discussion

Derivation of the Theory

The equations of GLET do not have to be proposed ad hoc. Instead, we can derive them starting with a few number of simple assumptions about the ether. As a consequence, relativistic symmetry — the Einstein Equivalence Principle (EEP) — is explained in GLET.

General degrees of freedom of an ether

The basic idea is that the "ether" is something similar to condensed matter. The question is how to transform this into axioms. Of course, the main difference between classical "ether theory" and relativity is the different concept of space and time. And the fact that our theory recovers the classical notions of space and time was one of the main reasons to name it an ether theory. Thus, we have the following

Postulate 1 (Newtonian background): We have a flat Euclidean space and absolute time.

This postulate is sufficiently simple. But we can, nonetheless, provide some justification even for this postulate: The argument for the existence of a predifined background comes from quantum gravity. Because quantum gravity is not accessible experimentally, all we can do is to consider thought experiments. But such a thought experiment allows to show that in a superpositional state of two quasiclassical gravitational fields one can observe if a particle is, in above gravitational fields, at the same position or not. But the notion of "the same position" for different metrics is completely undefined by the equations of GR — a consequence of the hole paradox of GR. To define it, one needs an additional equation which allows to connect the coordinates of different solutions. This is what the Newtonian background allows to do.

For the point that this background spacetime splits into absolute space and absolute time there is another strong argument — the violation of Bell's inequality.

For the description of the ether we use degrees of freedom which are usual in condensed matter theory:

Postulate 2 (Condensed matter variables): The ether is described by the following variables: 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 other "internal degrees of freedom" \(\varphi^m(\mathfrak{x}^i,\mathfrak{t})\), which will be identified with material fields.

It immediately follows that the field \(\rho(\mathfrak{x}^i,\mathfrak{t})\) should be positive everywhere. For the "stress tensor", we introduce another axiom:

Postulate 3 (Definiteness of pressure): The pressure or stress tensor \(\sigma^{ij}(\mathfrak{x}^i,\mathfrak{t})\) is negative definite.

Note: In classical condensed matter theory pressure is defined only modulo a constant. A microscopic theory allows to define this constant, so that in a gas pressure is always positive, and for a solid we have pressure zero in the stress-free reference state. But we have no microscopic model here, and our "pressure" may differ from usual pressure by a large constant. Thus, the unconventional negative-definiteness does not seem to be a real restriction.

The gravitational field

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

This formula is meaningful only in the preferred coordinates.

The positivity of the density, together with the negative definiteness of the stress tensor, explains the signature (1,3) of the gravitational field.

The matter fields

As GR, GLET does not specify type and number of matter fields, but defines only a general scheme how matter fields have to be incorporated. The assumption we make about matter fields is the following:

Postulate 4 (Universality of the ether): There is nothing except the ether.

Thus, in GLET all matter fields should be incorporated as inner variables or material properties of the ether. In this sense, the ether unifies all matter fields and gravity. GLET is therefore not a complete ether theory, only a general scheme for various different ether theories. The complete ether theory has to be a theory of everything. GLET defines only a few general properties.

The hypothesis that matter fields are inner degrees of freedom of an ether seems to be the most important physical axiom of GLET.

Conservation laws

Until now, the universality axiom looks very uncertain. Let's replace it now by an certain, mathematical formula. As usual in condensed matter theory we have conservation laws — a continuity equation for ether density which describes the conservation of "ether mass", and Newton's second law, which describes the conservation of momentum:

Postulate 4a (Conservation laws): The ether is conserved. We have the classical continuity equation as well as momentum conservation.

Now, looking at these conservation laws, we can distinguish external and internal degrees of freedom. Indeed, external degrees of freedom have separate energy and momentum densities, they exchange energy and momentum with the ether. Instead, energy and momentum densities related with internal degrees of freedom are simply part of the whole energy and momentum density of the ether. Thus, the conservation laws depend on the internal degrees of freedom only in an indirect way, by their influence on the fields \(\rho, v^i, \sigma^{ij}\).

Now the uncertain, verbal description given by the universality axiom we can replace by the following assumption:

Postulate 4b (Universality of the ether): The conservation laws do not contain any additional terms for interaction with external matter.

This postulate reduces the conservation laws to its standard form, the continuity and Euler equations for the ether alone: \[\begin{eqnarray} \frac{\partial}{\partial \mathfrak{t}} \rho + \frac{\partial}{\partial \mathfrak{x}^i} (\rho v^i) &=& 0,\\ \frac{\partial}{\partial \mathfrak{t}} (\rho v^j) + \frac{\partial}{\partial \mathfrak{x}^i} (\rho v^i v^j + \sigma^{ij}) &=& 0. \end{eqnarray}\]

Existence of a Lagrange formalism

Now we require that the equations are Euler-Lagrange equations. This is another non-trivial physical assumption. We have to note that this assumption is not very common in condensed matter theory - the equations are derived from more fundamental theories, not from a Lagrange mechanism.

Postulate 5 (Lagrange formalism) There exists a Lagrange density so that all equations are Euler-Lagrange equations for this Lagrange density.

Weak Covariant Formalism

It is well-known that a covariant formulation is not a special property of GR (as initially believed by Einstein), but exists for every physically meaningful theory. For example, a covariant formulation for SR has been given by Fock. Especially for comparison with GR, but also for simplicity of the following derivation, it is useful to use such a covariant formulation.

One way to obtain such a covariant formulation is a quite simple trick. We formally consider the preferred coordinates as additional fields \(\mathfrak{x}^i(x),\mathfrak{t}(x)=\mathfrak{x}^0(x)\), where the index i enumerates the "fields" \(\mathfrak{x}^i(x)\) and is no longer a spatial index. All what depends on the preferred coordinates, now depends on these fields. For example, the non-covariant term \(F^0\) becomes \(F^\mu \partial_\mu \mathfrak{t}(x)\).

In this formalism, initially covariant terms remain unchanged and do not depend on \(\mathfrak{x}^i(x)\) and \(\mathfrak{t}(x)\). Such terms we name strong covariant, to distiguish them from weak covariant terms — initially non-covariant terms, which now "look covariant", if we forget that the "fields" \(\mathfrak{x}^i(x)\) and \(\mathfrak{t}(x)\) have a special geometric meaning as preferred coordinates.

Applying this formal method to our Lagrangian L, we obtain a weak covariant Lagrangian.

Covariant Conservation Laws

But before we continue with the consideration of the Lagrangian, let's look at the conservation laws. We have already defined the symmetric tensor field \(g_{\mu\nu}(\mathfrak{x}^i,\mathfrak{t})\): \[\begin{eqnarray} g^{00}\sqrt{-g} &=& \rho\\ g^{i0}\sqrt{-g} &=& \rho v^i\\ g^{ij}\sqrt{-g} &=& \rho v^i v^j + \sigma^{ij} \end{eqnarray}\]

This formula itself is meaningful only in the preferred coordinates. But, together with these coordinates \(\mathfrak{x}^\mu(x)\) interpreted as "fields", we can use \(g_{\mu\nu}(x)\) to define the state of the ether uniquely. That means, we can describe the ether state by a Lorentz metric \(g_{\mu\nu}(x)\) together with four functions \(\mathfrak{x}^\mu(x)\) as well as the "matter fields" \(\varphi^m(x)\) which describe the other inner degrees of freedom of the ether.

Let's now rewrite the conservation laws in the new variables. The continuity and Euler equations give immediately an equation which is known as the harmonic condition for the metric \(g_{\mu\nu}(\mathfrak{x}^\mu)\): \[ \frac{\partial}{\partial \mathfrak{x}^\mu} (g^{\mu\nu} \sqrt{-g}) = 0.\] Remarkably, this equation for the metric \(g_{\mu\nu}(\mathfrak{x}^\mu)\) in the preferred coordinates is also equivalent to an equation for the preferred coordinates \(\mathfrak{x}^\mu(x)\) themself: \[ \square \,\mathfrak{x}^\nu = \left(\frac{\partial}{\partial \mathfrak{x}^\mu} g^{\mu\lambda} \sqrt{-g} \frac{\partial}{\partial \mathfrak{x}^\lambda}\right) \mathfrak{x}^\nu(\mathfrak{x}) = 0.\]

And the operator \(\square\) is simply the invariant Laplace operator of the metric \(g_{\mu\nu}(x)\). Thus, the equation for the preferred coordinates has already a covariant form.

Connection between Lagrangian and Conservation Laws

Now, we have a beautiful covariant formulation of the conservation laws, and we have assumed that we have a weak covariant Lagrangian. How are they connected?

The connection between Lagrange formalism and conservation laws is well-known as Noether's theorem. But it does not work as usual in our covariant formalism. If the Lagrangian is covariant, Noether's theorem does not give non-trivial conservation laws - the well-known problem with conservation laws in general relativity. But, on the other hand, we use only a slightly different formalism, the physics are classical physics on a Newtonian background. The conservation laws must be somewhere. Where are they hidden?

The answer is simple — in this formalism, the classical conservation laws are simply the Euler-Lagrange equations for the preferred coordinates \(\mathfrak{x}^\mu(x)\). The proof is even simpler in comparison with Noether's theorem — if the Lagrangian does not depend on the \(\mathfrak{x}^\mu(x)\) themself, we obtain immediately an equation of the form of a conservation law.

Postulate 6 (Conservation laws and Lagrangian) The conservation laws are the Euler-Lagrange equations for the preferred coordinates \(\mathfrak{x}^\mu(x)\).

Derivation of the general GLET Lagrangian

Now, we have all we need to derive the general Lagrangian:

What remains is surprisingly simple. To find a particular covariant Lagrangian which gives the harmonic equation for a scalar function is a triviality — we can use here the standard Lagrangian for scalar fields. Of course, there is some freedom of choice in form of a constant for each of the four scalar "fields" \(\mathfrak{x}^\mu(x)\), which gives \[L_0 = \Xi_i g^{\mu\nu}(x) \sqrt{-g}\partial_\mu \mathfrak{x}^i \partial_\nu \mathfrak{x}^i.\] But, once the factors \(\Xi_i\) have been chosen appropriately, one can consider the difference between the most general Lagrangian L and our particular choice \(L_0\). And this difference should no longer give any modification of the equation for the preferred coordinate — thus, it should not depend on the preferred coordinates, thus, should be covariant in the strong sense.

But the most general covariant Lagrangian is, essentially (if one restricts oneself to the lowest order term) the Lagrangian of general relativity. Thus, we have now: \[ L - L_0 = L_{GR}(g_{\mu\nu}) + L_{matter}(g_{\mu\nu},\varphi^m)\] where the right-hand side is covariant in the usual sense, that means, does not depend on the preferred coordinates \(\mathfrak{x}^\mu(x)\).

For the final step, we can use some hypothesis about the isotropy of the background space and introduce a conventional factor to justify the choice \(\frac{1}{8\pi G}\Xi = \Xi_1=\Xi_2=\Xi_3,\frac{1}{8\pi G}\Upsilon = \Xi_0\) to obtain \[ L = \frac{1}{8\pi G}(\Xi g^{ii} - \Upsilon g^{00})\sqrt{-g}+ L_{GR}(g_{\mu\nu}) + L_{matter}(g_{\mu\nu},\phi_{m}).\]

Einstein Equivalence Principle

The GLET Lagrangian is a Lagrangian of a metric theory of gravity: The matter Lagrangian fulfils the same requirements as the matter Lagrangian of GR — he has to be covariant. That means, for the matter Lagrangian the Einstein equivalence principle holds as well as in general relativity.

Given the fundamental importance of the EEP, it seems useful to understand what leads to the EEP in our derivation. Essentially it is the "action equals reaction" principle, which is a consequence of the Lagrange formalism. One can understand the "action equals reaction" principle as a rule that variational derivatives can be exchanged: \[ \frac{\delta}{\delta u} \frac{\delta}{\delta v} S(u,v) = \frac{\delta}{\delta v}\frac{\delta}{\delta u} S(u,v)\]

Here, \(\frac{\delta}{\delta v} S(u,v)\) is the Euler-Langrange equation for v, and if this equation depends on u, this is expressed by \(\frac{\delta}{\delta u}\) of the equation for v being nontrivial. So, the left hand side defines the "action" of u on v. Correspondingly, the right-handed side defines the action of v on u.

Now, the equations for the preferred coordinates are the conservation laws, thus, continuity and Euler equations, thus, depend only on density, velocity and stress tensor of the ether, which define the gravitational field, and not of any other inner degrees of freedom of the ether, which define the matter fields. Thus, there is no "action" of the matter fields on the preferred coordinates, and, therefore, there should be no "reaction", thus, the equations for the matter fields should not depend on the preferred coordinates. But this consequence is the EEP.

Conclusion

Thus, we have derived the general Lagrangian of GLET. The assumptions we have used are:

  1. a Newtonian framework of absolute space and time;
  2. the hypothesis that the ether is similar to classical condensed matter, thus, may be described with classical condensed matter variables (density, velocity, stress tensor) and classical conservation laws (continuity and Euler equations);
  3. the positivity of the density and the negative definiteness of the stress tensor;
  4. the hypothesis that there is nothing except the ether, therefore, our usual "matter fields" are material properties of the ether, internal degrees of freedom, with the consequence that ether conservation laws — the continuity and Euler equations — do not depend on matter fields;
  5. the requirement that the equations are Euler-Lagrange equations for some Lagrangian;
  6. and the special relation between conservation laws and preferred coordinates in the weak covariant Lagrange formalism: The conservation laws are the Euler-Lagrange equations for the preferred coordinates.

Note that here we have derived the Lagrangian of the whole theory from first principles. The most essential result which has been derived here is, of course, the Einstein Equivalence Principle (EEP). It appears that to derive the EEP even less is necessary. This makes a separate derivation of the EEP interesting too.