$$ \Delta_{\mathcal{J}(\text{rad})}^{\mu\nu} = \int \frac{\mathrm{d}u \mathrm{d}^2\Omega}{32\pi G} [4\hat{C}^{\rho[\mu}N^{\nu]}_{\rho} - 2N^{\rho\sigma}n^{[\mu}\delta^{\nu]}\hat{C}_{\rho\sigma} \\ - n^{[\mu}\tilde{n}^{\nu]}N^{\rho\sigma}\partial_{\mu}(u\hat{C}_{\rho\sigma}) \\ - N^{\rho\sigma}N_{\rho\sigma}(2\delta^{[\mu}C^{-} - C^{-}\tilde{n}^{[\mu]}n^{\nu]}], \quad (\text{A19}) $$

while its alternative form in Eq. (3.33) in terms of the invariant tensors $C_{AB}$ and $N_{AB}$ maps onto

$$ \Delta_{\mathcal{J}(\text{rad})}^{\mu\nu} = \int \frac{\mathrm{d}u \mathrm{d}^2\Omega}{32\pi G} [4C^{\rho[\mu}N^{\nu]}_{\rho} - 2N^{\rho\sigma}n^{[\mu}\delta^{\nu]}C_{\rho\sigma} \\ - n^{[\mu}\tilde{n}^{\nu]}N^{\rho\sigma}\partial_{\mu}(uC_{\rho\sigma})]. \quad (\text{A20}) $$

Finally, the static term in Eq. (3.25) becomes

$$ \Delta_{\mathcal{J}(\text{stat})}^{\mu\nu} = \int \frac{\mathrm{d}^2\Omega}{4\pi} M^{+} (2\delta^{[\mu}\Delta S - \Delta S \tilde{n}^{[\mu]}n^{\nu]}). \quad (\text{A21}) $$

## APPENDIX B: BOOSTED SCHWARZSCHILD METRIC

In Sec. III A of the main text, we motivated our definition of the mechanical angular momentum $\mathcal{J}$ by appealing to the explicit form of the Bondi metric components $\{M, N_A, C_{AB}\}$ for a boosted Schwarzschild space-time. Those expressions, which are given in Eq. (3.1), are derived in this appendix. We begin in Sec. B 1 by writing down the most general metric for a boosted Schwarzschild black hole in harmonic coordinates. The transformation to Bondi coordinates then proceeds in two stages. It is convenient to first transform the metric into Newman-Unti coordinates [95], which we do in Sec. B 2, before subsequently transforming to Bondi coordinates, which we do in Sec. B 3. (This same set of transformations is discussed in Ref. [67] in the context of the multipolar post-Minkowskian expansion, which can be used to describe, e.g., the space-time around an inspiraling binary.)

### 1. Harmonic coordinates

Let $\tilde{x}^{\mu} \equiv (\tilde{t}, \tilde{x}, \tilde{y}, \tilde{z})$ denote a set of Lorentzian coordinates that satisfy the harmonic condition $\tilde{\partial}_{\mu}(\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}) = 0$, where $\tilde{g}^{\mu\nu}$ are the components of the inverse metric in this coordinate chart, $\tilde{g}$ is the determinant of its inverse, and $\tilde{\partial}_{\mu} \equiv \partial/\partial\tilde{x}^{\mu}$ is the partial derivative with respect to these coordinates. Analogously to how the Bondi coordinates $(u, r, \theta^A)$ have corresponding Lorentzian coordinates $x^{\mu} \equiv (t, x, y, z)$, we can introduce the retarded coordinates $(\tilde{u}, \tilde{r}, \tilde{\theta}^A)$ via

$$ (\tilde{t}, \tilde{x}, \tilde{y}, \tilde{z}) = (\tilde{u} + \tilde{r}, \tilde{r} \sin \tilde{\theta} \cos \tilde{\phi}, \tilde{r} \sin \tilde{\theta} \sin \tilde{\phi}, \tilde{r} \cos \tilde{\theta}). \quad (\text{B1}) $$

This correspondence allows us to define the null vector $\tilde{n}^{\mu}$, the unit radial vector $\hat{r}^{\mu}$, and the two basis vectors $\tilde{e}_{A}^{\mu}$, which

are tangent to the unit 2-sphere, in the same way as how their Bondi counterparts $\{n^{\mu}, \hat{r}^{\mu}, \tilde{e}_{A}^{\mu}\}$ are defined in Appendix A, except that we here use the harmonic coordinates $\tilde{x}^{\mu}$ in place of the Bondi coordinates $x^{\mu}$. Just like their Bondi counterparts, these vectors are to be understood as living in the tangent bundle on $\mathfrak{F}^{+}$, and so their indices are to be raised and lowered with the Minkowski metric $\eta_{\mu\nu}$, not the full metric $\tilde{g}_{\mu\nu}$.

Now consider the space-time around a single Schwarzschild black hole of mass $m$, whose center of energy travels along the worldline $\tilde{x}^{\mu}(\tau) = b^{\mu} + p^{\mu}\tau/m$. (We can set $\eta_{\mu\nu}b^{\mu}p^{\nu} = 0$ without loss of generality.) As discussed in Sec. III A, this worldline is only inferred via extrapolation, since the harmonic coordinates do not actually extend past the event horizon. More rigorously, the displacement vector $b^{\mu}$ and the four-momentum $p^{\mu}$ are defined as the constant vectors whose components parametrize the Poincaré transformation that takes us from the black hole’s rest frame to this generic inertial frame.

By starting with the inverse metric in the rest frame [see, e.g., Eq. (5.172) of Ref. [96]] and then performing this Poincaré transformation, we find that we can write

$$ \tilde{g}^{\mu\nu} = - \left( \frac{\rho + Gm}{\rho - Gm} \right) \frac{p^{\mu} p^{\nu}}{m^2} + \frac{\rho^2}{(\rho + Gm)^2} \Pi^{\mu\nu} - \frac{G^2 m^2}{(\rho + Gm)^2} \hat{\rho}^{\mu} \hat{\rho}^{\nu}, \quad (\text{B2}) $$

where the projection operator $\Pi^{\mu\nu} = \eta^{\mu\nu} + p^{\mu}p^{\nu}/m^2$, the scalar function $\rho = \sqrt{\Pi_{\mu\nu}(\tilde{x}^{\mu} - b^{\mu})(\tilde{x}^{\nu} - b^{\nu})}$, and the spacelike vector $\hat{\rho}^{\mu} = \Pi_{\nu}^{\mu}(\tilde{x}^{\nu} - b^{\nu})/\rho$; the indices on $\Pi^{\mu\nu}$ are lowered with the Minkowski metric. It is worth stressing that the expression in Eq. (B2) is not generally covariant, as it holds only in harmonic coordinates; it is, however, Lorentz covariant. As a sanity check, note that if we undo the translation (i.e., send $\tilde{x}^{\mu} \mapsto \tilde{x}^{\mu} + b^{\mu}$) and then boost ourselves back into the black hole’s rest frame such that $p^{\mu} \to (m, 0)$, then $\rho \to \tilde{r}$ and $\hat{\rho}^{\mu} \to \hat{r}^{\mu}$, and we indeed recover Eq. (5.172) of Ref. [96].

### 2. Newman-Unti coordinates

As an intermediate step to obtaining the metric in Bondi coordinates, we shall first transform Eq. (B2) into Newman-Unti coordinates $(u, R, \theta^A)$. These differ from the Bondi coordinates $(u, r, \theta^A)$ of the main text only by the choice of radial coordinate [97]; the retarded time $u$ and the angular coordinates $\theta^A$ are the same in both cases. To determine the relation between the harmonic coordinates $\tilde{x}^{\mu}$ and these Newman-Unti coordinates $x^a \equiv (u, R, \theta^A)$, we use the standard transformation law

$$ g^{ab}(x) = \frac{\partial x^a}{\partial \tilde{x}^\mu} \frac{\partial x^b}{\partial \tilde{x}^\nu} \tilde{g}^{\mu\nu}(\tilde{x}) \quad (\text{B3}) $$

along with the gauge conditions
$$ \Delta_{\mathcal{J}(\text{rad})}^{\mu\nu} = \int \frac{\mathrm{d}u \mathrm{d}^2\Omega}{32\pi G} [4\hat{C}^{\rho[\mu}N^{\nu]}_{\rho} - 2N^{\rho\sigma}n^{[\mu}\delta^{\nu]}\hat{C}_{\rho\sigma} \\ - n^{[\mu}\tilde{n}^{\nu]}N^{\rho\sigma}\partial_{\mu}(u\hat{C}_{\rho\sigma}) \\ - N^{\rho\sigma}N_{\rho\sigma}(2\delta^{[\mu}C^{-} - C^{-}\tilde{n}^{[\mu]}n^{\nu]}], \quad (A19) $$

while its alternative form in Eq. (3.33) in terms of the invariant tensors $C_{AB}$ and $N_{AB}$ maps onto

$$ \Delta_{\mathcal{J}(\text{rad})}^{\mu\nu} = \int \frac{\mathrm{d}u \mathrm{d}^2\Omega}{32\pi G} [4C^{\rho[\mu}N^{\nu]}_{\rho} - 2N^{\rho\sigma}n^{[\mu}\delta^{\nu]}C_{\rho\sigma} \\ - n^{[\mu}\tilde{n}^{\nu]}N^{\rho\sigma}\partial_{\mu}(uC_{\rho\sigma})]. \quad (A20) $$

Finally, the static term in Eq. (3.25) becomes

$$ \Delta_{\mathcal{J}(\text{stat})}^{\mu\nu} = \int \frac{\mathrm{d}^2\Omega}{4\pi} M^{+} (2\delta^{[\mu} \Delta S - \Delta S \tilde{n}^{[\mu]} n^{\nu]}). \quad (A21) $$

## APPENDIX B: BOOSTED SCHWARZSCHILD METRIC

In Sec. III A of the main text, we motivated our definition of the mechanical angular momentum $\mathcal{J}$ by appealing to the explicit form of the Bondi metric components $\{M, N_A, C_{AB}\}$ for a boosted Schwarzschild space-time. Those expressions, which are given in Eq. (3.1), are derived in this appendix. We begin in Sec. B 1 by writing down the most general metric for a boosted Schwarzschild black hole in harmonic coordinates. The transformation to Bondi coordinates then proceeds in two stages. It is convenient to first transform the metric into Newman-Unti coordinates [95], which we do in Sec. B 2, before subsequently transforming to Bondi coordinates, which we do in Sec. B 3. (This same set of transformations is discussed in Ref. [67] in the context of the multipolar post-Minkowskian expansion, which can be used to describe, e.g., the space-time around an inspiraling binary.)

### 1. Harmonic coordinates

Let $\tilde{x}^{\mu} \equiv (\tilde{t}, \tilde{x}, \tilde{y}, \tilde{z})$ denote a set of Lorentzian coordinates that satisfy the harmonic condition $\tilde{\partial}_{\mu}(\sqrt{-\tilde{g}\tilde{g}^{\mu\nu}}) = 0$, where $\tilde{g}^{\mu\nu}$ are the components of the inverse metric in this coordinate chart, $\tilde{g}$ is the determinant of its inverse, and $\tilde{\partial}_{\mu} \equiv \partial/\partial\tilde{x}^{\mu}$ is the partial derivative with respect to these coordinates. Analogously to how the Bondi coordinates $(u, r, \theta^A)$ have corresponding Lorentzian coordinates $x^{\mu} \equiv (t, x, y, z)$, we can introduce the retarded coordinates $(\tilde{u}, \tilde{r}, \tilde{\theta}^A)$ via

$$ (\tilde{t}, \tilde{x}, \tilde{y}, \tilde{z}) = (\tilde{u} + \tilde{r}, \tilde{r} \sin \tilde{\theta} \cos \tilde{\phi}, \tilde{r} \sin \tilde{\theta} \sin \tilde{\phi}, \tilde{r} \cos \tilde{\theta}). \quad (B1) $$

This correspondence allows us to define the null vector $\tilde{n}^{\mu}$, the unit radial vector $\hat{r}^{\mu}$, and the two basis vectors $\tilde{e}_{A}^{\mu}$, which

are tangent to the unit 2-sphere, in the same way as how their Bondi counterparts $\{n^{\mu}, \hat{r}^{\mu}, \tilde{e}_{A}^{\mu}\}$ are defined in Appendix A, except that we here use the harmonic coordinates $\tilde{x}^{\mu}$ in place of the Bondi coordinates $x^{\mu}$. Just like their Bondi counterparts, these vectors are to be understood as living in the tangent bundle on $\mathcal{F}^{+}$, and so their indices are to be raised and lowered with the Minkowski metric $\eta_{\mu\nu}$, not the full metric $\tilde{g}_{\mu\nu}$.

Now consider the space-time around a single Schwarzschild black hole of mass $m$, whose center of energy travels along the worldline $\tilde{x}^{\mu}(\tau) = b^{\mu} + p^{\mu}\tau/m$. (We can set $\eta_{\mu\nu}b^{\mu}p^{\nu} = 0$ without loss of generality.) As discussed in Sec. III A, this worldline is only inferred via extrapolation, since the harmonic coordinates do not actually extend past the event horizon. More rigorously, the displacement vector $b^{\mu}$ and the four-momentum $p^{\mu}$ are defined as the constant vectors whose components parametrize the Poincaré transformation that takes us from the black hole’s rest frame to this generic inertial frame.

By starting with the inverse metric in the rest frame [see, e.g., Eq. (5.172) of Ref. [96]] and then performing this Poincaré transformation, we find that we can write

$$ \tilde{g}^{\mu\nu} = - \left( \frac{\rho + Gm}{\rho - Gm} \right) \frac{p^{\mu} p^{\nu}}{m^2} + \frac{\rho^2}{(\rho + Gm)^2} \Pi^{\mu\nu} - \frac{G^2 m^2}{(\rho + Gm)^2} \hat{\rho}^{\mu} \hat{\rho}^{\nu}, \quad (B2) $$

where the projection operator $\Pi^{\mu\nu} = \eta^{\mu\nu} + p^{\mu}p^{\nu}/m^2$, the scalar function $\rho = \sqrt{\Pi_{\mu\nu}(\tilde{x}^{\mu} - b^{\mu})(\tilde{x}^{\nu} - b^{\nu})}$, and the spacelike vector $\hat{\rho}^{\mu} = \Pi^{\mu}_{\nu}(\tilde{x}^{\nu} - b^{\nu})/\rho$; the indices on $\Pi^{\mu\nu}$ are lowered with the Minkowski metric. It is worth stressing that the expression in Eq. (B2) is not generally covariant, as it holds only in harmonic coordinates; it is, however, Lorentz covariant. As a sanity check, note that if we undo the translation (i.e., send $\tilde{x}^{\mu} \mapsto \tilde{x}^{\mu} + b^{\mu}$) and then boost ourselves back into the black hole’s rest frame such that $p^{\mu} \to (m, 0)$, then $\rho \to \tilde{r}$ and $\hat{\rho}^{\mu} \to \hat{r}^{\mu}$, and we indeed recover Eq. (5.172) of Ref. [96].

### 2. Newman-Unti coordinates

As an intermediate step to obtaining the metric in Bondi coordinates, we shall first transform Eq. (B2) into Newman-Unti coordinates $(u, R, \theta^A)$. These differ from the Bondi coordinates $(u, r, \theta^A)$ of the main text only by the choice of radial coordinate [97]; the retarded time $u$ and the angular coordinates $\theta^A$ are the same in both cases. To determine the relation between the harmonic coordinates $\tilde{x}^{\mu}$ and these Newman-Unti coordinates $x^a \equiv (u, R, \theta^A)$, we use the standard transformation law

$$ g^{ab}(x) = \frac{\partial x^a}{\partial \tilde{x}^\mu} \frac{\partial x^b}{\partial \tilde{x}^\nu} \tilde{g}^{\mu\nu}(\tilde{x}) \quad (B3) $$

along with the gauge conditions
$$g^{\mu\nu} = g^{\mu A} = 0, \quad g^{\mu R} = -1. \qquad (B4)$$

Sans serif indices {a, b, ...} are used to emphasize that we are working in a nonrectangular coordinate chart. The fact that it is easier to impose the condition $g^{\mu R} = -1$ and then later transform from $R$ to $r$, as opposed to imposing the Bondi gauge condition $\partial_r \det(g_{AB}/r^2) = 0$ directly, is why the Newman-Unti coordinates are a useful intermediate step.

The metric in Eq. (B2) is an exact solution to the vacuum Einstein equations, but to solve for $x^a$ as a function of $\tilde{x}^\mu$, it is helpful to perform a post-Minkowskian expansion. We can do this because we are ultimately interested only in the metric components {$M, N_A, C_{AB}$}, and so it suffices to determine just the first few terms in the $1/r$ expansion of the Bondi metric. Since a boosted Schwarzschild spacetime must admit smooth $u \to 0$ and $b^\mu \to 0$ limits, it follows from dimensional analysis that Newton's constant $G$ only ever appears in the Bondi metric as part of the dimensionless combination $Gm/r$. This means that terms that are of higher order in $G$ are also of higher order in the $1/r$ expansion. Because $M$ and $C_{AB}$ enter the Bondi metric starting at $O(1/r)$ [see Eq. (2.2)], we only need to work up to $O(G)$ to determine these two quantities exactly. However, we will need to work up to $O(G^2)$ to determine $N_A$ exactly, since this quantity first enters the metric starting at $O(1/r^2)$. We therefore write

$$\tilde{g}^{\mu\nu} = \eta^{\mu\nu} - \sum_{n \ge 1} G^n \tilde{h}_n^{\mu\nu}, \qquad (B5)$$

with the first two terms in this expansion given by

$$\tilde{h}_1^{\mu\nu} = \frac{2m}{\rho} \left( \eta^{\mu\nu} + \frac{2p^\mu p^\nu}{m^2} \right), \qquad (B6a)$$

$$\tilde{h}_2^{\mu\nu} = \frac{m^2}{\rho^2} \left( \hat{\rho}^{\mu} \hat{\rho}^{\nu} - 3\eta^{\mu\nu} - \frac{p^{\mu} p^{\nu}}{m^2} \right). \qquad (B6b)$$

The result for $\tilde{h}_1^{\mu\nu}$ can also be found in Refs. [31,54].

We now assume that the relation between $x^a$ and $\tilde{x}^\mu$ can also be expanded perturbatively in powers of $G$. Substituting the ansatz

$$u = \tilde{u} + \sum_{n \ge 1} G^n \zeta_n^u(\tilde{x}), \qquad (B7a)$$

$$R = \tilde{r} + \sum_{n \ge 1} G^n \zeta_n^R(\tilde{x}), \qquad (B7b)$$

$$\theta^A = \tilde{\theta}^A + \sum_{n \ge 1} G^n \zeta_n^A(\tilde{x}) \qquad (B7c)$$

into Eq. (B3) and then imposing the four Newman-Unti gauge conditions in Eq. (B4), we obtain a set of partial

differential equations for $\zeta_n^a$ at each order $n$ in $G$. We have that

$$\frac{\partial}{\partial \tilde{r}} \zeta_1^u = -\frac{1}{2} \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \tilde{n}_\nu, \qquad (B8a)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_1^R = \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \hat{r}_\nu + \hat{r}^\mu \tilde{\partial}_\mu \zeta_1^u, \qquad (B8b)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_1^A = \frac{1}{\tilde{r}} \tilde{e}_\mu^A (\tilde{h}_1^{\mu\nu} \tilde{n}_\nu + \tilde{\partial}^\mu \zeta_1^u) \qquad (B8c)$$

at first order in $G$, while at second order, we find that

$$\frac{\partial}{\partial \tilde{r}} \zeta_2^u = -\frac{1}{2} \tilde{h}_2^{\mu\nu} \tilde{n}_\mu \tilde{n}_\nu + \frac{1}{2} \tilde{\partial}^\mu \zeta_1^u \tilde{\partial}_\mu \zeta_1^u + \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \tilde{\partial}_\nu \zeta_1^u, \qquad (B9a)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_2^R = \tilde{h}_2^{\mu\nu} \tilde{n}_\mu \hat{r}_\nu + \hat{r}^\mu \tilde{\partial}_\mu \zeta_2^u + \tilde{\partial}^\mu \zeta_1^u \tilde{\partial}_\mu \zeta_1^R \\ + \tilde{h}_1^{\mu\nu} \tilde{n}_\nu \tilde{\partial}_\mu \zeta_1^R - \tilde{h}_1^{\mu\nu} \hat{r}_\mu \tilde{\partial}_\nu \zeta_1^u, \qquad (B9b)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_2^A = \frac{1}{\tilde{r}} \tilde{e}_\mu^A (\tilde{h}_2^{\mu\nu} \tilde{n}_\nu + \tilde{\partial}^\mu \zeta_2^u - \tilde{h}_1^{\mu\nu} \tilde{\partial}_\nu \zeta_1^u) \\ + \tilde{\partial}^\mu \zeta_1^u \tilde{\partial}_\mu \zeta_1^A + \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \tilde{\partial}_\nu \zeta_1^A. \qquad (B9c)$$

These two sets of equations can also be found in Ref. [67].

Notice that each line in Eqs. (B8) and (B9) is a linear differential equation whose most general solution must therefore be the sum of a particular integral and a complementary function. These complementary functions, which are all independent of $\tilde{r}$, account for two types of residual gauge freedoms: BMS transformations, and the freedom to shift the origin of the radial coordinate $R$ [67,97]. The latter gauge freedom is present because the Newman-Unti radius $R$ is the affine parameter along null geodesics with $du = d\theta^A = 0$; the Bondi radius $r$ exhibits no such residual freedom because it is not an affine parameter. Given that we eventually want to go into Bondi coordinates, we shall fix this residual gauge freedom in $R$ by imposing the boundary condition given above Eq. (2.4) of Ref. [97], which is tantamount to requiring that $\det(g_{AB}/R^2) = \det \Omega_{AB} + O(1/R^2)$. For the remaining complementary functions associated with the BMS group, we shall set them all to zero for the time being. The arbitrary supertranslation $\beta$, which plays a key role in the discussion of Sec. III, can be added in at the end of the calculation by using the results in Eq. (2.18) of Ref. [57].

The solution to Eq. (B8) subject to these boundary conditions is given by (see also Refs. [31,50,67])

$$\zeta_1^u = 2(\tilde{n} \cdot p) \log \tilde{r} + O(\tilde{r}^{-1}), \qquad (B10a)$$

$$\zeta_1^R = (-\tilde{n} \cdot p) + (\hat{r} \cdot p)(4 - 2 \log \tilde{r}) + O(\tilde{r}^{-1}), \qquad (B10b)$$

$$\tilde{r} \zeta_1^A = 2 \tilde{e}_\mu^A p^\mu (1 - \log \tilde{r}) + O(\tilde{r}^{-1}). \qquad (B10c)$$
$$g^{\mu\nu} = g^{\mu A} = 0, \quad g^{\mu R} = -1. \qquad (B4)$$

Sans serif indices {a, b, ...} are used to emphasize that we are working in a nonrectangular coordinate chart. The fact that it is easier to impose the condition $g^{\mu R} = -1$ and then later transform from $R$ to $r$, as opposed to imposing the Bondi gauge condition $\partial_r \det(g_{AB}/r^2) = 0$ directly, is why the Newman-Unti coordinates are a useful intermediate step.

The metric in Eq. (B2) is an exact solution to the vacuum Einstein equations, but to solve for $x^a$ as a function of $\tilde{x}^\mu$, it is helpful to perform a post-Minkowskian expansion. We can do this because we are ultimately interested only in the metric components {$M, N_A, C_{AB}$}, and so it suffices to determine just the first few terms in the $1/r$ expansion of the Bondi metric. Since a boosted Schwarzschild spacetime must admit smooth $u \to 0$ and $b^\mu \to 0$ limits, it follows from dimensional analysis that Newton's constant $G$ only ever appears in the Bondi metric as part of the dimensionless combination $Gm/r$. This means that terms that are of higher order in $G$ are also of higher order in the $1/r$ expansion. Because $M$ and $C_{AB}$ enter the Bondi metric starting at $O(1/r)$ [see Eq. (2.2)], we only need to work up to $O(G)$ to determine these two quantities exactly. However, we will need to work up to $O(G^2)$ to determine $N_A$ exactly, since this quantity first enters the metric starting at $O(1/r^2)$. We therefore write

$$\tilde{g}^{\mu\nu} = \eta^{\mu\nu} - \sum_{n \ge 1} G^n \tilde{h}_n^{\mu\nu}, \qquad (B5)$$

with the first two terms in this expansion given by

$$\tilde{h}_1^{\mu\nu} = \frac{2m}{\rho} \left( \eta^{\mu\nu} + \frac{2p^{\mu}p^{\nu}}{m^2} \right), \qquad (B6a)$$

$$\tilde{h}_2^{\mu\nu} = \frac{m^2}{\rho^2} \left( \hat{\rho}^{\mu} \hat{\rho}^{\nu} - 3\eta^{\mu\nu} - \frac{p^{\mu} p^{\nu}}{m^2} \right). \qquad (B6b)$$

The result for $\tilde{h}_1^{\mu\nu}$ can also be found in Refs. [31,54].

We now assume that the relation between $x^a$ and $\tilde{x}^\mu$ can also be expanded perturbatively in powers of $G$. Substituting the ansatz

$$u = \tilde{u} + \sum_{n \ge 1} G^n \zeta_n^u(\tilde{x}), \qquad (B7a)$$

$$R = \tilde{r} + \sum_{n \ge 1} G^n \zeta_n^R(\tilde{x}), \qquad (B7b)$$

$$\theta^A = \tilde{\theta}^A + \sum_{n \ge 1} G^n \zeta_n^A(\tilde{x}) \qquad (B7c)$$

into Eq. (B3) and then imposing the four Newman-Unti gauge conditions in Eq. (B4), we obtain a set of partial

differential equations for $\zeta_n^a$ at each order $n$ in $G$. We have that

$$\frac{\partial}{\partial \tilde{r}} \zeta_1^u = -\frac{1}{2} \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \tilde{n}_\nu, \qquad (B8a)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_1^R = \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \hat{r}_\nu + \hat{r}^\mu \tilde{\partial}_\mu \zeta_1^u, \qquad (B8b)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_1^A = \frac{1}{\tilde{r}} \tilde{e}_\mu^A (\tilde{h}_1^{\mu\nu} \tilde{n}_\nu + \tilde{\partial}^\mu \zeta_1^u) \qquad (B8c)$$

at first order in $G$, while at second order, we find that

$$\frac{\partial}{\partial \tilde{r}} \zeta_2^u = -\frac{1}{2} \tilde{h}_2^{\mu\nu} \tilde{n}_\mu \tilde{n}_\nu + \frac{1}{2} \tilde{\partial}^\mu \zeta_1^u \tilde{\partial}_\mu \zeta_1^u + \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \tilde{\partial}_\nu \zeta_1^u, \qquad (B9a)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_2^R = \tilde{h}_2^{\mu\nu} \tilde{n}_\mu \hat{r}_\nu + \hat{r}^\mu \tilde{\partial}_\mu \zeta_2^u + \tilde{\partial}^\mu \zeta_1^u \tilde{\partial}_\mu \zeta_1^R \\ + \tilde{h}_1^{\mu\nu} \tilde{n}_\nu \tilde{\partial}_\mu \zeta_1^R - \tilde{h}_1^{\mu\nu} \hat{r}_\mu \tilde{\partial}_\nu \zeta_1^u, \qquad (B9b)$$

$$\frac{\partial}{\partial \tilde{r}} \zeta_2^A = \frac{1}{\tilde{r}} \tilde{e}_\mu^A (\tilde{h}_2^{\mu\nu} \tilde{n}_\nu + \tilde{\partial}^\mu \zeta_2^u - \tilde{h}_1^{\mu\nu} \tilde{\partial}_\nu \zeta_1^u) \\ + \tilde{\partial}^\mu \zeta_1^u \tilde{\partial}_\mu \zeta_1^A + \tilde{h}_1^{\mu\nu} \tilde{n}_\mu \tilde{\partial}_\nu \zeta_1^A. \qquad (B9c)$$

These two sets of equations can also be found in Ref. [67].

Notice that each line in Eqs. (B8) and (B9) is a linear differential equation whose most general solution must therefore be the sum of a particular integral and a complementary function. These complementary functions, which are all independent of $\tilde{r}$, account for two types of residual gauge freedoms: BMS transformations, and the freedom to shift the origin of the radial coordinate $R$ [67,97]. The latter gauge freedom is present because the Newman-Unti radius $R$ is the affine parameter along null geodesics with $du = d\theta^A = 0$; the Bondi radius $r$ exhibits no such residual freedom because it is not an affine parameter. Given that we eventually want to go into Bondi coordinates, we shall fix this residual gauge freedom in $R$ by imposing the boundary condition given above Eq. (2.4) of Ref. [97], which is tantamount to requiring that $\det(g_{AB}/R^2) = \det \Omega_{AB} + O(1/R^2)$. For the remaining complementary functions associated with the BMS group, we shall set them all to zero for the time being. The arbitrary supertranslation $\beta$, which plays a key role in the discussion of Sec. III, can be added in at the end of the calculation by using the results in Eq. (2.18) of Ref. [57].

The solution to Eq. (B8) subject to these boundary conditions is given by (see also Refs. [31,50,67])

$$\zeta_1^u = 2(\tilde{n} \cdot p) \log \tilde{r} + O(\tilde{r}^{-1}), \qquad (B10a)$$

$$\zeta_1^R = (-\tilde{n} \cdot p) + (\hat{r} \cdot p)(4 - 2 \log \tilde{r}) + O(\tilde{r}^{-1}), \qquad (B10b)$$

$$\tilde{r} \zeta_1^A = 2 \tilde{e}_\mu^A p^\mu (1 - \log \tilde{r}) + O(\tilde{r}^{-1}). \qquad (B10c)$$
Only the leading terms in a $1/\tilde{r}$ expansion are presented for the sake of readability, but it is necessary to determine $\zeta_1^u$, $\zeta_1^R$, and $\tilde{r}\zeta_1^A$ down to $O(\tilde{r}^{-2})$ in order to correctly read off $\{M, N_A, C_{AB}\}$. For the second-order equations in Eq. (B9), we must also determine $\zeta_2^u$, $\zeta_2^R$, and $\tilde{r}\zeta_2^A$ down to $O(\tilde{r}^{-2})$.

Having solved for $\zeta_n^a$, we may now substitute the results back into Eq. (B7) to determine the inverse metric in Newman-Unti coordinates. We note, of course, that because Eq. (B7) gives us $x^a$ as a function of $\tilde{x}^\mu$, and because $\tilde{g}^{\mu\nu}$ is also a function of $\tilde{x}^\mu$, this substitution returns the components of the Newman-Unti metric as functions of the harmonic coordinates $\tilde{x}^\mu$. It then remains to perform the inverse of the transformation in Eq. (B7) to express these components properly as functions of the Newman-Unti coordinates. Since $(u, R, \theta^A)$ are equal to $(\tilde{u}, \tilde{r}, \tilde{\theta}^A)$ at zeroth order in $G$, we will need the inverse of Eq. (B7) only up to first order. Equations (B7a) and (B7b) are easy enough to invert, and we get

$$ \tilde{u} = u - G\zeta_1^u(x) + O(G^2), \quad (B11a) $$

$$ \tilde{r} = R - G\zeta_1^R(x) + O(G^2). \quad (B11b) $$

For the angular coordinates, what we really need to know is how the basis vectors in the two coordinate systems are related, since all of the angular dependence in Eqs. (B2), (B8), and (B9) arise from inner products like $(\tilde{n} \cdot p)$, $(\tilde{r} \cdot b)$, and so on. By first substituting Eq. (B7c) into the definition for $n^\mu$ in Eq. (2.8), we find that $n^\mu = \tilde{n}^\mu + G\zeta_1^A(\tilde{x})\partial\tilde{n}^\mu/\partial\tilde{\theta}^A + O(G^2)$. Inverting this relation then tells us that

$$ \tilde{n}^\mu = n^\mu - G\zeta_1^A(x)e_A^\mu + O(G^2), \quad (B11c) $$

after having used the definition of $e_A^\mu$ in Eq. (A1). Likewise,

$$ \hat{r}^\mu = \hat{r}^\mu - G\zeta_1^A(x)e_A^\mu + O(G^2). \quad (B11d) $$

In a similar way, we substitute Eq. (B7c) into Eq. (A1) and then use the chain rule to eventually find that

$$ \tilde{e}_\mu^A = e_\mu^A - G[\zeta_1^B(x)e_B^\nu \delta_\nu e_\mu^A - \zeta_1^A(x)\hat{r}_\mu] + O(G^2). \quad (B11e) $$

### 3. Bondi coordinates

Given the metric in Newman-Unti coordinates $(u, R, \theta^A)$, we define the Bondi radius $r$ via [97]

$$ r = \left( \frac{\det g_{AB}(u, R, \theta^C)}{\det \Omega_{AB}(\theta^C)} \right)^{1/4} . \quad (B12) $$

This relation could now be used to transform our explicit result for a boosted Schwarzschild metric into Bondi coordinates, but in practice, it is easier to use Eq. (B12)

to transform the general Bondi metric in Eq. (2.1) into Newman-Unti coordinates. Having done so, one finds that for any nonradiative spacetime with $N_{AB} = 0$ [67],

$$ g^{RR} = 1 - \frac{2GM}{R} + O(R^{-2}), \quad (B13a) $$

$$ R^2 g^{AB} = \Omega^{AB} - \frac{1}{R} C^{AB} + O(R^{-2}), \quad (B13b) $$

$$ R g^{RA} = \frac{1}{2R} D_B C^{AB} + \frac{2}{3R^2} G N^A - \frac{1}{2R^2} C^{AB} D^C C_{BC} + O(R^{-3}). \quad (B13c) $$

Comparing these general formulas with our explicit expressions for the boosted Schwarzschild metric allows us to read off the desired result for $\{M, N_A, C_{AB}\}$ as given in Eq. (3.1).

## APPENDIX C: DERIVING THE LOSS OF MECHANICAL ANGULAR MOMENTUM

This appendix is divided into three parts. We begin by listing a number of useful identities in Appendix C 1. The result in Eq. (3.22), which is an integral step in the derivation of the radiated flux $\Delta_J^{(\text{rad})}$, is then proved in Appendix C 2. Finally, in Appendix C 3, we prove the result in Eq. (4.10) for the static term $\Delta_J^{(\text{stat})}$.

### 1. Useful identities

To streamline our discussion, we begin by listing a number of identities that will be essential in later parts of this appendix. For starters, we have that

$$ (D_A D_B - D_B D_A) X_C = 2 \Omega_{A[C} \Omega_{D]B} X^D \quad (C1) $$

for any vector $X^A$. This identity follows from the definition of the Riemann tensor and the fact that $R_{ABCD} = 2\Omega_{A[C} \Omega_{D]B}$ in the case of a round 2-sphere.

Next up are several identities for the conformal Killing vector $Y^A$. First, we note that the contraction of Eq. (2.7) with any symmetric and traceless tensor $X^{AB}$ yields

$$ X^{AB} D_A Y_B = 0. \quad (C2) $$

Second, if we first differentiate Eq. (2.7), we get

$$ \begin{aligned} D_C D_B Y_A &= \Omega_{AB} D_C D_D Y^D - D_C D_A Y_B \\ &= \Omega_{AB} D_C D_D Y^D - D_A D_C Y_B - 2\Omega_{B[C} Y_{A]} . \end{aligned} \quad (C3) $$

where the second line follows from using Eq. (C1). Now, symmetrizing over the indices $B$ and $C$ and using Eq. (2.7), we obtain
Only the leading terms in a $1/\tilde{r}$ expansion are presented for the sake of readability, but it is necessary to determine $\zeta_1^u$, $\zeta_1^R$, and $\tilde{r}\zeta_1^A$ down to $O(\tilde{r}^{-2})$ in order to correctly read off $\{M, N_A, C_{AB}\}$. For the second-order equations in Eq. (B9), we must also determine $\zeta_2^u$, $\zeta_2^R$, and $\tilde{r}\zeta_2^A$ down to $O(\tilde{r}^{-2})$.

Having solved for $\zeta_n^a$, we may now substitute the results back into Eq. (B7) to determine the inverse metric in Newman-Unti coordinates. We note, of course, that because Eq. (B7) gives us $x^a$ as a function of $\tilde{x}^\mu$, and because $\tilde{g}^{\mu\nu}$ is also a function of $\tilde{x}^\mu$, this substitution returns the components of the Newman-Unti metric as functions of the harmonic coordinates $\tilde{x}^\mu$. It then remains to perform the inverse of the transformation in Eq. (B7) to express these components properly as functions of the Newman-Unti coordinates. Since $(u, R, \theta^A)$ are equal to $(\tilde{u}, \tilde{r}, \tilde{\theta}^A)$ at zeroth order in $G$, we will need the inverse of Eq. (B7) only up to first order. Equations (B7a) and (B7b) are easy enough to invert, and we get

$$ \tilde{u} = u - G\zeta_1^u(x) + O(G^2), \quad (B11a) $$

$$ \tilde{r} = R - G\zeta_1^R(x) + O(G^2). \quad (B11b) $$

For the angular coordinates, what we really need to know is how the basis vectors in the two coordinate systems are related, since all of the angular dependence in Eqs. (B2), (B8), and (B9) arise from inner products like $(\tilde{n} \cdot p)$, $(\tilde{r} \cdot b)$, and so on. By first substituting Eq. (B7c) into the definition for $n^\mu$ in Eq. (2.8), we find that $n^\mu = \tilde{n}^\mu + G\zeta_1^A(\tilde{x})\partial\tilde{n}^\mu/\partial\tilde{\theta}^A + O(G^2)$. Inverting this relation then tells us that

$$ \tilde{n}^\mu = n^\mu - G\zeta_1^A(x)e_A^\mu + O(G^2), \quad (B11c) $$

after having used the definition of $e_A^\mu$ in Eq. (A1). Likewise,

$$ \hat{r}^\mu = \hat{r}^\mu - G\zeta_1^A(x)e_A^\mu + O(G^2). \quad (B11d) $$

In a similar way, we substitute Eq. (B7c) into Eq. (A1) and then use the chain rule to eventually find that

$$ \tilde{e}_\mu^A = e_\mu^A - G[\zeta_1^B(x)e_B^\nu\delta_\nu e_\mu^A - \zeta_1^A(x)\hat{r}_\mu] + O(G^2). \quad (B11e) $$

### 3. Bondi coordinates

Given the metric in Newman-Unti coordinates $(u, R, \theta^A)$, we define the Bondi radius $r$ via [97]

$$ r = \left( \frac{\det g_{AB}(u, R, \theta^C)}{\det \Omega_{AB}(\theta^C)} \right)^{1/4} . \quad (B12) $$

This relation could now be used to transform our explicit result for a boosted Schwarzschild metric into Bondi coordinates, but in practice, it is easier to use Eq. (B12)

to transform the general Bondi metric in Eq. (2.1) into Newman-Unti coordinates. Having done so, one finds that for any nonradiative spacetime with $N_{AB} = 0$ [67],

$$ g^{RR} = 1 - \frac{2GM}{R} + O(R^{-2}), \quad (B13a) $$

$$ R^2 g^{AB} = \Omega^{AB} - \frac{1}{R} C^{AB} + O(R^{-2}), \quad (B13b) $$

$$ Rg^{RA} = \frac{1}{2R} D_B C^{AB} + \frac{2}{3R^2} G N^A - \frac{1}{2R^2} C^{AB} D^C C_{BC} + O(R^{-3}). \quad (B13c) $$

Comparing these general formulas with our explicit expressions for the boosted Schwarzschild metric allows us to read off the desired result for $\{M, N_A, C_{AB}\}$ as given in Eq. (3.1).

## APPENDIX C: DERIVING THE LOSS OF MECHANICAL ANGULAR MOMENTUM

This appendix is divided into three parts. We begin by listing a number of useful identities in Appendix C 1. The result in Eq. (3.22), which is an integral step in the derivation of the radiated flux $\Delta_J^{(\text{rad})}$, is then proved in Appendix C 2. Finally, in Appendix C 3, we prove the result in Eq. (4.10) for the static term $\Delta_J^{(\text{stat})}$.

### 1. Useful identities

To streamline our discussion, we begin by listing a number of identities that will be essential in later parts of this appendix. For starters, we have that

$$ (D_A D_B - D_B D_A) X_C = 2 \Omega_{A[C} \Omega_{D]B} X^D \quad (C1) $$

for any vector $X^A$. This identity follows from the definition of the Riemann tensor and the fact that $R_{ABCD} = 2\Omega_{A[C} \Omega_{D]B}$ in the case of a round 2-sphere.

Next up are several identities for the conformal Killing vector $Y^A$. First, we note that the contraction of Eq. (2.7) with any symmetric and traceless tensor $X^{AB}$ yields

$$ X^{AB} D_A Y_B = 0. \quad (C2) $$

Second, if we first differentiate Eq. (2.7), we get

$$ \begin{aligned} D_C D_B Y_A &= \Omega_{AB} D_C D_D Y^D - D_C D_A Y_B \\ &= \Omega_{AB} D_C D_D Y^D - D_A D_C Y_B - 2\Omega_{B[C} Y_{A]} . \end{aligned} \quad (C3) $$

where the second line follows from using Eq. (C1). Now, symmetrizing over the indices $B$ and $C$ and using Eq. (2.7), we obtain
$$D_{(B}D_{C)}Y_A = \Omega_{A(B}D_{C)}D_DY^D - \frac{1}{2}\Omega_{BC}D_A D_D Y^D \\ - \Omega_{BC}Y_A + \Omega_{A(B}Y_{C)}. \qquad (C4)$$

Contracting this with the symmetric and traceless tensor $X^{BC}$ then yields

$$X^{BC}D_B D_C Y^A = X^{AB}D_B D_C Y^C + X^{AB}Y_B. \qquad (C5)$$

A third identity involving three derivatives on $Y^A$ reads [65]

$$X^{AB}D_A D_B D_C Y^C = 0. \qquad (C6)$$

Finally, we will also make use of the fact that [49]

$$Y_{AB}D_C X^{BC} = 2Y^{BC}D_{[A}X_{B]C} \qquad (C7)$$

for any pair of symmetric and traceless tensors, $X^{AB}$ and $Y^{AB}$.

## 2. Radiative term

Here we prove the result for $j(\Delta M, C^-)$ in Eq. (3.22), which we use in the main text to obtain the radiated flux $\Delta_J^{(\text{rad})}$. Our starting point is Eq. (3.20), into which we substitute the expression for $\Delta M$ in Eq. (3.21). Having done so, we see that there are two types of terms in the result: those proportional to $\Delta C_{AB}$, and those proportional to $\Delta \mathcal{E}$. The former read

$$\int \frac{d^2 \Omega}{32 \pi G} \Delta C^{AB} D_A D_B (2 Y^C D_C C^- - C^- D_C Y^C), \qquad (C8)$$

where we have already integrated by parts twice to move the derivatives off $\Delta C_{AB}$. After using the product rule to distribute these derivatives, we find that several terms vanish or cancel one another due to the identities in Eqs. (C2), (C5), and (C6). For the terms that survive, we use Eq. (C1) to show that $D_A D_B D_C C^- = D_C D_A D_B C^- + 2\Omega_{A[B}D_C] C^-$, and thus find that Eq. (C8) is equivalent to

$$\int \frac{d^2 \Omega}{32 \pi G} \Delta C^{AB} (2 Y^C D_C D_A D_B C^- - D_C Y^C D_A D_B C^- + 4 D_A Y^C D_B D_C C^-). \qquad (C9)$$

Now use Eq. (3.8) to rewrite $D_A D_B C^-$ in terms of $C_{AB}^-$. Integrating by parts and using $\Delta C_{AB} = \int_{-\infty}^{\infty} N_{AB} du$ then yields

$$- \int \frac{du d^2 \Omega}{32 \pi G} Y^A \left( N^{BC} D_A C_{BC}^- - 2 D_B (N^{BC} C_{AC}^-) + \frac{1}{2} D_A (N^{BC} C_{BC}^-) \right). \qquad (C10)$$

To complete the derivation, we must add to Eq. (C10) the terms in $j(\Delta M, C^-)$ that are proportional to $\Delta \mathcal{E}$. They read

$$\begin{aligned} & - \int \frac{d^2 \Omega}{8 \pi} (2 Y^A D_A C^- - C^- D_A Y^A) \Delta \mathcal{E} \\ &= - \int \frac{du d^2 \Omega}{32 \pi G} \left( Y^A D_A C^- - \frac{1}{2} C^- D_A Y^A \right) N^2 \\ &= - \int \frac{du d^2 \Omega}{32 \pi G} Y^A \left( N^2 D_A C^- + \frac{1}{2} D_A (N^2 C^-) \right). \end{aligned} \qquad (C11)$$

The first equality follows from using the definition of $\Delta \mathcal{E}$ below Eq. (3.21) and writing $N^2 \equiv N^{AB} N_{AB}$ for brevity, while the second follows from integrating by parts. The sum of Eqs. (C10) and (C11) gives us the desired result in Eq. (3.22).

## 3. Static term

Here we show that the two expressions for $\Delta_J^{(\text{stat})}$ in Eqs. (3.25) and (4.10) are equivalent. We start by inserting the expression for $M^+$ in Eq. (4.7) into the former. After a straightforward integration by parts to move the derivatives off $f_{AB}^+$, we find that $\Delta_J^{(\text{stat})}$ is equal to

$$\int \frac{d^2 \Omega}{8 \pi} \left( \frac{1}{4G} f_{AB}^+ D^A D^B (2 Y^C D_C \Delta S - \Delta S D_C Y^C) + (3 \hat{r} - \hat{i}) \cdot P^+ (2 Y^C D_C \Delta S - \Delta S D_C Y^C) \right). \qquad (C12)$$

Now notice that the first line above is structurally identical to Eq. (C8), and so all of the steps taken in the previous part of this appendix follow through. The only key difference is that $\Delta Z$, which is contained in $\Delta S$, lives in the kernel of the differential operator $(2D_A D_B - \Omega_{AB} D^2)$, and so drops out when we integrate by parts. Ultimately, we arrive at the expression

$$\int \frac{du d^2 \Omega}{32 \pi G} Y^A \left( -f_{+}^{BC} D_A N_{BC} + 2 D_B (f_{+}^{BC} N_{AC}) - \frac{1}{2} D_A (f_{+}^{BC} N_{BC}) \right) + \int \frac{d^2 \Omega}{8 \pi} \left[ 3(\hat{r} \cdot P^+) (2Y^C D_C \Delta S - \Delta S D_C Y^C) + 3(\hat{i} \cdot P^+) (D_A Y^A) \Delta S \right]. \qquad (C13)$$

To get the final term, we have also performed one more integration by parts while using the fact that $(\hat{i} \cdot P^+) = \text{a constant}$. Equation (C13) is still not quite the desired result, however. Observe that we essentially have to exchange $(f_{AB}^+, N_{AB}) \mapsto (-N_{AB}, f_{AB}^+)$ to go from this to Eq. (4.10). We do so by using Eq. (C7) to show that

$$D_B(f_{+}^{BC} N_{AC}) = D_A(N^{BC} f_{BC}^+) - D_B(N^{BC} f_{AC}^+). \qquad (C14)$$

Inserting this into Eq. (C13) achieves the desired outcome.
$$D_{(B}D_{C)}Y_{A} = \Omega_{A(B}D_{C)}D_{D}Y^{D} - \frac{1}{2}\Omega_{BC}D_{A}D_{D}Y^{D} - \Omega_{BC}Y_{A} + \Omega_{A(B}Y_{C)}. \quad (C4)$$

Contracting this with the symmetric and traceless tensor $X^{BC}$ then yields

$$X^{BC}D_B D_C Y^A = X^{AB}D_B D_C Y^C + X^{AB}Y_B. \quad (C5)$$

A third identity involving three derivatives on $Y^A$ reads [65]

$$X^{AB}D_A D_B D_C Y^C = 0. \quad (C6)$$

Finally, we will also make use of the fact that [49]

$$Y_{AB}D_C X^{BC} = 2Y^{BC}D_{[A}X_{B]C} \quad (C7)$$

for any pair of symmetric and traceless tensors, $X^{AB}$ and $Y^{AB}$.

## 2. Radiative term

Here we prove the result for $j(\Delta M, C^{-})$ in Eq. (3.22), which we use in the main text to obtain the radiated flux $\Delta_{J}^{(\text{rad})}$. Our starting point is Eq. (3.20), into which we substitute the expression for $\Delta M$ in Eq. (3.21). Having done so, we see that there are two types of terms in the result: those proportional to $\Delta C_{AB}$, and those proportional to $\Delta \mathcal{E}$. The former read

$$\int \frac{d^2 \Omega}{32\pi G} \Delta C^{AB} D_A D_B (2Y^C D_C C^- - C^- D_C Y^C), \quad (C8)$$

where we have already integrated by parts twice to move the derivatives off $\Delta C_{AB}$. After using the product rule to distribute these derivatives, we find that several terms vanish or cancel one another due to the identities in Eqs. (C2), (C5), and (C6). For the terms that survive, we use Eq. (C1) to show that $D_A D_B D_C C^- = D_C D_A D_B C^- + 2\Omega_{A[B}D_C] C^-$, and thus find that Eq. (C8) is equivalent to

$$\int \frac{d^2 \Omega}{32\pi G} \Delta C^{AB} (2Y^C D_C D_A D_B C^- - D_C Y^C D_A D_B C^- + 4D_A Y^C D_B D_C C^-). \quad (C9)$$

Now use Eq. (3.8) to rewrite $D_A D_B C^-$ in terms of $C_{AB}^-$. Integrating by parts and using $\Delta C_{AB} = \int_{-\infty}^{\infty} N_{AB} du$ then yields

$$-\int \frac{du d^2 \Omega}{32\pi G} Y^A \left( N^{BC} D_A C_{BC}^- - 2D_B (N^{BC} C_{AC}^-) + \frac{1}{2} D_A (N^{BC} C_{BC}^-) \right). \quad (C10)$$

To complete the derivation, we must add to Eq. (C10) the terms in $j(\Delta M, C^{-})$ that are proportional to $\Delta \mathcal{E}$. They read

$$\begin{aligned} & -\int \frac{d^2 \Omega}{8\pi} (2Y^A D_A C^- - C^- D_A Y^A) \Delta \mathcal{E} \\ &= -\int \frac{du d^2 \Omega}{32\pi G} \left(Y^A D_A C^- - \frac{1}{2} C^- D_A Y^A\right) N^2 \\ &= -\int \frac{du d^2 \Omega}{32\pi G} Y^A \left(N^2 D_A C^- + \frac{1}{2} D_A (N^2 C^-)\right). \end{aligned} \quad (C11)$$

The first equality follows from using the definition of $\Delta \mathcal{E}$ below Eq. (3.21) and writing $N^2 \equiv N^{AB} N_{AB}$ for brevity, while the second follows from integrating by parts. The sum of Eqs. (C10) and (C11) gives us the desired result in Eq. (3.22).

## 3. Static term

Here we show that the two expressions for $\Delta_{J}^{(\text{stat})}$ in Eqs. (3.25) and (4.10) are equivalent. We start by inserting the expression for $M^+$ in Eq. (4.7) into the former. After a straightforward integration by parts to move the derivatives off $f_{AB}^+$, we find that $\Delta_{J}^{(\text{stat})}$ is equal to

$$\int \frac{d^2 \Omega}{8\pi} \left( \frac{1}{4G} f_{AB}^+ D^A D^B (2Y^C D_C \Delta S - \Delta S D_C Y^C) + (3\hat{r} - \hat{i}) \cdot P^+ (2Y^C D_C \Delta S - \Delta S D_C Y^C) \right). \quad (C12)$$

Now notice that the first line above is structurally identical to Eq. (C8), and so all of the steps taken in the previous part of this appendix follow through. The only key difference is that $\Delta Z$, which is contained in $\Delta S$, lives in the kernel of the differential operator $(2D_A D_B - \Omega_{AB} D^2)$, and so drops out when we integrate by parts. Ultimately, we arrive at the expression

$$\int \frac{du d^2 \Omega}{32\pi G} Y^A \left( -f_{+}^{BC} D_A N_{BC} + 2D_B (f_{+}^{BC} N_{AC}) - \frac{1}{2} D_A (f_{+}^{BC} N_{BC}) \right) + \int \frac{d^2 \Omega}{8\pi} \left[ 3(\hat{r} \cdot P^+) (2Y^C D_C \Delta S - \Delta S D_C Y^C) + 3(\hat{i} \cdot P^+) (D_A Y^A) \Delta S \right]. \quad (C13)$$

To get the final term, we have also performed one more integration by parts while using the fact that $(\hat{i} \cdot P^+)$ is a constant. Equation (C13) is still not quite the desired result, however. Observe that we essentially have to exchange $(f_{AB}^+, N_{AB}) \mapsto (-N_{AB}, f_{AB}^+)$ to go from this to Eq. (4.10). We do so by using Eq. (C7) to show that

$$D_B(f_{+}^{BC} N_{AC}) = D_A(N^{BC} f_{BC}^+) - D_B(N^{BC} f_{AC}^+). \quad (C14)$$

Inserting this into Eq. (C13) achieves the desired outcome.
APPENDIX D: OTHER DEFINITIONS OF ANGULAR MOMENTUM

Several references [49,51–53] have proposed a super-
translation-invariant definition of the angular momentum
that, in our notation, reads

$$
J_{(\text{inv})}(\sigma) := J(\sigma) - j(M(\sigma), C(\sigma)). \quad (\text{D1})
$$

Both arguments of the function $j(\cdot, \cdot)$, which is as defined
in Eq. (3.4), are to be evaluated on the cut $\sigma$. In the general
case, the shear tensor need not have the form in Eq. (3.2),
but the scalar potential $C \equiv C(u, \theta^A)$ can still be extracted
from $C_{AB}$ via the more general decomposition

$$
C_{AB} = -(2D_A D_B - \Omega_{AB} D^2) C + \epsilon_{C(A} D_{B)} D^C \bar{C}, \quad (\text{D2})
$$

where $\epsilon_{AB}$ is the volume form on $\sigma$ and $\tilde{C} \equiv \tilde{C}(u, \theta^A)$ is a
second scalar potential, sometimes called the magnetic-
parity piece of the shear [57]. (In this terminology, $C$ is the
electric-parity piece of the shear.)

From Eqs. (3.9) and (3.16a), we see that $J_{(\text{inv})}$ coincides
with our definition for the mechanical angular momentum $\mathcal{J}$
on the initial cut $\mathfrak{F}_{\pm}^{\pm}$, but $J_{(\text{inv})}$ and $\mathcal{J}$ are inequivalent on the
final cut $\mathfrak{F}_{+}^{\pm}$ because of the shift in reference point $\Delta Z$ that we
have corrected for. Consequently, If we let $\Delta_{(\text{inv})} := J_{(\text{inv})}^{-} -
J_{(\text{inv})}^{+}$ denote the total loss of $J_{(\text{inv})}$ between $\mathfrak{F}_{\pm}^{\pm}$ and $\mathfrak{F}_{+}^{\pm}$, then
the difference between $\Delta_{(\text{inv})}$ and our result for $\Delta_{\mathcal{J}}$ is

$$
\Delta_{\mathcal{J}} - \Delta_{(\text{inv})} = j(M^{+}, \Delta Z). \quad (\text{D3})
$$

Now appropriating the results of Appendix C 3, we find that
Eq. (D3) is equivalent to

$$
\int \frac{d^2\Omega}{8\pi} \left( 3(\hat{r} \cdot P^+) (2Y^C D_C \Delta Z - \Delta Z D_C Y^C) + 3(\hat{r} \cdot P^+) (D_A Y^A) \Delta Z \right). \quad (\text{D4})
$$

Reasoning similar to that below Eq. (4.10) of Sec. IV B tells us that the space-space components of $\Delta_{(\text{inv})}$ and $\Delta_{\mathcal{J}}$ agree at $O(G^2)$ in the binary's c.m. frame, but otherwise these two quantities are generally inequivalent. Moreover, as discussed in Sec. IVA, the quantity $j(M^+, \Delta Z)$ is not Lorentz covariant because projection operators like $\mathbb{P}_{\ell \le 1}$ and $\mathbb{P}_{\ell \ge 2}$ applied to $\Delta S$ do not commute with Lorentz boosts. Since we can verify explicitly that $\Delta_{\mathcal{J}}$ is Lorentz covariant, Eq. (D3) implies that $\Delta_{(\text{inv})}$ does not transform covariantly under boosts.

A different proposal for a supertranslation-invariant
definition of the angular momentum was put forward by
Javadinezhad and Porrati (JP) in Ref. [50]. Their definition
reads

$$
J_{(\text{JP})}(\sigma) = J(\sigma) - j(M(\sigma), C^{-}) + j(M(\sigma), C). \quad (\text{D5})
$$

{Actually, the definition in Ref. [50] is valid only for $D_A Y^A = 0$; in the Lorentz-tensor representation, this would correspond to defining just the space-space components $J_{(\text{JP})}^{ij}$. To facilitate a clearer comparison with our own result, we have offered a natural extension of their definition that holds in the more general case $D_A Y^A \neq 0$.

The effect of the second and third terms in Eq. (D5) is to
remove the contribution of the initial shear $C^{-}$ from the
Bondi angular momentum $J$ and replace it by the function
$C \equiv C(\theta^A)$, which is chosen in such a way that the space-
space components of $\Delta_{(\text{JP})} := J_{(\text{JP})}^{-} - J_{(\text{JP})}^{+}$ yield the same
result as the Bondi flux $F_J^{ij}$ when the latter is computed in
the intrinsic gauge. It then follows from the discussion
below Eq. (4.10) that $\Delta_{(\text{JP})}^{ij}$ also agrees with the result of $\Delta_{\mathcal{J}}^{ij}$
at $O(G^2)$ in the intrinsic c.m. frame, but generally $\Delta_{(\text{JP})}$ and
$\Delta_{\mathcal{J}}$ are inequivalent.

[1] A. Buonanno, M. Khalil, D. O'Connell, R. Roiban, M. P. Solon, and M. Zeng, Snowmass White Paper: Gravitational waves and scattering amplitudes, in 2022 Snowmass Summer Study (2022), arXiv:2204.05194.

[2] W. D. Goldberger, Effective field theories of gravity and compact binary dynamics: A Snowmass 2021 Whitepaper, in 2022 Snowmass Summer Study (2022), arXiv:2206.14249.

[3] T. Damour, Radiative contribution to classical gravitational scattering at the third order in $G$, Phys. Rev. D **102**, 124008 (2020).

[4] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Gravitational Bremsstrahlung from reverse unitarity, Phys. Rev. Lett. **126**, 201602 (2021).

[5] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Radiative classical gravitational observables at $O(G^3)$ from scattering amplitudes, J. High Energy Phys. 10 (2021) 148.

[6] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, The Eikonal approach to gravitational scattering and radiation at $O(G^3)$, J. High Energy Phys. 07 (2021) 169.

[7] N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté, and P. Vanhove, The amplitude for classical gravitational scattering at third Post-Minkowskian order, J. High Energy Phys. 08 (2021) 172.

[8] M. M. Riva and F. Vernizzi, Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity, J. High Energy Phys. 11 (2021) 228.
## APPENDIX D: OTHER DEFINITIONS OF ANGULAR MOMENTUM

Several references [49,51–53] have proposed a super-translation-invariant definition of the angular momentum that, in our notation, reads

$$J_{(\text{inv})}(\sigma) := J(\sigma) - j(M(\sigma), C(\sigma)). \quad (\text{D1})$$

Both arguments of the function $j(\cdot, \cdot)$, which is as defined in Eq. (3.4), are to be evaluated on the cut $\sigma$. In the general case, the shear tensor need not have the form in Eq. (3.2), but the scalar potential $C \equiv C(u, \theta^A)$ can still be extracted from $C_{AB}$ via the more general decomposition

$$C_{AB} = -(2D_A D_B - \Omega_{AB} D^2)C + \epsilon_{C(A} D_{B)} D^C \bar{C}, \quad (\text{D2})$$

where $\epsilon_{AB}$ is the volume form on $\sigma$ and $\bar{C} \equiv \bar{C}(u, \theta^A)$ is a second scalar potential, sometimes called the magnetic-parity piece of the shear [57]. (In this terminology, $C$ is the electric-parity piece of the shear.)

From Eqs. (3.9) and (3.16a), we see that $J_{(\text{inv})}$ coincides with our definition for the mechanical angular momentum $J$ on the initial cut $\mathfrak{F}_\pm$, but $J_{(\text{inv})}$ and $\mathcal{J}$ are inequivalent on the final cut $\mathfrak{F}_\pm$ because of the shift in reference point $\Delta Z$ that we have corrected for. Consequently, If we let $\Delta_{(\text{inv})} := J_{(\text{inv})}^-$, $J_{(\text{inv})}^+$ denote the total loss of $J_{(\text{inv})}$ between $\mathfrak{F}_\pm$ and $\mathfrak{F}_\pm$, then the difference between $\Delta_{(\text{inv})}$ and our result for $\Delta_{\mathcal{J}}$ is

$$\Delta_{\mathcal{J}} - \Delta_{(\text{inv})} = j(M^{+}, \Delta Z). \quad (\text{D3})$$

Now appropriating the results of Appendix C 3, we find that Eq. (D3) is equivalent to

$$\int \frac{d^2\Omega}{8\pi} (3(\hat{r} \cdot P^+) (2Y^C D_C \Delta Z - \Delta Z D_C Y^C) + 3(\hat{r} \cdot P^+) (D_A Y^A) \Delta Z). \quad (\text{D4})$$

Reasoning similar to that below Eq. (4.10) of Sec. IV B tells us that the space-space components of $\Delta_{(\text{inv})}$ and $\Delta_{\mathcal{J}}$ agree at $O(G^2)$ in the binary’s c.m. frame, but otherwise these two quantities are generally inequivalent. Moreover, as discussed in Sec. IVA, the quantity $j(M^+, \Delta Z)$ is not Lorentz covariant because projection operators like $\mathbb{P}_{\ell \le 1}$ and $\mathbb{P}_{\ell \ge 2}$ applied to $\Delta S$ do not commute with Lorentz boosts. Since we can verify explicitly that $\Delta_{\mathcal{J}}$ is Lorentz covariant, Eq. (D3) implies that $\Delta_{(\text{inv})}$ does not transform covariantly under boosts.

A different proposal for a supertranslation-invariant definition of the angular momentum was put forward by Javadinezhad and Porrati (JP) in Ref. [50]. Their definition reads

$$J_{(\text{JP})}(\sigma) = J(\sigma) - j(M(\sigma), C^{-}) + j(M(\sigma), C). \quad (\text{D5})$$

{Actually, the definition in Ref. [50] is valid only for $D_A Y^A = 0$; in the Lorentz-tensor representation, this would correspond to defining just the space-space components $J_{(\text{JP})}^{ij}$. To facilitate a clearer comparison with our own result, we have offered a natural extension of their definition that holds in the more general case $D_A Y^A \neq 0$.

The effect of the second and third terms in Eq. (D5) is to remove the contribution of the initial shear $C^-$ from the Bondi angular momentum $J$ and replace it by the function $C \equiv C(\theta^A)$, which is chosen in such a way that the space-space components of $\Delta_{(\text{JP})} := J_{(\text{JP})}^-$, $J_{(\text{JP})}^+$ yield the same result as the Bondi flux $F_J^{ij}$ when the latter is computed in the intrinsic gauge. It then follows from the discussion below Eq. (4.10) that $\Delta_{(\text{JP})}^{ij}$ also agrees with the result of $\Delta_{\mathcal{J}}^{ij}$ at $O(G^2)$ in the intrinsic c.m. frame, but generally $\Delta_{(\text{JP})}$ and $\Delta_{\mathcal{J}}$ are inequivalent.

[1] A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M. P. Solon, and M. Zeng, Snowmass White Paper: Gravitational waves and scattering amplitudes, in 2022 Snowmass Summer Study (2022), arXiv:2204.05194.

[2] W. D. Goldberger, Effective field theories of gravity and compact binary dynamics: A Snowmass 2021 Whitepaper, in 2022 Snowmass Summer Study (2022), arXiv:2206.14249.

[3] T. Damour, Radiative contribution to classical gravitational scattering at the third order in $G$, Phys. Rev. D 102, 124008 (2020).

[4] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Gravitational Bremsstrahlung from reverse unitarity, Phys. Rev. Lett. 126, 201602 (2021).

[5] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Radiative classical gravitational observables at $O(G^3)$ from scattering amplitudes, J. High Energy Phys. 10 (2021) 148.

[6] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, The Eikonal approach to gravitational scattering and radiation at $O(G^3)$, J. High Energy Phys. 07 (2021) 169.

[7] N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté, and P. Vanhove, The amplitude for classical gravitational scattering at third Post-Minkowskian order, J. High Energy Phys. 08 (2021) 172.

[8] M. M. Riva and F. Vernizzi, Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity, J. High Energy Phys. 11 (2021) 228.
[9] A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, Classical gravitational scattering from a gauge-invariant double copy, J. High Energy Phys. 10 (2021) 118.

[10] S. Mougiakakos, M. M. Riva, and F. Vernizzi, Gravitational Bremsstrahlung with tidal effects in the post-Minkowskian expansion, Phys. Rev. Lett. 129, 121101 (2022).

[11] G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, All things retarded: Radiation-reaction in worldline quantum field theory, J. High Energy Phys. 10 (2022) 128.

[12] C. Heissenberg, Angular momentum loss due to tidal effects in the post-Minkowskian expansion, Phys. Rev. Lett. 131, 011603 (2023).

[13] M. M. Riva, F. Vernizzi, and L. K. Wong, Gravitational Bremsstrahlung from spinning binaries in the post-Minkowskian expansion, Phys. Rev. D 106, 044013 (2022).

[14] G. U. Jakobsen and G. Mogull, Linear response, Hamiltonian and radiative spinning two-body dynamics, Phys. Rev. D 107, 044033 (2023).

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[16] C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Radiation reaction and gravitational waves at fourth post-Minkowskian order, Phys. Rev. Lett. 130, 101401 (2023).

[17] M. Ruf, Z. Bern, J. Parra-Martinez, R. Roiban, C.-H. Shen, M. Solon, and M. Zeng, Scattering amplitudes and conservative dynamics at the fourth post-Minkowskian order, Proc. Sci. LL2022 (2022) 051.

[18] Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, and F. Teng, Binary dynamics through the fifth power of spin at $O(G^2)$, Phys. Rev. Lett. 130, 201402 (2023).

[19] A. V. Manohar, A. K. Ridgway, and C.-H. Shen, Radiated angular momentum and dissipative effects in classical scattering, Phys. Rev. Lett. 129, 121601 (2022).

[20] P. Di Vecchia, C. Heissenberg, and R. Russo, Angular momentum of zero-frequency gravitons, J. High Energy Phys. 08 (2022) 172.

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[22] G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Classical gravitational Bremsstrahlung from a worldline quantum field theory, Phys. Rev. Lett. 126, 201103 (2021).

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[25] T. Damour, A. Nagar, D. Pollney, and C. Reisswig, Energy versus angular momentum in black hole binaries, Phys. Rev. Lett. 108, 131101 (2012).

[26] L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativity 17, 2 (2014).

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[15] G. Kälin, J. Neef, and R. A. Porto, Radiation-reaction in the effective field theory approach to post-Minkowskian dynamics, J. High Energy Phys. 01 (2023) 140.

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[31] G. Veneziano and G. A. Vilkovisky, Angular momentum loss in gravitational scattering, radiation reaction, and the Bondi gauge ambiguity, Phys. Lett. B 834, 137419 (2022).

[32] Z. Bern, J. P. Gatica, E. Herrmann, A. Luna, and M. Zeng, Scalar QED as a toy model for higher-order effects in classical gravitational scattering, J. High Energy Phys. 08 (2022) 131.

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[37] A. Ashtekar, T. De Lorenzo, and N. Khera, Compact binary coalescences: The subtle issue of angular momentum, Phys. Rev. D 101, 044005 (2020).

[38] O. M. Moreschi, Intrinsic angular momentum in general relativity, Classical Quantum Gravity 21, 5409 (2004).

[39] G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, J. High Energy Phys. 03 (2015) 033.

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