Do the Einstein field equations successfully predict/describe physical processes other than gravitational ones?
General relativity is a theory of gravity; as such, it makes predictions about gravity. However, general relativity does make predictions about time and physical entities such as black holes. Some of the predictions general relativity did make:
- Gravitational waves exist (proved by LIGO last year)
- Black holes exist
- Light bends (proved in 1919 by an expedition led by Sir Arthur Eddington)
- Mercury's orbit was predicted (Newton's predictions weren't perfect)
- Time is variable from person to person (the only constant is the speed of light)
- Other predictions are included at this website.
Hope this helps!
Certain condensed matter systems show emergent behavior that is similar to general relativity: see this for example. Also, in fluid mechanics, sound waves can become trapped behind an "event horizon" called an acoustic black hole. Finally, the Einstein field equations are essentially the only possible classical equations of motion for a massless spin-two particle, so any massless spin-two particle would "look like" a graviton even if it didn't actually mediate the gravitational force, as discussed in the question Would a spin-2 particle necessarily have to be a graviton?