Could the LIGO design be also used to detect spacetime distortions (curvature) caused by a small gold sphere?

Question

Suppose you put a gold sphere inside the LIGO interferometer, not in the path of the laser beams, but sufficiently close to  (in the vicinity of) one of the laser beams. Would the spacetime distortion caused by the gold sphere be detectable by the change in the interference patterns of the LIGO detector, or would it be too weak for detection? If the effect is detectable, this would be an experimental way to study the gravitational field of small objects. Some ballpark estimation of the magnitude of the effect would be appreciated.

Answer 1

LIGO doesn't detect gravitational fields, it detects curvature. Specifically, it detects the type of spacetime curvature associated with tidal forces. (This is why each detector has two arms.) Also, LIGO doesn't detect DC effects, it only detects AC signals within a certain range of frequencies. The most difficult thing about the LIGO experiment is that it also serves as the world's most sensitive vibration sensor. This is why they have two (now 3, IIRC?) facilities far apart. They look for correlated signals between the different facilities.

So if you move a dense object close to one of the detectors, as in the Cavendish experiment, what will happen is that that detector will be momentarily disabled by the vibration, just as it would if a truck drove by and stopped suddenly. Once the detector recovered from the insult, there would be no AC signal, probably no signal that would mimic the kind of tidal distortions it's looking for, and no correlation with the other site(s). So you would get no signal.

Answer 2

A static sphere of gold does not produce gravitational waves. Since LIGO is set up to detect perturbations in the arm lengths at frequencies of 30-3000 Hz then the presence of just another mass near the beam is just another (very) small DC offset that would not be detected.

Still, it is of interest to ask whether introducing a mass close to the beam would cause a change that needed any adjustments. I suppose you could model the perturbation as the Shapiro delay close to a spherically symmetric mass.

The round-trip delay for light travelling close to (grazing) a sphere next to the beam path would be $$\Delta t \simeq \frac{4GM}{c^3} \left(\ln\left[\frac{x_1 + (x_1^2 + r^2)^{1/2}}{-x_2 + (x_2^2 + r^2)^{1/2}} \right] - \frac{1}{2}\left[\frac{x_1}{(x_1^2 + r^2)^{1/2}} + \frac{x_2}{(x_2^2 + r^2)^{1/2}}\right]\right)\ , $$ where $r$ is the radius of the sphere and $x_1+x_2$ is the LIGO arm length of 4 km. If the sphere is halfway between the arm mirrors, such that $x_1=x_2$ and assuming $r\ll 4$ km, then $$\Delta t \simeq \frac{4GM}{c^3} \left(\ln\left[\frac{2 + r^2/2x^2}{r^2/2x^2} \right] - \left[\frac{1}{(1 + r^2/2x^2)}\right]\right)\ , $$

$$\Delta t \simeq \frac{4GM}{c^3} \left(\ln\left[\frac{4x^2}{r^2} \right] - 1 \right)\ , $$ where $x=2$ km.

If we let $r=1$ m (a lot of gold, surely lead would do!) then $M \simeq 8\times 10^4$ kg and $\Delta t \simeq 3\times 10^{-30}$ s, or an equivalent length change of $10^{-21}$ m.

This is many orders of magnitude smaller than the $\sim 10^{-18}$-m length changes that LIGO is sensitive to when they happen at frequencies of $30-1000$ Hz. It is also many more magnitudes smaller than the imposed offset in path length that takes the interferometer slightly off the dark fringe in normal operating conditions. I would conclude therefore that this offset is unobservable even under ideal conditions. Lower frequency changes are in any case obliterated by seismic noise, gravity gradient noise and various noisy instrumental control loops by many orders of magnitude more.