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Fig 2 Symmetry Requirement of Lensing Demonstrated 

 

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 Djoining the observer and the source. The astronomical distance DL is the distance from the observer to the lens and  DSLis 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 DL = DSL, 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 DL 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

 

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