By: Anes Palma, An Rodriguez (an@preferredframe.com)
Aug 2025
We derive the fundamental conservation laws of linear and angular momentum in a universe governed solely by source-free Maxwell equations. In this ontology, all matter consists of self-confined, structured electromagnetic fields. Without invoking particles, masses, or mechanical axioms, we demonstrate how momentum and angular momentum emerge directly from Maxwell dynamics. We show how their conservation arises under symmetry and closure conditions, and how deviations result from stress redistribution. This paper provides a complete mechanical foundation for conservation laws in a purely electromagnetic universe.
All classical mechanical behavior — including inertia, motion, and conservation laws — is shown to emerge rigorously from source-free Maxwell field dynamics alone.
Maxwell equations; electromagnetic ontology; momentum conservation; angular momentum; stress tensor; inertia; field mechanics; emergent motion; pure electromagnetism
Conservation laws form the foundation of classical mechanics, typically derived from Newton’s axioms or Noether’s theorem applied to action principles. However, in a universe where the only ontology is the electromagnetic field governed by source-free Maxwell equations in vacuum, there are no particles or external axioms. All physical entities are structured field configurations, such as standing waves or localized toroidal modes. The central question becomes: how do conservation laws such as linear and angular momentum emerge purely from field dynamics?
We approach this by analyzing the field-theoretic quantities that represent momentum, energy, and angular momentum, and showing their conservation follows directly from Maxwell’s equations and boundary conditions.
We assume SI units and source-free Maxwell equations:
\[ \nabla \cdot \vec{E} = 0, \quad \nabla \cdot \vec{B} = 0, \quad \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \]
These are the only laws assumed in this universe. All objects are field configurations, and all forces are electromagnetic.
The electromagnetic energy density is:
\[ u = \frac{1}{2} \left( \epsilon_0 |\vec{E}|^2 + \frac{1}{\mu_0} |\vec{B}|^2 \right) \]
The Poynting vector is:
\[ \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \]
The momentum density of the field is:
\[ \vec{g} = \frac{1}{c^2} \vec{S} = \epsilon_0 \vec{E} \times \vec{B} \]
The total linear momentum of the field in a volume \(V\) is:
\[ \vec{P} = \int_V \vec{g} \, d^3x = \epsilon_0 \int_V \vec{E} \times \vec{B} \, d^3x \]
This expression is exact and requires no particles.
We derive the time evolution using Maxwell’s equations:
\[ \frac{\partial \vec{g}}{\partial t} = \epsilon_0 \left( \frac{\partial \vec{E}}{\partial t} \times \vec{B} + \vec{E} \times \frac{\partial \vec{B}}{\partial t} \right) \]
Using:
\[ \frac{\partial \vec{E}}{\partial t} = \frac{1}{\mu_0 \epsilon_0} \nabla \times \vec{B}, \quad \frac{\partial \vec{B}}{\partial t} = -\nabla \times \vec{E} \]
we substitute and simplify:
\[ \frac{\partial \vec{g}}{\partial t} = \frac{1}{\mu_0} \left[ (\nabla \times \vec{B}) \times \vec{B} - \vec{E} \times (\nabla \times \vec{E}) \right] \]
This leads to the conservation equation:
\[ \frac{\partial \vec{g}}{\partial t} + \nabla \cdot \mathbf{T} = 0 \]
where \(\mathbf{T}\) is the Maxwell stress tensor:
\[ T_{ij} = \epsilon_0 \left( E_i E_j + c^2 B_i B_j - \frac{1}{2} \delta_{ij} (|\vec{E}|^2 + c^2 |\vec{B}|^2) \right) \]
This symmetric tensor governs the flow of momentum through space.
Integrating over volume:
\[ \frac{d\vec{P}}{dt} = -\int_{\partial V} \mathbf{T} \cdot d\vec{A} \]
Therefore, momentum is conserved if:
\[ \int_{\partial V} \mathbf{T} \cdot d\vec{A} = 0 \]
This holds when the region \(V\) encloses all field energy, and the stress tensor flux across the boundary vanishes (e.g., due to symmetry or confinement).
In a Maxwell universe, field lines are closed and energy circulates internally. Thus, momentum conservation holds locally by construction, and globally when configurations are symmetric and isolated.
Define total field energy:
\[ U = \int_V u \, d^3x \]
Define center of energy:
\[ \vec{R}(t) = \frac{1}{U} \int_V \vec{r} u(\vec{r}, t) \, d^3x \]
Time derivative:
\[ \frac{d\vec{R}}{dt} = \frac{1}{U} \int_V \vec{g} \, d^3x = \frac{\vec{P}}{U} \]
If \(\vec{P}\) is constant, then \(\vec{R}(t)\) moves at constant velocity. This is inertial motion emerging from internal field momentum, without requiring mass.
The angular momentum density is:
\[ \vec{\ell} = \vec{r} \times \vec{g} = \epsilon_0 \vec{r} \times (\vec{E} \times \vec{B}) \]
Total angular momentum:
\[ \vec{L} = \int_V \vec{\ell} \, d^3x = \epsilon_0 \int_V \vec{r} \times (\vec{E} \times \vec{B}) \, d^3x \]
Differentiate:
\[ \frac{d\vec{L}}{dt} = -\int_V \vec{r} \times (\nabla \cdot \mathbf{T}) \, d^3x \]
Using vector identity and symmetry of \(\mathbf{T}\):
\[ \frac{d\vec{L}}{dt} = -\int_{\partial V} \vec{r} \times (\mathbf{T} \cdot d\vec{A}) \]
Angular momentum is conserved if the net torque across the boundary vanishes:
\[ \int_{\partial V} \vec{r} \times (\mathbf{T} \cdot d\vec{A}) = 0 \]
This is typically satisfied for confined, symmetric field configurations (e.g. toroidal modes).
A laboratory wheel spins because it stores angular momentum in the
motion of its mass elements.
In a Maxwell universe the analogue is a self-confined field
structure that carries non-zero
\[ \vec{L} = \epsilon_0 \int_V \vec{r}\times(\vec{E}\times\vec{B})\,d^3x \; . \]
Definition of effective moment of inertia
Let \(\omega\) be the uniform
phase-rotation rate of the internal fields around a fixed axis (e.g. the
\(z\)–axis).
For a time-stationary configuration one may write
\[ \vec{L}=I_{\text{eff}}\;\vec{\omega},\qquad I_{\text{eff}}=\frac{1}{\omega^2}\, \epsilon_0\!\int_V\!|\vec{E}\times\vec{B}|_\perp\,d^3x , \]
where the subscript \(\perp\)
selects the component perpendicular to the axis.
\(\,I_{\text{eff}}\,\) plays the role
of a moment of inertia: it depends only on the internal
energy distribution, not on any postulated mass.
Persistence of rotation
Because source-free Maxwell equations contain no dissipative term, \(\partial_t\vec{L}=0\) whenever the boundary
torque vanishes.
Thus a “wheel” of electromagnetic energy keeps spinning indefinitely; an
external torque is required to change \(\vec{L}\) exactly as in ordinary
mechanics.
In a Maxwell universe, all physical structures are electromagnetic field configurations with finite energy. These structures — such as dipole-like or toroidal standing waves — exhibit behavior commonly attributed to “mass”, such as resistance to acceleration and persistent motion. We now explain these effects directly from Maxwell’s equations, without introducing mass as a fundamental concept.
Consider a localized electromagnetic field configuration (the “object”) occupying a finite region of space \(V\). A push is defined as the transient application of an additional electromagnetic field \(\vec{E}_{\text{ext}}(\vec{r}, t), \vec{B}_{\text{ext}}(\vec{r}, t)\) that temporarily overlaps and interacts with the original fields \(\vec{E}, \vec{B}\) in \(V\).
Mathematically, this modifies the local fields:
\[ \vec{E}_{\text{total}} = \vec{E} + \vec{E}_{\text{ext}}, \quad \vec{B}_{\text{total}} = \vec{B} + \vec{B}_{\text{ext}} \]
during a time interval \(t \in [t_0, t_1]\).
The total momentum density becomes:
\[ \vec{g}_{\text{total}} = \epsilon_0 (\vec{E}_{\text{total}} \times \vec{B}_{\text{total}}) \]
This changes the field’s momentum content and structure.
A deformation is the local change in field amplitude, phase, or spatial distribution due to the non-uniform overlap of the external and internal fields. Unlike a rigid-body response, Maxwell fields respond continuously and locally, according to the equations:
\[ \frac{\partial \vec{g}}{\partial t} = -\nabla \cdot \mathbf{T}, \quad \vec{g} = \epsilon_0 \vec{E} \times \vec{B} \]
The result is a redistribution of energy and momentum across the object’s volume \(V\).
In physical terms:
- The region receiving the strongest field overlap (i.e., “pushed”)
experiences an increase in local energy density and momentum.
- Adjacent regions adjust in response, establishing field gradients and
internal stresses.
- These internal adjustments propagate at speed \(c\), forming an electromagnetic analog of
mechanical strain.
Let the initial field energy and momentum before the push be:
\[ U_0 = \int_V u(\vec{r}, t_0) \, d^3x, \quad \vec{P}_0 = \int_V \vec{g}(\vec{r}, t_0) \, d^3x \]
After the external field ends (\(t > t_1\)), the total energy and momentum become:
\[ U_1 = \int_V u(\vec{r}, t_1) \, d^3x, \quad \vec{P}_1 = \int_V \vec{g}(\vec{r}, t_1) \, d^3x \]
The difference:
\[ \Delta \vec{P} = \vec{P}_1 - \vec{P}_0 \]
represents the net momentum transferred into the field configuration by the push.
This momentum is not localized at a point, but distributed across the internal field configuration. The original structure adjusts by altering the phase and shape of its internal modes — in effect, “absorbing” the impulse.
After the deformation and redistribution settle, the configuration possesses:
\[ \vec{R}(t) = \frac{1}{U} \int_V \vec{r} \, u(\vec{r}, t) \, d^3x \]
Its velocity is given by:
\[ \frac{d\vec{R}}{dt} = \frac{\vec{P}_1}{U_1} \]
This motion continues indefinitely (until further interactions), as no dissipation mechanism exists in source-free Maxwell equations.
The volume \(V\) of the object is defined as the smallest region satisfying:
This boundary is not arbitrary: it separates the dynamically active internal configuration from the rest of space. It is the natural field-theoretic analog of a material boundary.
A “push” is a finite-time overlap of external fields with an internal configuration. This induces a deformation, meaning a local and continuous redistribution of energy and momentum, governed by Maxwell’s equations.
The internal stress balances adjust, storing the impulse as coherent internal motion. Once the interaction ends, the structure moves with constant velocity determined by its internal momentum content.
This persistent motion — inertia — is thus fully explained by internal field dynamics, with no need to postulate mass or point particles.
Define the effective inertial mass
\[ m_{\text{eff}} = \frac{U}{c^{2}},\qquad U=\int_V u\,d^3x . \]
For any object the field linear momentum is
\[ \vec{P}=m_{\text{eff}}\;\vec{v},\qquad \vec{v}=\frac{d\vec{R}}{dt}. \]
Taking a total derivative gives
\[ \frac{d\vec{P}}{dt}=m_{\text{eff}}\;\frac{d\vec{v}}{dt} \;=\;m_{\text{eff}}\;\vec{a}, \]
where \(\vec{a}\) is the
acceleration of the center of energy.
But Maxwell dynamics already furnishes (section Global Momentum
Conservation)
\[ \frac{d\vec{P}}{dt} = -\!\int_{\partial V}\!\mathbf{T}\cdot d\vec{A}\;=\;\vec{F}_{\text{ext}} . \]
Hence
\[ \boxed{\;\vec{F}_{\text{ext}} = m_{\text{eff}}\;\vec{a}\;} \]
with no additional postulates: Newton’s second law appears as a bookkeeping identity coupling stress-tensor flux to the rate of change of field momentum.
Let us model a dipole-like field configuration aligned along the \(z\)-axis, consisting of two localized electromagnetic lobes separated by distance \(2d\), centered at \(\vec{r}_\pm = (0, 0, \pm d)\). The total field configuration satisfies:
\[ \vec{E}(\vec{r}, t) = \vec{E}_+(\vec{r} - \vec{r}_+) + \vec{E}_-(\vec{r} - \vec{r}_-), \quad \vec{B}(\vec{r}, t) = \vec{B}_+ + \vec{B}_- \]
Now suppose the dipole enters a background field with a gradient in energy density along \(z\):
\[ u_{\text{bg}}(z) = u_0 + \alpha z \]
This modifies the effective stress tensor in space. The net force on the structure is:
\[ \vec{F} = -\int_{\partial V} \mathbf{T}_{\text{total}} \cdot d\vec{A} \]
But we can also write the net internal field force as:
\[ \vec{F}_{\text{eff}} = \int_V \nabla \cdot \mathbf{T}_{\text{bg}}(\vec{r}) \, d^3x \]
Assuming that the background gradient is slow over the dipole scale (\(\alpha d \ll u_0\)), we can expand \(\mathbf{T}_{\text{bg}}\) near the center and approximate:
\[ \nabla \cdot \mathbf{T}_{\text{bg}} \approx \frac{d\mathbf{T}}{dz} \hat{z} \]
Therefore:
\[ \vec{F}_{\text{eff}} \approx \left( \frac{dT_{zz}}{dz} \right) \int_V d^3x \, \hat{z} = \alpha' V_{\text{eff}} \hat{z} \]
The dipole drifts along the gradient toward regions of higher \(u_{\text{bg}}\) if \(\alpha' < 0\). The drift velocity is given approximately by:
\[ \frac{d\vec{R}}{dt} = \frac{\vec{P}_{\text{field}}}{U} \]
where \(\vec{P}_{\text{field}}\) evolves due to asymmetric stress:
\[ \frac{d\vec{P}_{\text{field}}}{dt} = -\int_V \nabla \cdot \mathbf{T}_{\text{bg}} \, d^3x \approx \alpha' V_{\text{eff}} \hat{z} \]
This completes the derivation: a dipole in a gradient moves due to spatially varying field stresses, not due to an external force. The effect is mechanical, electromagnetic, and purely local.
In a universe described solely by source-free Maxwell equations:
Thus classical mechanics — linear motion, rotational motion, and Newton’s second law — is rigorously reproduced from pure electromagnetic dynamics, with no particles, masses, or spacetime curvature required.