# Principal equatorial null geodesic congruences in the Kerr metric, and their
quantum propagators

Josué G. Mateos Trujillo1 and Miguel Socolovsky1,2 1Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México, Cd. Universitaria, 04510,
Ciudad de México, México
2Instituto de Astronomía y Física del Espacio, Universidad de Buenos Aires-
CONICET, Argentina
<EMAIL_ADDRESS>
###### Abstract

Using the Raychaudhuri equation, we associate quantum probability amplitudes
(propagators) to equatorial principal ingoing and outgoing null geodesic
congruences in the Kerr metric. The expansion scalars diverge at the ring
singularity; however, the propagators remain finite, which is an indication
that at the quantum level singularities might disappear or, at least, become
softened.

Keywords: Kerr metric, principal null geodesics, propagators

##  1 Introduction

In a recent paper [1], Feynman propagators were associated to timelike and
null geodesic congruences in the Schwarzschild spacetime. The possibility of
this association lies in the Raychaudhuri equation [2] which describes the
flow of such congruences, even in the case of curves where an acceleration
term is present. In particular, the equation for the expansion scalar
$\Theta$, after a transformation of variables is done [3], becomes a
1-dimensional free harmonic oscillator equation for a function $F$ with a
“time” dependent frequancy (the rôle of time is played by the affine parameter
$\lambda$ of the geodesics). A Lagrangian can be immediately constructed, and
with initial and final values for $\lambda$ and for the function $F$
(basically representing the transverse area of the congruences), an exact path
integral
$K(F^{\prime\prime},\lambda^{\prime\prime};F^{\prime},\lambda^{\prime})$ of
the exponential of the corresponding action is obtained. The basic idea is
that this $K$ describes the quantum evolution of the congruences, even in the
absence, at present, of a final theory of quantum gravity [4].

In the present paper, we restrict the analysis to the simplest case in the
Kerr spacetime [5]: the forward and past directed, ingoing and outgoing,
principal null geodesics in the equatorial plane ($\theta=\pi/2$), and their
congruences. Since the Kerr’s solution in the Petrov’s classification [6] is
algebraically special, the Goldberg-Sachs theorem [7] implies that the shear
$\sigma_{\mu\nu}$ in the Raychaudhuri equation vanishes and, since an explicit
calculation shows that the rotation term $\omega_{\mu\nu}$ also vanishes, the
above mentioned Lagrangian, as in the Schwarzschild case, reduces to that of a
free non relativistic particle. At the ring singularity ($r=0,\ \theta=\pi/2$)
the expansion diverges as $1/r$. However, the relevant propagators $K$ remain
finite. This result, though does not constitute a proof that singularities
disappear in a quantum treatment of black holes, is however an indication that
at the quantum level they might indeed disappear or, at least, become
smoother.

In section 2, following the presentation of Ferrari et al [8], we integrate
the equations of motion of principal equatorial null geodesics p.e.n.g.’s,
identify the tangent vector field $k^{\mu}$ for them, and consequently
determine the affine parameter $\lambda$ as $r$ for outgoing geodesics and
$-r$ for ingoing ones. In the process one proves that the Carter constant [9]
reflecting a hidden symmetry of the Kerr solution, vanishes for p.e.n.g.’s. In
Figures 1 and 2 we plot $t(r)$ and $\psi(r)$ for forward and past directed
geodesics where $t$, $r$, and $\psi$ are the usual Boyer-Lindquist coordinates
($\theta=\pi/2$ is fixed at the equator). It is interesting to notice that in
the “shell horizon” [10], that is, the region between the event (outer) and
Cauchy (inner) horizons, precisely the only region where trapped surfaces
exist, all p.e.n.g.’s, both outgoing and ingoing, are past directed. In all
cases, the geodesics are asymptotic to at least one horizon.

In section 3 we construct the Raychaudhuri equation that adquires the simplest
form as in the Schwarzschild case due to the vanishing of both the shear and
the rotation terms (the second one because the tensor $B_{\mu\nu}=k_{\mu;\nu}$
turns out to be symmetric).

In section 4 we calculate the potentially divergent propagators associated
with geodesic congruences in the region $-\infty<r<r_{-}$ where the ring
singularity lies, and prove their finitness. Finally, section 5 is deserved to
conclusions and remarks.

We use the natural system of units: $G=c=\hbar=1$ and the metric signature
$(+,-,-,-)$.

##  2 Principal equatorial null geodesics

In Boyer-Lindquist coordinates
$x^{\mu}=(x^{0},x^{1},x^{2},x^{3})=(t,r,\theta,\psi)$ the stationary axial
symmetric Kerr metric is given by

$ds^{2}={{\Delta-a^{2}sin^{2}\theta}\over{\Sigma}}dt^{2}-{{\Sigma}\over{\Delta}}dr^{2}-\Sigma
d\theta^{2}-{{sin^{2}\theta}\over{\Sigma}}Ad\psi^{2}+{{2asin^{2}\theta}\over{\Sigma}}(r^{2}+a^{2}-\Delta)dtd\psi,$
(1)

where $\Delta=r^{2}+a^{2}-2Mr$, $\Sigma=r^{2}+a^{2}cos^{2}\theta$,
$A=(r^{2}+a^{2})^{2}-a^{2}sin^{2}\theta\Delta$, with $a=J/M<M$ where $M$ and
$J$ are the total energy and total angular momentum parameters of the
solution. Also, $\Delta=(r-r_{+})(r-r_{-})$ with $r_{+}=M+\sqrt{M^{2}-a^{2}}$
($r_{-}=M-\sqrt{M^{2}-a^{2}}$) the event or exterior horizon $h_{+}$ (the
Cauchy or interior horizon $h_{-}$). Respectively corresponding to the
stationarity and axial symmetries one has the Killing vectors $\partial_{t}$
and $\partial_{\psi}$.

The geodesic motion of particles in the metric (1) is described by the
Lagrangian

${\cal
L}={{1}\over{2}}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}={{1}\over{2}}({{\Delta-a^{2}sin^{2}\theta}\over{\Sigma}}\dot{t}^{2}-{{\Sigma}\over{\Delta}}\dot{r}^{2}-\Sigma\dot{\theta}^{2}-{{sin^{2}\theta}\over{\Sigma}}A\dot{\psi}^{2}+{{2asin^{2}\theta}\over{\Sigma}}(r^{2}+a^{2}-\Delta)\dot{t}\dot{\psi}),$
(2)

where $\dot{x}^{\mu}={{dx^{\mu}}\over{d\lambda}}$, with $\lambda=\tau$: proper
time for massive ($m$) particles, in which case ${\cal L}=m^{2}$, and
$\lambda$ an arbitrary (up to an affine transformation) affine parameter in
the massless case, in which case ${\cal L}=0$. The two conserved quantities
associated to the above mentioned symmetries are obtained from the Lagrange
equation

${{d}\over{d\lambda}}({{\partial{\cal
L}}\over{\partial\dot{x}^{\mu}}})={{\partial{\cal L}}\over{\partial
x^{\mu}}},$ (3)

where

$p_{\mu}={{\partial{\cal
L}}\over{\partial\dot{x}^{\mu}}}=g_{\mu\nu}\dot{x}^{\nu}.$ (4)

For $\mu=0,3$ the r.h.s. of (3) vanishes and one obtains

$p_{0}=E=({{1-2Mr}\over{\Sigma}})\dot{t}+{{2Marsin^{2}\theta}\over{\Sigma}}\dot{\psi}$
(5)

and

$p_{3}=L=-{{2Mrasin^{2}\theta}\over{\Sigma}}\dot{t}+{{sin^{2}\theta}\over{\Sigma}}((r^{2}+a^{2})^{2}-a^{2}sin^{2}\theta\Delta)\dot{\psi}^{2},$
(6)

for the energy and the angular momentum along the rotation axis. From (5) and
(6) one obtains

$\dot{t}=\dot{t}(r,\theta;M,a;E,L)={{CE-BL}\over{AC+B^{2}}}\ \ and\ \
\dot{\psi}=\dot{\psi}(r,\theta;M,a;E,L)={{BE+AL}\over{AC+B^{2}}},$ (7)

with

$A=g_{tt}=1-{{2Mr}\over{\Sigma}},\ \
B=g_{t\psi}={{2Mra}\over{\Sigma}}sin^{2}\theta,\ \
C=-g_{\psi\psi}=(r^{2}+a^{2}+{{2Mra^{2}}\over{\Sigma}})sin^{2}\theta,$ (8)

which can be integrated once the equation for $\dot{r}$ and $\dot{\theta}$ are
also found. To this aim, one appeals to the Hamilton-Jacobi equation [9]. The
Hamiltonian is

${\cal
H}(x^{\mu},p_{\nu})=p_{\mu}\dot{x}^{\mu}-{{1}\over{2}}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}={{1}\over{2}}g^{\mu\nu}p_{\mu}p_{\nu}=p^{2}/2=\kappa/2,$
(9)

where $\kappa=m^{2}$ for massive particles and $\kappa=0$ in the massless
case. The Hamilton principal function $S$ is defined by

${\cal H}(x^{\mu},{{\partial S}\over{\partial x^{\nu}}})+{{\partial
S}\over{\partial\lambda}}=0,$ (10)

where ${{\partial S}\over{\partial x^{\nu}}}=p_{\nu}$. Writing

$S=-{{1}\over{2}}\kappa\lambda+Et+L\psi+\tilde{S}(r,\theta)$ (11)

one looks for a separable solution

$\tilde{S}(r,\theta)=\tilde{S}_{1}(r)+\tilde{S}_{2}(\theta).$ (12)

A long but straightforward procedure [8] leads to the equality

$\Delta({{d\tilde{S}_{1}(r)}\over{dr}})^{2}+\kappa
r^{2}-{{r^{2}+a^{2}}\over{\Delta}}E^{2}-{{4Mra}\over{\Delta}}EL-{{a^{2}}\over{\Delta}}L^{2}+a^{2}E^{2}+L^{2}=-({{d\tilde{S}_{2}(\theta)}\over{d\theta}})^{2}+(\kappa+E^{2})a^{2}cos^{2}\theta-{{cos^{2}\theta}\over{sen^{2}\theta}}L^{2}$
(13)

where the l.h.s. (r.h.s.) depends only on $r$ ($\theta$). So, both sides of
(13) depend on a constant: -${\cal C}$, the Carter constant. ($[{\cal
C}]=[lenght]^{4}$.) The existence of this constant of motion is due to a
hidden symmetry of the Kerr metric, described by a covariant Killing 2-tensor
[11]. So one has the set of four constants

$\\{E,L,\kappa,{\cal C}\\}$ (14)

in terms of which the geodesic motion can be solved. Defining the functions

$T(\theta)={\cal
C}+cos^{2}\theta((\kappa+E^{2})a^{2}-{{1}\over{sin^{2}\theta}}L^{2})\ \ and\ \
R(r)=\Delta(\kappa r^{2}-(L-aE)^{2}-{\cal C})+(E(r^{2}+a^{2})-La)^{2}$ (15)

one obtains

$({{d\tilde{S}_{2}(\theta)}\over{d\theta}})^{2}=T(\theta)\ \ and\ \
({{d\tilde{S}_{1}(r)}\over{dr}})^{2}={{R(r)}\over{\Delta^{2}}};$ (16)

therefore

$S(t,\psi,r,\theta,\lambda;{\cal
C},\kappa,E,L)=-{{1}\over{2}}\kappa\lambda+Et+L\psi+\int^{r}dr^{\prime}{{\sqrt{R(r^{\prime})}}\over{\Delta(r^{\prime})}}+\int^{\theta}d\theta^{\prime}\sqrt{T(\theta^{\prime})}.$
(17)

For null geodesics (n.g.), $\kappa=0$ and so

$R(r)=-\Delta((L-aE)^{2}+{\cal C})+(E(r^{2}+a^{2})-La)^{2}\ \ and\ \
T(\theta)={\cal C}+cos^{2}\theta(E^{2}a^{2}-{{1}\over{sin^{2}\theta}}L^{2}).$
(18)

For equatorial null geodesics (e.n.g.), $\theta(\lambda)=\pi/2$ for all
$\lambda$, so $\Sigma=r^{2}$, $\dot{\theta}(\lambda)=0$, and $T(\theta)={\cal
C}$. From
$p_{\theta}=g_{\theta\nu}\dot{x}^{\nu}=g_{\theta\theta}\dot{\theta}=-\Sigma\dot{\theta}$,
$p_{\theta}^{2}=\Sigma^{2}\dot{\theta}^{2}=0=({{\partial
S}\over{\partial\theta}})^{2}=({{\partial\tilde{S}_{2}(\theta)}\over{\partial\theta}})^{2}=T(\theta)$,
it turns out that the Carter constant vanishes i.e. ${\cal C}=0$, and one has

$R(r)=-\Delta(L-aE)^{2}+(E(r^{2}+a^{2})-La)^{2}.$ (19)

The equation for $\dot{r}$ is obtained from
$p_{r}=g_{r\nu}\dot{x}^{\nu}=g_{rr}\dot{r}=-{{\Sigma}\over{\Delta}}\dot{r}={{\partial\tilde{S}_{1}(r)}\over{\partial
r}}$; so, from (16),

$\dot{r}^{2}={{R(r)}\over{\Sigma^{2}}}={{R(r)}\over{r^{4}}}.$ (20)

The principal equatorial null geodesics p.e.n.g. are defined by the condition
$L=aE$ i.e.

${{L}\over{E}}={{J}\over{M}}.$ (21)

That is, are the equatorial null geodesics in which the total angular
momentum/unit of energy equals the black hole angular momentum/unit of its
mass. In this case

$\dot{r}=\pm E$ (22)

and so

$\dot{t}={{r^{2}+a^{2}}\over{\Delta}}E\ \ and\ \
\dot{\psi}={{a}\over{\Delta}}E.$ (23)

Therefore, p.e.n.g. are characterized by the tangent vectors

$k^{\mu}=(\dot{t},\dot{r},\dot{\theta},\dot{\psi})=({{r^{2}+a^{2}}\over{\Delta}}E,\pm
E,0,{{a}\over{\Delta}}E).$ (24)

Rescaling the affine parameter $\lambda$ through the affine transformation
$\lambda=\bar{\lambda}/E$, we obtain

$k^{\mu}=(\dot{t},\dot{r},\dot{\theta},\dot{\psi})=({{r^{2}+a^{2}}\over{\Delta}},\pm
1,0,{{a}\over{\Delta}}).$ (25)

(The new affine parameter $\bar{\lambda}$ is again denoted by $\lambda$.) The
associated 1-forms are $k_{\mu}=(-1,\pm{{r^{2}}\over{\Delta}},0,a)$.

The + sign designs outgoing principal equatorial null geodesics (o.p.e.n.g.),
with affine parameter $\lambda=r$, while the - sign designs ingoing principal
equatorial null geodesics (i.p.e.n.g.), with affine parameter $\lambda=-r$.

From
${{dt}\over{d\lambda}}={{\dot{t}}\over{\dot{r}}}=\pm{{r^{2}+a^{2}}\over{\Delta}}\
\ and\ \
{{d\psi}\over{dr}}={{\dot{\psi}}\over{\dot{r}}}=\pm{{a}\over{\Delta}}$, one
has $dt=\pm{{r^{2}+a^{2}}\over{\Delta}}dr$, $d\psi=\pm{{a}\over{\Delta}}dr$
which imply

$t(r)=\pm\int^{r}dr^{\prime}{{{r^{\prime}}^{2}+a^{2}}\over{\Delta(r^{\prime})}}+const.,\
\ \psi(r)=\pm
a\int^{r}{{dr^{\prime}}\over{\Delta(r^{\prime})}}+const.^{\prime}$ (26)

Using (2.172) and (2.175.4) in [12], we finally obtain

$t(r)=\pm(r+{{Mr_{+}}\over{\sqrt{M^{2}-a^{2}}}}ln|{{r-r_{+}}\over{2M}}|-{{Mr_{-}}\over{\sqrt{M^{2}-a^{2}}}}ln|{{r-r_{-}}\over{2M}}|)+const.$
(27)

and

$\psi(r)=\pm{{a}\over{2\sqrt{M^{2}-a^{2}}}}ln|{{r-r_{+}}\over{r-r-}}|+const.^{\prime}$
(28)

The plot of $t=t(r)$ in the $(t,r)$ plane, with $t,r\in(-\infty,+\infty)$,
$r=0$ ($t$ axis) representing the ring singularity (which does not belong to
spacetime), shows that (Fig. 1):

Figure 1:  Principal equatorial null geodesics (p.e.n.g.’s) in the $t/r$ plane

i) For each point $q$ in the region I: $r_{+}<r$, pass two and only two
forward directed (one outgoing, $\alpha_{o}$, the other ingoing, $\alpha_{i}$)
principal equatorial null geodesics (f.d.o./i.p.e.n.g.) (no past directed
geodesics). Both $\alpha_{o}$ and $\alpha_{i}$ are inextendible i.e. future
and past inextendible.

ii) For each point $p$ in the region II: $r_{-}<r<r_{+}$, pass two and only
two past directed (one outgoing, $\gamma_{o}$, the other ingoing,
$\gamma_{i}$) principal equatorial null geodesics (p.d.o/i.p.e.n.g.) (no
future directed geodesics). Both $\gamma_{o}$ and $\gamma_{i}$ are
inextendible. Clearly, neither $\gamma_{o}$ nor $\gamma_{i}$ can be considered
physical geodesics since they propagate backwards in time.

iii) For each point $s$ in the region III: $-\infty<r<r_{-}$, pass two and
only two future directed (one outgoing, $\beta_{o}$), the other ingoing,
$\beta_{i}$) principal equatorial null geodesics (f.d.o./i.p.e.n.g.) (no past
directed geodesics). Both geodesics of each pair split at the singularity:
$\beta_{o}=(\beta_{o})_{1}\cup(\beta_{o})_{2}$,
$\beta_{i}=(\beta_{i})_{1}\cup(\beta_{i})_{2}$ with
$(\beta_{o})_{1}\cap(\beta_{o})_{2}=(\beta_{i})_{1}\cap(\beta_{i})_{2}=\phi$.
$(\beta_{o})_{1}$, $(\beta_{i})_{2}$, $(\beta_{o})_{2}$, and $(\beta_{i})_{1}$
are inextendible.

All these geodesics are asymptotic to an horizon: $\alpha_{i}$, $\alpha_{o}$,
and $\gamma_{i}$ to $h_{+}$; $\gamma_{0}$, $(\beta_{o})_{2}$, and
$(\beta_{i})_{2}$ to $h_{-}$. It is interesting to note that the region where
there are no f.d.p.e.n.g. is the unique region where trapped surfaces exist:
$r\in(r_{-},r_{+})$. In a Kruskal-Szekeres extension, this region corresponds
to the black and white holes.

The behavior of the azimuthal angle $\psi$ (eq.(28)) for each pair of the
geodesics discussed above is shown in Fig. 2. We notice that for forward
directed geodesics $\dot{\psi}(r)\to-\infty$ as $r\to(r_{+})_{+}$ and
$r\to(r_{-})_{-}$, and $\dot{\psi}(r)\to 0_{-}$ as $r\to\pm\infty$, in the
ingoing cases, while $\dot{\psi}(r)\to+\infty$ as $r\to(r_{+})_{+}$ and
$r\to(r_{-})_{-}$ and $\dot{\psi}(r)\to 0_{+}$ as $r\to\pm\infty$, in the
outgoing cases. For past directed geodesics, $\dot{\psi}(r)\to+\infty$ as
$r\to(r_{+})_{-}$ and $r\to(r_{-})_{+}$ in the ingoing case, and
$\dot{\psi}(r)\to-\infty$ as $r\to(r_{-})_{+}$ and $r\to(r_{+})_{-}$ in the
outgoing case.

Figure 2: Azimuthal angle as a function of $r$ of p.e.n.g.’s

##  3 Congruences, expansions, and Raychaudhuri equations

Varying the constants in (27) and (28) allows us to define geodesic
congruences, which are governed by the Raychaudhuri equation [2]

${{d\Theta}\over{d\lambda}}=-{{1}\over{2}}\Theta^{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}+\omega_{\mu\nu}\omega^{\mu\nu}-R_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$
(29)

where

$\Theta=k^{\mu}_{;\mu}={{1}\over{\sqrt{-g}}}\partial_{\mu}(\sqrt{-g}k^{\mu})$
(30)

is the expansion scalar ($g=det(g_{\mu\nu}))$; $\sigma_{\mu\nu}$, the shear,
is the traceless symmetric part of the tensor $B_{\mu\nu}=D_{\nu}k_{\mu}$;
$\omega_{\mu\nu}$, the rotation, is the antisymmetric part of $B_{\mu\nu}$;
and $R_{\mu\nu}$ is the Ricci tensor. In vacuum, and in the absence of a
cosmological constant, $R_{\mu\nu}=0$. Since the Kerr metric is an
algebraically special solution in the Petrov’s classification scheme [6], the
Goldberg-Sachs theorem [7] implies that $\sigma_{\mu\nu}=0$. A direct
calculation of $B_{\mu\nu}=k_{\mu,\nu}-\Gamma_{\mu\nu}^{\rho}k_{\rho}$ for
p.e.n.g.’s gives $B_{\mu\nu}=B_{\nu\mu}$, which implies $\omega_{\mu\nu}=0$. A
straightforward calculation of $\Theta$ gives

$\Theta_{\pm}=\pm{{2}\over{r}}$ (31)

with the upper (lower) sign for outgoing, $\lambda=r$ (ingoing, $\lambda=-r$)
geodesics. For (29) one has

${{d\Theta}\over{d\lambda}}+{{1}\over{2}}\Theta^{2}=0.$ (32)

For future and past directed, outgoing and ingoing geodesic congruences
defined by $\alpha$’s, $\beta$’s and $\gamma$’s, $\Theta(r)$ is finite for all
$r>0$. Instead, for:

$(\beta_{o})_{2}$: $r=0_{+}\to(r_{-})_{-}$, the f.d.o.p.n.g.’s birth at the
singularity at $r=0_{+}$, $\theta=\pi/2$ in the region III, and go
asymptotically towards $(r_{-})_{-}$; $\Theta_{+}\to+\infty$ as $r\to 0_{+}$;

$(\beta_{i})_{1}$: $r=0_{-}\to r=-\infty$, the f.d.i.p.n.g.’s birth at the
singularity at $r=0_{-}$, $\theta=\pi/2$ in the region III, and go
asymptotically towards $r=-\infty$ along the line $t=-r$;
$\Theta_{-}\to+\infty$ as $r\to 0_{-}$;

$(\beta_{i})_{2}$: $(r_{-})_{-}\to r=0_{+}$, the f.d.i.p.n.g.’s reach the
singularity at $r=0_{+}$, $\theta=\pi/2$ in the region III, coming
asymptotically from $(r_{-})_{-}$; $\Theta_{-}\to-\infty$ as $r\to 0_{+}$;

$(\beta_{o})_{1}$: $r=-\infty\to r=0_{-}$, the f.d.o.p.n.g.’s reach the
singularity at $r=0_{-}$, $\theta=\pi/2$ in the region III, coming
asymptotically along the line $t=r$ from $r=-\infty$; $\Theta_{+}\to-\infty$
as $r\to 0_{+}-$.

As expected, the only place where the expansions diverge is at the singularity
ring.

Defining the functions $F_{\pm}(\lambda)$ through [3]

$\Theta_{\pm}(\lambda)=2{{\dot{F}_{\pm}(\lambda)}\over{F_{\pm}(\lambda)}},$
(33)

(32) becomes the equation of a free non relativistic particle

$\ddot{F}=0$ (34)

with solution $F(\lambda)=v\lambda+w$, with $v$ and $w$ real constants. For
the cases of interest $(\beta_{i})_{s}$, $(\beta_{o})_{s}$, $s=1,2$,
$\Theta(0)=\pm\infty$ ($0=0_{+}$ or $0_{-}$) implies $F(0)=0$ if
$\dot{F}(0)\neq 0$. Then $w=0$ and $F(\lambda)=v\lambda$, $v\neq 0$.
(Essentially, $F$ is a measure of the transverse area of the congruence.)

The Lagrangian that reproduces (34) is ${{1}\over{2}}(\dot{F}(\lambda))^{2}$,
with an associated action

$S[F]={{1}\over{2}}\int_{\lambda^{\prime}}^{\lambda}(\dot{F}(\lambda))^{2}={{v^{2}}\over{2}}(\lambda^{\prime\prime}-\lambda^{\prime}).$
(35)

(Units: $[F]=[length]^{1/2}$ since $[S]=[length]^{0}$; then
$[v]=[length]^{-1/2}$.)

##  4 Quantum propagators

For each geodesic congruence associated to the geodesics $\alpha_{i}$,
$\alpha_{o}$, $\gamma_{i}$, $\gamma_{o}$, $(\beta_{i})_{2}$,
$(\beta_{i})_{1}$, $(\beta_{o})_{2}$, and $(\beta_{o})_{1}$, one can formally
associate the “quantum” (Feynman) propagator [13]

$K(F^{\prime\prime},\lambda^{\prime\prime};F^{\prime},\lambda^{\prime})={{1}\over{\sqrt{2\pi
i(\lambda^{\prime\prime}-\lambda^{\prime})}}}e^{{{i}\over{2}}v^{2}(\lambda^{\prime\prime}-\lambda^{\prime})}.$
(36)

The only propagators where divergences might appear as $r\to 0$ ($0=0_{+}$ or
$0_{-}$) are those associated to $(\beta_{i})_{s}$, $(\beta_{o})_{s}$,
$s=1,2$. We study these cases in detail.

$(\beta_{i})_{2}$: $F^{\prime\prime}=v_{-}\lambda^{\prime\prime}$,
$\lambda^{\prime\prime}=0_{+}$; $F^{\prime}=v_{-}\lambda^{\prime}$,
$\lambda^{\prime}=(r_{-})_{-}\equiv r_{-}$; then

$(K_{i})_{2}(0,0_{+};v_{-}r_{-},r_{-})={{e^{-iv_{-}^{2}r_{-}/2}}\over{\sqrt{-2\pi
ir_{-}}}}={{e^{-i(v_{-}^{2}r_{-}-\pi/2)/2}}\over{\sqrt{2\pi r_{-}}}},\ \ with\
\ |(K_{i})_{2}|=1/\sqrt{2\pi r_{-}}<+\infty.$ (37)

$(\beta_{i})_{1}$: $F^{\prime\prime}=v_{-}\tilde{r}$,
$\lambda^{\prime\prime}=\tilde{r}$; $F^{\prime}=v_{-}\lambda^{\prime}$,
$\lambda^{\prime}=0_{-}$; then

$(K_{i})_{1}(v_{-}\tilde{r},0_{+};0,0_{-})={{e^{iv_{-}^{2}\tilde{r}/2}}\over{\sqrt{2\pi
i\tilde{r}}}}={{e^{i(v_{-}^{2}\tilde{r}-\pi/2)/2}}\over{\sqrt{2\pi\tilde{r}}}},\
\ with\ \ |(K_{i})_{1}|=1/\sqrt{2\pi\tilde{r}}<+\infty.$ (38)

$(\beta_{o})_{2}$: $F^{\prime\prime}=v_{+}\lambda^{\prime\prime}$,
$\lambda^{\prime\prime}=(r_{-})_{-}\equiv r_{-}$;
$F^{\prime}=v_{+}\lambda^{\prime}$, $\lambda^{\prime}=0_{+}$; then

$(K_{o})_{2}(v_{+}r_{-},r_{-};0,0_{+})={{e^{iv_{+}^{2}r_{-}/2}}\over{\sqrt{2\pi
ir_{-}}}}={{e^{i(v_{+}^{2}r_{-}-\pi/2)/2}}\over{\sqrt{2\pi r_{-}}}},\ \ with\
\ |(K_{o})_{2}|=1/\sqrt{2\pi r_{-}}<+\infty.$ (39)

$(\beta_{o})_{1}$: $F^{\prime\prime}=v_{+}\lambda^{\prime\prime}$,
$\lambda^{\prime\prime}=0_{-}$; $F^{\prime}=\hat{r}v_{+}$,
$\lambda^{\prime}=\hat{r}$; then

$(K_{o})_{1}(0,0_{-};\hat{r}v_{+},\hat{r})={{e^{-iv_{+}^{2}\hat{r}/2}}\over{\sqrt{-2\pi
i\hat{r}}}}={{e^{-i(v_{+}^{2}\hat{r}-\pi/2)/2}}\over{\sqrt{2\pi\hat{r}}}},\ \
with\ \ |(K_{o})_{1}|=1/\sqrt{2\pi\hat{r}}<+\infty.$ (40)

The $\pm$ signs in $v_{\pm}$ indicate outgoing and ingoing geodesics;
$v_{\pm}\neq 0$ are arbitrary real constants; and in the
$\tilde{r},\hat{r}\to-\infty$ limit the corresponding propagators vanish since
the infinite oscillations of the exponentials are “killed” by the growing of
the denominators. The similarity between the propagators $(K_{i})_{2}$ and
$(K_{o})_{1}$ (($K_{i})_{1}$ and $K_{o})_{2}$) is due to the fact that the
congruences defined by $(\beta_{i})_{2}$ and $(\beta_{o})_{1}$ (
$(\beta_{i})_{1}$ and $(\beta_{o})_{2}$) arrive (birth) at the singularity.

The results (37)-(40) prove that, even if the expansion scalars $\Theta$ which
govern the classical evolution of the geodesic congruences diverge at the
singularity ring, the associated Feynman propagators remain finite, which is
an indication that at a quantum level singularities might disappear or at
least become softened.

##  5 Conclusion and final remarks

We showed that to any principal equatorial null geodesic congruence in Kerr
spacetime, can be asigned a quantum (Feynman) propagator describing its flow,
which is classically described by the Raychaudhuri equation for the expansion
scalar $\Theta$. In particular, the unique potentially divergent propagators,
those reaching the ring singularity at $(r=0,\theta=\pi/2)$, i.e. those in the
region $-\infty<r<r_{-}$, remain finite, in contradistinction with the
expansion scalars which diverge as $1/r$. This fact suggests that at the
quantum level, singularities which appear at the classical level, might not be
present or, at least, might be smoothened. The same conclusion is arrived by
S. Chakraborty and M. Chakraborty in their recent review [16].

As final remarks we want to mention two facts. First, the only part of the
Kerr solution which should represent the result of the collapse of a rotating
star, that is, the only physical part, is the previously called “shell
horizon” ($r_{-}<r<r_{+}$) which in particular also is the unique globally
hyperbolic region. In a Penrose diagram, it looks as a central “diamond” [14].
Paradoxically, it is free of singularities, even if it contains the black and
white holes! Moreover, in the asymptotically flat ($r\to-\infty$) regions
beyond $r_{-}$ and close to the singularity rings ($r<0$, $r\sim 0$), closed
timelike curves exist that violate causality [15]. This criterium for defining
the physical region should make the divergence of $\Theta$ as $r\to 0_{-}$
totally harmless. Second, a spacetime is considered singular, if,
inextendible, it has at least one causal (in particular null) inextendible
geodesic which does not admit an affine parameter extending from $-\infty$ to
$+\infty$. This is precisely the case of the inextendible geodesics
$\gamma_{o}$ and $\gamma_{i}$ with affine parameter $r\in(r_{-},r_{+})$. Any
other affine parameter must be of the form $\lambda=xr+y$ with
$x,y\in\mathbb{R}$, $x\neq 0$, which is always finite. Since both $\gamma_{o}$
and $\gamma_{i}$ are non physical, the physical spacetime (the diamond) can
still be considered non singular.

Acknowledgments

One of us (M.S.) thanks for hospitality to the Instituto de Astronomía y
Física del Espacio (IAFE) de la Universidad de Buenos Aires and CONICET,
Argentina, where this work was done during a sabbatical stay. The authors
thank Oscar Brauer at the University of Leeds, U.K., for the drawing of
Figures 1 and 2.

Conflicts of interest

The authors declare no conflicts of interest regarding the publication of this
paper.

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[1]Emanuele Sangiorgio

[1]Department of Social Sciences and Economics, Sapienza University of Rome,
P.le Aldo Moro, 5, 00185, Rome, Italy

2]Department of Computer Science, Sapienza University of Rome, Viale Regina
Elena, 295, 00161, Rome, Italy

3]GRANEM, Université d’Angers, SFR Confluences, F-49000, Angers, France

# Evaluating the effect of viral news on social media engagement

<EMAIL_ADDRESS>Niccolò Di Marco<EMAIL_ADDRESS>Gabriele Etta<EMAIL_ADDRESS>Matteo Cinelli
<EMAIL_ADDRESS>Roy Cerqueti<EMAIL_ADDRESS>Walter
Quattrociocchi<EMAIL_ADDRESS>*  [  [

###### Abstract

This study examines Facebook and YouTube content from over a thousand news
outlets in four European languages from 2018 to 2023, using a Bayesian
structural time-series model to evaluate the impact of viral posts. Our
results show that most viral events do not significantly increase engagement
and rarely lead to sustained growth. The virality effect usually depends on
the engagement trend preceding the viral post, typically reversing it. When
news emerges unexpectedly, viral events enhances users’ engagement,
reactivating the collective response process. In contrast, when virality
manifests after a sustained growth phase, it represents the final burst of
that growth process, followed by a decline in attention. Moreover, quick viral
effects fade faster, while slower processes lead to more persistent growth.
These findings highlight the transient effect of viral events and underscore
the importance of consistent, steady attention-building strategies to
establish a solid connection with the user base rather than relying on sudden
visibility spikes.

###### keywords:

Social media $|$ Virality $|$ Attention economy

##  1 Introduction

The advent and proliferation of social media have fundamentally altered the
information landscape [1, 2, 3], offering unprecedented opportunities for
content to achieve rapid and widespread attention. As these platforms have
become integrated into our daily lives [4], transforming into essential tools
for information diffusion [5, 6] and personal communication [7], they have
merged entertainment-driven business models with complex social dynamics [8],
raising significant concerns about their impact on society. This complex
interplay produced an environment in which information overload is the
foremost feature [9] and a wide range of content creators, from news
organizations to individual influencers [10], compete for the limited resource
that is users’ attention [11, 12, 13, 14].

Understanding the attention economy in the digital domain is paramount for
navigating this competitive information market, whereby the pursuit of
virality [15] plays a pivotal role in shaping how information sources design
their strategies of production and diffusion of content. Characterized by
content’s exceptional reach and engagement, virality is a core feature of this
environment, particularly when referring to viral news and their potential
impact on the public discourse. In today’s online ecosystem, it is crucial to
understand how collective attention responds to abrupt news diffusion and how
sudden spikes of visibility reverberate on the subsequent attention captured
by the source. While virality has been mainly investigated for its marketing
implications and received extensive coverage in the literature [16, 17, 18,
19, 20], the impact of viral news on collective attention has not obtained as
much consideration.

To address this gap in existing literature, this study aims to enhance our
comprehension of the attention economy through a data-driven approach by
exploring the dynamics of virality and its effects on users’ engagement on
different social media platforms. By analyzing data from Facebook and YouTube,
we examine attention patterns after viral events to assess how these events
influence users’ interactions over time.

In this study, we use a comparative interrupted time-series (CITS) design
implemented using a Bayesian structural time series model (BSTS) [21] to
evaluate the impact of viral events on users’ engagement. In our approach, we
apply the BSTS by using increasingly broader time windows to observe the
effect of the same viral event from a short-term to a long-term perspective.
Based on the BSTS’s results, we conduct our analysis first by examining the
magnitude of the impact and then its temporal dynamics to address the
following two research questions.

RQ1: Does virality induce engagement growth?

Our first research question aims to assess whether and how a viral event leads
to increased users’ attention received by the source. After that, our second
research question aims to analyze the temporal dynamics of these effects to
evaluate if the rapidity at which they occur influences their longevity.

RQ2: Do the faster-manifesting effects persist longer?

While our first research question aims to determine the actual impact of
virality and its magnitude, the second analysis provides valuable insights
into whether these events genuinely contribute to sustained growth or merely
act as transient spotlights. Our results indicate the presence of two
different types of viral events, a ‘loaded-type’ and a ‘sudden-type’ virality.
When virality follows a sustained growth phase, it represents the final burst
of that growth process, with users’ attention successively standing on lower
levels. Conversely, viral news boosts users’ engagement when occurring as a
sudden event, reactivating the collective response process. While virality can
temporarily boost user engagement, this effect is often short-lived. We
observe that quickly emerging viral effects rapidly fade out. On the other
hand, content achieving slower and sustained growth tends to show more
persistent effects on engagement. These results emphasize the importance of
continuous and consistent content strategies in establishing a solid and
enduring connection with the user base rather than relying on viral spikes.

The rest of this paper is organized as follows: Section 2, gives an overview
of the relevant literature. In Section 3, we outline our methodology for
virality detection and impact evaluation. In Section 4 we present our detailed
findings on the dynamics of the virality impact on users’ attention and its
persistence through time. In Section 5 we discuss the implications of our
findings, along with limitations and recommendation for future research.
Section 6 concludes the paper.

##  2 Literature review

###  2.1 Attention economy

The attention economy is central to today’s digital media landscape. In
information management, attention economics applies economic theory to human
attention, treating it as a scarce resource. The attention economy is defined
as a system of agents (senders) who aim to capture the attention of
individuals (receivers) by creating and sharing information packages (signals)
[22, 23, 24, 25, 11]. Once produced and disseminated, this information
undergoes a cognitive filtering process by the receivers, who select relevant
information and disregard the rest [26, 27, 28]. From a supply-side
perspective, the rise of social media platforms has led to an unprecedented
volume of available content and information. This overabundance, often
referred to as information overload in the literature [29, 30, 31, 9], is a
cardinal characteristic of the current attention market in the digital
landscape. In this competitive environment, a wide range of content creators,
from news organizations to individual influencers [10], compete for the
limited resource that is users’ attention. The concept of human attention as a
currency entails two key features [32]. Unlike real currencies, it can not be
accumulated but solely be spent. Additionally, its transient nature makes it
hard to trace and measure. Social media data help address this challenge, as
user interactions with content constitute precise engagement indicators. The
vast amount of tracked and aggregated data allows for studying large
populations, providing insights into collective behavior without the need for
experimental settings [33, 34]. Shifting the focus from individual users to a
broader community perspective highlights the dynamics of collective attention
[35, 36].

###  2.2 Human dynamics on social media

Originally designed for entertainment and personal connections, social media
platforms have become indispensable tools for information dissemination [37],
raising significant concerns about their potential impact on social dynamics.
Many works in the literature highlight how online users are prone to consume
information aligning with their existing beliefs [38, 39, 40] and commonly
ignore opposing viewpoints [28, 41, 42]. The creation and reinforcement of
online ‘echo chambers’ [43, 44], where shared narratives are collectively
shaped and solidified [39, 45], may exacerbate social divisions and foster
partisan and polarized communities [46, 47, 48], complicating the landscape of
public discourse [49] especially during sensitive periods such as global
elections [50]. Offering unprecedented opportunities for content to achieve
rapid and widespread attention [35], social media have become crucial
environments for the spread of information and misinformation during global
events [51, 52, 53], political events [54, 55], and discussions on emerging
technologies such as large language models [56]. The unprecedented amount of
available content distinguishes today’s digital ecosystem, considerably
complicating the search for information and giving rise to the phenomenon of
infodemics [57, 58], such as for the COVID-19 pandemic. These platforms may
also influence political landscapes, potentially affecting public opinion and
voter behavior during elections through the rapid dissemination and
amplification of political content. However, this influence is not
definitively proven [59, 60]. Research on the interplay between user behavior
and platforms’ byproduct presented both opportunities and challenges [46, 61,
62, 63], unveiling a multifaceted landscape where a prevalence of one over the
other has not yet emerged. As users’ behavior could exhibit persistent
patterns across different platforms, topics, and contexts [64], the
comparative analysis of diverse social media can isolate unaltered
consistencies of human dynamics in the digital ecosystem or underline
algorithmic and platform peculiarities.

###  2.3 Virality on social media

On social media platforms, virality refers to a piece of content having rapid
diffusion and high levels of users’ engagement. Viral events can differ
depending on the dissemination that leads to their emergence. Structural
virality distinguishes between broadcast and viral diffusion based on their
spreading patterns [65]. Broadcast diffusion follows a pattern where a single,
large parent node spreads information to several smaller entities. In
contrast, viral diffusion involves multiple smaller nodes, each contributing
to the spread by generating a limited number of infections.

One of the most complex issues is understanding why certain items achieve
sudden and widespread dissemination while other similar or higher-quality
items remain overlooked [66, 67]. Research suggests that the nature of the
content is a more significant factor in driving virality than the
characteristics of the spreading source [68]. For example, the source size in
terms of Followers does not affect the probability of going viral on Facebook
[14], with the growth dynamics of engagement exhibiting universal patterns in
the short run. Concerning content properties, [69] demonstrated that the type
of multimedia content on Twitter affects the volume and speed of retweeting,
[70] identified visual attributes that can predict relative virality using
Reddit data, while [71] characterized visual elements distinguishing viral
from non-viral memes.

On the other hand, the frantic search for attractiveness may translate into
severe drawbacks for the source itself. For instance, [72] evidenced how
readers often view clickbait as a manipulative tactic, which can reduce the
perceived competence and trustworthiness of the publisher. Information
overload leads to similar drawbacks also in business and managerial contexts.
The widespread practice of consumer-generated content can provoke social and
brand overload due to its quantity and poor quality, leading to negative
consequences such as brand disloyalty [73].

Beyond their media features, there is substantial agreement in the literature
that emotional resonance is a critical trigger of virality, particularly with
negative emotions and extreme or sensitive content. Extreme content, such as
sex, nudity, and violence, is more likely to become viral [74]. Hateful
content cascades tend to be larger, persist longer, and exhibit more
significant structural virality [75]. An initial burst of a topic’s diffusion
often correlates with increased negative reactions from users [76], and
negative messages are typically reposted more quickly and frequently than
positive or neutral ones [77].

The spread of misinformation and the rise of political partisanship are
critical issues due to their significant social implications. Conspiracy rumor
cascades tend to be more persistent and exhibit a positive relationship
between their lifetime and size [39]. During the COVID-19 pandemic,
misinformation was more likely to go viral than truthful information, often
due to the use of emotionally charged, other-condemning language [78]. A
similar mechanism is observed in political contexts, where posts about
political opponents are more likely to be shared on social media. This out-
group effect is more influential than other social media-sharing predictors,
such as emotional language [79]. Ultimately, the driving effects of virality
stem from the combination and mutual reinforcement of ideological segregation
and negative emotional resonance.

Besides cases of misinformation and harmful content, viral diffusion may also
have significant drawbacks in different social contexts. Social media might
influence risk perceptions from the general public [80], fueling moral panics
and amplifying threats posed by deviant behavior and ideas [81], so much so of
having tangible consequences in real contexts like inducing distortions and
detrimental effects on financial markets [82, 83].

##  3 Research design

We now outline our research design, starting from the data collection process
detailed in Section 3.1. In Section 3.2, we formally define virality along
with the measures to quantify it, while Section 3.3 accounts for the virality
detection methods. Lastly, in Section 3.4, we first illustrate our approach to
evaluate the virality effect, and then we separately outline our research
questions along with their related methods.

###  3.1 Data

We begin by selecting a list of news outlets from NewsGuard [84], an
organization known for monitoring news outlet activities internationally,
rating more than 35,000 news and information sources across several countries.
We select all the reported news outlets from Germany, France, Italy, and the
United Kingdom. NewsGuard provides several categories of descriptive metadata,
including the URLs of these news sources’ social media accounts, if available.
For Facebook, after identifying all the news outlets with active accounts
listed on NewsGuard, we use their Facebook URLs to gather data via
CrowdTangle. This Facebook-owned tool monitors interactions on public content
from Facebook pages, groups, and verified profiles [85]. Using the ‘Historical
Data’ feature in CrowdTangle, we download the complete content history of each
page, starting from January 1, 2018. Similarly, for YouTube, after selecting
all available accounts, we use the YouTube Data API [86] to collect their list
of published content. Given the fewer YouTube channels and videos compared to
Facebook, we extend the observation period for YouTube to begin on January 1,
2016.

Table 1 provides an overview of the dataset, detailing the total number of
Facebook Pages and YouTube channels used in our analysis for each country,
along with their respective number of posts and videos. The longitudinal
structure and temporal continuity of these page observations are crucial for
evaluating the impact of viral events. Therefore, the resulting dataset is
formatted consistently for both platforms, containing chronological
information about each posted and accessible item. This naturally excludes any
content that may have been removed by users or through platform moderation at
the time of download. Note that in what follows, we will refer to both
Facebook pages and YouTube channels as (information) sources and to Facebook
posts and YouTube videos as either content or posts, specifying the platform
in the latter case.

Table 1: Dataset overview | Facebook | YouTube
---|---|---
| 2018-2023 | 2016-2023
Country | Pages | Posts | Channels | Videos
Germany | 263 | 9,633,896 | 242 | 193,153
France | 252 | 8,705,928 | 177 | 204,665
Italy | 384 | 16,259,173 | 234 | 271,183
UK | 227 | 8,751,788 | 105 | 100,794
Total | 1,126 | 43,350,785 | 758 | 769,795

###  3.2 Defining virality

On social media platforms, virality represents the propensity of content to
achieve rapid diffusion and high interactions levels from users [15]. To
quantify virality, we use a bivariate approach based on two measures,
spreading and interactions. The following subsections provide the definition
of these two measures for each social media platform.

####  3.2.1 Spreading

For Facebook data, we adopt the metric provided by [87]. Originally defined on
Twitter data, the proposed metric can be applied to any social media platform
with equivalent Retweets and Followers metrics. On Facebook, Shares are
equivalent to Retweets, representing the number of users sharing certain
content. On both platforms, Followers represent the number of users subscribed
to a given page at the time of posting. Therefore, on Facebook, we can define
the Spreading $S$ of a given post $j$ of a Page $i$ as:

$S_{ijt}=ln\left(\frac{Shares_{ij}}{Followers_{it}}\right)$ (1)

where $Followers_{it}$ indicates the number of Followers of the Page $i$ at
time $t$, i.e. the time of posting.

For YouTube, we can define it using the video Views metric, which specifically
measures content spreading. Therefore, on YouTube, we define the Spreading $S$
of a given video $j$ published by a Channel $i$ at time $t$ as:

$S_{ijt}=ln\left(Views_{ij}\right).$ (2)

####  3.2.2 Interactions

On Facebook, interactions are measured by the Total Interactions, encompassing
the total number of users’ interactions with the post (the sum of Likes,
Comments, and Shares). Therefore, on Facebook, we define the Total
Interactions $TI$ of a given post $j$ of a Page $i$ as:

$TI_{ijt}=ln\left(Likes_{ij}+Comments_{ij}+Shares_{ij}\right)$ (3)

where $t$ is the date in which post $j$ appears.

Since YouTube does not provide a share count (and consequently its Shares
metric), we define the Total Interactions $TI$ of a given video $j$ of a
Channel $i$ published at day $t$ as:

$TI_{ijt}=ln\left(Likes_{ij}+Comments_{ij}\right).$ (4)

###  3.3 Detecting virality

We measure virality on social media as an (exceptional) over-performance of
content regarding spreading and users’ interactions. To detect virality, we
first apply a $z-$score normalization to both $S_{ijt}$ and $TI_{ijt}$ for
each piece of content $j$ from a source $i$. On Facebook, a post is considered
viral if both its $z-$scores exceed 3. We set the threshold for YouTube at 2.5
to ensure a comparable number of videos and channels as in Facebook.

Figure 1: Distributions of Spreading and Engagement $z-$scores for Facebook
and YouTube. The inset plots of each chart display the subset of values
exceeding the threshold for the given metric, showing its breakdown between
viral contents (which exceed the threshold in both metrics) and non-viral ones
(hence exceeding the threshold only in the displayed metric).

The distributions of the two $z-$scores for both platforms are shown in Fig.
1. The inset plots within each panel zoom on the subset of values that exceed
the threshold for the displayed metric, distinguishing between viral content
(which surpasses both thresholds) and non-viral content (which exceeds the
threshold only in the displayed one).

These insets reveal that the subset resulting from the combination of
overperformances constitutes a small fraction of the distribution tails,
especially on Facebook. In other words, content may induce high interactions
levels without widespread diffusion, or conversely, it may fail to engage
users despite extensive reach. This observation further emphasizes the
exceptional rarity of virality.

Fig. 2 presents the distributions of viral posts per source for Facebook and
YouTube. As the Figure shows, despite the differences in sample sizes, both
platforms exhibit similar heavy-tailed distributions over different scales
reflecting the peculiar dynamics of content virality. If, on the one hand, a
limited number of sources succeeded in achieving virality multiple times, it
represented a unique or rare event for most of them.

Figure 2: Distributions of viral posts per source on Facebook and YouTube.
Facebook sample sizes: Pages = 613, Posts = 9514; YouTube sample sizes:
Channels = 161, Videos = 3624.

###  3.4 Evaluating virality impact

After defining the procedure for detecting viral posts, we can now delve deep
into our approach for evaluating the impact of viral events on the performance
of Facebook pages and YouTube channels.

In terms of attention received, we quantify the performance of a given source
by calculating its daily engagement. We define the Engagement $E_{it}$ of a
source $i$ at day $t$ as the sum of the Total Interactions $TI$ (as in Eq. 3,
and 4) of all its $j$ posts published that day, i.e.:

$E_{it}=\sum_{j}TI_{ijt}.$

In assessing the total attention a news outlet receives, we interpret higher
engagement as indicative of greater user attention, regardless of the number
of posts published. This approach is based on the premise that increasing the
content volume only results in heightened engagement if the content
effectively captures users’ attention. Conversely, high engagement across
several posts implies high users’ attention, and calculating its average value
would potentially underestimate this effect [14].

Our analysis aims to evaluate the after-effects of a viral event on the
source’s performance in terms of attention caught. To inspect the attention
dynamics following the event, we first assess whether the engagement received
significantly changes after it. To detect a significant variation (either
positive or negative) on the engagement after a viral event, we use a
comparative interrupted time-series (CITS) design implemented using a Bayesian
structural time series model (BSTS, hereafter) [21], which has also been used
in previous research [88]. In our approach, we apply BSTS to each viral event,
using time windows associated to $n$ weeks ahead and before, with
$n=2,3,4,5,6$, constructed as follows: for any given $n$, we exclude the day
of the viral post and compare the engagement trend during the $n$ weeks
following the event to the expected trend based on the $n$ weeks preceding it
– hence, having a windows of $2n\times 7$ days – and controlling for the
presence of other viral posts. Based on the BSTS’s results, we then conduct
our analysis to address our two research questions outlined in the following
sections, first examining the magnitude of the impact and then its temporal
dynamics.

####  3.4.1 Research Question 1: Does virality induce engagement growth?

Our first research question aims to assess whether - and how - a viral event
increases users’ attention received by the source after its occurrence. We
start from the premise that each viral post can have a positive, null, or
negative effect on the subsequent engagement in the considered time window.

To evaluate the effect, we use two variables provided by the BSTS: 1) the
Average Absolute Effect, which indicates whether the impact was positive or
negative, along with its magnitude, and 2) the statistical significance of the
effect, which is captured by the $p$-value $p_{\alpha}$ of BSTS at a
confidence level $\alpha$. For the analysis, we set $\alpha=0.05$.

If the effect has no statistical significance at a confidence level $\alpha$,
we consider that virality did not have a discernible impact on users’
engagement - which we refer to as No Effect. Otherwise, if the effect is
statistically significant and the observed engagement significantly deviates
from the expected trend based on the examined $n$ weeks, we consider that
virality had a Growth or Decrease effect on users’ attention as quantified by
the Average Absolute Effect. For our analyses, we use the BSTS implementation
from the CausalImpact R package [89]. This research question is addressed in
Sec. 4.1.

####  3.4.2 Research Question 2: Do the faster-manifesting effects persist
longer?

To continue our investigation of virality dynamics, we aim to examine the
relationship between the speed at which effects manifest and their persistency
over time. By using increasingly wider time windows, we can observe the effect
of the same viral event from a short-term to a long-term perspective.

For each detected effect, its emergence and its subsequent persistency can be
determined through a classification based on the output from the BSTS. For
each viral post $j$ of a page $i$, we define the time of emergence of its
effect, denoted as $\tau^{(em)}_{ij}$, as follows:

$\tau^{(em)}_{ij}=\min\\{n=2,3,4,5:p_{ij}(n)<p_{\alpha}\\},$ (5)

where $p_{ij}(n)$ is the $p$-value of BSTS applied to the time window
associated to week $n$. Similarly, the fade-out time, $\tau^{(f-o)}_{ij}$, is
identified as the earliest time window in which the previously detected effect
no longer manifests, i.e.,

$\tau^{(f-o)}_{ij}=\min\\{n>\tau^{(em)}_{ij}:p_{ij}(n)>p_{\alpha}\\}.$ (6)

We notice that the sets associated to the definitions of $\tau^{(em)}_{ij}$
and $\tau^{(f-o)}_{ij}$ are not empty in the cases treated in this paper.

Notice also that the $\tau$’s lead to a partition of the collection of the
posts with the related pages. Specifically, fixed $\bar{n}$, we define

$\mathrm{T}_{\bar{n}}^{(em)}=\\{(i,j)\in\mathcal{I}\times\mathcal{J}:\tau^{(em)}_{ij}=\bar{n}\\}$
(7)

and

$\mathrm{T}_{\bar{n}}^{(f-o)}=\\{(i,j)\in\mathcal{I}\times\mathcal{J}:\tau^{(f-o)}_{ij}=\bar{n}\\}.$
(8)

We define the persistency $\phi_{kh}$ as the fraction of posts still having
persistent effects at a given week $h$ after their emergence time $k$, i.e.,

$\phi_{kh}=\frac{|\mathrm{T}_{k}^{(em)}|-\sum_{\ell=k+1}^{h}|\mathrm{T}_{\ell}^{(f-o)}|}{|\mathrm{T}_{k}^{(em)}|},$
(9)

where $|\bullet|$ is the cardinality of set $\bullet$, for each $k$
$\displaystyle\in[2,5]$ and $h$ $\displaystyle\in[k+1,6]$. See Sec. 4.2 for
further details, in which we address the Research Question 2.

By analyzing the persistency rates across all cases for each specified
emergence time, we can evaluate the relationship between the speed of
manifestation and the longevity of the effects. This analysis will help
determine whether effects that emerge more quickly tend to persist longer or,
conversely, if they vanish more rapidly than slow-emerging ones.

This approach helps us to understand the temporal behavior of viral effects,
as we can further extrapolate the observed dynamics to their unobservable
earlier phase by tracing back to the time of the event and evaluating the
dynamics as we approach it asymptotically. This analysis provides valuable
insights into how collective attention responds to sudden stress conditions,
like viral news events, and to determine whether these events genuinely
contribute to sustained growth or merely act as transient spotlights.

##  4 Results

Figure 3: Effects after virality with increasing time windows. The $y$-axis
represents the percentage of Growth, No Effect, and Decrease cases, for each
time scale on the $x$-axis. Links represent the effects flow between
consecutive weeks, color-coded according to the first observed effect (i.e.,
in the 2-week window).

Before discussing our research questions in depth, we first inspect the BSTS’s
results, which provide us with a useful initial overview to help delve into
the analysis. We begin by examining the proportion of cases that exhibit
positive, null, or negative impacts and assess their consistency across
various time windows.

The results are reported in Fig 3, which shows the percentage of Growth, No
Effect, and Decrease cases on the $y$-axis for each time scale on the
$x$-axis. The diagram includes links representing the effects flow between
consecutive weeks, color-coded according to the first observed effect to
maintain a clear visual trajectory through the data stream. A significant
observation from this analysis regards the high percentage of cases showing no
statistically significant impact on engagement performance. This first finding
suggests that virality does not necessarily enhance engagement, as it often
results in an indiscernible impact.

Figure 4: Density of the trend preceding the viral post and the average
absolute effect on engagement for the 2-week time window. Trend Pre Virality
is the $\beta 1$ coefficient of the regression estimated by the BSTS on the
weeks preceding virality. Given its previous trend, the average Absolute
Effect is the average effect on the Engagement after the viral event. Only
events with a statistically significant effect on Engagement are shown.

Secondly, the percentages of Growth and Decrease cases are comparable, with
slightly higher occurrences of negative impacts. This suggests that, apart
from being infrequent, the impact of a viral event on users’ attention can
even be detrimental. Contrary to common expectations [15, 90, 91], virality
rarely induces engagement growth. Furthermore, we observe that effects
typically emerge or shift between the second and fourth weeks, with the most
significant transitions happening from the third to the fourth week. Beyond
this period, the consistency of effects is broadly stable. Given the
inherently short-term nature of virality, effects observed in larger time
scales, such as 5- and 6-week windows, are less likely to be directly
attributable to the viral event. These longer time windows are primarily aimed
at assessing the persistency of earlier effects, enabling us to analyze the
dynamics from a short-term to a long-term perspective and to distinguish the
boundary between these periods.

###  4.1 RQ1: Dynamics of virality effect on Engagement

We now deepen the dynamics of the virality effect on Engagement by focusing
exclusively on the Growth and Decrease cases, as defined in Sec. 3.4.1.

In Fig. 4, we present the joint density of the slopes of the attention trends
preceding the viral posts and the absolute effects on engagement after the
event.

The ‘Trend Pre Virality’ on the y-axis represents the $\beta_{1}$ coefficient
of the regression estimated by the BSTS for the $n$ weeks preceding the viral
event. On the x-axis, we show the Average Absolute Effect on Engagement in the
examined $n$-week window after virality, accounting for its previous trend.

Fig. 4 reports the values for the 2-week window as an example, while Fig. 6 in
SI Appendix shows the results for other timescales on both platforms which
display consistent results.

As Fig. 4 shows, the density is split into two opposing quadrants: from growth
to decrease and from decrease to growth, highlighting a significant negative
correlation between the preceding trend and the consequent absolute effect.
This dynamic shows consistency on both platforms and across their timescales,
as shown in Tab 2, which reports the Spearman’s correlation coefficients
between the $\beta_{1}$ coefficient of the trend preceding virality and the
Average Absolute Effect on Engagement.

Table 2: Spearman’s correlation coefficients between the $\beta 1$ coefficient of the trend preceding virality and the Absolute Effect on Engagement for different platforms and time windows. Posts and Videos represent the number of events for which a Growth or Decrease effect is detected, according to the BSTS. | Facebook | YouTube
---|---|---
Timescale | $\boldsymbol{\rho}$ | Posts | $\boldsymbol{\rho}$ | Videos
2 Weeks | -0.92 | 3015 | -0.95 | 1100
3 Weeks | -0.91 | 2953 | -0.93 | 1048
4 Weeks | -0.90 | 3134 | -0.93 | 1049
5 Weeks | -0.88 | 3279 | -0.93 | 1028
6 Weeks | -0.87 | 3399 | -0.93 | 1016

Therefore, virality positively impacts users’ engagement when occurring as a
sudden event on a declining collective attention. Conversely, when virality
manifests following a sustained growth phase, it represents the final burst of
that growth process, with users’ attention successively standing on lower
levels than its previous phase.

These results shed light on the bounded yet elastic nature of collective
attention. While additional growth following a sustained growth phase is
extremely rare, viral events act as a booster when users’ attention is likely
to nearing its lower bound, reactivating the collective response process.

By focusing on viral news, these results potentially indicate the presence of
two different types of viral events. The first is a ‘loaded-type’ virality,
where attention progressively increases, culminating in the viral event. This
type could occur in scenarios where information is gradually revealed, such as
sequences of rumors, confirmations, and official announcements. The second
type represents a ‘sudden-type’ virality, with the news emerging unexpectedly
as an exogenous event. Based on the release patterns of information, this
interpretation could explain the differences between the two types, along with
the inverse relationship between the preceding attention pattern and the
after-effect of the viral news.

###  4.2 RQ2: Emergence and persistency of the virality effect

After assessing how virality affects users’ engagement and its extent, we now
inspect the temporal dynamics of the effect by examining the relationship
between its manifestation and longevity. In this section, the analysis will
specifically focus on the effect’s persistency based on its timing of
emergence and fade-out.

In Eq. 5, we defined the time of $\tau^{(em)}_{ij}$ as the shortest time
window during which a Growth or Decrease effect first becomes significant. For
example, if the impact of a viral event is significant within a 2-week time
window, we assign its emergence time as 2. Conversely, if the impact is not
significant within the 2-week window but becomes significant in the 3-week
window, the emergence time is then 3.

Similarly, we examine the duration for which the effect remains unchanged and
determine the window in which it ceases to be consistent. As we defined in Eq.
6, the fade-out $\tau^{(f-o)}_{ij}$ of the effect corresponds to the first
window where it no longer manifests. For example, if an event consistently
exhibits a Growth effect from the 2-week to the 4-week window and then ceases
to manifest in the 5-week window, the emergence time would be 2, and the fade-
out time would be 5.

Consequently, as we defined in Eq. 9, the persistency $\phi_{kh}$ is the
fraction of still persistent effects at $h$ weeks after their given emergence
time $k$, to which we generally refer here. Hence, we evaluate the effect’s
decay by calculating the fraction of still persistent effects after each week,
grouped by their time of emergence. For instance, over the total number of
cases where the impact appears two weeks after virality, we calculate the
proportion of persistent effects in the 3rd, 4th, 5th, and 6th week. This
method allows us to derive the effect’s decay curve for each time of emergence
$k$ $\displaystyle\in[2,5]$.

At this stage, we are able to observe the persistency rate up to a minimum of
two weeks after the viral event. By fitting a negative exponential function of
$k$ and power law in $h$, we can estimate the variation of the exponent of the
decay curves based on their time of emergence $k$ as:

$\leavevmode\resizebox{}{13.6572pt}{$\phi_{kh}=f(k,h)=h^{-\lambda+\frac{k}{\beta}}$}.$
(10)

This procedure allows us to describe the decay behavior of the effects based
on their emergence and to extend it to their unobservable earlier phases as if
we approach them asymptotically.

Figure 5: Persistency of the virality effect based on its time of emergence.
Solid lines represent the estimated decay for the observed curves - from the
2nd week to the 5th week of emergence time - along with their observed values.
The dotted lines represent the corresponding extrapolated decay curves for 0
and 1 week after virality as emergence time.

The results of the fitting procedure are reported in Tab. 3, while Fig. 5
shows the graphical representation of the decay curves for each emergence
time. Solid lines represent the estimated decay for the observed curves - from
the 2nd week to the 5th week of emergence time - along with their observed
persistency. The dotted lines represent the corresponding extrapolated decay
curves for 0 and 1 week after virality as emergence time. The curves
representing extrapolations inherently involve higher uncertainty and should
be considered useful asymptotic approximations for understanding the dynamics
during their unobservable phases.

Table 3: Paramaters estimation Platform | Parameter | Estimate | Std. error | p-value
---|---|---|---|---
Facebook | $\lambda$ | 1.09 | 0.02 | <0.001
Facebook | $\beta$ | 8.84 | 0.44 | <0.001
YouTube | $\lambda$ | 1.09 | 0.05 | <0.001
YouTube | $\beta$ | 17.44 | 4.55 | 0.0024

According to the estimations, in about 50% of cases, the impact either fades
out within the first week following the viral event or does not occur.
Moreover, the data consistently show that earlier emergences of viral impacts
are associated with faster decay across both platforms. This indicates that
faster processes tend to fade quicker, while slower ones exhibit more
persistence. This finding highlights the elastic nature of collective
attention when stretched to its limits. The volatile and fluctuating nature of
attention prevents it from being steadily focused, resulting in a trade-off
between the rapidity of the effect and its durability. These observations have
critical implications for content producers in the digital realm, underscoring
a clear distinction between short-term and long-term dynamics of collective
attention.

From a probabilistic perspective, sudden and disproportionate growth is rare
and rarely leads to a noticeable positive impact on engagement. Even when it
occurs, its effects tend to fade away swiftly. Conversely, organic and
sustained growth, though slower to manifest, tends to have more enduring
effects. This contrast emphasizes the transient nature of viral events
compared to the lasting effectiveness of consistent engagement, highlighting
the importance of gradual and continuous attention-building strategies rather
than relying on abrupt surges of visibility.

##  5 Discussion

###  5.1 Implications

Our findings highlight collective attention’s bounded and elastic nature under
sudden stress conditions, such as viral events, and have significant
implications for the information sources ecosystem in the digital domain.
Firstly, our analysis reveals that viral events rarely lead to engagement
growth, suggesting that the frantic pursuit of sudden visibility is often
unproductive. The study distinguishes between two primary types of virality,
each corresponding to a different mechanism of collective attention response.
The first type, ‘loaded-type virality,’ is characterized by a gradual increase
of engagement that culminates in a final burst of attention in the viral
event. The second type, ‘sudden-type virality,’ occurs when news emerges
unexpectedly, similar to an exogenous event, and is the only scenario in which
virality leads to an increased users’ response. Regardless of whether the
impact is positive or negative, our findings indicate that effects emerging
more quickly tend to fade faster, while slower-emerging processes are more
persistent over time. As well as being an extremely rare event, virality does
not turn out to be an effective long-term growth strategy. These insights
underscore the advantages of organic and continued growth strategies in
establishing a solid and enduring connection with the user base, yielding
positive and sustained responses regarding collective attention.

###  5.2 Limitations and Future Research

The findings of this study are specifically tied to the platforms analyzed,
the type of pages examined, and the period considered. Although the results
are consistent with general human dynamics observed in other collective
attention studies, they cannot be generalized across all social media
platforms. Each platform has distinct user bases and algorithmic
characteristics that influence how content is spread and engages users.
Additionally, the specificity of the sources analyzed — news outlets - and
their content type must be considered. Focusing exclusively on news items
offers the advantage of dealing with content inherently tied to specific
topics or events, providing a concrete subject matter. However, the dynamics
and peculiarities of the news agenda mean that these results may not
necessarily apply to the broader spectrum of content creators. Future research
should explore various social media platforms and content types to understand
better the dynamics of virality and collective attention across different
contexts. A more comprehensive picture of how virality functions in the
digital landscape can be developed by expanding the study to include a broader
range of content creators and user engagement patterns.

##  6 Conclusions

This study offers a detailed analysis of virality dynamics on social media
platforms. It examines how viral events affect users’ engagement and the
relationship between their rapid emergence and subsequent persistency. Our
findings challenge the common assumption that virality regularly enhances user
engagement. While viral events may momentarily capture attention, our
evaluation reveals that they rarely foster sustained engagement growth.
Indeed, the impact of virality typically depends on the preceding growth
trend, with a pronounced negative correlation observed between the two. We
categorize viral events into two distinct types: ‘loaded-type’ virality,
characterized by a gradual build-up of attention culminating in a viral burst,
and ‘sudden-type’ virality, which appears unexpectedly like an exogenous
shock.

When virality occurs following an ongoing growth phase, it represents its
final peak, resulting in diminished attention levels. In contrast, a viral
event reactivates the collective response process when arising suddenly after
a declining attention phase.

Additionally, our study highlights the elastic nature of collective attention,
demonstrating that effects emerging swiftly tend to fade quickly, whereas
slower-emerging effects show greater persistency. This finding highlights the
critical role of content producers in fostering consistent, high-quality
engagement rather than depending solely on transient viral spikes. The rapid
dissipation of viral impacts, particularly those that emerge quickly,
illustrates the volatility and fluctuation of collective attention. This
volatility implies a trade-off between the rapidity of an impact’s emergence
and its lasting presence, emphasizing content producers’ challenges in
capturing and maintaining users’ engagement. In conclusion, although virality
can temporarily surge users’ attention, its effects are usually short-lived,
making the frantic pursuit of sudden visibility an often fruitless strategy.

##  7 Author contributions statement

  * •

Emanuele Sangiorgio Conceptualization, Methodology, Validation, Data
collection, Data curation, Data analysis, Visualization, Writing.

  * •

Niccolò Di Marco Methodology, Validation, Data analysis, Writing.

  * •

Gabriele Etta Data collection, Data curation, Data analysis, Writing.

  * •

Matteo Cinelli Conceptualization, Methodology, Supervision, Writing.

  * •

Roy Cerqueti Conceptualization, Methodology, Supervision, Writing.

  * •

Walter Quattrociocchi Conceptualization, Supervision, Writing.

## Declaration of Competing Interest

The authors declare no competing interest.

##  8 Data availability

The data collection and analysis process are compliant with the terms and
conditions imposed by Crowdtangle [85]. Therefore, the results described in
this paper cannot be exploited to infer the identity of the accounts involved.
CrowdTangle does not include paid ads unless those ads began as organic, non-
paid posts that were subsequently “boosted” using Facebook’s advertising
tools. It also does not include activity on private accounts, or posts made
visible only to specific groups of followers.

##  9 SI Appendix Section

Figure 6: Density of the trend preceding the viral post and the average
absolute effect on engagement for the 3, 4, 5, and 6-week timescales. We
recall that Trend Pre Virality is the $\beta 1$ coefficient of the regression
estimated by the BSTS on the weeks preceding virality, and that the Average
Absolute Effect is the average effect on the Engagement after virality
estimated by the BSTS, given its previous trend. Only events with a
statistically significant effect on engagement are shown.

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