At the Long-baseline Interferometer Gravitationalwave Observatory (LIGO) facilities in Hanford, Washington, and Livingston, Louisiana, laser beams of wavelength \(550.0 \mathrm{nm}\) travel along perpendicular paths \(4.000 \mathrm{~km}\) long. Each beam is reflected along its path and back 100 times before the beams are combined and compared. If a gravitational wave increases the length of one path and decreases the other, each by 1.000 part in \(10^{21}\), what is the phase difference between the two beams as a result?

The Laser Interferometer Gravitational-Wave Observatory, commonly known as LIGO, is an engineering marvel tasked with the detection of gravitational waves. These elusive ripples in spacetime are generated by extraordinary cosmic events such as the collision of black holes or neutron stars.

LIGO operates two facilities in the United States—one in Hanford, Washington, and the other in Livingston, Louisiana. Both stations work in unison, using laser interferometry to measure minuscule changes in distance caused by passing gravitational waves. Each LIGO observatory features two arms arranged in an L-shape, each stretching about 4 kilometers long. Laser beams are bounced back and forth within these arms and are later combined to reveal any disturbances in their journey which could be indications of gravitational waves passing by.

As gravitational waves traverse the Earth, they stretch and compress space slightly but detectably. LIGO is designed to measure these minute changes in length on the order of a thousandth of the diameter of a proton, which is an incredibly small scale. This is equivalent to accurately measuring the distance to the nearest star outside our solar system (over 4 light-years away) to within the width of a human hair.

Interference is a fundamental property of light that arises when two or more light waves overlap. It results in a pattern of alternating bright and dark bands, known as interference fringes. This phenomenon occurs because light behaves as a wave, with its crests and troughs either reinforcing or canceling each other out.

When light waves meet in phase—that is, when the crests and troughs align—they constructively interfere to create a bright fringe. Conversely, when they meet out of phase, with the crests of one wave meeting the troughs of another, they destructively interfere and darken the area. This behavior is central to LIGO's ability to detect gravitational waves, as the interference pattern of the laser beams can indicate the minute spatial distortions caused by these waves.

This principle is harnessed in a device known as an interferometer, which splits a single light source into two beams to travel different paths before recombining them. By analyzing the resulting interference pattern, scientists can detect variations in the path lengths that are far too small to measure with conventional tools.

Wavelength is the distance between consecutive crests (or troughs) in a wave, which, for light, determines its color. In the context of LIGO, the wavelength of a laser beam is crucial as it sets the scale for measuring the tiny distance changes caused by gravitational waves.

Phase difference, on the other hand, is a way of expressing how 'in step' two waves are with each other. It's measured in radians, and when two waves are perfectly in sync (in phase), their phase difference is zero. A phase difference of \( \pi \) radians (or 180 degrees) means the waves are out of phase, leading to destructive interference.

In LIGO, detecting a gravitational wave involves looking for a shift in the interference pattern, which indicates a difference in the phase between the two laser beams. This phase difference arises because the gravitational wave stretches one arm and compresses the other, altering the distance each light beam travels, and hence, their phase relationship upon recombination. By using the wavelength of the laser light and the observed phase difference, scientists can calculate the infinitesimal changes in distance between the interferometer's mirrors, evidencing the presence of a gravitational wave.

Gravitational waves are measured by detecting the incredibly minor distortions they cause in the fabric of spacetime as they pass through Earth. In LIGO's interferometers, laser beams travel down the length of each arm and reflect back. The distance the lasers travel should be constant, but a gravitational wave will slightly lengthen one arm and shorten the other.

To put things in perspective, the changes in length we're talking about are smaller than 1/10,000th the width of a proton! It is precisely this change in length that causes a shift in the phase of the laser beams, resulting in a change in the interference pattern.

Detecting this requires exceptionally precise and accurate instruments. The mirrors at LIGO are some of the smoothest and most perfectly shaped in the world, and the instrument is isolated from all other environmental vibrations that could mask the signal of a gravitational wave. This requires complex systems for vibration isolation and laser stabilization, ensuring that only a genuine cosmic event can produce the appropriate signal in the LIGO detectors, confirming the presence of a gravitational wave and offering insights into the profound events that caused them.