discussion

Definition of the Theory

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.

Predefined Objects

There is a preferred Newtonian frame, with absolute space \(\mathbb{R}^3\), with flat Euclidean metric, described by three Cartesian coordinates \(\mathfrak{x}^i, 1 \le i \le 3\), and absolute time \(\mathbb{R}\) with time coordinate \(\mathfrak{x}^0 =\mathfrak{t}\).

Fields

In these coordinates the ether is described by the following fields: a positive density \(\rho(\mathfrak{x}^i,\mathfrak{t})\), a velocity \(v^i(\mathfrak{x}^i,\mathfrak{t})\), a negative-definite stress-tensor \(\sigma^{ij}(\mathfrak{x}^i,\mathfrak{t})\), and an unspecified number of "internal degrees of freedom" \(\varphi^m\), which will be identified with material fields.

The gravitational field \(g_{\mu\nu}(\mathfrak{x}^i,\mathfrak{t})\) is defined algebraically as: \[\begin{eqnarray} g^{00}\sqrt{-g} &=& \rho\\ g^{i0}\sqrt{-g} &=& \rho v^i\\ g^{ij}\sqrt{-g} &=& \rho v^i v^j + \sigma^{ij} \end{eqnarray}\]

This defines a Lorentz metric, which appears to be the effective metric measured by clocks and rulers. As a consequence of the positivity of the density \(\rho(\mathfrak{x}^i,\mathfrak{t})>0\), the absolute time \(\mathfrak{t}\) has to be a time-like coordinate.

Fixed parameters

The theory contains two additional cosmological constants \(\Xi\) and \(\Upsilon\) compared with general relativity (that means, we have also Einstein's cosmological constant \(\Lambda\) and an unspecified number of constants related with material fields, as in the related version of general relativity).

Lagrangian

The Lagrangian consists of the standard Lagrangian for the "effective metric" \(g_{\mu\nu}(\mathfrak{x}^i,\mathfrak{t})\) and the material fields \(\varphi^m(\mathfrak{x}^i,\mathfrak{t})\), and an additional part which depends on the preferred coordinates: \[ L \quad=\quad \frac{1}{8\pi G}(\Xi g^{ii} - \Upsilon g^{00})\sqrt{-g}\quad + \quad L_{GR}(g_{\mu\nu})\quad + \quad L_{matter}(g_{\mu\nu},\phi_{m}) \]

The additional term may be rewritten in a covariant form. To obtain this form, we have to handle the preferred coordinates \(\mathfrak{t}, \mathfrak{x}^i\) as additional variables: \(\mathfrak{t}(x), \mathfrak{x}^i(x)\) of some other, general spacetime coordinates x. Then, the non-covariant terms can be given a covariant form similar to that of a scalar field u(x) which would be \(g^{\mu\nu}(x)\partial_\mu u(x) \partial_\nu u(x)\sqrt{-g}\): \[ L\quad =\quad \frac{1}{8\pi G}\left(\Xi \delta_{ij}g^{\mu\nu}(x)\partial_\mu \mathfrak{x}^i(x) \partial_\nu \mathfrak{x}^j(x) - \Upsilon g^{\mu\nu}(x)\partial_\mu\mathfrak{t}(x) \partial_\nu \mathfrak{t}(x)\right)\sqrt{-g} \quad+\quad L_{GR}(g_{\mu\nu}) \quad+ \quad L_{matter}(g_{\mu\nu},\phi_{m}) \]

Field Equations

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 \(\mathfrak{t}(x)=\mathfrak{x}^0(x), \mathfrak{x}^i(x)\): \[ \square \mathfrak{x}^\nu = \frac{\partial}{\partial \mathfrak{x}^\mu} \left(g^{\mu\lambda} \sqrt{-g} \frac{\partial}{\partial \mathfrak{x}^\lambda} \mathfrak{x}^\nu\right) = \frac{\partial}{\partial \mathfrak{x}^\mu} \left(g^{\mu\nu} \sqrt{-g}\right) = 0.\]

Moreover, we obtain the Einstein equations of general relativity modulo an additional part in the energy-momentum tensor similar to that of additional scalar fields \(\mathfrak{t}(x)=\mathfrak{x}^0(x), \mathfrak{x}^i(x)\) without any interaction with other types of matter.

Thus, we obtain some special type of homogeneously distributed dark matter. For the choice \(\Xi>0, \Upsilon<0\) this additional "dark matter" is similar to a massless scalar dark matter field.

But one should not forget that there exists an important distinction between a usual scalar field and a preferred coordinate handled like a field: A usual field is nontrivial only in some small region, and becomes close to zero outside, while a coordinate becomes infinite in infinity. So, if one splits the variable preferred coordinate into some fixed coordinate and some small variable part, \(\mathfrak{x}^\mu(x) = \mathfrak{x}^\mu_0 + \delta\mathfrak{x}^\mu(x)\), then the fixed part gives some sort of cosmological term, while the variable part simply behaves like a scalar field, which has, additionally, to remain small so that the sum defines a valid system of coordinates.

Absolute Time and Proper Time

The equations for the matter fields are exactly the same as those of GR, because the matter Lagrangian is exactly the same. Thus, we have the usual equations known from GR: \[ \nabla_\mu T^{\mu\nu} = 0.\]

As a consequence, clocks and rulers which consist of matter behave exactly in the same way as in GR. That means, what clocks measure is GR proper time \[ \tau = \int_\gamma \sqrt{g_{\mu\nu}(\gamma(\mathfrak{t}))\frac{d \gamma^\mu(\mathfrak{t})}{d\mathfrak{t}} \frac{d \gamma^\nu(\mathfrak{t})}{d\mathfrak{t}}} d\mathfrak{t}.\]

All what changes for clocks and rulers mathematically is that the gravitational field they measure is slightly modified by the additional terms in the equations for the gravitational field. What changes radically is only the interpretation of the measurement results. What is named "proper time" in GR is not interpreted as some "proper time" in the ether interpretation - proper, true time would be, of course, absolute time \(\mathfrak{t}(x)\). But this is not what is measured by clocks. Thus, clocks measure, instead of true time, some distorted time, which depends on the particular velocity \(\frac{d \gamma^i(\mathfrak{t})}{d \mathfrak{t}}\) of the clock as well as the gravitational field \(g_{\mu\nu}(\gamma(\mathfrak{t}))\) at the actual position of the clock.

Completeness of the solution

The solution is complete, if it is defined for all values of values of the preferred coordinates, \(-\infty < \mathfrak{x}^i, \mathfrak{t} < \infty\).

On the other hand, geodesic completeness of the pseudo-Riemannian metric \(g_{\mu\nu}(\mathfrak{x}^i, \mathfrak{t})\) is not required. This is a consequence of the interpretation: One has to care only about the completeness in terms of true space and time. Once "proper time" is only a distorted measure of time, without any fundamental importance, it makes no sense to require geodesic completeness for the distorted clock showings. If, for whatever reasons, the clock is distorted so much that it effectively stops, so be it - that means, we have a clock in a frozen state.

Causality condition

The ether density \(\rho\) should be, as a density, always positive. This is equivalent to the requirement that \(\mathfrak{t}\) is time-like.

Domain of applicability

The ether interpretation suggests a strong analogy between usual condensed matter and the ether. We use this analogy to make a prediction about the boundaries of applicability of the equations.

The first hypothesis is that there is a microscopic ether theory which replaces GLET for small distances. The second hypothesis 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: \[ \rho(\mathfrak{x}^i, \mathfrak{t}) V_{cutoff} = \text{const.}\]