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Principle of Reciprocity applied to the Einstein Ring Calculation

the general and special cases

From Fig 2 the total deflection of the light ray, and thus the total angle of gravitational deflection representing the accumulative effect of gravitation on the light ray is give as

In this example the gravitating mass M is chosen to be positioned at the midpoint on the line of length D_S  joining the observer and the source. The astronomical distance D_L is the distance from the observer to the lens and  D_SLis the distance from the lens to the source. Also again, it is important to note that this case is only a simplified case, by setting D_L = D_SL, presented in most academic textbooks. There is no requirement at all that the lens be positioned exactly at the midpoint for an observation of a theoretical Einstein ring. Of course, this again would assume the validity of the light bending rule of General Relativity. This will become much more clearer in the next section that addresses the axis of symmetry of the lensed light ray for corresponding near- and far observers. It is readily seen that the axis of symmetry for a given light ray is perpendicular to the line joining the source and the observer only for the special case where the lens is positioned at the midpoint.

In this example, the radius R of the analytical Gaussian sphere enclosing the gravitating mass M is related to the angle h under which an astronomical object of that size might appear to an observer and is given by

where R and D_L are expressed in meters and the angle h is expressed in radians.  

The impact parameter for all the light rays that would be responsible for producing an image of an Einstein ring would simply be R. This is the nearest point of approach of the light rays to the center of the gravitating mass M. Since we are dealing with small angles, from Fig 2, the deflection of the light ray due to the gravitational effect on approach to the gravitating mass is just simply

From symmetry we have

Solving for the radius of the impact parameter of the light ray and thus the radius of the Einstein Ring expressed in radians we have

and finally

It is important to note that this is the simplified special case presented in most textbooks, where  D_L = D_SL , where D_L is the distance between the observer and the lens and D_SL is the distance between the light source and the lens, placing the lens exactly halfway between the light source and the observer.

Fig 2B Einstein Ring Calculation; the General Case 

Note that the angle of deflection for the approach segment, one half of the total gravitational effect, is exactly equal to the radius of the solved Einstein ring for the special case expressed in radians and is given as   

This principle, an essential Mathematical Physics principle on lensing, is often totally missed by researchers attempting to deal with this topic. From symmetry requirement, the gravitational effect on a light ray upon approach to a gravitating mass positioned exactly at the midpoint of a line joining the source and the observer, must equal that of the gravitational effect on the light ray upon receding the gravitating mass, whereby

and thus,

This is a rarely covered fundamental on gravitational lensing in modern academic textbooks. The accumulative effect must sum the effects on the light ray upon approach and receding the gravitating mass. From above, the total accumulative effect is thus

In this example, the radius R of the analytical Gaussian sphere enclosing the gravitating mass M is related to the angle h under which an astronomical object of that size might appear to an observer and is given by

where R and D_L are expressed in meters and the angle h is expressed in radians.  

The impact parameter for all the light rays that would be responsible for producing an image of an Einstein ring would in itself be a virtual ring of radius R, the same radius of the Gaussian sphere that encloses the mass M. This is the nearest point of approach of the light rays to the center of the gravitating mass M. Since we are dealing with astronomical distances and small angles, from Fig 2, the special case, the deflection of the light ray due to the gravitational effect on approach to the gravitating mass is just simply

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