# Tweet Moodifier: Towards giving emotional awareness to Twitter users

Belén Saldías F  MIT Media Lab
Massachusetts Institute of Technology
Cambridge, USA
<EMAIL_ADDRESS>Rosalind W. Picard  MIT Media Lab
Massachusetts Institute of Technology
Cambridge, USA
<EMAIL_ADDRESS>
###### Abstract

Emotional contagion in online social networks has been of great interest over
the past years. Previous studies have focused mainly on finding evidence of
affect contagion in homophilic atmospheres. However, these studies have
overlooked users’ awareness of the sentiments they share and consume online.
In this paper, we present an experiment with Twitter users that aims to help
them better understand which emotions they experience on this social network.
We introduce Tweet Moodifier (T-Moodifier), a Google Chrome extension that
enables Twitter users to filter and make explicit (through colored visual
marks) the emotional content in their News Feed. We compare behavioral changes
between 55 participants and 5089 of their public “friends.” The comparison
period spans from two weeks before installing T-Moodifier to one week
thereafter. The results suggest that the use of T-Moodifier might help Twitter
users increase their emotional awareness: T-Moodifier users who had access to
emotional statistics about their posts produced a significantly higher
percentage of neutral content. This behavioral change suggests that people
could behave differently while using real-time mechanisms that increase their
affect reflection. Also, post-experience, those who completed both pre- and
post-surveys could assert more confidently the main emotions they shared and
perceived on Twitter. This shows T-Moodifier’s potential to effectively make
users reflect on their News Feed.

###### Index Terms:

emotional awareness, social media, Twitter

##  I Introduction

Social media users are constantly creating content that connects them to
others, but are users aware of the emotional influence that social media has
on their moods or lives? While consuming and sharing information online is
advantageous to connecting people, it may pose a risk to ignore the emotions
that we (consciously or unconsciously) spread out to hundreds or thousands of
people with a single post.

Over the past few years, emotional contagion through social media has been of
great interest. Kramer et al. [17] argue, “Emotional states can be transferred
to others via emotional contagion, leading people to experience the same
emotions without their awareness. Emotional contagion is well established in
laboratory experiments, with people transferring positive and negative
emotions to others.” In addition, several studies, e.g., [17, 8, 6, 9], have
found evidence of affect contagion in homophilic atmospheres, where people
tend to connect with others who are socially similar to themselves.
Furthermore, Tromholt [28] and Hunt et al. [16] show that the content that we
consume on platforms like Twitter or Facebook affects not only the emotions
that we express on these platforms but also our general wellbeing, for
example, through consuming toxic content or experiencing cyberbullying attacks
[30, 24, 23].

Figure 1: T-Moodifier options. Framed statistics are only present in T1.

However, all of these insightful studies fail to address social media users’
emotional awareness, i.e., users’ ability to identify and differentiate
emotions [4, 15], of the content that they consume and share.  On the one
hand, we can leverage emotional awareness to help users improve the regulation
of their emotions on social media [7, 14] by assisting them with emotional
differentiation [14, 3]. On the other hand, it has been shown that users tend
to change their behavior on social media: They aim to deliver a positive
impression of themselves that they want others to perceive [1, 31, 29], self-
censor their posts if they cannot filter their audience [26], and try not to
involve themselves in conflicts that may give others an overall bad impression
[22]. In this study, we couple these users’ intentions with a new tool that
aims to help increase emotional awareness. We aim to understand to what extent
Twitter users are aware of or reflect on the impact caused by the emotional
content that they consume and create. In addition, we examine how users’
sharing patterns change after they use this new tool.

We introduce, build, deploy, and evaluate Tweet Moodifier (T-Moodifier), a
Google Chrome extension that enables Twitter users to explore the emotional
sentiment of posts in their News Feed (see figure 1). The extension is powered
by a machine learning algorithm that classifies Tweets into three different
sentiment categories: positive posts, which tend to use happy or surprising
language; negative posts, which tend to use sad, angry, or disgusting
language; and posts without strong emotional language, which are classified as
neutral. Using T-Moodifier, participants have access to three new views (which
can be regarded as different perspectives that can help users with emotional
differentiation [3]) of their Twitter feed: (1) highlight with colors the
emotional valence of each post in their News Feed, (2) filter out all negative
and neutral content to keep only positive posts, and (3) filter out all
positive and neutral content to keep only negative posts. Users also have the
opportunity to click an emoji (positive, neutral, or negative) below each
Tweet if they wish to indicate a different label for the automatically-labeled
valence. Our hypotheses are that by making users aware of the emotions that
surround them online, in this case through Tweets, users could:

  1. 1.

Change their emotional sharing patterns.

  2. 2.

Reflect more on, and consequently be more aware of, what they post before
doing so.

  3. 3.

Feel better or become mentally healthier.

From the studies mentioned above [6, 9], one may conclude that showing
positive content to people would make them happier. However, some researchers
in the mental health community [16, 21] propose that consuming only positive
emotions can lead people to experience a negative feeling of isolation. Since
it is not our primary goal to measure emotional contagion, T-Moodifier does
not suggest that participants should spend more time in one T-Moodifier view
than in another. Nevertheless, to prevent users from forgetting that they have
the negative emotions view activated, if a user spends more than 15 minutes
there, T-Moodifier shows a pop-up that blinks until the user restores his or
her original feed (this pop-up was tested in a 10-person pilot without
complaints, so we decided to keep it). To the best of our knowledge,
T-Moodifier is one of the first attempts to try to enhance and understand
social media users’ awareness of how Twitter affects them.

Results presented in this paper are based on 55 users who used T-Moodifier for
at least 7 days each, for a total of 21 days studied per user (from two weeks
before installing T-Moodifier to one week thereafter). Our findings suggest
that the use of T-Moodifier increases users’ emotional awareness. In addition,
participants tend to neutralize and reflect more on their content when they
have access to emotional statistics. This behavioral change suggests potential
benefits of creating real-time mechanisms that increase social media users’
awareness. Finally, participants who completed both pre- and post-experience
surveys asserted more accurately and confidently the primary emotions they
shared and perceived on Twitter.

The rest of this paper is organized as follows: In section II, we describe the
study protocol. In section III, we explain the classifier in charge of giving
emotional labels to users’ Tweets. In section IV, we disclose our privacy
policy. Finally, in sections V, VI, and VII we present our results,
conclusions, and discussion plus future work, respectively.

##  II Study protocol

First of all, we want to highlight that participants have full control over
the T-Moodifier experience: a) They choose to use the plugin (or not), b) They
have can stop using it at any time, and c) They are informed (and asked to
explicitly accept the Privacy Policy; more details in section IV) about which
kinds of data will be collected on their usage and sharing patterns.
Furthermore, we only use publicly available Twitter data.

Twitter users are invited to use T-Moodifier through its
website111https://tweetmoodifier.media.mit.edu, which, for this study, was
sporadically promoted by the MIT Media Lab’s social media accounts for two
weeks222 $\sim$ 450K Twitter followers.. Upon installing the extension
(participants are able to uninstall it at any time), they see a description of
the tool’s purpose and are prompted to answer a pre-experience survey and
continue only if they accept our privacy policy and usage conditions.

Next, participants are randomly assigned to one of two different treatment
groups, who are also compared to a control group based on public data.
Participants are shown a set of options (see figure 1) below their profile
picture in the Twitter homepage, that allows them to:

  * •

Treatment 1 (T1): see valences of Tweets in their News Feed, filter the
emotions of the Tweets that they consume, and see personal statistics
revealing the valence of what they tweeted.

  * •

Treatment 2 (T2): same as T1 but NOT seeing any statistics about Tweets they
generated.

  * •

Control: A random sample of “friends,” defined by Twitter as people our
participants follow, who do not install the plugin. This group could include
influencer/popular accounts whenever participants followed them.

We define the control group assuming that it might reflect a homophilic
atmosphere for participants. Then, if participants have been in synchrony with
their homophilic group until T-Moodifier is introduced, and only participants
change their behavior during treatment, we can say that T-Moodifier can
account for these behavioral changes and not their homophilic atmosphere.

When someone activates any of the T-Moodifier views, sentiment classification
is obtained using the Tweets emotion classifier described in section III. In
addition, T-Moodifier activates a visual mark as a reminder of the view they
have enabled; T-Moodifier displays green borders in positive Tweets, red
borders in negative Tweets, while neutral Tweets remain without a colored
mark. Finally, after a seven-day experience, participants are taken to a post-
survey. They are reminded of access to mental health resources (also available
in our Privacy Policy IV) each time they activate a T-Moodifier view (see
Appendix for visual details).

While participants were told that T-Moodifier aimed to help them better
understand which emotions they consume online, we also analyzed how they
behaved while using T-Moodifier. We hypothesized that, since participants in
T1 had access to personal statistics about their publicly observable behavior
on Twitter, they would be most likely to show a change in behavior regarding
the emotional distribution of the Tweets they create. This study was developed
under the Massachusetts Institute of Technology IRB Protocol #1810563376.

##  III Emotions classifier

Aiming to deliver an efficient application, based on predictions of an affect
sentiment classifier that is well known, relatively easy to interpret compared
to most recent techniques, and trained for Tweets (see Giachanou and
Crestani’s work for a survey [10]), we used the model by Go et al. [11]. Their
model uses Twitter APIs to generate its training and validation datasets. The
model analyzes Tweets’ text, but first, authors strip out all emojis in the
Tweets, to later try to predict these emojis as emotions (positive, neutral,
or negative) labels as a way of distant supervision. There are several emojis
that can be classified as positive (e.g. :), :-), and :D), and negative (e.g.
:(, :’(, and :@). The full list of emojis can be found in Go. et al.’s paper
[11]. Thus, their model uses self-labeled “non-verbal” emojis as its ground
truth for sentiment labeling. We used this pre-trained classifier through its
available API at http://help.sentiment140.com/api.

As with most machine learning systems, this classifier can make mistakes. Its
authors reported accuracy score levels above 80%. Unintended “accidents” can
occur and be potentially harmful for people [2]. Hence, we address this issue
by allowing participants to rectify the emotional classification of what they
observe (Tweets are relabeled only for them and not for all T-Moodifier
users). As we collect users’ re-labeled data, we could retrain the base model
to provide more accurate or personalized emotion labels. However, the results
reported here are under the assumption that by rectifying mislabeled Tweets,
users can reflect on more accurate valences of their Tweets; hence, we do not
retrain the base model.

##  IV Privacy Policy

We acknowledge that this line of research requires critical thinking about how
to exploit and interpret users’ data [20]. For Twitter experiments, some users
(private users) may have decided to make their Tweets only visible to their
Twitter followers. Hence, even though Twitter’s Privacy Policy would allow us,
T-Moodifier does not read or evaluate emotions of protected Tweets when
analyzing participants’ News Feeds. We make our privacy policy fully available
to the community and users at https://tweetmoodifier.media.mit.edu/privacy-
policy.

##  V Results

We present results for the hypotheses stated in Section I. This analysis is
based on 55 users; 20 female and 33 male participants reported their gender;
T1: 24, T2: 28, and protected: 3, who are only included in the survey
analysis. They all used T-Moodifier for at least 7 days days, yielding 385 of
T-Moodifier use. For the control group, we sample up to 100 friends of each
participant, reaching 5089 unique users. Data are collected using Twitter’s
Python API (TweePy333http://tweepy.org). A pre-experience survey also captured
the following information:

  * •

Eighty-two percent of users were aged 18 to 34 years.

  * •

Seventy-five percent of participants reported that they used Twitter at least
once a day, whereas 62% of all participants stated that they used it several
times a day.

  * •

What option best describes what you use Twitter for?: The most popular answer
(with a frequency of 51%) among 10 options was that participants used Twitter
to keep up with or share the news in general.

We use the t-test to compare between the treatments and control group and the
paired t-test to compare behavioral changes and pre- versus post-survey
responses. We use $p=0.05$ as the cutoff for significance.

###  V-A Sharing patterns

We analyzed the proportion of positive, negative, and neutral Tweets posted by
each member of T1 and T2; this analysis also includes what users wrote in
Tweets that retweeted others. We compared how the average distribution, for
public users, changed from before introducing T-Moodifier to one week
thereafter. Table I shows that the control group distribution of the emotions
they share remains steady from two weeks prior to T-Moodifier through the week
of treatment, with no statistically significant changes.

The number of Tweets pre/post experience did not change significantly, with an
average of 11 and 15 Tweets per user for treatments and the control group,
respectively. We define,

  * •

$W_{-2}$: the second week before treatments.

  * •

$W_{-1}$: the week before treatments.

  * •

$W_{0}$: the week of treatments.

  * •

T: All T-Moodifier participants.

TABLE I: Emotional content of control group’s Tweets in %. | $W_{-2}$ | $W_{-1}$ | $W_{0}$
---|---|---|---
Positive |  2 9.3 |  3 0.3 |  2 8.2
Neutral |  6 4.9 |  6 4.1 |  6 5.9
Negative |  5 .8 |  5 .7 |  5 .9
TABLE II: Emotional content of T-Moodifier users’ Tweets in %. |  $W_{-2}$, T |  $W_{-1}$, T |  $W_{-1}$, T1 |  $W_{0}$, T1
---|---|---|---|---
Positive |  2 6.8 |  2 8.7 |  3 4.7 |  1 6.9
Neutral |  6 8.2 |  6 6.3 |  6 2.8 |  7 9.7
Negative |  5 .0 |  5 .0 |  2 .5 |  3 .4
|  $W_{-1}$, T2 |  $W_{0}$, T2
---|---|---
Positive |  2 7.8 |  3 1.9
Neutral |  6 9.4 |  6 4.3
Negative |  2 .8 |  3 .8

As seen in Table II, the hypothesis that participants in T1 would change their
fraction of positive, neutral, and negative Tweets while using T-Moodifier
($W_{0}$, T1) was confirmed. In particular, they produced mainly neutral
content. This Table also illustrates that prior to introducing T-Moodifier
($W_{-1}$ and $W_{-2}$), participants (T) behaved similarly to the control
group.

On the one hand, from Tables II and III we observe significant differences for
T-Moodifier users under Treatment 1 (T1), and since there is no evidence to
claim that T1 participants behave differently from their friends prior to
using T-Moodifier, we can use their friends as a control group. Furthermore,
we see that while T1 participants use T-Moodifier they behave significantly
different from themselves prior to T-Moodifier. They also behave significantly
different from both the control and T2 groups during the treatment week. On
the other hand, we found that participants in T2 did not significantly change
their behavior as hypothesized. The only difference between treatments is that
participants in T1 had access to personal statistics about their posts’
valences.

TABLE III: P-values for differences between groups in tables I and II |  | Positive | Neutral
---|---|---|---
$W_{-1}$ |  $W_{-1}$, T1 | $0.4122$ | $0.8078$
$W_{0}$ |  $W_{0}$, T1 | 0.0370 | 0.0115
$W_{-1}$, T1 |  $W_{0}$, T1 | 0.0305 | 0.0351
$W_{0}$, T2 |  $W_{0}$, T1 | 0.0459 | 0.0482

Hence, the results suggest that T-Moodifier, especially with personal feedback
about the valence of what a user is posting, can potentially affect the
affective valence of what that user tends to share on social media. These
results resonate with Grosser’s [12] argument about Facebook users being
driven by social metrics.

Note that shifting toward a highly neutral distribution is not necessarily an
expected change. Social media are powerful platforms for discussing social
injustice issues that may make people experience negative emotions.
Introducing T-Moodifier is not an attempt to calm down those essential human
needs. Feeling negative or sharing negative information is sometimes very
healthy, as is feeling positive and sharing positive information.

Table III shows the results of testing for (and confirming) the statistical
significance of the hypothesized difference between $W_{0}$ and ($W_{0}$, T1).
We also see confirmed a significant difference between ($W_{-1}$, T1) and
($W_{0}$, T1) as well as between ($W_{0}$, T2) and ($W_{0}$, T1). We present a
summary of the standard deviations for these interesting sharing patterns
changes in Table IV.

TABLE IV: Standard deviations in % of results in Tables I and II | $W_{-1}$ |  $W_{-1}$, T1 | $W_{0}$ |  $W_{0}$, T1 |  $W_{0}$, T2
---|---|---|---|---|---
Positive | 26.2 | 30.7 | 26.5 | 19.4 | 31.5
Neutral | 26.1 | 30.8 | 26.7 | 21.3 | 31.1
Negative | 10.3 | 6.6 | 11.4 | 5.7 | 1.1

Finally, we looked at the participants’ engagement with T-Moodifier features.
Participants used T-Moodifier views 25.4% of the times they used Twitter (77%
of the time they preferred to highlight all three valences; 12% only positive
valences, and 11% only negative valences). While using the T-Moodifier views,
participants displayed 37.5 Tweets on a daily average (std: 57.5). Regarding
mislabeling, 35% of users relabeled 3% of the Tweets displayed. We did not
find tendencies toward specific mislabeled emotions.

###  V-B Self-reported awareness

How was awareness of mood assessed and was use of T-Moodifier associated with
any changes in self-reported awareness of how Twitter impacts mood? We
presented the participants with a pre- and post-questionnaire, each time
asking them the following questions, whose responses are discrete numbers from
1 to 100 according to the stated limits (e.g. “never - always,” “insignificant
- enormous”).

  1. (1)

Twitter influences my mood (never - always).

  2. (2)

My connections on Twitter influence the emotions that I experience (never -
always).

  3. (3)

Twitter has (insignificant - enormous) influence on the mood of others.

  4. (4)

I influence the emotions that others experience on Twitter (never - always).

  5. (5)

I am aware of (or reflect on) the emotions expressed in my Tweets before I
post them (never - always).

  6. (6)

My confidence in the response to the question: “Most of my Tweets are
emotionally: negative or neutral or positive” is (extremely weak - extremely
strong).

  7. (7)

My confidence in the response to the question: “Most of my friends’ Tweets are
emotionally: negative or neutral or positive” is (extremely weak - extremely
strong).

TABLE V: Pre-survey — Twitter emotional influence questions. | (1) | (2) | (3) | (4) | (5) | (6) | (7)
---|---|---|---|---|---|---|---
Pre, T |  |  |  |  |  |  |
Mean | 44.5 | 46.3 | 52.9 | 21.6 | 61.2 | 58.7 | 50.2
Std | 28.1 | 24.1 | 28.5 | 22.1 | 30.9 | 31.6 | 24.3
Note that row (Pre, T) are results for all 55 participant. |

####  Pre-survey results, T

In Table V, from Questions (1), (2), and (3) we see that T-Moodifier users
recognize that Twitter can influence emotions to some extent but only
occasionally. However, (see Question (4)) they strongly believe that their
Tweets or interactions on Twitter almost never influence the emotions of
others.

From Question (5), we see that the participants declare that they are somewhat
aware of the emotions of what they share. However, they declare in Question
(6) and (7) that they are not so sure about the emotional content they spread
through, or consume from, social media. These last three questions show that
users may recognize that what they consume or create has emotional content,
and they may reflect on it somewhat, but they are not very confident as to
whether the overall tone of what they are sending and receiving is more
negative or more positive.

We also asked them to point to the emoji that best describes how scrolling
through Twitter makes them feel; the results are shown in Figure 2. The large
bias toward neutral might be associated with the fact that 51% of the users
claimed that they use Twitter mostly to keep up with or share the news in
general (which could be thought as informative content) or it could also be
associated with recognizing that they have a lot of both positive and negative
emotions to things they see on Twitter, but they were forced to select only
one emoji.

Figure 2: Pre-questionnaire responses to: “The emoji that best describes how
scrolling through Twitter makes you feel is …”’.

#### Post-survey results, *

In this section, we present results regarding only those participants who
completed both pre-questionnaire (Pre, *) and post-questionnaire (Post, *)
responses. They constituted 28 people (T1: 16, T2: 12).

TABLE VI: Pre/Post-survey, * — Twitter emotional influence questions. | (1) | (2) | (3) | (4) | (5) | (6) | (7)
---|---|---|---|---|---|---|---
Pre, * |  |  |  |  |  |  |
Mean | 50.0 | 50.8 | 56.4 | 28.3 | 52.3 | 52.7 | 46.0
Std | 27.1 | 28.2 | 26.8 | 23.5 | 29.5 | 30.7 | 23.1
Post, * |  |  |  |  |  |  |
Mean | 60.7 | 60.6 | 69.4 | 40.9 | 63.5 | 70.8 | 58.5
Std | 24.3 | 24.5 | 14.3 | 16.9 | 25.5 | 23.0 | 20.1
P-value, * | 0.025 | 0.017 | 0.047 | 0.047 | 0.292 | 0.005 | 0.029
* participants who completed pre- and post-survey.

We analyzed post-survey responses for T1 and T2 altogether. We expected an
increase in T-Moodifier users’ perception of emotional contagion (Questions
(1)-(4)) while using T-Moodifier. As shown in Table VI, this perception
changed significantly in the expected direction for all four questions.

Regarding Questions (1) and (2), after the T-Moodifier experience, users could
recognize significantly more strongly that Twitter and their connections
influenced them, suggesting that they gained awareness thanks to T-Moodifier.
As for changes in Question (3), they might indicate that users perceived more
clearly what their friends shared and, hence, believed that what was happening
on Twitter influenced their friends’ reactions. In addition, changes in
Question (4) might reflect that users updated their belief regarding their
power to influence others.

Even though Question (5) did not change significantly, it moved in the
expected direction towards increasing users’ emotional reflection before
posting content.

On the subject of the users’ confidence regarding the emotions they shared and
consumed on Twitter, we observed the following in the responses to Questions
(6) and (7): the awareness of T-Moodifier users increased and they could
assert more confidently the main emotion they shared and perceived on Twitter.
For the emotions asked for Question (6), users changed from 41% in accuracy
before the experience to 50% in accuracy afterwards; for the emotions asked
for Question (7), they changed from 42% to 58%.

We conclude that these changes in awareness captured by the surveys, coupled
with the observed behavioral changes, reflect T-Moodifier’s potential to make
users reflect more on what they see and share, increasing their emotional
awareness.

#### Open-ended comments and design implications

Though we did not tell the participants that we would measure (publicly
visible) changes in their behavior, in the open-ended comments, some of them
acknowledged that they did reflect on how they reacted while using
T-Moodifier. For example, one T2 participant (who did not see their own
statistics), stated “[T-Moodifier] made me realize how I react to my feed’s
composition and people’s positive or negative news.” Furthermore, 75% of those
who returned the post-survey stated that T-Moodifier helped them better
understand the emotions they consumed on Twitter. Most of them explicitly
pointed to the tool’s design as the means for that. They said that “the
filtration system […] was a huge advantage,” as well as “the ability to toggle
between positive and negative emotions.” They also pointed to the labels and
colored borders as a way of differentiating and “giving more attention to
emotions,” or by giving them “a reason to hope they [Tweets] would be positive
before I even read them.” These are examples of how T-Moodifier provoked
emotional reflection, recognition, and awareness among social media users.

#### Why did people decide to use T-Moodifier?

Our pre-survey efforts included asking participants to comment on why they
were installing T-Moodifier. One-third of them indicated that they decided to
use T-Moodifier for various personal reasons. Another third of the
participants indicated that they started using T-Moodifier because they were
curious about the tool and how to use it. The final third stated that they
were very interested in understanding and exploring their News Feed and how it
affected them in greater depth.

Some people used the words “unconsciously” (ex: “[I am] interested to see what
I’ve followed unconsciously”) and “subconsciously” (ex: “I want to understand
what material I am subconsciously ingesting”). This indicated that they might
have known they were unconsciously perceiving emotions on social media, but
they did not have a tool, like T-Moodifier, to make those emotions explicit.

###  V-C Self-reported mental health

Previous works suggest that emotional awareness and recognition are crucial to
wellbeing [25, 14, 5] and can help to exploit emotions that improve one’s
mental health [19].

We designed T-Moodifier to help with these and hope that, under prolonged use,
it could help improve Twitter users’ emotional awareness and, consequently,
their mental wellbeing. To approach this assessment, we applied the Eight-Item
Personal Health Questionnaire (PHQ-8) [18] as a preliminary way to understand
whether tools like T-Moodifier could have an impact on users’ mental health.
Participants who completed both pre- and post-questionnaire responses scored
around 5.6 in both cases on average, with 7/55 (12.7%) of them scoring at
least 10 (which is considered a sign of depressive disorders); there were no
significant differences between the two treatments. We can draw only limited
conclusions based on the 28 returned post-surveys; however, we can say that
people in this target group are willing to use applications that are aimed at
helping them.

We do not have statistically significant evidence to support/refute our
hypothesis regarding T-Moodifier helping users’ wellbeing. For future studies,
we plan to use the Warwick-Edinburgh Mental Wellbeing Scale [27], a more
positively worded and less clinical-centric test, for assessment; this might
also prevent concerns about inducing negative emotions instead of preventing
them due to the application of a depression-centric test.

###  V-D Users’ inquiries and feedback about Tweet Moodifier

Prior to running the experiment reported here, we ran a pilot version to get
feedback on the T-Moodifier user experience. Some participants in the pilot
contacted us to ask for more details regarding the use of private information
being collected, to know how T-Moodifier classifies Tweets’ affective
valences, and to report mislabeled Tweets. Consequently, we updated
T-Moodifier and its website to provide all that information and implementation
details from the beginning of the reported T-Moodifier experience. These
questions show that transparency is key to reaching users and having them
participate in this type of studies. As stated by Amodei et al [2],
transparency is an emerging topic of concern among tech users.

##  VI Conclusions

We presented a new tool, Tweet Moodifier (T-Moodifier), a user-consented
Chrome extension that aims to help users better understand the emotional
valence of what they consume and share online when using Twitter. By allowing
users to filter and make explicit the emotional content in their News Feed,
T-Moodifier supports users in reflecting more confidently on the positive and
negative nature of what emotions they consume and share. While the survey
results in this work are preliminary (because of the small sample size
completing both pre- and post-questionnaires), the behavioral results showed a
significant association between receiving personalized statistical feedback on
Tweet valence and an increase in the percentage of neutral Tweets sent. This
result suggests that the use of T-Moodifier may help enhance Twitter users
emotional awareness and may also influence their Twitter behavior.

##  VII Discussion and future work

The reported results have several limitations that should be addressed in the
future to obtain stronger conclusions and to explore other aspects of Twitter
users’ emotional awareness that T-Moodifier currently does not capture.

While participation yielded 385 days of T-Moodifier use, a major limitation is
that the current sample of returned post-surveys is only 28, and the total
days of use per person is only 7. Thus, the conclusions based on self-report
are not as strong as those based on behavior, and the behavior is relatively
short-term. Our next step will entail shedding light on what the long-term
effects of T-Moodifier on a bigger audience might be. This could allow us to
generalize our conclusions and increase our understanding of how social media
affects us and how aware we are of that. Also, with having a bigger audience,
we could differentiate between clusters of users, such as strongly
negative-/positive-sharing users, or even focus the study on groups with
depressive disorders or social media dependency. Understanding diverse users’
awareness might be a significant step toward influencing them positively.

Another major limitation is that this study reduced the emotions of Tweets to
positive and negative valences. This reduction introduces noise mainly to
negative emotions, where we can find hate speech and compassionate expressions
of grief under the same negative label. It has been shown [8] that different
emotions produce different reactions in social media users. Hence, in further
versions, T-Moodifier should be able to break down both positive and negative
valences into more specific emotions. Also, we want to explore more
sophisticated models to improve the emotions classifier accuracy.

While T-Moodifier could potentially reach all Twitter users, an important
limitation is that its current version is only available for Google Chrome in
a desktop version for English Tweets. It is shown that most users access
social media using mobile devices [13], and that people’s sharing patterns are
different across devices. Therefore, it is highly probable that T-Moodifier
will not capture all representative patterns. We would also like to understand
the awareness of those users who are not willing to use applications like
T-Moodifier.

Furthermore, by using T-Moodifier’s current capabilities, we could try to
understand what would be a healthy emotional diet/regimen for people in social
media. So far, we explored participants organic behavior once they are aware
of the emotional content that they perceive. We plan on extending this to give
users the option to balance how much (percentage-wise) of each emotion they
want to receive [16, 21].

As for T-Moodifier’s design, we observed that the display of explicit feedback
(i.e., personal statistics) revealing the valence of what users posted
impacted them significantly. Also, users acknowledged that making emotions
salient via colored marks and emojis in each Tweet helped them to
differentiate and recognize emotions throughout their exposure to Twitter
better. For a future version, we plan on giving a more positive tone to the
availability of mental health resources by framing them as wellbeing
resources, which is what T-Moodifier aims to be.

Overall, we can see that T-Moodifier appears to be able to increase users’
emotional awareness. However, it is too soon to say whether T-Moodifier causes
a positive, negative or neutral effect in its users. We believe that prolonged
use of tools that subtly elicit user emotional awareness could reduce the
negative consequences of spending time in social media and help users take
better control over their affective well-being.

## Acknowledgment

The authors would like to thank the Network Computing Systems at the MIT Media
Lab for providing the hardware required for deploying Tweet Moodifier. This
research project was made possible thanks to the continuous guidance of the
Committee on the Use of Humans as Experimental Subjects.

## References

  * [1] June Ahn and Jinyoung Kim.  The show must go on: the presentation of self during interpersonal conflict on facebook.  In Proceedings of the 76th ASIS&T Annual Meeting: Beyond the Cloud: Rethinking Information Boundaries, page 94. American Society for Information Science, 2013.
  * [2] Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané.  Concrete problems in ai safety.  arXiv preprint arXiv:1606.06565, 2016.
  * [3] Lisa Feldman Barrett, James Gross, Tamlin Conner Christensen, and Michael Benvenuto.  Knowing what you’re feeling and knowing what to do about it: Mapping the relation between emotion differentiation and emotion regulation.  Cognition & Emotion, 15(6):713–724, 2001.
  * [4] Matthew Tyler Boden and Howard Berenbaum.  What you are feeling and why: Two distinct types of emotional clarity.  Personality and individual differences, 51(5):652–656, 2011.
  * [5] Matthew Tyler Boden, Jessica G Irons, Matthew T Feldner, Sarah Bujarski, and Marcel O Bonn-Miller.  An investigation of relations among quality of life and individual facets of emotional awareness and mindfulness.  Mindfulness, 6(4):700–707, 2015.
  * [6] Lorenzo Coviello, Yunkyu Sohn, Adam DI Kramer, Cameron Marlow, Massimo Franceschetti, Nicholas A Christakis, and James H Fowler.  Detecting emotional contagion in massive social networks.  PloS one, 9(3):e90315, 2014.
  * [7] Shannon M Erisman and Lizabeth Roemer.  A preliminary investigation of the effects of experimentally induced mindfulness on emotional responding to film clips.  Emotion, 10(1):72, 2010.
  * [8] Rui Fan, Jichang Zhao, Yan Chen, and Ke Xu.  Anger is more influential than joy: Sentiment correlation in weibo.  PloS one, 9(10):e110184, 2014.
  * [9] Emilio Ferrara and Zeyao Yang.  Measuring emotional contagion in social media.  PloS one, 10(11):e0142390, 2015.
  * [10] Anastasia Giachanou and Fabio Crestani.  Like it or not: A survey of twitter sentiment analysis methods.  ACM Computing Surveys (CSUR), 49(2):28, 2016.
  * [11] Alec Go, Richa Bhayani, and Lei Huang.  Twitter sentiment classification using distant supervision.  CS224N Project Report, Stanford, 1(12), 2009.
  * [12] Benjamin Grosser.  What do metrics want? how quantification prescribes social interaction on facebook.  Computational Culture, 2014.
  * [13] Youngsub Han, Beomseok Hong, Hyeoncheol Lee, and Kwangmi Kim.  How do we tweet? the comparative analysis of twitter usage by message types, devices, and sources.  The Journal of Social Media in Society, 6(1):189–219, 2017.
  * [14] Christina LM Hill and John A Updegraff.  Mindfulness and its relationship to emotional regulation.  Emotion, 12(1):81, 2012.
  * [15] Shanshan Huang, Howard Berenbaum, and Philip I Chow.  Distinguishing voluntary from involuntary attention to emotion.  Personality and Individual Differences, 54(8):894–898, 2013.
  * [16] Melissa G Hunt, Rachel Marx, Courtney Lipson, and Jordyn Young.  No more fomo: Limiting social media decreases loneliness and depression.  Journal of Social and Clinical Psychology, pages 751–768, 2018\.
  * [17] Adam DI Kramer, Jamie E Guillory, and Jeffrey T Hancock.  Experimental evidence of massive-scale emotional contagion through social networks.  Proceedings of the National Academy of Sciences, page 201320040, 2014.
  * [18] Kurt Kroenke, Tara W Strine, Robert L Spitzer, Janet BW Williams, Joyce T Berry, and Ali H Mokdad.  The phq-8 as a measure of current depression in the general population.  Journal of affective disorders, 114(1-3):163–173, 2009.
  * [19] Matthew D Lieberman, Tristen K Inagaki, Golnaz Tabibnia, and Molly J Crockett.  Subjective responses to emotional stimuli during labeling, reappraisal, and distraction.  Emotion, 11(3):468, 2011.
  * [20] Galen Panger.  Reassessing the facebook experiment: critical thinking about the validity of big data research.  Information, Communication & Society, 19(8):1108–1126, 2016.
  * [21] Brian A Primack, Ariel Shensa, Jaime E Sidani, Erin O Whaite, Liu yi Lin, Daniel Rosen, Jason B Colditz, Ana Radovic, and Elizabeth Miller.  Social media use and perceived social isolation among young adults in the us.  American journal of preventive medicine, 53(1):1–8, 2017.
  * [22] Lee Rainie, Amanda Lenhart, and Aaron Smith.  The tone of life on social networking sites.  Pew Internet Report, 2012.
  * [23] Hugo Rosa, N Pereira, Ricardo Ribeiro, Paula Costa Ferreira, João Paulo Carvalho, S Oliveira, Luísa Coheur, Paula Paulino, AM Veiga Simão, and Isabel Trancoso.  Automatic cyberbullying detection: A systematic review.  Computers in Human Behavior, 93:333–345, 2019.
  * [24] Anna Schmidt and Michael Wiegand.  A survey on hate speech detection using natural language processing.  In Proceedings of the Fifth International Workshop on Natural Language Processing for Social Media, pages 1–10, 2017.
  * [25] Nicola S Schutte and John M Malouff.  Emotional intelligence mediates the relationship between mindfulness and subjective well-being.  Personality and Individual Differences, 50(7):1116–1119, 2011.
  * [26] Manya Sleeper, Rebecca Balebako, Sauvik Das, Amber Lynn McConahy, Jason Wiese, and Lorrie Faith Cranor.  The post that wasn’t: exploring self-censorship on facebook.  In Proceedings of the 2013 conference on Computer supported cooperative work, pages 793–802. ACM, 2013.
  * [27] Ruth Tennant, Louise Hiller, Ruth Fishwick, Stephen Platt, Stephen Joseph, Scott Weich, Jane Parkinson, Jenny Secker, and Sarah Stewart-Brown.  The warwick-edinburgh mental well-being scale (wemwbs): development and uk validation.  Health and Quality of life Outcomes, 5(1):63, 2007.
  * [28] Morten Tromholt.  The facebook experiment: Quitting facebook leads to higher levels of well-being.  Cyberpsychology, behavior, and social networking, 19(11):661–666, 2016.
  * [29] Joseph B. Walther, Brandon Van Der Heide, Sang-Yeon Kim, David Westerman, and Stephanie Tom Tong.  The role of friends’ appearance and behavior on evaluations of individuals on facebook: Are we known by the company we keep?  Human Communication Research, 34, 2008.
  * [30] Ellery Wulczyn, Nithum Thain, and Lucas Dixon.  Ex machina: Personal attacks seen at scale.  In Proceedings of the 26th International Conference on World Wide Web, pages 1391–1399. International World Wide Web Conferences Steering Committee, 2017.
  * [31] Shanyang Zhao, Sherri Grasmuck, and Jason Martin.  Identity construction on facebook: Digital empowerment in anchored relationships.  Computers in human behavior, 24(5):1816–1836, 2008.

Figure 3: T-Moodifier updates its card to explain the activated view. Figure
4: “Moodified” Tweets.


# Nonlinear electromagnetic generalization of the Kerr-Newman solution with
cosmological constant

Oscar Galindo-Uriarte<EMAIL_ADDRESS>Nora Breton
<EMAIL_ADDRESS>Physics Department,
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico
Nacional (Cinvestav),
PO. Box 14-740, Mexico City, Mexico

###### Abstract

We present the two exact solutions of the Einstein-Nonlinear electrodynamics
equations that generalize the Kerr-Newman solution. We determined the
generalized electromagnetic potentials using the alignment between the tetrad
vectors of the metric and the eigenvectors of the electromagnetic field
tensor. It turns out that there are only two possible nonlinear
electromagnetic generalizations of the Kerr-Newman geometry, corresponding to
different electromagnetic potentials. The new solutions possess horizons and
satisfy physical energy conditions such that they can represent black holes
with nonlinear electromagnetic charges, characterized by the parameters of
mass, angular momentum, charge, and one nonlinear parameter; the nonlinear
parameter resembles the effect of a cosmological constant, negative or
positive, such that the solutions are asymptotically AdS or dS. The canonical
form of the electromagnetic nonlinear energy-momentum tensor is analyzed in
relation with the energy conditions; it is shown that the conformal symmetry
is broken by the electromagnetic nonlinear matter; the corresponding nonlinear
electromagnetic Lagrangian as a function of the coordinates is presented as
well.

##  I Introduction

There are current observations of gravitational waves detecting the collision
of two massive compact objects in the LIGO LIGO2016  and Virgo ALAV2021
interferometric facilities; this has lead to the assembly of catalogues of
colliding compact objects that result in a unique remnant. The astrophysical
compact objects are rotating and therefore in the context of the Einstein
exact solutions there is a great interest in stationary solutions since,
within some approximation, they resemble some features of celestial bodies.
Therefore the Kerr and Kerr-Newman stationary solutions of the coupled
Einstein-Maxwell equations are of the utmost relevance both theoretical and
astrophysically.

On the other hand, nonlinear electromagnetic (NLE) effects occur in the
vicinity of strongly magnetized compact objects, like magnetars or neutron
stars; the description of such effects require some extension of Maxwell
electrodynamics and one way is with Lagrangians that are nonlinear in the
electromagnetic invariants. Therefore exact solutions of the Einstein-NLE
equations can give insight of interesting properties of strongly magnetized
black holes (BH) as well as can be useful as test beds of numerical
simulations. Moreover, stationary solutions of the Einstein equations with NLE
fields may open a new perspective of the physics of rotating celestial bodies.
From the theoretical point of view there are several aspects for studying
stationary axisymmetric solutions that belong to the algebraic type D in the
Petrov classification. An interesting possibility of introducing NLE effects
in BH metrics is of avoiding the singularity, that for the static case there
is abundance of regular BHs sourced by some kind of nonlinear electrodynamics
AyonGarciaPRL . However the challenge of determining a NLE stationary solution
has been elusive until recently that a Euler-Heisenberg generalization of the
Kerr-Newman black hole was presented in BLM2022 .

There are several proposals of NLE Lagrangians, that are nonlinear functions
of the two Lorentz invariants $F$ and $G$ of the electromagnetic field,
$\mathcal{L}(F,G)$. In this paper we emphazise that even if we do not know
exactly the expression of the Lagrangian in terms of the electromagnetic
invariants, new NLE solutions that generalize Kerr-Newman solution can be
generated, such is the case in AGarcia2022  and AGarcia_Annals2022 , where a
stationary solution of the coupled Einstein-NLE equations was presented. This
exact solution is a Kerr-like geometry that describes a rotating BH endowed
with mass, angular momentum, cosmological constant, electric charge and an
electromagnetic nonlinear parameter. The electromagnetic tensors $F_{\mu\nu}$
and $P_{\mu\nu}$ of the solution fulfill a set of four generalized “Maxwell
equations” and two independent Einstein–NLE equations related with the two
independent eigenvalues of the NLE energy–momentum tensor. The NLE is
determined from a Lagrangian that is a function of the coordinates $r$ and
$\theta$, $\mathcal{L}(r,\theta)$, constructed from the two electromagnetic
invariants $F$ and $G$.

There are NLE theories that are Lorentz invariant and gauge invariant, these
theories were studied and classified by Plebański Pleb , and important
contributions are due to Boillat Boillat1970 . The propagation of light in NLE
environments is also of interest and it is known that for any theory of the
Plebański class the rays are the null geodesics of two optical metrics;
causality and signal propagation has been addressed in Perlick2016 . The
optical metric was rederived by Novello et al. Novello ; and, using a
different representation, by Obukhov and Rubilar Obukov2002 , that also
derived the Fresnel equation for the wave covectors and, for the class of
local nonlinear Lagrangian nondispersive models, it is demonstrated that the
quartic Fresnel equation factorizes, yielding the generic birefringence
effect.

The cosmological constant $\Lambda$ has acquired relevance lately related to
its interpretation as the dark energy in cosmological solutions of the
Einstein equations. Another aspect of interest are the anti–de Sitter (AdS)
spacetimes ($\Lambda<0$) related to the holographic correspondence between
gravity systems and the conformal field theory, the AdS/CFT duality Witten98 .
Moreover, BHs in AdS spacetimes admit a gauge duality description through a
thermal field theory. For these reasons we include the cosmological constant
in our study of the nonlinear electromagnetic stationary solutions.

In this paper we present the nonlinear electromagnetic generalizations of the
Kerr-Newman solutions. These new solutions are derived from aligning two
vectors of the metric tetrad with the two different eigenvectors of the
electromagnetic field tensor. The alignment conditions along with the
condition of integrability of the Lagrangian allows to derive a differential
equation for the electromagnetic potentials. Then we consider an Ansatz for
the electromagnetic potentials $A_{\mu}$ and ${}^{\star}{P}_{\mu}$ that
consists of a quotient of polynomials in the coordinates $r$ and $\theta$,
whose coefficients are constrained by a key equation.

The paper is organized as follows: In Sect. II we review the Kerr-Newman (KN)
metric emphasizing its electromagnetic fields. In Sect. III we present the NLE
equations and the alignment between two vectors of the metric tetrad with the
two different eigenvectors of the electromagnetic field tensor; it is also
derived the key equation that links the two electromagnetic potentials
$A_{\mu}$ and ${}^{\star}{P}^{\mu}$. In Sect. IV we derive the two possible
NLE generalizations of the KN spacetime. In Sect. V we present the main
features of the new NLE generalization of the KN solution, like the horizons,
ergosphere and energy conditions; the static limit that is a NLE
generalization of the Reissner-Nordstrom solution is presented as well. In
Sect. VI we examine the canonical form of the NLE energy-momentum tensor
$T_{\mu\nu}$ and the energy conditions satisfied by the NLE matter; it is
shown that the trace of the NLE energy-momentum tensor does not vanish and
then the introduction of the NLE field breaks the conformal symmetry. The
expression of the NLE Lagrangian as a function of the coordinates is
presented. Finally in Sect. VII Conclusions are given.

##  II The Kerr-Newman metric

It is not exaggerated to say that the Kerr-Newman (KN) solution of the coupled
Einstein-Maxwell equations is of the utmost importance, both, related to
recent black hole observations as well as a theoretical object which is not
completely understood in all its caveats, from thermodynamics to singularity
theory, while aspects like finding interior matter that sources the exterior
KN geometry are still challenges.

The Kerr-like metric with cosmological constant in Boyer-Lindquist coordinates
is

$\begin{split}ds^{2}&=\frac{a^{2}\sin^{2}{\theta}\Delta_{\theta}}{\Sigma\Xi^{2}}\left({\bf{dt}}-\frac{a^{2}+r^{2}}{a}{\bf
d\phi}\right)^{2}\\\ &+\frac{\Sigma}{\Delta_{r}}\,{\bf
dr}^{2}+\frac{\Sigma}{\Delta_{\theta}}\,{\bf
d\theta}^{2}-{\frac{\Delta_{r}}{\Sigma\Xi^{2}}}\left({\bf{dt}}-a\sin^{2}{\theta}{\bf
d\phi}\right)^{2},\end{split}$ (1)

$\begin{split}\Sigma(\theta,r)&:={{r^{2}+a^{2}\cos^{2}{\theta}}};\\\
\Delta_{\theta}(\theta)&=1+\frac{\Lambda}{3}a^{2}\cos^{2}\theta;\quad\Xi=1+\frac{\Lambda}{3}a^{2},\\\
\Delta_{r}(r)&=:K(r)-2mr+{r}^{2}+{a}^{2}-\frac{\Lambda}{3}r^{2}(r^{2}+a^{2}),\end{split}$

where the angular momentum $a$, mass $m$, cosmological constant $\Lambda$ and
the structural function $K(r)$ is to be determined from the coupled
NLE–Einstein equations; for the Kerr-Newman solution
$K(r)=Q_{e}^{2}+Q_{m}^{2}$, with $Q_{e}$, $Q_{m}$ being the BH electric and
magnetic charges, respectively; the determinant of the Kerr-like metric is
given by

$g=-g_{rr}g_{\theta\theta}\left(g_{t\phi}^{2}-g_{tt}g_{\phi\phi}\right)=-\frac{\Sigma^{2}\sin^{2}\theta}{\Xi^{4}}.$
(2)

Due to the existence of two Killing vectors $\partial_{t}$ and
$\partial_{\phi}$, in this spacetime the energy and angular momentum of a test
particle, $E$ and $L$, are conserved quantities, besides its mass, $\mu$.
Moreover, the existence of a Killing tensor (additional to $g_{\mu\nu}$)
implies a fourth motion constant, the Carter constant given by

$\displaystyle C$ $\displaystyle=$
$\displaystyle\Delta_{\theta}P^{2}_{\theta}+\cos^{2}\theta\left[a^{2}\left(\mu^{2}-\frac{\Xi
E^{2}}{\Delta_{\theta}}\right)+\frac{\Xi^{2}L^{2}}{\Delta_{\theta}\sin^{2}\theta}\right]$
(3) $\displaystyle+\frac{\Xi^{2}(aE-L)^{2}}{\Delta_{\theta}},$

where $P_{\theta}$ is the $\theta$\- component of the test particle momentum.
This property of the Kerr-like spacetimes allows the separability of the
Hamilton-Jacobi and the Klein-Gordon equations Frolov2007 , among other
remarkable features.

The alignment can be understood using the null tetrad $\bf{e^{a}}$ for the
Kerr-like metric given by

$\left.\begin{array}[]{cc}{\bf{e^{1}}}\\\
{\bf{e^{2}}}\end{array}\right\\}=\sqrt{\frac{\Sigma}{2\Delta_{\theta}}}\,{\bf
d\theta}\pm
i\,\frac{a\sin{\theta}}{\Xi}\sqrt{\frac{\Delta_{\theta}}{2\Sigma}}\left({\bf{dt}}-\frac{a^{2}+r^{2}}{a}{\bf
d\phi}\right),$

and

$\left.\begin{array}[]{cc}{\bf{e^{3}}}\\\
{\bf{e^{4}}}\end{array}\right\\}=\sqrt{\frac{\Delta_{r}}{2\Sigma\Xi^{2}}}\left({\bf{dt}}-a\sin^{2}{\theta}{\bf
d\phi}\right)\pm\,\sqrt{\frac{\Sigma}{2\Delta_{r}}}\,{\bf dr},$

the metric in the null tetrad is written as

$g=2{\bf{e^{1}}\bf{e^{2}}}-2{\bf{e^{3}}\bf{e^{4}}}=g_{\bf{a}\bf{b}}{\bf{e^{a}}\bf{e^{b}}},\quad\bf{e^{a}}=\bf{e^{a}}{}_{\mu}\,\bf{dx}^{\mu}.$
(4)

The null tetrad is associated to the eigenvector basis of $F_{\mu\nu}$ as

$F_{\mu\nu}=2F_{\bf{a}\bf{b}}{\bf{e^{a}}_{[\mu}\bf{e^{b}}_{\nu]}}=2F_{12}{e^{1}}_{[\mu}\,{e^{2}}_{\nu]}+2F_{34}{e^{3}}_{[\mu}\,{e^{4}}_{\nu]}.$
(5)

In the null tetrad the EM equations can be written in terms of a closed 2–form
$\omega$, as $d\omega=0$, with $\omega$ given by

$\begin{split}\omega&=\frac{1}{2}\left(F_{\mu\nu}+{{}^{\star}P_{\mu\nu}}\right)dx^{\mu}\,\wedge
dx^{\nu}\\\
&=\frac{1}{2}\left(F_{\bf{a}\bf{b}}+{{}^{\star}P_{\bf{a}\bf{b}}}\right)e^{\bf{a}}\wedge\,e^{\bf{b}}\\\
&=(F_{12}+P_{34})e^{1}\wedge e^{2}+(F_{34}+P_{12})e^{3}\wedge
e^{4}.\end{split}$ (6)

Throughout the paper we shall mainly use the coordinate components of the
electromagnetic quantities, $F_{\mu\nu}(r,\theta)$, etc.

###  II.1 Alignment conditions for the Kerr-like metric

The field tensor $F_{\mu\nu}$ is characterized by four nonvanishing
components: $F_{\theta\phi}$, $F_{\theta\,t}$, $F_{r\phi}$, $F_{rt}$. The
eigenvectors $V_{a}^{\mu}$ of the tensor $F_{\mu\nu}$ are determined by
solving the corresponding eigenvalue problem; the alignment of the
eigenvectors $V_{a}^{\mu}$, $a=1\cdots 4$ along the tetrad basis, or
equivalently, aligning the tetrad along the eigenvectors $V_{a}^{\mu}$ gives
rise to the alignment conditions:

$\begin{split}{F_{r\phi}}&=-a\,\sin^{2}{\theta}{F_{rt}},\\\ {F_{\theta
t}}&=-{\frac{{a}^{2}+{r}^{2}}{a}}{F_{\theta\phi}},\end{split}$ (7)

thus only two of the field components are independent, say $F_{rt}$ and
$F_{\theta t}$, while the remaining two ${F_{r\phi}}$ and $F_{\theta\phi}$ are
determined through the alignment conditions. Since the field tensor
${F_{\mu\nu}}$ is a curl, it can be determined from a vector potential
$A_{\mu}$, $F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$. The alignment conditions can
be integrated for the electromagnetic vector components $A_{t}$ and
$A_{\phi}$: replacing $F_{\theta\phi}=A_{\phi,\theta}$,
$F_{r\phi}=A_{\phi,r}$, $F_{rt}=A_{t,r}$, $F_{\theta t}=A_{t,\theta}$ in the
alignment conditions Eq. (7) one arrives at

$\begin{split}A_{\phi,r}+a\,\sin^{2}{\theta}A_{t,r}&=0,\\\
A_{\phi,\theta}+{\frac{{a}^{2}+{r}^{2}}{a}}A_{t,\theta}&=0;\end{split}$ (8)

while the integrability of $A_{\phi}$, $A_{\phi,r\theta}=A_{\phi,\theta r}$
leads to a partial differential equation for $A_{t}$

$A_{t,\theta
r}+\frac{2r}{\Sigma}A_{t,\theta}-\frac{2{a}^{2}\sin{\theta}\cos{\theta}}{\Sigma}A_{t,r}=0,$
(9)

whose general solution has the form

${A_{t}}={\frac{X(r)+Y({\theta})}{\Sigma}},$ (10)

where $X(r)$ and $Y(\theta)$ are arbitrary functions on their respective
variables. This solution for $A_{t}$ guarantees the integrability of
$A_{\phi}$, in Eq.(8), that leads to

$A_{\phi}=-a\sin^{2}{\theta}\frac{X(r)}{\Sigma}-\frac{\left({a}^{2}+{r}^{2}\right)}{a}\frac{Y({\theta})}{\Sigma}.$
(11)

For the KN solution the electromagnetic potentials are given by the simplest
polynomials, $X(r)=Q_{e}\,r$ and $Y(\theta)=Q_{m}\,a\cos{\theta}$, where
$Q_{e}$ and $Q_{m}$ are constants related to the electric and magnetic charges
respectively. Deriving $A_{\mu}$ we obtain the electromagnetic field
components,

$\begin{split}F_{rt}^{KN}&=-\frac{1}{\Sigma^{2}}\left[2Q_{m}ar\cos{\theta}+Q_{e}(r^{2}-a^{2}\cos{\theta}^{2})\right],\\\
F_{\theta
t}^{KN}&=\frac{a\sin{\theta}}{\Sigma^{2}}\left[2Q_{e}\,ar\cos{\theta}-Q_{m}(r^{2}-a^{2}\cos{\theta}^{2})\right],\end{split}$
(12)

Comparing with the Kerr-Newman solution we identify $Q_{e}$ and $Q_{m}$ with
the electric and magnetic BH charges, respectively.

##  III Nonlinear electromagnetic equations

If we consider a Lagrangian, $\mathcal{L}(F,G)$, that depends in general form
on the electromagnetic invariants $F$ and $G$, given by

$\displaystyle F$
$\displaystyle=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\left(B^{2}-E^{2}\right),$
(13) $\displaystyle G$
$\displaystyle=\frac{1}{4}F_{\mu\nu}{}^{\star}{F}^{\mu\nu}=-\vec{E}\cdot\vec{B},$
(14)

where
${}^{\star}{F}_{\mu\nu}=\frac{1}{2}\sqrt{-g}\varepsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$
is the dual stress-tensor; the contravariant dual
${}^{\star}F^{\mu\nu}=-\frac{1}{2}\frac{\varepsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}F_{\alpha\beta}$;
and $\varepsilon^{\mu\nu\alpha\beta}$ is the Levi-Civita symbol.

The variation of the action with minimal coupling between $\mathcal{L}(F,G)$
and Einstein General Relativity leads us to define a new skew symmetric tensor
$P_{\mu\nu}$ in terms of $F_{\mu\nu}$ and the derivatives of the Lagrangian
respect to the invariants, this equation is called the constitutive or
material equation,

$P_{\mu\nu}=\mathcal{L}_{F}F_{\mu\nu}+\mathcal{L}_{{G}}({}^{\star}{F}_{\mu\nu}),$
(15)

where $\mathcal{L}_{X}=\frac{\partial\mathcal{L}}{\partial X}$; in terms of
$P_{\mu\nu}$ and $F_{\mu\nu}$ the electromagnetic field (EM) equations are

$\displaystyle{P^{\mu\nu}}_{;\nu}=0\rightarrow{\left[\sqrt{-g}\left(\mathcal{L}_{F}\,{F^{\mu\nu}}+\mathcal{L}_{G}\,{{}^{\star}{F^{\mu\nu}}}\right)\right]_{,\nu}=0},$
(16)
$\displaystyle{{}^{\star}{F^{\mu\nu}}}_{;\nu}=0\rightarrow{(\sqrt{-g}{}^{\star}{F^{\mu\nu}})_{,\nu}=0},$
(17)

where $g$ is the determinant of the metric. Since $F_{\mu\nu}$ and
${{}^{\star}P_{\mu\nu}}$ are curls, both can be derived from two
electromagnetic potentials, $A_{\mu}$ and ${{}^{\star}P_{\mu}}$, namely,
$F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$ and
${{}^{\star}P_{\mu\nu}}={}^{\star}P_{\nu,\mu}-{}^{\star}P_{\mu,\nu}$. The
nonvanishing components of the dual tensor ${}^{\ast}P_{\mu\nu}$ in the Kerr-
like geometry (1) are ${}^{\ast}P_{\theta t}={}^{\ast}P_{t,\theta}$ and
${}^{\ast}P_{rt}={}^{\ast}P_{t,r}$. Then the EM field equations (16) become

$\left(\sqrt{-g}P^{\phi\theta}\right)_{,\theta}+\left(\sqrt{-g}P^{\phi
r}\right)_{,r}=0.$ (18)

The material equations (15) and the fact that
${}^{\star}P_{t,r\theta}={}^{\star}P_{t,\theta r}$ allows to write them in
matrix form as,

$\begin{pmatrix}\frac{F_{\theta t}}{a\sin\theta}&-F_{rt}\\\ a\sin\theta
F_{rt}&F_{\theta t}\end{pmatrix}\begin{pmatrix}\mathcal{L}_{F}\\\
\mathcal{L}_{G}\end{pmatrix}=\begin{pmatrix}{}^{\star}P_{t,r}\\\
-{}^{\star}P_{t,\theta}\end{pmatrix},$ (19)

from which we clear out the derivatives of the Lagrangian,

$\begin{pmatrix}\mathcal{L}_{F}\\\
\mathcal{L}_{G}\end{pmatrix}=\frac{1}{\left(\frac{F_{\theta
t}}{a\sin\theta}\right)^{2}+\left(F_{rt}\right)^{2}}\begin{pmatrix}\frac{F_{\theta
t}}{a\sin\theta}&\frac{F_{rt}}{a\sin\theta}\\\ -F_{rt}&\frac{F_{\theta
t}}{a^{2}\sin^{2}\theta}\end{pmatrix}\begin{pmatrix}{}^{\star}P_{t,r}\\\
-{}^{\star}P_{t,\theta}\end{pmatrix}.$ (20)

For $\mu=t$ in (16),
$\left(\sqrt{-g}P^{tr}\right)_{,r}+\left(\sqrt{-g}P^{t\theta}\right)_{,\theta}=0$,
we obtain the relations

$\displaystyle\sqrt{-g}P^{tr}$
$\displaystyle=\frac{\partial{}^{\star}P_{\phi}}{\partial\theta}=\left(\frac{r^{2}+a^{2}}{a}\right)\left(\mathcal{L}_{F}a\sin\theta\frac{\partial
A_{t}}{\partial r}+\mathcal{L}_{G}\frac{\partial
A_{t}}{\partial\theta}\right),$ (21) $\displaystyle\sqrt{-g}P^{t\theta}$
$\displaystyle=-\frac{\partial{}^{\star}P_{\phi}}{\partial
r}=a\sin^{2}\theta\left(\frac{\mathcal{L}_{F}}{a\sin\theta}\frac{\partial
A_{t}}{\partial\theta}-\mathcal{L}_{G}\frac{\partial A_{t}}{\partial
r}\right).$ (22)

Similarly for $\mu=\phi$ in (16), $\left(\sqrt{-g}P^{\phi
r}\right)_{,r}+\left(\sqrt{-g}P^{\phi\theta}\right)_{,\theta}=0$,

$\displaystyle\sqrt{-g}P^{\phi r}$
$\displaystyle=-\frac{\partial{}^{\star}P_{t}}{\partial\theta}=\mathcal{L}_{F}a\sin\theta\frac{\partial
A_{t}}{\partial r}+\mathcal{L}_{G}\frac{\partial A_{t}}{\partial\theta},$ (23)
$\displaystyle\sqrt{-g}P^{\phi\theta}$
$\displaystyle=\frac{\partial{}^{\star}P_{t}}{\partial
r}=\frac{\mathcal{L}_{F}}{a\sin\theta}\frac{\partial
A_{t}}{\partial\theta}-\mathcal{L}_{G}\frac{\partial A_{t}}{\partial r}.$ (24)

Comparing Eqs. (21) - (24) we arrive at the alignment conditions

$\displaystyle{{}^{\star}P_{\phi,r}}$ $\displaystyle=$
$\displaystyle-a\,\sin^{2}{\theta}{{}^{\star}P_{t,r}},$
$\displaystyle{{}^{\star}P_{\phi,\theta}}$ $\displaystyle=$
$\displaystyle-{\frac{{a}^{2}+{r}^{2}}{a}}{{}^{\star}P_{t,\theta}},$ (25)

that are similar to the ones for $F_{\mu\nu}$ in Eqs. (7). Solving the system
we arrive at the following solution for the vector potential
${}^{\star}P_{\mu}$,

${{}^{\star}P_{t}}=\frac{A(r)+B({\theta})}{\Sigma},$ (26)

${}^{\star}P_{\phi}=-{a\sin^{2}{\theta}\frac{A(r)}{\Sigma}}-\left(\frac{{a}^{2}+{r}^{2}}{a}\right)\frac{B({\theta})}{\Sigma},$
(27)

where $A(r)$ and $B(\theta)$ are arbitrary functions on their respective
variables.

Therefore we have derived the alignment conditions for $P_{\mu\nu}$ in a Kerr-
like metric, showing then that the alignment conditions are independent of the
kind of electrodynamics under study.

For the KN solution the ${}^{\star}P_{\mu}$ potential is given by the
polynomials, $A(r)=Q_{m}\,r$ and $B(\theta)=-Q_{e}\,a\cos{\theta}$. Deriving
${}^{\star}P_{\mu}$ we obtain the dual field components of
${}^{\star}P_{\mu\nu}$,

$\begin{split}{}^{\star}P_{rt}^{KN}&=\frac{1}{\Sigma^{2}}\left[2Q_{e}ar\cos{\theta}-Q_{m}(r^{2}-a^{2}\cos{\theta}^{2})\right],\\\
{}^{\star}P_{\theta
t}^{KN}&=\frac{a\sin{\theta}}{\Sigma^{2}}\left[2Q_{m}\,ar\cos{\theta}+Q_{e}(r^{2}-a^{2}\cos{\theta}^{2})\right].\end{split}$
(28)

We can found the next relations between the components of $F_{\mu\nu}$ and
${}^{\star}P_{\mu\nu}$

$\begin{split}{}^{\star}P_{\theta t}^{KN}&=-a\sin\theta F_{rt}^{KN}\\\
F_{\theta t}^{KN}&=a\sin\theta\,{}^{\star}P_{rt}^{KN}.\end{split}$ (29)

Some expressions that we shall use later are the invariants in terms of the
potential $A_{t}$,

$\begin{split}F&=\frac{\Xi^{2}}{2}\left[\left(\frac{F_{\theta
t}}{a\sin\theta}\right)^{2}-\left(F_{rt}\right)^{2}\right]\\\
&=\frac{\Xi^{2}}{2}\left[\left(A_{t,x}\right)^{2}-\left(A_{t,r}\right)^{2}\right],\end{split}$
(30)

$\begin{split}G&=-\Xi^{2}\left(\frac{F_{\theta
t}}{a\sin\theta}F_{rt}\right)\\\
&=\Xi^{2}\left(A_{t,x}\right)\left(A_{t,r}\right),\end{split}$ (31)

$\begin{split}\sqrt{F^{2}+G^{2}}&=\frac{\Xi^{2}}{2}\left[\left(\frac{F_{\theta
t}}{a\sin\theta}\right)^{2}+\left(F_{rt}\right)^{2}\right]\\\
&=\frac{\Xi^{2}}{2}\left[\left(A_{t,x}\right)^{2}+\left(A_{t,r}\right)^{2}\right],\end{split}$
(32)

where we have written $F$ and $G$ in terms of the coordinate $x=a\cos\theta$,
$dx=-a\sin\theta$. And the material or constitutive Eqs. (15) are equivalent
to:

$\displaystyle{}^{\ast}P_{t,r}$ $\displaystyle=$ $\displaystyle-
F_{rt}\mathcal{L}_{G}-F_{xt}\mathcal{L}_{F},$ $\displaystyle{}^{\ast}P_{t,x}$
$\displaystyle=$ $\displaystyle-F_{rt}\mathcal{L}_{F}-F_{xt}\mathcal{L}_{G}.$
(33)

###  III.1 The key equation for the NLE potentials $A_{\mu}$ and
${}^{\star}P_{\mu}$

The electromagnetic potentials $A_{\mu}$ and ${}^{\star}P_{\mu}$ are not
independent of each other but they are constrained by the constitutive Eq.
(15) that links $P_{\mu\nu}$ and $F_{\mu\nu}$. Moreover they must be in
agreement with the integrability condition or closure condition of the
Lagrangian, $d^{2}\mathcal{L}=0$, that amounts to

$\frac{\partial^{2}\mathcal{L}}{\partial{\theta}\partial
r}=\frac{\partial^{2}\mathcal{L}}{\partial{r}\partial{\theta}}.$ (34)

To determine the derivatives of the Lagrangian with respect to the coordinates
$r$ and $\theta$ we use the chain rule,

$\frac{\partial\mathcal{L}}{\partial x^{\alpha}}=\mathcal{L}_{F}\frac{\partial
F}{\partial x^{\alpha}}+\mathcal{L}_{G}\frac{\partial G}{\partial
x^{\alpha}},$ (35)

and the relations (20) and (30) to obtain

$\displaystyle\frac{\partial\mathcal{L}}{\partial r}$
$\displaystyle=\frac{\Xi^{2}}{a\sin\theta}\left[\frac{\partial{}^{\star}P_{t}}{\partial
r}\frac{\partial F_{\theta t}}{\partial
r}+\frac{\partial{}^{\star}P_{t}}{\partial\theta}\frac{\partial
F_{rt}}{\partial r}\right]$ (36)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\theta}$
$\displaystyle=\frac{\Xi^{2}}{a\sin\theta}\left[-\frac{\cos\theta}{\sin\theta}F_{\theta
t}\frac{{}^{\star}P_{t}}{\partial r}+\frac{\partial F_{\theta
t}}{\partial\theta}\frac{\partial{}^{\star}P_{t}}{\partial r}+\frac{\partial
F_{rt}}{\partial\theta}\frac{\partial{}^{\star}P_{t}}{\partial\theta}\right].$
(37)

Replacing the fields in terms of their potentials, $F_{rt}=A_{t,r}$,
$F_{\theta t}=A_{t,\theta}$, ${}^{\star}P_{\theta t}={}^{\star}P_{t,\theta}$
and ${}^{\star}P_{rt}={}^{\star}P_{t,r}$ in the closure condition (34) we
arrive at

$\left(\frac{\partial^{2}A_{t}}{\partial r\partial
r}\right)\frac{\partial}{\partial{\theta}}\left(\frac{1}{\sin{\theta}}\frac{\partial{}^{\star}P_{t}}{\partial{\theta}}\right)-\left(\frac{\partial^{2}{}^{\star}P_{t}}{\partial
r\partial
r}\right)\frac{\partial}{\partial{\theta}}\left(\frac{1}{\sin{\theta}}\frac{\partial
A_{t}}{\partial{\theta}}\right)=0.$ (38)

This equation has been called the key equation in AGarcia_Annals2022 . From
the key equation we obtain constraints for the functions $A(r)$, $B(\theta)$,
$X(r)$ and $Y(\theta)$ in Eqs. (10) and (26). Therefore, even if we do not
know the explicit dependence of the Lagrangian with respect to the
electromagnetic invariants $F$ and $G$, we derived a key equation that must be
fulfilled by the electromagnetic potentials of the electromagnetic fields in a
Kerr-like spacetime.

##  IV The nonlinear electromagnetic generalizations of the Kerr-Newman
metrics

From the expressions of the electromagnetic potentials $A_{t}$ and
${}^{\star}P_{t}$ in terms of the arbitrary functions $A(r)$, $B(\theta)$,
$X(r)$ and $Y(\theta)$ Eqs. (10) and (26), one could think at first that we
can find an infinite number of NLE solutions that generalize the Kerr-Newman,
just taking more terms in the polynomials $A(r)$, $B(\theta)$, $X(r)$ and
$Y(\theta)$. With that in mind, we try with polynomials for the
electromagnetic potential components $A_{t}=[X(r)+Y(\theta)]/\Sigma$ and
${}^{\star}P_{t}=[A(r)+B(\theta)]/\Sigma$, with $X(r)$, $A(r)$, $Y(\theta)$
and $B(\theta)$ according to the Ansatz,

$\begin{split}X(r)&=Q_{e}r+\sum_{n=-5}^{30}C_{n}r^{n},\\\
Y(\theta)&=Q_{m}\,a\cos{\theta}+\sum_{s=-5}^{30}D_{s}\cos^{s}{\theta},\\\
A(r)&=Q_{m}\,r+\sum_{l=-5}^{30}H_{l}r^{l},\\\
B(\theta)&=-Q_{e}\,a\cos{\theta}+\sum_{k=-5}^{30}G_{k}\cos^{k}{\theta},\end{split}$
(39)

where $C_{n}$, $D_{s}$, $H_{l}$ and $G_{k}$ are constants that are constrained
by the key equation (38). Substituting $A_{t}$ and ${}^{\star}P_{t}$ into the
key equation and equating powers it turns out that there are only two
nontrivial cases of new nonlinear electromagnetic generalizations of the Kerr-
Newman metric, that we have called the cubic vector potential and the quartic
vector potential. Numerically are obtained several cases with $A_{t}=({\rm
const}){}^{\star}P_{t}$ that are actually trivial since they reduce to the
Maxwell case.

###  IV.1 Case 1. The cubic vector potential

The electromagnetic potentials $A_{t}=[X(r)+Y(\theta)]/\Sigma$ and
${}^{\star}P_{t}=[A(r)+B(\theta)]/\Sigma$ of one NLE generalization of the KN
solution are given by,

$\begin{split}X(r)&=Q_{e}r(1-\beta r^{2}),\\\
Y(\theta)&=aQ_{m}\cos{\theta}(1+\beta a^{2}\cos^{2}\theta),\\\
A(r)&=Q_{m}r(1-\beta r^{2}),\\\ B(\theta)&=-aQ_{e}\cos{\theta}(1+\beta
a^{2}\cos^{2}\theta),\end{split}$ (40)

where $\beta$ is the nonlinear parameter. By deriving the electromagnetic
potentials, we obtain the electromagnetic field components, which can be
expressed as the KN field components plus the nonlinear contribution,

$\begin{split}F_{rt}&=F_{rt}^{KN}-\frac{\beta
r}{\Xi\Sigma^{2}}[2Q_{m}a^{3}\cos^{3}{\theta}\\\
&+Q_{e}r(r^{2}+3a^{2}\cos^{2}\theta)],\\\ F_{\theta t}&=F_{\theta
t}^{KN}-\frac{\beta}{\Xi\Sigma^{2}}a^{2}\cos{\theta}\sin{\theta}[2Q_{e}\,r^{3}\\\
&+Q_{m}\,a\cos{\theta}(3r^{2}+a^{2}\cos^{2}{\theta})],\\\
{}^{\star}P_{rt}&={}^{\star}P_{rt}^{KN}-\frac{\beta
r}{\Xi\Sigma^{2}}[-2Q_{e}a^{3}\cos^{3}{\theta}\\\
&+Q_{m}r(r^{2}+3a^{2}\cos^{2}\theta)],\\\ {}^{\star}P_{\theta
t}&={}^{\star}P_{\theta
t}^{KN}-\frac{\beta}{\Xi\Sigma^{2}}a^{2}\cos{\theta}\sin{\theta}[2Q_{m}\,r^{3}\\\
&-Q_{e}\,a\cos{\theta}(3r^{2}+a^{2}\cos^{2}{\theta})].\end{split}$ (41)

where the KN fields $F_{\mu\nu}^{KN}$ and ${}^{\star}P_{\mu\nu}$ are given in
Eqs. (12) and (28). The metric functions $\Delta_{r}$ and $K(r)$ in the Kerr-
like metric, Eq. (1), are given by

$\displaystyle\Delta_{r}$ $\displaystyle=$ $\displaystyle
r^{2}-2mr+a^{2}-\frac{\Lambda}{3}r^{2}(r^{2}+a^{2})+K(r),$ $\displaystyle
K(r)$ $\displaystyle=$ $\displaystyle({Q_{e}}^{2}+{Q_{m}}^{2})(1-\beta
r^{2})^{2},$ (42)

and $\beta=0$ corresponds to the Kerr-Newman solution. The cubic vector
potential was presented in AGarcia_Annals2022  and represents a rotating
nonlinearly charged BH characterized by its mass $m$, angular momentum $a$,
cosmological constant $\Lambda$, electric and magnetic charges $Q_{e}$ and
$Q_{m}$, and the nonlinear parameter $\beta$. Its asymptotics can be de Sitter
or anti-de Sitter, and even flatness, depending on the value of the nonlinear
parameter and of the cosmological constant. This BH can present one, two or
three horizons. the third one being the cosmological horizon in the de Sitter
case. Among the main differences between the NLE solution and the KN-BH are
the equatorial asymmetry that is enhanced by the NLE field and for charged
particles the access to one of the poles is forbidden; besides, a second
circular orbit in the neighborhood of the external horizon appears; the
presence of the nonlinear electromagnetic field increases the curvature
producing bounded orbits closer to the horizon more details can be found in
Breton2022 . In case the BH is static ($a=0$), the solution corresponds to a
NLE generalization of the Reissner-Nordstrom solution with cosmological
constant.

###  IV.2 Case 2. The quartic vector potential

The second NLE generalization of the KN solution is characterized by the
electromagnetic potentials,

$\begin{split}{\Xi\Sigma}A_{t}&=Q_{e}\,r+Q_{m}\,a\cos\theta-\frac{\xi
Q_{e}}{4}\left(r^{4}+a^{4}\cos^{4}\theta\right),\\\
{\Xi\Sigma}A_{\phi}&=-a\sin^{2}\theta
Q_{e}\,r\left(1-\xi\frac{r^{3}}{4}\right)\\\
&-(r^{2}+a^{2})\cos\theta\left(Q_{m}-\xi\frac{Q_{e}a^{3}\cos^{\theta}}{4}\right),\\\
{\Xi\Sigma}{}^{\star}P_{t}&=Q_{m}\,r-Q_{e}\,a\cos\theta-\frac{\xi
Q_{m}}{4}\left(r^{4}+a^{4}\cos^{4}\theta\right),\\\
{\Xi\Sigma}{}^{\star}P_{\phi}&=-a\sin^{2}\theta
Q_{m}\,r\left(1-\xi\frac{r^{3}}{4}\right)\\\
&+(r^{2}+a^{2})\cos\theta\left(Q_{e}+\xi\frac{Q_{m}\,a^{3}\cos^{\theta}}{4}\right),\end{split}$
(43)

where $\xi$ is the nonlinear electromagnetic parameter. These potentials
generate the fields $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
and
${}^{\star}P_{\mu\nu}=\partial_{\mu}({}^{\star}P_{\nu})-\partial_{\nu}({}^{\star}P_{\mu})$,
which are explicitly written as the KN field components plus the nonlinear
contribution,

$\begin{split}F_{rt}^{NLE}&=F_{rt}^{KN}+\xi\frac{Q_{e}\,r}{2\Xi}\\\ F_{\theta
t}^{NLE}&=F_{\theta
t}^{KN}+\xi\frac{a^{2}Q_{e}\,\sin\theta\cos\theta}{2\Xi},\\\
{}^{\star}P_{rt}^{NLE}&={}^{\star}P_{rt}^{KN}+\xi\frac{Q_{m}\,r}{2\Xi}\\\
{}^{\star}P_{\theta t}^{NLE}&={}^{\star}P_{\theta
t}^{KN}+\xi\frac{a^{2}Q_{m}\,\sin\theta\cos\theta}{2\Xi},\end{split}$ (44)

where the KN fields $F_{\mu\nu}^{KN}$ and ${}^{\star}P_{\mu\nu}$ are given in
Eqs. (12) and (28). Besides, the EM fields satisfy the alignments conditions
Eqs. (7) and (III).

This NLE generalization of the KN solution is new and we called it the quartic
vector potential solution and its metric is given by Eq. (1) with $\Delta_{r}$
given by

$\Delta_{r}=r^{2}-2mr+a^{2}-\frac{\Lambda
r^{2}}{3}(r^{2}+a^{2})+(Q_{e}^{2}+Q_{m}^{2})(1+\xi r^{3}),$ (45)

and the structural function $K(r)$

$K(r)=(Q_{e}^{2}+Q_{m}^{2})(1+\xi r^{3}),$ (46)

where $\xi$ is the nonlinear parameter. The other parameters are the mass $m$,
angular momentum $a$, electric and magnetic charges, $Q_{e},~{}Q_{m}$ and the
cosmological constant $\Lambda$. The Kerr-Newman solution is recovered by
making $\xi=0$. In the next section some features of this solution are
explored.

##  V The quartic vector potential NLE-KN

Since the quartic vector potential NLE generalization of the Kerr-Newman
solution has not been reported elsewhere, as far as we know, we show that it
represents a rotating black hole with a NLE field characterized by a NLE
parameter $\xi$ whose introduction induces a de Sitter effect, with a
cosmological horizon.

The metric function (45) is a fourth order polynomial that has at least one
real positive root, therefore it represents a BH. To analyze the horizons
determined by $\Delta_{r}(r)=0$ we restrict to the case $\Lambda=0$, that
corresponds to a cubic equation in $r$; we also restrict to a vanishing
magnetic charge, $Q_{m}=0$, but it can be recovered making $Q_{e}^{2}\to
Q_{e}^{2}+Q_{m}^{2}$.

###  V.1 Horizons and ergosphere without cosmological constant

Let us consider the cubic equation (45) with $\Lambda=0$,

$\Delta_{r}=Q_{e}^{2}\xi r^{3}+r^{2}-2mr+Q_{e}^{2}+a^{2}=0.$ (47)

First we show that the equation always has at least one positive real root.
Let us notice that at $r=0,\quad\Delta_{r}=a^{2}+Q_{e}^{2}>0$. Furthermore,
the dominant term is $Q_{e}^{2}\xi r^{3}$. In Section VI we show that $\xi<0$;
therefore, when $r\rightarrow\infty$ the function
$\Delta_{r}\rightarrow-\infty$. Since the function changes sign in the
interval $(0,\infty)$ it must cross the r-axis at least once and since the
polynomial is cubic with real coefficients it has at least one positive real
root; then the existence of an event horizon is guaranteed.

Figure 1:  Graphics of the behavior of the metric function $\Delta_{r}(r)$ for
the cases Schwarzschild, RN, Kerr, KN, NLE-RN and NLE-KN.

In this case, we will refer to the roots of $\Delta_{r}(r)=0$ as the event
horizon $(r_{+})$, inner horizon $(r_{-})$ and, a third horizon induced by
NLE, cosmological horizon $(r_{c})$; besides $r_{-}\leq r_{+}\leq r_{c}$. Fig.
(1) shows the behavior of $\Delta_{r}(r)$ for different sets of parameters
$(m,a,Q_{e},\xi)$. The relationships between the BH parameters and the roots
in Eq. (47), by the Cardano-Vieta formulas, are reduced to

$\displaystyle r_{-}$
$\displaystyle=\frac{2m(r_{+}+r_{c})-r_{+}r_{c}}{r_{+}+r_{c}-2m},$ (48)
$\displaystyle\xi$
$\displaystyle=\frac{2m-r_{+}-r_{c}}{Q_{e}^{2}(r_{+}^{2}+r_{+}r_{c}+r_{c}^{2})},$
(49) $\displaystyle a^{2}+Q_{e}^{2}$
$\displaystyle=\frac{r_{+}r_{c}\left[2m(r_{+}+r_{c})-r_{+}r_{c}\right]}{r_{+}^{2}+r_{+}r_{c}+r_{c}^{2}}.$
(50)

The cubic equation has three cases. The first one is of roots of multiplicity
three, i.e. the three horizons overlap $(r_{-}=r_{+}=r_{c})$; this corresponds
to $a^{2}+Q_{e}^{2}=\frac{4m^{2}}{3}$, $\xi=-\frac{1}{6mQ_{e}^{2}}$ and the
event horizon is at $r=2m$.

The other two cases are roots of multiplicity two. One of them is the
coincidence of the outer and cosmological horizon, $(0<r_{-}<r_{+}=r_{c})$,
that can be analyzed using

$r_{+}=2m-(r_{-})+\sqrt{{(r_{-})}^{2}-2m(r_{-})+4m^{2}}.$ (51)

The domain of this function is $0<r_{-}<2m$ and in this interval the repeated
roots are in the interval $(2m,4m)$. The case of coincidence of the inner and
outer horizons, $(0<r_{-}=r_{+}<r_{c})$, can be analyzed using

$r_{+}=2m-(r_{c})+\sqrt{{(r_{c})}^{2}-2m(r_{c})+4m^{2}}.$ (52)

The domain of this function is $2m<r_{c}<\infty$ and the repeated roots are in
the interval $(m,2m)$. For some values of the NLE parameter, $\xi$, there is
not an event horizon but an inner horizon.

To classify the type of roots, we analyze the discriminant of the cubic
equation,

$\displaystyle\Delta_{D}$
$\displaystyle=-27(a^{2}+Q_{e}^{2})^{2}Q_{e}^{4}\left(\xi-\xi_{+}\right)\left(\xi-\xi_{-}\right),$
(53)

where

$\xi_{\pm}=\frac{2\left[m(8m^{2}-9a^{2}-9Q_{e}^{2})\pm(4m-3a^{2}-3Q_{e}^{2})^{\frac{3}{2}}\right]}{27Q_{e}^{2}(a^{2}+Q_{e}^{2})^{2}}.$
(54)

According to the sign of the discriminant there are the following cases:
$\Delta_{D}=0$, repeated roots; $\Delta_{D}>0$, three different real roots;
and $\Delta_{D}<0$, one real root and two complex conjugated.

The case of repeated roots occurs when $\xi=\xi_{\pm}$ in (53). Taking into
account that $\xi$ must be real, the electric charge $Q_{e}$ and the angular
momentum $a$ are restricted to the interval
$0<a^{2}+Q_{e}^{2}\leq\frac{4m^{2}}{3}$. The function $\xi_{+}\in(0,\infty)$
for the values $0<a^{2}+Q_{e}^{2}<m^{2}$ and $\xi_{+}\in(-\infty,0]$ for the
values $m^{2}\leq a^{2}+Q_{e}^{2}\leq\frac{4m^{2}}{3}$. On the other hand, the
function $\xi_{-}\in(-\infty,0)$ for all values
$0<a^{2}+Q_{e}^{2}\leq\frac{4m^{2}}{3}$. Furthermore, the inequality
$\xi_{-}\leq\xi_{+}$ always holds.

To solve the equation (47), the transformation $r=z-\frac{1}{3Q_{e}^{2}\xi}$,
allows to depress the cubic equation to one without quadratic term,

$Q_{e}^{2}\xi\left(z^{3}+\mathcal{M}z+\mathcal{N}\right)=0,$ (55)

where

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle-\frac{1+6mQ_{e}^{2}\xi}{3Q_{e}^{4}\xi^{2}},$
$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\frac{2+18mQ_{e}^{2}\xi+27(a^{2}+Q_{e}^{2})Q_{e}^{4}\xi^{2}}{27Q_{e}^{6}\xi^{3}}.$
(56)

Next, with the change $z=2\sqrt{-\frac{\mathcal{M}}{3}}\cos\phi$ in (55), it
reduces to

$\cos 3\phi-\frac{3\mathcal{N}}{2\mathcal{M}}\sqrt{-\frac{3}{\mathcal{M}}}=0.$
(57)

Thus, the roots $r_{k}$ of (47) are

$\begin{split}r_{k+1}&=2\sqrt{-\frac{\mathcal{M}}{3}}\cos\left[\frac{1}{3}\arccos{\left(\frac{3\mathcal{N}}{2\mathcal{M}}\sqrt{-\frac{3}{\mathcal{M}}}\right)+\frac{2\pi
k}{3}}\right]\\\ &-\frac{1}{3Q_{e}^{2}\xi},\end{split}$ (58)

where $k=0,1,2.$ The previous solution demands that:

$\displaystyle 0$ $\displaystyle<-\frac{3}{\mathcal{M}},$ (59)
$\displaystyle-1$
$\displaystyle\leq\frac{3\mathcal{N}}{2\mathcal{M}}\sqrt{-\frac{3}{\mathcal{M}}}\leq
1.$ (60)

These restrictions reduce the ranges of the parameters as follows:
$a\in\left[0,\frac{2m}{\sqrt{3}}\right)$,
$Q_{e}\in\left(0,\frac{2m}{\sqrt{3}}\right]$ and
$\xi\in\left(-\infty,0\right)$. Analyzing the derivatives of the function
$\Delta_{r}(r)$, we can find a function for $\xi$,

$\Delta_{r}^{\prime}=3Q_{e}^{2}\xi r^{2}+2r-2m=0,\longleftrightarrow
Q_{e}^{2}\xi=\frac{2m}{3r^{2}}-\frac{2}{3r}.$ (61)

For $m\leq r$, we have that $-\frac{1}{6m}\leq Q_{e}^{2}\xi\leq 0$. By
substituting the term $Q_{e}^{2}\xi$ into $\Delta_{r}$ we can find that its
roots are given by the equation

$\Delta_{r}=\frac{r^{2}}{3}-\frac{4mr}{3}+a^{2}+Q_{e}^{2},$ (62)

whose internal horizons $(hr_{-})$ and external horizons $(hr_{+})$ are

$hr_{\pm}=2m\pm\sqrt{4m^{2}-3(a^{2}+Q_{e}^{2})},$ (63)

This expression for the horizon is only valid when
$a^{2}+Q_{e}^{2}\leq\left(\frac{2m}{\sqrt{3}}\right)^{2}\approx(1.1547m)^{2}$.
Additionally, the ergosphere is obtained from the roots of $g_{tt}=0$. In this
case, $g_{tt}=(a^{2}\sin^{2}\theta-\Delta_{r})$; so, using the previously
roots with $a^{2}\to a^{2}\cos^{2}\theta,$ the radius of the ergosphere is
given by

$r_{ergo}=2m\pm\sqrt{4m^{2}-3\left(a^{2}\cos^{2}\theta+Q_{e}^{2}\right)}$ (64)

On the other hand $\Delta_{r}^{\prime}=0$ (61) has two roots,

$r_{1,2}=\frac{1\pm\sqrt{1+6mQ_{e}^{2}\xi}}{-3Q_{e}^{2}\xi},$ (65)

considering that $\xi$ is negative, the two roots are positive, and
$r_{1}>r_{2}$. Evaluating the second derivative,

$\displaystyle\Delta_{r}^{\prime\prime}(r_{1})$ $\displaystyle=$
$\displaystyle-2\sqrt{1+6mQ_{e}^{2}\xi}<0,$
$\displaystyle\Delta_{r}^{\prime\prime}(r_{2})$ $\displaystyle=$
$\displaystyle+2\sqrt{1+6mQ_{e}^{2}\xi}>0,$ (66)

then the function $\Delta_{r}$ has a minimum at $r_{2}$ and a maximum at
$r_{1}$ with $r_{2}<r_{1}$, such that there are three horizons, being the
third horizon an effect of the nonlinear electromagnetic field. For some
particular values of the parameters $(a,Q_{e},m,\xi)$ the plots of ergosphere
and event horizon are shown in Fig. 2.

Figure 2:  The event horizon for KN (blue) and KN NLE (black), as well as the
ergosphere of KN (orange) and the one of KN NLE (red) are displayed. The BH
parameters are set to $Q_{e}=0.8m$, $a=0.6m$ and $\xi=-0.14/m^{3}$.

###  V.2 The static limit of the NLE Quartic vector potential solution

In the limit of vanishing rotation we obtain a NLE generalization of the
Reissner-Norsdtrom solution with cosmological constant, given by the static
metric,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}(\sin^{2}\theta
d\theta+d\phi^{2}),$ $\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2m}{r}+\frac{Q_{e}^{2}+Q_{m}^{2}}{r^{2}}(1+\xi
r^{3})+\frac{\Lambda}{3}r^{2},$ (67)

characterized by the electromagnetic fields

$\displaystyle{\Xi}F_{rt}$ $\displaystyle=$
$\displaystyle{\frac{Q_{e}}{2r^{2}}\left(\xi r^{3}-2\right)},$
$\displaystyle{\Xi}{}^{*}P_{rt}$ $\displaystyle=$
$\displaystyle{\frac{Q_{m}}{2r^{2}}\left(\xi r^{3}-2\right)},$ (68)

This NLE-RN also presents one cosmological horizon and other interesting
features, however, the analysis of this solution is beyond the scope of this
paper.

##  VI The nonlinear electromagnetic energy momentum tensor

To asses if the NLE energy-momentum tensor $T_{\mu\nu}$ is physically
reasonable, by this meaning that the local energy density measured by any
observer appears non-negative as well as the local energy flow vector be non-
spacelike, we check the energy conditions associated to the NLE generalization
of the KN. To this end we project $T_{\mu\nu}$ onto the orthonormal tetrad,
$\\{\bf{E^{a}},a=1,\ldots,4\\}$ associated to the metric (1), that is given
by:

$\displaystyle{\bf E^{1}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Sigma}{\Delta_{\theta}}}d\theta,\quad{\bf
E^{3}}=\sqrt{\frac{\Sigma}{\Delta_{r}}}dr,$ $\displaystyle{\bf E^{2}}$
$\displaystyle=$
$\displaystyle\frac{\sin\theta}{\Xi}\sqrt{\frac{\Delta_{\theta}}{\Sigma}}\left[adt-\left(r^{2}+a^{2}\right)d\phi\right],$
$\displaystyle{\bf E^{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Xi}\sqrt{\frac{\Delta_{r}}{\Sigma}}\left(dt-a\sin^{2}\theta
d\phi\right).$ (69)

Then projecting $T_{\mu\nu}$ onto the orthonormal basis its canonical form is
obtained,

$OT^{ab}=\begin{pmatrix}\frac{T_{\theta\theta}}{g_{\theta\theta}}&0&0&0\\\
0&\frac{T_{\theta\theta}}{g_{\theta\theta}}&0&0\\\
0&0&\frac{T_{rr}}{g_{rr}}&0\\\
0&0&0&-\frac{T_{rr}}{g_{rr}}\end{pmatrix}=\begin{pmatrix}p_{1}&0&0&0\\\
0&p_{2}&0&0\\\ 0&0&p_{3}&0\\\ 0&0&0&\mu\end{pmatrix}.$ (70)

Where $p_{1},~{}p_{2},~{}p_{3}$ represent the principal pressures in the three
spacelike directions $\mathbf{E^{\alpha}}(\alpha=1,2,3)$. The eigenvalue $\mu$
represents the energy density as measured by and observer whose world-line at
point $p$ has a unit tangent vector $\mathbf{E^{4}}$. Note that the $OT^{ab}$
tensor has the canonical form type I (Hawking ). In terms of the EM potentials
and the metric function $K(r)$ the $OT^{ab}$ components are

$\displaystyle-\mu$ $\displaystyle=$ $\displaystyle
p_{3}=\frac{T_{rr}}{g_{rr}}=-\sqrt{F^{2}+G^{2}}\frac{\mathcal{L}_{F}}{4\pi}+\frac{K^{\prime\prime}(r)}{32\pi\Sigma}$
$\displaystyle=$ $\displaystyle\frac{\Xi^{2}}{8\pi
a\sin\theta}\left(\frac{\partial A_{t}}{\partial
r}\frac{\partial{}^{\star}P_{t}}{\partial\theta}-\frac{\partial
A_{t}}{\partial\theta}\frac{\partial{}^{\star}P_{t}}{\partial
r}\right)+\frac{K^{\prime\prime}(r)}{32\pi\Sigma}$

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle
p_{2}=\frac{T_{\theta\theta}}{g_{\theta\theta}}=\sqrt{F^{2}+G^{2}}\frac{\mathcal{L}_{F}}{4\pi}+\frac{K^{\prime\prime}(r)}{32\pi\Sigma}$
$\displaystyle=$ $\displaystyle\frac{-\Xi^{2}}{8\pi
a\sin\theta}\left(\frac{\partial A_{t}}{\partial
r}\frac{\partial{}^{\star}P_{t}}{\partial\theta}-\frac{\partial
A_{t}}{\partial\theta}\frac{\partial{}^{\star}P_{t}}{\partial
r}\right)+\frac{K^{\prime\prime}(r)}{32\pi\Sigma},$

###  VI.1 Energy conditions

In what follows we determine the conditions on the $OT^{ab}$ components in
order to fulfil the weak, dominant and strong energy conditions, that are
summarized in Table 1.

Energy Cond. |  $OT_{ab}$ components |  | Inequalities
---|---|---|---
Weak | $T_{ab}u^{a}u^{b}\geq 0$ |  |  $\mu\geq 0$, $\mu+p_{\alpha}\geq 0$
Dominant |  $T_{ab}u^{a}u^{b}\geq 0$, $T^{ab}u_{b}\leq 0$ |  |  $\mu\geq 0$, $\mu\geq|p_{\alpha}|$
Strong | $(T_{ab}-\frac{1}{2}Tg_{ab})u^{a}u^{b}\geq 0$ |  |  $\mu+\sum_{\alpha=1}^{3}p_{\alpha}\geq 0$,
| $\mu+p_{\alpha}\geq 0$
Table 1: Energy conditions. The inequalities that the stress-energy tensor
$T_{ab}$ should satisfy to fulfill the energy conditions; $u^{a}$ is a
timelike vector; $\mu=-OT_{rr}/g_{rr}=-p_{3}$ and
$p_{1}=p_{2}=OT_{\theta\theta}/g_{\theta\theta}$.

####  VI.1.1  Weak Energy Condition (WEC)

If the energy momentum tensor is of type I, the Weak Energy Condition
(WEC)Hawking  holds if $\mu\geq 0$. Additionally, the energy density should
not be exceeded by any pressure, such that $\mu+p_{\alpha}\geq
0,\quad\alpha=1,2,3$. In the case of the quartic vector potential NLE-KN
solution WEC amounts to the following conditions,

$\mu=\frac{\tilde{Q}^{2}}{8\pi\Sigma^{2}}\left(1-2\xi r^{3}\right)\geq
0,\quad\mu+p_{3}=0.$

$\displaystyle\mu+p_{1,2}=\frac{\tilde{Q}^{2}}{8\pi\Sigma^{2}}\left(2+3\xi
ra^{2}\cos^{2}\theta-\xi r^{3}\right)\geq 0,$ (73)

where $\tilde{Q}^{2}=(Q_{e}^{2}+Q_{m}^{2})$ The conditions (VI.1.1) are
fulfilled if the NLE parameter is less than zero, $\xi<0$. If $\xi=0$ the KN
solution is recovered. If $\xi>0$ Eqs. (VI.1.1) reduce to $0\leq\xi
r^{3}\leq\frac{1}{2}$, such that as $r$ approaches infinity, the only value
that keeps within this range is $\xi=0$.

####  VI.1.2  Dominant Energy Condition (DEC)

The Dominant Energy Condition (DEC)Hawking  holds if $\mu\geq 0$, and $-\mu-
p_{\alpha}\leq 0\leq\mu-p_{\alpha}$. This leads to the following inequalities
for the NLE-KN

$\displaystyle-(\mu+p_{1,2})\leq$ $\displaystyle 0\leq\mu-
p_{1,2}=-\frac{T}{2}$
$\displaystyle\frac{\tilde{Q}^{2}}{8\pi\Sigma^{2}}\left[\xi r^{3}-3\xi
ra^{2}\cos^{2}\theta-2\right]\leq$ $\displaystyle
0\leq-\frac{3\tilde{Q}^{2}\xi r}{8\pi\Sigma}.$ (74)

The case for $p_{3}$ reduces to $0\leq 2\mu$. If $\xi\leq 0$ then (VI.1.2) are
automatically satisfied.

####  VI.1.3  Strong Energy Condition (SEC)

The SEC guarantees that matter is attractive, causing geodesics to converge.
In terms of the $OT_{ab}$ components SEC amounts to $\mu+p_{1}+p_{2}+p_{3}\geq
0$. For the quartic vector potential NLE-KN solution this reduces to

$0\leq\frac{\tilde{Q}^{2}}{4\pi\Sigma^{2}}\left(1+3\xi
ra^{2}\cos^{2}\theta+\xi r^{3}\right),$ (75)

the above inequality is satisfied for $\xi=0$ (KN case); however if $\xi<0$
the equation is violated when $r\rightarrow\infty$ by a large negative
pressure. The case $\xi>0$ causes WEC and DEC to be violated, therefore the
most sensible choice for the quartic potential generalization of the KN
solution is $\xi<0$. Note that even SEC is violated by the introduction of the
NLE parameter $\xi$, this does not make less meaningful the NLE-KN solution
since SEC can be violated by certain forms of matter such as a massive scalar
field and quantum fields can generically violate any of the energy conditions
Carrol2004 .

###  VI.2 Weyl conformal symmetry

From the geometric part of Einstein’s equations, that is, from the left hand
side of $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}$, the trace of the
energy-momentum tensor can be determined as

$\displaystyle 8\pi\tensor{T}{{}^{\mu}_{\mu}}$
$\displaystyle=\frac{1}{\Sigma}\left[\Delta_{r}{}^{\prime\prime}-2+\left(\frac{2}{3}a^{2}+4r^{2}\right)\Lambda\right]$
$\displaystyle=\frac{K^{\prime\prime}(r)}{\Sigma}.$ (76)

Therefore for the quartic vector potential NLE-KN solution, since
$K^{\prime\prime}(r)\neq 0$, then the conformal invaruance is broken by the
NLE field.

The Maxwell stress-tensor is trace-free; in agreement that for the Kerr-Newman
solution

$\displaystyle\Delta_{r}$ $\displaystyle=$ $\displaystyle
r^{2}-2mr+a^{2}+Q_{e}^{2}-\frac{\Lambda}{3}r^{2}(r^{2}+a^{2}),$
$\displaystyle\Delta_{r}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
2-\left(\frac{2}{3}a^{2}+4r^{2}\right)\Lambda,$ (77)

with $K(r)=Q_{e}^{2}$ then $K^{\prime\prime}(r)=0$ and the trace-free
condition is fulfilled. Moreover for any NLE that preserves conformal
invariance in a Kerr-like geometry, the condition to be trace-free demands
$K(r)^{\prime\prime}=0.$

For instance the ModMax generalization consists in transforming
$Q_{e}^{2}\mapsto e^{-\gamma}Q_{e}^{2}$, then preserving the $T_{\mu\nu}$
traceless.

###  VI.3 Lagrangian for the quartic vector potential solution

The Lagrangian for the Kerr-Newman metric in terms of the coordinates is given
by

$\mathcal{L}_{KN}=\frac{h_{1}(r,\theta)h_{2}(r,\theta)}{2(r^{2}+a^{2}\cos^{2}\theta)^{4}},$
(78)

where the functions $h_{1}(r,\theta)$ and $h_{2}(r,\theta)$ are

$\displaystyle h_{1}(r,\theta)$
$\displaystyle=r^{2}(Q_{e}-Q_{m})+2ra\cos\theta(Q_{e}+Q_{m})$
$\displaystyle-a^{2}\cos^{2}\theta(Q_{e}-Q_{m}),$ $\displaystyle
h_{2}(r,\theta)$ $\displaystyle=r^{2}(Q_{e}+Q_{m})-2ra\cos\theta(Q_{e}-Q_{m})$
$\displaystyle-a^{2}\cos^{2}\theta(Q_{e}+Q_{m}).$ (79)

While from the matter content of the stress-energy tensor,

$4\pi
T_{\mu\nu}=\mathcal{L}g_{\mu\nu}-\tensor{F}{{}_{\mu}^{\alpha}}P_{\nu\alpha}$,

the expression for the Lagrangian $\mathcal{L}$ is found as,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{\Xi^{2}}{2a\sin\theta}\left(\frac{\partial
A_{t}}{\partial
r}\frac{\partial{}^{\star}P_{t}}{\partial\theta}+\frac{\partial
A_{t}}{\partial\theta}\frac{\partial{}^{\star}P_{t}}{\partial
r}\right)+\frac{K^{\prime\prime}(r)}{8\Sigma}.$ (80)

Then for the quartic vector potential NLE-KN, the Lagrangian
$\mathcal{L}_{NLE}(r,\theta)$ is given by

$\displaystyle\mathcal{L}_{NLE}(F,G)$ $\displaystyle=$
$\displaystyle\mathcal{L}_{KN}+\xi\frac{\left[(Q_{e}^{2}+2Q_{m}^{2})r-Q_{e}Q_{m}\,a\cos\theta\right]}{2(r^{2}+a^{2}\cos^{2}\theta)}$
(81) $\displaystyle-\xi^{2}\frac{Q_{e}Q_{m}\,ra\cos\theta}{4}.$

And we recover $\mathcal{L}_{KN}$ making $\xi=0$.

##  VII Conclusions

We present in detail the method to find nonlinear electromangetic (NLE)
solutions in a Kerr-like metric, that previously was introduced in
AGarcia_Annals2022 . We then examine the general form of the electromagnetic
potentials to determine exact solutions of the coupled NLE-Einstein equations
in a Kerr-like geometry; and it was found that there are only two possible NLE
generalizations; one of them was already presented in AGarcia_Annals2022 . The
second case, that we called quartic vector potential solution, is new and
corresponds to a NLE generalization of the Kerr-Newman black hole
characterized by the introduction of a NLE parameter that induces a third
horizon, resembling a cosmological horizon. The electromagnetic fields,
horizons and ergosphere of the NLE quartic vector potential generalization of
the Kerr-Newman (KN) solution are presented. The static limit corresponds to a
NLE generalization of the Reissner-Nordstrom BH with cosmological constant.

Additionally, we determine the canonical form of the NLE stress-energy tensor
and set up the inequalities to fulfill the physically reasonable energy
conditions. Imposing the energy conditions to the found solution we find as a
constraint that the nonlinear parameter $\xi$ should be negative. The trace of
the NLE stress-energy tensor does not vanish, then the NLE field breaks
conformal invariance. Moreover, the expression of the NLE Lagrangian
$\mathcal{L}_{NLE}(r,\theta)$ is presented, its form consists of two terms,
the KN Lagrangian plus a term derived from the NLE contribution.

The nature and precise interpretation of the NLE field deserves further
investigation, for instance if there are electromagnetic multipoles associated
to the NLE fields; we leave this for a future work.

Acknowledgements

NB acknowledges partial support by CONAHCYT-Mexico project CBF-2023-2024-811.
The work of OG has been sponsored by CONAHCYT-Mexico through the Ph. D.
scholarship No. 815804. OG also acknowledges hospitality by Profr. C.
Laemmerzahl at the Universität Bremen where part of this work was done.

## References

  * (1) B. P. Abbot et al. [LIGO Scientific and Virgo Collaborations], Tests of general relativity with GW150914, Phys. Rev. Lett. 116, 221101 (2016).
  * (2) B. P. Abbot et al. [LIGO Scientific, Virgo Collaborationsand KAGRA Collaboration], GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run [arXiv: 2111.03606]
  * (3) E. Ayon-Beato, A. A. Garcia, Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics Phys. Rev. Lett. 80 5056 (1998)
  * (4) N. Breton, C. Laemmerzahl, A. Macias, Rotating structure of the Euler-Heisenberg black hole Phys. Rev. D 105, 104046 (2022).
  * (5) A. A. García-Díaz, Stationary Rotating Black Hole Exact Solution within Einstein-Nonlinear Electrodynamics, arXiv:2112.06302
  * (6) Alberto A. Garcia-Diaz, AdS-dS stationary rotating black hole exact solution within Einstein-nonlinear electrodynamics, Annals of Phys. 441 168880 (2022). see also arXiv:
  * (7) J. Plebanski, Electromagnetic Waves in Gravitational Fields. Phys. Rev 118, 1396–1408 (1960) doi:10.1103/PhysRev.118.1396
  * (8) G. Boillat, Nonlinear Electrodynamics: Lagrangians and Equations of Motion, J. Math. Phys. 11, 941-951 (1970)
  * (9) G. O. Schellstede, V. Perlick, C. Laemmerzahl, On causality in nonlinear vacuum electrodynamics of the Plebanski class, Ann. Phys. (Berlin) 528, 738–749 (2016)
  * (10) M. Novello, V. A. De Lorenci, J. M. Salim and R. Klippert, Geometrical aspects of light propagation in nonlinear electrodynamics, Phys. Rev. D 61, 45001 (2000).
  * (11) Y. N. Obukov, G. F. Rubilar, Fresnel analysis of wave propagation in nonlinear electrodynamics, Phys.Rev.D66 024042 (2002)
  * (12) V. P. Frolov, P. Krtous̆ and D. Kubizn̆ák, Separability of Hamilton-Jacobi and Klein-Gordon equations in general Kerr-NUT-AdS spacetimes, JHEP02 (2007)005.
  * (13) E. Witten, Anti De Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  * (14) S. W. Hawking and G. F. R. Ellis, The Large Scale structure of space-time, Cambridge Univ. Press 1973
  * (15) N. Breton, G. Gutierrez-Cano, A. A. Garcia-Diaz, Motion of the charged test particle in the spinning nonlinear electromagnetic black hole, Phys. Rev. D 106, 024056 (2022)
  * (16) S. Carrol, Spacetime and Geometry. An Introduction to General Relativity, Addison Wesley, 2004.