# Loop Special Relativity:
Kaluza-Klein area metric as a line element for stringy events

Aharon Davidson<EMAIL_ADDRESS>www.bgu.ac.il/~davidson Physics
Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Nadav Barkai<EMAIL_ADDRESS>Physics Department, Ben-Gurion University of
the Negev, Beer-Sheva 84105, Israel

###### Abstract

Let a physical event constitute a simple loop in spacetime. This in turn calls
for a generalized loop line element (= distance2 between two neighboring
loops) capable of restoring, at the shrinking loop limit, the special
relativistic line element (= distance2 between the two neighboring center-of-
masses, respectively). Sticking at first stage to a flat Euclidean/Minkowski
background, one is led to such a preliminary loop line element, where the role
of coordinates is played by the oriented cross-sections projected by the loop
event. Such cross-sections are generically center-of-mass independent, unless
(owing to a topological term) the loop events are intrinsically wrapped around
a Kaluza-Klein like compact fifth dimension. Serendipitously, it is the
Kaluza-Klein ingredient which, on top of its traditional assignments, is shown
to govern the extension of Pythagoras theorem to loop space. Associated with
$M_{4}\otimes S_{1}$ is then a 10-dim loop spacetime metric, whose 4-dim
center-of-mass core term is supplemented by a 6-dim Maxwell-style fine
structure. The imperative inclusion of a positive (say Nambu-Goto) string
tension within the framework of Loop Special Relativity is fingerprinted by a
low periodicity breathing mode. Nash global isometric embedding is conjectured
to play a major role in the construction of Loop General Relativity.

##  I Rationale, setting and plan

A mathematical event is by definition a point in spacetime. It marks, for
example, the location $x^{\mu}$ of a classical point-like particle at some
given instant. The square of the distance between two such infinitesimally
separated point-like events $x^{\mu}$ and $x^{\mu}+dx^{\mu}$, namely

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}~{},$ (1)

constitutes the special/general relativistic line element, where
$g_{\mu\nu}(x)$ is defined as the metric tensor of the underlying flat/curved
spacetime manifold. The line element eq.(1) has long been recognized as the
most fundamental geometrical tool in the service of theoretical physics.

There is no compelling reason, however, why must a classical physical event be
inherently point-like, stripped from any non-trivial (say stringy) micro
structure. With this idea in mind, let $x^{\mu}(\sigma)$ define the so-called
simple, that is devoid of self intersections or crossings, loop event (a
refined definition of simplicity will be given later). As the
$\sigma$-parameter varies from $0$ to $2\pi$, it traces the path of a
classical closed string at some given instant. The closed structure of the
loop event, formulated by

$x^{\mu}(\sigma+2\pi)=x^{\mu}(\sigma)~{},$ (2)

gets manifested by means of the Fourier series expansion

$x^{\mu}(\sigma)=x^{\mu}_{cm}+\ell\xi^{\mu}(\sigma)=x^{\mu}_{cm}+\ell\sum_{n\neq
0}\xi^{\mu}_{n}~{}e^{in\sigma}~{}.$ (3)

with $\xi_{-n}^{\mu}=\xi_{n}^{\mu\star}$. The coefficient $\ell$ sets the loop
length scale, leaving the various $\xi^{\mu}_{n}$ dimensionless.

The immediate question now is the following: Can one consistently construct,
using a covariant geometric formalism, a tenable loop line element $\delta S$
to measure the generalized distance between two such neighboring loop
configurations $x^{\mu}(\sigma)$ and $x^{\mu}(\sigma)+\delta x^{\mu}(\sigma)$?
The theoretical obstacle is threefold: conceptual, technical, and furthermore
dynamical.

Figure 1: Two neighbouring loops in spacetime are mapped into two infinitely
closed points in loop spacetime. A tenable loop line element $\delta S$ must
then reproduce, at the shrinking loop limit $\ell\rightarrow 0$, the
special/general relativistic line element measuring the distance between the
two infinitely separated center-of-masses, respectively.

$\bullet$ On the conceptual level: While obviously dealing with non-local
(stringy) configurations, the loop line element must nonetheless be local in
loop spacetime. To make geometrical sense out of such a requirement, let each
individual loop configuration be mapped into a certain point in loop
spacetime. And the more so, two neighboring (infinitesimally deformed) loops
residing in spacetime must be mapped into two infinitesimally separated points
in loop spacetime. This evidently calls for a tenable loop spacetime metric.
The precise identification of the entities which serve as loop spacetime
coordinates, the exact nature of the loop-to-point mapping, and eventually the
loop spacetime metric, are to be presented and discussed (see Chapter 2).

$\bullet$ On the technical level: There is a crucial condition which, on self
consistency grounds, must be met at the shrinking loop size limit. As
$\ell\rightarrow 0$, the loop line element $\delta S$ must reproduce the
special/general relativistic line element eq.(1), which will then measure the
distance between the two associated infinitesimally closed center-of-masses,
respectively, that is

$\lim_{\ell\rightarrow 0}\delta S^{2}\rightarrow
g_{\mu\nu}dx_{cm}^{\mu}dx_{cm}^{\nu}~{}.$ (4)

Fulfilling this limit is a non-trivial technical task. for example, the class
of so-called area metrics [1], which our preliminary loop line element shares
some common ingredients with, fails to deliver in this respect. The missing
ingredient called to the rescue is well known, albeit in a totally different
area of physics. We refer to a compact fifth dimension a la Kaluza-Klein [2],
which plays a novel role in our discussion (see Chapter 3).

$\bullet$ On the dynamical level: The rationale for replacing point-like
events by loop events would not make any sense if the length scale $\ell$ can
grow arbitrarily large. A dynamical physical mechanism, loop shape sensitive,
which would account for the natural shrinkage of loop events, seems to be in
order. Such a mathematical service can be provided by incorporating positive
loop event tension a la Nambu-Goto string theory [3]. Once loop dynamics is
introduced, one may expects the length scale $\ell$ to eventually acquire the
Planck scale. There is also room, and eventually a necessity, as mentioned
earlier, for a non-trivial spacetime topology to enter the game. In which
case, relevant for our discussion (see Chapter 3), loop events get wrapped
around a Kaluza-Klein-like cylinder.

A local realization of the loop line element idea is expected to pave the way
for a corresponding Loop Special Relativity (LSR) theory. With this in mind,
we first consider the case of a flat spacetime which admits a Cartesian or
pseudo Cartesian metric $\eta_{ij}$, where $x^{i}$ itself, rather than just
the differential $dx^{i}$, transforms as a vector. Unfortunately, one
immediately notices that the desired special relativity limit eq.(4) is
generically not reachable. Technically, it has to do with the geometrical fact
that the loop area is generically center of mass independent. The remedy we
offer requires a non-trivial topological touch, and counter intuitively
invokes the introduction of an extra dimension. To be specific, the loop event
must be wrapped around a spatial compact Kaluza-Klein like cylinder in order
to activate the explicit entrance of the otherwise hidden center-of-mass
coordinate into the loop metric. While our approach does not seem to have a
direct connection with the standard assignments [2] of the Kaluza-Klein idea,
it resembles some familiar features. For example, starting from an underlying
$M_{4}\otimes S_{1}$ spacetime, the emerging loop line element appears to be
10-dim, spanned by four essential center-of mass coordinates, accompanied by a
6-dim Maxwell style micro structure.

In Chapter 4 we pose the question whether string dynamics is imperative. In
other words, is it necessary to accompany kinematical LSR by (say) dynamical
Nambu-Goto, and we give a few examples to support our positive answer in a
Euclidean background. It turns out, however, see Chapter 5, that the inclusion
of Nambu-Goto action within the framework of LSR has a unique fingerprint in
the Lorentzian background, namely a low periodicity breathing mode.

The Loop Special Relativity (LSR) to Loop General Relativity (LGR)
generalization is still at large. One idea in this direction would be to
invoke the Nash embedding formalism [4], later adopted by Regge-Teitelboim [5]
in its local isometric version within the framework of geodesic brane gravity.
For example, given the constraint $x^{2}+y^{2}+z^{2}=1$, curved
$ds^{2}_{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ can be trivially embedded
within flat $ds^{2}_{3}=dx^{2}+dy^{2}+dz^{2}$, so that the distance between
two loops residing on the $S_{2}$ sphere gets translated (subject to the
constraint) into the distance between the two loops in the flat $E_{3}$ host.
A simple example is provided towards the end of Chapter 3. While every
arbitrary curved space metric is Nash embeddable, the embedding procedure
itself has several drawbacks: The troubles are that (i) The minimal embeddings
are in general local, not global, (ii) The embedding is not necessarily
unique, and (iii) The number of embedding dimensions is very much case
dependent. Another idea towards constructing LGR calls for geometric gauge
invariance of the second type [6]. Regretfully, this line of research lies
beyond the scope of the present paper. The same holds or Rosen’s bi-metric
approach [7].

LSR does not seem to show, at least at this stage, any compelling connection
with Loop Quantum Gravity (LQG) [8]. Still, encouraged by the 4-dim center of
mass resurrection in the loop area formalism, hereby considered as the
fingerprint of a compact 5-th dimension (see Chapter 3), establishing such a
bridge is certainly welcome.

##  II Preliminary loop line element

Let our starting point be a loop drawn in a flat plane characterized by the
Euclidean metric

$ds^{2}=dx_{1}^{2}+dx_{2}^{2}~{}.$ (5)

The area enclosed by a loop is given by

$A=\frac{1}{2}\oint(x_{1}dx_{2}-x_{2}dx_{1})=\frac{1}{2}\int_{0}^{2\pi}r^{2}d\sigma~{}.$
(6)

Being a global quantity, the enclosed area $A$ is not sensitive to the fine
local structure of the loop configuration. The mapping from the 2-dim
$\\{x_{1},x_{2}\\}$ plane onto the 1-dim $A$-axis is thus not one-to-one. And
most importantly, to be regarded a momentary drawback for our purposes, it has
nothing to do with the location of the center of mass of the loop.

Eq.(6) can be easily generalized for the case of a loop residing within a
larger flat space (or spacetime) equipped with a Cartesian (or pseudo
Cartesian) coordinate system. In which case, the projected areas $A^{ij}$ are
given by

$A^{ij}=\frac{1}{2}\oint(x^{i}dx^{j}-x^{j}dx^{i})=\frac{1}{2}\int_{0}^{2\pi}(x^{i}x^{\prime
j}-x^{j}x^{\prime i})d\sigma~{},$ (7)

where $\displaystyle{f^{\prime}\equiv\frac{\partial f}{\partial\sigma}}$. For
an $n$-dim spacetime, there are now $\frac{1}{2}n(n-1)$ such projected areas,
one per each pair of spacetime indices. It should be emphasized that it is
only for the case of a flat rectangular space (or spacetime) that (i) $x^{i}$
transforms as a vector itself, to be contrasted with the differential $dx^{i}$
which always does, and (ii) Owing to the global nature of the associated
Lorentz transformations, the integration over a tensor is mathematically
permissible. In other words, $A^{ij}$ as given by eq.(7) constitutes a rank-2
anti-symmetric tensor in flat spacetime.

Now, for any given $\sigma$ (keeping $\sigma$ untouched), consider a loop
variation

$x^{i}(\sigma)\rightarrow x^{i}(\sigma)+\delta x^{i}(\sigma)~{}.$ (8)

Following Euler-Lagrange, and subject to the periodicity condition eq.(2), we
find

$\delta A^{ij}=\oint(\delta x^{i}dx^{j}-\delta x^{j}dx^{i})~{}.$ (9)

The increment $\delta x^{i}(\sigma)$ can be controlled by some parameter
$\tau$. In which case, we have $\delta x^{i}=\dot{x}^{i}d\tau$, where
$\displaystyle{\dot{f}\equiv\frac{\partial f}{\partial\tau}}$. In turn, the
projected areas $A^{ij}$ get shifted by

$\delta A^{ij}=d\tau\int_{0}^{2\pi}(\dot{x}^{i}x^{\prime
j}-\dot{x}^{j}x^{\prime i})d\sigma~{}.$ (10)

Note that, owing to the built-in $i\leftrightarrow j$ anti-symmetry, the same
result would have been obtained had we started from the more general
expression $\delta x^{i}=\dot{x}^{i}d\tau+x^{\prime i}d\sigma$.

While the first derivative $\displaystyle{\frac{dA^{ij}}{dt}}$, involving the
troublesome contour integration, behaves as a tensor solely in flat spacetime,
it is the second derivative

$\frac{d^{2}A^{ij}}{d\tau d\sigma}=\dot{x}^{i}x^{\prime
j}-\dot{x}^{j}x^{\prime i}$ (11)

which appears to constitute a legitimate tensor even in curved spacetime. This
may be the point to start from when attempting to eventually generalize LGR
into LGR.

The anti-symmetry of the oriented cross sections is a fundamental feature, and
does not depend on the structure of the spacetime metric. The more so,

$\epsilon^{\alpha\beta}x^{i}_{,\alpha}x^{j}_{,\beta}\quad\quad\\{\alpha,\beta\\}=\tau,\sigma$
(12)

serves as a set of world sheet scalar densities associated with the world
sheet induced metric
$\gamma_{\alpha\beta}=\eta_{ij}x^{i}_{,\alpha}x^{j}_{,\beta}$. The integrant
within eq.(10) is thus re-parametrization invariant. We note in passing that
performing a proper re-parametrization transformation

$\Bigl{\\{}\begin{array}[]{lcc}\tau\rightarrow\tilde{\tau}(\tau,\sigma)=T(\tau)&&\vspace{3pt}\hfil\\\
\sigma\rightarrow\tilde{\sigma}(\tau,\sigma)=\sigma+\Sigma(\tau)&&\end{array}~{},$
(13)

prior to the integration, a transformation which fully respects the
$\Delta\tilde{\sigma}=\Delta\sigma=2\pi$, loop event periodicity, is
equivalent to posteriori gauge fixing $\tau\rightarrow T(\tau)$.

Once curvilinear coordinates are being used, while the integration over a
tensor is apparently forbidden, there is a simple way out. The trick is to
invoke the Vierbein formalism, where the curvilinear flat spacetime metric can
be written in the form

$g_{\mu\nu}=\eta_{ij}V^{i}_{,\mu}(x)V^{j}_{,\nu}(x)~{},$ (14)

notably involving total derivative Vierbeins. Eq.(10) can then be easily
generalized into

$\displaystyle\delta
A^{ij}=\displaystyle{d\tau\int_{0}^{2\pi}\epsilon^{\alpha\beta}V^{i}_{,\mu}V^{j}_{,\nu}x^{\mu}_{,\alpha}x^{\nu}_{,\beta}d\sigma}$
(15)
$\displaystyle=\displaystyle{d\tau\int_{0}^{2\pi}\epsilon^{\alpha\beta}V^{i}_{,\alpha}V^{j}_{,\beta}d\sigma}~{}$
$\displaystyle~{},$

and one is back to the original case, only with the rectangular coordinates
$V^{i}(x)$ replacing the curvilinear coordinates $x^{\mu}$.

The situation is totally different, however, for a curved spacetime
characterized by a non-vanishing Riemann tensor). Having in mind eq.(15), one
may prematurely expect that associated with a curved spacetime metric

$g_{\mu\nu}(x)=\eta_{ij}V^{i}_{\mu}(x)V^{j}_{\nu}(x)~{},$ (16)

where the Vierbeins have now matured into full gauge fields, are projected
loop area increments of the form

$\delta
A^{ij}=d\tau\int_{0}^{2\pi}\epsilon^{\alpha\beta}V^{i}_{\mu}V^{j}_{\nu}x^{\mu}_{,\alpha}x^{\nu}_{,\beta}d\sigma~{}.$
(17)

Unfortunately, with the exception of the 2-dim case, this formula does not
make sense mathematically. The problem is that it is impossible to erect any
single coordinate system that is locally inertial everywhere, unless the
spacetime continuum is flat. In other words, integration over a Lorentz tensor
does not make then any sense. It is only in 2-dim, owing to the corresponding
rank-2 Levi-Civita symbol $\epsilon_{ij}$, that $\delta
A^{12}=\frac{1}{2}\epsilon_{ij}\delta A^{ij}$, as defined by eq.(17), happens
to be a Lorentz scalar. From this point on, on both pedagogical and simplicity
grounds, while still lacking a proper formula for $\delta A^{\mu\nu}$ in a
curved background, and without invoking either (i) gauge invariance of the
second type, and/or Regge-Teitelboim like embedding technique as potential
remedies, we return to the comfortable choice of a Cartesian (or pseudo
Cartesian) spacetime.

Associated with each loop residing in an $n$-dim spacetime there is a
corresponding point in the so-called loop spacetime, where the
$\frac{1}{2}n(n-1)$ independent projected areas $A^{ij}$ play the role of
coordinates. Two such loops are then mapped into two points in loop spacetime,
and in principle, provided the corresponding loop metric is specified, a
geodesic trajectory can be drawn. If the two points are infinitesimally
closed, we pay tribute to the anti-symmetric structure of $A^{ij}$, imitate
the general relativistic structure of the Maxwell kinetic term, and
accordingly define the scalar loop line element

$\boxed{\delta S^{2}=\frac{\eta_{ik}\eta_{jl}}{16\pi^{2}\ell^{2}}\delta
A^{ij}\delta A^{kl}}~{},$ (18)

properly normalized for future assignments. Note that for the special case
$n=4$, there exists the option of supplementing eq.(18), or even replace it,
by the dual term proportional to $\epsilon_{ijkl}\delta A^{ij}\delta A^{kl}$.
Eq.(18) falls into the category of area metrices [1]. Note in passing that
areas also play an important role in Regge [10] calculus, and oriented areas
enclosed by string constitute a vital part of Clifford space metric [11].

Figure 2: Associated with every simple loop event (black) residing in an
$n$-dim flat space or spacetime, there are $\frac{1}{2}n(n-1)$ simple
projected areas (red, blue, green,…), thereby mapping a loop event into a
point $\\{A_{xy},A_{yz},A_{yz},...\\}$ in loop space.

Unfortunately, the special relativistic limit eq.(4) is still unattainable at
this stage. Substituting the Fourier expansion eq.(3) into eq.(10), we obtain

$\delta A^{ij}=\ell^{2}\int_{0}^{2\pi}(\dot{\xi}^{i}\xi^{\prime
j}-\dot{\xi}^{j}\xi^{\prime i})d\tau d\sigma~{},$ (19)

and immediately notice that the center of mass $x^{i}_{cm}(\tau)$ and its
$\tau$-derivative $\dot{x}^{i}_{cm}(\tau)$ do not enter the game. This poses a
major drawback, as all point-like ($=$ shrinking loop) events become
practically indistinguishable, and pile at the origin. Until the missing
ingredient is found, and the center of mass comes out of hiding, eq.(18) has
to be regarded incomplete.

##  III Kaluza-Klein to the rescue

###  III.1 Center of Mass resurrection

Counter intuitively, the remedy comes from topology. The idea is to supplement
spacetime by an extra Kaluza-Klein like closed dimension. Nothing to do though
with the original Kaluza-Klein idea. From some reason soon to be clarified,
this extra dimension must be space-like. On historical grounds we generically
refer to such an extra dimension as $x^{5}$, and consider loop events wrapped
around this cylindrical fifth dimension. The previously introduced spacetime
coordinates $x^{\mu}$ are now accompanied by a new $x^{5}$, subject to the
periodicity condition

$\Delta x^{5}=2\pi R~{}.$ (20)

The game changer element is then the presence of the topological term
$R\sigma$ in the modified Fourier expansion. To be contrasted with eq.(3), we
now have

$x^{5}(\sigma)=x^{5}_{cm}+R\sigma+\ell\xi^{5}(\sigma)~{}.$ (21)

where as usual $\xi^{5}(\sigma)=\displaystyle{\sum_{n\neq
0}}\xi^{5}_{n}~{}e^{in\sigma}$.

Owing to the slight yet significant modification in the corresponding partial
derivative expansions, that is

$\Bigl{\\{}\begin{array}[]{lcc}\dot{x}^{5}=\dot{x}^{5}_{cm}+\ell\dot{\xi}^{5}&&\vspace{4
pt}\hfil\\\ x^{\prime 5}=R+\ell\xi^{\prime 5}&&\end{array}$ (22)

in comparison with the former

$\Bigl{\\{}\begin{array}[]{lcc}\dot{x}^{i}=\dot{x}^{i}_{cm}+\ell\dot{\xi}^{i}&&\vspace{4
pt}\hfil\\\ x^{\prime i}=\ell\xi^{\prime i}&&\end{array}~{},$ (23)

the projected areas split into two distinguishable categories. While, as
before, the center of mass derivative $\dot{x}^{i}_{cm}$ is still absent from

$\frac{\delta
A^{ij}}{2\pi}=\left(\frac{\ell^{2}}{2\pi}\int_{0}^{2\pi}(\dot{\xi}^{i}\xi^{\prime
j}-\dot{\xi}^{j}\xi^{\prime i})d\sigma\right)d\tau~{},$ (24)

it makes its resurrection via

$\frac{\delta
A^{i5}}{2\pi}=\left(R\dot{x}^{i}_{cm}+\frac{\ell^{2}}{2\pi}\int_{0}^{2\pi}(\dot{\xi}^{i}\xi^{\prime
5}-\dot{\xi}^{5}\xi^{\prime i})d\sigma\right)d\tau~{}.$ (25)

Notably, as expected, and in contrast with the presence of $\dot{x}^{i}_{cm}$,
irrelevant $\dot{x}^{5}_{cm}$ stays completely out of the game.

Equipped with eqs.(24,25), one can now reconstruct the loop line element
eq.(18). In its modified version, reflecting the underlying $M_{n}\otimes
S_{1}$ spacetime, it takes the final form

$\boxed{\delta S^{2}=\frac{1}{8\pi^{2}R^{2}}\left(\eta_{ij}\eta_{55}\delta
A^{i5}\delta A^{j5}+\frac{1}{2}\eta_{ik}\eta_{jl}\delta A^{ij}\delta
A^{kl}\right)}$ (26)

Here, by requiring $\eta_{ij}\eta_{55}=\eta_{ij}$ on consistency grounds, in
order to account for the mandatory $\eta_{ij}x_{cm}^{i}x_{cm}^{j}$ term, we
are finally led to the tenable signature choice

$\eta_{55}=+1~{},$ (27)

a posteriori justifying the space-like nature of $x^{5}$ (nothing to do with
the original Kaluza-Klein theory). On the other hand, given the underlying
$\eta_{ij}$ metric, the coefficients $\eta_{ik}\eta_{jl}$ are uniquely fixed.

###  III.2 Pythagoras theorem in loop space

The special case we now discuss in detail is the simplest, yet the most
fundamental case in hands. And as such, has been moved from the Appendix level
to the main body of our paper. To be specific, let us calculate the distance
between two arbitrary loops residing in a flat 2-dim plane

$ds^{2}=dx^{2}+dy^{2}~{}.$ (28)

Following our prescription, our first step is to add the Kaluza-Klein
ingredient to the game, and deal with a larger yet flat 3-dim space governed
by

$d\bar{s}^{2}=ds^{2}+dx_{5}^{2}~{},\quad\Delta x_{5}=2\pi R~{}.$ (29)

On simplicity and pedagogical grounds, we consider the evolution (parametrized
by $\lambda$) of a circular loop. The loop is of radius $r(\lambda)$, is
centered at $\\{x_{cm}(\lambda),y_{cm}(\lambda)\\}$

$\begin{array}[]{lll}&x(\lambda,\sigma)=x_{cm}(\lambda)+r(\lambda)\cos\sigma&\vspace{3pt}\\\
&y(\lambda,\sigma)=y_{cm}(\lambda)+r(\lambda)\sin\sigma&\vspace{3pt}\\\
&x_{5}(\lambda,\sigma)=x_{5cm}(\lambda)+R\sigma&\end{array}~{},$ (30)

and is furthermore wrapped, precisely once at this stage, around the Kaluza-
Klein cylinder. The three projected areas are given explicitly by

$\displaystyle\frac{dA^{xy}}{d\lambda}=2\pi r\frac{dr}{d\lambda}~{},$ (31)
$\displaystyle\frac{dA^{x5}}{d\lambda}=2\pi R\frac{dx_{cm}}{d\lambda}~{},$
(32) $\displaystyle\frac{dA^{y5}}{d\lambda}=2\pi
R\frac{dy_{cm}}{d\lambda}~{}.$ (33)

Substituting the latter into eq.(26), we immediately find out that

$dS^{2}=dx_{cm}^{2}+dy_{cm}^{2}+\frac{r^{2}}{R^{2}}dr^{2}~{},$ (34)

and recalling the loop area $A^{xy}=\pi r^{2}$, brings us to the final
Pythagoras formula

$\boxed{dS^{2}=ds_{cm}^{2}+\left(\frac{dA}{2\pi R}\right)^{2}}$ (35)

We have thus reached a flat 3-dim loop space, with
$\displaystyle{\frac{A}{2\pi R}}$ serving as a third dimension. It is an open
dimension, to be contrasted with the compact nature of $x^{5}$, thus mimicking
the role of $z_{cm}$. Altogether, the geodesic distance $L$ between two loops,
of areas $A_{1}$ and $A_{2}$, centered at $\\{x_{1},y_{1}\\}$ and
$\\{x_{2},y_{2}\\}$ respectively, is given by

$L^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+\left(\frac{A_{2}-A_{1}}{2\pi
R}\right)^{2}~{}.$ (36)

One may now wonder under what circumstances is the (area difference)2 term
negligible? The trivial answer would be when $A_{1}=A_{2}$, the case of a
fixed loop area (the rigid loop case obviously included). The generic case
calls for $A^{xy}\sim\ell^{2}$ to pick up the only length scale floating
around. In general, however, as far as the geometry is concerned, the area
difference $A_{1}-A_{2}$ stays at this stage unbounded.

Figure 3: Generalized Pythagoras theorem: Let $L$ denote the geodesic distance
between two loops, demonstrated here for circles of opposite sides, so that
$A_{2}-A_{1}=\pi(r_{2}^{2}-(-r_{1}^{2}))$. While the shape of the loop stays
unconstrained in the absence of loop dynamics, the center of mass location, as
well as the loop area, evolve linearly.

###  III.3 Light cone in loop spacetime

Had one started from a Lorentzian spacetime $M_{2}$ enriched by a compact
($\Delta x_{5}=2\pi R$) fifth dimension

$d\bar{s}^{2}=-dt^{2}+dz^{2}+dx_{5}^{2}~{},$ (37)

and analogously considered the simple $\tau$-evolution

$\begin{array}[]{lll}&t(\tau,\sigma)=t_{cm}(\tau)+a(\tau)\cos\sigma&\vspace{3pt}\\\
&z(\tau,\sigma)=z_{cm}(\tau)+b(\tau)\sin\sigma&\vspace{3pt}\\\
&x_{5}(\tau,\sigma)=x_{5cm}(\tau)+R\sigma&\end{array}~{},$ (38)

he would have once again ended up with the fundamental eq.(35), only with
$A^{tz}=\pi ab$. Note that $A^{ij}$ is not sensitive to the time/spacelike
nature of the $ij$-indices.

Altogether, as long as loop dynamics is not switched on, the kinematical
evolution of a loop event, residing in $M_{2}$ and being wrapped around
$S_{1}$, is described by means of the geodesic evolution of a point-like event
in flat $M_{3}$. While the Poincare-like symmetry associated with the loop
spacetime metric

$dS^{2}=-dt_{cm}^{2}+dz_{cm}^{2}-\left(\frac{dA}{2\pi R}\right)^{2}$ (39)

is well established, it contains some novel elements such as a
$\displaystyle{\left\\{z_{cm},A\right\\}}$ boost and a
$\displaystyle{\left\\{t_{cm},A\right\\}}$ rotation. The timelike nature of
the $A^{tz}$-dimension stems from the opposite time/spacelike assignments of
$t$ and $z$, as expressed via $\eta_{tt}\eta_{zz}=-1$, in accordance with
$\eta_{tt}\eta_{55}=-1$. The accompanying energy/momentum relation takes then
the form

$m^{2}=E_{cm}^{2}+E_{A}^{2}-p_{cm}^{2}~{},$ (40)

where the total energy has now two independent sources
$E^{2}=E_{cm}^{2}+E_{A}^{2}$. The second energy operator

$E_{A}=-2\pi iR\frac{\partial}{\partial A}$ (41)

is identified as the generator of loop area expansions in the
$\\{t,z\\}$-plane. As a trivial consistency check one may verify that a loop
of a constant area, not necessarily of a constant shape, would return the
familiar formulae describing a point particle moving in $1+1$ dimensions.

###  III.4 From Special Relativity to Loop Special Relativity

The loop spacetime line element eq.(26) and the subsequent special cases
discussed bring us one step closer to our second goal, which is formulating a
Loop Special Relativity (LSR) theory capable of supporting the Special
Relativity (SR) limit

$\delta S^{2}=\eta_{ij}dx^{i}_{cm}dx^{j}_{cm}+O[\ell^{2}]~{}.$ (42)

The idea of trapping the loop around the compact fifth dimension can be
conveniently realized by executing the partial gauge choice
($\sigma$-redefinition)

$x^{5}(\tau,\sigma)=x_{cm}^{5}(\tau)+R\sigma~{}.$ (43)

This way, starting from the most general Fourier expansions for

$x^{i}(\tau,\sigma)=x^{i}_{cm}(\tau)+\ell\sum_{n\neq
0}\xi^{i}_{n}(\tau)e^{in\sigma}~{},$ (44)

we can significantly simplify the explicit expressions for the associated
infinitesimal projected areas. Absorbing for simplicity the $\frac{\ell}{R}$
ratio within $\xi^{i}$, we find

$\displaystyle\displaystyle{\frac{1}{2\pi
R}\frac{dA^{i5}}{d\tau}}=\frac{d}{d\tau}x^{i}_{cm}~{},$ (45)
$\displaystyle\displaystyle{\frac{1}{2\pi
R}\frac{dA^{ij}}{d\tau}}=2R\frac{d}{d\tau}\sum_{n>0}{\Im}(n~{}\xi^{i}_{n}\xi^{j\star}_{n})~{},$
(46)

which can finally be substituted into eq.(26). Altogether, the LSR line
element takes the elegant form

$\boxed{ds^{2}_{LSR}=ds^{2}_{SR}-R^{2}(ds_{E}^{2}-ds_{B}^{2})}$ (47)

Our anchor, the $3+1$ dimensional center of mass SR line element, is hereby
supplemented by a $3+3$ dimensional Maxwell-like structure. It should be noted
that (i) All $O[\ell]$ mixed pieces have been dropped away by the specific
gauge choice eq.(43), and consequently that (ii) $x_{cm}^{5}(\tau)$ appears
irrelevant as it has no realization in our formalism.

Naturally, attention is devoted to the light cone structure. The special
relativistic $ds^{2}_{SR}=0$ is replaced by its loop special relativistic
$ds^{2}_{LSR}=0$. Thus, from the point of view of an observer, unfamiliar with
(or just insensitive to) LSR, as is evident from

$ds^{2}_{LSR}=0\quad\Longrightarrow\quad
ds^{2}_{SR}=R^{2}(ds_{E}^{2}-ds_{B}^{2})~{},$ (48)

the center of mass light cone acquires a 6-dim (hopefully Planck scale)
Maxwell-like fine structure. Three of the extra dimensions are ’electric’
(timelike) and the other three are ’magnetic’ (spacelike). They are supposed
to fade away as $R\rightarrow 0$. This seems to constitute the main physics
fingerprint of LSR.

###  III.5 Towards Loop General Relativity: Nash global isometric Embedding

Attempting to go beyond flatness, one would now like to calculate the geodesic
distance $L$ between two loops which reside in a curved background, say two
parallel loops living on a 2-dim sphere of constant radius $\ell$. The crucial
point is that the 2-sphere can be globally embedded within a flat 3-dim space,
that is

$\begin{array}[]{lll}&x(\theta,\sigma)=\ell\sin\theta\cos\sigma&\vspace{3pt}\\\
&y(\theta,\sigma)=\ell\sin\theta\sin\sigma&\vspace{3pt}\\\
&z(\theta,\sigma)=\ell\cos\theta&\end{array}~{},$ (49)

subject to the global constraint $x^{2}+y^{2}+z^{2}=\ell^{2}$. The only non-
vanishing projected area increment is then

$\delta A^{xy}=2\pi\ell^{2}\sin\theta\cos\theta d\theta~{}.$ (50)

Following our prescription, one invokes the Kaluza-Klein topological term

$x_{5}(\theta,\sigma)=R\sigma~{},$ (51)

and consequently finds

$L=\frac{\ell}{R}\int_{\theta_{1}}^{\theta_{2}}\sqrt{\ell^{2}\cos^{2}\theta+R^{2}}\sin\theta
d\theta~{}.$ (52)

However, from a variety of reasons outlined towards the end of the
introduction, this is not necessarily a recipe for LGR. Still, one cannot rule
out the possibility, make it a conjecture, that Nash local/global embedding
may eventually play some role in constructing LGR.

##  IV Is loop dynamics imperative?

###  IV.1 Introducing Planck scale

So far, we have demonstrated how to map a loop event, residing in
$M_{n}\otimes S_{1}$ Kaluza-Klein spacetime, into a point-like event residing
in a larger $\frac{1}{2}n(n+1)$-dim flat loop spacetime whose center of mass
sub-metric is supplemented by a Maxwell style higher dimensional companion. We
are also aware of the topological advantage that, being wrapped around the
fifth dimension, the loop event cannot really shrink to a point-like event.
However, as it stands, while the mandatory SR-limit has been non-trivially
recovered, the overall picture is still not fully satisfactory. The reasons
are fourfold:

$\bullet$ Arbitrary loop shape: The loop-to-point mapping is unfortunately not
one-to-one, and only captures the projected areas involved. Eq.(26) is
incapable of telling one loop configuration from the other as long as their
projected areas are the same. The challenge would be to convert such a
residual degree of freedom into a physically tamed shape uncertainty.

$\bullet$ Unbounded loop size: Whereas the fifth dimension invoked is compact
by definition, characterized by its tiny Kaluza-Klein radius $R$, the loop
projected areas $A^{ij}$ can in principle take arbitrarily large values. In
turn, unless the projected areas are themselves of order $\mathcal{O}[R^{2}]$,
the loop line element $\delta S^{2}$ will be dominated by the Maxwell-like
term rather than by the center-of-mass term.

$\bullet$ Multiple timelike dimensions: Once $\eta_{ij}$ is specified, and
$x^{5}$ is assigned spacelike, the signatures of Maxwell-like terms are not a
matter of choice. To be specific, a single timelike spacetime coordinate $t$
gives rise to $(n-1)$ timelike loop spacetime coordinates $A^{it}$ (and of
course to $\frac{1}{2}(n-1)(n-2)$ spacelike loop spacetime coordinates
$A^{ij}$). This opens the door for problematic mathematical as well as
philosophical cause-and-effect issues, arguing that the behavior of physical
systems could not be predicted reliably. Saying this, note that several
theories, F-theory and 2T-theory [9] among them, do host multiple timelike
(and necessarily accompanied by spacelike) dimensions.

$\bullet$ Multiple KK wrappings: Recalling that $\pi_{1}(S_{1})={\cal Z}$, the
closed loop event can carry an arbitrary integer winding number $w=n\neq 0$.
We can of course always generalize our simple loop assumption, that is allow
no self intersections of the loop’s oriented projections (which surround the
projected areas $A^{ij}$), but it is not natural to do so in the presence of a
non-trivial topology. After all, a winding number $w=n\neq\pm 1$ loop gives
rise to $(|n|-1)$ self intersections on the KK cylinder. A presumably quantum
field theoretical self interaction mechanism, capable of decomposing a $w=n$
loop into $n$ separated $w=1$ simple loops, is certainly in order, but
unfortunately stays beyond the scope of the present paper.

Appreciating the above potential drawbacks, one is left with two apparently
contradictable options:

Option I: The exact configuration of the loop event does not show up at the
classical level at all. Had $R$ vanished as $\hbar\rightarrow 0$, such an
option would have been encouraged by quantum mechanics, with every loop
configuration carrying its own amplitude. In fact, this option favors the
entrance of the Planck scale into the game via

$R\sim\ell_{P}=\sqrt{\frac{\hbar G}{c^{3}}}~{},$ (53)

implying that $R$ should vanish at the $G\rightarrow 0$ limit as well and of
course as $c\rightarrow\infty$.

Option II: The exact configuration of the loop event does acquire a physical
meaning already at the classical level. The introduction of dynamics via some
string theoretical action seems then unavoidable, with the main goal being to
encourage loop events to shrink (positive string tension). Such a dynamical
approach would furthermore account for the assumption that $A^{ij}\sim R^{2}$
and that the perimeter $2\pi R$ of the fifth dimension share the one and the
same length scale.

While the second option may seem easier to utilize on technical grounds, it is
the first option which actually catches our imagination. Thus, we choose to
adopt the Planck length scale eq.(53), but without giving up the idea of loop
event self dynamics. In other words, we attempt to make a compromise, and
choose

Option I+II: Translated into the Lagrangian formalism, we propose

$\boxed{{\cal I}={\cal I}_{LSR}+\Lambda{\cal I}_{NG}}$ (54)

where ${\cal I}_{LSR}=\displaystyle{\int dS}$ is the loop spacetime geodesic
action, and ${\cal I}_{NG}=\displaystyle{\int\sqrt{-g_{2}}~{}d\tau d\sigma}$
stands for the familiar string theoretical Nambu-Goto action (or alternatively
for its Polyakov variant). The first ingredient contains the SR-limit, but is
insensitive to the detailed structure of the loop. The second ingredient does
not have an SR-limit, but forcefully governs the inner loop dynamics.
$\Lambda$ is a dimensionless coefficient, which may be eventually elevated in
some stage to the level of of a Lagrange multiplier. To show our point we
discuss now in some detail the simplest pedagogical case of sufficient
complexity, namely a 2-dim soap world-sheet embedded within a 3-dim Euclidean
space.

###  IV.2 No-go Nambu-Goto

We now prove, as was claimed before, that the Nambu-Goto action (as well as
its Polyakov vartiant) cannot consistently serve as a measure of the
’distance’ between two loops. To stand on familiar geometrical grounds, we
choose to make our point using the simplest pedagogical case of a 2-dim soap
worldsheet embedded within a 3-dim Euclidean space

$\left.\begin{array}[]{lll}x^{1}(z,\sigma)=r(z)\cos\sigma&&\\\
x^{2}(z,\sigma)=r(z)\sin\sigma&&\\\ x^{3}(z,\sigma)=z&&\end{array}\right.~{},$
(55)

connecting two circles of equal radii

$r(-h)=r(h)=\ell~{}.$ (56)

The corresponding Nambu-Goto action reads

$I_{NG}=2\pi\int_{-h}^{h}r(z)\sqrt{1+r^{\prime}(z)^{2}}~{}dz~{}.$ (57)

The Euler-Lagrange equation and the corresponding analytic solution are given
by

$rr^{\prime\prime}-r^{\prime 2}-1=0\quad\Longrightarrow\quad r(z)=\frac{\cosh
kz}{\cosh kh}\ell~{},$ (58)

subject to the symmetric boundary condition

$\cosh kh=k\ell~{}.$ (59)

It can be numerically verified that there is no solution for
$\displaystyle{\frac{\ell}{h}<1.508}$, with the critical value marking a phase
transition, a phenomenon which can be experimentally demonstrated with soap
films. The critical point is associated with the extra mathematical condition

$\sinh kh=\frac{\ell}{h}~{}.$ (60)

In other words, the Nambu-Goto action cannot be interpreted as ’distance’
between loops separated too far (beyond criticality) apart.

###  IV.3 Kaluza-Klein modified Nambu-Goto

We now show that the trick of adding a compact 5-th dimension and wrapping the
loop event once around it, that is

$\left.\begin{array}[]{lll}x^{1}(z,\sigma)=r(z)\cos\sigma&&\\\
x^{2}(z,\sigma)=r(z)\sin\sigma&&\\\ x^{3}(z,\sigma)=z&&\\\
x^{5}(z,\sigma)=R\sigma&&\end{array}\right.~{},$ (61)

while having some advantage over the plain NG case, is still quite
problematic. The former Nambu-Goto action eq.(57) gets now generalized into

$I_{NG+KK}=2\pi\int_{-h}^{h}\sqrt{(r(z)^{2}+R^{2})(1+r^{\prime}(z)^{2})}~{}dz~{},$
(62)

and subsequently, the modified Euler-Lagrange equation takes the form

$(r^{2}+R^{2})r^{\prime\prime}-r(1+r^{\prime 2})=0~{}.$ (63)

The analytic solution looks very much like the one given by eq.(58), save for
the modified boundary condition

$\cosh kh=\frac{k\ell}{\sqrt{1-k^{2}R^{2}}}~{},$ (64)

replacing the former eq.(59).

A critical $h_{c}=2.790R$ plays now a major role. For $h<h_{c}$, there is
exactly one real solution for every $a$. For $h>h_{c}$, there can still be a
single real solution provided $\ell<\ell_{min}$ or $\ell>\ell_{max}$. However,
for $h>h_{c}$ and $\ell_{min}<\ell<\ell_{max}$, where $\ell_{min,max}$ are the
local extrema values of eq.(64), there appear to be three real solutions. Not
only is uniqueness lost, but furthermore, one unexpectedly faces a hysteresis
phenomenon.

There are, however, some encouraging news to report on. This has to do with
the large loop separation region $h\gg R$, a region which plain ${\cal
I}_{NG}$ eq.(57) simply could not reach. In which case, we derive the long
distance behavior

$\frac{\cal I}{2\pi R}\simeq 2h+\frac{~{}\ell^{2}}{R}\tanh\frac{h}{R}+...~{},$
(65)

showing a small $\displaystyle{{\cal O}(\frac{~{}\ell^{2}}{R})}$ correction to
the classical large value of $2h$.

###  IV.4 Combining LSR with Nambu-Goto

Sticking to eqs.(61), we now substitute the various $x^{i}(z,\sigma)$ into the
action eq.(54) to arrive at

${\cal I}_{\Lambda}=2\pi\int_{-h}^{h}\left(\sqrt{R^{2}+r^{2}r^{\prime
2}}+\Lambda\sqrt{(R^{2}+r^{2})(1+r^{\prime 2})}\right)dz~{}.$ (66)

Here, $\Lambda$ is just a dimensionless coefficient, not a Lagrange
multiplier. $\Lambda\rightarrow\infty$ marks the NG-limit, whereas
$\Lambda\rightarrow 0$ takes us back to the LSR territory. At the first
glance, the associated Euler-Lagrange equation looks quite cumbersome, but
once recasted into the form $r^{\prime\prime}=f(r,r^{\prime})$, with $R$ and
$h$ serving as parameters, it can be numerically handled straight forwardly.

The numerical lesson is twofold:

(i) $\Lambda\geq 0$ on $z$-evolutionary grounds, as otherwise we are
necessarily driven into an undesirable $r=0$ collapse.

(ii) $\Lambda\ll 1$ on self-consistency grounds, as otherwise we loose track
of LSR, which becomes merely a perturbation on the classical string action.
For such a small mixing parameter $\Lambda$, we obtain

$\frac{{\cal I}_{\Lambda}}{2\pi
R}=2h\left(1+\sqrt{1+\frac{\ell^{2}}{R^{2}}}\Lambda\right)+...~{}\quad\mathsf{for}~{}\Lambda\ll
1~{},$ (67)

to be fully contrasted with

$\frac{{\cal I}_{\Lambda}}{2\pi
R}=2h\Lambda\xi+...~{}\quad\mathsf{for}~{}\Lambda\gg 1$ (68)

where $\xi$ is some geometrical ${\cal O}[1]$ factor.

Figure 4: From LSR to NG+KK: Soap branes connecting two identical circular
rings, residing in a 3-dim space and separated $2h$ apart. As $\Lambda$ grows,
while holding the KK radius $R$ fixed, the LSR cylinder transforms into an
NG+KK narrow waist candlestick (owing its stability to $R\neq 0$).

##  V Low frequency breathing mode

In a Lorentzian $M_{4}\otimes S_{1}$ background, with the following loop
assignments

$\left.\begin{array}[]{lll}x^{0}(\tau,\sigma)=\tau&&\\\
x^{1}(\tau,\sigma)=r(\tau)\cos\sigma&&\\\
x^{2}(\tau,\sigma)=r(\tau)\sin\sigma&&\\\ x^{3}(\tau,\sigma)=0&&\\\
x^{5}(\tau,\sigma)=R\sigma&&\end{array}\right.$ (69)

corresponding to a circular loop evolving in the $xy$-plane, the action
Eq.(66) is traded for

${\cal
I}_{\Lambda}=2\pi\int\left(\sqrt{R^{2}-r^{2}\dot{r}^{2}}+\Lambda\sqrt{(R^{2}+r^{2})(1-\dot{r}^{2})}\right)d\tau~{}.$
(70)

Associated with the latter action is the Euler-Lagrange equation

$\frac{\ddot{r}}{r}=-\frac{\dot{r}^{2}+\Lambda(1-\dot{r}^{2})m(r,\dot{r})}{r^{2}+\Lambda(R^{2}+r^{2})m(r,\dot{r})}~{},$
(71)

where $m(r,\dot{r})$ is given explicitly by the ratio

$m(r,\dot{r})=\frac{(R^{2}-r^{2}\dot{r}^{2})^{\frac{3}{2}}}{R^{2}(R^{2}+r^{2})^{\frac{1}{2}}(1-\dot{r}^{2})^{\frac{3}{2}}}~{}.$
(72)

Without loosing generality, the accompanying initial conditions are
$r(0)=\ell$, $\dot{r}(0)=0$.

The solutions $r(\tau)$ of eq.(71) are in general oscillatory in $\tau$, with
frequencies of the general form

$\omega_{\Lambda}=\frac{1}{R}f_{\Lambda}(\frac{\ell}{R})$ (73)

which we now proceed to extract. No oscillatory solutions, however, for

$-\frac{\ell^{2}}{R\sqrt{R^{2}+\ell^{2}}}\leq\Lambda\leq 0~{},$ (74)

an important observation when the small $\Lambda$ regime is one’s preference.

A remark is in order. The fact that the oscillatory solutions pass through
$r=0$, corresponding to an orientation flip of the loop in the $xy$-plane, is
of no special concern. Owing to the underlying topology, that is being wrapped
around the Kaluza-Klein cylinder, the loop cannot really shrink to a singular
point.

Three trivial cases can be immediately verified:

$\bullet$ $\Lambda=0$ marks the LSR limit. As explained earlier, the area
$\sim r^{2}$ evolves linearly with $\tau$, and given our initial conditions,
we face

$r(\tau)=a~{}.$ (75)

In light of the forthcoming analysis, we assign

$\omega_{LSR}=0~{}.$ (76)

$\bullet$ $\Lambda\rightarrow\infty$ is recognized as the NG+KK limit. In
which case, we recover radial oscillations via $r(\tau)=\ell\cos\omega\tau$,
with frequency

$\omega_{NG}=\frac{1}{\sqrt{R^{2}+\ell^{2}}}~{}.$ (77)

$\bullet$ The limit $\displaystyle\frac{\ell}{R}\rightarrow 0$ is trustfully
translated into $\displaystyle{\frac{r(\tau)}{R}\rightarrow 0}$. As the scale
of $r(\tau)$ shrinks away, one immediately notices that
$m(r,\dot{r})\rightarrow 1$. Altogether, as could have been expected, and for
any finite $\Lambda$, we are back to NG+KK only with

$\omega_{\Lambda}\rightarrow\frac{1}{R}~{}.$ (78)

For a positive string tension $\Lambda>0$, we now prove the existence of a
periodic breathing mode, and attempt to study the entire range
$\displaystyle{\omega_{LSR}<\omega_{\Lambda}<\omega_{NG}}$. Our interest lies
however with the low frequency mode associated with the LSR governed case
$\Lambda\ll 1$. The transition from the low to high frequency regimes is
depicted (for the special case $\ell=R$) on a log-log graph, see fig.(5).

Figure 5: High to law frequency transition as a function of the string tension
$\Lambda$ (plotted for the special case $\ell=R=1$). While $\omega_{\Lambda}$
is $\Lambda$-independent at the high-$\omega$ regime, it is
$\sim\sqrt{\Lambda}$ at the low-$\omega$ regime

$\bullet$ $\Lambda\ll 1$ is our case of interest. On simplicity and
pedagogical grounds, we choose to make our points using the prototype case
$\ell=R$. The periodic numerical solution then suggests
$\displaystyle{\omega_{\Lambda}\sim\frac{\sqrt{\Lambda}}{R}}$, a result which
we now extract semi-analytically. We start by noticing that $m(\tau)$ is a
shallow function of $\tau$. It starts at $m(0)=1/\sqrt{2}$ and stays below $1$
up to an extremely narrow yet finite peak when $r(\tau)$ passes through zero.
We then replace $m(\tau)$ by its $\tau$-average constant value
$\frac{1}{\sqrt{2}}<m<1$ (a tenable value is $m\simeq 0.8$). And finally,
throwing away two terms of tiny numerical contributions, we arrive at the
approximated equation of motion

$\frac{\ddot{r}}{r}\simeq-\frac{\dot{r}^{2}+\Lambda m}{r^{2}+R^{2}\Lambda
m}~{}.$ (79)

Not only does the latter equation admit an analytic solution, but it
forcefully captures the essence (shape, extrema, zeroes) of the original
numerical solution of eq.(71). The semi-analytic solution is given explicitly
by

$\frac{\tau}{R}=E\left[\frac{\pi}{2},\frac{-1}{\Lambda
m}\right]-E\left[\arcsin\frac{r(\tau)}{R},\frac{-1}{\Lambda m}\right]~{},$
(80)

where $E[\phi,\chi]\equiv\int_{0}^{\phi}\sqrt{1-\chi\sin^{2}\theta}~{}d\theta$
stands for the elliptic integral of the second kind.

The role of the constant term in eq.(80) is to reassure that $r(0)=R$.
Furthermore, the radial velocity

$\dot{r}(\tau)\simeq-\frac{\displaystyle{\sqrt{1-\frac{r^{2}}{R^{2}}}}}{\displaystyle{\sqrt{1+\frac{r^{2}}{R^{2}\Lambda
m}}}}$ (81)

not only confirms that $\dot{r}(0)=0$, but also correctly takes care of the
physical upper bound $|\dot{r}(\tau)|\leq 1$. Our main bonus now being the
frequency formula

$\omega_{\Lambda}=\frac{2\pi}{\Delta\tau}=\frac{\pi}{2R\displaystyle{E\left[\frac{\pi}{2},\frac{-1}{\Lambda
m}\right]}}~{}.$ (82)

Appreciating the fact that $\displaystyle{E[\frac{\pi}{2},\frac{-1}{x}]}$
behaves like $\displaystyle{\frac{1}{\sqrt{x}}}$ for $x\ll 1$, we end up with
the low frequency limit

$\boxed{\omega_{\Lambda}\simeq\frac{\pi\sqrt{\Lambda
m}}{2R}\quad~{}\mathsf{for}~{}\Lambda\ll 1}$ (83)

Note that for $\Lambda\gg 1$ we recapture eq.(78).

##  VI Epilogue

This paper focuses on an unexpected role played by the compact fifth
dimension, serving as the missing topological ingredient which allows for the
conversion of an area metric into a legitimate line element in loop spacetime.
Ironically, it is the Kaluza-Klein ansatz which paves the way for Loop Special
Relativity (LSR) to exhibit the indispensable Special Relativity (SR) limit.
Associated with $M_{4}\otimes S_{1}$ is then a 10-dim loop spacetime metric,
whose 4-dim center-of-mass core term is supplemented by a 6-dim Maxwell-style
Planck-scale fine structure which drops away at the shrinking loop limit.
There are, however, (as listed in Chapter 4) four flies in the LSR ointment,
so to speak. Being kinematical in nature, LSR suffers from arbitrary loop
shapes, unbounded loop sizes, multiple timelike dimensions, and multiple KK
wrappings. Some of these problems require a dynamical solution, suggesting the
presence of a positive string tension. But dynamical Nambu-Goto (or Polyakov)
by itself will not do either, as it clearly lacks the SR limit in the
technical sense of eq.(4). The action marriage eq.(54) is perhaps an elegant,
and by far the simplest, way out. It comes with a clear signature, that is a
low-frequency breathing mode. And finally, just a reminder that the
generalization of LSR (Loop Special Relativity) into LGR (Loop General
Relativity) is still in order. Nash global isometric embedding may play a key
role in constructing the latter theory.

###### Acknowledgements.

The idea presented here was shaped up back at 1996, during a visit to N.K.
Nielsen (University of Southern Denmark), and has later served to seed the
M.Sc. thesis of Nadav Barkai (Ben Gurion University, 2007, unpublished) which
dealt with ”Gauge invariance of the second type”.

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#  Evidence for two distinct populations of kilonova-associated Gamma Ray
Bursts

Dimple Aryabhatta Research Institute of Observational Sciences (ARIES), Manora
Peak, Nainital-263002, India.  Department of Physics, Deen Dayal Upadhyaya
Gorakhpur University, Gorakhpur-273009, India.  K. Misra Aryabhatta Research
Institute of Observational Sciences (ARIES), Manora Peak, Nainital-263002,
India.  K. G. Arun Chennai Mathematical Institute, Siruseri, 603103 Tamilnadu,
India.  Institute for Gravitation and the Cosmos, Department of Physics, Penn
State University, University Park,
Pennsylvania 16802, USA

###### Abstract

Identification of Gamma Ray Burst (GRB) progenitors based on the duration of
their prompt emission ($T_{90}$) has faced several roadblocks recently. Long-
duration GRBs (with $T_{90}>2s$) have traditionally been thought to be
originating from the collapse of massive stars, and the short-duration ones
(with $T_{90}<2s$) from compact binary mergers. However, recent observations
of a long GRB associated with a kilonova (KN) and a short GRB with supernova
(SN) association demand a more detailed classification of the GRB population.
In this Letter, we focus on GRBs associated with KNe, believed to be
originating from mergers of binaries involving neutron stars (NS). We make use
of the GRB prompt emission light curves of Swift/BAT 2022 GRB catalog and
employ machine learning algorithms to study the classification of GRB
progenitors. Our analysis reveals that there are five distinct clusters of
GRBs, of which the KN-associated GRBs are located in two separate clusters
indicating they may have been produced by different progenitors. We argue that
these clusters may be due to subclasses of binary neutron star (BNS) and/or
neutron star–black hole (NS-BH) mergers. We also discuss the implications of
these findings for future gravitational-wave (GW) observations and how those
observations may help in understanding these clusters better.

GRBs — Kilonovae – BNS mergers — NS-BH mergers — PCA — Machine Learning — tSNE
— UMAP

##  1 Introduction

GRBs are traditionally classified as long and short bursts based on the
bimodal distribution of their duration and their distinct spectral hardness
ratios (Kouveliotou et al., 1993). Long bursts, with $T_{\rm 90}$ $>2s$ (the
duration over which 90% of the energy in prompt emission is released), have
been argued to originate from the collapse of massive stars (Woosley, 1993;
Paczyński, 1998; Mészáros, 2006; Hjorth & Bloom, 2012; Woosley & Bloom, 2006).
This argument is supported by the detection of broad-lined type Ic SNe
associated with several of the long GRBs (Galama et al., 1998; Kulkarni et
al., 1998; Stanek et al., 2003; Cano et al., 2017). On the other hand, short
GRBs have been argued to arise from compact object mergers (Paczynski, 1986;
Eichler et al., 1989; Meszaros & Rees, 1992). The coincident discovery of GRB
170817A and GW 170817 confirmed that BNS mergers do produce short GRBs (Abbott
et al., 2017a; Goldstein et al., 2017). Kilonovae (KNe), argued to be produced
by r-process nucleosynthesis during compact object mergers (containing a NS),
have been observed in association with at least three of the short GRBs -
130603B (Tanvir et al., 2013; Berger et al., 2013), 160821B (Kasliwal et al.,
2017; Lamb et al., 2019; Troja et al., 2019), and 170817A (Valenti et al.,
2017; Abbott et al., 2017b), further strengthening the conjecture that mergers
of compact binaries can produce short GRBs.

However, there have been many instances where this canonical association of
long GRBs to massive star collapse and short GRBs to compact binary mergers
has been proven to be incorrect. For instance, GRB 200826A despite being
short, had associated SN (Rossi et al., 2022; Ahumada et al., 2021; Zhang et
al., 2021) and long GRBs 060614 and 060505 (Fynbo et al., 2006), had no
accompanying SNe, and even more interestingly GRBs 060614 (Yang et al., 2015)
and 211227A (Lü et al., 2022), yielded evidence for accompanying KNe. In
recent studies, collapsar signatures have been noticed in high-redshift short
GRBs (Dimple et al., 2022).

The recent discovery of a long burst, GRB 211211A, further complicates the
classification problem demanding a detailed re-examination of the paradigm. A
confirmed KN counterpart (Rastinejad et al., 2022) was identified with this
burst despite being a long GRB. The host galaxy properties were found to be
similar to short GRB hosts rather than long GRB hosts (Troja et al., 2022;
Yang et al., 2022). The observed large offset from the center of the host and
lack of spatial association with any star-forming region further strengthens
the claim of a compact binary merger as a progenitor to this burst (Fruchter
et al., 2006; Berger et al., 2013). Additionally, there is evidence of GeV
emission observed in the prompt phase of the burst. Mei et al. (2022)
suggested that this GeV emission could be caused by the inverse Compton
effect, where the jet interacts with an external source of photons. They
proposed that the KN emission observed could provide the necessary seed
photons for the GeV emission.

Various progenitor models have been put forward to explain the origin of GRB
211211A. Yang et al. (2022) proposed a neutron star-white dwarf (NS-WD) merger
with a rapidly spinning magnetar as the central engine. However, Gompertz et
al. (2022) suggest that the fast evolution of the burst light curves cannot be
explained by a NS-WD merger (or a SN), though their study supports a merger
origin of the burst. The presence of Quasi-periodic oscillations (QPOs) in the
prompt emission light curve of the burst has been argued to favor a magnetar
progenitor (Xiao et al., 2022). Troja et al. (2022) claim that the time scales
and colors of the magnetar-induced KN do not match the observed light curves,
proposing it to be a hybrid event (that shares similarity to both long and
short GRBs) originating from a compact object merger. It was revealed in the
studies by Barnes & Metzger (2023) and Kunert et al. (2023) that a collapsar
origin is unlikely, and the origin of the burst is consistent with a BNS or
NS-BH merger, with a mild preference for a BNS merger. On the other hand, Zhu
et al. (2022) reported the fallback signatures in GRB 211211A, suggesting a
NS-BH merger origin. Recent studies by Zhong et al. (2023) claim that GRB
211211A could arise from a NS-WD merger if it produces a magnetar as remnant.
They argue that the optical afterglows, along with KN-like emission, can be
well-modeled by combining the standard forward shock with the radioactive
decay power of $\rm\prescript{56}{}{Ni}$ adding a rotational power input from
the post-merger magnetar.

All these studies have strongly favored a merger origin for the long GRB
211211A in sharp contradiction to it being of collapsar origin as is
canonically thought for long GRBs. Previously, two other long GRBs 060614
(Yang et al., 2015) and 211227A (Lü et al., 2022) have been asserted to show
signatures of merger origin. It is, therefore, important to ask whether all
GRBs with a possible KNe-association share similar features and have a common
progenitor.

Arguably, the best approach to answer this question is to study a population
of GRBs detected by an instrument and analyze the corresponding prompt
emission light curves of the GRBs. The combined use of prompt emission light
curves and state-of-the-art machine learning techniques could help in
identifying any global structures within the GRB population and thereby give
us some vital clues on distinct progenitor classes of GRBs. However, analyzing
a large sample of GRB light curves can significantly increase the
dimensionality of the data, making analysis difficult. Traditional dimension
reduction methods, such as Principal Component Analysis (PCA), can address the
high dimensionality issue, but it has limitations when interpreting the
Principal Components (PCs) and using them for clustering. To overcome these
limitations, non-linear methods like t-distributed Stochastic Neighbor
Embedding (tSNE; van der Maaten & Hinton 2008; van der Maaten 2014) and
Uniform Manifold Approximation and Projection (UMAP; McInnes et al. 2018) can
be employed.

Jespersen et al. (2020) and Steinhardt et al. (2023) have used tSNE and UMAP,
respectively, to identify clustering in the population of GRBs. Luo et al.
(2022) also made use of supervised machine learning techniques to distinguish
between different progenitor systems of GRBs. However, these studies focused
more on the general classification of GRBs based on their light curves and did
not survey any specific subpopulation, such as that of the KN-associated GRBs.
Recently, Garcia-Cifuentes et al. (2023) has used tSNE to investigate the
extended emission GRBs. However, all these approaches used tSNE or UMAP alone,
which preserves the global structures but not the local structures in the
data. Instead, Kobak & Linderman (2021) have shown that initializing tSNE/UMAP
with PCA can improve its ability to capture both the global and local
structures in the data.

Motivated by the recent advances in the development of machine learning
algorithms and the puzzles posed by GRB 211211A and similar GRBs, in this
Letter, we analyze the landscape of KN-associated GRBs in the Swift/BAT
catalog111https://swift.gsfc.nasa.gov/results/batgrbcat/ (Lien et al., 2016)
using UMAP and tSNE with a PCA initialization. Unlike the previous works, our
goal is to identify subclusters in the GRB population. Specifically, we are
interested in identifying subclasses of KN-associated GRBs and understanding
their possible progenitors.

##  2 Methodology

###  2.1 Use of Machine Learning tools

Unsupervised machine learning techniques are widely used for clustering and
dimensionality reduction in diverse areas of science. Such methods have the
ability to reveal hidden patterns in the data associated with complex
phenomena. In our context, the underlying physics associated with the prompt
emission of GRBs is extremely complex and not well understood. It is possible
to use some of the well-known GRBs, whose properties are studied extensively,
to assist the interpretation of the classification derived from these
algorithms. This is the approach we take in this work where the techniques of
tSNE and UMAP with PCA initialization are used to cluster a set of GRB prompt
emission light curves, and then the known properties of a subset of the GRBs
are incorporated to provide a basis for the interpretation.

tSNE is a non-linear dimensionality reduction technique that helps in
visualizing the data using stochastic embedding of neighborhoods (van der
Maaten & Hinton, 2008; van der Maaten, 2014). It models the probability
distributions of pairwise similarities in high and lower-dimensional spaces
and minimizes their differences using Kullback-Leibler divergence (Kullback &
Leibler, 1951). Initially, the algorithm randomly positions points in the
high-dimensional space, then iteratively moves each point towards a position
in the lower-dimensional space that best aligns with its neighboring points.
This movement is guided by a divergence function that is minimized when the
distribution of points in the lower-dimensional space matches that in the
high-dimensional space. The hyperparameters of the algorithm control the
visualization of the data, particularly the perplexity parameter, which
determines the number of nearest neighbors to consider when calculating
similarity distribution. A higher perplexity value takes more neighbors into
account, producing a smoother-looking embedding with greater emphasis on
global structure. Conversely, a lower perplexity value produces an embedding
with a more detailed local structure but less emphasis on global structure.
Choosing the right perplexity value is crucial for capturing both local and
global structure. For a given problem, one tune this parameter to get the
balance between local and global structures that is necessary for the
visualization of the data.

Uniform Manifold Approximation and Projection, on the other hand, uses more
abstract mathematical concepts from topology for dimension reduction, and
therefore the method is more deterministic than tSNE, especially when it comes
to defining notions of distance (McInnes et al., 2018). The algorithm aims to
reduce the dimensionality of data while preserving its local structure. The
algorithm consists of two main steps: first, constructing a high-dimensional
representation of the data by creating a similarity matrix between all points
in the data set. Second, reducing the dimensionality of the data while
retaining its local structure by finding a low-dimensional projection that
preserves the distances between all points in the similarity matrix. The
algorithm has several parameters, but the number of nearest neighbors
(n_neighbors, similar to perplexity) and minimum distance (min_dist, how
closely the method packs points which are close to each other) are
particularly important in balancing local and global structure in the final
projection. Increasing n_neighbors connects more neighboring points, resulting
in a more precise representation of the overall structure of data, but may not
capture the local structures in the data and therefore increase computational
complexity. On the other hand, increasing min_dist results in a more spread-
out representation of the data, reducing data clumping but potentially leading
to a loss of global structure. By construction, UMAP has a better ability to
preserve global distances compared to tSNE. Further, it is found that UMAP is
more time-efficient than t-SNE with regard to computation time.

In this work, we analyze the landscape of Swift GRBs utilizing the prompt
emission light curves in four energy bins with machine learning technique;
PCA-tSNE/UMAP. Similarities between the light curves are used by these
algorithms to produce two-dimensional maps of the GRBs, which are further
clustered using AutoGMM (a clustering algorithm to optimize the number of
clusters in a given data). Then, we superimpose the known KNe/SNe cases on top
of the identified clusters and infer interesting correlations and discuss
their implications for multi-messenger astronomy involving GWs. Next, we turn
our attention to the technical aspects of the analysis.

###  2.2 Data standardization

The Swift/BAT catalog 2022 consists of prompt emission light curves of 1525
GRBs detected between 17 December 2004 and 15 July 2022. The diverse data from
Swift/BAT reveals varying durations, fluence, and start times of the bursts,
which can lead to distraction in the algorithm. Hence, it is necessary to
homogenize the data format before employing any machine learning program. To
do this, we followed the standardizing procedure described in Jespersen et al.
(2020). The main steps involved are i) normalizing the light curves with the
fluence, ii) shifting the light curves to the same start time, iii) padding
the light curves with zero so that they have a similar length, and iv)
performing the Discrete-time Fourier Transform (DTFT) to preserve the time
delay information between the different light curves. The duration and fluence
(corresponding to the best-fit model) values were adopted from the summary
files provided in the Swift/BAT catalog. For some GRBs, the $T_{\rm 90}$ or
fluence information was missing. Due to this, the standardization procedure
could be carried out only for 1450 GRBs. We saved this standardized data in
the form of a matrix with dimensions [1450 x 25883], where each row represents
a GRB and contains corresponding light curves in four energy channels with a
binning of 64 ms.

Figure 1: Scree plot presenting the contribution of PCs to explain the
variance of the data. The leading four components explain $\sim 82$% of the
variance (inset). The first 500 components explain 99% of the variance.

###  2.3 Dimensionality reduction

Following the standardization of data, the subsequent step involves
dimensionality reduction. To accomplish this, we first performed the PCA and
generated the corresponding scree plot, which is a graphical representation
explaining the variance of each PC. Fig. 1 illustrates the scree plot, it can
be observed that around 4 leading PCs account for approximately 80% of the
variance in the data. However, in order to avoid biases due to the
incompleteness of the data, we utilized the leading 500 PCs, which preserves
approximately 99% of the variance. This resulted in a matrix with dimensions
of [1450 x 500], which we used to generate two-dimensional embeddings using
tSNE and UMAP algorithms.

As previously stated, the tSNE and UMAP algorithms are sensitive to specific
hyperparameters, and the proper selection of these key parameters is essential
for obtaining a meaningful representation of the data in a two-dimensional
space. We tuned these parameters after several iterations, we gradually
increased the perplexity from 5 to 50 (in increments of 5) in the case of tSNE
and chose a value of 25, for which we get a clear separation of various
clusters. Fig. 2 shows the embeddings obtained using PCA-tSNE (top left) and
the respective density plot (top right).

For UMAP, we iteratively increased the n_neighbor from 5 to 50 (in increments
of 5) and decreased min_dist from 0.1 to 0.005. We chose the parameters where
we get a clear separation of various clusters. We fixed the n_neighbors to 25
and min_dist to 0.01. The 2-dimensional embedding (bottom left), along with
the density plots (bottom right) obtained using these parameters, is shown in
Fig. 2. It should be noted that various choices of hyperparameters simply
translate into different projections of the data from higher dimensions to two
dimensions. Those projections that are more useful for visualization of the
data, depending on the context, are employed in the analysis.

Figure 2: Top: PCA-tSNE results depicting the two-dimensional embedding (left)
and the respective density map (right, bluer regions correspond to a higher
density of the bursts clustered at different locations on the map). Bottom:
Results for PCA-UMAP depicting embedding and density map.

###  2.4 Clustering with AutoGMM

For a more flexible and accurate representation of the data and to identify
the various clusters within the map, we used the AutoGMM module available in
Python (Athey et al., 2019). AutoGMM is an automated approach to determine the
optimal number of clusters in the data using GMM that postulates that data
points originate from a combination of several Gaussian distributions. The
algorithm begins by running the GMM clustering process for a range of
different cluster numbers. The algorithm then tries to estimate the parameters
of the Gaussian distributions that best fit the data, including the number of
clusters in the dataset. The application of AutoGMM to the two-dimensional
embeddings obtained from PCA-tSNE and PCA-UMAP resulted in the identification
of five clusters in the embedding (see Fig. 3).

Since both the algorithms PCA-tSNE and PCA-UMAP work independently and are
governed by several hyperparameters, it is crucial to compare the results
obtained from both methods. Therefore, we assess the similarity between the
embeddings generated from these algorithms using flameplot, as outlined in
Appendix A.

Figure 3: Left: The locations of KN-associated GRBs on a two-dimensional
embedding obtained using PCA-tSNE. The coral-colored labels represent the GRBs
with confirmed KN association. The turquoise-colored labels are the GRBs with
probable KN candidates. The KN-associated GRBs with long and short duration
occupy two locations on the embedding (right and left corners). The brown-
colored symbols show the location of GRB-SNe on the embedding. Right: Same
using PCA-UMAP. The embeddings are color-coded with respect to the five
clusters identified by AutoGMM.

##  3 Results and Discussion

Figure 4: PCA-tSNE and PCA-UMAP embedding color-coded with $T_{\rm 90}$ (top)
and $t_{\rm mvts}$ (bottom). The encircled regions indicate the locations of
KN-associated GRBs in the embeddings.

Fig. 3 displays the two-dimensional embeddings obtained using PCA-tSNE and
PCA-UMAP, color-coded with the clusters identified by AutoGMM. It is
conspicuous from the figure that the algorithm identifies five distinct
clusters in both embeddings, indicating five different classes of GRBs in the
Swift/BAT catalog 2022. This suggests that the results obtained from both the
algorithms (PCA-tSNE and PCA-UMAP) are similar, which is also evident from
their similarity score (see Fig. A.1). Here, we will focus only on those GRBs
with KN counterparts, and a detailed analysis of the full landscape will be
reported elsewhere (Dimple et al., in preparation).

We locate the confirmed and probable candidates of the KN-associated GRBs on
the PCA-tSNE/UMAP embeddings. The sample of confirmed KN-associated cases
include GRBs 130603B (Berger et al., 2013; Tanvir et al., 2013), 160821B
(Kasliwal et al., 2017; Lamb et al., 2019; Troja et al., 2019) and 211211A
(Rastinejad et al., 2022; Troja et al., 2022). Although GRB 170817A has a
confirmed KN-association, we do not have it in our sample as it was not
detected by Swift/BAT. GRBs 060614 (Yang et al., 2015), 061006 (Gao et al.,
2017), 070714B (Gao et al., 2017), 070809 (Jin et al., 2019), 200522A (Fong et
al., 2021; O’Connor et al., 2021) and 211227A (Lü et al., 2022) were discussed
to be probable KN candidates and constitute the sample in this work.

The KN-associated GRBs are seen to occupy two distinct regions on both
embeddings (see Fig. 3). Six GRBs lie on the left side of the embedding (left
cluster henceforth), and three others are on the right side (right cluster
henceforth). Interestingly, the GRBs in the left cluster have been
traditionally classified as short and those lying in the right cluster as long
based on the $T_{\rm 90}$ duration (see Fig. 4). Further, to better understand
the landscape of GRBs, we track all the GRBs with an established SN-
association 222Thirteen GRBs (GRBs 060218A, 071112C, 100316D, 111209A,
111228A, 120714B, 120729A, 130215A, 130831A, 161219B, 171010A, 190114C and
190829A) in the Swift/BAT catalog were reported to have a confirmed SN-
association (Bufano et al., 2012; Cano et al., 2014, 2017; Kann et al., 2019;
Klose et al., 2019; Melandri et al., 2022; Hu et al., 2021). However, GRBs
060218, 071112C, and 171010A were excluded from the sample due to the non-
availability of Swift/BAT data. GRB 111209A has incomplete data in Swift/BAT
and was excluded. The final sample has nine GRB-SNe. in these embeddings. The
SN-associated GRBs occupy the central region of the embedding (central
cluster), well separated from the KN-associated GRBs. The only exception is
GRB 070714B with a probable KN counterpart, which overlaps with the SN-
associated GRBs. Given our sample size and the tentative nature of the
detection, it is difficult to investigate on this GRB further. A detailed
analysis of the SN-associated GRBs will be the subject of a future paper.

Zhu et al. (2022) proposed that the KN-associated long GRBs share similar
progenitors via analyses of individual GRBs and the features in their light
curves. Our analysis using the prompt emission light curves agrees with Zhu et
al. (2022) with all KN-associated long GRBs lying in the same cluster. The
sub-clustering seen in Fig. 3 could be suggestive of something profound. If
the KN-associated GRBs in the two clusters are representative of other GRBs in
those clusters, then this classification clearly unravels two distinct
populations of such GRBs, potentially having different progenitors or
formation pathways.

###  3.1 Inference of the physical parameters of the clusters of KN-associated
GRBs

The GRB prompt emission light curves are of paramount importance in
understanding the characteristics of the central engine. The $T_{\rm 90}$
duration describes the time scale over which the central engine is active, and
the minimum variability time scale ($t_{\rm mvts}$) gives an idea about the
compactness of the emitting region (Sari & Piran 1997; Golkhou et al. 2015;
Sonbas et al. 2015, see also Camisasca et al. 2023 for a recent work that
studied the correlation of $t_{\rm mvts}$ of GRBs with quantities such as peak
luminosity and Lorentz factor). We, therefore, explored the effect of these
two parameters on the clusters of KN-associated GRBs. The $T_{\rm 90}$ values
are extracted from the Swift/BAT catalog. The $t_{\rm mvts}$ values are
estimated using the methods described in Golkhou et al. (2015) and Dimple et
al. (2022).

Fig. 4 shows the PCA-tSNE/UMAP embeddings color-coded with $T_{\rm 90}$ and
$t_{\rm mvts}$. The two clusters of KN-associated GRBs have a significant
difference in their $T_{\rm 90}$ values. The GRBs lying on the left cluster of
the embeddings are of consistently shorter duration than those on the right
cluster. On the other hand, both the KN-associated GRB clusters have similar
$t_{\rm mvts}$ values. This suggests that the compactness of the central
engines for both populations may be comparable, but they differ in the
duration for which the central engine is active.

In addition, we plot the light curves of KN-associated GRBs belonging to
different subclusters identified by AutoGMM to examine the similarities
between structures of the light curves. Fig. B.1 shows the light curves in the
15-350 keV band. By simply visualizing these light curves, it is evident that
KN-associated GRBs within the same cluster share similar characteristics, such
as duration and overall shapes of the light curves, which is further supported
by Fig. 4. However, the algorithm uses the light curves in four energy bands
and performs a more thorough job of utilizing the finer structures in the
light curves while generating the final maps.

###  3.2 Possible progenitors of the KN-associated GRBs

BNS and/or NS-BH mergers are the most plausible candidates for producing a KN
and a GRB (Metzger et al., 2010). There are several possible pathways in these
two types of mergers that may lead to very different end states (see Fig. 1 of
Bartos et al. 2013). These outcomes depend on quantities such as the extent of
tidal disruption, masses of the NS and BH that are involved, the maximum mass
of the NS that is supported by the equation of state, and spins of the compact
binaries involved.

For example, Gao et al. (2022) argued that the mergers of BNSs in which at
least one of the NSs has a very high magnetic field, with a magnetic flux of
about $10^{29}cm^{2}\,G$, can lead to halting of the accretion flow well
before the innermost stable circular orbit and therefore produce longer
duration GRBs. Recent studies by Lu & Quataert (2022) also predicted a long-
lived accretion disk after the BNS mergers. Similarly, if the BNS merger leads
to the formation of a stable hypermassive magnetized NS, the corresponding GRB
may be different than those produced from the mergers which directly produces
a BH as the remnant (see, for instance, Metzger et al. 2008; Gompertz et al.
2014). What is even more interesting is that the magnetar in the latter case
may be formed not necessarily by a BNS merger but by a double WD merger or
accretion-induced collapse of a WD.

In the case of NS-BH mergers, tidal disruption is favored when BH mass is low,
BH spin is prograde and large radii of neutron stars (Foucart et al., 2018;
Foucart, 2020) (see Fig. 2 of Foucart 2020). An efficient tidal disruption
produces enough material around the remnant to accrete. Energy injection from
the fallback material of the tidally-disrupted NS in a NS-BH merger (Rees,
1988; Desai et al., 2019) has been suggested as a mechanism that can lead to a
longer episode of accretion and hence longer duration GRBs. This may offer an
alternative explanation for the GRBs in our right cluster. For instance, Zhu
et al. (2022) invoked fallback accretion to the remnant BH to explain long
durations of GRBs 060614, 211211A, and 211227A.

If these low mass ratio NS-BH mergers are frequent, a relatively nearby
population of them may be detectable by LIGO/Virgo detectors in their future
observing runs. Though the first LIGO detections of NS-BHs (Abbott et al.,
2021) did not have any accompanying EM counterparts, it has been argued by
Sarin et al. (2023) that if a LIGO-like detector were operational, GRB 211211A
might have been a multi-messenger source if a compact binary merger was its
progenitor. It will be interesting to see whether the upcoming fourth
observing run (O4) of the LIGO/Virgo interferometers will register the first
multi-messenger NS-BH merger. When we have a handful of multimessenger compact
binary mergers with associated GRBs/KNe, a more robust test of this
classification hypothesis can be carried out.

As the next-generation GW detectors, such as Voyager (Adhikari et al., 2020),
Cosmic Explorer (Evans et al., 2021), Einstein Telescope etc. (Sathyaprakash
et al., 2012; Kalogera et al., 2021), become operational, the large population
of BNS and NS-BH mergers they will detect should help in uncovering the bigger
landscape of the GRB population and answer several open problems related to
the progenitor physics. More sensitive optical instruments, such as Vera Rubin
Observatory, may also enhance our horizons for KN detections much beyond what
we have today (Gupta et al., 2023).

In summary, in this Letter, we show there exist two distinct classes of KN-
associated GRBs. Only future multimessenger observations can reveal whether
they are two classes of BNS or NS-BH mergers or something else. There are
several possible future directions of this work. Analysis of the Fermi catalog
of GRBs, similar to Steinhardt et al. (2023) but tracking the presence of GRBs
associated with SNe and KNe to gain insights and the inclusion of afterglow
information into the current analysis are two of them. As machine-learning-
based methods continue to evolve and improve, they will likely play an
increasingly important role in understanding the progenitors of GRBs.

## acknowledgments

We thank the referee for the critical reading of the manuscript and the
comments that have improved the presentation of the manuscript. We thank L.
Resmi for the delightful discussions and very useful comments on the
manuscript. KGA thanks Ish Gupta, Arnab Dhani, and Rahul Kashyap for useful
discussions. K.G.A. acknowledges the Swarnajayanti grant
DST/SJF/PSA-01/2017-18 of the Department of Science and Technology and SERB,
India, and support from Infosys Foundation.

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##  Appendix A Similarity between PCA-tSNE/UMAP

Although both algorithms, PCA-tSNE/UMAP, excel in dimensionality reduction,
they are sensitive to hyperparameters, which can significantly affect the
embeddings produced. To ensure optimal performance, we carefully tuned the
hyperparameters for each algorithm separately. However, it is crucial to
compare the results of both algorithms to ensure their consistency and
reliability. We quantify the similarities between the two maps using the
python module flameplot developed by Taskesen (2020). Flameplot works by
quantifying the local similarities across two maps or embeddings. For the
comparison of two embeddings, X and Y, the algorithm compares the kX and kY
nearest neighbors of each embedding. The algorithm first computes the
Euclidean distance between samples in each embedding and ranks the sample
using Euclidean distance with the smallest distance at the top. Next, it
compares the ranks of embedding X to the ranks of embedding Y for the nearest
neighbors of kX and kY. Finally, it quantifies the score for the overlap
between ranks. The score is a measure of random similarity between cards. If
the two maps completely overlap, the score is 1.

The heatmap in Fig. A.1 shows the similarity score (a measure of random
similarity, where a higher score means more similar embeddings) for the 15
nearest neighbors. Despite a decrease in score with fewer neighbors, it
remains above 0.5, demonstrating that PCA-initialized tSNE and UMAP embeddings
are similar.

Figure A.1: The heatmap illustrates the similarity score between PCA-tSNE and
PCA-UMAP embeddings, which is close to 1 for a high value of nearest
neighbors. Despite a decrease in score with fewer neighbors, it still stays
above 0.5, suggesting that the two embeddings, initialized by PCA, are
similar.

##  Appendix B Light curves of KN-associated GRBs

The algorithm processes four light curves and captures collective images of
these curves across different energy bands. However, to avoid the complexity,
we display only the light curves of the KNe groups in the 15-350 keV band.
Fig. B.1 shows the light curves of the kilonovae belonging to various
subgroups ([GRB 130603B, 160821B, and 200522A, upper left group, turquoise
colored], [GRB 060614, 211211A and 211227A, upper right group, grey colored],
[GRB 061006, 070714B and 070809, group colored as green] in Fig. 3).

Figure B.1: The light curves of KN-associated GRBs belonging to various
subclusters as identified by AutoGMM.