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A nonzero whole-space plane wave has infinite total energy

Physical scope

An exact computation for one explicit ideal plane-wave configuration on $R^{3} \times R$ : the positivity and uniformity of the period-averaged energy density, the divergence of the box integrals, and the contrast with compactly supported finite-energy data. No measured quantity, no finite beam and no empirical claim.

Example

Assume Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the Lebesgue-energy and compact-source comparison branches (ii), (iii), as required by their measure and wave-Cauchy suppliers. The explicit plane-wave fields of (i) use no choice.

Fix $K > 0$ , $ω : = c K$ , $k : = K e_{3}$ , $E_{0} > 0$ (unit $V m^{- 1}$ ), the amplitude $e : = E_{0} e_{1}$ and $b : = k \times e / ω = (E_{0} / c) e_{2}$ , and let $(E, B)$ be the corresponding plane-wave solution of Dispersion, transversality and the two polarization conditions for plane-wave solutions (Real plane-wave fields encoded by complex amplitudes) on $R^{3} \times R$ with energy density $u$ of Vacuum electromagnetic energy density and energy flux (the Poynting vector); write $θ (x, t) = K x_{3} - ω t$ . Then:

(i) [Explicit density] $E = E_{0} \cos θ e_{1}$ , $B = (E_{0} / c) \cos θ e_{2}$ and $u (x, t) = ε_{0} E_{0}^{2} \cos^{2} θ (x, t)$ , a nonnegative function vanishing exactly on the nodal planes $\cos θ = 0$ ; the period average at every fixed $x$ is $⟨ u ⟩ = ε_{0} E_{0}^{2} / 2 > 0$ , uniformly in $x$ and $t$ (Period-averaged energy density, Poynting vector and momentum flux of a plane wave(a)).

(ii) [Unbounded box integrals] Fix $t \in R$ and let $R \geq (ω ∣ t ∣ + π / 4) / K$ . On the sub-box $Q_{R} : = [- R, R]^{2} \times [(ω t - π / 4) / K, (ω t + π / 4) / K]$ of the box $[- R, R]^{3}$ one has $\cos^{2} θ \geq 1 / 2$ , hence $u \geq ε_{0} E_{0}^{2} / 2$ there; both boxes are product boxes with finite volume (Axis-parallel rectangles in $R^{m}$ and their volume), $u$ is continuous on them and therefore Riemann integrable (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set), and monotonicity of the multidimensional Darboux integral (The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , The Riemann integral of a bounded function over a bounded Jordan measurable set) gives $\int_{[- R, R]^{3}} u (x, t) d x \geq \int_{Q_{R}} u (x, t) d x \geq \frac{ε_{0} E_{0}^{2}}{2} v o l (Q_{R}) = \frac{π ε_{0} E_{0}^{2}}{K} R^{2} ⟶ + \infty (R \to \infty) .$ Consequently the box integrals have no finite limit as $R \to \infty$ : the ideal plane wave has infinite total energy in the box-exhaustion sense, in contrast to the total energy of a single wave 'mode' being finite on any bounded box.

(iii) [Contrast with compactly supported data] If instead the constrained data of Constrained smooth vacuum Maxwell initial-value theory, finite-energy conservation and finite propagation vanish off a compact set, then by clause (d) of that theorem the fields have compact spatial support at each time, and the instantaneous total energy is the integral of the continuous density $u$ over a bounding closed box containing its support, hence finite. So finite total energy is available for compactly supported data, while the ideal plane wave of (i)-(ii) is not a finite-energy configuration.

(iv) [Scope] The example is an exact computation for the ideal plane wave; it reports no measurement, and "infinite total energy" means the divergence statement (ii), not a claim about any laboratory field or finite beam.

Facts & Assumptions

Given: Countable Choice ${A C}_{ω}$ for clauses (ii), (iii); The constants $K > 0$ , $c > 0$ , $ω = c K$ , $k = K e_{3}$ , $E_{0} > 0$ in $V m^{- 1}$ , the amplitude $e = E_{0} e_{1}$ , the magnetic amplitude $b = k \times e / ω = (E_{0} / c) e_{2}$ , the phase $θ (x, t) = K x_{3} - ω t$ , the plane-wave fields $(E, B)$ , and the energy density $u$ .

[F1]

Plane-wave data (Real plane-wave fields encoded by complex amplitudes, Dispersion, transversality and the two polarization conditions for plane-wave solutions): the fields are $E = Re (e \exp (i θ))$ , $B = Re (b \exp (i θ))$ , and with $∣ k ∣ = ω / c$ , $k \cdot e = 0$ , $b = k \times e / ω$ the pair lies in the source-free smooth Maxwell branch.

[F2]

Energy density (Vacuum electromagnetic energy density and energy flux (the Poynting vector)): $u = \frac{ε_{0}}{2} ∣ E ∣^{2} + \frac{1}{2 μ_{0}} ∣ B ∣^{2} \geq 0$ is $C^{\infty}$ and $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

[F3]

Plane-wave averages (Period-averaged energy density, Poynting vector and momentum flux of a plane wave(a)): for this class of plane waves $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ e ∣^{2}$ , independent of $x$ and of the starting time, where the average is taken over the period $T = 2 π / ω$ ; in particular the real-amplitude identity $\frac{1}{T} \int_{0}^{T} \cos^{2} (ϕ_{0} - ω s) d s = \frac{1}{2}$ is the scalar case of Period averages of products of real parts of complex exponentials.

[F4]

Integration over boxes and Jordan sets (Axis-parallel rectangles in $R^{m}$ and their volume, The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set): a product box $[a, b] = \prod_{j} [a_{j}, b_{j}]$ has volume $\prod_{j} (b_{j} - a_{j})$ ; a continuous function on a compact Jordan measurable set is Riemann integrable there; and the Darboux integral of a nonnegative function is monotone under inclusion of product boxes.

[F5]

Compact-support propagation (Constrained smooth vacuum Maxwell initial-value theory, finite-energy conservation and finite propagation(d)): for initially constraint-satisfying $C^{\infty}$ data vanishing off a compact $K \subseteq R^{3}$ with $K \subseteq {\overline{B}}_{ρ} (0)$ , the solution fields satisfy $E (x, t) = B (x, t) = 0$ whenever $∣ x ∣ > ρ + c (t - t_{0})$ .

[F6]

Euclidean structure (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): the norm and inner product are the Euclidean ones, so for real vectors $v$ , $∣ v ∣^{2} = ⟨ v, v ⟩$ .

Verification

technique · direct

1.1givenF1F2F6algebra

Explicit fields and density. Since $k \times e = (K e_{3}) \times (E_{0} e_{1}) = K E_{0} e_{2}$ , one has $b = (E_{0} / c) e_{2}$ ; by [F1] and Euler's formula in the real-part encoding, $E = Re (E_{0} e_{1} e^{i θ}) = E_{0} \cos θ e_{1}$ and $B = Re ((E_{0} / c) e_{2} e^{i θ}) = (E_{0} / c) \cos θ e_{2}$ . By [F2] and [F6], $u = \frac{ε_{0}}{2} E_{0}^{2} \cos^{2} θ + \frac{1}{2 μ_{0}} \frac{E_{0}^{2}}{c^{2}} \cos^{2} θ = ε_{0} E_{0}^{2} \cos^{2} θ$ because $1 / (μ_{0} c^{2}) = ε_{0}$ ; the function is nonnegative and vanishes exactly where $\cos θ = 0$ , the union of the nodal planes $K x_{3} - ω t = π / 2 + π Z$ .

2.1step 1.1F3algebra

Period average. At fixed $x$ , $u (x, \cdot)$ is the $\cos^{2}$ profile $ε_{0} E_{0}^{2} \cos^{2} (ϕ_{0} - ω s)$ with $ϕ_{0} = K x_{3}$ ; the scalar case of [F3] gives $⟨ u ⟩ = ε_{0} E_{0}^{2} \cdot \frac{1}{2} = \frac{ε_{0}}{2} E_{0}^{2}$ , which equals $\frac{ε_{0}}{2} ∣ e ∣^{2}$ and is strictly positive, independent of $x$ and of the starting time.

2.2step 1.1F4algebra

Unbounded box integrals. Fix $t \in R$ and $R \geq (ω ∣ t ∣ + π / 4) / K$ , and let $Q_{R}$ be the sub-box of $[- R, R]^{3}$ with $x_{3} \in [(ω t - π / 4) / K, (ω t + π / 4) / K]$ and $x_{1}, x_{2} \in [- R, R]$ . Each point of $Q_{R}$ satisfies $∣ K x_{3} - ω t ∣ \leq π / 4$ , so $\cos^{2} θ \geq 1 / 2$ and $u \geq ε_{0} E_{0}^{2} / 2$ there, while $Q_{R} \subseteq [- R, R]^{3}$ because $∣ (ω t \pm π / 4) / K ∣ \leq R$ ; both are product boxes with finite volume by [F4], and $u$ is continuous on the compact boxes, hence Riemann integrable by [F4], so monotonicity of the Darboux integral gives $\int_{[- R, R]^{3}} u \geq \int_{Q_{R}} u \geq \frac{ε_{0} E_{0}^{2}}{2} v o l (Q_{R})$ . The volume is $v o l (Q_{R}) = (2 R)^{2} \cdot \frac{π}{2 K} = \frac{2 π R^{2}}{K}$ by [F4], so $\int_{[- R, R]^{3}} u \geq \frac{π ε_{0} E_{0}^{2}}{K} R^{2} \to + \infty$ as $R \to \infty$ ; hence the box integrals are unbounded and the box-exhaustion total energy $\sup_{R > 0} \int_{[- R, R]^{3}} u$ is $+ \infty$ .

3.1step 2.2F4F5algebra

Compactly supported data. Suppose the constrained $C^{\infty}$ data of [F5] satisfy $div E_{0} = div B_{0} = 0$ and vanish off a compact $K \subseteq {\overline{B}}_{ρ} (0)$ , and let $(E, B)$ be the corresponding solution; for fixed $t$ the fields vanish off the closed ball ${\overline{B}}_{ρ + c (t - t_{0})} (0)$ , a compact set, so $u (\cdot, t)$ is a continuous function with compact support there; choose a nondegenerate closed box containing that ball. By [F4] the continuous density is Riemann integrable on the box with a finite integral, and it vanishes outside the box. All larger box integrals have this same finite value. Thus the instantaneous total energy of the compactly supported configuration is finite, in contrast with the divergence of step 2.2 for the ideal plane wave.

4.1step 1.1step 2.1step 2.2step 3.1given∎

Conclusion and scope. Step 1.1 is clause (i) except for the average, which is step 2.1; step 2.2 is clause (ii) including the divergence of the box integrals; step 3.1 is clause (iii); and clause (iv) is a scope qualification: all quantities are exact ideal-field quantities of the specified plane wave, no laboratory measurement or finite beam is described, and the phrase "infinite total energy" refers to the divergence statement of step 2.2 in the box-exhaustion sense, not to any finite measured energy.

Remarks

Idealisation, not evidence. The computation shows that the ideal whole-space plane wave is not a finite-energy configuration; it makes no claim about any laboratory field, and the compact-support branch of Constrained smooth vacuum Maxwell initial-value theory, finite-energy conservation and finite propagation is the source of the finite-energy contrast.

A nonzero whole-space plane wave has infinite total energy

Example

Facts & Assumptions

Verification

Remarks

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