Physical–Observation Dual-Axis(PODA) Relativity: Continuous Observation Fields, Recovery of General Relativity, and Experimental Tests

1. Introduction

To assign a frequency to light, one must specify the clock that receives it. To assign an energy to a particle, one must specify the reference state against which its four-momentum is resolved. Clocks and rods thus belong to the physical meaning of relativistic quantities, and general relativity already describes this dependence.[1,2] The question pursued here concerns the dynamics of those conditions of measurement. If an observation state evolves as a physical field, what equations govern its motion, what stress-energy does it carry, and how does that stress-energy enter the geometry in which measurements are made?

The Physical–Observation dual-axis(PODA) structure writes the formation of a record as $r=F(X,s)$. The state under observation is X; the objective observation state through which it produces a record is s. An observing system is characterized by physically distinguishable conditions of record formation, without reference to human consciousness. The distinction between the two inputs follows the origin of a change in a record: the measured state may vary, or the conditions of its measurement may vary. For constant s, the formation law describes fixed observation conditions. For a field that varies over spacetime, it must be supplemented by a rule for comparing neighboring observation states and by equations that determine their evolution.[3]

A fixed observation state selects a section of the joint state space. Restricting the connection to that section is a geometric operation; requiring the section to remain fixed under evolution is a dynamical condition. The latter has physical content. An initially constant observation field subject to a nonzero potential gradient will begin to move. It can remain constant only where its own field equation permits this. The recovery of general relativity must therefore be established from the coupled equations as well as from the restriction of the connection.[3,4]

We study the classical branch in which the spacetime metric and observation field are independent variables. The observation field takes values in a space with a positive-definite kinetic metric and enters a minimally coupled two-derivative action together with gravity and matter. Its kinetic metric is induced by the geometry of response projectors; variation of the action determines its propagation and stress-energy. The relation to the parent response action, including the conditions for a consistent reduction, is developed in Section 3. The construction preserves the four-dimensional character of physical spacetime and specifies a relativistic model whose field equations can be investigated within those conditions.[4,5]

The connection with measurement then follows the same sequence as the dynamics. A solution first determines the metric and observation state; the metric determines clock rates and light propagation, while the formation law determines the recorded response. After proving recovery on a fixed section and deriving the initial-value evolution, we follow these two effects into coherent phase measurements and differential clock comparisons. This separation identifies what each experiment measures and how the same observation field enters both.

Unless units are explicitly restored, we set $c=1$, use signature $(-+++)$, and write ${\kappa }_E=8\pi G_N$. The four-dimensional physical spacetime is M; the observation space is $\mathcal{S}$, with local coordinates $s^a$. The kinetic metric $G_{ab}$ and the Einstein tensor $G_{\mu \nu }$ are distinguished by their index types. Matter variables are denoted collectively by $\mathsf{\Psi }$, the observation formation law by F, and connection curvature by $\mathcal{F}$. The number of observation-space coordinates does not change the dimension of physical spacetime.

2. Continuous Dual-Axis Geometry and Restriction to a Section

2.1. Two Kinds of State and Continuous Histories

A record depends both on what is measured and on the physical conditions that make the measurement possible. Physical–Observation theory expresses this dependence through the formation law $y=F(X,s)$, where X is the physical state under comparison and s is the objective observation state. Clocks, frames, reference motion, and response apparatus realize observation states that can be distinguished by their response behavior. The distinction between state and spacetime must be kept equally precise: X denotes a physical state, whereas $x^{\mu }$ denotes a spacetime coordinate. A single field configuration may specify data at infinitely many spacetime points; its parameters and those points therefore belong to different spaces.[3]

On a smooth regular stratum, changes of the two inputs define the joint domain and its tangent decomposition,

$${\mathcal{B}}^{\times }\subseteq \mathcal{X}\times \mathcal{S},T{\mathcal{B}}^{\times }=H_X\oplus H_S.$$

(2.1)

The two tangent directions describe different roles in a change of state; their separation does not increase the dimension of physical spacetime. Relativistic propagation takes place on a four-dimensional manifold M. To follow the observation conditions continuously, assign a section of the observation-state bundle over M, written locally as $s:M\to \mathcal{S}$. Every worldline $\gamma :\tau \mapsto x\left(\tau \right)$ then carries the observation history $s_{\gamma }=s\circ \gamma$, whose rate of change is

$${\displaystyle \frac{\mathrm{d}{s}_{\gamma }^a}{\mathrm{d}\tau }}=u^{\mu }{\partial }_{\mu }{s}^a,u^{\mu }={\displaystyle \frac{\mathrm{d}{x}^{\mu }}{\mathrm{d}\tau }}.$$

(2.2)

On a differentiable branch of the record, a joint history $q\left(\tau \right)=\left(X\right(\tau ),s(\tau \left)\right)$ gives

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}\tau }}=D_{X}F\dot{X}+D_{S}F\dot{s}.$$

(2.3)

The first term follows the change of the physical state; the second follows the change of the conditions under which it is recorded. Finite exposure and discrete sampling act on this continuous history to produce the measured record.

Holding the observation state fixed defines the map ${\iota }_{s_0}:X\mapsto (X,s_0)$. For a given $s_0$, this is a section of the joint domain with formation law $F\circ {\iota }_{s_0}$. The map $X\mapsto (X,s(X\left)\right)$ also defines a section, but its tangent generally has an observation-direction component. A fixed-observation section thus holds the observation conditions on a specified branch; it is a section of the state domain, not an instantaneous spatial slice. This distinction leaves intact the treatment of moving reference frames in general relativity. Recovery requires two results: the geometric restriction must be exact, and the equations of motion must preserve the restricted states.

2.2. Connections, Mixed Curvature, and Comparison Along a History

Let $m:{\mathcal{B}}^{\times }\to M$ be the structural map of the relativistic realization, and let $O^{+}(M,g_L)$ be the oriented and time-oriented orthonormal frame bundle. The Levi–Civita connection $\omega$ pulls back along m to ${\omega }_B=m^{*}\omega$, whose curvature satisfies

$${\mathsf{\Omega }}_B=\mathrm{d}{\omega }_B+{\omega }_B\wedge {\omega }_B=m^{*}\mathsf{\Omega }.$$

(2.4)

In local coordinates $(X^i,s^a)$ adapted to the two axes,

$$\begin{array}{cc}\hfill {\omega }_B& ={\omega }_i\mathrm{d}{X}^i+{\omega }_a\mathrm{d}{s}^a,\hfill \\ \hfill {\mathsf{\Omega }}_B& =\frac{1}{2}{\mathsf{\Omega }}_{ij}\mathrm{d}{X}^i\wedge \mathrm{d}{X}^j+{\mathsf{\Omega }}_{ia}\mathrm{d}{X}^i\wedge \mathrm{d}{s}^a+\frac{1}{2}{\mathsf{\Omega }}_{ab}\mathrm{d}{s}^a\wedge \mathrm{d}{s}^b,\hfill \\ \hfill {\mathsf{\Omega }}_{ia}& ={\partial }_i{\omega }_a-{\partial }_a{\omega }_i+[{\omega }_i,{\omega }_a].\hfill \end{array}$$

(2.5)

Mixed curvature measures the dependence of local comparison on the order of the two changes: physical state followed by observation state, or observation state followed by physical state. Along a joint history, both changes enter the parallel transport of a column V of frame components,

$${\displaystyle \frac{\mathrm{d}V}{\mathrm{d}\tau }}+\left({\omega }_i{\dot{X}}^i+{\omega }_a{\dot{s}}^a\right)V=0.$$

(2.6)

For a smoothly varying section $s=s\left(X\right)$, the pulled-back connection and curvature have components

$$\begin{array}{cc}\hfill {\omega }_i^{\left(s\right)}& ={\omega }_i+{\omega }_a{\partial }_i{s}^a,\hfill \\ \hfill {\mathsf{\Omega }}_{ij}^{\left(s\right)}& ={\mathsf{\Omega }}_{ij}+{\mathsf{\Omega }}_{ia}{\partial }_j{s}^a-{\mathsf{\Omega }}_{ja}{\partial }_i{s}^a+{\mathsf{\Omega }}_{ab}{\partial }_i{s}^a{\partial }_j{s}^b.\hfill \end{array}$$

(2.7)

Substituting $\mathrm{d}{s}^a={\partial }_i{s}^a\mathrm{d}{X}^i$ into Eq. (2.5) gives these expressions directly. On a fixed section whose tangent space lies in $H_X$, the pulled-back terms containing $\mathrm{d}{s}^a$ vanish. The remaining connection and curvature are exactly ${\iota }_{s_0}^{*}{\omega }_B={\iota }_{s_0}^{*}{\omega }_X$ and ${\iota }_{s_0}^{*}{\mathsf{\Omega }}_B={\iota }_{s_0}^{*}{\mathsf{\Omega }}_{XX}$.[3]

These relations establish how to compare one geometric object along a dual-axis history. Their physical content depends on the structural map and the response dynamics. If m is constant in the observation directions, the pullback has no new curvature in those directions. A pure gauge change of local frame likewise leaves the physics unchanged. Nontrivial observation response consequently requires a nontrivial realization in the structural map and response fields; introducing the total connection alone supplies a transport rule, not a second gravitational field.

2.3. Clocks, Spatial Response, and Lorentzian Geometry

The response measures changes of the visible subspace. Let $\mathcal{E}\to M$ be the parent Hermitian bundle, represent this subspace by a fixed-rank projector $P^2=P=P^{†}$, and equip the bundle with a unitary connection $D=\mathrm{d}+\mathbf{A}$. The covariant change of the projector defines the quadratic form

$$D_{v}P=v\left(P\right)+[\mathbf{A}\left(v\right),P],g_R(v,w)={\displaystyle \frac{1}{2}}ReTr\left(D_{v}P{D}_{w}P\right).$$

(2.8)

Hermiticity of $D_{v}P$ gives $g_R(v,v)\ge 0$, with a possible kernel along directions that leave the response unchanged. A temporal direction with negative norm must therefore be supplied by the clock structure.

Represent the clock by a nowhere-vanishing one-form ${\alpha }_{\tau }$, and choose its future-directed flow T with ${\alpha }_{\tau }\left(T\right)=1$. The complementary directions form $H=ker{\alpha }_{\tau }$. When ${DP|}_H$ is injective, the response resolves all these directions and $h_R=g_R{|}_{H\times H}$ is positive definite. Extend this spatial form by $h_R(T,·)=0$ and introduce a positive clock scale $c_{\tau }$. The resulting metric is

$$g_L=-c_{\tau }^2{\alpha }_{\tau }\otimes {\alpha }_{\tau }+h_R,TM=\mathbb{R}T\oplus H.$$

(2.9)

The unique decomposition $v=aT+v_H$ gives $g_L(v,v)=-c_{\tau }^2{a}^2+h_R(v_H,v_H)$. Thus $g_L$ is nondegenerate and has Lorentzian signature $(-,+,+,+)$: the clock fixes the temporal scale, and the response fixes the positive spatial metric.[4]

Clock and response must also agree with propagation. If propagation already defines a smooth Lorentzian quadratic cone with conformal representative $q_L$, agreement requires ${\alpha }_{\tau }$ to be a $q_L$-timelike covector and the temporal and spatial parts to satisfy

$$\begin{array}{cc}\hfill T& ={\displaystyle \frac{q_L^{♯}{\alpha }_{\tau }}{q_L^{-1}({\alpha }_{\tau },{\alpha }_{\tau })}},\hfill \\ \hfill h_R& ={\mathsf{\Omega }}^2{q}_L{|}_H,c_{\tau }^2=-{\mathsf{\Omega }}^2{q}_L(T,T),\mathsf{\Omega }>0.\hfill \end{array}$$

(2.10)

The first relation makes the temporal flow orthogonal to the spatial directions, $q_L(T,H)=0$. The second gives both blocks the same scale factor, so that $g_L={\mathsf{\Omega }}^2{q}_L$. These are local geometric conditions. A global clock representation ${\alpha }_{\tau }=\mathrm{d}t$ need not exist, and a global initial-value problem requires more than these local conditions. The evolution analysis will therefore be carried out in a globally hyperbolic region with a compatible spacelike initial hypersurface.

The metric acquires its observational meaning through a reference state. A reference four-velocity u of unit norm assigns to a four-momentum p the energy

$$E=-p_{\mu }{u}^{\mu },g_{L,\mu \nu }{u}^{\mu }{u}^{\nu }=-1,{\displaystyle \frac{\mathrm{d}E}{\mathrm{d}\tau }}=-u^{\mu }{\left({\nabla }_{u}p\right)}_{\mu }-p_{\mu }{a}^{\mu },a={\nabla }_{u}u.$$

(2.11)

Metric compatibility and the product rule give the last equality. The recorded energy changes through the evolution of the momentum and through the acceleration of the reference. To calculate that change, the coupled dynamics must determine both p and u.

3. The Origin of the Observation Kinetic Metric and the Independent Variables

3.1. From the Parent Response Projector to Observation-State Coordinates

To determine how response geometry evolves, one must specify which quantities are varied. In the parent-field branch these are $(g_L,\mathbf{A},P,\mathsf{\Psi })$, with $\mathsf{\Psi }$ denoting matter and other response variables and P constrained to the manifold of fixed-rank projectors. A real, local, parity-even action retaining the lowest-order kinetic terms is

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\mathrm{par}}={\int }_M\sqrt{-g_L}{\mathrm{d}}^{4}x[& {\displaystyle \frac{R\left(g_L\right)-2{\mathsf{\Lambda }}_X}{2{\kappa }_E}}-{\displaystyle \frac{{\langle {\mathbf{F}}_{\mu \nu },{\mathbf{F}}^{\mu \nu }\rangle }_{\mathfrak{u}}}{4g_{★}^2}}\hfill \\ \hfill & -{\displaystyle \frac{f^2}{2}}ReTr\left(D_{\mu }P{D}^{\mu }P\right)-U_C(P,\mathsf{\Psi })+{\mathcal{L}}_{\mathrm{m}}],\hfill \end{array}$$

(3.1)

where $\mathbf{F}=\mathrm{d}\mathbf{A}+\mathbf{A}\wedge \mathbf{A}$ and ${\langle A,B\rangle }_{\mathfrak{u}}=-ReTr\left(AB\right)$ is the inner product on the anti-Hermitian curvature of the parent connection. The constants ${\kappa }_E$ and ${\mathsf{\Lambda }}_X$ set the gravitational normalization. The response realization and its internal dynamics determine the coefficients and potential; the decomposition into physical and observation states alone leaves their values undetermined.[4,5]

Choose local coordinates $s^a$ on the projector manifold, write $P=P\left(s\right)$, and set $P_a=\partial P/\partial s^a$. In a region where the parent gauge group preserves this response manifold, its generators $t_{\alpha }$ define tangent vector fields $k_{\alpha }^a$ such that

$$[t_{\alpha },P]=P_a{k}_{\alpha }^a,{\mathcal{D}}_{\mu }{s}^a={\partial }_{\mu }{s}^a+{\mathbf{A}}_{\mu }^{\alpha }{k}_{\alpha }^a,D_{\mu }P=P_a{\mathcal{D}}_{\mu }{s}^a.$$

(3.2)

Substitution into the projector kinetic term yields

$$\begin{array}{cc}\hfill G_{ab}\left(s\right)& =f^{2}ReTr\left(P_a{P}_b\right),\hfill \\ \hfill -{\displaystyle \frac{f^2}{2}}ReTr\left(D_{\mu }P{D}^{\mu }P\right)& =-{\displaystyle \frac{1}{2}}{G}_{ab}\left(s\right){\mathcal{D}}_{\mu }{s}^a{\mathcal{D}}^{\mu }{s}^b.\hfill \end{array}$$

(3.3)

The target-space metric $G_{ab}$ is therefore inherited from the kinetic energy of the projector. Its sign follows directly: for every real tangent vector $v^a$, $G_{ab}{v}^a{v}^b=f^{2}ReTr\left({\left(v^a{P}_a\right)}^2\right)\ge 0$. It is positive definite wherever $P\left(s\right)$ is an immersion. A direction with $v^a{P}_a=0$ changes the coordinates without changing the response; it lies in the kernel of the parameterization. Such directions must be removed by passage to the corresponding quotient before the kinetic metric can be nondegenerate. The two metrics serve distinct contractions: $g_{L,\mu \nu }$ acts on spacetime propagation directions, and $G_{ab}$ on observation-state directions. Their indices belong to different spaces throughout.

3.2. Two Representations of the Same Observation Field

After parameterizing P by s, the independent variables become $(g_L,\mathbf{A},s,\mathsf{\Psi })$. The chain rule for variations gives

$$\delta P=P_a\delta s^a,{\displaystyle \frac{\delta \mathcal{S}}{\delta s^a}}=〈{\displaystyle \frac{\delta \mathcal{S}}{\delta P}},P_a〉.$$

(3.4)

If the $P_a$ span the tangent space of the projector manifold, these components vanish precisely when the tangential Euler derivative with respect to P vanishes. The variables s and $P\left(s\right)$ thus describe one observation field, with one kinetic contribution and one variational equation expressed in two ways. Restricting the coordinates to a proper submanifold changes the conclusion: Eq. (3.4) then enforces only the tangent equations. The omitted transverse Euler components must also vanish for the restricted field to solve the parent system.

The same geometry constrains the potential. Since the unitary group acts transitively on projectors of a fixed rank, a pointwise potential depending only on P and invariant under all unitary conjugations is constant. Nontrivial dependence in $U_C$ must come from covariant response structures or matter couplings in the theory. Writing the potential as $U\left(s\right)$ expresses that dependence in coordinates; it does not create it.

On a branch where the parent connection acts trivially on the selected response directions, or where a consistent reduction has already been carried out, ${\mathcal{D}}_{\mu }{s}^a$ reduces to ${\partial }_{\mu }{s}^a$. One then obtains the continuous observation-field action used below,

$${\mathsf{\Gamma }}_{\mathrm{obs}}[g_L,s]=-{\int }_M\sqrt{-g_L}{\mathrm{d}}^{4}x\left[{\displaystyle \frac{1}{2}}{G}_{ab}\left(s\right){\partial }_{\mu }{s}^a{\partial }^{\mu }{s}^b+U\left(s\right)\right].$$

(3.5)

The equations of the eliminated fields remain conditions on this reduction. A varying projector generally carries a gauge current, for example, and setting $\mathbf{A}$ to zero is admissible only when the corresponding parent equation is satisfied. Under the reduction conditions, Eq. (3.5) describes a branch of the parent system. Alternatively, it can define a chosen macroscopic response theory in its own right. Its solutions lift to parent solutions only when those same reduction conditions hold.

3.3. The Variational Relation Between the Response Metric and the Gravitational Metric

The clock–response construction gives the metric its meaning in terms of clocks and rulers. Its dynamical role follows from the choice of independent variables in the action. In the independent parent-field branch and its consistent reduction, $g_L$ and the observation state are varied separately; Eq. (2.9) relates the resulting field solutions to reference clocks and rulers. Taking $g_L$ as an independent field is essential to this variational problem. A composite metric instead leads to a different set of equations.

For a purely composite realization $g_L=\mathfrak{G}[P,{\alpha }_{\tau },T,\dots ]$, substitute the defining relation into the action before varying. A change of the projector now changes the metric as well, giving the schematic chain rule

$${\displaystyle \frac{\delta {\mathcal{S}}_{\mathrm{comp}}}{\delta P}}={\left.{\displaystyle \frac{\delta \mathcal{S}}{\delta P}}\right|}_{g_L}+{\left(D_P\mathfrak{G}\right)}^{*}{\displaystyle \frac{\delta \mathcal{S}}{\delta g_L}}.$$

(3.6)

When $\mathfrak{G}$ contains derivatives, the star denotes the formal adjoint obtained by integration by parts. Since arbitrary independent variations $\delta g_L$ are no longer available, this composite variation does not separately yield the Einstein equation. The distinction fixes how observation kinetic energy, energy–momentum, and gravitational backreaction are related. The following derivation, including recovery on a fixed-observation section, uses the independent-metric branch of Eq. (3.5).

4. Common Action and Covariant Field Equations

4.1. A Low-Derivative Action for the Continuous Observation Field

We henceforth write $g=g_L$ and use the regular two-derivative observation action. The field $s:M\to \mathcal{S}$ assigns an observation state to every spacetime point, with kinetic metric $G_{ab}\left(s\right)=Z\left(s\right){\overline{h}}_{ab}\left(s\right)$. In the branch where G is positive definite, U depends only on s, and the matter action has no direct dependence on s, the common action is[5]

$$\begin{array}{cc}\hfill \mathsf{\Gamma }[g,s,\mathsf{\Psi }]={\int }_M\sqrt{-g}[& {\displaystyle \frac{R-2{\mathsf{\Lambda }}_X}{2{\kappa }_E}}-{\displaystyle \frac{1}{2}}{G}_{ab}\left(s\right)g^{\mu \nu }{\partial }_{\mu }{s}^a{\partial }_{\nu }{s}^b-U\left(s\right)]{\mathrm{d}}^{4}x+{\mathsf{\Gamma }}_X[g,\mathsf{\Psi }].\hfill \end{array}$$

(4.1)

The constant ${\mathsf{\Lambda }}_X$ represents any constant term inherited from the physical-axis action; it vanishes when that action has no such term. The functions $G,U$ encode the reduced observation dynamics. Together with compatible initial and boundary data, they determine the field throughout spacetime. A prescribed observation history is therefore a candidate solution of the field equations, rather than an independently assignable value at each point.

Equation (18) is varied with respect to $g,s,\mathsf{\Psi }$ independently. Its domain of validity is the regime in which these effective variables and the low-derivative expansion describe the response. Parent-connection or projector modes that remain dynamical belong in the parent action; contributions already absorbed into $G,U$ are counted once in the effective action. Compactly supported variations give the bulk equations below. On a domain with boundary, the gravitational action also includes the appropriate metric boundary term, with the boundary data held fixed.

4.2. Variation of the Observation State

At fixed g, the first variation of the observation action is

$$\begin{array}{cc}\hfill {\delta }_s{\mathsf{\Gamma }}_{\mathrm{obs}}=-\int \sqrt{-g}[& G_{ab}{\partial }^{\mu }{s}^b{\partial }_{\mu }\delta s^a\hfill \\ \hfill & +\left({\displaystyle \frac{1}{2}}{\partial }_a{G}_{bc}{\partial }_{\mu }{s}^b{\partial }^{\mu }{s}^c+U_{,a}\right)\delta s^a]{\mathrm{d}}^{4}x.\hfill \end{array}$$

(4.2)

Integrating the first term by parts and using the Levi–Civita connection of the kinetic metric,

$$\mathsf{\Gamma }{{\left[G\right]}^a}_{bc}={\displaystyle \frac{1}{2}}{G}^{ad}({\partial }_b{G}_{dc}+{\partial }_c{G}_{db}-{\partial }_d{G}_{bc}),$$

(4.3)

gives

$${\delta }_s{\mathsf{\Gamma }}_{\mathrm{obs}}=\int \sqrt{-g}{G}_{ab}{\mathcal{E}}_s^b\delta s^a{\mathrm{d}}^{4}x,{\mathcal{E}}_s^a={\square }_g{s}^a+\mathsf{\Gamma }{{\left[G\right]}^a}_{bc}{\partial }_{\mu }{s}^b{\partial }^{\mu }{s}^c-G^{ab}{U}_{,b}.$$

(4.4)

Since the interior variation is arbitrary, ${\mathcal{E}}_s^a=0$. Each term has a distinct geometric role. The operator ${\square }_g$ propagates the field through spacetime; $\mathsf{\Gamma }\left[G\right]$ compares its values at neighboring points of observation space; the target-space gradient ${\nabla }_{G}U$ supplies the potential force. Derivatives of a state-dependent Z are already contained in $\mathsf{\Gamma }\left[G\right]$. The observation equation is thus a covariant wave equation for a map into observation space.

A change of observation coordinates $s^{\prime a}=f^a\left(s\right)$ transforms G as a tensor and leaves the equation unchanged in geometric content. It describes the same field in different coordinates. Evolution of $s\left(x\right)$ has a different physical meaning: it changes the conditions of formation and hence the response distinguished by the formation law.

4.3. Metric Variation and Observation Stress-Energy

Variation of the inverse metric gives $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }$. The kinetic contraction and volume element contribute the directional and trace terms, respectively, yielding

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}& :=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta {\mathsf{\Gamma }}_{\mathrm{obs}}}{\delta g^{\mu \nu }}}\hfill \\ \hfill & =G_{ab}{\partial }_{\mu }{s}^a{\partial }_{\nu }{s}^b-g_{\mu \nu }\left({\displaystyle \frac{1}{2}}{G}_{ab}{\partial }_{\alpha }{s}^a{\partial }^{\alpha }{s}^b+U\right).\hfill \end{array}$$

(4.5)

Define $T_{\mu \nu }^X$ in the same way. The interior variation of the gravitational action produces the Einstein tensor, so the coupled system is

$$\begin{array}{cc}\hfill G_{\mu \nu }+{\mathsf{\Lambda }}_X{g}_{\mu \nu }& ={\kappa }_E(T_{\mu \nu }^X+{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}),\hfill \\ \hfill {\square }_g{s}^a+\mathsf{\Gamma }{{\left[G\right]}^a}_{bc}{\partial }_{\mu }{s}^b{\partial }^{\mu }{s}^c& =G^{ab}{U}_{,b},\hfill \\ \hfill {\displaystyle \frac{\delta {\mathsf{\Gamma }}_X}{\delta \mathsf{\Psi }}}& =0.\hfill \end{array}$$

(4.6)

These equations describe reciprocal gravitational influence. Motion of the observation field changes its stress-energy; that stress-energy changes the metric; and the metric enters the propagation of both the observation field and matter. The interaction through geometry is present even when the action contains no direct exchange term between the two sectors. Its normalization is fixed by the variation: the metric derivative defines observation stress-energy, and multiplication by ${\kappa }_E$ gives its contribution to the curvature source.

The energy carried by the observation field has a direct origin in the action. Temporal variation costs kinetic energy, spatial variation costs gradient energy, and the potential assigns an energy to the local state. Once included in the action, these contributions couple to geometry on the same footing as the energy and stress of other fields. Their magnitude is set by the kinetic scale and by the field solution. The clock comparison in Section 8 makes this magnitude an observable question.

To identify the energy and pressure, choose any future-directed unit timelike vector n and use ${\gamma }_{\mu \nu }=g_{\mu \nu }+n_{\mu }{n}_{\nu }$ for spatial projection. Writing ${\mathsf{\Pi }}^a=n^{\mu }{\partial }_{\mu }{s}^a$ and denoting spatial derivatives by $D_i{s}^a$, we obtain

$${\rho }_{\mathrm{obs}}:={\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}{n}^{\mu }{n}^{\nu }={\displaystyle \frac{1}{2}}{G}_{ab}{\mathsf{\Pi }}^a{\mathsf{\Pi }}^b+{\displaystyle \frac{1}{2}}{G}_{ab}{D}_i{s}^a{D}^i{s}^b+U.$$

(4.7)

The energy density separates into temporal, spatial, and potential contributions. Spatial homogeneity removes the gradient term while allowing the state to continue evolving in time. For any null vector k,

$${\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}{k}^{\mu }{k}^{\nu }=G_{ab}\left(k^{\mu }{\partial }_{\mu }{s}^a\right)\left(k^{\nu }{\partial }_{\nu }{s}^b\right)\ge 0.$$

(4.8)

The null energy condition in this branch follows directly from the positive definiteness of G. If $U\ge 0$ as well, Eq. (4.7) also gives a nonnegative local energy density.

4.4. Conservation from the Common Action

Take the covariant divergence of Eq. (4.5). Commutation of second derivatives on scalars, metric compatibility of G, and the product rule collect the terms into

$${\nabla }^{\mu }{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}=G_{ab}{\mathcal{E}}_s^a{\partial }_{\nu }{s}^b.$$

(4.9)

For a single field with $G_{ss}=1$, this identity reads $({\square }_{g}s-U^{\prime }){\partial }_{\nu }s$ on both sides. For general $G\left(s\right)$, derivatives of the kinetic metric assemble into the target-space connection in Eq. (4.4). The field equation therefore implies conservation of observation stress-energy. The matter equations give ${\nabla }^{\mu }{T}_{\mu \nu }^X=0$, and both conservation laws are consistent with the geometric identity ${\nabla }^{\mu }{G}_{\mu \nu }=0$.

If the underlying action also contains a direct exchange term ${\mathsf{\Gamma }}_{\mathrm{int}}[g,s,\mathsf{\Psi }]$, that term must enter the variations of both field sectors. With an explicit convention for assigning interaction contributions, one may write

$${\nabla }^{\mu }{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}=J_{\nu }^{\mathrm{ex}},{\nabla }^{\mu }{T}_{\mu \nu }^X=-J_{\nu }^{\mathrm{ex}},{\nabla }^{\mu }(T_{\mu \nu }^X+{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}})=0.$$

(4.10)

The subsequent theorems and exact solution concern Eq. (4.1), for which there is no direct exchange. In this branch, the same action determines the evolution equations and the conservation laws that make their gravitational coupling consistent.

5. Exact Recovery of General Relativity on a Fixed Section

5.1. From Geometric Restriction to Dynamical Invariance

On a fixed-observation section, $\mathrm{d}{s}^a=0$, so the pulled-back connection and curvature retain only their physical tangential components. To determine whether the section supports the full relativistic dynamics, the constant observation state must also be substituted into Eq. (4.6). The observation equation gives

$$U_{,a}\left(s_0\right)=0,{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}{|}_{s_0}=-U\left(s_0\right)g_{\mu \nu }.$$

(5.1)

The derivatives of the constant field vanish, while its potential energy remains in the stress-energy tensor. Define $U_0=U\left(s_0\right)$ and

$${\mathsf{\Lambda }}_{\mathrm{eff}}={\mathsf{\Lambda }}_X+{\kappa }_E{U}_0.$$

(5.2)

Theorem 1 

(Dynamical recovery on a fixed-observation section). Suppose the observation kinetic metric in Eq. (4.1) is nondegenerate and the matter action has no direct dependence on s. A constant section $s=s_0$ supports a solution of the full coupled system if and only if $s_0$ is a stationary point of U and $(g,\mathsf{\Psi })$ satisfies the Einstein–matter equations with cosmological constant ${\mathsf{\Lambda }}_{\mathrm{eff}}$. Under these conditions, solutions on the fixed section correspond one-to-one to the associated general-relativistic solutions.

Proof. 

All derivative terms in the observation equation vanish for constant $s_0$. Since G is nondegenerate, that equation is equivalent to $U_{,a}\left(s_0\right)=0$. Equation (4.5) gives observation stress-energy $-U_0{g}_{\mu \nu }$, so the coupled Einstein equation becomes

$$G_{\mu \nu }+{\mathsf{\Lambda }}_{\mathrm{eff}}{g}_{\mu \nu }={\kappa }_E{T}_{\mu \nu }^X.$$

(5.3)

The matter equations are unchanged. Conversely, any $(g,\mathsf{\Psi })$ satisfying Eq. (5.3) and the matter equations, together with a constant stationary state $s_0$, satisfies every equation of the full coupled system. For fixed $s_0$, adjoining that state and forgetting it are inverse maps between the two descriptions, establishing the correspondence. □

Stationarity of the potential is what makes the geometric restriction a sector of the field equations. The constant observation state can then accompany an exact general-relativistic solution; varying observation states belong to the larger solution space. A physical realization of the fixed section must also meet the conditions on the formation domain and reference structure in the foundational recovery theorem.[3]

5.2. Contributions Retained on a Varying Section

Write

$${\widehat{\mathsf{\Delta }}}_{\mu \nu }:={\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}+U_0{g}_{\mu \nu }.$$

(5.4)

The full geometric equation becomes

$$G_{\mu \nu }+{\mathsf{\Lambda }}_{\mathrm{eff}}{g}_{\mu \nu }={\kappa }_E(T_{\mu \nu }^X+{\widehat{\mathsf{\Delta }}}_{\mu \nu }).$$

(5.5)

The kinetic energy, gradients, and potential energy of s relative to the stationary state now contribute to the source. The response connection also samples observation and mixed directions along the history. Both changes arise from the joint state. The stress-energy enters the equation for the geometry, while the connection enters the comparison of responses along that geometry.

5.3. Perturbations About a Stationary State

Let $g=\overline{g}+\epsilon h$ and $s=s_0+\epsilon \eta +O\left({\epsilon }^2\right)$, where $(\overline{g},s_0)$ satisfies the recovery theorem. In target-space normal coordinates near $s_0$, the first-order observation equation is

$$\overline{\square }{\eta }^a-{{\mathsf{M}}^a}_b{\eta }^b=0,{{\mathsf{M}}^a}_b=G^{ac}\left(s_0\right){\nabla }_c{\nabla }_{b}U\left(s_0\right).$$

(5.6)

The background has no observation-state gradient. Consequently, the variation $\delta {\square }_g$ acting on $s_0$ vanishes, and stationarity removes the first derivative of the potential. At first order, the only change in observation stress-energy is the metric variation of its constant vacuum term:

$$\delta {\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}=-U_0{h}_{\mu \nu },\delta {\widehat{\mathsf{\Delta }}}_{\mu \nu }=0.$$

(5.7)

After $U_0$ has been absorbed into the cosmological constant, the additional observation source begins at second order:

$$\begin{array}{cc}\hfill {\widehat{\mathsf{\Delta }}}_{\mu \nu }={\epsilon }^2& \{G_{ab}\left(s_0\right){\overline{\nabla }}_{\mu }{\eta }^a{\overline{\nabla }}_{\nu }{\eta }^b\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\overline{g}}_{\mu \nu }\left[G_{ab}\left(s_0\right){\overline{\nabla }}_{\alpha }{\eta }^a{\overline{\nabla }}^{\alpha }{\eta }^b+U_{;ab}\left(s_0\right){\eta }^a{\eta }^b\right]\}+O\left({\epsilon }^3\right).\hfill \end{array}$$

(5.8)

There are therefore first-order observation waves without a first-order additional metric source on this stationary, minimally coupled background. Their gravitational backreaction begins at second order, so they do not add polarizations to the linearized metric waves merely by being present. On a varying background, ${\partial }_{\mu }{\overline{s}}^a\ne 0$, cross terms such as $\partial \overline{s}\partial \eta$ already occur at first order and couple the two perturbation sectors.

The operator $\mathsf{M}$ is self-adjoint in the G inner product. Positive-definite kinetic energy excludes observation modes with negative kinetic energy, and a positive-semidefinite potential Hessian excludes negative mass-squared modes near a flat stationary background. These statements characterize the local kinetic and linearized spectra. Long-time nonlinear stability on a specified spacetime requires the corresponding initial-value analysis.

6. Initial-Value Constraints and a Common Causal Structure

6.1. Geometric and Observation Data on a Common Spacelike Slice

The geometry and the observation field cannot be prescribed independently on an initial surface: their energy and momentum are tied by the gravitational constraints. To see this relation explicitly, let ${\Sigma }_t$ be a spacelike slice with future-directed unit normal n. A positive lapse N measures the separation of neighboring slices along the normal, while a spatial shift $\beta$ describes the tangential part of the time flow. Our conventions are

$$\begin{array}{cc}\hfill \mathrm{d}{\ell }^2& =-N^2\mathrm{d}{t}^2+{\gamma }_{ij}(\mathrm{d}{x}^i+{\beta }^i\mathrm{d}t)(\mathrm{d}{x}^j+{\beta }^j\mathrm{d}t),\hfill \\ \hfill K_{ij}& =-{\displaystyle \frac{1}{2}}{\mathcal{L}}_n{\gamma }_{ij},K={\gamma }^{ij}{K}_{ij},{\mathsf{\Pi }}^a=n^{\mu }{\partial }_{\mu }{s}^a.\hfill \end{array}$$

(6.1)

Here $D_i$ is the Levi–Civita derivative of $\gamma$; the target-space connection remains explicit as $\mathsf{\Gamma }\left[G\right]$. The pair $({\gamma }_{ij},K_{ij})$ specifies the intrinsic geometry of the slice and its rate of change along the normal. The pair $(s^a,{\mathsf{\Pi }}^a)$ specifies the observation state and its normal velocity. Together with the matter data, these quantities constitute admissible initial data only when they satisfy the constraints below.

Write $Q_i^a=D_i{s}^a$, with squared normal velocity ${\mathsf{\Pi }}_G^2=G_{ab}{\mathsf{\Pi }}^a{\mathsf{\Pi }}^b$ and squared spatial gradient $Q_G^2=G_{ab}{\gamma }^{ij}{Q}_i^a{Q}_j^b$. Projecting the observation stress-energy along and orthogonal to the normal gives the energy density, momentum density, and spatial stress measured in this frame:

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{obs}}& ={\displaystyle \frac{1}{2}}({\mathsf{\Pi }}_G^2+Q_G^2)+U,\hfill \\ \hfill j_i^{\mathrm{obs}}& :=-{{\gamma }_i}^{\mu }{n}^{\nu }{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}=-G_{ab}{\mathsf{\Pi }}^a{Q}_i^b,\hfill \\ \hfill S_{ij}^{\mathrm{obs}}& :={{\gamma }_i}^{\mu }{{\gamma }_j}^{\nu }{\mathsf{\Delta }}_{\mu \nu }^{\mathrm{obs}}=G_{ab}{Q}_i^a{Q}_j^b+{\gamma }_{ij}\left[{\displaystyle \frac{1}{2}}({\mathsf{\Pi }}_G^2-Q_G^2)-U\right].\hfill \end{array}$$

(6.2)

Adding the ordinary-matter contributions gives the total $\rho ,j_i,S_{ij}$, with spatial trace $S={\gamma }^{ij}{S}_{ij}$. The Gauss and Codazzi relations express the normal projections of the Einstein equations entirely in terms of data on the slice:

$$\begin{array}{cc}\hfill {}^{\left(3\right)}R+K^2-K_{ij}{K}^{ij}& =2{\kappa }_E\rho +2{\mathsf{\Lambda }}_X,\hfill \\ \hfill D_j{K^j}_i-D_{i}K& ={\kappa }_E{j}_i.\hfill \end{array}$$

(6.3)

Thus both spatial variation of the observation state and its normal velocity constrain the initial geometry. Their contributions are fixed by the stress-energy tensor derived from the action. The sign of the momentum constraint follows from $j_i=-{{\gamma }_i}^{\mu }{n}^{\nu }{T}_{\mu \nu }$ and the extrinsic-curvature convention in Eq. (6.1).

6.2. The Three-Plus-One Form of Continuous Evolution

Once the constraints hold on the initial slice, the tangential projections of the field equations determine how the data evolve. Define ${\partial }_{\perp }={\partial }_t-{\mathcal{L}}_{\beta }$ to remove the change due to the shift; on s and $\mathsf{\Pi }$, the Lie derivative is ${\beta }^i{\partial }_i$. The metric and extrinsic curvature then obey

$$\begin{array}{cc}\hfill {\partial }_{\perp }{\gamma }_{ij}& =-2N{K}_{ij},\hfill \\ \hfill {\partial }_{\perp }{K}_{ij}& =-D_i{D}_{j}N+N[{}_{\left(3\right)}R_{ij}+K{K}_{ij}-2K_{ik}{K^k}_j\hfill \\ \hfill & -{\kappa }_E\left(S_{ij}-{\displaystyle \frac{1}{2}}{\gamma }_{ij}(S-\rho )\right)-{\mathsf{\Lambda }}_X{\gamma }_{ij}].\hfill \end{array}$$

(6.4)

The observation equation has an equally direct decomposition. The identities ${\nabla }_{\mu }{n}^{\mu }=-K$ and $a_i=D_{i}lnN$ account, respectively, for the expansion of the normal congruence and its acceleration. They give

$${\square }_g{s}^a=-n\left({\mathsf{\Pi }}^a\right)+K{\mathsf{\Pi }}^a+D_i{Q}^{ia}+a_i{Q}^{ia}.$$

(6.5)

Inserting this expression into the second line of Eq. (4.6) gives the evolution of the observation state and its normal velocity:

$$\begin{array}{cc}\hfill {\partial }_{\perp }{s}^a& =N{\mathsf{\Pi }}^a,\hfill \\ \hfill {\partial }_{\perp }{\mathsf{\Pi }}^a& =N\left[D_i{Q}^{ia}+K{\mathsf{\Pi }}^a+\mathsf{\Gamma }{{\left[G\right]}^a}_{bc}(Q_i^b{Q}^{ic}-{\mathsf{\Pi }}^b{\mathsf{\Pi }}^c)-G^{ab}{U}_{,b}\right]+\left(D_{i}N\right)Q^{ia}.\hfill \end{array}$$

(6.6)

The term $K{\mathsf{\Pi }}^a$ is the response to a changing spatial volume element. Spatial gradients and normal velocities enter the target-space connection term with opposite signs, as required by the Lorentzian contraction in the covariant equation. For a homogeneous expanding background, $K=-3H$, $N=1$, and $Q_i=0$, so that the same equation reads ${\ddot{s}}^a+3H{\dot{s}}^a+\mathsf{\Gamma }{{\left[G\right]}^a}_{bc}{\dot{s}}^b{\dot{s}}^c+G^{ab}{U}_{,b}=0$. The expansion term therefore follows from the geometry of the foliation; no separate damping law has been introduced.

A first-order formulation can treat $Q_i^a$ as a separate variable, with evolution and constraints

$${\partial }_{\perp }{Q}_i^a=D_i\left(N{\mathsf{\Pi }}^a\right),Q_i^a-D_i{s}^a=0,D_{[i}{Q}_{j]}^a=0.$$

(6.7)

The last two relations impose its definition and integrability. They ensure that $Q_i^a$ remains a gradient throughout the evolution, so this first-order formulation introduces no additional observation field.

6.3. Gauge Reduction, the Common Characteristic Cone, and Constraint Propagation

The three-plus-one equations expose the geometric meaning of the initial data and constraints. For local well-posedness, it is more convenient to use generalized harmonic coordinates, in which the Einstein equations form a quasilinear wave system. Freezing the coefficients, the second-order principal symbols acting on metric and observation-field variations are

$${\left[{\sigma }_2\left({\mathcal{E}}_g\right)\left(\zeta \right)h\right]}_{\alpha \beta }=-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\zeta }_{\mu }{\zeta }_{\nu }{h}_{\alpha \beta },{\left[{\sigma }_2\left({\mathcal{E}}_s\right)\left(\zeta \right)\eta \right]}^a=g^{\mu \nu }{\zeta }_{\mu }{\zeta }_{\nu }{\eta }^a.$$

(6.8)

The coupling does not mix the second-order principal blocks: the observation stress-energy contains only first derivatives of s, and the spacetime connection in the observation equation contains only first derivatives of g. Both sectors consequently have the characteristic condition

$$g^{\mu \nu }{\zeta }_{\mu }{\zeta }_{\nu }=0.$$

(6.9)

The minimally coupled metric and observation fields thus share a causal cone. A nonzero observation-field mass changes wave-packet dispersion without changing this highest-order characteristic condition. The propagation fronts are governed by g in both sectors.

Suppose the initial data are smooth and regular, G is uniformly positive definite, the slices are spacelike, and the matter equations admit a compatible hyperbolic closure. In such a region, the wave system admits a first-order reduction whose principal energy is a sum of squares of time derivatives and spatial gradients. Positive-definite estimates hold in a sufficiently small regular neighborhood. The local energy iteration then yields an evolution in the chosen gauge that depends continuously on the initial data. The geometric solution is unique up to diffeomorphism. This is a local result in a regular region, with the shared propagation and strong hyperbolicity required in the underlying observation geometry.[4]

It remains to check that gauge reduction preserves the original equations. If $C_{\mu }$ denotes the harmonic gauge constraint, the reduced Einstein equations and total stress-energy conservation give

$${\square }_g{C}_{\nu }+{R_{\nu }}^{\mu }{C}_{\mu }=0.$$

(6.10)

This propagation equation is homogeneous. Compatible initial data set C and its normal derivative to zero, and local uniqueness then preserves $C=0$. The reduced solution therefore satisfies the original Einstein equations, including the constraints (38), throughout its local evolution. The same uniqueness argument gives the dynamical content of the recovery theorem: if the initial data have $s=s_0$ and $\mathsf{\Pi }=0$, with $U_{,a}\left(s_0\right)=0$, the constant observation field solves the initial-value problem and is its unique local continuation. The fixed-observation section is consequently preserved by the field equations.

The local propagating degrees of freedom in this regular branch are two gravitational tensor modes and $dim\mathcal{S}$ nondegenerate observation-field modes, together with the original matter modes. Gauge freedom and gravitational constraints remove redundant descriptions of the geometry. They do not remove those variations of the observation state that are identifiable through the formation law; these variations evolve according to the observation-field equation.

7. Continuous Relativistic Records and an Exact Solution

7.1. Clock Frequencies and Redshift from a Common Geometric Solution

Einstein’s analysis of light in a gravitational field identified the frequency shift produced by a difference in gravitational potential.[6] In the metric formulation of general relativity, clock rates and light propagation are determined by the spacetime geometry.[2] Here these relations are evaluated on the metric of the coupled field solution.

To pass from a field solution to a frequency record, consider light propagation in the geometric-optics branch. The ray wavevector $k^{\mu }$ obeys

$$k^{\mu }{k}_{\mu }=0,k^{\nu }{\nabla }_{\nu }{k}^{\mu }=0.$$

(7.1)

An observing system with unit four-velocity u measures frequency by comparing the wave phase with its proper time. Its angular frequency and the redshift between emission and reception are

$$\omega =-u_{\mu }{k}^{\mu },1+z={\displaystyle \frac{{(-u·k)}_{\mathrm{em}}}{{(-u·k)}_{\mathrm{rec}}}}.$$

(7.2)

Three physical ingredients enter the recorded frequency. The metric determines the ray and the proper time along the clock’s worldline; the motion of the receiving system fixes the local frequency through the contraction above; and the observation response forms the instrument’s record. A prediction therefore depends on the coupled field solution, the specified worldlines, and a formation law F appropriate to the observation state.[3,4]

For the static, spherically symmetric vacuum branch of the recovery theorem, let $M_0$ be the mass parameter. The metric is

$$\mathrm{d}{\ell }^2=-f\left(r\right)\mathrm{d}{t}^2+{\displaystyle \frac{\mathrm{d}{r}^2}{f\left(r\right)}}+r^2\mathrm{d}{\mathsf{\Omega }}^2,f\left(r\right)=1-{\displaystyle \frac{2G_N{M}_0}{r}}-{\displaystyle \frac{{\mathsf{\Lambda }}_{\mathrm{eff}}{r}^2}{3}}.$$

(7.3)

Where $f>0$, a static reference system has $u=f^{-1/2}{\partial }_t$. The conserved quantity associated with time-translation symmetry is $E_k=-k_t$, so the frequency along the ray is $\omega \left(r\right)=E_k/\sqrt{f\left(r\right)}$. The emission and reception frequencies then give

$$1+z=\sqrt{{\displaystyle \frac{f\left(r_{\mathrm{rec}}\right)}{f\left(r_{\mathrm{em}}\right)}}}.$$

(7.4)

This redshift law is unchanged on the fixed-observation section. A varying observation field affects it through any resulting change in stress-energy: the coupled solution for g and s changes, and Eq. (7.2) evaluates the frequency on that geometry. The additional source contribution thus enters through the field solution and its continuous evolution.

7.2. Transport Along the Complete History and Noncommuting Observation Compensation

A frequency at one event does not by itself specify how coherent records at successive events are compared. That comparison is determined by the connection on the response bundle. Denoting the total connection by $\mathcal{A}$, its restriction to a joint history gives

$${\mathcal{A}}_t={\mathcal{A}}_i{\dot{X}}^i+{\mathcal{A}}_a{\dot{s}}^a={\mathcal{A}}_{X,t}+{\mathcal{A}}_{S,t},\dot{U}=-{\mathcal{A}}_{t}U,U\left(t_0\right)=I.$$

(7.5)

The solution is a path-ordered exponential, which retains the order in which changes of physical and observation state occur. When the connection is noncommuting, the two contributions cannot in general be integrated as independent exponentials. Their separation can nevertheless be made in a transported frame. Let $U_X$ satisfy ${\dot{U}}_X=-{\mathcal{A}}_{X,t}{U}_X$ with $U_X\left(t_0\right)=I$, and put $U=U_{X}V$. Substitution gives

$$\dot{V}=-\left(U_X^{-1}{\mathcal{A}}_{S,t}{U}_X\right)V,V\left(t_0\right)=I.$$

(7.6)

The conjugated connection is the observation contribution expressed in the frame carried by physical transport. Integrating it in that frame preserves the matrix order and gives the observation response relative to the physical history. Conjugation is immaterial in the Abelian scalar branch. With the real-phase convention of the phase transport fundamental equation (PTFE), the accumulated observation phase and its inverse compensation reduce to

$${\mathsf{\Gamma }}_S(t,t_0)=exp\left(i{\int }_{t_0}^t{A}_a{\dot{s}}^a\mathrm{d}t\right),r_{\mathrm{comp}}={\mathsf{\Gamma }}_S^{-1}r.$$

(7.7)

Where the coherent response is nonzero, inverse transport removes the phase accumulated along the specified observation path and leaves the rest of the response intact. The gravitational effect of observation stress-energy remains: compensating a readout phase does not change the spacetime in which the signal propagated. Although the source and the phase belong to the same theory, the former acts through the field equations and the latter through the formation map.[3,4]

A finite exposure records an integral of this continuously evolving response. For a calibrated linear instrument,

$$R_n={\int }_{t_n}^{t_n+\mathsf{\Delta }{t}_n}{w}_n\left(t\right)r\left(t\right)\mathrm{d}t,\int w_n\left(t\right)\mathrm{d}t=1.$$

(7.8)

Changing the sampling interval changes the time resolution of the record. It leaves the underlying initial-value history of the fields and connection unchanged; discrete records sample a continuous evolution.

7.3. An Exact Coupled Solution with a Continuously Evolving Observation Field

The gravitational effect of a moving observation state can be seen in an exact solution. Take the single-observation-coordinate branch of Eq. (4.1), with a coordinate chosen so that $G_{ss}=1$. Set $U=0$ and ${\mathsf{\Lambda }}_X=0$, and omit ordinary matter. For a spatially flat, homogeneous and isotropic geometry,

$$\mathrm{d}{\ell }^2=-\mathrm{d}{t}^2+a{\left(t\right)}^2\mathrm{d}{\mathbf{x}}^2,s=s\left(t\right),H=\dot{a}/a.$$

(7.9)

the coupled field equations reduce to

$$3H^2={\displaystyle \frac{{\kappa }_E}{2}}{\dot{s}}^2,\dot{H}=-{\displaystyle \frac{{\kappa }_E}{2}}{\dot{s}}^2,\ddot{s}+3H\dot{s}=0.$$

(7.10)

The observation equation integrates to $a^3\dot{s}=C$: expansion dilutes the normal velocity in inverse proportion to the spatial volume. The two gravitational equations give $\dot{H}=-3H^2$. On the expanding branch, a shift of the time origin gives the following exact solution for $t>0$:

$$\begin{array}{cc}\hfill a\left(t\right)& =a_0{\left({\displaystyle \frac{t}{t_0}}\right)}^{1/3},H\left(t\right)={\displaystyle \frac{1}{3t}},\hfill \\ \hfill s\left(t\right)& =s_0\pm \sqrt{{\displaystyle \frac{2}{3{\kappa }_E}}}ln{\displaystyle \frac{t}{t_0}},\hfill \\ \hfill {\rho }_{\mathrm{obs}}\left(t\right)& =p_{\mathrm{obs}}\left(t\right)={\displaystyle \frac{1}{3{\kappa }_E{t}^2}}.\hfill \end{array}$$

(7.11)

Substitution into the field equations verifies both the gravitational constraint and the observation evolution, while energy conservation takes the form ${\dot{\rho }}_{\mathrm{obs}}+3H({\rho }_{\mathrm{obs}}+p_{\mathrm{obs}})=0$. The curvature provides a separate check: tracing the Einstein equations yields

$$R=-{\displaystyle \frac{2}{3t^2}},$$

(7.12)

which agrees with $R=6(\dot{H}+2H^2)$ obtained from $a\left(t\right)$. Spatial homogeneity has not made the observation field stationary. Its continued temporal motion supplies energy density and pressure, and these sustain the spacetime curvature.

The solution therefore describes evolution away from a fixed-observation section, with backreaction determined by the observation motion itself. Its scope is the branch with neither a potential nor ordinary matter; a particular cosmological application requires the relevant matter content and observation initial data. The same action also admits the Minkowski branch when $\dot{s}=0$ and the vacuum initial geometry is flat. These two evolutions arise from different joint initial data, one with a moving observation field and one without. Both take place in spacetime of the same dimension.

8. Experimental Tests: Continuous Transport and Observation Stress-Energy

8.1. From Field Variables to Experimental Controls

An experimental prediction requires a physical identification of the observation state. The coordinates of a response manifold acquire dynamical meaning only when the conditions that prepare the field, and the response through which it is measured, have been specified together. Denote the independently controlled and recorded parameters of the apparatus by $q^A\left(t\right)$, and its calibrated material, geometric, and response coefficients by $\vartheta$. The required correspondence is

$$s^a\left(x\right)={\mathcal{S}}^a[q;\vartheta ]\left(x\right),r=F(X,s;\vartheta ),$$

(8.1)

Here $\mathcal{S}$ determines the field produced under the preparation conditions, while F gives the measured response of that same state. The field s satisfies its equation of motion with compatible initial and boundary data. A receiver basis, polarization, or set of array weights may provide coordinates for the response state; their identification with the dynamical variables of the action also requires a preparation mechanism and a normalization of the field energy.

The normalization matters because response geometry and gravitational strength carry different information. Geometric calibration can determine the shape of the response manifold while leaving the kinetic scale f in Eq. (3.3) undetermined. Changing the units of s and transforming G accordingly leaves the stress-energy and the measured response unchanged. A measured phase change therefore does not, on its own, determine a gravitational source. That connection requires independent dynamical information.

In the action (4.1), matter has no direct exchange term with s. Preparation within this branch is consequently expressed through compatible initial data, boundary conditions, or the common metric. A direct drive of s, if required by a particular apparatus, would have to follow from an actual interaction in the parent action, with the stress-energy of the apparatus, its boundaries, and the observation field included together. These considerations lead to two complementary measurements. Coherent transport probes how a changing observation state enters a record; a clock comparison probes the geometric backreaction of its stress-energy. Both refer to the same s.

8.2. A Differential Experiment for Continuous Coherent Transport

A common coherent input provides a natural reference against which to measure the phase associated with observation-state evolution. The input is divided between two channels with a shared frequency reference and sampling time base. The reference channel remains in a prescribed response state, while the test channel follows a predetermined continuous history $q\left(t\right)$. Independent calibration of their propagation delays, gains, and fixed phase difference allows the reference channel to record the physical phase of the input. The following expressions apply when the physical-direction phases of the channels coincide, or when their difference has been independently corrected using the full connection. Within a region of nonzero scalar response, the predicted observation phase is

$${\varphi }_S\left[q\right]\left(t\right)={\int }_{t_0}^t{A}_a\left(X\left(t^{\prime }\right),s\left(t^{\prime }\right)\right){\dot{s}}^a\left(t^{\prime }\right)\mathrm{d}{t}^{\prime },$$

(8.2)

with A determined by the formation law and independent calibration data. After the known channel phases have been removed, a separate set of test data gives the residual

$$e_{\varphi }\left(t\right)=Arg\left[r_{\mathrm{test}}\left(t\right)r_{\mathrm{ref}}{\left(t\right)}^{*}exp\{-i{\varphi }_S\left[q\right]\left(t\right)\}\right].$$

(8.3)

The prediction is that $e_{\varphi }$ contains no remaining phase contribution associated with the prescribed observation path. The common physical phase cancels in this differential comparison; it is preserved when ${\mathsf{\Gamma }}_S^{-1}$ is instead applied to the test channel alone. The distinction is essential to the measurement: a differential residual tests the relative observation phase, whereas compensation of an individual record retains its physical phase. Independence of calibration and test data is equally essential. Constructing A from the phase derivative of a record and using it to remove that same phase establishes an identity, not a prediction.

A fixed-state record supplies the reference case. Slow variation, faster variation, and reverse traversal then examine the predicted transport over different histories, including paths absent from calibration. The sampling bandwidth is chosen to resolve the actual state evolution. Intervals near a response zero are identified by a signal-to-noise threshold fixed beforehand, so that their inclusion does not depend on the apparent success of compensation. Finite exposure introduces a further physical distinction between the instantaneous response and the recorded quantity. For a measured exposure $R_n$, the appropriate prediction is

$$R_n^{\mathrm{pred}}={\int }_{t_n}^{t_n+\mathsf{\Delta }{t}_n}{w}_n\left(t\right)r_X\left(t\right)exp\left\{i{\varphi }_S\left[q\right]\left(t\right)\right\}\mathrm{d}t.$$

(8.4)

where $r_X$ contains the calibrated amplitude response and the physical phase. If the state varies appreciably during the exposure, integration and phase correction cannot be interchanged: a single phase factor at the window midpoint does not exactly compensate the integrated record.

For multicomponent responses, Eq. (7.6) supplies the ordered transport, and independent input states determine the transfer matrix. Closed paths require particular care in interpreting the resulting phase. A globally single-valued, nonzero scalar response admits the normalization $z=r/\left|r\right|$, for which $A=Im\left(z^{*}\mathrm{d}z\right)=\mathrm{d}argz$ and $exp(i\oint A)=1$. Nontrivial closed-path transport therefore depends on the appropriate response subspace, curvature, or nontrivial bundle structure; it does not follow from scalar phase winding alone.

The physical content of this measurement lies in the prediction of an independently recorded history from a specified formation law. Its power to distinguish descriptions is established by comparing the same records with the electromagnetic propagation and conventional calibration models of the apparatus. Only a difference predicted before measurement can serve that purpose. Agreement with a transport correction by itself establishes no additional gravitational contribution.

8.3. Weak-Field Clock Differences Sourced by Observation Stress-Energy

The gravitational test follows a different chain: the prepared field determines stress-energy, stress-energy determines the metric perturbation, and the metric determines the relative rates of clocks. Restoring c, taking $x^0=ct$, and expressing stress-energy in units of energy density gives the coupling $8\pi G_N/c^4$. Within a region much smaller than the background curvature radius, a weak field varying slowly relative to the laboratory has

$$g_{00}=-1-{\displaystyle \frac{2\mathsf{\Phi }}{c^2}},\left|{\displaystyle \frac{\mathsf{\Phi }}{c^2}}\right|\ll 1.$$

(8.5)

With the constant $U_0$ included in the background cosmological term, the additional source relevant to the clock potential is the local orthonormal-frame combination

$${\mathcal{E}}_{\mathrm{obs}}={\widehat{\mathsf{\Delta }}}_{\widehat{0}\widehat{0}}+{\displaystyle \sum _{i=1}^3}{\widehat{\mathsf{\Delta }}}_{\widehat{i}\widehat{i}}.$$

(8.6)

Indeed, the quasistatic relation $R_{00}\simeq {\nabla }^2\mathsf{\Phi }/c^2$ and the trace-reversed Einstein equation $R_{00}=4\pi G_N(T_{\widehat{0}\widehat{0}}+{\sum }_i{T}_{\widehat{i}\widehat{i}})/c^4$ identify the potential generated by this source. Subtraction of the known matter and apparatus contributions, evaluated under the same preparation conditions, leaves

$${\nabla }^2{\mathsf{\Phi }}_{\mathrm{obs}}={\displaystyle \frac{4\pi G_N}{c^2}}{\mathcal{E}}_{\mathrm{obs}},{\mathsf{\Phi }}_{\mathrm{obs}}\left(\mathbf{x}\right)=-{\displaystyle \frac{G_N}{c^2}}\int {\displaystyle \frac{{\mathcal{E}}_{\mathrm{obs}}\left({\mathbf{x}}^{\prime }\right)}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}}{\mathrm{d}}^3{x}^{\prime }.$$

(8.7)

The integral applies to a localized source after subtraction of the known boundary background, with the perturbing potential vanishing at infinity. A bounded experimental region instead requires the corresponding Green function and boundary solution. The approximation also depends on the time scale: a source of characteristic frequency $\mathsf{\Omega }$ at propagation distance L is quasistatic when $\mathsf{\Omega }L/c\ll 1$. Otherwise the observable follows from the retarded metric perturbation and Eq. (7.2) evaluated along the actual light path.

The source in Eq. (8.6) can be evaluated directly from the observation stress-energy in Eq. (4.5). With $\dot{s}$ denoting differentiation with respect to local proper time and ${|\nabla s|}_G^2$ the squared spatial gradient, its components are

$$\begin{array}{cc}\hfill {\widehat{\mathsf{\Delta }}}_{\widehat{0}\widehat{0}}& ={\displaystyle \frac{G_{ab}{\dot{s}}^a{\dot{s}}^b}{2c^2}}+{\displaystyle \frac{1}{2}}{|\nabla s|}_G^2+U-U_0,\hfill \\ \hfill \sum _i{\widehat{\mathsf{\Delta }}}_{\widehat{i}\widehat{i}}& ={\displaystyle \frac{3G_{ab}{\dot{s}}^a{\dot{s}}^b}{2c^2}}-{\displaystyle \frac{1}{2}}{|\nabla s|}_G^2-3(U-U_0),\hfill \\ \hfill {\mathcal{E}}_{\mathrm{obs}}& ={\displaystyle \frac{2G_{ab}{\dot{s}}^a{\dot{s}}^b}{c^2}}-2(U-U_0).\hfill \end{array}$$

(8.8)

The cancellation of the spatial-gradient terms belongs to this particular trace-reversed combination. Those gradients still enter the field equation, other metric components, and boundary stresses. Pressure is consequently part of the gravitational source: the energy density in Eq. (4.7) divided by $c^2$ is not sufficient by itself. A confined field also carries the stresses of its confining apparatus into the total source. Including them preserves conservation of the complete stress-energy.

For stationary clocks A and B, normalized to the same reference transition, the link-corrected comparison measures

$$y_{AB}={\displaystyle \frac{\mathrm{d}{\tau }_A/\mathrm{d}t}{\mathrm{d}{\tau }_B/\mathrm{d}t}}-1\simeq {\displaystyle \frac{\mathsf{\Phi }\left({\mathbf{x}}_A\right)-\mathsf{\Phi }\left({\mathbf{x}}_B\right)}{c^2}}.$$

(8.9)

The observation-field contribution to the fractional frequency difference is thus

$$\delta y_{AB}^{\mathrm{obs}}=-{\displaystyle \frac{G_N}{c^4}}\int {\mathcal{E}}_{\mathrm{obs}}\left({\mathbf{x}}^{\prime }\right)\left({\displaystyle \frac{1}{|{\mathbf{x}}_A-{\mathbf{x}}^{\prime }|}}-{\displaystyle \frac{1}{|{\mathbf{x}}_B-{\mathbf{x}}^{\prime }|}}\right){\mathrm{d}}^3{x}^{\prime }.$$

(8.10)

This expression connects the field solution and its stress-energy to the differential clock observable. It also shows why the clock positions matter. A potential variation common to both positions disappears from the comparison; the selected geometry must have a nonzero response to the predicted spatial distribution.

8.4. Perturbation Reversal, Amplitude Scans, and Blind Analysis

The order at which the new source appears determines the symmetry of the experiment. Near a stationary point $s_0$, suppose that preparation produces the compatible family

$$s_{\epsilon }^a=s_0^a+\epsilon {\eta }^a+O\left({\epsilon }^2\right),U_{,a}\left(s_0\right)=0,$$

(8.11)

with the first-order profile $\eta$ and its boundary normalization fixed as $\epsilon$ varies. The second-order stress-energy in Eq. (5.8) then yields

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\mathrm{obs}}& ={\epsilon }^2{\mathcal{E}}_2+O\left({\epsilon }^3\right),\hfill \\ \hfill {\mathcal{E}}_2& ={\displaystyle \frac{2G_{ab}\left(s_0\right){\dot{\eta }}^a{\dot{\eta }}^b}{c^2}}-U_{;ab}\left(s_0\right){\eta }^a{\eta }^b,\hfill \\ \hfill \delta y_{AB}^{\mathrm{obs}}& ={\epsilon }^2{C}_{AB}\left[\eta \right]+O\left({\epsilon }^3\right),\hfill \end{array}$$

(8.12)

Here $C_{AB}$ is the integral in Eq. (8.10) with ${\mathcal{E}}_2$ replacing ${\mathcal{E}}_{\mathrm{obs}}$. Its value, including the possibility of zero, depends on the field solution and the clock geometry and is calculated for the chosen apparatus.

The natural comparison alternates among $0,+\epsilon ,-\epsilon$. If $d\left(\epsilon \right)$ denotes the fractional frequency difference after subtraction of known gravitational contributions and systematic clock and link offsets, these measurements determine

$$\begin{array}{cc}\hfill d_{\mathrm{even}}\left(\epsilon \right)& =\frac{1}{2}[d(+\epsilon )+d(-\epsilon )]-d\left(0\right),\hfill \\ \hfill d_{\mathrm{odd}}\left(\epsilon \right)& =\frac{1}{2}[d(+\epsilon )-d(-\epsilon )].\hfill \end{array}$$

(8.13)

Analytic dependence of the solution family and response on $\epsilon$ gives the observation-field terms

$$d_{\mathrm{even}}^{\mathrm{obs}}=C_{AB}{\epsilon }^2+O\left({\epsilon }^4\right),d_{\mathrm{odd}}^{\mathrm{obs}}=O\left({\epsilon }^3\right).$$

(8.14)

The desired leading signal is even under perturbation reversal. Direct subtraction of the positive and negative states would remove it, whereas their mean relative to the unperturbed state retains it. An amplitude scan then asks whether one and the same $C_{AB}$ describes all records. Quadratic scaling alone is insufficient: heating, magnetic fields, and mechanical displacement can share that dependence. The temporal relations and the spatial pattern across clock positions predicted by the field equation provide the additional comparisons.

Such a comparison is meaningful when the prediction is fixed independently of the residual it is intended to explain. Independent data determine $G,U,\mathcal{S},F$ and the error model before the clock positions, amplitudes, and sampling windows are chosen. During measurement, the three states are randomly interleaved, with temperature, electromagnetic fields, optical power, clock elevations, and link delays recorded alongside them. A control reproducing the known apparatus disturbances without the target field variation separates those disturbances from the proposed source. Concealed state labels or a prespecified blind offset keep the signal assignment hidden until data selection, systematic corrections, and the uncertainty budget are fixed. Measurements at amplitudes or clock positions excluded from parameter determination then provide the independent test.

The same separation of prediction and adjustment has a simple statistical expression. Let $\mathbf{d}$ collect the residuals over times and clock pairs, and let $\mathbf{h}$ be the fixed theoretical signal after integration over the actual exposure windows. With the declared drift and environmental responses in the columns of B, and noise covariance C, the measurement model is

$$\mathbf{d}=\alpha \mathbf{h}+B\mathbf{b}+\mathbf{n},\mathbb{E}\mathbf{n}=0,\mathbb{E}\left(\mathbf{n}{\mathbf{n}}^{\mathsf{T}}\right)=C.$$

(8.15)

The amplitude $\alpha$ measures agreement with the prediction: $\alpha =1$ is the fixed dual-axis field model, while $\alpha =0$ is the baseline with no additional source. For positive-definite C and full-column-rank B, elimination of $\mathbf{b}$ gives

$$\begin{array}{cc}\hfill W_{\perp }& =C^{-1}-C^{-1}B{\left(B^{\mathsf{T}}{C}^{-1}B\right)}^{-1}{B}^{\mathsf{T}}{C}^{-1},\hfill \\ \hfill \widehat{\alpha }& ={\displaystyle \frac{{\mathbf{h}}^{\mathsf{T}}{W}_{\perp }\mathbf{d}}{{\mathbf{h}}^{\mathsf{T}}{W}_{\perp }\mathbf{h}}},{\sigma }_{\alpha }^2={\displaystyle \frac{1}{{\mathbf{h}}^{\mathsf{T}}{W}_{\perp }\mathbf{h}}}.\hfill \end{array}$$

(8.16)

These expressions follow from the generalized least-squares normal equations for Eq. (8.15); under Gaussian noise they also determine confidence intervals. A vanishing denominator means that the fitted systematic terms can reproduce the target signal, so that a different geometry or control history is required to separate them. Calibration uncertainties enter both the predicted signal and its confidence interval. The role of $\alpha$ is lost if G and U are selected anew after the residuals are seen in order to recover $\alpha =1$.

8.5. Measurement Scales and the Interpretation of Experimental Results

Optical clocks already resolve gravitational variations over laboratory distances. Bothwell et al. measured a frequency gradient consistent with gravitational redshift within a millimetre-scale strontium sample; Zheng et al. carried out a blind differential test with five atomic ensembles spanning one centimetre.[7,8] These measurements establish methods for differential clock comparison in the Earth’s existing gravitational field. The strength of a separately prepared observation-field source remains a question for its field solution.

A monopole estimate outside a compact source makes the scale explicit. The effective gravitational mass and the corresponding clock difference are

$$M_{\mathrm{act}}={\displaystyle \frac{1}{c^2}}\int {\mathcal{E}}_{\mathrm{obs}}{\mathrm{d}}^{3}x,\delta y_{AB}^{\mathrm{obs}}\simeq -{\displaystyle \frac{G_N{M}_{\mathrm{act}}}{c^2}}\left({\displaystyle \frac{1}{R_A}}-{\displaystyle \frac{1}{R_B}}\right).$$

(8.17)

where both $R_A$ and $R_B$ are much larger than the source dimensions. As a sensitivity conversion, a target fractional frequency difference of ${10}^{-18}$ and $|R_A^{-1}-R_B^{-1}|=1{\mathrm{m}}^{-1}$ correspond to $|M_{\mathrm{act}}|\simeq 1.35\times {10}^9\mathrm{kg}$. This is a hypothetical source scale, not a predicted observation-field mass. It expresses the weakness of gravitation: ordinary laboratory energies generate extremely small local clock shifts even at optical-clock precision. Experimental feasibility is therefore determined by the actual field solution, its spatial distribution, and the stress-energy of the full apparatus as well as by the clocks.

The two measurements answer distinct physical questions. Continuous records test a specified formation law and its transport rule. Clock differences test the gravitational contribution of a specified field solution, with a null result constraining its parameter range. The local dynamics of Eq. (4.1) have the form of Einstein gravity coupled to a nonlinear sigma field, so an extra gravitational signal alone does not identify a dual-axis origin. That identification depends on the independently determined response geometry and preparation law predicting the measurements together. No new experimental data are reported here; the results of this section are the observable relations through which the proposed dynamics can be tested.

9. Discussion and Conclusions

A dynamical observation state carries more information than the settings of an isolated measurement. It has initial data, propagates through spacetime, and contributes stress-energy. In the classical independent-metric branch studied here, these properties follow from a common action with a positive-definite observation kinetic metric and minimal two-derivative coupling. The target-space connection compares neighboring observation states; the spacetime connection governs their propagation. A field solution and its formation law then determine the record.

The relation to general relativity follows from the observation equation itself. A constant state at a stationary point of the potential contributes only a vacuum stress. Absorbing that stress into the cosmological constant leaves the Einstein–matter equations, and uniqueness of the evolution preserves the fixed section for compatible initial data. Once the state varies, its kinetic, gradient, and potential terms contribute to the source. The fixed-section solutions and the evolving observation fields remain solutions of one system with a common causal structure.

The experimental consequences depend on where this evolution enters the measurement. A change in response transport is tested by comparing coherent records with predictions made after independent calibration. A change in stress-energy is tested by solving for the metric and following its effect on clock rates and light propagation. In the weak-field limit, the clock source contains pressure as well as energy density. Near a stationary background, it is quadratic in the observation perturbation. An unperturbed reference, combinations even under reversal of the perturbation, and amplitude scans are therefore dictated by the source equation.

The local form of the action is that of Einstein gravity coupled to a nonlinear sigma field. A dual-axis interpretation acquires empirical content through the independent identification and preparation of the observation field, and through the requirement that one choice of $G,U,F$ predict records not used to determine those functions. Once the preparation, energy normalization, and geometry are fixed, the prediction has a definite amplitude and a definite dependence on time and position. Agreement across these observables supports the realization; disagreement rules out the tested realization under the same conditions.

The field equations, recovery theorem, initial-value evolution, and measurement relations thus answer successive parts of one physical question. They describe how an observation state moves, how its energy and stress affect spacetime, and how that change reaches a clock or a coherent record. This is the sense in which continuous observation enters the relativistic dynamics developed here.