# Artificial Intelligence in Music and Performance: A Subjective Art-Research
Inquiry

Baptiste Caramiaux  Université Paris-Saclay, CNRS, Inria, LRI  Marco
Donnarumma  Independent Researcher

###### Abstract

This article presents a five-year collaboration situated at the intersection
of Art practice and Scientific research in Human-Computer Interaction (HCI).
At the core of our collaborative work is a hybrid, Art and Science methodology
that combines computational learning technology – Machine Learning (ML) and
Artificial Intelligence (AI) – with interactive music performance and
choreography. This article first exposes our thoughts on combining art,
science, movement and sound research. We then describe two of our artistic
works Corpus Nil and Humane Methods – created five years apart from each other
– that crystallize our collaborative research process. We present the
scientific and artistic motivations, framed through our research interests and
cultural environment of the time. We conclude by reflecting on the methodology
we developed during the collaboration and on the conceptual shift of
computational learning technologies, from ML to AI, and its impact on Music
performance.

##  1 Introduction

In many contemporary societies the pervasiveness of technology is constantly
expanding. From communication to social networks, digital health and welfare
services, every aspect of social life in industrialized societies is being
captured by technology with the objective of human enhancement, optimized
services, or automated management. Among these technologies, Machine Learning
(ML) and the broader field of Artificial Intelligence (AI) received
considerable attention in the past decades. In this chapter, we use an
autoethnographic approach to present and discuss the hybrid methodology that
we developed in five years of collaborative research across computation,
science and art. The analysis we offer here combines insight from academic
research in Human-Computer Interaction (HCI), in particular body-based
interaction, and from aesthetic research in the performing arts to inspect and
question the role of ML and AI in our practices.

The past decade has shown that data-driven learning-based algorithms can
succeed in many tasks that were unthinkable not too long ago. Since the deep
learning breakthrough in 2012 [44], these algorithms have been shown to
recognize images as well as humans do, to acquire motor control capacities
from few observations, and to understand and respond to several kinds of human
languages. These breakthroughs prompted major investments in the field, thus
accelerating technological advance at an exponential pace (the number of
attendees at major conferences or the number of published papers at online
repository like arxiv show the radical increase of interest in the topic). A
consequence of the impressive advance of the field is a frantic race to embed
ML-based algorithms in most digital services, without offering ways for users
to acquire a basic literacy of the technology. The general paradigm behind the
development of these technologies is based on centralized ownership by tech
companies or institutions. By interacting with these technologies, people
produce large amount of usage data, which, in turn, is used to train the
underlying algorithms. This type of technology is therefore detached from
people’s control and understanding.

Artists have historically been among the first to question technological
innovations (see, for instance, the organization Experiments in Art and
Technology, E.A.T., founded in 1967). They have often been early adopters and
disruptors of new technological tools, and ML is not an exception [13]. On one
hand, many artists use ML to enrich the way they work with their preferred
media. On the other, some artists use ML, and in particular AI, to shed light
onto certain facets of these same tools which can be invisible or taken for
granted by public opinion, media, or institutions.

Our own path, which we elaborate next, has taken different directions
throughout the past five years of collaboration. Initially, our approach was
to use ML as a tool to design improvised and markedly physical musical
performance, exploring the relation between computation and corporeality.
Eventually, our approach shifted to utilize and reflect on AI as an actor in a
performance, an entity whose functioning can be used to question the
understanding of computational intelligence in Western society. As a team
doing research in overlapping fields, we tried to combine our respective
methodologies in a zone of encounter, where _practice-based research_ meets
_artistic intervention in research_. The former involves the use of artistic
practice as a means of research. The latter entails that a creative, artistic
process acts upon scientific research objectives. Naturally, such a hybrid
methodology has been explored by others before us [28, 26].

In this chapter, we intend to extract the subjective aspects of our
collaborative works and to discuss the methodological perspective that our
particular mode of collaboration offers to the study of ML and AI, as well as
to the study of their impact on HCI and the performing arts. The chapter is
structured as follows. First, we provide general thoughts on combining
scientific and artistic practices; these observations will help us give
context to the methodology of our collaborative work. The next two sections
discuss the conception and development of our collaborative artworks, two
hybrid performances of computational music and choreography, _Corpus Nil_ and
_Humane Methods_. We present the scientific and artistic drives behind each
project, highlighting scientific, artistic and cultural contexts. Finally, we
provide a closing discussion where we bring together the core findings emerged
from our research.

##  2 Combining Art, Science and Sound Research

A dialogue between scientific research and artistic practice is inclined to
generate multiple and often contrasting perspectives, rather than producing an
agreement. Important insight can emerge from those contrasts, insight that
would have been obscured by a unified and monodisciplinary approach. This is
the methodological grounding of the present chapter. We do not strive for
objective knowledge. Art, even in its practice-based forms, does not offer any
provable truth but creates alternate visions and, sometimes, it is called upon
to “contemplate the dark without drawing a resolutely positive lesson”, as
Fuller and Goriunova put it [33].

###  2.1 Practice-based Research and Objective Knowledge

The kind of art and science coupling we discuss here is not the use of
scientific findings for the creation of an artwork, for that strategy requires
an attentive process of translation, as stressed by Stengers [53], and – based
on our subjective experience – it can easily feel constraining. As we will
elaborate later when discussing our first collaborative project _Corpus Nil_ ,
valid insight or correct measurements achieved in a laboratory setting may not
be useful when applied to a musical performance. The motivation lies in the
experimental limitations of particular scientific methods found in HCI and
other research fields combining natural science, design and engineering.
Contrary to the social sciences, for instance, HCI experiments are often
conducted in a laboratory context. In controlled studies, every aspect of such
studies is thoroughly directed by the experimenters. Whereas the protocols of
such HCI experiments aim to ensure scientific rigour and reproducibility,
those same protocols create the specific context wherein the collected data
and related findings are meaningful. Outside of the specified context, data
and findings may not be as meaningful, especially when applying them to an
unpredictable and ever-changing situation such as a live artistic performance
before a public audience.

For example, during a live performance, sensor measurements that have been
validated through a particular experiment may very well vary according to a
range of factors, including room temperature, magnetic interferences and the
like. Those variables - contrary to a laboratory experiment situation - cannot
be controlled. More importantly, the performer experiences her own body, a
technological musical instrument and the context in subjective ways; during an
artistic performance a performer is driven by instincts, desires and feelings
that arise from the relation with the audience, other performers and a
particular musical instrument. Artistic expression in musical performance
emerges from the interaction of human actions and desires with the
capabilities of a musical instrument.

This emphasises the need for, on one hand, a careful selection and
interpretation of the methods of art and science collaboration, and, on the
other, a precise definition of the shared knowledge needed by artist and
scientist to operate collaboratively; an entanglement that Roger Malina aptly
calls “deep art-science coupling” [46]. Practice-based research represents a
good framework for the exploration of art and science coupling, for it
generates knowledge from action, self-reflection and empirical experimentation
[11, 6], methods that are equally fitting in HCI as in music and the
performing arts. Crucially, we believe that the goal of art and science
collaboration is not the production or re-staging of the limits of a normative
type of science, but rather is about “working outside current paradigms,
taking conceptual risks”, following Malina, so as to mindfully merge
disciplines towards new practices of experimentation. This can result in
contrasts and disimilarities which must be considered and which, in fact,
produce the richness of art and science practice.

###  2.2 Artistic Intervention in Scientific Research

As technology has become more ubiquitous, spreading from workplaces to
everyday life, the field of HCI has adopted new ways to investigate how we
interact with such pervasive technology [39, 23]. The so-called third-wave of
HCI looks at notions such as experience, emotions, aesthetics and embodiment
[9]. The field thus embraces methodological resources from a broader range of
fields such as cognitive science, social science, science studies, or the
Arts. In this context, artistic intervention encourages research practitioners
to look at their subject of study under an alternative “hypothesis”, through
different facets. It is important to underline that, differently from
practice-based research, artistic intervention in research emphasizes the idea
of intervention: the artistic process _acts_ upon the research objectives. For
example, artistic intervention in scientific research, and more specifically
in HCI, can help address a problem (or study a phenomenon) in radically new
ways [5]. Such as in the case of Fdili Alaoui [28] who, working between HCI
and dance, questions the academic culture of HCI through her expertise in
interactive dance performance.

In our research, we use artistic intervention to build and study interactions
between people and intelligent systems. This methodological choice is
motivated by two factors. On one hand, scientific research practice can feel
normative and standardised. As researchers, we employ established methods to
structure our investigations on a particular question, as well as to
disseminate the outcomes of the research. While this standardisation has its
pragmatic rationale (for instance, to help its dissemination), it can be
detrimental to scientific research practice in many ways, such as limiting
exploration or surprise. A detailed articulation of the pros and cons of it
goes beyond the scope of this paper. Here we stress how artistic intervention
in research can complement standard scientific methods in leaving more room
for unexpected challenges and nuanced conceptual questioning. As an example,
which will be developed in the following section, our study of gestural
expressivity in musical performances has shifted the emphasis from
characterizing gestural variations in kinematic space to characterizing
variations in muscle coordination space.

On the other hand, the specific field of HCI has its own culture and
shortcomings. While the field examines and generates the systems underpinning
the technological world with which humans interact daily, its political
dimension – that is, the varying balances of power between who produces the
technology, who designs it, who uses it, and what socio-cultural impact it may
have – is rarely addressed [40, 4, 42]. By avoiding to explicitly address the
politics of technological innovation, one risks to contribute to the creation
of forms of technological solutionism. Technological solutionism refers to the
idea that technology can solve any problem, including issues which may not
exist or which ignore both socio-cultural and political contexts [8]. We found
that one method to prevent (at least partially) our research from entering a
solutionist discourse is to invoke artistic intervention within the process of
scientific research. Critical artistic intervention can explicitly provoke
self-reflection about the involvement of a particular technology. Through an
analytical and self-reflexive engagement, a critical artistic intervention can
question the role of technology and its capacities of meaning-making, forcing
artists and researchers to face the inner beliefs that ultimately motivated
their choices.

In the following, we discuss how we applied the methods and insight described
above to two separate research projects, which led to the creation of two
artistic performances and scientific publications. This also serves to
illustrate how our collaboration developed iteratively through the
intertwining of practice-based research in the art field and artistic
intervention in the scientific research.

##  3 Machine Learning as a Tool for Musical Performance

Our first project stemmed from a collaboration started in 2014. The outcomes
of this particular project were a performance entitled _Corpus Nil_ (premiered
in 2016 at ZKM, Center for Art and Media, Germany and still touring
internationally) and academic publications [12, 22].

###  3.1 _Corpus Nil_

The piece is a twenty-minute dance and music performance exploring hybrid
forms of identity and musicianship. It is an intense and ritualistic
interaction between an autonomous musical instrument, a human body, and sound.
Figure 1 depicts a picture from a live performance. The theater space is
completely dark. The player (Donnarumma in this case), whose body is partly
naked and partly painted in black, performs a tense choreography which
gradually morphs his body. Two types of wearable biosensors transmit data from
his body to our machine learning-based software. Chip microphones capture
sounds from muscles and internal organs (mechanomyogram or MMG) and electrodes
capture muscle voltages (electromyogram or EMG). The software uses special
filters to generate a description of the amplitude and frequencies of all
sounds produced within the performer’s body (between 1-40 Hz), as well as
their variations over time. Then, it re-synthesises those sounds by
orchestrating a feedback network of twenty digital oscillators. Because the
choreography demands slow, subtle and iterative physical movements, the
resulting music is equally slow and recursive, mutating across microtonal
variations of a minimal set of pitches.

Figure 1: Picture from the performance _Corpus Nil_. Photo courtesy of ONUK
Fotografie. The piece evolves through different levels of muscular tension,
torsion and articulations of the upper limbs. Aspects of the movements
captured by means of two different muscle sensors placed on the arms.

The instrument – here intended as a combination of our chosen hardware and
software – analyses the collected data to learn the nuances of the performer’s
movement (muscular tension, gesture abruptness, rate of relaxation). Upon
observing a certain nuance, it chooses whether to mute or activate particular
oscillators, how to regulate volumes, phases, glissandos and multi-channel
diffusion, and how to adjust feedback amounts within the network. The player
cannot control the instrument, but can only learn how to affect it and be
affected by it. The piece thus discards conventional performer-instrument
relationships – founded on the performer’s full control of the instrument – in
favour of an unstable corporeal engagement between the two. Through the rhythm
of sound, vibration and light, the performer’s body and the instrument mutate,
physically and conceptually, into something “other”; an unfamiliar creature
defying common definition of the human.

The software used in _Corpus Nil_ , collaboratively created by the authors,
consists of a set of algorithms linked to one another forming a recursive
network. The algorithms have different tasks, such as biosignal analysis,
learning of movement nuances (through linear regression and statistical
methods), and sound resynthesis. A group of algorithms extracts a set of high-
level features from the biosignals collected from the player’s body (change
rate, damping and spectral centroid). This feature set does not represent the
movement per se, for it does not account for the shape of movement in space.
Rather, it indicates specific traits, such as muscular force, abruptness of a
contraction, and damping of the muscular tissues, that characterise the
articulation of the movement. The combination of analysis, learning and sound
synthesis algorithms into a recursive network makes the instrument dynamic
enough to provide highly unexpected responses to the player’s movements. This,
in turn, forces the performer to adapt the choreography “on the fly”,
establishing thus a continuous and adaptive dialogue with the instrument.

###  3.2 Scientific and artistic drives

Here we present the scientific and artistic contexts, specific to each author,
in which our collaboration on _Corpus Nil_ took place.

#### Gesture Expressivity and Interactive Machine Learning

When we started the collaboration in 2014, my (first author) research focused
on capturing movement expressivity to be creatively used in music performance.
A hypothesis was that aspects of the physical expressivity of a performer
could be observed by analysing intentional variations in the execution of a
movement. A system which would capture such variations could then be used to
drive sound synthesis engines. However capturing those variations proved not
trivial. Our approach at the time, developed with colleagues at Ircam (Paris)
and Goldsmiths College (London), relied on a machine learning-based algorithm
– called Gesture Variation Follower (GVF) [14] – which tracks, in real-time,
continuous movement variations across space and time. In order to allow
artists with radically different movement vocabularies and expressivity to use
the system, this was designed to be rapidly calibrated.

This research belongs to a broader line of endeavour where machine learning
algorithms are considered as tools to design expressive musical instruments
and interactions [30, 15, 36, 29, 34, 32]. In this type of research, the
significant advantage of machine learning is that it leverages advances in
computation and data analysis to allow musicians to tackle fairly complex
musical scenarios [29]. More precisely, instead of writing rules that govern
the criteria of interaction between performed movements and resulting sound,
one can provide the algorithmic system with demonstrations of those criteria,
which will be automatically learned by the system [31]. This approach involves
tight interactions between the performer and the ML algorithm, formalised
under the discipline of Interactive Machine Learning (IML) [55, 27, 1]. Using
machine learning as a design tool seemed a natural approach when considering
movement inputs to an interactive system. Writing rules that govern the
analysis of movement inputs is clearly unfeasible when we consider that the
system is expected to handle a wide scope of complex movements, mostly
subjective to each performer. In addition, movement literacy is tacit, which
means it cannot be easily formalised through words and lines of code of a
programming language in a design context [38]. A machine learning-based
approach where one can configure a system by demonstration seems a much more
efficient approach [35]. In addition, such a system can be more inclusive,
meaning that it can be used by novices and people with diverse physical
abilities, who may not have previous experience in movement analysis or
computer science [41].

#### Deconstructing a Performer’s Body

By the time our collaboration began, I (second author) had been working for
four years in the field of gestural music as a researcher, composer and
performer; in particular, I had been developing a musical performance practice
known as _biophysical music_ [21]. With this term, I refer to live music
pieces based on a combination of physiological technology and markedly
physical, gestural performance. The approach of biophysical music differs
significantly from previous strands of biosignal-based musical performance
[43, 49, 54] which are largely rooted in the use of physiological signal as a
means of control over electronic or digital interfaces. The practice of
biophysical music focuses, instead, on designing ways and tools through which
the physical and physiological properties of a performer’s body are interlaced
with the material and computational qualities of the electronic instruments,
with varying degrees of mutual influence. Through this method, which I term
human-machine _configuration_ [20], musical expression arises from an intimate
and, often, not fully predictable negotiation of human bodies, instruments,
and programmatic musical ideas (an anthology of biophysical music projects can
be viewed in [19]).

After several years exploring potentialities and limits of gestural music
research, I started questioning – in both my artistic and scientific processes
– the kind of human body at the center of conventional gestural music
performances. My concern arouse from both my own work experience in the field
and my participation to the HCI community through conferences and research
exchanges, as well as from my study of feminist theory, in particular body
theory [7] and disability studies [51]. Soon, it became clear to me that the
kind of body shown in the performances of gestural music, and new musical
instruments in general, was most times a heavily normalised body, that is, a
body fitting particular criteria of normality. This triggered me to
investigate how elements of gestural music performance and HCI could be
exploited – in combination with research on movement and aesthetic – to create
a different kind of performance practice; a practice that would account for
the human body as an ever-changing, fluid entity with multiple potentials and
varying forms of embodiment, instead of a static, controlled and controlling
subject as established by our societal regime. Thus, drawing on the artistic
work of performers such as Kō Murobushi [47] and Maria Donata D’Urso [25],
among others, I developed a particular methodology of movement research that
emphasised the combined use of unconventional choreographic methods and
intense somatic experimentation towards the physical manipulation and symbolic
deconstruction of a performer’s body.

The movement experiments I conducted at the time were based on a gesture
vocabulary composed of complex torsions, flexions, and contractions of
shoulders, upper arms, and neck. No lower arm or hands gesture figured in the
movement vocabulary I had designed for the piece (which then became _Corpus
Nil_); hands, lower arms and even the head were, in fact, hidden from view and
rendered purposely useless. Because the literature and tools in regard to this
particular mode of gestural interaction are scarce, we decided to explore
muscle activity at a very low level – through choreographic methods and
resources from biomedical engineering. This, in turn, led us to investigate
complementary sensor modality, with the goal to understand physical expression
outside of a frame focused on “control”.

###  3.3 Development and Observations

The initial seed of the collaboration was an intuition to investigate
applications of interactive machine learning to the analysis of movement in a
body-based musical performance. The performance in question, however, relied
on the expression of bodily and motion qualities that were uncharted by
previous work in the field. This spawned a research of muscle activity, the
sensorimotor system and how, through different sensing methods, aspects of
expressivity could be observed in human-computer interaction.

#### Understanding Movement Expressivity

Initially, the analysis of movement expressivity as conveyed by the specific
kind of muscle torsions, flexions and contractions described in the previous
section seemed a good challenge for the Gesture Variation Follower algorithm
(GVF) [14]. GVF was originally designed to track variations of movement
trajectories in space and time, so we explored fundamental questions about
movement expressivity at the level of the muscle activation. What is the
trajectory of muscle activity in this context? To what extent information on
muscle temporal activity is meaningful to a performer, and how muscle temporal
activity can be controlled?

We began by experimenting with mechanomyogram (MMG) sensors, capturing the
mechanical activity of the muscle, built by Donnarumma in previous work [18].
We organised sessions of data capture and post-analysis. It became quickly
clear that MMG data captured dynamic transitions between separate muscle
contractions and the resulting trajectories calculated by GVF did not make
sense. We added a second sensor based on electromyogram (EMG), which measures
the electrical discharges activating the muscle. Here a new problem emerged,
following Donnarumma’s feedback about the resulting movement-sound
interaction: controlling EMG trajectories was not aesthetically convincing,
for it could not allow a highly dynamic mode of interaction, one of our main
musical goal. Therefore, we worked on a new way to analyse muscle biosignals.
We developed a different tracking system where the tracked parameters were not
linked to the movement trajectory but the to the parameters of an underlying
second order dynamical system with damping (symbolically representing muscle
dynamic as an oscillating string).

How to define gesture expressivity when the gestures in questions operate
through a set of symbolic and aesthetic signs? What kind of expressivity
emerges from a performer’s body that is physically constrained by a given
choreography? And how can a ML system actively and subtly support a semi-
improvisational physical performance of music where the very notion of
“control” is put into question? We felt that these questions were to be
tackled through a strategy that combined scientific and artistic methods. Our
research forked in two parallel streams of investigation. On one hand, we
deepened our understanding of expressivity to take into account involuntary
aspects of whole-body gestures. Using real-time analysis of complementary
muscle biosignals, EMG and MMG, we began isolating a set of features that
could describe relative levels of muscular effort without the need to define
in advance the type or the timing of a gesture. This would support the open-
ended and semi-improvised nature of the choreography that was being developed.
On the other hand, we started exploring the design of a computational music
system that would not be tied to conventional mapping techniques, but instead
would reflect, through sound, levels of effort intensity, movement abruptness
and degree of complexity of particular gestures – i.e., the amount of muscle
groups activated by a given gesture.

We finally conducted a controlled experiment to study gestural expressivity
from the point of view of motor control (i.e., the ability of users to control
gesture variations), using both EMG and MMG interfaces [12]. The main study
brought together 12 participants for a session of 45 minutes each. In these
sessions, participants were asked to perform a certain number of gestures
(with or without tangible feedback) and then to vary one or more dimensions of
expressivity. We showed that the participants consistently characterized
dimensions of implicit expressivity (force, tension, etc.) and that the
physiological interfaces used made it possible to describe these dimensions.
Finally, we showed that participants could control these dimensions under
certain conditions linked to constraints from the laws of motion.

#### Highlighting algorithm’s limitation with IML

GVF was designed to be quickly trained (or calibrated) by providing one
example for each movement that the system has to recognise, and its capacity
to track movement variation was meant to give to users a means to explore a
movement space. However its use in a tight interaction loop with the
performer’s body during experiments and rehearsals felt like a failure. One
major factor was that the system felt restrictive rather than extensive. The
movement possibilities seemed to shrink rather than extend. In the typical IML
workflow, the performer is engaged in a close interaction loop with the
movement-based machine learning system. However, if the algorithm has low-
capacity (failing to handle a wide range of input movements) with respect to
the complexity of the input movement, the performer may ultimately adapt to
the system limitations, hence constraining her own movement qualities.
Consequently, this adaptation would make the interaction converge towards
simplistic movements and interactions.

This phenomenon may have also been emphasized by the absence of clear
explanations about the system’s prediction behavior. Incorrect analysis of the
GVF were hard to grasp by the performer (second author). Typically, the
algorithm could have failed because the beginning of the movement is too
different from the pre-recorded ones, but this information was not clearly fed
back to the performer. As most temporal gesture recognition systems, GVF
recognises a gesture (or movement) from a starting point until an end point.
Establishing what defines the beginning of a movement and its ending point
during the artistic performance was not trivial and it could definitely not be
handled by the algorithm. Moreover, the system was designed to handle
continuous and rather slow variations, whereas Donnarumma’s muscle activity
during the choreography presented types of dynamics that could not be tracked
by the system. We needed a more flexible approach and thus we developed a new
tracking system. This was designed to estimate in real-time the parameters of
a physical model that mimicked the performer’s muscular activity, as mentioned
above. In doing so, we were not tracking gesture trajectories but movement
“regimes”. Thus, this method allowed us to explore alternative representations
of movement using the input data at hand.

##  4 Artificial Intelligence as Actor in Performance

In 2019, we initiated a second collaboration, which actively involves a group
of 13 people, including artists working with new media, biological materials,
and performance, as well as designers, AI and neurorobotics scientists. The
project outcome is a dance-theater production entitled _Humane Methods_. The
first version of the performance premiered at Romaeuropa Festival and, at the
time of writing, the project is currently ongoing. As for the previous
collaborative project, we begin by describing the piece and then situating it
within distinct scientific and artistic contexts.

###  4.1 _Humane Methods_

_Humane Methods_ is a dance-theater production exploring the multilayered
nature of today’s violence. The project departs from the assumption that –
through the physical and psychological brutalization of people and non-human
beings – the combination of digital technologies and capitalistic urge has
driven the natural ecosystem towards impending destruction. The project then
aims to dissect the violence of algorithmic societies, where power structures,
knowledge creation and normative criteria become means of manipulation. Being
an evening-length production, _Humane Methods_ is composed of a multitude of
elements, including robotic prosthetic limbs driven by their own neural
networks, dead plant specimen and fungi growing on costumes, uncompleted or
destroyed architectural structures, chimeric creatures, experiments in trauma,
actions as rituals. For the sake of clarity, here we focus on the
technological aspects of the piece and how they relate to the techniques and
implications of AI. Figure 2 shows a picture from the performance.

Figure 2: Photo courtesy of Giada Spera. Picture from a live performance of
Humane Methods at Romaeuropa Festival. Portrayed is one of the iterations of
the ritual at the core of the performance. It is possible to observe in detail
the lighting generated by the AI.

In a landscape of ruined nature and technological relics, a community of nine
semi-human figures lives a primal and stern existence. They perform a ritual
of prayer in a loop, but with every repetition of the prayer, something in
their interpersonal relations or in the world surrounding changes. As
variations accumulate, the praying loop is drastically altered and the
storyline branches out into a constellation of stories, of hints, of untold
events, playing out various degrees of intolerance, isolation and
exploitation. Throughout the action, a deep reinforcement learning algorithm
tries to learn a meaningless sequence of ten digits.

Our motivation in choosing to work with an AI algorithm performing meaningless
calculations lies in a conceptual shift from conventional uses of deep
learning techniques in music and performance. What interests us is not the
capacity of the algorithm to reach its target, but rather the ways in which
the inner (obsessive) logic of this type of computation can be made
perceivable at an aesthetic and sensorial level to both performers and
audience. To that end, we designed an audiovisual system that links the
actions of the AI algorithm to synaesthetic patterns of music and light, so
that the algorithm “speaks”, as it were, of its actions and choices through
sound and light. The result is a dogged, hypnotic stream of sound and light
that inundate the venue: the calculations of the AI are turned into
perceivable material, auditory and visual pulsating patterns that literally
mark the choreographic actions on stage, as well as the spectators’ experience
of the piece.

###  4.2 Scientific and Artistic Drives

#### Human - AI Interaction

ML and AI algorithms have known an exponential development in the past
decades. Several important works have shown the capacity of learning-based
algorithms to outperform humans in specific cognitive tasks such as playing
the game Go [52], recognising images [44] or understanding natural language
[37].

Simultaneously, the fields of HCI, Humanities, and Social Sciences have
started to take seriously the study of the socio-cultural impact of AI and to
propose design guidelines or good practices. Several works have recently been
published along this line. Among them, some aim to propose a set of guidelines
for designers of AI-powered interactive systems [2]. In this context, AI is
seen as a tool to improve users’ services and generally empower humans [10].
As an extension, AI can be seen as a material that can be appropriated by
designers to improve user experience in AI-based applications [24]. However,
most of these attempts have highlighted the inherent challenges of this
approach due to the difficulty of grasping what AI can and cannot do [56]. In
my work (first author), I am exploring how the computational learning
mechanisms themselves can become interactive in order to foster exploration
and human learning, as we recently explored in the specific domain of sound
design [50].

On the other hand, recent works have investigated the complex consequences of
the inherent biases in AI-powered applications and their impact when used in
health care, education or culture [3, 45, 17]. This last branch of research is
a necessary undertaking that I am currently pushing in my recent research
interest, which looks at the socio-cultural impact of technology. Therefore,
at a scientific level, the collaboration presented in this section was
motivated by an intention to investigate how people understands AI technology
and which are the founding beliefs supporting particular forms of such
understanding.

#### Motivations and Artistic Needs

The project _Humane Methods_ is a collaborative endeavour that involves, aside
from the collaboration between myself (second author) and Caramiaux, a shared
authorship between Margherita Pevere [48], myself and video artist Andrea
Familari. In particular, the artistic concept for the piece was jointly
created by Pevere and myself, and then furthered through a close collaboration
with Caramiaux. As artists working respectively with biotechnology and AI
technology, the first question that Pevere and I asked ourselves was how to
situate our artistic practices within the current historical period, one
characterised by a generally intolerant and polarised socio-political forces,
converging with a serious, technologically-driven disruption of the natural
environment. This question led our efforts towards an analysis of the
multifaceted nature of human violence, as it is exerted between humans as well
as it is enforced by humans on the lives of non-human entities and the natural
environment.

Turning to the main topic at hand – AI technology and its use in collaborative
art and science projects – we chose to deploy AI algorithms in an
unconventional way. Instead of using AI to aid human performers in the
creation of sound – as we had done for _Corpus Nil_ – or using an AI
generative capacity to create a musical automata, we chose to make tangible
the “brute force” computing mechanism of AI algorithms. In other words, we
chose to highlight how many of the most common learning algorithms being
presently developed attempt to reach their targets using a mechanism that is
obsessive and raw.

###  4.3 Development and Observations

The collaboration developed through personal exchanges on the general theme of
technology and society and particularly on how technology induces violence or
empathy on people, to other people and to their environment. We wanted to
create an algorithmic behavior that obsessively learned something and which
could autonomously explore different ways to reach its goal. The first trials,
in terms of scenography, music, light and algorithms happened in June 2019.

#### Machine Behaviour

The algorithm designed and implemented for _Humane Methods_ is based on deep
reinforcement learning, which means that the algorithm explores a high-
dimensional continuous space and receives positive rewards when approaching
the goal. These rewards are used to exploit certain moves within this space
that would help the algorithm reach the goal. In _Humane Methods_ , the
algorithm’s target is an arbitrary set of parameter values which – purposely –
holds no meaning. The reward is given by its distance to the target. Its
actions are moves in the parametric space. As the algorithm moves in the
parametric space, it outputs a list of 10 values – indicating the degree of
proximity of its calculations in respect to its target on an integer scale
between 1 and 3. Each of the 10 values is then mapped to a musical pattern –
which are predefined arpeggios of acoustic piano samples – and a light
pulsation pattern. Sonic and light patterns follow relentlessly one after the
other, creating the perception of an audiovisual entity morphing continually
through a multitude of syncopated variations. This dynamic morphing of sound
and light is made more complex by a parallel method linking the algorithm
degree of proximity to the target to, on one hand, the loudness of each single
musical pattern, and on the other, to the brightness intensity of physical
analog lights. The closer a value is to the corresponding target, the quieter
is the volume of the related musical pattern and the dimmer is the brightness
of the corresponding light pattern.

The algorithm moves within the space are entirely inferred by the learning
process. This process consequently makes apparent to the spectators the
behaviour of the algorithm. The algorithm becomes thus an actor, a performer
on its own. Because neither the human performers nor the technical team behind
the piece can control the computation of the algorithm, the latter is the sole
entity responsible for the audiovisual dramaturgy of the piece. Interestingly,
this role taken by the algorithm made us personify it: we began referring to
it by a name, or discussing its contribution to the piece as an actor on its
own. Journalists have also asked questions along this line, after seeing the
show. This has been all the more striking for us as we were very careful not
to anthropomorphize AI, a process that we find counter-creative for it
typically impedes a precise understanding of technology by imposing a human
worldview onto something which is not human.

#### Controlling Artificial Intelligence and Being Controlled by It

Following our first experiments with the algorithm and, thus, observing the
modalities of its interaction with music, light and choreography, it became
more interesting to us _how_ the algorithm learned rather than what it
learned. We tested several strategies to map, more or less directly, the
behaviour of the algorithm (the how) to the choreography executed by the human
performers – discarding, in the process, any information regarding what the
algorithm learned. In doing so, we soon faced a different challenge: how to
find a balance between the control of the AI over the piece’s narrative and
the influence of the performers over the AI’s learning process. We explored
different strategies. One was to have the algorithm dictate the start and end
points of each choreographic action. This, however, felt narratively
constraining and aesthetically poor. Another attempt required the AI to
describe the movement qualities with which each loop was to be performed by
the human actors. This proved cumbersome, for a reliable, real-time
communication between the AI and the performers was too complex to design.

At this point we changed our approach. Instead of mapping the AI’s behaviour
to the actions on stage, we thought of implementing the opposite strategy,
that is, to map the actions on stage to the behaviour of the AI. Thus the
choreography was re-conceived to mimic the temporal structure of the AI’s
learning process. One single action (a hybrid form of gestural prayer) is
repeated in a loop, but with each repetition comes a more or less drastic
variation. This loop structure mirrors the episodic learning of the algorithm:
the algorithm learns through episodes – each can take a few minutes – and each
episode adds variations to the next iteration of learning. This solution meant
to limit the explicit interaction between actors’ performance and AI learning
process. The interaction between the two elements became thus implicitly
intertwined: at conceptual and aesthetic levels the performers’ action and the
algorithm’s learning process were connected, but this connection was not
defined by explicit mapping of control signals. While we successfully
performed the piece before a public audience in this format, we consider the
issue still open. In our view, the piece would have a stronger impact if the
performers’ actions would be linked _both_ implicitly and explicitly to the
learning process of the AI.

#### Audience Perception of Artificial Intelligence

In June 2019, we had the opportunity to collect subjective feedback from the
audience during an open rehearsal at the Centre des Arts Enghien les Bains
(CDA), a co-producer of _Humane Methods_ together with Romaeuropa Festival,
Italy. Our encounter with the audience in France was important, for it allowed
us to open our aesthetics and research to a general public, most of whom had
little or no familiarity with the ongoing project or our previous works. Here,
we focus on the feedback given by the audience about their perception of the
AI, as they observed it during the excerpts of the piece we performed for
them. It is important to note that, before the performance, we purposely
avoided informing the audience about the presence of an algorithm driving
lights and music.

The two most significant comments of the audience were that some spectators
did not realize that music and light were driven by an AI algorithm, and that
another spectator suggested us to create more explicit interactions between
the algorithm and the performers on stage. These comments are perhaps
unsurprising if we consider the issue of implicit/explicit interaction
discussed above. Nevertheless, they confirmed our early intuition about the
challenges of making an AI perceivable to a public audience. Our algorithm
manifests itself only indirectly, through lights and music and it lacks a
physical embodiment. This was a choice we made in order to reinforce our
critical stand on the impact of AI in society; the AIs regulating social
media, welfare and warfare are pervasive and ubiquitous, while simultaneously
invisible and implicit, integrated seamlessly as they are in institutional
structures and interpersonal lives. Our AI was designed with this in mind, and
is therefore omnipresent and unobservable.

However, concept aside, on an actual theater stage it is indeed a challenge to
manifest the AI’s presence and expressivity while completely avoiding a
physical representation of it (and potentially fall into naive
anthropomorphism). Thus, while we still believe the concept we explored is
meaningful to the overall piece, the audience feedback made us realize that,
in order to be perceived as a “real” presence, the AI in _Humane Methods_
needs to be embodied and its agency has to be manifested, in a way or another.
This is still ongoing research, thus we can only speculate here, but we
believe this issue opens a fascinating challenge regarding the fragile balance
between “AI in control” versus “humans in control” elaborated in the previous
section.

##  5 Discussion

This chapter focused on possible methodological approaches to the use of ML
and AI in the context of music and performance creation. We have offered an
autoethnographic perspective on a research and creation process spanning five
years. Our aim was to open up our process and proffer insight into one
possible way of coupling art and science in the application of AI and ML. In
closing this chapter, we discuss three main aspects of the research. The first
relates to how we approached AI and ML in relation to music and sound
creation. The second concerns a shift in terminology that accompanied, and
became manifested through, our projects. The third and final one addresses the
hybrid methodological approach we developed progressively and contextually to
each artwork.

###  5.1 Artificial Intelligence and Music

Many music-related fields are currently facing important changes due to the
intervention of machine learning and artificial intelligence technology in
music processing. These changes occur at different levels: creation,
consumption, production and diffusion [13]. While AI technologies offer
increasingly broad options to researchers and artists alike, with the present
contribution we stress the importance of acknowledging the historicity of
these technologies – where and how they emerged and developed – as well as the
increasingly complex implications of their interaction with multiple layers of
humans’ societies. As we have demonstrated through the review and discussion
of our specific collaborative projects, the use of ML and AI in music-related
research does not have to be constrained by a paradigm of control – of humans
over algorithms, of humans over musical forms, or of humans over other humans.
Rather, by nurturing in-depth and long-term art and science collaborations it
is possible to define new modalities of interaction across ML, scientific
research and artistic creation.

At a musical level, both _Corpus Nil_ and _Humane Methods_ foster
improvisational and unconventional types of interaction between algorithms,
human bodies and musical forms. What should be emphasised, however, is that
these kinds of musical interactions emerged from the combination of particular
scientific and artistic drives which, in turn, evolved out of specific socio-
cultural contexts. Artistic intervention into scientific research was coupled
with practice-based research in ways that ramified our thoughts and strategies
in multiple directions. Our own idea of what computational music can be was
heavily affected by the particular paths we chose throughout the past five
years of collaborative work. Whereas this allowed the germination of nuanced
and technically rigorous interaction systems, it also made the development of
satisfying outcomes slower than it could have been, had either of chosen to
work alone. Thus, the same combination of practices and approaches which
affords us with the capacity to deeply couple art and science, is a factor
that impedes fast turnarounds and requires mid-large timescales in order to
reveal artistically and scientifically valid outcomes.

###  5.2 From Machine Learning to Artificial Intelligence

Our work witnessed a change in terminology that has been occurring since the
deep learning breakthrough in 2012, that is, the shift from “machine learning”
to “artificial intelligence”. Machine learning used to – and, to some extent,
still does – refer to the technological underpinnings that allow computational
systems to “learn” from data, where learning means extracting structural
components from a dataset. The term artificial intelligence is however more
ambiguous: it was originally coined in the 1950’s to designate a symbolic
approach for modeling human cognition, but nowadays what is called AI mostly
relies on vector-space generated by deep neural networks (which is not a
symbolic approach) [16]. The term has its advantage to communicate about the
technology and to create research and business incentives. Then, we also
believe that the term imposed itself following the recent capacities of
machine learning techniques. When AlphaGo [52] won against the world’s best Go
player, the idiom “machine learning” did not seem comprehensive enough to
designate the technology that made it possible, its computational capacities,
or the new prospect that such technology had opened up.

While, today, the HCI community often uses the term Artificial Intelligence,
in our own research we preferably adopt the term Machine Learning to designate
the type of algorithms we develop and use. Nevertheless, in our view, the
adoption of the term AI triggers interesting questions into the concepts that
are most commonly associated with the term. Speaking of AI triggers a specific
and ambiguous imaginary that is worth understanding in detail in order to make
better use (or no use at all) of this technology as practitioners and
researchers in HCI.

The dialectical shift can be traced in our own work. Our first project,
_Corpus Nil_ , relied on a low-capacity movement recognition system. In this
case, we were certain that Machine Learning was the right designation because
the algorithm did not manifest a distinctive behavior. Rather, its role was to
recognize and learn given aspects of a performer’s physical expressivity, so
as to emphasize elements of human corporeality. On the contrary, the second
project, _Humane Methods_ , involved a radically different algorithm chosen
not for its capacity to achieve a certain goal, but rather for the particular
computational behavior driving its learning mechanism. In this case, the term
Machine Learning felt constraining, for it precluded an understanding of the
algorithm as a socio-cultural phenomenon, beyond its computational
capabilities. The algorithm had almost become a black box, interpretable only
by observing its choices and subsequent actions. In this sense, it became
observable as an actor.

###  5.3 Hybrid Methodology

By observing the varied methods we deployed in the research presented in this
chapter, what comes to the fore is a hybrid methodology, an adaptive range of
methods which evolved according to the nature and context of the particular
problem at hand. This is illustrated by how, in _Corpus Nil_ and _Humane
Methods_ , learning algorithms were used in two fundamentally different ways,
and how, in each case, the coupling of the respective needs of art and science
led to unconventional and effective results.

In _Corpus Nil_ , we used an algorithmic system to design expressive, body-
based musical interactions between a human performer and algorithms. We set
out to explore Machine Learning algorithms as a _tool_ , therefore our focus
was less on the algorithm itself and more on what results the algorithm could
achieve during the performance. The particular choreography created for the
piece revealed limits of available computational approaches for the design of
interactions with sound, and led us to find an ad hoc method based on
physiological sensing. This method – triggered by artistic intervention into
scientific research – yielded a new approach to the analysis of gesture
expressivity, one based on complimentary aspects of muscle physiological data.
This allowed us to capture dimensions of expressivity that greatly differ from
those obtained through computer vision interfaces, which are the most commonly
used in the literature.

In _Humane Methods_ , we designed a computational framework which eventually
became an actor, an independent entity directing sound and light in a dance-
theatre performance according to its own internal functioning. Contrarily to
_Corpus Nil_ , at the core of this piece was an investigation of AI as a
cultural and social _concept_ , therefore we were less concerned with the
results the algorithm could achieve and more preoccupied with how to manifest
the functioning of the algorithm itself and to exhibit its behavior. On one
hand, this shift of focus proved challenging because it forced our research to
confront a broad set of (scientific, artistic and popular) expectations
surrounding the notion of AI. On the other hand, though, the challenge has
opened before us a vast field of investigation, which extends across HCI,
performing art and into the socio-political impact of AI. We feel that the
prospect for a further entanglement of science, art and politics is, today,
perhaps more urgent than ever.

##  6 Acknowledgements

We would like to thank Frédéric Bevilacqua (IRCAM, Sorbonne Université), Sarah
Fdili Alaoui and Margherita Pevere for their helpful comments on the paper.
This research was partly supported by the European Research Council under the
European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant
Agreement n. 283771; by the CNRS under the programme PEPS / project INTACT;
and by La Diagonale Paris-Saclay, under the programme Phares 2019 / project
Humane Methods.

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PASA 2024

# The One-Way Speed of Light and the Milne Universe

Geraint F. Lewis1 and Luke A. Barnes2
1Sydney Institute for Astronomy, School of Physics, A28, The University of
Sydney. NSW 2006, Australia
2Western Sydney University, Locked Bag 1797, Penrith South DC, NSW 2751,
Australia<EMAIL_ADDRESS>
###### Abstract

In Einstein’s Special Theory of Relativity, all observers measure the speed of
light, $c$, to be the same. However, this refers to the round trip speed,
where a clock at the origin times the outward and return trip of light
reflecting off a distant mirror. Measuring the one-way speed of light is
fraught with issues of clock synchronisation, and, as long as the average
speed of light remains $c$, the speeds on the outward and return legs could be
different. One objection to this anisotropic speed of light is that views of
the distant universe would be different in different directions, especially
with regards to the ages of observed objects and the smoothness of the Cosmic
Microwave Background. In this paper, we explore this in the Milne universe,
the limiting case of a Friedmann-Robertson-Walker universe containing no
matter, radiation or dark energy. Given that this universe is empty, it can be
mapped onto flat Minkowski space-time, and so can be explored in terms of the
one-way speed of light. The conclusion is that the presence of an anisotropic
speed of light leads to anisotropic time dilation effects, and hence observers
in the Milne universe would be presented with an isotropic view of the distant
cosmos.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

cosmology: theory

##  1 INTRODUCTION

Central to Einstein’s Special Theory of Relativity is that all inertial
observers will measure an identical value of the speed of light, $c$
(Einstein, 1905). However, as noted by Einstein himself, this refers to the
average of a round-trip journey for light that is reflected off a distant
mirror, and, as long as the average speed is $c$, the outward and inward
velocities could be different (see extensive review in Anderson et al., 1998).
Whilst this might seem strange, anisotropy in the speed of light would result
in anisotropy in time dilation effects, ensuring that synchronisation of
distant clocks remains fraught. Hence, no experimental measurement of the one-
way speed of light is possible.

An objection to differing one-way speeds of light might be observations of the
distant universe, where we clearly have, on average, an isotropic view, seeing
young galaxies at high redshift, and the smoothness and uniformity of the
Cosmic Microwave Background over the sky; surely the anisotropy in the speed
of light would be imprinted on this view? In this paper, we will tackle this
question by considering an idealised cosmological model, the Milne universe,
and will explore an extreme case where the speed of light is infinite in one
direction, and $c/2$ in the other. The layout of this paper is as follows: In
Section 2 we present the mathematics of differing one-way speeds of light, and
will present the Milne universe in Section 3. We discuss the Milne universe
with differing one-way speeds of light in Section 4, whilst presenting our
conclusions in Section 5. In the following, we will set $c$, the average round
trip speed of light, to unity.

Figure 1: Space-time diagram for the situation where the speed of light is
equal in both directions (left) and the limiting case where the speed of light
is $c/2$ in one direction, and infinite in the other (right). The green dashed
line represents a future lightcone for the observer at the origin of space and
time, whereas the grey lines represent the worldlines of massive objects
moving relative to the coordinate system. Through synchronising all clocks at
the origin, the blue lines represent light rays emitted from the massive
objects after a fixed amount of time has passed for both. Clearly, the
observer at the origin sees the massive objects at the same age when the light
rays are seen.

##  2 ONE-WAY SPEED OF LIGHT

In considering differing one-way speed of light models, the underlying
transformations of coordinates are modified. In the following, we follow the
mathematical formalism of Anderson et al. (1998). We consider differing one-
way speeds of light related to $c$ by

$c_{\pm}=\frac{c}{1\mp\kappa}\equiv\frac{1}{1\mp\kappa}$ (1)

Setting $\kappa=0$ corresponds to an isotropic speed of light, whereas
$\kappa=1$ presents the extreme case where the $c_{+}=\infty$ and $c_{-}=1/2$.
To preserve the observations of special relativity, a coordinate velocity
$v=\frac{dx}{dt}$ in the isotropic $c$ case ($\kappa=0$) is mapped to a new
velocity

$\tilde{v}=\frac{d\tilde{x}}{d\tilde{t}}=\frac{\gamma}{\tilde{\gamma}}v$ (2)

where

$\gamma=\frac{1}{\sqrt{1-v^{2}}}\ \ \ {\rm and}\ \ \
\tilde{\gamma}=\frac{1-{\kappa v}}{\sqrt{1-v^{2}}}$ (3)

The relative time dilation between two observers in the case where the one-way
speed of light is given by

$\frac{d\tilde{t}^{\prime}}{d\tilde{t}}=\frac{1}{\tilde{\gamma}}$ (4)

Figure 1 presents an illustration of the impact of the anisotropic speed of
light. The left-hand side of this figure presents the familiar case where the
speed of light is equal in both directions with the green-dashed line
representing a light cone for an observer at the origin. The grey lines
represent the worldlines of massive objects moving relative to the observer at
the origin at $v=0.7$, with their clocks synchronised at the origin. The blue
lines represent light rays emitted from massive objects after a fixed about of
proper time has passed. Given the symmetry of the situation, the light rays
arrive at the origin at the same time, and the observer sees the moving
objects showing the same amount of proper time passing at the same instant.

##  3 THE MILNE UNIVERSE

The discussion in the previous section is wrapped in the language of special
relativity, whereas the cosmological description of the universe relies on
Einstein’s General Theory of Relativity. Whilst the typical approach to
studying cosmology is to begin with the Friedmann-Robertson-Walker (FRW)
metric (e.g. Hobson et al., 2006), this is just a convenient choice of
coordinates and other choices can be made. Infeld & Schild (1945) demonstrated
that cosmological models can be cast in a kinematic form, where by motion
through space, coupled with gravitational potentials, replaces the picture of
expanding space (e.g. Lewis et al., 2007).

The focus of this paper will be the limiting case of the FRW metric, namely
the Milne universe in which the universe is empty, devoid of any matter,
radiation or energy (Milne, 1933). A key feature of of the Milne universe is
that, while it is spatially curved, its space-time is flat and can, therefore,
be directly mapped into the Minkowski metric (see Chodorowski, 2005)

Figure 2: Space-time diagram for the Milne universe in FRW coordinates. The
horizontal dashed grey line denotes now in cosmic time, whilst the sold grey
lines are comoving objects at $x=1,5,10,50$ and $100$. The blue lines
represent the past light cone for an observer at the origin today and the time
where they cross the comoving objects is the age we observe them at today;
clearly, due to the symmetry of the situation, the view in opposite directions
will be the same, with more distant objects appearing younger. The two black
dots denote emission from $x=5$ that is observed at the origin today, whilst
the red dashed line represents the age of the universe when the light from
these sources is emitted.

We begin by describing the Milne universe in Minkowski space-time. At $t=0$, a
collection of massive test particles (quaintly referred to as _galaxies_) are
ejected in all directions from the origin ($x=0$) with a range of velocities.
In the limiting case of an empty universe, we disregard gravity (the effect of
these galaxies on space-time), and so the galaxies maintain a constant
velocity. Because faster galaxies move further in a given period of time, the
further away we look, the faster the galaxies are moving and the more their
light is redshifted; this gives the Hubble law for any of the galaxies.

Assuming an isotropic speed of light, consider two galaxies that are emitted
with the same speed in opposite directions. In order to measure their
positions, we (still at the origin) send a light beam after them at time
$t_{1}$. The light bounces off the galaxy at $(x_{g},t_{g})$ and returns to us
at $t_{2}$. With an isotropic speed of light, the light reached the galaxy
halfway between $t_{1}$ and $t_{2}$. The distance to the galaxy is half of the
total light travel time:

$\displaystyle t_{g}$ $\displaystyle=\frac{1}{2}(t_{2}+t_{1})$ (5)
$\displaystyle x_{g}$ $\displaystyle=\frac{1}{2}(t_{2}-t_{1})~{}.$ (6)

This second expression effectively defines the radar distance to an object
(c.f. Lewis et al., 2008). If the galaxy started a clock as it departed the
origin, the time on that clock when our photon arrives is given by,

$\tau_{p}=\sqrt{x_{g}^{2}-t_{g}^{2}}=\sqrt{t_{1}~{}t_{2}}~{}.$ (7)

From this, we infer that the galaxy is moving with speed
$v_{g}=x_{g}/t_{g}=(t_{2}-t_{1})/(t_{2}+t_{1})$.

Now, suppose that the speed of light is $c_{+}=\infty$ in the positive
$x$-direction and $c_{-}=1/2$ in the negative $x$-direction, as shown in
Figure 1 (right). As before, we send a beam of light after the galaxy at
$t_{1}$ and it returns to us at $t_{2}$.

### Right-hand Galaxy (moving in the positive direction):

The light travels instantaneously to the galaxy at $t_{1}$, and returns to us
at speed $1/2$,

$\displaystyle t_{gr}$ $\displaystyle=t_{1}$ (8) $\displaystyle x_{gr}$
$\displaystyle=\frac{1}{2}(t_{2}-t_{1})~{}.$ (9)

For this anisotropic universe, the formula to calculate the proper time is,

$\tau_{gr}=\sqrt{t_{gr}^{2}+2x_{gr}t_{gr}}=\sqrt{t_{1}~{}t_{2}}~{},$ (10)

as above. The inferred speed of the galaxy is
$v_{gr}=x_{gr}/t_{gr}=-\frac{1}{2}(t_{2}-t_{1})/t_{1}$, which is related to
the inferred velocity for the isotropic case by Equation (2).

### Left-hand Galaxy (moving in the negative direction):

The light travels at speed $c/2$ to the galaxy at $t_{1}$, and returns
instantaneously to us,

$\displaystyle t_{gl}$ $\displaystyle=t_{2}$ (11) $\displaystyle x_{gl}$
$\displaystyle=-\frac{1}{2}(t_{2}-t_{1})~{}.$ (12)

For this anisotropic universe, the formula to calculate the proper time is,

$\tau_{gl}=\sqrt{t_{gl}^{2}+2x_{gl}t_{gl}}=\sqrt{t_{1}~{}t_{2}}~{},$ (13)

as above. The inferred speed of the galaxy is
$v_{gl}=|x_{gl}|/t_{gl}=\frac{1}{2}(t_{2}-t_{1})/t_{2}$, which is related to
the inferred velocity for the isotropic case by Equation (2).

Importantly, the redshift of the returning light that is observed by us is the
same in all three cases: $1+z_{g}=\sqrt{t_{2}/t_{1}}$, so the observed
universe is identical whether the speed of light is identical in all
directions, or is anisotropic. However, for the anisotropic advocate, galaxies
with the same redshift in different directions are located at the same
distance, but emitted the light we observe at different times. They have
different velocities, and so they conclude that the left side of the universe
is expanding faster than the right hand side.

Figure 3: The Milne universe presented in Figure 2, but now mapped into the
flat space-time coordinates. The comoving objects have been mapped into sloped
lines, whereas synchronised lines of constant cosmological time have been
mapped into hyperbola. There are clear similarities between this and the left-
hand space-time diagram presented in Figure 1.

What does this Milne universe look like in an expanding space framework? We
begin with the the FRW metric,

$ds^{2}=-dt^{2}+a^{2}(t)\left[dx^{2}+R_{o}^{2}S_{k}^{2}\left(x/R_{o}\right)d\Omega^{2}\right]$
(14)

where $a(t)$ is the normalised scale factor, such that $a(t_{o})=1$ and
$t_{o}$ is the present age of the universe. The function, $S_{k}(x)$ is
$\sin(x)$, $x$, and $\sinh(x)$ for a spatially closed, flat and open universe
respectively. The angular terms, which are related to the surface of a
3-sphere, are given by $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\ d\phi^{2}$.
The present day scale factor, $R_{o}$, in an open universe is given by

$R_{o}=\frac{1}{H_{o}}\frac{1}{\sqrt{1-\Omega_{o}}}$ (15)

where $H_{o}$ is the present day Hubble Constant and $\Omega_{o}$ is the
present day total energy density. For the Milne universe, $\Omega_{o}=0$, and
so $R_{o}=H_{o}^{-1}$, and the normalised scale factor $a(t)=t/t_{o}$.

It is instructive to construct a space-time diagram for the Milne universe
(Figure 2) which shows the instantaneous proper distance to a comoving
observer at a spatial coordinate, $x$, given by $D(t)=a(t)x$ verses the
cosmological time, $t$; as an illustration, comoving observers are presented
at $x=1,5,10,50$ and $100$. Also, presented in blue, is the past lightcone for
an observer at the spatial origin at the present time. The path of a light ray
in these coordinates is governed by

$\frac{dx}{dt}=\pm\frac{t_{o}}{t}$ (16)

Remembering that in these coordinates, the proper times of comoving observers
are synchronised with the cosmic time, $t$, the observer at the origin will
see distant objects with an age given by their crossing of the past light
cone, and, given the symmetry of the situation, the origin observers view will
be symmetrical, with more distant objects appearing younger.

##  4 MILNE AND SPECIAL RELATIVITY

Examining Figure 2 suggests that the Milne universe is very different to the
flat space-time of special relativity. However, given that it has no material
content, the underlying space-time structure of the two are the same, and so
we should be able to undertake a coordinate transformation between the two.
Note that this is different to the conformal representation of FRW universes
(e.g. Harrison, 1991), which straightens light rays to $45^{o}$, as there can
be a complex relationship between universal conformal time and the experienced
proper time.

We follow Chodorowski (2007) by firstly defining $\chi=x/R_{o}$ and
$dt=R_{o}ad\eta$ such that Equation 14 can be written as

$ds^{2}=R_{o}^{2}a^{2}(\eta)\left[-d\eta^{2}+d\chi^{2}+\sinh^{2}(\chi)d\Omega^{2}\right]$
(17)

Figure 4: The Milne universe presented in Figure 2, but mapped into the case
with an anisotropic speed of light. Again, the grey dashed line corresponds to
the present time in the cosmological time of the Milne universe, whereas the
the red dashed line is the cosmological time for a pair of emitters on either
side of the sky. As can be seen, the observer at the origin is presented with
an isotropic view of the sky, even through the speed of slight is anisotropic.

At this stage, we define a coordinate transformation from $(\eta,\chi)$ to new
coordinate $(T,R)$ through

$\displaystyle T=Ae^{\eta}\cosh(\chi)$ (18) $\displaystyle
R=Ae^{\eta}\sinh(\chi)$ (19)

where $A$ is a constant. With these

$Ae^{\eta}=\sqrt{T^{2}-R^{2}}\ \ \ {\rm and}\ \ \ \tanh(\chi)=\frac{R}{T}$
(20)

with these transformations, and a little algebra, Equation 17 can be written
as

$ds^{2}=-dT^{2}+dR^{2}+R^{2}d\Omega^{2}$ (21)

which is just the flat space-time of special relativity with polar coordinates
over the spatial part. Clearly, in these new coordinates, light rays travel at
$45^{o}$, comoving observers are represented as straight lines with a slope
given by $\tanh(\chi)$. Additionally, the relationship between the proper time
experienced by an observer at the origin, $dT$, and the comoving observer,
$d\tau$, is given by

$\frac{d\tau}{dT}=\sqrt{1-\left(\frac{dR}{dT}\right)^{2}}=\sqrt{1-\tanh^{2}(\chi)}$
(22)

To illustrate this, we map the situation presented in Figure 2 into the
$(T,R)$ coordinates. This is illustrated in Figure 3. As noted above, the
comoving observers to sloped lines, and as can be seen, these asymptote to
$45^{o}$ as all motion is bounded by the speed of light. In these coordinates,
the comoving observers in the Milne universe, which are moving apart due to
expanding space, are transformed into objects moving with velocities relative
to each other through flat space-time. The key points that synchronous lines
in the Milne universe, which are lines of constant universal time, have been
mapped into hyperbola in the $(T,R)$ coordinates.

The remaining question is: how do lines of constant cosmological time map onto
the situation where the one-way speed of light is anisotropic? Hence we
undertake the transformation of the situation in Figure 3 through the
mathematics present in Section 2, noting that the velocity is given by
$v=\frac{dR}{dT}$, and present the result in Figure 4. Again, the extreme case
is considered, so the speed of light in one direction is $1/2$ and the other
is infinite.

Figure 5: The line of simultaneity in the emitter in the FRW coordinates
(Figure 2), mapped into the anisotropic velocity of light coordinates (Section
4) from $\kappa=0$, the isotropic case represented as a hyperbola, to the
extreme case, with $\kappa=1$, both presented in bolder red, with intermediate
cases, in steps of $\kappa=0.2$, presented in lighter red. The filled circles
represent the location of the emitter in these coordinates for each of the
cases. Note that the spatial location of the emitter in the $\tilde{R}$
coordinate is independent of $\kappa$.

As seen in Figure 1, the comoving observers are now asymmetric about the
observer at the origin, but also added to Figure 4 are lines of synchronicity
of the FRW universe, with the grey dashed line being now, whereas the red
dashed line is the time that two sources emit their light to be observed by
the observer at the origin (see Figure 2). As can be seen, these are now
asymmetric due to the asymmetric action of time dilation. In fact, the
difference in the one-way speed of light and the resultant time dilation
conspire to ensure that the view of the observer at the origin is identical
irrespective of the one-way speed of light. Hence, at least in the Milne
universe, whether the speed of light is identical in all directions, or is
anisotropic, results in a uniform view of the distant universe.

So far, we have considered two cases, where either the speed of light is
isotropic, or the extreme case where the anisotropic speed is $1/2$ in one
direction, and infinite in the other. The question remains whether this holds
true in general case, for an arbitrary $\kappa$. To consider this, we explore
the form of the synchronous lines in the FRW universe (lines of constant
cosmological time) in the Milne universe when we consider the anisotropic
speed of light. In Figure 5 the emitters from the previous discussion are
presented as filled circles, but now values of $\kappa$ of $0$ to $1$ in steps
of $0.2$ are considered; as can be seen, as $\kappa$ increases, the shape of
this synchronous lines becomes more asymmetrical. Also shown, as filled
circles, is the location of the emitters for each of the considered values of
$\kappa$; importantly, it should be noted that their location in $\tilde{R}$
is fixed.

We can explore the impact of differing values of $\kappa$ in terms of geometry
in the space-time diagram. Consider an observer located at
$\tilde{T}=\tilde{t}_{o}$ at the origin, and emitters who have experienced a
proper-time, $\tilde{\tau}_{e}$ since leaving the origin. For an arbitrary
value of $\kappa$, and noting that the location of the emitter when a photon
is emitted, $(\tilde{t}_{r},\tilde{r}_{e})$, is given by

$\tilde{r}_{e}=\tilde{v}\ \tilde{t}_{e}$ (23)

and noting that, from Equation 4, that

$\tilde{t}_{e}=\frac{1-kv}{\sqrt{1-v^{2}}}\ \tilde{\tau}_{e}$ (24)

then, using the definition of the velocity in these coordinates (Equations 2),

$\tilde{r}_{e}=\frac{v}{\sqrt{1-v^{2}}}\ \tilde{\tau}_{e}$ (25)

which is independent of $\kappa$, as expected (for more explanation, see the
detailed discussion of the transformations presented in Anderson et al.,
1998). Hence, the speed of a light ray connecting the emitter and the observer
is given by

$\frac{d\tilde{r}}{d\tilde{t}}=\frac{\Delta\tilde{r}}{\Delta\tilde{t}}=\frac{\tilde{r}_{e}}{\tilde{t}_{o}-\tilde{t}_{e}}$
(26)

Noting that this speed equal $\pm 1$ for the case where $\kappa=0$, this
expression becomes

$\frac{d\tilde{r}}{d\tilde{t}}=\frac{-sgn(v)}{1+sgn(v)\kappa}$ (27)

where $sgn(v)$ is the sign of the velocity. When $\kappa=1$, this recovers the
velocity of light as being $1/2$ and infinite (c.f. Equation 1), as expected,
but also illustrates that for all other values of $\kappa$, the observer will
see an isotropic universe.

##  5 CONCLUSION

In his formulation of the special theory of relativity, Einstein chose the
convention that the speed of light is isotropic and so equal in all
directions. He also acknowledged that the physical predictions of his theory
will be unchanged if the speed of light was anisotropic, as long as the
average round-trip speed is equal to $c$. In this paper, we have considered
the question of the impact of the one-way speed of light on cosmological
observations, addressing the suggestion we should observe different sides of
the sky possessing different ages if light speed was unequal. By examining the
simplest FRW universe, namely the empty Milne universe, it is seen that the
anisotropic speed of light results in anisotropic time dilation effects that
compensate for the differing light travel times. In this universe, any
observer would see an isotropic universe around them, even if the speed of
light was not.

The anisotropic speed of light advocate must conclude that galaxies that are a
given distance away have a faster recession speed in one direction than in the
other, and the universe is expanding faster to the right or to the left.
However, the dependence of redshift on the speed of light means that this does
not change the appearance of the night sky. One side of the sky is not
significantly more redshifted than the other. The initial velocities given to
the galaxies at the beginning of the universe had just the right degree of
anisotropy to balance the effect of anisotropic redshift. Perhaps at this
point, the anisotropic speed of light advocate will suspect a fix, but there
is nothing internally inconsistent or in contradiction with data about this
model.

Of course, the Milne universe, as a limiting case of FRW cosmologies, is
special in that it can be directly mapped onto the flat space-time of special
relativity, and hence the question of the one-way speed of light can be
directly addressed. For more general cosmological models, where the presence
of mass and energy results in curved space-time, the picture is more
complicated as there is no simple mapping of the modified Lorentz
transformations into the general relativistic picture. We leave this
discussion for a future contribution.

###### Acknowledgements.

We thank the anonymous referee for their positive comments on this paper. GFL
thanks Derek Muller (@veritasium) and Petr Lebedev for discussions on the one-
way speed of light that sparked this current study. These were part of a
consultation that led to the preparation of an informative YouTube
presentation on the topic (https://youtu.be/pTn6Ewhb27k).

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