The right answer is seldom as important as the right question.
Gravitational waves will bring us exquisitely accurate maps of black holes - maps of their space-time. Those maps will make it crystal clear whether or not what were dealing with are black holes as described by general relativity.
Closed timelike curve is the jargon for time travel. It means you go out, come back and meet yourself in the past.
A black hole really is an object with very rich structure, just like Earth has a rich structure of mountains, valleys, oceans, and so forth. Its warped space whirls around the central singularity like air in a tornado.
If you have a wormhole, then you can turn them into time machines for going backward in time.
When gravitational waves reach the earth, the waves stretch and squeeze space. This is a tiny stretch and squeeze. Far too small to detect with ordinary human senses.
Whether you can go back in time is held in the grip of the law of quantum gravity. We are several decades away from a definitive understanding, 20 or 30 years, but it could be sooner than that.
