# Underwater Intention Recognition using Head Motion and
Throat Vibration for Supernumerary Robotic Assistance*

Yuqin Guo1,2,#, Rongzheng Zhang3,#, Wanghongjie Qiu2, Harry Asada4,
Fang Wan1,3,∗, and Chaoyang Song2,5,∗ *This work was partly supported by the
SUSTech-MIT Joint Centers for Mechanical Engineering Research and Education,
the Science, Technology, and Innovation Commission of Shenzhen Municipality
[JCYJ20220818100417038, 20200925155748006], National Science Foundation of
China [62206119], Guangdong Provincial Key Laboratory of Human-Augmentation
and Rehabilitation Robotics in Universities. Y. Guo and R. Zhang are co-first
authors. F. Wan and C. Song are co-corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>Key Laboratory of Intelligent Robotics and Flexible
Manufacturing, Southern University of Science and Technology, Shenzhen,
Guangdong 518055, China.2Department of Mechanical and Energy Engineering,
Southern University of Science and Technology, Shenzhen, Guangdong 518055,
China.3School of Design, Southern University of Science and Technology,
Shenzhen, Guangdong 518055, China.4Department of Mechanical Engineering,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA.5SUSTech
Institute of Robotics, Southern University of Science and Technology,
Shenzhen, Guangdong 518055, China.

###### Abstract

This study presents a multi-modal mechanism for recognizing human intentions
while diving underwater, aiming to achieve natural human-robot interactions
through an underwater superlimb for diving assistance. The underwater
environment severely limits the divers’ capabilities in intention expression,
which becomes more challenging when they intend to operate tools while keeping
control of body postures in 3D with the various diving suits and gears. The
current literature is limited in underwater intention recognition, impeding
the development of intelligent wearable systems for human-robot interactions
underwater. Here, we present a novel solution to simultaneously detect head
motion and throat vibrations under the water in a compact, wearable design.
Experiment results show that using machine learning algorithms, we achieved
high performance in integrating these two modalities to translate human
intentions to robot control commands for an underwater superlimb system. This
study’s results paved the way for future development in underwater intention
recognition and underwater human-robot interactions with supernumerary
support.

##  I Introduction

Diving with Self-Contained Underwater Breathing Apparatus (SCUBA) is a popular
activity for exploring the ocean, which involves a series of professional
equipment wearable on the human body for life-support and body movement [1].
However, the level of intelligence of these diving gears remains primarily
mechanical by design. There remains a research gap in introducing robotic
solutions toward autonomous, natural interactions between human divers and the
underwater environment, where novel designs in wearable robots and interactive
mechanisms need further exploration [2].

Before introducing wearable robots to assist human divers, intention
recognition underwater becomes a critical issue due to challenges brought by
the aquatic environment. Currently, hand gestures are the most effective
method for diver communication [3]. However, when submerged underwater, the
divers must constantly move all limbs to maintain body postures against the
water, making it physically demanding and mentally exhausting to spare extra
attention to hand gestures or tool operations. The water greatly limited
divers’ sense of the environment while restricting regular verbal
communications or facial expressions, making it urgently necessary to develop
novel solutions for intention recognition underwater for effective human-robot
interactions [4].

Figure 1: Summary of the underwater intention recognition method using
wearable IMU and throat microphone. (A) Diver with standard diving equipment,
including a diving backpack, diving mask, diving computer, diving suit,
flipper, oxygen cylinder, etc. (B) The IMU headband can collect the motion
information of the six types of head motion, including extension/flexion,
bending left/right, and rotating left/right. (C) The throat microphone can
acquire the throat vibration, including the first five musical scales (“do”,
“re”, “mi”, “fa”, and “so”). (D) Based on the servo angle definition of the
superlimb, we define five types of motion modes. $\delta_{1}$ and $\delta_{2}$
are the angles of the left and right thrusters. $n_{1},n_{2}$ and
$T_{1},T_{2}$ are the rotational speed and the thrust force of the left and
right thruster, which can be controlled continuously via PWM. (E) The five
motion modes mapping from the classification token with head motion or/and
throat vibration.

One way to drive novel designs for aquatic systems is by drawing inspiration
from on-land systems to underwater applications, where the integration of head
motion and throat vibration seems viable. For example, recent work by Yang [5]
demonstrated an artificial throat to check vocal vibrations to recognize
everyday words vaguely spoken by a patient after a laryngectomy. Wang [6]
proposed a method through detecting eye motions and throat vibrations to
interpret the intention of patients with amyotrophic lateral sclerosis (ALS).
Severin [7] developed a system using inertial sensors to detect head movement
for intention recognition. Machangpa [8] designed a wheelchair controlled by
head gestures for quadriplegic patients. Although the divers’ limbs are busy
maintaining body postures and the mouth is filled with the breathing
mouthpiece, we can leverage such limitations to use the head and throat to
express intentions for controlling wearable robots underwater, such as an
underwater superlimb [9].

In this paper, we propose a novel solution for underwater intention
recognition by simultaneously detecting the diver’s head motion and throat
vibration, as shown in Fig. 1, to enable multi-modal human-robot interactions
with an underwater supernumerary robotic limb designed for providing
propulsion assistance. The design features customization of the headband with
a waterproof IMU sensor mounted on the top and a throat microphone on the neck
for hands-free interaction. The system determines the diver’s intention by
sensing the diver’s head motion through the IMU sensor, or confirms the
control commands by detecting the diver’s vocal vibration through the throat
microphone using learning algorithms. By designing mapping commands to the
underwater superlimb, the system recognizes the diver’s intention for posture
control underwater, aiming at reducing the diver’s physical load and mental
fatigue for nature interactions without using hands. The contributions of this
study are as the following:

  * •

Proposed a novel design for underwater intention recognition by sensing the
diver’s head motion and throat vibration in a compact form factor for diving
scenarios.

  * •

Developed a multi-modal, real-time classification algorithm based on five
musical scales and six head motion types for intention recognition underwater.

  * •

Verified the feasibility of the proposed method for controlling an underwater
superlimb prototype with continuous motion commands for underwater propulsion
assistance.

The rest of this paper is organized as the following. Section II presents the
diver intention recognition method of the wearable sensing device, including
the engineering design and classification algorithm for head motion and throat
vibration. Section III reports the experiment results using head motion,
throat vibration, and combined modalities for superlimb control. Section IV
discusses the experiment results and implications. The conclusion,
limitations, and future work are in the final section.

##  II Method

###  II-A Engineering Design

We designed a multi-modal sensing system as shown in Fig. 2 for underwater
intention detection. The IMU sensor can be fixed on the head using the
headband, as shown in Figs. 2A&C, picking up Euler angles and accelerations of
head motion at up to 500Hz in 16 bits. The throat microphone is wearable on
the neck to detect throat vibration at 16k or 60kHz in 16 bits, as shown in
Fig. 2B. To protect the IMU from water eroding, we sealed the waterproof shell
with silicone and sealant (Epoxy sealant for seawater from ROVMAKER). We
modified the mask design (M8038 from SMACO), as shown in Fig. 2D, to adjust
the IMU sensor’s angle by turning the knob of the joint on top. The design of
the IMU headband is compatible with the full-face diving mask for SCUBA
divers, which connects the oxygen tank with a regulator for SCUBA diving.

Figure 2: Engineering design of the wearable sensing devices to collect the
head motion and throat vibration data underwater. (A) IMU headband with a
waterproof shell integrated with a 9-axis IMU (3DX-GX3-25 from Parker
Hannifin), sealed by a silicone layer and a silicone sealant layer. (B) Throat
microphone (Z033 from WADSN Corp.) smeared with a polyurethane waterproofing
spray (from SKSHU). (C) A test user is wearing the waterproof IMU sensor with
the headband (on land). (D) The IMU sensor mounted on a full-face diving mask
(for underwater).

###  II-B Detecting Head Motions and Throat Vibrations

We used different methods to process the two time-series data, as shown in
Fig. 3. We adopted the Dynamic time warping (DTW) algorithm to distinguish six
types of head motion (Bending left/right, Rotating left/right,
Extension/Flexion), commonly used to process IMU data [10]. The raw data from
the IMU (Accelerations and Euler Angles) was smoothed by a low-pass filter.
Then, a self-adapting threshold segmentation method extracted the segment with
the practical meaning of instructions. The DTW algorithms maximize the
difference between different head motion types and minimize the distance
between those of the same kind [11]. Since the Adaptive DTW barycenter
averaging (ADBA) algorithm can average the motion data sequences in time and
space, this time-series averaging template method has a higher recognition
accuracy than a randomly averaged method. It was used to generate the data
sequence template for the six head motion types [12]. 1,436 head motion
samples were collected from two male and two female participants (Bending
left/right: 280/288, Extension/Flexion: 270/266, Rotating left/right:
296/306). Half of the dataset was used to generate the head motion template,
and the rest was used for testing. Results show that the head motion
recognition accuracy is measured at an average of $94\%$, as shown in Fig. 4A.

Figure 3: Underwater interaction method based on the wearable sensing devices
integrated with throat microphone and IMU. (A) The IMU can sense the head
motion information, including acceleration and Euler angles. After the
endpoint detection method based on adaptive thresholds, segments would be
matching using DTW algorithm to distinguish the head motion types based on the
head motion templates. (B) The throat microphone can acquire the vibration of
the throat. After noise reduction, the significant fragments of the raw signal
are extracted through the endpoint detection method. Mel-filter bank analysis
transforms these fragments into Mel-frequency cepstral coefficients (MFCCs).
After LSTM processing and classification recognition, the command index can be
mapped to a user-defined motion mode sent to control the superlimb.

However, the result was unsatisfactory when we applied the same method for
throat vibration data. Instead, we adopted the Mel-frequency cepstral
coefficients (MFCCs) to extract features in speech recognition [13].
Alternatively, we can also use Long Short Term Memory (LSTM) as a candidate
algorithm for acoustic modeling of speech [14]. We collected throat vibration
signals using a throat microphone and the Mel filter banks to transform the
audio signal to MFCCs. After pre-processing the raw data, we obtained a 20
$\times$ 20 matrix by cutting off the MFCCs matrix or padding the zeros matrix
into the time dimension of the MFCCs matrix, which describes the response of
the human auditory system for the specific audio signal. Then, we fused the
MFCCs matrix as the input of LSTM to get a classification result for the
throat vibration. Ten participants (seven males, and three females) were
invited to collect the throat vibration signal for data acquisition. They were
asked to phonate musical scales with the throat microphone shown in Fig. 2B.
We collected a dataset of 3,253 musical scale audio segments in WAV format
containing 647 “do”, 660 “re”, 594 “mi”, 668 “fa”, and 684 “so”, which were
then split with 70% for training and 30% for testing. The model’s average
accuracy for testing is about $86\%$. The confusion matrix of the
classification results is shown in Fig. 4B.

Figure 4: Accuracy of recognition-related performance in the classification
experiments. (A) Confusion matrix of head motion classification with IMU. (B)
Confusion matrix of throat vibration classification with a throat microphone.

##  III Results

###  III-A Intention Recognition via Head Motion

We divided the head motions into two groups to control the speeds and angles
of the two thrusters, respectively, as shown in Table LABEL:tab:mapping_exp_1.

TABLE I: Mapping between the head motions with superlimb control.

Head Motion | Euler Angle | Thruster Speeds4
---|---|---
Flexion | $\alpha$ | $(-K_{1}\ast\alpha,-K_{1}\ast\alpha)$
Extension | $\alpha$ | $(K_{1}\ast\alpha,K_{1}\ast\alpha)$
Head Motion | Euler Angle | Angle of Servos1
Left Bending | $\beta$ | $(K_{2}\ast\beta,K_{2}\ast\beta)$
Right Bending | $\beta$ | $(-K_{2}\ast\beta,-K_{2}\ast\beta)$
Left Rotation | $\gamma$ | $(-K_{3}\ast\gamma,K_{3}\ast\gamma)$
Right Rotation | $\gamma$ | $(K_{3}\ast\gamma,-K_{3}\ast\gamma)$

  * 1

Angle of Servos are those of the left and right ones.

  * 2

$K_{1}$ is the degree coefficient for $\alpha$.

  * 3

$K_{2}$ is the degree coefficient for $\beta$.

  * 4

Thruster Speeds are those of the left and right ones.

  * 5

$K_{3}$ is the coefficient converting $\gamma$ to the thrusters speed.

Fig. 5 demonstrates the human-robot interactions experiments. The time series
of accelerations and Euler angles along the $x/y/z$ axis recorded by IMU are
shown in Figs. 5A&B. The corresponding control command sequence is shown in
Fig. 5C, where the mapping of head motions to control command index is
(Bending right/left, Extension/Flexion, Rotating left/right) $\mapsto$ command
index: $(1,2,3,4,5,6)$. We executed four actions in each of the three DoFs of
the rotation. The Euler angles range smoothly within [45°,50°] for Flexion,
[70°,80°] for Extension, [35°,45°] for Bending left/right, and [65°,75°] for
Rotating left/right, respectively. The system recognized all 12 head motions
correctly, and the corresponding control commands were sent to the superlimb
afterward.

Figure 5: Control of the superlimb using head motion. (A) Accelerations along
the $x-y-z$ axis of the head motion from the IMU during the control process of
the superlimb. (B) Euler angles along the $x-y-z$ axis of the head motion from
the IMU during the control process of the superlimb. (C) According to the
motion information of the data sequence by the head motion recognition
algorithm, the head motion types were classified and the mapped motion control
command was sent to the control unit of the superlimb.

Fig. 6 compares the control command and actual feedback of the servos and
thrusters from the superlimb, indicating that the pipeline can detect human
intentions and achieve robot control continuously with low latency (less than
one second). However, to achieve precise control of the thrusters through the
Euler angles of the head motion, an operator would need training and practice
to obtain muscle memories of finer-grained mapping between head motions and
robot control. Such activity involves humans in the loop as a human-robot
system though the robot system alone is an open control loop.

Figure 6: Experimental results of superlimb control with head motion
recognition experiment: comparing theoretical output and actual feedback of
the superlimb . (A) The Control command and the actual feedback of the left
servo. (B) The Control command and the actual feedback of the right servo. (C)
The PWM command is sent to the left and right thruster based on the
classification token output from the head motion recognition algorithm.

###  III-B Intention Recognition via Throat Vibration

We designed a mapping between musical scales and both thrusters’ angle and
speed for the throat vibration signal in Table LABEL:tab:mapping_exp_2. Three
types of musical scales and lengths distinguish six kinds of control commands.
Meanwhile, the amplitude $A$ of the musical scale signal (within 64 $ms$) is
detected in real-time continuously and is used as a coefficient in each of the
six commands. For example, a short type of “do” is mapped to rotating both
thrusters to positive angles (show in Fig .1 E($v$)) determined by its
amplitude $A_{1}$.

TABLE II: mapping superlimb control with throat vibration recognition
experiment

Musical Scale | Duration | Amplitude | Angle of Servo1
---|---|---|---
do | $t<500ms$ | $A_{1}$ | $(A_{1}\ast K_{4},\enspace\,\,A_{1}\ast K_{4})$
do | $t>500ms$ | $A_{2}$ | $(-A_{2}\ast K_{4},-A_{2}\ast K_{4})$
re | $t<500ms$ | $A_{3}$ | $(-A_{3}\ast K_{4},\enspace\,\,A_{3}\ast K_{4})$
re | $t>500ms$ | $A_{4}$ | $(A_{4}\ast K_{4},-A_{4}\ast K_{4})$
Musical Scale | Duration | Amplitude | Thruster Speeds3
mi | $t<500ms$ | $A_{5}$ | $(A_{5}\ast K_{5},\enspace\,\,A_{5}\ast K_{5})$
mi | $t>500ms$ | $A_{6}$ | $(-A_{6}\ast K_{5},\enspace\,\,-A_{6}\ast K_{5})$

  * 1

Angle of Servos are those of the left and right ones.

  * 2

$K_{4}$ is the degree coefficient for throat vibration amplitude.

  * 3

Thruster Speeds are those of the left and right ones.

  * 4

$K_{5}$ is the coefficient converting throat vibration amplitude to thruster
speed.

We demonstrate the human-robot interaction through throat vibration. Fig. 7
shows the raw signal (solid purple line) and the recognized musical scales.
Every intention consists of two sequential waveform segments, with the first
indicating the type of command (pink shaded areas) and the second for the
amplitude of action (blue shaded areas). Although one could still express
control intentions with the throat vibration, the user must be trained in
vocal control for the system to recognize the intention effectively. Fig. 8
shows the theoretical and actual feedback of the servo angles and the control
command of PWM (Range from $[1100,1900]$) sent to the control module of the
thrusters. The blue shaded area is the action execution of the superlimb based
on the corresponding motion mode shown in Fig. 7.

Figure 7: Waveform of throat vibration acquired by throat microphone during
superlimb control with throat vibration recognition experiment. Marked in pink
shaded areas are the different types of throat vibration, and marked in blue
shaded areas are the throat vibrations used to continuously control the servo
angle and thruster speed of the superlimb according to the amplitude of the
vibration signals. Figure 8: Experimental results of superlimb control with
Throat vibration recognition experiment: comparing theoretical output and
actual feedback of the superlimb. (A) The Control command and the actual
feedback of the left servo. (B) The Control command and the actual feedback of
the right servo. (C) The PWM command is sent to the left and right thruster
based on the classification token output from the throat vibration recognition
algorithm.

###  III-C Multi-modal Intention Recognition and Interactions

In this experiment, we test the feasibility of using head motion and throat
vibration simultaneously as a multi-modal mechanism for controlling the
underwater superlimb based on intention recognition. Table
LABEL:tab:mapping_exp_3 defines the action vectors to describe the diver’s
robot control intentions mapped to the throat vibration and head motion.

TABLE III: Mapping of multi-modal intention recognition and interactions
experiment.

Action vector | Left thruster (rpm) | Right thruster (rpm)
---|---|---
(do,short,null) | Accelerate $k$ | null
(do,long,null) | Accelerate -$k$ | null
(re,short,null) | null | Accelerate $k$
(re,long,null) | null | Accelerate $-k$
(mi,short,null) | Stop | Stop
(mi,long,null) | Stop | Stop
(fa,short,null) | Accelerate $k$ | Accelerate $k$
(fa,long,null) | Accelerate $-k$ | Accelerate $-k$
(so,short,null) | Switch control mode | Switch control mode
(so,long,null) | Switch control mode | Switch control mode
Action vector | Left servo (degree) | Right servo (degree)
(null,null,left rotation) | $-90$ | null
(null,null,right rotation) | $\enspace\,\,90$ | null
(null,null,left bending) | $\enspace\,\,90$ | null
(null,null,right bending) | $-90$ | null
(null,null,extension) | $-90$ | $-90$
(null,null,flexion) | $\enspace{\,\,}90$ | $\enspace\,\,90$

  * 1

$null$ means tester has no action or superlimb doesn’t change its motion.

  * 2

$k$ means the acceleration coefficient of the thrusters.

In this experiment, the musical scale “so” was defined as the mode switch from
the control of servo angle to the rotational speed of the thruster. Fig. 9
compares theoretical output and actual measurement of the superlimb. We made
two observations in the multi-modal integration experiment. The noticeable
latency of the system could increase to two seconds when controlling the
servos using head motion because of the different frequencies between these
two recognition methods. The misalignment in Fig. 9C is caused by the system
transforming $(re,long,null)$ to $(so,long,null)$. The other observation is
when the extension was not correctly recognized in Fig. 9, which was caused by
an unexpected servo angle during the experiment. After all, the head motion
recognition module has to lower the frequency of data collection and
classification to meet the frequency of the throat vibration recognition
module.

Figure 9: Experimental results of multi-modal intention recognition and
interactions experiment: comparing theoretical output and actual feedback of
the superlimb using head motion and throat vibration. (A) The control command
and the actual feedback of the left servo. (B) The control command and the
actual feedback of the right servo. (C) The PWM command sent to the left and
right thrusters based on the token output from the intention recognition
algorithm.

##  IV Discussion

###  IV-A Towards Underwater Intention Recognition

This study presents the engineering design and experiment results of an
underwater multi-modal interaction mechanism for intention recognition using
head motions and throat vibrations. Although the reported system is still a
lab prototype that requires further testing underwater, the simple design in a
compact form factor makes the proposed solution promising for human-robot
interactions underwater. For on-land scenarios, the head motion or the throat
vibration has been demonstrated effective in intention recognition for
different applications with no need for integration. In this study, due to the
need for aquatic interaction, we propose to combine these two modalities for
intention recognition underwater. The high classification performance reported
in this study aligns with the literature. Our results further demonstrated
that when combining these two modalities, they formulate an intuitive
mechanism for intention expression that is effective in intention recognition
through learning algorithms, which could be a practical solution for
controlling an underwater superlimb robot.

This study tested only three pairs of head motion and five musical scales for
intention expression and recognition. However, one can quickly expand the
vocabulary by extending the head motion to the total five degree-of-freedoms
of the head to include the two translational motions. Some users may need
further training before being able to do so fluently. On the other hand, one
can also expand the musical scales to a broader range or develop the system to
recognize a sequence of them. For example, it is easier for people to remember
a piece of tune rather than the specific musical scale for differentiating
different meanings. The artificial throat [5] provides an excellent
inspiration to expand this work towards a more natural expression of
intentions for the system to recognize effectively, which we intend to further
explore by using sensors of more compact sizes [15].

###  IV-B Learning Intentions via Head Motion & Throat Vibration

In this study, we proposed a multi-modal learning framework for integrating
head motion and throat vibration in intention recognition. For on-land
scenarios with a clear voice, one could directly use conventional methods to
differentiate the volume, pitch, and tune from voice signals, a mature
technology already used in commercial products. However, the experiments in
this study specifically chose a learning approach as the signal detected was
through the throat, which had more noise than those collected from the mouth.
On the other hand, when submerged underwater, the noise signal from the water
would further reduce the quality of the sound signals, making it a challenging
task to classify different vocal signals clearly. However, our experiment
results show that the learning algorithms effectively classified the different
musical notes hymned by the test user. We intend to further test the proposed
system by collecting training data from underwater to refine the model for a
more realistic scenario.

On the other hand, the experiment results of the combined modalities in
controlling the superlimb robot were successful, real-time, and continuous
(please refer to the supplementary materials for demonstration). Throughout
the experiment, only the head movement and throat vibrations were used to
support hands-free interaction with the underwater superlimb. Further testing
in the aquatic environment is needed in future work to test the proposed
system’s performance thoroughly. On the other hand, one can easily extend the
application of the proposed method to interact with other underwater robots,
such as UAVs and robotic fishes, on-land robots, such as robotic manipulators,
legged robots, aerial robots, or mobile robots, or common Internet-of-Things
(IoTs) devices in domestic scenarios. Another application area is for people
with vocal impairment, where slight modifications could make the system
wearable for users with disabilities.

###  IV-C Human-Robot Interactions for an Underwater Superlimb

Humans are not biologically evolved for natural activities underwater, which
influenced the priority of design considerations when developing diving gear.
The current diving devices mainly provide life-support and swimming assistance
in an aquatic environment while being wearable to the diver’s body forms. With
the limited space, the complexity of need, and the waterproof requirement, two
significant challenges remain in introducing robotic intelligence to diving
gear. One is the design problem for a wearable robot compatible with the
existing diving gear while providing meaningful assistance underwater to
reduce the physical load of the body limbs. In our previous work [9], we
developed a reconfigurable underwater jetpack to enable wearable propulsion
with compatible connections to the current Buoyancy control device (BCD)
system, aiming at sharing the burdens of manual posture control underwater so
that the diver can spare their hands for tool operation. We named it an
underwater superlimb due to its functionalities in providing supernumerary
limb support for manual posture control during diving and swimming, similar to
other superlimb designs for on-land scenarios [16].

Intention recognition is the other problem that is yet to be solved when
developing an intelligent wearable underwater robot, which is shared by the
other superlimbs for on-land scenarios. The proposed solution, while being
inspired by many recent works for on-land scenarios, is found to be practical
for underwater human-robot interactions. With limited sensory feedback and
limited eyesight underwater, divers usually need to turn their heads
constantly for a better inspection of the surrounding environment, providing a
more explicit cue for expressing intentions without using their hands. While
the diver’s mouth is usually filled with breathing tubes of the regulator,
common language expression becomes a challenge. Even with a full-face mask,
there are still difficulties in underwater signal or voice transmission.
However, as demonstrated in this study, one can still leverage the throat
vibration to express a wide range of vocal commands for intuitive and direct
interaction.

##  V Conclusions

In this study, we proposed a novel mechanism for underwater intention
recognition using head motion and throat vibration. Experiment results showed
that the system accurately classified different intention expressions coded
through these two signals. The proposed multi-modal learning algorithm
effectively recognized the intentions of the test user and controlled the
underwater superlimb robot through various commons.

This study still needs to be improved in collecting data from the underwater
environments for training, which we intend to conduct once the water pools are
open for testing. The current system was tested on a breadboard, which needs
further integration with the underwater superlimb’s controller for an
integrated system design. Further refinement of the command mapping between
these two modalities and the robot control commands could be arranged.
Nevertheless, the results of this study paved the foundation for future
development in underwater intention recognition and underwater human-robot
interactions with supernumerary support.

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# On the inclusion of statistical matter in the non-relativistic limit of NS-
NS supergravity

Eric Lescano<EMAIL_ADDRESS>University of Wroclaw, Faculty of
Physics and Astronomy, Maksa Borna 9, 50-204 Wroclaw, Poland

###### Abstract

We combine techniques from kinetic theory and string dualities to couple
statistical matter to a non-relativistic (NR) supergravity background so that
the theory can be reformulated in a T-duality invariant form. While in the
relativistic case the many-strings system requires a viscous fluid to describe
the statistical matter, the NR formulation of ST simplifies this issue and
therefore the system can be described using the perfect fluid technology.

##  I Introduction

T-duality [1] is an exact symmetry [2, 3] of string theory (ST) and its effect
on the non-relativistic (NR) limit of ST [4, 5, 6, 7, 8, 9] is that a space-
like longitudinal direction of the theory is mapped to a null one, underlying
a discrete light-cone quantization [10, 11, 12]. Remarkably, this leads to a
drastic simplification of ST while its low energy limit is still given by a
supergravity with a finite action principle [13, 14, 15, 16]. In the
relativistic case, the energy-momentum tensor of a many-strings system is
given by

$\displaystyle T^{ij}=\frac{1}{4\pi\alpha^{\prime}}\int
d\sigma\frac{d\tau}{dX^{0}}(\partial_{\tau}X^{i}\partial_{\tau}X^{j}-X^{\prime
i}X^{\prime j})\,,$ (1)

which can be effectively described by a fluid with viscosity [17], the latter
appearing to compatibilize T-duality with the matter description, i.e.,

$\displaystyle\tilde{p}=0\quad\tilde{\eta}\neq 0$ (2)

where $\tilde{p}$ is the pressure and $\tilde{\eta}$ is the share viscosity
after a T-duality transformation. The physics behind this statistical model is
the angular stone of string cosmology [18].

In this work we will construct a duality invariant model for statistical
strings under the NR limit of ST and we will show that the viscosity
contribution is not required. The vacuum description of the NR supergravity is
given by a transverse vielbein $\tau_{\mu}{}^{a}$, a longitudinal vielbein
$e_{\mu}{}^{a}$, a two-form $b_{\mu\nu}$ and a scalar field $\phi$. All these
fields and their dynamics can be rewritten in a Double Field Theory (DFT) [19,
20, 21, 22] formalism 111For reviews see [23, 24, 25] and the second lecture
in [26]. where the fundamental fields obey a convergent c-expansion.
Interestingly enough, in the supergravity case, all the bosonic fields contain
divergences in their expansions and therefore the finitness of the action and
the equations of motion can be easily recognized from the DFT prescription
[27]. Moreover, the DFT rewriting is a particular case of a larger family of
non-Riemannian double geometries [28, 29, 30, 31, 32, 33, 34]. We can
summarize these concepts in the following diagram,

$\displaystyle\begin{array}[]{ccccc}\textrm{supergravity
}&\longleftrightarrow&\textrm{bosonic DFT }&&\\\
(\hat{g},\hat{B},\hat{\Phi})&&(\hat{\cal H},\hat{d})&&\\\
\downarrow&&\downarrow\\\ \text{{non}-finite $c$-expansion}&&\text{finite
$c$-expansion}\\\ (\hat{g},\hat{B},\hat{\Phi})&&(\hat{\mathcal{H}},\hat{d})\\\
\downarrow&&\downarrow&&\\\ \textrm{finite NR-
supergravity}&\longleftrightarrow&\text{finite NR-DFT}&&\end{array}$ (10)

where the $\leftrightarrow$-arrow means

$\displaystyle\rightarrow$ : DFT rewriting, $\displaystyle\leftarrow$ :
breaking the duality group.

The tension between the generalized frame formalism [35] and generalized
metric formalism [36] is also present when the generalized fields are expanded
using a c-expansion [37]. The former contains divergences in its expansion
while the generalized metric is convergent. Therefore, both the action and the
equations of motion converge, which indeed forces a very particular form for
the generalized energy-momentum tensor.

In this work we distinguish statistical matter from ordinary matter. The
former is described by a distribution function which satisfies the Boltzmann
equation while the latter is given by a matter Lagrangian. In the context of
relativistic hydrodynamics, the equilibrium distribution function for a
perfect fluid is given by [38]

$\displaystyle f(x,p)\propto e^{-p_{\mu}\beta^{\mu}}$ (11)

with $\beta^{\mu}=\frac{u^{\mu}}{T}$ and $T$ the temperature of the fluid.
Then, the integration $\int_{x,p}gf(x,p)p^{\mu}p^{\nu}$ results in

$\displaystyle T^{\mu\nu}=(e+p)u^{\mu}u^{\nu}+pg^{\mu\nu}\,.$ (12)

While the inclusion of the perfect fluid in the NR supergravity is given by
the Einstein equation

$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{k}{c^{4}}T_{\mu\nu}\,,$ (13)

with $\kappa=8\pi G_{D}$, there is not a unique way to perform the c-expansion
on the hydrodynamics variables. Therefore, in this work we propose to use DFT
as a guide principle to construct the expansions and give the explicit form of
energy-momentum tensor. This means that we will use the $O(D,D)$ symmetry to
constrain the form of the expansion of the hydrodynamics variables.

The inclusion of statistical matter in the DFT framework was given in [39] and
further study in [40, 41, 42]. In these series of works, the generalized
energy-momentum for a perfect fluid coupled to the double geometry was
constructed from a kinetic theory perspective. This tensor is fully compatible
with the generalized Einstein equation [43, 44], and also is in correspondence
[45] with the dynamics of a massless generalized scalar field. The fluid
description of the statistical matter constitutes, in general, a simple toy
model to study the string hydrodynamics/thermodynamics and cosmological
scenarios, among others. In those cases, the fluid might represent a string
gas [46] or a more general many-strings system. The main goal of this work is
to construct the explicit form of the energy-momentum tensor for these kind of
scenarios, in such a way that the whole NR supergravity can be embedded in a
T-duality invariant formalism. We will construct the energy-momentum tensor
from the DFT framework in order to ensure that the final result is compatible
with $O(D,D)$ symmetry. We will follow the same strategies implemented in [47,
48, 49, 50], but starting from the double geometry and respecting the
constraints of the latter. The main results of this work are:

  * •

We construct the expansion of the generalized velocity $U_{M}$ in order to
satisfy the usual DFT constraints $U_{M}{\cal H}^{MN}U_{N}=-c^{2}$,
$U_{M}\eta^{MN}U_{N}=0$. We also verify that the generalized distribution
function $f(X,{\cal P})$ [42] remains finite after considering the NR limit.
This result can be found in (III.1) and (IV).

  * •

We study the generalized correspondence [41] when the fundamental fields are
expanded under the NR expansion. In this case, the correspondence is no longer
compatible meaning that the generalized scalar field is not in correspondence
with a perfect fluid for NR double backgrounds. This result can be found in
(III.2).

  * •

We construct the generalized energy-momentum tensor for a perfect fluid, which
leads to a fluid model in supergravity with no dilaton or B-field sources
($\sigma=0$ and $J^{\mu\nu}=0$). While in the relativistic proposal the model
is constrained by $\sigma=2p$ [40] which leads to the viscosity inclusion
given in [42], in the NR limit this constraint is modified due to the rhs of
the generalized dilaton equation, $\frac{p}{c^{4}}{\cal H}_{MN}\rightarrow 0$
and the dynamics do not require viscosity. The explicit form of both the
generalized energy-momentum tensor and its parametrization is the main topic
of section (IV).

##  II Coupling matter in the non-relativistic formulation of NS-NS
supergravity

Our starting point is the standard NS-NS supergravity formulation with action
principle given by

$\displaystyle\\!\\!\\!S_{0}=\int{\mathrm{d}}^{D}x\,\operatorname{e}^{-2\hat{\Phi}}\sqrt{-\hat{g}}\left[R(\hat{e})+4\,(\partial\hat{\Phi})^{2}-\frac{1}{12}\,\hat{H}^{2}\right],$

where the indices $\mu,\nu=0,\dotsc,D-1$ and
$\hat{H}_{\mu\nu\rho}=3\partial_{[\mu}\hat{B}_{\nu\rho]}$. The fundamental
fields are expanded in powers of $c$ as

$\displaystyle\hat{e}_{\mu}{}^{a}$ $\displaystyle=$ $\displaystyle c\
\tau_{\mu}{}^{a}\,,\quad\hat{e}_{\mu}{}^{a^{\prime}}=e_{\mu}{}^{a^{\prime}}\,$
(14) $\displaystyle\hat{B}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-c^{2}\epsilon_{ab}\tau_{\mu}{}^{a}\tau_{\nu}{}^{b}+b_{\mu\nu}\,,$
(15) $\displaystyle\hat{\Phi}$ $\displaystyle=$ $\displaystyle\ln(c)+\phi\,,$
(16)

where $\hat{a}=(a,a\textasciiacute)$ is the decomposition of the Lorentz
index, $a=0,1$ are the transverse directions and
$a\textasciiacute=2,\dots,D-1$ are the longitudinal ones. The new fundamental
fields obey the following relations

$\displaystyle\tau_{\mu}{}^{a}e^{\mu}{}_{a^{\prime}}$
$\displaystyle=\tau^{\mu}{}_{a}e_{\mu}{}^{a^{\prime}}=0\,,\qquad
e_{\mu}{}^{a^{\prime}}e^{\mu}{}_{b^{\prime}}=\delta^{a^{\prime}}_{b^{\prime}}\,,$
(17) $\displaystyle\tau_{\mu}{}^{a}\tau^{\mu b}$
$\displaystyle=\eta^{ab}\,,\qquad\tau_{\mu}{}^{a}\tau^{\nu}{}_{a}+e_{\mu}{}^{a^{\prime}}e^{\nu}{}_{a^{\prime}}=\delta_{\mu}^{\nu}\,.$
(18)

The limit $c\rightarrow\infty$ at the level of the Lagrangian is convergent,
meaning that all the divergences coming from the Ricci scalar are cancelled by
contributions coming from the curvature of the B-field. On the other hand, we
couple matter considering a variational principle. In that case, the Einstein
equations can be schematically written as

$\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta S_{0}}{\delta h^{\mu\nu}}$
$\displaystyle=$ $\displaystyle T_{\mu\nu}\,,\ \
\frac{2}{\sqrt{-g}}\frac{\delta S_{0}}{\delta e_{\mu}{}^{a}}=T_{\mu}{}^{a}\,,$
(19) $\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta S_{0}}{\delta b_{\mu\nu}}$
$\displaystyle=$ $\displaystyle J^{\mu\nu}\,,\ \
\frac{1}{\sqrt{-g}}\frac{\delta S_{0}}{\delta\phi}=-\sigma\,,$ (20)

and the right hand side is given by the matter sources defined as

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{-2}{\sqrt{-g}}\frac{\delta S_{\rm mat}}{\delta
h^{\mu\nu}}\,,\ \ T_{\mu}{}^{a}=\frac{-2}{\sqrt{-g}}\frac{\delta S_{\rm
mat}}{\delta\tau^{\mu}{}_{a}}\,,\ \ $ (21) $\displaystyle J^{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{-2}{\sqrt{-g}}\frac{\delta S_{\rm
mat}}{\delta b_{\mu\nu}}\,,\sigma=\frac{-1}{\sqrt{-g}}\frac{\delta S_{\rm
mat}}{\delta\phi}\,,$ (22)

where $T_{\mu\nu}$ is the spacial energy-momentum tensor, $T_{\mu}{}^{a}$ is
the energy current. On the other hand, we couple statistical matter
considering the Einstein equation,

$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{k}{c^{4}}T_{\mu\nu}\,,$ (23)

where $T_{\mu\nu}$ might be, for example, the energy-momentum for a perfect
fluid. Then, for a supergravity background after considering the NR limit, we
know that the lhs of the equation is finite. Therefore, the energy-momentum
tensor requires the following expansion,

$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
c^{4}T^{(4)}_{\mu\nu}+\mathcal{O}(c^{3})\,,$ (24)

while $J^{\mu\nu}$ and $\sigma$ can be coupled in a similar fashion, but
considering the equation of motion of $b_{\mu\nu}$ and $\phi$.

##  III Construction of the model

Recently [27], a formulation of NR DFT (compatible with the non-Riemannian
formulation of DFT) has been constructed. The idea of this work is to include
statistical matter to this model, in order to obtain statistical matter
coupled to a NR supergravity upon parametrization. For simplicity we will
restrict our study to the bosonic case, meaning that the duality group is
$O(D,D)$. The invariant DFT metric is given by

$\displaystyle{\eta}_{{MN}}=\left(\begin{matrix}0&\delta_{\mu}^{\nu}\\\
\delta^{\mu}_{\nu}&0\end{matrix}\right)\,,$ (25)

with $M,N,\dots=0,\dots,2D-1$ in the fundamental of the duality group, while
the generalized metric is expanded as

$\displaystyle\hat{\cal H}_{MN}={\cal H}^{(0)}_{MN}+\frac{1}{c^{2}}{\cal
H}^{(-2)}_{MN}\,,$ (26)

where we use the suffix (i) to indicate the order of a given term. Moreover,
in order to match with the NR limit of supergravity, the parametrization of
the previous metric is given by

$\displaystyle{\cal H}^{(0)}_{MN}$
$\displaystyle=\left(\begin{matrix}h^{\mu\nu}&\epsilon_{ab}\tau_{\nu}{}^{b}\tau^{\mu
a}-b_{\rho\nu}h^{\rho\mu}\\\ \epsilon_{ab}\tau_{\mu}{}^{b}\tau^{\nu
a}-b_{\rho\mu}h^{\rho\nu}&h_{\mu\nu}+b_{\rho\mu}h^{\rho\sigma}b_{\sigma\nu}-2\epsilon_{cd}\tau_{(\mu|}{}^{d}b_{\sigma|\nu)}\tau^{\sigma
c}\,\end{matrix}\right)\quad{\cal H}^{(-2)}_{MN}$
$\displaystyle=\left(\begin{matrix}\tau^{\mu
a}\tau^{\nu}{}_{a}&-b_{\rho\nu}\tau^{\rho c}\tau^{\mu}{}_{c}\\\
-b_{\rho\mu}\tau^{\rho c}\tau^{\nu}{}_{c}&b_{\rho\mu}\tau^{\rho
c}b_{\sigma\nu}\tau^{\sigma}{}_{c}\end{matrix}\right)\,.$ (27)

As we can easily observe, the main difference between NR DFT and NR
supergravity is that in the former, the dynamical metric is finite while in
the latter the expansion of the metric (and the expansion of the B-field)
contains divergences.

As in the relativistic case, a crucial consistency equation of the DFT
formulation is given by the strong constraint

$\displaystyle\partial_{M}(\partial^{M}\star)=0\,\quad(\partial_{M}\star)(\partial^{M}\star)=0\,,$
(28)

where $\partial_{M}=(\partial_{\mu},\tilde{\partial}^{\mu})$ refers to
derivatives with respect to the ordinary/dual coordinates, respectively, and
$\star$ represents any combination of generalized fields and/or gauge
parameters. Thanks to the strong constraint the closure of the generalized
diffeomorphism transformations infinitesimally by $\xi^{M}$ through the
generalized Lie derivative

$\displaystyle{\cal
L}_{\xi}V_{M}=\xi^{N}\partial_{N}V_{M}+(\partial_{M}\xi^{N}-\partial^{N}\xi_{M})V_{N}\,,$
(29)

is guaranteed. The DFT Jacobiator is not trivial and therefore the algebraic
structure of DFT is given by an $L_{\infty}$-algebra with a non-trivial
$l_{3}$ product [51]-[54], which measures the failure of the Jacobi identity
in the double geometry. The usual solution to (28) in order to recover the
supergravity framework is $\tilde{\partial}^{\mu}=0$ . In this formulation the
Lorentz transformations act trivially, but we refer to [27] for details on the
parametrization of the double Lorentz parameter.

On the other hand, it is worth observing that ${\cal H}^{(0)}_{MN}$ is an
element of the duality group, i.e.,

$\displaystyle{\cal H}^{(0)}_{MP}\eta^{PQ}{\cal H}^{(0)}_{QN}$
$\displaystyle=$ $\displaystyle\eta_{MN}\,.$ (30)

Therefore, we can construct the leading order of the generalized projectors in
the following way,

$\displaystyle P^{(0)}_{MN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\eta_{MN}-{\cal H}^{(0)}_{MN}\right)\,,$ (31)
$\displaystyle\overline{P}^{(0)}_{MN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\eta_{MN}+{\cal H}^{(0)}_{MN}\right)\ ,$ (32)

and they satisfy the following properties

$\displaystyle{\overline{P}}^{(0)}_{{MQ}}{\overline{P}}^{(0)Q}{}_{N}={\overline{P}}^{(0)}_{{MN}}\,,$
$\displaystyle\quad{P}^{(0)}_{{MQ}}{P}^{(0)Q}{}_{N}={P}^{(0)}_{{MN}},$
$\displaystyle{P}^{(0)}_{{MQ}}{\overline{P}}^{(0)Q}{}_{N}={\overline{P}}^{(0)}_{{MQ}}{P}^{(0)Q}{}_{N}=0\,,$
$\displaystyle\quad{\overline{P}}^{(0)}_{{MN}}+{P}^{(0)}_{{MN}}=\eta_{{MN}}\,.$

The generalized dilaton has a finite $c$-expansion, as the expansion of the
dilaton $\hat{\Phi}$ and the metric determinant $\hat{g}$ cancel,

$\displaystyle
e^{-2\hat{d}}=e^{-2\hat{\Phi}}\sqrt{-\hat{g}}=e^{-2\phi}\sqrt{f(\tau,h)}=e^{-2d},$
(33)

where $f(\tau,h)=-\frac{\hat{g}}{c^{4}}$.

Since both $\hat{\cal H}_{MN}$ and $\hat{d}$ do not contain divergences, all
the leading contributions to the generalized curvatures is given by ${\cal
H}^{(0)}_{MN}$ and $d$, i.e.,

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle\frac{1}{8}{\cal
H}^{(0)MN}\partial_{M}{\cal H}^{(0)KL}\partial_{N}{\cal H}^{(0)}_{KL}+4{\cal
H}^{(0)MN}\partial_{M}d\partial_{N}d$ (34) $\displaystyle-\frac{1}{2}{\cal
H}^{(0)MN}\partial_{N}{\cal H}^{(0)KL}\partial_{L}{\cal H}^{(0)}_{MK}$
$\displaystyle-2(\partial_{M}{\cal
H}^{(0)MN})\partial_{N}d+\mathcal{O}\left(\frac{1}{c}\right)\,,$

and

$\displaystyle{\cal R}_{MN}=P^{(0)}_{M}{}^{P}{\cal
K}_{PQ}{\overline{P}}^{(0)Q}{}_{N}+{\overline{P}}^{(0)}_{M}{}^{P}{\cal
K}_{PQ}P^{(0)Q}{}_{N}\,,$ (35)

with

$\displaystyle{\cal K}_{MN}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\partial_{M}{\cal H}^{(0)KL}\partial_{N}{\cal
H}^{(0)}_{KL}$
$\displaystyle-\frac{1}{4}\left(\partial_{L}-2\partial_{L}d\right)\left({\cal
H}^{(0)LK}\partial_{K}{\cal H}^{(0)}_{MN}\right)$
$\displaystyle+2\partial_{M}\partial_{N}d-\frac{1}{2}\partial_{(M}{\cal
H}^{(0)KL}\partial_{L}{\cal H}^{(0)}_{N)K}$
$\displaystyle+\frac{1}{2}\left(\partial_{L}-2\partial_{L}d\right)\left({\cal
H}^{(0)KL}\partial_{(M}{\cal H}^{(0)}_{N)K}\right.$ $\displaystyle\left.+{\cal
H}^{(0)K}{}_{(M}\partial_{K}{\cal H}^{(0)L}{}_{N)}\right)\,.$

The generalized Einstein equation is given by [43, 44],

$\displaystyle\mathcal{G}_{MN}=\frac{k}{c^{4}}\mathcal{T}_{MN}\,,$ (36)

with the generalized Einstein tensor defined as

$\displaystyle\mathcal{G}_{MN}=\mathcal{R}_{MN}-\frac{1}{2}\,\mathcal{R}\,\hat{\cal
H}_{MN}\,.$ (37)

Considering that the generalized energy-momentum tensor can be factorized as

$\displaystyle\mathcal{T}_{MN}=\hat{\mathcal{T}}_{MN}-\frac{1}{2}\,{\cal
T}\,\hat{\cal H}_{MN}\,,$ (38)

where both ${\cal R}_{MN}$ and $\hat{{\cal T}}_{MN}$ contain only mixed
components with respect to the full DFT projectors, then (36) reads

$\displaystyle\mathcal{R}_{MN}$
$\displaystyle=\frac{\kappa}{c^{4}}\hat{\mathcal{T}}_{MN}\,,$ (39)
$\displaystyle\mathcal{R}$ $\displaystyle=\frac{\kappa}{c^{4}}\mathcal{T}\,.$
(40)

The explicit form of the generalized energy-momentum tensor for a perfect
fluid configuration is given by [40] 222This tensor is a particular case of
the recent proposal of [42], where the authors included non-perfect terms as
well. Also, it is the commutative version of the one in [55].,

$\displaystyle{\mathcal{T}}_{MN}=(e+p)\left(\hat{U}_{\overline{M}}\,\hat{U}_{\underline{N}}+\hat{U}_{\overline{N}}\,\hat{U}_{\underline{M}}\right)+p\,\hat{\cal
H}_{MN}\,,$ (41)

where $\hat{U}_{M}$ is a generalized velocity which satisfies

$\displaystyle\hat{U}_{M}\hat{\cal H}^{MN}\hat{U}_{N}$ $\displaystyle=$
$\displaystyle-c^{2}$ (42) $\displaystyle\hat{U}_{M}\eta^{MN}\hat{U}_{N}$
$\displaystyle=$ $\displaystyle 0\,.$ (43)

The previous constraints ensure that the generalized velocity describes the
ordinary D-velocity of the perfect fluid hydrodynamics. From the previous
relations it is natural to follow a similar expansion to the one in [49], but
in the double geometry,

$\displaystyle\hat{U}_{M}=c^{2}U^{(2)}_{M}+U^{(0)}_{M}+\mathcal{O}(\frac{1}{c^{2}})\,.$
(44)

The previous expansion requires a particular form for the parametrization of
$U^{(0)}$ and $U^{(2)}$ since the generalized velocity needs to satisfy the
constraints (42), which can be written as

$\displaystyle U^{(2)}_{M}{\cal H}^{(0)MN}U^{(2)}_{N}$ $\displaystyle=$
$\displaystyle 0\,,$ (45) $\displaystyle U^{(2)}_{M}{\cal
H}^{(-2)}_{MN}U^{(2)}_{N}$ $\displaystyle=$ $\displaystyle-1\,,$ (46)
$\displaystyle U^{(0)}_{M}{\cal H}^{(0)MN}U^{(0)}_{N}\,$ $\displaystyle=$
$\displaystyle 0\,,$ (47) $\displaystyle U^{(2)}_{M}\eta^{MN}U^{(2)}_{N}$
$\displaystyle=$ $\displaystyle 0\,,$ (48) $\displaystyle
U^{(0)}_{M}\eta^{MN}U^{(2)}_{N}$ $\displaystyle=$ $\displaystyle 0\,,$ (49)
$\displaystyle U^{(0)}_{M}\eta^{MN}U^{(0)}_{N}$ $\displaystyle=$
$\displaystyle 0\,.$ (50)

Besides (45)-(50), we need to ensure that the generalized distribution
function related to double phase space/kinetic theory formalism is also
finite. We will explore this point in the next subsection. Finally, at this
point we have learned information about the hydrodynamics quantities involved
in the generalized energy-momentum tensor. Since the generalized velocity
contains $c^{2}$ contributions and the projectors are at most order $1$, then
the generalized energy density and pressure admit an expansion of the form,

$\displaystyle e$ $\displaystyle=$ $\displaystyle
e^{(0)}+\mathcal{O}(\frac{1}{c})\,,\quad
p=p^{(0)}+\mathcal{O}(\frac{1}{c})\,.$ (51)

These expansions are not in agreement with the generalized correspondence
between a perfect fluid and a scalar field, so we will briefly discuss it in
the subsection (III.2).

###  III.1 Statistical point of view

The statistical description of the perfect fluid was originally described in
[39]. In this section we inspect the conditions to obtain a finite
distribution function for the perfect fluid when the fields are expanded using
the NR ansatz.

The double phase space coordinates are given by

$\displaystyle\Big{\\{}X^{M},{\cal P}^{M}\Big{\\}}\,,$ (52)

where ${\cal P}^{M}$ is an $O(D,D)$ vector. The infinitesimal generalized
diffeomorphisms acting on a double phase space vector $V^{Q}=V^{Q}(X,{\cal
P})$ are defined as

$\displaystyle\delta_{\xi}V^{Q}={\cal L}_{\xi}V^{Q}+{\cal
P}^{N}\frac{\partial\xi^{M}}{\partial X^{N}}\frac{\partial
V^{Q}}{\partial{\cal P}^{M}}\,,$ (53)

where ${\cal L}_{\xi}$ is the generalized Lie derivative and
$\xi_{M}=\xi_{M}(X)$ is an infinitesimal parameter. The section condition is
given by

$\displaystyle(\frac{\partial}{\partial{\cal
P}^{M}}\star)(\frac{\partial}{\partial{\cal P}_{M}}\star)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial{\cal
P}^{M}}(\frac{\partial}{\partial{\cal P}_{M}}\star)=0\,$ (54)
$\displaystyle(\frac{\partial}{\partial
X^{M}}\star)(\frac{\partial}{\partial{\cal P}_{M}}\star)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial X^{M}}(\frac{\partial}{\partial{\cal
P}_{M}}\star)=0\,.\,$ (55)

We solved the strong constraint of the double phase space with the solution

$\displaystyle\frac{\partial}{\partial{\cal P}_{\mu}}=0\,.$ (56)

and ${\cal P}_{M}=(p^{\mu},0)$. The closure of (53) is given by the C-bracket,
and the covariant derivative of the double phase space is given by the
generalized Liouville operator $D_{M}$ defined as

$\displaystyle D_{M}=\nabla_{M}-\Gamma_{MN}{}^{Q}{\cal
P}^{N}\frac{\partial}{\partial{\cal P}^{Q}}\,.$ (57)

The generalized Boltzmann equation considering a generalized distribution
function $F=F\big{(}X,{\cal P}\big{)}$ is given by [39],

$\displaystyle{\cal P}^{M}{\cal D}_{M}F={\cal C}[F]\,,$ (58)

where ${\cal C}[F]$ is a generalized collision term and the operator ${\cal
D}_{M}$ contains a generalized dilaton dependence,

$\displaystyle{\cal D}_{M}=D_{M}-{\cal U}_{M}\,,$ (59)

with ${\cal U}_{M}={\cal U}_{M}(X)=2\,\partial_{M}\hat{d}.$

The generalized equilibrium distribution function is given by [42],

$\displaystyle F_{\rm eq}(X,{\cal P})=\operatorname{e}^{2\hat{d}-{\cal
P}^{M}\hat{\cal H}_{MN}\beta^{N}}$ (60)

where $\beta_{N}=\hat{U}_{N}/T$. Considering the NR expansion on the
generalized dilaton and the generalized metric, and a generic expansion for
the velocity we find

$\displaystyle F_{\rm eq}(X,{\cal P})=\operatorname{e}^{2d-\frac{1}{T}({\cal
P})({\cal H}^{(0)}_{MN}+{\cal H}^{(-2)}_{MN})(U^{(2)N}+U^{(0)N}})\,.$

On the other hand, the expansion of the generalized momentum can be induced
from the expansion of the generalized metric,

$\displaystyle{\cal P}_{M}={\cal P}^{(0)}_{M}+{\cal P}^{(-2)}_{M}\,.$ (61)

Therefore, we observe that in order to have a finite distribution function we
need

$\displaystyle{\cal P}^{(0)M}{\cal H}^{(0)}_{MN}U^{(2)N}=0\,.$ (62)

In this work we are not considering an expansion for the generalized
temperature, but we can easily see that the temperature may include
divergences that do not affect the behaviour of the generalized distribution
function. At this point we have found the sufficient condition to construct a
finite kinetic theory, and we will continue our discussion about the
distribution function after parametrizing the fields, in section IV. In the
next subsection we will study the generalized correspondence between the
perfect fluid and a generalized scalar field in the double geometry.

###  III.2 Generalized correspondence with the scalar field

The scalar field-perfect fluid correspondence [45] is a formal correspondence
between the energy-momentum tensor of a massless scalar field and the energy-
momentum tensor of a perfect fluid. The latter can be constructed from the
former considering a map between the scalar field and the hydrodynamics
quantities. From this relation it is possible to write a matter Lagrangian in
terms of the fluid variables. Particularly, ${\cal L}_{\rm m}=p$, where $p$ is
a function of the scalar field.

The generalization of the fluid velocity and scalar field derivative
correspondence at the double geometry level is given by [41]

$\displaystyle\hat{U}_{M}=\frac{\partial_{M}\Phi}{\sqrt{|\hat{\cal
H}^{PQ}\partial_{P}\Phi\partial_{Q}\Phi}|}\,,$ (63)

while the generalized pressure and energy density are given by

$\displaystyle p$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\hat{\cal
H}^{PQ}\partial_{P}\Phi\partial_{Q}\Phi-V(\Phi)\,,$ (64) $\displaystyle e+p$
$\displaystyle=$ $\displaystyle|\hat{\cal
H}^{PQ}\partial_{P}\Phi\partial_{Q}\Phi|\,.$ (65)

These relations turn

$\displaystyle{\cal T}_{MN}$ $\displaystyle=$
$\displaystyle-4\,\overline{P}_{K(M}\,P_{N)L}\,\partial^{K}\Phi\partial^{L}\Phi-\frac{1}{2}\hat{\cal
H}_{MN}\hat{\cal H}^{RQ}\partial_{R}\Phi\partial_{Q}\Phi$ (66)
$\displaystyle-\hat{\cal H}_{MN}\,V(\Phi)\,.$

into (41). When the fields are expanded under a NR expansion, we can easily
observe that the generalized correspondence does not hold anymore. The
generalized velocity requires a $c^{2}$ contribution, meaning that the
generalized scalar field also contains such expansion, and then the pressure
receive contributions as well. However, this includes divergences in the
matter Lagrangian, which cannot be regulated, since we have learned in (51)
that the hydrodynamics quantities require a convergent expansion.

##  IV Breaking the duality group: the supergravity model

The parametrization of the generalized velocity is given by

$\displaystyle
U^{(2)}_{\mu}=-\tau_{\mu}{}^{a}\tau_{\rho}{}^{b}\epsilon_{ab}u^{\rho},\quad
U^{(2)\mu}=0$ (67)

and

$\displaystyle U^{(0)}_{\mu}=-b_{\mu\rho}u^{\rho},\quad U^{(0)\mu}=u^{\mu}\,,$
(68)

where $u^{\mu}$ is the ordinary D-velocity, which satisfies
$u^{\mu}h_{\mu\nu}=u_{\mu}h^{\mu\nu}=0$. Now it is a convenient moment to
return to the equation (62). The parametrization of the generalized velocity
is in agreement with this constraint on the generalized distribution function,
i.e.,

$\displaystyle\frac{mc^{2}}{T}h^{\mu\rho}{u}_{\rho}{\cal
H}^{(0)}_{\mu}{}^{\nu}U_{(2)\nu}=0\,,$ (69)

where we have used $p^{\mu}=m\hat{g}^{\mu\nu}u_{\nu}$. Therefore the
distribution function is finite as expected.

On the other hand, the decomposition of (38) gives

$\displaystyle\hat{{\cal T}}_{MN}$
$\displaystyle=(e+p)\left(\hat{U}_{\overline{M}}\,\hat{U}_{\underline{N}}+\hat{U}_{\overline{N}}\,\hat{U}_{\underline{M}}\right)\,,$
(70) $\displaystyle\mathcal{T}$ $\displaystyle=-2\,p\,.$ (71)

After considering the NR limit, the generalized Einstein equation is

$\displaystyle\mathcal{R}_{MN}$ $\displaystyle=$
$\displaystyle\kappa(e^{(0)}+p^{(0)})\left(U^{(2)}_{\overline{M}}\,U^{(2)}_{\underline{N}}+U^{(2)}_{\overline{N}}\,U^{(2)}_{\underline{M}}\right)\,,$
(72) $\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle 0\,,$ (73)

where the indices are projected with $P^{(0)}_{MN}$ and
$\overline{P}^{(0)}_{MN}$. From the last equation we learn that the source for
the dilaton vanishes $\sigma=0$. When exploring the independent components of
the rhs of (72) we obtain

$\displaystyle\hat{\cal T}^{(0)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\kappa(e^{(0)}+p^{(0)})(\tau_{\mu}{}^{a}\epsilon_{ab}u^{b}\tau_{\nu}{}^{c}\epsilon_{cd}u^{d}$
(74) $\displaystyle-\tau_{\mu a}u^{a}\tau_{\nu b}u^{b})\,,$
$\displaystyle\hat{\cal T}^{(0)}_{\mu}{}^{\nu}$ $\displaystyle=$
$\displaystyle\hat{\cal T}^{(0)\mu\nu}=0\,,$ (75)

where $u^{a}=\tau_{\mu}{}^{a}u^{\mu}$. From the last split we learn that also
the source of the B-field is trivial i.e., $J^{\mu\nu}=0$, similarly to the
relativistic situation [40]. Finally, using

$\displaystyle\epsilon_{ab}\epsilon_{cd}=-\eta_{ab}\eta_{cd}+\eta_{ad}\eta_{bc}$
(76)

we can simplify the generalized Einstein tensor as

$\displaystyle\hat{\cal T}^{(0)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa(e^{(0)}+p^{(0)})\tau_{\mu a}u^{a}\tau_{\nu
b}u^{b}\,.$ (77)

When we compare the previous object with the ordinary energy-momentum tensor
for perfect fluids

$\displaystyle T_{\mu\nu}=(e+p)\hat{u}_{\mu}\hat{u}_{\nu}+p\hat{g}_{\mu\nu}\,$
(78)

we infer that the T-duality invariant model is constructed considering

$\displaystyle\hat{u}^{\mu}$ $\displaystyle=$ $\displaystyle
c^{2}\tau_{\mu}{}^{a}u_{a}+\mathcal{O}(1)\,,$ (79) $\displaystyle e$
$\displaystyle=$
$\displaystyle-\frac{1}{2}e^{(0)}+\mathcal{O}(\frac{1}{c})\,,$ (80)
$\displaystyle p$ $\displaystyle=$
$\displaystyle-\frac{1}{2}p^{(0)}+\mathcal{O}(\frac{1}{c})\,.$ (81)

Therefore, we have obtained the form of the energy-momentum tensor for a
perfect fluid coupled to a NR supergravity compatible with the $O(D,D)$
symmetry.

##  V Conclusions

The generalized metric formalism of DFT is a useful framework to explore the
NR limit of NS-NS supergravity due to its simplicity and the possibility to
take the limit $c\rightarrow\infty$ before breaking the duality group. In this
work we have extended this formalism and we have included statistical matter
following the prescription given in [39]-[41]. Once more, T-duality proves to
be a powerful symmetry to construct effective models related to string theory.
The finiteness of the generalized Ricci tensor and scalar forces a particular
generalized energy-momentum tensor. The generalized velocity solves the
typical DFT constraints ($\hat{U}_{M}\hat{H}^{MN}\hat{U}_{N}=-c^{2}$,
$\hat{U}_{M}\eta^{MN}\hat{U}_{N}=0$) but also requires to be compatible with a
finite generalized distribution function. All these constraints are solved
considering the expansion (44) and the parametrization (67)-(68). Using this
divergent expansion, the generalized energy density and pressure have to be
expanded in a convergent form according to (51).

We can summarize our findings in the following diagram,

$\displaystyle\begin{array}[]{ccccc}\textrm{supergravity
}&\longleftrightarrow&\textrm{bosonic DFT }&&\\\
(\hat{g},\hat{B},\hat{\Phi},\hat{u},p,e)&&(\hat{\cal
H},\hat{d},\hat{U},p,e)&&\\\ \downarrow&&\downarrow\\\ \text{{non}-finite
$c$-expansion}&&\text{{non}-finite $c$-expansion}\\\
(\hat{g},\hat{B},\hat{\Phi},\hat{u})&&(\hat{U})\\\ \downarrow&&\downarrow&&\\\
\textrm{finite NR-supergravity}&\longleftrightarrow&\text{finite NR-
DFT.}&&\end{array}$ (89)

We prove that the generalized perfect fluid-scalar field correspondence is no
longer valid when we consider the NR expansion on the fundamental fields.
Finally, the main result of this work is the construction of the model (74),
which does not require any particular source for the B-field and the dilaton
in order to be compatible with the $O(D,D)$ symmetry. This is an interesting
point since in the relativistic case, the only way to describe the perfect
fluid without imposing any particular source is to include viscosity in the
starting DFT model [42], in such way that the (generalized version of the)
entropy production in the double geometry is trivial. However, here we show
that when expanding the fields under the NR expansion, it is possible to
rewrite/construct the T-duality invariant perfect fluid model without the help
of any particular B-field or dilaton source, and viscosity. Similarly to (2),
the effective description is pressureless after a redefinition of the DFT
hydrodynamics variables $e^{(0)}+p^{(0)}\rightarrow p$, and due to the
cancellation of the rhs of (73) the viscosity contributions are not required.
The results of this work pave the way to the study of NR cosmological
backgrounds in ST.

### Acknowledgements

We thank D. Marques for his comments on the first version of the draft, and N.
Miron-Granese for enlightening discussions. This work is supported by the
SONATA BIS grant 2021/42/E/ST2/00304 from the National Science Centre (NCN),
Poland.

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