timmdeeg said:
This reminds me of Joseph Webers's cylinder, which should be squeezed and stretched as gravitational waves pass through. So, it seems that this cylinder isn't an ideal ruler. You say "the Earth can be considered as rigid as far as the gravity wave goes. The gravity wave does induce tiny stresses in the Earth, but they're insignificant." Are these "tiny stresses" comparable to those induced in said cylinder? Or put it another way, will the Earth be squeezed and stretched (but because of the noise not detectable) if a gravitational wave passes, thereby causing distance changes between fixed points on it which are comparable to those between the freely moving mirrors of the interferometer?
Well, in contrast to the cylinder, the Earth doesn't consist of a homogeneous rigid material. Does this eventually make a difference regarding the distortion?
The story of rigid objects in GR turns out to be hoarier than I thought when I wrote that post. A more careful examination of the math, which developed out of that discussion, shows that there just can't be a 3-dimensional congruence of worldlines that maintain the Born rigidity condition in a plane-wave GW space-time. So if we adopt the Born criterion for a rigid object, the mathematical solution satisfying the rigidity conditions doesn't exist. This means we can't compare the behavior of the Earth to some 3-d Born rigid model, because the 3-d Born rigid model doesn't exist.
So I'd describe the status of rigid (Born-rigid) objects in GR this way. When they exist, they are a handy, intuitive tool - but they don't always exist.
The good news is that we don't need to create a 3-d rigid congruence to have something useful. A 2-d rigid congruence will suffice. There doesn't appear to be any mathematical difficulty in creating such a 2 dimensional "rigid plane" that satisfies the Born rigidity conditions, assuming some reasonable constraints on the size of our "rigid plane". I'd have to dig to recall where I worked out & posted the necessary congruence, the Born rigidity condition can be described concisely (though abstractly and using highly specialized vocabulary) as saying that the Lie derivative of the spatial projection of the metric along the vector field that represents a congruence must vanish for the congruence to be rigid.
The Ligo detector is essentially 2 dimensional. As actually implemented it has some "thickness", but this thickness isn't critical to the operation of the detector. So the existence of a 2 dimensional Born-rigid plane is basically enough to save the intuitive picture of gravitational waves using our "rigid plane" as a reference system.
Using this rigid plane then, we can say that the lengths of rulers in our rigid plane don't change when a GW passes. What happens is that the freely floating test masses move relative to our rigid plane.
Now is a good time to add the following caveat. We noted that our 2-d "rigid plane" wasn't infinite. The size constraints are necessary, as our model eventually would otherwise wind up with test masses exceeding the speed of light relative to our "rigid plane". This would be nonsensical, so we need auxillary assumption that the velocity of the test masses relative to the "rigid plane" is small enough not to cause significant Lorentz contraction.
Trying to estimate the strains in the Earth itself, as a 3-d object, due to the passage of a gravitational wave becomes hard, but - we don't need to know how the Earth actually deforms. The design of Ligo makes the freely floating test masses float freely, one of the demanidng technical achivements of Ligo was to isolate the motion of the test masses from the motion of their environment to the necessary extent. Thus for our easy-to-imagine mental picture, it's sufficient to know how the masses move relative to our idealized "rigid plane".
This leaves open the question of how to deal with situations in which rigid planes don't exist. In the gravitational wave (GW) example, the idealized plane wave is highly symmetrical, it has enough symmetry for our "rigid plane" to exist. This won't always be the case.
The answer to this issue gets a bit complicated - basically it revolves around noting the fact that curvature tensor has only a second-order effect on distances. It causes geodesics (for example) to accelerate away from each other - a second order effect - but not to move away from each other - a first order effect.
When we consider a small enough object, the second order effects basically don't matter, and we can apply our intuition about distances for small things. For a large object, the only good treatment I'm aware of is to do the math.
