# Linguistic Cues of Deception in a Multilingual April Fools’ Day Context

Katerina Papantoniou1,2, Panagiotis Papadakos2, Giorgos Flouris2, Dimitris
Plexousakis1,2
1\. Computer Science Department, University of Crete, Greece
2\. Institute of Computer Science, FORTH, Greece
{papanton, papadako, fgeo<EMAIL_ADDRESS>
###### Abstract

In this work we consider the collection of deceptive April Fools’ Day (AFD)
news articles as a useful addition in existing datasets for deception
detection tasks. Such collections have an established ground truth and are
relatively easy to construct across languages. As a result, we introduce a
corpus that includes diachronic AFD and normal articles from Greek newspapers
and news websites. On top of that, we build a rich linguistic feature set, and
analyze and compare its deception cues with the only AFD collection currently
available, which is in English. Following a current research thread, we also
discuss the individualism/collectivism dimension in deception with respect to
these two datasets. Lastly, we build classifiers by testing various
monolingual and crosslingual settings. The results showcase that AFD datasets
can be helpful in deception detection studies, and are in alignment with the
observations of other deception detection works.

00footnotetext: Copyright ©2021 for this paper by its authors. Use permitted
under Creative Commons License Attribution 4.0 International (CC BY 4.0).

##  1 Introduction

April Fools’ Day (for short AFD) is a long standing custom, mostly in Western
societies. It is the only day of the year when practical jokes and deception
are expected. This is the case for all social interactions, including
journalism, which is generally considered to aim at the presentation of truth.
Every year on this day, newspapers and news websites take part in an
unofficial competition to invent the most believable, but untrue story. In
this respect, AFD news articles fall into the deception spectrum, as they
satisfy widely acceptable definitions of deception as in Masip et al. [Masip
et al., 2005].

The massive participation of news media in this custom establishes a rich
corpus of deceptive articles from a diversity of sources. Although AFD
articles may exploit common linguistic instruments with satire news, like
exaggeration, humour, irony and paralogism, they are usually considered a
distinct category. This is mainly due to the fact that they also employ other
mechanisms which characterize deception in general, like sophisms, and changes
in cognitive load and emotions [Hauch et al., 2015] to deceive their audience.
AFD articles are often believable, and there exist cases where sophisticated
AFD articles have been reproduced by major international news agencies
worldwide111https://www.nationalgeographic.com/history/article/150331-april-
fools-day-hoax-prank-history-holiday.

This motivated us to extend our previous work on linguistic cues of deception
and their relation to the cultural dimension of individualism and collectivism
[Papantoniou et al., 2021], in the context of the AFD. That work examines if
differences in the usage of linguistic cues of deception (e.g., pronouns)
across cultures can be identified and attributed to the
individualism/collectivism divide.

Specifically, the contributions of this work are:

  * •

A new corpus that includes diachronic AFD and normal articles from Greek
newspapers and news websites222The collection is available in:
https://gitlab.isl.ics.forth.gr/papanton/elaprilfoolcorpus, adding one more
AFD collection to the currently unique one in English [Dearden and Baron,
2019].

  * •

A study and discussion of the linguistic cues of deception that prevail in the
Greek and English collection, along with their similarities.

  * •

A discussion on whether the consideration of the individualism/collectivism
cultural dimension in the context of AFD aligns with the results of our
previous work.

  * •

An examination of the performance of various classifiers in identifying AFD
articles, including multilanguage setups.

##  2 Related work

The creation of reliable and realistic ground truth datasets for the deception
detection task is a challenging task [Fitzpatrick and Bachenko, 2012].
Crowdsourcing, in the form of online campaigns in which people express
themselves in truthful and/or deceitful manner for a small payment are a well
established way to collect deceptive data [Ott et al., 2011]. Real-life
situations such as trials [Soldner et al., 2019] or the use of data from board
games have also been employed [Peskov et al., 2020]. Also a popular approach
is the reuse of content from sites that debunk articles like fake news and
hoaxes [Wang, 2017, Kochkina et al., 2018]. Lastly, satire news are another
way to collect deceptive texts, but with some particularities due to humorous
deception [Skalicky et al., 2020].

The only work that explores AFD articles is that of Dearden et al. [Dearden
and Baron, 2019]. They collected 519 AFD and 519 truthful stories and articles
in English for a period of 14 years. A large set of features was exploited to
identify deception cues in AFD stories. Structural complexity and level of
detail were among the most valuable features while the exploitation of the
same feature set to a fake news dataset resulted in similar observations.

To the best of our knowledge, the only deception related dataset for the Greek
language is that of Karidi et al. [Pla Karidi et al., 2019]. This work
proposed an automatic process for the creation of a fake news and hoaxes
articles corpus, but unfortunately the created corpus over Greek websites is
not available. If we also consider that the creation of a Greek dataset for
deception through crowdsourcing is a cumbersome and expensive task, that is
further hindered by the exceptionally limited number of native Greek crowd
workers, it is easy to understand why there is a lack of datasets.

Regarding the individualism/collectivism cultural dimension, it constitutes a
well-known division of cultures that concerns the degree in which members of a
culture value more individual over group goals and vice versa. In
individualism, ties between individuals are loose and individuals are expected
to take care of only themselves and their immediate families, whereas in
collectivism ties in society are stronger. In Papantoniou et al. [Papantoniou
et al., 2021] there is an preliminary effort driven by prior work in
psychology discipline [Taylor et al., 2017] to examine if deception cues are
altered across cultures and if this can be attributed to this divide. Among
the conclusions were that people from individualistic cultures employ more
third and less first person pronouns to distance themselves from the deceit
when they are deceptive, whereas in the collectivism group this trend is
milder, signalling the effort of the deceiver to distance the group from the
deceit. In addition, in individualistic cultures positive sentiment is
employed in deceptive language, whereas in collectivists there is a restraint
of expression of sentiment both in truthful and deceptive texts.

To this end, this work explores the deception-related characteristics of a new
Greek corpus based on AFD articles from a variety of sources, and compares
them with the English ones333We also experimented with data from the limited
number of satirical and hoaxes sources of the Greek Web. We do not discuss
them here though, since the classifiers reported excellent accuracy showcasing
the lack of diversity and the existence of domain specific information in the
collected data.. Further, since related studies [Triandis and Vassiliou, 1972,
Hofstede, 1980, Koutsantoni, 2005] describe Greece as a culture with more
collectivistic characteristics (by using country as proxy from culture), we
also discuss differences in deception cues along this cultural dimension.

##  3 Corpus creation

The AFD articles have been hand gathered because a crawling based collection
approach was not applicable in our case. Since the news web sites industry in
Greece is not huge to establish an acceptable number of crawled AFD articles,
we had to additionally collect articles from the press, including articles
from the pre-WWW era. Specifically, we visited the local library that
maintains a printed archive of newspapers and searched for disclosure articles
in the issues after the 1st April, took photos of the AFD articles, and then
used OCR and manual inspection to extract the text. In addition we contacted
national and local news media providers to get access in their digitalized
archives. The rest were gathered from the Web.

The articles were categorized thematically into the following five categories:
society, culture, politics, world, and sports. If no category was provided by
the original source, we manually annotated the articles. For each article we
kept the title, the main body, the published date, the name, the type of the
source (newspaper or news website), and (if available) the caption, the
subtitle and the author. As preprocesing steps we applied spellcheck and
normalization. The correction of spelling mistakes was necessary primarily for
articles extracted through OCR tools, although spelling errors were identified
in other articles too. Normalization was performed for homogeneity reasons in
the texts retrieved from the 80’s, since we observed language differences in
some forms (e.g., in the suffix of genitive case), which are remains of an old
form of Modern Greek444https://en.wikipedia.org/wiki/Katharevousa.

For the truthful collection we used the same manual procedure and we tried to
have a balanced dataset in terms of thematic categories. The truthful
collection consists of articles that have been published in days relatively
close to the 1st of April in order to have articles that do not differ
significantly in respect to their topics, mentioned named entities, etc.

Since the AFD tradition is vivid in Greece, we were able to locate a lot of
such articles from various newspapers and new websites for our corpus (112
different sources). Specifically, we managed to collect 254 truthful and 254
deceptive articles spanning over the period 1979 - 2021. In Tables 1 to 2 some
statistics of the corpus are depicted.

Table 1: Overview of the dataset. Measure | Truthful | Deceptive
---|---|---
Num. of articles | 254 | 254
Avg. length | 336 | 255
Min. length | 57 | 33
Max. length | 1347 | 1163
Table 2: Distribution of articles per topic. Topic | Truthful | Deceptive
---|---|---
culture | 20 | 24
politics | 85 | 78
society | 86 | 118
sports | 22 | 29
world | 41 | 5

##  4 Features analysis

For the analysis of AFD articles we adapt and build upon the feature set used
in Papantoniou et al. [Papantoniou et al., 2021], but for the Greek language.
The resulting feature set consists of 64 features for the Greek language and
75 for the English, due to the smaller availability of linguistic resources
for Greek (e.g., in sentiment lexicons). For the analysis we performed the
non-parametric Mann–Whitney U test (two-tailed) with a 99% confidence interval
(CI) and $\alpha=0.01$. Table 3 depicts the results of this analysis for elAFD
and enAFD datasets555All the features are described in
https://gitlab.isl.ics.forth.gr/papanton/elaprilfoolcorpus.

In both datasets, positive sentiment is related to the deceptive articles,
while negative sentiment with the truthful articles. The only exception
concerns the enAFD dataset, where for the NRC lexicon the opposite holds (NRC
is one of the six sentiment lexicons used for features in English). In
addition, negative emotions like anger, fear and sadness are related to
truthful news articles in both datasets. The use of positive emotive language
during deception may be a strategy for deceivers to maintain social harmony as
noticed also by other studies [Newman et al., 2003, Pérez-Rosas et al., 2018].
The difference in the use of emotional language between truthful and deceptive
news is more intense in the enAFD dataset, where five out of the eight
emotions in the NRC lexicon are found statistical significant. This is in
alignment with the results in Papantoniou et al. [Papantoniou et al., 2021]
for individualistic and collectivistic cultures.

Further, deceptive texts seem to be related with an increased use of adverbs
in both datasets. This can be related to the less concreteness of deceptive
texts as discussed in Kleinberg et al. [Kleinberg et al., 2019] and it is in
line with many theories of deception like the Reality Monitoring [Johnson et
al., 1998], Criteria based Content Analysis [Undeutsch, 1989] and
Verifiability Approach [Nahari et al., 2014]. This also explains the
prevalence of the number of named entities, spatial related words,
conjunctions and WDAL imagery score in truthful texts in the enAFD dataset and
the use of more motion verbs in deceptive texts in the elAFD dataset.
According to cognitive load theory [Sweller, 2011] in deceptive texts the
language is less specific and consists of simpler constructs. The same holds
for modality, another common feature among the datasets, that is considered a
signal of subjectivity that provides a degree of uncertainty. In addition,
hedges in enAFD dataset, also express some feeling of doubt or hesitancy.

Lexical diversity as expressed by the token-type ratio (TTR), that is the
ratio of unique words to the total number of tokens, is related to the
deceptive texts. This seems to contradict all the above, but could be
attributed to the fact that deceptive texts are shorter. Although this is more
evident in the case of the enAFD dataset, it also holds for elAFD dataset (see
Table 1).

Boosters, which are words that express confidence (e.g., certainly) are quite
discriminative for deceptive texts for the enAFD dataset. Moreover we observe
the connection of the future tense with deception and of the past with truth.
The above were also marked in Papantoniou et al. [Papantoniou et al., 2021] in
different domain from the news articles domain.

Finally, first personal pronouns have been found to be rather discriminative
of deceptive texts in various deception detection and cultural studies,
including Papantoniou et al. [Papantoniou et al., 2021]. However, in this
study pronouns are statistical important only for the enAFD dataset. This
probably reflects idiosyncrasies of the news domain, since articles mainly
present objectively facts and not opinions, and as a result the use of first
personal pronouns is avoided. This holds for the elAFD dataset that includes
AFD articles from the news sites and the press, and not for the enAFD dataset
that consists of various types of AFD articles and stories collected from the
web through crowdsourcing666https://aprilfoolsdayontheweb.com/2004.html.

Table 3: The statistical significant features (p<0.1) with at least a small
effect size (r>0.1) for the elAFD and enAFD datasets. The features are in
ascending p value order. We also report the effect size. Features with
moderate effect size (r>0.3) are bold, while common features between the
datasets are underlined. pp denotes personal pronouns.

Deceptive | Truthful
---|---
elAFD
adverbs (0.31) | punctuation (-0.17)
adj. & adv. (0.27) |  nrc sadness(-0.17)
TTR (0.27) | plosives (-0.16)
pos. sentiment (0.21) |  nrc anger (-0.15)
modal verbs (0.17) |  nrc fear (-0.14)
motion verbs (0.117) | vowels (-0.14)
|  consonants (-0.14)
enAFD
boosters (0.39) | NE num. (-0.27)
modal verbs (0.35) | spatial num. (-0.26)
TTR(0.31) | conjuctions (-0.24)
future (0.27) |  nrc fear (-0.23)
adverbs (0.2) | past (-0.23)
1st pers. pp (0.2) |  nrc sadness (-0.23)
mpqa pos. (0.2) |  nrc anger (-0.21)
nrc neg.* (-0.2) | nrc trust (-0.21)
2nd pers. pp (0.19) | avg. word len. (-0.17)
1st pers. pp pl. (0.18) | collectivism (-0.16)
sentiwordnet pos. (0.17) |  nrc pos.* (-0.16)
demonstrative (0.17) | wdal imagery (-0.15)
hedges (0.17) | mpqa neg. -0.14)
adj & adv (0.16) | nasals (-0.14)
present (0.15) | fbs neg. (-0.14)
vader sentiment (0.14) |  consonants (-0.13)
verb num. (0.14) | anew arousal (-0.13)
pers. pron. (0.12) | prepositions (-0.12)
total pronouns (0.11) | fricatives (-0.11)
| 3rd per. pp sg. (-0.11)
| avg. preverb len. (-0.11)
| nrc disgust (-0.1)

##  5 Classification

We evaluated the predictive performance of different feature sets and
approaches for AFD datasets, including logistic regression experiments777We
employ the Weka API [Hall et al., 2009] and fine-tuned monolingual BERT models
for each language888We used tensorflow 2.2.0, keras 2.3.1, and the bert-for-
tf2 0.14.4 implementation of google-research/bert, over an AMD Radeon VII card
and the ROCm 3.7 platform. [Devlin et al., 2019, Koutsikakis et al., 2020]. We
also performed cross lingual experiments by exploiting the multilingual BERT
model (mBERT) to examine if there are similarities among AFD datasets captured
by the BERT.

A stratified split to the datasets was used to create training, testing, and
validation subsets with a 70-20-10 ratio. For the cross lingual experiment we
trained and validated a model over the 80% and 20% of a language specific
dataset respectively, and then tested the performance of the model over the
other dataset. We report the results on test sets, while validation subsets
were used for fine-tuning the hyper-parameters of the algorithms. For the
logistic regression the tuned through brute force parameters were: a) Weka
algorithm ($SimpLog|Log$: simple logistic [Landwehr et al., 2005] or logistic
[Le Cessie and Van Houwelingen, 1992]) b) all n-grams of size in $[a,b]$, with
$a\geq b$ and $a$, $b$ $\in$ $[1,3]$ ($(a,b)$), c) stemming ($stem$), d)
attribute selection ($attrsel$) (applicable only to $Log$ algorithm since it
is the default for $SimpLog$ ), e) stopwords removal ($stop$) and, f)
lowercase conversion ($lowercase$). For the BERT experiments, the
hyperparameters were tuned by random sampling 60 combinations of values,
keeping the combination that gave the minimum validation loss. Early stopping
with patience 4 was used and the max epochs number was set to 20. The tuned
hyperparameters were: learning rate, batch size, dropout rate, max token
length, and randomness seeds.

In all cases, we report Recall ($R$), Precision ($P$), F-measure ($F$),
Accuracy ($A$) and AUC ($A^{\prime}$). Since the datasets are balanced the
majority baseline is 50%. The input for the models consists of the
concatenation of the title, the subtitle, the body of the articles and the
caption text. Since titles are important for deception detection [Horne and
Adali, 2017] and BERT processes texts of up to 512 wordpieces, we placed the
title first.

###  5.1 Logistic regression experiments

The examined features sets were: a) the features presented in section 4
(ling), b) n-grams features i.e., phoneme-gram (ph-gram), character-gram
(char-gram), word-gram (w-gram), POS-gram (pos-gram), and syntactic-gram (sn-
gram) (the latter for the enAFD only), and c) the linguistic+ model that
represents the best model that combines the linguistic features with any of
the n-gram features. The results are presented in Tables 4 and 5. With * we
mark the setups with a statistically significant difference to the best setup
regarding accuracy, based on a two proposition z-test (1-tailed) with a 99%
CI. We observe that the combination of lingustic features with uni/bi/tri-
grams for the elAFD dataset and the unigrams for the enAFD are the best
setups. For the enAFD dataset, the second best model is the combination of
linguistic features with trigrams. $SimpLog$ seems to perform better, while
stemming, lowercase conversion and stopwords removal are generally
beneficiary.

Table 4: Logistic regression results for el AFD .

Best setup | R | P | F | A’ | A
---|---|---|---|---|---
ling.SimpLog | 62 | 76 | 68 | 82 | 71
ph-gram(1,2),attrsel,Log* | 70 | 67 | 68 | 77 | 68
char-gram(3,3),SimpLog* | 72 | 68 | 70 | 76 | 69
w-gram(1,2),SimpLog | 68 | 73 | 71 | 80 | 72
pos-gram(2,3),SimpLog* | 72 | 65 | 68 | 75 | 67
ling.+word,(1,3),stop, |  |  |  |  |
lowercase,SimpLog | 74 | 79 | 76 | 85 | 77

Table 5: Logistic regression results for en AFD .

Best setup | R | P | F | A’ | A
---|---|---|---|---|---
ling.Log* | 66 | 80 | 72 | 87 | 75
ph-gram(1,1),SimpLog | 80 | 77 | 78 | 84 | 78
char-gram(1,3),attrsel,Log* | 76 | 72 | 74 | 80 | 73
w-gram(1,1),stem,SimpLog | 79 | 81 | 80 | 87 | 80
pos-gram(3,3),SimpLog* | 71 | 69 | 70 | 76 | 69
sn-gram(2,2),SimpLog* | 80 | 68 | 73 | 77 | 71
ling.+Word,(3,3),stop, |  |  |  |  |
lowercase,SimpLog | 74 | 80 | 77 | 87 | 78

Table 6: BERT models evaluation results.  | R | P | F | A’ | A
---|---|---|---|---|---
elbert | 76 | 83 | 79 | 90 | 80
elbert+ling | 72 | 80 | 76 | 86 | 77
elmbert | 70 | 73 | 71 | 81 | 72
elmbert+ling | 50 | 81 | 62 | 83 | 69
enbert | 88 | 85 | 87 | 94 | 86
enbert+ling | 74 | 89 | 81 | 91 | 83
enmbert | 50 | 95 | 66 | 91 | 73
enmbert+ling | 50 | 86 | 63 | 86 | 71
en$\rightarrow$el mbert | 38 | 76 | 50 | 72 | 63
el$\rightarrow$en mbert | 24 | 87 | 37 | 71 | 60

###  5.2 BERT experiments

In these experiments, we fine-tuned BERT by adding a task-specific linear
classification layer on top, using the sigmoid activation function. We also
combined BERT with linguistics features by concatenating the embedding of the
[CLS] token with the linguistic features, and pass the resulting vector to the
task-specific classifier (with a slightly modified architecture). The results
of the experiments are presented in Table 6. Although it outperformed logistic
regression experiments in both datasets, the differences are not statistical
significant. In addition, the combination with linguistic features is not
beneficial. Multilingual BERT models perform worse, especially for Greek. In
the cross lingual experiments the classifiers performance is limited to about
60% accuracy in both experiments, showcasing that the BERT layers are not able
to capture language agnostic information from our datasets.

##  6 Conclusion and Future work

We introduced a new dataset with AFD news articles in Greek and analyzed and
compared its deception cues with another English one. The results showcased
the use of emotional language, especially of positive sentiment, for deceptive
articles which is even more prevalent in the individualistic English dataset.
Further, deceptive articles use less concrete language, as manifested by the
increased use of adverbs, hedges, and boosters and less usage of named
entities, spatial related words and conjunctions compared to the truthful
ones. The future and past tenses were correlated with deceptive and truthful
articles respectively. All the above, mainly align with previous work
[Papantoniou et al., 2021], except from some differences in the usage of
pronouns for the Greek dataset, which is attributed to the idiosyncrasies of
the news domain. The accuracy of the deployed classifiers offered adequate
performance, with no statistically significant differences between the best
logistic regression and the BERT models.

In the future we aim at creating even more crosslingual datasets for deception
detection tasks through crowdsourcing and by employing the Chattack platform
[Smyrnakis et al., 2021].

## Acknowledgement

This work has received funding from the Hellenic Foundation for Research and
Innovation (HFRI) and the General Secretariat for Research and Technology
(GSRT), under grant agreement No 4195.

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# Odd-parity stability of black holes in Einstein-Aether gravity

Shinji Tsujikawa1, Chao Zhang2,3,4, Xiang Zhao2, and Anzhong Wang2 1Department
of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
2GCAP-CASPER, Physics Department, Baylor University, Waco, Texas, 76798-7316,
USA
3Institute for Theoretical Physics & Cosmology, Zhejiang University of
Technology, Hangzhou, 310023, China
4United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310023, China

###### Abstract

In Einstein-Aether theory, we study the stability of black holes against odd-
parity perturbations on a spherically symmetric and static background. For
odd-parity modes, there are two dynamical degrees of freedom arising from the
tensor gravitational sector and Aether vector field. We derive general
conditions under which neither ghosts nor Laplacian instabilities are present
for these dynamical fields. We apply these results to concrete black hole
solutions known in the literature and show that some of those solutions can be
excluded by the violation of stability conditions. The exact Schwarzschild
solution present for $c_{13}=c_{14}=0$, where $c_{i}$’s are the four coupling
constants of the theory with $c_{ij}=c_{i}+c_{j}$, is prone to Laplacian
instabilities along the angular direction throughout the horizon exterior.
However, we find that the odd-parity instability of high radial and angular
momentum modes is absent for black hole solutions with $c_{13}=c_{4}=0$ and
$c_{1}\geq 0$.

###### pacs:

04.50.Kd,95.30.Sf,98.80.-k

##  I Introduction

General Relativity (GR) is a fundamental theory of gravity well tested by
solar-system experiments. With the dawn of gravitational-wave astronomy, it is
now possible to probe the validity of GR around black holes (BHs) and neutron
stars Abbott2016 ; GW170817 . Recently, there has been growing interest in
searching for extra degrees of freedom beyond GR and standard model of
particle physics in such a strong gravity regime Berti ; Barack . The
existence of new degrees of freedom is also motivated by the firm
observational evidence of dark matter and dark energy Bertone:2004pz ;
Copeland:2006wr ; Clifton:2011jh .

The construction of GR is based on Lorentz invariance (LI), which is a
continuous symmetry invariant under the 4-dimensional diffeomorphism. In
discrete spacetime that can arise from the quantization of gravity, the
Lorentz symmetry can be broken at very high energy. The violation of LI in
standard model fields is tightly limited from various experiments
Kostelecky:2008ts ; Mattingly:2005re , but the Lorentz violation in the
gravity sector is much less constrained Flowers:2016ctv ; Bourgoin:2017fpo .
Hořava gravity Horava:2009uw ; Blas:2009qj  is an example of allowing for
gravitational Lorentz violation at high energy, in which a Lifshitz-type
anisotropic scaling is introduced to realize a power-counting renormalizable
theory of gravity (for a recent review of Hořava gravity, see, for example,
Wang:2017brl  and references therein).

There is yet the other type of a gravitational Lorentz-violating scenario
dubbed Einstein-Aether theory Jacobson:2000xp ; Jacobson . In this scenario
there is a unit time-like vector (Aether) field $u^{\alpha}$ at every point in
spacetime characterized by the metric tensor $g_{\alpha\beta}$, so it breaks
local Lorentz symmetry under a rotation. This is a subclass of vector-tensor
theories possessing two derivative terms of the Aether field. The existence of
a unit Aether field is ensured by the constraint
$g_{\alpha\beta}u^{\alpha}u^{\beta}=-1$ [with the metric signature
$(-,+,+,+)$], which appears as the Lagrange multiplier
$\lambda(g_{\alpha\beta}u^{\alpha}u^{\beta}+1)$ in the action. We note that
generalized Proca theories with a broken $U(1)$ gauge symmetry Heisenberg ;
Tasinato ; Allys ; Jimenez2016 ; Heisenberg:2016eld  do not have such a
constraint, so the vector-field dynamics is generally different from that in
Einstein-Aether theory.

In Einstein-Aether theory there are scalar, transverse vector, and tensor
perturbations, whose propagation speeds $c_{S}$, $c_{V}$, $c_{T}$ on the
Minkowski background are generally different from that of light
Jacobson:2004ts . To ensure the stability of Minkowski spacetime, we require
that all of $c_{S}^{2}$, $c_{V}^{2}$, and $c_{T}^{2}$ are positive. Moreover,
the observations of gravitational Cerenkov radiation Elliott:2005va , solar
system tests Foster:2005dk , big-bang nucleosynthesis Carroll:2004ai , binary
pulsars Foster:2007gr ; Yagi:2013ava , and gravitational waves Gong:2018cgj ;
Oost:2018tcv  put constraints on the dimensionless coupling constants
$c_{1,2,3,4}$ of Aether derivative interactions. In particular, the
gravitational-wave event GW170817 GW170817  together with the gamma-ray burst
170817A Goldstein  placed the upper limit $|c_{T}-1|\lesssim 10^{-15}$, which
translates to $|c_{13}|\lesssim 10^{-15}$ Gong:2018cgj ; Oost:2018tcv , where
$c_{ij}:=c_{i}+c_{j}$. However, there are still theoretically viable parameter
spaces in which all the observational constraints are satisfied.

In Einstein-Aether theory, the existence and properties of spherically
symmetric vacuum solutions have been extensively studied in the literature
Eling:2006ec ; Eling:2007xh ; Tamaki:2007kz ; Barausse:2011pu ; Blas:2011ni ;
Berglund:2012bu ; Gao:2013im ; Lin:2014eaa ; Ding:2015kba ; Ding:2016wcf ;
Ding:2018whp ; Chan:2019mdn ; Chan:2020amr ; Khodadi:2020gns . Some of them
were already excluded by the combination of observational bounds mentioned
above. However, the recent papers Zhang:2020too ; Oost:2021tqi  have shown the
presence of spherically symmetric and static BH solutions compatible with
current observational constraints. Since the speeds of scalar and transverse
vector perturbations can be arbitrarily large, there exists a universal
horizon corresponding to a causal boundary of any large speeds of propagation
Blas:2011ni ; Berglund:2012bu ; Wang:2017brl . The universal horizon can exist
inside the event horizon, so that particles can cross the event horizon to
escape toward infinity. It is expected that this unique feature of Einstein-
Aether BHs may leave some distinguished signatures in the gravitational-wave
measurements of binary BHs.

In this paper, we study the stability of spherically symmetric and static BHs
against odd-parity perturbations in Einstein-Aether theory. We first identify
two dynamical gauge-invariant perturbations corresponding to the tensor and
vector propagations. Then, we obtain the second-order action of odd-parity
perturbations and explicitly derive stability conditions for the absence of
ghosts and Laplacian instabilities. The tensor and vector propagation speeds
along the radial and angular directions are different from those in Minkowski
spacetime. Thus, our analysis of BH perturbations in the odd-parity sector
provides new stability conditions for Einstein-Aether BHs. We also note that
our general formulation of odd-parity perturbations will be useful to study
the propagation of gravitational waves during the inspiral and ringdown phases
of binary BHs.

We apply our conditions to the Einstein-Aether BH solutions known in the
literature. We show that an exact Schwarzschild BH present for the couplings
$c_{13}=0$ and $c_{14}=0$ is excluded by the Laplacian instability along the
angular direction. The BH solutions with $c_{13}=0$, $c_{14}\neq 0$, and
$c_{4}\neq 0$ are prone to the ghost instability by imposing a superluminal
propagation of the transverse vector mode ($c_{4}<0$) to avoid the
gravitational Cerenkov radiation. However, provided that $c_{1}\geq 0$, the BH
solutions with $c_{13}=0$ and $c_{4}=0$ are stable against odd-parity
perturbations with high radial and angular momentum modes. Thus, our general
stability conditions are sufficiently powerful to distinguish between unstable
and stable BHs in Einstein-Aether theory.

##  II Background equations of motion

We begin with the Einstein-Aether theory given by the action Jacobson:2004ts

${\cal S}=\frac{1}{16\pi G_{\ae}}\int\sqrt{-g}\,{\rm d}^{4}x\left[R+{\cal
L}_{\ae}+\lambda(g_{\alpha\beta}u^{\alpha}u^{\beta}+1)\right],$ (1)

where $G_{\ae}$ is a constant, $R$ is the Ricci scalar, $g$ is the determinant
of metric tensor $g_{\alpha\beta}$, $\lambda$ is a Lagrange multipler,
$u^{\alpha}$ is the Aether vector field, and

${\cal
L}_{\ae}=-{M^{\alpha\beta}}_{\mu\nu}\nabla_{\alpha}u^{\mu}\nabla_{\beta}u^{\nu}\,,$
(2)

with

${M^{\alpha\beta}}_{\mu\nu}:=c_{1}g^{\alpha\beta}g_{\mu\nu}+c_{2}\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}+c_{3}\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu}-c_{4}u^{\alpha}u^{\beta}g_{\mu\nu}\,.$
(3)

Here, the Greek indices represent from 0 to 3, $\nabla_{\alpha}$ is a
covariant derivative operator with respect to the metric tensor $g_{\mu\nu}$,
and $c_{i}$’s are four dimensionless coupling constants.

Variation of the action (1) with respect to $\lambda$ leads to

$u^{\alpha}u_{\alpha}+1=0\,.$ (4)

This constraint ensures the existence of a time-like unit vector field, so
that there is a preferred frame responsible for the breaking of LI. Varying
Eq. (1) with respect to $u^{\mu}$, it follows that

$\nabla_{\mu}{J^{\mu}}_{\alpha}+\lambda
u_{\alpha}+c_{4}\dot{u}^{\mu}\nabla_{\alpha}u_{\mu}=0\,,$ (5)

where

$\displaystyle{J^{\mu}}_{\alpha}$ $\displaystyle:=$
$\displaystyle{M^{\mu\nu}}_{\alpha\beta}\nabla_{\nu}u^{\beta}\,,$ (6)
$\displaystyle\dot{u}^{\mu}$ $\displaystyle:=$ $\displaystyle
u^{\beta}\nabla_{\beta}u^{\mu}\,.$ (7)

Multiplying Eq. (5) by $u^{\alpha}$ and using Eq. (4), the Lagrange multiplier
can be expressed as

$\lambda=u^{\alpha}\nabla_{\mu}{J^{\mu}}_{\alpha}+c_{4}\dot{u}^{\mu}\dot{u}_{\mu}\,.$
(8)

For the general line element ${\rm d}s^{2}=g_{\mu\nu}{\rm d}x^{\mu}{\rm
d}x^{\nu}$, the gravitational field equations derived by the variation of (1)
with respect to $g_{\mu\nu}$ are

$\displaystyle G_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\left[u_{(\alpha}{J^{\mu}}_{\beta)}+u^{\mu}J_{(\alpha\beta)}-u_{(\alpha}{J_{\beta)}}^{\mu}\right]$
(9)
$\displaystyle+c_{1}\left(\nabla_{\alpha}u^{\nu}\nabla_{\beta}u_{\nu}-\nabla^{\nu}u_{\alpha}\nabla_{\nu}u_{\beta}\right)$
$\displaystyle+c_{4}\dot{u}_{\alpha}\dot{u}_{\beta}+\frac{1}{2}g_{\alpha\beta}{\cal
L}_{\ae}+\lambda u_{\alpha}u_{\beta}\,,$

where $G_{\alpha\beta}$ is the Einstein tensor.

In general, the theory admits three different species of gravitons, the
spin-0, spin-1, and spin-2 ones. According to the perturbative analysis on the
Minkowski background, their squared speeds are given by Jacobson:2004ts

$\displaystyle c_{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{c_{123}(2-c_{14})}{c_{14}(1-c_{13})(2+c_{13}+3c_{2})}\,,$
(10) $\displaystyle c_{V}^{2}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}\,,$ (11)
$\displaystyle c_{T}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{1-c_{13}}\,,$ (12)

where $c_{ijk}:=c_{i}+c_{j}+c_{k}$, and $c_{S,V,T}$ represent the speeds of
the spin-0, spin-1, and spin-2 gravitons, respectively. If we require that the
theory: (i) be self-consistent, such as free of ghosts and instability; and
(ii) be compatible with all the observational constraints obtained so far, it
was found that the parameters $c_{i}$’s must satisfy the conditions
Oost:2018tcv

$\displaystyle\left|c_{13}\right|\lesssim 10^{-15},$ (13) $\displaystyle
0<c_{14}\leq 2.5\times 10^{-5},$ (14) $\displaystyle c_{14}\leq c_{2}\leq
0.095,$ (15) $\displaystyle c_{4}\leq 0.$ (16)

It should be noted that the above conditions assure $c_{S,V,T}\geq 1$, that
is, all the propagation speeds are not subluminal, in order to avoid the
gravitational Cerenkov radiation Elliott:2005va . Later, we shall come to this
point again when we study the odd-parity stability of BHs in Sec. V.

With the above in mind, let us consider a spherically symmetric and static
background given by

${\rm d}s^{2}=-f(r){\rm d}t^{2}+h^{-1}(r){\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}\right)\,,$ (17)

where $f$ and $h$ depend on the distance $r$ from the center of symmetry. The
Aether-field profile compatible with the background (17) is of the form

$u^{\mu}=\left(a(r),b(r),0,0\right)\,,$ (18)

where $a$ and $b$ are functions of $r$. The constraint (4) gives the following
relation

$b=\epsilon\sqrt{(a^{2}f-1)h}\,,$ (19)

where $\epsilon=\pm 1$. The existence of the Aether-field profile (19)
requires that $(a^{2}f-1)h\geq 0$.

Under the constraint (19), there are three independent background equations of
motion following from (5) and (9), with the Lagrange multiplier $\lambda$
determined by Eq. (8). Then, the $\alpha=0$ component of Eq. (5) and
$(\alpha,\beta)=(1,1),(2,2)$ components of Eq. (9) lead to

$\displaystyle
2h\alpha_{1}a^{\prime\prime}+\alpha_{1}h^{\prime}a^{\prime}+2h\alpha_{2}f^{\prime\prime}+\alpha_{2}f^{\prime}h^{\prime}+\alpha_{3}a^{\prime
2}+\alpha_{4}f^{\prime 2}$
$\displaystyle+\alpha_{5}a^{\prime}f^{\prime}+\alpha_{6}f^{\prime}+\alpha_{7}h^{\prime}+\alpha_{8}a^{\prime}+\alpha_{9}=0\,,$
(20) $\displaystyle
2h\beta_{1}a^{\prime\prime}+\beta_{1}h^{\prime}a^{\prime}+2h\beta_{2}f^{\prime\prime}+\beta_{2}f^{\prime}h^{\prime}+\beta_{3}a^{\prime
2}+\beta_{4}f^{\prime 2}$
$\displaystyle+\beta_{5}a^{\prime}f^{\prime}+\beta_{6}f^{\prime}+\beta_{7}h^{\prime}+\beta_{8}a^{\prime}+\beta_{9}=0\,,$
(21) $\displaystyle
2h\mu_{1}a^{\prime\prime}+\mu_{1}h^{\prime}a^{\prime}+2h\mu_{2}f^{\prime\prime}+\mu_{2}f^{\prime}h^{\prime}+\mu_{3}a^{\prime
2}+\mu_{4}f^{\prime 2}$
$\displaystyle+\mu_{5}a^{\prime}f^{\prime}+\mu_{6}f^{\prime}+\mu_{7}h^{\prime}+\mu_{8}a^{\prime}=0\,,$
(22)

where a prime represents the derivative with respect to $r$. The explicit form
of $\lambda$ as well as the coefficients $\alpha_{1},\cdots\alpha_{9}$,
$\beta_{1},\cdots,\beta_{9}$, and $\mu_{1},\cdots\mu_{8}$ are given in
Appendix A. We note that Eqs. (20)-(22) hold irrespective of the sign of
$\epsilon$ in Eq. (19). For given coupling constants $c_{i}$’s, the variables
$f$, $h$, and $a$ are known by integrating Eqs. (20)-(22) with appropriate
boundary conditions.

##  III Second-order action of odd-parity perturbations and general stability
conditions

In this section, we derive the second-order action of dynamical perturbations
in the odd-parity sector to study the stability of spherically symmetric and
static BH solutions in Einstein-Aether theory. Analogous to the analysis
performed in Ref. Kase:2018voo  in the context of generalized Proca theories,
we consider metric perturbations $h_{\mu\nu}$ on the background (17) as well
as the perturbation of the Aether field. We express the perturbations in terms
of the sum of spherical harmonics $Y_{lm}(\theta,\varphi)$.

For $l\geq 2$, we choose the Regge-Wheeler gauge in which the components
$h_{ij}$, where $i$ and $j$ correspond to either $\theta$ or $\varphi$, vanish
Regge:1957td ; Zerilli:1970se . For the dipole ($l=1$), the metric components
$h_{ij}$ vanish identically, so we need to handle this case separately. In the
following, we first study the case $l\geq 2$ and then proceed to the
discussion for $l=1$.

###  III.1 $l\geq 2$

In the Regge-Wheeler gauge, the nonvanishing components of metric
perturbations are given by

$\displaystyle h_{ti}$ $\displaystyle=$
$\displaystyle\sum_{l,m}Q_{lm}(t,r)E_{ij}\partial^{j}Y_{lm}(\theta,\varphi)\,,$
(23) $\displaystyle h_{ri}$ $\displaystyle=$
$\displaystyle\sum_{l,m}W_{lm}(t,r)E_{ij}\partial^{j}Y_{lm}(\theta,\varphi)\,,$
(24)

where the subscripts $i$, $j$ represent either $\theta$ or $\varphi$ with the
notation $\partial^{j}Y_{lm}=\partial Y_{lm}/\partial x_{j}$, and $Q_{lm}$ and
$W_{lm}$ are functions of $t$ and $r$. The tensor $E_{ij}$ is defined by
$E_{ij}=\sqrt{\gamma}\,\varepsilon_{ij}$, where $\gamma=\sin^{2}\theta$ is the
determinant of two dimensional metric $\gamma_{ij}$ on the sphere and
$\varepsilon_{ij}$ is the anti-symmetric symbol with
$\varepsilon_{\theta\varphi}=1$.

In the presence of odd-parity perturbations, the covariant Aether field is
expressed as

$u_{\mu}=\left(-a(r)f(r),\frac{b(r)}{h(r)},u_{\theta},u_{\varphi}\right)\,,$
(25)

where the $i=\theta,\varphi$ components are

$u_{i}=\sum_{l,m}\delta
u_{lm}(t,r)E_{ij}\partial^{j}Y_{lm}(\theta,\varphi)\,.$ (26)

The perturbation $\delta u_{lm}$ is a function of $t$ and $r$. We expand the
action (1) up to second order in odd-parity perturbations. In doing so, we can
set $m=0$ without loss of generality and multiply the action $2\pi$ for the
integral with respect to $\varphi$ Kase:2018voo . In the following, we also
omit the subscripts “$lm$” from the variables $Q_{lm}$, $W_{lm}$, and $\delta
u_{lm}$ for the simplification of notation. On using the background Eqs.
(20)-(22), the resulting second-order action of odd-parity perturbations is
expressed in the form

${\cal S}_{\rm odd}=\sum_{l,m}L\int{\rm d}t{\rm d}r\,{\cal L}_{\rm odd}\,,$
(27)

where

$L:=l(l+1)\,,$ (28)

and

$\displaystyle{\cal L}_{\rm odd}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{16\pi
G_{\ae}}\sqrt{\frac{f}{h}}\biggl{[}C_{1}\left(\dot{W}-Q^{\prime}+\frac{2}{r}Q\right)^{2}+2\left(C_{2}\dot{\delta
u}+C_{3}\delta u^{\prime}+C_{4}\delta
u\right)\left(\dot{W}-Q^{\prime}+\frac{2}{r}Q\right)+C_{5}\dot{\delta
u}^{2}+C_{6}\dot{\delta u}\delta u^{\prime}$ (29) $\displaystyle+C_{7}{\delta
u}^{\prime 2}+(L-2)\left(C_{8}W^{2}+C_{9}W\delta
u-aC_{9}WQ+C_{10}Q^{2}+C_{11}Q\delta u\right)+(LC_{12}+C_{13})\delta
u^{2}\biggr{]}\,,$

with a dot being the derivative with respect to $t$. The coefficients
$C_{i}$’s in Eq. (29) are given in Appendix B. Even with the unit-vector
constraint $u^{\alpha}u_{\alpha}+1=0$ in Einstein-Aether theory, the
Lagrangian (29) is of the same form as that derived for generalized Proca
theories Kase:2018voo , with the correspondence of the temporal vector
component $A_{0}\to-af$. The difference appears only for the coefficients
$C_{i}$’s, so we can resort to the prescription exploited in Ref. Kase:2018voo
for the derivation of stability conditions of dynamical perturbations.

Let us consider the gauge transformation $x_{\mu}\to x_{\mu}+\xi_{\mu}$, where

$\xi_{t}=\xi_{r}=0\,,\qquad\xi_{i}=\sum_{lm}\Lambda(t,r)E_{ij}\partial^{j}Y_{lm}(\theta,\varphi)\,.$
(30)

Then, the perturbations $Q$, $W$, and $\delta u$ transform as Kobayashi ; EFT
; Kase:2018voo

$\displaystyle Q$ $\displaystyle\to$ $\displaystyle Q+\dot{\Lambda}\,,$ (31)
$\displaystyle W$ $\displaystyle\to$ $\displaystyle
W+\Lambda^{\prime}-\frac{2\Lambda}{r}\,,$ (32) $\displaystyle\delta u$
$\displaystyle\to$ $\displaystyle\delta u\,.$ (33)

Besides the Aether perturbation $\delta u$, we consider the following gauge-
invariant combination

$\chi=\dot{W}-Q^{\prime}+\frac{2}{r}Q+\frac{C_{2}\dot{\delta u}+C_{3}\delta
u^{\prime}+C_{4}\delta u}{C_{1}}\,,$ (34)

which is associated with the tensor perturbation in the odd-parity gravity
sector. The gauge-invariant perturbation (34) is introduced to combine the
first and second contributions to the square brackets of Eq. (29). We express
the Lagrangian (29) in the form

$\displaystyle{\cal L}_{\rm odd}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{16\pi
G_{\ae}}\sqrt{\frac{f}{h}}\biggl{[}C_{1}\biggl{\\{}2\chi\biggl{(}\dot{W}-Q^{\prime}+\frac{2}{r}Q+\frac{C_{2}\dot{\delta
u}+C_{3}\delta u^{\prime}+C_{4}\delta
u}{C_{1}}\biggr{)}-\chi^{2}\biggr{\\}}-\frac{(C_{2}\dot{\delta u}+C_{3}\delta
u^{\prime}+C_{4}\delta u)^{2}}{C_{1}}+C_{5}\dot{\delta u}^{2}$ (35)
$\displaystyle+C_{6}\dot{\delta u}\delta u^{\prime}+C_{7}{\delta u}^{\prime
2}+(L-2)\left(C_{8}W^{2}+C_{9}W\delta u-aC_{9}WQ+C_{10}Q^{2}+C_{11}Q\delta
u\right)+(LC_{12}+C_{13})\delta u^{2}\biggr{]}\,,$

where $\chi$ is regarded as a Lagrange multiplier independent of the fields
$W$ and $Q$ in Eq. (35). The similar treatment was also performed in the
context of scalar-tensor theories DST ; Kobayashi ; EFT  and generalized Proca
theories Kase:2018voo .

Varying Eq. (35) with respect to $W$ and $Q$, it follows that

$\displaystyle 2C_{1}\dot{\chi}-(L-2)\left[2C_{8}W+C_{9}\left(\delta
u-aQ\right)\right]=0\,,$ (36) $\displaystyle
2C_{1}\chi^{\prime}+\frac{2rfhC_{1}^{\prime}+(8fh+rf^{\prime}h-rfh^{\prime})C_{1}}{rfh}\chi$
$\displaystyle-(L-2)\left(aC_{9}W-2C_{10}Q-C_{11}\delta u\right)=0\,.$ (37)

These equations can be solved for $W$ and $Q$ to express them in terms of
$\chi$, $\dot{\chi}$, $\chi^{\prime}$, and $\delta u$. Substituting them into
Eq. (35) and integrating it by parts, we obtain the reduced Lagrangian

$\displaystyle{\cal L}_{\rm odd}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{16\pi
G_{\ae}(L-2)}\sqrt{\frac{f}{h}}(\dot{\vec{\mathcal{X}}}^{t}{\bm{K}}\dot{\vec{\mathcal{X}}}+\dot{\vec{\mathcal{X}}}^{t}{\bm{R}}\vec{\mathcal{X}}^{\prime}$
(38) $\displaystyle+\vec{\mathcal{X}}^{\prime
t}{\bm{G}}\vec{\mathcal{X}}^{\prime}+\vec{\mathcal{X}}^{t}{\bm{M}}\vec{\mathcal{X}})\,,$

where

$\vec{\mathcal{X}}^{t}=\left(\chi,\delta u\right)\,,$ (39)

and ${\bm{K}},{\bm{R}},{\bm{G}},{\bm{M}}$ are $2\times 2$ symmetric matrices.
We note that the contributions to Eq. (38) of the forms
$\dot{\vec{\mathcal{X}}}^{t}{\bm{T}}\vec{\mathcal{X}}$ and
$\vec{\mathcal{X}}^{\prime t}{\bm{S}}\vec{\mathcal{X}}$, which appear in
generalized Proca theories Kase:2018voo , vanish in Einstein-Aether theory.
The Lagrangian (38) can now be used to study the stability of dynamical fields
$\chi$ and $\delta u$.

The nonvanishing components of ${\bm{K}}$ are given by

$K_{11}=q_{1}\,,\qquad K_{22}=(L-2)q_{2}\,,$ (40)

where

$q_{1}:=\frac{4C_{1}^{2}C_{10}}{a^{2}C_{9}^{2}-4C_{8}C_{10}}\,,\qquad
q_{2}:=\frac{C_{1}C_{5}-C_{2}^{2}}{C_{1}}\,.$ (41)

To avoid the appearance of ghosts, we require that

$\displaystyle q_{1}>0\,,$ (42) $\displaystyle q_{2}>0\,,$ (43)

where the former and latter correspond to the no-ghost conditions of gravity
and vector-field sectors, respectively.

The matrices ${\bm{R}}$ and ${\bm{G}}$ have the following nonvanishing
components

$\displaystyle R_{11}={\cal R}_{11}q_{1},\qquad R_{22}=(L-2){\cal R}_{22}\,,$
(44) $\displaystyle G_{11}={\cal G}_{11}q_{1}\,,\qquad G_{22}=(L-2){\cal
G}_{22}\,,$ (45)

where

$\displaystyle{\cal R}_{11}:=-\frac{aC_{9}}{C_{10}}\,,\qquad{\cal
R}_{22}:=\frac{C_{1}C_{6}-2C_{2}C_{3}}{C_{1}}\,,$ (46) $\displaystyle{\cal
G}_{11}:=\frac{C_{8}}{C_{10}}\,,\qquad{\cal
G}_{22}:=\frac{C_{1}C_{7}-C_{3}^{2}}{C_{1}}\,.$ (47)

To derive the dispersion relation along the radial direction, we assume the
solutions of Eq. (39) in the form
$\vec{\mathcal{X}}^{t}=\vec{\mathcal{X}}_{0}^{t}e^{i(\omega t-kr)}$, where
$\vec{\mathcal{X}}_{0}^{t}$ is a constant vector, and $\omega$ and $k$ are the
constant frequency and wavenumber respectively. In the limits $k\to\infty$ and
$\omega\to\infty$, the existence of nonvanishing solutions of
$\vec{\mathcal{X}}^{t}$ requires that ${\rm det}(\omega^{2}{\bm{K}}-\omega
k{\bm{R}}+k^{2}{\bm{G}})=0$. Since there are no off-diagonal components in
${\bm{K}}$, ${\bm{R}}$, and ${\bm{G}}$, it follows that

$\displaystyle\omega^{2}-\omega k{\cal R}_{11}+k^{2}{\cal G}_{11}=0\,,$ (48)
$\displaystyle\omega^{2}q_{2}-\omega k{\cal R}_{22}+k^{2}{\cal G}_{22}=0\,.$
(49)

In terms of the proper time $\tau=\int\sqrt{f}\,{\rm d}t$ and the rescaled
radial coordinate $r_{*}=\int{\rm d}r/\sqrt{h}$, the propagation speed of
perturbations along the radial direction is given by $c_{r}={\rm d}r_{*}/{\rm
d}\tau=\hat{c}_{r}/\sqrt{fh}$, where $\hat{c}_{r}={\rm d}r/{\rm d}t=\omega/k$
is the propagation speed in the coordinates $t$ and $r$. Substituting
$\omega=k\sqrt{fh}\,c_{r}$ into Eqs. (48) and (49), the solutions to $c_{r}$
are given, respectively, by

$\displaystyle c_{r1}$ $\displaystyle=$ $\displaystyle\frac{{\cal
R}_{11}\pm\sqrt{{\cal F}_{1}}}{2\sqrt{fh}}\,,$ (50) $\displaystyle c_{r2}$
$\displaystyle=$ $\displaystyle\frac{{\cal R}_{22}\pm\sqrt{{\cal
F}_{2}}}{2q_{2}\sqrt{fh}}\,,$ (51)

where

$\displaystyle{\cal F}_{1}$ $\displaystyle:=$ $\displaystyle{\cal
R}_{11}^{2}-4{\cal G}_{11}\,,$ (52) $\displaystyle{\cal F}_{2}$
$\displaystyle:=$ $\displaystyle{\cal R}_{22}^{2}-4q_{2}{\cal G}_{22}\,.$ (53)

The speeds $c_{r1}$ and $c_{r2}$ correspond to the radial sound speeds
associated with the propagation of gravity and vector-field sectors,
respectively. Depending on the direction of radial propagation, the signs of
$c_{r1}$ and $c_{r2}$ can be either positive or negative. As long as $c_{r1}$
and $c_{r2}$ are real, we have $c_{r1}^{2}\geq 0$ and $c_{r2}^{2}\geq 0$.
Hence the absence of Laplacian instabilities along the radial direction
requires that

$\displaystyle{\cal F}_{1}\geq 0\,,$ (54) $\displaystyle{\cal F}_{2}\geq 0\,.$
(55)

The propagation speed $c_{\Omega}$ along the angular direction can be derived
by taking the limits $L\to\infty$ and $\omega\to\infty$ in Eq. (38). In these
limits, the dominant contributions to the matrix components of ${\bm{M}}$ are
given by

$M_{11}=-LC_{1}\,,\qquad M_{22}=L^{2}D_{1}\,,$ (56)

where

$D_{1}:=C_{12}+\frac{C_{8}C_{11}^{2}+C_{9}^{2}(C_{10}+aC_{11})}{4C_{1}^{2}C_{10}}q_{1}\,.$
(57)

There are also the matrix components $M_{12}\,(\,=M_{21})$ proportional to
$L$, but they do not affect the angular sound speeds derived below.
Substituting the solution of the form
$\vec{\mathcal{X}}^{t}=\vec{\mathcal{X}}_{0}^{t}e^{i(\omega t-l\theta)}$ into
the perturbation equations following from (38), we obtain the dispersion
relation ${\rm det}(\omega^{2}{\bm{K}}+{\bm{M}})=0$. There are no off-diagonal
components of ${\bm{K}}$ and ${\bm{M}}$, so that

$\displaystyle\omega^{2}q_{1}-LC_{1}=0\,,$ (58)
$\displaystyle\omega^{2}q_{2}+LD_{1}=0\,.$ (59)

The angular propagation speed in proper time is given by $c_{\Omega}=r{\rm
d}\theta/{\rm d}\tau=\hat{c}_{\Omega}/\sqrt{f}$, where $\hat{c}_{\Omega}=r{\rm
d}\theta/{\rm d}t$ satisfies $\omega^{2}=\hat{c}_{\Omega}^{2}l^{2}/r^{2}$.
Taking the limit $L\to\infty$ and substituting the relation
$\omega^{2}=c_{\Omega}^{2}fL/r^{2}$ into Eqs. (58) and (59), the solutions to
$c_{\Omega}^{2}$ are given, respectively, by

$\displaystyle c_{\Omega 1}^{2}=\frac{C_{1}r^{2}}{fq_{1}}\,,$ (60)
$\displaystyle c_{\Omega 2}^{2}=-\frac{D_{1}r^{2}}{fq_{2}}\,.$ (61)

To avoid the Laplacian instabilities along the angular direction, we require
that

$\displaystyle c_{\Omega 1}^{2}\geq 0\,,$ (62) $\displaystyle c_{\Omega
2}^{2}\geq 0\,,$ (63)

which translate to $C_{1}/q_{1}\geq 0$ and $D_{1}/q_{2}\leq 0$, respectively,
outside the horizon ($f>0$).

So far, we have considered the stabilities of perturbations $\chi$ and $\delta
u$ along the radial and angular directions by separately taking the limits
$k\to\infty$ or $L\to\infty$. We will also study the propagation of inclined
modes where the limits $k\to\infty$ and $L\to\infty$ are taken, with the ratio

$\xi:=\frac{k}{\sqrt{L}},$ (64)

being constant. In this case, we substitute the solution
$\vec{\mathcal{X}}^{t}=\vec{\mathcal{X}}_{0}^{t}e^{i(\omega t-kr-l\theta)}$
into the perturbation equations following from Eq. (38). Then, the dispersion
relation yields ${\rm det}(\omega^{2}{\bm{K}}-\omega
k{\bm{R}}+k^{2}{\bm{G}}+{\bm{M}})=0$, so that

$\displaystyle q_{1}\left(\omega^{2}-\omega k{\cal R}_{11}+k^{2}{\cal
G}_{11}\right)-LC_{1}=0\,,$ (65) $\displaystyle\omega^{2}q_{2}-\omega k{\cal
R}_{22}+k^{2}{\cal G}_{22}+LD_{1}=0\,.$ (66)

Solving Eqs. (65) and (66) for $\omega$ respectively, we obtain the dispersion
relations for the perturbations $\chi$ and $\delta u$, as

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\frac{\sqrt{L}}{2}\left[{\cal R}_{11}\xi\pm\sqrt{{\cal
F}_{1}\xi^{2}+\frac{4C_{1}}{q_{1}}}\right]\,,$ (67) $\displaystyle\omega$
$\displaystyle=$ $\displaystyle\frac{\sqrt{L}}{2q_{2}}\left[{\cal
R}_{22}\xi\pm\sqrt{{\cal F}_{2}\xi^{2}-4q_{2}D_{1}}\right]\,,$ (68)

where ${\cal F}_{1}$ and ${\cal F}_{2}$ are defined by Eqs. (52) and (53). The
absence of Laplacian instabilities for the perturbations $\chi$ and $\delta u$
can be ensured under the conditions

$\displaystyle{\cal F}_{1}\xi^{2}+\frac{4C_{1}}{q_{1}}\geq 0\,,$ (69)
$\displaystyle{\cal F}_{2}\xi^{2}-4q_{2}{\cal D}_{1}\geq 0\,,$ (70)

respectively. Let us first consider the stability of the perturbation $\chi$.
In the limit $\xi\to\infty$, there is no Laplacian instability for ${\cal
F}_{1}\geq 0$, see Eq. (54). In the other limit $\xi\to 0$, the angular
propagation speed squared is given by Eq. (60), so the stability is ensured
for $C_{1}/q_{1}\geq 0$. Under these two conditions, the inequality (69) holds
for any arbitrary values of $\xi$. On using Eq. (70), we also find that the
same property holds for the perturbation $\delta u$, i.e., the conditions (55)
and (63) are sufficient to ensure the Laplacian stability of the inclined
mode. Hence the inclined mode does not give rise to additional conditions to
those derived for purely radial and angular modes.

In summary, for $l\geq 2$, the stabilities of perturbations for high radial
and angular momentum modes are ensured under the conditions (42), (43), (54),
(55), (62) and (63). We caution that these conditions are derived in the large
$k$ or (and) $l$ limits, so they are not sufficient to guarantee all the
stabilities for finite values of $k$ and $l$. Due to the complexity of matrix
components of ${\bm{M}}$, we do not consider the stability of perturbations
for such an intermediate range of $k$ and $l$.

In addition, instabilities might arise when we consider the spectrum of
$\vec{\mathcal{X}}^{t}$, by solving the corresponding differential equations
for $\chi$ and $\delta u$ with boundary conditions. However, such studies are
out of the scope of the current paper, and we wish to return to this important
issue in another occasion.

###  III.2 $l=1$

Since the metric components $h_{ij}$ vanish identically for the dipole
perturbation ($l=1$), there is a gauge degree of freedom to be fixed. In this
case, we choose the gauge $W=0$. From Eq. (32), the gauge-transformation
scalar $\Lambda(t,r)$ is constrained to be

$\Lambda=-r^{2}\int{\rm
d}\tilde{r}\,\frac{W(t,\tilde{r})}{\tilde{r}^{2}}+r^{2}{\cal C}(t)\,,$ (71)

where ${\cal C}(t)$ is a function of $t$. The Lagrangian (29) has been derived
for $l\geq 2$ with a nonvanishing $W$, but it is also valid for $l=1$ by
setting $W=0$ for the above gauge choice. An alternative procedure to be taken
for $l=1$ is that we literally exploit the Lagrangian (29), vary it with
respect to $W$ and $Q$, and set $W=0$ at the end. This process leads to

$\dot{\cal E}=0\,,\qquad\left(r^{2}{\cal E}\right)^{\prime}=0\,,$ (72)

where

${\cal
E}:=r^{2}\sqrt{\frac{f}{h}}\left[C_{1}\left(Q^{\prime}-\frac{2}{r}Q\right)-\left(C_{2}\dot{\delta
u}+C_{3}\delta u^{\prime}+C_{4}\delta u\right)\right].$ (73)

From Eq. (72), we obtain the integrated solution

${\cal E}=\frac{{\cal E}_{0}}{r^{2}}\,,$ (74)

where ${\cal E}_{0}$ is a constant. On using Eq. (73), the perturbation $Q$
can be expressed as

$\displaystyle Q$ $\displaystyle=$ $\displaystyle r^{2}\int{\rm
d}\tilde{r}\frac{1}{C_{1}\tilde{r}^{2}}\left(\frac{{\cal
E}_{0}}{\tilde{r}^{4}}\sqrt{\frac{h}{f}}+C_{2}\dot{\delta u}+C_{3}\delta
u^{\prime}+C_{4}\delta u\right)$ (75) $\displaystyle+r^{2}{\cal C}_{2}(t)\,,$

where ${\cal C}_{2}$ is a function of $t$. From Eqs. (31) and (75), the
residual gauge mode ${\cal C}(t)$ in Eq. (71) can be removed by setting

${\cal C}(t)=\int{\rm d}\tilde{t}\,{\cal C}_{2}(\tilde{t})\,.$ (76)

On using Eq. (74) with Eq. (73), we can eliminate the terms containing
$-Q^{\prime}+2Q/r$ in Eq. (29). This process leads to the reduced Lagrangian

$\displaystyle{\cal L}_{\rm odd}=$ $\displaystyle\frac{r^{2}}{16\pi
G_{\ae}}\sqrt{\frac{f}{h}}\Bigg{[}\frac{C_{1}C_{5}-C_{2}^{2}}{C_{1}}\dot{\delta
u}^{2}+\left(C_{6}-\frac{2C_{2}C_{3}}{C_{1}}\right)\dot{\delta u}\delta
u^{\prime}+\left(C_{7}-\frac{C_{3}^{2}}{C_{1}}\right)\delta u^{\prime 2}$ (77)
$\displaystyle\hskip 56.9055pt-\frac{2C_{3}C_{4}}{C_{1}}\delta
u^{\prime}\delta u+\left(2C_{12}+C_{13}-\frac{C_{4}^{2}}{C_{1}}\right)\delta
u^{2}+\frac{h{\cal E}_{0}^{2}}{C_{1}r^{8}f}\Bigg{]}\,.$

This shows that only the Aether perturbation $\delta u$ propagates. From the
coefficient of $\dot{\delta u}^{2}$ in Eq. (77), we find that the ghost is
absent for $(C_{1}C_{5}-C_{2}^{2})/C_{1}>0$, which is equivalent to the
condition $q_{2}>0$ derived for $l\geq 2$. From the first three terms in Eq.
(77), we can also show that the radial propagation speed is equivalent to
$c_{r2}$ given by Eq. (51). This is analogous to the result found for
generalized Proca theories Kase:2018voo . In summary, for $l=1$, there are
neither ghost nor Laplacian instabilities under the conditions $q_{2}>0$ and
${\cal F}_{2}\geq 0$.

##  IV Stability conditions in Einstein-Aether theories

To study the odd-parity stabilities of BHs, we use the explicit forms of
$C_{i}$’s given in Appendix B together with the value of $b$ constrained by
Eq. (19). Then, the no-ghost conditions (42) and (43) translate to

$q_{1}=\frac{h(1-c_{13})(1-c_{13}a^{2}f)}{2f^{2}}>0\,,$ (78)
$q_{2}=\frac{c_{1}+c_{4}a^{2}f}{r^{2}f}-\frac{c_{13}^{2}(a^{2}f-1)}{2r^{2}f(1-c_{13})}>0\,.$
(79)

The conditions (54) and (55), which ensure the absence of Laplacian
instabilities along the radial direction, are given by

$\displaystyle{\cal F}_{1}$ $\displaystyle=$
$\displaystyle\frac{4fh(1-c_{13})}{(1-c_{13}a^{2}f)^{2}}\geq 0\,,$ (80)
$\displaystyle{\cal F}_{2}$ $\displaystyle=$
$\displaystyle\frac{2hc_{14}[2c_{1}-c_{13}(2c_{1}-c_{13})]}{r^{4}f(1-c_{13})}\geq
0\,.$ (81)

The conditions (62) and (63) for the absence of Laplacian instabilities along
the angular direction translate to

$\displaystyle c_{\Omega 1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{1-c_{13}a^{2}f}\geq 0\,,$ (82) $\displaystyle c_{\Omega
2}^{2}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2(1-c_{13})(c_{1}+c_{4}a^{2}f)-c_{13}^{2}(a^{2}f-1)}\geq
0.$ (83)

For the dipole ($l=1$), only the two conditions (79) and (81) need to be
satisfied for the Aether perturbation $\delta u$.

On the Minkowski background characterized by the metric components $f=h=1$,
the Aether field is given by $u^{\mu}=(+1,0,0,0)$ and hence $a=1$ and $b=0$.
Then, in Minkowski spacetime, the stability conditions (78)-(83) reduce,
respectively, to

$\displaystyle(q_{1})_{\rm Min}$ $\displaystyle=$
$\displaystyle\frac{(1-c_{13})^{2}}{2}>0\,,$ (84) $\displaystyle(q_{2})_{\rm
Min}$ $\displaystyle=$ $\displaystyle\frac{c_{14}}{r^{2}}>0\,,$ (85)
$\displaystyle({\cal F}_{1})_{\rm Min}$ $\displaystyle=$
$\displaystyle\frac{4}{1-c_{13}}\geq 0\,,$ (86) $\displaystyle({\cal
F}_{2})_{\rm Min}$ $\displaystyle=$
$\displaystyle\frac{2c_{14}[2c_{1}-c_{13}(2c_{1}-c_{13})]}{r^{4}(1-c_{13})}\geq
0\,,$ (87) $\displaystyle(c_{\Omega 1}^{2})_{\rm Min}$ $\displaystyle=$
$\displaystyle\frac{1}{1-c_{13}}\geq 0\,,$ (88) $\displaystyle(c_{\Omega
2}^{2})_{\rm Min}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}\geq 0\,,$
(89)

which are satisfied for

$\displaystyle c_{13}<1\,,$ (90) $\displaystyle c_{14}>0\,,$ (91)
$\displaystyle 2c_{1}-c_{13}(2c_{1}-c_{13})\geq 0\,.$ (92)

These conditions coincide with those derived in Refs. Jacobson:2004ts ;
Oost:2018tcv  by expanding the action (1) up to quadratic order in tensor and
vector perturbations on the Minkowski background. On using the fact that the
coefficients $C_{9}$, $C_{2}$, and $C_{6}$ vanish in Eqs. (50) and (51), the
radial sound speed squares in Minkowski spacetime are given by

$\displaystyle(c_{r1}^{2})_{\rm Min}$ $\displaystyle=$
$\displaystyle\frac{1}{1-c_{13}}\,,$ (93) $\displaystyle(c_{r2}^{2})_{\rm
Min}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}\,.$ (94)

Then we have that $(c_{r1}^{2})_{\rm Min}=(c_{\Omega 1}^{2})_{\rm Min}$ and
$(c_{r2}^{2})_{\rm Min}=(c_{\Omega 2}^{2})_{\rm Min}$, while this equality
does not generally hold on the curved background (17).

As shown in Refs. Jacobson:2004ts ; Oost:2018tcv , $(c_{r1}^{2})_{\rm Min}$
and $(c_{r2}^{2})_{\rm Min}$ correspond to the propagation speed squares of
tensor and vector perturbations on the Minkowski background, respectively, see
Eqs. (12) and (11). From the gravitational-wave event GW170817 GW170817
together with its electromagnetic counterpart Goldstein , the speed squared of
tensor perturbations is in the range $-3\times 10^{-15}<(c_{r1})_{\rm
Min}-1<7\times 10^{-16}$, so the coupling $c_{13}$ is constrained to be

$|c_{13}|\lesssim 10^{-15}\,.$ (95)

We note that there are also stability conditions in the Minkowski spacetime
arising from scalar perturbations Jacobson:2004ts ; Oost:2018tcv . They can be
derived by considering even-parity perturbations on the curved background (17)
and taking the Minkowski limit $f\to 1$, $h\to 1$, and $a\to 1$.

In this paper, we will not carry out the analysis of even-parity
perturbations, but we will show in Sec. V that the stability analysis based on
odd-parity perturbations alone is sufficiently powerful to exclude some BH
solutions in Einstein-Aether theory.

##  V Stability of Einstein-Aether black holes

Let us consider the stability of spherically symmetric and static BH solutions
known in the literature. Performing the transformation

${\rm d}v={\rm d}t+\frac{{\rm d}r}{\sqrt{fh}}\,,$ (96)

the line element (17) is transformed to the Eddington-Finkelstein coordinate
of the form

${\rm d}s^{2}=-f(r){\rm d}v^{2}+2B(r){\rm d}v{\rm d}r+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}\right)\,,$ (97)

where

$B(r)=\sqrt{\frac{f}{h}}\,.$ (98)

From Eq. (25), the nonvanishing components of the background Aether field
$u_{\mu}$ are given by $u_{t}=-af$ and $u_{r}=b/h$, where $b$ is constrained
as Eq. (19). On using Eq. (96), we have

$u_{t}{\rm d}t+u_{r}{\rm d}r=u_{v}{\rm d}v+\tilde{u}_{r}{\rm d}r\,,$ (99)

where

$u_{v}=-af\,,\qquad\tilde{u}_{r}=a\sqrt{\frac{f}{h}}+\frac{b}{h}\,.$ (100)

Since $g^{vv}=0$, $g^{vr}=g^{rv}=\sqrt{h/f}$, and $g^{rr}=h$, the nonvanishing
components of $u^{\mu}$ in the $(v,r)$ coordinate are given by

$u^{v}=a+\frac{b}{\sqrt{fh}}\,,\qquad\tilde{u}^{r}=b\,.$ (101)

In the notation of Ref. Zhang:2020too , the variable $A$ is used for $u^{v}$,
in which case we have

$u^{v}=A=a+\frac{b}{\sqrt{fh}}\,,\qquad\tilde{u}^{r}=b=\frac{fA^{2}-1}{2AB}\,.$
(102)

Since we would like to consider the case in which the gravitational-wave bound
(95) is satisfied, we will focus on the BH solutions satisfying the conditions

$c_{13}=0\,,$ (103)

in the following analysis.

###  V.1 Stealth Schwarzschild solution

We first consider the coupling constants satisfying

$c_{14}=0\,.$ (104)

For this choice we have $(q_{2})_{\rm Min}=0$ from Eq. (85), so there is a
strong coupling problem on the Minkowski background. In curved spacetime the
stability conditions are different from those on the Minkowski background, so
we will study whether BH solutions satisfying the condition (104) are stable
or not.

The background Eqs. (20)-(22) admit the existence of an exact stealth BH
solution characterized by

$\displaystyle f=h=1-\frac{r_{s}}{r}\,,$ (105) $\displaystyle
a=\frac{\sqrt{4r^{3}(r-r_{s})+w_{2}^{2}}}{2r(r-r_{s})}\,,\qquad
b=\epsilon\frac{w_{2}}{2r^{2}}\,,$ (106)

where $r_{s}$ is the Schrawarzschild radius, $\epsilon=\pm 1$, and $w_{2}$ is
a positive constant. From Eqs. (98) and (102), we have

$A=\frac{\epsilon
w_{2}+\sqrt{4r^{3}(r-r_{s})+w_{2}^{2}}}{2r(r-r_{s})}\,,\qquad B=1\,,$ (107)

so that $A\to 1$ as $r\to\infty$. For $\epsilon=+1$, the temporal vector
component diverges as $u^{v}=A\propto(r-r_{s})^{-1}$ around $r=r_{s}$. In this
case, the quantity $J:=A^{2}f$ is in the range $J>1$ outside the horizon and
it exhibits the divergence $J\to(w_{2}^{2}/r_{s}^{3})(r-r_{s})^{-1}$ for $r\to
r_{s}$. On the other hand, for $\epsilon=-1$, the expansion of $A$ around
$r=r_{s}$ gives

$A=\frac{r_{s}^{2}}{w_{2}}-\frac{r_{s}(r_{s}^{4}-2w_{2}^{2})}{w_{2}^{3}}(r-r_{s})+{\cal
O}((r-r_{s})^{2})\,,$ (108)

and hence $A$ is finite at $r=r_{s}$. In this case, $J<1$ outside the horizon
and $J=0$ at $r=r_{s}$. For this latter branch ($b<0$), the above exact BH
solution with $w_{2}=3\sqrt{3}r_{s}^{2}/8$ gives rise to a universal horizon
at $r=3r_{s}/4$ Zhang:2020too .

From Eq. (106), the temporal vector component $u^{t}$ in the $(t,r)$
coordinate has the divergent behavior $u^{t}=a\propto(r-r_{s})^{-1}$ as $r\to
r_{s}$, irrespective of the signs of $b$. Defining the quantity

$j:=a^{2}f\,,$ (109)

we have

$j=\frac{w_{2}^{2}}{4r^{3}(r-r_{s})}+1\,,$ (110)

and hence $j>1$ outside the horizon. There is the divergence $j\to\infty$ as
$r\to+r_{s}$, with the asymptotic behavior $j\to 1$ at spatial infinity.

From Eqs. (78)-(83), the quantities associated with the stability conditions
are given by

$\displaystyle q_{1}=\frac{1}{2f}\,,\qquad
q_{2}=-\frac{c_{1}(j-1)}{r^{2}f}\,,$ (111) $\displaystyle{\cal
F}_{1}=4f^{2}\,,\qquad{\cal F}_{2}=0\,,$ (112) $\displaystyle c_{\Omega
1}^{2}=1\,,\qquad c_{\Omega 2}^{2}=-\frac{1}{j-1}\,.$ (113)

The conditions $q_{1}>0$, ${\cal F}_{1}\geq 0$, ${\cal F}_{2}\geq 0$, and
$c_{\Omega 1}^{2}\geq 0$ are satisfied for $r>r_{s}$. Since $j>1$ outside the
horizon, we have $c_{\Omega 2}^{2}<0$ and hence there is a Laplacian
instability along the angular direction. In particular, as $r$ increases from
$r_{s}$ to spatial infinity, $c_{\Omega 2}^{2}$ changes from $-0$ to
$-\infty$. For $c_{1}>0$, we have $q_{2}<0$ outside the horizon, so the ghost
instability is also present. From Eqs. (50) and (51), the radial propagation
speed squares are given by

$c_{r1}^{2}=1\,,\qquad c_{r2}^{2}=\frac{j}{j-1}\,.$ (114)

Since both $c_{r1}^{2}$ and $c_{r2}^{2}$ are positive outside the horizon, the
Laplacian instabilities are absent along the radial direction.

In summary, the stealth Schwarzschild solution with the vector-field profile
(106) is unstable due to the Laplacian instability associated with the
negative propagation speed squared $c_{\Omega 2}^{2}$ outside the horizon. In
addition, for $c_{1}>0$, the ghost instability for the Aether perturbation
also exists. It is interesting to note that stealth Schwarzschild solutions
present in the context of generalized Proca theories Chagoya:2016aar ;
Heisenberg:2017xda ; Heisenberg:2017hwb  are also unstable against odd-parity
perturbations Kase:2018voo .

###  V.2 BH solutions with $c_{14}\neq 0$

We proceed to study the stability of BH solutions for the couplings

$c_{14}\neq 0\,.$ (115)

Then, the quantities in Eqs. (78)-(83) reduce to

$\displaystyle q_{1}=\frac{h}{2f^{2}}\,,\qquad
q_{2}=\frac{c_{1}+c_{4}j}{r^{2}f}\,,$ (116) $\displaystyle{\cal
F}_{1}=4fh\,,\qquad{\cal F}_{2}=\frac{4hc_{1}c_{14}}{r^{4}f}\,,$ (117)
$\displaystyle c_{\Omega 1}^{2}=1\,,\qquad c_{\Omega
2}^{2}=\frac{c_{1}}{c_{1}+c_{4}j}\,,$ (118)

where $j$ is defined by Eq. (109). Since $f>0$ and $h>0$ outside the horizon,
the conditions $q_{1}>0$, ${\cal F}_{1}\geq 0$, and $c_{\Omega 1}^{2}\geq 0$
are satisfied. The condition ${\cal F}_{2}\geq 0$ translates to

$c_{1}c_{14}\geq 0\,.$ (119)

Taking the asymptotically flat (Minkowski) limit $a\to 1$, $f\to 1$, and $h\to
1$ in Eqs. (116) and (118), it follows that $q_{2}\to c_{14}/r^{2}$ and
$c_{\Omega 2}^{2}\to c_{1}/c_{14}$. Then, the stability conditions $q_{2}>0$
and $c_{\Omega 2}^{2}\geq 0$ translate to

$c_{14}>0\,,\qquad c_{1}\geq 0\,,$ (120)

which are compatible with Eq. (119). In the Minkowski limit the propagation
speed squares of the transverse vector mode along both radial and angular
directions are $(c_{r2}^{2})_{\rm Min}=(c_{\Omega 2}^{2})_{\rm
Min}=c_{1}/c_{14}$, so the propagation is subluminal (or superluminal) for
$c_{4}>0$ (or for $c_{4}<0$). In Einstein-Aether theory, the gravitational
Cerenkov radiation can occur for the subluminal propagation of transverse
vector mode. For an interaction between a fermion and a graviton studied in
Ref. Elliott:2005va , the emission rate $\Gamma$ from a fermion for the
transverse vector mode is proportional to $c_{13}^{2}[1-(c_{r2}^{2})_{\rm
Min}]$ Elliott:2005va , so that $\Gamma=0$ for $c_{13}=0$. When
$(c_{r2}^{2})_{\rm Min}<1$, however, there may be a possibility that other
higher-order interactions give rise to the gravitational Cerenkov radiation
even for $c_{13}=0$. In the superluminal range realized by the coupling
$c_{4}<0$, there is no constraint arising from the gravitational Cerenkov
radiation.

To discuss the stability of BH solutions around the horizon, we search for
background solutions where the temporal vector component $u^{v}=A$ in the
($v,r$) coordinate is regular at $r=r_{s}$ like Eq. (108). In doing so, we
expand $A$, $f$, $h$ around $r=r_{s}$ in the forms

$\displaystyle A$ $\displaystyle=$ $\displaystyle
A_{0}+A_{1}(r-r_{s})+A_{2}(r-r_{s})^{2}+\cdots\,,$ (121) $\displaystyle f$
$\displaystyle=$ $\displaystyle
f_{1}(r-r_{s})+f_{2}(r-r_{s})^{2}+f_{3}(r-r_{s})^{3}\cdots\,,$ (122)
$\displaystyle h$ $\displaystyle=$ $\displaystyle
h_{1}(r-r_{s})+h_{2}(r-r_{s})^{2}+h_{3}(r-r_{s})^{3}\cdots\,,$ (123)

where $A_{i}$, $f_{i}$, $h_{i}$ are constants. On using Eqs. (98) and (102),
there is the following relation

$a=\frac{A^{2}f+1}{2Af}\,.$ (124)

Then, we can express Eqs. (20)-(22) as the differential equations for $A$,
$f$, $h$, instead of those for $a$, $f$, $h$. Substituting Eqs. (121)-(123)
into such differential equations, we find that there are solutions where the
coefficients $A_{1,2,3,\cdots}$ and metric components are related to the
constant $A_{0}$. For the special case with $c_{14}=0$, we confirmed that the
iterative solutions derived by this prescription coincide with those obtained
by expanding Eqs. (105) and (107) around $r=r_{s}$.

From Eqs. (121)-(123), the temporal metric component $a$ in the ($t,r$)
coordinate and the quantity $j=a^{2}f$ have the following dependence around
the horizon:

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{2A_{0}f_{1}}\left(r-r_{s}\right)^{-1}+{\cal
O}((r-r_{s})^{0})\,,$ (125) $\displaystyle j$ $\displaystyle=$
$\displaystyle\frac{1}{4A_{0}^{2}f_{1}}\left(r-r_{s}\right)^{-1}+{\cal
O}((r-r_{s})^{0})\,.$ (126)

Since $j$ diverges at $r=r_{s}$, the quantities $q_{2}$ and $c_{\Omega 2}^{2}$
around the horizon can be estimated as

$\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{c_{4}}{4A_{0}^{2}r_{s}^{2}f_{1}^{2}}\left(r-r_{s}\right)^{-2}+{\cal
O}((r-r_{s})^{-1})\,,$ (127) $\displaystyle c_{\Omega 2}^{2}$ $\displaystyle=$
$\displaystyle\frac{4c_{1}A_{0}^{2}f_{1}}{c_{4}}\left(r-r_{s}\right)+{\cal
O}((r-r_{s})^{2})\,.$ (128)

From Eq. (127) the ghost is absent for

$c_{4}>0\,.$ (129)

Provided that $c_{1}\geq 0$, we also have $c_{\Omega 2}^{2}\geq 0$ around
$r=r_{s}$. Indeed, for $c_{1}\geq 0$ and $c_{4}>0$, the two conditions
$q_{2}>0$ and $c_{\Omega 2}^{2}\geq 0$ hold throughout the horizon exterior,
since $j$ is positive. We note that the odd-parity stability about the
Minkowski spacetime, which is satisfied under the conditions (120), does not
necessarily require that $c_{4}>0$ (unless the superluminality of $c_{\Omega
2}^{2}$ is imposed). For BHs the term $c_{4}j$ in Eq. (116) dominates over
$c_{1}$ around the horizon, so the positivity of $q_{2}$ demands that
$c_{4}>0$. In other words, the inequality (129) is a new stability condition
derived by the analysis on the curved background.

The BH solution with $c_{4}<0$ and $c_{1}>0$ is plagued by the ghost
instability as well as the Laplacian instability around the horizon. If we
restrict the superluminal propagation of transverse vector mode, we have
$c_{4}<0$ and hence the BH solution in this case is unstable.

So far, we have performed the expansion of Taylor series of $A$ as Eq. (121)
with a finite value of $A$ at $r=r_{s}$. Suppose that there is a solution of
$A$ diverging at $r=r_{s}$ in the form

$A=\frac{A_{0}}{(r-r_{s})^{p}}\,,$ (130)

where $A_{0}$ and $p$ are constants. Here we are considering positive values
of $p$, but we also include the case $p=0$ in the analysis below. Analogous to
the discussion in scalar-tensor theories Hui:2012qt , we consider the scalar
product $J^{\mu\nu}J_{\mu\nu}$ for the current tensor ${J^{\mu}}_{\alpha}$
defined in Eq. (6) and impose the regularity of $J^{\mu\nu}J_{\mu\nu}$ at
$r=r_{s}$. On using Eq. (130) and regular expansions of $f$ and $h$ as those
in Eqs. (122) and (123), the scalar product $J^{\mu\nu}J_{\mu\nu}$ diverges at
$r=r_{s}$ apart from the special cases $p=0$ and $p=1$. For $p$ close to be 0
or 1, there is the power-law dependence $J^{\mu\nu}J_{\mu\nu}\propto
1/(r-r_{s})^{q}$ with $q$ close to 2. The powers $p=0$ and $p=1$ are the
special cases in which $J^{\mu\nu}J_{\mu\nu}$ is regular at $r=r_{s}$. For
$p>1$, the scalar product diverges as $J^{\mu\nu}J_{\mu\nu}\propto
1/(r-r_{s})^{2p}$ at $r=r_{s}$.

The expansion of $A$ performed in Eq. (121) corresponds to the power $p=0$, in
which case the regularity of $J^{\mu\nu}J_{\mu\nu}$ is ensured at the horizon.
For $p=1$, expanding the quantities $q_{2}$ and $c_{\Omega 2}^{2}$ around
$r=r_{s}$ gives

$\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{c_{4}A_{0}^{2}}{4r_{s}^{2}}\left(r-r_{s}\right)^{-2}+{\cal
O}((r-r_{s})^{-1})\,,$ (131) $\displaystyle c_{\Omega 2}^{2}$ $\displaystyle=$
$\displaystyle\frac{4c_{1}}{c_{4}A_{0}^{2}f_{1}}\left(r-r_{s}\right)+{\cal
O}((r-r_{s})^{2})\,.$ (132)

Under the superluminal condition $c_{4}<0$, there is the ghost instability
($q_{2}<0$) as well as the Laplacian instability ($c_{\Omega 2}^{2}<0$) for
$c_{1}>0$. As in the case of $p=0$, we require the conditions (120) and (129)
to ensure the odd-parity stability of BHs, but in this case the propagation of
transverse vector mode is subluminal.

###  V.3 BH solutions with $c_{4}=0$

Let us finally discuss the stability of BH solutions for the coupling

$c_{4}=0\,.$ (133)

In this case, the quantities $q_{1}$, ${\cal F}_{1}$, and $c_{\Omega 1}^{2}$
are the same as those given in Eqs. (116), (117), and (118), which are all
positive outside the horizon. For $c_{1}\neq 0$, the other quantities are
given by

$q_{2}=\frac{c_{1}}{r^{2}f}\,,\qquad{\cal
F}_{2}=\frac{4hc_{1}^{2}}{r^{4}f}\,,\qquad c_{\Omega 2}^{2}=1\,.$ (134)

Provided that $c_{1}>0$, the ghost is absent.

When $c_{1}=0$, both the denominator and numerator of $c_{\Omega 2}^{2}$ in
Eq. (118) vanish. This reflects the fact that, for $c_{1}=0$, the vector
perturbation does not propagate as in the case of GR. The coupling constant
$c_{2}$ does not appear in any of the stability conditions obtained in Sec.
IV, so the case $c_{1}=0$ can be regarded as the GR limit for the couplings
under consideration now (i.e., $c_{1}=0$, $c_{3}=0$, and $c_{4}=0$). In this
case, we only need to consider the stability conditions $q_{1}>0$, ${\cal
F}_{1}\geq 0$, and $c_{\Omega 1}^{2}\geq 0$ in the odd-parity sector, all of
which are trivially satisfied outside the horizon.

In summary, for $c_{4}=0$, the stability of BHs against odd-parity
perturbations with large values of $k$ and $l$ is ensured for

$c_{1}\geq 0\,.$ (135)

There are numerically obtained BH solutions consistent with this range of
couplings Eling:2006ec ; Zhang:2020too .

##  VI Conclusions

In this paper, we studied the stability of spherically symmetric and static
BHs against odd-parity perturbations in Einstein-Aether theory. On the
background (17), the presence of a unit vector constraint (4) gives the
relation (19) between the temporal and radial components of the Aether field.
At the background level, there are three independent Eqs. (20)-(22) to be
solved for $a$ and the metric components $f$ and $h$.

In Sec. III, we derived the second-order action of odd-parity perturbations by
using the expansion in terms of the spherical harmonics
$Y_{lm}(\theta,\varphi)$. Choosing the Regge-Wheeler gauge for $l\geq 2$, we
obtained the second-order Lagrangian of the form (29) and identified $\chi$
and $\delta u$ as the two dynamical perturbations associated with the gravity
sector and the Aether field, respectively. After the integration by parts, the
Lagrangian of these dynamical fields is given by Eq. (38). We showed that
there are neither ghost nor Laplacian instabilities under the conditions (42),
(43), (54), (55), (62) and (63) for large values of $k$ and $l$. For the
dipole ($l=1$) the propagating degree of freedom is the Aether perturbation
$\delta u$ alone, which does not give additional constraints to those derived
for $l\geq 2$.

Using the explicit forms of coefficients $C_{i}$’s given in Appendix B, the
stability conditions in Einstein-Aether theory reduce to Eqs. (78)-(83). In
the limit of the Minkowski spacetime, we also showed in Sec. IV that the
propagation speeds along both radial and angular directions coincide with
those of tensor and vector perturbations already derived in the literature.
The combination of coupling constants $c_{13}=c_{1}+c_{3}$ is tightly
constrained to be $|c_{13}|\lesssim 10^{-15}$ from the GW170817 event together
with 170817A.

In Sec. V, the odd-parity stabilities of BHs in Einstein-Aether theory were
studied for the couplings satisfying

$c_{13}=0\,.$ (136)

This choice is consistent with the constraint (95) on the speed of tensor
perturbations in the range $-3\times 10^{-15}<(c_{r1})_{\rm Min}-1<7\times
10^{-16}$ obtained from the gravitational-wave event GW170817 GW170817  and
its electromagnetic counterpart Goldstein . In doing so, we used the relations
of metric and vector-field components between the two different coordinates
(17) and (97). In Sec. V.1, we considered the exact Schwarzschild solution
present for $c_{14}=0$ and found that there is a Laplacian instability along
the angular direction throughout the horizon exterior. Moreover, for
$c_{1}>0$, the ghost instability also exists for the Aether perturbation. In
Sec. V.2, we discussed the BH solutions for $c_{14}\neq 0$ and showed that
their stabilities require the conditions (120) and $c_{4}>0$. In this case,
the propagation of the vector perturbation is subluminal in the asymptotically
flat regime, so there is a possibility for the gravitational Cerenkov
radiation to occur. In other words, the superluminal propagation of transverse
vector mode occurring for $c_{4}<0$, under which the gravitational Cerenkov
radiation is avoided, is incompatible with the BH stability conditions. In
Sec. V.3, we showed that the BH solutions with

$c_{4}=0,\qquad c_{1}\geq 0\,,$ (137)

are stable against odd-parity perturbations for high radial and angular
momentum modes. Clearly, if we demand the odd-parity stability of BHs, the
viable region of the parameter space of Eqs. (13)-(16), obtained recently in
Ref. Oost:2018tcv , is reduced further.

It is interesting to note that the instability of perturbations mainly happens
in the Aether field, represented by $\delta u$, while the metric part,
represented by $\chi$, behaves well for the coupling with $c_{13}=0$.

The Lagrangian (29) of odd-parity perturbations can be applied to the
computation of quasi-normal modes of BHs. Moreover, the analysis of even-
parity perturbations will provide us additional stability conditions of BHs to
those derived in this paper. We leave these issues, together with the analyses
of their corresponding quasi-normal mode spectra, for future separate
publications.

## Acknowledgements

We thank Ted Jacobson and Shinji Mukohyama for useful discussions and
comments. ST is supported by the Grant-in-Aid for Scientific Research Fund of
the JSPS No. 19K03854. This work is also partially supported by the National
Key Research and Development Program of China under the Grant No.
2020YFC2201503, and the National Natural Science Foundation of China under the
grant No. 11975203.

## Appendix A: Qnatities in background equations

On the spherically symmetric and static background (17), the Lagrange
multiplier (8) is given by

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\bigg{\\{}fr^{2}\left[a^{2}\left(6c_{1}+3c_{2}+2c_{13}-8c_{14}\right)h\left(f^{\prime}\right)^{2}+c_{2}\left(2hf^{\prime\prime}+f^{\prime}h^{\prime}\right)\right]$
(A.1)
$\displaystyle+2a^{2}f^{4}\left[-2a^{2}\left(2\left(c_{2}+c_{13}\right)h-c_{2}rh^{\prime}\right)+4\left(c_{14}-c_{1}\right)hr^{2}\left(a^{\prime}\right)^{2}+a\left(-c_{1}+c_{2}+c_{13}\right)r\left(2hra^{\prime\prime}+a^{\prime}\left(rh^{\prime}+4h\right)\right)\right]$
$\displaystyle-f^{2}\Big{[}rh^{\prime}\left(a^{2}\left(-2c_{1}+3c_{2}+2c_{13}\right)rf^{\prime}-4c_{2}\right)$
$\displaystyle+2h\big{(}a^{4}\left(3c_{1}+c_{2}+c_{13}-4c_{14}\right)r^{2}\left(f^{\prime}\right)^{2}+a^{2}r\left(\left(-2c_{1}+3c_{2}+2c_{13}\right)rf^{\prime\prime}+2\left(-2c_{1}+c_{2}+2c_{13}\right)f^{\prime}\right)$
$\displaystyle-a\left(11c_{1}-5c_{2}-5c_{13}-8c_{14}\right)r^{2}a^{\prime}f^{\prime}+4\left(c_{2}+c_{13}\right)\big{)}\Big{]}-\left(2c_{2}+c_{13}\right)hr^{2}\left(f^{\prime}\right)^{2}$
$\displaystyle+2f^{3}\Big{[}a^{4}r\left(\left(-c_{1}+c_{2}+c_{13}\right)rf^{\prime}h^{\prime}+2h\left(\left(-c_{1}+c_{2}+c_{13}\right)rf^{\prime\prime}+\left(-2c_{1}+c_{2}+2c_{13}\right)f^{\prime}\right)\right)$
$\displaystyle+a^{2}\left(8\left(c_{2}+c_{13}\right)h-4c_{2}rh^{\prime}\right)-2\left(-2c_{1}+c_{2}+c_{13}+2c_{14}\right)hr^{2}\left(a^{\prime}\right)^{2}$
$\displaystyle+a^{3}\left(-11c_{1}+3c_{2}+3c_{13}+8c_{14}\right)hr^{2}a^{\prime}f^{\prime}+a\left(c_{1}-c_{2}-c_{13}\right)r\left(2hra^{\prime\prime}+a^{\prime}\left(rh^{\prime}+4h\right)\right)\Big{]}\bigg{\\}}/[4f^{2}r^{2}\left(a^{2}f-1\right)],$

where $c_{ij}=c_{i}+c_{j}$.

Introducing the following quantity

$\beta:=(c_{2}+c_{3}-c_{4})fa^{2}+c_{14}\,,$ (A.2)

Then, the coefficients in Eqs. (20)-(22) are expressed as

$\displaystyle\alpha_{1}=-2\beta(a^{2}f-1)r^{2}f^{2}\,,$
$\displaystyle\alpha_{2}=(c_{2}+c_{13}-2\beta)(a^{2}f-1)r^{2}af\,$
$\displaystyle\alpha_{3}=4(c_{2}+c_{13})r^{2}f^{3}ah\,,$
$\displaystyle\alpha_{4}=-[(c_{2}+c_{13}-2\beta)a^{2}f-2(c_{2}+c_{13}-\beta)]r^{2}ah\,,$
$\displaystyle\alpha_{5}=2[(2c_{2}+2c_{13}-3\beta)a^{2}f+3\beta]r^{2}fh\,,$
$\displaystyle\alpha_{6}=4(c_{2}a^{2}f+c_{13}-2\beta)(a^{2}f-1)rafh\,,$
$\displaystyle\alpha_{7}=-4c_{2}(a^{2}f-1)^{2}raf^{2}\,,$
$\displaystyle\alpha_{8}=-8\beta(a^{2}f-1)rf^{2}h\,,$
$\displaystyle\alpha_{9}=8(c_{2}+c_{13})(a^{2}f-1)^{2}af^{2}h\,,$ (A.3)
$\displaystyle\beta_{1}=4\beta(a^{2}f-1)r^{2}af^{3}\,,$
$\displaystyle\beta_{2}=-2(c_{2}+c_{13}-2\beta)(a^{2}f-1)r^{2}a^{2}f^{2}\,,$
$\displaystyle\beta_{3}=-4[2(c_{2}+c_{13})a^{2}f-\beta]r^{2}f^{3}h\,,$
$\displaystyle\beta_{4}=[2(c_{2}+c_{13}-2\beta)a^{4}f^{2}-8(c_{2}+c_{13}-\beta)a^{2}f$
$\displaystyle\qquad\,+c_{2}+c_{13}]r^{2}h\,,$
$\displaystyle\beta_{5}=-4[(2c_{2}+2c_{13}-3\beta)a^{2}f+c_{2}+c_{13}+\beta]r^{2}af^{2}h\,,$
$\displaystyle\beta_{6}=-4(a^{2}f-1)[2+2c_{2}-2(2c_{2}-c_{13}+2\beta)a^{2}f$
$\displaystyle\qquad+2c_{2}a^{4}f^{2}]rfh\,,$
$\displaystyle\beta_{7}=8c_{2}(a^{2}f-1)^{2}rf^{3}a^{2}\,,$
$\displaystyle\beta_{8}=16(c_{2}+\beta)(a^{2}f-1)raf^{3}h\,,$
$\displaystyle\beta_{9}=4[2-\\{2+4c_{2}+2c_{13}-2(4c_{2}+3c_{13})a^{2}f$
$\displaystyle\qquad+4(c_{2}+c_{13})a^{4}f^{2}\\}h]f^{2}(a^{2}f-1)\,,$ (A.4)
$\displaystyle\mu_{1}=-4c_{2}(a^{2}f-1)raf^{3}\,,$
$\displaystyle\mu_{2}=2(1+c_{2}-2c_{2}a^{2}f)(a^{2}f-1)rf\,,$
$\displaystyle\mu_{3}=4(2c_{2}+\beta-2c_{2}a^{2}f)rf^{3}h\,,$
$\displaystyle\mu_{4}=[2+3c_{2}+c_{13}-2(1+c_{2}+2c_{13}-2\beta)a^{2}f$
$\displaystyle\qquad-4c_{2}a^{4}f^{2}]rh\,,$
$\displaystyle\mu_{5}=4(6c_{2}-7c_{2}a^{2}f-c_{13}+2\beta)raf^{2}h\,,$
$\displaystyle\mu_{6}=4(a^{2}f-1)[1+2c_{2}+c_{13}-3(2c_{2}+c_{13})a^{2}f]fh\,,$
$\displaystyle\mu_{7}=4(a^{2}f-1)[1+2c_{2}+c_{13}-(2c_{2}+c_{13})a^{2}f]f^{2}\,,$
$\displaystyle\mu_{8}=-16(2c_{2}+c_{13})(a^{2}f-1)af^{3}h\,.$ (A.5)

## Appendix B: Coefficients in perturbation equations

The coefficients in Eq. (29), which should be evaluated on the background
(17), are

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{(1-c_{13})h}{2r^{2}f}\,,$ $\displaystyle C_{2}$
$\displaystyle=$ $\displaystyle-\frac{c_{13}b}{2r^{2}f}\,,$ $\displaystyle
C_{3}$ $\displaystyle=$ $\displaystyle-\frac{c_{13}ah}{2r^{2}}\,,$
$\displaystyle C_{4}$ $\displaystyle=$
$\displaystyle\frac{[(2c_{14}-c_{13})(fa^{\prime}+af^{\prime})r+2c_{13}af]h}{2r^{3}f}\,,$
$\displaystyle C_{5}$ $\displaystyle=$
$\displaystyle\frac{c_{1}+c_{4}a^{2}f}{r^{2}f}\,,$ $\displaystyle C_{6}$
$\displaystyle=$ $\displaystyle\frac{2c_{4}ab}{r^{2}}\,,$ $\displaystyle
C_{7}$ $\displaystyle=$
$\displaystyle\frac{[c_{4}(a^{2}f-1)-c_{1}]h}{r^{2}}\,,$ $\displaystyle C_{8}$
$\displaystyle=$ $\displaystyle-\frac{[c_{13}(a^{2}f-1)+1]h}{2r^{4}}\,,$
$\displaystyle C_{9}$ $\displaystyle=$ $\displaystyle\frac{c_{13}b}{r^{4}}\,,$
$\displaystyle C_{10}$ $\displaystyle=$
$\displaystyle\frac{1-c_{13}a^{2}f}{2r^{4}f}\,,$ $\displaystyle C_{11}$
$\displaystyle=$ $\displaystyle\frac{c_{13}a}{r^{4}}\,,$ $\displaystyle
C_{12}$ $\displaystyle=$ $\displaystyle-\frac{c_{1}}{r^{4}}\,,$ $\displaystyle
C_{13}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{r^{2}}-\frac{c_{13}[(rh^{\prime}+2h-2)f+rhf^{\prime}]}{2r^{4}f}$
(B.1) $\displaystyle-\frac{2c_{4}(fa^{\prime}+f^{\prime}a)ah}{r^{3}}\,,$

where $b$ is given by Eq. (19).

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