A binary system of black holes merging into a single black hole provides one of the most interesting systems that can be probed by gravitational waves.
One traditionally distinguishes three phases in the phenomenon, often called black hole coalescence. They are depicted in the following diagram, drawn by the renowned black hole specialist, Kip Thorne.
The two black holes initially form a binary system in rotation. This is mass in motion: it thus loses energy in the form of gravitational waves (the frequency of the gravitational waves is directly related to the frequency of rotation). The two black holes thus get closer and rotate faster. This first phase is called the inspiral phase. The gravitational attraction between the black holes remains small and one can apply standard gravitational methods to compute the gravitational wave signal emitted.
At some point, the two black holes become so close that their horizons touch. The horizon of the black hole is the spherical surface that corresponds to the surface of no return: once crossed, impossible to go backward and tell what we have observed, one is fatally drawn to the centre of the black hole. Once the horizons have touched, one is left with a single black hole. This is the merger phase.
Because gravitational effects close to the horizon are strong, one needs to solve Einstein’s equation in the strong regime. This is done using numerical methods and was a major achievement of the field called numerical relativity in the last ten years (this computation of the form of the gravitational waves during merger was even called the “grand challenge” in the late 1990s). Probing this phase will lead to tests of general relativity in the strong regime (i.e. when gravity is strong), a real premiere.
The final phase is called ringdown. Once the new black hole has been formed, with a rather irregular sort of horizon (made of the two previous horizons), it will shake off its unwanted characteristics through a series of resonant oscillations and emission of gravitational waves. The oscillations are depending on the parameters of the black holes (the initial ones and the final one) and the gravitational waves emitted carry all this information away, encoded in their shape. Again, one will gain precious information on black holes and general relativity by studying the waves produced during ringdown.
The following video shows (credit: NASA/C. Henze) presents a simulation of the merger of two black holes and the resulting emission of gravitational radiation. The coloured fields represent a component of the curvature of space-time. The outer red sheets correspond directly to the outgoing gravitational radiation that may be detected by gravitational-wave observatories.
A final question : how long is a signal of black hole coalescence seen in a detector?
To answer this question, one must realize that detectors function in a given range of frequencies, typically between 10 and 1000 Hz for ground detectors, and between 1/10 000 and 1/10 Hz for space detectors. The gravitational wave signal is seen in the detector only if it falls in this range. But as the two black holes rotate around one another, their rotation frequency increases and thus the frequency of the emitted gravitational waves as well.
The binary system has been evolving for ages but the frequency enters the detector range only a short time before the final plunge (the black holes are then close to one another, the gravitational attraction is stronger and thus the amplitude of the gravitational waves is larger). Typically, ground detectors pick up the signal a few seconds before the merger, and space detectors a few months before.