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# Getting Gender Right in Neural Machine Translation

Eva Vanmassenhoveα
Christian Hardmeierβ

α ADAPT, School of Computing, Dublin City University, Dublin, Ireland
<EMAIL_ADDRESS>

β Department of Linguistics and Philology, Uppsala University, Uppsala, Sweden
<EMAIL_ADDRESS>Andy Wayα

###### Abstract

Speakers of different languages must attend to and encode strikingly different
aspects of the world in order to use their language correctly Sapir (1921);
Slobin (1996). One such difference is related to the way gender is expressed
in a language. Saying “I am happy” in English, does not encode any additional
knowledge of the speaker that uttered the sentence. However, many other
languages do have grammatical gender systems and so such knowledge would be
encoded. In order to correctly translate such a sentence into, say, French,
the inherent gender information needs to be retained/recovered. The same
sentence would become either “Je suis heureux”, for a male speaker or “Je suis
heureuse” for a female one. Apart from morphological agreement, demographic
factors (gender, age, etc.) also influence our use of language in terms of
word choices or even on the level of syntactic constructions Tannen (1991);
Pennebaker et al. (2003). We integrate gender information into NMT systems.
Our contribution is two-fold: (1) the compilation of large datasets with
speaker information for 20 language pairs, and (2) a simple set of experiments
that incorporate gender information into NMT for multiple language pairs. Our
experiments show that adding a gender feature to an NMT system significantly
improves the translation quality for some language pairs.

##  1 Introduction

In the field of linguistics, the differences between male and female traits
within spoken and written language have been studied both empirically and
theoretically, revealing that the language used by males and females differs
in terms of style and syntax Coates (2015). The increasing amount of work on
automatic author classification (or ‘author profiling’) reaching relatively
high accuracies on domain-specific data corroborates these findings Rangel et
al. (2013); Santosh et al. (2013). However, determining the gender of an
author based solely on text is not a solved issue. Likewise, the selection of
the most informative features for gender classification remains a difficult
task Litvinova et al. (2016).

When translating from one language into another, original author traits are
partially lost, both in human and machine translations Mirkin et al. (2015);
Rabinovich et al. (2017). However, in the field of Machine Translation (MT)
one of the most observable consequences of this missing information are
morphologically incorrect variants due to a lack of agreement in number and
gender with the subject. Such errors harm the overall fluency and adequacy of
the translated sentence. Furthermore, gender-related errors are not just
harming the quality of the translation as getting the gender right is also a
matter of basic politeness. Current systems have a tendency to perpetuate a
male bias which amounts to negative discrimination against half the population
and this has been picked up by the
media.111https://www.theguardian.com/technology/2017/apr/13/ai-programs-
exhibit -racist-and-sexist-biases-research

Human translators rely on contextual information to infer the gender of the
speaker in order to make the correct morphological agreement. However, most
current MT systems do not; they simply exploit statistical dependencies on the
sentence level that have been learned from large amounts of parallel data.
Furthermore, sentences are translated in isolation. As a consequence, pieces
of information necessary to determine the gender of the speakers, might get
lost. The MT system will, in such cases, opt for the statistically most likely
variant, which depending on the training data, will be either the male or the
female form. Additionally, in the field of MT, training data often consists of
both original and translated parallel texts: large parts of the texts have
already been translated, which, as studied by Mirkin et al. (2015), does not
preserve the original demographic and psychometric traits of the author,
making it very hard for a Neural MT (NMT) system to determine the gender of
the author.

With this in mind, a first step towards the preservation of author traits
would be their integration into an NMT system. As ‘gender’ manifests itself
not only in the agreement with other words in a sentence, but also in the
choice of context-based words or on the level of syntactic constructions, the
sets of experiments conducted in this paper focus on the integration of a
gender feature into NMT for multiple language pairs.

The structure of the paper is the following: related work is described in
Section 2; Section 3 describes and analyses the datasets that were compiled;
the experimental setup is discussed in Section 4; the results are presented in
Section 5; finally, we conclude and provide some ideas for future work in
Section 6.

##  2 Related Work

Differences in the language between male and female speakers have been studied
within various fields related to linguistics, including Natural Language
Processing (NLP) for author profiling, conversational agents, recommendation
systems etc. Mirkin et al. (2015) motivated the need for more personalized MT.
Their experiments show that MT is detrimental to the automatic recognition of
linguistic signals of traits of the original author/speaker. Their work
suggests using domain-adaptation techniques to make MT more personalized but
does not include any actual experiments on the inclusion of author traits in
MT.

Rabinovich et al. (2017) conducted a series of experiments on preserving
original author traits, focusing particularly on gender. As suggested by
Mirkin et al. (2015), they treat the personalization of Statistical MT (SMT)
systems as a domain-adaptation task treating the female and male gender as two
different domains. They applied two common simple domain-adaptation techniques
in order to create personalized SMT: (1) using gender-specific phrase-tables
and language models, and (2) using a gender-specific tuning set. Although
their models did not improve over the baseline, their work provides a detailed
analysis of gender traits in human and machine translation.

Our work is, to the best of our knowledge, the first to attempt building a
speaker-informed NMT system. Our approach is similar to the work of Sennrich
et al. (2016) on controlling politeness, where some sentence of the training
data are followed with an ‘informal’ or ‘polite’ tag indicating the level of
politeness expressed.

##  3 Compilation of Datasets

One of the main obstacles for more personalized MT systems is finding large
enough annotated parallel datasets with speaker information. Rabinovich et al.
(2017) published an annotated parallel dataset for EN–FR and EN–DE. However,
for many other language pairs no sufficiently large annotated datasets are
available.

To address the aforementioned problem, we published online a collection of
parallel corpora licensed under the Creative Commons Attribution 4.0
International License for 20 language pairs Vanmassenhove and Hardmeier
(2018).222https://github.com/evavnmssnhv/Europarl-Speaker-Information

We followed the approach described by Rabinovich et al. (2017) and tagged
parallel sentences from Europarl Koehn (2005) with speaker information (name,
gender, age, date of birth, euroID and date of the session) by retrieving
speaker information provided by tags in the Europarl source files. The
Europarl source files contain information about the speaker on the paragraph
level and the filenames contain the data of the session. By retrieving the
names of the speakers together with meta-information on the members of the
European Parliament (MEPs) released by Rabinovich et al. (2017) (which
includes among others name, country, date of birth and gender predictions per
MEP), we were able to retrieve demographic annotations (gender, age, etc.). An
overview of the language pairs as well as the amount of annotated parallel
sentences per language pair is given in Table 1.

Languages | # sents | Languages | # sents
---|---|---|---
EN–BG | 306,380 | EN–IT | 1,297,635
EN–CS | 491,848 | EN–LT | 481,570
EN–DA | 1,421,197 | EN–LV | 487,287
EN–DE | 1,296,843 | EN–NL | 1,419,359
EN–EL | 921,540 | EN–PL | 478,008
EN–ES | 1,419,507 | EN–PT | 1,426,043
EN–ET | 494,645 | EN–RO | 303,396
EN–FI | 1,393,572 | EN–SK | 488,351
EN–FR | 1,440,620 | EN–SL | 479,313
EN–HU | 251,833 | EN–SV | 1,349,472
Table 1: Overview of annotated parallel sentences per language pair

###  3.1 Analysis of the EN–FR Annotated Dataset

We first analysed the distribution of male and female sentence in our data. In
the 10 different datasets we experimented with, the percentage of sentences
uttered by female speakers is very similar, ranging between 32% and 33%. This
similarity can be explained by the fact that Europarl is largely a
multilingual corpus with a big overlap between the different language pairs.

We conducted a more focused analysis on one of the subcorpora (EN–FR) with
respect to the percentage of sentences uttered by males/females for various
age groups to obtain a better grasp of what kind of data we are using for
training. As can be seen from Figure 1, with the exception of the youngest age
group (20–30), which represents only a very small percentage of the total
amount of sentences (0.71%), more male data is available in all age groups.
Furthermore, when looking at the entire dataset, 67.39% of the sentences are
produced by male speakers. Moreover, almost half of the total number of
sentences are uttered by the 50–60 age group (43.76%).

Figure 1: Percentage of female and male speakers per age group

The analysis shows that indeed, there is a gender unbalance in the Europarl
dataset, which will be reflected in the translations that MT systems trained
on this data produce.

##  4 Experimental Setup

###  4.1 Datasets

We carried out a set of experiments on 10 language pairs (the ones for which
we compiled more than 500k annotated Europarl parallel sentences): EN–DE,
EN–FR, EN–ES, EN–EL, EN–PT, EN–FI, EN–IT, EN–SV, EN–NL and EN–DA. We augmented
every sentence with a tag on the English source side, identifying the gender
of the speaker, as illustrated in ( ()). This approach for encoding sentence-
specific information for NMT has been successfully exploited to tackle other
types of issues, multilingual NMT systems (e.g., Zero Shot Translation Johnson
et al. (2017)), domain adaptation Sennrich et al. (2016), etc.

  * ()

    * “FEMALE Madam President, as a…”
---

For each of these language pairs we trained two NMT systems: a baseline and a
tagged one. We evaluated the performance of all our systems on a randomly
selected 2K general test set. Moreover, we further evaluated the EN–FR systems
on 2K male-only and female-only test sets to have a look at the system
performance with respect to gender-related issues. We also looked at two
additional male and female test sets in which the first person singular
pronoun appeared.

###  4.2 Description of the NMT Systems

We used the OpenNMT-py toolkit Klein et al. (2017) to train the NMT models.
The models are sequence-to-sequence encoder-decoders with LSTMs as the
recurrent unit Bahdanau et al. (2014); Cho et al. (2014); Sutskever et al.
(2014) trained with the default parameters. In order to by-pass the OOV
problem and reduce the number of dictionary entries, we use word-segmentation
with BPE Sennrich (2015). We ran the BPE algorithm with 89,500 operations
Sennrich (2015). All systems are trained for 13 epochs and the best model is
selected for evaluation.

##  5 Results

In this section we discuss some of the results obtained. We hypothesized that
the male/female tags would be particularly helpful for French, Portuguese,
Italian, Spanish and Greek, where adjectives and even verb forms can be marked
by the gender of the speaker. Since, according to the literature, women and
men also make use of different syntactic constructions and make different word
choices, we also tested the approach on other languages that do not have
morphological agreement with the gender of the speaker such as Danish (DA),
Dutch (NL), Finnish (FI), German (DE) and Swedish (SV).

First, we wanted to see how our tagged systems performed on the general test
set compared to the baseline. In Table 2, the BLEU scores for 10 baseline and
10 gender-enhanced NMT systems are presented.

Systems | EN | EN-TAG
---|---|---
FR | 37.82 | 39.26*
ES | 42.47 | 42.28
EL | 31.38 | 31.54
IT | 31.46 | 31.75*
PT | 36.11 | 36.33
DA | 36.69 | 37.00*
DE | 28.28 | 28.05
FI | 21.82 | 21.35*
SV | 35.42 | 35.19
NL | 28.35 | 28.22
Table 2: BLEU scores for the 10 baseline (denoted with EN) and the 10 gender-
enhanced NMT (denoted with EN-TAG) systems. Entries labeled with * present
statistically significant differences (p $<$ 0.05). Statistical significance
was computed with the MultEval tool Clark et al. (2011).

While most of the BLEU-scores Papineni et al. (2002) in Table 2 are consistent
with our hypothesis, showing (significant) improvements for the NMT systems
enriched with a gender tag (EN-TAG) over the baseline systems (EN) for French,
Italian, Portuguese and Greek, the Spanish enriched system surprisingly does
not (–0.19 BLEU). As hypothesized, the Dutch, German, Finnish and Swedish
systems do not improve. However, the Danish (EN–DA) enriched NMT system does
achieve a significant +0.31 BLEU improvement.

We expected to see the strongest improvements in sentences uttered by female
speakers as, according to our initial analysis, the male data was over-
represented in the training. To test this hypothesis, we evaluated all systems
on a male-only and female-only test set. Furthermore, we also experimented on
test sets containing the pronoun of the first person singular as this form is
used when a speaker refers to himself/herself. The results on the specific
test set for the EN–FR dataset are presented in Table 3. As hypothesized, the
biggest BLEU score improvement is observed on the female test set,
particularly, the test sets containing first person singular pronouns (F1).

Test Sets | EN | EN-TAG
---|---|---
FR (M) | 37.58 | 38.71*
FR (F) | 37.75 | 38.97*
FR (M1) | 39.00 | 39.66*
FR (F1) | 37.32 | 38.57*
Table 3: BLEU-scores on EN–FR comparing the baseline (EN) and the tagged
systems (EN–TAG) on 4 different test sets: a test set containing only male
data (M), only female data (F), 1st person male data (M1) and first person
female data (F1). All the improvements of the EN-TAG system are statistically
significant (p $<$ 0.5), as indicated by *.

We had a closer look at some of the translations.333We used the tool provided
by Tilde https://www.letsmt.eu/Bleu.aspx to see where the BLEU score between
the baseline and our tagged systems varied the most. There are cases where the
gender-informed (TAG) system improves over the baseline (BASE) due to better
agreement. Interestingly, in ( ()) the French female form of vice-president
(vice-présidente) appears in the translation produced by the BASE system while
the male form is the correct one. The gender-informed system does make the
correct agreement by using the female variant. In ( ()) the speaker is female
but the baseline system outputs a male form of the adjective ‘happy’
(‘heureux’).

  * ()

    * (Ref) | En tant que _vice-président_ …
---|---
(BASE) | En tant que _vice-présidente_ …
(TAG) | En tant que _vice-président_ …

  * ()

    * (Ref) | … je suis _heureuse_ que…
---|---
(BASE) | … je suis _heureux_ que…
(TAG) | … je suis _heureuse_ que…

However, we also encountered cases where the gender-informed system fails to
produce the correct agreement, as in ( ()), where both the BASE and the TAG
system produce a male form (‘embarassé’) instead of the correct female one
(‘embarassée’ or ‘gênée’).

  * ()

    * (Ref) | je suis _gênée_ que…
---|---
(BASE) | je suis _embarassé_ que…
(TAG) | je suis _embarassé_ que…

For some language pairs the gender-informed system leads to a significant
improvement even on a general test set. This implies that the improvement is
not merely because of better morphological agreement, as these kinds of
improvements are very hard to measure with BLEU, especially given the fact
that Europarl consists of formal spoken language and does not contain many
sentences using the first person singular pronoun. From our analysis, we
observe that in many cases the gender-informed systems have a higher BLEU
score than the baseline system due to differences in word choices as in ( ())
and ( ()), where both translations are correct, but the gender-informed system
picks the preferred variant.

The observations with respect to differences in word preferences between male
and female speakers are in accordance with corpus linguistic studies, which
have shown that gender does not only have an effect on morphological
agreement, but also manifests itself in other ways as males and females have
different preferences when it comes to different types of constructions, word
choices etc. Newman et al. (2008); Coates (2015). This also implies that, even
for languages that do not mark gender overtly (i.e. grammatically), it can
still be beneficial to take the gender of the author/speaker into account.

  * ()

    * (Ref) | Je pense que …
---|---
(BASE) | Je crois que…
(TAG) | Je pense que…

Although more research is required in order to draw general conclusions on
this matter, from other linguistic studies, it appears that it is indeed the
case that there is a relation between the use of the word “pense” (“think”) /
“crois” (“believe”) and the gender of the speaker. To see whether there is a
difference in word choice and whether this is reflected in our data, we
compiled a list of the most frequent French words for the male data and the
female data. Our analysis reveals that “crois” is, in general, used more by
males (having position 303 in the most frequent words for males, but only
position 373 for females), while “pense” is found at a similar position in
both lists (position 151 and 153). These findings are in accordance with other
linguistic corpus studies on language and gender stating that women use less
assertive speech Newman et al. (2008). “Croire” and “penser” are both verbs of
cognition but there is a difference in the degree of confidence in the truth
value predicated: the verb “croire” denotes more confidence in the truth of
the complement clause than the verb “penser” does. In the future, we would
like to perform a more detailed analysis of other specific differences in
lexical choices between males and females on multiple language pairs.

  * ()

    * (Ref) | J’ ai plusieurs remarques…
---|---
(BASE) | J’ ai un nombre de commentaires…
(TAG) | J’ ai plusieurs remarques…

##  6 Conclusions and Future Work

In this work, we experimented with the incorporation of speaker-gender tags
during the training of NMT systems in order to improve morphological
agreement. We focused particularly on language pairs that express grammatical
gender but included other language pairs as well, as linguistic studies have
shown that the style and syntax of language used by males and females differs
Coates (2015).

From the experiments, we see that informing the NMT system by providing tags
indicating the gender of the speaker can indeed lead to significant
improvements over state-of-the-art baseline systems, especially for those
languages expressing grammatical gender agreement. However, while analyzing
the EN–FR translations, we observed that the improvements are not always
consistent and that, apart from morphological agreement, the gender-aware NMT
system differs from the baseline in terms of word choices.

In the future, we would like to conduct further manual evaluation on the
translations to further analyze the differences with the baseline system.
Furthermore, we aim to experiment with other ways of integrating speaker
information. We envisage working on gender classification techniques in order
to work on other types (more informal) of corpora that are more likely to
express speaker characteristics.

## Acknowledgements

This work has been supported by COST action IS1312, the Dublin City University
Faculty of Engineering & Computing under the Daniel O’Hare Research
Scholarship scheme and by the ADAPT Centre for Digital Content Technology,
which is funded under the SFI Research Centres Programme (Grant 13/RC/2106).
Christian Hardmeier was supported by the Swedish Research Council under grant
2017-930.

We would also like to thank the anonymous reviewers for their insightful
comments and feedback.

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# On the evolution of compact binary black holes

G.G. Adamian1, N.V. Antonenko1, H. Lenske2, and V.V. Sargsyan1 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia,
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany

###### Abstract

Based on the consideration of potential energy of the di-black-hole as a
function of mass asymmetry (transfer) collective coordinate, the possibility
of matter transfer between the black holes in a binary system is investigated.
The sensitivity of the calculated results is studied to the value of the total
mass of binary system. The conditions for the merger of two black holes are
analyzed in the context of gravitational wave emission.

###### pacs:

26.90.+n, 95.30.-k
Keywords: close binary stars, mass transfer, mass asymmetry

##  I Introduction

The general point of view is that the binaries of compact object, composed of
white dwarfs, neutron stars and black holes, eventually merge through
gravitational wave emission, as observed recently with the LIGO and Virgo
interferometers Abbott . There is also a general consensus that the physical
processes during the merger belong to the biggest unsolved problems of stellar
evolution.

Mass transfer is an important process for close binary systems, because of its
decisive role for the fate of the system. Hence, it is meaningful and
necessary to study the evolution of binary stellar systems in the mass
asymmetry coordinate $\eta=(M_{1}-M_{2})/M$ [$M_{k}$ ($k$=1,2) are the object
masses and $M=M_{1}+M_{2}$ is the total mass of system]. In our previous works
IJMPE ; IJMPE2 ; IJMPE4 , we have analyzed the total potential energy
$U(\eta)$ and the orbital period as functions of $\eta$ at fixed total mass
$M=M_{1}+M_{2}$ and orbital angular momentum $L$ of the contact binary stars
and galaxies. We have shown that the mass asymmetry (transfer) collective
degree of freedom plays a comparable important role in macroscopic objects,
being of comparable importance as in microscopic dinuclear systems. In close
binary galaxies and stars, the mass asymmetry coordinate governs the
asymmetrization (the transfer of mass from the lighter component to the heavy
one) and symmetrization (the transfer of mass from the heavier component to
the light one) of the system. As found, the symmetrization of binary object is
the most favorable evolution channel. The symmetrization of binary galaxies
and stars leads to the release of a large amount of energy. For example, in
the case of binary galaxies, the released energy is about $10^{48-52}$ J, thus
reaching the energy release of novae or even close to supernovae events IJMPE2
. Thus, the symmetrization of close binary galaxy due to the mass transfer is
one of the important sources of the transformation of the gravitational energy
to other types of energy, like radiation energy, in the Universe.

By this article, we intend to add a few aspects highlighting the role of mass
transfer for binary black holes (BBH). Under all stellar binaries, BBH are
undoubtfully a special class of objects. The presence of the event horizon
inhibits the flow of mass between the core systems. On a speculative level,
quantum tunneling as the Hawking radiation Hawking  may play a faint, but most
likely negligible, role in view of the yet pending firm confirmation. However,
BBH are embedded typically in a cloud of remnant matter, left over from the
progenitor stars. Hence, a BBH system must always be considered together with
the surrounding accretion disk. The material of that disk may serve for mass
transfer between the core systems by the so–called _sloshing effect_ as
illustrated clearly by the hydrodynamical calculations in Ref. Bowen:2016mci .

In order to understand the conditions for matter exchange in a BBH system, we
investigate the potential energy of a di-black-hole system as a function of
mass asymmetry coordinate $\eta$. We consider the possibility of this transfer
based on the calculation of the potential energy of BBH system – always to be
understood together with its accretion disk – is calculated as a function of
mass asymmetry. Note that in addition to its own extreme strong gravitational
fields of two black holes, the potential energy contains the interaction
energy between two black holes. We use Newtonian mechanics, being well aware
of the strong gravitational fields. As found in Bowen:2016mci , at the
distance post–Newtonian effects may alter the binary potential not more than
25% which will not affect the overall BBH properties. Of course, general
relativity will become essential at separations close to touching which is a
scenario beyond the scope of this work.

In section II we discuss the theoretical approach and the derived predictions,
based on a fix point analysis of the dependence of the total BBH energy
functional on the mass asymmetry coordinate. The work is summarized in section
III.

##  II Theoretical Method and Its Application

The total potential energy of the BBH system

$\displaystyle U=U_{1}+U_{2}+V_{1R}+V_{2R}+V$ (1)

is given by the sum of the potential $U_{k}$ ($k=1,2$) and rotational energies
$V_{kR}$ ($k=1,2$) of the two nonzero spin black holes, and black-hole-black-
hole interaction potential $V$. The energy of the black hole ”$k$” is

$\displaystyle U_{k}=-\frac{GM_{k}^{2}}{R_{k}},$ (2)

where $G$, $M_{k}$, and $R_{k}$ are the gravitational constant, mass, and
radius of the black hole, respectively. The rotation energy of black-hole is

$\displaystyle V_{kR}=\frac{S_{k}\omega_{k}}{2}=\frac{M_{k}c^{2}}{2},$ (3)

where $S_{k}=M_{k}R_{k}c$ and $\omega_{k}=c/R_{k}$ are the spin and rotational
frequency of the black hole, respectively. Here, $c$ is the speed of light.

The radius of the event horizon (distance from the gravitating mass $M_{k}$ at
which the particle velocity becomes equal to the speed of light $c$) Murad1 ;
Murad2 ; Murad3

$\displaystyle R_{k}=\frac{GM_{k}}{c^{2}}$ (4)

is derived from the energy conservation law for a particle with mass $m$:
$-GM_{k}m/R_{k}+mc^{2}=0$, where $mc^{2}$ is the sum of kinetic $mc^{2}/2$ and
rotation $mc^{2}/2$ energies. Note that the derivation of the radius $R_{k}$
is based on classical mechanics and Newtonian law of gravity. Similarly, in
Ref. Michell , an expression for the radius $R_{k}=2GM_{k}/c^{2}$ was obtained
in the case of a non-rotating black-hole. Employing Eqs. (2) and (4), we
obtain

$\displaystyle U_{k}$ $\displaystyle=$ $\displaystyle-M_{k}c^{2}.$ (5)

Because the two black holes rotate with respect to each other around the
common center of mass, the black-hole-black-hole interaction potential $V(R)$
contains, together with the gravitational energy of interaction $V_{G}$ of two
black holes, the kinetic energy of orbital rotation $V_{R}$:

$\displaystyle V(R)=V_{G}+V_{R}=V_{G}+\frac{L^{2}}{2\mu R^{2}}=V_{G}+\frac{\mu
v^{2}}{2},$ (6)

where $v=(GM[2/R-1/R_{m}])^{1/2}$ and $R_{m}$ are the speed and the semimajor
axis of the elliptical relative orbit, respectively, $L$ is the orbital
angular momentum of the di-black-hole system, and
$\mu=M_{1}M_{2}/(M_{1}+M_{2})=M_{1}M_{2}/M$ is the reduced mass. At $R\geq
R_{t}=R_{1}+R_{2}$, the black-hole-black-hole interaction potential is

$\displaystyle V_{G}(R)=-\frac{GM_{1}M_{2}}{R}.$ (7)

From the conditions $\partial V/\partial R|_{R=R_{m}}=0$ and
$\partial^{2}V/\partial R^{2}|_{R=R_{m}}>0$, we find the relative equilibrium
distance between two black holes corresponding to the minimum of $V$:

$\displaystyle R_{m}=\frac{L^{2}}{G\mu^{2}M}.$ (8)

We assume that the total angular momentum $J=GM^{2}/c$ of di-black-hole system
Murad1 ; Murad2 ; Murad3  is conserved during the conservative mass transfer
and the orbital angular momentum of the di-black-hole is

$\displaystyle L=J-S_{1}-S_{2}=\frac{G(M^{2}-M_{1}^{2}-M_{2}^{2})}{c}.$ (9)

Here the orbital and spin angular momenta are the parallel and the orbital
angular momentum is minimal. Employing Eqs. (8) and (9), we derive the
expression for relative distance

$\displaystyle R_{m}=\frac{4GM}{c^{2}}.$ (10)

As seen, the larger $M$ the larger $R_{m}$ is. Because
$R_{m}/(R_{1}+R_{2})=4$, the separation of components increases at $M_{1}\to
M_{2}$. From Eqs. (6), (7), and (10) one can obtain the simple formula

$\displaystyle
V(R_{m})=-\frac{GM_{1}M_{2}}{2R_{m}}=-\frac{M_{1}M_{2}c^{2}}{8M}=-\frac{\mu
c^{2}}{8}$ (11)

for the interaction potential. Here, $V(R_{m})$ depends only on the reduced
mass $\mu$ and velocity of light.

So, using Eqs. (5), (3), and (11), we obtain the final expression

$\displaystyle
U=-\frac{Mc^{2}}{2}\left(1+\frac{M_{1}M_{2}}{4M^{2}}\right)=-\frac{Mc^{2}}{2}\left(1+\frac{\mu}{4M}\right)$
(12)

for the total potential energy of the binary black hole system. For the BBH
considered, $v(R_{m})=(GM/R_{m})^{1/2}=c/2$ and with a satisfactory accuracy
one can neglect the relativistic effects and use the Newtonian law of gravity.

Using the mass asymmetry coordinate $\eta$ instead of masses
$M_{1}=\frac{M}{2}(1+\eta)$ and $M_{2}=\frac{M}{2}(1-\eta)$, we rewrite the
expressions (9) and (12) for the orbital angular momentum

$\displaystyle L=\frac{GM^{2}}{2c}\left(1-\eta^{2}\right)$ (13)

and the total potential energy

$\displaystyle U=-\frac{Mc^{2}}{2}\left(1+\frac{1-\eta^{2}}{16}\right).$ (14)

Since the solution of equation

$\frac{\partial U}{\partial\eta}=\frac{Mc^{2}}{16}\eta=0$

gives $\eta=\eta_{m}=0$ and

$\frac{\partial^{2}U}{\partial\eta^{2}}|_{\eta=\eta_{m}}=\frac{Mc^{2}}{16}>0,$

the potential landscape has an oscillator shape and a global minimum at
$\eta=\eta_{m}=0$ for the arbitrary total mass $M$ of the BBH system (Fig. 1).
So, the transfer of mass between the black holes in the binary system is
energetically favorable and, correspondingly, can occur. The initial
asymmetric system is driven without additional driving energy to the symmetric
BBH configuration (towards a global minimum). As seen, this conclusion does
not depend on the choice of parameters, and the losses of the total mass and
orbital angular momentum do not influence the symmetrization (the transfer of
mass from the heavier component to the light one) of system. Due to the
interaction energy between two black holes with their own extreme gravity
fields, it becomes possible to transfer matter between them. However, the
evolution of binary system in mass asymmetry coordinate depends also on the
mass parameter in this degree of freedom. Since the surfaces of two black
holes are spaced $R_{1}+R_{2}<R_{m}$ from each other, the mass parameter in
$\eta$ is very large and, accordingly, to some extent prohibits the
symmetrization of the di-black-hole system.

Figure 1: The potential energy of the BBH (14) divided by $Mc^{2}$ as a
function of mass asymmetry $\eta$.  Figure 2: The potential energy of the BBH
(16) divided by $Mc^{2}$ as a function of mass asymmetry $\eta$.

The asymmetrization (merger due to the transfer of mass from the lighter
component to the heavy one) of the binary black holes considered is not
energetically favorable process and, correspondingly, the merger channel in
mass asymmetry is strongly suppressed for di-black-hole. Thus, the question
remains open on the mechanism of the merger of two black holes and the origin
of gravitational waves Abbott .

In the case of the antiparallel orbital and spin angular momenta, the orbital
angular momentum

$\displaystyle L=J+S_{1}+S_{2}=\frac{G(M^{2}+M_{1}^{2}+M_{2}^{2})}{c}$ (15)

is the maximal. For the binary black hole system with the orbital angular
momentum (15), the total potential energy is

$\displaystyle
U=-\frac{Mc^{2}}{2}\left(1+\frac{(1-\eta^{2})^{3}}{16(3+\eta^{2})^{2}}\right)$
(16)

and $R_{m}/(R_{1}+R_{2})\geq 36$, $v(R_{m})=(GM/R_{m})^{1/2}\leq c/6$,
$\eta_{m}=0$, $\partial^{2}U/\partial\eta^{2}|_{\eta=\eta_{m}}>0$, and all
conclusions given above are also valid in this case. As seen from the
comparison of Fig. 2 with Fig. 1, the same conclusions are also obtained in
the case when the black holes of binary system are spinning in opposite
directions.

##  III Summary

Employing the Newtonian law of gravity and considering the potential energy of
the di-black-hole as a function of mass asymmetry (transfer) collective
coordinate $\eta$, we have shown the possibility of matter transfer between
the black holes in a binary system. The evolution of asymmetric binary system
to the symmetric configuration with $\eta=\eta_{m}=0$ is energetically
favorable process. Although black holes have their own strong gravitational
fields, the transfer of matter between two black holes arises from the energy
of interaction between them. A di-black-hole system does not send any signals
during its evolution. One can indirectly observe only the result of this
evolution. In the course of time evolution the maximum of the distribution of
expected BBH mass ratios shifts towards binary systems with equal masses.

The symmetrization of initially asymmetric BBH leads to the decrease of
potential energy $U$, thus transforming the potential energy into internal
kinetic energy. For example, for the binary black-holes
$4M_{\odot}+2M_{\odot}$ ($\eta_{i}=0.33$) and $36M_{\odot}+29M_{\odot}$
($\eta_{i}=0.11$), the internal energies of black-holes will increase during
symmetrization by the amount $\Delta
U=U(\eta_{i})-U(\eta=0)=Mc^{2}\eta_{i}^{2}\approx 10^{47}$ J. For the
comparison, in the cases of compact binary stars and compact binary galaxies,
the released energies are about $10^{41}$ J and $10^{48-52}$ J, respectively
IJMPE ; IJMPE2 . So, the BBH is the source of thermal energy.

The transfer of matter from a lighter component to a heavy one, leading to the
merger of black holes in a binary system, is not an energetically favorable
process and, accordingly, the question of the origin of gravitational waves
remains open.

In the frame of our model, one can also perform the dynamic calculations of
the evolution of binary system in mass asymmetry coordinate. But this
extension of our model is the subject of future studies.

## Acknowledgements

This work was partially supported by Russian Foundation for Basic Research
(Moscow) and DFG (Bonn). V.V.S. acknowledge the partial supports from the
Alexander von Humboldt-Stiftung (Bonn).

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