Polarized spatial radiation

Summary: Both complex pulse polarizations determine an exact outgoing quadrupole source, whose full spatial spectral measure fixes radiated energy and a low-frequency suppression law.

A response channel of the same spatial field

The spatial displacement action uses a declared stationary component field with two ordinary bound coordinates, an existing propagation coordinate $y$, size $\lambda$ and tension $\mathcal T$. Two weak incoming displacement pulses are $X=A(u)+B(v)$, where $u=t-y$, $v=t+y$. Their derivatives $a=A'$ and $b=B'$ have independent real spatial waveforms and complex Fourier polarizations.

The local collision calculation gives conservative reciprocal delays. The full spatial field can also carry energy away in its complementary quadrupole modes. This note calculates that channel to leading order in the weak incoming slopes. It assumes no independent incoming complementary excitation. These modes belong to the same component field; no mediator particle is added. Its identification with the physical master field remains conditional.

Keep both complex frequency polarizations

The real trace-free source has components

$$S_1=2(a_1b_1-a_2b_2),\qquad S_2=2(a_1b_2+a_2b_1). \tag{1}$$

Use Fourier conventions

$$\widetilde S(\Omega,q)= \int dt\,dy\,e^{i\Omega t-iqy}S(t,y),\qquad \widetilde a(\alpha)=\int du\,e^{i\alpha u}a(u). \tag{2}$$

Since $dt\,dy=\tfrac12du\,dv$, the product source is exactly

$$\boxed{ \widetilde S(\Omega,q) =C\!\left( \widetilde a\!\left(\frac{\Omega+q}{2}\right), \widetilde b\!\left(\frac{\Omega-q}{2}\right) \right),\qquad C(A,B)= \begin{pmatrix} A_1B_1-A_2B_2\\ A_1B_2+A_2B_1 \end{pmatrix}.} \tag{3}$$

The pairing is complex bilinear; inserting a conjugate would describe a different frequency combination. For an outgoing propagating quadrupole, $\Omega>0$ and $\Omega^2-q^2>0$. Thus both arguments $\alpha=(\Omega+q)/2$ and $\beta=(\Omega-q)/2$ are positive. The emitted mode samples the sum of two incoming positive frequencies.

For arbitrary complex vectors, define the signed circular component $h(A)=2\operatorname{Im}(A_1^*A_2)$. Direct expansion gives

$$\boxed{\|C(A,B)\|^2 =\|A\|^2\|B\|^2+h(A)h(B).} \tag{4}$$

Every amplitude and relative phase of both objects is present. Exchanging the pulses leaves $C$ unchanged. A common rotation through $\theta$ in their actual bound plane rotates $C$ through $2\theta$, as required for a quadrupole.

Distinguish a radiation zero from no interaction

For unit positive-frequency circular vectors in fixed bound-plane axes,

$$e_+=\frac{(1,i)}{\sqrt2},\qquad e_-=\frac{(1,-i)}{\sqrt2}, \tag{5}$$

equation (3) gives the following strengths:

These signs label rotation senses relative to the same physical axes, not an already-established photon-helicity assignment. Because the pulses propagate oppositely, a helicity convention tied to each propagation direction would require a separate sign conversion.

If the positive-frequency spectra have these fixed circular vectors, opposite senses cancel the entire leading timelike quadrupole source. Real waveforms retain the conjugate negative-frequency components. Their difference-frequency source can remain nonzero and spacelike. Consequently the conservative interaction and its delay can survive while this outgoing radiation channel vanishes.

An exactly one-sided polarization statement concerns the full Fourier spectrum. A Gaussian envelope times a real circular carrier generally has a small opposite-frequency tail; its cancellation is correspondingly approximate. Narrow-band intuition must not silently remove that tail.

Derive the emitted energy from the actual kinetic norm

The retarded response defines a positive dimensionless radial spectral measure by

$$F(z)=\int_0^\infty\frac{d\mu(\ell)}{\ell-z^2}, \qquad \int\frac{d\mu(\ell)}{\ell}=1. \tag{6}$$

Its radial operator is the full spatial quadrupole Hessian, with $f(r)=8/(1+r^2)$ and $d\mu=\tfrac12d\langle f,P_\ell f\rangle_w$. No radiation weight is fitted.

In a normalized spatial eigenmode with eigenvalue $\ell$, the amplitude $h_\ell(t,q)$ obeys

$$\ddot h_\ell+\left(q^2+\frac{\ell}{\lambda^2}\right)h_\ell =\frac{f_\ell}{\lambda^2}S(t,q). \tag{7}$$

Its full-field kinetic energy has coefficient $\mathcal T\lambda^2/4$ for each real quadrupole component. Solving (7) with zero incoming data and evaluating its outgoing energy therefore gives

$$\boxed{ E_{\rm out} =\frac{\mathcal T}{2\lambda^2} \int_{-\infty}^{\infty}\frac{dq}{2\pi} \int_0^\infty d\mu(\ell)\, \left\| \widetilde S\!\left( \sqrt{q^2+\ell/\lambda^2},q \right) \right\|^2.} \tag{8}$$

This is the leading emitted energy, fourth order in the incoming weak slopes. The integral is positive and symmetric under exchanging the two pulses, which sends $q$ to $-q$. It includes the full radial continuum and both quadrupole polarizations.

For a real forced oscillator, the same normalization follows directly: after the source has ceased, $|\dot h|^2+\omega^2|h|^2=|\widetilde f(\omega)|^2$. Combining the $q$ and $-q$ components gives (8) for real spatial fields. This also fixes the factors of two from Fourier and kinetic conventions.

Retain the energy lost by each incoming pulse

Write $dE_{\rm out}(\ell,q)$ for the positive measure in (8), including both quadrupole components, and put $\Omega=\sqrt{q^2+\ell/\lambda^2}$. The outgoing normal momentum is $P_{\rm out}=\int(q/\Omega)\,dE_{\rm out}$. The same source coupling fixes the two leading pulse energy losses:

$$\boxed{ \Delta E_A=\int\frac{\Omega+q}{2\Omega}\,dE_{\rm out}, \qquad \Delta E_B=\int\frac{\Omega-q}{2\Omega}\,dE_{\rm out}.} \tag{8a}$$

Both are nonnegative. Their sum is $E_{\rm out}$, while their difference is $P_{\rm out}$, as required by the opposite incoming propagation directions. Equivalently, their weights are $\alpha/(\alpha+\beta)$ and $\beta/(\alpha+\beta)$.

This partition can be checked directly from the full coupling. In dimensionless radial coordinates its term is $\tfrac{\mathcal T}{2}\sum_{c=1}^2\langle h_c,fS_c\rangle_w$. Varying the first incoming waveform by its time translation gives $\partial_uS_c$; varying the second gives $\partial_vS_c$. Thus the integrated work removed from the first pulse is $-\tfrac{\mathcal T}{2}\sum_{c=1}^2\int dt\,dy\, \langle h_c,f\partial_uS_c\rangle_w$, and similarly for the second. Their sum integrates by parts to the work on the complementary field. Using its retarded solution supplies exactly (8a). There is no independently assigned absorption coefficient for either pulse.

The losses refer to the outgoing complement at leading fourth order in weak slopes, with the retained right- and left-moving wave energies computed from their reciprocal response. The local conservative delays are part of the same expansion, but the local action alone omits these finite-width losses.

Obtain the long-pulse law

The outgoing low-frequency response derived from the complete spatial Hessian is $\operatorname{Im}F(\omega+i0)=\pi\omega^2/4+o(\omega^2)$. Thus the continuum density near threshold is $d\mu/d\ell=\ell/4+o(\ell)$. For pulses concentrated at frequencies small compared with $1/\lambda$, substitute this into (8) and change variables to $\alpha=(\Omega+q)/2$, $\beta=(\Omega-q)/2$:

$$\boxed{ \frac{E_{\rm out}}{\mathcal T} =\frac{\lambda^2}{\pi} \int_0^\infty d\alpha\int_0^\infty d\beta\, \alpha\beta(\alpha+\beta) \|C(\widetilde a(\alpha),\widetilde b(\beta))\|^2 +\hbox{higher-frequency corrections}.} \tag{9}$$

The Jacobian is $d(\Omega^2-q^2)\,dq=4(\alpha+\beta)\,d\alpha\,d\beta$. The powers in (9) follow from the spatial spectral density and source derivatives, rather than a presumed frequency filter.

For equal Gaussian displacement pulses in real unit directions,

$$A(u)=\epsilon L e^{-u^2/(2L^2)}e_a,\qquad B(v)=\epsilon L e^{-v^2/(2L^2)}e_b,\qquad L\gg\lambda, \tag{10}$$

the derivative spectrum satisfies $|\widetilde a(\alpha)|^2= 2\pi\epsilon^2L^4\alpha^2e^{-L^2\alpha^2}$. The elementary Gaussian moments in (9) give

$$\boxed{ \frac{E_{\rm out}}{\mathcal T} =\frac{3\pi^{3/2}}2\, \frac{\epsilon^4\lambda^2}{L} +o\!\left(\frac{\epsilon^4\lambda^2}{L}\right),\qquad \frac{E_{\rm out}}{E_A+E_B} =\frac{3\pi}{2}\epsilon^2 \left(\frac{\lambda}{L}\right)^2 +o\!\left(\epsilon^2\frac{\lambda^2}{L^2}\right).} \tag{11}$$

The small-frequency limit here is $L/\lambda\to\infty$ at fixed weak slope, within the leading weak-field radiation calculation. Higher orders in the slope are separate corrections. The incoming pair energy is $E_A+E_B=\mathcal T\epsilon^2L\sqrt\pi$.

For unequal widths $L_A,L_B$ and weak slopes $\epsilon_A,\epsilon_B$ in real unit directions, the corresponding leading individual losses are

$$\boxed{ \frac{\Delta E_A}{\mathcal T} =\frac{3\pi^{3/2}}4 \frac{\lambda^2\epsilon_A^2\epsilon_B^2}{L_A}, \qquad \frac{\Delta E_B}{\mathcal T} =\frac{3\pi^{3/2}}4 \frac{\lambda^2\epsilon_A^2\epsilon_B^2}{L_B}.} \tag{12}$$

The shorter pulse contributes more emitted energy at fixed slopes. Swapping the complete incoming profiles swaps the two results.

For $\lambda=1$, the complete radial spectral calculation gives $E_{\rm out}/E_{\rm soft}$ approximately $0.99717$, $0.99986$ and $0.999992$ at $L=5,10,20$, with agreement under radial refinement. No coefficient is changed between these comparisons.

This establishes a concrete frequency-dependent suppression and a two-polarization cancellation for one specified outgoing channel. Its frequency is the actual incoming spatial waveform frequency, as distinguished in Source frequencies in spatial coupling. The calculation does not assign a neutrino mode, an electric charge, or a measured electromagnetic or weak rate.

Verification

polarized_pulse_collision_checks.py checks the complex source identity, exchange symmetry, common spatial rotation and both circular senses. Independent Fourier transforms of real multi-frequency waveforms retain all conjugate components and check the timelike cancellation alongside its nonzero spacelike source. Direct forced-mode time integration checks the emitted-energy normalization against an independent temporal Fourier integral. Unequal-width pulses additionally test both direct source-work integrals against (8a) and the actual outgoing mode energy. A separate full radial eigensystem and source measure evaluate (8), with radial resolution and outer-domain refinement, and recover (11). The companion nonlinear evolution tests the conservative pulse delays. Reports have timestamps and checks survive Python optimization.