# Analysing the Impact of Audio Quality on the Use of Naturalistic Long-Form
Recordings for Infant-Directed Speech Research

María Andrea Cruz Blandón<EMAIL_ADDRESS>
Unit of Computing Sciences, Tampere University, P.O. Box 553, FI-33101,
Finland Alejandrina Cristia<EMAIL_ADDRESS>
Département d’études cognitives, ENS, EHESS, CNRS, PSL University, 29 rue
d’Ulm, 75005 Paris, France Okko Räsänen<EMAIL_ADDRESS>
Unit of Computing Sciences, Tampere University, P.O. Box 553, FI-33101,
Finland

###### Abstract

Modelling of early language acquisition aims to understand how infants
bootstrap their language skills. The modelling encompasses properties of the
input data used for training the models, the cognitive hypotheses and their
algorithmic implementations being tested, and the evaluation methodologies to
compare models to human data. Recent developments have enabled the use of more
naturalistic training data for computational models. This also motivates
development of more naturalistic tests of model behaviour. A crucial step
towards such an aim is to develop representative speech datasets consisting of
speech heard by infants in their natural environments. However, a major
drawback of such recordings is that they are typically noisy, and it is
currently unclear how the sound quality could affect analyses and modelling
experiments conducted on such data. In this paper, we explore this aspect for
the case of infant-directed speech (IDS) and adult-directed speech (ADS)
analysis. First, we manually and automatically annotated audio quality of
utterances extracted from two corpora of child-centred long-form recordings
(in English and French). We then compared acoustic features of IDS and ADS in
an in-lab dataset and across different audio quality subsets of naturalistic
data. Finally, we assessed how the audio quality and recording environment may
change the conclusions of a modelling analysis using a recent self-supervised
learning model. Our results show that the use of modest and high audio quality
naturalistic speech data result in largely similar conclusions on IDS and ADS
in terms of acoustic analyses and modelling experiments. We also found that an
automatic sound quality assessment tool can be used to screen out useful parts
of long-form recordings for a closer analysis with comparable results to that
of manual quality annotation.

Keywords: infant-directed speech; benchmarking; long-form recordings; acoustic
analyses; computational modelling; early language acquisition

## Introduction

Computational modelling of early language acquisition aims to understand how
infants bootstrap their language skills. Development of plausible models and
using them for cognitive hypothesis testing requires attention on three
largely independent aspects: models themselves, the language data used as
inputs for the models, and evaluation protocols for assessing linguistic
competence or performance of the models.

Given the advances in machine learning and data collection methods,
computational models have become increasingly capable of learning from semi-
naturalistic large-scale auditory (?, ?) or audiovisual data (?, ?). This is
in contrast to earlier studies with simplified inputs, such as phonetic
transcripts (?, ?), pre-segmented units of speech (e.g., ?, ?), and small-
scale speech corpora often associated with such studies. This development has
been largely facilitated by the research on so-called self-supervised learning
(SSL; e.g., ?, ?). These methods enable autonomous speech representation
discovery from low-level sensory inputs without strong linguistic priors to
guide the learning process. Therefore, they are also highly relevant also as
potential models of language learning. As a result, recent research has shown
successes in modelling phonemic and lexical learning from acoustic speech only
(?, ?), including also models learning from naturalistic child-centred long-
form audio recordings (?, ?).

In terms of model evaluation, there is also ongoing progress in working
towards more realistic model evaluation protocols with respect to experimental
or observational findings on infant language development. For instance,
earlier studies that have compared models against individual behavioural
experiments (?, ?) or analysed developmental patterns (?, ?, ?). ? (?) also
recently proposed an evaluation framework where models are subjected to
computational tests comparable to those used with real infants in lab,
allowing comparison of model language capabilities to those of infants as a
function of model training data and infant age.

Notably, behavioural studies and hence their computational replications
typically employ stimuli that have been carefully curated so that researchers
can control the dimensions of the stimulus that are of interest for the study.
This, in turn, can potentially lead to less authentic and simpler stimuli to
what infants are actually exposed to in real world. There is evidence that
using in-lab data might overestimate the effect of linguistic capabilities
when contrasted with more natural/spontaneous data (?, ?). Therefore, a
logical step would be to extend computational model evaluation towards more
naturalistic data, such as use of audio stimuli extracted from child-centred
long-form recordings from children’s everyday environments (e.g., ?, ?).

However, one central problem with using naturalistic recordings is their
highly variable and on average very low audio quality compared to cleaner
corpora recorded in controlled settings (e.g., almost 0 dB average signal-no-
noise ratio reported by ?, ?). The noise in audio may influence results of
acoustic analyses and computational modelling studies, and also introduces a
potential mismatch between datasets used to train and evaluate models. Hence,
it would be important to separate the effects of moving from simplified to
naturalistic input from those of moving from clean data to potentially more
noisy data. While the former is relevant for motivating the use of more
naturalistic data, the latter is simply a potential nuisance factor in
analyses, unless the utilised acoustical analyses and computational models are
specifically equipped against noise (which is rarely the case).

The present paper uses analysis of infant-directed speech (IDS) and adult-
directed speech (ADS) as a case study to address the above issues. We
investigate whether 1) basic properties of naturalistic IDS and ADS speech
data differ from those of in-lab stimuli, and 2) investigate whether
computational modelling findings with naturalistic are different from in-lab
data. Importantly, we perform both investigations with different sound quality
screening methods applied to the data in order to test how varying sound
quality criteria impact study findings, and in order to identify the best
practice for creating high-quality stimulus sets from longform audio
recordings. As our computational model, we employ a self-supervised algorithm
for statistical learning investigated by ? (?).

## Methodology

The aim of the study was to create a naturalistic test set of IDS and ADS
utterances that can be used to benchmark computational models of language
acquisition in their capability to process IDS and ADS differentially, and to
study its properties compared to in-lab data as a function of different sound
quality criteria.

We chose to study IDS due to its pivotal role in language bootstrapping,
guiding infants’ attention to the relevant parts of utterances and making
contrasts more easy to identify (?, ?, ?). IDS has been found to be widely
used across different languages and communities, and to display common
features that varies in strength depending on the language (?, ?). Typically
IDS will have a more salient pitch contour, longer vowels and simpler grammar
than ADS (?, ?, ?). In addition, there exists carefully curated in-lab data
with well-established research findings for IDS and ADS comparison from the
ManyBabies (MB) consortium (?, ?) as well as long-form recordings annotated
for speech addressee (?, ?, ?). Finally, there are earlier modelling findings
using both in-lab and naturalistic IDS data, where the authors reported
smaller effects for naturalistic data compared to the in-lab data (?, ?).
However, the authors did not control for sound quality in their experiments,
which could potentially explain the difference in their findings between the
two data types.

What then makes long-form child-centred recordings typically noisier than
their in-lab counterparts is that they include background noises (e.g., radio
or television noise), overlapped speech, baby cries and screams among other
noises in addition to speech varying from near- to far-field recordings.
Hence, long-form recordings are more challenging to perceive and process for
annotators and computational models alike. Therefore, in this study, we extend
the comparison of long-form recordings against in-lab data by introducing the
audio quality as a tentative factor moderating the results. Additionally, we
run the analyses for two rhythmically different languages: English and French.
Finally, we also evaluate the impact of the audio quality in model evaluation
by using the developed test set with self-supervised models for predictability
of IDS and ADS utterances (see, e.g., ?, ?, ? for earlier related modelling
approaches).

### Corpora for Benchmarking IDS and ADS Segments from Long-Form Recordings

For creation of a naturalistic test set of IDS and ADS stimuli, we employed
the hand-annotated IDS and ADS segments of the English part of the ACLEW
project data (?, ?, ?) (hereafter, SEEDLings data) and the annotated segments
of the Lyon project data (?, ?) for the French dataset (hereafter, Lyon data).
Both SEEDLings and Lyon data were collected by placing a LENA audio recorder
device in the clothes of the infant and recording a whole day. The annotated
segments were obtained by randomly sampling two-minute (SEEDLings) or ten-
minute (Lyon) segments from a full recording of an infant, and for ten infants
($3-20$ months of age) in the SEEDLings data and 16 infants ($3-41$ months) in
the Lyon data. Authors of these datasets had the clips manually annotated for
utterances111In contrast to ? (?) who used LENA-based automatic processing
followed by manual verification of labels. and their speakers and addressees.

For our experiments, we used the utterances from adult speakers (female or
male) with adult addressee as ADS and utterances from adult speakers with
infant addressee as IDS. This yielded a total of 1124 ADS utterances and 952
IDS utterances for the SEEDLings data, and 3915 ADS utterances and 863 IDS
utterances for the Lyon data. These utterances were further subjected to sound
quality assessment, as explained in the next subsection.

In order to compare the utterances from the long-form recordings against
stimuli used in behavioural experiments, we chose the MB project stimulus set
as the reference, consisting of 120 IDS and 120 ADS utterances (?, ?). The MB
recordings were carried out using a LENA device or a mobile device, but
conducted in a fixed lab environment with an elicitation task related to
object description to an infant or adult (see ?, ?, for details).

### Assessing the Audio Quality of IDS and ADS Segments

To determine the audio quality of the naturalistic IDS and ADS segments, we
used two strategies: manual and automatic annotation. For the manual strategy,
three labels were utilised: good, medium and bad quality. The quality was
defined in terms of perceived background noise, overlapping speech, baby cries
and screams, and loudness and reverberation of the target speech. The
distinction between good and medium quality was based on the loudness and
distance to the microphone (i.e., good quality required high loudness and
little reverberation). One annotator (the first author) annotated all the
samples in both datasets across several days.

Automatic sound quality ratings were obtained from a recent Signal-to-Noise
Ratio (SNR) estimator system (?, ?). SNRs (in dBs) of all utterances were
first estimated. The manual annotations were then used to establish the SNR
threshold that best divides the data into good and medium/poor quality data.
This procedure yielded thresholds of 14.50 dB for SEEDLings and 14.03 dB for
Lyon data.

Given the ratings, we defined three subsets of data for our experimental
analyses: Strict quality, consisting of manual good quality utterances.
Relaxed quality, consisting of utterances with good and medium quality manual
labels, and SNR quality utterances whose automatic estimates were above the
set SNR thresholds. Figure 1 shows the resulting sample counts of IDS and ADS
utterances in the different quality subsets. As seen from the figure, there is
a tradeoff between sound quality and test set size with SNR approach resulting
in least strict quality requirements with the largest sample size.

Figure 1: Distribution of IDS and ADS segments according to the determined
quality of the segments for the Lyon (top) and SEEDLings (bottom) datasets.

### Acoustic Analyses of IDS and ADS Characteristics

Following ? (?), we performed acoustic analyses to compare long-form
recordings and in-lab recordings in terms of their basic IDS/ADS
characteristics. For this, the mean of logarithm of fundamental frequency
(log-F0) and log-F0 variability (std log-F0) were extracted for all utterances
with PRAAT222By means of the python library wrapper: Parselmouth (?, ?).
Additionally, the speech delivered to infants tend to be shorter than the
speech delivered to other adults (?, ?). Thus, we included utterance duration
(in seconds) in our analyses. Finally, there is evidence that speakers use an
exaggerated articulation for IDS (?, ?) to help the processing of the message.
Although extraction of vowel space measures from data without phonetic
annotation is not possible, we used mean spectral tilt as a proxy for general
vocal effort that also reflects potential differences in speaking style. The
first Mel-frequency cepstral coefficient (MFCC) coefficient was used for this
purpose.

Overall, we expected to replicate ? (?) findings for both languages, i.e.,
higher pitch and variability for IDS than ADS utterances. For duration, we
expected the duration to be longer for in-lab recordings, and to be
independent of the subset quality on naturalistic data. For tilt, we expected
greater tilt for utterances with greater vocal effort. Unpaired t-tests were
used to identify if there were any significant differences between IDS and ADS
acoustic features, and we also report the magnitude of the difference using
Cohen’s d (?, ?).

### Computational Modelling Analyses

The aim of the modelling experiments was to test if 1) experiments on
naturalistic data replicate behaviour observed on in-lab data, and 2) whether
this is moderated by quality screening applied to the data. While the former
is not necessarily expected (the reason to use naturalistic data in the first
place), it is important to know if models’ behaviours are greatly affected by
sound quality of the test data.

#### Self-Supervised Model for Speech Representation Learning

We used the self-supervised statistical learning neural network model
introduced in (?, ?) as a model of early language acquisition. The model
learns directly from the raw speech signal in an unsupervised manner. The
learning process is centred on optimisation of a predictive process that
attempts to predict future speech observations from current speech input.
However, instead of predicting the acoustic signal directly, the model learns
to predict its own latent representations of the future speech, allowing it to
also invent these representations as seen fit for the prediction task (cf.
brains predicting their own neural activations for sensory input instead of
the actual physical signals). The reader is recommended to see ? (?) for
technical details of the learning mechanism.

The model was trained on read audiobooks from LibriVox and LibriLight,
training a model for English and French (?, ?, ?) separately. We used the
model versions from ? (?) with a total training of 50, 100, 200, 400, 800,
1600 or 3200 h to simulate different amounts of language experience for the
learner.

#### Predictability Analysis

Predictability of speech is linked to the encoding of speech structure. Models
that are better at predicting the input exhibit better comprehension of the
speech signal, whereas models that find the input difficult to predict can be
viewed as more ”surprised” with the stimuli. Previous work has linked lower
predictability of IDS (compared to ADS) to heightened attentional preference
towards IDS (?, ?, ?; see also ?, ?). However, here we do not take a position
in the familiarity/novelty coding of attention, but simply use predictability
of input as a measure of model’s competence with the given speech style. This
allows us then to compare potential mismatches between datasets in terms of
naturalness and noise levels.

In the case of these models trained on audiobooks, they have not seen examples
of IDS or ADS styles. Nevertheless, based on the previous findings (?, ?), we
expect the models to have better speech representations and higher prediction
accuracy as they become trained with more data. If the models are learning
speech representations that are encompassing features prevalent across the
different speech styles, we expect to have similar predictability for both IDS
and ADS. Another option is to have better predictability for ADS than IDS
utterances as ADS is more similar to the read speech the models have seen
during training. If the quality of the test subset does not affect the
behaviour, we expect to have the same predictability pattern across all three
qualities. We expect that predictability findings between in-lab and long-form
data might differ. This is since the models are focusing on the entire
acoustic signals instead of a small set of chosen characteristics, and hence
data with acoustically similar measures may still have notable structural
differences from the perspective of speech perception.

## Results

### Acoustic Analyses

Table 1: Results of the acoustic analysis: mean pitch (F0), and variability of the pitch in log scales, spectral tilt, and duration (s) for IDS and ADS segments are reported for all three audio quality subsets. Effect size is reported as Cohen’s d, asterisk indicating all significant differences (at $\alpha=0.05$). Bold font denotes the larger values in cases of significant difference between the IDS and ADS. | mean(log-F0) | std(log-F0) | Spectral Tilt | Duration (s)
---|---|---|---|---
IDS | ADS | d | IDS | ADS | d | IDS | ADS | d | IDS | ADS | d
SEEDLings | Strict | 5.31 | 5.07 | 0.80* | 0.22 | 0.20 | 0.16 | 135.71 | 137.23 | 0.04 | 1.22 | 1.46 | 0.27
Relaxed | 5.32 | 5.10 | 0.77* | 0.23 | 0.22 | 0.07 | 133.67 | 133.75 | 0 | 1.37 | 1.86 | 0.43*
SNR | 5.39 | 5.16 | 0.81* | 0.23 | 0.23 | 0.03 | 135.72 | 135.09 | 0.02 | 1.12 | 1.20 | 0.10
Lyon | Strict | 5.46 | 5.58 | 0.43* | 0.25 | 0.22 | 0.20 | 82.42 | 85.09 | 0.07 | 2.25 | 1.99 | 0.21
Relaxed | 5.45 | 5.50 | 0.18* | 0.25 | 0.26 | 0.05 | 81.98 | 90.81 | 0.23* | 2.51 | 2.24 | 0.19*
SNR | 5.46 | 5.56 | 0.35* | 0.25 | 0.24 | 0.06 | 84.91 | 91.52 | 0.17* | 2.15 | 1.86 | 0.24*
ManyBabies | 5.39 | 5.19 | 1.18* | 0.28 | 0.29 | 0.03 | 180.16 | 169.08 | 0.31* | 2.97 | 5.14 | 1.00*

Table 1 shows the measurements of acoustic features obtained for the two
speech styles, two corpora, sound quality subsets, and with the MB data as a
reference. The MB data follow the predictions of the literature for three
acoustic features (see also ?, ?), where IDS utterances exhibit higher F0,
shorter duration, and higher spectral tilt than what is in ADS.

As for the in-lab versus naturalistic data, the effect sizes (Cohen’s d) of
acoustic feature differences are in general larger for the MB data than for
the long-form recordings, also replicating the findings of ? (?). This adds to
the evidence that in-lab recordings tend to be more distinctive between IDS
and ADS than at-home recordings. However, when we look at the patterns found
for the SEEDLings and Lyon data, only the SEEDLings dataset replicates the
majority of findings of the MB data, except for the spectral tilt, where no
significant difference is found between IDS and ADS, and duration, for which
only the Relaxed quality replicates the pattern. In contrast to the English
data, the French data (Lyon) exhibits a higher mean F0 and tilt for the ADS
utterances and longer utterances for IDS than ADS.

The results for the Lyon corpus contrast with previous findings where the
pitch was found higher for IDS than ADS (?, ?). During the manual inspection
of the audio quality, we found that some of the utterances were annotated as
ADS when the adult was in fact talking to another child in the room. This
inconsistency in the given annotations could contribute general accuracy of
the measurements. However, it cannot explain the reversal of the acoustic cue
values in IDS and ADS. Therefore, the reason for the result remains currently
unclear, and exemplifies how the analysis of naturalistic data may result in
findings that are not expected based on analyses conducted on more controlled
datasets.

Regarding the audio quality of the subsets, we observed that in both
naturalistic corpora, the measurements of the same acoustic feature slightly
vary across the three audio qualities. Generally, the effect sizes are larger
for cleaner data (see Cohen’s d in Table 1). However, main effects and their
directions observed for the strict audio quality are likely to be found in the
less stringent quality screening cases. Hence, from the acoustic analysis
point of view, the quality of the recordings does not drastically change the
characteristics studied. Since there are no remarkable differences in the
results between the manual and automatic quality screening, the findings also
suggest that automatic screening is applicable to long-form audio data for
basic acoustic analyses. We further investigate whether this finding also
holds in the predictability analysis with a computational model.

### Modelling Analyses

Figure 2: Accuracy of predictions on strict audio quality data for the
different model training data sizes (x-axis). Top: SEEDLings data. Middle:
Lyon data Bottom: MB data. Left: English-trained model, Right: French-trained
model. ADS in red and ADS in green. Significant differences between IDS and
ADS are denoted with asterisks or ns for no difference ($*$ stands for
$p<0.05$, $**$ for $p<0.01$, and $***$ for $p<0.001$).

Figure 2 presents the prediction accuracy of the self-supervised models for
the SEEDLings, Lyon and MB data for the strict quality. The results are shown
for each of the 7 training data amounts (checkpoints) separately. Figures 3
and 4 show the results for the relaxed and SNR qualities for the SEEDLings and
Lyon data.

The results on the in-lab MB data indicate that the model reaches above-chance
level accuracy ($>0.02$) already at 50 h of training, and the predictability
improves as the models use more training data; this is regardless of the
language used for training. Also, the model exhibits higher predictability of
the ADS utterances than those of IDS. However, this difference between the two
styles on in-lab data becomes smaller with more training.

When comparing MB data against the long-form recordings in the strict quality
subset (Fig. 2), we find that only the increasing accuracy pattern is
replicated by the SEEDLings and Lyon data. In general, the predictability
improves with more training data, again despite of the language of training.
However, for the SEEDLings data, there is no significant difference between
IDS and ADS. For the Lyon data, the pattern is the opposite from MB, the model
obtaining higher predictability for the IDS compared to ADS utterances. These
results demonstrate that modelling outcomes on in-lab data and long-form data
can differ, as we further discuss in the final section.

When analysing the results across the three audio quality (see Figs. 3 and 4),
we can observe that the predictability patterns found for each corpus apply
across all the three qualities. As with the acoustic analyses, the
predictability becomes slightly worse with more noisy data (e.g., going from
accuracy of $\approx 0.24$ for the 50 h English model for the SEEDLings data
in the strict subset to $\approx 0.22$ for the relaxed subset to $\approx
0.20$ for the SNR subset). However, the basic pattern of the relative
predictability of IDS and ADS as a function of training data amount is not
affected by the sound quality.

Figure 3: Predictability of IDS and ADS utterances in the relaxed quality
subset. See the caption of Fig. 2 for details of the plot. Figure 4:
Predictability of IDS and ADS utterances in the SNR quality subset. See the
caption of Fig. 2 for details of the plot.

Overall, despite being trained on audiobooks, the model is capable of
predicting both speech styles with a comparable accuracy, showing no notable
difference predicting either style (differences between IDS and ADS do not go
greater than $0.1$). The result implies that the model is learning speech
representations that are tuned to the properties of spoken language, but the
model does not appear to treat IDS/ADS styles in a particularly distinct
manner. This happens independently of whether the model is trained on English
or French language, although with slightly lower accuracy for the non-native
predictions. This may be due to models were not trained on natural ADS or IDS,
but on read audiobooks, which may cause a comparable mismatch in speech
properties when moving from the read speech to naturalistic speech, no matter
whether the naturalistic speech is IDS or ADS.

## Discussion and Conclusion

In order to develop more naturalistic evaluation settings for computational
modelling, it is necessary to understand the differences between in-lab and
at-home recordings, as well as the implications of sound quality differences
in different datasets. This paper investigated sound quality for the case of
IDS/ADS comparison by extracting three audio quality for long-form recordings
and contrasting their acoustic features against the features of curated
recordings for lab testing. We also used predictability analysis to
investigate whether the audio quality could affect modelling findings.

Overall, there are several differences between in-lab data and the long-form
data: contents and duration of the recording (short controlled sessions vs.
uncontrolled full-day everyday activities), social contexts in which the
utterances were spoken (see also ?, ?), and with in-lab data displaying more
larger acoustic differences between IDS and ADS than those found in long-form
data. This suggests that the input infants face in natural contexts may
consist of less salient features in comparison to those encountered in-lab
contexts. As a consequence, all the differences between the speech styles
found in in-lab data might not replicate in at-home data, both in terms of
acoustic analysis and computational modelling results (e.g., lack of
difference between the spectral tilt of IDS versus ADS). And even in the case
of having similar acoustic patterns (e.g., pitch and variability of the pitch
for the MB and SEEDLings data), in-lab and at-home data could lead to
different conclusions when evaluating computational models (see e.g.,
predictability analysis).

However, it is also possible, if not even likely, that the saliency of
different speaking styles may vary more in naturalistic environments. The
average acoustic properties of naturalistic IDS may not reflect language input
to children in those communicative scenarios or even certain time-points in
utterances where attention control by stylistic means truly matters for
learning, attentional control or affective regulation (see also ?, ? for a
discussion). When creating evaluation test sets for models under naturalistic
settings, the above-mentioned aspects need to be considered for
interpretation. Ultimately, the aim of the testing protocol also matters: the
use of naturalistic data enables to test model behaviours “in-the-wild”, and
can lead to novel empirical hypotheses about the role and characteristics of
IDS and ADS in real-life learning scenarios. However, this should be preceded
by model validation on data for which comparable infant data on infant
behaviour exists, which currently is largely limited to in-lab data from
controlled experiments.

Concerning the audio quality, our results suggest that comparable and
acceptable audio quality is reachable by manual annotation of the clips but
also by using automatic tools for SNR estimation (at least for the acoustic
features and modelling task investigated here). The practical implication is
that automatic tools, such as that of ? (?), can be effectively used to screen
out subsets of data that fit desired audio quality needs. This is especially
useful when used in conjunction with other automated tools to process long-
form datasets, as manual quality annotation does not scale up to hundreds or
thousands of hours of speech.

To sum up, the present study adds to the shared knowledge on how the recording
conditions (in-lab versus at-home) play a role in analysis and computational
modelling outcomes, in this case for IDS and ADS speech. Ideally, use of both
in-lab and at-home data in tandem would enable a more comprehensive analysis
of the phenomenon of interest. In addition, determining the audio quality of
long-form recordings can be achieved with an automatic tool instead of tedious
annotations, and with a low risk of missing the salient findings that are also
present in data annotated as good quality by human listeners. In future work,
a more comprehensive testing of the potential domain mismatch between model
training and testing data is needed, for instance, by also training the models
on naturalistic instead of clean data. However, this is left to future work.

## Acknowledgement

Authors MACB and OR were supported by Academy of Finland grants no. 314602,
320053, and 345365, and Kone Foundation grant L-SCALE. AC was funded by Agence
Nationale de la Recherche (ANR-16-DATA-0004 ACLEW), the ERC (ExELang,
101001095), and a J. S. McDonnell Foundation Understanding Human Cognition
Scholar Award. We thank Marvin Lavechin and Maureen de Seyssel for their
insights that benefited this research at the early stages. The measurements
and scripts can be found on
https://github.com/SPEECHCOG/ids_audio_quality_analysis

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# Charged spherically symmetric and slowly rotating charged black hole
solutions in bumblebee gravity

Jia-Zhou Liu1 2, Wen-Di Guo1 2, Shao-Wen Wei1 2, and Yu-Xiao Liu1 2
111Corresponding author. E-mail<EMAIL_ADDRESS>1Lanzhou Center for
Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,
and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou
University, Lanzhou, Gansu 730000, China,
2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China

###### Abstract

In this paper, we present charged spherically symmetric black hole solutions
and slowly rotating charged solutions in bumblebee gravity with and without a
cosmological constant. The static spherically symmetric solutions describe the
Reissner-Nordström-like black hole and Reissner-Nordström-(anti) de Sitter-
like black hole, while the stationary and axially symmetric soltuions describe
Kerr-Newman-like black hole and Kerr-Newman-(anti) de Sitter-like black hole.
We utilize the Hamilton-Jacobi formalism to study the shadows of the black
holes. Additionally, we investigate the effect of the electric charge and
Lorentz-violating parameters on the radius of the shadow reference circle and
the distortion parameter. We find that the radius of the reference circle
decreases with the Lorentz-violating parameter and charge parameter, while the
distortion parameter increases with the Lorentz-violating parameter and the
charge parameter.

Black Holes, Lorentz Symmetry Breaking,Black Hole Shadow

###### Contents

  1. I INTRODUCTION
  2. II  EINSTEIN-BUMBLEBEE THEORY
  3. III CHARGED SPHERICALLY SYMMETRIC BLACK HOLE SOLUTIONS
    1. III.1  Case A: $V(X)=\frac{\lambda}{2}X^{2}$ and $\Lambda=0$
    2. III.2 Case B: $V(X)=\frac{\lambda}{2}X$ and $\Lambda\neq 0$
  4. IV  SlOWLY ROTATING CHARGED BLACK HOLE SOLUTIONS
    1. IV.1 Case A: $V(X)=\frac{\lambda}{2}X^{2}$ and $\Lambda=0$
    2. IV.2 Case B: $V(X)=\frac{\lambda}{2}X$ and $\Lambda\neq 0$
  5. V  SHADOW OBSERVABLES
    1. V.1 The photon orbit
    2. V.2 Observables
  6. VI  CONCLUSIONS

##  I INTRODUCTION

In the area of classical physics, gravitational phenomena are well-explained
by general relativity, which has endured rigorous experimental and theoretical
validation. On the quantum front, the Standard Model of particle physics has
been successfully described the other three fundamental interactions. These
two theories offer a comprehensive and understanding of the natural world.
Therefore, the quest to unify these two theories is fundamental, and its
achievement will inevitably lead us to a deeper comprehension of nature.

Several quantum gravity theories have been proposed Birrell and Davies (1984);
Maldacena (1998); Aharony et al. (2000); Gubser et al. (1998); Alfaro et al.
(2000, 2002); Rovelli and Smolin (1990), but it is needed the experiments at
the Planck scale ($\sim 10^{19}$ GeV) to observe the quantum effect, this is
beyond the ability of current experiments. However, some quantum gravity
theories assume that Lorentz symmetry might be broken in the gravitational UV
regime which is at our presently accessible low-energy scales. Stemming from
the concept of spontaneous Lorentz symmetry breaking (LSB) in string theory
Kostelecky and Samuel (1989a); Kostelecky (2004), the Standard Model extension
was introduced as a suitable framework for violating Lorentz symmetry Colladay
and Kostelecky (1997, 1998); Kostelecky (2004). The Standard Model extension
has been widely studied Colladay and Kostelecky (2001); Kostelecky and Lehnert
(2001); Kostelecky and Mewes (2001); Colladay and McDonald (2009); Berger et
al. (2016); Carroll et al. (1990); Andrianov and Soldati (1995, 1998); Lehnert
and Potting (2004); Baeta Scarpelli (2012); Brito et al. (2013).

In 1989, Kostelecky and Samuel introduced the prototype of the bumblebee model
Kostelecky and Samuel (1989b, a), a string-inspired framework featuring
tensor-induced spontaneous Lorentz symmetry breaking. In the framework of
bumblebee gravity, the spontaneous Lorentz violation arises from a potential
$V(B^{\mu}B_{\mu})$ acting on an vector field $B_{\mu}$ Kostelecky (2004);
Kostelecky and Tasson (2011). Not just in Minkowski’s spacetime, The bumblebee
model can also be explored in various spacetime geometries, including Riemann,
Riemann-Cartan spacetimes Bailey and Kostelecky (2006); Maluf et al. (2013);
Li and Övgün (2020); Bluhm et al. (2008a); Escobar and Martín-Ruiz (2017);
Maluf et al. (2015); Hernaski (2014); Carroll et al. (2009).

The pursuit of black hole solutions holds significance for any theory of
gravity. In 2017, Casana et al. obtained an exact Schwarzschild-like black
hole solution in the bumblebee gravity Casana et al. (2018). Subsequently,
researchers have independently proposed exact solutions for traversable
wormholes Övgün et al. (2019), Schwarzschild-anti-de Sitter (AdS)-like black
holes Maluf and Neves (2021), slowly rotating Kerr-like black holes Ding et
al. (2020), and other solutions Santos et al. (2015); Jha and Rahaman (2021);
Filho et al. (2023); Xu et al. (2023); Ding et al. (2023) within bumblebee
gravity. Moreover, in other scenarios within the framework of the Standard
Model Extension, such as when considering the nonminimally coupled Kalb-Ramond
field with a nonzero vacuum expectation value, certain black hole solutions
have also been obtained Lessa et al. (2020); Kumar et al. (2020); Yang et al.
(2023); Duan et al. (2023); Do and Kao (2020); Liu et al. (2024a). And the
properties of the blakc holes were also studied Guo et al. (2023); Du et al.
(2024); Kuang and Övgün (2022); Oliveira et al. (2019); Liu et al. (2023);
Gomes et al. (2020); Kanzi and Sakallı (2019); Güllü and Övgün (2022);
Oliveira et al. (2021); Kanzi and Sakallı (2021); Hosseinifar et al. (2024);
Liu et al. (2024b, a).

In 1966, Synge initiated the examination of the shadow cast by a Schwarzschild
black hole Synge (1966). Subsequently, Luminet Luminet (1979) studied the
influence of a thin accretion disk on the black hole shadow, while Bardeen
Bardeen (1973) investigated the shadow of a Kerr black hole. Over the decades,
significant progress has been made in the study of black hole shadows de Vries
(1999); Takahashi (2005); Bambi et al. (2012); Atamurotov et al. (2013);
Perlick et al. (2018); Meng et al. (2023); Sui et al. (2023). Recently, direct
imaging of the black holes M87* at the core of the Virgo A galaxy and Sgr A*
at the core of the Milky Way galaxy has been achieved by the sub-millimeter
Event Horizon Telescope (EHT) employing very-long baseline interferometry
Akiyama et al. (2019a, b, c, d); Akiyama et al. (2022). Therefore,
investigating the black hole shadow is highly desirable.

Our aim in this paper is to obtain Charged spherically symmetric and slowly
rotating charged black hole solutions. Subsequently, using the equations
governing the motion of photons, we analyze the shadows cast by rotating black
holes, exploring the impact of Lorentz violation on the size and deformation
of these shadows. Calculations are then conducted to determine the radius and
distortion observables.

This paper is structured as follows: Sec. II provides a review of the
Einstein-Bumblebee theory. In Sec. III, we present the charged bumblebee black
hole solution in both the presence and absence of a cosmological constant. And
in Sec. IV, we extend this solution to the case of rotating black holes. Sec.
V is devoted to investigating the shadow of rotating black holes and computing
relevant parameters. Finally, Sec. VI presents a summary and discussion of the
work.

##  II  EINSTEIN-BUMBLEBEE THEORY

As discussed in the preceding section, the bumblebee model extends general
relativity by adding a vector field called bumblebee field which couples with
gravity nonminimally. The bumblebee vector field $B_{\mu}$ obtains a nonzero
vacuum expectation value through a specified potential, resulting in the
spontaneous breaking of Lorentz symmetry in the gravitational sector. The
action is described by Kostelecky (2004)

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\bigg{[}\frac{1}{2\kappa}\left(R-2\Lambda\right)+\frac{\xi}{2\kappa}B^{\mu}B^{\nu}R_{\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-V(B^{\mu}B_{\mu}\pm
b^{2})\bigg{]}$ (1) $\displaystyle+\int d^{4}x\sqrt{-g}\mathcal{L}_{M},$

where $\Lambda$ denotes the cosmological constant, $\kappa=\frac{8\pi
G}{c^{4}}$ is the gravitational coupling constant, and $\xi$ represents the
nonminimal coupling constant between gravity and the bumblebee field.
Analogous to the electromagnetic field, the bumblebee field strength is given
by

$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$ (2)

The potential $V$ provides a nonvanishing vacuum expectation value for
$B_{\mu}$ and could take the functional form of $V(B^{\mu}B_{\mu}\pm b^{2})$,
implying that the condition $B^{\mu}B_{\mu}=\mp b^{2}$ must be satisfied. In
essence, the field $B_{\mu}$ acquires a nonvanishing vacuum expectation value
$\left\langle B_{\mu}\right\rangle=b_{\mu}$, where the vector $b_{\mu}$ is a
function of the spacetime coordinates and satisfies $b^{\mu}b_{\mu}=\mp
b^{2}=\mathrm{const}$. For convenience, we define $X=B^{\mu}B_{\mu}\pm b^{2}$
and $V^{\prime}=\frac{\partial V}{\partial X}$, which will be used in the
subsequent discussions.

We consider the matter field to be an electromagnetic field nonminimally
coupled with the bumblebee vector field Lehum et al. (2024). Its Lagrangian
density expression reads as

$\displaystyle\mathcal{L}_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}(F^{\mu\nu}F_{\mu\nu}+{\gamma}B^{\mu}B_{\mu}F^{\alpha\beta}F_{\alpha\beta}),$
(3)

where the electromagnetic tensor related to the electromagnetic field is

$\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$
$\displaystyle A=(\phi(r),0,0,0),$ (4)

and ${\gamma}$ presents the coupling coefficient. The gravitational field
equations in the framework of bumblebee gravity can be derived by varying the
action (1) with respect to the metric tensor $g^{\mu\nu}$:

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T^{B}_{\mu\nu}+\kappa T^{M}_{\mu\nu},$
(5)

where

$\displaystyle T^{B}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\xi}{\kappa}\left[\frac{1}{2}B^{\alpha}B^{\beta}R_{\alpha\beta}g_{\mu\nu}-B_{\mu}B^{\alpha}R_{\alpha\nu}-B_{\nu}B^{\alpha}R_{\alpha\mu}\right.+\frac{1}{2}\nabla_{\alpha}\nabla_{\mu}\left(B^{\alpha}B_{\nu}\right)$
(6)
$\displaystyle+\frac{1}{2}\nabla_{\alpha}\nabla_{\nu}\left(B^{\alpha}B_{\mu}\right)\left.-\frac{1}{2}\nabla^{2}\left(B_{\mu}B_{\nu}\right)-\frac{1}{2}g_{\mu\nu}\nabla_{\alpha}\nabla_{\beta}\left(B^{\alpha}B^{\beta}\right)\right]$
$\displaystyle+2V^{\prime}B_{\mu}B_{\nu}+B_{\mu}^{\
\alpha}B_{\nu\alpha}-\left(V+\frac{1}{4}B_{\alpha\beta}B^{\alpha\beta}\right)g_{\mu\nu},$

and

$T^{M}_{\mu\nu}=\frac{1}{2\kappa}\left[(1+\gamma
b^{2})(2F_{\mu\alpha}F_{\nu}^{\alpha}-\frac{1}{2}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta})+\gamma
B_{\mu}B_{\nu}F^{\alpha\beta}F_{\alpha\beta}\right].$ (7)

For computational convenience, we can express the field equations in the
following form:

$R_{\mu\nu}=\Lambda g_{\mu\nu}+\kappa\mathcal{T}_{\mu\nu},$ (8)

where

$\displaystyle\mathcal{T}^{M}_{\mu\nu}$ $\displaystyle=$
$\displaystyle{T}^{M}_{\mu\nu}-\frac{1}{2}{T}^{M}g_{\mu\nu}$
$\displaystyle\mathcal{T}^{B}_{\mu\nu}$ $\displaystyle=$
$\displaystyle{T}^{B}_{\mu\nu}-\frac{1}{2}{T}^{B}g_{\mu\nu}$ (9)

and

$\displaystyle\kappa\mathcal{T}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\kappa\mathcal{T}^{M}_{\mu\nu}+\kappa\mathcal{T}^{B}_{\mu\nu}$
(10) $\displaystyle=$
$\displaystyle\kappa\left(T^{M}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T^{M}\right)+\kappa\left[V^{\prime}\left(2B_{\mu}B_{\nu}-b^{2}g_{\mu\nu}\right)+B_{\mu}^{\
\alpha}B_{\nu\alpha}+Vg_{\mu\nu}-\frac{1}{4}B_{\alpha\beta}B^{\alpha\beta}g_{\mu\nu}\right]$
$\displaystyle+\xi\left[\frac{1}{2}B^{\alpha}B^{\beta}R_{\alpha\beta}g_{\mu\nu}-B_{\mu}B^{\alpha}R_{\alpha\nu}-B_{\nu}B^{\alpha}R_{\alpha\mu}+\frac{1}{2}\nabla_{\alpha}\nabla_{\mu}\left(B^{\alpha}B_{\nu}\right)\right.+\frac{1}{2}\nabla_{\alpha}\nabla_{\nu}\left(B^{\alpha}B_{\mu}\right)$
$\displaystyle\left.-\frac{1}{2}\nabla^{2}\left(B_{\mu}B_{\nu}\right)\right].$

Similarly, by varying the action (1) with respect to the bumblebee vector
field and the electromagnetic field, we can obtain the equations of motion for
the corresponding fields:

$\displaystyle\nabla_{\mu}B^{\mu\nu}-2\left(V^{\prime}B^{\nu}-\frac{\xi}{2\kappa}B_{\mu}R^{\mu\nu}-\frac{1}{2\kappa}\gamma
B^{\nu}F^{\alpha\beta}F_{\alpha\beta}\right)$ $\displaystyle=$ $\displaystyle
0,$ (11) $\displaystyle\nabla_{\mu}\left(F^{\mu\nu}+\gamma
B^{\alpha}B_{\alpha}F^{\mu\nu}\right)$ $\displaystyle=$ $\displaystyle 0.$
(12)

##  III CHARGED SPHERICALLY SYMMETRIC BLACK HOLE SOLUTIONS

We consider the metric ansatz for a static and spherically symmetric
spacetime, which is expressed as

${d}{s}^{2}=-A(r){dt}^{2}+S(r){dr}^{2}+r^{2}{\leavevmode\nobreak\
d}\Omega^{2},$ (13)

where ${\leavevmode\nobreak\ d}\Omega^{2}={\leavevmode\nobreak\
d}\theta^{2}+\sin^{2}\theta{d}\varphi^{2}$.

In the previous study conducted by Casana et al., they formulated an exact
black hole solution without a cosmological constant and the matter field
$A_{\mu}$ Casana et al. (2018). In the scenario where the bumblebee field
$B_{\mu}$ maintains its vacuum expectation value $b_{\mu}$, the bumblebee
field is given by Bertolami and Paramos (2005)

$B_{\mu}=b_{\mu}.$ (14)

Similar to Ref. Casana et al. (2018), we consider a spacelike background
$b_{\mu}$ with the form

$b_{\mu}=\left(0,b_{r}(r),0,0\right).$ (15)

Utilizing the aforementioned condition $b_{\mu}b^{\mu}=b^{2}=const$, we can
derive

$b_{r}(r)=b\sqrt{S(r)}.$ (16)

For convenience, we set $\ell=\xi b^{2}$ as Lorentz-violating parameter. By
combining the spherically symmetric metric, we can obtain the specific field
equations:

$\displaystyle\frac{(2+\ell)A^{\prime}(r)^{2}}{8A(r)S(r)}-\frac{(2+\ell)A^{\prime}(r)S^{\prime}(r)}{8S(r)^{2}}-\frac{(1+\ell)A^{\prime}(r)}{rS(r)}+\frac{(2+\ell)A^{\prime\prime}(r)}{4S(r)}$
(17) $\displaystyle-\frac{\ell
A(r)S^{\prime}(r)}{2rS(r)^{2}}-(1+2b^{2}\gamma)A(r)\phi^{\prime}(r)^{2}+\Lambda
A(r)+\kappa(V^{\prime}b^{2}+V)A(r)$ $\displaystyle=$ $\displaystyle 0,$ (18)
$\displaystyle\frac{A^{\prime}(r)S^{\prime}(r)}{4A(r)S(r)}-\frac{A^{\prime\prime}(r)}{2A(r)}+\frac{A^{\prime}(r)^{2}}{4A(r)^{2}}+\frac{S^{\prime}(r)}{rS(r)}$
$\displaystyle+\frac{2(1+2b^{2}\gamma)\phi^{\prime}(r)^{2}S(r)}{(2+3\ell)}-\frac{2\Lambda
S(r)}{(2+3\ell)}-2\kappa\frac{(V^{\prime}b^{2}+V)S(r)}{(2+3\ell)}$
$\displaystyle=$ $\displaystyle 0,$ (19) $\displaystyle
1-\frac{1+\ell}{S(r)}+\frac{rS^{\prime}(r)}{2S(r)^{2}}+\frac{\ell
r^{2}A^{\prime\prime}(r)}{4A(r)S(r)}-\frac{\ell
r^{2}A^{\prime}(r)S^{\prime}(r)}{8A(r)S(r)^{2}}-\frac{\ell
r^{2}A^{\prime}(r)^{2}}{8A(r)^{2}S(r)}$
$\displaystyle-\frac{(1+\ell)rA^{\prime}(r)}{2A(r)S(r)}-(1+2b^{2}\gamma)\phi^{\prime}(r)^{2}r^{2}-\Lambda
r^{2}-\kappa(V^{\prime}b^{2}+V)r^{2}$ $\displaystyle=$ $\displaystyle 0,$ (20)
$\displaystyle\frac{A^{\prime\prime}(r)}{2A(r)}-\frac{A^{\prime}(r)^{2}}{4A(r)^{2}}-\frac{A^{\prime}(r)S^{\prime}(r)}{4A(r)S(r)}-\frac{2\gamma\phi^{\prime}(r)^{2}}{\xi\left(1+\ell\right)A(r)}-\frac{S^{\prime}(r)}{rS(r)}+\frac{2\kappa}{\xi}S(r)V^{\prime}$
$\displaystyle=$ $\displaystyle 0,$ (21)
$\displaystyle\frac{d}{dr}\left[r^{2}\frac{F_{tr}}{\sqrt{A(r)S(r)}}\right]$
$\displaystyle=$ $\displaystyle 0.$ (22)

Next, we will solve the above equations and give spherically symmetric black
hole solutions under different scenarios.

###  III.1  Case A: $V(X)=\frac{\lambda}{2}X^{2}$ and $\Lambda=0$

In the absence of the cosmological constant, similar to the work of Casana et
al., we impose the vacuum conditions $V=0$ and $V^{\prime}=0$ Casana et al.
(2018). An illustrative example of the potential satisfying these conditions
is readily presented by a smooth quadratic form:

$V(X)=\frac{\lambda}{2}X^{2},$ (23)

where $\lambda$ is a constant.

Using Eq. (52), we can derive $F_{tr}=\sqrt{A(r)S(r)}\phi^{\prime}(r)$.
Substituting the expressions of $F_{tr}$ here and $V$ in Eq. (23) into above
conditions into the field equations (18) - (22), and considering the value of
the coupling parameter $\gamma$ to be $\frac{\xi}{2+\ell}$, the field
equations are given by:

$\displaystyle\frac{(2+\ell)A^{\prime}(r)^{2}}{8A(r)S(r)}-\frac{(2+\ell)A^{\prime}(r)S^{\prime}(r)}{8S(r)^{2}}-\frac{(1+\ell)A^{\prime}(r)}{rS(r)}$
$\displaystyle+\frac{(2+\ell)A^{\prime\prime}(r)}{4S(r)}-\frac{\ell
A(r)S^{\prime}(r)}{2rS(r)^{2}}-\frac{A(r)\phi^{\prime}(r)^{2}}{\left(1+\ell\right)}$
$\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle\frac{A^{\prime}(r)S^{\prime}(r)}{4A(r)S(r)}-\frac{A^{\prime\prime}(r)}{2A(r)}+\frac{A^{\prime}(r)^{2}}{4A(r)^{2}}+\frac{S^{\prime}(r)}{rS(r)}+\frac{2\phi^{\prime}(r)^{2}S(r)}{(2+\ell)}$
$\displaystyle=$ $\displaystyle 0,$ (25) $\displaystyle
1-\frac{1+\ell}{S(r)}+\frac{rS^{\prime}(r)}{2S(r)^{2}}+\frac{\ell
r^{2}A^{\prime\prime}(r)}{4A(r)S(r)}-\frac{\ell
r^{2}A^{\prime}(r)S^{\prime}(r)}{8A(r)S(r)^{2}}-\frac{\ell
r^{2}A^{\prime}(r)^{2}}{8A(r)^{2}S(r)}$
$\displaystyle-\frac{(1+\ell)rA^{\prime}(r)}{2A(r)S(r)}-\frac{(2+3\ell)\phi^{\prime}(r)^{2}r^{2}}{(2+\ell)}$
$\displaystyle=$ $\displaystyle 0,$ (26)
$\displaystyle\frac{A^{\prime\prime}(r)}{2A(r)}-\frac{A^{\prime}(r)^{2}}{4A(r)^{2}}-\frac{A^{\prime}(r)S^{\prime}(r)}{4A(r)S(r)}-\frac{\phi^{\prime}(r)^{2}}{\left(1+\ell\right)A(r)}-\frac{S^{\prime}(r)}{rS(r)}$
$\displaystyle=$ $\displaystyle 0,$ (27)
$\displaystyle\frac{d}{dr}[r^{2}\phi^{\prime}(r)]$ $\displaystyle=$
$\displaystyle 0,$ (28)

by simplifying Eq. (24) and Eq. (25), we obtain:

$[S(r)A(r)]^{\prime}=0,$ (29)

which implies that $S(r)=\frac{C_{1}}{A(r)}$, where $C_{1}$ is a constant.
Similar to the Schwarzschild-like black hole solution in bumblebee gravity, we
set the value of $C_{1}$ to be $1+\ell$ Casana et al. (2018). Building upon
this, we solve the modified Maxwell’s equations to obtain:
$\phi(r)=\frac{C_{2}}{r}+C_{3}$. Here, we set $C_{2}=Q_{0}$ and $C_{3}=0$
(i.e., $\phi(r)=\frac{Q_{0}}{r}$), and we ultimately arrive at the final
solution:

$\displaystyle A(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{2(1+\ell)Q_{0}^{2}}{(2+\ell)r^{2}},$ (30) $\displaystyle
S(r)$ $\displaystyle=$ $\displaystyle\frac{1+\ell}{A(r)},$ (31)
$\displaystyle\phi(r)$ $\displaystyle=$ $\displaystyle\frac{Q_{0}}{r}.$ (32)

Through the modified Maxwell’s equations, the conserved current is modified to
be $J^{\nu}=\nabla_{\mu}\left(F^{\mu\nu}+\gamma
B^{\alpha}B_{\alpha}F^{\mu\nu}\right)$. Then, we can obtain the integration
constants, the Maxwell’s equations was also modified.

$\displaystyle Q$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int_{\Sigma}dx^{3}\sqrt{\gamma^{(3)}}n_{\mu}J^{\mu}$
(33) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int_{\partial\Sigma}d\theta
d\phi\sqrt{\gamma^{(2)}}n_{\mu}\sigma_{\nu}\left(F^{\mu\nu}+\frac{\xi}{l+2}B^{\mu\nu}B^{\alpha\beta}F_{\alpha\beta}\right)$
$\displaystyle=$ $\displaystyle\left(1+b^{2}\frac{\xi}{(l+2)}\right)Q_{0}$
$\displaystyle=$ $\displaystyle\frac{2(1+\ell)}{2+\ell}Q_{0}.$

Here $\Sigma$ represents a 3-dimensional spacelike region with the induced
metric $\gamma_{ij}^{(3)}$, the boundary $\partial\Sigma$ is a two-sphere
located at spatial infinity with the induced metric
$\gamma_{ij}^{(2)}=r^{2}{\leavevmode\nobreak\ d}\Omega^{2}$, while
$n_{\mu}=(1,0,0,0)$ and $\sigma_{\mu}=(0,1,0,0)$ denote the unit normal
vectors associated with $\partial\Sigma$ and $\Sigma$, respectively. The
solution of $A(r)$ can also be rewritten as

$A(r)=1-\frac{2M}{r}+\frac{(2+\ell)Q^{2}}{2(1+\ell)r^{2}},$ (34)

This solution is similar to the Reissner-Nordström (RN) solution. As $\ell$
tends to 0, it recovers to the RN solution. Additionally, it is evident that
the metric functions approach to $A(r)\rightarrow 1$ and $S(r)\rightarrow
1+\ell$ at infinity, indicating that the spacetime is not asymptotically
Minkowski. Similar to the Reissner-Nordström black hole, this solution
exhibits two horizons:

$r_{\pm}=M\pm\sqrt{M^{2}-\frac{2(\ell+1)}{\ell+2}Q_{0}^{2}}.$ (35)

It is obvious that the mass and charge parameter of the black hole should
satisfy

$\frac{Q_{0}^{2}}{M^{2}}\leq\frac{(2+\ell)}{2(1+\ell)}.$ (36)

Another aspect of this solution is the physical singularity which can be
studied through the Kretschmann scalar $K$, derived from the Riemann tensor:

$\displaystyle K$ $\displaystyle=$ $\displaystyle
R_{\alpha\beta\delta\gamma}R^{\alpha\beta\delta\gamma}$ (37) $\displaystyle=$
$\displaystyle\frac{4\left(\ell^{2}r^{2}+4lMr+12M^{2}\right)}{(\ell+1)^{2}r^{6}}-\frac{16Q_{0}^{2}\left((\ell+2)r(\ell
r+12M)-14(\ell+1)Q_{0}^{2}\right)}{(\ell+1)(\ell+2)^{2}r^{8}}.$ (38)

From this we can see that, the singularity only appears at $r=0$ (assuming
$0<\left|\ell\right|\ll 1$). This is similar to the RN black hole.

###  III.2 Case B: $V(X)=\frac{\lambda}{2}X$ and $\Lambda\neq 0$

Next, we consider the scenario involving a nonzero cosmological constant. Our
purpose is to derive an analytical black hole solution in this context. The
geometry of the solution is similar to the RN-Ads black hole or the RN-ds
black hole. Based on the groundwork by R.V. Maluf et al. Maluf and Neves
(2021), conditions are different from those in the previous discussion. The
straightforward option for the potential can be a linear function:

$V(X)=\frac{\lambda}{2}X,$ (39)

where $\lambda$ is a Lagrange multiplier Bluhm et al. (2008b). Note that the
equation of motion for the Lagrange multiplier ensures the vacuum condition
$X=0$, which also ensures that $V=0$ for any field $\lambda$ on-shell.
However, unlike the vacuum condition in Case A, $V^{\prime}\neq 0$ is valid
when the field $\lambda$ is nonzero. Thus, the potential $V$ contributes to
the field equations. The constraint for the Lagrange multiplier $\lambda$ is
providedthe field Eqs (10) and (11) .

Combining the spherically symmetric metric and the form of the potential $V$,
we can derive the specific field equations:

$\displaystyle\frac{(2+\ell)A^{\prime}(r)^{2}}{8A(r)S(r)}-\frac{(2+\ell)A^{\prime}(r)S^{\prime}(r)}{8S(r)^{2}}-\frac{(1+\ell)A^{\prime}(r)}{rS(r)}$
$\displaystyle+\frac{(2+\ell)A^{\prime\prime}(r)}{4S(r)}-\frac{\ell
A(r)S^{\prime}(r)}{2rS(r)^{2}}-\frac{A(r)\phi^{\prime}(r)^{2}}{\left(1+\ell\right)}+\frac{\Lambda
A(r)(2+\ell)}{2\ell}$ $\displaystyle=$ $\displaystyle 0,$ (40)
$\displaystyle\frac{A^{\prime}(r)S^{\prime}(r)}{4A(r)S(r)}-\frac{A^{\prime\prime}(r)}{2A(r)}+\frac{A^{\prime}(r)^{2}}{4A(r)^{2}}+\frac{S^{\prime}(r)}{rS(r)}+\frac{2\phi^{\prime}(r)^{2}S(r)}{(2+\ell)}-\frac{\Lambda
S(r)}{1+\ell}$ $\displaystyle=$ $\displaystyle 0,$ (41) $\displaystyle
1-\frac{1+\ell}{S(r)}+\frac{rS^{\prime}(r)}{2S(r)^{2}}+\frac{\ell
r^{2}A^{\prime\prime}(r)}{4A(r)S(r)}-\frac{\ell
r^{2}A^{\prime}(r)S^{\prime}(r)}{8A(r)S(r)^{2}}-\frac{\ell
r^{2}A^{\prime}(r)^{2}}{8A(r)^{2}S(r)}$
$\displaystyle-\frac{(1+\ell)rA^{\prime}(r)}{2A(r)S(r)}-\frac{(2+3\ell)\phi^{\prime}(r)^{2}r^{2}}{(2+\ell)}-\frac{\Lambda(2+\ell)r^{2}}{2(1+\ell)}$
$\displaystyle=$ $\displaystyle 0,$ (42)
$\displaystyle\frac{A^{\prime\prime}(r)}{2A(r)}-\frac{A^{\prime}(r)^{2}}{4A(r)^{2}}-\frac{A^{\prime}(r)S^{\prime}(r)}{4A(r)S(r)}-\frac{\phi^{\prime}(r)^{2}}{\left(1+\ell\right)A(r)}-\frac{S^{\prime}(r)}{rS(r)}+\frac{\Lambda
S(r)}{1+\ell}$ $\displaystyle=$ $\displaystyle 0,$ (43)
$\displaystyle\frac{d}{dr}[r^{2}\phi^{\prime}(r)]$ $\displaystyle=$
$\displaystyle 0.$ (44)

An analytical solution is be possible if and only if

$\Lambda=\frac{\kappa\lambda}{\xi}(1+\ell).$ (45)

Under the above condition, we can obtain

$\displaystyle A(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{2(1+\ell)Q_{0}^{2}}{(2+\ell)r^{2}}-\frac{1}{3}(1+\ell)r^{2}\Lambda_{e},$
(46) $\displaystyle S(r)$ $\displaystyle=$
$\displaystyle\frac{1+\ell}{1-\frac{2M}{r}+\frac{2(1+\ell)Q_{0}^{2}}{(2+\ell)r^{2}}-\frac{1}{3}(1+\ell)r^{2}\Lambda_{e}},$
(47) $\displaystyle\phi(r)$ $\displaystyle=$ $\displaystyle\frac{Q_{0}}{r}.$
(48)

For convenience, we set $\Lambda_{e}=\frac{\kappa\lambda}{\xi}$ as an
effective cosmological constant, which satisfies the relation
$R_{\mu\nu}=\Lambda_{e}g_{\mu\nu}$ at the boundary of the spacetime and is
independent of the Lorentz-violating parameter.

As $\ell$ approaches to 0, this solution becomes RN-(A)ds black hole solution.
Similarly, we compute the Kretschmann scalar in order to examine the
singularity study the singularity. The expression for $K$ is

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{8}{3}\Lambda_{e}^{2}+\frac{l}{(l+1)r^{2}}\Lambda_{e}-\frac{16Q_{e}^{2}\left((\ell+2)r(\ell
r+12M)-14(\ell+1)Q_{e}^{2}\right)}{(\ell+1)(\ell+2)^{2}r^{8}}$ (49)
$\displaystyle+$ $\displaystyle\frac{4\left(\ell^{2}r^{2}+4\ell
Mr+12M^{2}\right)}{(\ell+1)^{2}r^{6}},$

which shows that, the RN-(A)ds-like singularity is only present at $r=0$
(assuming $0<\left|\ell\right|\ll 1$).

##  IV  SlOWLY ROTATING CHARGED BLACK HOLE SOLUTIONS

In this section, we will give slowly rotaing charged solutions through solving
Einstein-bumblebee equations in both the cases with and without a cosmological
constant. The forms of these solutions are very similar to the Kerr-Newman
(KN) and Kerr-Newman (KN)-(A)ds solutions.

In 1963, Kerr discovered the rotating black hole solution Kerr (1963). A few
years later, Newman obtained rotating charged black hole solution Newman et
al. (1965). In bumblebee gravity, some rotating solutions were also provided
Ding et al. (2020); Ding and Chen (2021); Jha and Rahaman (2021). Here, we
consider slowly rotating charged black hole solution with and without a
cosmological constant. The general form of the corresponding metric is given
by

$ds^{2}=-A(r,\theta)dt^{2}+S(r,\theta)dr^{2}+2F(r)H(\theta)adtd\phi+\rho(r,\theta)^{2}d\theta^{2}+h(r,\theta)^{2}\sin^{2}\theta
d\phi^{2}.$ (50)

The bumblee field and the $U(1)$ vector field are assumed as

$\displaystyle b_{\mu}$ $\displaystyle=$
$\displaystyle(0,b_{r}(r,\theta),0,0),$ (51) $\displaystyle F_{\mu\nu}$
$\displaystyle=$ $\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$
$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle(A_{0}(r,\theta),0,0,A_{\phi}(r,\theta)),$ (52)

###  IV.1 Case A: $V(X)=\frac{\lambda}{2}X^{2}$ and $\Lambda=0$

It is noted that when the Lorentz-violating parameter is 0, this solution
should reduce to the Kerr-Newman solution. Similarly, when $a=0$, we should
obtain the spherically symmetric black hole solution derived in the previous
section. Therefore, we consider the product of $a$ and $\ell$ is a small
quantity.

Ding et al. obtained a slowly rotating black hole solution within bumblebee
gravity Ding et al. (2020); Ding and Chen (2021). Following their approach, we
obtained a slowly rotating charged black hole solution within bumblebee
gravity. The metric in Boyer-Lindquist coordinates reads:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\rho^{2}}\Big{(}\mathrm{d}t-a\sqrt{1+\ell}\sin^{2}\theta\mathrm{d}\phi\Big{)}^{2}+(1+\ell)\frac{\rho^{2}}{\Delta_{r}}\mathrm{d}r^{2}+{\rho^{2}}\mathrm{d}\theta^{2}$
(53)
$\displaystyle\quad+\frac{\sin^{2}\theta}{\rho^{2}}\Big{(}a\sqrt{1+\ell}\mathrm{d}t-(r^{2}+a^{2}(1+\ell))\mathrm{d}\phi\Big{)}^{2},$
$\displaystyle A_{0}(r,\theta)$ $\displaystyle=$
$\displaystyle-\frac{Q_{0}r}{\rho^{2}},$ (54) $\displaystyle
A_{\phi}(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{Q_{0}a\sqrt{1+\ell}r\sin^{2}\theta}{\rho^{2}},$ (55)
$\displaystyle b_{r}(r)$ $\displaystyle=$ $\displaystyle
b\rho\sqrt{\frac{\left(1+\ell\right)}{\Delta_{r}}},$ (56)

where

$\displaystyle\Delta_{r}$ $\displaystyle=$
$\displaystyle(1+\ell)a^{2}+r^{2}-2Mr+\frac{2(1+\ell)Q_{0}^{2}}{(2+\ell)},$
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sqrt{r^{2}+(1+\ell)a^{2}\text{cos}^{2}\theta}.$ (57)

When $\ell\rightarrow 0$, it recovers to the usual Kerr-Newman metric and when
$a\rightarrow 0$, it becomes to metric (13) with Eqs. (30) and (31). By
solving $\Delta_{r}=0$, we can obatin the radius of the black hole’s event
horizon

$r_{\pm}=M\pm\sqrt{M^{2}-\frac{2(\ell+1)Q_{0}^{2}+(\ell+2)a^{2}}{\ell+2}}.$
(58)

It becomes evident that the mass and spin parameters of the black hole must
satisfy.

$\frac{2(\ell+1)Q_{0}^{2}+(\ell+2)a^{2}}{\ell+2}\leq M^{2}.$ (59)

Next, we discuss the small perturbations introduced by this rotating solution
to the field equations, and impose certain constraints on the parameters
through these perturbations. We set

$\displaystyle\Delta_{\mu\nu}=$ $\displaystyle R_{\mu\nu}$
$\displaystyle-\kappa\left[V^{\prime}\left(2B_{\mu}B_{\nu}+b^{2}g_{\mu\nu}\right)+B_{\mu}^{\
\alpha}B_{\nu\alpha}+Vg_{\mu\nu}\frac{1}{4}B_{\alpha\beta}B^{\alpha\beta}g_{\mu\nu}\right]$
(60) $\displaystyle-$
$\displaystyle\kappa(T^{M}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T^{M})-\xi\left[\frac{1}{2}B^{\alpha}B^{\beta}R_{\alpha\beta}g_{\mu\nu}-B_{\mu}B^{\alpha}R_{\alpha\nu}-B_{\nu}B^{\alpha}R_{\alpha\mu}\right.$
$\displaystyle+$
$\displaystyle\frac{1}{2}\nabla_{\alpha}\nabla_{\mu}\left(B^{\alpha}B_{\nu}\right)+\frac{1}{2}\nabla_{\alpha}\nabla_{\nu}\left(B^{\alpha}B_{\mu}\right)\left.-\frac{1}{2}\nabla^{2}\left(B_{\mu}B_{\nu}\right)\right],$

naturally,for finite $Q_{0}$, when the field equations hold, we have
$\Delta_{\mu\nu}=0$. Substituting the metric into the right hand side of Eq.
(60), our calculations yield:

$\Delta_{\mu\nu}=\left\\{\begin{aligned}
\mathcal{O}(a^{2}\ell)&,&\quad\mu\nu=tt,rr,\phi\phi,\theta\theta,r\theta,t\phi\\\
0&,&\quad\text{otherwise},\end{aligned}\right.$ (61)

and the remaining components of $\bar{R}_{\mu\nu}=0$ are all zero. At the same
time, we derive the equations of motion for the bumblebee field and the
electromagnetic field. Only the $r$ component nonvanishing:

$\displaystyle\left(\nabla_{\mu}B^{\mu
r}-2(V^{\prime}B^{r}-\frac{\xi}{2\kappa}B_{\mu}R^{\mu r}-\gamma
B^{r}F^{\alpha\beta}F_{\alpha\beta})\right)b$ $\displaystyle=$
$\displaystyle\mathcal{O}(a^{2}\ell),$ $\displaystyle\nabla_{\mu}\left(F^{\mu
r}+\gamma B^{\alpha}B_{\alpha}F^{\mu r}\right)$ $\displaystyle=$
$\displaystyle\mathcal{O}(a^{2}\ell).$ (62)

Combining the above equations, we find that the rotating solution needs to
satisfy the condition $\mathcal{O}(a^{2}\ell)$ being a small quantity. This
implies that when considering this solution,$a$ and $\ell$ cannot both take
large values simultaneously. Thus, for our solution, we need the condition
that $a^{2}\ell$ is small quantities ($\frac{a^{2}\ell}{M^{2}}\ll 1$) for
finite $Q_{0}$ , respectively.

###  IV.2 Case B: $V(X)=\frac{\lambda}{2}X$ and $\Lambda\neq 0$

Next, we consider the case of $V(X)=\frac{\lambda}{2}X$ and $\Lambda\neq 0$.
Similar to the case A, we can get the following solution:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\rho^{2}}\Big{(}\mathrm{d}t-\frac{a\sqrt{1+\ell}\sin^{2}\theta}{\Xi}\mathrm{d}\phi\Big{)}^{2}+(1+\ell)\frac{\rho^{2}}{\Delta_{r}}\mathrm{d}r^{2}+\frac{\rho^{2}}{\Delta_{\theta}}\mathrm{d}\theta^{2}$
(63)
$\displaystyle\quad+\frac{\sin^{2}\theta\Delta_{\theta}}{\rho^{2}}\Big{(}a\sqrt{1+\ell}\mathrm{d}t-\frac{r^{2}+a^{2}(1+\ell)}{\Xi}\mathrm{d}\phi\Big{)}^{2},$
$\displaystyle A_{0}(r,\theta)$ $\displaystyle=$
$\displaystyle-\frac{Q_{0}r}{\rho^{2}},$ (64) $\displaystyle
A_{\phi}(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{Q_{0}ar\sin^{2}\theta}{\Xi\rho^{2}},$ (65) $\displaystyle
b_{r}(r)$ $\displaystyle=$ $\displaystyle
b\rho\sqrt{\frac{\left(1+\ell\right)}{\Delta_{r}}},$ (66)

where

$\displaystyle\Delta_{\theta}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{3}(1+\ell)^{2}\Lambda_{e}a^{2}\text{cos}^{2}\theta,$
$\displaystyle\Delta_{r}$ $\displaystyle=$
$\displaystyle\left((1+\ell)a^{2}+r^{2}\right)\left(1-\frac{1}{3}(1+\ell)\Lambda_{e}r^{2}\right)-2Mr+\frac{2(1+\ell)Q_{0}}{(2+\ell)},$
$\displaystyle\Xi$ $\displaystyle=$ $\displaystyle
1+\frac{1}{3}(1+\ell)^{2}\Lambda_{e}a^{2},$ $\displaystyle\rho$
$\displaystyle=$
$\displaystyle\sqrt{r^{2}+(1+\ell)a^{2}\text{cos}^{2}\theta}.$ (67)

From the form of the solution, it is a KN-(A)dS-like solution. Note that when
the charge parameter $Q_{\text{0}}=0$, the solution is reduced to a slowly
rotating Kerr-(A)dS-like black hole solution, which is not found in previous
work. Otherwise, when $\ell=0$ and $a=0$, we get the usual KN-(A)dS black hole
solution and the metric (13) with (46) and (47), respectively.

By calculating the errors in the field equations, we find that the results are
the same as case A. Similarly, the rotating solution must satisfy the
condition that $a^{2}\ell$ is a small quantity.

##  V  SHADOW OBSERVABLES

The black hole shadow is formed by the strong gravitational field of the black
hole. The shadow appears as a two-dimensional dark region to a distant
observer. Shadows serve as one of the most intuitive pieces of evidence for
the presence of black holes. Particularly, the unveiling of an image provides
even stronger evidence for black hole’s existence Akiyama et al. (2019a, b, c,
d); Akiyama et al. (2022). The shadow of the Kerr-like black hole in bumblebee
gravity has been extensively studied in previous works Wang and Wei (2022);
Ding et al. (2020); Kuang and Övgün (2022); Chen et al. (2020). In this
section, we aim to investigate the shadow of the KR-AdS like black holes in
the framework of Einstein-bumblebee gravity.

###  V.1 The photon orbit

In this section, we will derive the equations of motion of photons orbiting
the black holes. From Eqs. (61) and (62), it is apparent that the field
equations hold when the rotating black hole solution satisfies that
$a\ell^{2}$ is a small quantity. Therefore, in the subsequent part, all the
analysies are under this condition. The motion of a photon in the background
of the black hole is given by the geodesics. However, the process of solving
geodesic equations is very difficult. we choose to use the Hamilton-Jacobi
approach to solve it. The Lagrangian for the photon is

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$ (68)

where $\dot{x}$ is defined as $\dot{x}^{\mu}=dx^{\mu}/d\lambda=u^{\mu}$, with
the four-velocity of the photon $u^{\mu}$ and the affine parameter $\lambda$.
To facilitate the study of photon orbits and the black hole shadow observed by
a distant observer, we introduce two parameters:

$\xi=\frac{L_{z}}{E}\,,\quad\eta=\frac{K}{E^{2}}\,,$ (69)

where $E$, $L_{z}$, and $K$ represent the energy, the axial component of the
angular momentum, and the Carter constant Carter (1968), respectively. The
expressions of the energy and the $z$-component of the angular momentum is
given by

$E=-\frac{\partial{\mathcal{L}}}{\partial\dot{t}}=-g_{\varphi
t}\dot{\varphi}-g_{tt}\dot{t},$ (70)

$L_{z}=\frac{\partial{\mathcal{L}}}{\partial\dot{\varphi}}=g_{\varphi\varphi}\dot{\varphi}+g_{\varphi
t}\dot{t}.$ (71)

Using the Hamilton-Jacobi equation, we can get the null geodesic equation of
the photon around the slowly rotating black hole:

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{E}{\rho^{2}}\Bigg{[}\frac{\left(r^{2}+(1+\ell)a^{2}\right)\left[\left(r^{2}+(1+\ell)a^{2}\right)-\sqrt{1+\ell}a\xi\Xi\right]}{\Delta_{r}}$
$\displaystyle+\frac{\sqrt{1+\ell}a\left(\xi\Xi-\sqrt{1+\ell}a\sin^{2}\theta\right)}{\Delta_{\theta}}\Bigg{]},$
$\displaystyle\dot{\varphi}$ $\displaystyle=$
$\displaystyle\frac{E\Xi}{\rho^{2}}\left[\frac{\sqrt{1+\ell}a(\left(r^{2}+(1+\ell)a^{2}\right)-\sqrt{1+\ell}a\xi\Xi)}{\Delta_{r}}+\frac{\xi\Xi-\sqrt{1+\ell}a\sin^{2}\theta}{\sin^{2}\theta\Delta_{\theta}}\right],$
(73) $\displaystyle(1+\ell)(\rho^{2}\dot{r})^{2}$ $\displaystyle=$
$\displaystyle
E^{2}\left[\left[\left(r^{2}+(1+\ell)a^{2}\right)-\sqrt{1+\ell}a\xi\Xi\right]^{2}-\Delta_{r}\Big{[}\frac{\left(\sqrt{1+\ell}a-\xi\Xi\right)^{2}}{\Delta_{\theta}}+\eta\Big{]}\right]$
(74) $\displaystyle\equiv$ $\displaystyle(1+\ell)\mathcal{R}(r),$
$\displaystyle\rho^{2}\dot{\theta}$ $\displaystyle=$ $\displaystyle
E^{2}\left[\eta\Delta_{\theta}-\cos^{2}\theta\left(\frac{\xi^{2}\Xi^{2}}{\sin^{2}\theta}-(1+\ell)a^{2}\right)\right]\equiv\Theta(\theta).$
(75)

Figure 1:  The effective potential $V_{\text{eff}}$ for the photon. The
paramengts are set as $M=1,a=0.1$ , $Q_{0}=0.1$ , $l=0,0.1$ and $\xi_{c}$ is
an the value of $\xi$ for unstable circular null orbit.

Photons with sufficiently large orbital angular momentum can escape from the
black hole, while those with small angular momentum will fall into the black
hole. Now we focus on the critical case, the circular null orbit, where the
photons will neither escape from nor fall into the black hole. The radial
motion can also be expressed as

$(\rho^{2}\dot{r})^{2}+V_{eff}=0,$ (76)

where the effective potential $V_{\text{eff}}$ reads

$V_{eff}=-\frac{E^{2}}{(1+\ell)}\left[\left[\left(r^{2}+(1+\ell)a^{2}\right)-\sqrt{1+\ell}a\xi\Xi\right]^{2}-\Delta_{r}\big{[}\frac{\left(\sqrt{1+\ell}a-\xi\Xi\right)^{2}}{\Delta_{\theta}}+\eta\big{]}\right].$
(77)

In Fig. 1, we depict the effective potential $V_{\text{eff}}$ for a rotating
charged black hole with $\ell=0$ and $\ell=0.1$. From Fig. 1, we can observe
that Lorentz-violating parameter greater than 0 can increase the effective
potential Wei et al. (2015). For the unstable circular null orbit, the
effective potential satisfies

${V}(r)=\frac{d{V}(r)}{dr}=0,\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \frac{d^{2}V(r)}{dr^{2}}<0,$ (78)

Then we can obtain

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{r^{2}\left(16(1+\ell)a^{2}\Delta_{r}\Delta_{\theta}-\left(r\Delta_{r}^{\prime}-4\Delta_{r}\right)^{2}\right)}{(1+\ell)a^{2}\Delta_{\theta}\Delta_{r}^{\prime
2}},$ (79) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{\left(r^{2}+(1+\ell)a^{2}\right)\Delta_{r}^{\prime}-4r\Delta_{r}}{\sqrt{1+\ell}a\Xi\Delta_{r}^{\prime}}.$
(80)

(a) $M=1,a=0.1,\ell=0.01,Q_{0}=0.2,\theta=\frac{\pi}{2}.$ (b)
$M=1,a=0.9,\ell=0.01,\Lambda_{e}=-0.1,Q_{0}=0.3.$

(c) $M=1,a=0.1,\Lambda_{e}=-0.1,Q_{0}=0.2.$ (d)
$M=1,a=0.647,\ell=0.01,\Lambda_{e}=-0.2,\theta=\frac{\pi}{2}.$

Figure 2: The shadows for the KN-(A)dS-like black hole with different
parameters.

###  V.2 Observables

We now aim to determine the apparent shape of the slowly rotating charged
black hole’s shadow. The apparent shape of the black hole shadow for an
observer far away from the black hole can be described using the celestial
coordinates $x$ and $y$:

$\displaystyle x$ $\displaystyle=$
$\displaystyle\lim_{r\to\infty}\frac{rp^{(\theta)}}{p^{(t)}}=\sqrt{\eta+a^{2}\cos^{2}\theta-\xi^{2}\cot^{2}\theta},$
(81) $\displaystyle y$ $\displaystyle=$
$\displaystyle\lim_{r\to\infty}\frac{-rp^{(\varphi)}}{p^{(t)}}=-\xi\csc\theta.$
(82)

The coordinate $x$ denotes the apparent perpendicular distance from the
shape’s projection on the equatorial plane, while $y$ represents the apparent
perpendicular distance of the shape from the axis of symmetry. The boundary
curve of the shadow is determined by the unstable circular null orbit.
Substituting the metric into Eqs. (79)and (80), we have

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{4r^{2}\left(\left(a^{2}(\ell+1)+r^{2}\right)\left(1-\frac{1}{3}(\ell+1)r^{2}\Lambda_{e}\right)-2Mr+\frac{(1+\ell)Q_{0}^{2}}{(1+\frac{\ell}{2})}\right)}{\left(\frac{1}{3}(\ell+1)r\Lambda_{e}\left(a^{2}(\ell+1)+2r^{2}\right)+M-r\right)^{2}}$
(83)
$\displaystyle-\frac{r^{2}\left(-\frac{1}{3}a^{2}(\ell+1)r^{2}\Lambda_{e}+2a^{2}-\frac{3Mr}{1+\ell}+\frac{4Q_{0}^{2}}{(2+\ell)}+r^{2}\right)^{2}}{a^{2}\left(\frac{1}{3}a^{2}(\ell+1)^{2}\Lambda_{e}\cos^{2}(\theta)+1\right)\left(\frac{1}{3}(\ell+1)r\Lambda_{e}\left(a^{2}(\ell+1)+2r^{2}\right)+M-r\right){}^{2}},$
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{3}a^{2}(\ell+1)^{2}r\Lambda_{e}\left(a^{2}(\ell+1)+r^{2}\right)+a^{2}(\ell+1)(M+r)+r\left(r(r-3M)+\frac{4Q_{0}^{2}}{(2+\ell)}\right)}{a\sqrt{\ell+1}\left(\frac{1}{3}a^{2}(\ell+1)^{2}\Lambda_{e}+1\right)\left(\frac{1}{3}(\ell+1)r\Lambda_{e}\left(a^{2}(\ell+1)+2r^{2}\right)+M-r\right)}.$
(84)

(a) $\theta=\frac{\pi}{2}$ (b) $\theta=\frac{\pi}{4}$

(c) $\theta=\frac{\pi}{6}$

Figure 3:  The radius $R_{s}$ of the shadow against the Lorentz-violating
parameter $\ell$.

(a) $\theta=\frac{\pi}{2}$ (b) $\theta=\frac{\pi}{4}$

(c) $\theta=\frac{\pi}{6}$

Figure 4: The distortion parameter $\delta_{s}$ against the Lorentz-violating
parameter $\ell$.

It is easy to see that both the coordinates $x$ and $y$ are the functions of
$(M,Q_{eff},a,\Lambda_{e},\ell,\theta)$. We set $M=1$, and show the shapes of
the shadow in Fig. 2. Similar to the Kerr-Newman-AdS black hole, the shadow
radius increases with the negative effective cosmological costant
$\Lambda_{e}$, as shown in Fig. 2(a). From Fig. 2(b) we can see that, the
position of the shadow depends on the observation angle $\theta$. The Lorentz-
violating parameter $\ell$ also affects the shaow size which can be seen from
Fig. 2(c). Besides, the deformation of the shadow becomes more pronounced when
the charge parameter $Q_{0}$ increases, which can be seen from Fig. 2(d).

(a) $\theta=\frac{\pi}{2}$ (b) $\theta=\frac{\pi}{4}$

(c) $\theta=\frac{\pi}{6}$

Figure 5: The radius $R_{s}$ of the shadow against the charge parameter
$Q_{0}$.

(a) $\theta=\frac{\pi}{2}$ (b) $\theta=\frac{\pi}{4}$

(c) $\theta=\frac{\pi}{6}$

Figure 6:  The distortion parameter $\delta_{s}$ against the charge parameter
$Q_{0}$.

To effectively analyze astronomical observational data, it is essential to
establish easily measurable and reliable astronomical observables. In this
study, we will examine the following observables, the radius $R_{s}$ and the
distortion parameter $\delta_{s}$ which are defined by Hioki and Maeda Hioki
and Maeda (2009). The parameter $R_{s}$ represents the radius of the reference
circle, which is expected to pass through the top point $(x_{t},y_{t})$, the
bottom point $(x_{b},y_{b})$, and the right point $(x_{r},0)$ of the shadow.
Meanwhile, the distortion parameter quantifies the distortion of the black
hole shadow in comparison with the reference circle, and they can be expressed
as

$R_{s}=\frac{(x_{t}-x_{r})^{2}+y_{t}^{2}}{2(x_{r}-x_{t})},\quad\delta_{s}=\frac{x_{l}-\tilde{x}_{l}}{R_{s}},$
(85)

where $(x_{l},0)$ and $(\tilde{x}_{l},0)$ represent the left points of the
shadow and the reference circle, respectively.

For a given spin parameter, we illustrate the radius $R_{s}$ and the
distortion parameter $\delta_{s}$ as functions of the Lorentz-violating
parameter $\ell$ for
$\theta=\frac{\pi}{2},\theta=\frac{\pi}{4},\theta=\frac{\pi}{6}$. The behavior
of $R_{s}$ and $\delta_{s}$ is exhibited in Fig. 4 and Fig. 4. For fixed $a$
and $\theta$, the radius $R_{s}$ decreases with $\ell$, but $\delta_{s}$
increases with $\ell$ increases. Particularly noteworthy is the observation
that for observers situated far from the equatorial plane, the shadow of a
rapidly rotating black hole appears smaller compared to that of a slower one
with the same $\ell$. However, this discrepancy diminishes as the observation
angle increases, eventually resulting in a convergence of all curves as
$\theta$ approaches $\frac{\pi}{2}$. Furthermore, as the observer
progressively approaches the equatorial plane, the distortion of the shadow
becomes increasingly pronounced.

Similarly, we investigate the relationship between the value of the charge
parameter and $R_{s}$ and $\delta_{s}$ which is shown in Fig. 6 and Fig. 6. We
find that $R_{s}$ decreases with the charge parameter, whereas $\delta_{s}$
increases with the charge parameter. Another notable observation in our study
is that for small values of the spin parameter $a$, the deformation of the
shadow is nearly imperceptible.

##  VI  CONCLUSIONS

In this paper, by considering the coupling between the electromagnetic field
and the bumblebee vector field, we obtained the RN-like black hole solutions
and the RN-(A)dS-like black hole solutions in bumblebee gravity. When the
Lorentz-violating parameter vanishes $(\ell=0)$, these solutions reduce to the
RN and RN-(A)dS black hole solutions. In the stationary and axially symmetric
scenario, we obtained solutions for slowly rotating charged black holes, i.e.,
the KN-(A)dS-like black hole, within the framework of bumblebee gravity. We
further analyzed the series expansion of small quantities.

Because the nonminimal interaction term between the electromagnetic field and
bumblebee field was introduced, the Maxwell’s equations was also modified. The
charge term of the black hole solution is related to the Lorentz-violating
parameter. When the electromagnetic field does not exist, our solutions are
then reduced to another bumblebee black hole solutions in previous works.

Moreover, in the background of the KN-(A)dS-like black hole, we derived null
geodesics using the Hamilton-Jacobi equation. Utilizing the conditions for
unstable circular null orbits, we derived two conserved parameters, $\xi$ and
$\eta$, which govern the photon’s motion. Using these parameters, we
determined the celestial coordinates of the shadow. Subsequently, we
investigated the size and deformation of the shadow. By varying the
observation angle $\theta=\frac{\pi}{2},$ $\theta=\frac{\pi}{4},$
$\theta=\frac{\pi}{6}$, we calculated the shadow’s radius $R_{s}$ for
different parameters. We observed that for fixed spin $a$ and Lorentz-
violating parameter $\ell$, observers situated at lower latitudes consistently
observe a larger shadow. Meanwhile, we found that the radius of the reference
circle decreases with the Lorentz-violating parameter and electric charge ,
while the distortion parameter increases with the Lorentz-violating parameter
and electric charge.

##  Acknowledgements

We are grateful to Tao-tao Sui and Ke Yang for useful discussions. This work
was supported by the National Key Research and Development Program of China
(Grant No. 2021YFC2203003), the National Natural Science Foundation of China
(Grants No. 12205129, No. 12147166, and No. 12247101), the China Postdoctoral
Science Foundation (Grant No. 2021M701529), the 111 Project (Grant No.
B20063), the Major Science and Technology Projects of Gansu Province, and
Lanzhou City’s scientific research funding subsidy to Lanzhou University.

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