Gravitational waves do not impart energy to stationary objects.

Gravity in general relativity is described by the geometry of spacetime. For example, the current universe is accurately represented by a homogeneous and isotropic expansion model of its geometry. If we reduce space to two dimensions, this can be visualized as an expanding balloon.

The physical distance between two points A and B on the surface of an expanding balloon increases over time. The same thing happens in a universe where three-dimensional space is expanding. The physical velocity of particles moving through this cosmic space decreases steadily due to this spatial expansion. Furthermore, particles that were stationary at a point from the beginning remain stationary during the expansion.

Let us reverse time in this phenomenon. The universe then contracts rapidly. The velocity of particles that were moving slightly increases as we go back in time. However, particles that were stationary from the beginning remain stationary even as the universe contracts. As a result, even after the contraction of the universe stops, those particles remain at rest in their original locations. In other words, during the contraction process of the universe, those stationary particles receive no energy at all. A similar phenomenon occurs with the mirrors of laser interferometer devices used for gravitational wave observation.

Even in the laser interferometers of gravitational wave observatories, when gravitational waves from distant celestial bodies reach the Earth's surface, the physical distance between the mirrors changes over time, just as in an expanding and contracting universe. As a result, when two laser beams traveling along two different orthogonal paths are made to interfere, changes can also be seen in the interference fringes of the laser light. By performing this measurement, we can observe the gravitational waves that have reached the Earth's surface.

Although this interferometer has mirrors that reflect lasers, the mirrors themselves do not receive energy from ordinary gravitational waves coming to Earth. The mirrors are stationary before the gravitational wave arrives, and they remain stationary after the gravitational wave has passed. This is the same phenomenon as particles that were stationary within an expanding and contracting universe. The space simply stretched and shrank, but the mirrors did not oscillate.

For example, a gravitational wave traveling in the positive z-axis direction is written with the following metric tensor in the transverse-traceless (TT) gauge.

The function representing the waveform in the components of this metric tensor is given by the following in a unit system where the speed of light c is 1.

It is written as follows. Let us consider the geodesic equation for the interferometer mirrors in this TT gauge.

First,

From the gauge condition, the Christoffel symbols have the following property:

This property holds. Considering this property, if the velocity of the mirror in this coordinate system was 0 before the gravitational wave arrived, the geodesic equation becomes

and it is guaranteed that the mirror's velocity remains 0.

Therefore, gravitational waves that pass through do not impart energy to interferometer mirrors that are stationary in TT gauge coordinates.

However, the physical distance of the laser light path reflecting between the two mirrors changes as follows while the gravitational wave is passing through.

For this reason, changes also occur in the interference fringes created by two lasers coming from different orthogonal paths, which means that gravitational waves can actually be observed.

However, if the interferometer mirrors were oscillating before the gravitational wave arrived, they would exchange energy with the same gravitational wave, and the mirror's velocity would change after the gravitational wave passed. It is only in the case of mirrors that were stationary in the TT gauge coordinate system that the mirror's velocity does not change.

In the analysis above, we use the point-mass approximation, where the size of the mirror is sufficiently small compared to the wavelength of the gravitational wave. Since actual mirrors have a finite size, there are vibration modes that occur due to the excitation of the mirror's internal degrees of freedom while the gravitational wave is passing through. However, quantitatively, this effect is extremely small and can be ignored in current gravitational wave observations.

Also, here we treated the stationary mirror as a perfect classical system. When considering quantum zero-point oscillations regarding the mirror's position, gravitational waves generally increase those quantum fluctuations. This is an effect where the mirror is not stationary even initially but is fluctuating quantum-mechanically. After the gravitational wave passes, an increase in the mirror's energy corresponding to those amplified quantum fluctuations also occurs, albeit very slightly. That quantum energy increment after passage is supplied from the gravitational wave through spacetime curvature. However, these quantum effects are negligibly small in current gravitational wave interferometers.

Furthermore, the analysis here adopts the approximation that the mass of the interferometer mirrors is overwhelmingly small. This holds true in actual experimental situations, but if one were to consider a hypothetically very heavy mirror, one would also need to consider the effect of the mirror itself distorting spacetime. Analysis in such cases would also involve considering the nonlinearity of the Einstein equations.