Relativistic covariant aether model

QFT, QED, Allgemeine Relativität, elastisch, Materialgesetz, GUT, Gravitation, Wechselwirkungen

Idea: In this chapter, the four-dimensional, relativistic covariant aether model is established and introduced. First off, the transition from very small lattice structures to the macroscopic description is sketched. The description is sufficient to interpret both quantum mechanical and macroscopic physical constants as elastic constants. As a first result, the gravitational constant can be calculated from quantum mechanical quantities.

 

Starting point

The standard model of particle physics and general relativity

Today’s physics is successfully described by the standard model of particle physics and general relativity. Some aspects of these models are summarized in a separate article.

Difficulties with the status quo

Although very successful in various fields, today’s physical model does not fulfil certain requirements:
A mathematical unification of the standard model and general relativity has so far not been accomplished.

As a second point, both theories can’t explain what matter really is. They explain clearly how matter interacts, but not why it interacts.

Finally, the point which is essential for this work: both theories assume a vacuum, an “emptiness”, in which matter (“particles”) move. With this assumption it has so far been impossible to find an explanation for some measurable effects:

• Vacuum fluctuations: In the vacuum particles and antiparticles are continuously formed and annihilated, with detectable effects. Vacuum is therefore not empty, as required in the earliest formulation.

KANN MAN DAS ABER NICHT MIT DER UNSCHÄRFERELATION ERKLÄREN (ODER MINDESTENS ERLAUBEN)? 

• Higgs field: The discrepancy is even clearer when introducing the Higgs field in the theory. It is a field who’s expected value is nonzero in all of spacetime. For an interpretation this means clearly that there is “something” everywhere, for vacuum’s “nothingness” could only be equated with an expected value of zero.
• Dark energy: In general relativity, current measurements of the universe’s expansion can only be explained by introducing a constant which has a nonzero value throughout all of spacetime, the cosmological constant. Since it is completely unclear where this constant comes from in the current model, the underlying energy is termed “dark energy”.

 

Idea

To overcome the difficulties mentioned above, a model is created where certain properties are assigned to the vacuum; In that sense, the vacuum is actually replaced by a substance, as in an aether model. The new model, however, must at the same time be relativistically covariant and is therefore called an RC aether model.

It is possible to attain relativistic covariance by developing a consistent four-dimensional material model and incorporating the considerations in Chapter 1. The details are explained in the next section.

(Current, disproven aether theories are not discussed here. Interested readers will for example find a comprehensive, up-to-date overview with many further references in the introduction of ; an historic overview is given by ; as a side note, the following article is also interesting: [9].

 

Model theoretic considerations

When looking for a RC aether, there are initially infinitely many degrees of freedom. To find a good model, it is therefore important to address all possibilities and choose the model that works with the least number of parameters (Occam's razor). It is likely that any RC aether model could reproduce the Standard Model. However, if the RC aether model needs more parameters than the standard model, the previous model should still be preferred from the stand point of model theory.

On the other hand, if an RC model is found to have less parameters than the Standard Model, but reproduces it fully, then the RC model should be favoured.

With the above in mind, one can now work out a sensible starting point in the search for an appropriate RC aether model. It turns out that with a few simple considerations and observations of nature, the number of parameters can be significantly reduced.

Relativistic Covariance

An RC aether model must have a constitutive equation which reproduces the equations of the current Standard Model of physics. If possible, existing concepts should be used. This is where the first problem already arises:

The physics of traditional substances – i.e. gases, liquids, solids – is treated in continuum mechanics. All of continuum mechanics is however limited to Euclidean (or at least positively definite) spaces. For a theory of space-time, however, it is imperative that it be relativistically covariant, and thus defined on a (not positively definite) Minkowski space.

Because of the importance of this topic, a separate article has been dedicated to this point, showing how this apparent contradiction can be resolved. In Chapter 1 it is shown that the Minkowski metric is a direct consequence of a temporally flat observer measuring (observing) Euclidean space-time.

Briefly summarized, the following hypothesis is presented: the physics of spacetime can very well take place in a four-dimensional Euclidean space, yet the measurement by man - who can only overlook a single point in time - leads to a measurement rule, which maps the Euclidean space onto the Minkowski space.

The underlying space and the measurement rule must therefore be separated in a clear way. The concept of a temporally flat observer (i.e. man), who makes measurements in the Euclidean space and thereby sees a Minkowski space, allows concepts of continuum mechanics to be applied to the underlying Euclidean space without contradiction, and thus to make use of existing, efficient concepts in the description of an RC aether.    

Dimension

When considering the number of parameters, the model that works with the smallest number of dimensions should be favoured. One- and two-dimensional models are out of the question, as they can barely match as a model of our real three-dimensional perception of space.

Three dimensional models (with time as an extra dimension outside of the metric) are out of the question, because models with a three dimensional aether correspond to the traditional aether models, which were experimentally refuted 100 years ago since they are not relativistically covariant.

Four-dimensional models seem to be the ones with the lowest valid dimension. This fits well with today's understanding of spacetime and is hence considered for the construction of a model. If a for four dimensional RC aether model can be excluded, then five dimensional, later six dimensional, etc. models need to be tested.

Constitutive equation

The question arises as to how «something» is defined as opposed to vacuum. It could basically have characteristics of any known or unknown substance/medium. Of course, it would be desirable to create a model that corresponds with our experience.

By observing nature, two conclusions can be drawn:

Transversal waves
Electromagnetic waves - that is, light - are transverse waves. The model must therefore be able to describe transverse waves. Transverse stress (also called shear stress) must therefore be allowed.

In the known basic models, transverse stress terms are only allowed in elastic mediums, but not in gas or fluids. It therefore makes sense to equip the constitutive equation with an elastic term. More terms can be added later on if required, this is simply a starting point.

Expanding universe
The accelerated expansion of the universe results in the medium not being in an equilibrium state, but under some sort of universal tension. This aspect should be incorporated into the constitutive equation by means of an underlying stress term.

 

Summary of the model theory considerations: The starting point for the development of an RC aether model is given by the model with the least parameters: a four dimensional elastic medium under tension. If these should not be sufficient, and only in this case, further parameters should be considered. Relativistic covariance is provided through the measurement of a temporally flat observer.

 

Constitutive equation and energy conservation

The theory of elasticity consists of two main components. On the one hand a constitutive equation, which determines the material-specific properties by describing how a volume element of the material reacts to external forces: It describes the qualities of the material.

On the other hand the overlying general mechanics can be used to define a local energy conservation in the form of a Lagrangian and thus an equation of motion. The material-specific properties are incorporated in this energy conservation from the constitutive equation.

Furthermore, within the theory of elasticity, continuous substances can be understood as the limit of a smaller lattice structure made of springs and point masses, that is, an elastic lattice, as long as the transition does not lead to any fundamentally different behaviours.

The following graphic summarises the individual elements of the theory:

Diagram of the classification and structure of elasticity theory, including the transition from continuum to elastic lattice. 

Hypothesis

All above considerations lead to the following hypotheses:

Hypothesis 1:
The equations of electroweak theory with symmetry breaking for leptons can be interpreted as the energy conservation (Lagrangian) of a point defect in a four-dimensional elastic lattice, observed by a temporally flat observer.

Hypothesis 2:
The Einstein field equations of general relativity correspond to the constitutive equation of an elastic four-dimensional medium in the continuum limit, observed by a temporally flat observer.

Remarks:

• The case of the temporally flat observer was treated in Chapter 1.
• A major difficulty when comparing the equations of the theory of elasticity and particle physics is that they have been developed using different mathematical methods historically. Therefore, a whole separate article was devoted to the derivation of a system of equations for the energy conservation of a moving point defect in the methodology of particle physics (Chapter 2).

Conclusion 1:
If these hypotheses are true, there must be a well-defined relationship between these two systems of equations:
The constitutive equation and the Lagrangian are both functions of the elastic properties of the underlying medium.

A mechanism allowing to convert these constants into each other is presented as a first conclusion from the above considerations.

  

Mathematically

Relativistic covariance, dimension and metric

The metric tensor of a four-dimensional medium is a 4x4 matrix, which, in the stress free case, is orthonormal. According to the arguments in Chapter 1 on how a flat observer sees the world, the sign of one of the metric elements is the opposite of the others. It is to begin with undefined which elements are negative or positive. There are therefore two possible metrics, which can be represented as follows:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ g_{\mu\nu}=[S1]\cdot\left(\begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) }[/math] </equation>

Where the sign [math]\displaystyle{ [S1]=\pm{}1 }[/math] can be chosen depending on the convention. (Nomenclature according to , available in [10])

 

Constitutive equation

The constitutive equation of a four-dimensional isotropic elastic medium in the small defect approximation is :

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \left[ S3 \right] \cdot \mathbf{ε} = {2\mu}^\prime \cdot \mathbf{σ} + {\lambda}^\prime \cdot \mathrm{tr}(\mathbf{σ})\mathbf{g} }[/math] </equation>

Where the sign [math]\displaystyle{ [S3] }[/math] can be chosen depending on the convention.

[math]\displaystyle{ \mathbf{ε} }[/math] is the deformation tensor, [math]\displaystyle{ \mathbf{σ} }[/math] is the stress tensor, [math]\displaystyle{ {2\mu}^\prime }[/math] and [math]\displaystyle{ \lambda^\prime }[/math] are called first and second Voigt coefficients. The constitutive equation states that deformations are proportional to stresses in the material. [math]\displaystyle{ \mathbf{g} }[/math] is once again the metric.

The constitutive equation can also be rewritten with an adapted stress tensor:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot \mathbf{ε}={2\mu{}}^\prime\cdot\mathbf{σ}^\prime }[/math] </equation>

Where

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \mathbf{σ}^\prime = \mathbf{σ}+\frac{{\lambda}^\prime}{{2\mu}^\prime}\cdot \mathrm{tr}(\mathbf{σ})\mathbf{g} }[/math] </equation>

Divergence/ Ricci tensor
If there are no external forces, the divergence of the stress tensor is zero by definition:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \mathbf{\nabla σ} := -\mathbf{F} =0 }[/math] </equation>

Hence, the divergence fo the deformation tensor must also be zero. The deformation tensor has thus the following properties:

1. Tensor of second order in four dimensions
2. Measure for the curvature and metric of the space.
3. Dissapearing divergence in the force free case.

Thus the trace-reversed Ricci tensor (Einstein tensor) is the appropriate tensor which fulfils all these required properties:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \varepsilon_{ij}=G_{ij}=R_{ij}-2Rg_{ij} \qquad \qquad \qquad \mathbf{ε}=\mathbf{G}=\mathbf{R} - 2R\mathbf{g} }[/math] </equation>

Where, in the definition of the Ricci tensor, a further sign can be arbitrarily determined depending on the convention (not visible here, [math]\displaystyle{ [S2] }[/math]). The definition of the components of the Ricci tensor can be found in the appendix.

Comparison with the Einstein field equations
The constitutive equation:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot{\varepsilon}_{\mu\nu}={2\mu}^\prime\cdot{\sigma^\prime}_{\mu\nu} }[/math] </equation>

Can be compared with the Einstein field equations:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle [S3]\cdot G_{\mu\nu}= \frac{8\pi G}{c^4}\cdot T_{\mu\nu} }[/math] </equation>

Where the Einstein tensor [math]\displaystyle{ G_{\mu\nu} }[/math] can be identified with the deformation tensor and the energy-momentum tensor [math]\displaystyle{ T_{\mu\nu} }[/math] with the stress tensor of the constitutive equation. Literature: .

Consequently, the first Voigt parameter relates to the Gravitational G constant as follows:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle {2\mu}^\prime c^2= \frac{8\pi G}{c^2} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \left[\mu^\prime \right] = \frac{1}{N} }[/math] </equation>

 

Lagrangian of a defect

In Chapter 2 the Lagrange density for a defect in a four-dimensional lattice was derived. In contrast with traditional text books on defect dynamics in elastic lattices, the focus was put on describing the problem mathematically by means of local gauge theories, such as to allow a comparison with quantum mechanics. The approach pays off. The Lagrange density for an intrinsic defect (comprising of a misplaced and vacant position) moving in an isotropic medium is:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle L_{\mathbf{m}} = \mathbf{\overline{W}_\alpha} \left( -\xi + 2 \overset{\approx}{M} \cdot i \cdot \gamma^j \cdot \partial_j - 4 \overset{\approx}{K} \cdot \gamma^j v_j \right) \mathbf{W_\alpha} - \overset{\approx}{K} \cdot V_{ij} V^{ij} - \overset{\approx}{\Phi}_0 }[/math] </equation>

To simplify, the (massive) [math]\displaystyle{ SU(2) }[/math] interaction terms were left out. This system of equations is sufficient as an initial comparison.

The symbols above are:

• [math]\displaystyle{ \mathbf{W_\alpha} }[/math]: combined deflection field of an intrinsic defect pair [math]\displaystyle{ \mathbf{\alpha} }[/math] (misplaced and vacant position). In this notation, the adjunct field [math]\displaystyle{ \mathbf{\overline{W}_\alpha} }[/math] is considered to be an independent field.
• [math]\displaystyle{ \xi }[/math]: Relative defect strength in the elastic lattice.
• [math]\displaystyle{ 2 \overset{\approx}{M} \cdot i \cdot \gamma^j \cdot \partial_j }[/math]: Term for the motion of the combined defect, which models diffusion.
• [math]\displaystyle{ \overset{\approx}{M} }[/math]: Longitudinal modulus of the elastic lattice. It gives the interaction strength of the directed motion of the defect in the lattice.

DIESES TEIL (OBEN) IM DEUTSCHEN? ICH GLAUBE DA IST EIN FEHLER.
ICH WÜRDE AUCH NOCH KURZ GAMMA ALS DIRAC TENSOR UND DELTA ALS ABLEITUNG DEFINIEREN

• [math]\displaystyle{ V_{ij}=\partial_i v_j - \partial_j v_i }[/math]: Non massive (skew symmetric, [math]\displaystyle{ U(1) }[/math]) terms of the deflection field of the unperturbed lattice, interacting with the defect as a perturbing term. Massive terms are not included for simplicity.
• [math]\displaystyle{ \overset{\approx}{K} }[/math]: Compression modulus of the elastic lattice, the interaction strength of the defect with isotropic oscillation modes in the unperturbed lattice.

In the continuum limit, this Lagrangian can be compared to the Lagrange density of quantum-electro-dynamics, and the components can be interpreted one to one as properties of a point defect in the four-dimensional elastic lattice, as measured by a flat observer with respect to time.

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \mathcal{L} = \overline{\mathbf{ψ}}_i \left( -mc^2 + i \cdot \hslash c \cdot \gamma^\mu \cdot \partial_\mu - \frac{e^2}{\varepsilon_0} \cdot \gamma^\mu A_\mu \right) \mathbf{ψ}_i - \frac{e^2}{4 \varepsilon_0} \cdot F_{ij} F^{ij} }[/math] </equation>

Where the electromagnetic constants were written explicitly as separate factors before the four potential.

By comparing both systems, two elasticity constants of the elastic lattice can be directly determined: The compression modulus [math]\displaystyle{ \overset{\approx}{K} }[/math] and the longitudinal modulus [math]\displaystyle{ \overset{\approx}{M} }[/math]:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \overset{\approx}{K}=\frac{e^2}{{4\varepsilon{}}_0} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \overset{\approx}{M}=\frac{\hslash{}c}{2} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \left[\overset{\approx}{K}\right]=\left[\overset{\approx}{M}\right]=Pa \cdot m^4=Jm }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \left[\mathbf{{W}_\alpha} \right]=1 \qquad \qquad \left[\mathbf{∂{W}_\alpha} \right]=m^{-1} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \left[\mathbf{v} \right]=m^{-1} \qquad \qquad \left[\mathbf{∂v} \right]=m^{-2} }[/math] </equation>

If one neglects the symmetry breaking of space by the elastic lattice, thus assuming an isotropic lattice, both quantities can be converted, up to a constant factor, into the two Voigt coefficients [math]\displaystyle{ \mu^\prime }[/math] and [math]\displaystyle{ \lambda^\prime }[/math] in the constitutive equation. The constant factor corresponds to the normalizing condition for the elastic lattice.

  [math]\displaystyle{ {\overset{\sim}{x}} }[/math]

Relationship between the constitutive equation and the Lagrangian

Longitudinal und compression moduli
Form the Voigt coefficients [math]\displaystyle{ \mu^\prime }[/math] and [math]\displaystyle{ \lambda^\prime }[/math] other elastic coefficients can be determined. Of interest here are the longitudinal modulus [math]\displaystyle{ M }[/math] for the description of directional deformations and the compression modulus [math]\displaystyle{ K }[/math] for the description of isotropic deformations, which are used for the microscopic descriptions of the elastic lattice.

Both moduli can be determined by rewriting the constitutive equation. Yet a basic problem arises from the application of the microscopic theory of the elastic lattice:

 

Constitutive equation with an inverted sign metric

The constitutive equation determined in this manner assumes a direct deformation of the metric. The model of the linear lattice describes the compression of springs based on a fixed space with a fixed metric. The constitutive equation must therefore be adapted to describe not a covariant deformation of the metric, but a contravariant deformation of the space lattice.

This is possible following the insights from Chapter 1, by changing the sign of the metric. First, remember the original constitutive equation:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \left[ S3 \right] \cdot \mathbf{ε} = {2\mu}^\prime \cdot \mathbf{σ} + {\lambda}^\prime \cdot \mathrm{tr}(\mathbf{σ})\mathbf{g} }[/math] </equation>

The right side can be separated into trace free and homogeneous parts:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot\mathbf{ε}=2\mu^\prime \cdot \mathbf{σ_\perp} + \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathbf{σ_\parallel} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \mathbf{σ_\parallel} = \frac{1}{4}\cdot \mathrm{tr}\left(\mathbf{σ} \right) \cdot \mathbf{g} \qquad \qquad \mathrm{tr}\left(\mathbf{σ_\perp} \right) = 0 \qquad \qquad \mathbf{σ}=\mathbf{σ_\parallel} + \mathbf{σ_\perp} }[/math] </equation>

Where, by contraction, the trace of the equation can be found:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot \mathrm{tr}\left(\mathbf{ε}\right)= [S3]\cdot g^{ab}ε_{ab} = \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot g^{ab}σ_{ab} = \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathrm{tr}\left(\mathbf{σ}\right) }[/math] </equation>

 

The constitutive equation with an inverted sign in the metric is:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \left[ S3 \right] \cdot \mathbf{\overset{\smile}{ε}} = {2\mu}^\prime \cdot \mathbf{\overset{\smile}{σ}} + {\lambda}^\prime \cdot \mathrm{tr} \left(\mathbf{\overset{\smile}{σ}} \right) \mathbf{\overset{\smile}{g}} }[/math] </equation>

Where the individual components transform as follows:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \mathbf{\overset{\smile}{g}} = - \mathbf{g} \qquad \qquad \mathbf{\overset{\smile}{ε}} = \mathbf{ε} \qquad \qquad \mathbf{\overset{\smile}{σ}} = \mathbf{σ} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \mathrm{tr}\left(\mathbf{\overset{\smile}{ε}}\right) = \overset{\smile}{g}^{ab}\overset{\smile}{ε}_{ab} = - \mathrm{tr}\left(\mathbf{ε}\right) \qquad \qquad \mathrm{tr}\left(\mathbf{\overset{\smile}{σ}}\right) = \overset{\smile}{g}^{ab}\overset{\smile}{σ}_{ab} = - \mathrm{tr}\left(\mathbf{σ}\right) }[/math] </equation>

The transformation properties of the components are studied im more detail in the appendix.

Once again, the inverted metric constitutive equation can be separated and contracted:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot\mathbf{\overset{\smile}{ε}}=2\mu^\prime \cdot \mathbf{\overset{\smile}{σ}_\perp} + \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathbf{\overset{\smile}{σ}_\parallel} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \mathbf{\overset{\smile}{σ}_\parallel} = \frac{1}{4}\cdot \mathrm{tr}\left(\mathbf{\overset{\smile}{σ}} \right) \cdot \mathbf{\overset{\smile}{g}} \qquad \qquad \mathrm{tr}\left(\mathbf{\overset{\smile}{σ}_\perp} \right) = 0 \qquad \qquad \mathbf{\overset{\smile}{σ}}=\mathbf{\overset{\smile}{σ}_\parallel} + \mathbf{\overset{\smile}{σ}_\perp} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot \mathrm{tr}\left(\mathbf{\overset{\smile}{ε}}\right)= [S3]\cdot \overset{\smile}{g}^{ab}\overset{\smile}{ε}_{ab} = \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \overset{\smile}{g}^{ab}\overset{\smile}{σ}_{ab} = \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathrm{tr}\left(\mathbf{\overset{\smile}{σ}}\right) }[/math] </equation>

The point of interest for the comparison with the linear lattice is the original deformation state [math]\displaystyle{ \mathbf{ε} }[/math], displayed in the converted constitutive equation with the metric [math]\displaystyle{ \mathbf{\overset{\smile}{g}} }[/math]. So the original deformation state, according to its type of transformation, is inserted in the changed constitutive equation by inverting the sign of the metric:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \left[ S3 \right] \cdot \mathbf{ε} = {2\mu}^\prime \cdot \mathbf{\overset{\smile}{σ}} + {\lambda}^\prime \cdot \mathrm{tr} \left(\mathbf{\overset{\smile}{σ}} \right) \mathbf{\overset{\smile}{g}} }[/math] </equation>

And in contracted form:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ [S3]\cdot \mathrm{tr}\left(\mathbf{ε}\right)= - \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathrm{tr}\left(\mathbf{\overset{\smile}{σ}}\right) }[/math] </equation>

Notice the minus sign.

 

Conversion of the constitutive equation

The further conversion follows the standard procedure of continuum mechanics yet, by changing the sign, one gets different results.

The contracted constitutive equation can still be reformulated a bit:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle [S3]\cdot \frac{-1}{\left( 2\mu^\prime + 4\lambda^\prime \right)} \cdot \mathrm{tr}\left(\mathbf{ε}\right)= \mathrm{tr}\left(\mathbf{\overset{\smile}{σ}}\right) }[/math] </equation>

And introducing this in the constitutive equation:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \left[ S3 \right] \cdot \mathbf{ε} = {2\mu}^\prime \cdot \mathbf{\overset{\smile}{σ}} - [S3]\cdot \frac{{\lambda}^\prime}{\left( 2\mu^\prime + 4\lambda^\prime \right)} \cdot \mathrm{tr}\left(\mathbf{ε}\right) \cdot \mathbf{\overset{\smile}{g}} }[/math] </equation>

The trace term can be moved to the other side of the equation:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \left[ S3 \right] \cdot \left( \mathbf{ε} +\frac{{\lambda}^\prime}{\left( 2\mu^\prime + 4\lambda^\prime \right)} \cdot \mathrm{tr}\left(\mathbf{ε}\right) \cdot \mathbf{\overset{\smile}{g}} \right) = {2\mu}^\prime \cdot \mathbf{\overset{\smile}{σ}} }[/math] </equation>

[S3] is now chosen to be positive and the conversion is finished:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \frac{1}{{2\mu}^\prime} \cdot \mathbf{ε} + \frac{{\lambda}^\prime}{{2\mu}^\prime \left( 2\mu^\prime + 4\lambda^\prime \right)} \cdot \mathrm{tr}\left(\mathbf{ε}\right) \cdot \mathbf{\overset{\smile}{g}} = \mathbf{\overset{\smile}{σ}} }[/math] </equation>

This form of the constitutive equation now allows the determination of the longitudinal and compression moduli.

 

Lamé parameter
The relationship between the Voigt [math]\displaystyle{ \left(\mu^\prime,\lambda^\prime \right) }[/math] and the Lamé parameters [math]\displaystyle{ (\mu,\lambda) }[/math] is directly deducible. The Lamé parameters are defined as follows:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ 2\mu \cdot \mathbf{ε} + \lambda \cdot \mathrm{tr}\left(\mathbf{ε}\right) \cdot \mathbf{\overset{\smile}{g}} = \mathbf{\overset{\smile}{σ}} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \Rightarrow 2\mu = \frac{1}{{2\mu}^\prime} \qquad \qquad \Rightarrow \lambda = \frac{{\lambda}^\prime}{{2\mu}^\prime \left( 2\mu^\prime + 4\lambda^\prime \right)} }[/math] </equation>

 

Longitudinal modulus
The longitudinal modulus [math]\displaystyle{ M }[/math] is defined as the stress response of a material with respect to a uniaxial deformation. W.l.o.g the initial deformation is chosen :

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \mathbf{ε}:= \left(\begin{array}{cccc} ε_{00} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) }[/math] </equation>

And is inserted in the constitutive equation:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \Rightarrow σ_{00} = 2\mu \cdot ε_{00} + \lambda \cdot ε_{00} \qquad \qquad \Rightarrow σ_{11} = σ_{22} = σ_{33} = \lambda \cdot ε_{00} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ M := \frac{σ_{00}}{ε_{00}} = 2\mu + \lambda = \frac{{2\mu}^\prime+{5\lambda}^\prime}{{2\mu}^\prime \left( 2\mu^\prime + 4\lambda^\prime \right)} \qquad \qquad \nu^\prime := \frac{-σ_{11}}{σ_{00}} = \frac{-\lambda}{2\mu + \lambda} = \frac{-\lambda^\prime}{2\mu^\prime + 5\lambda^\prime} }[/math] </equation>

The accents were omitted in the components of the stress tensor.

 

Compression modulus
The compression modulus [math]\displaystyle{ K }[/math] describes the behavior of a material by shape preserving volume changes :

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \mathbf{ε}:= \left(\begin{array}{cccc} dε & 0 & 0 & 0 \\ 0 &-dε& 0 & 0 \\ 0 & 0 &-dε& 0 \\ 0 & 0 & 0 &-dε \end{array}\right) }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \Rightarrow dσ = σ_{00} = σ_{11} = σ_{22} = σ_{33} = 2\mu \cdot dε + 4\lambda \cdot dε = \left(2\mu + 4\lambda\right) \cdot dε =: 4K \cdot dε }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \Rightarrow K = \frac{1}{2}\mu + \lambda = \frac{1}{4} \frac{{2\mu}^\prime+{8\lambda}^\prime}{{2\mu}^\prime \left( 2\mu^\prime + 4\lambda^\prime \right)} }[/math] </equation>

 

Microscopic longitudinal and compression moduli and connection

The longitudinal and compression modules [math]\displaystyle{ M }[/math] and [math]\displaystyle{ K }[/math] determined from the material law and the Voigt coefficients must now coincide with the microscopic modules [math]\displaystyle{ \overset{\approx}{M} }[/math] and [math]\displaystyle{ \overset{\approx}{K} }[/math] of the elastic lattice up to a constant normalization factor. The normalization factor follows from the fact that the Lagrangian of the elastic lattice is only determined up to a constant prefactor.

The normalization factor used here is:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ k = {4\mu}^\prime c^2 \left( {2\mu}^\prime + {4\lambda}^\prime \right)\cdot e^* }[/math] </equation>

[math]\displaystyle{ [k] = m^2 }[/math] and therefore [math]\displaystyle{ e^* = N^2 s^2 }[/math]

The factor is justified in the appendix.

Now, by considering the normalizing factor when transitioning to the elastic lattice:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \overset{\approx}{K} = k \cdot K = \frac{c^2}{2}\left( 2\mu^\prime + 8\lambda^\prime \right) \cdot e^* }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \overset{\approx}{M} = k \cdot M = 2c^2 \left( 2\mu^\prime + 5\lambda^\prime \right) \cdot e^* }[/math] </equation>

And therefore:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle 2\mu^\prime c^2 = \left( \frac{8}{6}\overset{\approx}{M} - \frac{5}{6} 4\overset{\approx}{K} \right) \cdot e^{*-1} }[/math] </equation>

On the other hand:

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle 4\overset{\approx}{K}=\frac{e^2}{{\varepsilon{}}_0} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle \overset{\approx}{M}=\frac{\hslash{}c}{2} }[/math] </equation>

<equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle {2\mu}^\prime c^2= \frac{8\pi G}{c^2} }[/math] </equation>

And therefore finally: <equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle G = \frac{c^2}{8\pi} \left( \frac{2}{3}\hslash{}c - \frac{5}{6} \frac{e^2}{{\varepsilon{}}_0} \right) \cdot e^{*-1} = 6.673\, 192\, 3(3) \cdot {10}^{-11} \frac{m^3}{kg \cdot s^2} }[/math] </equation> as a result for the gravitational constant, which is two orders of magnitude more precise than the experimentally measured value: <equation id="eqn:su3.01" shownumber> [math]\displaystyle{ \displaystyle G_{Exp.} = 6.674\, 08(93) \cdot {10}^{-11} \frac{m^3}{kg \cdot s^2} }[/math] </equation> The 3σ-error interval is given. The error calculation is in the appendix.

 

Open questions

Many open question remain.

Micro <-> macro transition

• The arguments of the transition are plausible, but they are still very blurred. Effects arising from the transition to the continuum and from the isotropic to the anistropic medium must be examined further.

Approximations

• Corrections with higher order?
• Effects when completely adapting to GRT?
• In general, the equations have to be defined more precisely: Treatment of poles, definition and value domains.

Physical constants and units

• By correlating the various constants and interpreting them, the physical units must be rethought and reinterpreted.
• Influence of the negative lambda?

Field equations

• Does the adaptation of the stress tensor to compare it with Einstein's field equations have an influence? (Influence on vacuum solutions, inner solutions, cosmological models?)

Defect types

• Internal degrees of freedom
• Influence of the 0-potential

Metric and sign inversion

• Systematise the approach.
• What are other effects?
• Can one determine the absolute sign of the metric?

Commonalities/ differences with other theories

• Overlapping, e.g. with quantum gravity, string theory
• Comparison with work on 4dim elasticity

 

References

 

Appendix
Appendix 3A: Transition macro [math]\displaystyle{ ↔ }[/math] micro
Appendix 3B: Components with sign inversion of the metric tensor

 

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