3. Electromagnetism in the SCC

3.1 Electromagnetic Wave Propagation

Problem Statement:

Propagation of an electromagnetic wave in a vacuum.

Maxwell's Equations in SCC:

1. Faraday's Law:

   $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial \chi}$$

2. Ampère's Law (no currents):

   $$\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial \chi}$$

3. Gauss's Laws:

   $$\nabla \cdot \mathbf{E} = 0, \quad \nabla \cdot \mathbf{B} = 0$$

Wave Equations:

Taking the curl of Faraday's Law and substituting from Ampère's Law:

$$\nabla \times (\nabla \times \mathbf{E}) = - \frac{\partial}{\partial \chi} (\nabla \times \mathbf{B}) = - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial \chi^2}$$

Using vector identity:

$$\nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} = - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial \chi^2}$$

Since $\nabla \cdot \mathbf{E} = 0$:

$$\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial \chi^2}$$

Similarly for $\mathbf{B}$:

$$\nabla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{B}}{\partial \chi^2}$$

Solution:

Assume plane wave solutions:

$$\mathbf{E}(\mathbf{r}, \chi) = \mathbf{E}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega \chi)}$$ $$\mathbf{B}(\mathbf{r}, \chi) = \mathbf{B}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega \chi)}$$

Substitute into the wave equation:

$$- k^2 \mathbf{E} = \mu_0 \epsilon_0 (-\omega^2) \mathbf{E}$$

Simplify:

$$k^2 = \mu_0 \epsilon_0 \omega^2$$

Dispersion Relation:

$$\omega = \frac{k}{\sqrt{\mu_0 \epsilon_0}} = k c$$

 

 where $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$is the speed of light.

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Interpretation:

• Electromagnetic waves propagate with speed $c$ with respect to the change parameter $\chi$.
• The form of the equations and solutions is identical to standard electromagnetism with $t$ replaced by $\chi$.
  

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4. Discussion and Comparison with Standard Physics

4.1 Recovering Time Dependence

In all the examples, the equations and solutions mirror those in standard physics, with $t$ replaced by $\chi$. To compare with experimental results, we need to relate $\chi$ to observable time $t$.

Possible Relationship:

• If $\chi$ is proportional to $t$, i.e., $\chi = \alpha t$, where $\alpha$ is a constant (possibly unity), then the SCC equations reduce to standard equations.

Implications:

• The SCC framework reproduces known results when $\chi$ and $t$ are linearly related.
• This suggests that, at least for these systems, the SCC is consistent with standard physics under appropriate conditions.

4.2 Interpretation of the Change Parameter $\chi$

• Operational Time:

  • Clocks measure change through periodic processes (e.g., oscillations in a quartz crystal).
  • In the SCC, these periodic processes progress with respect to $\chi$.

• Experiments:

  • Time measurements are essentially tracking the progression of $\chi$ via physical systems.
  • This supports the idea that time $t$ is emergent from change $\chi$.