]https://aovgun.com

# Testing Symmergent gravity through the shadow image and weak field photon
deflection by a rotating black hole using the M87∗ and Sgr. A∗ results

Reggie C. Pantig<EMAIL_ADDRESS>Physics Department, Mapúa University,
658 Muralla St., Intramuros, Manila 1002, Philippines  Ali Övgün
<EMAIL_ADDRESS>[ Physics Department, Eastern Mediterranean University,
Famagusta, 99628 North Cyprus, via Mersin 10, Turkey  Durmuş Demir
<EMAIL_ADDRESS>Faculty of Engineering and Natural Sciences,
Sabancı University, 34956 Tuzla, İstanbul, Turkey

###### Abstract

In this paper, we study rotating black holes in symmergent gravity, and use
deviations from the Kerr black hole to constrain the parameters of the
symmergent gravity. Symmergent gravity induces the gravitational constant $G$
and quadratic curvature coefficient $c_{\rm O}$ from the flat spacetime matter
loops. In the limit in which all fields are degenerate in mass, the vacuum
energy $V_{\rm O}$ can be wholly expressed in terms of $G$ and $c_{\rm O}$. We
parametrize deviation from this degenerate limit by a parameter
${\hat{\alpha}}$ such that the black hole spacetime is dS for
${\hat{\alpha}}<1$ and AdS for ${\hat{\alpha}}>1$. In constraining the
symmergent parameters $c_{\rm O}$ and ${\hat{\alpha}}$, we utilize the EHT
observations on the M87* and Sgr. A* black holes. We investigate first the
modifications in the photon sphere and shadow size, and find significant
deviations in the photonsphere radius and the shadow radius with respect to
the Kerr solution. We also find that the geodesics of time-like particles are
more sensitive to symmergent gravity effects than the null geodesics. Finally,
we analyze the weak field limit of the deflection angle, where we use the
Gauss-Bonnet theorem for taking into account the finite distance of the source
and the receiver to the lensing object. Remarkably, the distance of the
receiver (or source) from the lensing object greatly influences the deflection
angle. Moreover, $c_{\rm O}$ needs be negative for a consistent solution. In
our analysis, the rotating black hole acts as a particle accelerator and
possesses the sensitivity to probe the symmergent gravity.

Black hole; Modified gravity; Geodesics; Weak gravitational lensing; Shadow;
Deflection angle.

###### pacs:

95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.Kd

##  I Introduction

In the Wilsonian sense, quantum field theories (QFTs) are characterized by a
classical action and an ultraviolet (UV) cutoff $\Lambda$. Quantum loops lead
to effective QFTs with loop momenta cut at $\Lambda$. The effective QFTs
suffer from UV oversensitivity problems: The scalar and gauge boson masses
receive ${\mathcal{O}}(\Lambda^{2})$ corrections. The vacuum energy, on the
other hand, gets corrected by ${\mathcal{O}}(\Lambda^{4})$ and
${\mathcal{O}}(\Lambda^{2})$ terms. The gauge symmetries get explicitly
broken. The question is simple: Can gravity emerge in a way restoring the
explicitly broken gauge symmetries? Asking differently, can gravity emerge in
a way alleviating the UV oversensitivities of the effective QFT? This question
has been answered affirmatively by forming a gauge symmetry-restoring emergent
gravity model Demir (2021, 2019, 2016). This model, briefly called as
symmergent gravity, has been built by the observation that, in parallel with
the introduction of Higgs field to restore gauge symmetry for a massive vector
boson (with Casimir invariant mass) Anderson (1963); Englert and Brout (1964);
Higgs (1964), spacetime affine curvature can be introduced to restore gauge
symmetries for gauge bosons with loop-induced (Casimir non-invariant) masses
proportional to the UV cutoff $\Lambda$ Demir (2021, 2019, 2016). Symmergent
gravity is essentially emergent general relativity (GR) with a quadratic
curvature term. It exhibits distinctive signatures, as revealed in recent
works on static black hole spacetimes Çimdiker (2020); Çimdiker  _et al._
(2021); Rayimbaev  _et al._ (2022); Riasat Ali and Rimsha Babar and Zunaira
Akhtar and Ali Övgün (2023). In the present work, we use observational data on
M87* and Srg.A* black holes to study symmergent gravity. We focus on rotating
black hole spacetimes as these black holes can act as a richer test and
analysis laboratory. What we are doing can be viewed as “doing particle
physics via black holes” as it will reveal salient properties of the QFT
sector through various black hole features.

In 1919 Arthur Eddington led an expedition to prove Einstein’s theory of
relativity using gravitational lensing and since then lensing has become an
important tool in astrophysics Virbhadra and Ellis (2000, 2002); Virbhadra
_et al._ (1998); Virbhadra and Keeton (2008); Virbhadra (2009); Adler and
Virbhadra (2022); Bozza  _et al._ (2001); Bozza (2002); Perlick (2004); He
_et al._ (2020). In astrophysics, distances are very important when
determining the properties of astrophysical objects. But Virbhadra showed that
just the observation of relativistic images without any information about the
masses and distances can also accurately give value of the upper bound on the
compactness of massive dark objects Virbhadra (2022a). Furthermore, Virbhadra
proved that there exists a distortion parameter such that the signed sum of
all images of singular gravitational lensing of a source identically vanishes
(tested with Schwarzschild lensing in weak and strong gravitational fields
Virbhadra (2022b)).

In 2008, Gibbons and Werner used the Gauss-Bonnet theorem (GBT) on the optical
geometries in asymptotically flat spacetimes, and calculated weak deflection
angle for first time in literature Gibbons and Werner (2008). Afterwards, this
method has been applied to various phenomena Övgün (2018, 2019a, 2019b); Javed
_et al._ (2019); Werner (2012); Ishihara  _et al._ (2016, 2017); Ono  _et al._
(2017); Li and Övgün (2020); Li  _et al._ (2020a); Belhaj  _et al._ (2022);
Javed  _et al._ (2023, 2022); Pantig and Övgün (2023). One of the aims of the
present paper is to probe symmergent gravity through the black hole’s weak
deflection angle using the GBT and also the shadow silhouette as perceived by
a static remote observer. In essence, by the very nature of the symmergent
gravity, we will be studying the numbers of fermions and bosons and related
effects in the vicinity of a spinning black hole, which has the potential to
affect the motions of null and time-like geodesics as well as their spin
parameter $a$. (By definition, $a=cJ/GM^{2}$ where $J$ and $M$ are the angular
momentum and mass of the rotation black hole.) Recently, shadow coming from
different black hole models has been analyzed extensively in the literature as
a probe of the imprints of the astrophysical environment affecting it Vagnozzi
_et al._ (2022); Chen  _et al._ (2022); Dymnikova and Kraav (2019); Pantig and
Övgün (2022a); Uniyal  _et al._ (2023); Kuang and Övgün (2022); Meng  _et al._
(2022); Tang  _et al._ (2022); Kuang  _et al._ (2022); Wei  _et al._ (2019);
Xu  _et al._ (2018); Hou  _et al._ (2018); Bambi  _et al._ (2019); Tsukamoto
(2018); Kumar  _et al._ (2020, 2019); Wang  _et al._ (2017, 2018); Amarilla
and Eiroa (2018); Peng-Zhang  _et al._ (2020); Tsupko  _et al._ (2020); Hioki
and Maeda (2009); Konoplya (2019); Okyay and Övgün (2022); Belhaj  _et al._
(2020); Li  _et al._ (2020b); Ling  _et al._ (2021); Belhaj  _et al._ (2021);
Cunha and Herdeiro (2018); Gralla  _et al._ (2019); Perlick  _et al._ (2015);
Nedkova  _et al._ (2013); Li and Bambi (2014); Khodadi  _et al._ (2021);
Khodadi and Lambiase (2022); Cunha  _et al._ (2017); Shaikh (2019); Allahyari
_et al._ (2020); Yumoto  _et al._ (2012); Cunha  _et al._ (2016a); Moffat
(2015); Cunha  _et al._ (2016b); Zakharov (2014); Hennigar  _et al._ (2018);
Chakhchi  _et al._ (2022); Pantig and Övgün (2022b); Pantig  _et al._ (2022);
Lobos and Pantig (2022); Pantig and Övgün (2022c) Calculation of the shadow
cast of a non-rotating black hole was pioneered by Synge (1966) and Luminet
(1979), and later on extended by Bardeen (1973) Chandrasekhar (1998) to an
axisymmetric spacetime. Recently, after the detection of gravitational waves
in 2015 Abbott  _et al._ (2016), the first image of the black hole in M87* was
formed by the Event Horizon Telescope (EHT) using its electromagnetic spectrum
Akiyama  _et al._ (2019a). At the time of writing of this paper, new milestone
has been achieved since the EHT revealed the shadow image of the black hole in
our galaxy, Sgr. A* Akiyama  _et al._ (2022). This new result indicates that
black hole theory and phenomenology is a hot research topic in vivew of the
rapidly developing observational techniques.

The paper is organized as follows. In Sec. II, we give a detailed discussion
of the symmergent gravity. We give in Sec. III the rotating metric, indicating
dependence on the symmergent parameters. In Sec. IV, we study the effects of
symmergent gravity on null geodesics, which in turn influence the shadow
radius and observables associated with it. We also examine the said effect on
time-like orbits in the same section. We devote Sec. V to investigation of the
weak deflection angle at finite-distance, and in Sec. V.1 we calculate the
center-of-mass energy (CM) of two particles and study particle acceleration
near rotating symmergent black hole background. In Sec. VI, we conclude and
give future prospects.

##  II Symmergent Gravity

In this section, we give a brief description of the symmergent gravity in
terms of its fundamental parameters. The starting point is quantum field
theories (QFTs). Quantum fields are endowed with mass and spin as the Casimir
invariant s of the Poincaré group. Fundamentally, QFTs are intrinsic to the
flat spacetime simply because they rest on a Poincaré-invariant (translation-
invariant) vacuum state Macias and Camacho (2008); Wald (2018). Flat spacetime
means the total absence of gravity. Incorporation of gravity necessitates the
QFTs to be taken to curved spacetime, but this is hampered by Poincaré
breaking in curved spacetime Dyson (2013); Wald (2018). This hamper and
absence of a quantum theory of gravity ’t Hooft and Veltman (1974) together
lead one to emergent gravity framework Sakharov (1967); Visser (2002);
Verlinde (2017) as a viable approach.

In general, loss of Poincaré invariance could be interpreted as the emergence
of gravity into the QFT Froggatt and Nielsen (2005). In a QFT, curvature can
emerge at the Poincaré breaking sources. One natural Poincaré breakings source
is the hard momentum cutoff on the QFT. Indeed, an ultraviolet (UV) cutoff
$\Lambda$ Polchinski (1984) limits momenta $p_{\mu}$ within
$-\Lambda^{2}\leq\eta^{\mu\nu}p_{\mu}p_{\nu}\leq\Lambda^{2}$ interval as the
intrinsic validity edge of the QFT Polchinski (1984). Under the loop
corrections up to the cutoff $\Lambda$, the action $S[\eta,\phi,V]$ of a QFT
of scalars $S$ and gauge bosons $V_{\mu}$ receives the correction (with
$(+,-,-,-)$ metric signature appropriate for QFTs)

$\displaystyle\delta S[\eta,\phi,V]=\int d^{4}x\sqrt{-\eta}\left\\{-V_{\rm
O}-c_{\rm
O}\Lambda^{4}-\sum_{i}c_{m_{i}}m_{i}^{2}\Lambda^{2}-c_{S}\Lambda^{2}S^{\dagger}S+c_{V}\Lambda^{2}V_{\mu}V^{\mu}\right\\}$
(1)

in which $\eta_{\mu\nu}$ is the flat metric, $V_{\rm O}$ is the vacuum energy
which is not power-law in $\Lambda$, $m_{i}$ stands for the mass of a QFT
field $\psi_{i}$ (summing over all the fermions and bosons), and $c_{\rm O}$,
$c_{m}$, $c_{S}$ and $c_{V}$ are respectively the loop factors describing the
quartic vacuum energy correction, quadratic vacuum energy correction,
quadratic scalar mass correction, and the loop-induced gauge boson mass
Umezawa  _et al._ (1948); Kallen (1949). As revealed by the gauge boson mass
term $c_{V}\Lambda^{2}V_{\mu}V^{\mu}$, the UV cutoff $\Lambda$ breaks gauge
symmetries explicitly since $\Lambda$ is not a particle mass, that is,
$\Lambda$ is not a Casimir invariant of the Poincaré group. The loop factor
$c_{V}$ (as well as $c_{S}$) depends on the details of the QFT. (It has been
calculated for the standard model gauge group in Demir (2021, 2019).)

In Sakharov’s induced gravity Sakharov (1967); Visser (2002), the UV cutoff
$\Lambda$ is associated with the Planck scale, albeit with explicitly-broken
gauge symmetries and Planckian-size cosmological constant and scalar masses.
In recent years, Sakharov’s setup has been approached from a new perspective
in which priority is given to the prevention of the explicit gauge symmetry
breaking Demir (2021). For this aim, one first takes the effective QFT in (1)
to curved spacetime of a metric $g_{\mu\nu}$ such that the gauge boson mass
term is mapped as $c_{V}\Lambda^{2}\eta^{\mu\nu}V_{\mu}V_{\nu}\longrightarrow
c_{V}V_{\mu}\left(\Lambda^{2}g^{\mu\nu}-R^{\mu\nu}(g)\right)V_{\nu}$ in
agreement with the fact that the Ricci curvature $R^{\mu\nu}(g)$ of the metric
$g_{\mu\nu}$ can arise only in the gauge sector via the covariant derivatives
Demir (2021, 2019, 2016). One next inspires from the Higgs mechanism to
promote the UV cutoff $\Lambda$ to an appropriate spurion field. Indeed, in
parallel with the introduction of Higgs field to restore gauge symmetry for a
massive vector boson (Poincare-conserving mass) Anderson (1963); Englert and
Brout (1964); Higgs (1964), one can introduce spacetime affine curvature to
restore gauge symmetries for gauge bosons with loop-induced (Poincare-breaking
mass) masses proportional to $\Lambda$ Vitagliano  _et al._ (2011); Karahan
_et al._ (2013); Demir (2021, 2019). Then, one is led to the map

$\displaystyle\Lambda^{2}g^{\mu\nu}\rightarrow{\mathbb{R}}^{\mu\nu}(\Gamma)$
(2)

in which ${\mathbb{R}}^{\mu\nu}(\Gamma)$ is the Ricci curvature of the affine
connection $\Gamma^{\lambda}_{\mu\nu}$, which is completely independent of the
curved metric $g_{\mu\nu}$ and its Levi-Civita connection Vitagliano  _et al._
(2011); Karahan  _et al._ (2013); Demir and Puliçe (2020). This map is
analogue of the map $M_{V}^{2}\rightarrow\phi^{\dagger}\phi$ of the vector
boson mass $M_{V}$ (Poincare conserving) into the Higgs field $\phi$. Under
the map (2) the effective QFT in (1) takes the form

$\displaystyle\delta S[g,\phi,V,{\mathbb{R}}]=\int
d^{4}x\sqrt{-\eta}\left\\{-V_{\rm O}-\frac{c_{\rm
O}}{16}{\mathbb{R}}^{2}(g)-\sum_{i}\frac{c_{m_{i}}}{4}m_{i}^{2}{\mathbb{R}}(g)-\frac{c_{S}}{4}{\mathbb{R}}(g)S^{\dagger}S+c_{V}V_{\mu}\left({\mathbb{R}}^{\mu\nu}(\Gamma)-R^{\mu\nu}(g)\right)V_{\nu}\right\\}$
(3)

in which ${\mathbb{R}}(g)\equiv g^{\mu\nu}{\mathbb{R}}_{\mu\nu}(\Gamma)$ is
the scalar affine curvature Demir (2021). This metric-Palatini theory contains
both the metrical curvature $R(g)$ and the affine curvature
${\mathbb{R}}(\Gamma)$. From the third term, Newton’s gravitational constant
$G$ is read out to be

$\displaystyle G^{-1}=4\pi\sum_{i}c_{m_{i}}m_{i}^{2}\xrightarrow{\rm one\
loop}\frac{1}{8\pi}{\rm str}\\!\left[{\mathcal{M}}^{2}\right]$ (4)

where ${\mathcal{M}}^{2}$ is the mass-squared matrix of all the fields in the
QFT spectrum. In the one-loop expression, ${\rm str}[\dots]$ stands for super-
trace namely ${\rm str}[{\mathcal{M}}^{2}]=\sum_{i}(-1)^{2s_{i}}(2s_{i}+1){\rm
tr}[{\mathcal{M}}^{2}]_{s_{i}}$ in which ${\rm tr}[\dots]$ is the usual trace
(including the color degrees of freedom), $s_{i}$ is the spin of the QFT field
$\psi_{i}$ ($s_{i}=0,1/2,\dots$), and $[{\mathcal{M}}^{2}]_{s_{i}}$ is the
mass-squared matrix of the fields having that spin (like mass-squared matrices
of scalars ($s_{i}=0$), fermions $(s_{i}=1/2)$ and so on). One keeps in mind
that ${\rm tr}[\dots]$ encodes degrees of freedom $g_{i}$ (like color and
other degrees of freedom) of the particles.

It is clear that known particles (the standard model spectrum) cannot generate
Newton’s constant in (4) correctly (in both sign and size). It is necessary to
introduce therefore new particles. Interesting enough, these new particles do
not have to couple to the known particles since the only constraint on them is
the super-trace in (4) Demir (2021, 2019).

The action (3) remains stationary against variations in the affine connection
provided that

$\displaystyle{}^{\Gamma}\nabla_{\lambda}{\mathbb{D}}_{\mu\nu}=0$ (5)

such that ${}^{\Gamma}\nabla_{\lambda}$ is the covariant derivative of the
affine connection $\Gamma^{\lambda}_{\mu\nu}$, and

$\displaystyle{\mathbb{D}}_{\mu\nu}=\left(\frac{1}{16\pi
G}+\frac{c_{S}}{4}S^{\dagger}S+\frac{c_{\rm
O}}{8}g^{\alpha\beta}{\mathbb{R}}_{\alpha\beta}(\Gamma)\right)g_{\mu\nu}-c_{V}V_{\mu}V_{\nu}$
(6)

is the field-dependent metric. The motion equation (5) implies that
${\mathbb{D}}_{\mu\nu}$ is covariantly-constant with respect to
$\Gamma^{\lambda}_{\mu\nu}$, and this constancy leads to the exact solution

$\displaystyle\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}({\mathbb{D}}^{-1})^{\lambda\rho}\left(\partial_{\mu}{\mathbb{D}}_{\nu\rho}+\partial_{\nu}{\mathbb{D}}_{\rho\mu}-\partial_{\rho}{\mathbb{D}}_{\mu\nu}\right)={}^{g}\Gamma^{\lambda}_{\mu\nu}+\frac{1}{2}({\mathbb{D}}^{-1})^{\lambda\rho}\left(\nabla_{\mu}{\mathbb{D}}_{\nu\rho}+\nabla_{\nu}{\mathbb{D}}_{\rho\mu}-\nabla_{\rho}{\mathbb{D}}_{\mu\nu}\right)$
(7)

in which ${}^{g}\Gamma^{\lambda}_{\mu\nu}$ is the Levi-Civita connection of
the curved metric $g_{\mu\nu}$. The Planck scale in (4) is the largest scale
and therefore it is legitimate to make the expansions

$\displaystyle\Gamma^{\lambda}_{\mu\nu}$ $\displaystyle=$
$\displaystyle{}^{g}\Gamma^{\lambda}_{\mu\nu}+8\pi
G\left(\nabla_{\mu}{\mathbb{D}}^{\lambda}_{\nu}+\nabla_{\nu}{\mathbb{D}}^{\lambda}_{\mu}-\nabla^{\lambda}{\mathbb{D}}_{\mu\nu}\right)+{\mathcal{O}}\left(G^{2}\right)$
(8)

and

$\displaystyle{\mathbb{R}}_{\mu\nu}(\Gamma)$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}(g)+8\pi
G\left(\nabla^{\alpha}\nabla_{\mu}{\mathbb{D}}_{\alpha\nu}+\nabla^{\alpha}\nabla_{\nu}{\mathbb{D}}_{\alpha\mu}-\Box{\mathbb{D}}_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}{\mathbb{D}}_{\alpha}^{\alpha}\right)+{\mathcal{O}}\left(G^{2}\right)$
(9)

so that both $\Gamma^{\lambda}_{\mu\nu}$ and ${\mathbb{R}}_{\mu\nu}(\Gamma)$
contain pure derivative terms at the next-to-leading
${\mathcal{O}}\left(G\right)$ order Demir (2019, 2016). The expansion in (8)
ensures that the affine connection $\Gamma^{\lambda}_{\mu\nu}$ is solved
algebraically order by order in $G$ despite the fact that its motion equation
(5) involves its own curvature ${\mathbb{R}}_{\mu\nu}(\Gamma)$ through
${\mathbb{D}}_{\mu\nu}$ Vitagliano  _et al._ (2011); Karahan  _et al._ (2013).
The expansion (9), on the other hand, ensures that the affine curvature
${\mathbb{R}}_{\mu\nu}(\Gamma)$ is equal to the metrical curvature
$R_{\mu\nu}(g)$ up to a doubly-Planck suppressed remainder. In essence, what
happened is that the affine dynamics took the affine curvature ${\mathbb{R}}$
from its UV value $\Lambda_{\wp}^{2}$ in (2) to its IR value $R$ in (9). This
way, the GR emerges holographically ’t Hooft (1993); Cohen  _et al._ (1999)
via the affine dynamics such that loop-induced gauge boson masses get erased,
and scalar masses get stabilized by the curvature terms. This mechanism
renders effective field theories natural regarding their destabilizing UV
sensitivities Demir (2021, 2019). It gives rise to a new framework in which
(i) the gravity sector is composed of the Einstein-Hilbert term plus a
curvature-squared term, and (ii) the matter sector is described by an
${\overline{MS}}$-renormalized QFT Demir (2021, 2019). We call this framework
gauge symmetry-restoring emergent gravity or simply symmergent gravity to
distinguish is from other emergent or induced gravity theories in the
literature.

It is worth noting that symmergent gravity is not a loop-induced curvature
sector in curved spacetime Visser (2002); Birrell and Davies (1984). In
contrast, symmergent gravity arises when the flat spacetime effective QFT is
taken to curved spacetime Demir (2021, 2019) in a way reviving the gauge
symmetries broken explicitly by the UV cutoff. All of its couplings are loop-
induced parameters deriving from the particle spectrum of the QFT (numbers and
masses of particles). It is with these loop features that the GR emerges. In
fact, the metric-Palatini action (3) reduces to the metrical gravity theory

$\displaystyle\int d^{4}x\sqrt{-g}\Bigg{\\{}\\!$ $\displaystyle-V_{\rm
O}-\frac{{\mathbb{R}}(g)}{16\pi G}-\frac{c_{\rm
O}}{16}\left({\mathbb{R}}(g)\right)^{2}+\frac{c_{S}}{4}S^{\dagger}S\,{\mathbb{R}}(g)+c_{V}{\rm
tr}\left[V^{\mu}\\!\left({\mathbb{R}}_{\mu\nu}(\Gamma)-R_{\mu\nu}({}^{g}\Gamma)\right)\\!V^{\nu}\right]\\!\Bigg{\\}}$
(10) $\displaystyle\xrightarrow{\rm equation\,(\ref{expand-curv})}\int
d^{4}x\sqrt{-g}\Bigg{\\{}\\!-V_{\rm O}-\frac{R}{16\pi G}-\frac{c_{\rm
O}}{16}R^{2}-\frac{c_{S}}{4}S^{\dagger}SR+{\mathcal{O}}\\!\left(G\right)\\!\Bigg{\\}}$

after replacing the affine curvature ${\mathbb{R}}_{\mu\nu}(\Gamma)$ with its
solution in (9). Of the parameters of this emergent GR action, Newton’s
constant $G$ was already defined in (4). The loop factor $c_{S}$ depends on
the underlying QFT. (It reads $c_{S}\simeq 0.29$ in the standard model.) The
loop factor $c_{\rm O}$, which was associated with the quartic ($\Lambda^{4}$)
corrections in the flat spacetime effective QFT in (1), turned to the
coefficient of quadratic-curvature $(R^{2})$ term in the symmergent GR action
in (10). At one loop, it takes the value

$\displaystyle c_{\rm O}=\frac{n_{\text{B}}-n_{\text{F}}}{128\pi^{2}}$ (11)

in which $n_{\text{B}}$ ($n_{\text{F}}$) stands for the total number of
bosonic (fermionic) degrees of freedom in the underlying QFT (including the
color degrees of freedom). Both the $n_{\text{B}}$ bosons and $n_{\text{F}}$
fermions contain not only the known standard model particles but also the
completely new particles. As was commented just above (5), it is a virtue of
symmergence that these new particles do not have to couple to the known ones,
non-gravitationally.

The last parameter of the symmergent GR action (10) is the vacuum energy
density $V_{\rm O}$. It belongs to the non-power-law sector of the flat
spacetime effective QFT in (1). At one loop, it takes value (${\rm tr}[\dots]$
involves all degrees of freedom $g_{i}$ of particles, like color)

$\displaystyle V_{\rm O}=\frac{{\rm
str}\left[{\mathcal{M}}^{4}\right]}{64\pi^{2}}$ (12)

after discarding a possible tree-level contribution. Being a loop-induced
quantity, Newton’s constant in (4) involves super-trace of $({\rm
masses})^{2}$ of the QFT fields. In this regard, the potential energy $V_{\rm
O}$, involving the super-trace of $({\rm masses})^{4}$ of the QFT fields, is
expected to expressible in terms of $G$. To see this, it proves useful to
start with mass degeneracy limit in which each and every boson and fermion
possess equal masses, $m_{b}=m_{f}=M_{0}$, for all $b$ and $f$. Needless to
say, $M_{0}$ is essentially the characteristic scale of the QFT. (Essentially,
$M_{0}$ is the mean value of all the field masses.) Under this degenerate mass
spectrum the potential $V_{\rm O}$ can be expressed as follows:

$\displaystyle V_{\rm O}=\frac{{\rm
str}\left[{\mathcal{M}}^{4}\right]}{64\pi^{2}}=\frac{1}{64\pi^{2}}\left(\sum_{\rm
B}m^{4}_{\rm B}-\sum_{\rm F}m^{4}_{\rm F}\right)\xrightarrow{\rm
mass\,degeneracy}\frac{M_{0}^{4}}{64\pi^{2}}(n_{\text{B}}-n_{\text{F}})=\frac{M_{0}^{2}}{8\pi
G}=\frac{1}{2(8\pi G)^{2}c_{\rm O}}$ (13)

where use have been made of the $G$ formula in (4) and $c_{\rm O}$ formula in
(11). Now, the problem is to take into account realistic cases in which the
QFT fields are not all mass-degenerate. For a QFT with characteristic scale
$M_{0}$ but with no detailed knowledge of the mass spectrum, realistic cases
might be represented by parametrizing the potential energy as

$\displaystyle V_{\rm O}$ $\displaystyle=$
$\displaystyle\frac{1-{\hat{\alpha}}}{(8\pi G)^{2}c_{\rm O}}$ (14)

in which the new parameter ${\hat{\alpha}}$ is introduced as a measure of the
deviations of the boson and fermion masses from the QFT characteristic scale
$M_{0}$. Clearly, ${\hat{\alpha}}=1/2$ corresponds to the degenerate case in
(13). Alternatively, ${\hat{\alpha}}=1$ represents the case in which
$\sum_{\rm B}m^{4}_{\rm B}=\sum_{\rm F}m^{4}_{\rm F}$ in (13). In general,
${\hat{\alpha}}>1$ (${\hat{\alpha}}<1$) corresponds to the boson (fermion)
dominance in terms of the trace $({\rm masses})^{4}$.

Symmergence makes gravity emerge from within the flat spacetime effective QFT.
Fundamentally, as follows from the action (1), Newton’s constant $G$ in (4),
the quadratic curvature coefficient $c_{\rm O}$ in (11), and the vacuum energy
$V_{\rm O}$ in (12) transpired in the flat spacetime effective QFT from the
matter loops. (The loop factor $c_{S}$ and similar parameters couple the
matter fields.) Once the gravity emerges as in (10), however, $G$, $c_{\rm O}$
and $V_{\rm O}$ gain a completely new physical meaning as the curvature sector
parameters (not involving the matter fields). In this sense, they are
symmergent gravity parameters in the sense that they became curvature sector
parameters via the symmergence.

A glance at the second line of (10) reveals that symmergent gravity is an
$R+R^{2}$ gravity theory with non-zero cosmological constant. In fact, it can
be put in the form

$\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left(R+f(R)\right)$
(15)

after leaving aside the scalars $S$ and the other matter fields, after
switching to $(-,+,+,+)$ metric signature (appropriate for the black hole
analysis in the sequel), and after introducing the $f(R)$ gravity function

$\displaystyle f(R)=-\pi Gc_{\rm O}R^{2}-16\pi GV_{\rm O}$ (16)

comprising the vacuum energy $V_{\rm O}$ and the quadratic-curvature term
proportional to $c_{\rm O}$. The Einstein field equations arising from the
action (15)

$\displaystyle\mathit{R}_{\mu\nu}(1+f^{\prime}(R))-\frac{1}{2}g_{\mu\nu}(R+f(R))+(\nabla_{\mu}\nabla_{\nu}-\Box
g_{\mu\nu})f^{\prime}(R)=0$ (17)

become $R_{0}(f^{\prime}(R_{0})-1)-2f(R_{0})=0$ upon contraction at constant
curvature ($R=R_{0}$), and possess the solution Cembranos  _et al._ (2014);
Dastan  _et al._ (2022)

$\displaystyle R_{0}=32\pi GV_{\rm O}=\frac{1-{\hat{\alpha}}}{2\pi Gc_{\rm
O}}$ (18)

after using the vacuum energy formula in (14) in the second equality. It is
clear that this $R_{0}$ value would vanish if the vacuum energy were not
nonzero. Indeed, for a quadratic gravity with $f(R)=bR^{2}$ one gets the
solution $R_{0}=0$. But for a quadratic gravity like $f(R)=\hat{a}+bR^{2}$ one
finds $R_{0}=-2\hat{a}\neq 0$, which reduces to the $R_{0}$ in (18) for
$\hat{a}=-16\pi GV_{\rm O}$.

The Einstein field equations (17) possess static, spherically-symmetric,
constant-curvature solutions of the form Cembranos  _et al._ (2014); Dastan
_et al._ (2022)

$\displaystyle
ds^{2}=-h(r)dt^{2}+\frac{dr^{2}}{h(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (19)

in which

$\displaystyle h(r)=1-\frac{2GM}{r}-\frac{(1-{\hat{\alpha}})}{24\pi Gc_{\rm
O}}r^{2}$ (20)

is the lapse function following from (18). In the sequel, we will analyze this
constant-curvature configuration ($R=R_{0}\neq 0$) in both the dS ($V_{\rm
O}>0$ namely ${\hat{\alpha}}<1$) and AdS ($V_{\rm O}<0$ namely
${\hat{\alpha}}>1$) spacetimes in the case of rotating black holes. We will
use rotating black hole properties to determine or constrain the model
parameters $c_{\rm O}$ and ${\hat{\alpha}}$.

##  III Rotating Black holes in symmergent Gravity

Appropriate for a rotating black hole geometry, we use Boyer-Lindquist
coordinates with a metric free of the coordinate singularities in the
spacetime both exterior to the black hole and interior to the cosmological
horizon. The metric is given by Cembranos  _et al._ (2014); Perez  _et al._
(2013); Suvorov (2019); Myung (2013); Nashed and Nojiri (2021); Nashed (2021)

$\mathrm{d}s^{2}=\frac{\rho^{2}}{\Delta_{r}}\mathrm{~{}d}r^{2}+\frac{\rho^{2}}{\Delta_{\theta}}\mathrm{d}\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left[a\mathrm{d}t-\left(r^{2}+a^{2}\right)\frac{\mathrm{d}\phi}{\Xi}\right]^{2}-\frac{\Delta_{r}}{\rho^{2}}\left(\mathrm{d}t-a\sin^{2}\theta\frac{\mathrm{d}\phi}{\Xi}\right)^{2}$
(21)

with the various quantities

$\displaystyle\Delta_{r}=\left(r^{2}+a^{2}\right)\left(1-\frac{r^{2}}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\rm O}}\right)-2Mr\,,$ (22)
$\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta\,,$ (23)
$\displaystyle\Delta_{\theta}=1+\frac{r^{2}}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\rm O}}a^{2}\cos^{2}\theta\,,$ (24)
$\displaystyle\Xi=1+\frac{r^{2}}{3}\frac{(1-\hat{\alpha})}{8\pi c_{\rm
O}}a^{2}\,.$ (25)

Here, $a$ is the black hole spin parameter, and ${\hat{\alpha}}$ and $c_{\rm
O}$ are the parameters of the symmergent gravity. The physical energy $E$ and
angular momentum $J$ of the black hole are related to the parameters $M$ and
$a$ via the relations Gibbons  _et al._ (2005); Gao  _et al._ (2022)

$E=\frac{M}{\Xi^{2}},\quad J=\frac{aM}{\Xi^{2}}$ (26)

in which $M$ is the root of the equation $\Delta_{r}\left(r_{+}\right)=0$
namely

$\displaystyle
M=\frac{1}{2r_{+}}\left(r_{+}^{2}+a^{2}\right)\left(1-\frac{r_{+}^{2}}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\rm O}}\right)$ (27)

where $r_{+}$ is the radius of the event horizon. In this way, one can express
energy and angular momentum in terms of the parameters $r_{+}$, $a$, and $G$.
(One here notes that $\frac{1}{l^{2}}={\frac{(\hat{a}-1)}{24\pi G\,c_{\rm
O}}}$, where the curvature radius $l$ is related to the negative cosmological
constant as $\Lambda=-3l^{-2}$). As a result, one finds

$E=\frac{1}{2\Xi^{2}r_{+}}\left(r_{+}^{2}+a^{2}+\frac{r_{+}^{4}}{l^{2}}+\frac{a^{2}r_{+}^{2}}{l^{2}}\right),\quad
J=\frac{a}{2\Xi^{2}r_{+}}\left(r_{+}^{2}+a^{2}+\frac{r_{+}^{4}}{l^{2}}+\frac{a^{2}r_{+}^{2}}{l^{2}}\right)$
(28)

as the explicit expressions for the definitions in (26).

Numerically, horizons can be determined by analyzing the lapse function, that
is, by solving $\Delta_{r}=0$ using its definition in (22). Besides, the
location of the ergoregions can be determined numerically by plotting the
metric component $g_{tt}$ and looking for points satisfying $g_{tt}=0$. In
fact, plotted in Fig. 1 are $\Delta_{r}$ and $g_{tt}$ as functions of $r$ for
the integration constant ${\hat{\alpha}}$ of $0.90$ and $1.10$, black hole
spin parameter $a=0.9\,M$, and the various values of $c_{\rm O}$. In general,
symmergent black hole can mimic the dS (${\hat{\alpha}}>1$) or AdS-Kerr
(${\hat{\alpha}}<1$) black holes depending on the sign of ${\hat{\alpha}}$.
But, as already shown by Pogosian and Silvestri Pogosian and Silvestri (2008),
consistent f(R) gravity theories require $\frac{dF(R)}{dR}>0$ in order to
remain stable the (non-tachyonic scalaron). This constraint imposes the
condition $c_{\rm O}<0$ on the loop-induced quadratic curvature coefficient,
and restricts viable solutions to the AdS-Kerr type. In view of the underlying
QFT, $c_{\rm O}<0$ implies that $n_{\text{F}}>n_{\text{B}}$ – a mostly
fermionic QFT Demir (2021, 2019, 2016). In the extreme case of no new bosons
(in agreement with $n_{\text{F}}>n_{\text{B}}$), one finds out that the new
particle sector remains inherently stable thanks to the fact that fermion
masses do not have any power-law sensitivity to the UV cutoff.

Usually, value of the cosmological constant is scaled to a higher value to see
its overall effect. In this sense, assumed values of $\hat{\alpha}$ and
$c_{\rm O}$ are also rescaled similarly. As is seen from Fig. 1, null
boundaries are greatly affected by the values of the symmergent parameters
$\hat{\alpha}$ and $c_{\rm O}$. Indeed, $\hat{\alpha}>1$ leads to three
horizons where, in the context of the dS geometry, the third horizon
corresponds to the cosmic outer boundary. It shifts farther as $c_{\rm O}$
gets more negative. The two remaining null boundaries are the Cauchy and event
horizons. In constrast to the dS solution, the symmergent AdS solution gives
only two horizons such that symmergent effects are nearly vanishingly small on
the inner horizon compared to the outer horizon. As for the ergoregions,
$\hat{\alpha}>1$ leads to three ergoregions, and it can be seen that the
farthest one turns out to be even beyond the cosmic horizon.

Figure 1: Variations of the functions $\Delta_{r}$ (left panel) and $g_{tt}$
(right panel) with the radial coordinate $r$ for the black hole spin parameter
$a=0.9\,M$. The plots show the null boundaries, with solid (dashed) lines
mimicking the behavior of AdS (dS) black holes. We consider only negative
values of $c_{\rm O}$ in agreement with Pogosian and Silvestri (2008).

The Hawking temperature is given by

$T=\frac{\kappa}{2\pi}$ (29)

such that the surface gravity

$\kappa^{2}=-\frac{1}{2}\triangledown^{\mu}\chi^{\nu}\triangledown_{\mu}\chi_{\nu}$
(30)

involves the null Killing vectors $\chi^{\nu}$. The two Killing vectors
$\xi^{\nu}=\partial_{t}$ and $\zeta^{\nu}=\partial_{\varphi}$ in the metric
are associated with the time translation and rotational invariance,
respectively. Consequently, we take

$\chi^{\nu}=\xi^{\nu}+\Omega\zeta^{\nu}$ (31)

and determine $\Omega$ under the condition that $\chi^{\nu}$ is a null vector.
This leads to the constraint

$\chi^{\nu}\chi_{\nu}=g_{tt}+2\Omega
g_{t\varphi}+\Omega^{2}g_{\varphi\varphi}=0,$ (32)

from which one gets

$\Omega=-\frac{g_{t\varphi}}{g_{\varphi\varphi}}\pm\sqrt{\left(\frac{g_{t\varphi}}{g_{\varphi\varphi}}\right)^{2}-\frac{g_{tt}}{g_{\varphi\varphi}}}$
(33)

which reduces to

$\Omega_{+}=\frac{a\Xi}{r_{+}^{2}+a^{2}}$ (34)

at the event horizon $\Delta\left(r_{+}\right)=0$. For this $\Omega$ value one
then gets

$\kappa=\frac{1}{2\left(r_{+}^{2}+a^{2}\right)}\left.\frac{d\Delta_{r}}{dr}\right|_{r=r_{+}},$
(35)

for the surface gravity,

$T=\frac{r_{+}}{4\pi\left(r_{+}^{2}+a^{2}\right)}\left(1+\frac{a^{2}}{l^{2}}+\frac{3r_{+}^{2}}{l^{2}}-\frac{a^{2}}{r_{+}^{2}}\right)=\frac{3r_{+}^{4}+\left(a^{2}+l^{2}\right)r_{+}^{2}-l^{2}a^{2}}{4\pi
l^{2}r_{+}\left(r_{+}^{2}+a^{2}\right)}$ (36)

for the Hawking temperature, and

$S=\frac{A\left[1+f^{\prime}(R_{0})\right]}{4}=\frac{\pi\left(r_{+}^{2}+a^{2}\right)}{\Xi}$
(37)

for the Bekenstein-Hawking entropy. Needless to say, all thermodynamic
quantities are necessarily non-negative.

The rotating symmergent black hole satisfies therefore the first law of
thermodynamics

$dE=TdS+\Omega dJ$ (38)

with

$\displaystyle\Omega=\frac{a\left(1+r_{+}^{2}l^{-2}\right)}{r_{+}^{2}+a^{2}}\,.$
(39)

##  IV Geodesics around rotating symmergent black holes

In this section we give a detailed analysis of the geodesics and orbits around
the symmergent black holes.

###  IV.1 Null geodesic and shadow cast

We start the analysis with null geodesics. The Hamilton-Jacobi equation gives

$\frac{\partial S}{\partial\lambda}=-H,$ (40)

where $S$ is the Jacobi action in terms of the affine parameter $\lambda$
(proper time) and coordinates $x^{\mu}$. The Hamiltonian is given by

$H=\frac{1}{2}g^{\mu\nu}\frac{\partial S}{\partial x^{\mu}}\frac{\partial
S}{\partial x^{\nu}},$ (41)

in the GR so that

$\frac{\partial S}{\partial\lambda}=-\frac{1}{2}g^{\mu\nu}\frac{\partial
S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu}}$ (42)

as follows from (40) above. Using the separability ansatz for the Jacobi
function

$S=\frac{1}{2}\mu^{2}\lambda-Et+L\phi+S_{r}(r)+S_{\theta}(\theta),$ (43)

with the particle mass $\mu$, one is led to the following first-order motion
equations Slany and Stuchlik (2020)

$\displaystyle\Sigma\frac{dt}{d\lambda}=\frac{\Xi(r^{2}+a^{2})P(r)}{\Delta_{r}}-\frac{\Xi
aP(\theta)}{\Delta_{\theta}},$
$\displaystyle\Sigma\frac{dr}{d\lambda}=\sqrt{R(r)},$
$\displaystyle\Sigma\frac{d\theta}{d\lambda}=\sqrt{\Theta(\theta)},$
$\displaystyle\Sigma\frac{d\phi}{d\lambda}=\frac{\Xi
aP(r)}{\Delta_{r}}-\frac{\Xi P(\theta)}{\Delta_{\theta}\sin^{2}\theta}$ (44)

after introducing the functions

$\displaystyle R(r)=P(r)^{2}-\Delta_{r}(\mu^{2}r^{2}+K),$ $\displaystyle
P(r)=\Xi E(r^{2}+a^{2})-\Xi aL,$
$\displaystyle\Theta(\theta)=\Delta_{\theta}(K-\mu^{2}a^{2}\cos^{2}\theta)-\frac{P(\theta)^{2}}{\sin^{2}\theta},$
$\displaystyle P(\theta)=\Xi(aE\sin^{2}\theta-L).$ (45)

From the third equation in (IV.1) above, constants of motion can be correlated
via the relation $K=\Xi^{2}(aE-L)^{2}$, which is a consequence of a hidden
symmetry in the $\theta$-coordinate Slany and Stuchlik (2020); Carter (1968).

First, we study null geodesic and shadow cast for massless particles
($\mu=0$). Photon circular orbits, which are always unstable, must then
satisfy the following condition

$R(r)=\frac{dR(r)}{dr}\mid_{r=r_{\text{o}}}=0.$ (46)

The null geodesic is important is an important quantity for symmergent black
holes. We need, in particular, the photon region in constant $r_{0}$, the so-
called photonsphere, which is an unstable orbit. The shadow cast ultimately
depends on the photon region. With $\mu=0$, it proves convenient to define
these two impact parameters

$\xi=\frac{L}{E}\quad{\rm and}\quad\eta=\frac{K}{E^{2}}.$ (47)

These parameters are found to possess the following explicit expressions

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{\Delta_{r}^{\prime}(r^{2}+a^{2})-4\Delta_{r}r}{a\Delta_{r}^{\prime}},$
(48) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{-r^{4}\Delta_{r}^{\prime
2}+8r^{3}\Delta_{r}\Delta_{r}^{\prime}+16r^{2}\Delta_{r}(a^{2}-\Delta_{r})}{a^{2}\Delta_{r}^{\prime
2}}$ (49)

after a lengthy algebra. The photonsphere radius $r_{\text{ph}}$ can then be
determined by solving $\eta(r)=0$ for $r$. The analytical solutions are well-
known for both Schwarzschild and Kerr black holes. In our case, we plot (49)
in Fig. 2 to get the qualitative impression about the location of
$r_{\text{ph}}$.

Figure 2: Location of the photonsphere (left panel), and location of the
photonsphere crossing at $\phi=0$ or along the equatorial plane (right panel).
We take $a=0.90M$ in both plots ($a$ is the black hole spin parameter).

The left panel of Fig. 2 concerns location of the photonsphere along the
equatorial plane. As we would expect, the Schwarzschild case gives
$r_{\text{ph}}=3M$. The extreme Kerr case gives two possible locations, that
is, the prograde orbit at $r_{\text{ph}}=M$ and the retrograde orbit at
$r_{\text{ph}}=4M$. The plot shows the locations for $a=0.90M$, which fall
near the said extreme values. For the symmergent effect, the dS type (dashed
lines) gives slightly lower values in the retrograde case while the AdS type
gives a larger one. It is clear that the more negative the $c_{\rm O}$ the
closer the photon ring to the Kerr case. The symmergent effect is also present
in the prograde case though the deviation is smaller than in the retrograde
case. In the right panel of Fig. 2, we consider photons with zero angular
momentum, that is, those that traverse the equatorial plane in a perpendicular
manner (the so-called nodes). The retrograde case gives higher values for such
an orbit than the prograde case. The deviation caused is barely evident,
especially in the prograde case (the inset highlights the deviation). As a
final remark, we note that the symmergent effect mimics dS or AdS cases, where
such a behavior does not occur in the Schwarzschild case. The coupling between
the spin and symmergent parameters made it possible to affecy the behavior of
the photonsphere.

Escaping photons in the unstable orbit gives the possibility for remote
observers to backward-trace and obtain a shadow cast by using the celestial
coordinates $(r_{\text{obs}},\theta_{\text{obs}})$. Such an observer is also
known as the Zero Angular Momentum Observer (ZAMO). Also, in this type of co-
moving frame, without loss of generality, one makes the approximation
$r_{\text{obs}}\rightarrow\infty$ and takes $\theta_{\text{obs}}=\pi/2$. The
celestial coordinates are defined as Johannsen (2013)

$\displaystyle\mathcal{X}$
$\displaystyle=-r_{\text{obs}}\frac{\xi}{\zeta\sqrt{g_{\phi\phi}}\left(1+\frac{g_{t\phi}}{g_{\phi\phi}}\xi\right)},$
$\displaystyle\mathcal{Y}$
$\displaystyle=r_{\text{obs}}\frac{\pm\sqrt{\Theta(i)}}{\zeta\sqrt{g_{\theta\theta}}\left(1+\frac{g_{t\phi}}{g_{\phi\phi}}\xi\right)},$
(50)

and the condition $r_{\text{obs}}\rightarrow\infty$ leads to the simplified
relations

$\displaystyle\mathcal{X}$ $\displaystyle=-\xi\csc\theta_{\text{obs}},$
$\displaystyle\mathcal{Y}$
$\displaystyle=\pm\sqrt{\eta+a^{2}\cos^{2}\theta_{\text{obs}}-\xi^{2}\cot^{2}\theta_{\text{obs}}}.$
(51)

These expressions further simplify to $\mathcal{X}=0$ and
$\mathcal{Y}=\pm\sqrt{\eta}$ when $\theta_{\text{obs}}=\pi/2$. If,
furthermore, $a=0$ then shadow cast of a Schwarzschild black hole (a circle)
is obtained. The plot of $\mathcal{Y}$ vs. $\mathcal{X}$ is shown in Fig. 3
for the black hole spin parameter value of $a=0.90M$.

Figure 3: Shadow plot for different inclination angles for the black hole spin
parameter $a=0.90M$ (aspect ratios of figures are set to obtain the actual
view).

Overall, we observe the D-shaped nature of the shadow cast due to the spin
parameter. Considering first the effect of the inclination angle, we see that
as we decrease the value of $\theta$ (going up from the equatorial plane), we
see that the shape changes. Remarkably, in the upper right figure, the shape
becomes more compressed in the $\alpha$ axis, and as one continues to go up
(lower left figure), it goes back to an almost spherical shape. Finally, near
the pole, a drastic change in the shadow shape occurs due to stretching of the
$\alpha$-axis. For the symmergent effects, we note that the dS type tends to
decrease the photonsphere radius. Nevertheless, as photons travel in the
intervening space under the effect of the symmergent parameter the shadow size
is seen to increase. We can see the contrast in the AdS type symmergent effect
in that while the photonsphere increases the shadow size tends to decrease.

In the azimuthal plane, we see how the shadow becomes “D-shaped” when the
balck hole spin parameter $a$ is near extremal. The numerical value of the
shadow radius associated with this shape Hioki and Maeda (2009) can be
calculated Dymnikova and Kraav (2019) by using

$R_{\text{s}}=\frac{\mathcal{Y}_{\text{t}}^{2}+(\mathcal{X}_{\text{t}}-\mathcal{X}_{\text{r}})^{2}}{2|\mathcal{X}_{\text{t}}-\mathcal{X}_{\text{r}}|}$
(52)

which we plot in Fig. 4 (upper left), along with the shadow’s angular radius
$\theta_{\text{sh}}$ (upper right), where the latter is defined by

$\theta_{\text{sh}}=9.87098\times 10^{-3}\frac{R_{\text{s}}M}{D}$ (53)

after taking the black hole mass $M$ in units of $M_{\odot}$ and $D$ in
parsecs. We emphasize that these behaviors are consistent with Fig. 3. Other
observables that can be derived from the shadow are the distortion parameter
$\delta_{\text{s}}$ and the energy emission rate $\frac{d^{2}E}{d\omega dt}$,
which are defined as

$\delta_{\text{s}}=\frac{d_{\text{s}}}{R_{\text{s}}}=\frac{\tilde{\mathcal{X}}_{\text{l}}-\mathcal{X}_{\text{l}}}{R_{\text{s}}},$
(54) $\frac{d^{2}E}{d\omega
dt}=2\pi^{2}\frac{\Pi_{ilm}}{e^{\omega/T}-1}\omega^{3}$ (55)

such that the energy absorption cross-section can be approximated as
$\Pi_{ilm}\sim\pi R_{\text{s}}^{2}$ for an observer at
$r_{\text{obs}}\rightarrow\infty$. These observables are plotted in Fig. 4 in
the lower two panels. From these panels we are able to see how the spin
parameter increases the distortion of the shadow. We can also see that even if
the dS symmergent effect tends to increase the shadow, the distortion it gives
is smaller than the AdS symmergent effect.

Figure 4: Shadow observables associated with the upper left panel of Fig. 3.
Shown here are shadow radius (upper-left panel), angular diameter in $\mu$as
for M87* (upper-right panel), distortion parameter (lower-left panel), and
energy emission rate (lower-right panel).

In the lower right panel of Fig. 4, we comparatively show $a=0.25M$ and
$a=0.90M$ for the Kerr, dS and AdS symmergent black holes. For the Kerr case,
we see that low values of $a$ leads to higher values for energy emission rate
and peak frequency. Adding the symmergent effect at low values of $a$ and
$\hat{\alpha}=0.90$, we see that a less negative $c_{\rm O}$ provides the
highest peak frequency and energy emission rate. For $\hat{\alpha}=1.1$, this
very same $c_{\rm O}$ gives the lowest emission rate and peak frequency. For
$a=0.90M$, we observe in the inset plot that the roles are reversed near the
peak of the curve. We see that there is a point in the plot where the behavior
flips as the frequency $\sigma$ increases whilst the emission rate decreases.

We now investigate observational constraints on the symmergent parameter
$c_{\rm O}$ using the black hole shadow data collected from M87* Akiyama  _et
al._ (2019b) and Sgr. A* Akiyama  _et al._ (2022). The data is tabulated in
Table 1.

Black hole |  Mass ($M_{\odot}$) |  Angular diameter: $2\theta_{\text{sh}}$ ($\mu$as) |  Distance (kpc)
---|---|---|---
Sgr. A* |  $4.3\pm 0.013$x$10^{6}$ (VLTI) |  $48.7\pm 7$ (EHT) |  $8.277\pm 0.033$
M87* |  $6.5\pm 0.90$x$10^{9}$ |  $42\pm 3$ |  $16800$
Table 1: Black hole observational constraints.

With these data, one can determine diameter of the shadow size in units of the
black hole mass with

$d_{\text{sh}}=\frac{D\theta}{M}.$ (56)

In this sense, the diameter of the shadow image of M87* and Sgr. A* are
$d^{\text{M87*}}_{\text{sh}}=(11\pm 1.5)m$ and $d^{\text{Sgr.
A*}}_{\text{sh}}=(9.5\pm 1.4)m$, respectively. Meanwhile, the theoretical
shadow diameter can be obtained via
$d_{\text{sh}}^{\text{theo}}=2R_{\text{sh}}$ with the use of equation (52).

Figure 5: Constraint on the symmergent gravity parameter $c_{\rm O}$ by the
Sgr. A* data (left-panel) and M87* data (right-panel).

Based on Fig. 5, we can conclude that the parameters we used in describing the
shadow cast fall within the upper and lower bounds at the $1\sigma$ level
Akiyama  _et al._ (2019b, 2022); Kocherlakota  _et al._ (2021); Vagnozzi  _et
al._ (2022). Where this happens, the range in $c_{\rm O}$ is more extensive in
M87* than in Sgr. A*. We also observe that less negative $c_{\rm O}$
introduces more sensitivity to deviation than more negative $c_{\rm O}$.

Sgr. A* | $2\sigma$ | $1\sigma$ | observed $R_{\text{sh}}$
---|---|---|---
spin $a$ (${\hat{\alpha}=0.90}$) | upper | lower | upper | lower | mean
$0.25M$ | - | - | - | -0.057 | -0.184
$0.90M$ | - | - | - | -0.060 | -0.196
spin $a$ (${\hat{\alpha}=1.10}$) | upper | lower | upper | lower | mean
$0.25M$ | -0.127 | - | -0.397 | - | -
$0.90M$ | -0.137 | - | -0.431 | - | -

M87* | $2\sigma$ | $1\sigma$ | observed $R_{\text{sh}}$
---|---|---|---
spin $a$ (${\hat{\alpha}=0.90}$) | upper | lower | upper | lower | mean
$0.25M$ | - | -0.054 | - | -0.196 | -
$0.90M$ | - | -0.057 | - | -0.185 | -
spin $a$ (${\hat{\alpha}=1.10}$) | upper | lower | upper | lower | mean
$0.25M$ | -0.081 | - | -0.117 | - | -0.336
$0.90M$ | -0.089 | - | -0.128 | - | -0.364

Table 2: Bounds on $c_{\rm 0}$ as read off from the curves in Fig. 5.

###  IV.2 Symmergent gravity effects on time-like orbits: Effective potential
and ISCO radii

We now turn our attention to time-like orbits. Here, we will still use the
equations of motion from the Hamilton-Jacobi approach but this time set
$\mu=1$. We are interested in determining the radius of the innermost stable
circular orbit (ISCO), which is very important in the study of matter
accretion disks and in the study also of the effective potential for getting a
qualitative description of bound and unbound orbits under the symmergent
effects.

To determine the qualitative behavior of massive particle’s motion around the
symmergent black hole (i.e. bound, stable and unstable circular orbits), we
use the effective potential formula Bautista-Olvera  _et al._ (2019)

$V_{\pm}=\frac{g^{t\phi}}{g^{tt}}L\pm\Bigg{\\{}\left[\left(\frac{g^{t\phi}}{g^{tt}}\right)^{2}-\frac{g^{\phi\phi}}{g^{tt}}\right]L^{2}-\frac{1}{g^{tt}}\Bigg{\\}}^{1/2}.$
(57)

written in terms of the inverse metric $g^{\mu\nu}$ and angular momentum $L$.
Shown in Fig. 6 is the effective potential for $L=\pm 3.50M$.

Figure 6: The effective potential as a function of $r/M$ for $L>0$ (left-
panel) and $L<0$ (right-panel). In both plots, the black hole spin parameter
is set to $a=0.90M$.

In this figure, maxima and minima correspond to the unstable and stable
circular orbits, respectively. One notes that for $a=0$ (Schwarzschild) case,
the maxima of the effective potential curve are way lower compared to when
$a=0.90M$ (Kerr case). We can see that the energy for an unstable circular
orbit is higher in the AdS symmergent effect, where less negative $c_{\rm 0}$
giving the highest energy. Its minima, which represents the stable circular
orbit, has a greater radius for more negative $c_{\rm 0}$. Peculiar as it may
seem, particles with energies higher that than the peak energy tend to get
deflected back when it reaches at some radial point near the cosmic horizon
(one recalls that we scaled the Symmergent dS and AdS parameters). As for the
dS symmergent effect, we can see that the least negative $c_{\rm O}$ gives the
lowest peak energy in the unstable circular orbit. Particles with low energy
are deflected back to $r\rightarrow\infty$ when they reach low values of $r$.
While the AdS type can have elliptical bound orbits the same way as in the
Kerr case, we see that dS type does not admit such orbit as far as the value
of the angular momentum used herein is concerned. In Fig. 6 (right-panel), we
plotted also the effective potential in which the particle attains a negative
angular momentum. It is known that it does not happen in the Schwarzschild
case. As we can see in the right-panel, particles with negative energies can
be created near the rotating black hole’s event horizon. Clearly, symmergent
gravity affects the particle’s energy, which in turn contributes to the
Penrose process and black hole evaporation.

The ISCO radius occurs when the maximum and minimum of the effective potential
curve merge in an inflection point for a given value of $L$. It is clear that
we have two separate cases for ISCO determination since we need to consider
both the prograde and retrograde motion of the time-like particles. Besides,
such a circular orbit is only marginally stable, which means that any inward
perturbation will lead the particle to spiral toward the event horizon. In
locating the ISCO radius, we use equation (46) and solve for the energy
$E_{\text{cir}}$. One starts with Slaný  _et al._ (2013); Slany and Stuchlik
(2020),

$X^{2}=L-aE=\frac{r^{3}\left(\Delta_{r}^{\prime}-2E_{\text{cir}}^{2}r\right)}{-2a^{2}-r\Delta_{r}^{\prime}+2\Delta_{r}}.$
(58)

Differentiating equation (58) with respect to $r$ and after some algebra we
get ($E_{\text{cir}}=E_{\text{isco}}$)

$E_{\text{isco}}^{2}=\frac{1}{Br}\biggl{\\{}a^{2}\left[-\left(r\Delta_{r}^{\prime\prime}+3\Delta_{r}^{\prime}\right)\right]+r\Delta_{r}\Delta_{r}^{\prime\prime}-2r\Delta_{r}^{\prime
2}+3\Delta_{r}\Delta_{r}^{\prime}\biggr{\\}},$ (59)

where
$B=-8a^{2}+r\left(r\Delta_{r}^{\prime\prime}-5\Delta_{r}^{\prime}\right)+8\Delta_{r}$.
The ISCO radius can then be found by solving the equation Pantig and Rodulfo
(2020)

$\displaystyle\eta(r)_{\text{isco}}$ $\displaystyle=\pm
2\Delta_{r}\left(a^{2}-\Delta_{r}\right)^{2}\pm\frac{9}{4}r\Delta_{r}\left(a^{2}-\Delta_{r}\right)\Delta_{r}^{\prime}\pm\frac{1}{16}r^{3}\Delta_{r}^{\prime}\left(\Delta_{r}\Delta_{r}^{\prime\prime}-2\Delta_{r}^{\prime
2}\right)$
$\displaystyle\pm\frac{1}{16}r^{2}\bigl{[}4\Delta_{r}\left(a^{2}-\Delta_{r}\right)\Delta_{r}^{\prime\prime}+\left(15\Delta_{r}-4a^{2}\right)\Delta_{r}^{\prime
2}\bigr{]}$
$\displaystyle+a\Delta_{r}\sqrt{4a^{2}+2r\Delta_{r}^{\prime}-4\Delta_{r}}\bigl{[}-a^{2}+\frac{1}{8}r\bigl{(}r\Delta_{r}^{\prime\prime}-5\Delta_{r}^{\prime}\bigr{)}+\Delta_{r}\bigr{]}=0$
(60)

which redues to the Kerr case

$3a^{2}\mp 8a\sqrt{m}\sqrt{r}+r(6m-r)=0$ (61)

when $\Delta(r)=r^{2}-2mr+a^{2}$. The upper and lower signs in equation (IV.2)
correspond to the prograde and retrograde orbit of the massive particle at the
ISCO radius, respectively. We determine ISCO radius numerically by plotting
equation (IV.2) in Fig. 7.

Figure 7: Location of ISCO radius for counter-rotation of a massive particle
(left-panel) and co-rotation of a massive particle (right-panel). We set
$a=0.90M$ in both panels.

First, we note that in the Schwarzschild case, the ISCO radius is located at
$r=6M$. For the extreme Kerr case, on the other hand, the prograde ISCO is at
$r=M$ (which coincides with the prograde photonsphere) and retrograde ISCO is
at $r=9M$. Thus, in Fig. 7, the left-panel represents the retrograde ISCO
where we can see clearly how the symmergent parameters affect the radius. The
innermost radius, which determines the accretion disk’s innermost region, is
smaller in the AdS type whilst the dS type is slightly larger in the Kerr
case. We note that the first root of $\eta_{\text{isco}}$ is unphysical since
those radii are smaller than the photonsphere in the retrograde case. Finally,
we see the same symmergent effects in the prograde ISCO (right-panel of Fig.
7). Before we close this section, we remark that we used lower values of
$c_{\rm 0}$ in time-like geodesics than when analyzing the null boundaries and
geodesics. Thus, there must be some different constraints to be considered in
studying time-like geodesics. If we used the same parameters for the null
case, the results would be unphysical. Nonetheless, even if $c_{\rm 0}$ are
more negative, we see that time-like particles are more sensitive to
symmergent effects.

##  V Weak deflection angle from rotating symmergent black holes using Gauss-
bonnet theorem

In this section, we investigate the deflection angle of light rays from the
rotating symmergent black hole in the weak field limit using the methodology
given in Ishihara  _et al._ (2016); Ono  _et al._ (2017). Below, we briefly
discuss the finite distance method. First of all, the deflection angle is
defined as

$\hat{\theta}=\Psi_{\text{R}}-\Psi_{\text{S}}+\phi_{\text{RS}}$ (62)

in which $\phi_{\text{RS}}$ is the longitude separation angle,
$\Psi_{\text{R}}$ is the angle between light rays and radial direction, and
$\Psi_{\text{S}}$ is the angle between the observer and the source. Unit
tangential vector $e^{i}$ is used to write the above angle as follows

$(e^{r},e^{\theta},e^{\phi})=\epsilon\left(\frac{dr}{d\phi},0,1\right)$ (63)

where $\epsilon$ is a radial quantity. For a stationary spacetime described by
the metric

$ds^{2}=-A(r,\theta)dt^{2}+B(r,\theta)dr^{2}+C(r,\theta)d\theta^{2}+D(r,\theta)d\phi^{2}-2H(r,\theta)dtd\phi,$
(64)

one straightforwardly finds

$\epsilon=\frac{A(r)D(r)+H^{2}(r)}{A(r)(H(r)+A(r)b)},$ (65)

where $b$ is the impact parameter. We can write conserved quantities energy
and the angular momentum per unit mass in the form

$E=A(r)\dot{t}+H(r)\dot{\phi}\qquad L=D(r)\dot{\phi}-H(r)\dot{t},$ (66)

where the overdot denotes the derivative of the coordinates $t$ and $\phi$
with respect to the affine parameter $\lambda$ (proper time). In the
equatorial plane ($\theta=\frac{\pi}{2}$), the line element reduces to

$dl^{2}\equiv\gamma_{ij}dx^{i}dx^{j}$ (67)

with the spatial metric $\gamma_{ij}$. It is then possible to define

$\cos\Psi\equiv\gamma_{ij}e^{i}R^{j}.$ (68)

as the angle between the direction $e^{i}$ and the radial vector
$R^{j}=(\frac{1}{\sqrt{\gamma_{rr}}},0,0)$. Having $\cos\Psi$ at hand, we use
equations (63) and (68) to get

$\sin\Psi=\frac{H(r)+A(r)b}{\sqrt{A(r)D(r)+H^{2}(r)}}.$ (69)

Now, using this methodology we can calculate the deflection angle of the
symmergent black hole in the weak field limit. Indeed, with the impact
parameter $b=\frac{L}{E}$ and radial coordinate $1/u$ in place of $r$ we get

$\displaystyle\left(\frac{du}{d\phi}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{b^{2}}-\frac{4aMu}{b^{3}}-u^{2}+2Mu^{3}+\frac{1-\hat{\alpha}}{24\pi
c_{\rm 0}}-\frac{2a}{3b^{3}u^{2}}\frac{(1-\hat{\alpha})}{8\pi c_{\rm
0}}-\frac{8aM}{3b^{3}u}\frac{(1-\hat{\alpha})}{8\pi c_{\rm
0}}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi c_{\rm
0}}\right)^{2},a^{2}\right]$ (70)

where we considered only the retrograde solution. For the prograde solution,
we just flip the sign of the spin parameter $a$. It is convenient to start by
computing the separation angle integral. This can be done by writing
$\left(\frac{du}{d\phi}\right)^{2}=F(u)$ and extracting the angle

$\phi_{\text{RS}}=\int^{\text{R}}_{\text{S}}d\phi=\int^{u_{0}}_{u_{\text{S}}}\frac{1}{\sqrt{F(u)}}du+\int^{u_{0}}_{u_{\text{R}}}\frac{1}{\sqrt{F(u)}}du,$
(71)

where $u_{\text{S}}$ ($u_{\text{R}}$) is inverse distance to the source
(observer), and $u_{0}$ is the inverse of the closest approach $r_{0}$.
Considering the weak field and slow rotation approximations, the impact
parameter can be related to $u_{0}$ as

$b=\frac{1}{u_{0}}+M-2aMu_{0}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi
c_{\text{0}}}\right)^{2},a^{2}\right].$ (72)

Then, performing the necessary calculations we find

$\displaystyle\phi_{\text{RS}}$ $\displaystyle=$
$\displaystyle\phi_{\text{RS}}^{\text{Kerr}}+\left(\frac{u_{\text{R}}}{\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{u_{\text{S}}}{\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{b^{3}}{6}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\left(\frac{b\left(2-3b^{2}u_{\text{R}}^{2}\right)}{2\left(1-b^{2}u_{\text{R}}^{2}\right){}^{3/2}}+\frac{b\left(2-3b^{2}u_{\text{S}}^{2}\right)}{2\left(1-b^{2}u_{\text{S}}^{2}\right){}^{3/2}}\right)\frac{M}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}$ (73) $\displaystyle+$
$\displaystyle\left(\frac{1-2b^{2}u_{\text{R}}^{2}}{u_{\text{R}}\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{1-2b^{2}u_{\text{S}}^{2}}{u_{\text{S}}\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{a}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi
c_{\text{0}}}\right)^{2},a^{2}\right],$

where the Kerr term is given by

$\displaystyle\phi_{\text{RS}}^{\text{Kerr}}$ $\displaystyle=$
$\displaystyle\phi_{\text{RS}}^{\text{SW}}-\left(\frac{1}{\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{1}{\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{2aM}{b^{2}}.$
(74)

In this equation, the Schwarzschild term is

$\displaystyle\phi_{\text{RS}}^{\text{Schw}}=\pi-\arcsin\left(bu_{\text{R}}\right)-\arcsin\left(bu_{\text{S}}\right)+\left(\frac{2-b^{2}u_{\text{R}}^{2}}{\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{2-b^{2}u_{\text{S}}^{2}}{\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{M}{b}.$
(75)

To get the remaining terms appearing in the light deflection angle expression,
one should identify the $\Psi$ terms. We find

$\displaystyle\sin\Psi$ $\displaystyle=$ $\displaystyle bu-
bMu^{2}+2aMu^{2}-\left(\frac{bM}{6}-\frac{a}{3u}+\frac{b}{6u}\right)\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi
c_{\text{0}}}\right)^{2},a^{2}\right].$ (76)

This relation produces

$\displaystyle\Psi_{\text{R}}$ $\displaystyle-$
$\displaystyle\Psi_{\text{S}}=\Psi_{\text{R}}^{\text{Kerr}}-\left(\frac{1}{u_{\text{R}}\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{1}{u_{\text{S}}\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{b}{6}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}-\left(\frac{2b^{2}u_{\text{R}}^{2}-1}{\left(1-b^{2}u_{\text{R}}^{2}\right){}^{3/2}}+\frac{2b^{2}u_{\text{S}}^{2}-1}{\left(1-b^{2}u_{\text{S}}^{2}\right){}^{3/2}}\right)\frac{bM}{6}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}$ (77) $\displaystyle+$
$\displaystyle\left(\frac{1}{u_{\text{R}}\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{1}{u_{\text{S}}\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{a}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi
c_{\text{0}}}\right)^{2},a^{2}\right],$

where one has

$\displaystyle\Psi_{\text{R}}^{\text{Kerr}}$ $\displaystyle-$
$\displaystyle\Psi_{\text{S}}^{\text{Kerr}}=\Psi_{\text{R}}^{\text{Schw}}-\Psi_{\text{R}}^{\text{Schw}}+\left(\frac{u_{\text{R}}^{2}}{\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{u_{\text{S}}^{2}}{\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)2aM,$
(78)

and where one has found

$\displaystyle\Psi_{\text{R}}^{\text{Schw}}-\Psi_{\text{R}}^{\text{Schw}}=\left(\arcsin\left(bu_{\text{R}}\right)+\arcsin\left(bu_{\text{S}}\right)-\pi\right)-\left(\frac{u_{\text{R}}^{2}}{\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{u_{\text{S}}^{2}}{\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)Mb.$
(79)

Combining the above equations, we get an expression of the light deflection
angle given by

$\displaystyle\hat{\theta}$ $\displaystyle=$
$\displaystyle\left(\sqrt{1-b^{2}u_{\text{R}}^{2}}+\sqrt{1-b^{2}u_{\text{S}}^{2}}\right)\frac{2M}{b}-\left(\sqrt{1-b^{2}u_{\text{R}}^{2}}-\sqrt{1-b^{2}u_{\text{S}}^{2}}\right)\frac{2aM}{b^{2}}$
(80) $\displaystyle-$
$\displaystyle\left(\frac{1-b^{2}u_{\text{R}}^{2}}{u_{\text{R}}\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{1-b^{2}u_{\text{S}}^{2}}{u_{\text{S}}\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{b}{6}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\left(\frac{1}{\sqrt{1-b^{2}u_{\text{R}}^{2}}}+\frac{1}{\sqrt{1-b^{2}u_{\text{S}}^{2}}}\right)\frac{bM}{6}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}$ $\displaystyle+$
$\displaystyle\left(\frac{\sqrt{1-b^{2}u_{\text{R}}^{2}}}{u_{\text{R}}}+\frac{\sqrt{1-b^{2}u_{\text{S}}^{2}}}{u_{\text{S}}}\right)\frac{2a}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi
c_{\text{0}}}\right)^{2},a^{2}\right].$

This form can be reduced to a simplified one using certain convenable
approximations. Taking $u_{\text{S}}b<<1$ and $u_{\text{R}}b<<1$, we can get
an expression involving divergent terms coupled to the cosmological
contributions. These terms should be existed to show the cosmological
background dependence. The desired deflection angle of light rays is found to
be

$\displaystyle\hat{\theta}$ $\displaystyle=$
$\displaystyle\frac{4M}{b}-\frac{4aM}{b^{2}}+\frac{bM}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}-\left(\frac{1}{u_{\text{R}}}+\frac{1}{u_{\text{S}}}\right)\frac{b}{6}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\left(\frac{1}{u_{\text{R}}}+\frac{1}{u_{\text{S}}}\right)\frac{2a}{3}\frac{(1-\hat{\alpha})}{8\pi
c_{\text{0}}}+\mathcal{O}\left[M^{2},\left(\frac{1-\hat{\alpha}}{8\pi
c_{\text{0}}}\right)^{2},a^{2}\right].$ (81)

An examination shows that this expression recovers many previous findings.
Without the cosmological contributions, we get the results of the ordinary
Symmergent black hole solutions investigated in Rayimbaev  _et al._ (2022).

Figure 8: Weak deflection angle of photons as a function of $b/m$ for various
values of the spin and symmergent parameters. The solid lines correspond to
$a>0$ (prograde) and dashed lines correspond to $a<0$ (retrograde). In making
these plots we take $u_{\text{S}}=u_{\text{R}}=0.5b^{-1}$ (left),
$u_{\text{S}}=u_{\text{R}}=0.01b^{-1}$ (right), and $\hat{\alpha}=0.90$. The
horizontal dotted lines represent critical impact parameters in each case.

Plotted in Fig. 8 are the equations (80) (left-panel) and (81) (right-panel).
The solid curves represent photons co-rotating with the black hole, while the
dashed curves represent the counter-rotating photons. For the left-panel,
which holds for finite distance, the results show that the co-rotating photons
produce weak deflection angle at lower impact parameters compared to the
counter-rotating photons. If we compare $\hat{\theta}$ for a specific orbital
direction of photons, we see that the symmergent AdS type gives a higher value
for $\hat{\theta}$ at large impact parameters. However, AdS type in the
counter-rotation case gives a higher $\hat{\theta}$ than the co-rotation case.
Finally, the right-panel stands for the case where we used the large distance
approximation. Since the symmergent parameters are scaled, the location of the
receiver in this case is comparable to the cosmic horizon. In this case, the
symmergent effect is strong, giving the effects shown in the plot. We could
see that there are cases where $\hat{\theta}$ is repulsive. In general, weak
deflection angle is seen to have a strong sensitivity to the symmergent
parameters.

###  V.1 Particle acceleration near rotating Symmergent black hole backgrounds

As Banados, Silk and West (BSW) show that collisions in equatorial plane near
a Kerr black hole can occur with an arbitrarily high center of mass-energy and
Kerr black holes can serve as particle accelerators Banados  _et al._ (2009).
Then the BSW mechanism has been studied different black hole spacetimes
Halilsoy and Ovgun (2017a, b); Yang  _et al._ (2014); Li  _et al._ (2011);
Zhang  _et al._ (2019); Wang  _et al._ (2022); Dymnikova  _et al._ (2022). In
the present work, we find that the symmergent parameter $c_{\rm 0}$ has an
important effect on the result. To do that first we consider the particle
motion on the equatorial plane ($\theta=\frac{\pi}{2}$, $\rho^{2}=r^{2}$). In
fact, generalized momenta $P_{\mu}$ can be written as
($\mu,\nu=t,r,\phi,\theta$)

$P_{\mu}=g_{\mu\nu}\dot{x}^{\nu}$ (82)

where the dot stands for derivative with respect to the affine parameter
$\lambda$. Then, one can write the generalized momenta $P_{t}$ (test
particle’s energy per unit mass $E$) and $P_{\phi}$ (the angular momentum per
unit mass $L$ parallel to the symmetry axis)

$\displaystyle P_{t}$ $\displaystyle=$ $\displaystyle
g_{tt}\dot{t}+g_{t\phi}\dot{\phi},$ (83) $\displaystyle P_{\phi}$
$\displaystyle=$ $\displaystyle g_{\phi\phi}\dot{\phi}+g_{t\phi}\dot{t}$ (84)

where one keeps in mind that $P_{t}$ and $P_{\phi}$ are constants of motion.

Our goal is to study the CM energy of the two-particle collision in the
background spacetime of the rotating symmergent black holes. To this end, we
start with the CM energy $E_{\text{CM}}$ Banados  _et al._ (2009)

$\displaystyle
E_{\text{CM}}^{2}=2m_{0}^{2}\left(1-g_{\mu\nu}u_{1}^{\mu}u_{2}^{\nu}\right)$
(85)

in which $u^{\mu}_{1},u^{\nu}_{2}$ are the 4-velocity vectors of the two
particles ($u=(\dot{t},\,\dot{r},\,0,\,\dot{\phi})$). Correspondingly,
particle $i=1,2$ has angular momentum per unit mass $L_{i}$ and energy per
unit mass $E_{i}$. We also consider particles with the same rest mass $m_{0}$.
Initially the two particles are at rest at infinity ($E_{1}/m_{0}=1$ and
$E_{2}/m_{0}=1$) and then they approach the black hole and collide at a
distance $r$. Then, their CM energy takes the form

$E_{\text{CM}}^{2}=\frac{2m_{0}^{2}}{\Delta_{r}r^{2}}\left[\left(r^{2}+a^{2}\right)^{2}-a\left(L_{1}+L_{2}\right)\left(r^{2}+a^{2}-\Delta_{r}\right)+L_{1}L_{2}\left(a^{2}-\Delta_{r}\right)+\Delta_{r}\left(r^{2}-a^{2}\right)-X_{1}X_{2}\right]$
(86)

with the functions

$X_{i}=\sqrt{\left(aL_{i}-r^{2}-a^{2}\right)^{2}-\Delta_{r}\left(\left(L_{i}-a\right)^{2}+\mu^{2}r^{2}\right)}.$
(87)

Collisions occur at the horizon of the black hole so that $\Delta_{r}=0$ in
equation (86) so CM energy could diverge when the particles approach the
horizon. It is not difficult to see that the denominator of
$E_{\text{CM}}^{2}$ is zero there. Evidently, the maximal energy of the
collision occurs if $L_{1}$ and $L_{2}$ are opposite (such as head-on
collision). Plotted in Fig. 9 is $E_{\text{CM}}$ as a function of the radial
distance $r$. As is seen from the figure, CM energy blows up at the horizons.
It demonstrates the radial dependence of the CM energy of the particles moving
along circular orbits with the different symmergent parameters. The symmergent
parameter decreases the CM energy.

Figure 9: Dependence of the CM energy $E_{\text{CM}}$ on the radial distance
$r$ for different values of the spin and symmergent parameters. We consider
two parameter sets $m=M=1$, $L_{1}=-4$,$c_{0}=-0.5$, $\hat{\alpha}=0.9$ and
$a=0.5$ (left-panel) as well as $m=M=1$, $L_{1}=-4$,$L_{2}=50$,
$\hat{\alpha}=0.9$ and $a=0.5$ (right-panel). It is clear that the CM energy
blows up at the horizons.

##  VI Conclusion

In this paper, we have studied rotating black holes in symmergent gravity, and
used deviations from the Kerr black hole to constrain the parameters of the
symmergent gravity. Symmergent gravity generates parameters of the
gravitational sector from flat spacetime loops. It induces the gravitational
constant $G$ and quadratic-curvature coefficient $c_{\rm O}$. In the limit in
which all fields are degenerate in mass, the vacuum energy $V_{\rm O}$ can be
expressed in terms of $G$ and $c_{\rm O}$. We parametrize deviation from this
degenerate limit by the parameter ${\hat{\alpha}}$. The black hole spacetime
is dS for ${\hat{\alpha}}<1$ and AdS for ${\hat{\alpha}}>1$. In constraining
the symmergent parameters $c_{\rm O}$ and ${\hat{\alpha}}$, we utilize the EHT
observations on the M87* and Sgr. A* black holes.

We have analyzed symmergent gravity parameters for different spinning black
hole properties. We first investigated the effects of symmergent gravity on
the photonsphere, black hole shadow, and certain observables related to them.
Our findings show that location of the photonsphere depends on the symmergent
effects due to its coupling to the black hole spin parameter. We have seen how
the instability of the photon radii depends on and gets affected by $c_{\rm
O}$ and $\hat{\alpha}$. Also, symmergent gravity effect is evident, even for
observers deviating from the equatorial plane. Interesting effects are also
seen in the energy emission rate, which may directly affect the black hole’s
lifetime. Constraints on symmergent parameters from the shadow radius data
coming from recent observations are also discussed. For a low and high spin
parameter $a$ values (the parameter $\hat{\alpha}$ dictates whether the
symmergent effect mimics the dS and AdS type), we found that the parameter
$c_{\rm 0}$ fits within $\pm 2\sigma$ at $95\%$ confidence level for M87* than
in Sgr. A*. It means that the true value is somewhere within such confidence
level. The plot in Fig. 5 reveals that the detection of symmergent effects can
be achieved more easily in M87* than in Sgr. A*, for the obvious reason that
the farther the galaxy is the stronger the symmergent effects are. Finally,
for comparison, we also briefly analyzed such an effect on the time-like
geodesic. Our findings indicate that massive particles are more sensitive to
deviations caused by the symmergent parameter than null particles.

In addition, we investigated the symmergent effects on photons’ weak
deflection angle as they traverse near the black hole. We also considered the
finite distance effects. We have found that the deflection angle depends on
$\hat{\alpha}$ but the deviations are also caused by how far the receiver is
from the black hole. This effect is clearly shown in Fig. 8. It implies that
the weak deflection angle can detect symmergent effects more easily for black
holes that are significantly remote compared to our location.

Lastly, we study the particle collision near the rotating symmergent black
hole background and we analyze the possibility that rotating symmergent black
holes could act as particle accelerators. We show that rotating symmergent
black holes can serve as particle accelerators because the center-of-mass
energies blow up at the horizons. As a result, the BSW mechanism depends on
the value of the Symmergent parameter $c_{\text{0}}$ of the black hole.

Future research direction may include investigation of the shadow, and showing
its dependence on the observer’s state. One may also study the spherical
photon orbits or the stability of time-like orbits.

###### Acknowledgements.

We thank to conscientious referee for their comments, questions and
criticisms. We thank Xiao-Mei Kuang for useful discussions. A. Ö. acknowledges
hospitality at Sabancı University where this work was completed. A. Ö. and R.
P. would like to acknowledge networking support by the COST Action CA18108 -
Quantum gravity phenomenology in the multi-messenger approach (QG-MM).

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