Diskussion

Derivation of the Theory

The equations of GLET are not proposed ad hoc. Instead, we derive them starting with a few number of simple assumptions about the ether. As a consequence, relativistic symmetry - the Einstein equivalence principle - 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

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

We do not introduce them ad hoc. Instead, there are serious reasons to make these assumptions. For using a fixed space-time we have a justifcation based on a thought experiment for quantum gravity, and absolute time is sufficiently justified by the violation of Bell's inequality.

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

Assumption 2 (Condensed matter variables): The ether is described by the following variables: the density ρ(x,t), the average velocity vi(x,t), the stress (or pressure) tensor σij(x,t) (which is in general a symmetrical tensor field), and an unspecified number of "internal degrees of freedom" φm.

It immediately follows that the field ρ(x,t) should be positive everywhere. For the "stress tensor", we introduce another axiom:

Assumption 3 (Definiteness of pressure): The pressure σij(x,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 definiteness does not seem to be a real restriction.

The gravitational field

The gravitational field gmn in ether theory is defined algebraically by the following formulas:

g00(-g)12   =   ρ
gi0(-g)12   =   ρ vi
gij(-g)12   =   ρvivj + σij

This formula is meaningful only in the preferred coordinates.

The positivity of the density, together with the positive 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:

Assumption 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 is the 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:

Assumption 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 ρ(x,t), vi(x,t), σij(x,t).

This influence is described by the material laws of the ether. The conservation laws are fixed, common, and independent of special assumptions about the material properties of the ether:

∂T ρ + ∂Xi (ρvi)  =  0
∂T (ρvj) + ∂Xi (ρvivjij)  =  0

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

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

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.

Assumption 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.

To obtain such a covariant formulation is very easy, a simple trick. We formally consider the preferred coordinates as additional fields T(x), Xi(x), where the index i enumerates the "fields" Xi(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 F0 becomes TFμ.

In this formalism, initially covariant terms remain unchanged and do not depend on T and Xi. 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" T(x) or Xi(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 gmn:

g00(-g)12   =   ρ
gi0(-g)12   =   ρ vi
gij(-g)12   =   ρvivj - σij

This formula itself is meaningful only in the preferred coordinates. But, together with these coordinates interpreted as "fields", we can use gmn to define the state of the ether uniquely. That means, we describe the ether state by a Lorentz metric gmn(x) and four functions T(x), Xi(x) - the preferred coordinates. Of course, we have to add the inner degrees of freedom too.

Now, we have to rewrite the conservation laws in the new variables gmn(x), T(x), Xi(x). But this is really simple. We obtain immediately the harmonic coordinate condition — a well-known, beautiful, covariant equation for four of our new fields T(x), Xi(x):

(∂Xμ gμλ(-g)12 ∂Xλ)   Xν   =  ∂Xμ (gμν(-g)12)   =   0

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 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 T(x), Xi(x). The proof is even simpler in comparison with Noether's theorem - if the Lagrangian does not depend on the T(x), Xi(x) themself, we obtain immediately an equation of the form of a conservation law.

Assumption 6 (Conservation laws and Lagrangian) The conservation laws are the Euler-Lagrange equations for the preferred coordinates T(x), Xi(x).

Derivation of the general GLET Lagrangian

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

What remains is surprisingly simple. A particular covariant Lagrangian which gives the harmonic equation for T and Xi as equations for T and Xi is the sum of four standard scalar Lagrangians. The remaining part - the difference between the full Lagrangian and our particular guess - should no longer give any non-trivial terms for the equation for T and Xi.

But that means the remaining part fulfils the general requirement for the Lagrangian of general relativity. Thus, we obtain the general Lagrangian of GLET as the sum of our particular Lagrangian depending on T and Xi and the general Lagrangian of general relativity.

Because of the rotational symmetry of the Newtonian background space we choose the same coefficient Ξ for the three spatial components Xi. The factor Υ for time T is different. For convenience we introduce also a common factor used in the GR Lagrangian and obtain the following general GLET Lagrangian:

L  =  -(8πG)-1(Υg00-Ξgii) (-g)1/2  +  LGR(gμν)  +  Lmatter(gμνm)

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. That means, for the matter Lagrangian the Einstein equivalence principle holds as well as in general relativity.

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 and classical conservation laws;
  3. 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 do not depend on matter fields;
  5. the requirement that the equations are Euler-Lagrange equations for some Lagrangian;
  6. and the relation between conservation laws and preferred coordinates in the weak covariant Lagrange formalism.