Longitudinal–Transverse Coupling

01Why a transverse core needs a longitudinal channel

ASD connects transverse shear energy and longitudinal volume state through coupling terms in a common action. Chapter 5 gives both variational sources, the energy-exchange identity and a local energy bound; these ingredients can then be solved in a specified background and material configuration.

If matter cores mainly involve closed shear and orientation while mass and gravity involve longitudinal states, the two need a mutual-driving channel. Independent free-wave equations describe propagation but cannot show how one core produces a longitudinal background and receives feedback.

Coupling must supply both source and reaction, with compatible energy, symmetry, dimensions and boundaries. Chapter 5 provides a variational lowest-order cubic candidate. This page keeps that structure without treating the physical requirement as proof of a unique interaction.

02Density and transverse curl in the candidate

Use compression-positive $\sigma=-\nabla\cdot\mathbf u_L$ and transverse curl $\boldsymbol\Omega_T=\nabla\times\mathbf u_T$. Both are dimensionless for length-dimensional displacement. Curl is axial and its square a rotational scalar. Manuscript (5.4.19) gives

$g_{LT}$ has $\mathrm{J\,m^{-3}}$ units and specifies both density-dependent transverse restoration and transverse driving of volume response. With no time derivatives, interaction energy is $+g_{LT}\sigma|\boldsymbol\Omega_T|^2$, opposite to the Lagrangian sign.

Rotational invariance and correct units permit this local candidate without uniquely fixing its coefficient or microscopic origin. Local curl is not automatically a complete topological core. The selected variables and truncation define its applicability.

03Reciprocal sources from one variation

Using $\delta\sigma=-\nabla\cdot\delta\mathbf u_L$, longitudinal variation is $g_{LT}\int|\boldsymbol\Omega_T|^2\nabla\cdot\delta\mathbf u_Ldt\,d^3x$. For constant coupling and compatible boundaries, integration by parts gives $-g_{LT}\nabla|\boldsymbol\Omega_T|^2$.

Transverse variation uses $\delta\boldsymbol\Omega_T=\nabla\times\delta\mathbf u_T$, with the factor two from differentiating the square:

Integrating curl by parts and removing the specified surface term gives

Both have generalized force-density units. A gradient source is curl-free; a curl source is divergence-free, preserving sector constraints. These nonlinear sources are not electric currents, which need a separate conserved-flow identification and coupling.

Adding them to the sector equations supplies driving and reaction from the same candidate. Spatially varying coupling would produce coefficient derivatives; the displayed expressions assume constant $g_{LT}$.

04Energy conversion is distinct from dissipation

Each sector has power $\mathbf J_A^{\rm nl}\cdot\partial_t\mathbf u_A$. With the same boundary integration, longitudinal power integrates to $-g_{LT}\int|\boldsymbol\Omega_T|^2\partial_t\sigma d^3x$, while transverse power integrates to $-2g_{LT}\int\sigma\boldsymbol\Omega_T\cdot\partial_t\boldsymbol\Omega_Td^3x$. Their sum is

With no external source or net boundary flow, free-sector energies plus interaction energy are conserved. A sector can lose energy through transfer, temporary interaction storage or outward flux; none automatically means irreversible dissipation. Dissipation requires the coarse-grained environmental or thermal degrees of freedom that receive energy.

The cubic term does not change the quadratic spectrum around $\sigma=0$, $\mathbf u_T=0$. Nonzero backgrounds, cores and appropriately phase-matched finite-amplitude waves can permit conversion, whose rate and background spectrum still require a specified calculation.

05Completing the local energy square

The cubic term alone does not bound large-amplitude energy. The manuscript includes $K_s\sigma^2/2$ and $\lambda_T|\boldsymbol\Omega_T|^4/4$, with energy-density coefficients $K_s>0$ and $\lambda_T$:

The first square is nonnegative. For $\lambda_T\ge2g_{LT}^2/K_s$, the remaining quartic term is also nonnegative, providing a sufficient lower-bound condition for this local truncation. At equality, additional stabilizers must be checked for the residual flat direction.

$K_s$ belongs to this density energy and must match the full action; it is not automatically any gradient coefficient or ordinary elastic modulus. A pointwise lower bound does not prove a localized solution exists or check all gradients, splitting, orientation constraints and boundary modes. Stable three-dimensional structures require further topological energy and scaling checks.

06Closing the matter and gravitational interfaces

The candidate has reciprocal sources, energy exchange and a local bound that can test a specified response. It has not determined a matter core, its source strength or the gravitational far field. Local volume and long-range scale must remain distinct; ordered-vacuum feedback needs the full spectrum rather than uncoupled speed matching.

Chapter 5 section 6 separately introduces $g_LsF$ and $g_T\mathbf w\cdot\mathbf T$ in a displacement/order-parameter candidate. Its equations also derive from one functional. The master-equation page develops its actual variables, without equating distinct candidates before a mapping.

OP-SD-001 examines common dynamics, and OP-SD-006 requests source, far field and mass relations from one matter configuration. The local results provide concrete inputs to those tasks.