Are there aspects of General Relativity that have yet to be tested?

Good evening everyone, I am new in the field of General Relativity and I have been reading and learning about the subject in recent months.

For example, I read several articles about experiments designed to test the Principle of Equivalence and other aspects of this theory.

So I asked myself:

Are there aspects of General Relativity that have yet to be tested? Are there any experiments or projects under development? How can I contribute as an engineering student?

My questions are not innocent. This year I have to work on a project of initiation to research in my engineering school and I wondered in what type of project I could start, and above all if it is wise to get into such a field at my level.

I am very curious and wish to deepen my knowledge in the field. Last year I studied the Three-body problem from a newtonian point of view and in particular the calculation of Lagrange points and the study of their stability in the Sun-Earth orbital plane. Would there be one aspect of this problem that needs to be explored from a relativistic point of view as much in mathematics, physics and computer science?

I hope my English is sufficiently well written and understandable.

Answer

A good general resource on this kind of thing is the review article by Will, The Confrontation between General Relativity and Experiment, https://arxiv.org/abs/1403.7377 . He updates it every few years. The most recent version predates the direct detection of gravitational waves (although they were already verified observationally based on systems like the Hulse-Taylor pulsar [PSR B1913+16]).

If objects like Sag A* are correctly modeled as Kerr (i.e., spinning) black holes, then we have not yet observed whether they have event horizons, as predicted by GR. As safesphere noted in a comment, this may happen in the fairly near future.

GR incorporates the equivalence principle, and therefore it predicts that nothing special happens at an event horizon in terms of the local properties of space -- no firewalls or anything else crazy. This has not been tested.

There is quite a bit of work by theorists like Joshi suggesting that astrophysical collapse might not actually lead to a black hole but rather to some other object such as a timelike singularity. That is, the cosmic censorship conjecture is looking weaker and weaker. This has not been tested.

If it turns out that people like Joshi are right in their suspicions, then we could potentially observe singularities (because they wouldn't be hidden behind event horizons). This would allow us to test a prediction of GR, which is that singularities should be a generic thing that happens for most initial conditions. If we don't ever get such an opportunity, then the only singularity we'll ever have a chance to observe is the big bang.