# Ecosystem for Closed Timelike Curves: An Energy Conditions Perspective

Sashideep Gutti<EMAIL_ADDRESS>Birla Institute of
Technology and Science, Pilani—Hyderabad Campus, Hyderabad, India 500 078
Shailesh Kulkarni<EMAIL_ADDRESS>bDepartment of Physics,
Savitribai Phule Pune University, Ganeshkhind, Pune 411 007, India  Vaishak
Prasad<EMAIL_ADDRESS>c Inter-University Centre for Astronomy and
Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India  Sheldon Suresh
<EMAIL_ADDRESS>Saint Xaviers, Fort, Mumbai, Maharashtra 400001, India

###### Abstract

In this article, we explore the relationship between the existence of closed
timelike curves and energy conditions that occur in the Kerr-Newman spacetime.
To quantify the dependence, we define a correlation index between energy
conditions and closed timelike curves. Based on the inputs from Hawking’s
chronology protection conjecture, we analyze two popular variants of Kerr-
Newman spacetime: Non-commutative and Rastall Kerr-Newman spacetimes. These
two models provide complementary scenarios that aid in analyzing Hawking’s
statements regarding the correlation of closed timelike curves and energy
conditions from a local and a global perspective. We report the results
outlining the possible role played by violations of energy conditions in
eliminating the closed timelike curves in two contrasting situations, namely
in spacetimes with and without curvature singularities.

##  I Introduction

The formulation of the basic laws of physics intrinsically assumes the
preservation of causality (the causal ordering of events). A Closed Timelike
Curve (CTC) is a closed curve whose tangent is everywhere timelike with
topology $S^{1}$. The existence of CTCs in a spacetime, therefore, implies the
violation of cause/effect relations that are sacrosanct to predictability in
physics Morris:1988tu . To deal with the pathology, Hawking came up with the
Chronology Protection Conjecture which states that The laws of physics do not
allow the appearance of closed timelike curves. From a classical general
relativity perspective, many spacetimes have the pathology of the CTCs; these
are bonafide solutions to the Einstein’s field equations e.g Gött, Gödel, Kerr
family of spacetimes Gott:1998re ; Gott:1990zr ; godel1  (see also
Hawking:1973uf  and references therein). More prominent are the CTCs in the
Kerr-Newman spacetimes. The CTCs are usually present in the interior of the
inner horizon in the Kerr-Newman spacetime (they are also present in the Kerr-
Newman spacetime that has a globally naked singularity). The CTCs in this
spacetime are in causal contact with the singularity and therefore the buck of
unpredictability rests with the singularity. The quotation by Hawking
reiterates the fact. Quoting Hawking from his famous paper Hawking:1991nk ,
"In spacetimes exhibiting closed timelike curves, either there exist
singularities (like in the Kerr spacetime), pathological behaviour at infinity
(as in the Gödel and Gött spacetimes) or the weak energy condition is
violated".

The singularity of spacetime is a well known and more dangerous pathology that
the theory of relativity is plagued with. There is an extensive study on
reconciling the singularities with predictability in physics. The cosmic
censorship conjecture by Penrose penrose  states that the singularities are
always hidden behind a horizon, therefore, protecting large portions of
spacetime from the unpredictability that arises due to causal communication
with the singularity. Though there is no formal proof yet regarding the
conjecture, there are many models proposed with the intention of
proving/disproving the same. A review of the status of the problem can be
found in krolak1 ; krolak2 ; joshi1 ; joshi2 ; tpsingh ; landsman . The models
that present the possibilities of naked singularities (where the causal rays
from the singularity can reach non-singular points in spacetime) are not
generic enough to conclude against the conjecture. It is well accepted that
once we have a quantum theory of gravity, the singularities are expected to be
smeared away into a high curvature region suggesting that the singularities
are the artefacts of classical general relativity thiemann ; bojowald ;
modestosingularity ; hussain .

Although substantial efforts have been made (e.g. see Visser:1995cc ;
Friedman:2006tea ) to understand the issue related to CTCs, it still lacks the
rigorous status as the singularity theorems associated with curvature
singularities. The existence, formation and avoidance of CTCs in spacetime are
poorly understood. One factor that is detrimental to exhaustive research in
the area of CTCs is the lack of an adequate number of models that present the
pathology of CTCs. Even though a lot of work is carried out for regular black
hole geometries where there are models with the singularities are smeared out,
the elimination of CTCs are usually not addressed. Our recent work
Prasad:2018hdj  is the first attempt to address the issue of eliminating CTCs
using variants of gravity. The ecosystem for the existence/emergence of the
CTCs in spacetime is not well understood and the field of research is in its
nascent stage. Since quantum gravity is expected to smear out the
singularities. It is maybe reasonable to expect that the solutions within the
correct quantized theory of gravity theory will also be free of the CTCs. Some
peculiarities in the spacetimes endowed with CTC that appears at the semi-
classical level have been studied extensively in Boulware:1992pm ;
Visser:2002ua . We, in this article, address this elephant in the Kerr-Newman
space-time and track its ecosystem from the point of view of energy
conditions.

In the Kerr spacetime, the CTCs exist in an environment where the energy
conditions are preserved, but in the presence of a curvature singularity.
However, various variants of the Kerr family of solutions have been found in
modified gravity models that are interesting for their own reasons. In
Prasad:2018hdj , we have analyzed the non-commutative version of the Kerr-
Newman spacetime presented in Modesto:2010rv . It is shown that there exists a
parameter space that constitutes the Kerr-Newman spacetime, ($M,Q,a,\Theta$)
for which the pathology due to the presence of CTC exists and a parameter
space in which the CTCs are eliminated. We showed that for a given set of
parameters $(M,Q,a)$, we can choose the non-commutative parameter
$\Theta=\Theta_{0}$ such that CTCs are eliminated. The paper also describes
the analysis of the Kerr-Newman version in $f(R)$ gravity and discusses the
possibility of elimination of the CTCs within the parameters present in the
model. A systematic study of CTCs, their existence and elimination is,
therefore, a gap in current research. It is well known that the energy
conditions are violated in some quantum field theoretic contexts (e.g Casimir
energy). It is also fairly certain that for eliminating the spacetime
singularities, quantum gravity is instrumental. Energy condition violation is
therefore expected to prevent the formation of singularities. A lot of work on
the regular black holes and energy conditions have been carried out in nves ;
bambi ; tom  with proposals for singularity free spacetimes. Most of these
models are based on insights from modified general relativity.

One can ask the following questions about the CTCs and their elimination. Does
avoidance of singularities automatically lead to the elimination of CTCs as
well? Are energy condition violations needed if the final aim is to eliminate
CTCs that are present in the solutions for classical general relativity? Can
we preserve energy conditions and eliminate CTCs? Among the different energy
conditions (weak, strong, null, dominant, etc), which of them is crucial for
the preservation and elimination of CTCs?

Variants of Kerr-Newman spacetimes present us with an excellent laboratory to
test out the dependence of CTCs on the various energy conditions and provide
answers to the above questions. It can be expected that the prevention or
elimination of CTCs might be possible provided we compromise one (or more)
energy conditions.

In this work, we study the correlation between energy conditions and CTCs. We
study this correlation in two ways, local and global perspectives. Hawking’s
statement connects the violation of the energy conditions and the presence of
CTCs. We explore the possibility of refining this connection in the following
way. Suppose we consider a point in spacetime through which a CTC passes. Some
of the energy conditions may not be violated locally in the region of CTCs
(i.e. at the location of CTCs), but rather globally. It is therefore
worthwhile to investigate the interplay between the energy conditions and CTCs
from the local as well as global points of view. The understanding of this
correlation can be very useful in many contexts. Firstly, when one considers
models for modified gravity, the selection of one model over the other is
usually based on consistency with cosmological observations. One more
parameter for this selection criteria could be the CTC elimination. If one
constructs equivalent models of certain solutions, e.g the Kerr-Newman
spacetime, using phenomenological quantum gravity models, the status of
causality preservation due to elimination of CTCs could be yet another
independent criterion for choosing one model over the other.

We study two cases of modified Kerr-Newman spacetime: The non-commutative
(regular) Kerr-Newman spacetime and the Kerr-Newman version found in Rastall
gravity. Hawking’s statement brings out two distinct situations. CTCs near
singularity and CTCs in the absence of singularities. The two models presented
here cover both these scenarios. The Rastall version has curvature singularity
and therefore CTCs exist in the presence of a curvature singularity. We show
that one can choose parameters in this model to eliminate CTCs. The non-
commutative counterpart is regular and the curvature singularity is smeared
out. So this presents a model that has CTCs but no curvature singularity. Both
these models, therefore, present excellent theoretical testing grounds to
examine the ecosystem of CTCs.

The paper is organized as follows. In Sec. II, we propose a general formalism
for quantifying the association of energy conditions with the existence of
closed timelike curves. In Secs. III and IV we apply this procedure to the
variants of Kerr-Newman spacetimes in non-commutative and Rastall gravity and
discuss the correlations between the energy conditions and CTCs. In Sec. V we
make concluding remarks.

Throughout, we use geometric units in which $G=c=1$. We also set the ADM mass
of the spacetime to be 1 i.e. $M=1$. For algebraic computations, we use
_Mathematica_ math . Although we deal with solutions that are used to describe
the exterior geometries of black holes, we will refer to them as spacetimes as
for some of the parameters, the spacetime may or may not contain horizons.

##  II Correlation index for the CTCS

We explore in the subsequent sections, the correlation of the CTCs with the
energy conditions. Our goal is to explore the models in variants of general
relativity. To make the estimate quantitative, we give the prescription for
defining a correlation index in this section. We observe a common feature
among the variants of Kerr-Newman spacetime solutions. The variants are in
general, dependent on few parameters $(\mu_{1},\mu_{2}...\mu_{n})$ that are
specific to the model of gravity chosen. The Einstein gravity corresponds to
particular values of these parameters. The Kerr-Newman metric has an axial
symmetry corresponding to a Killing vector $\xi_{(\phi)}^{\alpha}$ presented
in a coordinate chart say $(x^{\alpha})$ (Here the subscript $\phi$ is a label
indicating the axial Killing vector field). The CTCs in these coordinates can
be singled out by examining the sign of the norm of this Killing vector
$\xi_{(\phi)}^{\alpha}\xi_{(\phi)\alpha}$. Wherever
$\xi_{(\phi)}^{\alpha}\xi_{(\phi)\alpha}$ becomes negative, we have CTCs. Now
this expression for the norm is a scalar function of the coordinates
($x^{\alpha}$). The modified Kerr-Newman solution is generally dependent on
the various parameters $(\mu_{1},\mu_{2}...\mu_{n})$ and so is the norm of
$\xi_{(\phi)})^{\alpha}$. When one computes the expression for the energy
conditions, one gets a function of the form
$f(x^{\alpha},\mu_{1},\mu_{2}...\mu_{n})$. The sign of that function dictates
whether a particular energy condition is violated or not. Motivated by this we
associate the index for correlation between the existence of CTCs and energy
conditions as

$\mathcal{C}=\frac{\bigint\frac{f}{|f|}\left(\frac{\xi_{(\phi)}^{\alpha}\xi_{(\phi)\alpha}}{|\xi_{(\phi)}^{\alpha}\xi_{(\phi)\alpha}|}-1\right)\sqrt{h}\
dX\
d\mu}{\bigint\left(\frac{\xi_{(\phi)}^{\alpha}{\xi_{(\phi)}}_{\alpha}}{|\xi_{(\phi)}^{\alpha}\xi_{(\phi)\alpha}|}-1\right)\sqrt{h}\
dX\ d\mu}$ (1)

Here $dX$ is a short notation to the relevant spacetime volume element, $d\mu$
stands for volume in the parameter space $d\mu_{1}d\mu_{2}\cdots d\mu_{n}$ and
$\sqrt{h}$ is the infinitesimal volume element on the relevant hypersurface
orthogonal to a timelike Killing vector field $\xi_{(t)}$. If we use the
standard Boyer-Lindquist coordinate system ($t,r,\theta,\phi$), the norm
$\xi^{\alpha}_{(\phi)}\xi_{(\phi)\alpha}$ is then equal to the metric
component $g_{\phi\phi}$. In the models that we consider below, we restrict
our analysis to the equatorial plane and hence we set $\theta=\pi/2$. Due to
axial symmetry we can further set $\phi=0$. Consequently, the index now
assumes the form,

$\mathcal{C}=\frac{\bigint\frac{f}{|f|}\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)\sqrt{h}\
dX\ d\mu}{\bigint\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)\sqrt{h}\
dX\ d\mu}$ (2)

The motivation for the above definition of the index is stated below. The term
$\displaystyle\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)$ is zero
wherever CTCs are not present and is equal to $-2$ in regions where CTCs are
present. The term $f/|f|$ is $+1$ whenever the relevant energy condition is
preserved, as we vary the spacetime coordinates as well as the parameters
$\mu_{i}$. The term $f/|f|$ will then have a value $-1$ wherever the relevant
energy condition is violated. Hence, the numerator of the integrand in Eq. (2)
is positive for the parameters for which the CTCs are present and relevant
energy condition is violated. It is zero wherever CTCs are absent. It is $-2$
wherever CTCs are present and the relevant energy conditions are preserved.
The denominator of the above expression integrates over the entire parameter
range which includes CTCs and it gives vanishing contribution from the points
where CTCs are absent. Consequently, the index is ill-defined in situations
when there are no CTCs exist anywhere in the spacetime for all parameter
values. The resultant index, therefore, takes values between $-1$ and $+1$. If
the index is negative, one can infer that CTCs prefer to exist in the region
where the relevant energy condition, denoted by
$f(x^{\alpha},\mu_{1},\mu_{2}...\mu_{n})$ is violated. A positive value of the
index implies that CTCs prefer to live in an energy condition preserved
environment suggesting that elimination of CTCs might require violation of
energy conditions.
In the models below, we evaluate the index given by Eq. (2) in a sub-region of
the parameter space, to systematically study the correlation between the
existence of CTCs and the weak energy condition. To avoid numerical artefacts,
while implementing the index formula, care must be taken such that CTCs are
present in the domain of integration.

##  III Non-commutative Kerr-Newman spacetime

We consider charged rotating black hole spacetimes inspired by non-commutative
geometry. The motivation behind introducing the non-commutativity in the usual
black holes setup is to cure pointlike curvature singularities (e.g. the
Dirac-delta like the singularity of Schwarzschild BH, the ring singularity of
the Kerr BH) by effectively replacing it by appropriate mass distribution
Nicolini:2005vd ; Nicolini:2009gw ; Spallucci:2008ez . Within the framework of
coordinates coherent state approach together with expected first order quantum
gravitational correction to classical general relativity it can be shown that
the Dirac-delta like singularities will be replaced by Gaussian distributions
Modesto:2010rv . For the point particle at the origin, the non-commutative
nature of coordinates reflects into a modification to usual delta functions
distribution $\delta({\bf x})$ as

$\rho_{0}({\bf x})=\frac{1}{2\pi\Theta}e^{-\frac{{\bf x}^{2}}{2\Theta}}$ (3)

The width of the Gaussian distribution is now characterized by the Non-
Nommutative (NC) parameter $\Theta$. It has been shown that Cembranos:2011sr
the effective corrections to the Einstein’s-field equations due to the above
replacement can be modelled by replacing the pointlike sources by a suitable
Gaussian distribution while keeping the differential operators unchanged. The
non-commutative solution for the Kerr-Newman (KN) spacetime can be written as
Modesto:2010rv

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}}dt^{2}-2a\sin^{2}\theta\Big{(}1-\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}}\Big{)}dtd\phi$
(4)
$\displaystyle+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\Sigma^{2}}{\rho^{2}}\sin^{2}\theta
d\phi^{2}$

with

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
r^{2}-2m(r)r+a^{2}+q^{2}(r)$ $\displaystyle\Sigma^{2}$ $\displaystyle=$
$\displaystyle(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta$ (5)
$\displaystyle\rho^{2}$ $\displaystyle=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta$

where the mass and charge functions take the form

$m(r)=M\frac{\gamma(3/2,r^{2}/(4\Theta))}{\Gamma(3/2)}$ (6) $\displaystyle
q^{2}(r)$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{\pi}\Big{[}\gamma^{2}(1/2,r^{2}/(4\Theta))-\frac{r}{\sqrt{2\Theta}}\gamma(1/2,r^{2}/(2\Theta))$
(7) $\displaystyle+$ $\displaystyle
r\sqrt{\frac{2}{\Theta}}\gamma(3/2,r^{2}/(4\Theta))\Big{]}$

Using the Einstein equations for the above metric the expression for
components of energy-momentum tensor read as

$\displaystyle T^{\mu}_{\nu}=\begin{bmatrix}T^{t}_{t}&0&0&T^{t}_{\phi}\\\
0&T^{r}_{r}&0&0\\\ 0&0&T^{\theta}_{\theta}&0\\\
T^{\phi}_{t}&0&0&T^{\phi}_{\phi}\end{bmatrix}$ (8)

At this point, we would like to make the following observations. The energy-
momentum tensor is off-diagonal in the $t-\phi$ sector. To extract the
information about the energy conditions we shall seek a tetrad frame in which
$t-\phi$ sector is diagonalised while keeping the other components intact
since other components are already in diagonal form. It is worth mentioning
that our energy-momentum tensor Eq. (8) falls under the type-I of Hawking-
Ellis classification Hawking:1973uf  which is based on the extent to which the
orthonormal components of the energy-momentum tensor can be expressed in a
diagonal form Martin-Moruno:2017exc . General expressions for the components
of energy momentum tensor are complicated and are given in the appendix. Since
the region near to the equator ($\theta=\pi/2$) is densely populated with
CTCs, we shall restrict ourselves in the near equatorial region111This point
is explained with plots at the beginning of Sec. III.1. For $\theta=\pi/2$,
the energy momentum tensor can be written as

$\displaystyle 8\pi T^{t}_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{4r^{6}}\Big{[}-4a^{2}r^{3}m^{\prime\prime}(r,\Theta)+\Big{(}8a^{2}r^{2}+8r^{4}\Big{)}m^{\prime}(r,\Theta)+$
$\displaystyle
2a^{2}r^{2}q^{\prime\prime}(r,\Theta,Q)+2r\Big{(}-4a^{2}-2r^{2}\Big{)}q^{\prime}(r,\Theta,Q)+\Big{(}8a^{2}+4r^{2}\Big{)}q(r,\Theta,Q)\Big{]}$
$\displaystyle 8\pi T^{r}_{r}$ $\displaystyle=$
$\displaystyle-\frac{r\left[2rm^{\prime}(r,\Theta)-q^{\prime}(r,\Theta,Q)\right]+q(r,\Theta,Q)}{r^{4}}$
$\displaystyle 8\pi T^{\theta}_{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{4r^{4}}\Big{[}2r^{2}\left(-4m^{\prime}(r,\Theta)-2rm^{\prime\prime}(r,\Theta)+q^{\prime\prime}(r,\Theta,Q)\right)$
$\displaystyle+4\left(r\left(2rm^{\prime}(r,\Theta)-q^{\prime}(r,\Theta,Q)\right)+q(r,\Theta,Q)\right)\Big{]}$
$\displaystyle 8\pi T^{\phi}_{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{4r^{6}}\Big{[}2r\left(2r^{2}\left(a^{2}+r^{2}\right)m^{\prime\prime}(r,\Theta)+\left(4a^{2}+2r^{2}\right)q^{\prime}(r,\Theta,Q)\right)-$
$\displaystyle
8a^{2}r^{2}m^{\prime}(r,\Theta)-2r^{2}\left(a^{2}+r^{2}\right)q^{\prime\prime}(r,\Theta,Q)+2\left(-4a^{2}-2r^{2}\right)q(r,\Theta,Q)\Big{]}$
$\displaystyle 8\pi T^{t}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{4r^{6}}\Big{[}a\left(a^{2}+r^{2}\right)\Big{(}-4r^{3}m^{\prime\prime}(r,\Theta)+8r^{2}m^{\prime}(r,\Theta)+2r^{2}q^{\prime\prime}(r,\Theta,Q)$
$\displaystyle-8rq^{\prime}(r,\Theta,Q)+8q(r,\Theta,Q)\Big{)}\Big{]}$
$\displaystyle 8\pi T^{\phi}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{4r^{6}}\Big{[}2ar^{2}\left(4m^{\prime}(r,\Theta)+2rm^{\prime\prime}(r,\Theta)-q^{\prime\prime}(r,\Theta,Q)\right)$
$\displaystyle-8a\left(r\left(2rm^{\prime}(r,\Theta)-q^{\prime}(r,\Theta,Q)\right)+q(r,\Theta,Q)\right)\Big{]}$

Next, we carry out the reduction of the energy momentum tensor to the
canonical form. This requires us to find the eigenvalues of the expression

$det|T_{\mu\nu}-\lambda g_{\mu\nu}|=0$ (9)

Using the above one can reduce the energy-momentum tensor to the canonical
form. If the eigenvalues are real, then the energy-momentum tensor belongs to
type-I according to Hawking and Ellis classification. type-I implies that the
energy-momentum tensor and the metric are simultaneously diagonalizable. This
implies that one can find a tetrad basis in which the canonical energy-
momentum tensor is $T_{ab}=T_{\mu\nu}e^{\mu}_{a}e^{\nu}_{b}$ where
($e^{\mu}_{a}$) are the tetrads which are the simultaneous eigenvectors of
type-I energy-momentum tensor and the metric. Thus, the eigenvalues represent
the principle pressures and density. For the NCKN case, we show that the
energy-momentum tensor can be canonically decomposed to type-I by Hawking
classification on the equatorial plane $\theta=\pi/2$. We evaluate the
eigenvalues and eigenvectors ($e_{(t)}$, $e_{(\phi)}$) by focussing on the
$t-\phi$ sector of the energy-momentum tensor (the rest of the components are
already in the canonical form). The eigenvector $e_{(t)}$ is timelike while
$e_{(\phi)}$ is spacelike. The explicit form of these tetrads is furnished by
the following expressions

$\displaystyle e^{t}_{(t)}$ $\displaystyle=\frac{a^{2}+r^{2}}{a}\ ;\
e^{\phi}_{(t)}=1\ ;$ (10) $\displaystyle e^{t}_{(\phi)}$ $\displaystyle=a\ ;\
e^{\phi}_{(\phi)}=1\ ;\ e^{r}_{(t)}=\ ;\ e^{\theta}_{(t)}=\ ;\
e^{r}_{(\phi)}=\ ;\ e^{\theta}_{(\phi)}=0$ (11)

Also, as expected, the eigenvectors are orthogonal to each other. After
normalizing the eigenvectors and re-expressing the components of the energy-
momentum tensor in the $t-\phi$ sector, we finally obtain the following
components

$\displaystyle 8\pi\tilde{T}^{t}_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{4}}\Big{[}q(r,\Theta,Q)+2r^{2}m^{\prime}(r,\Theta)-rq^{\prime}(r,\Theta,Q)\Big{]}$
$\displaystyle 8\pi\tilde{T}^{\phi}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{4}}\Big{[}q(r,\Theta,Q)-r^{3}m^{\prime\prime}(r,\Theta)-rq^{\prime}(r,\Theta,Q)$
$\displaystyle+$ $\displaystyle 2r^{2}q^{\prime\prime}(r,\Theta,Q)\Big{]}$
$\displaystyle 8\pi\tilde{T}^{t}_{\phi}$ $\displaystyle=$ $\displaystyle
8\pi\tilde{T}^{\phi}_{t}=0$ (12)

while other components remain unchanged.

###  III.1 Correlation of energy conditions with presence of CTCs

As demonstrated in Prasad:2018hdj , for a given charge, angular momentum and
mass of a black hole, we can choose the non-commutative parameter
($\Theta>\Theta_{critical}$) such that the CTCs are eliminated. In contrast,
the usual Kerr-Newman spacetime has CTCs in its spacetime for the same set of
parameters (this can be achieved by taking the limit $\Theta=0$).

In Fig. 1 we have plotted $g_{\phi\phi}$ against the re-scaled radial
coordinate $r$ for three different values of non-commutative parameter
$\Theta$ while keeping $Q$ and $a$ fixed. For $\Theta=10^{-3}$ (orange curve)
and $10^{-5}$ (green curve) there are regions of CTC, with different radial
extent and depth. We see that CTCs are absent close to $r=0$ and develop for a
larger radius. Though not apparent from the figure, the CTCs are again absent
for a larger radius $r$ since the metric approaches the usual Kerr-Newman
spacetime asymptotically Prasad:2018hdj ; Modesto:2010rv . We also observe
that the effect of non-commutativity is more prominent at small radii and
indicates that the non-commutativity is conducive to eliminating the CTCs. In
Fig. 2, we show that the CTCs are more prominent on the equatorial plane. On
the top panel of Fig. 2, the $x$-axis is radial coordinate $r$ and $y$-axis is
the angle $\theta$ in radians. The shaded portion is the region where CTCs are
present. At the bottom panel, we plot $\Theta$ on $x-$ axis and $\theta$ on
$y-$ axis for fixed radius $r$. This plot illustrates that CTCs are
concentrated more near $\theta=\pi/2$.

Figure 1: Behaviour of $g_{\phi\phi}$ in the equatorial plane ($\theta=\pi/2$)
for the fixed value of charge $Q=0.6$ and $a=0.5$ with $\Theta=10^{-2}$
(blue), $\Theta=10^{-3}$ (orange) and $\Theta=10^{-5}$ (green). Region (in $r$
space) containing the closed time like curves exists for orange and blue
curves.

Figure 2: Top: CTC region in $r-\theta$ space. Bottom: CTC region in
$\theta-\Theta$ space

For correlating this information with the energy condition, we similarly
obtain the violation details of the energy conditions as a function of $r$. We
then superimposed on the same plot, the CTC region and the energy condition
violation region. The correlation obtained is therefore a local one where we
address the question that for a given point, the presence of CTCs is
correlated with the violation of a particular energy condition or not. We
present here the observations regarding the possible correlation of the CTCs
and energy conditions. For the type I matter field the various energy
conditions are given by

$\displaystyle\rho\geq 0\ ,\rho+p_{i}\geq 0\qquad\text{WEC}$
$\displaystyle\rho+\sum_{i=1}^{3}p_{i}\geq 0\qquad\qquad\text{SEC}$
$\displaystyle\rho+p_{i}\geq 0\ \qquad\qquad\quad\text{NEC}$
$\displaystyle\rho-|p_{i}|\geq 0\qquad\qquad\quad\text{DEC}$

We now illustrate the interplay between the weak energy condition and CTCs.
Fig. 3 illustrates the various quantities , $\rho$, $\rho+p_{i}$ as a function
of radius for the fixed $\Theta,Q$ and $a$. We superimpose the graphs for
various expressions that dictate the status of the weak energy condition with
the existence of CTCs i.e. $g_{\phi\phi}$. For definiteness we use the
parameter set chosen for green curve in Fig. 1. We note that the energy
density $\rho$ is negative for small values of radius, turns positive and
eventually becomes zero. Intuitively this is expected since the NCKN is a
regular spacetime where singularity is smeared out. We expect a repulsive
effect would sustain an equilibrium scenario that has smeared mass in place of
a singularity. The other curves are $\rho+p_{i}$, we see that they become
negative closer to the singularity and become positive for a larger radius
before eventually becoming zero. It can be seen from the figure that there is
an overlap between the radii where $g_{\phi\phi}$ and other weak energy
expressions are simultaneously negative.

Figure 3: Behaviour of $g_{\phi\phi}$ and weak energy conditions against $r$.
The parameters are set to: $\Theta=10^{-5},Q=0.6$ and $a=0.5$

In Fig. 4, we superimpose regions in $r$ and $\Theta$ space where weak energy
conditions are preserved and CTCs are present. In the plots, the top one is
for parameter values $Q=0.6$ and $a=0.5$ while bottom is panel has $Q=0.7$ and
$a=0.5$. The $x$-axis is the re scaled radius $r/M$ and $y-$ axis represents
the non-commutative parameter $\Theta$. The shaded area in both the plots
represents the region where CTCs are present and Weak Energy conditions are
preserved. Fig. 5 illustrate the common region where CTC’s exist and various
the energy conditions are satisfied in the $r-\Theta$ space (with
$Q=0.6,a=0.5$ and $Q=0.7,a=0.5$, respectively). The coloured regions in the
parameter space indicate the preservation of the respective energy condition
(weak, strong, null).

Figure 4: Region containing CTC where weak energy condition is preserved for
the parameters i) $Q=0.6,a=0.5$ (top) and ii) $Q=0.7,a=0.5$ (bottom).

Figure 5: Region containing CTC where weak, strong and null energy conditions
are preserved for the parameters i) $Q=0.6,a=0.5$ (top) and ii)
$Q=0.7,a=0.5$(bottom).

Few observations from the plots are in order. We can make a local statement
regarding the relation between energy conditions and CTCs. At a given
spacetime point, one can have a CTC passing through it and the weak energy
condition at that point may or may not be violated. For a point at radius $r$,
we observe that weak energy condition is preserved for small $\Theta$ but get
violated as we increase $\Theta$. This trend is also displayed for the
elimination of CTCs. This seems to indicate that a violation of weak energy
conditions favours the elimination of CTCs. This can be seen from Fig. 5 that
the boundary of the CTC region occurs beyond the region of validity of the
weak energy condition, strongly suggesting that in the absence of
singularities, the violation of weak energy condition help us to eliminate
CTCs. The large $r$ behaviour of CTC boundary might not contribute to this
correlation since even for KN spacetime, the CTCs die down as radius
increases. We now recall Hawking’s statement which states that for the
spacetimes with no singularities and no pathological behaviour at future
infinity, the presence of CTCs requires the violation of weak energy
condition. The conclusions drawn from our analysis seems to suggest that the
above statement is valid globally but not locally. It means it might be
possible that the weak energy condition could be violated at some region of
the spacetime but not necessarily at the same location as that of CTCs. To
elucidate this point further, we scan the entire parameter space and look for
the regions containing CTCs where weak energy conditions are also preserved.
In the first two plots of Fig. 6, we highlight the region in the parameter
space: $r-\Theta-a$ and $r-\Theta-Q$ (right) where the CTCs are present and
also the weak energy condition is preserved. For completeness, we have also
presented the $3D-$ regions of co-existence of CTCs and null, strong energy
conditions in the last two plots of Fig. 6.

Figure 6: i) $3D-$ region showing the co-existence of CTCs and weak energy
condition in $r,\Theta,a$ space with $Q=0.6$ (first plot) and $r,\Theta,Q$
space with $a=0.5$ (second plot). ii) $3D-$ region showing the co-existence of
the CTCs and the null (third plot) and strong (fourth plot) energy conditions
in $r,\Theta,Q$ space with $a=0.5$.

These plots indicate that at a given radius, one can make a local statement
that for the spacetimes with no singularities and no pathological behaviour at
future infinity, the presence of CTCs does not imply the violation of weak
energy conditions. Consequently, Hawking’s chronological conjecture holds in
the global sense but not locally.

###  III.2 Correlation index

Here we evaluate the correlation index for the NCKN case for fixed point
($t,r,\theta=\pi/2,\phi$) on the equatorial plane in spacetime.

$\mathcal{C}_{r}=\frac{\bigint\frac{f}{|f|}\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)dadQd\Theta}{\bigint\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)dadQd\Theta}$
(13)

with the function $f$ given as

$f=Min[\rho,\rho+p_{1},\rho+p_{2},\rho+p_{3}]$

.

Figure 7: Indices for correlation between WEC and CTCs in the NCKN geometry
for values of x ranging $0.03$ to $0.200$ (scaled by factor $10^{2}$). The
index is calculated over $a$, $Q$ and $\Theta$.

In the Fig. 7, we present the correlation index at few values of $x=r/M$. The
index is calculated by integrating over the parameter space involving
$a,Q,\Theta$. The range of $x$ chosen is such that the CTCs are present in
some parameter range. The normalized radius $(x=r/M)$ is varied from $0.03$ to
$0.12$. The index is evaluated by carrying out an integration over $a$, $Q$
and $\Theta$. We see that for smaller values of x, the index is close to $-1$
and for relatively larger values of x, the index smoothly varies till it
becomes $+1$ for larger values of $x$. This implies that for smaller values of
x, the CTCs exist in an environment where WEC is violated. The effect is more
prominent in smaller values of x because the WEC violation is required to
smoothen out the singularity. The NCKN spacetime asymptotes to the standard
Kerr-Newman spacetime for larger $r$ where the energy conditions are not
broken. So for larger x, the index smoothly goes to the WEC energy preserving
environment.

##  IV Rastall gravity

We now consider the interesting Kerr-Newman solution in Rastall gravity. This
model presents a contrasting situation where singularity is present and
parameter range can be chosen where CTCs are present and another parameter
range where they are eliminated. Rastall gravity is set up based on the
following framework. The conservation of energy-momentum tensor plays an
important role in classical general relativity. In Rastall gravity, there is a
relaxed assumption of conservation condition of the energy-momentum tensor
Rastall . Here the divergence of energy-momentum tensor is given as,

$\displaystyle\nabla_{\nu}T^{\mu\nu}=\lambda\nabla^{\mu}R$ (14)

Where R is the Ricci scalar. We see that in the flat spacetime where the Ricci
scalar vanishes, one recovers the usual conservation law. $\lambda$ is called
the Rastall parameter and measures a deviation from the standard classical
general relativity’s conservation law. The field equation in Rastall gravity
is given by,

$\displaystyle G_{\mu\nu}+\kappa\lambda g_{\mu\nu}=\kappa T_{\mu\nu}$ (15)

When $\lambda\rightarrow 0$, we get $\kappa=8\pi G_{N}$ which is the Newtonian
coupling constant while for non zero value of $\lambda$, it is called Rastall
coupling constant. By solving the modified field equation we can get the
metric for different types of spacetimes.

###  IV.1 Kerr-Newman spacetime in Rastall gravity

The complete derivation of the Reisnner-Nordstrom spacetime in Rastall gravity
surrounded by perfect fluid have been studied in Darabi . The metric for Kerr-
Newman spacetime in Rastall gravity surrounded by a perfect fluid can be given
by using the

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2Mr}{\rho^{2}}+\frac{Q^{2}}{\rho^{2}}-\frac{N_{s}r^{\frac{1+3\omega_{s}-6\kappa\lambda(1+\omega_{s})}{1-3\kappa\lambda(1+\omega_{s})}}}{\rho^{2}}\right)dt^{2}-2a\sin^{2}\theta\left(\frac{2Mr-Q^{2}+N_{s}r^{\frac{1+3\omega_{s}-6\kappa\lambda(1+\omega_{s})}{1-3\kappa\lambda(1+\omega_{s})}}}{\rho^{2}}\right)dtd\phi$
(16) $\displaystyle+$
$\displaystyle\frac{\rho^{2}}{\rho^{2}-2Mr+Q^{2}-N_{s}r^{\frac{1+3\omega_{s}-6\kappa\lambda(1+\omega_{s})}{1-3\kappa\lambda(1+\omega_{s})}}+a^{2}sin^{2}\theta}dr^{2}+\rho^{2}d\theta^{2}$
$\displaystyle+$ $\displaystyle
sin^{2}\theta\left[r^{2}+a^{2}+a^{2}sin^{2}\theta\left(\frac{2Mr-Q^{2}+N_{s}r^{\frac{1-3\omega_{s}}{1-3\kappa\lambda(1+\omega_{s})}}}{\rho^{2}}\right)\right]d\phi^{2}$

where $\omega_{s}$ is the state parameter of the fluid surrounding the black
hole and $N_{s}$ represents the surrounding field structure parameter.
Therefore the Horizon function from Eq. (16) is given as

$\displaystyle\Delta(r)=\rho^{2}-2Mr+Q^{2}-N_{s}r^{\frac{1+3\omega_{s}-6\kappa\lambda(1+\omega_{s})}{1-3\kappa\lambda(1+\omega_{s})}}+a^{2}sin^{2}\theta$
(17)

Similar derivation can also be found in Xu  where the Kerr-Newman spacetime is
derived in Anti-De Sitter spacetime. To find the CTC we require the
$g_{\phi\phi}$ term. From eqn(13), the $g_{\phi\phi}$ in Rastall gravity
surrounded by a perfect fluid is given as

$\displaystyle
g_{\phi\phi}=sin^{2}\theta\left[r^{2}+a^{2}+a^{2}sin^{2}\theta\left(\frac{2Mr-Q^{2}+N_{s}r^{\frac{1-3\omega_{s}}{1-3\kappa\lambda(1+\omega_{s})}}}{\rho^{2}}\right)\right]\
;\ \rho^{2}$ $\displaystyle=$ $\displaystyle r^{2}+a^{2}cos^{2}\theta$ (18)

We see that there are two independent parameters that can be examined in the
context of CTCs ($\kappa\lambda$ and $\omega$). We now look for CTC in Kerr-
Newman black hole for various equations of state. The value of $\omega_{s}$
characterizes each fluid (for e.g dust ($\omega_{s}=0$), dark energy
($\omega_{s}=-\frac{2}{3}$) and perfect dark matter
($\omega_{s}=-\frac{1}{3}$)). We substitute this value into Eq. (18) and plot
it in the equatorial plane against $x=r/M$ as shown in Fig. 8 for various
values of $\kappa\lambda$. We see that for each value of $\omega$, we can find
values of $\kappa\lambda$ for which CTCs are there since $g_{\phi\phi}$
becomes negative and also values of $\kappa\lambda$ for which $g_{\phi\phi}$
stays positive. We can therefore see that the model is equipped with a natural
protection mechanism against causality violation.

In the second part , Fig. 9, we keep $\kappa\lambda$ constant and plot
$g_{\phi\phi}$ component for different values of $\omega_{s}$. As we can see
from Fig. 9, for $\kappa\lambda=0.2$ there are CTC for every value of
$\omega_{s}$ portrayed in the figure, but when we increase the value of
$\kappa\lambda$ to 1.33 then CTC is seen only for $\omega_{s}=-0.8$. This
implies that if we go on increasing the value of $\kappa\lambda$ there is a
value above which for a given $\omega_{s}$ value there shouldn’t be any CTC.
This behaviour is not seen for $\omega_{s}=-1$ for which how much bigger the
value of $\kappa\lambda$ is there is always a CTC. This similar behaviour is
also seen for the negative value of $\kappa\lambda$. We see that while varying
$\omega$, we can have the possibility of eliminating CTCs. We, therefore, have
two handles that can be tweaked to choose scenarios that aid in eliminating
CTCs.

Figure 8:  CTC plot on the equatorial plane for the parameters: i) $a=0.5M$,
$q=0.25M$, $\omega_{s}=0$ and $N_{s}=1$ ii) $a=0.5M$, $q=0.25M$,
$\omega_{s}=-\frac{2}{3}$ and $N_{s}=1$ and iii) $a=0.5M$, $q=0.25M$,
$\omega_{s}=-\frac{1}{3}$ and $N_{s}=1$.

Figure 9:  CTC plot on the equatorial plane for the parameters: i) $a=0.5M$,
$q=0.25M$, $\kappa\lambda=0.2$ and $N_{s}=1$ ii) $a=0.5M$, $q=0.25M$,
$\kappa\lambda=1.33$ and $N_{s}=1$ and iii) $a=0.5M$, $q=0.25M$,
$\kappa\lambda=-5$ and $N_{s}=1$

###  IV.2 Correlation index of CTCs with weak energy condition

The expressions for the weak energy condition in the model under consideration
is given in Darabi ; Xu . The condition has a simple form given by
$(3\kappa\lambda(1+\omega)-3\omega)(1-4\kappa\lambda)\geq 0$. If the above
inequality is met, it implies WEC is preserved and WEC is broken in the
parameter space where the inequality is not met. We now adapt the index we had
defined in Eq. (2) to the case of Rastall gravity. Firstly we fix
$\theta=\pi/2$ and $\phi=0$. Then for a fixed value of radius (normalized with
the mass parameter M) $x=r/M$, we scan the entire ecosystem (parameter space)
consisting of angular momentum $a$, the charge parameter $Q$, equation of
state parameter $\omega$ and Rastall parameter $\kappa$. We then define the
index as,

$\mathcal{C}_{r}=\frac{\bigint\frac{f}{|f|}\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)dadQd\omega
d\kappa}{\bigint\left(\frac{g_{\phi\phi}}{|g_{\phi\phi}|}-1\right)dadQd\omega
d\kappa}$ (19)

The index $r$ indicates that this calculation is at a fixed radial coordinate
$r$. $f$ is the function whose sign decides whether the weak energy condition
is broken or not. We note that when we freeze $r$, it does not necessarily
imply that we are performing the integral over a fixed spacetime point. The
reason is that the spacetime manifolds are different when we choose a
different set of parameters, and there is little meaning in identifying
spacetime points across different spacetimes. So in a sense, we are averaging
over different spacetime solutions. Given below is the value of
$\mathcal{C}_{r}$ over few values of $r$ that definitely have CTCs in some
parameter range. Otherwise the integral becomes undefined since both
denominator and numerator go to $zero$, in which case numerical analysis
yields spurious values. The observations are the following. The index whose
allowed values are in the interval $[-1,1]$ is seen to be positive in Fig. 10.

Figure 10: Indices for correlation between WEC and the existence of CTCs in
the Kerr-Newman spacetime of Rastall gravity for values of x (scaled by factor
$10^{2}$) from $0.01$ to $0.2$. Figure 11: Indices for correlation between WEC
and the existence of CTCs in the Kerr-Newman spacetime of Rastall gravity a
fixed value of $x=0.15$ and Q goes from $0.3$ to $0.6$ in steps of 0.02

This strongly suggests that in the vicinity of a curvature singularity, the
CTCs tend to flourish when the WEC is preserved and if the goal is to
eliminate CTCs, the breaking of WEC is effective. In Fig. 11, we fix the value
of $r$ ($x=0.15$, where the CTCs are definitely present for a large volume in
the parameter space). We tabulate the value of the index for few values of
$Q$. The above index Eq. 19 is modified such that the integral is over $a$,
$\omega$ and $\kappa$ and not Q. We may define this index as $C_{rQ}$ with the
subscript indicating that we have evaluated the index for a fixed r and Q. In
Fig. 11, it is seen that the index is positive, indicating again that in the
vicinity of a singularity, a violation of WEC helps in the elimination of
CTCs.

##  V Conclusions

It was conjectured by Hawking that nature prevents the appearance of CTCs. The
problem due to the presence of CTCs is a fundamental pathology in a manner
similar to the pathology of singularity. A concrete proof to the question of
whether nature prevents CTCs might have to wait for inputs from Quantum
Gravity in a manner similar to the cosmic censorship hypothesis. We in this
work address few aspects that add to our understanding regarding CTCs. The
guiding principle was Hawking’s statement regarding the relation between CTCs,
singularities and energy conditions. In particular, we study the local
validity of Hawking’s statement regarding CTCs and energy conditions by
defining and computing a correlation index. We analyze two models that provide
contrasting situations. The non-commutative Kerr-Newman black hole is non-
singular whereas the Rastall gravity is singular. These two models provide an
excellent platform to understand the dependence between CTCs and energy
conditions. To quantify the correlations, we define a correlation index
between CTCs and energy conditions that averages over all the parameters of
the model in the region containing CTCs

Based on the analysis, we could demonstrate that Hawking’s statement can have
a local and global version. We could show that Hawking’s statement can be true
globally but need not hold locally. This is manifest in the cases analyzed in
this work on both non-commutative and Rastall versions of Kerr-Newman black
holes. The analysis of the non-commutative Kerr-Newman black hole yields that
in the absence of a singularity, the weak energy condition may or may not be
violated at the same spacetime point that has a CTC passing through it. The
violation of WEC is conducive in smearing out the singularity. We observe
that, based on the correlation index, in the absence of a singularity, the
CTCs thrive in a WEC violating environment close to the smeared out
singularity. For CTCs that are away from the smeared out singularity, the
thriving environment for the CTC is a WEC preserving situation. A violation of
WEC favours elimination of CTCs in these regions. An analysis of Rastall
gravity presents a contrasting situation where a curvature singularity is in
causal contact with the spacetime points having CTCs. Based on calculating the
correlation index, it is clear that the thriving environment for the CTCs is a
WEC preserving environment. This analysis is also suggestive that violation of
energy conditions aid CTC elimination. In effect, this is contrary to the
local version of Hawking’s statement wherein the WEC is violated in spacetimes
that has CTCs but has no singularities or pathological behaviour at infinity.

One can explore other models of Kerr-Newman spacetimes (the study in progress)
and other spacetimes admitting the CTCs to gain further understanding of the
interplay between CTCs and weak energy condition. The study can provide
insights and directions towards a more systematic and rigorous study for
characterizing CTCs. One can expect these studies to eventually lead to
developing theorems for CTCs in a manner similar to the singularity theorems.
Another area that shall be worth studying is how to extrapolate the
aforementioned correlation at the semi-classical level. This might be achieved
by replacing $T^{\mu\nu}$ with its expectation value corresponding to
appropriately chosen quantum states. We would like to address these issues in
near future.

## Acknowledgements:

S.K. is supported by University Grants Commission’s Faculty Recharge Programme
(UGC-FRP), Govt. of India, New Delhi, India. V.P. is supported by Shyama
Prasad Mukherjee Fellowship, CSIR, India. Some computations were carried out
using the resources of IUCAA, Pune, India.

All the authors contributed equally to this work and the authors’ names are
mentioned according to the alphabetical order of their last names.

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[jacob] Jacob Fox
Department of Mathematics, Stanford University
Stanford, CA, USA
jacobfoxstanfordedu
https://stanford.edu/ jacobfox/ [yuval] Yuval Wigderson
School of Mathematics, Tel Aviv University
Tel Aviv 69978, Israel
yuvalwigtauextauacil
http://www.math.tau.ac.il/ yuvalwig/


[1]Avik Roy 11footnotetext: on behalf of the FAIR4HEP collaboration:
https://fair4hep.github.io

# FAIR Principles for data and AI models in high energy physics research and
education

###### Abstract

In recent years, digital object management practices to support findability,
accessibility, interoperability, and reusability (FAIR) have begun to be
adopted across a number of data-intensive scientific disciplines. These
digital objects include datasets, AI models, software, notebooks, workflows,
documentation, etc. With the collective dataset at the Large Hadron Collider
scheduled to reach the zettabyte scale by the end of 2032, the experimental
particle physics community is looking at unprecedented data management
challenges. It is expected that these grand challenges may be addressed by
creating end-to-end AI frameworks that combine FAIR and AI-ready datasets,
advances in AI, modern computing environments, and scientific data
infrastructure. In this work, the FAIR4HEP collaboration explores the
interpretation of FAIR principles in the context of data and AI models for
experimental high energy physics research. We investigate metrics to quantify
the FAIRness of experimental datasets and AI models, and provide open source
notebooks to guide new users on the use of FAIR principles in practice.

##  1 Introduction

In order to make datasets findabile, accessibile, interoperable, and reusable
(FAIR), a set of data principles have been defined so that scientific datasets
could be readily reused by both humans and machines [1]. Originally envisioned
for preservation of scientific datasets, the FAIR principles have been
interpreted in the context of different kinds of digital objects, including
research software [2], notebooks [3], custom digital libraries [4], and
machine learning (ML) models [5, 6]. This work summarizes the multifaceted
interpretation and applications of the FAIR principles explored by the
FAIR4HEP collaboration in the context of High Energy Physics (HEP).

##  2 FAIR principles for HEP data

Interpretation and application of FAIR principles for HEP datasets has been
explored in Ref. [7]. Taking the publicly available $H\to b\bar{b}$ dataset
from CMS Open Data portal, this work performs a thorough evaluation of the
FAIR-readiness of this dataset using domain-agnostic metrics. Having the
established the FAIRness of this dataset, this work demonstrates how adhering
to these principles allow this dataset to be AI-ready. This dataset has been
made available in formats widely used by the broader ML community. Pedagogical
examples of usage of this dataset shows implementation of cutting edge methods
in ML for jet classification. Guided by the FAIR principles, the FAIR4HEP
collaboration has published multiple datasets for a variety of problems in
HEP.

  * •

Super Cryogeinc Dark Matter Search (Super CDMS) dataset [8] was obtained from
a prototype detector (Figure 1a) for recording phonon scattering from an
interacting particle. With six operating channels (Figure 1b), this detector
operates at 30 mK and records phonon pulse amplitude as a function of time in
each channel (Figure 1c). This dataset consists of timing information about
these pulses for 7000 individual measurements taken over 13 different impact
locations on the detector. This dataset is being used to reconstruct the
impact location based on these timing information and to obtain generative
models for simulation of particle intraction.

(a)

(b)

(c)

Figure 1: LABEL:sub@fig:CDMS-det A Super CDMS detector prototype and
LABEL:sub@fig:det-3d illustration of its different channels.
LABEL:sub@fig:waveform shows the waveforms recorded by different channels for
an example particle interaction. These images are taken from https://fair-
umn.github.io/FAIR-UMN-CDMS/

  * •

Laser Response from Electromagnetic Calorimeter (ECal) crystals at CMS [9] is
obtained from the thousands of calibrations performed during the course of
Run-2 between 2016 and 2018. The ECal calorimeter at CMS consists of almost
758k lead tungstate ($PbWO_{4}$) crystals. Exposure to prolonged radiation
during data taking modifies the transparency of these crystals, which is
partially recovered when such exposure is taken away (Figure 2). This dataset
contains information about the Ecal crystal’s laser response and can be used
to model this behavior as a function of integrated luminosity to predict
future behavior.

Figure 2: Transparency of CMS ECal crystals in different regions of
pseudorapidity $(\eta)$ during Run-2. This image is taken from Ref. [10]

##  3 FAIR Principles for AI Models

Application of the FAIR principles for AI models requires incorporating their
dependence on computing environments into the interpretation of FAIR
principles. We use the Interaction Network (IN) model [11] for classification
of jets from the aforementioned $H\to b\bar{b}$ dataset. Figure 3 shows the IN
model architecture and tabulates the default hyperparameters and data
dimensions. This network is built to train on graph data structure whose nodes
comprise of $N_{p}$ particle tracks, each with $P$ features, and $N_{v}$
secondary vertices, each with $S$ features, associated with the jet. The
physical description of each feature is given in Appendix C of ref. [11]. It
creates a fully connected directed graph with $N_{pp}=N_{p}(N_{p}-1)$ edges
for the particle tracks. A separate graph with $N_{vp}=N_{v}N_{p}$ generates
all possible connections between the particle tracks and the secondary
vertices.

Default IN hyperparameters
---
$(P,N_{p},S,N_{v})$ | $(30,60,14,5)$
Hidden layers | 3
Hidden layer | 60
dimension
$(D_{e},D_{o})$ | (20, 24)
Activation | ReLU

Figure 3: A Schematic diagram of the network architecture and dataflow in the
IN model. This image is taken from Ref. [11]. The choice of model
hyperparameters and input data dimensions for the baseline model is given in
the accompanying table.

The node level features for the track-track (track-vertex) graph are
transformed to edge level features via a couple of interaction matrices,
transformed via fully connected NNs, and finally transferred back to the
track-level representations by the interaction matrices. The trainable dense
MLP creates the post-interaction internal representation that are summed over
the tracks and then linearly combined to produce a two-dimensional, softmax-
transformed output to predict individual class probabilities.

###  3.1 Interoperability of the IN mode

To inspect that the model can infer under multiple deep learning frameworks,
we convert the IN model to ONNX [12] format and then to TensorRT [13] engine
from the ONNX format model. The ONNX format is designed to help the model
become portable between the commonly used deep learning formats like PyTorch,
TensorRT, TensorFlow etc. We also quantitatively evaluate the inference
performance before and after the conversion by measuring inference accuracy,
running time, and Area under the Curve (AUC) for the Receiver Operating
Characteristic (ROC) curve.

We separately infer three test groups using the PyTorch and ONNX model with a
batch size of 128. With all three methods of inference, the model’s
performance in terms of accurancy and the ROC-AUC score was identical. When
comparing the average time needed for inference per batch, the PyTorch model
took 1.226 ms, while inference with ONNX and TensorRT models took 12.64 ans
12.31 ms respectively. This order of magnitude difference can be understood
from the data transfer between the GPU device hosting these models and the
actual host of the data. The running time/epoch refers to the time used to run
one batch, including the data transfer between the two sides (device, host)
and the inference part in the GPU device. When we increase the batch size from
1 to 128, the running time becomes larger, because the time used for data
transfer also increases. The running time difference between the ONNX and
TensorRT models can be attributed to the differences in hardware acceleration.

Figure 4: GPU utility and throughput for inference with TensorRT engine with
different batch size

Here we also test whether different batch size will influence the inference
results using the model. We record the performance of the GPU utility and
throughput, i.e the number of inferences performed per second, with the change
of batch size in TensorRT format. The model’s performance was stable across
batch sizes. Figure 4 shows the results. GPU utility steadily increases up to
a batch size of 1000 and saturates at 100% beyind that. The performance of
throughput also matches with the GPU utility, it reaches maximum when batch
size reaches around 1200, which is the almost same period when the GPU utility
is fully occupied.

###  3.2 Streamlining FAIR AI Development

As AI models are often developed as an intricate software repository, an
automation tool to allow a streamlined organization of such repositories has
been developed. Cookiecutter Data Science [14] provides one such template,
specifically oriented at data science projects. It consists of a logical,
reasonably standardized, but flexible project structure hosted on GitHub for
performing and sharing data science work. We took inspiration from this and
created a fork of this template generator, called cookiecutter4fair [15], with
additional features to promote the adoption of our FAIR principles. It allows
structural organization of machine learning model repositories, development of
containers, and publishing the codebase with persistent identifiers like
digital object identifier (DOI). It includes comprehensive instructions to
build the necessary environments and allows coherently incorporating notebooks
for various studies. This tool has been used to restructure the codebase for
the IN model [16] and include notebooks that facilitate studies on its
interoperbility and explainability.

##  4 Pedagogical aspects of FAIR in HEP

FAIR principles can facilitate education in data science and machine learning.
Developing classroom contents aligned with the FAIR principles can amplify
their reach and effectiveness. On the other hand, introducing students with
FAIR principles and their interpretation for data and ML can help standardize
modern practices in digital object management.

  * •

Pedagogical Introduction to FAIR222https://github.com/yorkiva/FAIR-Exercises
is a series of notebooks developed based on the previously mentioned Super
CDMS dataset. It contains an introduction to FAIR principles, evaluation of
FAIRness of datasets, and builds up ML models, both traditional and neural
network based, and demonstrates their organization and management in line with
the FAIR principles. These notebooks include the analysis of the impact
location estimation problem with the CDMS dataset using regularized linear
regression, principal component analysis, deep neural netowrks, and
variational autoencoders.

  * •

Data Science for Physics333https://github.com/MIT-8s50/course is a course
designed to introduce modern concepts of machine learning via data analysis
problems in HEP. This course was originally developed as an online substitute
to the junior lab at MIT during the COVID pandemic and will be offered as a
full, independent class starting from Spring 2023. It includes as series of
lectures on different topics of physics, statistics, and data analysis and
requires analysis of physics data from LIGO, CMS, and CHIME experiments. An
open source version of this course will also be featured on the
MITx444https://mitxonline.mit.edu platform.

##  5 Conclusion

With a view to inspiring the modern community-wide standards for preservation
and management of digital objects, FAIR4HEP is developing HEP-specific
interpretation of FAIR and active implementation by developing FAIR datasets,
models, and tools. This work summarizes the ongoing efforts of the project
along these ventures.

## Acknowledgements

This work was supported by the FAIR Data program of the U.S. Department of
Energy, Office of Science, Advanced Scientific Computing Research, under
contract number DE-SC0021258.

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