Gravitational Deflection of Light

[kevmac]

I'm looking for an expression for the deflection of light in a static gravitational field. Referring to 'deflection of star light past the sun' in Sean Carroll's "Spacetime and Geometry" - equation 7.80 for the "transverse gradient": 

 

 

Deflection angle is 

 

 

As far as I understand it, the transverse gradient is only valid for weak fields/small deflection. And I'm not looking for a general integral solution - I'd like to plot photon paths in strong fields, so I'm looking for the instantaneous deflection, which I'll plot/integrate numerically, based on mass, radial distance from mass, and angle of photon trajectory. It should not use the Schwarzschild solution/metric, because I don't want the singularity at r=Rs and it only needs to be in 2 dimensions, because of spherical symmetry. So, is there an expression for the polar coordinates r2, θ2 and trajectory a2, for a photon travelling from p1 to p2, using M, r1, θ1, a1, L? Below is a diagram which I hope illustrates it.

Many thanks

 

Edited September 6, 2017 by kevmac
Inline image didn't work, so attaching file.

[Mordred]

Under GR you have 3 classes of solutions.

1)vacuum

2) weak field limit

3) strong field limit

The weak field limit includes gravity of all stars short of a BH. (provided no relativistic effects)

The strong field as you noted typically uses the Schwartzchild metric. 

This paper has the deflection angle formulas for the strong field limit, note that the strong field limit is based on the Schwartzchild metric. This class of solutions works whenever you have relativistic effects regardless if there is a singularity or not.

https://arxiv.org/abs/gr-qc/0208075

Think of it this way, the deflection angle formula for the strong field needs an effective cutoff ( under the Schwartzchild) that cutoff is provided by the event horizon. Hence why the Schwartzchild metric is appropriate for the above application.

Anyways this paper shows the technique for what your looking for

Edited September 6, 2017 by Mordred

[kevmac]

On 9/6/2017 at 1:05 PM, Mordred said:

 

This paper has the deflection angle formulas for the strong field limit, note that the strong field limit is based on the Schwartzchild metric. This class of solutions works whenever you have relativistic effects regardless if there is a singularity or not.

https://arxiv.org/abs/gr-qc/0208075

Think of it this way, the deflection angle formula for the strong field needs an effective cutoff ( under the Schwartzchild) that cutoff is provided by the event horizon. Hence why the Schwartzchild metric is appropriate for the above application.

Anyways this paper shows the technique for what your looking for

Hi Mordred, thank you for the link to that paper.  However, what I want to do is plot photon paths as they cross the Schwarzschild radius - these will vary with M and will be smooth, because the Schwarzschild radius is just a co-ordinate singularity or  'cutoff'.

[Mordred]

The geodesic equations for the Schwartzchild is the valid equations from at rest, to EH. So I fail to see why you cannot use them.

Are you trying to model beyond EH, if so then your observer location will be an issue. Not to mention infinite redshifts beyond the photon sphere.

Why do you feel the Schwartzchild won't work outside the EH cutoff?

The ds^2 line element of the Schwartzchild is the Worldline path from observer at rest to EH boundary. Well assuming a static non rotating BH.

If rotating charged or uncharged use the Kerr metric,

If you want to confirm the above look specifically at the strong field geodesic equations of the strong field lensing papers. (They use the same ds^2 line element worldline)

Edited September 13, 2017 by Mordred

[kevmac]

Yes, I want to model the photon path across and beyond the EH. Perhaps using Kruskal–Szekeres, which lack the Schwarzschild coordinate singularity at r=Rs?

[Mordred]

Ah ok that is a different scenario then, as the lens equation stops at the Schwartzchild radius.

Kruskal may work for you I will have to dig a bit at the different coordinate systems.

I believe what you need is Bozzas technique for the weak field/strong field multiple image.

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/gr-qc/0102068&ved=0ahUKEwja8M6Qo6LWAhVhr1QKHYtCDNgQFggfMAA&usg=AFQjCNFRwiIbOVQOAqG1GaKJJ-ITTK3qNg

 

often described as Bozzas technique in numerous strong lensing papers. Beyond that I am not aware of any solutions. At the very least you can get a direction on the weak field/strong field image strength boundaries for multiple images due to the strong field. However that is a region just outside the photon sphere to the single image weak field.

Edited September 13, 2017 by Mordred