tylerfarzam said:
1) Does energy have mass? It seems like energy does not have mass but i thought e=mc2 proved that it does.
No. Mass and energy are properties of a system, not things of their own. It would be more appropriate to say that a system which has mass also has a corresponding amount of energy, given by ##E = mc^2##.
2) Does gravity have an effect on energy?
Again, energy is not a thing but a property of a physical system. That said, gravitation (in general relativity) couples not only to mass but to all forms of energy, momentum, and stress. The reason Newtonian gravitation works so well for daily use is that, in all the settings that you will be familiar with, the amount of energy corresponding to the mass is much larger than any of the other sources of gravitation.
3) If photons do not have mass, how are they affected by the curvature of space time?
See (2). Gravitation does not couple to mass only.
4) If the curvature of space time can transmit information at the same speed as photons, is it possible that the electromagnetic force is the carrier of the gravitational force?
No. Electromagnetism and gravitation have properties that are very different. In relativity, any massless particle will travel at the speed of light.
5) Is it possible to become an accomplished theoretical physics researcher if you do not have an actual research job?
I would say no. Mainly due to the fact that you need a lot of time and you would have to make your living somehow, but also because you will find it hard to stay connected to the field and some resistance to getting your work published if you are not affiliated to a research institute (for better or worse).
6) What do researchers in general relativity study and do?
The obvious answer would be "general relativity". A more intricate answer would say that you can study the mathematical foundations, the implications, new ways to produce analytic solutions, ways of finding solutions numerically, ways in which gravity can be quantised, the relation to other fields. The list goes on.
Isn't relativity done and solved?
You cannot "solve" a physics theory. It does not work like that. You can always ask new questions about the theory and its implications that are worthwhile. The combined world knowledge about GR today is far beyond what it was in 1920. That being said, we do know a lot about how GR works and what its implications are. After all, we have studied it for more than 100 years.