Gravity · Spatial Structure and Observables

01Understanding spatial structure through gravitational redshift

Place two atomic clocks of the same kind at different heights. After accounting for signal transfer and environmental effects, their frequencies still show a systematic offset associated with the gravitational potential difference. A clock records repetitions of a physical process, so this observation also raises a question about matter and space: why does the same standard process change with position? ASD approaches gravity through this metrological question, tracing frequency changes to the spatial length state on which the standard process depends.

The physical proposition of Chapter 2 is that matter produces static longitudinal stretching of surrounding space: an extension of length along the gradient of the spatial state. This stretching changes the characteristic length and associated wavelength of a standard process; with the local vacuum propagation speed $c$ unchanged, its frequency and period change accordingly. When the stretching varies with position, the interaction between a body and surrounding space is no longer balanced. The gradient of the longitudinal spatial state then produces gravitational action. Frequency comparisons read out scale differences between positions, while gravitational acceleration describes the local variation of that state.

The objects studied here include the matter source that establishes the background, test bodies within it, propagating light, and the clocks and reference conditions used to compare processes. ASD places them in a common picture of spatial dynamics: matter changes the spatial state, and the changed space acts on matter's motion and on light propagation. Time enters the calculation as a process-counting parameter in a common reference; changes in clocks are assigned to the scale response of their actual periodic processes.

Macroscopic constitutive relations make this interpretation calculable. Here a constitutive relation is a physical input specifying how the spatial state couples to length, standard processes and energy. Conservation laws and variation then yield results from those inputs. The picture of matter as a localized spatial structure guides the source-coupling problem. Calculations in this module use a specified macroscopic mass source; they do not require an already obtained stable topological solution for a particular particle.

02From scale variation to the gravitational field

First consider an isolated, static, spherical, uncharged source, neglecting the test body's backreaction on it. Let $M$ denote the source mass and $G_0$ the gravitational coupling calibrated in this background. The radial coordinate $r$ is defined by the sphere's area $A=4\pi r^2$ and is called the areal radius. It must be distinguished from the accumulated true spatial length along a radial path. Define the gravitational length $\ell_g=G_0M/c^2$. The model used here applies outside the matter source, in the region $r>2\ell_g$.

Let $\Omega(r)$ be the linear expansion coefficient, normalized by $\Omega\to1$ in the far field. For standard processes of the same kind, already expressed in a common reference, the manuscript adopts $L(r)/L_0=\Omega(r)$ and $f(r)/f_0=\Omega(r)^{-1}$. Here $L_0$ and $f_0$ are the far-field reference characteristic length and frequency; $L(r)$ and $f(r)$ are the corresponding quantities for the process at position $r$. This process coupling translates spatial stretching into comparable frequencies. It is not a response law for arbitrary devices deduced from unit definitions alone. The spherical exterior scale law is

The larger value of $\Omega$ near the source represents stronger stretching of the corresponding linear scale. Its inverse $\Phi$ is a dimensionless line-density coefficient. Here $\Phi$ does not denote the effective wave field used in the quantum module, nor is it the three-dimensional medium density $\rho_s$, which has mass-density dimensions. Volume density also depends on deformation in every direction. If $\mathbf F_s$ denotes the deformation gradient mapping a small reference line element into its actual counterpart, its determinant $\det\mathbf F_s$ gives the local volume ratio. A conserved medium obeys $\rho_s=\rho_0/\det\mathbf F_s$, where $\rho_0$ is the reference volume density. Only when all three principal directions stretch by the same $\Omega$ does $\rho_s/\rho_0=\Phi^3$ follow. Radial stretching requires the volume ratio of its actual deformation.

To enter the source and motion equations, define the dimensionless variable $X=\Phi^2$ and denote the gravitational potential, with dimensions of speed squared, by $\Psi$:

The exterior of an attractive source has $\Psi<0$, with $\Psi\to0$ at large distance. In a weak field, for a slowly moving test body with negligible self-gravity, adopt the leading external potential energy $U_{\rm ext}=m_0\Psi$. Here $m_0$ is the inertial and passive gravitational mass parameter normalized through universal coupling. From $\mathbf F=-\nabla U_{\rm ext}$, the gradient action per unit mass is

This result writes familiar Newtonian gravity as the leading dynamical expression of a spatial-state gradient. A body near Earth's surface, for example, encounters slightly different longitudinal spatial states on its nearer and farther sides. ASD interprets the net action through the body's stress coupling to this nonuniform background. In free fall the net action changes the body's momentum; static weighing measures the support force needed to balance it. The full pressure function and interface coupling belong to the deeper field-dynamics problem and cannot be read off from the gradient formula alone.

For a general weak static mass distribution, the manuscript adopts the following source equation and far-field flux relation. $\rho_g$ is the effective gravitational source density, distinct from the spatial medium's volume density $\rho_s$ introduced above; $d\mathbf S$ is the outward area vector:

The gradient and Laplacian here are defined in reference spatial coordinates. In a spherical source-free region, $\nabla^2X=0$ reduces to $(r^2X')'=0$, where a prime denotes differentiation with respect to $r$. Integrating twice gives $X=C_0+C_1/r$. Far-field normalization fixes $C_0=1$, and the flux fixes $C_1=-2G_0M/c^2$, connecting to the exterior background above. This shows the consistent normalization of source strength and exterior scale. The weak static source equation and weak-field redshift do not, by themselves, determine a complete constitutive law for dynamical strong fields.

The scale and normalization of gravitational coupling

The manuscript organizes $G_0$ through background density, propagation speed and an intrinsic length, tying the macroscopic coupling to a possible microscopic calculation:

Here $\rho_0$ has mass-density units, $\ell_0$ is an intrinsic length, and the dimensionless $\xi_G$ describes structural coupling and longitudinal–transverse conversion efficiency. The combination $c^2/(\rho_0\ell_0^2)$ has the units of $G_0$; the numerical efficiency must still follow from a specified structure. With a prescribed slowly varying $G(\mathbf x)$, the static extension is $\nabla\cdot[G(\mathbf x)^{-1}\nabla\Psi]=4\pi\rho_g$. If $G$ instead depends on a dynamical field, its variation must enter the action. Subsequent exterior calculations keep the same calibrated $G_0$ fixed.

03Propagation and motion in the same spatial background

Knowing the scale change at a point is not yet enough to describe a path through the background. Spherical symmetry does not make radial and tangential responses identical: the central source's gradient selects the radial direction, while symmetry relates tangential directions on the same sphere. The manuscript adopts the true radial length $d\ell_r=\Omega\,dr$. Because $r$ is already defined by the actual sphere area, the tangential arc length is $r\,d\varphi$. In the orbital plane, the two contributions combine as

Here $d\ell_s$ is the true spatial line element and $\varphi$ is the azimuthal angle. The radial coefficient is an explicit constitutive input. The tangential term already contains the information encoded by the areal radius and is not multiplied by $\Omega$ again. This division expresses ASD's static spatial anisotropy and supplies the common background for calculating light paths and material trajectories.

Spatial stretching affects both the path and the counting standard. Establish the counting parameter $t$ with a far-field standard. A local standard process accumulates $dN=f_0\Phi\,dt$ cycles during $dt$, so the speed calibrated by local length and cycles is $v_*=f_0d\ell_s/dN=\Phi^{-1}d\ell_s/dt$. Setting the local speed of light to $v_*=c$ converts true spatial length into a propagation optical path in the common reference:

The reference optical path $d\ell_{\rm opt}$ must be distinguished from $d\ell_s$. For radial propagation, one factor of $\Omega$ comes from stretching the path itself, and another comes from the wavelength stretching of the standard process and the conversion of counts. Propagation along a sphere retains the corresponding tangential coefficient. Integrating $d\ell_{\rm opt}/c$ along a light path gives the propagation counting interval. With endpoints fixed, a small change of the actual path produces no first-order change in optical path. This is Fermat's stationary-path condition, which leads to the deflection and delay calculations.

Material motion additionally requires the local inertial relation. The Length Vector Effect (LVE) describes the directional length response induced by motion. Combining it with a separately stated constitutive weighting of inertia, Chapter 2 obtains the momentum of the core and accompanying field as a whole, $\mathbf p=m_0\gamma\mathbf v$. The work relation and rest-energy normalization then give the free Lagrangian $\mathcal L_0(v)=-m_0c^2\sqrt{1-v^2/c^2}$. Here $\gamma=(1-v^2/c^2)^{-1/2}$ represents the inertial gain of the moving structure. In a static gravitational background, use the local speed $v_*$ and adopt universal static energy coupling for a fixed internal state, $\mathcal L\,dt=\mathcal L_0(v_*)\Phi\,dt$. This yields

A dot denotes differentiation with respect to the common-reference counting parameter $t$. The quantity $\mathcal L$ is the Lagrangian; integrating $\int\mathcal L\,dt$ along a path gives the action, whose stationary paths satisfy the equations of motion. Spatial length determines the local speed along the path, while energy coupling determines how the local structure's energy is expressed in the common reference. Both enter the action. Its symmetries give energy and angular-momentum conservation; eliminating $t$ then yields the orbit equation. Material dynamics thus continues from ASD's LVE and constitutive inertia, without replacing that physical starting point by standard spacetime field transformations.

04From spatial relations to observables

Gravitational redshift is the most direct local comparison. Standard processes of the same kind at two positions obey $f(r_2)/f(r_1)=\Phi(r_2)/\Phi(r_1)$. The weak-field expansion gives the relative frequency shift $[\Psi(r_2)-\Psi(r_1)]/c^2$. Raising a clock moves it into a less stretched region: its associated wavelength shortens and its common-reference frequency increases. The redshift article separates spatial scale, wavelength, cycle counts and actual readings step by step.

Light deflection and Shapiro delay measure, respectively, the transverse variation and the accumulated contribution of the same optical-path structure. The optical-path gain is larger on the side nearer the source, continuously turning the wavefront toward it. Accumulating that gain along the path between specified endpoints gives an additional propagation counting interval. For distant endpoints and small deflection in a static weak field, the deflection angle is $4G_0M/(c^2b)$, where $b$ is the impact parameter normalized at large distance. The delay is a logarithmic term relative to propagation without gravity between the same endpoints in the same reference coordinates; endpoint and round-trip link conditions cannot be omitted. The explicit integrations are developed in light deflection and propagation delay.

Planetary orbits connect static spatial stretching with the inertia of motion. As a planet approaches or recedes from the central source, the path scale, local speed and structural energy change continuously. The common action makes a radial oscillation and an azimuthal revolution no longer match by exactly $2\pi$. Retaining the first weak-field and speed-squared corrections relative to Newtonian dynamics, namely first post-Newtonian accuracy, gives a periapsis advance per orbit of $6\pi G_0M/[a(1-e^2)c^2]$, where $a$ and $e$ are the semimajor axis and eccentricity of the zeroth-order ellipse. Actual ephemeris comparisons also include perturbations from other bodies and source multipoles. This first-order result does not supply complete second-order precession. The orbital article retains the continuous derivation from action through conserved quantities to the precession term.

The same exterior optical path also describes critical light trajectories and capture scales around static compact objects. Areal radius, true radial length and reference optical path must be kept distinct, as they have different limits near the boundary. The manuscript further proposes a strong-field boundary where the continuous spatial state terminates. This physical hypothesis requires dynamics of boundary formation and stability; a zero of $X$ alone does not prove it. An actual bright-ring image additionally requires emission, absorption and instrument-response models.

05Different physical interpretations of the same observational relations

ASD first specifies the spatial length state, standard-process response and energy coupling, then uses conservation and variation to calculate consequences. General relativity organizes the corresponding relations through spacetime geometry, proper time and geodesics. Chapter 2 explicitly notes that, under this module's static spherical constitutive relations, the spatial line element, optical path and orbital action have the same mathematical form as the corresponding Schwarzschild exterior expressions.

This identity of form means that the redshift, deflection, delay and precession relations above can yield the same numerical values when coordinates, reference conditions and approximation order are matched. Experiments test final observables; these identical results alone cannot select an underlying physical interpretation. ASD must further explain the spatial-dynamical origin of its constitutive relations. A discriminating prediction additionally requires a difference generated by independently determined dynamics, together with a check that the difference cannot be absorbed into mass, distance or other fitted parameters.

Observational material, calculation scope and discriminating power for each phenomenon are collected in the gravity tests matrix. It separates reproduced results, interpretive differences and experimental questions still requiring calculation. An identical formula is not automatically counted as an independent confirmation.

06Scope and relations requiring further closure

This module establishes a macroscopic constitutive theory in a static spherical exterior background at fixed $G_0$. The scale law, radial length law, standard-process coupling and energy coupling each have distinct roles, and the results are conditional on their respective inputs. The scope of the weak-field source equation and first-order approximations is stated at each step. Rotation, dynamical sources, strong self-gravity or different kinds of standard require the corresponding background and device responses to be formulated anew.

Following this physical interpretation further requires tracing the macroscopic inputs to a common spatial dynamics: how localized matter structures generate longitudinal fields, how standard processes respond to those fields, and how complete strong-field constitutive relations follow from field equations and boundary conditions. The corresponding tasks concern microscopic source coupling, strong-field closure, nondegenerate orbital observables and rotating backgrounds. They identify what must connect macroscopic calculations to microscopic structural solutions.

07Shared upstream theory and derivation sources

Space Dynamics places gravity's longitudinal structure and electromagnetism's transverse response within a common continuum-field problem. Further reading begins with the longitudinal field and continuum dynamics. FCC remains a candidate microscopic realization within that discussion, not a prerequisite for this page's macroscopic calculations.

The metrological and kinematic premises used here are set out in metrology, time and processes and dynamics. Detailed manuscript locations, equation numbers and version mappings follow.