11institutetext: University of Minnesota
11email<EMAIL_ADDRESS>
# Knots, Music and DNA

Maria Mannone I would like to thank the mathematician and musician Franck
Jedrzejewski for his important suggestion to study the implications of knot
theory for the mathematical description of musical gestures. I also would like
to thank the mathematicians and musicians Emmanuel Amiot and Tom Collins, the
composer Dimitri Papageorgiou, the mathematicians Anar Akhmedov and Ian
Whitehead for the interesting feedback on the manuscript, and the
mathematician Giuseppe Metere for the insightful discussions about
2-categories.

###### Abstract

Musical gestures connect the symbolic layer of the score to the physical layer
of sound. I focus here on the mathematical theory of musical gestures, and I
propose its generalization to include braids and knots. In this way, it is
possible to extend the formalism to cover more case studies, especially
regarding conducting gestures. Moreover, recent developments involving
comparisons and similarities between gestures of orchestral musicians can be
contextualized in the frame of braided monoidal categories. Because knots and
braids can be applied to both music and biology (they apply to knotted
proteins, for example), I end the article with a new musical rendition of DNA.

###### Keywords:

category theory, musical gesture, composition, visual art, synesthesia

##  1 Introduction

The study of musical gestures is a growing field of research in performance
Leman (2007); Fortuna (2017); Mazzola Mannone (2016), composition Antoniadis
(2016), mathematics Visi . (2014), physics Bouënard . (2012), cognitive
sciences Schutz Lipscomb (2007), as well as signal processing and computer
sciences Antoniadis Bevilacqua (2016); Bergsland Wechsler (2015), and the
comparisons among different definitions of musical gestures Cadoz Wanderley
(2000). The mathematical definition of musical gestures uses the formalism of
category theory Mac Lane (1978). A musical gesture is a mapping from a
directed graph (skeleton) to a system of curves in space and time111Indicated
in the literature as a topological space Mazzola Andreatta (2007). (body)
Mazzola Andreatta (2007); see Section 2.1. Such a definition has been extended
to conducting and to singing. In fact, singing can be described as the result
of inner – partially conscious and non-conscious – movements Mazzola Mannone
(2016); Mannone (20171). Recently, criteria to compare gestures have been
proposed, mainly using homotopy Antoniadis Bevilacqua (2016) and
mathematically defining similarities among homotopically equivalent gestures
producing similar effects on the final sound spectra Mannone (to appear
2018).222Two gestural curves are called homotopically equivalent if there
exists a continuous mapping that transforms one into the other; if one can be
continuously deformed into the other. Such a theory also allows comparisons
between music and visual arts in light of the same gestural generators,
highlighting similarities between gestures leading to music, and gestures
leading to visuals, such as drawing and sculpting and even writing, as
proposed in cadoz.

Gesture studies can borrow the formalism of networks recently applied to the
comparison of visuals Guibas (2014), where concepts from category theory such
as colimit and limit are used Mac Lane (1978). We can define networks of
images as well as networks of sounds and connect them Mannone (20172). We can
describe visuals as the result of some drawing gesture(s), and thus we can
define similarity between music and visual arts Mannone (to appear 2018). In
this way, we can refine a technique to translate music into images, and vice
versa – tridimensional images into music Mannone (2011).333With the mapping:
height to pitch, depth to time and length to loudness. We can argue that the
most successful examples of translation between different art fields verify
crossmodal correspondences Spence (2011); Mannone (20171). We may refer to
several studies in the field of crossmodal correspondences in psychology,
psychophysics, neuroscience and linguistics Köhler (1929); Nobile (2013);
Gentilucci Bernardis (2007); Zbikowski (2002, 2017); Roffler Butler (1968) to
support this idea. Other applications of networks in music also refer to
cognitive models Popoff . (2015). Moreover, visuals and sound can often be
seen as parts of some unique, unitary entity, the audio-visual object and the
theory of indispensable attributes Kubovy Schutz (2010). In Kubovy’s studies,
it is highlighted that the main function of the visual channel is detecting
shape, while the main function of the auditory channel is localizing moving
sound sources. According to Kubovy, the indispensable attribute for visuals is
shape, and the indispensable attribute for sound is pitch. This may constitute
a conceptual basis for a sonification of essential lines through pitch
variations, one of the possible techniques.

Following the input of recent studies Jedrzejewski (2008, ); Popoff (2013), I
try to advance the mathematical theory of musical gestures using concepts from
knot theory Adams (1994) and braided monoidal categories Yetter (2001). Knots
and braids can describe several musical situations, such as conducting
gestures, and hand-crossing piano playing – let us think of Webern’s
Variations Op. 27 No. 2 (1922). The formalism of functors and n-category
theory Baez Lauda (2009) can be easily extended to embrace these applications
Yetter (2001). Moreover, knots and braids are successfully applied inside
theoretical models in biological studies, such as DNA Austin (2017). The
reason is that the molecule of DNA, the Deoxyribonucleic Acid, is formed by
“pairs of molecular strands that are bonded together” and that “spiral around
each other” (Adams, 1994, p. 181). Thus, the famous double helix can be
mathematically modelled as a pair of curves mutually twisting, easily
described as consecutive braids. Moreover, DNA molecules are then packed and
knotted several times. The entire double helix can be turned, closed in
circles and coiled; it can thus be studied as a topic of topology and knot
theory Buck (2009). Knots can also be found in other molecules Lim Jackson
(2015). Among the connections of DNA, knot theory and braids, we may consider
Lissajous knots Boogle . (1994); Vaugans Przytycki (1998), but first of all
knotted proteins Sułkowska . (2013); Vinson (2016). Musicians may already be
familiar with Lissajous for his experiments with tuning forks Quereda .
(2011), and for the patterns obtained with Chladni plates Rossing (1982).

Joining the concept of musical and visual gestures with knots and biology, I
end the article with the description of an original musical piece whose
structure is derived from the shape of DNA. DNA has already been the topic of
musical transpositions and algorithm developments Sternheimer (1996).444See
also the webpage https://www.yourdnasong.com. Here the approach is different,
but it can be interpreted as complementary. I use the same gestural generator
for both the visual shape of DNA and the musical patterns, as I will describe
in Section 6. In fact, the mathematical theory of musical gestures has both a
descriptive role (for analysis), and a prescriptive role (for composition of
new works); see origin for an example of a creative technique. In fact, these
examples show how inputs from theory and analytical approaches can be employed
to enhance musical creativity. I will discuss this more fully in the
conclusion. Sections 2.1 and 2.2 concern the mathematical definition of
gestures, hypergestures and some basic definitions of 2-category theory.
Section 3 introduces knots in the framework of the mathematical theory of
musical gestures. Section 4 deals with monoidal categories and references to
quantum mechanics. Section 5 presents a generalized approach to networks using
tools from category theory. Section 6 analyses an original musical rendition
of DNA structure. The Conclusion summarizes the results and discusses further
research.

##  2 Mathematical definitions

###  2.1 Gestures and Hypergestures

The mathematical theory of gestures has been developed to explain the
embodiment of music in performance. In fact, in order to produce sounds,
musicians have to interact physically with instruments, transforming the
symbolic indications contained in the scores, in a sequence of precise
movements. The first definition is given in jmm. Fig. 1 shows the most simple
case: a mapping from a skeleton constituted by a directed graph with only two
points and an arrow connecting them, to a body given by a curve connecting two
points in a topological space. This is completely general, and thus the scheme
can be applied to any musical (but not only musical) gesture. The skeleton is
indicated with capital Greek letters, and the curves in the topological space
$X$ with the symbol $\vec{X}$. Thus, a gesture can be notated as
$Hom(\Delta,\vec{X})$. In jmm, global and gest_sim it is used as the non-
standard notation $\Delta@\vec{X}$.

In a nutshell, the simplest gesture is a path connecting two points in the
space. The entire gesture can be seen as a single point in a space of higher
dimensions. A hypergesture is a path connecting two points, but these points
are two gestures. Each point in the hypergestural curve is a gesture, so, in
principle, it is possible to transform a gesture into another via an infinite
collection of intermediate gestures. As a more practical example, we might
think of a pianist making a movement to play a note. When the pianist moves
the arm to play notes in several points of the keyboard, he or she is
connecting gestures, and thus making a hypergesture.

The entire gesture can be considered as a point in the space of hypergestures;
a curve connecting two points in such a space is a hypergesture.

Figure 1:  A gesture is a mapping from a directed graph (left) to a system of
continuous curves in a topological space (right). The image shows the simplest
case, where we have only one arrow, and then only one curve.

In general, skeleta and bodies have more arrows. Recently, skeleta also
involving branching have been studied Mazzola Mannone (2016), and extensions
of the gesture theory to singing and conducting have been developed Mannone
(20181, 20182). In singing, there are inner movements, only partially
controlled by the performer. External movements, such as posture changes and
hand movements/face expressions, as well as visual metaphors, help the singers
to shape their vocal system correctly in order to pick the desired pitch and
timbre. Comparisons between gestures of different orchestral performers, as
well as between conductor and orchestra, are described in gest_sim, where the
formalism of 2-categories is involved.

###  2.2 2-Categories

Hypergestures can be more naturally described by means of 2-categories Mannone
(to appear 2018). In a nutshell, a 2-category is a category with morphisms
between morphisms Baez Lauda (2009). More precisely, it is possible to talk
about equivalence classes of hypergestures to have a 2-category; see Theorem
4.2 for more details.

Fig. 2 shows a smiling simplified face, Fig. 3 shows a 2-category and Fig. 4
shows a nested structure with a 2-category whose objects are other
2-categories. In Fig. 3, there are two objects, with two different morphisms
connecting them. The two objects are two categories, and the morphisms are
functors. The double arrow transforming one functor into the other is a
natural transformation. In principle, we can have n-categories and infinite-
categories Lurie (2008). 2- and n-categories are also important for
theoretical physics Baez Lauda (2009).

Let us now examine musical examples of 2-categorical thinking. One can think
of two different ways to perform the same musical sequence, one with loudness
piano, the other with forte. One can transform the first version into the
second one, via a crescendo. However, there are different ways to make a
crescendo: slower or faster, for example. One can define a natural
transformation “tempo” to transform the slower crescendo into the faster
crescendo. A “tempo” transformation would thus be an arrow between arrows. A
fast crescendo can be played by strings, or by winds; the same is true for a
slow crescendo. There can also be transitions from strings to winds in
orchestration, and one can describe this via another arrow, and so on. All
these musical structures can be easily described using the formalism of 2- and
n-categories.

Figure 2:  A smiling face; a humour depiction of a morphism connecting two
objects.

Figure 3:  An example of a 2-category.

Figure 4:  A 2-category containing two 2-categories, giving a 4-category.

##  3 Why (k)not?

Jedrzejewski Jedrzejewski () raised a question concerning the absence of any
formalism, in Mazzola’s mathematical theory of gestures, to describe gestures
that inherit some characteristics from other gestures. The problem of gestural
inheritance had already been highlighted years before the development of such
a mathematical theory Cadoz Wanderley (2000). A way to overcome this
difficulty may be the use of monoidal tensor categories Yetter (2001) and of
knot theory Adams (1994). Due to its nature, a musical gesture – let us say,
the objectivized trajectory of a point, for example the tip of the conducting
baton – may contain repetitions of the same points in the path. This fact, as
pointed out by Jedrzejewski, can be easily framed in knot theory. One can
imagine a gesture as a closed path, and we can investigate whether a gestural
curve may be reduced to the unknot, or … not. A very first and intuitive
example of this is the conducting gesture of the right hand, closed and
cyclic. Knot theory is related to braids, and braided (monoidal) categories
Baez Lauda (2009) are a powerful tool of analysis that can be used in gesture
theory Jedrzejewski (, 2006). Another comparison between knot theory and
gesture theory may come from billiard knots, i.e., the study of the trajectory
of a ball on a billiard table. It is possible to study the correspondence
between the straight lines reflected in the wall, and a curve bending on
itself described in knot theory Vaugans Przytycki (1998). One might compare
the billiard ball’s trajectory with the skeleton of the gesture, and the
smooth knot path with the body of the gesture. There is not only one knot
corresponding to a billiard ball’s path, just as there is not only one body
(of a gesture) corresponding to a skeleton (of a gesture).

Moreover, one can extend the comparison between knot theory and gesture theory
through the concept of link: for example, the trajectories of the two hands of
an orchestral conductor do not intersect, but may be linked. This would
constitute a development of analysis of conducting gestures in mathematical
terms.

In summary, a description of musical gestures closer to their physical reality
and their incredible complexity joins several approaches so far envisaged:
branching and global skeleta Mazzola Mannone (2016), braided monoidal
categories Baez Lauda (2009); Jedrzejewski (), knots Adams (1994);
Jedrzejewski (2008), 2-and n-categories Mannone (to appear 2018); Lurie
(2008), and networks Mannone (20172).555The connection between different
theories and approaches can go on and on. For example, it is possible to frame
the world-sheets Mazzola Mannone (2016) in Cobordism of complex spaces
$(Cob_{C})$. For example, the skeleton of a gesture can be branched Mazzola
Mannone (2016), but also twisted in a braid, and the body will consequently be
twisted; see Fig. 5. Also, we can have knots when the skeleton is “simple”;
see Fig. 6. Analysing conducting gestures, it seems that they are (all?)
realizations of the unknot; see Fig. 7.

In principle, we can also use concepts from physics, such as QFT (Quantum
Field Theory), fields (as defined in physics), and path integrals for each
couple of points, to describe the possible realizations of a skeleton once the
initial and final points (within the topological space) have been chosen – in
all my examples, I chose the subspace topology of knots.

Figure 5:  A skeleton that is first branched and then braided.

Figure 6:  The three-point skeleton can be mapped either in the unknot, or in
a trefoil knot (if we see the body of the gesture as a closed line).

Figure 7:  The skeleton of a quaternary conducting movement, and its
realization as a system of continuous curves. If it is seen as a unique closed
curve of knot theory, conducting gestures correspond to the unknot.

One can imagine developing a ribbon category into a conducting gesture with
modulo 3; see Fig. 8. The definition itself of gesture can be extended and
improved in knot frameworks Jedrzejewski (). Returning to the first
mathematical definition of a gesture Mazzola Andreatta (2007), and to the use
of 2- and n-categories Baez Lauda (2009); Lurie (2008) applied to music
Mannone (to appear 2018), one can further define the concept of hypergesture.

Figure 8:  The ribbon-category useful to describe the graph on the left can be
closed, giving a ternary-time conducting gesture, modulo 3.

##  4 Hypergestures, Monoidal Categories and possible openings to Quantum
Mechanics and Music

###  4.1 Hypergestures

In the space of hypergestures, points are gestures. Let $\vec{X}$ be a space
of gestures, let $\Delta,\Gamma$ be two skeleta, and let $g_{1}$ and $g_{2}$
be two gestures. In the space $\vec{Y}$, the hypergesture $l_{1}$ connects
point $g_{1}$ with point $g_{2}$, where

$g_{1}=Hom(\Delta,\vec{X}),\,\,g_{2}=Hom(\Gamma,\vec{X}).$ (1)

But gestures, and consequently also hypergestures, are mappings from a
skeleton to a system of curves in a topological space: so, given a skeleton
$\Xi$, there is $h_{1}$ that maps $\Xi$ into $\vec{Y}$. But the curve in Y is
$l_{1}$ connecting $g_{1}$ and $g_{2}$, if the skeleton $\Xi$ is an arrow
connecting two points. One can thus describe the hypergesture,
$h_{1}=Hom(\Xi,\vec{Y})$, as a mapping from $\Xi$ to $l_{1}=Hom(g_{1},g_{2})$,
i.e., $h_{1}=Hom(\Xi,(Hom(g_{1},g_{2})))$. If there are two possible paths,
$h_{1}^{\alpha}$ and $h_{1}^{\beta}$ from $\Xi$ to $\vec{Y}$, one can define a
morphism of morphisms $\alpha\beta$ connecting them. The same argument can be
applied to the gestures from $\Delta$ to $\vec{X}$ and from $\Gamma$ to
$\vec{X}$, as well as to the hypergestural curve(s) in $\vec{Y}$ space. Using
the formalism of 2-categories, we obtain the diagram shown in Fig. 9.

Figure 9:  For a given skeleton $\Xi$ corresponding to the simple structure
point-arrow-point, we sketch here a 2- (and 4-) categorical description of
morphisms of morphisms. We have a recursive structure of arrows and arrows
between arrows.

Following the intuitive concept of nested gestures given by $Hom(Hom(...))$,
one may argue for the existence of gestural modulations, the gestural
analogues of frequency modulations.

One can summarize this last result in a theorem:

###### Theorem 4.1

Gestures’ and hypergestures’ structures can be defined recursively.

###### Proof

Let $\Delta,\Gamma,\Xi$ be skeleta, and $\vec{X}$ a topological space.
Gestures $g_{1},\,g_{2}$ are defined as $g_{1}=Hom(\Delta,\vec{X})$ and
$g_{2}=Hom(\Gamma,\vec{X})$. Let $\Delta=\Gamma=\Xi=$ point-arrow-point. Thus,
both $g_{1}$ and $g_{2}$ are lines connecting two points. A line in $\vec{Y}$,
the space of hypergestures, connects $g_{1}$ and $g_{2}$; see Fig. 10. If we
generalize such a structure to N-dimension, we will still have two points
connected. A N-hypergesture with point-arrow-point skeleton connects
(N-1)-hypergestures, each (N-1)-hypergesture with point-arrow-point skeleton
connects (N-2)-hypergestures, and so on. Finally, the N-structure can be
reduced to a collection of pairs of points in the $X$ space ($\vec{X}$ being
the space of curves with points in $X$).

Figure 10:  Recursive structure of gesture and hypergesture structure. There
is a slight abuse of notation here, because with $g_{1},\,g_{2}$ we indicate
the mapping from skeleta to the body of gestures but also the body of gestures
themselves. For this reason, we indicate the latter with
$g^{\prime}_{1},\,g^{\prime}_{2}$. Let $p_{1},\,p_{2}$ be two points connected
by the first gesture, and let $p_{3},p_{4}$ be two points connected by the
second gesture. We have: $g^{\prime}_{1}=Hom(p_{1},p_{2})$ and
$g^{\prime}_{2}=Hom(p_{3},p_{4})$. Thus,
$l_{1}=Hom(g^{\prime}_{1},g^{\prime}_{2})=Hom(Hom(p_{1},p_{2}),Hom(p_{3},p_{4}))$.
A higher-order hypergesture can be written as
$Hom(Hom(...,Hom(p_{1},p_{2}),Hom(p_{3},p_{4})))$.

Exploring modulation in the sense of a gradual change, one can also refer to
tempo modulation, tonal modulation and intensity modulation. In this way, it
is possible to develop a categorical approach to musical concepts with
generalized functors and natural transformations.666In initial literature
about mathematical gesture theory Mazzola Andreatta (2007) contravariant
functors are often used.

To determine if it is possible to apply such a recursion to the 2-category
structure, one can see more details about the associativity of hypergestures
using their quotient space.777The quotient set $X/\sim$ is the set of all
possible equivalence classes in $X$. If $X$ is a topological space, then
$X/\sim$ is the quotient space.

###### Theorem 4.2

The composition of hypergestures (paths) is associative up to a path of paths.

###### Proof

To define 2-categories, we need to prove that the associativity property is
satisfied. Let $X$ be a topological space, let $\vec{X}$ (or $X^{I}$,
$I\rightarrow X$) be the space of paths within $X$, with $I=[0,1]$ the unitary
interval. Let $f,\,g$ be two paths that connect points, and let each point be
a gesture. The composition of $f,\,g$ needs a reparametrization: $f(x)$ if
$0<x<1/2$, and $g(x)$ if $1/2<x<1$. While composing it with another path $h$,
$h(gf)=(hg)f$, the parametrization is not the same, and we no longer have a
category. However, the problem seems to be solved if we define the
reparametrization $r(t,x)$ as $r(0,x)=h(gf)(x)$ and $r(1,x)=(hg)(f(x))$. The
composition of two paths is associative up to a path of paths. If we try to
compose two paths of paths, we find that it is associative up to a path of
paths of paths. We can use, instead of paths, equivalence classes of paths. In
fact, the multiplication of equivalence classes is associative. Provided that
objects (points) are gestures, they are called 0-cells. Hypergestures (paths
between points) are 1-cells, and hyper-hypergestures (paths of paths) are
2-cells. We conclude that the composition of 1-cells is associative. If we are
looking at paths and 2-cells (and not higher-order combinations), we can
redefine 2-cells, taking equivalence classes up to a 3-cell.

In summary, we can redefine hypergestures as equivalence classes of
hypergestures to formally have a 2-category. The alternative solution is not
to limit our analysis to 2-categories but to look instead at N-categories,
being aware that the associativity is verified up to a (N+1)-cell, with
$N\rightarrow\infty$.

###  4.2 Tensor products

Let us consider a gesture $g_{1}$, represented by a curve in $\vec{X}$. In the
space $\vec{Y}$ of hypergestures, $g_{1}$ is a point. To indicate the
embedding in the higher dimensional space, we can introduce tensor products
here, choosing the notation $g_{1}\in\vec{X}$, and $g_{1}\otimes 1\in\vec{Y}$.
The same argument can be applied to another gesture; let us say $g_{2}$:
$g_{2}\in\vec{X}$, $g_{2}\otimes 1\in\vec{Y}$. Curves $g_{1}$ and $g_{2}$ are
connected by a surface in $\vec{X}$, that is a line in $\vec{Y}$; let us call
it $l_{1}$.888The described surface is defined as a world-sheet in mcm15 and
global. Thus, $l_{1}\in\vec{Y}$. $l_{1}$ is the path that connects $g_{1}$ and
$g_{2}$: $l_{1}=path(g_{1},g_{2})\in\vec{Y}$. We have the following theorem:

###### Theorem 4.3

Let $g_{1}$ and $g_{2}$ be two gestures in $\vec{X}$. The hypergesture
connecting them, i.e., the path $l_{1}$ connecting them, lives in
1-dimensional higher space $\vec{Y}$, and thus the tensor product of $g_{1}$
and $g_{2}$ lives in $\vec{Y}$ too:

$g_{1}\otimes_{l_{1}}g_{2}\in\vec{Y}.$ (2)

###### Proof

The gesture $g_{1}$ is represented by a curve in $\vec{X}$, but it is a point
in $\vec{Y}$. Thus, it can be considered as a trivial curve in $\vec{Y}$, and
we write this as $g_{1}\otimes 1\in\vec{Y}$. Similarly, $g_{2}\otimes
1\in\vec{Y}$. The path $l_{1}$ connecting $g_{1},\,g_{2}$ is a curve in
$\vec{Y}$, and thus $l_{1}\in\vec{Y}$. If we connect the two trivial curves
$g_{1}\otimes 1$ and $g_{2}\otimes 1$, we get a curve in $\vec{Y}$:

$[(g_{1}\otimes 1)\rightarrow(g_{2}\otimes
1)\in\vec{Y}]\Rightarrow[(g_{1}\rightarrow g_{2})\otimes 1]\in\vec{Y}.$ (3)

Because $l_{1}=path(g_{1},g_{2})$, we have
$l_{1}=path(g_{1},g_{2})=(g_{1}\rightarrow g_{2})\otimes 1$. If we re-write
$(g_{1}\rightarrow g_{2})\otimes_{l_{1}}1$ (where the label $l_{1}$ indicates
the specific choice of path) as $g_{1}\otimes_{l_{1}}g_{2}$, we have
$g_{1}\otimes_{l_{1}}g_{2}\in\vec{Y}$. As an alternative proof, we may
consider the collection of (infinite) points from $g_{1}$ to $g_{2}$ as a
collection of trivial curves in $\vec{Y}$:
$\left\\{g_{1},g_{1n},g_{1(n+1)},...,g_{2}\right\\}\otimes 1$. They constitute
the path $l_{1}\in\vec{Y}$, and they are already in $\vec{Y}$ by definition.

This opening toward tensor products helps the introduction of braided
categories. When one starts with a couple of gestures $g_{1}$ and $g_{2}$, and
a path $p_{1}$ brings the first into the second, and a second path $p_{2}$
does the opposite, one can describe the situation as $g_{1}\otimes
g_{2}\rightarrow g_{2}\otimes g_{1}$, with the two paths being
$p_{1}:g_{1}\rightarrow g_{2}$ and $p_{2}:g_{2}\rightarrow g_{1}$, that
compose the braid and are in $\vec{Y}$.

In summary, in $\vec{X}$ the gestures $g_{1}$ and $g_{2}$ are curves
(1-dimension), and the connecting path is a surface (2-dimension) embedded in
a tridimensional space in a first approximation; in $\vec{Y}$, $g_{1}$ and
$g_{2}$ are points (0-dimension), and their connecting path is a curve
(1-dimension). Moving toward higher levels in the hierarchy, the dimension
decreases. In music, the space $\vec{X}$ can be the space of hands’ movements,
and $\vec{Y}$, the space of arms’ movements, for example. Finally, the topic
of cohomology discussed in global can be reviewed in this context of tensor
(monoidal) categories.

###  4.3 Density Matrices for Gestures?

Because quantum mechanics can also be described in terms of monoidal
categories Baez Lauda (2009), especially in reference to qubits Nielsen Chuang
(2000), one may describe the transition from some gestural states (as points
in the spacetime) to other gestural states in terms of probability matrices.
For example, some gestural curves may be more suitable than others, and then
their probability is expected to be higher. Matrices would be useful in
gestural similarity comparisons, both quantifying the gestures themselves, as
well as their final spectral results (as separated matrices).

The general idea is that a matrix can characterize a gestural configuration,
or the probability transition from a set of parameters to another. The matrix
formalism can simplify the comparisons between gestures of different artistic
codes. In the literature, there are matrices to describe transitions between
homometric states Amiot (2016).

Let us consider, as a minimal gesture, two points connected by an arrow. One
can simply think of the two points as the two energy levels of a 2-level atom,
and the arrow between them, as a process in the atom. A probability matrix
would help to describe the likelihood of such a process to happen – for
example, with the transition from up to down spin state.

Criteria to quantify memory in quantum systems have been defined Mannone .
(2013). Thus, following the analogies between quantum mechanical matrices and
musical matrices, criteria to quantify gestural similarity can be developed.
There have been attempts to adapt the formalism of non-Markovianity (the
amount of memory in quantum systems) to music Mannone Compagno (2014); such a
gestural extension would be an ideal development and improvement of that
former work.

Another reference to quantum mechanics can be given by comparisons of 2-level
atoms with the generalized intervals of Lewin’s theory Lewin (1993), with the
general definition of distance between an element and another. One more
reference is the concept itself of interval in physics, involving space and
time: it can be compared with a multi-parameter comprehensive notation of a
musical interval.

Another connection is given by Dirac notation applied to networks, as
described in kevin – where the problem of memory and neural networks is also
discussed. Category theory has been applied to simple qubit states Coecke
(2015), and one can easily envisage further research developments involving
category theory and network theory applied to music, using Dirac notation for
musical generalized intervals. I will deal with networks in Section 5.
Finally, further developments of the mathematical theory of musical gestures
should also involve the relations between hand movements, magnetic field
variation and sound production in theremin playing. The theremin is already an
object of scientific research Skeldon . (1998), and such a new study would
connect once more physics, category theory and art.

##  5 Functorial approach to Networks

The use of braids and knots can be extended to networks of musical gestures.
Networks and categories for music are already a topic of research, with the
transformation of K-nets within a categorical framework Popoff . (2015). Once
a musical network is defined, one can define another network, for example a
visual one Mannone (20172). A network of images contains a collection of
images, one transformed into another, as proposed by leonidas and tulsiani.
Musical gestures can be compared with gestures that generate visual artworks
via gestural similarity techniques Mannone (to appear 2018). What can be done
for single, isolated gestures, can be transferred to structures of gestures
such as networks Mannone (20172). For this reason, one can connect musical
networks with visual networks via functors. In the formalism of 2-categories,
one can transform a functor $F_{1}$ into a functor $F_{2}$ via a natural
transformation. One is thus defining networks of gestures, and hypergestures
of networks. Fig. 11 summarizes these concepts.

Figure 11:  A network mapped onto another. One can map musical networks onto
visuals, and vice versa. Here, the structure of each of them is arbitrary. The
network on the left shows branching, braiding and knots.

One can connect objects inside a network via morphisms, but one can also
connect networks to other networks and transform morphisms of a network into
morphisms of another network.999One can define functors between networks, that
map objects of a network onto objects of another network, and arrows of a
network onto arrows of the other network, and preserve the composition of
morphisms and identity morphisms. Fig. 12 shows the transformations
$\alpha,\,\beta,\,\gamma$ between morphisms in two different networks. This is
another way to approach the problem of gestural similarity, inside and between
networks. Thus, one can also have knots, branching and braiding in networks:
some structures that one finds in gestures and hypergestures may be replicated
on a larger scale, as happens with fractals. Here, I will not deal with
fractals, but with other interdisciplinary applications.

The scope of this section is not concerned with unfolding details, but with
envisaging some guidelines to create a general, abstract model, one that may
help modeling complex musical systems using simple ideas.

In summary, I have shown how to generalize the mathematical theory of
gestures, including elements of knot theory and monoidal braided categories.
One can find these elements inside musical networks. One can connect musical
networks with networks of visual arts, for example, and the connections
between these networks can be investigated via category theory. Future
developments may also involve gestural K-nets, and all combinations of these
elements. These ideas may be useful not only for analytical purposes, but also
for creative ones. In the following section, I discuss some possible ways to
translate into music the structure of DNA, topoisomerase and knotted proteins,
starting from a simplification of their shape, and of course using their
mathematics.

Figure 12:  Morphisms between the arrows of two separated networks. I can also
define morphisms from $\alpha$ to $\beta$, from $\beta$ to $\gamma$, and their
composition.

##  6 From DNA to Music

Topics in biology such as DNA and knotted proteins are also the object of
mathematical studies Buck (2009); Austin (2017). In fact, there are
topological problems linked with proteins and enzymes, such as the
topoisomerase. By investigating first the topological meaning of actions and
reactions of proteins, and seeing them in light of gesture theory and
generalized interval theory, it is possible to envisage a musical description
of these biological topics. This would also give a natural, philosophical
meaning and foundation to some compositional strategies, highlighting the link
between nature and music.

We can also imagine DNA as a small category, and music as another category;
the sonification of DNA is made possible via a functor translating data and
strings of bases into musical gestures. I will give later some detail about a
possible way to realize such a project.

In previous work related to DNA, the technique implied different chords for
combinations of proteins, and notes associated with protein sequences
Sternheimer (1996). Here, I want to see the double-helix structure resulting
from drawing gestures, using analogies between sound and visuals. Fig. 13
shows the technique I followed to write a piece. This piece, titled DNA, has
the following instrumentation: chimes, glockenspiel, celesta, harp, string
quartet and double bass. The bases A, C, T and G are rendered with different
intervals (A as a minor third A-C, C as a major third C-E, T as a minor third
B-D following the “T” name used in American solfeggio for B, and G as a major
third G-B) and played by the percussionists. The connections between bases are
represented by harp glissandi.

Figure 13:  A flow diagram showing the technique behind the piece DNA. The
piece contains the simultaneous musical rendition of three main elements the
image was simplified into: the general shape of the double helix, the bases
and their connection.

How can one render the double helix musically? I have two melodic lines,
mutually exchanging and intertwining. The notes, given to strings, are found
as sampling of sinusoids, then approximated, rescaled and transformed into
musical notes, all via MathematicaTM software. The exchanging points are
musical unisons. The melodic-line structures are meant to create, in the mind
of the listener, an intertwining structure, with some highlighted points (the
bases). Cases with 3-and 4-helix DNA are also taken into account later in the
piece. For this reason, the melodic lines representing the double helix were
modified: their duration values were increased (augmentation) and their
pitches transposed. The original and modified melodic lines are then played
together to represent a 3- and 4-helix DNA, creating an effect of
counterpoint. Fig. 14 shows a fragment from the score.

A more detailed description would involve knotted protein Sułkowska . (2013);
Vinson (2016) and topoisomerase Wang (2002). The topological problems involved
in the topoisomerase are supercoiling, knotting and concatenation. A possible
musical rendition of knotting (in DNA) has been described above. Concatenation
can be musically rendered with superposed sequences, time reduction and
deformation of the patterns to obtain more superposition. This may be compared
with counterpoint transformations, voice leading and voice exchange
procedures.

Fig. 15 shows a possible musical rendition of coiling. The repeated unison
notes correspond to the superposition points in the diagram. One can compare
this topic with studies about topological transformations of musical motives.
One can also see the crossing as the result of a gestural operator on the
musical pattern.

In chromosomes, there is supercoiling of coiled DNA structures. With a slight
abuse of notation, one can compare coiling and supercoiling with gesture and
hypergesture. I propose another musical rendition of coiling with a melody
returning periodically on the same note, and, for supercoiling, one can think
of several melodies with the same structure, that are repeated in repeated
dynamic patterns: from piano to mezzoforte to fortissimo to mezzoforte to
piano again, and so on. Thus, I use pitch to characterize coiling, and
loudness to characterize supercoiling; see Fig. 16. Conceptually, it is a
gesture in pitch, inside a larger gesture in loudness. One can compare all
these steps, and the use itself of elementary gestures between two points, as
an extension of Lewin’s generalized intervals Lewin (1993), in a
tridimensional space environment.

In summary, the topological meaning of action and reaction of proteins can be
the object of musical composition via gestural concepts. This opens the way to
a broader discussion on how nature can influence composition, and how artistic
strategies give us a more direct way to understand nature.

Figure 14:  An extract from DNA. The curved line indicates the double helix,
which then becomes a triple helix, and later a quadruple helix. The isolated
intervals indicate the bases A, C, G and T of DNA, as described in the text.

Figure 15:  A possibile musical rendition of coiling.

Figure 16:  A possibile musical rendition of supercoiling.

##  7 Conclusion

In this article I started from recent work on the mathematical theory of
musical gestures, taking into account new contributions and open questions,
and trying to build up the basis for a more general and coherent model. The
results of several previous studies are extended via knot theory and braided
monoidal category theory. All the old and new elements are included in the
definition of musical networks and their interaction with visual networks.

Due to the interdisciplinary nature of studies about knot theory and braided
categories, I ended the text with a description of a new musical composition
based on the DNA structure. The examples discussed stress the relationship
between musical structures, mathematics and nature, and this can potentially
open the way to new connections and mutual exchanges between art, mathematics
and biology.

The main goal is not only about showing particular, specific case studies; I
have also attempted here to extend mathematical thinking to other fields, and
to analyse compositional strategies. In fact, I aim not only to develop a
descriptive theoretical model but also a prescriptive model, in order to
create new artworks. New analytical strategies also constitute a powerful tool
to enhance musical creativity: once a method is envisaged, there are in
principle endless applications and variations. Moreover, the identification of
steps in flow diagrams allows composers to reduce the complexity of the
construction of a score, breaking it down into more simple elements. For this
reason, the approach to the composition of complex music may be easier, faster
and more rationally based. The accomplished composer, as well as the student
composer, may find within this theory helpful guidelines to build the
structure of musical pieces, and creativity can be liberated within this
flexible and creative structure.

Finally, further research might address the application of the discussed model
to more analytical cases. Computer programs to analyse networks, quantifying
the transformations at each step or branch, may be developed for this scope.
Artificial intelligence developments may benefit from a mathematical and in
particular categorical framework, as well as from artistic correlations
supported by cognitive studies.

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11institutetext: Department of Physics, University of Maryland, Baltimore
County, Baltimore, MD 21250, USA

# Boosting thermodynamic performance by bending space-time

Emily E. Ferketic  Sebastian Deffner E-mail<EMAIL_ADDRESS>
###### Abstract

Black holes are arguably the most extreme regions of the universe. Yet, they
are also utterly inaccessible to experimentation, and even just indirect
observation poses significant technical challenges. The phenomenological
approach of thermodynamics is uniquely suited to explore at least some of the
physical properties of such scenarios, and this has motivated the study of so-
called holographic engines. We show that the efficiency of an endoreversible
Brayton cycle is given by the Curzon-Ahlborn efficiency if the engine is
fueled by a 2-dimensional ideal gas; and that the efficiency is higher, if the
working medium is a (2+1)-dimensional BTZ black hole. These findings may be
relevant not only in the quest to unlock the mysteries of black holes, but
also for potential technological applications of graphene.

As pretentious as it might sound, every single time we navigate our cars
towards a gas station we are forced to make a decision of non-trivial
thermodynamic consequences – which fuel should we buy? Typically, gas stations
offer the mundane garden variety of rather similar chemical compositions,
which offer only small differences in maximal power and engine efficiency.

The situation is a lot more interesting at the forefront of technological
development, where more and more quantum fuels are being investigated [1]. In
fact, it has been recognized early on that engines operating with quantum
fuels can do things that classical engines cannot [2, 3], and that this is
perfectly legal under the laws of thermodynamics [4]. As a phenomenological
theory, thermodynamics only has very limited knowledge of the underlying
microscopic structure of the considered systems [5]. Any differences in engine
performance are thus fully derived from the fundamental relation, or a
complete set of equations of state. This fundamental relation can be obtained
empirically as, for instance, for the ideal gas, or it can be derived from
microscopic theories such as statistical mechanics or quantum field theory
[6].

Owing to the universality of the thermodynamic laws, deriving statements about
the thermodynamic characteristics becomes then almost only an exercise.
However, such an approach does permit to unveil further statements about,
e.g., the question which fuel has the best performance in what scenario. In
particular, we have recently shown that endoreversible Otto engines can
outperform classical engines, if they operate with quantum fuels [7, 8],
whereas corrections from special relativity do not support an additional boost
of thermodynamic performance [9].

Thus, the rather natural question arises whether in even more extreme
scenarios thermodynamic performance is governed by the properties of space-
time. Such a question is not quite as esoteric as it might appear, since black
hole thermodynamics [10, 11] has been a particular fruitful approach in
elucidating the properties of such mysterious regions of space in the
universe. The central quantity is the Bekenstein-Hawking entropy [12, 13, 14],
which is the amount of entropy that must be assigned to a black hole in order
for it to comply with the laws of thermodynamics as they are interpreted by
external observers. To further investigate the thermodynamic properties of
such black holes, it has now become an almost common practice to analyze so-
called _holographic engines_ [15]. These are engine cycles, in which the
cosmological constant, i.e, the energy density of the considered region of
space [16], is treated as thermodynamic variable akin to pressure. Such
research is often motivated by attempts to develop a more complete
understanding of the universe [15]. Yet, it is somewhat unlikely that our
civilization will ever reach a state in which one would use cosmological black
holes as fuel in technological applications [17]. However, special and general
relativity have found close analogies in condensed matter [18] and optical
systems [19, 20], which are clearly going to find near-term applications in
nanotechnology.

In the present analysis, we thus investigate whether singularities in space-
time can be exploited as a thermodynamic resource. More specifically, we show
that the efficiency at maximum power of an endoreversible Brayton cycle is
higher, when operated with a Bañados-Teitelboim-Zanelli (BTZ) black hole [6]
than when fueled by an ideal gas. The Brayton cycle consists of two isentropic
and two isobaric strokes, and hence it is most conveniently treated in the
entalphy representation [5]. Since also the fundamental relation of black
holes is particularly accessible in enthalpy form [21, 22], the Brayton cycle
appears as the somewhat natural choice for holographic engines. As main
results, we find that for the 2-dimensional ideal gas the efficiency at
maximum power is identical to the Curzon-Ahlborn efficiency [23], and that for
the BTZ black hole the efficiency is larger. All results are derived
analytically and in closed form.

##  1 Endoreversible Brayton cycle

We start by establishing notions and notations. The ideal _Brayton or Joule
cycle_ is consists of two isentropic and two isobaric strokes [5]. It is
closely related to the Otto cycle, in that the two isochoric strokes are
replaced by two isobaric strokes, which is more apt as a design principle for,
e.g., gas turbines. The endoreversible Brayton cycle can then be constructed
in full analogy to the endoreversible Otto cycle [7]. In particular, we have:

###  1.1 Isobaric heating (A$\rightarrow$ B)

For the isobaric strokes it is most convenient to work with the enthalpy
representation of the fundamental equation, $H=H(S,P)$. Recall that in
differential form we have [5],

$dH=T\,dS+V\,dP\quad\text{and\, thus}\quad Q_{A\rightarrow B}=H_{B}-H_{A}\,.$
(1)

Correspondingly, we can express the work done during isobaric heating as

$W_{A\rightarrow B}=\Delta E_{A\rightarrow B}-Q_{A\rightarrow
B}=P_{A}\left(V_{A}-V_{B}\right)$ (2)

which is nothing but the first law of thermodynamics, and where we used the
thermodynamic identity $H=E+PV$, and that $P_{A}=P_{B}$.

In complete analogy to the endoreversible Carnot [23] and Otto [7, 8, 9]
cycles, we now assume that the working substance is in a state of _local_
equilibrium at a temperature below the temperature of the hot reservoir. Thus
we write,

$T(0)=T_{A}\quad\mathrm{and}\quad T(\tau)=T_{B}\quad\mathrm{with}\quad
T_{A}<T_{B}\leq T_{\mathrm{hot}}\,,$ (3)

where $\tau$ is the finite duration of the stroke. In linear approximation,
the time-dependence of the temperature is then given by Fourier’s law [7],

$\frac{dT}{dt}=-\alpha\left(T(t)-T_{\mathrm{hot}}\right)$ (4)

where $\alpha$ is a constant depending on the heat conductivity and heat
capacity of the working substance.

Equation (4) can be solved exactly, and we obtain

$T_{B}-T_{\mathrm{hot}}=\left(T_{A}-T_{\mathrm{hot}}\right)\,\exp{\left(-\alpha\,\tau\right)}\,.$
(5)

###  1.2 Isentropic expansion (B $\rightarrow C)$

During the isentropic strokes the working substance is decoupled from the
environment. Therefore, the endoreversible description is identical to the
equilibrium cycle. Again from the first law of thermodynamics, $\Delta E=Q+W$,
we simply have,

$Q_{B\rightarrow C}=0\quad\mathrm{and}\quad W_{B\rightarrow C}=E_{C}-E_{B}$
(6)

where $Q_{B\rightarrow C}$ is the heat exchanged, and $W_{B\rightarrow C}$ is
the work performed during the expansion.

###  1.3 Isobaric cooling (C $\rightarrow$ D)

Heat and work during the isobaric cooling read,

$Q_{C\rightarrow D}=H_{D}-H_{C}\quad\mathrm{and}\quad W_{C\rightarrow
D}=P_{C}\left(V_{C}-V_{D}\right)\,,$ (7)

and we now have

$T(0)=T_{C}\quad\mathrm{and}\quad T(\tau)=T_{D}\quad\mathrm{with}\quad
T_{C}>T_{D}\geq T_{\mathrm{cold}}\,.$ (8)

For the sake of simplicity, we assume that the stroke time of the isobaric
cooling is identical to the stroke time of the isobaric heating, $\tau$. The
generalization to different strokes times is straight forward, but does
significantly increase the clutter in the formulas, cf Ref. [7, 8] for the
corresponding Otto cycle.

Similarly to above (4) the heat transfer is described by Fourier’s law

$\frac{dT}{dt}=-\alpha\left(T(t)-T_{\mathrm{cold}}\right)\,,$ (9)

where $\alpha$ is a constant characteristic for the cold stroke., which again
we choose to be identical to the heating stroke. From the solution of Eq. (9)
we now obtain

$T_{D}-T_{\mathrm{cold}}=\left(T_{C}-T_{\mathrm{cold}}\right)\,\exp{\left(-\alpha\,\tau\right)}\,,$
(10)

which describes the cooling from $T_{C}$ to $T_{D}$.

###  1.4 Isentropic compression (D $\rightarrow$ A)

The cycle is completed by isentropic expansion, for which we have

$Q_{D\rightarrow A}=0\ \quad\mathrm{and}\quad W_{D\rightarrow
A}=E_{D}-E_{A}\,.$ (11)

###  1.5 Brayton efficiency

As always, the efficiency of a thermodynamic cycle is defined as ratio of work
output, $W_{\mathrm{out}}=(Q_{A\rightarrow B}-Q_{C\rightarrow D})$, and (hot)
heat input, $Q_{A\rightarrow B}$. For the present Brayton cycle we immediately
find

$\eta=1+\frac{Q_{C\rightarrow D}}{Q_{A\rightarrow
B}}=1-\frac{H_{C}-H_{D}}{H_{B}-H_{A}}\,,$ (12)

which cannot be further simplified without knowledge of the fundamental
equation, $H=H(S,P)$.

##  2 The ideal gas

As a point of reference and to build intuition we begin with an endoreversible
Brayton cycle that operates with an ideal gas as working medium. The well-
known equation of state, aka the ideal gas law reads

$PV=Nk_{B}T\,,$ (13)

where $k_{B}$ is Boltzmann’s constant. Expressing the entropy as a function of
temperature [5]

$S=S_{0}+cNk_{B}\ln{\left(\frac{T}{T_{0}}\right)}+Nk_{B}\ln{\left(\frac{V}{V_{0}}\right)}$
(14)

we obtain the corresponding PV and TS-diagrams depicted in Fig. 1. Note that
$S_{0}$, $T_{0}$, $V_{0}$, and $c$ are constants that only depend on the
dimension of the system. Below we will be considering the BTZ black hole,
which lives in $(2+1)$-dimensional space. For a fair comparison, we continue
in only 2 spatial dimensions, and hence $c=1$ [5].

Figure 1:  Pressure-volume (top) and temperature-entropy (bottom) diagrams of
the Brayton cycle for an ideal gas.

From the expression of the entropy (17) and the ideal gas law (13), we can
also directly read off that

$\frac{T_{C}}{T_{B}}=\frac{T_{D}}{T_{A}}=\frac{P_{C}}{P_{A}}\equiv\kappa$ (15)

where $\kappa$ is the compressor ratio.

It is also convenient to express the enthalpy as a function of temperature. We
have

$H=E+PV=2Nk_{B}T=c_{P}T\,,$ (16)

where we introduced the isobaric heat capacity $c_{P}\equiv 2Nk_{B}$. Hence we
obtain for the efficiency

$\eta=1-\frac{T_{C}-T_{D}}{T_{B}-T_{A}}=1-\kappa\,.$ (17)

where we used Eq. (15). Thus, we are now interested in determining the value
of $\kappa$, for which the power output of the endoreversible engine is
maximum.

The power output is given by the work output divided by the total cycle time
$\tau_{\mathrm{cyc}}=\gamma\tau$,

$\mathcal{P}=\frac{W_{\mathrm{out}}}{\tau_{\mathrm{cyc}}}=\frac{c_{P}}{\gamma\tau}\left(T_{B}-T_{A}+T_{D}-T_{C}\right)\,.$
(18)

where we used again $W_{\mathrm{out}}=(Q_{A\rightarrow B}-Q_{C\rightarrow D})$
and Eq. (15). Now further exploiting Eqs. (4) and (9) it is a little tedious,
but straight forward to eliminate the temperatures $T_{A}$, $T_{B}$, $T_{C}$,
and $T_{D}$. We obtain

$\mathcal{P}=\frac{c_{P}\tanh\left(\alpha\tau/2\right)}{\gamma\tau}\,\frac{\kappa-1}{\kappa}\left(T_{\mathrm{hot}}\kappa-
T_{\mathrm{cold}}\right)\,.$ (19)

Notice that the expression for the power (19) factorizes, and thus
$\mathcal{P}$ can be maximized as a function of $\kappa$ for all values of
$\alpha$ and $\tau$. This expression for the power (19) has a rather simple
dependence on $\kappa$ with a unique maximum.

Figure 2:  Compressor ratio $\kappa_{0}$, for which $\mathcal{P}(\kappa_{0})$
is maximum as a function of $T_{\mathrm{cold}}/T_{\mathrm{hot}}$ (blue, solid
line), cf. Eq. (20). The gray dashed lines correspond to $\mathcal{P}(\kappa)$
(19) in units of $c_{P}\tanh\left(\alpha\tau/2\right)/\gamma\tau$ for
different values of $T_{\mathrm{hot}}$ and serve to demonstrate the well-
defined maximum.

As usual, the maximum of $\mathcal{P}(\kappa)$ is determined by

$\frac{\partial\mathcal{P}}{\partial\kappa}=0\quad\text{if\, and\, only\,
if}\quad\kappa=\kappa_{0}=\sqrt{\frac{T_{\mathrm{cold}}}{T_{\mathrm{hot}}}}\,.$
(20)

This is illustrated in Fig. 2.

In conclusion, we obtain that the efficiency of an endoreversible Brayton
cycle at maximal power is given for a 2-dimensional ideal gas by

$\eta=1-\sqrt{\frac{T_{\mathrm{cold}}}{T_{\mathrm{hot}}}}$ (21)

which is nothing but the Curzon-Ahlborn efficiency [23]. This result is
remarkable, since in the original treatment of Curzon and Ahlborn [23] the
analysis was restricted to endoreversible Carnot cycles.

However, we have also already shown that the endoreversible Otto engines have
the same efficiency at maximal power, as long as the internal energy is a
linear function of the temperature [8]. In the present case, also the enthalpy
is linear in temperature and thus Eq. (21) is mathematically not all that
surprising. In fact, the ideal gas is rather special, since generally the
enthalpy is not simply proportional to the internal energy, and we would thus
expect the efficiency to intimately depend on the corresponding equations of
state.

However, Eq. (21) does pose the interesting question what properties of
working mediums can be exploited to enhance thermodynamic efficiency. In the
following, we will demonstrated that “bending space-time” is such a
“thermodynamic resource”.

##  3 The BTZ black hole

Rather than continuing the analysis for a realistic model with curved space-
time, we analyze a pedagogical example of some physical significance. In fact,
the BTZ black hole, named after Bañados, Teitelboim, and Zanelli [6], has
proven to be particularly instructive [24, 25, 26, 27] due to its mathematical
simplicity. This “black hole” is a solution for (2+1)-dimensional topological
gravity with a negative cosmological constant [6]. Remarkably, many of the
physical properties of the BTZ black hole mirror the properties of more
involved models, such as the (3+1)-dimensional Schwarzschild and Kerr black
holes [24]. In particular, it supports a version of the _no hair theorem_ [25]
and it has a consistent thermodynamic description [26].

As usual in black hole thermodynamics [11] one identifies the mass with the
thermodynamic entropy, and the cosmological constant plays the role of
pressure [26]. Note that the cosmological constant is not a “constant” in the
strict sense, but rather represents the energy density of the considered space
[16]. Hence, this black hole pressure can be considered a thermodynamic
variable, that can be varied in black hole heat engines [17, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37] and so-called holographic engines [15, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50].

We will now explore whether the efficiency of an endoreversible Brayton cycle
with such a BTZ black hole as a working medium has a higher efficiency than we
found above for the ideal gas. The BTZ enthalpy is simply given by [26]

$H(S,P)=\frac{4\ell^{2}}{\pi}S^{2}P\,,$ (22)

where $\ell\equiv\hbar G_{N}$ with the gravitational constant $G_{N}$.

As before, we first determine the $PV$\- and $TS$-diagrams for the
corresponding Barton cycle. Using thermodynamic identities [5] we have for the
volume,

$V(S,P)=\frac{\partial H}{\partial
P}\bigg{|}_{S}=\frac{4\ell^{2}}{\pi}S^{2}\,.$ (23)

Hence, the $PV$-diagram becomes a simple rectangle, cf. Fig. 3 in the top
panel.

Now exploiting the Euler relation [5]

$E=TS-PV\quad\text{and}\quad H=TS$ (24)

we obtain the mechanical equation of state

$P\sqrt{V}=\frac{2\ell}{\sqrt{\pi}}T\,,$ (25)

which replaces the ideal gas law (13). The latter also permits to express the
entropy as a function of temperature, $T$, and pressure, $P$. We have

$S=\frac{\pi}{4\ell^{2}}\frac{T}{P}\,,$ (26)

and thus isobaric processes are straight lines in the $TS$-diagram, cf. Fig. 3
in the bottom panel. Thus, while deriving the enthalpy from general relativity
is a somewhat involved task [26], the resulting thermodynamic cycle could
hardly be any simpler.

Figure 3:  Pressure-volume (top) and temperature-entropy (bottom) diagrams of
the Brayton cycle for a BTZ black hole.

From Eq. (26) it is also easy to see that we again have

$\frac{T_{C}}{T_{B}}=\frac{T_{D}}{T_{A}}=\frac{P_{C}}{P_{A}}\equiv\kappa\,.$
(27)

Accordingly, the total work $W_{\mathrm{out}}=(Q_{A\rightarrow
B}-Q_{C\rightarrow D})$ can be written as

$W_{\mathrm{out}}=\frac{\pi}{4\ell^{2}}\frac{1-\kappa}{P_{A}}\left(T_{A}^{2}-T_{B}^{2}\right)\,.$
(28)

Similarly, we find

$Q_{A\rightarrow
B}=H_{B}-H_{A}=\frac{\pi}{4\ell^{2}}\frac{1}{P_{A}}\left(T_{A}^{2}-T_{B}^{2}\right)\,,$
(29)

and hence the efficiency can again be written as

$\eta=\frac{W_{\mathrm{out}}}{Q_{A\rightarrow B}}=1-\kappa\,.$ (30)

The task is now to determine the value of $\kappa$ for which the power output
is maximum.

As before, we now exploit Fourier’s law (4) and (9) to eliminate $T_{A}$ and
$T_{B}$. After straight forward manipulations we obtain

$\mathcal{P}=\frac{\pi\,\tanh\left(\alpha\tau/2\right)}{8\ell^{2}\,\gamma\,\tau\,P_{A}}\,\frac{\kappa-1}{\kappa^{2}}\left(T_{\mathrm{hot}}^{2}\kappa^{2}-T_{\mathrm{cold}}^{2}\right)\,,$
(31)

which is similar to what we found for the ideal gas (19).

Remarkably, the expression for the power (31) also factorizes, and hence
$\mathcal{P}$ can be maximized as a function of only the compressor ratio
$\kappa$. However, whereas for the ideal gas the resulting polynomial was only
quadratic (20), for the present case the situation is a little more involved.
In fact, the power has extrema for all $\kappa$ that solve

$\kappa^{3}\,T_{\mathrm{hot}}^{2}+(\kappa-2)\,T_{\mathrm{cold}}^{2}=0\,.$ (32)

This third order polynomial can be solved analytically, but only one solution
is real (the other two are complex). For this unique, real solution the power
(31) as a well-defined maximum, cf. Fig. (4).

Figure 4: Compressor ratio $\kappa_{0}$, for which $\mathcal{P}(\kappa_{0})$
is maximum as a function of $T_{\mathrm{cold}}/T_{\mathrm{hot}}$ (blue, solid
line), cf. Eq. (33). The gray dashed lines correspond to $\mathcal{P}(\kappa)$
(31) in units of
$\pi\,\tanh\left(\alpha\tau/2\right)/8\ell^{2}\,\gamma\,\tau\,P_{A}$ for
different values of $T_{\mathrm{hot}}$ and serve to demonstrate the well-
defined maximum.

This maximum is attained for $\kappa=\kappa_{0}$, where

$\kappa_{0}=\frac{-3^{1/3}\,T^{2/3}_{\mathrm{cold}}+\left(9\,T_{\mathrm{hot}}+\sqrt{3}\sqrt{T_{\mathrm{cold}}^{2}+27\,T_{\mathrm{hot}}^{2}}\right)^{2/3}}{3^{2/3}\,T_{\mathrm{hot}}T_{\mathrm{cold}}^{-2/3}\left(9\,T_{\mathrm{hot}}+\sqrt{3}\sqrt{T_{\mathrm{cold}}^{2}+27\,T_{\mathrm{hot}}^{2}}\right)^{1/3}}\,.$
(33)

However, the resulting expression is involved enough that it is not
immediately clear how it compares to our findings for the ideal gas (20).

The corresponding efficiency at maximum power is given by

$\eta=1-\kappa_{0}\,.$ (34)

This is plotted together with the efficiency for the ideal gas (21) in Fig.
(5).

Figure 5:  Efficiency of the endoreversible Brayton cycle at maximum power for
the ideal gas (21) (blue dashed, line), and the BTZ black hole (34) (red,
solid line). The dashed, purple line depicts the Carnot efficiency,
$\eta_{\mathrm{CA}}=1-T_{\mathrm{cold}}/T_{\mathrm{hot}}$, for comparison.

We observe that the efficiency for the BTZ black hole (34) is larger than the
Curzon-Ahlborn efficiency, which we found for the endoreversible Brayton cycle
operating with an ideal gas. As a consistency check, we also compare our
results with Carnot efficiency, $\eta_{\mathrm{CA}}$. The endorervesible
results are clearly below $\eta_{\mathrm{CA}}$ as they should. In conclusion,
we have found that the endoreversible Brayton cycle is indeed more efficient
at maximum power, if a BTZ black hole is used as working medium. Since the BTZ
black hole shares all important characteristics of more realistic black holes,
we may also conclude that singularities in space-time can, indeed, enhance
thermodynamic performance.

##  4 Concluding remarks

Thermodynamics was invented at the beginning of the Industrial Revolution in
an attempt to understand and optimize newly emerging technologies – steam
engines. Therefore, most of the thermodynamic language is phrased for devices
that operate with gases, and observables such as pressure are the central
thermodynamic variables. However, the mindset and approach of thermodynamics
are as universal as they could possibly be. As a phenomenological theory, it
is often the only approach that permits to investigate physical systems that
are not accessible to experimentation. This is particularly true for black
holes, with the caveat that then the interpretation of the results may become
somewhat involved. For instance, one may want to ponder what identifying a
negative cosmological constant with a variable pressure really means.

However, it also has become more and more apparent that many inaccessible
objects in the universe have close analogies in condensed matter, and in
particular in 2-dimensional materials. For instance, holograms of black holes
have been reported in graphene flakes [51], “hairy” versions of the BTZ black
hole have been suggested to exist in curved graphene sheets [52], and even
wormholes seem to exist between neighboring sheets of graphene [53]. Due to
the attractive and promising applications of 2-dimensional materials in
nanotechnology, it is thus not a surprise that engines fueled with such
materials are a subject of intense study [54, 55, 56]. Our present results
contribute to this vivid and exciting field of research, and they may pave the
way for more investigations of thermodynamic consequences of curved space-
time.

###### Acknowledgements.

E.F. gratefully acknowledges support from UMBC through an Undergraduate
Research Award. S.D. acknowledges support from the John Templeton Foundation
under Grant No. 62422.

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