Background

On the 14th of September 2015 the twin detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) made the first direct observation of gravitational waves. According to Einstein's theory of general relativity, when massive objects are accelerated they emit gravitational waves – ripples in spacetime that spread at the speed of light.

And when the objects are several times as massive as our sun, and move at extremely high speeds, they are strong enough that we have a chance to detect them. Mergers of black holes and of neutron stars provide exactly the right conditions for the emission of strong gravitational waves.

The source of the first detected signal, named GW150914, was the collision and merger of two black holes in a distant galaxy, 440 megaparsec (=1435000000 light years) from earth. The masses of the two black holes were about 35 and 30 times that of the sun. The resulting black hole had a mass of only 62 solar masses, with the difference, 3 solar masses (!!), being emitted as gravitational waves.

Gravitational waves are moving distortions of spacetime itself. When they pass through objects which are not rigidly bound together, such as a ring of masses floating in space (or the suspended test masses in the LIGO detectors), they change the distance between them.

Imagine a ring of masses freely floating around the earth. When a gravitational wave passes through it distorts the ring, stretching it in one direction and compressing it in the other. The distortion rotates with the same frequency as the black holes that created the gravitational wave orbited each other. As the black holes get closer together they orbit faster and faster, and the gravitational waves, and therefore the distortion of our ring of masses gets stronger.

The models

The models here show the time evolution of this distortion during the last one tenth of a second before the black holes merge. Each horizontal slice corresponds to one instant of time. The distortion is calculated for rings at the location of the earth and shown exaggerated by a factor of 5×10^20, that is 500000000000000000000! In reality the effect is absolutely minuscule, changing the length of the 4km (=2.5 miles) LIGO detectors by less than the diameter of a proton. (Here is a video giving a glimpse into how the LIGO detectors can measure such tiny variations in distance.)

There are 6 models, one for each gravitational wave signal detected until now. In the first image above you can see the sources for each of the signals, the size of the circles corresponds to the size (and therefore mass) of the objects.

The first 5 come from collisions of black holes. The amplitude of the signal depends on both the masses of the black holes, their distance from earth, and their orientation. The frequency of the signal depends on the mass of the black holes, with the first one, GW150914 being the heaviest and GW170608 the lightest.

The source of the last signal, GW170817, was the merger of two neutron stars, which (probably) formed a black hole. As neutron stars are a lot lighter than black holes the frequency is very high. The strength of the emitted gravitational waves was a lot lower than that of the black hole mergers, but here the source was more than 10 times closer than that of GW150914.

All the models are at the same scale, both in time (10.8 millisecond per centimeter) and amplitude (5×10^20 times the actual effect size), so you can compare the different types of signals.

You can see how the frequency and amplitude of the signal rises as the two black holes spiral closer and closer to each other (this is called the inspiral phase), then the instant at which they merge, and finally the ringdown where the amplitude decays exponentially as the newly formed single black hole wobbles until it settles down to its final shape.

Design Process

Data Source

These models are based on data from LIGO. The data coming from the detectors themselves are too noisy to generate models like this, so I'm taking the templates that are used to search for the signal and to extract parameters, such as the masses of the black holes, from it.

For GW150914, GW151226, GW170104 I used templates directly from LIGO. These are not the exact ones that are used for the scientific analyses, but they should be close enough for visualisations. For the remaining events I used pycbc to generate waveforms with the parameters that LIGO determined for these sources. The waveform for the neutron star collision GW170817 was also generated with a model for black hole mergers, so the last portion of the signal is probably not accurate, but that part is basically impossible to print anyway :D

Processing

From these waveforms, the distance to the source, and the inclination (the angle between the rotation axis of the source and the line connecting it to earth), I then calculated the distortions of a ring of masses at the location of the earth, pointing towards the source.

The models show a bit more data then the LIGO detectors can measure accurately, specifically the inclination. (The two detectors are almost parallel, so they can't really distinguish between higher inclination and higher distance.) I just used the maximum likelihood estimate for the inclination, but the real signal could look substantially different. Once more detectors come online in the next years we should get new signals with much better determined inclination.

Modelling

The slices (between 150 and 500 for each model) were assembled into a 3d model in Fusion 360, using a python script.