1) The document investigates photon geodesics and the shadow cast by a Kerr black hole modified by the generalized uncertainty principle (GUP). Key characteristics like the horizon size and photon sphere radius are calculated. 2) The Kerr metric is modified to incorporate GUP corrections and rotation. Photon trajectories are studied to determine the visual characteristics and apparent shape of the black hole shadow. 3) Figures show the shadows for different values of the black hole spin parameter a and GUP parameter σ. Future work is proposed to relate observables to astrophysical properties of black holes that could be measured by the Event Horizon Telescope.

1. Theory behind black hole shadows
The apparent shape of the black hole is obtained by plotting 𝛽 𝑣𝑠 𝛼. Everything that lies inside the unstable circular orbits represents
photons trapped by the black hole, while everything outside is light that will reach the observer at infinity. Therefore, the contour
represents the shadow casted by black hole. The distorted disk feature corresponds to the direct circular orbit radius decreasing faster
than what the retrograde’s radius increases. Consequently, the left endpoint displaces more to the right than what the right endpoint
does, hence distorting more significantly the left hand side.
Shadow of a GUP Kerr Black Hole
Photon geodesics
Generally, geodesic motion in a stationary axisymmetric space-time will allow two integrals of motion: the energy 𝐸 and the angular
momentum 𝐿 𝑧. Besides, the norm of the four-velocity will also be conserved by virtue of its parallel propagation. However, these three
conservation laws do not suffice to reduce the problem of solving geodesic equations to one involving quadratures only. Nevertheless,
with Carter’s discovery of the existence of a fourth constant of motion 𝒦, the Hamilton-Jacobi equation become solvable. Now we solve
the equation governing geodesic motion:
𝜕𝑆
𝜕λ
=
−1
2
𝑔 𝜇𝜈
𝜕𝑆
𝜕𝑥 𝜇
𝜕𝑆
𝜕𝑥 𝜈
where S denotes Hamilton’s principal function, 𝑔 𝜇𝜈
the inverse metric in modified Kerr Geometry and λ an affine parameter.
Exploiting the separability of the equation, we seek a solution of the form:
𝑆 =
1
2
𝑚2λ − 𝐸𝑡 + 𝐿 𝑧 𝜙 + 𝑆𝑟 𝑟 + 𝑆 𝜃 𝜃
Using the fact that
𝑑𝑆 𝑟
𝑑𝑟
= 𝑝 𝑟 = 𝑔 𝑟𝑟 𝑟,
𝑑𝑆 𝜃
𝑑𝜃
= 𝑝 𝜃 = 𝑔 𝜃𝜃 𝜃 and m=0 for photons we obtain decoupled equations:
𝜌2
𝑑𝑟
𝑑λ
= ℛ, 𝜌2
𝑑𝜃
𝑑λ
= Θ, 𝜌2
𝑑𝑡
𝑑λ
= 𝑎 𝐿 𝑧 − 𝑎𝐸𝑠𝑖𝑛2
𝜃 +
𝑟2
+ 𝑎2
Δ
𝑟2
+ 𝑎2
𝐸 − 𝑎𝐿 𝑧 , 𝜌2
𝑑𝜙
𝑑λ
=
𝐿 𝑧
sin2 𝜃
− 𝑎𝐸 +
𝑎
Δ
𝑟2
+ 𝑎2
𝐸 − 𝑎𝐿 𝑧
ℛ = 𝑟2
+ 𝑎2
𝐸 − 𝑎𝐿 𝑧
2
− Δ 𝒦 + 𝐿 𝑧 − 𝑎𝐸 2
, Θ = 𝒦 + cos2
𝜃 𝑎2
𝐸2
−
𝐿 𝑧
2
sin2 𝜃
Although the 𝑡 and 𝜙 equations are not of importance for us given that the black hole studied is static and axisymmetric.
We proceed to determine the unstable photon orbits of constant radius (photo sphere) by requiring ℛ 𝑟 = 0 = 𝑑ℛ 𝑟 𝑑𝑟.
The impact parameters that fulfill these equations are 𝜉 =
𝐿 𝑧
𝐸
and 𝜂 =
𝒦
𝐸2 , and unstable orbits of constant radius are determined by:
ℛ = 𝑟4
+ 𝑎2
− 𝜉2
− 𝜂2
𝑟2
+ 2𝑀𝐴𝐷𝑀 𝜂 + 𝜉 − 𝑎 2
𝑟 − 𝑎2
𝜂 = 0
𝜕ℛ 𝜕𝑟 = 4𝑟3
+ 2 𝑎2
− 𝜉2
− 𝜂 𝑟 + 2𝑀𝐴𝐷𝑀 𝜂 + 𝜉 − 𝑎2 2
= 0
which can be solved for the impact parameters to give:
𝜉 =
𝑀𝐴𝐷𝑀 𝑟2 − 𝑎2 − 𝑟Δ
𝑎 𝑟 − 𝑀𝐴𝐷𝑀
, 𝜂 =
𝑟3 4𝑀𝐴𝐷𝑀Δ − 𝑟 𝑟 − 𝑀𝐴𝐷𝑀
2
𝑎2 𝑟 − 𝑀𝐴𝐷𝑀
2
Finally, we relate this to the celestial coordinates of the image as seen by an observer at infinity:
𝛼 = lim
𝑟0⟶∞
−𝑟0
2
sin 𝜃0
𝑑𝜙
𝑑𝑟
and 𝛽 = lim
𝑟0⟶∞
𝑟0
2
𝑑𝜃
𝑑𝑟
to get 𝛼 = −𝜉 csc 𝜃0 , 𝛽 = ± 𝜂 + 𝑎2 cos2 𝜃0 − 𝜉2 cot2 𝜃0
and in the equatorial plane 𝜃0 =
𝜋
2
we get 𝛼 = −𝜉 , 𝛽 = ± 𝜂
Luciano Manfredi and Jonas Mureika, Department of Physics, Loyola Marymount University
Introduction
The shadow of a black hole — i.e. the region interior to the photosphere — is a characteristic determined
exclusively by the object's mass and rotation, and thus presents a novel test of the underlying gravitational theory.
The Event Horizon Telescope (EHT, comprised of the orbital radio telescopes RadioAstron, Millimetron, and the X-
Ray interferometer MAXIM) is gearing up to observe the shadow of Sagittarius A*, the supermassive black hole at
the center of our galaxy. Consequently, theoretical models will be compared to experimental results, making it
possible to distinguish valid theories from those that should be rejected. Hence, this provides a direct way to test
General Relativity and alternative theories of gravity.
Characteristics of photon geodesics in a class of GUP black holes that mimic dimensional reduction at the Planck scale are investigated. This is achieved by theoretical derivations as well as computer calculations and simulations to display the results.
Moreover, the static metric is modified to account for rotation. Specifically, gravitational lensing effects and morphological characteristics of the photon sphere are studied in detail. Finally, to provide experimental verifiability, deviations from standard general
relativistic (GR) predictions will be determined, and the likelihood of observing such effects in the Event Horizon Telescope (EHT) will be addressed, thus providing a direction for future research in this area. While such corrections are likely to be small and
confined to the near-horizon regime, it is anticipated that the projected resolution of the EHT will be able to provide a first-order glimpse of such potential deviations from GR.
JM et al. propose in [1] a quantum correction to the Schwarzschild metric of the form:
𝑑𝑠2
= 𝐹 𝑟 𝑑𝑡2
− 𝐹 𝑟 −1
𝑑𝑟2
− 𝑟2
𝑑Ω2
𝐹 𝑟 = 1 −
2
𝑀 𝑃𝑙
2
𝑀
𝑟
1 +
𝜎
2
𝑀 𝑃𝑙
2
𝑀2
Which remains Schwarzschild like given that the modification factor is coordinate independent.
Furthermore, the self-dual metric nature under M ⟷ 𝑀−1
naturally implies a GUP with linear form ∆𝑥 ~
1
∆𝑝
+ ∆𝑝.
Finally, natural dimensional reduction features such as gravitational radius and thermodynamics of sub Planckian
objects resemble that of (1+1)-D gravity.
The horizon size, effective potential and radius of the photosphere were calculated to be:
𝑟 𝐻 =
2
𝑀 𝑃𝑙
2
𝑀2+
𝜎
2
𝑀 𝑃𝑙
2
𝑀
, 𝑊𝑒𝑓𝑓 𝑟 =
1
𝑟2 1 −
2𝐺𝑀
𝑟
1 +
𝜎
2𝐺𝑀2 , 𝑟𝛾 = 3𝐺𝑀 +
3
2
𝜎
𝑀
Next, rotation was incorporated by replacing 𝑀 ⟶ 𝑀𝐴𝐷𝑀 = 𝑀 1 +
𝜎
2
𝑀 𝑃𝑙
2
𝑀2 in the Kerr metric:
𝑑𝑠2
= 1 −
2𝐺𝑀𝐴𝐷𝑀 𝑟
𝜌2
𝑑𝑡2
−
𝜌2
Δ
𝑑𝑟2
− 𝜌2
𝑑𝜃2
− 𝑟2
+ 𝑎2
+
2𝐺𝑀𝐴𝐷𝑀 𝑟𝑎2
𝜌2
sin2
𝜃 sin2
𝜃 𝑑𝜙2
+
4𝐺𝑀𝐴𝐷𝑀 𝑟𝑎 sin2 𝜃
𝜌2
𝑑𝑡𝑑𝜙
𝜌2
= 𝑟2
+ 𝑎2
cos2
𝜃 , Δ = 𝑟2
− 2𝐺𝑀𝐴𝐷𝑀 𝑟 + 𝑎2
, 𝑎 =
𝒥
𝑀 𝐴𝐷𝑀
This metric exposes an outer horizon when
1
𝑔 𝑟𝑟
= 0 at 𝑟+ = 𝐺𝑀 +
𝜎
2𝑀
+ 𝐺𝑀 2 − 𝑎2 + 𝜎𝐺 +
𝜎2
4𝑀2, and in the extremal
case when 𝑟± degenerates, 𝑟 = 𝐺𝑀 +
𝜎
𝐺𝑀
. Also, the outer boundary of the ergoregion corresponds to the infinite-
redshift surface determined by 𝑔𝑡𝑡 = 0 at 𝑟𝑒 =
4𝐺𝑀+𝜎+ 𝜎+4𝐺𝑀 2−16𝑎2 cos2 𝜃
4
A new quantum-corrected black hole solution
References:
[1] B. Carr, J. Mureika, P. Nicolini, “Sub-
Planckian Black Holes and the Generalized
Uncertainty Principle”, JHEP 1507:052 (2015)
[2] S. Chandrasekhar, The mathematical theory
of black holes (1992)
[3] K. Hioki and K.I. Maeda, Phys. Rev. D 80,
024042 (2009)
[4] L. Amarilla and E. F. Eiroa, Phys. Rev. D 85,
064019 (2012)
[5] S. V´asquez and E. Esteban, Nuovo Cim. B
119, 489 (2004)
Photon Geodesics of a Generalized Uncertainty Principle Black Hole
To determine the visual characteristics of these objects, we first have to
define the apparent shape of a shadow. Here we assume light sources
come from infinity and are uniformly distributed in all directions. Therefore,
the shadow is obtained by solving the scattering problem of photons
injected from any points at infinity with any and every impact parameters.
We also assume that the observer stays at infinity, and has an inclination
angle 𝜃0 = 𝜋 2 defined to be the angle between the rotation axis and the
line of sight of the observer. The galactic supermassive black hole is also
expected to lie close to 𝜋 2. Next, the celestial coordinates 𝛼, 𝛽 of the
observer are the apparent angular distances of the image on the celestial
sphere measured from the direction of the line of sight, perpendicular and
parallel to the projected rotation axis onto the celestial sphere,
respectively. Then, we define the shadow by considering the following
cases. First, suppose light rays are emitted at infinity and pass near the
collapsed object. If they reach the observer at infinity after scattering, then
that direction is not dark. Otherwise, if the photons fall into the event
horizon, the observer will never see them again: such direction is dark.
We define the apparent shape of the black hole by the boundary of the
shadow, determined by the radius of unstable circular orbits.
FIG. 1: Geometry of the rotating gravitational lens. An
observer can set up a reference coordinate system (x,
y, z) with the black hole at the origin. The Boyer-
Lindquist coordinates coincide with this system only at
infinity. The reference frame is chosen so that, as
seen from infinity, the black hole is rotating around the
z axis. In this system, the line joining the origin and
observer is normal to the α-β plane. The tangent
vector to an incoming light ray defines a straight line,
which intersects the α-β plane at the point 𝛼𝑖, 𝛽𝑖 .
a)
𝑎
𝑀
= 0 b)
𝑎
𝑀
= 0.5 c)
𝑎
𝑀
= 1. 𝜎 = 0 (blue line), 𝜎 = 0.5 (green line), 𝜎 = 1 (red line)
In addition to what is presented above, one could find expressions for the
observables 𝑅 𝑠 (radius of a reference circle) and 𝛿𝑠 = 𝐷 𝑅 𝑠 (D is difference
between the endpoints of the circle and shadow), such that upon experimental
observation, astrophysical properties of black holes can be obtained.
Future Work
FIG. 2: The Shadows of GUP Kerr Black Hole.
The celestial coordinates 𝛼, 𝛽 are measured
in the unit of the black hole mass M. In this
case, M=1 and inclination angle 𝜃0 = 𝜋 2.
𝛼[𝑀]𝛼[𝑀]𝛼[𝑀]