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| Units and Symbols : |
| G = c = hbar = 1 |
| M is the mass of the black hole. |
| E is the energy of the incident particle. |
| v is the speed of the particle in units of c. For a massless particle, v = 1. |
| The product ME may be thought of as the ratio of the black hole horizon to the wavelength of the quantum particle. It is a measure of the coupling strength of the gravitational interaction. |
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| Photon geodesics around a Schwarzschild black hole. The critical impact parameter is at b = \sqrt{27}M = 5.196M. The inner circle shows the event horizon at r = 2M, and the outer circle shows the unstable photon orbit at r = 3M. |
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| Deflection angle as a function of impact parameter. The plot compares the scattering angle with (i) the Einstein deflection angle 4M / b, valid when b >> bc, and (ii) Darwin's approximation, b ~ 3 \sqrt{3} M + 3.48 e^{-\phi}, valid when b ~ bc. |
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| Massless. Shows the absorption cross section of a massless scalar [solid] and a massless fermion [dashed]. The classical limit at $27 \pi$ is also marked. | Massive. The solid line is the quantum absorption cross section for a massive fermion with $mM = 0.2$, and the dotted line is the classical cross section. |
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| Glory and Spiral Scattering for Massless Spinor Wave. This plot shows the log of the scattering cross section as a function of angle for a range of black hole masses. |
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| Glory approximation for the massless spinor and scalar waves. This plot compares numerical glory oscillations (solid line) in the backward direction with the path integral approximation (dotted line). |
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| Glory oscillations for the massless spinor and scalar waves. This plot compares the glory oscillations for the scalar and spinor waves at Mw = 4.0. The upper and lower plots show the same thing, but the lower plot uses a log scale. |
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| Waves with mass are partially polarised by the scattering interaction. In the context of EM scattering this is known as Mott polarisation. |
| The plot below compares the numerically-determined polarisation with a second-order Born series result. Absorption of the low-l partial waves enhances the polarisation effect. |
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| Mott polarisation as a function of scattering angle. Waves with mass are partially-polarised by the interaction. As $ME$ increases, oscillations arise in the polarisation. Left: For v = 0.9. Right : For v = 0.2. | |
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