Seminar yefak 2019
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Weak Gravitational Lensing and Gauss-Bonnet Theorem discusses using the Gauss-Bonnet theorem to calculate light deflection angles in various spacetime geometries, including: 1. Calculating the deflection angle of light near a Schwarzschild black hole using the optical geometry and Gaussian curvature. 2. Computing the deflection of light by dyonic wormholes in Einstein-Maxwell-Dilaton theory and showing the deflection depends on the electric, magnetic, and dilaton charges. 3. Finding the deflection angle of light is inversely proportional to the Rindler acceleration for a Rindler modified Schwarzschild black hole.
![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).](https://image.slidesharecdn.com/seminaryefak2019-190130185650/85/Seminar-yefak-2019-12-320.jpg)
Seminar yefak 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?
- 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.