is described by the
Schwarzschild-dS geometry,

$\begin{split}&ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega_{2}^{2},\\\
&f(r)=1-\frac{2GM}{r}-H^{2}r^{2}.\end{split}$ (7)

Since $f(r)$ becomes negative infinity as $r\to 0$ and $r\to\infty$, one finds
that $f(r)>0$ in which $t$ is along the timelike direction is satisfied only
for specific range of $r$ and the specific parameter $GMH$. Indeed, one of
three solutions to $f(r)=0$ is negative, which is not physical. When another
two solutions $r_{1},~{}r_{2}$ to $f(r)=0$ are real and positive
($r_{1}<r_{2}$), $f(r)$ is positive for $r_{1}<r<r_{2}$ then $r_{1}$ and
$r_{2}$ are interpreted as the black hole horizon and the cosmological
horizon, respectively.

Figure 1: The behavior of $F(x)=c-x+x^{3}$ for two extreme cases, $c=0$ and
$c=2/3^{3/2}$. For $0<c<2/3^{3/2}$, $F(x)$ lies between two curves and both
the black hole horizon and the cosmological horizon are well defined. We note
that for any value of $c$, $F(x)$ has a local maximum at $x=1/\sqrt{3}$.

From this observation, Banks:2019arz  suggested the dS CKN bound by simply
requiring $f(r)>0$. Here we investigate this bound more quantitatively and
discuss what happens when the SDC is additionally taken into account. For this
purpose, we use the dimensionless parameter $x=Hr$, such that $f(r)$ is
rewritten as

$\begin{split}f(r)=-\frac{1}{x}\Big{(}2GMH-x+x^{3}\Big{)}\equiv\frac{F(x)}{x},\end{split}$
(8)

thus $f(r)>0$ for $F(x)>0$. As depicted in Fig. 1, we find that $F(x)=0$, or
equivalently, $f(r)=0$ has two positive solutions for
$0<2GM<(2/3^{3/2})H^{-1}$. When $2GM=(2/3^{3/2})H^{-1}$, $F(x)$ vanishes at a
local maximum $x=1/\sqrt{3}$ so two real solutions become the same. This shows
that as the mass of black hole increases, the cosmological horizon decreases
until it coincides with the black hole horizon. Introducing a parameter
$\theta$ defined as

$\begin{split}\cos\theta=-3^{3/2}GMH,\end{split}$ (9)

two horizons are given by

$\begin{split}Hr_{1}=\sin\frac{\theta}{3}-\frac{1}{\sqrt{3}}\cos\frac{\theta}{3},\quad\quad&({\rm
black~{}hole})\\\
Hr_{2}=\frac{2}{\sqrt{3}}\cos\frac{\theta}{3}\quad\quad&({\rm
cosmological}).\end{split}$ (10)

We note that as $\theta$ increases over the region $\pi/2<\theta<\pi$, which
is equivalent to the increase in $2GM$ from $0$ to $(2/3^{3/2})H^{-1}$,
$r_{1}$ monotonically increases from $0$ to $(1/\sqrt{3})H^{-1}$ whereas
$r_{2}$ monotonically decreases from $H^{-1}$ to $(1/\sqrt{3})H^{-1}$. That
is, $r_{2}$ is always larger than $r_{1}$. In perfect dS, in which the
cosmological horizon size is just $H^{-1}$, the vacuum energy is the only
contribution to the energy of spacetime. Since matter does not exist, black
hole is not produced as $r_{1}=0$ indicates. When we add matter to this
background, its backreaction results in the deformation of the cosmological
horizon down to some value smaller than $H^{-1}$ and the geometry is no longer
perfect dS. If matter forms black hole, the new cosmological horizon size will
be $r_{2}$.

For the EFT for matter in (quasi-)dS background to be valid without producing
black hole, matter is required to be distributed over the region of the size
$L$ larger than $r_{1}$, the radius of black hole whose mass is given by the
corresponding energy. The conjecture in Banks:2019arz  quantifies this no-
black-hole-formation condition by $f(L)>0$, or

$\begin{split}r_{1}(\theta)<L<r_{2}(\theta),\end{split}$ (11)

where the energy is contained in $\theta$ through (9). We note that $r_{2}$,
the cosmological horizon determined by black hole, behaves like the upper
bound of the cosmological horizon for a fixed $M$. This may be justified from
the fact that given $M$, black hole has the highest energy concentration. Then
we expect that the components of the energy-momentum tensor decay quickly as
we move away from the black hole horizon and their values around the
cosmological horizon are small compared to those of other form of the matter
distribution. This results in the small local backreaction on the horizon.
222Unlike the black hole horizon, the (quasi-)dS cosmological horizon (in the
absence of black hole) is an observer dependent object. Even in this case, if
the (quasi-)dS geometry is deformed, any observer must find the local
backreaction of the energy-momentum tensor on the horizon (see, e.g.,
Gong:2020mbn ). Of course, in the presence of black hole, the cosmological
horizon becomes observer-independent, as it is defined by the distance from
the black hole center. Another way to see this is to consider the case of
$GMH\ll 1$, in which $r_{2}$ is very close to $H^{-1}$. Suppose that matter is
distributed almost homogeneously inside the horizon. If such a homogeneous
distribution extends beyond $r_{2}$ to reach $H^{-1}$, the energy density of
matter is indistinguishable from that of the vacuum energy. Then the region
inside $H^{-1}$ keeps expanding exponentially. This is contradict to our
expectation, the contraction of the cosmological horizon by the gravitational
attraction of matter. If the fluctuation of the matter distribution is
sufficiently small, in the absence of the dimensionful parameter comparable to
$H$, the mean energy is the same as that in the homogeneous case up to some
constant, giving the similar result. This indeed is the way to treat the
cosmological perturbations over the almost homogeneous and isotropic mean
energy distribution. This issue will be visited in the next section, in which
we find that the condition $L<r_{2}$ excludes the eternally inflating
spacetime, supporting the argument above.

##  IV Deformation of dS space

As we discussed in the previous section, the dS CKN bound describes the
deformation (more precisely, contraction) of the cosmological horizon without
invalidating the EFT in the presence of matter. In this section, we combine
the dS CKN bound given by (11) with the SDC such that the energy inside the
cosmological horizon is increased by the increasing number of the particle
species ${\cal N}$ along the inflaton trajectory. Here we use the energy
density given by ${\cal N}T^{4}\sim{\cal N}H^{4}$ as justified in the
discussion around (6).

For the clear interpretation, we take the limit $GMH\ll 1$ of (11). This
obviously includes the flat space limit in which the cosmological horizon is
much larger than the black hole size, but also describes the small deformation
of the cosmological horizon by the small amount of matter. Expanding $\theta$
defined in (9) around $\pi/2$, one finds that

$\begin{split}r_{1}\simeq 2GM,\quad\quad r_{2}\simeq H^{-1}-GM\end{split}$
(12)

for $GMH\ll 1$. Therefore, the dS CKN bound (11) reads

$\begin{split}GM<{\rm min}\Big{[}\frac{L}{2},\frac{1}{H}-L\Big{]}.\end{split}$
(13)

This shows that $GM$ is bounded by $L/2$ for $L<(2/3)H^{-1}$ which includes
the flat space limit and by $H^{-1}-L$ for $L>(2/3)H^{-1}$, which describes
the small deformation of the cosmological horizon, respectively. 333 The bound
(13) tells us that if we set $L^{-1}$, the bound (13) only allows for $M=0$,
excluding any finite mass. This may imply that $L$ in fact cannot be
arbitrarily fixed, but determined by the backreaction from matter in dS. In
the absence of any matter, i.e., $M=0$, the horizon does not change at all, so
cosmological size is just given by $H^{-1}$. This motivates us to interpret
the CKN bound as a horizon size deformed by the matter in dS. In any case, the
bound on $GM$ is smaller than $(3H)^{-1}$, hence by setting $M={\cal
N}H^{4}L^{3}$ we expect the naive bound 444While we take the mass density
${\cal N}H^{4}$ for the radiation, one may consider the mass density ${\cal
N}H^{2}M_{\rm Pl}^{2}$ motivated by the energy density of $\varphi$, the dark
energy. The latter gives more stringent bound ${\cal N}\leq 1$, restricting
$\Delta\varphi$ to the value close to zero.

$\begin{split}{\cal N}\lesssim\frac{M_{\rm Pl}^{2}}{3H^{5}L^{3}}.\end{split}$
(14)

Then for $(2/3)H^{-1}\ll L\lesssim H$, i.e., when $L$ is very close to the dS
horizon, we obtain the bound (5) with the help of (4). However, this
estimation lacks the shrinking bound on $M$ in the perfect dS limit $L\simeq
H^{-1}$, in which case the upper bound on $M$ vanishes. Therefore, we
concentrate on the explicit bound given by (13).

Before discussing the deformation of the cosmological horizon, we briefly
address the case of $L<(2/3)H^{-1}$. This includes the flat space limit, in
which the energy $M={\cal N}H^{4}L^{3}$ is concentrated on the small region
deep inside the cosmological horizon. Then the bound $GM<L/2$ gives

$\begin{split}{\cal N}\lesssim\frac{M_{\rm Pl}^{2}}{H^{4}L^{2}},\end{split}$
(15)

which is trivially satisfied for $L<(2/3)H^{-1}$ since the SDC imposes

$\begin{split}{\cal N}\sim
e^{\frac{2}{3}\lambda\Delta\varphi}\lesssim\Big{(}\frac{M_{\rm
Pl}}{H}\Big{)}^{2}.\end{split}$ (16)

We now move onto the case of $L>(2/3)H^{-1}$, in which $L$ is very close to
$H^{-1}$. The dS CKN bound in this case can be combined with the SDC by
considering the matter production in quasi-dS space induced by the excursion
of $\varphi$ as well as the interaction between matter and the background. As
$\varphi$ moves, ${\cal N}$ is drastically increased by the descent of a tower
of states. As the backreaction from the produced matter on the background
deforms the cosmological horizon, matter is distributed inside the new
cosmological horizon of the size $L$ which is slightly smaller than $H^{-1}$.
The dS CKN bound imposes that $L$ is smaller than $r_{2}$, the change in the
size of cosmological horizon when the produced matter forms black hole.

To see this more explicitly, we note that time and length scales in our
discussion such as the constant dS cosmological horizon size $H^{-1}$ are
written in terms of the static coordinates. Meanwhile, for the comparison with
the inflationary cosmology in which the slow-roll parameters and the matter
fluctuation are defined in terms of the flat coordinates $(t,r)$, we need the
relation between two coordinates,

$\begin{split}&e^{-Ht_{s}}=e^{-Ht}\sqrt{1-H^{2}r_{s}^{2}},\\\
&r_{s}=re^{Ht}.\end{split}$ (17)

From this, we find that at fixed $r_{s}$, say, $r_{s}=H^{-1}(1-\gamma)$ with a
constant $\gamma$, we have $t_{s}=t-\log(2\gamma)/2$ thus $dt_{s}=dt$. This
indicates that so far as we are interested in the fixed $r_{s}$, $dt$ and
$dt_{s}$ can be used interchangeably. Moreover, as can be inferred from the
fact that the time evolution of $\varphi$ breaks the dS isometry associated
with the time translation, the quantum fluctuation of the inflaton $\varphi$
is identified with the fluctuation of the flat time coordinate, $\Delta t$.
Under such fluctuation, the energy density of matter inside the horizon
$\rho={\cal N}H^{4}$ fluctuates to give
$\Delta\rho=\rho[\Delta(e^{Ht})/e^{Ht}]=({\cal N}H^{4})\times H\Delta t$. Now
suppose we are in the empty dS space with the almost constant horizon size
$H^{-1}$. From this, we separate the dS background from matter produced by the
density fluctuation, to quantify how the backreaction from matter deforms the
background geometry. Then for the given energy density $\rho={\cal N}H^{4}$,
the mass produced through the fluctuation becomes

$\begin{split}M={\cal N}H^{4}(H\Delta
t)(\frac{4\pi}{3}H^{-3})=\frac{4\pi}{3}{\cal N}H^{2}\Delta t.\end{split}$ (18)

If $H$ is constant, the fluctuation leading to $\Delta t$ results in the
increase of $M$ inside the horizon as ${\cal N}$ increases by the SDC, the
backreaction of which leads to the shrink of the cosmological horizon by
$H^{-1}-L$.

Meanwhile, in the inflationary cosmology, $\varphi$ not just increases ${\cal
N}$ through the fluctuation giving $\Delta t$, but also slow-rolls on the
potential, resulting in the decrease of $H$ in time. Then two effects that
change the size of the cosmological horizon compete : the shrink by $H^{-1}-L$
through the matter production, and the growth through the slow-roll of
$\varphi$, which is parametrized by the slow-roll parameter. If the former
dominates over the latter, inflation never ends, realizing the eternal
inflation. From the expression for the slow roll parameter $\epsilon$,

$\begin{split}\epsilon=-\frac{\dot{H}}{H^{2}}=\frac{4\pi{\dot{\varphi}}^{2}}{H^{2}},\end{split}$
(19)

one obtains

$\begin{split}\frac{dH}{d\varphi}=-H\sqrt{4\pi\epsilon}.\end{split}$ (20)

Treating $\epsilon$ as almost constant, the increase in the horizon size
induced by the change in $\varphi$ is given by

$\begin{split}\Delta
H^{-1}_{\varphi}=H^{-1}\sqrt{4\pi\epsilon}\Delta\varphi=\epsilon\Delta
t\end{split}$ (21)

where for the last equality, the relation $\Delta\varphi={\dot{\varphi}}\Delta
t=[\epsilon/(4\pi)]^{1/2}H\Delta t$ is used (see (19)).

Given the total mass (18), the shrink of the horizon $H^{-1}-L$ is at most
$GM$, as constrained by the dS CKN bound $GM<H^{-1}-L$. For the inflation to
end, such a shrink must be compensated by the increase in the horizon size by
the slow-roll given by (21), giving $\Delta H^{-1}_{\varphi}>GM$, or

$\begin{split}{\cal N}\lesssim\sqrt{4\pi\epsilon}\frac{M_{\rm
Pl}^{2}}{H^{3}}\frac{\Delta\varphi}{\Delta t}=\epsilon\frac{M_{\rm
Pl}^{2}}{H^{2}}.\end{split}$ (22)

To see the physical meaning of this bound, we assume $\epsilon\ll 1$ such that
$L$ is still close to $H^{-1}$ after a single $e$-fold, $H\Delta t\sim 1$.
This means that $t\sim H^{-1}$ can be regarded as the early stage of the
matter production, i.e., $\Delta\varphi\ll 1$ thus UV degrees of freedom do
not descend below $H$ yet and ${\cal N}\sim{\cal O}(1)$. From ${\cal N}\simeq
1+(2/3)\lambda\Delta\varphi+\cdots$ one finds that (22) provides the bound on
$\epsilon$, 555While an expansion of ${\cal N}$ contains ${\cal O}(\epsilon)$
term given by $(2/9)\lambda^{2}(\Delta\varphi)^{2}$, a coefficient of
$\epsilon$ is ${\cal O}(1)$, which is much smaller than $M_{\rm Pl}^{2}/H^{2}$
in (22). Thus, this term does not affect the result significantly.

$\begin{split}\epsilon\gtrsim\frac{H^{2}}{M_{\rm
Pl}^{2}}+\frac{\lambda}{6\sqrt{\pi}}\frac{H^{3}}{M_{\rm
Pl}^{3}}+\cdots.\end{split}$ (23)

Here the $\lambda$ dependence is suppressed as $H\ll M_{\rm Pl}$ and we are
left with the model-independent inequality $\epsilon\gtrsim(H/M_{\rm
Pl})^{2}$. This bound is nothing more than the well-known condition that the
eternal inflation does not take place Steinhardt:1982 ; Vilenkin:1983xq ;
Linde:1986fc ; Linde:1986fd ; Goncharov:1987ir  (see also Starobinsky:1986fx
). 666The conventional condition $\epsilon\gtrsim(H/M_{\rm Pl})^{2}$
prohibiting the eternal inflation comes from the width of the probability
distribution for the deviation of $\varphi$ from the classical trajectory.
Since the exact coefficient attached to $(H/M_{\rm Pl})^{2}$ is obtained from
the Fokker-Planck equation, which may be model dependent, we just consider the
parametric dependence, by assuming ${\cal N}_{0}$ is not too large. For recent
discussions along the line of this article are given in, for example,
ArkaniHamed:2007ky ; Kinney:2018kew ; Brahma:2019iyy ; Rudelius:2019cfh . If
the universe is in the quantum mechanically excited state, the eternal
inflation may take place for $\epsilon$ larger than the integer multiples of
$H^{2}/M_{\rm Pl}^{2}$ Seo:2020ger . The eternal inflation occurs when the
cosmological fluctuations, the wavelength of which is stretched beyond the
cosmological horizon (thus the low energy degrees of freedom below $H$),
enhance the probability that the inflaton does not roll down its potential.
Recalling the discussion at the end of previous section, one finds the dS CKN
bound $L<r_{2}$ imposes that the descent of states claimed by the SDC induces
the contraction of the horizon by their gravitational attraction, so it is
natural that the eternal inflation is not favored.

As $\varphi$ keeps moving on, the number of low energy degrees of freedom
exponentially increases in time. Since $\Delta t$ is no longer small, the
matter produced fills the region inside the horizon almost homogeneously,
which is in thermal equilibrium with the background. This results in the
increase in $M$ toward $(3^{3/2}GH)^{-1}$ such that $\theta$ gets close to
$\pi$. Around $\theta=\pi$, the matter distribution must satisfy both
$L\simeq(1/\sqrt{3})H^{-1}$ and $GM=(1/3^{3/2})H^{-1}$. From $M={\cal
N}H^{4}L^{3}\simeq{\cal N}H$ we find that ${\cal N}\simeq(M_{\rm Pl}/H)^{2}$,
which is close to the saturation of the bound (5), and Ooguri:2018wrx  claims
that $\epsilon$ gradually increases and eventually becomes ${\cal O}(1)$. More
concretely, once the bound ${\cal N}\leq(M_{\rm Pl}/H)^{2}$ is saturated, the
horizon is strongly deformed not to violate the bound. The shrink of the
horizon is forbidden by the rapid increase in $H^{-1}$, as $\epsilon\sim{\cal
O}(1)$ implies. This is achieved when the rate of change in ${\cal N}$ is
still smaller than that of $(M_{\rm Pl}/H)^{2}$, i.e., $(d{\cal
N}/d\varphi){\dot{\varphi}}\lesssim(-2{\dot{H}}/H^{3})M_{\rm Pl}^{2}$. Since
$d{\cal N}/d\varphi=(2/3)\lambda{\cal N}=(2/3)\lambda(M_{\rm Pl}/H)^{2}$ at
the saturation, from (19) we obtain (see also Seo:2019wsh )

$\begin{split}\epsilon\gtrsim\frac{\lambda^{2}}{36\pi}.\end{split}$ (24)

As estimated in Seo:2019wsh ; Cai:2019dzj , the saturation takes place after
about ${\cal N}\sim(1/\langle\epsilon\rangle)^{1/2}\log(M_{\rm Pl}/H)$
$e$-folds, where $\langle\epsilon\rangle$ is the averaged value of $\epsilon$
in the quasi-dS regime hence smaller than unity. Since $H/M_{\rm Pl}<10^{-5}$,
if $\langle\epsilon\rangle$ is as small as $10^{-2}$, $\varphi$ slow-rolls
over $60~{}e$-folds, which is sufficient to resolve the horizon problem or the
flatness problem Seo:2019wsh .

Furthermore, the refined dSC Andriot:2018mav ; Garg:2018reu ; Ooguri:2018wrx
imposes another condition $\dot{\epsilon}/(H\epsilon)\sim{\cal O}(1)$ for
$\epsilon\ll 1$. If it were the case, even if $\epsilon\simeq 0$ at some
moment, it soon increases as $\varphi$ moves. This in fact can be frequently
found in the supergravity models of the inflation, which is known as the eta
problem Copeland:1994vg .

##  V Conclusion

So far, we have discussed the implications of the CKN bound when we take the
SDC into account. In quasi-dS space, the exponential descent of the low energy
degrees of freedom as conjectured by the SDC leads to the rapid increase in
the energy inside the cosmological horizon. The dS CKN bound states that for
the EFT not to be spoiled by the production of black hole, matter must be
distributed between the black hole radius and the cosmological horizon
determined by black hole of the same energy. By identifying the maximal size
of the matter distribution with the cosmological horizon deformed by the
production of matter, we find that the condition for the SDC-assisted dS CKN
bound to forbid the eternal inflation coincides with the well-known condition
on the slow-roll parameter, $\epsilon\gtrsim H^{2}/M_{\rm Pl}^{2}$. Our
discussion is supplementary to the entropy argument supporting the dSC, as it
enables us to investigate more detailed dynamics of the contracting
cosmological horizon.

Meanwhile, the SDC seems to be consistent with other swampland conjectures,
such as the weak gravity conjecture (WGC) ArkaniHamed:2006dz . The weak
gravity conjecture quantifies the quantum gravity obstruction to restoring a
global symmetry by taking vanishing limit of the unbroken U(1) gauge coupling
(for a recent discussion relevant to the CKN bound, see, e.g.,
Davoudiasl:2021aih  and for discussions on the WGC in dS space, see, e.g.,
Huang:2006hc ; Montero:2019ekk ; Montero:2021otb ). The correlation between
the WGC and the SDC can be found from, for example, the restoration of the
global shift symmetry in the limit of infinite field value of the modulus,
where non-perturbative effects become suppressed Palti:2017elp ; Grimm:2018ohb
; Lee:2018urn ; Lee:2018spm ; Gendler:2020dfp . The SDC claims that this can
be prevented by cutting off $\Delta\varphi$ at the Planckian value. The
quantum gravity implications of the connection between the SDC, the dSC, the
weak gravity conjecture and the CKN bound will be a subject of further
investigation.

###### Acknowledgements.

## References

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# Half period theorem of binary black holes

Masaru<EMAIL_ADDRESS>and Daisuke Ida

###### Abstract

Merging event horizons of the binary black holes is investigated. While recent
development of the numerical study of the binary black hole coalescence has
shown that their apparent horizons can orbit for many periods, we study the
orbital motion of the event horizon. We discuss how many periods their event
horizons orbit before their coalescence. Then, we find that they soon merge
into one and the black holes cannot orbit for a half period while the apparent
horizons can orbit many times.

Department of Physics, Tokyo Institute of Technology,
Oh-Okayama 1-12-1, Megro-ku, Tokyo 152-8550, Japan
Department of Physics, Gakushuin University, Tokyo 171-8588, Japan

##  1 Introduction

Recently the black hole binary system is much paid attention to as a candidate
for gravitational wave source. Indeed numerical studies of the binary black
hole coalescence advances so that it predicts the time profile of the
gravitational radiation. Usually in numerical simulation, the black hole
formation is examined by determining an apparent horizon. This is because the
existence of the apparent horizon strongly suggests that of the event horizon.
The event horizon is meaningful for an asymptotic observer, since he can just
observe the outside of the event horizon, but not the apparent horizon.
Moreover the event horizon is gauge invariant concept, while apparent horizon,
even its existence depends on timeslicing. Hence, we would like to discuss the
event horizon for the binary black hole system.

Some authors have made sure that two apparent horizons can orbit around each
other in a case of the binary black hole coalescence[1][2]. Then we attempt to
discuss whether this picture also holds for the event horizon. Reason why we
suspect the behavior of event horizon differs from that of the apparent
horizon is that, in some studies of black hole formation, it is insisted that
the event horizon soon settles to a single sphere[5][4]. In other words, we
speculate that there does not remain sufficient time for the two black holes
to orbit around each other.

It is known that the event horizon is generated by null geodesics without a
future endpoint [3]. In particular, the event horizon is a null hypersurface
which may not be smooth at the past endpoints of the null geodesic generators.
The set of the past endpoints of the null geodesic generators of the event
horizon will be called the crease set [6][4]. Since the crease set is a subset
of the null hypersurface, it is an achronal set(any two points are not
connected by a timelike curve) as illustrated in Fig.1 [8][4]. That the crease
set is achronal roughly implies two event horizons coalesce in superluminal
speed.

Figure 1: An event horizon and its crease set. The crease set is achronal and
roughly implies two event horizon coalesce in superluminal speed.

As a rough estimation, let two black holes be located with separation $2R$.
They will coalesce within the time-interval of order of $\Delta t=R/c$ while
we cannot expect any circular motion with this time duration. Hence we expect
there is any upper bound of a rotation angle of the binary black holes before
their coalescence.

In the present article, we assert the existence of the upper bound of the
rotation angle and demonstrate a half period theorem that binary black holes
cannot orbit for a half period in terms of their event horizons assuming the
reflection symmetry with respect to the orbital surface. This assumption is
required for technical reason to formulate the notion of the half period of
the binary black holes without ambiguities. Clearly, it is meaningless to say
that the binary black holes go around or half around in terms of a coordinate
system. We should formulate the half period of the binary black holes without
referring to a specific coordinate system.

The outline of our argument is as follows. We first consider the foliation of
the spacetime by a family of timelike curves and choose a reference timelike
curve as a center of the binary motion. Next we determine the opposite
direction, which we call the light ray opposite, of each black hole with
respect to the center. Then, whether the black hole orbits for a half period
from the initial configuration or not can be determined in terms of the light
ray opposite. Finally, we show that each black hole cannot orbit for a half
period irrespective of the choice of the foliation and the center. This gives
a possible way to measure the amount of the binary motion in a coordinate-free
way.

We first explain more details of our scheme in the next section. In the third
section we give several definitions and set up the situation. The main theorem
is stated in the fourth section. In the last section, we give some remarks and
implications.

##  2 schematic picture

Now, we try schematic discussion based on the topological notion of event
horizon to illustrate that a half period of the binary rotation typically
bounds duration of binary rotating era. From Newtonian picture, one may think
that the coalescence of binary black hole event horizons occurs after two
black hole event horizons orbit around each other for several periods. A
simple picture of the orbiting event horizons, however, makes us suspect them
of merging before the several periods have elapsed.

Here we simply consider two black holes with identical mass fated to coalesce.
They are on a binary orbit shrinking by the energy loss caused by
gravitational radiation. In a coordinate system $(t,x^{i})$ such that each
$x^{i}={\rm const.}$ line is timelike, every comoving observer inside a black
hole at some moment will stay within the black hole region, while the black
hole move on the binary orbit. This implies that a Newtonian picture of binary
motion as in Fig.2 is incorrect.

We expect two possible scenarios for binary black hole coalescence, which we
call the quasi-stationary scenario and the rapid coalescence scenario, as
follows. The quasi-stationary scenario is what we expect to occur in the
quasi-stationary binary motion of black holes. In the quasi-stationary
scenario, the event horizons form a toroidal event horizon along the binary
orbit in a half period of binary motion (See Fig.2). The rapid coalescence
scenario is expected to occur when the binary black holes don’t have orbital
angular momentum enough. In this case, two black holes coalesce without
forming a toroidal horizon (See Fig.2). In both scenarios, binary black holes
do not orbit for a half period before coalescence.

In the above argument, the term “a half period” is loosely used. It is in
general difficult to determine whether the binary black holes go around
without specific coordinate system. The present work is our attempt to seek
for a rigorous statement corresponding to the above schematic discussion. For
this purpose, we should study it in a generally covariant manner incorporating
a technique based on the causal studies of general relativity.

Figure 2: The figures in the left side illustrate a Newtonian picture of
binary motion. The others are two possible scenarios for binary black hole
coalescence. The center is the quasi-stationary scenario and the right is the
rapid coalescence scenario.

##  3 set up

First of all, we have to consider the amount of orbital rotation of the binary
black holes. This will be determined by introducing a global angular
coordinate function. However, it is in general difficult due to the lack of
the standard coordinate system.

For this reason, we give up considering one radian to be an upper bound of
orbital rotation angle, and attempt to put a mark corresponding to a half
period in order to check orbiting of black holes. We ought to be able at least
to discuss a half period by introducing a ‘straight’ curve passing the
antipodal point, even without the angular coordinate.

Nevertheless, each black hole does not always pass the opposite side of the
straight curve in general cases. For simplicity in the present work, we assume
reflection symmetry such that the plane of symmetry, which we call the orbital
surface, intersects binary black holes. With the reflection symmetry, it would
be enough to discuss the motion of the object only on the orbital surface.

We define a light ray opposite as the mark of a half of the orbital period.
Suppose a spacetime $(M,g)$ is globally hyperbolic. Then, $(M,g)$ admits
timeslicing $\\{\Sigma(t)\\}$ in terms of a global time function $t:M\to{\bf
R}$ and there will be a timelike vector field $T$ without zero points. Let the
vector field $T$ be normalized such that $\langle T,dt\rangle=1$ holds.

Let us consider a cylindrical region ${\mathscr{U}}$ generated by the vector
field $T$ with a timelike side-boundary $B_{{\mathscr{U}}}$ as illustrated in
figure 3. Then the timelike vector field $T$ determines a natural projection

$\pi_{t}:{\mathscr{U}}\mapsto{\mathscr{U}}\cap\Sigma(t)$

of the closed subset ${\mathscr{U}}$ of $M$ into each timeslice $\Sigma(t)$
along the integrated curve of $T$.

######  Definition 1 (Comoving ball with the origin).

Let $(M,g)$ be a globally hyperbolic space-time. Let $t:M\to{\bf R}$ be a
global time function in $M$. The sequence $\\{\Sigma(t)\\}$ of the $t={\rm
const.}$ hypersurface $\Sigma(t)$, $t\in[t_{i},t_{f}]$ will be called the
timeslicing. The space-time $M$ will admit a future directed timelike vector
field $T$ such that $\langle T,dt\rangle=1$ holds. Let
$U\subset\Sigma(\overline{t})$ be a topological 3-ball embedded in
$\Sigma(\overline{t})$, $\overline{t}\in(t_{i},t_{f})$. Let ${\mathscr{U}}$ be
the closed subset of $M$ generated by $T$ such that
${\mathscr{U}}\cap\Sigma(\overline{t})=U$ holds. Let
$B_{{\mathscr{U}}}\subset\dot{\mathscr{U}}$ be the closed subset of
$\dot{\mathscr{U}}$, which is the product set $\dot{U}\times[t_{i},t_{f}]$
generated by $T$, and let $o:[t_{i},t_{f}]\to M;t\mapsto o(t)$ be the integral
curve of $T$ which pass through an interior point $o(\overline{t})$ of $U$.

The 4-tuple $(\\{\Sigma(t)\\},{\mathscr{U}},T,o)$ will be called the comoving
ball with the origin in $(M,g)$. ∎

Figure 3: A cylinder defined by the 4-tuple $(\Sigma(t),{\mathscr{U}},T,o(t))$
is the comoving ball with the origin.

It is difficult to give a precise notion of orbital plane of binary black
holes or period of binary motion in general settings. A possible way to
overcome this difficulty is to impose a reflection symmetry of $(M,g)$.
Although this restriction might be rather stringent, we could still consider a
large class of space-times describing binary black holes. In the rest of the
present paper, we will assume that the space-time $(M,g)$ admits a reflection
symmetry with respect to a timelike hypersurface ${\mathscr{O}}$ in $M$. The
fixed point set ${\mathscr{O}}$ under this isometry will be called the orbital
surface and the orbital surface at the time $t$ will be denoted as
${\mathscr{O}}_{t}={\mathscr{O}}\cap\Sigma(t)$. All the settings including the
global time function $t$ and the comoving ball with the origin
$(\\{\Sigma(t)\\},{\mathscr{U}},T,o)$ are taken as respecting the reflection
symmetry.

Here we introduce a notion of the opposite side of a point $p$ beyond the
origin $o$, which we call the light ray opposite (abbreviated to LRO) of $p$,
in terms of a null geodesic generator of $J^{+}(p)$ (see Fig. 4).

######  Definition 2 (The light ray opposite).

Let $(M,g)$ be a reflection symmetric spacetime with respect to the timelike
hypersurface ${\mathscr{O}}$. Let $(\\{\Sigma(t)\\},{\mathscr{U}},T,o)$ be a
comoving ball with the origin in $M$ respecting the reflection symmetry. Let
us call ${\mathscr{O}}_{t}={\mathscr{O}}\cap\Sigma(t)$ the orbital surface at
the time $t$. For a point $p\in{\mathscr{O}}_{t_{1}}\cap{\mathscr{U}}$,
$t_{1}\in(t_{i},t_{f})$, $\gamma_{p}$ is defined to be the geodesic generator
of $\dot{J}^{+}(p)$ from $p$ which pass through $o$ at the time
$t=t_{c}\in(t_{1},t_{f})$, if there is exactly one null geodesic generator of
$\dot{J}^{+}(p)$ from $p$ through $o$. The null geodesic $\gamma_{p}$ will be
within ${\mathscr{O}}$ due to the reflection symmetry. Then, for a point $p$
and a time $t_{2}\in(t_{1},t_{f})$, the light ray opposite (LRO)
$\lambda(p,t_{2})$ of $p$ at the time $t=t_{2}$ is defined by

$\displaystyle\lambda(p,t_{2})=\pi_{t_{2}}\left[\gamma_{p}\cap\bigcup_{t\in[t_{c},t_{f})}\Sigma(t)\right]$

where $\pi_{t_{2}}$ denotes the projection ${\mathscr{U}}\to\Sigma(t_{2})$
naturally defined by the timelike vector field $T$, and the LRO of $p$ is
defined to be the empty set if $\gamma_{p}$ is not defined.

∎

######  Remark 1.

The LRO is the empty set when the origin is far from the reference point $p$
such that $o~{}\cap J^{+}(p)=\emptyset$. Besides there might possibly occur
the situation where $o~{}\cap J^{+}(p)\neq\emptyset$ holds but
$o~{}\cap\dot{J}^{+}(p)=\emptyset$, when the congruence of the light rays from
$p$ to $o$ have caustics due to the local gravitational effects. We have just
precluded such a possibility for simplicity.

Figure 4: A light ray opposite $\lambda$ on $\Sigma(t_{2})$ with respect to
$p$ is given by $T$-projection of $\gamma(t)$.

Next, we introduce the notion of coalescing binary black holes as follows
(Fig. 5).

######  Definition 3 (The binary black hole coalescence system).

Let $(M,g)$ be a black hole spacetime and let
$(\\{\Sigma(t)\\},{\mathscr{U}},T,o(t))$ be a comoving ball with the origin in
$M$. A binary black hole coalescence system $(H,\\{\Sigma(t)\\})$ is defined
to be the pair consiting of the event horizon $H$ in $M$ and a timeslicing
$\\{\Sigma(t)\\}$, $t\in[t_{i},t_{f}]$, such that there is a coalescence time
$t^{\prime}\in(t_{i},t_{f})$ decomposing $H\cap{\mathscr{U}}$ into a pre-
coalescence part

$H_{pr}=H\cap{\mathscr{U}}\cap\left[\bigcup_{t\in[t_{i},t^{\prime})}\Sigma(t)\right]$

which has a pair of connected components, and a post-coalescence part

$H_{po}=H\cap{\mathscr{U}}\cap\left[\bigcup_{t\in(t^{\prime},t_{f}]}\Sigma(t)\right]$

which is connected, by the spatial hypersurface $\Sigma(t^{\prime})$. ∎

Figure 5: This is a binary black hole coalescence system which is composed of
a disconnected pre-coalescence part and a connected post-coalescence part.

It should be noted that the concept of the black hole coalescence depends on
the choice of the timeslicing. In a different timeslicing the black hole
coalescence system can always be regarded as the formation of a single black
hole [4, 10]. Here we consider the typical case $S^{2}\sqcup S^{2}\to S^{2}$
for the transition of the horizon topology.

Since the reflection symmetry of $M$ is imposed, it is enough to consider the
causal structure of the orbital surface ${\mathscr{O}}$. In other words, we
will concentrate on the section of the event horizon by ${\mathscr{O}}$. In
the following, we assume that $H_{pr}\cap{\mathscr{O}}_{t}$ consist of a pair
of circles $S^{1}\sqcup S^{1}$ and $H_{po}\cap{\mathscr{O}}_{t}$ consists of a
single circle $S^{1}$ in the binary black hole coalescence system.

We want to define the half cycle of the binary black hole system by saying
that that a black hole go around a half cycle if all the infinitesimal area
elements of the black hole go around a half. For this purpose, we need to
specify each trajectory of a point on the black hole. At first sight, the null
geodesic generators of the event horizon seems to naturally determine each
orbit. However, this is not appropriate, since new null geodesic generators
emerge incessantly. Instead, we consider arbitrarily chosen one parameter
family of homeomorphisms, $\phi_{t}:S^{1}\to S^{1}$ between the
$H_{pr}\cap{\mathscr{O}}_{t_{i}}$ and $H_{pr}\cap{\mathscr{O}}_{t}$,
$(t\in[t_{i},t^{\prime}))$, continuous with respect to $t$. Such one parameter
family of homeomorphisms $\phi_{t}$ determines the motion of the infinitesimal
area element of the black hole. Note that each orbit determined by $\phi_{t}$
necessarily exceeds or equals the speed of light, for it lies on the null
hypersurface.

The following is the definition of a half period of binary black holes in an
orbital surface, where each black hole event horizon moves as shown in
figure6.

######  Definition 4 (A half period of the binary coalescence system).

Let $(H,\\{\Sigma(t)\\})$ be a binary black hole coalescence system with the
reflection symmetry with respect to the orbital surface ${\mathscr{O}}$ with
the coalescence time $t^{\prime}\in(t_{i},t_{f})$. Let the pre-coalescence
part $H_{pr}\cup{\mathscr{O}}$ in ${\mathscr{O}}$ consists of a disconnected
sum $H_{pr}\cup{\mathscr{O}}=H_{\rm I}\sqcup H_{\rm II}$ such that each of
$H_{\rm I}\cup{\mathscr{O}}_{t}$ and $H_{\rm II}\cup{\mathscr{O}}_{t}$,
$t\in[t_{i},t^{\prime})$ is a circle. Let
$\phi_{t}^{A}:H_{A}\cap{\mathscr{O}}_{t_{1}}\to H_{A}\cap{\mathscr{O}}_{t}$
($A={\rm I}$, ${\rm II}$) be a one parameter family of homeomorphisms such
that $\phi_{t_{1}}^{A}$ is the identity map of $H_{A}$ and $\phi_{t}^{A}$ is
continuous with respect to $t$. For $t_{i}<t_{1}<t_{2}<t_{f}$, we say that a
half period of the binary coalescence system has elapsed during
$(t_{1},t_{2})$, if the following statement hold both for $A={\rm I}$ and
${\rm II}$; There is a point $p$ on $H_{A}\cap O_{t_{1}}$ such that every
orbit of a point on $H_{A}\cup{\mathscr{O}}_{t_{1}}$ intersects the LRO
$\lambda(p,t)$ of $p$ during $t_{1}<t<t_{2}$. ∎

Figure 6: The definition of a half period of the binary coalescence system.

Note that whether the half period has elapsed or not does not depend on the
choice of the correspondences $\\{\phi^{\rm I}_{t},\phi^{\rm II}_{t}\\}$.

##  4 The half period theorem

Now we show that a half period of the binary coalescence system does not
elapse before the coalescence. Firstly, it is easily seen that two black holes
before the coalescence are causally separated with each other as illustrated
in figure 7.

######  Proposition 1.

Let ${\mathscr{B}}$ be the black hole region in the binary black hole system
with reflection symmetry, and let ${\mathscr{O}}_{pr}$ be the part of
${\mathscr{O}}\cap{\mathscr{U}}$ before the coalescence time defined by
${\mathscr{O}}_{pr}=\bigcup_{t\in[t_{i},t^{\prime})}{\mathscr{O}}(t)\cap{\mathscr{U}}$,
so that the black hole region $B={\mathscr{B}}\cap{\mathscr{O}}_{pr}$ in the
orbital surface is composed of two black hole regions $B_{\rm I}\simeq
D^{2}\times[t_{i},t^{\prime})$ and $B_{\rm II}\simeq
D^{2}\times[t_{i},t^{\prime})$ without intersection, where $D^{2}$ denotes the
closed 2-disk. Then,

$\displaystyle\forall p_{\rm I}\in B_{\rm I},\forall p_{\rm II}\in B_{\rm
II},\quad{\mathscr{O}}_{pr}\cap J^{+}(p_{\rm I})\cap J^{+}(p_{\rm
II})=\emptyset$

holds.

###### Proof.

Since $p_{A}$ ($A={\rm I}$, ${\rm II}$) is a point in $B_{A}$, and $B_{A}$ is
a future set in ${\mathscr{O}}_{pr}$,
$J^{+}(p_{A})\cap{\mathscr{O}}_{pr}\subset B_{A}$ holds. Then, $B_{\rm I}\cap
B_{\rm II}=\emptyset$ implies $(J^{+}(p_{\rm
I})\cap{\mathscr{O}}_{pr})\cap(J^{+}(p_{\rm
II})\cap{\mathscr{O}}_{pr})=\emptyset$. ∎

Figure 7: Two black holes are causally separated.

This proposition just reflects the fact that a black hole region is a future
set. The following corollary immediately follows.

######  Corollary 1.

A timelike curve $o$ in ${\mathscr{O}}$ does not intersect both with
$J^{+}(p_{\rm I})\cap{\mathscr{O}}_{pr}$ and $J^{+}(p_{\rm
II})\cap{\mathscr{O}}_{pr}$ for any $p_{\rm I}\in B_{\rm I}$ and $p_{\rm
II}\in B_{\rm II}$.

Next, we show the following lemma.

######  Lemma 1.

For every pair of points $p_{\rm I}\in B_{\rm I}\cap{\mathscr{O}}_{t_{1}}$ and
$p_{\rm II}\in B_{\rm II}\cap{\mathscr{O}}_{t_{1}}$, either of the following
statements holds.

  1. 1.

The LRO $\lambda(p_{\rm I},t_{2})$ of $p_{\rm I}$ does not intersect with
$J^{+}(p_{\rm I})$ for any $t_{2}\in(t_{1},t^{\prime})$.

  2. 2.

The LRO $\lambda(p_{\rm II},t_{2})$ of $p_{\rm II}$ does not intersect with
$J^{+}(p_{\rm II})$ for any $t_{2}\in(t_{1},t^{\prime})$.

###### Proof.

It follows from the Cor. 1 that, for $p_{\rm I}\in B_{\rm
I}\cap{\mathscr{O}}_{t_{1}}$ and $p_{\rm II}\in B_{\rm
II}\cap{\mathscr{O}}_{t_{1}}$, $o$ does not intersect both with $J^{+}(p_{\rm
I})\cap{\mathscr{O}}_{t_{1}}$ and $J^{+}(p_{\rm
II})\cap{\mathscr{O}}_{t_{1}}$. Assume $J^{+}(p_{\rm
I})\cap{\mathscr{O}}_{t_{1}}$ does not intersect $o$. Let the LRO
$\lambda(p_{\rm I},t_{2})$ have an intersection with $J^{+}(p_{\rm
I})\cap{\mathscr{O}}_{pr}$, then $\lambda(p_{\rm I},t_{2})$ starts from
$o(t_{2})$ and extends to the point $q$ on $\dot{J}^{+}(p_{\rm
I})\cap{\mathscr{O}}_{pr}$. It follows from the definition of the LRO that $q$
is on the past directed timelike curve generated by $T$ starting from the
point $p$ on $\dot{J}^{+}(p_{\rm I})$. By slightly deforming the causal curve
from $p_{\rm I}$ to $p$ obtained by joining the null geodesic generator from
$p_{\rm I}$ to $q$ and the timelike curve from $q$ to $p$, one can construct
the timelike curve from $p_{\rm I}$ to $p$. It follows that there is an open
neighborhood $U$ of $p$, such that $U$ is contained in the chronological
future $I^{+}(p_{\rm I})$ of $p_{\rm I}$. This contradicts the fact that the
neighborhood $U$ of the boundary point $p$ of $J^{+}(p_{\rm I})$ necessarily
contains an exterior point of $J^{+}(p_{\rm I})$. ∎

Figure 8: The LRO of a point $p$, will not belong to the causal future of $p$
itself.

We are now at the position to state the main theorem.

######  Theorem 1 (The half period theorem).

If the half period of the binary coalescence system elapses during
$(t_{1},t_{2})$, two black holes merge into a single black hole at a time
$t^{\prime}\in(t_{1},t_{2})$.

###### Proof.

First note that the restriction $H\cap{\mathscr{O}}$ of $H$ on ${\mathscr{O}}$
is a null hypersurface in ${\mathscr{O}}$ generated by null geodesics, each
without a future end point. By the definition of the half period of the binary
coalescence system, there is a point $p$ on $H_{\rm
I}\cap{\mathscr{O}}_{t_{1}}$ such that the trajectory of its LRO
$\bigcup_{t\in(t_{1},t_{2})}\lambda(p,t)$ intersects with the orbit
$\bigcup_{t\in(t_{1},t_{2})}\phi_{t}^{I}(p^{\prime})$ of every point
$p^{\prime}$ on $H_{\rm I}$. This implies that a null geodesic generator of
$J^{+}(p)$, which is also a null geodesic generator of $H_{\rm
I}\cap{\mathscr{O}}$ through $p$, intersects with its own LRO
$\lambda(p,t_{p})$ at a time $t_{p}\in(t_{1},t_{2})$. In the same way, there
is a point $q$ on $H_{\rm II}\cap{\mathscr{O}}_{t_{1}}$ such that a null
geodesic generator of $J^{+}(q)$ intersects with the LRO $\lambda(q,t_{q})$ of
$q$ at a time $t_{q}\in(t_{1},t_{2})$. That is impossible is an immediate
consequence of the Lemma 1.

∎

##  5 Discussion

First, we comment on the general covariance of our result. The set up of the
problem here might seem to depend on a specific coordinate system. In fact, we
prepare a time function and a timelike vector field to define the LRO. This
obviously corresponds to a specific choice of the time coordinate and
$x^{i}={\rm const.}$ $(i=1,2,3)$ lines. Then, the LRO of a point $p$ is
regarded as the line $x^{i}=x^{i}(s)$ on $t={\rm const.}$ surface obtained by
projecting points on the future directed null geodesic $x^{\mu}=x^{\mu}(s)$
$(\mu=0,1,2,3)$ starting from $p$ into the $t={\rm const.}$ surface along the
$x^{i}={\rm const.}$ lines. In this sense, the definition of the LRO, hence
that of the half period of the binary black hole system, depends on the
coordinate system chosen. Nevertheless, the half period theorem is formulated
in a covariant manner in the sense that it holds for arbitrary choice of such
ordinary coordinate system, where ordinary coordinate we mean is such that a
$t={\rm const.}$ surface is a spacelike hypersurface and an $x^{i}={\rm
const.}$ line is a timelike curve.

There are at least a couple of shortcomings in the definition of the LRO. One
is that the LRO of a point $p$ can be empty set when the congruence of the
light rays from $p$ to the orbit $o$ of the origin has caustics before
reaching $o$. Another is that the LRO might not be a sufficiently long curve,
so that it cannot be used as a goal line of the halfway around.

The statement of the half period theorem might sound extraordinary, since it
seems to contradict an existence of the quasi-stationary phase of binary black
hole system. However, we would like to emphasis that it states an ordinary
thing that a comoving observer of a black hole does not exceed a speed of the
light, and that it does not contradict the quasi-stationary motion of the
binary black holes. Recently, there have appeared numerical results [1, 2]
showing that binary black holes go around many times, which apparently
contradicts our theorem. In these numerical computations, apparent horizons,
not the event horizons, are searched in the numerical space-time. There are
two possibilities explaining this apparent contradiction. The first one is the
following. We expect that each apparent horizon is surrounded by the event
horizon. Then, there is a possibility that a pair of apparent horizons going
around each other is already enclosed by a single event horizon. In other
words, while apparent horizons go around many times, their event horizons
quickly merge into one. The second possibility is that the coordinate system
used in the numerical simulation is superluminal, where $x^{i}={\rm const.}$
lines become space-like.

We expect that at least the second possibility is correct. First, the apparent
horizon, when regarded as a dynamical surface, become a spacelike hypersurface
in the space-time. This means that the apparent horizon has a confinement
property. Therefore, each comoving observer, once entered the trapped region,
never exit unless the apparent horizon disappears. Hence, if the apparent
horizons seem to go around many times, the coordinate system describing this
will be a superluminal one.

Finally, we speculate on implications of our theorem to astrophysical
observation. We cannot directly observe the event horizon. What are observed
are light rays which miss the event horizon. Let us consider the situation
where we can directly observe shadows of a binary system due to the existence
of bright background light. This will be possible at least in principle. If
the binary system consists of ordinary dark stars other than black holes and
we observe it in the orbital surface (or with the maximal inclination angle),
we will see two shadows intersect many times each other. While if it consist
of black holes, we just observe two shadows merging into one exactly once.

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