Continuum Degrees of Freedom

01Reference position and state change

ASD describes space through related volume, shape and orientation states rather than identifying physical change with coordinate relabelling. Small displacement describes compatible deformation; independent frames and compact orientation supply degrees of freedom beyond that linear description.

Consider a small region marked in an approximately homogeneous background. It can translate, change volume, or change shape and orientation at nearly constant volume. Reference position $\mathbf x$ identifies the region, while $\mathbf u(\mathbf x,t)$ describes its displacement-type change from the ground state. Small displacement describes a configuration generated continuously by a single-valued field; the manuscript also allows equivalent phase-refresh or spatial-state changes under explicit normalization.

The field has length units, and $t$ is the process-evolution parameter. Relabelling coordinates and changing a physical state have different roles: passive transformations rewrite components, whereas physical changes alter energy, propagation or material response through constitutive relations.

A constant $\mathbf u$ translates the whole region without local deformation. Differences between neighbours are described by $\partial_j u_i$. In a compatible deformation model, $F_{ij}=\delta_{ij}+\partial_j u_i$. This dimensionless matrix $F$ differs from the amplitude-source function $F(\Theta)$ in the master-equation candidate.

02Volume change and density convention

For small deformation, $\det F=1+\partial_i u_i+O(|\nabla\mathbf u|^2)$. Conserving constituents or connections in a reference volume makes number density decrease as volume grows, motivating

The dimensionless $s$ is dilation strain; $\sigma$ is compression-positive linear relative density. Number density $n_s$ and its reference value $n_{s,0}$ differ from the kinetic coefficient $\rho_s$. Chapter 4 uses $\eta$ for compression, whereas Chapter 5 and this module use $\sigma$. Chapter 5 also uses $\eta$ for a core perturbation, whose meaning is defined in its own equation.

This relation is the linear Jacobian limit. Finite deformation requires $n_s/n_{s,0}=1/\det F$. An anisotropic background's volume cannot be inferred from elongation along a single direction. The longitudinal field further separates local density from the gravitational scale background.

03Shear and rotation have separate definitions

The symmetric displacement gradient is the small-strain tensor. Removing its trace gives $e^{\rm dev}_{ij}$, the shape change without linear volume change. The antisymmetric part describes local orientation rotation. The convention is

The transverse curl $\boldsymbol\Omega_T$ and small-rotation variable $\boldsymbol\omega$ are dimensionless. Here $\boldsymbol\omega$ is not an angular frequency in $\mathrm{s^{-1}}$ or the scalar gravitational scale factor $\Omega$. Curl is twice local rotation; the symmetric traceless strain contains a different part of deformation.

Transverse response can involve shear and rotation without identifying them pointwise. A local rotation does not specify global closure, a target space or a topological classification, which require additional structures and boundaries.

04Organizing sectors by propagation direction

For linear propagation, volume and transverse channels can be organized as

The Helmholtz decomposition extracts curl-free and divergence-free parts for fields with appropriate decay or compatible boundaries. For nonzero Fourier wave number,

The wave vector $\mathbf k$ has inverse-length units. $P_L$ extracts the parallel component, and $P_T$ the perpendicular components. Direct multiplication gives $P_L^2=P_L$, $P_T^2=P_T$, $P_LP_T=0$, supporting separate mode calculations in constant-coefficient problems.

Zero-wave-number translation and harmonic components in finite regions require global conditions. In inhomogeneous or coupled systems, projections need not commute with evolution. Geometric decomposition alone does not imply dynamically independent equations.

05Orientation and amplitude beyond small displacement

A strongly nonlinear core can contain large closed orientation changes that its exterior small displacement cannot fully describe. Chapter 4 introduces compact $Q\in SU(2)$ and real amplitude $\chi$:

Here $SU(2)$ is the candidate matrix orientation space with the displayed constraints; $\chi$ measures ordering. The matrix $\Psi$ combines amplitude and orientation, and differs from the Quantum module's slow envelope with the same name. Its units follow the selected action normalization. A local phase can coordinatize a section, but one scalar phase cannot contain all three-dimensional orientation and particle information.

Chapter 5 also uses dimensionless real components $\Theta_A$ for a local dual-field candidate. Those components, $\chi Q$ and small displacement have different constraints. Comparing them requires target spaces, inner products, energies and a mapping. Choosing $SU(2)$ does not itself derive the weak-interaction gauge theory.

The external matter-wave phase $\Xi$ must also remain distinct from matrix orientation. Equation (5.6.30) matches $\hbar\nabla\Xi$ to the momentum of the complete dressed state. Multiplying $Q$ by arbitrary $e^{i\Xi}$ generally changes its determinant and leaves $SU(2)$; introduce phase through an allowed group action or an independent mode amplitude.

06An extended frame for scale, shape and orientation

Chapter 5 illustrates keeping these states separate with a local frame. Internal index $a$ labels frame directions and spatial index $i$ labels coordinates. Its polar decomposition is

Dimensionless $\varphi$ describes isotropic logarithmic linear stretch, $S$ volume-preserving shear, and $R$ orientation. The direction-even length quadratic form is

The rotation drops out of this length expression. A physical cost for orientation gradients needs a separate connection energy. For a compatible frame $e_i{}^a=\delta_i{}^a+\partial_i u^a$, linear order gives $\varphi=s/3$, while $S-I$ and $R-I$ correspond to the symmetric traceless and antisymmetric gradients. General frames need not arise from global displacement; independent stretch and incompatible orientation add degrees of freedom requiring constraints and energy.

This spatial frame is not automatically a four-dimensional metric. Its length quadratic form treats opposite directions equally and cannot by itself represent the direction-odd term of LVE. The defined variables delimit the physical tasks they can carry.

07Define variables before selecting a realization

These definitions supply continuum variables and boundaries without selecting a unique lattice. Continuum dynamics supplies actions and sources; wave propagation calculates linear spectra. Microscopic models must explain how their links generate the effective coefficients and allowed branches.

Stable localized structures additionally require asymptotic orientation, whether amplitude may vanish, and a defined topological quantity. The stable-topology task then asks for existence and perturbation stability; decomposition or local curl alone cannot answer it.