The discovery of gravitational waves from two merging black holes has raised a number of questions about what would happen if two black holes merged near our solar system. While the real answer is complex, we can do a back of the envelope calculation.
The observed merger released three solar masses of energy as gravitational waves in a fraction of a second. That’s a huge amount of energy, but it was released 1.3 billion light years away from us. Like light, the energy of a gravitational wave decreases by the square of the distance, so very little energy actually reached us. The basic design of advanced LIGO is a set of mirrors spaced 4 kilometers apart. When the gravitational waves passed through LIGO, the separation of the mirrors shifted by less than a hundredth of a proton’s width, or one part in 1021.
The amount of shift caused by a gravitational wave is due to its amplitude, not its energy. While the energy of gravitational waves follow the inverse square relation, the amplitude of gravitational waves follows the inverse distance relation. In other words, if we were half as far away from the merger we’d have seen four times the energy, but only twice the shift. As long as we aren’t too close to the merger where things become complicated and nonlinear, this relation will give us a good idea of just how strongly the gravitational waves will affect us.
For example, suppose the two black holes were a billion times closer. At a distance of 1.3 light years the gravitational waves of the merger would be a billion times greater, raising the shift to one part in 1012. The arm of advanced LIGO would have shifted by four nanometers, or about half the width of a hydrogen atom. Huge by optical standards, but not really noticeable. The entire Earth would shift in diameter by about a hundredth of a millimeter. Such a shift might trigger some seismic activity that was on the edge of happening anyway, but it wouldn’t be the end of the world. If we put the black holes even closer, their mass alone would start to disrupt the Oort cloud, regardless of any gravitational waves. So we can safely say that merging black holes will never have a serious effect on us.
But just for fun let’s look at how close we could get. Before the merger the two black holes have a diameter of about 212 and 170 kilometers respectively. After the merger the final black hole has a diameter of about 365 kilometers. If we were really close to the black holes, the tidal forces alone would kill us, so let’s assume we’re at least 10,000 kilometers from ground zero. At that distance the shift caused by the gravitational waves would be about one part in a thousand. If you were floating in space you would likely feel that, since a person would experience a shift of a millimeter or two. Would it hurt, or possibly harm you? That’s hard to say. It would really depend on how resilient humans are to gravitational wave distortion, and we don’t have any experimental data on that. If I were to guess I’d say as long as your space suit held up you’d be fine.
If you were really 10,000 kilometers from two orbiting solar mass black holes their gravity would pose a much greater threat than any gravitational waves. While gravitational waves can carry a great deal of energy, they only interact weakly with matter. In many ways it’s amazing that we can detect gravitational waves at all.