Is there any problems to be solved in general relativity?
I found one for you: go to this page and in section 8.2 you'll find a box in the right margin.
Quantum gravity is not a problem in general relativity.
For instance, people are working on (and finally having some success) with how to apply numerical methods to get valid solutions for physical events like black hole mergers.
Realistic gravitational collapse has a host of theoretical questions that need to be answered, especially rotating collapse.
And there are still people interested in finding new exact analytical solutions to Einstein's field equations.
And these are probalby not the important open questions, they are just the ones that interest me at the moment.
What happens to a rotating disk?
There are a lot of papers on the question of rotating disk in relativity. For instance, check out the sci.physics.faq
As a result of these papers, it is known that the disk cannot be rigid. Finding a consistent model of a disk that can deform (as it must), but still rotate at relativistic velocities is a challenge. One can certainly model a disk made out of normal matter, with varying degrees of sophistication, and find out that it will explode before there are significant relativistic effects other than the sagnac effect.
The sagnac effect on rotating disks is reasonably well tested and understood in the literature, though there are different approaches as to how to handle the rotating disk. There are also a few maverick papers which are published and peer reviewed but unfortunately rather confused, illustrating that peer review does not always guarantee that a paper is correct. Appealing to experiment, though, it should be clear that papers that papers that predict that ring laser gyroscopes don't function can not be consistent with experiment. The literature could use a good pedagogical paper on this age-old problem, but there isn't any one paper that has risen to stand above the rest, leaving the situation a bit of a muddle for the student. This is probably more disturbing to amateurs than people who are versed in the subject well enough to form their own opinions with some degree of self-confidence.
There are some non-peer reviewed papers that take up this challenge of non-rigid spinning disks, one of them http://www.gregegan.net/SCIENCE/Rings/Rings.html appears to me to be quite reasonable, but unfortunately I'm not aware of any peer-reviewed papers that address this question in as much detail as the above article does.
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