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We choose to present our results in terms of the distance of closest approach r0, as in. Another convention is to write the results in terms of the impact parameter b, which has the advantage of being an invariant of the light ray with respect to r0. For the Schwarzschild metric the relation between these two quantities is: r0=2b3cos [13arccos (-33GMb)] which is given as a root of cubic polynomial. For GM b, r0=b(1+O(GM/b)) and therefore can be identified at leading order. The differences of the Einstein angle expressed in form r0 or b are therefore in the next-to-leading order terms, i.e. O(G2M2/r02) or O(G2M2/b2) (see fourth reference of);
-
We choose to present our results in terms of the distance of closest approach r0, as in. Another convention is to write the results in terms of the impact parameter b, which has the advantage of being an invariant of the light ray with respect to r0. For the Schwarzschild metric the relation between these two quantities is: r0=2b3cos [13arccos (-33GMb)] which is given as a root of cubic polynomial. For GM b, r0=b(1+O(GM/b)) and therefore can be identified at leading order. The differences of the Einstein angle expressed in form r0 or b are therefore in the next-to-leading order terms, i.e. O(G2M2/r02) or O(G2M2/b2) (see fourth reference of);
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For SdS the impact parameter b is not directly J/E (J and E being the angular momentum and the energy, respectively). The relation between r0 and J/E (the latter being the invariant of the light ray) for SdS is: r0=2J3E2+ΛJ2cos [13arccos (-33E2+ΛJ2GMJ)]. Again r0 J/E in the SdS case at leading order for GME/J 1, ΛJ2/E2 1.
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Note that the light bending in SdS written in terms of J/E (rather than r0) contains Λ because of the relation between r0 and J/E in.
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