I use a Metabones adapter for Canon lenses on a Sony body. Why doesn't the lens adapter, which sets the lens further from the sensor, have the same effect as using extension tubes, allowing closer focus?
Because the Canon EF mount lens "expects" to be further from the sensor than a Sony E mount lens; this is known as the flange focal distance or the registration distance - a Canon EF lens focuses the incoming light on a plane 44mm behind the lens, while a Sony E lens focuses it on a plane 18mm behind the lens.
If you somehow bodged it horribly so that a Canon lens was mounted in the same place as the Sony lens, it would be focusing everything on a point 26mm behind the sensor and it would all be a bit of a disaster really. The EF to E lens mount adapter ensures that the Canon lens is mounted 44mm from the sensor so that the incoming light is focused in the correct place.
Why don't lens mount adapters have the same effect as extension tubes?
They do. But whether it limits far-focus capability depends on one factor: the difference in distances from the image plane to the back of the lens mount flange between the two systems.
If the adapted lens's mount is deeper than the camera's mount, the adapter tube can make up the difference so that the lens sits exactly where it needs to in order to have full range of focus.
If the adapted lens's mount is shallower than the camera's mount, then the adapter is working exactly like an extension tube and will reduce far-focusing capabilities, if there's no glass element. If there is a glass element in the adapter, the adapter no longer acts like an extension tube, but is now acting like a converter lens, where the added glass has changed the apparent focal length to regain focus-to-infinity. The Speedbooster acts like a wide converter, which decreases the focal length (and increases the DoF and max. aperture). But other adapters often use a teleconverter lens to increase the focal length (which decreases the max. aperture).
The Metabone adapter is an optical accessory that mounts between the camera body and the camera lens. Such a mounting relocates the lens further forward. Such repositioning corresponds precisely like the action of extension rings or tubes in common use. However rings and tubes are void of lenses whereas the Metabone adapter contains lenses that modify the focal length of attached lens.
Supplemental optics similar to this device has been in use almost from the beginnings camera / lens design. We can mount such devices before or after the camera lens. These devices alter close focusing distance or they modify the lens, converting it to a telephoto or a wide-angle.
Many such supplemental lenses have been or are currently being marketed. The Metabone adapter revises the focal length of the attached lens, it shortens it about 70%. In other words, say a 50mm lens is combined with a Metabone adapter. The 50mm focal length is altered, it converts to become35mm. This action is analogous to add-on wide-angle converters commonly available. However, most such devices mount before the camera lens whereas the Metabone adapter mounts after the camera lens. Again, not a new idea.
How does the Metabone adapter realize a lens speed increase? The photographic industry uses the f-number system to specify the relative speed of a lens. The camera lens acts like a funnel in that it gathers light. The greater the working diameter of the lens (aperture) the brighter the projected image (speed of the lens). Entwined with aperture is focal length. Both are contributors as to how bright the projected image will be. Say the aperture diameter is untouched and the focal length doubled the projected image dims to 25% of its original brightness (2 f-stops). Conversely, if the focal length cut in half, the image brightens 2 f-stops.
The Math: 50mm lens 12.5mm aperture = f/4 --- 100mm lens 12.5mm aperture = f/8 --- 25mm 12.5mm aperture = f/2
Suppose you set your 50mm to f/8 achieved with a 6.25mm aperture. Now you mount a Metabone that shortens the focal length 70%. The revised focal length is35mm. The aperture remains unchanged at 6.25mm, the revised f-number is 35 ÷ 6.25 = 5.6 (written f.5.6). Simply stated, the Metabone shortened the focal length, this act increased the angle-of-view (more wide-angle), the image brightened as a result, the revised f-stop is f/5.6 = 1 f-stop speed gain.
The cleverness of the Metabone is, it shortens the focal length, this gives rise to a speed gain plus it adapts a lens designed for a larger format camera to operate on smaller format cameras mitigating the crop factor that would normally charge. Remarkable is the fact that back-focus distance (flange focus distance) of the lenses is corrected taken into account allowing lens interchangeability.
Also, Meabone is credited with suppressing some aberrations normally realized when a supplemental lens is added. (If all true, hats off to the optician