Dipole and Additional-Field Radiation

An additional long-range field can radiate a lower multipole than the tensor gravitational field. For compact bodies carrying unequal scalar charge per unit mass, the center-of-mass scalar dipole is nonzero and its flux is larger than the tensor quadrupole by two powers of orbital velocity. The enhancement is conditional: charge symmetry, a mass threshold, screening, or a missing healthy radiative mode can remove it.

Helpful background. Einstein Equations fixes the tensor baseline; Gravitational Actions and Boundary Terms fixes radiative stress-energy; Formulations of General Relativity supplies constraint versus radiative modes; Relativistic Two-Body Dynamics supplies center-of-mass variables and balance laws; Gravitational Radiation and Waveforms supplies tensor multipoles; and Gravity Beyond General Relativity supplies the extra-field viability conditions.

Scalar multipoles from the retarded field

In the weak-field radiation zone, the scalar convention of this chapter gives

$$\Box\varphi=4\pi G_*\rho_s, \qquad \rho_s=\sum_A q_A\delta^3(\boldsymbol{x}-\boldsymbol{x}_A), \qquad q_A=m_A\alpha_A.$$

The retarded solution is $\varphi=-G_*\int d^3x'\,\rho_s(t-|\boldsymbol{x}-\boldsymbol{x}'|,\boldsymbol{x}')/|\boldsymbol{x}-\boldsymbol{x}'|$. At radius $R$ much larger than the source,

$$\varphi_{\rm rad}(t,R\boldsymbol n) =-\frac{G_*}{R}\left[ Q_{\rm rad}+n_i\dot D^i+\frac12n_in_j\ddot I^{ij}+\cdots \right]_{t-R},$$

Here $Q=\sum_Aq_A$, $D^i=\sum_Aq_Ax_A^i$, and $I^{ij}=\sum_Aq_Ax_A^{\langle i}x_A^{j\rangle}$. Angle brackets denote the symmetric trace-free part. The trace cannot simply be dropped in scalar radiation: with $J=\sum_Aq_A|\boldsymbol x_A|^2$, decomposing the raw second moment gives $Q_{\rm rad}=Q+\ddot J/6+\cdots$. In a relativistic theory, corrections to the source also contribute to this radiative monopole at the same slow-motion order. The scalar kinetic term in the declared action has stress tensor

$$T^{(\varphi)}_{\mu\nu}=\frac{1}{4\pi G_*} \left(\partial_\mu\varphi\partial_\nu\varphi -\frac12g_{\mu\nu}(\partial\varphi)^2\right).$$

Using $\int n_in_jd\Omega=4\pi\delta_{ij}/3$ and the corresponding rank-four identity gives the positive outgoing power

$$\mathcal F_\varphi =G_*\left[\dot Q_{\rm rad}^{\,2}+\frac13\ddot D_i\ddot D^i +\frac1{30}\dddot I_{ij}\dddot I^{ij}+\cdots\right].$$

These are the leading slow-motion contributions at each displayed multipole. The ellipsis includes retardation corrections to lower moments and higher multipoles, so the formula is not an all-orders separation of three exact channels.

For constant adiabatic charges, the leading term $\dot Q$ vanishes. This does not eliminate all scalar monopole radiation: $\dddot J/6$ can be nonzero on an eccentric orbit, and relativistic source corrections can vary as well. For the constant-charge circular Newtonian example below, $J$ is constant and that trace contribution also vanishes. The separation into scalar monopole, dipole, and quadrupole channels agrees with the compact-body balance-law decomposition in Damour and Esposito-Farèse 1996, § V, Eq. (5.4).

Why charge asymmetry produces dipole radiation

In the Newtonian center-of-mass frame,

$$\boldsymbol{x}_1=\frac{m_2}{M}\boldsymbol r, \qquad \boldsymbol{x}_2=-\frac{m_1}{M}\boldsymbol r.$$

Therefore

$$\boldsymbol D =m_1\alpha_1\boldsymbol{x}_1+m_2\alpha_2\boldsymbol{x}_2 =\mu(\alpha_1-\alpha_2)\boldsymbol r.$$

For a circular orbit, $\ddot{\boldsymbol D}=-\Omega^2\boldsymbol D$, and hence

$$\mathcal F_\varphi^{\rm dip} =\frac{G_*}{3}\mu^2(\Delta\alpha)^2r^2\Omega^4, \qquad \Delta\alpha=\alpha_1-\alpha_2.$$

The leading tensor quadrupole power in the same Einstein-frame normalization is

$$\mathcal F_T^{\rm quad}=\frac{32}{5}G_*\mu^2r^4\Omega^6.$$

Thus, with $v=r\Omega$,

$$\frac{\mathcal F_\varphi^{\rm dip}}{\mathcal F_T^{\rm quad}} =\frac{5}{96}\frac{(\Delta\alpha)^2}{v^2}.$$

The coefficient belongs to the scalar normalization declared above; field rescalings move factors between $\alpha_A$ and the flux. The invariant statement is the $v^{-2}$ enhancement and the squared charge asymmetry. Mirshekari and Will derive the associated mechanical-energy loss and identify it as $-1$PN relative to tensor quadrupole radiation in Mirshekari and Will 2013, § VI D, eqs. (6.14)–(6.16), p. 26. Their $\mathcal S_-$ is one half of a convention-dependent charge difference; see Mirshekari and Will 2013, § VII A, eqs. (7.2)–(7.3), p. 28.

Energy balance fixes the sign:

$$\frac{dE_{\rm orb}}{dt}=-\mathcal F_\infty-\mathcal F_{\rm horizon},$$

where each ordinary outgoing channel contributes positively to $\mathcal F_\infty$. Since $E\sim-\mu v^2/2$, scalar dipole flux scales as $v^8$ while tensor quadrupole flux scales as $v^{10}$. The resulting Fourier-phase correction begins as $v^{-7}$ rather than the GR $v^{-5}$ term. This pre-Newtonian scaling and its placement in a generic waveform are described in Berti, Yagi, and Yunes 2018, § 3.1, eqs. (18)–(19), pp. 13–15.

Other fields and thresholds

There is no theory-independent “extra-field flux coefficient.” One must diagonalize the quadratic action, retain only healthy radiative eigenmodes, normalize their kinetic terms, and project the compact-body sources onto them.

Radiative sector	Possible leading moment	What can eliminate it?
massless scalar	radiative monopole if $Q_{\rm rad}$ evolves; charge dipole	constant radiative monopole; $\alpha_1=\alpha_2$ removes the leading dipole; screening
constrained vector	transverse current dipole or preferred-frame multipoles	a universal conserved charge-to-mass ratio; vector constraints; vanishing sensitivity difference
additional tensor	quadrupole, or lower moments if sourced through extra constraints	decoupling from matter; nondynamical or ghostlike mode removed from the spectrum
massive mode	first allowed multipole above threshold	harmonic frequency $\omega_n\le m_s$; small group velocity near threshold

For a massive scalar, $\omega^2=k^2+m_s^2$. A binary harmonic with $\omega_n=n\Omega$ propagates only when $\omega_n\gt m_s$; below threshold it is evanescent. Above threshold its group velocity $k/\omega_n$ and additional powers of $k$ suppress the multipole flux near turn-on. A massless PN series cannot be extrapolated through this nonanalytic threshold. Berti, Yagi, and Yunes discuss theory-specific generation mappings and such validity boundaries in Berti, Yagi, and Yunes 2018, Appendix B, Eq. (44), p. 35, PDF.

Checks and limitations

Dimensional analysis gives $[\mathcal F]=M/L$ for both displayed fluxes, equivalent to mass squared when $\hbar=c=1$. Exchange $1\leftrightarrow2$ sends $\Delta\alpha\to-\Delta\alpha$ and $\boldsymbol r\to-\boldsymbol r$, leaving $\boldsymbol D$ and its power physically unchanged. The GR limit $\alpha_A\to0$ removes all scalar moments. The equal-charge limit removes the dipole but generally leaves scalar quadrupole radiation and conservative scalar exchange.

The circular-orbit flux comparison assumes constant adiabatic charges, a massless canonical scalar, quasi-circular slow motion, a wave zone inside the EFT domain, and no incoming radiation. The multipole expansion also applies to eccentric slow-motion sources, but their time-dependent moments must be evaluated and averaged along the eccentric orbit. Dynamical scalarization, mode mixing, massive thresholds, horizon absorption, and cosmological propagation require separate matching rather than an unqualified coefficient substitution.

Exercises

Dipole diagnostic for a mixed binary. Let $m_1=2m_2$, $\alpha_1=a$, and $\alpha_2=0$. Derive $\boldsymbol D$ and find the orbital speed at which the leading scalar dipole and tensor quadrupole powers are equal in the normalization above.

Solution

Here $M=3m_2$ and $\mu=2m_2/3$, so $\boldsymbol D=(2m_2a/3)\boldsymbol r$. The flux ratio is $5a^2/(96v^2)$. Equality therefore occurs at $v=|a|\sqrt{5/96}$. This is a scaling diagnostic, not automatically a controlled physical point: it must also satisfy $v\ll1$, the massless propagation condition, adiabatic charge matching, and the field-theory cutoff.

References

• Berti, Emanuele, Kent Yagi, and Nicolás Yunes. “Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (I) Inspiral–Merger.” General Relativity and Gravitation 50 (2018): 46. DOI; Open PDF.
• Damour, Thibault, and Gilles Esposito-Farèse. “Tensor-Scalar Gravity and Binary-Pulsar Experiments.” Physical Review D 54 (1996): 1474–1491. DOI; Open PDF.
• Mirshekari, Saeed, and Clifford M. Will. “Compact Binary Systems in Scalar-Tensor Gravity: Equations of Motion to 2.5 Post-Newtonian Order.” Physical Review D 87 (2013): 084070. DOI; Open PDF.