Note: the definition given here is a formulation appropriate for comparison with GR and experiment. It does not mean that we introduce the equations, terms of the Lagrangian, global restrictions etc. given here as ad hoc axioms. The derivation is given at another place.
There is a preferred Newtonian frame with a flat Euclidean metric in spatial slices. It is described by orthonormal coordinates T, Xi, 1 ≤ i ≤ 3.
In these coordinates the ether is described by the following fields: density ρ(X,T), average velocity vi(X,T), stress-tensor σij(X,T), and an unspecifeid number of "internal degrees of freedom" φm, which will be identified later with material fields.
The physical metric gmn(X,T) is defined algebraically as:
|gij(-g)1⁄2||=||ρvivj - σij|
Two additional cosmological constants Ξ and Υ compared with general relativity (that means, we have also Einstein's cosmological constant Λ and an unspecified number of constants related with material fields, as in the related version of general relativity).
The Lagrangian consists of the standard Lagrangian for the "effective metric" gμν(x) and the material fields φm(x), and an additional part which depends on the preferred coordinates:
L = -(8πG)-1(Υg00-Ξgii) (-g)1/2 + LGR(gμν(x),φm(x))
The additional term may be rewritten in a "weak" covariant form — we consider the preferred coordinates as additional variables: T(x), Xi(x):
L = -(8πG)-1(ΥgμνT,μT,ν - ΞδijgμνXi,μXj,ν) (-g)1/2 + LGR(gμν(x),φm(x))
The field equations are the Euler-Lagrange equations of this Lagrangian. For the derivation it is useful to handle the preferred coordinates like additional matter fields. This leads immediately to the harmonic equation for the preferred coordinates Xμ = (T,Xi):
∂⁄∂Xμ gμλ(-g)1⁄2 ∂⁄∂Xλ Xν = ∂⁄∂Xμ (gμν(-g)1⁄2) = 0
Moreover, we obtain the Einstein equations of general relativity modulo an additional part in the energy-momentum tensor similar to additional scalar fields T(x), Xi(x) without any interaction with other types of matter.
Thus, we obtain some special type of homogeneously distributed dark matter.
The solution is complete, if it is defined for all values of T, Xi. The ether density ρ should be always positive. As a consequence, T should be time-like.
On the other hand, completeness of the physical metric gμν(x) is not required.
The ether paradigm suggests a strong analogy between usual condensed matter and the ether. We use this analogy to make a prediction about the boundaries of the continuous approximation.
The first prediction is that there is a microscopic ether theory which replaces GLET for small distances. The second prediction is that the ether density is really something like the number of microscopic ether particles per volume. In this case, we have only one unknown number — the Avogadro number of the ether. Modulo this number we can predict the critical volume in dependence from ether density:
ρ(x) Vcutoff = const.
Clocks which consist of matter show proper time. This is a known consequence of the Einstein equations, independent of the cosmological effects which may be caused by the additional cosmological terms.