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Is the fact that light bends twice as much as matter does in Schwarzschildian gravitational fields related to the implication of the second Friedmann equation (in which $\ddot{a}\propto\rho + 3p$, where $p=\rho/3$ for photons) that light induces twice the gravitational acceleration that matter does?

Sten
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Yuan Liu
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    No, that has nothing to do with it, the gravitational field of the photon is neglible compared to that of the dominant mass anyway. The reason for the 2x angle in the weak field is the 3x downward acceleration d²y/dT², if you integrate that over a long distance along x that gives 2x the angle, see https://physics.stackexchange.com/a/758504/24093 – Yukterez Aug 10 '23 at 15:25
  • @Yukterez Thank you very much for the reply. Could you share me a link that has the maths of the detailed calculation? And back to my original question, don't you think the Scharzschild metric should be seamlessly connected to the FLRW metric at the verge of the mass point (supposing the point is actually a homogeneous sphere)? Please kindly give me your thoughts. – Yuan Liu Aug 11 '23 at 01:56
  • I put in in the 2nd simulator from here which uses the geodesic equation from here and plotted the y acceleration and position, so the detailed calculation was numeric but if I find the time I'll see if I can make it shorter by removing all the unecessary parts. – Yukterez Aug 11 '23 at 04:23
  • If you're looking for the y acceleration on the path along x see here, for the initial conditions of the light ray (τ is then the photon's affine parameter) set μ=0, vₓ=1 at position x=0, y=large. For the deflection integrate the acceleration to get the position and finally get the angle with α=arctan[(y-y₀)/(x-x₀)]. – Yukterez Aug 11 '23 at 04:46
  • You can connect the Schwarzschild and FLRW metric, the easiest example with a mass in an expanding universe with constant H is the SSdS metric but that is only relevant on much larger scales than the deflection of light around the sun. In our example we can neglect all gravity other than that of the sun. – Yukterez Aug 11 '23 at 04:53

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