My question concerns things like the ultra wide angle adapter for the Sony Alpha/Nex/e-mount line 16mm f/2.8 prime pancake lens, which changes the focal length of said lens from 16mm to 12mm. Since aperture sizes are expressed as a fraction of the focal length, does the effective f-stop change when the focal length is changed by such an adapter?
There is a product called Metabones Speedbooster, which is a lens mount adapter that allegedly increases the speed of a lens while widening its field of view. Is the speed increase a trivial consequence of the focal length change or is there some other principle at play there?
The ratio f/2.8 means the diameter of the entrance pupil is equal to the focal length divided by 2.8.
The key thing to note about the above is that the entrance pupil is the image of the aperture stop as seen through the front of the lens, the ratio does not depend on the physical size of the aperture itself.
A rear-mounted 2x teleconverter, such as you would use with a telephoto lens, changes the focal length whilst the entrance pupil remains the same size, so the aperture number goes from 2.8 to 5.6.
A front mounted tele/wideconverter will change the focal length, but it will also change how large the aperture appears, meaning the entrance pupil changes in line with the focal length so the f-ratio stays the same.
A rear mounted focal reducer such as the Metabones SpeedBooster will give you a brighter f-ratio as the entrance pupil remains the same size whilst the focal length gets shorter.