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# Can Einstein Field Equations Be Confused with Constitutive Relations?

In his 2018 Hamilton Lecture, Exploring the universe with gravitational waves, Thorne raised the question of what is the Young modulus (~bulk modulus) of spacetime. Weiss supposedly came up with a Young modulus of spacetime of the order Y~c^2f^2/G, where c is the speed of light, f the frequency of the GW wave and G is Newton's gravitational constant. The corresponding energy density of the GW is u~Y h^2, where h denotes strain. "Strain" of what might be an embarrassing question though.

This definitely implies a localization of energy, something Einstein and all his followers could not account for. Y has dimensions of pressure, force F per unit area A,

F/A=Y,

and the claim is that Y=(cf)^2/G. This would apparently come from Einstein's field equations

G_{ij}-(1/2)g_{ij}G=-(8 pi f^2/Y^2 c^2 T_{ij}. (1)

So now they have confused a generalized Poisson equation with a constitutive relation! The Einstein tensor, G_{ij}, is not a strain, since it has dimensions, i.e. 1/d^2. And there is no regime where the Young modulus is independent frequency, thereby contradicting any hope of obtaining a Hookean law in an asymptotic limit. And why should the propagation speed,

c=(Y/rho)^1/2,

be the speed of light in which rho is the corresponding mass density. Density of what?

We have stated that the Young modulus is a force per unit area. As such, it is a pressure acting on a surface. Where is this surface?

How can you talk about GWs carrying energy, when energy is not defined in GR? For example, the observed change in the orbital periods of binary pulsars is due, so we have been hammered with, their orbital energy being carried off in the form of GWs. How can this be if the gravitational field has no energy-momentum.

The energy stress tensor in Einstein's equations (1) contain contributions from matter and electromagnetic fields--but not gravitational energy. In the presence of GWs, it is not the divergence of T_{ij} that must be zero, but the divergence of t_{ij}+T_{ij} that must vanish, where t_{ij} is the pseudo-tensor that can be made to vanish by a coordinate transform. If this term is added to the right-hand side of (1), it would make the equation completely meaningless (if it weren't already) because you would be equating bananas with oranges. On the left-hand side you would have a tensor, which on the right-hand side you would have a mixture of tensor and non-tensor.

To save what is salvageable, they contend that what is carried away by GW fields is a nonlocal measure of energy that would require regions of spacetime much greater than the wavelength of a GW wave. But, that wouldn't allow us to define something that we could stick into Einstein's equations. In fact, the gravitational field is zero locally so that anything we could imagine that would represent a gravitational field would also be zero, or it could be put equal to zero, as in the case of the Einstein pseudo tensor, by a coordinate change. The whole idea of Einstein's field equations as describing GW waves and radiation is a complete sham.

Since there is no energy-momentum tensor of the gravitational field, anything that could be rung out of Einstein's field equations cannot be associated with GWs. To say that GWs carry energy and momentum is sheer ludicrousness.

Maxell's stress tensor establishes a balance between electric field lines in the direction of the field and a pressure normal to them at the bounding surface. Since electric field lines are not subjugated to compression or expansion, no longitudinal perturbations can propagate. The divergence of the electric field vanishes far from its source. That is why radio waves, for example, are independent of the oscillating dipoles that produce it.

Since there is no gravitational energy-stress tensor, we cannot talk about a balance between off diagonal shear stress components and diagonal pressure components as we can in the Maxwell stress tensor. There, the time rate of change of the momentum of matter and that of the EM field is equal to a surface integral of the stress tensor,

(d/dt) Int (volume) (p+g) dV= Int (surface) dS T,

where p is the density of momentum of matter and g is the density of the momentum of the EM field. The right hand side is a surface integral over S enclosing the volume V. This represents the pressure which balances the inner stresses. Where is this bounding surface in Thorne's supposed analogy with Young's modulus?

Even if it could be done, there would be nothing on the left-hand side of the equation because the energy-stress tensor of the gravitational field is zero!