[math]\Box = \partial^{\mu}\partial_{\mu}[/math]

[math]\Box = \partial^{\mu} \partial_{\mu} + i (\sigma \partial^{\mu} \Gamma_{\mu})[/math]

[math]\Box \phi = \partial^{\mu} \partial_{\mu}\phi + i (\sigma \partial^{\mu} \Gamma_{\mu} \phi) = \rho[/math]

Nordstrom derived a simpler version, but he argued the equation says matter depends on the gravitational field - but it is also inversely true that the gravitational field (curved spacetime) depends on the density of matter. So the relationship can be interpreted a few different ways.

[math]\mathbf{G}_{\mu \nu} = \frac{8 \pi G}{c^4} \mathbf{T}_{\mu \nu} = \partial_{\mu} \cdot \mathbf{D}_{\nu} + i \sigma \cdot (\Gamma^{\mu} \times \mathbf{D}_{\nu})[/math]

[math]\mathbf{F} = \mathbf{A}^{\mu \nu} \mathbf{T}_{\mu \nu} = \mathbf{A}^{\mu \nu}[\partial_{\mu} \cdot \mathbf{D}_{\nu} + i \sigma \cdot (\Gamma_{\mu} \times \mathbf{D}_{\nu})][/math]

because

[math]\Box \phi = \frac{Gm}{r^3}[/math]

being edited:

There is also an identity from the main equation that can be obtained, known as the shear stress: A shear stress is when a fluid possesses a motion. In classical physics, this motion is often attributed to the particles which make up the fluid. Aside from zero point energy, there cannot be any particle associated to the vacuum fundamentally - in other words, unification attempts to describe say an aether made of particles, should be forbidden by principles of relativity. It is easy to argue from relativity that thiings like gravitons should probably not exist, due to gravity being strictly a pseudo force. It's not so easy to throw away the idea though that there is some ''particular aether'' associated to quantum mechanics, especially in light of field theory which involves the creation and annihilation of particles in the quantum realm. To say motion is forbidden from relativity, may be a harsh statement, so perhaps we can state:

1. There is no [detectable] motion can be associated to the aether field

That is

2. Until we find evidence of particle creation and annihilation on scales much smaller than an atom.

There is a lot of energy out there in the vacuum, in fact physics predicts the energy scale as [math]10^{120}[/math] which is how many orders of magnitude in which exists the discrepancy: We do not measure this massive amount of energy, so where is it? Zero point energy is not observable, at least not yet - they are known as off shell particles, and in theory are treated in such a way that they are not described by Hermitian operators (which is the way to create ''observables'' in quantum theory) - off shell particles [are] virtual particles. Might it be there is no such discrepancy and dark energy is really off-shell zero point fluctuations? Why can we not measure this energy? The answer may be surprisingly simple - zero point fluctuations do not generally live long enough to interact with real matter in the vacuum so the presence of this energy is completely shielded from our experimental prowess.

In the case of shear stress of a vacuum and the idea of a spacetime tension, have to be gravitational features and analogues of quantum mechanical types. Viewing space like a fluid makes wonderful predicts, as our own cosmological physics is based primarily from derivations involving the ideal fluid solution which also gives rise to the continuity equation (time evolution) within the Friedmann model, something which general relativity ironically enough lacks. Some physicists have considered whether spacetime itself is a type of superfluid! When we say spacetime tension and shear stress have to be gravitational features, has a strong connection with the gravitational aether theory, in which not only is spacetime not nothing, but it predicts that the speed of light and the gravitational constant is variable. It even provides logical reasons out of the information paradox.

The speed of light being variable, could very well be true, but we mean this variability in a different sense to thinking it varies without reason. Phenomenon like the Shapiro effect suggests that frame dragging makes light move a longer distance when traveling in the opposite direction of earths spin. In a similar stance, [we know] light does not always move at light speed in a general theory of relativity, in fact it strictly states in relativity, that light moves at c in an empty vacuum (ie when gravity is sufficiently weak) so it remains to be only a special case. However, gravity can affect the speed in which a photon moves, due to the thickness of space cause by a gravitational field - this is why light has to travel that extra bit more when coupling to the curvature of some source of gravity. Now... does a photon then still move at light speed? Of course, it probably does, but relative to someone outside the system watching this happen, would suggest spacetime is the medium and the speed of light varies proportional to the field strength. In gravitational aether theory, the speed of light can only approach zero speeds, therefore light cannot be bent in such a way that there is a point of no return, concerning black holes. It might take a photon billions of years to travel from inside the system and back out again, similar to how a combination of nuclear events and gravity prevents a photon from leaving the inner core of the sun and will take roughly 40,000 years to make its escape!

As for variability in Newton's so-called ''constant'' [math]G[/math] surely this is all just pseudoscience I hear you say perhaps? Well no, the implications of a varying [math]G[/math] have been speculated upon for a while, even back to Dirac's large number hypothesis. Historical attempts to measure the value of [math]G[/math] to current day has shown remarkable discrepancies, that vary on either spectrum.

It has been argued that a black hole cannot contain a surface tension because it is ''not a thing,'' - that the area boundary of a black hole is not special in the sense it should have a tension, but this depends on the way you might view this. Certainly in analogy, you could theoretically place something on the boundary - and even though this is not about inter-molecular bonds in the way of Van der Waals forces, this does have something to do with the dynamic feature of spacetime itself. Just like placing an object with some surface on water taking extra force to remove it due to the waters surface tension, there is also a force pulling on the system that has touched the event horizon - and so additional force theoretically would be needed to ''pull it out,'' if such a thing could even be possible. I see this as an analogue to a type of spacetime tension. All we need to do, is think of spacetime, not only as a fluid, but a special fluid that varies around massive bodies, like planets, stars and black holes due to the presence of gravity.

Back to shear stress - as a I stated, shear stress arises from fluids that are in motion - the Ricci flow is the heat equation for a Riemannian manifold. It is a simple proposition, that curvature itself can flow and this remarkable feature allows a spacetime to take on fluid like features from nevertheless, a heat equation written in a form which uses gravitational physics with a usual diffusion constant. The Ricci flow of curvature should be the missing piece to explain the motion of a spacetime as if it were acting like a perfect fluid in motion. So how do we get the shear stress from the modified EFE?

[math]\mathbf{G}_{\mu \nu} = \frac{8 \pi G}{c^4} \mathbf{T}_{\mu \nu} = \partial_{\mu} \cdot \mathbf{D}_{\nu} + i \sigma \cdot (\Gamma^{\mu} \times \mathbf{D}_{\nu})[/math]

Well what we will find out is the shear stress must be related to the stress energy tensor as:

[math]\tau_{\mu \nu} = \frac{c^4}{8 \pi G} \mathbf{G}_{\mu \nu} = \mathbf{T}_{\mu \nu} = \frac{c^4}{8 \pi G}[\partial_{\mu} \cdot \mathbf{D}_{\nu} + i \sigma \cdot (\Gamma^{\mu} \times \mathbf{D}_{\nu})][/math]

This has dimensions of force over area, which is the same dimensions for the shear stress. The relationship of sheer stress to the stress energy tensor has been well-known for a while, whether in this form or not, the stress energy tensor does contain off-diagonal elements which describes the sheer stress of a system from the momentum density tensor.

The flux of relativistic mass across a surface is equivalent to the density of the i'th component of linear momentum,

[math]T^{0i} = T^{i0}[/math]

and the components

[math]T^{ik}[/math]

represents the flux of linear momentum and the remaining component after [math]T^{ii}[/math] which represents the pressure, then

[math]T^{ik}[/math]

represents the shear stress. Knowing this we can write it under standard convention:

[math]g_{ik} \tau = \frac{c^4}{8 \pi G} \mathbf{G}_{i k} = \mathbf{T}_{i k} = \frac{c^4}{8 \pi G}[\partial_{i} \cdot \mathbf{D}_{k} + i \sigma \cdot (\Gamma_{i} \times \mathbf{D}_{k})][/math]

**Edited by Dubbelosix, 27 December 2018 - 11:34 AM.**