Editing Æther Research & Development (section)

== Equations Describing Æther Operations ==
# '''Wave Equation for Æther Propagation:'''
    <math>\nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0</math>
    <math>\nabla^2 \mathbf{B} - \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0</math>
   - These equations describe the propagation of electromagnetic waves through Æther, where <math>\mathbf{E}</math> and <math>\mathbf{B}</math> represent the electric and magnetic fields, respectively, and <math>c</math> is the speed of light in Æther.
# '''Continuity Equation for Æther Flux:'''
    <math>\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0</math>
   - This equation represents the conservation of Æther flux, where <math>\mathbf{J}</math> is the flux density vector and <math>\rho</math> is the Æther density.
# '''Field Equations with Æther as a Medium:'''
   - Maxwell's equations modified to account for the presence of Æther:
      <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}</math>
      <math>\nabla \cdot \mathbf{B} = 0</math>
      <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}</math>
      <math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}</math>
   - These equations describe the behavior of electric and magnetic fields within Æther, where <math>\varepsilon_0</math> and <math>\mu_0</math> are the permittivity and permeability of Æther, respectively. Note: there is no modification needed, as Maxwell's equations were already compatible wit Æther and fundimentally based on the concept that Æther is the medium which electric and magnetic field propagate.
# '''Modified Einstein Field Equations:'''
    <math>R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu}</math>
   - These are the modified Einstein field equations incorporating Æther as a medium, where <math>R_{\mu\nu}</math> is the Ricci curvature tensor, <math>R</math> is the scalar curvature, <math>g_{\mu\nu}</math> is the metric tensor, <math>G</math> is Newton's gravitational constant, <math>c</math> is the speed of light, and <math>T_{\mu\nu}</math> is the stress-energy tensor.
# '''Scalar Field Equations for Æther Dynamics:'''
    <math>\Box \phi + V'(\phi) = 0</math>
   - This equation represents the dynamics of a scalar field <math>\phi</math> within Æther, where <math>\Box</math> is the d'Alembertian operator and <math>V'(\phi)</math> is the potential function governing the scalar field's behavior.

# '''Schrödinger Equation with Æther Potential:'''
   - <math> i\hbar \frac{\partial}{\partial t} \Psi = \left( -\frac{\hbar^2}{2m} \nabla^2 + V_{\text{eff}}(\mathbf{r},t) \right) \Psi </math>
   - The Schrödinger equation is modified to include an effective potential <math> V_{\text{eff}}(\mathbf{r},t) </math> arising from interactions with Æther, influencing the quantum behavior of particles.

# '''Fluid Dynamics Equations for Æther Flow:'''
** Conservation equations for Æther flow, including continuity equation, momentum equation, and energy equation, adapted from fluid dynamics principles to describe the motion and behavior of Æther as a fluid-like medium.
   - Continuity equation:
   <math> \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 </math>

   - Momentum equation:
   <math> \rho \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} </math>

   - Energy equation:
   <math> \frac{\partial e}{\partial t} + \nabla \cdot (\mathbf{v} e) = -p \nabla \cdot \mathbf{v} + \nabla \cdot (\mathbf{q} + \rho \mathbf{v} \cdot \mathbf{v}) + \mu \nabla^2 \mathbf{v}^2 </math>

# '''Entropy Equations for Æther Thermodynamics:'''
** Entropy equations describe the thermodynamic behavior of Æther, accounting for its temperature, pressure, and entropy evolution within physical systems.
    - First law of thermodynamics:
      <math> dU = \delta Q - \delta W </math>

    - Second law of thermodynamics:
      <math> dS \geq \frac{\delta Q}{T} </math>

# '''Nonlinear Wave Equations for Æther Perturbations:'''
** Nonlinear wave equations describe the propagation of Æther perturbations, accounting for nonlinear interactions and self-interference phenomena that arise due to Æther's dynamic nature.
    - Nonlinear Schrödinger equation:
      <math> i\frac{\partial \psi}{\partial t} + \frac{1}{2} \nabla^2 \psi + |\psi|^2 \psi = 0 </math>

# '''Quantum Field Theory Formulation with Æther Fields:'''
** Quantum field theory is adapted to include Æther fields as fundamental entities, introducing new interaction terms and field equations to describe their quantum behavior and interactions with matter and other fields.
    - Klein-Gordon equation for a scalar field:
      <math> (\Box + m^2) \phi = 0 </math>

    - Dirac equation for a fermionic field:
      <math> (i \gamma^\mu \partial_\mu - m) \psi = 0 </math>

# '''Equations of State for Æther:'''
** Equations of state describe the thermodynamic properties of Æther, relating its pressure, density, and temperature under different conditions, providing crucial information for modeling its behavior in various physical contexts.
    - Ideal gas law:
      <math> PV = nRT </math>

    - Van der Waals equation:
      <math> (P + \frac{a}{V^2})(V - b) = RT </math>