Longitudinal Field: Density and Gravitational Stretch

01Starting from local volume

The ASD longitudinal sector connects local compression, material sources and long-range gravitational stretch. Local density and the dimensionless long-range scale potential are defined separately; their action fixes sources, energy and feedback before they enter wavelength and clock-frequency readings.

The longitudinal field first describes local volume change. Let the displacement-type state $\mathbf u$ have length units, reference positions be $\mathbf x$, and $n_s$ be constituent or connection number density. Conserving reference counts in an invertible, orientation-preserving region gives the volume Jacobian relation

The dimensionless deformation gradient has $\det F>0$. The isotropic example separately takes $F=e^\varphi I$. Here $\varphi$ is logarithmic linear scale for that deformation, and $e^\varphi$ is the dilation factor called $\Omega$ in Chapter 4 (4.15). At $|\nabla\mathbf u|\ll1$, $\sigma=-\nabla\cdot\mathbf u_L$ is positive for compression and negative for rarefaction.

General deformation requires the determinant. Different radial and tangential stretches cannot be replaced by the cube of one directional scale. A gravitational scale read through light or a standard process must first be constitutively matched to actual deformation before it enters this density relation.

For the weak, static, long-wave and approximately isotropic frame matching in Chapter 5, $\sigma=-3\varphi$ is the linear part of $e^{-3\varphi}-1$. This algebraic mapping travels with the deformation, source and energy assumptions; it does not turn the high-frequency density kinetic kernel into a local kinetic term for independent stretch.

02Density evolution from displacement dynamics

In a homogeneous constant-coefficient background, longitudinal displacement obeys $Z_L(\partial_t^2-c^2\nabla^2)\mathbf u_L=\mathbf J_L$. The positive kinetic normalization has $[Z_L]=\mathrm{kg\,m^{-3}}$ for length-dimensional displacement. The generalized force density has $[\mathbf J_L]=\mathrm{N\,m^{-3}}$. The common speed is supplied by the effective matching.

Taking minus the divergence and using $\sigma=-\nabla\cdot\mathbf u_L$ gives

For the legacy formula ASD-WEB-WAV-001, this page identifies $\psi_L$ with the source-free density channel $\sigma$: this is a name for the same linear free equation, not an additional independent long-range gravitational potential.

The source $s_\sigma$ has $\mathrm{s^{-2}}$ units and is the divergence response of a displacement source, not ordinary matter density or an independent arbitrary input. Boundaries and zero-wave-number constraints must follow the variable change. Source-free propagation, steady sourcing and matter-comoving response have different solutions despite sharing an operator.

The free longitudinal waves page treats initial data and reception. Here the next question is whether density variables preserve the original energy.

03A shared wave equation does not permit arbitrary energy

With Fourier convention $e^{i\mathbf k\cdot\mathbf x}$, longitudinal displacement is parallel to $\mathbf k$, so $\sigma=-i\mathbf k\cdot\mathbf u_L$. Inverting at $\mathbf k\ne0$ and substituting into the original kinetic and gradient energy gives

The same transformation gives the source-free Lagrangian in manuscript (5.4.9):

The density time derivative now carries a $1/k^2$ weight. In real space, the kinetic term contains $(-\nabla^2)^{-1}$, defined with the appropriate boundaries and zero-mode treatment. The source action must also transform $\mathbf J_L\cdot\mathbf u_L$ using the same inverse relation.

Replacing it by a constant $Z_\sigma$ times $[(\partial_t\sigma)^2-c^2|\nabla\sigma|^2]/2$ introduces a different local scalar constitutive model. Identical free speeds do not preserve energy at fixed density amplitude, source normalization or static response, so that replacement is not an equivalent action.

04Which static response does a local source produce?

The local dual-field candidate couples $s=\nabla\cdot\mathbf u$ to $F(\Theta)=\sum_A\Theta_A^2-v_0^2$ through energy $g_LsF$. The real components are dimensionless, $v_0^2$ is vacuum amplitude squared, and $F$ here is a function rather than the deformation matrix. Neglecting additional terms and transverse sources, the constant-$K_u>0$ static equation is $-K_u\nabla^2\mathbf u=g_L\nabla F$.

Its divergence gives

The implication requires smoothness on all space, localized $F$, no extra singular sources, and a combination vanishing at infinity. The harmonic combination is then zero, so local amplitude drives compression. Where $F=0$ outside the core, density response also vanishes. Displacement may retain a tail without density or scale potential inheriting that tail.

Forcing $s\propto1/r$ out to infinity while retaining local energy $K_us^2/2$ with a positive asymptotically constant coefficient makes the volume integral diverge as $\int^\infty dr$. Local volume response is established, but Chapter 2's gravity needs an extended longitudinal constitutive structure. Fixed-modulus orientation can make $F$ identically zero, also requiring a properly defined alternative source.

05Long-range response from independent stretch

The manuscript's candidate extension retains dimensionless stretch potential $\varphi$ as an independent field, with gradient energy rather than a $1/r$ local density. Its action and variational equation are

The coefficient has $[Z_\varphi]=\mathrm{J\,s^2m^{-3}}$, and $\mathcal J_g[\Theta,\nabla\Theta]$ has energy-density units without time derivatives in this candidate. The static integral assumes a localized integrable source and zero potential at infinity. Integration position $\mathbf r'$ differs from observation position $\mathbf r$, with their reference distance in the denominator. The leading far field is proportional to $\int\mathcal J_gd^3r'/r$; its $|\nabla\varphi|^2\propto r^{-4}$ energy tail is integrable.

This provides a concrete far-field route. Matter must still generate the source functional, and stretch must feed back on the core. Keeping both stretch and displacement-density descriptions requires independent-field counting, constraints or scale separation. Time-derivative source dependence requires new canonical momenta and the full energy. The source-coupling task tracks these matches.

Source support matters here. For the spherical source in Chapter 4 (4.37)–(4.38), an exactly source-free exterior beyond $R_{\rm core}$ gives the exact potential $\gamma_\rho Q_\rho/(4\pi r)$. Absolute integrability $\int|s_\rho|d^3r<\infty$ alone still allows a source tail: finite-radius solutions retain inner and outer shells, and $1/r$ is then the far-field asymptote. The full integral is retained rather than treating a finite source integral as sufficient for an exact exterior solution.

06How spatial scale enters gravity readings

Chapter 5 uses weak-field conventions $\Phi_N=-c^2\varphi$ and $\mathbf g=c^2\nabla\varphi=-\nabla\Phi_N$. The dimensional potential, acceleration and dimensionless scale potential must match Chapter 2's references, scales and boundaries. Local number density is not an instrument's length or frequency reading merely because dimensions can be assigned.

ASD's measurement chain still starts from spatial scale. The background couples to propagating wavelength and atomic or other standard-process response; phase or cycle counting then forms a frequency comparison. The redshift page follows this sequence. Chapter 4 also notes that identical propagation and absorber scaling may cancel in a resonance ratio, requiring joint calculations of source, propagation and standard response.

Longitudinal dynamics supplies sources and constitutive interfaces to that chain, without replacing it by an independently changing time-flow rate. Static backgrounds, free density waves and detected radiation polarizations still follow their own equations and receivers.