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\title{Momentum Conservation and Effective Stress--Energy Transfer in Emergent Vacuum Electrodynamics}

\author{
Erick Sangalang\\
ORCID: 0009-0009-4039-2815
}

\date{2026}

\begin{document}

\maketitle

\begin{center}
Version 1.0
\end{center}

\vspace{0.5em}

\noindent\textbf{Keywords:}
emergent vacuum response;
stress--energy transfer;
vacuum electrodynamics;
momentum conservation;
nonequilibrium electrodynamics;
effective field phenomenology;
resonant cavity systems;
electromagnetic momentum;
Maxwell stress tensor;
cavity metrology

\vspace{1em}

\begin{abstract}

Previous investigations in this series explored phenomenological emergent vacuum-response models under resonant nonequilibrium electromagnetic conditions, including constraint analysis and nonlinear dynamical behavior. The present work examines a central consistency question for any such framework:

\begin{quote}
If an emergent vacuum-response sector exists, how are momentum conservation and stress--energy transfer maintained?
\end{quote}

The paper develops a conservative phenomenological treatment in which any observable anomalous-force-like behavior must arise through internally conserved stress--energy exchange within a coupled electromagnetic-vacuum effective system rather than through violation of conservation laws. Possible mechanisms examined include transient field-geometry asymmetry, delayed nonequilibrium relaxation, resonant stress redistribution, and effective momentum exchange between coherent electromagnetic modes and an emergent collective response sector.

The framework remains intentionally constrained and does not claim established reactionless propulsion, antigravity, or experimentally verified momentum nonconservation. Instead, the work argues that any physically viable emergent electrodynamic framework must remain globally energy bounded, compatible with center-of-mass conservation, Lorentz compatible at ordinary scales, and fully consistent with conservation principles under experimentally accessible conditions.

\end{abstract}

\section{Introduction}

A central requirement of all physically viable theories is consistency with conservation laws.

Previous papers in this series explored:

\begin{itemize}
    \item emergent vacuum susceptibility,
    \item resonant enhancement,
    \item nonlinear nonequilibrium dynamics,
    \item and constraint compatibility with precision electrodynamics.
\end{itemize}

However, an immediate consistency question naturally follows:

\begin{quote}
If an emergent vacuum-response sector exists, how is momentum conserved?
\end{quote}

Historically, many speculative electromagnetic propulsion proposals fail precisely because they violate global conservation principles or implicitly assume reactionless momentum generation.

The present paper therefore approaches the issue conservatively. Rather than proposing momentum creation, the framework examined here assumes that any physically meaningful anomalous-force-like behavior must emerge through internally conserved stress--energy redistribution within a coupled effective system.

The analysis remains phenomenological and intentionally restrained. No claim is made regarding experimentally established propulsion effects, violation of conservation laws, or modification of General Relativity.

\section{Classical Electromagnetic Momentum}

In classical electrodynamics, electromagnetic fields possess momentum density:

\begin{equation}
\mathbf{g}
=
\epsilon_0 \mathbf{E} \times \mathbf{B}.
\end{equation}

The associated momentum flux is described by the Maxwell stress tensor:

\begin{equation}
T_{ij}
=
\epsilon_0
\left(
E_i E_j
-
\frac12 \delta_{ij} E^2
\right)
+
\frac{1}{\mu_0}
\left(
B_i B_j
-
\frac12 \delta_{ij} B^2
\right).
\end{equation}

Momentum conservation follows from the local continuity equation:

\begin{equation}
\partial_t g_i
+
\partial_j T_{ij}
=
-f_i,
\end{equation}

where $f_i$ represents force density acting on matter.

Under ordinary conditions, closed electromagnetic systems cannot generate net external momentum without interaction with external matter, radiation, or fields.

\section{Effective Emergent Response Sector}

Earlier work introduced a phenomenological emergent susceptibility coordinate $\phi$.

The present paper interprets this quantity not as a confirmed fundamental field, but as an effective coarse-grained collective response parameter associated with coherent nonequilibrium electromagnetic organization.

A simplified effective interaction term may be written:

\begin{equation}
\mathcal{L}_{\mathrm{int}}
=
g \phi \mathbf{E}\cdot\mathbf{B}.
\end{equation}

This coupling does not itself imply momentum nonconservation. Instead, it introduces the possibility that coherent electromagnetic energy may transiently redistribute stress and momentum internally within a coupled effective system.

The response sector should therefore be understood as an effective bookkeeping device for possible collective nonequilibrium susceptibility dynamics rather than evidence for a new independently verified physical medium.

\section{Stress--Energy Redistribution}

A central principle of the present framework is:

\begin{quote}
Observable force-like behavior, if it exists, must arise through stress redistribution rather than net momentum creation.
\end{quote}

Under highly asymmetric resonant excitation, coherent electromagnetic configurations may produce:

\begin{itemize}
    \item transient internal pressure asymmetry,
    \item delayed relaxation dynamics,
    \item anisotropic field confinement,
    \item or effective momentum storage within nonequilibrium field structure.
\end{itemize}

In this interpretation, any measurable force-like signal would represent internal stress redistribution rather than true reactionless propulsion.

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{momentum_flow_diagram.png}
\caption{Conceptual momentum-flow diagram for the coupled effective system. The arrows represent internal redistribution and environmental coupling rather than net momentum creation.}
\end{figure}

\section{Effective Stress Tensor Extension}

Phenomenologically, the total stress tensor may be represented as:

\begin{equation}
T^{\mu\nu}_{\mathrm{eff}}
=
T^{\mu\nu}_{\mathrm{EM}}
+
T^{\mu\nu}_{\phi}
+
T^{\mu\nu}_{\mathrm{int}},
\end{equation}

where:

\begin{itemize}
    \item $T^{\mu\nu}_{\mathrm{EM}}$ is the ordinary electromagnetic contribution,
    \item $T^{\mu\nu}_{\phi}$ is an effective emergent-response contribution,
    \item and $T^{\mu\nu}_{\mathrm{int}}$ represents interaction terms.
\end{itemize}

Global conservation requires:

\begin{equation}
\partial_\mu T^{\mu\nu}_{\mathrm{eff}} = 0.
\end{equation}

This condition is fundamental. Any phenomenological framework failing to satisfy global conservation would be considered physically inconsistent.

\section{Covariant Consistency and Frame Independence}

A conservation-compatible framework must not rely on a preferred laboratory interpretation of momentum that fails under a change of frame. At the effective-field-theory level, the stress--energy tensor must be interpretable covariantly, even if a specific experiment selects a preferred laboratory frame through boundary conditions, static magnetic fields, cryogenic preparation, or resonant cavity geometry.

The distinction is important. Laboratory configurations can break symmetry operationally without implying fundamental Lorentz violation. A cavity, magnet, support structure, or thermal boundary condition defines a preferred experimental frame, but the underlying bookkeeping of stress--energy exchange must still remain compatible with relativistic conservation.

No physically preferred universal frame is introduced by the framework; only experimentally selected boundary conditions define effective laboratory asymmetry.

Thus, any viable emergent-response model must satisfy:

\begin{equation}
\partial_\mu T^{\mu\nu}_{\rm total}=0
\end{equation}

when all relevant sectors are included, including fields, effective response coordinates, material supports, radiation leakage, and environmental coupling.

\section{Center-of-Mass Consistency}

A closed realization of the framework must preserve total center-of-mass motion. This requirement is a direct consequence of momentum conservation.

For an isolated complete system,

\begin{equation}
\frac{d\mathbf{P}_{\rm total}}{dt}=0,
\end{equation}

where

\begin{equation}
\mathbf{P}_{\rm total}
=
\mathbf{P}_{\rm EM}
+
\mathbf{P}_{\phi}
+
\mathbf{P}_{\rm matter}
+
\mathbf{P}_{\rm radiation}.
\end{equation}

Any apparent one-directional force in a subsystem must therefore be balanced by momentum stored or transported elsewhere. Possible repositories include fields, supports, radiation, thermal gradients, residual gas, or delayed internal response modes.

This condition prevents the framework from being interpreted as reactionless propulsion. If a proposed configuration predicts net acceleration of a closed isolated system without compensating momentum exchange, then that configuration is rejected as physically inconsistent.

\section{Stress Geometry in Asymmetric Cavities}

Highly asymmetric resonant geometries are frequently discussed in speculative propulsion literature. Within the present framework, cavity asymmetry alone is insufficient to produce net momentum.

However, asymmetry may influence:

\begin{itemize}
    \item local stress concentration,
    \item mode localization,
    \item phase synchronization,
    \item field-boundary pressure gradients,
    \item and nonequilibrium energy storage.
\end{itemize}

The framework therefore interprets cavity asymmetry as a modifier of internal stress geometry rather than a direct propulsion mechanism.

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{stress_tensor_geometry.png}
\caption{Schematic stress-tensor geometry in an asymmetric resonant cavity. The diagram illustrates localized stress concentration and anisotropic pressure distribution without implying net external momentum generation.}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{stress_cancellation.png}
\caption{Illustration of stress cancellation versus asymmetric stress geometry. Local stress imbalance may occur within subsystems while total momentum conservation remains globally enforced.}
\end{figure}


\section{Nonequilibrium Momentum Storage}

One possible interpretation explored phenomenologically is transient momentum storage within coherent nonequilibrium field structure.

Examples from known physics include:

\begin{itemize}
    \item electromagnetic cavity pressure,
    \item delayed resonator relaxation,
    \item phonon momentum storage,
    \item optomechanical field-pressure coupling,
    \item and transient stress accumulation in driven media.
\end{itemize}

The present framework asks whether highly coherent electromagnetic configurations could similarly support temporary nonequilibrium stress organization.

Importantly, this would not imply permanent net momentum generation. Instead, any observable signal would necessarily remain bounded, reversible, and consistent with global conservation.

\section{Delayed Relaxation Dynamics}

If coherent nonequilibrium stress organization exists, delayed relaxation effects may occur.

A simplified phenomenological relaxation equation may be written:

\begin{equation}
\partial_t \phi
=
-\Gamma \phi
+
S(t),
\end{equation}

where $\Gamma$ is the relaxation rate and $S(t)$ represents coherent electromagnetic driving.

Transient stress imbalance during excitation and relaxation could potentially produce:

\begin{itemize}
    \item phase-lagged force signals,
    \item delayed recoil,
    \item or hysteresis-like response.
\end{itemize}

However, total momentum conservation remains globally enforced.

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{relaxation_recoil_plot.png}
\caption{Illustrative phase-lagged stress response following finite-duration electromagnetic excitation. The delayed response and relaxation tail are schematic and do not represent experimentally calibrated force data.}
\end{figure}


\section{Boundary Conditions and Real Systems}

Real electrodynamic systems are never perfectly isolated. Practical cavity experiments include:
\begin{itemize}
\item support structures,
\item finite wall conductivity,
\item dielectric interfaces,
\item thermal expansion,
\item cable coupling,
\item imperfect reflectivity,
\item and environmental vibration.
\end{itemize}

These effects can alter local stress distribution and produce apparent force asymmetries even when total momentum remains conserved. Consequently, boundary conditions play a central role in interpreting any claimed anomalous-force signal.

A physically meaningful analysis therefore requires complete bookkeeping of both internal field dynamics and environmental coupling channels.

\section{Radiation and Environmental Coupling}

Any experimentally measurable momentum transfer must ultimately involve coupling to:

\begin{itemize}
    \item external radiation,
    \item material supports,
    \item thermal gradients,
    \item residual gas interaction,
    \item or environmental fields.
\end{itemize}

This point is crucial. Many historical anomalous-force claims are later explained through thermal drift, Lorentz coupling, vibration, electrohydrodynamic effects, or measurement artifacts.

The present framework therefore emphasizes artifact discrimination as mandatory.

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{momentum_leakage_table.png}
\caption{Representative momentum leakage and artifact channels relevant to precision cavity-force measurements.}
\end{figure}

Any claimed anomalous-force signal must survive:
\begin{itemize}
\item vacuum isolation,
\item thermal inversion tests,
\item orientation reversal,
\item cable-decoupling tests,
\item blind calibration,
\item and independent replication.
\end{itemize}


\section{Conservation Closure}

The full system may include electromagnetic fields, effective response modes, matter, supports, radiation leakage, and environmental degrees of freedom. Conservation closure requires that all such channels be included in the momentum accounting.

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{conservation_closure_diagram.png}
\caption{Conservation-closure diagram. Momentum exchange between sectors is permitted, but net creation of momentum in the closed system is forbidden.}
\end{figure}

A useful schematic expression is:

\begin{equation}
\mathbf{P}_{\rm total}
=
\mathbf{P}_{\rm EM}
+
\mathbf{P}_{\phi}
+
\mathbf{P}_{\rm matter}
+
\mathbf{P}_{\rm radiation}
+
\mathbf{P}_{\rm environment}.
\end{equation}

For a closed system:

\begin{equation}
\frac{d\mathbf{P}_{\rm total}}{dt}=0.
\end{equation}

For an open system, apparent momentum changes must be traced to exchanged momentum flux across the boundary.

\section{Known Analog Systems}

The present framework is not intended to claim that vacuum response behaves exactly like any known material medium. However, known systems provide useful analogies for disciplined momentum bookkeeping.

\begin{figure}[H]
\centering
\includegraphics[width=0.82\textwidth]{known_analog_systems.png}
\caption{Known analog systems illustrating that fields, media, boundaries, and collective modes can exchange stress and momentum without violating conservation laws.}
\end{figure}

Relevant analog systems include:

\begin{itemize}
    \item radiation pressure in optical cavities,
    \item optomechanical momentum exchange,
    \item phonon momentum transport,
    \item electromagnetic stress in dielectric media,
    \item superconducting cavity wall pressure,
    \item and effective-medium response in metamaterials.
\end{itemize}

These analogies do not prove the emergent vacuum framework. They only demonstrate that stress redistribution, delayed relaxation, and field-boundary momentum exchange are familiar conservation-compatible concepts.

\section{Illustrative Effective Dynamics}

A simplified effective coupled system may schematically take the form:

\begin{align}
\partial_\mu T^{\mu\nu}_{\mathrm{EM}}
&=
-J^\nu_{\mathrm{int}},
\\
\partial_\mu T^{\mu\nu}_{\phi}
&=
+J^\nu_{\mathrm{int}},
\end{align}

where $J^\nu_{\mathrm{int}}$ represents internal stress--energy exchange.

The total conserved tensor remains:

\begin{equation}
\partial_\mu
\left(
T^{\mu\nu}_{\mathrm{EM}}
+
T^{\mu\nu}_{\phi}
\right)
=
0.
\end{equation}

This framework therefore preserves overall conservation while permitting transient internal redistribution dynamics.

A compact momentum-accounting expression may be written schematically as:

\begin{equation}
\Delta P_{\rm EM}
+
\Delta P_{\phi}
+
\Delta P_{\rm env}
+
\Delta P_{\rm rad}
=0.
\end{equation}

The expression emphasizes that any apparent momentum imbalance in one sector must be compensated elsewhere within the complete coupled system.


\section{Simple Stress-Scale Estimate}

A useful order-of-magnitude stress scale for electromagnetic fields is the energy density:

\begin{equation}
u_{\rm EM}
=
\frac12 \epsilon_0 E^2
+
\frac{1}{2\mu_0}B^2.
\end{equation}

The corresponding pressure scale is roughly:

\begin{equation}
p_{\rm EM} \sim u_{\rm EM}.
\end{equation}

For a strong magnetic field of order:

\begin{equation}
B_0 \sim 10\,{\rm T},
\end{equation}

the magnetic energy density scale is:

\begin{equation}
u_B
=
\frac{B_0^2}{2\mu_0}
\sim
4\times 10^7\,{\rm J\,m^{-3}}.
\end{equation}

This is a real electromagnetic stress scale, but in a symmetric closed system its integrated contributions cancel. A measurable net force requires asymmetry, boundary coupling, radiation leakage, environmental interaction, or incomplete cancellation. Therefore, large internal stress does not by itself imply net propulsion.

Even under optimistic assumptions, any experimentally accessible residual effect would likely remain extremely small relative to ordinary electromagnetic and thermal background forces.


\section{Effective Energy Functional}

A useful phenomenological bookkeeping structure is an effective total energy functional:

\begin{equation}
\mathcal{E}_{\rm eff}
=
\mathcal{E}_{\rm EM}
+
\mathcal{E}_{\phi}
+
\mathcal{E}_{\rm int}.
\end{equation}

Physical consistency requires the total effective energy remain globally bounded under experimentally accessible conditions. The framework therefore assumes:
\begin{itemize}
\item no runaway amplification,
\item no perpetual energy growth,
\item dissipative relaxation under decoherence,
\item and eventual return toward ordinary Maxwellian behavior outside resonance.
\end{itemize}

The effective energy functional is not presented as a microscopic derivation, but rather as a phenomenological consistency structure enforcing bounded nonequilibrium dynamics.

\section{Experimental Implications}

The framework suggests several possible observables:

\subsection{Phase-Lagged Force Response}

Transient force-like signals may exhibit delay relative to drive excitation.

\subsection{Relaxation Recoil}

Small reverse relaxation signals may appear during cavity ringdown.

\subsection{Resonance-Dependent Stress Localization}

Observable behavior may occur only near narrow coherence windows.

\subsection{Strong Environmental Sensitivity}

Any effect should remain highly sensitive to temperature, vibration, cavity geometry, phase coherence, and electromagnetic shielding.

\section{Null Interpretation}

Null results are scientifically meaningful within the present framework.

If all measured force-like signals vanish under vacuum-isolated, thermally controlled, electromagnetically shielded, and mechanically decoupled null tests, then at least one of the following conclusions follows:

\begin{enumerate}
    \item the effective coupling is below experimental sensitivity,
    \item the required coherence regime was not reached,
    \item the proposed emergent response sector is absent,
    \item or the phenomenological model is incomplete or incorrect.
\end{enumerate}

In all cases, null results constrain the allowed parameter space and improve the scientific utility of the framework.

\section{No Reactionless Propulsion Claim}

The present framework does not claim:

\begin{itemize}
    \item reactionless propulsion,
    \item momentum creation,
    \item violation of Newtonian mechanics,
    \item free-energy extraction,
    \item or experimentally verified anomalous thrust.
\end{itemize}

Instead, the paper explores whether highly coherent nonequilibrium electrodynamic systems could exhibit internally conserved stress redistribution phenomena under specialized resonant conditions.

Any experimentally viable realization must remain fully compatible with conservation principles.

A perfectly isolated closed system is not expected to exhibit sustained net acceleration within the present framework.

\section{Discussion}

The central purpose of this paper is not to argue for propulsion claims, but to establish a conservation-compatible phenomenological framework for analyzing possible nonequilibrium electromagnetic stress dynamics.

The framework attempts to move beyond simplistic ``reactionless thrust'' narratives by emphasizing global conservation, center-of-mass consistency, bounded dynamics, environmental coupling, decoherence, and falsifiability.

The analysis remains speculative and phenomenological. At present, no experimental evidence exists demonstrating stress--energy transfer effects of the type explored here.

Even robust null results would provide valuable constraints on hypothetical emergent nonequilibrium electrodynamic sectors.

\section{Conclusion}

This work examined momentum conservation and effective stress--energy transfer within a phenomenological emergent electrodynamic framework.

The analysis argues that any physically viable emergent-response sector must preserve:

\begin{itemize}
    \item global momentum conservation,
    \item center-of-mass consistency,
    \item energy boundedness,
    \item and compatibility with ordinary electrodynamics.
\end{itemize}

Possible phenomenology explored includes:

\begin{itemize}
    \item transient stress redistribution,
    \item delayed relaxation dynamics,
    \item resonance-dependent force-like response,
    \item conservation-closed environmental coupling,
    \item and nonequilibrium momentum storage.
\end{itemize}

The framework remains intentionally conservative and does not claim experimentally verified propulsion phenomena or conservation-law violation.

Future investigations should focus on:

\begin{itemize}
    \item precision cavity force metrology,
    \item time-domain relaxation measurements,
    \item numerical stress-tensor simulations,
    \item finite-element cavity stress mapping,
    \item and experimental artifact discrimination.
\end{itemize}

\begin{thebibliography}{99}

\bibitem{jackson}
J. D. Jackson,
\textit{Classical Electrodynamics},
3rd Edition, Wiley, 1998.

\bibitem{landau}
L. D. Landau and E. M. Lifshitz,
\textit{The Classical Theory of Fields},
Butterworth-Heinemann, 1975.

\bibitem{milonni}
P. W. Milonni,
\textit{The Quantum Vacuum},
Academic Press, 1994.

\bibitem{griffiths}
D. J. Griffiths,
\textit{Introduction to Electrodynamics},
4th Edition, Cambridge University Press, 2017.

\bibitem{brevik}
I. Brevik,
``Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,''
\textit{Physics Reports}, 52, 133--201 (1979).

\bibitem{pfeifer}
R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop,
``Colloquium: Momentum of an electromagnetic wave in dielectric media,''
\textit{Reviews of Modern Physics}, 79, 1197--1216 (2007).

\bibitem{ashkin}
A. Ashkin,
``Acceleration and trapping of particles by radiation pressure,''
\textit{Physical Review Letters}, 24, 156--159 (1970).

\bibitem{aspelmeyer}
M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,
``Cavity optomechanics,''
\textit{Reviews of Modern Physics}, 86, 1391--1452 (2014).

\bibitem{padamsee}
H. Padamsee,
\textit{RF Superconductivity for Accelerators},
Wiley, 2009.

\bibitem{taflove}
A. Taflove and S. Hagness,
\textit{Computational Electrodynamics: The Finite-Difference Time-Domain Method},
Artech House, 2005.

\bibitem{wilczek}
F. Wilczek,
``Two Applications of Axion Electrodynamics,''
\textit{Physical Review Letters}, 58, 1799 (1987).

\bibitem{strogatz}
S. H. Strogatz,
\textit{Nonlinear Dynamics and Chaos},
Westview Press, 2015.

\end{thebibliography}

\end{document}
