Essentially, there is no modern understanding of ether theory. There is the classical Lorentz ether, where the metric is only the Minkowski metric. Here the [math]\eta^{0i}=0[/math] so that the velocity is zero and the density [math]\rho=g^{00}\sqrt{-g}[/math] is constant. And there is the Leyden lecture, which is nothing than a popular lecture, which has been essentially ignored by the mainstream which made "ether" a bad word. All this essentially a century old. There are ether freaks who don't even understand SR, unpublishable for good reasons. There is Jacobson's "Einstein ether" which is based on completely different concepts. What else?

Just a small question how are the dimensions being constructed on the right hand side here;

v_i(x, t) = g_[oi](x, t) / g_[oo](x, t)

I know the definition 9f the metric terms, so this is just a general question about the dimensions alone. Thanks in advance.

Usually I follow the c=1 convention, so that a velocity becomes a dimensionless number (simply v/c). If one would have to handle it with c, then one would have to include the appropriate factors of c to everything. In order to avoid giving the different components of [math]g^{\mu\nu}[/math] different dimensions, the appropriate choice would be to use the [math]x^0 = ct[/math] so that all the coordinates [math]x^\mu[/math] have the same dimension of length. Then [math]v_i(x, t)/c = g^{0i}(x, t) / g^{oo}(x, t)[/math] would be the correct dimensionless expression.

BTW, the components are the ones with upper indices, the coefficients used in the wave equation [math]\square = \partial_\mu g^{\mu\nu}\sqrt{-g} \partial_\nu[/math] and the harmonic condition [math]\square x^\nu = \partial_\mu g^{\mu\nu}\sqrt{-g} = 0[/math]. These have to be identified with the continuity and Euler equations of condensed matter theory, applied to the ether:

[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]

in such a way as to get the appropriate factors of c into it.

**Edited by Schmelzer, 12 November 2019 - 01:41 AM.**