# A subtle misconception about how LIGO works

If a gravitational wave contracts and expands space, shouldn’t the light wavelength be affected in the same way as the interferometer arm distance? If so, how is the wave detected at all?

When the first ever direct gravitational wave detection was announced by LIGO in February 2016, excitement rippled through the physics community. The detection had been hinted at for months, but this wasn’t the first time such rumors had circulated. So to have confirmation, finally, was almost as much a moment of relief as it was of triumph. Now there are confirmed fourth and fifth detections, both of these, thrillingly, also measured by Virgo–vastly improving our ability to triangulate the location of the source of the waves.

The LIGO (Laser Interferometer Gravitational-Wave Observatory) project has been of particular interest to me since I spent time as an undergraduate doing research at LASTI, the MIT center for LIGO research. LIGO was still almost 10 years from its first detection and the scope and ambition of the project boggled the mind.

The goal was simple: to make the first direct detection of a gravitational wave.

The principle was simple: a gravitational wave distorts space, therefore a gravitational wave passing through a Michelson interferometer will change the path lengths of each arm in such a way as to produce an interference pattern characteristic of the wave.

In practice, of course, there’s nothing simple about it. The LIGO interferometers (there are 2 of them) are a masterful feat of engineering. Each arm of each interferometer is 4 km long, with massive 40 kg mirrors suspended on the end of 360 kg quadruple pendula. The sensitivity is such that they can detect distance changes 1000 times smaller than the size of a proton. They do this by employing many different tricks, including a Fabry-Perot cavity along each arm that makes its effective length more like 1120 km (about the distance between New York and Chicago). The passive and active noise canceling technologies are the best in the world.

Even still, LIGO can only detect some of the most violent sources of gravitational waves in the universe: the collision and merger of massive bodies, predominantly black holes. The most recent black hole merger involved two bodies with about 25 and 30 times the mass of our sun.

While discussing the first detection with some fellow Miller Fellows, we stumbled into a seemingly simple question, but one that turned out to be quite subtle:

If a gravitational wave contracts and expands space, shouldn’t the light wavelength be affected in the same way as the interferometer arm distance? If so, how is the wave detected at all?

We were all tripped up by the question, which was a bit embarrassing considering the table included a guy who studies the CMB, a guy who studies black holes, and me, who actually briefly worked on LIGO. However we were perhaps redeemed in that it appears to be a question that comes up a lot and has produced a handful papers on the subject. (This article is based on two of them: V. Faraoni, Gen Relativ Gravit 39, 677 (2007) and especially Peter R. Saulson, Am Jour Phys 65, 501 (1997).)

Here I’ll attempt to give a satisfying answer without resorting to equations. I highly recommend the Faraoni paper for a quantitative treatment. It can also be found on the arXiv.

### How LIGO works

First, we work under the assumption that the wavelength of the gravitational wave is much longer than the length of the interferometer arm (and the light wavelength). Thus each photon only samples a single, constant value of the gravitational wave distortion (a constant spacetime metric). This is a safe assumption: black hole mergers release radiation on the wavelength scale of 100-1000 km, long compared to LIGO’s 4 km arms and its laser’s central wavelength of 1064 nm.

What happens to the interferometer arm when the gravitational wave arrives? The beamsplitter and Fabry-Perot mirrors act like free masses. So as space expands and contracts, the distance along the arm changes along with it.

The change in the path length is linearly proportional to the gravitational wave’s contribution to the spacetime metric. This is a tiny length change–recall from above that we’re looking for deviations of 1000th of a proton (10^{-18} m) on a path length of 4 km (10^{3} m).

Let’s send a pulse of photons into the interferometer.

Since the speed of light is constant, the return time of the photons will be determined by the distance they travel. Without a gravitational wave, the photons traveling each arm return to where they were emitted at the same time. A gravitational wave, having a quadrupole moment, will cause each arm to contract or expand differently. Under the assumption we made–that the wavelength of the gravitational wave is long compared with the interferometer–we can treat the length of each arm as constant during the time the photons are traveling. The time difference in the arrival time of the photons from each arm will be proportional to the length difference of the arms. Thus from the arrival time, we can measure the strength of the gravitational wave.

In practice, we trade a pulse of photons for a constant, coherent stream of them. Instead of measuring the difference in arrival time of each pulse, we measure the different arrival times of the “crests” of each beam (the phase difference between each beam). Intuitively, this should be the same as measuring the photon arrival time.

### Gravitational redshift?

Yet it *is* true that the expansion of of the universe causes a redshift in light.

We can measure it directly in fact–nearby galaxies appear to have a Doppler-like redshift at all wavelengths. Changing the spacetime metric does indeed change the wavelength of light.

Imagine a light wave, where at one instant in time, each crest of the wave aligns with a freefalling test mass. If in that instant, the spacetime metric is changed–spacetime expands for example–then the distance between the test masses will grow accordingly. These test masses define our comoving coordinate system–the light crests must still align with the test masses. The light wavelength has increased.

So how do we escape this apparent contradiction?

### Resolving the contradiction

Immediately at the time the gravitational wave arrives, there will indeed be no apparent shift in the phase of the light. If a “crest” of light is at the beamsplitter when the wave arrives, it will still be there afterward (think of the beamsplitter acting as a coordinate marker the same way our test masses did).

However at later times, the picture changes. Each crest of light now has further to travel through the interferometer. The pulses closest to returning to the beamsplitter have the least amount of “added” distance, those that just entered the arm have the most. The wave crests still propagate at the speed of light. So the wave crest arrivals are still delayed compared to what they would be without the added path length. This is illustrated below with an example of a single Michelson arm.

What about light that enters the arm after the gravitational wave arrives? To first order, the change in the wavelength of the light is determined by the *variation* in the spacetime metric along the arm, which is zero in our approximation. This is different than the case of the gravitational redshift, which is due to the *continuous* expansion of the universe. (For the curious, the next order contributions to the change in the light’s wavelength go as the square of the gravitational wave’s contribution to the metric, many many orders of magnitude smaller than the change in the arm length!)

A passing gravitational wave distorts (stretches and contracts) the wavelength of light due primarily to the

changein the spacetime metric and distorts the LIGO interferometer arm length due to thevalueof the metric.

So subsequent photons generated after the gravitational wave arrives are not stretched. The effect on the phase shift is just a constant phase difference from the un-stretched case. That is, a “dc” response proportional to the magnitude of the gravitational wave in that moment. This is illustrated below.

So there we have it! A passing gravitational wave distorts (stretches and contracts) the wavelength of light due primarily to the *change* in the spacetime metric and distorts the LIGO interferometer arm length due to the *value* of the metric. The net result is an interference pattern such as the ones shown below, elegantly telling the tale of a violent duet of massive proportions.

LIGO works after all, a gravitational eye gazing into the cosmos.