Now that gravitational waves have been detected, theoreticians have been furiously speculating about what we might learn from our gravitational wave observatories. Now that we have a couple of observed black hole collisions under our belt, it is time to consider what we might study. There's some speculation that, depending on the sort of physics at play, the event horizon of a black hole might be studied through gravitational waves.

For this to work, the gravitational wave signal has to change depending on what type of black holes are merging. A recent paper in Physical Review Letters indicates that, unfortunately, reality will probably not cooperate.

No bells on a black hole

When two black holes collide, gravitational waves are emitted as a result of a kind of relaxation process. So far, we have mostly been excited about the waves generated just before the merger, where the objects are spiralling into each other and emitting waves as they plunge through space at very high speed.

But, just after the merger, there are more oscillations, called relaxation oscillations. These might tell us something about the event horizon of the merged object. In one sense, you can think of this like dropping a stone in a pool: waves are generated and expand outward, but the process doesn't continue endlessly. The few waves that are generated help the water relax back to a flat surface.

A black hole goes through a similar process. But what determines the frequency of the waves generated during relaxation?

One thought is that the waves might depend on the shape of the event horizon. To investigate this, a trio of researchers calculated the oscillations of event horizons and the gravitational waves emitted by mergers. They then compared these for two cases: the gravitational wave signature from the relaxation of the event horizon after an object has passed through a worm hole and an equivalently sized object dropping into a black hole.

To perform the calculation, the researchers consider the shape of the event horizon for the black hole and the wormhole. They use the shape to calculate the frequencies at which the event horizon would ring.

It's relatively easy to understand how this calculation works, even if you've never done it before. A black hole has an event horizon that is basically a sphere. If you could somehow give the event horizon a gigantic whack with a hockey stick, what would we observe? Well, the whole sphere can contract and expand, as if it were breathing. But it can also contract along one axis while expanding along a second axis 90 degrees from the first.

These different modes of expansion and contraction all take different amounts of energy to excite, so they emit different tones of gravitational waves. But, thanks to the shape of the black hole, the tones also die away very quickly. The result is that a black hole's event horizon doesn't ring like a bell. Instead it rings like a lead cannonball.

A similar calculation performed on a wormhole results in a different answer. The researchers found that a particular frequency of gravitational wave could travel back and forth between the two horizons of a wormhole, causing them both to expand and contract repeatedly. These waves and the event horizon oscillations could persist for long periods of time, giving the wormhole a sharp and distinct tone. In this case, wormholes ring like a bell.

The bells are muted by the fog of space-time

Based on these results, you would think that we would be celebrating. Clearly, wormholes and black holes have different gravitational wave spectra. Except they don't. When the researchers calculated what a distant observer would measure for the wormhole and short time scales, the measurement looked exactly like that of the black hole: an oscillation that died away very sharply.

This similarity is not just due to how gravitational waves are produced but also to how they escape from their local environment. In general relativity everything travels in straight lines, but space bends and curves. The oscillation that produces the gravitational wave also destabilizes these trajectories, trapping and damping out the output.

The authors conclude that it is an accident that the relaxation of a black hole looks like the ringing of the event horizon (this had been suspected for some time). However, this calculation really shows that, even for two objects that are completely different, the gravitational wave signature of the relaxation at short time scales will be the same. Indeed, their result shows that an event horizon is not even required as long as there is sufficient destabilization of space-time.

The calculations do leave us with a tiny bit of hope. It is likely that at longer time periods, the differences in relaxation oscillations will be visible. Even better, the oscillations may well be detectable—though their amplitude is much smaller than the peak from just before the merger—since the generated gravitational waves are still within the frequency range that current gravitational wave detectors can pick up.

One important point of this paper is that, even for the events detected so far, we cannot conclusively state from the gravitational wave signature that the objects involved had event horizons. Sure, a black hole is by far the most likely object, but the relaxation oscillations post-merger are not conclusive.

On the other hand, the signals observed so far do contain strong evidence for light rings—the name given to the destabilization of trajectories—which is itself quite interesting.

Sometimes, I think that theoreticians hold back on us. Ever since we detected gravitational waves, there has been an avalanche of papers about what we might learn from gravitational wave observations. I find it hard to believe that all of this has been done in just a few months—the theoreticians seem to have been waiting for the right moment to release them.

Of course, all of these papers just tell us that there is a lot of interesting stuff to look for.

Physical Review Letters, 2016, DOI: 10.1103/PhysRevLett.116.171101