ISTANBUL UNIVERSITESI 30-31 OCAK 2019

1. Weak Gravitational Lensing and Gauss-Bonnet
Theorem
AL˙I ¨OVG¨UN
*Doktor ¨O˘gretim ¨Uyesi, Do˘gu Akdeniz ¨Universitesi (DA¨U), Gazima˘gusa, KKTC
*FONDECYT Doktora Sonrası Ara¸stırmacı, Instituto de F´ısica,
Pontificia Universidad Cat´olica de Valpara´ıso (PUCV), S¸˙IL˙I
30-31 Jan. 2019, ˙Istanbul ¨Universitesi, T¨urkiye

2. References:
Figure 1: QR Code

4. Figure 2: How can we open gate into space-time?
• How can we connect two regions of space-time?
• Can we make stable and traversable wormholes?

5. Figure 3: ER=EPR

6. Figure 4: How to realize Wormholes in real life

7. DEFLECTION OF LIGHT BY A POINT MASS
Figure 5: Deflection of a light ray passing within a distance b of a point mass
M (arXiv:astro-ph/0304438)
A ray that passes within a distance b (the impact parameter) of this mass
feels a Newtonian acceleration component perpendicular to its direction
of motion of
g⊥ =
GMb
(b2 + z2)
3/2
(1)

8. (provided the deflection is small) which results in a total integrated
velocity component v⊥ = g⊥dt = g⊥dz/c = 2GM/bc. The resulting
deflection angle is then
α = v⊥/c =
2GM
bc2
(Newton). (2)
General Relativity (GR) predicts exactly twice the deflection angle of
Newtonian theory—it was in fact this factor of two that was used as a
test of GR during Eddington’s solar eclipse expedition of 1930—so that
the deflection angle for a ray with impact parameter b near a point mass
M is
α =
4GM
bc2
(GR) provided α 1. (3)

9. Figure 6: Stars lying behind the gravitational lens no longer appear to be in
their real position for an observer but seem to have slightly shifted.

10. Gravitational lensing in a broad sense: heir of a classical
astronomy
At least in principle, fundamental to observational astronomy
Applications in three interconnected fields:
• Theoretical physics: testing fundamental theory of gravity
• Astronomy: dark matter distribution, extrasolar planets
• Mathematics: singularity theory, topology
Three approaches to gravitational lensing theory:
• Geometry of null geodesics in Lorentzian manifolds
• Geometry of spatial light rays: optical geometry, also called Fermat
geometry, optical reference geometry
• Framework used in astronomy: impulse approximation

11. Figure 7: Gravitational lensing

12. OPTICAL GEOMETRY
Also called Fermat geometry and optical reference geometry:
Metric manifold whose geodesics are the spatial projections of spacetime
null geodesics, by Fermat’s principle Useful for the study of
• Inertial forces in general relativity [e.g., Abramowicz, Carter &
Lasota (1988)]
• Gravitational lensing: deflection angle, multiple images and
topology, using Gauss-Bonnet.
• No glory in cosmic string theory.
G. W. Gibbons. Phys. Lett. B 308 237–39 (1993).
• -Static spacetime: Riemannian optical geometry
G. W. Gibbons and M. C. Werner.
Classical Quantum Gravity 25, 235009 (2008)
YOUTUBE: Lecture 16 and 17: Optical Geometry I (International
Winter School on Gravity and Light 2015) by M.Werner.
• -Stationary spacetime: Finslerian optical geometry
M. C. Werner. Gen. Relativ. Gravit. 44, 3047 (2012).

13. • Massive particle : Jacobi’s metrics.
G W Gibbon. Class. Quantum Grav. 33 (2016) 025004.
• Gravitational bending angle of light for finite distance and the
Gauss-Bonnet theorem for Schwarzschild-de Sitter
(non-asymptotically flat).
Asahi Ishihara, Yusuke Suzuki, Toshiaki Ono, Takao Kitamura,
Hideki Asada.
Phys.Rev. D94 (2016) no.8, 084015.
• Weak lensing in a plasma medium and gravitational deflection of
massive particles using Gauss-Bonnet theorem.
Gabriel Crisnejo, Emanuel Gallo. Phys.Rev. D97 (2018) no.12,
124016.

14. Figure 8: 50 citations in 10 years. (topic of the PhD thesis of M.C. Werner.)

15. OPTICAL GEOMETRY OF STATIC SPACETIMES
Consider a static spacetime with line element
ds2
= gµνdxµ
dxν
. (4)
The coordinate time along spatial projections of null curves obeys
with optical metric
dt2
= gabdxa
dxb
(5)
with optical metric gab = gab
(−gtt ) , whose geodesics are spatial light rays, by
Fermat’s principle.
Figure 9: The Schwarzschild black hole

16. OPTICAL GEOMETRY OF SCHWARZSCHILD
Given the line element of the Schwarzschild solution:
ds2
= − 1 −
2µ
r
dt2
+ 1 −
2µ
r
−1
dr2
+ r2
dθ2
+ sin2
θdφ2
(6)
the metric of the optical geometry can be read off from (θ = π/2):
dt2
= 1 −
2µ
r
−2
dr2
+ 1 −
2µ
r
−1
r2
dφ2
(7)
Figure 10: Isometric embedding of the equatorial plane (θ = π/2) , thick line
indicates the photon sphere at r = 3µ.

17. LENSING IN THIS OPTICAL GEOMETRY
Geodesics on this surface correspond to spatial light rays. However, the
Gaussian curvature at every point
K < 0 (8)
so geodesics must locally diverge.
Then how can two light rays from a light source refocus at the observer,
so that the two images of the Schwarzschild lens are obtained?

18. • This method relies on the fact that the deflection angle can be
calculated using a domain outside of the light ray.
• It is known that the effect of lensing strongly depends on the mass
of the enclosed region body on spacetime.
• to calculate the Gaussian curvature of K, so that the GBT is found
as follows:
A
KdS +
∂A
κdt +
i
αi = 2πχ(A). (9)
• κ stands for the geodesics curvature of ∂A : {t} → A
• αi is the exterior angle with the ith
vertex.
• Euler characteristic of χ
• a Riemannian metric of g
• This technique is for asymptotically flat observers and sources.
• The resulting deflection angle is expected to be too small, which is
also a joint point in astronomy.

22. The asymptotic deflecting angle of α:
ˆα = −
A∞
KdS. (10)
Note that we use the infinite region of the surface A∞ bounded by the
light ray to calculate our integral.
To obtain the deflection angle of the light, we use the zero-order
approximation of the light ray, and the deflection angle of ˆα is obtained
in leading-order terms.

23. The optical Schwarzschild metric is:
dt2
= 1 −
2µ
r
−2
dr2
+ 1 −
2µ
r
−1
r2
dφ2
(11)

28. Light Deflection by Charged Wormholes in
Einstein-Maxwell-Dilaton theory PHYSICAL REVIEW D,
ARXIV:1707.01416
jointly with K. Jusufi (PhD Student) and A. Banerjee (Post.Doc.)
Dyonic wormholes in the Einstein-Maxwell-Dilaton Theory
P. Goulart arXiv:1611.03164
The dyonic wormholes in the EMD theory is :
ds2
= −
r2
r2 + 2PQ
dt2
+
r2
+ 2PQ
r2 + Σ2 + 2PQ
dr2
+ (r2
+ 2QP)(dθ2
+ sin2
θdϕ2
).
(12)
In 4- D, because of electromagnetic duality, it is possible to construct a
black hole which carries both electric and magnetic charges. Such black
hole solution is called dyonic black hole. Q is the electric charge, P is the
magnetic charge.
It is worth noting that by letting Σ = 0, the radius of the throat is found
to be Rthro. =
√
2PQ.

29. Weak deflection limit with GBT and Gaussian Optical Curvature
Goulart’s wormhole solution considering the null geodesic ds2
= 0, with
the deflection angle of light in the equatorial plane θ = π/2, we obtain
the optical metric of CW as follows:
dt2
=
(r2
+ 2PQ)2
r2(r2 + Σ2 + 2PQ)
dr2
+
(r2
+ 2PQ)2
r2
dϕ2
. (13)
Since we are interested in the weak limit, we can approximate the optical
Gaussian curvature as
K ≈ −
16PQ
r4
+
Σ2
r4
−
16PQΣ2
r6
+
32P2
Q2
r6
. (14)
Deflection angle can be recast in the following from
ˆα = −
π
0
∞
b
sin ϕ
KdS. (15)

30. Then the integral reduce to following form
ˆα = −
π
0
∞
b
sin ϕ
−
16PQ
r4
+
Σ2
r4
−
16PQΣ2
r6
+
32P2
Q2
r6
det ˜g drdϕ.
One can easily solve this integral in the leading order terms to find the
following result
ˆα
3πPQ
2b2
−
πΣ2
4b2
+ O(P2
, Q2
, Σ2
). (16)
• The deflection angle is affected by the magnetic charge, electric
charge, and the dilaton charge.
• Tthe magnetic and electric charges increase the deflection angle.
• On the other hand, the dilaton charge decreases the deflection angle.

31. Deflection of Light from Rindler Modified
Schwarzschild Black Hole
EPL 118 (2017) 60006, arXiv:1702.04636. jointly with I. Sakalli
Rindler Modified Schwarzschild Black Hole (Gr¨umiller’s BH)
D. Grumiller, Phys. Rev. Lett. 105, 211303 (2010).
The spacetime of the RMSBH is
ds2
= −fdt2
+
1
f
dr2
+ r2
dθ2
+ sin2
θdφ2
, (17)
with
f = 1 −
2M
r
+ 2ar (18)
KdA ≈ −
1
2
√
2
ar3 − 3
2
drdφ (19)
α = −
D2
KdA ≈
1
2
√
2
π
0
∞
b/ sin φ
ar3 − 3
2
drdφ (20)

32. α
0.126127529
√
a3b7
(21)
• The above result shows the RMSBH’s gravitational lensing
deflection angle in weak field limits.
• Evidently, the deflection angle is inversely proportional to the Rindler
acceleration, that is, gravitational lensing of an accelerated RMSBH
is less than the almost non-accelerated RMSBH.
• The latter remark implies that observing RMSBH will be more
difficult than observing Schwarzschild BH.

33. Effect of Lorentz Symmetry Breaking on the Deflection
of Light in a Cosmic String Spacetime
Phys. Rev. D 96, 024040 (2017), arXiv:1705.06197
jointly with I. Sakalli and K. Jusufi
Noninertial effects on the ground state energy of a massive scalar
field in the cosmic string spacetime
H. F. Mota and K. Bakke Phys. Rev. D 89, 027702 (2014).
The effective cosmic string spacetime is
ds2
= −dt2
+ dr2
+ r2
dθ2
+ η2
r2
(1 + ) sin2
θdϕ2
where η = 1 − 4µ is the parameter of the cosmic string. Using Taylor
series in η and , we can approximate the result for the deflection angle as
ˆα 4µπ −
π
2
− 2πµ + O µ2
, η2
.
• The first term is just the deflection angle by a static cosmic string.
• Interestingly, due to the Lorentz symmetry breaking by the
parameter , we find that the deflection angle decreases.

34. Light deflection by Damour-Solodukhin wormholes and
Gauss-Bonnet theorem
Phys. Rev. D 98, 044033 (2018), arXiv:1805.06296
jointly with my LAPTOP.
Wormholes as black hole foils
T. Damour and S. N. Solodukhin.
Phys. Phys. Rev. D 76, 024016 (2007).
The spacetime metric of the Schwarzschild-like wormhole’s solution is:
ds2
= −f (r)dt2
+
dr2
g(r)
+ r2
dΩ2
(2) (22)
where f (r) = 1 − 2M
r and g(r) = 1 −
2M(1+λ2
)
r .
The deflection angle as
ˆα
4M
b
+
2Mλ2
b
• The deflection angle by DSW is increased with the ratio of the
parameter λ with compared to Schwarzschild BH.