Can gravitational waves be waylaid by gravity?

Yesterday, I learnt the answer is ‘yes’. Gravitational waves can be gravitationally lensed. It seems obvious once you think about it, but not something that strikes you (assuming you’re not a physicist) right away.

When physicists solve problems relating to the spacetime continuum, they imagine it as a four-dimensional manifold: three of space and one of time. Objects exist in the bulk of this manifold and visualisations like the one below are what two-dimensional slices of the continuum look like. This unified picture of space and time was a significant advancement in the history of physics.

While Hendrik Lorentz and Hermann Minkowski first noticed this feature in the early 20th century, they did so only to rationalise empirical data. Albert Einstein was the first physicist to fully figure out the why of it, through his theories of relativity.

Specifically, according to the general theory, massive objects bend the spacetime continuum around themselves. Because light passes through the continuum, its path bends along the continuum when passing near massive bodies. Seen head-on, a massive object – like a black hole – appears to encircle a light-source in its background in a ring of light. This is because the black hole’s mass has caused spacetime to curve around the black hole, creating a cosmic mirage of the light emitted by the object in its background (see video below) as seen by the observer. By focusing light flowing in different directions around it towards one point, the black hole has effectively behaved like a lens.

So much is true of light, which is a form of electromagnetic radiation. And just the way electrically charged particles emit such radiation when they accelerate, massive particles emit gravitational waves when they accelerate. These gravitational waves are said to carry gravitational energy.

Gravitational energy is effectively the potential energy of a body due to its mass. Put another way, a more massive object would pull a smaller body in its vicinity towards itself faster than a less massive object would. The difference between these abilities is quantified as a difference between the objects’ gravitational energies.

Such energy is released through the spacetime continuum when the mass of a massive object changes. For example, when two binary black holes combine to form a larger one, the larger one usually has less mass than the masses of the two lighter ones together. The difference arises because some of the mass has been converted into gravitational energy. In another example, when a massive object accelerates, it distorts its gravitational field; these distortions propagate outwards through the continuum as gravitational energy.

Scientists and engineers have constructed instruments on Earth to detect gravitational energy in the form of gravitational waves. When an object releases gravitational energy into the spacetime continuum, the energy ripples through the continuum the way a stone dropped in water instigates ripples on the surface. And just the way the ripples alternatively stretch and compress the water, gravitational waves alternatively stretch and compress the continuum as they move through it (at the speed of light).

Instruments like the twin Laser Interferometer Gravitational-wave Observatories (LIGO) are designed to pick up on these passing distortions while blocking out all others. That is, when LIGO records a distortion passing through the parts of the continuum where its detectors are located, scientists will know it has just detected a gravitational wave. Because the frequency of a wave is directly proportional to its energy, scientists can use the properties of the gravitational wave as measured by LIGO to deduce the properties of its original source.

(As you might have guessed, even a cat running across the room emits gravitational waves. However, the frequency of these waves is so very low that it is almost impossible to build instruments to measure them, nor are we likely to find such an exercise useful.)

I learnt today that it is also possible for instruments like LIGO to be able to detect the gravitational lensing of gravitational waves. When an object like a black hole warps the spacetime continuum around it, it lenses light – and it is easy to see how it would lens gravitational waves as well. The lensing effect is the result not of the black hole’s ‘direct’ interaction with light as much as its distortion of the continuum. Ergo, anything that traverses the continuum, including gravitational waves, is bound to be lensed by the black hole.

The human body evolved eyes to receive information encoded in visible light, so we can directly see lensed visible-light. However, we don’t possess any organs that would allow us to do the same thing with gravitational waves. Instead, we will need to use existing instruments, like LIGO, to detect these particular distortions. How do we do that?

When two black holes are rapidly revolving around each other, getting closer and closer, they shed more and more of their potential energy as gravitational waves. In effect, the frequency of these waves is quickly increasing together with their amplitude, and LIGO registers this as a chirp (see video below). Once the two black holes have merged, both frequency and amplitude drop to zero (because a solitary spinning black hole does not emit gravitational waves).

In the event of a lensing, however, LIGO will effectively detect two sets of gravitational waves. One set will arrive at LIGO straight from the source. The second set – originally sent off in a different direction – will become lensed towards LIGO. And because the lensed wave will effectively have travelled a longer distance, it will arrive a short while after the direct wave.

However, LIGO will not register two chirps; in fact, it will register no chirps at all. Instead, the direct wave and the lensed wave will interfere with each other inside the instrument to produce a characteristically mixed signal. By the laws of wave mechanics, this signal will have increasing frequency, as in the chirp, but uneven amplitude. If it were sonified, the signal’s sound would climb in pitch but have irregular volume.

A statistical analysis published in early 2018 (in a preprint paper) claimed that LIGO should be able to detect gravitationally lensed gravitational waves at the rate of about once per year (and the proposed Einstein Telescope, at about 80 per year!). A peer-reviewed paper published in January 2019 suggested that LIGO’s design specs allow it to detect lensing effects due to a black hole weighing 10-100,000-times as much as the Sun.

Just like ‘direct’ gravitational waves give away some information about their sources, lensed gravitational waves should also give something away about the objects that deflected them. So if we become able to use LIGO, and/or other gravitational wave detectors of the future, to detect gravitationally lensed gravitational waves, we will have the potential to learn even more about the universe’s inhabitants than gravitational-wave astronomy currently allows us to.

Thanks to inputs from Madhusudhan Raman, @ntavish, @alsogoesbyV and @vaa3.