The stress tensor of accelerations is
T = mu E + lambda div g (1)
where mu and lambda are the Lame coefficients (mu measures stress and lambda compression) and
E = grad^S g = (1/2)(grad g + g grad) (2)
is the strain and g is the acceleration vector. Under the equilibrium condition, div T =0,
g= w^2 r - m/r^2, (3)
where w is the angular speed, and m, the gravitational parameter, GM.
When the equilibrium condition,
div T = mu div grad g + (mu+lambda) grad (div g) , (4)
is combined with Newton's second law, we would come out the the wave equation
rho d^2 g/dt^2 = div T=(lambda+2mu){g"+2g'/r-2g/r^2, (5)
if the Lame parameters are constant, and where rho is the mass density. It is easy to see that the time independent solutions to (5) are given by (3). This means that Newton's law applies, as well as Hooke's, iff the gravitational acceleration is time-independent.
If we take the divergence of both sides of Eqn (5), the dilatation div g = D, satisfies
d^2 D/dt^2= v_p^2 D (6)
where the phase velocity, v_p={(2mu+lambda)/rho}^1/2. The subscript "p" stands for primus since these waves are the first to arrive after an earthquake has occurred. They are compressive, and involve a change in volume. of the material as the wave passes through.
Although it is claimed that gravitational waves are transverse, they, nevertheless cause changes in a volume element as they pass through according to
dV≈(1−16 R_(jk) x_jx_k+O(x^3))dV_f
where R_(jk) is the Ricci tensor and V_f is the initial volume element. It is also claimed that the Ricci tensor measures volume changes whereas shape changes are accounted for by the Weyl tensor. They confuse polarizations modes with volume changes.
According to the relevant wikipedia article:
As a gravitational wave passes an observer, that observer will find spacetime distorted by the effects of strain. Distances between objects increase and decrease rhythmically as the wave passes, at a frequency equal to that of the wave. The magnitude of this effect is inversely proportional to the distance from the source.
Interpretations of the LIGO interferometry experiments claim that [Profolus 2/1/21]
Based on the concept of interferometry, these lasers measure the changes in the size of a plane or a flat two-dimensional surface. Take note that this plane is just a simple representation of spacetime fabric. Gravitational waves stretch and squeeze the plane, causing it to expand or shrink in size. Lasers can accurately measure this expansion or shrinkage in the plane using the speed of light as a reference. An expanded plane would make light travel longer from one side to another as opposed to a shrunk plane.
If this is not the description of a compressive-rarefaction wave, nothing is!
If gravitational waves travel at the speed of light and are polarizable, just like light, we can take the curl of (5) to get
rho d^2 C/dt^2 = mu div grad C, (6)
where C=curl g. Equation (6) contains only the rigidity parameter, mu,
since the compressional part vanishes because curl grad =0. The wave velocity is
v_s=(mu/rho)^(1/2),
where s stands for secondary in reference to p- and s- waves in earthquakes. The s- waves are transverse that involve no displacements in the direction of the motion.
The two types of waves can be distinguished insofar as for p- waves we require curl g =0, while for s- waves, the dilatation vanishes, div g =0. Thus, for s- waves
div T = - mu curl curl g = -2 mu G,
where G is the Einstein tensor. Introducing Newton's second law as in (5) we get
(1/v_s^2)d^2/dt^2 g = G. (7)
Equation (7) is identical to the wave equation since div g = 0. In electromagnetism, the shearing is caused by the interaction of the electric and magnetic fields. In gravity, there is nothing comparable to curl g. The fact that the presence of the magnetic field invalidates the property that the electric field is a gradient. This appears explicitly by the appearance of the time rate of change of the vector potential in the expression for the electric field.
The presence of a magnetic-like field is due to the linearized Einstein tensor, G. However, its vanishing is none other that the Saint Venant compatibility condition for the existence of a strain interpretation of the gravitational acceleration field which steps in for the displacement field of elasticity theory.
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