Revisiting the gravitational lensing with Gauss Bonnet theorem

Transcription

1 Gravity and Cosmology 2018 YITP 27 Feb 2018 Revisiting the gravitational lensing with Gauss Bonnet theorem Hideki Asada (Hirosaki) Ishihara, Ono, HA, PRD 94, (2016) PRD 95, (2017) Ono, Ishihara, HA, PRD 96, (2017)

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3 Figure 5: Measurements of the coefficient (1 + )/2 from light deflection and time delay measurements. Its GR value is unity. The arrows at the top denote anomalously large values from early eclipse expeditions. The Shapiro time-delay measurements using the Cassini spacecraft yielded an agreement with GR to 10 3 percent, and VLBI light deflection measurements have reached 0.02 percent. Hipparcos denotes the optical astrometry satellite, which reached 0.1 percent.

4 Gravitational deflection angle of light provides a powerful tool Gravitational Lens NASA/HST

5 Sahu et al., Science 356, (2017) 9 June 2017 Fig. 1. Hubble Space Telescope image showing the close passage of the nearby white dwarf Stein 2051 B in front of a distant source star. This color image was made by combining the F814W (orange) and F606W (blue) frames, obtained at epoch E1. The path of Stein 2051 B across the field due to its proper motion toward southeast, combined with its parallax due to the motion of Earth around the Sun, is shown by the wavy cyan line. The small blue squares mark the position of Stein 2051 Bateachofoureightobservingepochs,E1 through E8. Its proper motion in 1 year is shown by an arrow. Labels give the observation date at each epoch. The source is also labeled; the motion of the source is too small to be visible on this scale. Linear features are diffraction spikes from Stein 2051 B and the red dwarf star Stein 2051 A, which falls outside the lower right of the image. Stein 2051 B passed arcsec from the source star on 5 March Individual images taken at all the eight epochs, and an animated video showing the images at all epochs are shown in fig. S1 and movie S1 (24).

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7 Derivation of Standard formula (at textbook level) = 4GM bc 2 assumes asymptotic source and observer(receiver). r R,r S r RS =

8 static and spherically symmetric (SSS) spacetime ds 2 = A(r)dt 2 + B(r)dr 2 + r 2 dω 2.

9 Optical metric ds 2 = 0, dt 2 = γ ij dx i dx j = B(r) A(r) dr2 + r2 A(r) dω2, Note ij = g ij We consider a space defined by optical metric. Light ray dt =0 Fermat s principle ij dx i dt dx j dt dt =0 In this space with ij, light rays are spatial geodesic.

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11 We define α Ψ R Ψ S + φ RS This definition seems to make no sense, because 1) Two Ψ s are angles at different positions. 2) Φ is merely an angular coordinate. We examine this definition in more detail.

12 Gauss-Bonnet theorem T KdS + T gd + N a=1 a =2

13 Asymptotically flat spacetime Euclidean space α = Ψ R Ψ S + φ RS = R S KdS. coordinate-invariant Ishihara et al. (2016) See also Gibbons&Werner (2008) for r= case (R and S are in Euclid space)

14 III. EXAMPLES we assume r R and r S. Then, Ψ R = 0 and Ψ S = π α = φ RS π. agrees with the textbook calculations

15 B. Approximations Schwarzschild metric Correction by finite distance δα = α α For both weak and strong deflection limits, δα O ( Mb r 2 + Mb ) S r 2 R

16 Examples Sun δα Mb r R arcsec. ( M M )( b R )( 1AU r R ) 2 Sgr A* δα Mb r S arcsec. ( M M )( b 3M )( 0.1pc r S ) 2

17 . 8: δα given by Eq. (23) for the Sun. The vertical axis denotes the deflection angle of light and the horizontal axis denotes the receiver [arcsec ] 10 4 b = R b = 10R e in color) and dashed one (red in color) correspond to b = R a dotted 10 5 line (yellow in color) corresponds to 10 micro arcseconds r R [ km ] [arcsec ] 10-4 Sun

18 otes the finite-distance correcs the source distance r S. The 10-2 nd to b =6M and b = 10 2 M, icro arcseconds [arcsec ] b =6M 10 micro arcseconds r S [ ] Sgr A*

19 OOtthheerr BBHH mmooddeellss Kottler (Schwarzschild de-sitter) in GR α = r ] g [ 1 b b 2 u 2R + 1 b 2 u 2 S [ Λb ] 1 b2 u 2 R 1 b2 u 2 S + 6 ( u R u S ) [ ] + r gλb O(r 12 g, 2 Λ 2 ). [ 1 b2 u 2 R 1 b2 u 2 ] S Weyl conformal gravity ( 1 b 2 u 2R + ) α = 2m b mγ ( bu R 1 b2 u 2 R + 1 b 2 u 2 S bu S 1 b2 u 2 S ) + O(m 2, γ 2 )

20 Ishihara et al. (2017) SSttrroonngg ddeefflleeccttiioonn lliimmiitt > 2π Darwin(1959), Bozza(2002) and so on 1 loop case FIG. 3: One-loop diagram for the photon trajectory in M opt.

21 By induction, one can prove for any winding number the coordinate invariance of α = Ψ Ψ + φ R S RS

22 Ono et al. (2017) Stationary, axisymmetric spacetime Lewis(1932), Levy and Robinson (1963), Papapetrou (1966) ds 2 =g µν dx µ dx ν = A(y p,y q )dt 2 2H(y p,y q )dtdφ + F (y p,y q )(γ pq dy p dy q )+D(y p,y q )dφ 2 p, q =1, 2 We choose spherical coordinates (Cylindrical coordinates => Weyl-Lewis-Papapetrou form)

23 ds 2 = A(r, θ)dt 2 2H(r, θ)dtdφ + B(r, θ)dr 2 + C(r, θ)dθ 2 + D(r, θ)dφ 2. dt = γ ij dx i dx j + β i dx i, Induced by rotation cf. charged particle in magnetic field L = 1 2 mv2 qv A

24 Lorentz (Lorentz-like) force is direction-dependent B or B g

25 Let us consider the photon orbits on the equatorial plane. Again, we define α Ψ R Ψ S + φ RS We use the Gauss-Bonnet theorem...

26 S α = KdS κ g dl, R S R New correction caused by rotation (gravitomagnetic effect) coordinate-invariant

27 Prograde α prog = 2M ( 1 b2 u S2 + 1 b b 2 u R 2) 2aM ( 1 b2 u b 2 R2 + ) ( ) 1 b 2 u 2 M 2 ( ) S + O b 2 ( α prog 4M ( 4aM ( ) ) ( ) M 2 ( + O ) b b 2 ) b ( 2 ) infinity limit Retrograde ( ) α retro = 2M ( 1 b2 u S2 + 1 b b 2 u R 2) + 2aM ( 1 b2 u b 2 R2 + ) 1 b 2 u 2 S ( ) infinity limit α retro 4M + 4aM ( ) M 2 + O b b 2 b 2 agrees with the known result + O ( ) M 2 b 2 agrees with the known result

28 Summary The gravitational deflection angle of light by using the GB theorem stationary and axisymmetric Extensions are future work

29 TThhaannkk yyoouu!!

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