11institutetext:  Univ. Bordeaux, Inserm, Bordeaux Population Health Research
Center, team ERIAS, UMR 1219, F-33000 Bordeaux, France
<EMAIL_ADDRESS>22institutetext: CHU de Bordeaux, Pôle de santé
publique, Service d’information médicale, Informatique et Archivistique
Médicales (IAM), F-33000 Bordeaux, France

# IAM at CLEF eHealth 2018: Concept Annotation and Coding in French Death
Certificates

Sébastien Cossin 1122 Vianney Jouhet 1122 Fleur Mougin 11 Gayo Diallo 11
Frantz Thiessard 1122

###### Abstract

In this paper, we describe the approach and results for our participation in
the task 1 (multilingual information extraction) of the CLEF eHealth 2018
challenge. We addressed the task of automatically assigning ICD-10 codes to
French death certificates. We used a dictionary-based approach using materials
provided by the task organizers. The terms of the ICD-10 terminology were
normalized, tokenized and stored in a tree data structure. The Levenshtein
distance was used to detect typos. Frequent abbreviations were detected by
manually creating a small set of them. Our system achieved an F-score of 0.786
(precision: 0.794, recall: 0.779). These scores were substantially higher than
the average score of the systems that participated in the challenge.

###### Keywords:

Semantic annotation Entity recognition Natural Language Processing Death
certificates

##  1 Introduction

In this paper, we describe our approach and present the results for our
participation in the task 1, i.e. multilingual information extraction, of the
CLEF eHealth 2018 challenge [1]. More precisely, this task consists in
automatically coding death certificates using the International Classification
of Diseases, 10th revision (ICD-10) [2].

We addressed the challenge by matching ICD-10 terminology entries to text
phrases in death certificates. Matching text phrases to medical concepts
automatically is important to facilitate tasks such as search, classification
or organization of biomedical textual contents [3]. Many concept recognition
systems already exist [3, 4]. They use different approaches and some of them
are open source. We developed a general purpose biomedical semantic annotation
tool for our own needs. The algorithm was initially implemented to detect
drugs in a social media corpora as part of the Drugs-Safe project [5]. We
adapted the algorithm for the ICD-10 coding task. The main motivation in
participating in the challenge was to evaluate and compare our system with
others on a shared task.

##  2 Methods

In the following subsections, we describe the corpora, the terminology used,
the steps of pre-processing and the matching algorithm.

###  2.1 Corpora

The data set for the coding of death certificates is called the CépiDC corpus.
Three CSV files (AlignedCauses) were provided by task organizers containing
annotated death certificates for different periods : 2006 to 2012, 2013 and
2014. This training set contained $125383$ death certificates. Each
certificate contains one or more lines of text (medical causes that led to
death) and some metadata. Each CSV file contains a "Raw Text" column entered
by a physician, a "Standard Text" column entered by a human coder that
supports the selection of an ICD-10 code in the last column. Table 1 presents
an excerpt of these files. Zero to multiples ICD-10 codes can be assigned to
each line of a death certificate.

Raw Text | Standard Text | ICD-10 code
---|---|---
SYNDROME DE GLISEMENT AVEC GRABATISATION DEPUIS OCTOBRE 2012 | syndrome glissement | R453
SYNDROME DE GLISEMENT AVEC GRABATISATION DEPUIS OCTOBRE 2012 | grabatisation 2 mois | R263
?tablename? 1: One raw text sample with three selected columns of the training
data. Raw Text: text entered by a physician (duplicated in the file when
multiple codes are assigned).
Standard Text: text entered by a human coder to support the selection of the
ICD-10 code

###  2.2 Dictionaries

We constructed two dictionaries based on ICD-10. In practice, we selected all
the terms in the "Standard Text" column of the training set to build the first
one which was used in the second run. In the first run, we added to this
previous set of terms the 2015 ICD-10 dictionary provided by the task
organizers. This dictionary contained terms that were not present in the
training corpus. When a term was associated with multiple ICD-10 codes in our
dictionary, we kept the most frequent one (Table 2).

The first dictionary contained $42439$ terms and 3,539 ICD-10 codes (run2) and
the second one $148448$ terms and 6,392 ICD-10 codes (run1).

Metadata on death causes were not used (age, gender, location of death).

Standard Text | ICD-10 code | number of occurrences
---|---|---
avc | F179 | 1
avc | I64 | 260
avc | I640 | 1,635
avc | T821 | 1
avc | Z915 | 1
avc | I489 | 1
?tablename? 2: Some terms like "avc" were associated with multiple ICD-10
codes in our dictionary. We kept the most frequent ICD-10 code, I640 in this
case.
Standard Text: text entered by a human coder to support the selection of an
ICD-10 code.

###  2.3 Terms pre-processing

All the terms were normalized through accents (diacritical marks) and
punctuation removal, lowercasing and stopwords removal (we created a list of
25 stopwords for this task). Then, each term was tokenized and stored in a
tree data structure. Each token of a N-gram term is a node in the tree and
N-grams correspond to different root-to-leaf paths [6] (Figure 1).

?figurename? 1: Terms in our dictionaries are normalized, tokenized and stored
in a tree data structure. Each dark blue node corresponds to a term. In this
figure, five terms are described: "insuffisance cardiaque", "insuffisance
cardiaque aigue", "insuffisance cardiaque congestive", "insuffisance
respiratoire" and "insuffisance respiratoire aigue". The token "insuffisance"
is the first token of many terms but it does not match any term by itself
(light blue).

###  2.4 Matching algorithm

?figurename? 2: Algorithm pipeline. The text was normalized and tokenized. For
each token, the algorithm used three matching techniques to detect a token in
the dictionary. Possible dictionary tokens depended on the current location in
the tree.
Raw Text: a death cause entered by a physician. Solid arrows: path taken by
the algorithm. Dotted arrows: other available tokens at this depth. Dark blue
nodes : a term in the dictionary. Light blue nodes: token that does not match
any term by itself

The goal of our algorithm was to recognize one or many dictionary entries in a
raw text. An example is given in Figure 2. For each raw text entry, the same
normalization steps described above were performed first. The raw text was
then tokenized. For each token, the algorithm looked for an available
dictionary token depending on where it currently was in the tree. For example,
the token "cardiaque" was possible after the token "insuffisance" but was not
available at the root of the tree.
For each token, the algorithm used three matching techniques: perfect match,
abbreviation match and Levenshtein match. The abbreviation match technique
used a dictionary of abbreviations. We manually added nine frequent
abbreviations after looking at some examples. The Levenshtein matching
technique used the Levenshtein distance. It corresponds to the minimum number
of single-character edits (insertions, deletions or substitutions) required to
change one token into the other. The Lucene™implementation of the Levenshtein
distance was used.
In Figure 2, the algorithm used these three techniques to match the tokens
"ins", "cardiaqu", "aigue" to the dictionary term "insuffisance cardiaque
aigue" whose ICD-10 code is I509. As the following token "detresse" was not a
dictionary entry at this depth, the algorithm saved the previous and longest
recognized term and restarted from the root of the tree. At this new level,
"detresse" was detected but as no term was associated with this token alone,
no ICD-10 code was saved. Finally, only one term was recognized in this
example.
Besides unigrams, bigrams were also indexed in Lucene™to resolve composed
words. For example, "meningoencephalite" matched the dictionary entry
"meningoencephalite" by a perfect match and "meningo encephalite" thanks to
the Levensthein match (one deletion). Therefore, the algorithm entered two
different paths in the tree (Figure 3). By combining these different matching
methods for each token, the algorithm was able to detect multiple lexical
variants. The program was implemented in Java and the source code is on
Github111https://github.com/scossin/IAMsystem.

?figurename? 3: The token "meningoencephalite" was matched to the unigram
"meningoencephalite" by the perfect match method and to the bigram "meningo
encephalite" by the Levenshtein method. The algorithm explored different paths
in the tree. It detected the term "meningo encephalite virale"
("meningoencephalite virale" did not exist). Only the longest term was kept.

##  3 Results

We submitted two runs on the CépiDC test set, one used all the terms entered
by human coders in the training set only (run 2), the other (run 1) added the
2015 ICD-10 dictionary provided by the task organizers to the set the terms of
run 1. We obtained our best precision (0.794) and recall (0.779) with run 2.

Table 3 shows the performance of our system with median and average scores of
all participants in this task.

System | Precision | Recall | F-measure
---|---|---|---
run1 | 0.782 | 0,772 | 0.777
run2 | 0.794 | 0.779 | 0.786
average score | 0.712 | 0.581 | 0.634
median score | 0.771 | 0.545 | 0.641
?tablename? 3: System performance on the CépiDC test set

##  4 Discussion

Surprisingly, adding more terms (run 1) did not improve the recall, which
appears to be even slightly worse. The results were quite promising for our
first participation in this task, using a general purpose annotation tool.

A limitation of the proposed algorithm that impacted recall was the absence of
term detection when adjectives were isolated. For example, in the sentence
"metastase hepatique et renale", "metastase renale" was not recognized even
though the term existed. This situation seemed to be quite frequent.

Some frequent abbreviations were manually added to improve the recall in this
corpora. Improvement at this stage may be possible by automating the
abbreviation detection or by adding more entries manually.

In the past, other dictionary-based approaches performed better [7]. In 2016,
the Erasmus system [8] achieved an F-score of 0.848 without spelling
correction techniques. In 2017, the SIBM team [9] used a dictionary-based
approach with fuzzy matching methods and phonetic matching algorithm to obtain
an F-score of 0.804.

Further improvement may be possible by using a better curated terminology. We
are currently investigating frequent irrelevant codes that may have impacted
the precision. A post-processing filtering phase could improve the precision.

We also plan to combine machine learning techniques with a dictionary-based
approach. Our system can already detect and replace typos and abbreviations to
help machine learning techniques increase their performance.

##  5 Affiliation

DRUGS-SAFE National Platform of Pharmacoepidemiology, France

##  6 Funding

The present study is part of the Drugs Systematized Assessment in real-liFe
Environment (DRUGS-SAFE) research platform that is funded by the French
Medicines Agency (Agence Nationale de Sécurité du Médicament et des Produits
de Santé, ANSM). This platform aims at providing an integrated system allowing
the concomitant monitoring of drug use and safety in France. The funder had no
role in the design and conduct of the studies; collection, management,
analysis, and interpretation of the data; preparation, review, or approval of
the manuscript; and the decision to submit the manuscript for publication.
This publication represents the views of the authors and does not necessarily
represent the opinion of the French Medicines Agency.

## References

  * [1] Suominen, H. and Kelly, L. and Goeuriot, L. and Kanoulas, E. and Azzopardi, L. and Spijker, R. and Li, D. and Névéol, A. and Ramadier, L. and Robert, A. and Palotti, J. and Jimmy and Zuccon, G.: Overview of the CLEF eHealth Evaluation Lab 2018. CLEF 2018 - 8th Conference and Labs of the Evaluation Forum, Lecture Notes in Computer Science (LNCS). Springer. (2018).
  * [2] Névéol, A. and Robert, A. and Grippo, F. and Lavergne, T. and Morgand C. and Orsi, C. and Pelikán L. and Ramadier, L. and Rey, G. and Zweigenbaum,P: CLEF eHealth 2018 Multilingual Information Extraction task Overview: ICD10 Coding of Death Certificates in French, Hungarian and Italian. CLEF 2018 Evaluation Labs and Workshop: Online Working Notes, CEUR-WS, September, 2018.
  * [3] Jovanović, J. and Bagheri, E.: Semantic Annotation in Biomedicine: The Current Landscape. Journal of Biomedical Semantics (2017). 10.1186/s13326-017-0153-x
  * [4] Tseytlin, E. and Mitchell, K. and Legowski, E. and Corrigan, J. and Chavan, G. and Jacobson, RS.: BMC bioinformatics (2016). NOBLE - Flexible Concept Recognition for Large-Scale Biomedical Natural Language Processing. 10.1186/s12859-015-0871-y
  * [5] Bigeard, E.: Construction de lexiques pour l’extraction des mentions de maladies dans les forums de santé. TALN (2017).
  * [6] Pibiri, GE. and Venturini, R.: Efficient Data Structures for Massive N-Gram Datasets. In: 40th International ACM SIGIR Conference on Research and Development in Information Retrieval, pp. 615–624. ACM, Shinjuku, Tokyo, Japan (2017) 10.1145/3077136.3080798
  * [7] Névéol, A. and Cohen, KB. and Grouin, C. and Hamon, T. and Lavergne, T. and Kelly, L. and Goeuriot, L. and Rey, G. and Robert, A. and Tannier, X. and Zweigenbaum, P.: Clinical Information Extraction at the CLEF eHealth Evaluation Lab 2016. CEUR workshop proceedings. (2016).
  * [8] Van Mulligen, E. and Afzal, Z. and Akhondi, S. and Vo, D. and Kors, J.: Erasmus MC at CLEF eHealth 2016: Concept Recognition and Coding in French Texts. Online Working Notes. CEUR-WS. (2016).
  * [9] Cabot, C. and Soualmia, LF. and Darmoni, S.: SIBM at CLEF eHealth Evaluation Lab 2017: Multilingual Information Extraction with CIM-IND. CEUR-WS. (2017).


# Extremal Myers-Perry black holes coupled to Born-Infeld electrodynamics in
five dimensions

Masoud<EMAIL_ADDRESS>José P. S.
<EMAIL_ADDRESS>Ahmad<EMAIL_ADDRESS>1
Centro Multidisciplinar de Astrofísica - CENTRA, Departamento de Física,
Instituto Superior Técnico - IST, Universidade Técnica de Lisboa - Av. Rovisco
Pais 1, 1049-001 Lisboa, Portugal.
2 Physics Department and Biruni Observatory, College of Sciences, Shiraz
University, Shiraz 71454, Iran
3 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran

###### Abstract

Abstract

We construct a new class of perturbative extremal charged rotating black hole
solutions in five dimensions in the Einstein–Born-Infeld theory. We start with
an extremal five-dimensional Myers-Perry black hole seed in which the two
possible angular momenta have equal magnitude, and add Born-Infeld electrical
charge keeping the extremality condition as a constraint. The perturbative
parameter is assumed to be the electric charge $q$ and the perturbations are
performed up to 4th order. We also study some physical properties of these
black holes. It is shown that the perturbative parameter $q$ and the Born-
Infeld parameter $\beta$ modify the values of the physical quantities of the
black holes. The solution suggests that the magnetic moment and gyromagnetic
ratio of the black hole spacetime change sign when the Born-Infeld character
of the solution starts to depart strongly from the Maxwell limit. We venture
an interpretation for the effect.

##  I Introduction

The field of black hole solutions in higher dimensions was started in 1963 by
Tangherlini Tan . After some discussion of the old problem of the
dimensionality of space he proceed to find static spherical symmetric
solutions of the $d$ dimensional Einstein-Maxwell equations, with $d\geq 4$,
generalizing thus the Schwarzschild and the Reissner-Nordström solutions.
Tangherlini also included a cosmological term finding the corresponding de
Sitter (dS) and anti-de Sitter (AdS) $d$-dimensional black holes Tan .

Since then, with the belief that at a more fundamental level Einstein’s theory
has to be modified and extra dimensions might be crucial in the process,
static higher-dimensional black holes solutions have been devised in
gravitational theories such as Kaluza-Klein, Gauss-Bonnet theory and its
Lovelock extension, scalar-tensor gravity, supergravity, and low-energy string
theory. Likewise, there has been attempts to modify Maxwell’s electromagnetism
in order to deal in a more consistent manner with the point particle problem
and its field divergences. An important modification is given by the Born-
Infeld nonlinear electrodynamics. Although it became less prominent with the
appearance of quantum electrodynamics and its renormalization scheme, the
Born-Infeld theory has been now occurring repeatedly with the development of
string theory, where the dynamics of D-branes is governed by a Born-Infeld
type action. It is thus natural to study higher dimensional black holes in
these extended gravity-electromagnetic set ups.

For the static case, several examples of higher dimensional black holes can be
given. Black holes in Lovelock gravity and Maxwell electromagnetism were
studied in banteitzadilemos . Black hole solutions in Einstein-Born-Infeld
gravity are less singular in comparison with the Tangherlini-Reissner-
Nordström solution and such solutions with and without a cosmological constant
have been discussed in general relativity Garcia1 ; Wiltshire1 ; Aiello1 ;
Tamaki1 ; Dey1 ; Gunasekaran , and in Gauss-Bonnet GB  and Lovelock Lov
gravities. The extension to the cases where the horizon has zero or negative
curvature has also been considered Cai1 . Scalar-tensor theories of gravity
coupled to Born-Infeld black hole solutions have also been studied in yaz3 .
Attempts to find solutions in general relativity coupled to other fields, such
as dilaton, rank three antisymmetric tensor, and Born-Infeld and others, have
been performed to construct exact solutions YI ; Tam ; yaz ; she . In these
theories the dilaton field can be thought of as coming from a scalar field of
low energy limit of string theory. The appearance of the dilaton field changes
the asymptotic behavior of the solutions to be neither asymptotically flat nor
de Sitter or Anti-de Sitter.

Introducing rotation makes the search for solutions much more difficult.
Following the example of the extension of the Schwarzschild and Reissner-
Nordström solutions to higher dimensions in different theories, it was natural
to do the same for the Kerr and the Kerr-Newman solutions. The generalization
of the Kerr solution to higher dimensional Einstein gravity was performed by
Myers and Perry Myer . These Myers-Perry solutions include the non-trivial
cases of several modes of rotation due to the existence of other rotation
planes in higher dimensions. The inclusion of a cosmological constant on these
higher dimensional solutions was performed in many . Rotating black hole
solutions in other gravity theories like Gauss-Bonnet or Lovelock are not
know.

Finally, the generalization of the Kerr-Newman black holes to higher
dimensional Einstein-Maxwell theory has not been found. Likewise, the
generalization of Kerr with some other gauge field, different from Maxwell, to
higher dimensions has not had success. In order to study this problem one has
to resort to approximations, as was done in Aliev2 , when studying charged
rotating black hole solutions in higher dimensions with a single rotation
parameter in the limit of slow rotation (see also Aliev3 ; Aliev4 ; kunz1 ).
Further study, relying on perturbative or numerical methods, to construct
those solutions in asymptotically flat backgrounds has been performed in Kunz2
and in asymptotically anti-de Sitter spacetimes in Kunz3 . Employing higher
perturbation theory with the electric charge as the perturbation parameter,
solutions and properties of charged rotating Einstein-Maxwell asymptotically
flat black holes have been constructed in five dimensions Navarro . Focusing
on extremal black holes, this perturbative method was also applied to obtain
Einstein-Maxwell black holes with equal magnitude angular momenta in odd
dimensions Allahverdi1 . A generalization to include a scalar field in a
Einstein-Maxwell-dilaton theory was performed in Allahverdi2  where black
holes with equal magnitude angular momenta in general odd dimensions were
obtained.

In this paper, we use this perturbative approach to find extremal rotating
Einstein-Born-Infeld black holes in a five dimensional spacetime. Starting
from the Myers-Perry black holes Myer , we evaluate the perturbative series up
to 4th order in the Born-Infeld electric charge parameter $q$. We determine
the physical properties of these black holes for general Born-Infeld coupling
constant $\beta$. In fact, we study the effects of the perturbative parameter
$q$ and the Born-Infeld parameter $\beta$ on the mass, angular momentum,
magnetic moment, gyromagnetic ratio, and horizon radius of these rotating
black holes.

The outline of this paper is as follows: In section II, we present the basic
field equations of nonlinear Born-Infeld theory in Einstein gravity and obtain
a new class of perturbative charged rotating solutions in five dimensions. In
section III, we calculate the physical quantities of the solutions and discuss
their properties. In section IV we study the mass formula for these black
holes. Section V is devoted to conclusions.

##  II Metric and Gauge Potential

We start with the Einstein-Hilbert action coupled to the Born-Infeld nonlinear
gauge field in five dimensions

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
dx^{5}\sqrt{-g}\left(\frac{R}{16\pi G_{5}}\text{ }+L(F)\right),$ (1)

where $G_{5}$ is five dimensional Newtonian constant, ${R}$ is the curvature
scalar and $L(F)$ is the Lagrangian of the nonlinear Born-Infeld gauge field
given by

$\displaystyle L(F)$ $\displaystyle=$ $\displaystyle
4\beta^{2}\left(1-\sqrt{1+\frac{F^{\mu\nu}F_{\mu\nu}}{2\beta^{2}}}\right),$
(2)

where, in turn, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is
the electromagnetic field tensor, $A_{\mu}$ is the electromagnetic vector
potential, and $\beta$ is the Born-Infeld parameter with unit of mass. In the
limit $\beta\rightarrow\infty$, $L(F)$ reduces to the Lagrangian of the
standard Maxwell field, $L(F)={F^{\mu\nu}F_{\mu\nu}}$. The equations of motion
can be obtained by varying the action with respect to the gravitational field
$g_{\mu\nu}$ and the gauge field $A_{\mu}$. This procedure yields the
gravitational field equations

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{2}g_{\mu\nu}L(F)+\frac{2F_{\mu\eta}F_{\nu}^{\text{
}\eta}}{\sqrt{1+\frac{F^{\mu\nu}F_{\mu\nu}}{2\beta^{2}}}},$ (3)

and the electromagnetic equation

$\partial_{\mu}{\left(\frac{\sqrt{-g}F^{\mu\nu}}{\sqrt{1+\frac{F^{\mu\nu}F_{\mu\nu}}{2\beta^{2}}}}\right)}=0\,.$
(4)

Our aim here is to find perturbative extremal charged rotating black hole
solutions of the above field equations in five dimensions. Using coordinates
$(t,r,\theta,\varphi_{1},\varphi_{2})$, the $5$-dimensional Myers-Perry
solution Myer  restricted to the case where the two possible angular momenta
have equal magnitude, and following Navarro  for the parametrization of the
metric, one has

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{tt}dt^{2}+\frac{dr^{2}}{W}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}_{1}+\cos^{2}\theta
d\varphi^{2}_{2}\right)+N\left(\varepsilon_{1}\sin^{2}\theta
d\varphi_{1}+\varepsilon_{2}\cos^{2}\theta d\varphi_{2}\right)^{2}$ (5)
$\displaystyle-2B\left(\varepsilon_{1}\sin^{2}\theta
d\varphi_{1}+\varepsilon_{2}\cos^{2}\theta d\varphi_{2}\right)dt,$

where $\varepsilon_{k}$ denotes the sense of rotation in the $k$-th orthogonal
plane of rotation, such that $\varepsilon_{k}=\pm 1$, $k=1,2$. For the
mentioned Myers-Perry solution Myer  one has
$g_{tt}=-1+\frac{2\hat{M}}{r^{2}}$,
$W=1-\frac{2\hat{M}}{r^{2}}+\frac{2\hat{J}^{2}}{\hat{M}r^{4}}$, and
$N=\frac{2\hat{J}^{2}}{\hat{M}r^{2}}$, where $\hat{M}$ and $\hat{J}$ are two
constants, namely, the mass and angular momentum parameters, respectively,
related to the mass $M$ and angular momenta $J$ of the Myers-Perry solution,
through the relations $\hat{M}=\frac{16\pi G_{5}}{3A}M$ and
$\hat{J}=\frac{4\pi G_{5}}{A}J$, where $A$ is the area of the unit $3$-sphere.
An adequate parametrization for the gauge potential is given by

$\displaystyle A_{\mu}dx^{\mu}$ $\displaystyle=$ $\displaystyle
a_{0}+a_{\varphi}\left(\varepsilon_{1}\sin^{2}\theta
d\varphi_{1}+\varepsilon_{2}\cos^{2}\theta d\varphi_{2}\right).$ (6)

We further assume the metric functions $g_{tt}$, $W$, $N$, $B$ and also the
two functions $a_{0}$, $a_{\varphi}$ for the gauge field, depend only on the
radial coordinate $r$.

We consider perturbations around the Myers-Perry solution, with a Born-Infeld
electric charge $q$ as the perturbative parameter, so that $q$ is much less
than $\hat{M}$ or $\hat{J}^{\,2/3}$. Taking into account the symmetry with
respect to charge reversal and the seed solution, the metric and gauge
potentials take the form

$\displaystyle
g_{tt}=-1+\frac{2\hat{M}}{r^{2}}+q^{2}g^{(2)}_{tt}+q^{4}g^{(4)}_{tt}+O(q^{6}),$
(7) $\displaystyle
W=1-\frac{2\hat{M}}{r^{2}}+\frac{2\hat{J}^{2}}{\hat{M}r^{4}}+q^{2}W^{(2)}+q^{4}W^{(4)}+O(q^{6}),$
(8) $\displaystyle
N=\frac{2\hat{J}^{2}}{\hat{M}r^{2}}+q^{2}N^{(2)}+q^{4}N^{(4)}+O(q^{6}),$ (9)
$\displaystyle B=\frac{2\hat{J}}{r^{2}}+q^{2}B^{(2)}+q^{4}B^{(4)}+O(q^{6}),$
(10) $\displaystyle a_{0}=qa^{(1)}_{0}+q^{3}a^{(3)}_{0}+O(q^{5}),$ (11)
$\displaystyle
a_{\varphi}=qa^{(1)}_{\varphi}+q^{3}a^{(3)}_{\varphi}+O(q^{5})\,.$ (12)

Here $g^{(2)}_{tt}$ and $g^{(4)}_{tt}$ are the second and fourth order
perturbative terms, respectively. The other perturbative terms are defined
similarly.

We now fix the angular momenta at any perturbative order, and impose the
extremal condition in all orders. We also assume that the horizon is regular.
With these assumptions we are able to fix all constants of integration. To
simplify the notation we introduce a parameter $\nu$ through the equations

$\hat{M}=2\nu^{2}\quad,\quad\hat{J}=2\nu^{3}\,,$ (13)

meaning that the extremal Myers-Perry solution holds in five dimensions. Then,
using the field equations (3)-(4), the perturbative solutions up to 4th order
can be written as,

$\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle-1+\frac{4\nu^{2}}{r^{2}}+\frac{(r^{2}-4\nu^{2})q^{2}}{3\nu^{2}r^{4}}+\Bigg{\\{}{\frac{11}{135}}\,{\frac{1}{{\nu}^{2}{\beta}^{2}{r}^{8}}}-{\frac{16}{45}}\,{\frac{{\nu}^{2}}{{\beta}^{2}{r}^{12}}}+{\frac{58}{135}}\,{\frac{1}{{\beta}^{2}{r}^{10}}}$
(14)
$\displaystyle-\,{\frac{8\,{\nu}^{2}{\beta}^{2}+3}{9{\beta}^{2}{\nu}^{6}{r}^{4}}}+{\frac{1}{540}}\,{\frac{79+240\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{4}{r}^{6}}}+{\frac{1}{720}}\,{\frac{79+220\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{8}{r}^{2}}}$
$\displaystyle{\frac{4\left({r}^{2}-2\,{\nu}^{2}\right)^{2}}{27{\beta}^{2}{\nu}^{10}{r}^{4}}}\,\left({\nu}^{2}{\beta}^{2}+\frac{3}{8}\right)\ln\left(1-2\,{\frac{{\nu}^{2}}{{r}^{2}}}\right)\Bigg{\\}}q^{4}+O(q^{6}),$
$\displaystyle W$ $\displaystyle=$ $\displaystyle
1-\frac{4\nu^{2}}{r^{2}}+\frac{4\nu^{4}}{r^{4}}-\frac{q^{2}(r^{2}-2\nu^{2})}{3\nu^{2}r^{4}}$
(15)
$\displaystyle+\Bigg{\\{}{\frac{1}{2160}}\,{\frac{522+480\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{10}}}+{\frac{1}{2160}}\,{\frac{-2420\,{\nu}^{4}{\beta}^{2}-2607\,{\nu}^{2}}{{\beta}^{2}{\nu}^{10}{r}^{2}}}+{\frac{1}{2160}}\,{\frac{3840\,{\nu}^{4}+3620\,{\beta}^{2}{\nu}^{6}}{{\beta}^{2}{\nu}^{10}{r}^{4}}}$
$\displaystyle+{\frac{1}{2160}}\,{\frac{-1032\,{\nu}^{6}-1280\,{\nu}^{8}{\beta}^{2}}{{\beta}^{2}{\nu}^{10}{r}^{6}}}-\frac{1}{5}\,{\frac{1}{{r}^{8}{\nu}^{2}{\beta}^{2}}}-{\frac{23}{45}}\,{\frac{1}{{\beta}^{2}{r}^{10}}}+{\frac{44}{45}}\,{\frac{{\nu}^{2}}{{\beta}^{2}{r}^{12}}}-{\frac{56}{45}}\,{\frac{{\nu}^{4}}{{\beta}^{2}{r}^{14}}}$
$\displaystyle+\frac{\left({r}^{2}-2\,{\nu}^{2}\right)^{3}}{9{\beta}^{2}{\nu}^{12}{r}^{4}}\,\left({\frac{87}{80}}+{\nu}^{2}{\beta}^{2}\right)\ln\left(1-2\,{\frac{{\nu}^{2}}{{r}^{2}}}\right)\Bigg{\\}}q^{4}+O(q^{6}),$
$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{4\nu^{4}}{r^{2}}-\frac{2q^{2}(r^{2}+2\nu^{2})}{3r^{4}}+\Bigg{\\{}-{\frac{1}{360}}\,{\frac{\left(87+80\,{\nu}^{2}{\beta}^{2}\right){r}^{2}}{{\beta}^{2}{\nu}^{10}}}+{\frac{52}{135}}\,{\frac{{\nu}^{2}}{{\beta}^{2}{r}^{10}}}+{\frac{4}{135}}\,{\frac{1}{{\beta}^{2}{r}^{8}}}-{\frac{16}{45}}\,{\frac{{\nu}^{4}}{{\beta}^{2}{r}^{12}}}$
(16)
$\displaystyle+\frac{1}{45}\,{\frac{1+20\,{\nu}^{2}{\beta}^{2}}{{r}^{6}{\nu}^{2}{\beta}^{2}}}-{\frac{1}{90}}\,{\frac{61+40\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{4}{r}^{4}}}+{\frac{1}{360}}\,{\frac{87+80\,{\nu}^{2}{\beta}^{2}}{{\nu}^{8}{\beta}^{2}}}+{\frac{1}{180}}\,{\frac{144+35\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{6}{r}^{2}}}$
$\displaystyle-\,\ln\left(1-2\,{\frac{{\nu}^{2}}{{r}^{2}}}\right)\left({\frac{87}{80}}\,{r}^{6}+{r}^{6}{\nu}^{2}{\beta}^{2}-{\frac{57}{20}}\,{\nu}^{4}{r}^{2}+{\nu}^{6}+\frac{8}{3}\,{\nu}^{8}{\beta}^{2}\right)\frac{\left({r}^{2}-2\,{\nu}^{2}\right)}{9{\beta}^{2}{\nu}^{12}{r}^{4}}\Bigg{\\}}q^{4}+O(q^{6}),$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{4\nu^{3}}{r^{2}}-\frac{4\nu}{3r^{4}}q^{2}+\Bigg{\\{}{\frac{1}{1440}}\,{\frac{480\,{\nu}^{2}{\beta}^{2}+864}{{\beta}^{2}{\nu}^{7}{r}^{2}}}+{\frac{2}{27}}\,{\frac{1}{{\beta}^{2}\nu\,{r}^{8}}}+{\frac{52}{135}}\,{\frac{\nu}{{\beta}^{2}{r}^{10}}}-{\frac{1}{60}}\,{\frac{29+40\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{5}{r}^{4}}}$
(17)
$\displaystyle+{\frac{1}{90}}\,{\frac{9+40\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{3}{r}^{6}}}-{\frac{16}{45}}\,{\frac{{\nu}^{3}}{{\beta}^{2}{r}^{12}}}-\frac{1}{45}\,{\frac{9+5\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{9}}}+(-{\frac{1}{270}}\,{\frac{27+15\,{\nu}^{2}{\beta}^{2}}{{\beta}^{2}{\nu}^{11}}}$
$\displaystyle+{\frac{1}{270}}\,{\frac{54\,{\nu}^{2}+30\,{\nu}^{4}{\beta}^{2}}{{\beta}^{2}{\nu}^{11}{r}^{2}}}-{\frac{1}{270}}\,{\frac{80\,{\beta}^{2}{\nu}^{6}+30\,{\nu}^{4}}{{\beta}^{2}{\nu}^{11}{r}^{4}}})\left({r}^{2}-2\,{\nu}^{2}\right)\ln\left(1-2\,{\frac{{\nu}^{2}}{{r}^{2}}}\right)\Bigg{\\}}q^{4}+O(q^{6}),$
$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{q}{r^{2}}+\Bigg{\\{}{\frac{1}{180}}\,{\frac{40\,{\nu}^{2}{\beta}^{2}+15}{{\beta}^{2}{\nu}^{6}{r}^{2}}}-{\frac{1}{180}}\,{\frac{15\,{\nu}^{2}+40\,{\nu}^{4}{\beta}^{2}}{{\beta}^{2}{\nu}^{6}{r}^{4}}}-{\frac{11}{180}}\,{\frac{1}{{r}^{6}{\nu}^{2}{\beta}^{2}}}-{\frac{26}{45}}\,{\frac{1}{{\beta}^{2}{r}^{8}}}+\frac{2}{3}\,{\frac{{\nu}^{2}}{{\beta}^{2}{r}^{10}}}$
(18)
$\displaystyle+\frac{1}{9}\,\left({\nu}^{2}{\beta}^{2}+\frac{3}{8}\right)\left({r}^{2}-2\,{\nu}^{2}\right)\ln\left(1-2\,{\frac{{\nu}^{2}}{{r}^{2}}}\right){\beta}^{-2}{\nu}^{-8}{r}^{-2}\Bigg{\\}}q^{3}+O(q^{5}),$
$\displaystyle a_{\varphi}$ $\displaystyle=$ $\displaystyle-\frac{\nu
q}{r^{2}}+\Bigg{\\{}-{\frac{1}{720}}\,{\frac{80\,{\nu}^{2}{\beta}^{2}+30}{{\nu}^{7}{\beta}^{2}}}-{\frac{1}{720}}\,{\frac{40\,{\nu}^{4}{\beta}^{2}+27\,{\nu}^{2}}{{\nu}^{7}{\beta}^{2}{r}^{2}}}-{\frac{1}{720}}\,{\frac{-160\,{\beta}^{2}{\nu}^{6}-88\,{\nu}^{4}}{{\nu}^{7}{\beta}^{2}{r}^{4}}}+{\frac{7}{60}}\,{\frac{1}{\nu\,{\beta}^{2}{r}^{6}}}$
(19)
$\displaystyle+{\frac{26}{45}}\,{\frac{\nu}{{\beta}^{2}{r}^{8}}}-\frac{2}{3}\,{\frac{{\nu}^{3}}{{\beta}^{2}{r}^{10}}}-{\frac{1}{144}}\,\left(3+8\,{\nu}^{2}{\beta}^{2}\right)\left(\frac{r^{2}}{\beta^{2}\nu^{9}}-\frac{4}{r^{2}\beta^{2}\nu^{5}}\right)\ln\left(1-2\,{\frac{{\nu}^{2}}{{r}^{2}}}\right)\Bigg{\\}}q^{3}+O(q^{5}).$

It is seen that there are the usual $1/r^{n}$ polynomial expressions as well
as terms involving logarithmic functions. It is also worth mentioning that the
Born-Infeld parameter $\beta$ appears in terms which are of 3rd and 4th order
in the electric charge parameter $q$. One may note that in Maxwell’s limit,
$\beta\longrightarrow\infty$, these perturbative solutions reduce to the five
dimensional perturbative charged rotating black holes in Einstein-Maxwell
theory presented in Navarro . A consistent check of these solutions can be
provided by Smarr’s formula.

##  III Physical Quantities

The mass $M$, the angular momenta $J$, the electric charge $Q$, and the
magnetic moment $\mu_{\rm mag}$ can be read off the asymptotic behavior of the
metric and the gauge potential Kunz2 . The asymptotic forms are,

$\displaystyle g_{tt}=-1+\frac{\tilde{M}}{r^{2}}+...,\quad
B=\frac{2\tilde{J}}{r^{2}}+...,\quad a_{0}=\frac{\tilde{Q}}{r^{2}}+...,\quad
a_{\varphi}=\frac{\tilde{\mu}_{\rm mag}}{r^{2}}+...,$ (20)

where $\tilde{M}$, $\tilde{J}$, $\tilde{Q}$, and $\tilde{\mu}_{\rm mag}$ are
the mass, angular momentum, electric charge, and magnetic moment parameters,
respectively, and we have defined $\tilde{Q}\equiv q$ for notational
consistency. These parameters are related to the real mass $M$, angular
momentum $J$, electric charge $Q$, and magnetic moment ${\mu}_{\rm mag}$,
through the relations,

$\displaystyle\tilde{M}$ $\displaystyle=$ $\displaystyle\frac{16\pi
G_{5}}{3A}M,\ \ \quad\tilde{J}=\frac{4\pi G_{5}}{A}J,$
$\displaystyle\quad\tilde{Q}$ $\displaystyle=$ $\displaystyle\frac{4\pi
G_{5}}{2A}Q,\quad\tilde{\mu}_{\rm mag}=\frac{4\pi G_{5}}{2A}\mu_{\rm mag},$
(21)

Note that when the perturbative parameter $q$ is equal to zero, the tilde
quantities of Eq. (20) reduce to the hat quantities of Eq. (13), i.e.,
$\tilde{M}=\hat{M}$, $\tilde{J}=\hat{J}$, $\tilde{Q}=\hat{Q}=0$, and
$\tilde{\mu}_{\rm mag}=\hat{\mu}_{\rm mag}=0$.

Now, comparing the expansions in Eqs. (20) and Eq. (III) with the asymptotic
behavior of the solutions given in Eqs. (14)-(19), we obtain

$\displaystyle M=\frac{3\pi\nu^{2}}{2}+\frac{\pi q^{2}}{8\nu^{2}}+\frac{\pi
q^{4}\left(20\nu^{2}\beta^{2}-3\right)}{5760\beta^{2}\nu^{8}}+O(q^{6}),$ (22)
$\displaystyle J=\pi\nu^{3},$ (23) $\displaystyle Q=\pi q,$ (24)
$\displaystyle\mu_{\rm mag}=\pi\nu q-\frac{\pi
q^{3}\left(40\nu^{2}\beta^{2}+3\right)}{720\nu^{5}\beta^{2}}+O(q^{5}),$ (25)

The gyromagnetic ratio $g$ is then given by

$\displaystyle g=2\,\frac{\mu_{\rm mag}/Q}{J/M}=\frac{2M\mu_{\rm
mag}}{QJ}=3+\frac{q^{2}\left(20\nu^{2}\beta^{2}-3\right)}{240\nu^{6}\beta^{2}}-\frac{q^{4}\left(10\nu^{2}\beta^{2}+3\right)}{1440\nu^{10}\beta^{2}}+O(q^{6})\,.$
(26)

The horizon radius $r_{H}$ is given by

$\displaystyle
r_{H}=\sqrt{2}\nu+{\frac{{q}^{2}\sqrt{2}}{24{\nu}^{3}}}+{\frac{11}{1152}}\,{\frac{{q}^{4}\sqrt{2}}{{\nu}^{7}}}+O(q^{6})\,.$
(27)

All these quantities are worth commenting.

The mass $M$ of the black holes as a function of the Born-Infeld parameter
$\beta$ has an interesting behavior, as shown in Fig. 1, see also Eq. (22).
The mass $M$ increases with increasing $\beta$ and as
$\beta\rightarrow\infty$, i.e., in the Maxwell limit, the mass takes the value
$M=\frac{3\pi\nu^{2}}{2}+\frac{\pi q^{2}}{8\nu^{2}}+\frac{\pi
q^{4}}{288\nu^{6}}+O(q^{6})$, exactly the result obtained for the five-
dimensional perturbative Einstein-Maxwell black hole Navarro . For very small
$\beta$ the mass turns negative. We do not attach any significance to this
result since the value of $\beta$ for which the mass is zero uses a
perturbative $q^{2}$ term which is much larger than than the zeroth order term
in the expressions for the magnetic moment and gyromagnetic ratio. In fact,
the values of $\beta$ for which the results make thorough sense are values of
$\beta$ larger than the ones which yield the zeros of the magnetic moment and
gyromagnetic ratio.

The angular momenta $J$ in Eq. (23) and the charge $q$ in Eq. (24) are fixed
and do not depend on $\beta$, following our approach. The magnetic dipole
moment $\mu_{\rm mag}$ given in Eq. (25) appears due to the rotation of the
electrically charged black hole. The first order term $\pi\,q\,\nu\ =Q\,\nu$
is equivalent to the magnetic moment of a point particle rotating around an
axis, with charge $q_{\rm particle}=3\pi Q$, and with the same angular
momentum $j_{\rm particle}$ and mass $m_{\rm particle}$ of the black hole,
i.e., $j_{\rm particle}=J$ and $m_{\rm particle}=M$, since for such a system
$\mu_{\rm mag}=\frac{1}{2}\,q_{\rm particle}\frac{j_{\rm particle}}{m_{\rm
particle}}=Q\,\nu$. The higher order terms presumably come from the spacetime
curvature. From Fig. 2 we find out that the magnetic dipole moment $\mu_{\rm
mag}$ increases with increasing $\beta$. In the limit
$\beta\rightarrow\infty$, the mass, the magnetic moment $\mu_{\rm mag}=\pi\nu
q-\frac{\pi q^{3}}{18\nu^{3}}+O(q^{5})$ which is exactly the result obtained
for the five-dimensional perturbative Einstein-Maxwell black hole Navarro .
For small $\beta$ the magnetic moment is negative. We analyze this result
below, when we comment on the gyromagnetic ratio.

Figure 1: The black hole mass $M$ versus the Born-Infeld parameter $\beta$ for
$\nu=1.16$ and $q=0.09$.

Figure 2: The black hole magnetic moment $\mu_{\rm mag}$ versus the Born-
Infeld parameter $\beta$ for $\nu=1.16$ and $q=0.09$.

Figure 3: The black hole gyromagnetic ratio $g$ versus the Born-Infeld
parameter $\beta$ for $\nu=1.16$ and $q=0.09$.

The dimensional gyromagnetic ratio for a given system defined in Eq. (26) is
twice the ratio of the magnetic moment divided by the charge to the angular
momentum divided by the mass. It has the value $1$ for a classical body with
uniform mass and uniform charge distribution rotating about an axis of
symmetry. For an electron it has the value of $2.00$ plus small quantum
corrections, and for the proton $5.59$ and neutron has the value $5.59$ and
$-3.826$, respectively. From Eq. (26) we see that the perturbative parameter
$q$ and the parameters $\beta$ and $\nu$ modify the gyromagnetic ratio of
asymptotically flat five-dimensional charged rotating black holes as compared
to the uncharged extremal Myers-Perry black holes. We want to study in more
detail this modification of the gyromagnetic ratio when one varies the Born-
Infeld parameter $\beta$, see Fig. 3. From the figure we find that the
gyromagnetic ratio $g$ increases with increasing $\beta$, and in the limit
$\beta\rightarrow\infty$, the gyromagnetic ratio reduces to
$g=3+\frac{q^{2}}{12\nu^{4}}-\frac{q^{4}}{144\nu^{8}}+O(q^{6})$ which is
exactly the result obtained for the five-dimensional perturbative Einstein-
Maxwell black hole Navarro . Now, from Fig. 3 one finds that for some low
value of $\beta$ the gyromagnetic ratio is zero, and then turns negative. This
change of sign comes from a perturbative $q^{2}$ term, and thus the result
might not hold for the full exact solution. On the other hand the result is
sufficiently intriguing that deserves some attention. One can speculate that
it is at least qualitatively correct, and perhaps expected. Let us see why. It
is known that the Born-Infeld theory is different from Maxwell’s theory when
the electromagnetic fields are very strong. The Born-Infeld theory gives a
finite total energy $E$ for the field around a point particle with charge
$q_{\rm particle}$, indeed $E\simeq\sqrt{q_{\rm particle}^{3}\,\beta}$. It
also gives an effective radius $r_{0}$ for the charge distribution,
$r_{0}=\sqrt{\frac{q_{\rm particle}}{\beta}}$. Curiously, the reversal of the
gyromagnetic ratio $g$ in Fig. 3 (concomitant to the reversal of the magnetic
dipole moment $\mu_{\rm mag}$ in Fig. 2) happens when
$r_{0}=\sqrt{\frac{Q}{\beta}}$ is of the order or larger than the horizon
radius $r_{H}$. Indeed, for the values of $\nu$ and $q$ used in the figures,
one finds $r_{0}=8.79$ and $r_{H}=1.65$. This reversal could then be
interpreted as follows. For large $\beta$ much of the electrical charge is
distributed in a point-like manner, as in Maxwell’s theory. For small $\beta$
the charge distribution is extended, and for sufficiently small $\beta$ it
even extends outside the horizon. It is known that an object with magnetic
moment ${\vec{\mu}}_{\rm mag}$ placed on a magnetic field $\vec{B}$ suffers a
torque given by ${\vec{\mu}}_{\rm mag}\times\vec{B}$. A black hole with
magnetic moment is also subjected to this kind of torque. So, for large
$\beta$, i.e., the point like case, when a magnetic field is applied to the
black hole spacetime the resultant torque on the black hole tends to rotate it
in the expected sense, and thus the magnetic dipole moment $\mu_{\rm mag}$ and
$g$ are positive, On the other hand, when $\beta$ is small, i.e., the charge
distribution outside the black hole case, it is the region external to the
black hole horizon that is effectively charged and it is this very region that
upon application of a magnetic field tends to rotate in the expected sense. So
here, the black hole rotates in the opposite sense, giving a negative magnetic
dipole moment $\mu_{\rm mag}$ and thus a negative gyromagnetic ratio $g$.

The horizon radius $r_{H}$ in Eq. (27) has the unexpected feature that it does
not depend on $\beta$ at least up to 4th order in the charge. Up to this order
$r_{H}$ is equal to the Einstein-Maxwell case Navarro .

##  IV The mass formula

Define $\xi$ as the timelike Killing vector and $\eta_{k}$, $k=1,2$, as the
two azimuthal Killing vectors. The two equal horizon constant angular
velocities $\Omega$ can then be defined by imposing that the Killing vector
field

$\displaystyle\chi=\xi+\Omega\sum^{2}_{k=1}\epsilon_{k}\eta_{k},$ (28)

is null on the horizon and orthogonal to it as well. This yields,

$\displaystyle\Omega=\frac{1}{2\nu}-\frac{q^{2}}{24\nu^{5}}-\frac{q^{4}\left(5\nu^{2}\beta^{2}-1\right)}{1440\beta^{2}\nu^{11}}+O(q^{6})\,.$
(29)

The 3-area of the horizon $A_{H}$ and the electrostatic potential at the
horizon $\Phi_{H}$ are given by

$\displaystyle A_{H}=8\pi^{2}\nu^{3}+O(q^{6})\,.$ (30)
$\displaystyle\Phi_{H}=\frac{q}{4\nu^{2}}+\frac{q^{3}\left(20\beta^{2}\nu^{2}\right)}{1440\beta^{2}\nu^{8}}+O(q^{5})\,.$
(31)

The surface gravity $\kappa$ is defined by
$\kappa^{2}=-\frac{1}{2}(\nabla_{\mu}\chi_{\nu})(\nabla^{\mu}\chi^{\nu})$.
Taking into account the conserved quantities obtained in the last section, one
can check that these quantities satisfy the Smarr mass formula up to 4th order
Gunasekaran . Indeed, in general the formula is

$\displaystyle M=\frac{3\kappa_{sg}A_{H}}{16\pi G_{D}}+3\Omega
J+\Phi_{H}Q-\frac{\beta}{2}\frac{\partial M}{\partial\beta}.$ (32)

For an extremal solution with $\kappa=0$, the Smarr mass formula reduces to

$\displaystyle M=3\Omega J+\Phi_{H}Q-\frac{\beta}{2}\frac{\partial
M}{\partial\beta}\,.$ (33)

Taking into account that the mass $M$ of the black holes is given by Eq. (22),
one can determine the last term in (33), and find

$\displaystyle M=3\Omega J+\Phi_{H}Q+\Phi_{\beta}Q^{4},$ (34)

where

$\displaystyle\Phi_{\beta}=-\frac{\beta}{2Q^{4}}\frac{\partial
M}{\partial\beta}=-{\frac{1}{1920}}\,{\frac{1}{{\pi}^{3}{\nu}^{8}{\beta}^{2}}}\,.$
(35)

##  V Conclusions

In conclusion, we have presented a new class of perturbative charged rotating
black hole solutions in five dimensions in the presence of a nonlinear Born-
Infeld gauge field. This class of solutions is restricted to the extremal
black holes with equal angular momenta. At infinity, the metric is
asymptotically locally flat. Our strategy for obtaining these solutions was
through a perturbative method up to the $4$th order for the perturbative
parameter $q$. We have started from rotating Myers-Perry black hole solutions
Myer  in five dimensions, and then studied the effects of adding a charge
parameter to the solutions. We have calculated the conserved quantities of the
solutions such as mass, angular momentum, electric charge, magnetic moment,
gyromagnetic ratio, and horizon radius. We found that the Born-Infeld
parameter $\beta$ modifies the values of all the physical quantities, except
the horizon radius, relative to the corresponding Einstein-Maxwell five
dimensional rotating solutions. For large $\beta$ the solutions reduce to the
perturbative rotating Einstein-Maxwell solutions Navarro , as we expected. We
also speculated on what might happen for these solutions in the strong
electromagnetic regime, i.e., when $\beta$ is small. The generalization of the
present work to all higher dimensional is quite an interesting subject which
will be addressed elsewhere.

###### Acknowledgements.

The support of the Fundação para a Ciência e a Tecnologia of Portugal Project
PTDC/FIS/098962/2008 and PEst-OE/FIS/UI0099/2011 is gratefully acknowledged.
M. Allaverdizadeh is supported by a FCT grant. The work of A. Sheykhi has been
supported financially by Research Institute for Astronomy and Astrophysics of
Maragha (RIAAM), Iran.

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