The Physics of Energy Flow
Deriving All Known Physics from Electromagnetic Energy
Deriving All Known Physics from Electromagnetic Energy
2026-03-14
There is an old observation, so obvious it is easy to overlook: for two things to interact, they must share something that can be mutually affected.
The fact that we register interaction at all implies a shared substrate. Throughout this book, we call that substrate energy.
All that is required to recover known physics is the concept of energy. Static energy, in the sense developed in what follows, would be inert and could not be experienced. What reaches us are the effects of its redistribution. By registering those effects, as in experiment, we infer its continuous flow.
This is not a limitation to be overcome. What we have access to is the consequence of its flow. And it turns out that is enough.
The pages that follow ask a single question, pursued with as few assumptions as possible: if something exists and redistributes continuously, what must follow?
The answer is, as we shall see, at least all of known physics.
The minimal starting point of physical description is the bare fact that something exists and is seen to change. We notice change because we register differences: a scratch on a table, a shifted shadow, a moved object.
Call this something \(u\). Its presence can be at least partially described relative to itself by writing \(u(\mathbf{r})\), where \(\mathbf{r}\) is a label for relative position within the extent of \(u\):
\[ u(\mathbf{r}) \geq 0. \]
The name we give to \(u\) is energy.
If apparently distinct things exist and interact, they are not fundamentally independent substances. They are configurations of \(u\). Interaction is therefore always only between \(u\) and itself.
In this book, \(u\) names that one interacting substance: all that physically exists.1 This single assumption - that something real is present as a nonnegative magnitude across its extent - is the ontological foundation of all known physics. Nothing more primitive is assumed in what follows.
A single registered distribution of energy,
\[ u(\mathbf{r}), \]
says only how energy is distributed across the extent of \(u\).
If another registered distribution differs from it, we may label the two records
\[ u_1(\mathbf{r}), \qquad u_2(\mathbf{r}). \]
Change is the acknowledgement of a difference between distributions.
What is observed is not disappearance in one part of the extent of \(u\) and reappearance in another, but continuous reconfiguration. Energy present in one part is registered in another. We interpret that observed reconfiguration as flow.
To describe energy flow is to describe its continuous redistribution within itself.
We write it as
\[ \mathbf{S}(\mathbf{r};1,2). \]
The flow lines of \(\mathbf{S}\) trace the reconfiguration by which \(u_1\) becomes \(u_2\).
The energy field \(u\) tells us how much energy is present. The flow \(\mathbf{S}\) tells us how that energy redistributes.
Reconfiguration in a region of \(u\) is a fact of \(u\) itself. It is not private to one part of the extent. The surrounding extent must accord with that change. If it did not, the reconfiguration could not be said to have occurred in \(u\) at all. That makes an ordering of registrations possible. Label two such ordered registrations \(1\) and \(2\). Then \(u_2\) is related to \(u_1\) through a flow field \(\mathbf{S}(\mathbf{r};1,2)\), which encodes how one registered distribution changes into another.
The difference
\[ u_2(\mathbf{r})-u_1(\mathbf{r}) \]
is understood as the result of a continuous redistribution of energy within itself, described by a flow connecting the two registrations. Energy in a region changes only by crossing its boundary to a neighboring region.
In one direction, say the x-direction, the statement is
\[ u_2-u_1+\partial_x S_{12}=0. \]
Here \(S_{12}\) refers to the redistribution flow connecting registrations \(1\) and \(2\).
This is accounting of energy, not yet its dynamics. It is like accounting for the brightness of the pixels on a screen without yet recognizing the image they compose.2
Continuity is therefore the minimal consistency requirement for transport. It gives closed bookkeeping.
We now turn to exploring the implied consequences of this energy accounting in free space.
Across the extent of \(u\), continuity holds. Energy transported across a region does not create or destroy energy there. Rather, it changes how much energy is stored there by moving it across regions.
Since \(u\) is what is given to exist, its total quantity is given with it and can therefore be considered fixed. A primitive source or sink would require either that total to change or that disconnected regions compensate one another without transport between them.
This is the source-free continuity statement itself: no added creation term, no added destruction term, only transport.
Continuity gives local accounting. We now explore what it implies for the shape of energy reorganization as a complete process.
Although \(\mathbf{S}\) handles a directional accounting of how energy is transported, even when taken as a whole it still fails to reveal the structure of that transport. The next chapters develop that structure.
We next consider the transport of energy across empty space.
Regional conservation uses the flow \(\mathbf{S}\). It gives region by region numerical accounting. The shape of the reorganization as a complete process comes next. For that we introduce a flow field \(\mathbf{F}\).
In empty space, a source-free transport cannot begin or end at an isolated point. If energy leaves one small region, it must pass into another neighboring one. Looked at as a whole, the transport forms closed loops rather than disconnected starts and stops.
This is the geometric content of calling the flow divergence-free. For the fundamental flow field, that condition is
\[ \nabla \cdot \mathbf{F} = 0. \]
Source-free transport, understood as a complete pattern, has no primitive endpoints. Local gain or loss of stored energy is still tracked by the regional accounting of chapter 4 through \(\mathbf{S}\). What is added here is the shape of the process as a whole, described by \(\mathbf{F}\).
Locally, the picture is circulation. Circulation lines are closed. The next question is how local evolution of \(\mathbf{F}\) must be described in order to preserve this source-free structure.
Divergence-free language is therefore not the origin of anything. It is the mathematical encoding of a prior physical fact: source-free flow has no primitive beginnings or endings. The connected structure comes first. The vector equation is the language we later use to write it down.
Recall the energy flow field, \(\mathbf{F}(\mathbf{r})\).
To preserve the source-free character of the transport seen in experiments, local evolution must allow the flow of energy without introducing primitive endpoints. We therefore ask what kinds of local update can reorganize \(\mathbf{F}\) while maintaining its source-free nature.
To express more precisely the idea of accounting for flow across a boundary, take any region \(V\) with closed boundary \(\partial V\). Gauss’s theorem gives
\[ \int_V \nabla \cdot (\Delta \mathbf{F})\,dV = \oint_{\partial V} \Delta \mathbf{F} \cdot d\mathbf{A}. \]
This says that divergence measures the net transport across a closed boundary. In the source-free case, every such boundary must give zero net flow. No separate charges, masses, sources, or sinks are inserted into the accounting: there is only energy being transported. The divergence must therefore remain identically zero.
A purely algebraic change, such as rescaling
\[ \mathbf{F} \mapsto \lambda \mathbf{F}, \]
can strengthen or weaken what is already there, but it does not explain how the flow turns or reorganizes in space in more complex ways. Furthermore, it leaves zeros where they are and adds no new spatial structure.
If the evolution of \(\mathbf{F}\) is written as the gradient of some field \(\phi\),
\[ \Delta \mathbf{F} = \nabla \phi. \]
then, taking the divergence \(\nabla \cdot\),
\[ \nabla \cdot (\Delta \mathbf{F}) = \nabla^2 \phi, \]
which is generally nonzero.
Such an update can compress, expand, begin, or end the transport. It does not preserve source-free reorganization.
What does preserve the source-free condition identically is a rotation.
For any vector field \(\mathbf{A}\), we can express source-free evolution of \(\mathbf{F}\) as
\[ \Delta \mathbf{F} = \nabla \times \mathbf{A} \qquad\Longrightarrow\qquad \nabla \cdot (\Delta \mathbf{F}) = 0. \]
To make this explicit, in three dimensions, write
\[ \mathbf{A} = (A_x,A_y,A_z). \]
Then, by definition,
\[ \nabla \times \mathbf{A} = ( \partial_y A_z - \partial_z A_y,\; \partial_z A_x - \partial_x A_z,\; \partial_x A_y - \partial_y A_x ), \]
and therefore
\[ \nabla \cdot (\Delta \mathbf{F}) = \partial_x\partial_y A_z - \partial_x\partial_z A_y + \partial_y\partial_z A_x - \partial_y\partial_x A_z + \partial_z\partial_x A_y - \partial_z\partial_y A_x = 0. \]
The mixed derivatives cancel pairwise. That is why curl preserves the source-free condition identically.
Curl therefore preserves source-free structure identically. It is the differential form of source-free reorganization: continuous turning, with no tearing and no start or end points introduced by the evolution itself.
Chapter 6 established that the fundamental flow \(\mathbf{F}\) must evolve by curl if it is to preserve its divergence-free structure. That still leaves a narrower question: what is the simplest curl-based closure that makes transport possible?
A single self-curl relation,
\[ \partial_t \mathbf{F} = k\,\nabla \times \mathbf{F}, \]
preserves source-free turning. But one local rotation only recirculates the flow. It turns the structure around itself, but it does not yet carry energy from one region into another. Transport requires a second, complementary rotation. A point rotated once circulates. Rotated twice in a complementary way, it advances.
A second step cannot be a gradient, because chapter 6 already showed that a gradient update generically introduces divergence. The next admissible possibility is therefore a second source-free rotation:
\[ \nabla \times (\nabla \times \mathbf{F}). \]
In the source-free case,
\[ \nabla \cdot \mathbf{F} = 0 \qquad\Longrightarrow\qquad \nabla \times (\nabla \times \mathbf{F}) = -\nabla^2 \mathbf{F}. \]
Locally, a transporting configuration of \(\mathbf{F}\) can therefore be expressed by an axis of advance together with two complementary transverse degrees of freedom of the same flow. The same local point serves as the fulcrum for both turns, but the turns occur in different transverse orientations. That difference is what makes them complementary rather than redundant. Call these two aspects
\[ \mathbf{F}_{+}, \qquad \mathbf{F}_{-}. \]
The minimal first-order local relation that realizes this double turning is
\[ \partial_t \mathbf{F}_{+} = k\,\nabla \times \mathbf{F}_{-}, \qquad \partial_t \mathbf{F}_{-} = -k\,\nabla \times \mathbf{F}_{+}, \]
with
\[ \nabla\cdot\mathbf{F}_{+}=0,\qquad \nabla\cdot\mathbf{F}_{-}=0. \]
Now each aspect changes by the curl of the other. This is the minimal real closed transport relation: no sources, no action at a distance, no extra fields, and no higher-order operators.
Transport becomes legible only after the flow is resolved into these complementary aspects, because the coupled update acts on the pair as a whole. If we write one complete local state as
\[ C = (\mathbf{F}_{+},\mathbf{F}_{-}), \]
then one application of the coupled update maps one configuration \(C_1\) into another configuration \(C_2\). Repeated application therefore generates an ordered chain of configurations:
\[ (\mathbf{F}_{+},\mathbf{F}_{-})_1,\; (\mathbf{F}_{+},\mathbf{F}_{-})_2,\; (\mathbf{F}_{+},\mathbf{F}_{-})_3,\;\dots \]
That order is abstracted from the mapping itself. Only afterward is it conveniently labeled by a parameter and written in differential form. In that later form, the double rotation yields the wave equation for each aspect, as shown in Appendix 203. For the present argument, the essential point is simpler: one rotation recirculates, while two complementary rotations transport.
This is the simplest transport closure built from two complementary closed turns. In its local ideal form it gives the plane-wave idealization of transport: the same organized advance repeated from point to point, without yet requiring helical closure. Such an idealization already picks out a local axis of advance, but because it extends everywhere it does not yet describe a bounded from-here-to-there transfer. One may picture the transverse frame as precessing about that local axis, so that the forward projection remains nonzero while the transverse orientation oscillates. Helical and toroidal forms are global closures of the same local geometry.
Only at this stage, after choosing conventional scale factors and absorbing them into the electromagnetic normalization, are the two aspects named
\[ \mathbf{E} \equiv k_{+}\,\mathbf{F}_{+}, \qquad \mathbf{B} \equiv k_{-}\,\mathbf{F}_{-}. \]
They are not two substances. They are two complementary aspects of the same organized flow \(\mathbf{F}\). Their cross relation fixes the local direction of transport:
\[ \mathbf{E}\cdot\mathbf{B}=0,\qquad \mathbf{E}\times\mathbf{B}\parallel\mathbf{S}. \]
The scale choices are then absorbed into the usual constitutive constants \(\varepsilon_0\) and \(\mu_0\), and the energy flow can be written in Maxwell form:
\[ \mathbf{S}=\frac{1}{\mu_0}\,\mathbf{E}\times\mathbf{B}. \]
Maxwell theory appears here as the minimal two-aspect closure of source-free rotational transport.
Minimal does not mean unique. It means the weakest local closure that actually propagates source-free flow.
The vacuum Maxwell equations are symmetric under the duality rotation
\[ \mathbf{E} \to c\mathbf{B}, \qquad c\mathbf{B} \to -\mathbf{E}. \]
This reflects the complementary status of the two fields within one transport relation. The symmetry does not collapse \(\mathbf{E}\) and \(\mathbf{B}\) into a single field, and it does not erase their distinct roles in a given solution. In a propagating configuration they remain two transverse aspects of the same organized flow \(\mathbf{F}\), whose cross relation determines the direction of transport.
When source-free flow closes on itself, closure imposes integer winding. Integer winding yields discrete mode families. Discreteness is geometric before it is spectral.
A closed surface does not automatically support smooth continuous circulation. On a sphere, a continuous nowhere-vanishing tangential flow is impossible: any attempt to comb it smoothly leaves at least one zero or defect, by the hairy-ball theorem. A torus avoids that obstruction. It is the simplest closed surface on which continuous circulation can close on itself without enforced stagnation points.
On a closed loop of such a surface, a continuous pattern must match itself after one circuit. That requires
\[ n \lambda = L, \]
where \(L\) is the circuit length and \(n\) is an integer. On a torus there are two independent non-contractible cycles, so a closed flow is labeled by an integer pair \((m,n)\).
Those winding classes do more than label allowed modes. They classify the circulation itself. The same closed winding that forces discrete matching also carries angular momentum about the center of the bounded mode. If \(\mathbf{X}\) is that center, write the relative position within the mode as
\[ \boldsymbol{\rho}=\mathbf{x}-\mathbf{X}. \]
Then the angular momentum of the closed circulation is
\[ \mathbf{L}=\int \boldsymbol{\rho}\times\frac{\mathbf{S}}{c^2}\,dV. \]
Spin is this angular momentum of the self-closing toroidal circulation. Charge, introduced in chapter 10, is a different global aspect of the same mode: the signed through-hole flux across the torus aperture. The torus therefore supports more than one discrete global aspect of one circulation, because it has more than one non-contractible cycle.
The sign of spin is set by handedness: the orientation of the circulation relative to the direction of advance. Different winding classes \((m,n)\), together with that handedness, define different discrete circulation classes.
The allowed wavelengths and frequencies are therefore discrete. Different integers label different global modes. The key point comes before any specific spectrum: once source-free transport closes on itself, continuity and single-valuedness force integer classes of solutions.
Specific spectral laws require additional geometry and come later. The present step is narrower and stronger: quantization begins as closure.
Mass is what confined energy becomes when part of its momentum is trapped in closed circulation and no longer available for straight translation.
Consider a localized flow whose energy moves along a smooth closed trajectory \(X(s)\), parameterized by arclength \(s\). Locally, the propagation speed is still \(c\). Choose a macroscopic direction of motion \(\hat{\mathbf{z}}\), and let \(\hat{\mathbf{t}}(s)\) be the unit tangent to the flow. Define the local pitch angle by
\[ \cos\theta(s)=\hat{\mathbf{t}}(s)\cdot\hat{\mathbf{z}}. \]
Over a segment \(ds\), the forward displacement is
\[ dz=\cos\theta(s)\,ds, \]
while the elapsed time is
\[ dt=\frac{ds}{c}. \]
So the local forward speed is
\[ v_{\text{forward}}(s)=\frac{dz}{dt}=c\cos\theta(s). \]
Over one full circuit of length \(L\), the effective forward speed is therefore
\[ v_{\text{eff}} = c\,\left\langle\cos\theta\right\rangle, \]
where
\[ \left\langle\cos\theta\right\rangle := \frac{1}{L}\int_0^L \cos\theta(s)\,ds. \]
If the path were everywhere straight, \(\langle\cos\theta\rangle=1\) and the energy would simply propagate at \(c\). But once the trajectory has persistent transverse winding, part of the motion is no longer available for forward translation.
Electromagnetic energy of total energy \(E\) carries momentum of magnitude
\[ P=\frac{E}{c}. \]
Only the component aligned with \(\hat{\mathbf{z}}\) contributes to forward motion. Integrating around the loop gives
\[ P_z=\frac{E}{c}\left\langle\cos\theta\right\rangle. \]
The rest is trapped in closed transverse circulation:
\[ P_{\perp,\text{eff}} := \sqrt{P^2-P_z^2} = \frac{E}{c}\sqrt{1-\left\langle\cos\theta\right\rangle^2}. \]
This trapped momentum is the reason the configuration resists redirection. To change the macroscopic motion of the object, one must reorient the circulating momentum throughout the whole closed path, not merely push a point.
That is inertia. Its measure is
\[ m_{\text{eff}} := \frac{P_{\perp,\text{eff}}}{c} = \frac{E}{c^2}\sqrt{1-\left\langle\cos\theta\right\rangle^2}. \]
In the rest frame of the confined configuration, the net translational momentum vanishes, so \(\langle\cos\theta\rangle=0\). Then
\[ m=\frac{E_0}{c^2}, \]
where \(E_0\) is the rest energy of the closed mode.
In this framework, this states what mass is: the energy trapped in circulation, measured in the frame where the closed flow has no net translation.
Why does the energy not simply straighten its path and eliminate its mass? Because the circulation is topologically closed. Once winding exists, removing it would require reconnection of the flow itself. Mass is kinematic delay locked in by topology.
Unconfined propagation carries momentum \(p=E/c\) but no inertial rest mass. Mass appears when topology confines part of the momentum into persistent circulation.
Electric charge is the far-field projection of conserved topological winding: a \(1/r^2\) intensity falloff that results from a fixed total circulation spread over an expanding sphere.
In source-free Maxwell dynamics, \(\nabla \cdot \mathbf{E} = 0\) everywhere. No electric field lines originate or terminate. Yet we observe a \(1/r^2\) falloff in field intensity around what we call a “charged particle.”
There is no contradiction. Consider a toroidal energy configuration with a fixed total circulation, with winding numbers \((m, n)\) characterizing the closed flow. The total is conserved.
A torus has a distinguished aperture. Choose a spanning surface \(\Sigma\) across that aperture. The closed circulation carries a signed through-hole flux
\[ \Phi_\Sigma=\int_\Sigma \mathbf{S}\cdot d\mathbf{A}. \]
This is not a source or sink. It is the oriented through-hole moment of the closed circulation. Its sign reverses with handedness.
Because the circulation closes in integer winding classes, this through-hole flux is not arbitrary. For a stable toroidal mode it comes in discrete classes set by the winding itself.
Far from the torus, the detailed local winding is no longer resolved. What remains visible is the projection of that conserved oriented quantity. Now enclose the configuration in a sphere of radius \(r\) much larger than the torus. The sphere’s area is \(4\pi r^2\).
A fixed quantity, spread over a growing area, produces an average projected intensity that falls as:
\[ \text{Intensity} \propto \frac{1}{r^2}. \]
This yields the inverse-square far-field scaling from projection geometry. Charge is the name we give to the conserved oriented quantity whose projection we are measuring. In this sense, charge is quantized before any force law is written: its sign and class are fixed by the discrete closed circulation. Spin and charge are therefore not separate ontologies. They are different global aspects of the same toroidal mode: charge measures the signed through-hole flux, while spin measures the angular momentum of the closed circulation about the mode’s center.
Opposite charge signs correspond to opposite senses of winding, equivalently to opposite signs of the through-hole flux. The present chapter identifies the geometric far-field character of charge. The detailed interaction between such configurations belongs later, when momentum transfer and flux accounting are made explicit.
The Schrodinger equation appears here as a controlled envelope limit of Maxwell transport. Chapters 8 and 9 already gave two things needed for that limit: discrete stable modes and an emergent mass scale. The remaining task is to describe slow modulation of one such mode.
Each Cartesian component \(f(\mathbf{r},t)\) of \(\mathbf{E}\) or \(\mathbf{B}\) satisfies the vacuum wave equation:
\[ \left(\nabla^2-\frac{1}{c^2}\partial_t^2\right)f=0. \]
Select the positive-frequency part of the field near a stable carrier frequency \(\omega_0\), and demodulate the carrier:
\[ \psi(\mathbf{r},t)=e^{i\omega_0 t}f^{(+)}(\mathbf{r},t). \]
The field is narrow-band when
\[ \varepsilon = \frac{\Delta\omega}{\omega_0}\ll 1, \]
so the envelope \(\psi\) varies slowly compared with the carrier. After separating the carrier and the base-mode contribution, and using the fact that the carrier already satisfies the dispersion relation of the underlying stable mode, the exact envelope identity is
\[ i\partial_t\psi = -\frac{c^2}{2\omega_0}\nabla^2\psi +\frac{1}{2\omega_0 c^2}\partial_t^2\psi. \]
The last term is the difference between exact Maxwell transport and the Schrodinger limit. For spectral width \(\Delta\omega\), it is controlled by
\[ \left\|\frac{1}{2\omega_0 c^2}\partial_t^2\psi\right\| \le \frac{\Delta\omega^2}{2\omega_0 c^2}\|\psi\| = O(\varepsilon^2)\|\psi\|. \]
So, to leading order in the narrow-band parameter,
\[ i\partial_t\psi = -\frac{c^2}{2\omega_0}\nabla^2\psi +O(\varepsilon^2). \]
Now define the emergent constants from the carrier mode itself:
\[ \hbar=\frac{E_0}{\omega_0},\qquad m=\frac{E_0}{c^2}, \]
where \(E_0\) is the rest energy of the underlying stable mode. Then
\[ \frac{c^2}{2\omega_0}=\frac{\hbar}{2m}. \]
Multiplying by \(\hbar\) gives
\[ i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m}\nabla^2\psi +O(\varepsilon^2). \]
This is the free Schrodinger equation. It arises as the narrow-band envelope equation of a stable Maxwell mode.
The interaction case uses the same ontology. In structured backgrounds, the envelope accumulates additional region-dependent phase. The double-slit treatment later represents such interaction regions by localized potentials \(V_j\) that rotate the relative phase of the propagation channels. The potential term is therefore a summary of background interaction in the same envelope dynamics. This chapter, however, derives only the free narrow-band case.
Superposition, interference, and uncertainty enter because the envelope remains a wave field. Quantum mechanics is the effective theory of slowly varying Maxwell envelopes.
Chapter 11 derived the free Schrodinger equation as the narrow-band envelope of a stable Maxwell mode. Interaction enters when that envelope propagates through a structured background.
In the envelope description, such a background appears as a local potential:
\[ i\hbar\,\partial_t\psi = \left( -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \right)\psi. \]
\(V\) is a compact summary of how the background changes the phase accumulated by the envelope during propagation.
The propagator makes this explicit:
\[ K(x,T;x_0,0) = \int\mathcal Dq\; \exp\!\left[ \frac{i}{\hbar}\int_0^T dt \left( \frac{1}{2}m\dot q^2 - V(q,t) \right) \right]. \]
All interaction enters through the action in the exponential. The background rotates the phase of the wave.
This is especially clear in the double-slit case. If two spatial channels experience different interaction backgrounds, represented by \(V_1\) and \(V_2\), the physically relevant quantity is the action difference between them:
\[ \Delta\phi = \frac{1}{\hbar}\int_0^T dt\,(V_1-V_2). \]
The total amplitude at the screen is then
\[ \psi(x) = \psi_1^{(0)}(x) + e^{i\Delta\phi}\psi_2^{(0)}(x), \]
where \(\psi_j^{(0)}\) is the reference amplitude for free or symmetric propagation.
If \(V_1 = V_2\), then \(\Delta\phi = 0\) and the full interference pattern is recovered. If the two backgrounds differ, the relative phase changes and the pattern changes with it. What disappears is coherent phase relation.
This closes the loop left open in chapter 11. The potential term in Schrodinger’s equation summarizes background interaction in the same wave-envelope dynamics.
Newton’s second law is the integrated continuity law for momentum in a nearly stable localized electromagnetic configuration.
Chapter 7 identified the energy flow with the Maxwell realization
\[ \mathbf{S}=\frac{1}{\mu_0}\,\mathbf{E}\times\mathbf{B}. \]
The same transport therefore carries momentum. The electromagnetic momentum density is
\[ \mathbf{g}=\frac{\mathbf{S}}{c^2} = \epsilon_0\,\mathbf{E}\times\mathbf{B}. \]
To track how momentum moves through a surface, we use the Maxwell stress tensor:
\[ T_{ij} = \epsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}\mathbf{E}^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}\mathbf{B}^2\right). \]
Differentiate \(\mathbf{g}\) in time and substitute Maxwell’s equations. The first result is
\[ \partial_t\mathbf{g} = \frac{1}{\mu_0}(\nabla\times\mathbf{B})\times\mathbf{B} - \epsilon_0\,\mathbf{E}\times(\nabla\times\mathbf{E}). \]
Using the standard vector identity for \((\nabla\times\mathbf{A})\times \mathbf{A}\) and the source-free conditions \(\nabla\cdot\mathbf{E}=0\), \(\nabla\cdot\mathbf{B}=0\), this rearranges into the exact local momentum continuity law
\[ \partial_t g_i + \partial_j T_{ij} = 0, \]
or, in vector form,
\[ \partial_t\mathbf{g}+\nabla\cdot\mathbf{T}=0. \]
This is the momentum analogue of Poynting’s theorem. Momentum does not appear or disappear. It changes only by flux through the boundary of a region.
Integrate over a localized region \(K\) containing a nearly stable bounded configuration:
\[ \mathbf{P}_K=\int_K \mathbf{g}\,d^3x. \]
Then
\[ \frac{d}{dt}\mathbf{P}_K = -\int_{\partial K}\mathbf{T}\cdot\mathbf{n}\,dA. \]
The right-hand side is what later language calls force. It is the net rate at which momentum crosses the boundary.
To connect this with motion of the object as a whole, define the energy in the region and its center of energy:
\[ E_K=\int_K u\,d^3x, \qquad \mathbf{X}_K=\frac{1}{E_K}\int_K \mathbf{x}\,u\,d^3x. \]
When boundary leakage is small and the mode remains coherent,
\[ E_K\,\dot{\mathbf{X}}_K \approx \int_K \mathbf{S}\,d^3x, \qquad \mathbf{P}_K \approx \frac{E_K}{c^2}\dot{\mathbf{X}}_K. \]
For such a bounded configuration, define the effective inertial mass
\[ m_K := \frac{E_K}{c^2}. \]
If the energy of the localized configuration is roughly constant, then
\[ m_K\,\ddot{\mathbf{X}}_K \approx -\int_{\partial K}\mathbf{T}\cdot\mathbf{n}\,dA. \]
This is Newton’s second law in its effective form for a stable bounded mode:
\[ \mathbf{F}=\frac{d\mathbf{P}}{dt}. \]
It describes momentum bookkeeping for a bounded region of field.
A mediating force field inserted between supposedly independent substrates does not rescue their independence. If interaction is real, a common structure is already present, and force is only the accounting of that coupling.
Particles are localized regions. Forces are boundary integrals. Newton’s second law is momentum continuity applied to a stable electromagnetic knot.
Gravity appears here as electromagnetic refraction: the bending of energy transport paths caused by a spatially varying propagation speed induced by concentrated energy density.
The propagation speed of electromagnetic energy in vacuum is:
\[ c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}. \]
A dielectric does not work by letting light pass through inert matter. The incident field drives an existing electromagnetic organization in the medium, and that response emits a secondary electromagnetic field. The macroscopic summary of that response is a change in the constitutive coefficients. Linearity does not forbid this. It means only that a probe field propagates linearly through coefficients set by a background.
The same idea is used here. Mass is self-confined energy. A massive body is therefore a concentrated background organization of electromagnetic energy flow. A passing field perturbs that background, and the background response contributes a secondary electromagnetic field. In chapter 7, \(\mathbf{E}\) and \(\mathbf{B}\) were identified as complementary aspects of one organized flow. The constitutive closure adopted here therefore shifts both constitutive channels together:
\[ \varepsilon_\text{eff}(r)=\varepsilon_0\bigl(1+2\eta(r)\bigr), \qquad \mu_\text{eff}(r)=\mu_0\bigl(1+2\eta(r)\bigr), \]
with
\[ \eta(r)=\frac{GM}{rc^2}. \]
This symmetric change preserves the local vacuum impedance,
\[ Z_\text{eff}=\sqrt{\frac{\mu_\text{eff}}{\varepsilon_\text{eff}}}=Z_0, \]
while lowering the local propagation speed:
\[ c_\text{local}(r)=\frac{1}{\sqrt{\varepsilon_\text{eff}(r)\mu_\text{eff}(r)}} =\frac{c}{1+2\eta(r)}. \]
The corresponding refractive index is therefore
\[ n(r)=\frac{c}{c_\text{local}(r)} =1+\frac{2GM}{rc^2}. \]
If only one constitutive channel were perturbed, this first-order shift would be halved, giving the Newtonian half-value. Within this symmetric electromagnetic constitutive closure, the full factor of two follows.
In optics, when a wave propagates through a medium of varying speed, it bends toward the slower region. This is refraction.
Gravity, in this framework, is that refraction applied to all energy transport. The trajectory of any moving configuration curves toward regions of high energy density, because those are the regions of lower propagation speed.
For a ray passing a body with impact parameter \(b\), the weak-field bending is
\[ \theta \approx \int_{-\infty}^{\infty}\nabla_\perp n\,dz = \frac{4GM}{bc^2}. \]
At the solar limb this is about \(1.75\) arcseconds.
On this reading, light bending follows from index-of-refraction arguments within the transport picture.
Spacetime curvature, in this reading, is a geometric restatement of the same refraction. The geometry follows from the transport, not the other way around.
The thirteen chapters above establish the physical spine of the argument.
The short chapters that follow do not add new dynamics. They state conceptual consequences about cause, time, space, and force that are consistent with the reconstruction but are not needed to derive it.
Cause is not directly observed. What is directly observed is registered change.
We register configurations. We compare records. If the records differ, change has been noticed.
Because physical registrations do not present contradictions — a given record does not assign multiple incompatible values to the same quantity at the same location — we infer an order in the reconfiguration. That persistent order is what later receives causal language: this before that; this producing that.
Cause and consequence are therefore not primitive furniture of the world. They are compact descriptions of ordered, registered reconfiguration. Change is primary. Cause is the interpretation added afterward.
Time is the ordering of persistent flow, counted by recurrence.
A stable flow registers change by continuing to reconfigure. When records of that reconfiguration recur, the recurrence can be counted. Any recurrent pattern of flow can therefore serve as a clock. Time is the count of such recurrent steps.
In this framework, the coordinate \(t\) in the equations labels ordered registrations of the changing present. Physical time is the operational order of change.
Space is the collection of distances defined by ordered signal separation.
A distance is a count of how many recurrent signal steps separate two configurations. Once such steps are ordered, they can also be described as causal steps. The total set of such separations forms the relational structure we call space.
Geometry is descriptive. It summarizes the organization of flow. When the resulting distance relations can be represented coherently in three dimensions, we use that representation and call it space.
Force is a derived description of reconfiguration and momentum transfer.
When a bounded region changes its motion, what has changed physically is the flux of momentum through its boundary. “Force” is the compact name for that boundary accounting.
This also clarifies the status of so-called force fields. If two supposedly independent substrates interact, they are not truly independent. A mediating field does not preserve the split. It exposes a deeper common structure in which the coupling is taking place.
Force is descriptive. It is bookkeeping for coupled change within one underlying physical organization.
The appendices fix later technical points without interrupting the main chain.
The geometry developed in chapter 7 has an immediate open-form analogue in the standard Lorentz description of motion in a magnetic field. The point here is not to derive that law from first principles, but to pin down the same transport pattern in familiar notation.
Take a prescribed magnetic field with unit direction \(\hat{\mathbf{b}}\) and write the conventional magnetic Lorentz equation
\[ m\,\dot{\mathbf{v}} = q\,\mathbf{v}\times\mathbf{B}. \]
Decompose the velocity into components parallel and perpendicular to the field:
\[ \mathbf{v}=\mathbf{v}_{\parallel}+\mathbf{v}_{\perp}, \qquad \mathbf{v}_{\parallel}=(\mathbf{v}\cdot\hat{\mathbf{b}})\hat{\mathbf{b}}. \]
Then
\[ \mathbf{v}_{\parallel}\times\mathbf{B}=0, \]
so the parallel component is carried forward unchanged, while the perpendicular component obeys
\[ m\,\dot{\mathbf{v}}_{\perp} = q\,\mathbf{v}_{\perp}\times\mathbf{B}. \]
This is pure transverse turning. The magnitude of \(\mathbf{v}_{\perp}\) stays fixed, but its direction rotates around \(\hat{\mathbf{b}}\). At the same time, the nonzero parallel component advances the motion along the axis. The result is a helix.
So the familiar Lorentz helix already exhibits the same geometry isolated in the main text:
In the language of this book, the helix is the open form of double-rotation transport. The torus is the same structure after closure over itself. What appears in one case as guided propagation appears in the other as trapped circulation.
This is why the helical aspect of Lorentz motion matters here. It shows, in a standard physical setting, that transport does not require a primitive push along a line. It can arise from persistent transverse turning together with a nonvanishing forward component.
The double-curl transport closure of chapter 7 should not be composed with itself naively. Applying
\[ \nabla\times(\nabla\times\mathbf{F}) \]
again acts on field structure. It produces a higher spatial operator. It does not by itself describe one transport riding on another.
Nested transport belongs to kinematics, not to repeated application of the field operator. The question is this: if one transport process is nested inside another, and both respect the same invariant transport speed \(k\), what is the resulting law of composition?
For clarity, consider one spatial direction \(x\). The invariant transport speed \(k\) means that the distinguished transport lines are
\[ x = \pm kt. \]
These are the boundaries of the transport cone. Any admissible change of frame must preserve them.
Assume two frames are related by uniform relative motion along \(x\). Because the transport background is homogeneous, the change of coordinates must be linear. Instead of \((t,x)\), write the null coordinates
\[ u = t + \frac{x}{k}, \qquad w = t - \frac{x}{k}. \]
Then the transport lines are simply
\[ u = 0 \qquad\text{or}\qquad w = 0. \]
Preserving the transport cone means preserving these null directions. So the most general admissible linear transformation is
\[ u' = \lambda u, \qquad w' = \lambda^{-1} w, \]
for some positive constant \(\lambda\).
Returning to \((t,x)\) coordinates gives
\[ t' = \frac{u'+w'}{2} = \frac{\lambda+\lambda^{-1}}{2}\,t + \frac{\lambda-\lambda^{-1}}{2k}\,x, \]
and
\[ x' = \frac{k(u'-w')}{2} = \frac{k(\lambda-\lambda^{-1})}{2}\,t + \frac{\lambda+\lambda^{-1}}{2}\,x. \]
Now choose the sign convention so that the primed origin moves with speed \(v\) in the unprimed frame. Writing
\[ \lambda = e^{-\eta}, \]
we have
\[ \frac{\lambda+\lambda^{-1}}{2} = \cosh \eta, \qquad \frac{\lambda-\lambda^{-1}}{2} = -\sinh \eta. \]
Therefore
\[ t' = \cosh\eta\,t - \frac{\sinh\eta}{k}\,x, \]
\[ x' = -k\sinh\eta\,t + \cosh\eta\,x. \]
If \(x'=0\), then the primed origin satisfies
\[ x = k\tanh\eta \, t. \]
So the relative speed is
\[ v = k\tanh\eta. \]
This is the hyperbolic parametrization of velocity.
Now compose two such frame changes, with parameters \(\eta_1\) and \(\eta_2\). In null coordinates,
\[ u'' = e^{-\eta_2}u' = e^{-(\eta_1+\eta_2)}u, \]
\[ w'' = e^{\eta_2}w' = e^{\eta_1+\eta_2}w. \]
So the parameters add:
\[ \eta_{\mathrm{tot}} = \eta_1 + \eta_2. \]
Since
\[ v = k\tanh\eta, \]
the composed speed is
\[ v_{\mathrm{tot}} = k\tanh(\eta_1+\eta_2) = \frac{v_1+v_2}{1+v_1v_2/k^2}. \]
This is the hyperbolic composition law.
So the distinction is exact:
The train-and-passenger image is therefore valid, but only at the kinematic level. One transport process may be nested inside another. The resulting composition is hyperbolic because the same invariant transport speed \(k\) is preserved at each level.
This appendix gives the mathematical step used structurally in chapter 7.
The result is the following.
A single real first-order self-curl evolution of a divergence-free field does not produce neutral propagating transport. The minimal local propagating closure in this class requires two coupled divergence-free fields.
The point is not to postulate Maxwell’s equations, but to show why they appear as the minimal propagating closure of source-free rotational transport.
Let
\[ \mathbf{F}(\mathbf{r},t) \]
be a vector field on three-dimensional space.
Source-free transport means
\[ \nabla \cdot \mathbf{F} = 0. \]
This expresses the absence of primitive beginnings or endings of the flow.
Assume a local first-order evolution relation
\[ \partial_t \mathbf{F} = \mathcal{D}(\mathbf{F}), \]
where \(\mathcal{D}\) is a spatial differential operator.
To preserve the source-free condition we require
\[ \nabla \cdot (\partial_t \mathbf{F}) = 0. \]
Substituting the evolution relation gives
\[ \nabla \cdot \mathcal{D}(\mathbf{F}) = 0 \]
for every divergence-free field \(\mathbf{F}\).
A natural local first-order differential operator with this property is curl, since
\[ \nabla \cdot (\nabla \times \mathbf{A}) = 0 \]
for any vector field \(\mathbf{A}\).
So a natural divergence-preserving self-update is
\[ \partial_t \mathbf{F} = k\,\nabla \times \mathbf{F}. \]
We now examine whether this relation yields propagating transport.
Consider plane-wave modes
\[ \mathbf{F}(\mathbf{r},t) = \mathbf{f}\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}. \]
Because \(\nabla\cdot\mathbf{F}=0\), the amplitude must satisfy
\[ \mathbf{k}\cdot\mathbf{f}=0. \]
Substituting into
\[ \partial_t \mathbf{F} = k\,\nabla \times \mathbf{F} \]
gives
\[ -i\omega\mathbf{f} = ik(\mathbf{k}\times\mathbf{f}). \]
Rearranging,
\[ \omega\mathbf{f} = -k(\mathbf{k}\times\mathbf{f}). \]
On the transverse plane, the operator \(\mathbf{k}\times\) has eigenvalues
\[ \pm i|\mathbf{k}|. \]
Therefore
\[ \omega = \pm i\,k|\mathbf{k}|. \]
The time dependence becomes
\[ e^{-i\omega t} = e^{\pm k|\mathbf{k}|t}. \]
So the modes either grow or decay exponentially.
A single self-curl evolution therefore does not produce neutral wave propagation. It generates unstable rotational modes.
Now introduce two divergence-free fields
\[ \mathbf{F}_+, \qquad \mathbf{F}_-. \]
Consider the coupled evolution
\[ \partial_t \mathbf{F}_+ = k\,\nabla \times \mathbf{F}_- \]
\[ \partial_t \mathbf{F}_- = -k\,\nabla \times \mathbf{F}_+. \]
Taking a time derivative of the first equation,
\[ \partial_t^2 \mathbf{F}_+ = k\,\nabla \times (\partial_t \mathbf{F}_-). \]
Substituting the second equation,
\[ \partial_t^2 \mathbf{F}_+ = -k^2\,\nabla \times (\nabla \times \mathbf{F}_+). \]
Using the vector identity
\[ \nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla\cdot\mathbf{F})-\nabla^2\mathbf{F}, \]
and the divergence-free condition
\[ \nabla\cdot\mathbf{F}_+ = 0, \]
we obtain
\[ \partial_t^2 \mathbf{F}_+ = k^2\nabla^2\mathbf{F}_+. \]
Thus \(\mathbf{F}_+\) satisfies the wave equation
\[ \partial_t^2\mathbf{F}_+ - k^2\nabla^2\mathbf{F}_+ = 0. \]
The same derivation holds for \(\mathbf{F}_-\).
The analysis shows:
So the minimal propagating closure in this class is
\[ \partial_t \mathbf{F}_+ = k\,\nabla \times \mathbf{F}_- \]
\[ \partial_t \mathbf{F}_- = -k\,\nabla \times \mathbf{F}_+. \]
These equations preserve
\[ \nabla\cdot\mathbf{F}_+ = 0,\qquad \nabla\cdot\mathbf{F}_- = 0. \]
Now define
\[ \mathbf{E} \equiv \mathbf{F}_+,\qquad \mathbf{B} \equiv \mathbf{F}_-/k. \]
Then the coupled equations become
\[ \partial_t \mathbf{E} = k^2\nabla \times \mathbf{B} \]
\[ \partial_t \mathbf{B} = -\nabla \times \mathbf{E}. \]
With conventional constants absorbed into the normalization of \(k\), these correspond to the source-free Maxwell equations.
The two fields are not independent substances.
They are two complementary transverse aspects of the same organized source-free transport. Their mutual curl coupling yields the minimal propagating structure compatible with divergence-free flow.
Starting from divergence-free transport:
Maxwell dynamics therefore appears here as the minimal propagating closure of source-free rotational transport.
The Physics of Energy Flow presents a monistic derivation of physics from continuous electromagnetic energy transport.
Instead of assigning independent ontologies to matter, charge, force, gravity, and curved spacetime, this book starts from one substrate only and asks what must follow if energy exists, flows, closes, and organizes.
The result is a reconstruction in which Maxwell dynamics is the minimal local closure of transport, quantization begins as topological closure, mass is trapped energy, force is bookkeeping for momentum transfer, and gravity is refraction.
This is not a denial of mathematics. It is a refusal to mistake mathematical maps for the territory they describe.
These are caption-first figure concepts for TPOEF. Each caption is written to be usable both as a book caption and as the basis for image generation.
Complementary electric and magnetic aspects of one advancing flow. A local energy flow \(\mathbf{S}\) with transverse \(\mathbf{E}\) and \(\mathbf{B}\), orthogonal to one another, and \(\mathbf{E}\times\mathbf{B}\) aligned with the direction of transport.
Double rotation as self-advancing transport. One field rotating the other, and being rotated in return, shown as a helical advance rather than a single recirculating swirl.
Sphere versus torus for continuous circulation. A sphere with an unavoidable defect in any smooth tangential flow, beside a torus supporting continuous closed circulation without enforced stagnation points.
Two independent non-contractible winding cycles on a torus. A toroidal flow labeled by integer winding numbers \((m,n)\), with one cycle around the tube and one around the hole.
Spin as angular momentum of toroidal circulation. A self-closing toroidal flow with angular momentum vector \(\mathbf{L}\) about the mode’s center, where winding class and handedness fix the sign and class of the circulation.
Gravity as symmetric electromagnetic slowing. A concentrated background energy configuration surrounded by a halo of raised effective \(\varepsilon_{\text{eff}}\) and \(\mu_{\text{eff}}\), with a passing ray bending toward the slower region.
Electric-only half-medium versus full electromagnetic medium. Two side-by-side schematics: one with only \(\varepsilon\) perturbed giving the Newtonian half-value, and one with both \(\varepsilon\) and \(\mu\) perturbed giving the full deflection.
There cannot, by definition, be another kind of existence if it is to interact with the physical.↩︎
As Plato and many others observed, one can become skilled at recognizing images, patterns, and regular sequences of appearance while remaining ignorant of what produces them. Physics can encode repeatable regularities without thereby laying hold of the underlying causes. Even so, such encoding is far better than treating the screen as uniform brightness alone.↩︎