Can I tempt you at all to revise this velocity dispute? You can preserve a type of motion to the metric asWhile special relativity has to different interpretations - the original one by Lorentz and Poincare known as the "Lorentz ether" and the "spacetime interpretation" proposed by Minkowski - for GR, only one interpretation is widely known, the spacetime interpretation.

The non-existence of a generalization of the Lorentz ether interpretation to gravity is certainly a decisive argument against the Lorentz ether. In such a situation, one should not wonder that the mainstream follows the spacetime interpretation and rejects the Lorentz ether.

But this argument no longer holds. There exists a surprisingly simple and beautiful Lorentz ether interpretation of the Einstein equations of GR in harmonic coordinates. Let's see. What is the most popular coordinate condition used in GR? It is the harmonic condition. It simplifies the Einstein equations in an essential way. What are these conditions? They look like conservation laws, or, alternatively, like a wave equation for the preferred coordinates [math]X^{\nu}[/math]:

[math]\partial_\mu (g^{\mu\nu} \sqrt{-g} ) = 0 \text{ resp. } \square X^{\nu} = 0 [/math]

What are the most important classical equations for condensed matter? They are the continuity and the Euler equations:

[math]\partial_t \rho + \partial_i (\rho v^i) = 0.[/math]

[math]\partial_t (\rho v^j) + \partial_i(\rho v^i v^j - \sigma^{ij}) = 0.[/math]

Compare them and there will be a straightforward Lorentz ether fulfilling continuity and Euler equations as reasonable for an ether:

[math] \rho = g^{00}\sqrt{-g}, \quad \rho v^i = g^{0i}\sqrt{-g},\quad \rho v^i v^j - \sigma^{ij} = g^{ij}\sqrt{-g}.[/math]

Let's note an especially interesting property of this interpretation: The condition [math]\rho>0[/math] translates into the preferred time coordinate [math]T=X^0[/math] being really a time-like coordinate. What does this mean for solutions of the Einstein equations with causal loops? It means that they don't allow for such an ether interpretation. The Einstein equations interpreted as equations for the Lorentz ether would, of course, become invalid and meaningless if the ether density becomes zero. This would be the boundary of the ether, the boundary conditions have not been defined, so, the boundary is simply not covered by the Einstein equations.

Solutions of the Einstein equations which are also valid solutions of the Lorentz ether have an ether density greater zero everywhere, thus, they have a global time-like coordinate as the preferred time coordinate.

For more details see https:ilja-schmelzer.de/ether

[math] \partial_t g_{ij}= - 2R_{ij} [/math]

In this, geometry can flow but we haven't assigned a velocity because motion in the aether is forbidden from first principles.

**Edited by Dubbelosix, 04 November 2019 - 08:37 AM.**