Low Multipoles, Parameter Shifts, and Gauge Modes

Monopoles and dipoles describe changes of mass, spin, and reference frame, not the freely propagating gravitational waves carried by higher multipoles. The distinction is not “physical versus unphysical”: a stationary change of mass or angular momentum is physical even though it is not radiation. For smooth vacuum perturbations of Schwarzschild, we will solve all three low-mode sectors in a chart regular at the future horizon. Their complete content is a mass variation, three spin variations, and a smooth coordinate perturbation. Whether that last term is removable also depends on the boundary conditions. A separately normalized Reissner–Nordström example shows how electric charge enters without claiming a complete classification of charged perturbations.

Required background. PDEs and Green Functions for Gravitation supplies the regularity and boundary conditions that define the solution and gauge classes.

Helpful background. Black-Hole Geometry and Mechanics fixes the stationary families and charges; Formulations of General Relativity separates constraints from gauge; Killing Fields and Isometry Groups explains when a coordinate change leaves the background unchanged.

Monopoles and dipoles in a regular chart

Write $\mu=GM\gt0$ and use ingoing Eddington–Finkelstein coordinates,

$$g_{\alpha\beta}dx^\alpha dx^\beta =-f\,dv^2+2\,dv\,dr+r^2d\Omega^2, \qquad f=1-\frac{2\mu}{r}.$$

Take the domain to be $I\times(r_0,\infty)\times S^2$, where $I$ is a connected open interval and $0\lt r_0\lt2\mu$. Smoothness means smoothness of the tensor at every finite point of this domain, including $r=2\mu$, not bounded coordinate components at the poles of spherical coordinates. It does not yet impose a condition at infinity.

Let $p_{\alpha\beta}=\delta g_{\alpha\beta}$ solve $\delta G_{\alpha\beta}[p]=0$ and contain only $\ell=0,1$ harmonics. Use the preceding pages’ inverse-identification gauge convention

$$p'_{\alpha\beta} =p_{\alpha\beta}-(\mathcal L_Xg)_{\alpha\beta}.$$

Thus $X$ is a generator that removes a coordinate perturbation $p=\mathcal L_Xg$; it is the negative of the active generator used in Infinitesimal Diffeomorphisms and Gauge Transformations. With no restriction on the asymptotic behavior of smooth $X$, the result is

$$p=p^{(M)}(\delta M)+p^{(J)}(\delta J_i)+\mathcal L_Xg, \qquad \partial_v\delta M=\partial_v\delta J_i=0.$$

Here $p^{(M)}$ and $p^{(J)}$ are the regular representatives constructed below. The statement holds on the whole product domain, not only on a small neighborhood of a point. It is a statement about smooth solutions modulo smooth coordinate changes; a boundary-restricted quotient is a second question.

The following table separates the intrinsic constants from the frame data that can remain in the explicitly Poincaré-asymptotic sector discussed below.

Sector	Content modulo all smooth coordinate changes	Additional Poincaré frame data
Even $\ell=0$	Mass variation $\delta M$	None
Odd $\ell=1$	Spin variations $\delta J_i$	None at first order about nonspinning Schwarzschild
Even $\ell=1$	No intrinsic field	Momentum $\delta P_i$ and initial mass dipole $\delta D_i(0)$

There is no odd monopole: $X_A^{00}=0$. The trace-free harmonics $Y_{AB}^{1m}$ and $X_{AB}^{1m}$ vanish, whereas the angular trace $\Omega_{AB}Y^{\ell m}$ survives. Consequently the radiative spin-two gravitational master variables are undefined for $\ell\lt2$. An undefined variable is not a field with value zero. Use the actual low-mode harmonic basis, not a substitution into an $\ell\ge2$ master formula.

The mass monopole

For one normalized monopole $Y=Y^{00}=1/\sqrt{4\pi}$, the only amplitudes are

$$p_{ab}=h_{ab}Y,\qquad p_{aB}=0,\qquad p_{AB}=r^2K\Omega_{AB}Y.$$

Lower-case orbit indices run over $v,r$; upper-case indices refer to the unit sphere. A contravariant gauge vector $X=T(v,r)Y\partial_v+R(v,r)Y\partial_r$ gives $h'_{rr}=h_{rr}-2\partial_rT$ and $K'=K-2R/r$. Choose a fixed reference radius $\rho\gt r_0$ and set

$$T(v,r)=\frac12\int_\rho^r h_{rr}(v,s)\,ds+a_0(v), \qquad R(v,r)=\frac r2K(v,r).$$

These are smooth finite-radius operations, with no division by $f$. They set $h_{rr}=K=0$ across the future horizon. Write the remaining amplitudes as $a=h_{vv}$ and $b=h_{vr}$. The $rr$ equation is

$$\frac{\delta G_{rr}}{Y}=\frac{2}{r}\partial_rb=0.$$

Therefore $b=b(v)$. Residual gauge $X=\alpha(v)Y\partial_v$ changes $b$ by $-\dot\alpha$ and $a$ by $2f\dot\alpha$; a primitive on $I$ sets $b=0$. Dots denote $\partial_v$. The next two equations are

$$\begin{aligned} \frac{\delta G_{vr}}{Y} &=-\frac{\partial_r(ra)}{r^2}=0,\\ \frac{\delta G_{vv}}{Y} &=\frac{f\,\partial_r(ra)}{r^2}+\frac{\partial_va}{r}=0. \end{aligned}$$

The first gives $a=c(v)/r$; the second gives $\dot c=0$. The angular trace equation is then satisfied, since $\partial_r^2a+2\partial_ra/r=0$. Thus the surviving physical component is

$$p^{(M)}_{vv}=\frac{2G\,\delta M}{r}, \qquad h_{vv}=\frac{2\sqrt{4\pi}\,G\,\delta M}{r}.$$

The factor $\sqrt{4\pi}$ belongs to the harmonic amplitude, not to the metric component. Differentiating the exact Schwarzschild metric at fixed $v,r,\theta,\phi$ gives precisely this solution. More generally, differentiating any exact family $E[g(\lambda)]=0$ gives $E_g^{(1)}[\partial_\lambda g]=0$.

There is also a geometric check. For a spherically symmetric metric with areal radius $r$, define the geometrized mass $m_{\mathrm{ar}}=r(1-\nabla_ar\nabla^ar)/2$. Its background value is the constant $\mu$, and the solution has $\delta m_{\mathrm{ar}}=G\,\delta M$. A coordinate change cannot alter the perturbation of this constant background scalar. The family horizon has area $16\pi\mu^2$, so its variation is $32\pi\mu G\,\delta M$; holding the old coordinate sphere fixed would not compute the horizon-area variation.

This completes the monopole integration in Martel and Poisson 2005, arXiv v1, § VIII.A, Eqs. (8.1)–(8.4), p. 10, PDF. That section uses an unnormalized constant harmonic; the component result agrees after the normalization above.

The odd dipole and spin

For each $\ell=1$ harmonic, use $X_A=-\epsilon_A{}^B D_BY$, with $\epsilon_{\theta\phi}=\sin\theta$. Only $p_{aB}=h_aX_B$ is present. A covariant angular gauge component $X_B^{\mathrm{gauge}}=\xi(v,r)X_B$ gives

$$h'_a=h_a-\partial_a\xi+\frac{2r_a}{r}\xi, \qquad r_a=\partial_ar.$$

The following choice sets $h_r=0$ smoothly:

$$\xi(v,r)=r^2\left[ \int_\rho^r\frac{h_r(v,s)}{s^2}\,ds+\alpha(v) \right].$$

To see exactly what the field equations retain, define

$$\mathcal J =r^4\left[ \partial_v\!\left(\frac{h_r}{r^2}\right) -\partial_r\!\left(\frac{h_v}{r^2}\right) \right].$$

The gauge terms cancel by equality of mixed derivatives. Direct evaluation of the linearized Einstein tensor gives its only nonzero projections,

$$\delta G_{aB}=\frac12P_aX_B,\qquad P_r=-\frac{\partial_r\mathcal J}{r^2}, \qquad P_v=\frac{\partial_v\mathcal J+f\partial_r\mathcal J}{r^2}.$$

Hence vacuum requires both radial and time constancy of $\mathcal J$. In radial gauge, $P_r=0$ becomes $\partial_r^2h_v-2h_v/r^2=0$, with complete solution $h_v=c_1(v)r^2+c_2(v)/r$. Residual $\xi=r^2\alpha(v)$ removes the $r^2$ term. The other equation gives $\dot c_2=0$, and $\mathcal J=3c_2$. There is no angular tensor equation to add at this multipole.

For the axisymmetric normalized harmonic, write $Y^{10}=N\cos\theta$, $N=\sqrt{3/(4\pi)}$; then $X_\phi=-N\sin^2\theta$. The spin representative is

$$h_v=\frac{2G\,\delta J}{Nr},\qquad h_r=0, \qquad p^{(J)}_{v\phi}=-\frac{2G\,\delta J}{r}\sin^2\theta, \qquad \mathcal J=\frac{6G\,\delta J}{N}.$$

This is a normalized angular-momentum variation, not merely a term “proportional to spin.” To verify the horizon gauge, differentiate Kerr at zero spin, with $a_{\mathrm K}=J/M$. In the usual ingoing azimuth $\bar\phi$, the linear metric has $p_{r\bar\phi}=-\delta a_{\mathrm K}\sin^2\theta$. The smooth parameter-dependent change $\bar\phi=\phi-a_{\mathrm K}/r$ contributes the opposite radial component. Differentiation at fixed $v,r,\theta,\phi$ then gives the displayed $p_{r\phi}=0$ and $p_{v\phi}$. No horizon-singular Schwarzschild-time transformation is needed.

Rotating this solution supplies the other two real dipoles. If $n_i=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, their component form is

$$p^{(J)}_{vA} =-\frac{2G}{r}\epsilon_A{}^B\delta J_iD_Bn_i, \qquad p^{(J)}_{rA}=0.$$

The vector $\delta J_i$ is intrinsic spin at first order about Schwarzschild. In contrast, $h_v=\Omega_0r^2$, $h_r=0$ has $\mathcal J=0$ and is a rotating coordinate frame, not a Kerr parameter shift. Its growth is also incompatible with a fixed asymptotically inertial chart. These integrations and the Cartesian identification are given in Martel and Poisson 2005, arXiv v1, § VIII.B, Eqs. (8.5)–(8.9), pp. 10–11, PDF.

The even dipole is a smooth coordinate perturbation

Now $p_{ab}=h_{ab}Y$, $p_{aB}=j_aD_BY$, and $p_{AB}=r^2K\Omega_{AB}Y$ for $\ell=1$. There is no trace-free $G$ amplitude. Use the contravariant vector

$$X=T Y\partial_v+R Y\partial_r+Z D^AY\partial_A.$$

Its relevant transformations are $h'_{rr}=h_{rr}-2\partial_rT$, $j'_r=j_r-T-r^2\partial_rZ$, and $K'=K-2R/r+2Z$. The constructive gauge choice is

$$\begin{aligned} T(v,r)&=\frac12\int_\rho^r h_{rr}(v,s)\,ds+a_1(v),\\ Z(v,r)&=\int_\rho^r\frac{j_r(v,s)-T(v,s)}{s^2}\,ds+a_2(v),\\ R(v,r)&=\frac r2\bigl[K(v,r)+2Z(v,r)\bigr]. \end{aligned}$$

It sets $h_{rr}=j_r=K=0$ and is again smooth at every finite radius. The residual vectors have $T=\alpha_1(v)$, $Z=\alpha_1(v)/r+\alpha_2(v)$, and $R=\alpha_1(v)+r\alpha_2(v)$. As in the monopole, $\delta G_{rr}=0$ makes $b=h_{vr}$ independent of $r$. Residual gauge changes it by $-\dot\alpha_1-\alpha_2$, so set $b=0$. Thereafter $\alpha_2=-\dot\alpha_1$ preserves that choice.

Put $a=h_{vv}$ and $j=j_v$. Define $\delta G_{ab}=E_{ab}Y$ and $\delta G_{aB}=Q_aD_BY/2$. The two mixed equations are

$$\begin{aligned} Q_r&=\partial_r^2j-\frac{2j}{r^2}=0,\\ Q_v&=\partial_ra-f\partial_r^2j-\partial_v\partial_rj +\frac{2}{r}\partial_vj-\frac{4\mu}{r^3}j=0. \end{aligned}$$

First integrate the radial equation, then the equation for $\partial_ra$:

$$j=k_1(v)r^2+\frac{k_2(v)}r,\qquad a=-\frac{k_2(v)}{r^2}+\frac{3\dot k_2(v)}r+k_3(v)+2k_1(v)r.$$

The remaining orbit equations do not allow three independent functions. They give $E_{vr}=-k_3/r^2=0$ and, after $k_3=0$,

$$E_{vv}=\frac{3}{r^2}\bigl(2\mu k_1+\ddot k_2\bigr)=0.$$

Thus write $k_2=2\mu k$ and $k_1=-\ddot k$. The complete remaining metric is

$$h_{vv}=-2r\ddot k+\frac{6\mu}{r}\dot k-\frac{2\mu}{r^2}k, \qquad j_v=-r^2\ddot k+\frac{2\mu}{r}k.$$

There is no untested angular condition: its trace projection is $Q^\flat=-\partial_r^2a-2\partial_ra/r+2\partial_rj/r^2$ and vanishes for these expressions. All other components either have been used or vanish by the harmonic decomposition.

The remaining arbitrary function is exactly gauge. In covariant components, the removal vector is

$$\begin{aligned} X_v&=\left(-r\dot k+\frac{2\mu}{r}k\right)Y,& X_r&=kY,\\ X_A&=\left(-r^2\dot k+rk\right)D_AY. \end{aligned}$$

Substitution gives $p-\mathcal L_Xg=0$, including every vanishing component. For any smooth $k(v)$ it is regular at the future horizon. This proves the even-dipole part of the decomposition and its converse: every such Lie derivative solves the linearized vacuum equation because the background Einstein tensor vanishes. The construction works separately for all three real dipoles. See Martel and Poisson 2005, arXiv v1, § VIII.C, Eqs. (8.10)–(8.20), pp. 11–12, PDF.

Proper gauge, momentum, and center of mass

The integrations above prove smoothness, not decay at infinity. In particular, an arbitrary $k(v)$ is not automatically asymptotically flat. For a prescribed boundary problem, $p=\mathcal L_Xg$ is removable precisely when there is a proper admissible generator $Z$ with

$$\mathcal L_{X-Z}g=0.$$

In words, $X$ must differ from an allowed proper gauge vector by a background Killing field. Merely finding one singular or nondecaying generator does not exclude a different admissible representative.

Here is a concrete boundary setting. Take a domain containing complete asymptotically spacelike slices; for example, allow all $v\in\mathbb R$. Choose their time $\tau$ to agree with $t=v-r_*$ at large radius and to extend smoothly across the horizon. A bounded interval of advanced time need not contain such a slice, and $r\to\infty$ at fixed $v$ is not the ADM spatial-infinity limit.

In an asymptotically Cartesian chart, the original Regge–Teitelboim conditions take the form

$$\begin{aligned} \gamma_{ij} &=\delta_{ij}+\frac{\bar h_{ij}(n)}r +\frac{h^{(2)}_{ij}(n)}{r^2}+o(r^{-2}),\\ \pi^{ij} &=\frac{\bar\pi^{ij}(n)}{r^2} +\frac{\pi^{(2)ij}(n)}{r^3}+o(r^{-3}),\\ \bar h_{ij}(-n)&=\bar h_{ij}(n),\qquad \bar\pi^{ij}(-n)=-\bar\pi^{ij}(n). \end{aligned}$$

The coefficients are smooth on the sphere and the remainders have the corresponding differentiated falloff. Here $\gamma_{ij}$ is the spatial metric and $\pi^{ij}$ its canonical momentum density. These conditions make the Hamiltonian surface charges finite. Proper gauge means that the boundary generator vanishes as a function on this allowed phase space, not just that its numerical value happens to vanish on Schwarzschild.

Choose the parameter representatives to obey this same asymptotic identification before classifying the residual vector. For example, the fixed-$v$ mass derivative above has $p_{tr}=2G\delta M/(rf)$ on the fixed background $t$ slices. Its induced canonical momentum does not obey the displayed parity condition. At large radius choose a time vector $Z$ with $\partial_rZ^t=-2G\delta M/(rf^2)$, extended smoothly to zero near the horizon. Then $p^{(M)}-\mathcal L_Zg$ has $p_{tr}=0$ there and is the fixed-asymptotic-time mass representative. The logarithmic growth of $Z^t$ is a change of the family identification, not a claim that $Z$ is proper RT gauge. Horizon-regular and asymptotically fixed coordinates can be joined in a finite transition region.

For clarity, restrict the residual coordinate changes in this example to the allowed Poincaré-asymptotic class. Their normal and tangential components on a fixed slice have the form

$$\begin{aligned} X^\perp&=b_ix^i+c+\alpha_{\mathrm{odd}}(n)+O(r^{-1}),\\ X^i&=\omega^i{}_jx^j+d^i+\beta^i_{\mathrm{odd}}(n)+O(r^{-1}), \qquad \omega_{ij}=-\omega_{ji}. \end{aligned}$$

The angle-dependent displayed remainders are odd under $n\mapsto-n$. They and the decaying terms are proper gauge; the constants, rotations, and boosts generate Poincaré transformations. Also fix the asymptotic inertial lapse to one and shift to zero, with differentiated $O(r^{-1})$ perturbations. The fixed-slice parameters above are extended to spacetime Poincaré generators: $c,b_i,\omega_{ij}$ are constant, and the translation coefficient at time $t$ is $d^i(0)+t b^i$. They are not arbitrary time-dependent functions. Conditions on $\gamma_{ij},\pi^{ij}$ alone would not fix the asymptotic clock. These are the original parity conditions reviewed in Henneaux and Troessaert 2018, arXiv v1, § 2.1, Eqs. (2.1)–(2.2), (2.7)–(2.10), pp. 3–5, PDF, not that paper’s later alternative BMS boundary conditions. The full charge construction is developed in Poincaré Symmetry and ADM Charges.

Time translations and rotations are Schwarzschild Killing fields; they produce no first-order metric change. Spatial translations and boosts are not. To construct an unambiguous example, use at large radius

$$\Xi=d^i\partial_i+ b^i\bigl(x_i\partial_t+t\partial_i\bigr), \qquad p=-\mathcal L_\Xi g.$$

Multiply this vector by a smooth radial function equal to zero near the horizon and one sufficiently far away. It then has a smooth horizon extension without changing its asymptotic translation and boost. The minus sign specifies an actively displaced black hole, with center $d^i+b^it$ in the fixed asymptotic coordinates. At first order,

$$\delta E=0,\qquad \delta P_i=M b_i,\qquad \delta D_i(t)=M d_i+t\,\delta P_i, \qquad \delta B_i=\delta D_i-t\,\delta P_i.$$

Here $D_i=E C_i$ is the mass dipole, $C_i$ the center-of-mass position, and $B_i$ the conserved boost charge in the displayed sign convention. For the background, $E=M$ and $P_i=D_i=0$. The transformation law follows from the Lorentz-vector energy–momentum and angular-momentum tensor; the mass dipole evolves with $\partial_tD_i=P_i$. See Szabados 2003, arXiv v2, § 4.1, pp. 26–29, especially Eq. (4.1.5), PDF. A boost changes momentum at first order but the energy of a resting mass only at second order.

Accordingly, in this Poincaré-asymptotic frame sector, fixing $\delta P_i=0$ and $\delta D_i(0)=0$ removes the six displacement and boost data. That is a rest/centered sector restriction. Quotienting all Poincaré frame changes would be a different choice. Neither operation removes $\delta M$ or the intrinsic $\delta J_i$.

The extra asymptotic hypothesis is essential: the smooth radial-gauge construction alone does not prove that every perturbation satisfying some asymptotic falloff has a residual generator in the displayed class. For a different boundary problem, use the exact Killing-field criterion and check that generator’s asymptotics. Do not promote the finite-radius classification to an unrestricted asymptotic-flatness theorem. Even choosing $k(v)$ affine in $v$ in the earlier radial gauge is not by itself an ADM boost construction: conversion to Schwarzschild time can introduce logarithmic spatial terms.

Electric charge as an exact-family direction

The following example is Einstein–Maxwell, not Schwarzschild vacuum. Fix the electric normalization by

$$S=\int d^4x\,\sqrt{-g}\left[ \frac{R}{16\pi G}-\frac14F_{\alpha\beta}F^{\alpha\beta} \right], \qquad F=dA,\qquad Q=\int_{S^2}*F.$$

With $\epsilon_{vr\theta\phi}=+r^2\sin\theta$, the ingoing electric Reissner–Nordström family is

$$\begin{aligned} f_Q&=1-\frac{2\mu}{r}+\frac{GQ^2}{4\pi r^2},\\ A&=-\frac{Q}{4\pi r}\,dv,\qquad F_{vr}=-\frac{Q}{4\pi r^2},\qquad (*F)_{\theta\phi}=\frac{Q}{4\pi}\sin\theta. \end{aligned}$$

Use a nonextremal member, $\mu^2\gt GQ^2/(4\pi)$, on a domain crossing its future outer horizon, and keep magnetic charge fixed. Differentiating at fixed ingoing coordinates gives

$$\begin{aligned} p_{vv}&=\frac{2\,\delta\mu}{r} -\frac{GQ\,\delta Q}{2\pi r^2},\\ \delta A_v&=-\frac{\delta Q}{4\pi r},\qquad \delta F_{vr}=-\frac{\delta Q}{4\pi r^2}. \end{aligned}$$

The metric and Maxwell variations must be kept together. Differentiating both exact equations gives $\delta G_{\alpha\beta}=8\pi G\,\delta T_{\alpha\beta}$ and the linearized Maxwell equation, including its metric variation. The charge variation is directly $\delta\int*F=\delta Q$. At $Q=0$ a pure electric-charge direction has $p=0$ and $\delta T=0$ but $\delta F\ne0$: the stress tensor first changes quadratically in the new field. A metric-only test would miss this physical monopole. The Reissner–Nordström background and coupled charged perturbations supply the wider context. These family derivatives do not classify all charged low modes or magnetically charged sectors.

Curvature tests and their limits

A vanishing wave diagnostic is compatible with a nonzero parameter change. To make the diagnostic precise, choose the regular Schwarzschild principal null tetrad

$$l=\partial_v+\frac f2\partial_r,\qquad n=-\partial_r,\qquad m=\frac{\partial_\theta+i\csc\theta\,\partial_\phi}{\sqrt2\,r}, \qquad l\cdot n=-1,\quad m\cdot\bar m=1.$$

Use the same minus-Weyl contractions as the Kinnersley tetrad page:

$$\Psi_0=-C(l,m,l,m),\qquad \Psi_2=-C(l,m,\bar m,n),\qquad \Psi_4=-C(n,\bar m,n,\bar m).$$

In the site’s curvature convention, Schwarzschild has $\Psi_2=\mu/r^3$ and $\Psi_0=\Psi_4=0$. For first variations use a metric-compatible tetrad; for example, each background leg $e^\alpha$ can be varied by $\delta e^\alpha=-g^{\alpha\gamma}p_{\gamma\beta}e^\beta/2$. The variation of each Weyl contraction includes the curvature change and all four tetrad-leg terms. This yields, for the regular mass and axisymmetric spin representatives above,

$$\begin{aligned} \delta\Psi_0&=\delta\Psi_4=0,\\ \delta\Psi_2^{(M)}&=\frac{G\,\delta M}{r^3},\qquad \delta\Psi_2^{(J)}=\frac{3iG\,\delta J\cos\theta}{r^4}. \end{aligned}$$

The extreme scalars also vanish for a coordinate perturbation with the tetrad transported consistently. Infinitesimal Lorentz changes of the normalized tetrad do not change their first variations on this type-D background. This is the appropriate linear gauge-invariant wave test; a fixed, unvaried coordinate tetrad is not the definition.

Do not apply the same argument to the Coulomb scalar. For $p=-\mathcal L_\Xi g$ with transported legs,

$$\delta\Psi_2=-\mathcal L_\Xi\!\left(\frac{\mu}{r^3}\right) =\frac{3\mu}{r^4}\Xi^r.$$

It need not vanish because its background value is not constant. Thus nonzero $\delta\Psi_2$ alone does not distinguish a mass shift from a coordinate displacement; the geometric mass or asymptotic charges do. For the low modes proved above, vanishing extreme scalars does not mean that their charges vanish.

The conclusion is linear, vacuum, and boundary-dependent. Matter can carry dipole data; a changed outer boundary defines a different problem. At second order, products of radiative modes can source mass, momentum, and spin changes. None of the present statements is a nonlinear-stability theorem.

Exercises

A mass function is not an arbitrary vacuum solution

In the regular monopole gauge let $p_{vv}=2G\,m_1(v)/r$. Which field equation forces $m_1$ to be constant? Would checking only the radial equation establish a vacuum solution?

Solution

The radial equation gives $\partial_r(rp_{vv})=0$ for any $m_1(v)$, but the remaining equation is

$$\delta G_{vv}=\frac{2G\,\dot m_1(v)}{r^2}.$$

Vacuum therefore requires $\dot m_1=0$. A time-dependent mass function would require a source; satisfying the radial constraint alone is insufficient. This is why the classification uses all remaining equations, not only the expected radial profile.

Proper gauge or a change of frame?

Compare (a) the smooth cutoff translation/boost above; (b) an even dipole generated by a proper smooth gauge vector; and (c) a proposed removal vector that diverges at the future horizon. Are any of these arguments by themselves evidence of radiation? What data must be fixed to select the rest/centered Poincaré sector?

Solution

(a) changes $\delta P_i=M b_i$ and $\delta D_i(0)=M d_i$ in the fixed asymptotic frame. It is not proper gauge merely because it is a Lie derivative. (b) is removable: its radiative Weyl data vanish; an $\ell=1$ radiative spin-two master variable is not defined. For (c), the proposed generator is inadmissible. This does not prove nonremovability: another generator may differ from it by a Killing field, or the original perturbation may itself fail the required regularity. Check the perturbation and the exact removal criterion.

None of these facts establishes a propagating gravitational wave. Within the specified Poincaré-asymptotic frame sector, impose $\delta P_i=0$ and $\delta D_i(0)=0$ to select a resting, centered member. Mass and intrinsic spin variations are separate parameters and remain.

A charge with no first-order metric change

At $Q=0$, differentiate the electric Reissner–Nordström family with $\delta\mu=0$ and $\delta Q\ne0$. Find $p$, $\delta F$, and the electric flux variation. Why does this not contradict Einstein’s equation?

Solution

The factor $Q\,\delta Q$ vanishes, so $p_{\alpha\beta}=0$, while

$$\delta F=-\frac{\delta Q}{4\pi r^2}\,dv\wedge dr, \qquad \int_{S^2}\delta(*F)=\delta Q.$$

The background Maxwell field is zero and its stress tensor is quadratic in $F$. Consequently $\delta T_{\alpha\beta}=0$ and the first-order Einstein equation has zero on both sides. Maxwell’s equation still has a nontrivial source-free Coulomb solution on the exterior domain. Nonzero flux through a sphere does not require a charge density in that domain; its interior is not part of the exterior problem.

References

• Henneaux, Marc, and Cédric Troessaert. “BMS Group at Spatial Infinity: The Hamiltonian (ADM) Approach.” Journal of High Energy Physics 2018, no. 3 (2018): 147. DOI. Open PDF, arXiv:1801.03718v1; the original Regge–Teitelboim conditions are reviewed in § 2.1.
• Martel, Karl, and Eric Poisson. “Gravitational Perturbations of the Schwarzschild Spacetime: A Practical Covariant and Gauge-Invariant Formalism.” Physical Review D 71 (2005): 104003. DOI. Open PDF, arXiv:gr-qc/0502028v1. The cited §§ VIII.A–C and pp. 10–12 belong to this 21-page preprint, not the shorter journal edition.
• Szabados, László B. “On the Roots of the Poincaré Structure of Asymptotically Flat Spacetimes.” Classical and Quantum Gravity 20 (2003): 2627–2662. DOI. Open PDF.