# In-vacuo dispersion from $\kappa$-anti-de Sitter algebra

Iarley P. Lobo<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Chemistry and Physics, Federal University of Paraíba, Rodovia BR 079 - km 12,
58397-000 Areia-PB, Brazil  Physics Department, Federal University of Lavras,
Caixa Postal 3037, 37200-900 Lavras-MG, Brazil

###### Abstract

Based on deformed translations in $\kappa$-anti-de Sitter algebra, we derive a
delay in the time of detection between a soft and a hard photon which are
simultaneously emitted at a distant event, at first order in the quantum
gravity parameter. In the basis analyzed, the trajectories are undeformed and
the effect depends exclusively on the symmetry properties of the quantum
algebra. The time delay depends linearly on the energy of the hard particle
and has a sinusoidal dependence on the redshift of the source.

##  I Introduction

The quantization of gravity is one of the most challenging problems of
science, and that has been examined by some generations of physicists and
mathematicians in the past decades. Despite some success in constructing
theories that capture different aspects of the expected nature of a quantum
spacetime or of the quantum nature of the gravitational interaction, like
canonical quantum gravity Kiefer:2004xyv , string theoryPolchinski:1998rq ;
Polchinski:1998rr , loop quantum gravity Ashtekar:2021kfp , causal dynamical
triangulation Loll:2019rdj  and asymptotic safety Eichhorn:2018yfc , no
experimental verification that validates any of these approaches has been
found yet.

A common result in many of these approaches is that in the asymptotic limit
that approaches the low gravity regime or low energy scale, the physical
observables are given by perturbations of equations of special/general
relativity or quantum mechanics Amelino-Camelia:2008aez . Furthermore, that
such corrections are suppressed by a scale that is usually proportional to
planckian units. As this regime is so far reaching when we consider typical
energies of elementary particles in accelerators, the quantization of the
gravitational interaction has been assumed as a purely mathematical and
philosophical curiosity for many years. However, with the development of the
area of astroparticle physics, this picture has changed, since we are not only
able to detect particles with higher energies, but also the length in which
these particles propagate are much longer than in accelerators: a property
that is responsible for amplifying the tiny planckian corrections Amelino-
Camelia:2008aez ; Addazi:2021xuf . These new inputs, along with improvement in
detection precision by observatories, have led to the dawn of the area of
quantum gravity phenomenology by the end of the 20th century, whose capability
of constraining quantum gravity-inspired corrections with Planck scale
sensitivity has been achieved in the past few years fermigbmlat:2009nfe  and
presents many appealing windows to be explored in the next years
AlvesBatista:2023wqm .

The most explored of these avenues is the phenomenon called in-vacuo
dispersion Addazi:2021xuf , which is the dependence of massless particles’
speed with their energy, and is a prediction of several approaches to quantum
gravity Amelino-Camelia:2008aez ; Gambini:1998it ; Amelino-Camelia:2016gfx
and other forms of Lorentz Invariance Violation (LIV) scenarios, like
extensions of the standard model theories Kostelecky:2003fs . The standard
assumption in the derivation of in-vacuo dispersion bounds is the Jacob-Piran
formula Jacob:2008bw ; Ellis:2002in , that gives the dependence on particles
energies and propagation distance dependence for the time delay from the
arrival of massless particles when they are simultaneously emitted in a given
source.

As was shown in Amelino-Camelia:2012vzf ; Rosati:2015pga , the main
assumptions of the Jacob-Piran formula Jacob:2008bw  is the minimal coupling
between the particles’ momentum and local violation of Poincaré symmetry
(rotations, boosts and translations). The relaxation of these hypotheses has
been explored in some recent works on curvature-triggered effects Amelino-
Camelia:2020bvx ; Amelino-Camelia:2022pja ; Amelino-Camelia:2023srg  and other
ansatze for defining such delay Pfeifer:2018pty . For instance, if instead of
violation, one incorporates in the derivation a deformation of local Poincaré
symmetry that preserves the quantum gravity departures of relativistic
equations, one actually finds corrections to Jacob-Piran formula, that could
actually better fit the data from astrophysical events, as has been shown in
the preliminary work Bolmont:2022yad . This deformation has been initially
shown for the special case of assuming that the spacetime has a Planck-scale-
deformed de Sitter structure, where a deformation of the de Sitter algebra has
been the background of the derivation, which besides presenting deformed
worldlines (that coincide with Jacob-Piran’s ansatz), presents also deformed
translation transformations, which governs the communication between the
distant frames given by the emission and detection events. We should stress
that calculating such kind of effect in a Deformed Special Relativity (DSR)
scenario Amelino-Camelia:2000stu  using modified translations (following the
lessons of the principle of relative locality Amelino-Camelia:2011lvm ) is of
utmost importance to avoid non-local pathologies pointed out by different
authors Schutzhold:2003yp ; Hossenfelder:2010tm , whose solutions were
discussed in Amelino-Camelia:2011uwb .

Although the de Sitter case has been extensively explored in the literature,
deformations of the anti-de Sitter (AdS) algebra from the point of view of
time delays has been neglected, despite the dedicated theoretical work that
has been devoted to rigorously formulating the features of a quantum
deformation of anti-de Sitter algebra Ballesteros:2016bml , in which not only
the full set of isometries are explored, but also the co-product structure in
AdS space for multiparticle systems and its carrollian and galilean limits
Ballesteros:2021dob .

In this paper, we fill this gap and derive a time delay formula, that depends
on the energy of a hard (high energy) massless particle simultaneously emitted
with a soft one (low energy) in a given spacetime event, following the recipe
discussed in Amelino-Camelia:2012vzf , but applied to the quantum deformation
of the anti-de Sitter algebra described in Ballesteros:2016bml , at first
order in the quantum gravity scale.

The paper is organized as follows. In section II, we revise the classical
mechanics of anti-de Sitter spacetime by the derivation of the conserved
charges from the Killing equations in cosmological coordinates and the
associated Casimir that is the mass shell. In section III, we find the
dependence of the symmetry generators and Casimir charge of the $\kappa$-AdS
algebra in terms of the classical ones. In section IV, we express the
translation generators in conformal coordinates, which are easier to handle
computationally. In section V, we use the deformed translations to derive the
time delay of the $\kappa$-AdS spacetime from the basis rigorously explored in
Ballesteros:2016bml . We draw our final remarks in section VI. We are assuming
natural units, such that $\hbar=c=1$.

##  II Classical Mechanics and Symmetries in Anti-de Sitter Space

Before we proceed to the derivation of the deformed kinematics of particles in
anti-de Sitter geometry, we revisit its simpler, classical counterpart. In
this section, we are mostly relying on purely algebraic tools to derive our
results, however our starting point is the derivation of symmetry generators
in AdS space. As is usually the case of similar analyses, we shall rely on the
$1+1$ dimensional example. The generators of symmetries are conserved charges,
therefore they should be found from the Killing vectors in a given spacetime.
For AdS space, we consider the metric given by the following line element of
FLRW type Magueijo:2009ff

$ds^{2}=dt^{2}-\cos^{2}(\lambda t)dx^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,,$ (1)

where the parameter $\lambda$ is related to the anti-de Sitter curvature as
$\Lambda=-\lambda^{2}$. We solve the Killing equation
$\nabla_{\mu}K_{\nu}+\nabla_{\mu}K_{\mu}=0$, for the Killing vector $K_{\mu}$,
which in $1+1$ dimensions presents only three solutions, which are responsible
for describing time translation $K^{\mu}_{(0)}$, space translation
$K^{\mu}_{(1)}$ and spacetime boost $K^{\mu}_{(2)}$. The energy $E_{0}$,
momentum $P_{0}$ and boost generator $N$ are conserved charges, given by

$\displaystyle E_{\circ}$ $\displaystyle=$ $\displaystyle
K^{\mu}_{(0)}p_{\mu}=\cosh{(\lambda x)}p_{0}+\sinh{(\lambda x)}\tan{(\lambda
t)}p_{1}\,,$ (2) $\displaystyle P_{\circ}$ $\displaystyle=$ $\displaystyle
K^{\mu}_{(1)}p_{\mu}=p_{1}\,,$ (3) $\displaystyle N_{\circ}$ $\displaystyle=$
$\displaystyle K^{\mu}_{(2)}p_{\mu}=\frac{\sinh{(\lambda
x)}}{\lambda}p_{0}+\cosh{(\lambda x)}\frac{\tan{(\lambda
t)}}{\lambda}p_{1}\,,$ (4)

where we use the subscript “$\circ$” in these expressions to stress that we
are considering the classical anti-de Sitter generators.

Considering that the generalized coordinates $(t,x,p_{0},p_{1})$ are
canonical, i.e., that

$\\{t,p_{0}\\}=1=\\{x,p_{1}\\}\,,$ (5)

and zero otherwise, we verify that the above charges define a Lie algebra as
follows

$\displaystyle\\{E_{\circ},P_{\circ}\\}$ $\displaystyle=$
$\displaystyle\lambda^{2}N_{\circ}=-\Lambda N_{\circ}\,,$ (6)
$\displaystyle\\{N_{\circ},P_{\circ}\\}$ $\displaystyle=$ $\displaystyle
E_{\circ}\,,$ (7) $\displaystyle\\{N_{\circ},E_{\circ}\\}$ $\displaystyle=$
$\displaystyle P_{\circ}\,,$ (8)

where the Poisson brackets of two phase space quantities $F$ and $G$ are

$\\{F,G\\}=\frac{\partial F}{\partial x^{\mu}}\frac{\partial G}{\partial
p_{\mu}}-\frac{\partial F}{\partial p_{\mu}}\frac{\partial G}{\partial
x^{\mu}}=\frac{\partial F}{\partial\eta}\frac{\partial G}{\partial
p_{0}}+\frac{\partial F}{\partial x}\frac{\partial G}{\partial
p_{1}}-\frac{\partial F}{\partial p_{0}}\frac{\partial G}{\partial
t}-\frac{\partial F}{\partial p_{1}}\frac{\partial G}{\partial t}$ (9)

This is the known algebra of anti-de Sitter generators in $1+1$-dimensions,
$\text{AdS}_{\Lambda}$ Ballesteros:2016bml . The Casimir operator related to
this algebra is given by Eq.(2.2) of Ballesteros:2016bml  and in cosmological
coordinates read

$C_{\circ}=E_{\circ}^{2}-P_{\circ}^{2}-\lambda^{2}N_{\circ}^{2}=p_{0}^{2}-\sec^{2}{(\lambda
t)}p_{1}^{2}\,,$ (10)

which is simply the norm $g^{\mu\nu}p_{\mu}p_{\nu}$ using the metric defined
by the line element (1).

###  II.1 Classical Trajectories

The trajectories can be easily found using the Casimir (10) as the generator
of the parametrization evolution (here “dot” means derivative with respect to
an affine parameter)

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\\{t,C_{\circ}\\}=2p_{0}\,,$ (11) $\displaystyle\dot{x}$
$\displaystyle=$ $\displaystyle\\{x,C_{\circ}\\}=-2p_{1}\sec^{2}{(\lambda
t)}\,,$ (12) $\displaystyle\dot{p_{0}}$ $\displaystyle=$
$\displaystyle\\{p_{0},C_{\circ}\\}=2\lambda p_{1}^{2}\sec^{2}{(\lambda
t)}\tan^{2}{(\lambda t)}\,,$ (13) $\displaystyle\dot{p_{1}}$ $\displaystyle=$
$\displaystyle\\{p_{1},C_{\circ}\\}=0\,.$ (14)

To derive an equation of the kind $x(t)$, we need to express the equation
$dx/dt=\dot{x}/\dot{t}$ using Eqs.(11) and (12). This gives an equation that
depends on $t$ and $p_{0}$. However, we can also express $p_{0}$ as a function
of $t$ for the trajectories of particles with mass $m$, i.e., those for which
$C_{\circ}=m^{2}$ in (10). This gives

$\frac{dx}{dt}=-\frac{p_{1}\sec^{2}{(\lambda
t)}}{\sqrt{m^{2}+\sec^{2}{(\lambda t)}p_{1}^{2}}}\,.$ (15)

This equation can be solved analytically for any mass $m$. For the massless
case $m\rightarrow 0$ and out-going particles $p_{1}<0$, we find

$x(t)=x_{0}+\frac{\operatorname{arctanh}{[\sin(\lambda t)]}}{\lambda}\,.$ (16)

This is the affine null geodesic of AdS spacetime.

###  II.2 Classical Translations

Using the canonical brackets (5), we can construct the infinitesimal
translations in time and space, and the boost by the action of the generators
$E_{\circ}$ (2), $P_{\circ}$ (3) and $N_{\circ}$ (4)

$\displaystyle\\{t,E_{\circ}\\}$ $\displaystyle=$ $\displaystyle\cosh{(\lambda
x)}\,,$ (17) $\displaystyle\\{t,P_{\circ}\\}$ $\displaystyle=$ $\displaystyle
0\,,$ (18) $\displaystyle\\{t,N_{\circ}\\}$ $\displaystyle=$
$\displaystyle\frac{\sinh{(\lambda x)}}{\lambda}\,,$ (19)
$\displaystyle\\{x,E_{\circ}\\}$ $\displaystyle=$ $\displaystyle\sinh{(\lambda
x)}\tan{(\lambda t)}\,,$ (20) $\displaystyle\\{x,P_{\circ}\\}$
$\displaystyle=$ $\displaystyle 1\,,$ (21) $\displaystyle\\{x,N_{\circ}\\}$
$\displaystyle=$ $\displaystyle\cosh{(\lambda x)}\frac{\tan{(\lambda
t)}}{\lambda}\,.$ (22)

Similar rules could be found for momenta, but since we will not use them in
this paper, we shall not dwell with them here. In any case, endowed with these
tools, we have finite transformations $T_{G;a}$ generated by a charge $G$,
parametrized by the quantity $a$, acting of a phase space quantity
$X=X(t,x,p_{0},p_{1})$ as (called an exponentiation)

$T_{G;a}\triangleright X=e^{-aG}\triangleright
X=\sum_{n=0}^{\infty}\frac{(-a)^{n}}{n!}\\{X,G\\}_{n}\,,$ (23)

where $\\{X,G\\}$ is recursively defined as

$\displaystyle\\{X,G\\}_{0}$ $\displaystyle=$ $\displaystyle X\,,$ (24)
$\displaystyle\\{X,G\\}_{n}$ $\displaystyle=$
$\displaystyle\\{\\{X,G\\}_{n-1},G\\}\,.$ (25)

This allows us to translate individual events and trajectories in AdS
spacetime, for example. In the following, we shall consider the deformed case
and derive similar analytical mechanics results using a deformed version of
the AdS algebra.

##  III Quantum Deformation of Anti-de Sitter Algebra

In this section, we rely mainly on results reported in Ballesteros:2016bml ,
that describes the $\kappa$-AdS algebra, as a deformation of the anti-de
Sitter algebra by the presence of a parameter with dimensions of energy
$\kappa^{-1}\doteq\ell$, which supposedly is proportional to the invarian
quantum gravity energy scale. In $1+1$-dimensions, this algebra presents the
following Poisson brackets

$\displaystyle\\{N,E\\}$ $\displaystyle=$ $\displaystyle P\,,$ (26)
$\displaystyle\\{E,P\\}$ $\displaystyle=$ $\displaystyle\lambda^{2}N\,,$ (27)
$\displaystyle\\{N,P\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell}(1-e^{-2\ell
E})-\frac{\ell}{2}P^{2}-\frac{\ell\lambda^{2}}{2}N^{2}\approx
E-\ell\left[E^{2}-\frac{P^{2}}{2}-\frac{\lambda^{2}}{2}N^{2}\right]\,,$ (28)

where we are considering only first order terms in $\ell$. Besides that, the
deformed Casimir is

$C=\frac{1}{2\ell^{2}}(\cosh{(\ell E)}-1)-e^{\ell
E}(P^{2}+\lambda^{2}N^{2})\approx
E^{2}-P^{2}-\lambda^{2}N^{2}+\ell\left(EP^{2}-\lambda^{2}EN^{2}\right)\,.$
(29)

As can be seen, we can treat this formalism as $\kappa$-deformation of the AdS
algebra (which is recovered when $\ell=\kappa^{-1}\rightarrow 0$), or as a
$\Lambda$-deformation of the $\kappa$-Poincaré algebra Ballesteros:2021dob
(which is recovered when $\lambda=\sqrt{-\Lambda}\rightarrow 0$).

Considering the former viewpoint, we can also treat the generators $(E,P,N)$
as $\kappa$-deformations of the AdS ones considered in the previous section
$(E_{\circ},P_{\circ},N_{\circ})$. At this point, we need to make some
simplificatons and bear our intuition on similar approaches previously studied
in the literature. Due to the isotropy of this approach, we shall assume that
the space translation generator is not modified by the $\kappa$-corrections
and is given by the anti-de Sitter one considered in the previous section,
i.e., $P=P_{\circ}$. We should stress that this same property applies to the
$\kappa$-de Sitter algebra considered by other authors, as can be seen in
Eq.(19) of Amelino-Camelia:2012vzf . As we will see, this is a valid
representation of this algebra. The rest of the analysis is done according to
the following ansatz

$\displaystyle N$ $\displaystyle=$ $\displaystyle N_{\circ}+\ell
f(E_{\circ},P_{\circ},N_{\circ})\,,$ (30) $\displaystyle E$ $\displaystyle=$
$\displaystyle E_{\circ}+\ell(\alpha E_{\circ}^{2}+\beta
E_{\circ}P_{\circ}+\gamma P_{\circ}^{2}+\delta\lambda
E_{\circ}N_{\circ}+\epsilon\lambda
P_{\circ}N_{\circ}+\phi\lambda^{2}N_{\circ}^{2})\,,$ (31) $\displaystyle P$
$\displaystyle=$ $\displaystyle P_{\circ}\,,$ (32)

where $g$ is a function of the undeformed charges, and
$(\alpha,\beta,\gamma,\delta,\epsilon,\phi)$ are dimensionless parameters that
shall be fixed a posteriori. We can find the form of the function $f$ by
substituting Eqs.(30)-(32) in the relation (27). Using the underformed
brackets (6), (7), (8) in this expression, we find that the function $g$ must
have the following form

$f=2\alpha N_{\circ}E_{\circ}+\beta
N_{\circ}P_{\circ}+\frac{\delta}{\lambda}(\lambda^{2}N_{\circ}^{2}+E_{\circ}^{2})+\frac{\epsilon}{\lambda}P_{\circ}E_{\circ}+2\phi
N_{\circ}E_{\circ}\,.$ (33)

Now, we use this form of $f$ into $N$ (40) and consider the bracket
$\\{N,E\\}$ in (26) (where we also use the brackets involving the underformed
generators (6), (7), (8)). Isolating the factors that multiply the product of
independent charges
$E_{\circ}^{2},\,N_{\circ}^{2},\,P_{\circ}^{2},\,N_{\circ}P_{\circ},\,N_{\circ}E_{\circ},\,E_{\circ}P_{\circ}$,
we find the following set of conditions

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\delta=0\,,$ (34)
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle-2\alpha-\gamma\,,$ (35)

and $\epsilon$ remains an arbitrary parameter. We consider this simplification
in the definition of $N$ and $E$, given by (30) and (31), to analyze the
bracket $\\{N,P\\}$. Therefore, using (6), (7), (8) into (28), we isolate the
factors that multiply the products of underformed charges to further constrain
the remaining parameters as

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\beta=\delta=0\,,$ (36)
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Rightarrow\phi=-\frac{1}{2}\,,$ (37)
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle\text{arbitrary}\,.$
(38)

This leads us to the following expression of the deformed charges in terms of
the undeformed ones

$\displaystyle E$ $\displaystyle=$ $\displaystyle
E_{\circ}+\ell\left(\frac{P_{\circ}^{2}}{2}-\frac{\lambda^{2}}{2}N_{\circ}^{2}+\epsilon\lambda
P_{\circ}N_{\circ}\right)\,,$ (39) $\displaystyle N$ $\displaystyle=$
$\displaystyle
N_{\circ}+\ell\left(\frac{\epsilon}{\lambda}P_{\circ}E_{\circ}-N_{\circ}E_{\circ}\right)\,,$
(40) $\displaystyle P$ $\displaystyle=$ $\displaystyle P_{\circ}\,.$ (41)

To check the consistency of this result, it is sufficient to check that
Eqs.(26), (27), (28) are satisfied (using the undeformed expressions (6), (7),
(8)). We can express these charges in cosmological coordinates
$(t,x,p_{0},p_{1})$ by using Eqs.(2), (3), (4). It is interesting to notice
the presence of an arbitrary parameter $\epsilon$, that as we will see, will
play a non-trivial role in the behavior of the time delay. Also, we would like
to remark that in other approaches that analyze the de Sitter case, for
example in Amelino-Camelia:2012vzf , arbitrary parameters are also present.

Endowed with these expressions, we can calculate the deformed Casimir operator
in terms of the undeformed charges as well. Substituting Eqs.(39), (40), (41)
into the Casimir charge (29), we find

$C=E^{2}-P^{2}-\lambda^{2}N^{2}+\ell\left(EP^{2}-\lambda^{2}EN^{2}\right)+{\cal
O}(\ell^{2})=E_{\circ}^{2}-P_{\circ}^{2}-\lambda^{2}N_{\circ}^{2}+{\cal
O}(\ell^{2})=C_{\circ}+{\cal O}(\ell^{2})\,.$ (42)

Surprisingly, we do not find first order corrections on the Casimir operator
when we express it in terms of the undeformed charges (and this is a
coordinate independent statement). For this reason, we would find, for
example, that the dispersion relation of particles in this scenario is given
by the undeformed expression (10). This implies that the trajectory of a
massless particle is unmodified and is given by (16). A similar issue on the
difference between the symmetry generators and the energy and momentum of
photons is also discussed in footnote 4 of Carmona:2022pro .

As we shall see, despite the absence of a deformed trajectory, we will be able
to find a time delay at first order in $\ell$ due to the presence of deformed
translations generated by $E$ and $P$. In fact, we can express the
infinitesimal transformation generated by these charges by using their
dependence with the undeformed ones and the actions already calculated in
(17)-(22).

##  IV Translations in Conformal Coordinates

Some lessons considered in previous investigations by other authors on the de
Sitter case can be applied to ours, as for example, the fact that the use
conformally flat coordinates simplifies a lot the computations needed to find
finite translations. The AdS line element can be cast in conformally flat
coordinates by the definition of a conformal time $\eta$

$d\eta=\sec{(\lambda
t)}dt\Rightarrow\eta=\frac{\operatorname{arctanh}{[\sin(\lambda
t)]}}{\lambda}\,,$ (43)

in which the line element (1) assumes the form
$ds^{2}=\operatorname{sech}^{2}{(\lambda\eta)}(d\eta^{2}-dx^{2})$. As the
undeformed charges (2)-(4) are scalars from the point of view of coordinate
transformations, then in order to express them in coordinates $(\eta,x)$, we
must simply perform the transformation (43) on the time coordinate and
correspondingly express the momenta $p_{\mu}$ in the novel coordinates (which
gives the same result of transforming $K^{\mu}$ to new coordinates as
calculating $K^{\prime}{}^{\mu}p^{\prime}_{\mu}$). Let us dub the energy and
momentum in these new coordinates as $(\Omega,\Pi)$, respectively. One can
easily verify that the following relation is valid

$\displaystyle p_{0}$ $\displaystyle=$
$\displaystyle\frac{d\eta}{dt}\Omega=\cosh{(\lambda\eta)}\Omega\,,$ (44)
$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle\Pi\,.$ (45)

For this reason, the undeformed conserved charges read

$\displaystyle E_{\circ}$ $\displaystyle=$
$\displaystyle\cosh{(\lambda\eta)}\cosh{(\lambda
x)}\Omega+\sinh{(\lambda\eta)}\sinh{(\lambda x)}\Pi\,,$ (46) $\displaystyle
P_{\circ}$ $\displaystyle=$ $\displaystyle\Pi\,,$ (47) $\displaystyle
N_{\circ}$ $\displaystyle=$
$\displaystyle\frac{\cosh{(\lambda\eta)}\sinh{(\lambda
x)}\Omega}{\lambda}+\frac{\sinh{(\lambda\eta)}\cosh{(\lambda
x)}\Pi}{\lambda}\,.$ (48)

In fact, using this representation of the charges, the Casimir (29) takes the
form

$C_{\circ}=\cosh^{2}{(\lambda\eta)}(\Omega^{2}-\Pi^{2})\,,$ (49)

as expected. Besides that, as the analysis done in the previous section
regarding the $\kappa$-AdS algebra was performed in a coordinate-invariant
way, the same results discussed previously are preserved in these conformally
flat coordinates. In these coordinates, the massless trajectories are simply
straight lines.

##  V Time delay in $\kappa$-Anti-de Sitter Space

In order to connect distant events by the emission and detection of light
rays, we need to consider these events as if they were local to relatively
distant observers, for instance Alice and Bob. We shall consider a thought
experiment in which massless particles with different energies are emitted in
a single event that coincides with the origin of the coordinate system
associated with the observer Alice and are detected at the $x$-axis of the
observer Bob with a delay in time coordinates.

In order to find this delay, we need to translate the geodesics from Alice’s
coordinates to Bob’s coordinates, which include translating the inicial
conditions from Alice’s coodinate system to Bob’s. A difference in the
location of the emission event will appear in Bob’s coordinates due to
deformed translations and this is intimately related the principle of relative
locality, in which the location, in spacetime, of a distant event is described
by a distant observer as if nonlocal.111More precisely, relative locality is
related to modified interactions, which is absent in our case, however we use
it as synonym of considering deformed translations, since they lead to effects
in which the location of events is an observer-dependent statement. This is
just an apparent nonlocality, since for an observer close enough to the
emission event (Alice), it takes place objectively at a single spacetime
event.

For this reason, we will need to translate the origin of Alice’s coordinate
system to express it in terms of Bob’s coordinates. This is done by acting
with a finite translation as defined in (23) as222We define the translation in
spacetime as a translation in time followed by a space translation.

Bob $\displaystyle=$ $\displaystyle e^{-a_{x}P}\triangleright
e^{-a_{\eta}E}\triangleright\text{Alice}\Rightarrow$ (50)
$\displaystyle(\eta_{0}^{B},x_{0}^{B})$ $\displaystyle=$ $\displaystyle
e^{-a_{x}P}\triangleright e^{-a_{\eta}E}\triangleright(0,0)\,.$ (51)

In this representation, the coordinates $(\eta,x,\Omega,\Pi)$ are canonical
since the charges are scalar quantities and the Poisson brackets are invariant
under coordinate transformations. From the definition of $E$ and $P$ in (39),
(41), along with the undeformed charges in conformally flat coordinates (46),
(47), (48), we notice that space translations generated by $P$ are undeformed,
do not affect the time coordinate and simply shifts the spatial one, while
time translations presents an energy-dependence. Performing the series of
operations defined in (23), we exponentiate the translations, and after a
tedious computation, we find the following result when applied to the point
$(0,0)$ (verified until 6th order perturbation in $a_{\eta}$)

$\displaystyle\eta_{0}^{B}$ $\displaystyle=$
$\displaystyle\eta_{0}^{B}|_{\ell=0}+{\cal O}(\ell^{2})\,,$ (52)
$\displaystyle x_{0}^{B}$ $\displaystyle=$ $\displaystyle-
a_{x}-\frac{\ell\Pi}{\lambda}\sin{(\lambda a_{\eta})}[1-\epsilon\tan{(\lambda
a_{\eta})}]+{\cal O}(\ell^{2})\,,$ (53)

where $\eta_{0}^{B}|_{\ell=0}$ is the undeformed translated origin, in Bob’s
coordinates, in classical anti-de Sitter space, that we shall not dwell with
in this paper. This is the central set of equations of this paper, and from it
the time delay is calculated straightforwardly.

In fact, in this gedankenexperiment, two massless particles (that we simply
call photons) are emitted simultaneously from Alice’s origin: a soft photon
with momentum $\Pi_{s}$, whose kinematics is unaffected by
$\kappa$-corrections, and a hard photon with momentum $\Pi_{h}$, whose
kinematics presents Planck-scale corrections. As we calculated that both
particles present the same dispersion relation at first order in $\ell$ (in
terms of classical generators), their trajectories are the same concerning the
soft and hard photons. Since Alice and Bob’s origin are connected by the soft
photon trajectory, we set $\eta_{0}^{B}|_{\ell=0}=-a_{x}$. Below, we express
the photons trajectories in the conformally flat patch using Alice and Bob’s
coordinates, where we use the superscript “$A$” and “$B$” to represent them

$\displaystyle x^{A,B}_{s}$ $\displaystyle=$ $\displaystyle
x^{A,B}_{0;s}+\eta^{A,B}_{s}-\eta^{A,B}_{0;s}\,,$ (54) $\displaystyle
x^{A,B}_{h}$ $\displaystyle=$ $\displaystyle
x^{A,B}_{0;h}+\eta^{A,B}_{h}-\eta^{A,B}_{0;h}+{\cal O}(\ell^{2})\,.$ (55)

The time delay is a difference in the time coordinates of both particles when
they reach the same location in Bob’s coordinates, i.e., when
$x^{B}_{s}=x^{B}_{h}$ in (54), (55). We can assume, without loss of
generality, that the soft particle reaches the origin of Bob’s coordinate
system, i.e., $\eta^{B}_{s}=0$ in (54). Using these conditions, along with
(52), (53) we subtract Eqs.(54) and (55) to find

$\Delta\eta^{B}=\eta^{B}_{h}-\eta^{B}_{s}=\eta^{B}_{h}=x^{B}_{0;s}-x^{B}_{0;h}=\frac{\ell}{\lambda}\Pi_{h}\sin{(\lambda
a_{\eta})}[1-\epsilon\tan{(\lambda a_{\eta})}]\,.$ (56)

From this expression, we identify a problem with the freedom to consider the
parameter $\epsilon$, as it introduces a discontinuous behavior in the time
delay whenever $\epsilon\neq 0$. The reason is that the tangent function
diverges at $\pi/2$, (therefore at $\lambda a_{\eta}=\pi/2)$). As such
discontinuity brings a nonphysical behavior (since there’s nothing special
such value of $a_{\eta}$), we set $\epsilon=0$. We depict the relative
locality effect in conformal coordinates in Fig.1. As can be seen, the
emission event (local in Alice’s coordinate) appears nonlocal in Bob’s
coordinates. This is the origin of the time delay effect even with undeformed
trajectories.

Figure 1: $\kappa$-AdS trajectories of photons in Bob’s conformal coordinates.
The red/dashed line describes the soft photon emitted at
$x^{B}_{0;s}=-a_{x}=-a_{\eta}=-5$ in Bob’s coordinates. Although the emission
occurs at the same spacetime location in Alice’s coordinates
$(t^{A}_{0;s},x^{A}_{0;s})=(t^{A}_{0;h},x^{A}_{0;h})=(0,0)$, it seems
displaced in Bob’s coordinates due to relative locality. The blue/solid line
describes the trajectory of the hard photon for $|p|=1$, $\ell=0.2$ and
$\lambda=1$.

Due to its oscillatory behavior, the time delay can be positive or negative,
depending on the distance between the detector and the source. In Fig.2, we
describe the conformal time delay as function of $a_{\eta}$, where we located
the detector in the origin of Bob’s coordinate system. This oscillatory
behavior does not obey exactly the intuition of the time delays literature, in
which the effect should grow with the distance between the source and the
detector, but the point is that the trajectories in this case are classical,
so we depend exclusively on the symmetry properties of the anti-de Sitter
space, which as we know presents an exotic behavior, like the presence of
closed timelike curves and that some geodesics usually converge, expand and
reconverge, so such oscillatory behavior should not be a complete surprise.
Such general properties are discussed in section 5.2 of Hawking:1973uf .

Figure 2: Conformal time delay as a function of the translation parameter
$a_{\eta}$ for $\lambda=1=|p|$, $\ell=0.2$. We can have positive or negative
delays due to the oscillatory behavior of $\kappa$-AdS symmetries.

In order to compare this result with the Jacob-Piran ansatz, we express it in
terms of the redshift function and the cosmological time. By Eq.(43), we have
that the relation between the redshift $z$, cosmological scale factor $a(t)$
and conformal time is the following

$\displaystyle\frac{1}{1+z}$ $\displaystyle=$ $\displaystyle
a(t)=\cos{(\lambda t)}=\operatorname{sech}{(\lambda\eta)}\Rightarrow$ (57)
$\displaystyle\eta(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\operatorname{arcsech}{\left(\frac{1}{1+z}\right)}\,.$
(58)

For this reason, the translation parameter is related to the redshift as
$a_{\eta}=\lambda^{-1}\operatorname{arccosh}{(1+z)}$.

Considering that $t=\lambda^{-1}\arcsin{(\tanh{(\lambda\eta)})}$ and that, as
we discussed below Eq.(15), we have $\Pi=p_{1}\doteq p<0$, we express the time
delay in cosmological coordinates as (we see that $\Delta\eta=\Delta t+{\cal
O}(\ell^{2})$)

$\displaystyle\Delta
t=-\frac{\ell|p|}{\lambda}\sin{\left(\operatorname{arccosh}{(1+z)}\right)}\,.$
(59)

This is the main result of our paper. It is the description of a time delay
from the $\kappa$-anti-de Sitter algebra. It does not present deformed
trajectories, yet due to the presence of deformed translations, a time delay
is presented.

For small redshifts, this expression assumes the form

$\Delta t|_{z\ll
1}\approx-\frac{\ell|p|}{\lambda}\left(\sqrt{2z}-\frac{5z^{3/2}}{6\sqrt{2}}\right)\,.$
(60)

The same kind of powers in redshift would appear had one considered the Jacob-
Piran formula Jacob:2008bw  for an anti-de Sitter space with scale factor
$a(t)=\cos{(t)}$ and a modified dispersion relation of the kind
$p_{0}^{2}-\sec^{2}{(\lambda t)}p^{2}+\ell\sec^{3}{(\lambda t)}|p|^{3}=0$
(where $H(t)=a(t)^{-1}da(t)/dt$)

$\Delta t_{JP,z\ll
1}=\ell|p|\int_{0}^{z}\frac{1+\bar{z}}{H(\bar{z})}d\bar{z}\approx-\frac{\ell|p|}{\lambda}\left(\sqrt{2z}-\frac{z^{3/2}}{2\sqrt{2}}\right)\,.$
(61)

In Fig.3, we depict the $\kappa$-AdS time delay of Eq.(59) in blue, and the
Jacob-Piran time delay of Eq.(61) in orange. We see that both formulas agree
for small delays, but then, the oscillatory behavior of the $\kappa$-AdS time
delay generated by the translations of AdS space pulls back the time delay.
Usually, one expects that the relative locality effects should grow with the
distance between the source and the detector, but we are showing that this is
not always the case. The delay that we presented is bounded despite of the
relative distance between the events and can also be null. A similar behavior
can be found in more recent analyses carried out in Amelino-Camelia:2023srg ,
where such richer class of delays is presented.

Figure 3: Ratio between the absolute value of time delay in cosmological
coordinates and the hard photon’s momentum, as function of the redshift, for
$\ell^{-1}=E_{\text{Planck}}=1.2\times 10^{19}\,\text{GeV}$, $\lambda=5\times
10^{-18}\,\text{s}^{-1}$. The blue/solid line describes the $\kappa$-AdS time
delay given by Eq.(59) without MDR and just with relative locality effects.
While the orange/dashed line describes a completely differenc scenario, with a
MDR in cosmological coordinates and we depict only the time delay of the
Jacob-Piran formula (61).

The accelerated expansion of the universe implies that for small redshifts we
can actually describe the spacetime geometry by a de Sitter metric and the
$\kappa$-de Sitter algebra could actually be used to constrain the quantum
gravity parameter at the dark energy dominated epoch. But we cannot say the
same of our case. There is no evidence that we have had an anti-de Sitter
epoch in the universe, which in fact, has some unphysical features, like the
existence of closed timelike curves, besides other features, like the absence
of Cauchy surfaces (see section 5.2 of Hawking:1973uf ). Nevertheless, the
main point of this paper is the construction of a time delay from deformation
of translations in a class of quantum algebras that was not explored before to
derive this observable. We expect that in the future we can use these results
as a starting point to construct a more realistic scenario that glues pieces
of anti-de Sitter propagation to construct an FLRW-based time delay that is
compatible with the DSR principles in a similar way of what was done in
Rosati:2015pga  for the $\kappa$-de Sitter case, in order to extend the class
of DSR-inspired time delays formulas.

##  VI Final Remarks

We found the relation between the generators of the $\kappa$-anti-de Sitter
algebra in $1+1$ dimensions Ballesteros:2016bml  and the undeformed anti-de
Sitter algebra at first order in the quantum gravity parameter. We found that
in the basis used in Ballesteros:2016bml , the boost and spacetime
translations are deformed. However, we discovered that there is no deformation
of the Casimir charge when we express it in terms of the undeformed
quantities. This means that the trajectories are undeformed at first order of
correction.

In a Lorentz invariance violating scenario, this means that the presence of
classical trajectories implies in the absence of a time delay when particles
with different energies are emitted simultaneously in a distant source and
detected on earth Addazi:2021xuf . However, if we assume that the relativity
principle is preserved, i.e., that the the local frames of the emission of the
detection events are related by a deformed translation, then a reminiscent
first order effect remains. This result has been derived using the $\kappa$-de
Sitter algebra in the past Amelino-Camelia:2012vzf . However for the first
time, we perform a derivation using the $\kappa$-anti-de Sitter algebra.

A novel result is the emergence of a sinusoidal time delay, which is related
to the oscillatory behavior of the anti-de Sitter spacetime, which is a
property that is evident in cosmological coordinates, but it is in fact a
coordinate-independent feature of this geometry, as discussed in section 5.2
of Hawking:1973uf . Since the time delays phenomenology is based on the
$\Lambda$-CDM cosmology, which does not present an anti-de Sitter epoch
Staicova:2023vln ; Staicova:2024ljn , we do not envisage a direct applications
of this result to observations.

Nevertheless, the $\kappa$-de Sitter results has been used as the cornerstone
for the derivation of a time delay formula that would be compatible with local
relativistic principles, through the creative construction of a propagation of
signals and communication frames in a quantum FLRW spacetime by infinitesimal
delays based on $\kappa$-de Sitter algebra with different curvatures
Rosati:2015pga . Since there is no privileged role of positive curvature in
comparison to the negative one from the geometric and algebraic perspectives,
we wonder what kind of result would be found had one used the $\kappa$-AdS
algebra as the fundamental bricks to build such delay in the effective FLRW
spacetime.

The main challenge of this approach is how to handle the composition of
coordinates of $\kappa$-AdS algebra which is given by trigonometric functions,
while in $\kappa$-dS algebra the cosmological and conformal coordinates
furnish computationally simpler functions. Despite these difficulties, a first
and necessary step towards this analysis is the derivation of such delay for a
single curvature, which was carried out in this paper. In the future, we aim
to extend the results of this paper in two fronts: the exploration of
different bases of $\kappa$-AdS algebra in order to derive more general
results that include deformed trajectories; and to construct the effective
FLRW time delay based on the $\kappa$-AdS algebra building bricks for the
slicing technique.

## Acknowledgments

I. P. L. was partially supported by the National Council for Scientific and
Technological Development - CNPq grant 306414/2020-1 and by the grant
3197/2021, Paraíba State Research Foundation (FAPESQ). I. P. L. acknowledges
the networking support by the COST Action QGMM (CA18108), supported by COST
(European Cooperation in Science and Technology).

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