Moment tracking and their coordinate transformations for macroparticles with an application to plasmas around black holes

To study the dynamics of plasma, calculations are performed in 7 dimensional phase space. Because of this, several different summation conventions for indices are needed: summations over just space, just velocity, space and velocity, time and space but not velocity, and space, time and velocity. In order to account for all these different summations, this article uses the convention that Latin indices $a, b, c$ represent summations over $0, \ldots, 6$; Greek indices $\mu, \nu, \rho$ represent summations from $0,1,2,3$; and an underlined index means there is no summation over the 0 index, i.e. $\underline{a} = 1, \ldots, 6$ and $\underline{\mu} = 1,2,3$. The coordinates used are $(t, x^{\underline{\mu}}, u^{\underline{\mu}})$, where t is the global time, $x^{\underline{\mu}}$ is a space coordinate, and $u^{\underline{\mu}}$ is a spatial component of the 4-velocity. The notation $(\xi^a)$ will be used to represent a general coordinate, such that

$$\begin{equation} \xi^0 = t, \quad \xi^{\underline{\mu}} = x^{\underline{\mu}}, \quad \xi^{\underline{\mu} + 3} = u^{\underline{\mu}} \end{equation} \tag{ 1 }$$

i.e. $\xi^{4} = u^1$. Dimensions are used such that $c = G = 1$. The notation $f\,|_p$ will be used to represent the evaluation of a function at a point, i.e. $f\,|_p$ is f evaluated at p. After their introduction, the arguments of functions will not be written, unless needed for emphasis.

To calculate the complex dynamics of plasmas, calculations are performed in 7 dimensional phase space. These dimensions are time, 3 spatial dimensions and 3 proper velocity (4-velocity) dimensions, where t is the global coordinate time. As previously stated, these will be represented in coordinates as $(t, x^{\underline{\mu}}, u^{\underline{\mu}})$. The time component of 4-velocity, $u^0(t,x^{\underline{\mu}},u^{\underline{\mu}})$, is a function in phase space, defined as the positive solution to the quadratic equation

$$\begin{equation} g_{\mu \nu} u^\mu u^\nu = -1 \end{equation} \tag{ 2 }$$

where $g(t, x^{\underline{\mu}})$ is the metric with signature $(-,+,+,+)$. In the specific example of Minkowski spacetime with Cartesian coordinates, u⁰ is the Lorentz factor in special relativity, given by

$$\begin{equation} u^0 = \left(1 + \sum_{\underline{\mu} = 1}^3 u^{\underline{\mu}} u^{\underline{\mu}}\right)^{\frac{1}{2}}. \end{equation} \tag{ 3 }$$

Consider the dynamics of a group of charged particles, each with charge q and mass m with distribution function $f(t, x^{\underline{\mu}}, u^{\underline{\mu}})$. By assuming these particles are collisionless, they can be modelled by the Vlasov equation. To find the Vlasov equation, consider an arbitrary spacetime with metric $g_{\mu \nu}(t, x^{\underline{\mu}})$, Christoffel symbols $\Gamma^{\mu}_{\nu \rho}(t, x^{\underline{\mu}})$ and electromagnetic 2-form $F_{\mu \nu}(t, x^{\underline{\mu}})$. Let C(t) be a world line parameterised by t, the same parameter as the global time ξ⁰. Let $\eta(t)$ be the prolongation of C(t), that is, a curve such that

$$\begin{align} \eta^0\left(t\right) = C^0\left(t\right) = t, \quad \eta^{\underline{\mu}}\left(t\right) = C^{\underline{\mu}}\left(t\right), \quad \eta^{\underline{\mu} + 3}\left(t\right) = \frac{dC^{\underline{\mu}}\left(t\right)}{dt}.\nonumber\\ \end{align} \tag{ 4 }$$

For a given value of t, $\eta(t)$ is a point in 7-dimensional phase space. In terms of coordinate functions $(t, x^{\underline{\mu}}, u^{\underline{\mu}})$, this can be represented as

$$\begin{equation} \begin{aligned} \eta^0\left(t\right) &= C^0\left(t\right) = t, \quad \eta^{\underline{\mu}}\left(t\right) = C^{\underline{\mu}}\left(t\right) = x^{\underline{\mu}}|_{\eta\left(t\right)}, \\ \eta^{\underline{\mu} + 3} &= \frac{dC^{\underline{\mu}}\left(t\right)}{dt} = \left. \frac{u^{\underline{\mu}}}{u^0} \right|_{\eta\left(t\right)}. \end{aligned} \end{equation} \tag{ 5 }$$

In general, $\eta(t)$ is a curve in position-velocity phase space. In this work $\eta(t)$ will be the curve moments are taken around. In particle accelerators, the ideal orbit is a natural choice for $\eta(t)$. This choice does not exist in plasmas; in these cases choices for $\eta(t)$ could be the position of the macroparticle, or a trajectory based on the initial centre of charge of the macroparticle.

The Vlasov vector field, W, has the integral curves η, defined as

$$\begin{equation} W^a|_{\eta\left(t\right)} = \frac{d}{dt} \eta^a\left(t\right). \end{equation} \tag{ 6 }$$

Finding these derivatives gives 7 ODEs,

$$\begin{equation} \begin{aligned} W^0|_{\eta} &= \left. \frac{dC^0\left(t\right)}{dt} \right|_{\eta} = 1, \quad W^{\underline{\mu}}|_{\eta} = \frac{dC^{\underline{\mu}}\left(t\right)}{dt} = \left.\frac{u^{\underline{\mu}}}{u^0} \right|_{\eta}, \\ W^{\underline{\mu} + 3}|_{\eta} &= \frac{d^2 C^{\underline{\mu}}\left(t\right)}{dt^2}. \end{aligned} \end{equation} \tag{ 7 }$$

The acceleration can be found using the pregeodesic equation combined with the (pre-)Lorentz force,

$$\begin{equation} \begin{aligned} &\left. \frac{d^2 C^\mu\left(t\right)}{dt^2} \right|_{\eta} + \left. \left( \Gamma^\mu_{\nu \rho} \frac{dC^\nu \left(t\right)}{dt} \frac{dC^\rho \left(t\right)}{dt} \right) \right|_{\eta} \\ &\quad = \left. \left( \kappa \frac{dC^\mu\left(t\right)}{dt} \right) \right|_{\eta} + \left. \left( \frac{q}{m} \frac{1}{u^0} F_{\nu \rho} g^{\underline{\mu} \nu} \frac{d C^{\rho}\left(t\right)}{dt} \right) \right|_{\eta} \end{aligned} \end{equation} \tag{ 8 }$$

where the $1/u^0$ term in the Lorentz force arises when coordinate time is the parameterisation, rather than proper time, and $\kappa(t)$ arises from the parameterisation. $\kappa(t)$ can be found by noting $C^0 = t$, hence $d^2 C^0/dt^2 = 0$, solving this gives

$$\begin{equation} \kappa = \Gamma^0_{\nu \rho} \frac{dC^\nu \left(t\right)}{dt} \frac{dC^\rho \left(t\right)}{dt} - \frac{q}{m} \frac{1}{u^0} F_{\nu \rho} g^{0 \nu} \frac{d C^{\rho}\left(t\right)}{dt}. \end{equation} \tag{ 9 }$$

From this, solving equation (7) for all η gives

$$\begin{equation} \begin{aligned} W^0 &= 1, W^{\underline{\mu}} = \frac{u^{\underline{\mu}}}{u^0}, \\ W^{\underline{\mu} + 3} &= - \Gamma^{\underline{\mu}}_{\nu \rho} \frac{u^\nu}{u^0} \frac{u^\rho}{u^0} + \frac{q}{m} \frac{1}{u^0} F_{\nu \rho} g^{\underline{\mu} \nu} \frac{u^\rho}{u^0} \\ &\quad+ \Gamma^0_{\nu \rho} \frac{u^\nu}{u^0} \frac{u^\rho}{u^0} \frac{u^{\underline{\mu}}}{u^0} - \frac{q}{m} \frac{1}{u^0} F_{\nu \rho} g^{0 \nu} \frac{u^\rho}{u^0} \frac{u^{\underline{\mu}}}{u^0}. \end{aligned} \end{equation} \tag{ 10 }$$

The Vlasov equation describes the motion of the whole particle distribution,

$$\begin{equation} W^a \partial_{a} f = 0. \end{equation} \tag{ 11 }$$

where $\partial_a = \partial/\partial \xi^a$.

This article will work exclusively in coordinate time frames: these are frames where the parameterisation of η is the same as the time coordinate. In any coordinate time frame, the Vlasov vector field has the form of equation (10). This formulation of the Vlasov vector field is distinct from the $3+1$ formalism used in other approaches for simulating plasma in general relativity [21, 22], although the moment tracking and coordinate transformations presented in this work can be used in both formalisms.

When parameterising with a global time, the dynamics of the moments can be calculated by differentiating (15) with respect to time [1, 2],

$$\begin{equation} \frac{dV^{\underline{a}_{1}, \ldots \underline{a}_{n}}}{dt} = \frac{1}{n!} \frac{d}{dt} \int \left(\xi^{\underline{a}_{1}} - \eta^{\underline{a}_{1}}\right) \ldots \left(\xi^{\underline{a}_{n}} - \eta^{\underline{a}_{n}}\right) f \, d^3x \, d^3u. \end{equation} \tag{ 16 }$$

By using the Vlasov equation and Taylor expanding around η, the dynamics can be found. As an example, the differential equation for the dipole is

$$\begin{equation} \frac{dV^{\underline{a}}}{dt} = \sum_{m = 1}^\infty \frac{1}{m!} V^{\underline{b}_1 \ldots \underline{b}_m} \partial_{\underline{b}_1} \ldots \partial_{\underline{b}_m} W^{\underline{a}} |_{\eta}. \end{equation} \tag{ 17 }$$

This differential equation is an infinite series of higher order moments, and a choice must be made to truncate this series at some point. Truncating at quadrupole (second) order, the differential equations for the dipole and quadrupole are given by

$$\begin{equation} \begin{aligned} \frac{d V^{\underline{a}}}{dt} &= V^{\underline{b}} \partial_{\underline{b}} W^{\underline{a}}|_{\eta} + \frac{1}{2} V^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} W^{\underline{a}}|_{\eta}, \\ \frac{d V^{\underline{a} \underline{b}}}{dt} &= V^{\underline{c} \underline{b}} \partial_{\underline{c}} W^{\underline{a}}|_{\eta} + V^{\underline{a} \underline{c}} \partial_{\underline{c}} W^{\underline{b}}|_{\eta}. \end{aligned} \end{equation} \tag{ 18 }$$

The dipole generated from the quadrupole can clearly be seen. All moment tracking methods suffer from this error—higher order moments generate lower order moments. The importance of the choice of truncation on the accuracy of the model is discussed in section 6.

As moments are defined through integrals, they are highly coordinate dependent objects. For coordinate transformations where the time coordinate is unchanged, this definition of the moments can be used to find the coordinate transformation. Consider a new coordinate system denoted by hatted coordinates $(\hat{t},\hat{\xi}^{\hat{\underline{a}}})$ where $\hat{t} = t$ (since the time coordinate stays the same) and use hatted indices to represent a summation in the new coordinate system. The distribution function f transforms as a scalar density [27] of weight 1, i.e.

$$\begin{equation} \hat{f} = f \frac{d^7 \xi}{d^7 \hat{\xi}} \end{equation} \tag{ 19 }$$

where a hatted variable represents that variable in the new coordinate system, and $d^7 \xi/d^7 \hat{\xi}$ is the Jacobian. By expanding ξ in terms of $\hat{\xi}$ around $\hat{\eta}$,

$$\begin{align} {\xi}^{\underline{a}} = \sum_{n = 0}^{\infty} \frac{1}{n!} \left(\hat{\xi}^{\underline{b}_{1}} - \hat{\eta}^{\underline{b}_{1}} \right) \ldots \left(\hat{\xi}^{\underline{b}_{n}} - \hat{\eta}^{\underline{b}_{n}} \right) \frac{\partial}{\partial \hat{\xi}^{\underline{b}_{1}}} \ldots \frac{\partial}{\partial \hat{\xi}^{\underline{b}_{n}}} {{\xi}}^{\underline{a}}|_{{\eta}}, \nonumber\\ \end{align} \tag{ 20 }$$

and noting $\hat{\eta}(\hat{t}) = \eta(t)$, the coordinate transformations can be found. These are infinite sums in terms of all higher order moments. Truncating the expansion at quadrupole order, the coordinate transformations up to the quadrupole order for the dipole and quadrupole are given by

$$\begin{equation} \begin{aligned} \hat{V}^{\underline{a}} &= V^{\underline{b}} \frac{\partial \hat{\xi}^{\underline{a}}}{\partial \xi^{\underline{b}}}+ \frac{1}{2} V^{\underline{b} \underline{c}} \frac{\partial^2 \hat{\xi}^{\underline{a}}}{\partial \xi^{\underline{b}} \partial \xi^{\underline{c}}} \\ \hat{V}^{\underline{a} \underline{b}} &= V^{\underline{c} \underline{d}} \frac{\partial \hat{\xi}^{\underline{a}}}{\partial \xi^{\underline{c}}} \frac{\partial \hat{\xi}^{\underline{b}}}{\partial \xi^{\underline{d}}}. \end{aligned} \end{equation} \tag{ 21 }$$

For coordinate transformations that mix the space and time coordinates, this method for coordinate transformations will not work. The difficulty in transforming coordinates using this method arises from the important dependence on the choice of time. In order to find the moments, the coordinate systems defines a time slicing, which is a hypersurface of constant t (figure 3). The integrals in the definition for the moments in equation (15) are over these hypersurfaces. When performing a coordinate transformation that mixes the time and space coordinates, then the time slicing will also change. Changing the time slicing will give different moments. The integral representation of moments in equation (15) cannot be transformed between coordinate systems that mix time and space coordinates. In this article we present a new representation of moments using Schwartz distributions. These distributions are introduced in section 4, and the use of this language to find the coordinate transformations for the moments is presented in section 5.

Rather than defining moments explicitly through spatial integrals (equation (15)), an alternative representation of moments is through derivatives of Dirac delta functions. This means that rather than modelling a macroparticle as just a Dirac delta function and using a shape function to deposit charge (the cloud-in-cell approach), the macroparticle is represented as a set of derivatives of Dirac delta functions. By using the language of the Schwartz distributions presented in this section, any coordinate transformation of the moments can be calculated, including those mixing space and time coordinates, which, as previously stated in section 3.2, cannot be done through equation (15). By defining a moment through the method presented below, there is no dependence on the time slicing in the definition of the moment, and as such, the spacetime coordinate transformation can be found. Representing a multipole expansion using derivatives of a Dirac delta function is known as the Ellis representation of a multipole [6, 25, 28]. It may also be known as the Schwartz distribution representation of a multipole. In terms of Dirac delta functions the expansion to the second order (the quadrupole) is

$$\begin{equation}\begin{aligned} \mathscr{J}^a &= \frac{1}{2} \int_{\mathbb{R}} \dot{\eta}^a \, V^{\underline{b} \underline{c}} \left( \partial_{\underline{b}} \partial_{\underline{c}} \delta^{\left(6\right)} \left(\xi - \eta\right) \right) dt \\ &\quad - \int_{\mathbb{R}} \delta^a_{\underline{b}} \, X^{\underline{b} \underline{c}} \left( \partial_{\underline{c}} \delta^{\left(6\right)} \left(\xi-\eta\right) \right) dt \\ &\quad - \int_{\mathbb{R}} \dot{\eta}^a \, V^{\underline{b}} \left( \partial_{\underline{b}} \delta^{\left(6\right)}\left(\xi-\eta\right) \right) dt + \int_{\mathbb{R}} \delta^a_{\underline{b}} \, X^{\underline{b}} \, \delta^{\left(6\right)}\left(\xi-\eta\right) dt \\ &\quad + \int_{\mathbb{R}} \dot{\eta}^a \, q \, \delta^{\left(6\right)} \left(\xi-\eta\right) dt \end{aligned} \end{equation} \tag{ 22 }$$

where

$$\begin{equation} \delta^{\left(6\right)} \left(\xi-\eta\right) = \delta\left(\xi^1 - \eta^1\right) \ldots \delta\left(\xi^6 - \eta^6\right) \end{equation} \tag{ 23 }$$

and

$$\begin{equation} \begin{aligned} V^{\underline{b} \underline{c}}\left(t\right) &= \int \left(\xi^{\underline{b}} - \eta^{\underline{b}}\right) \left(\xi^{\underline{c}} - \eta^{\underline{c}}\right) \, f \, d^6 \xi, \\ V^{\underline{b}}\left(t\right) &= \int \left(\xi^{\underline{b}} - \eta^{\underline{b}}\right) \, f \, d^6 \xi, \quad q = \int f \, d^6 \xi, \\ X^{\underline{a} \underline{b}}\left(t\right) &= V^{\underline{a} \underline{c}} \partial_{\underline{c}} W^{\underline{b}}, \quad X^{\underline{a}}\left(t\right) = V^{\underline{b}} \partial_{\underline{b}} W^{\underline{a}} + \frac{1}{2} V^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} W^{\underline{a}}. \end{aligned} \end{equation} \tag{ 24 }$$

where $d^6 \xi = d^3x d^3u$. Note q has no dependence on time due to the conservation of charge.

The terms $V^{\underline{a}}, X^{\underline{a}}, V^{\underline{a} \underline{b}}$ and $X^{\underline{a} \underline{b}}$ are called the components of a multipole. As this definition involves derivatives of Dirac delta functions, they are defined by their action on test functions $(\phi_0, \ldots, \phi_6)$ which are the components of a covector, and have compact support. Equation (22) acting on φ_a gives

$$\begin{align} \int \mathscr{J}^a \phi_a d^7 \xi &= \frac{1}{2} \int_{\mathbb{R}} \dot{\eta}^a V^{\underline{b} \underline{c}} \left( \partial_{\underline{c}} \partial_{\underline{b}} \phi_a|_{\eta} \right) dt \nonumber\\ &\quad + \int_{\mathbb{R}} X^{\underline{a} \underline{b}} \left( \partial_{\underline{b}} \phi_{\underline{a}}|_{\eta} \right) dt + \int_{\mathbb{R}} \dot{\eta}^a V^{\underline{b}} \left( \partial_{\underline{b}} \phi_a|_{\eta} \right) dt \nonumber\\ &\quad + \int_{\mathbb{R}} X^{\underline{a}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \dot{\eta}^a \, q \, \phi_a|_{\eta} dt. \end{align} \tag{ 25 }$$

The evaluation of the test form at η will not be explicitly written in future representations of $\mathscr{J}^a$, but is implicitly present.

The components in (24) are only over $(1,\ldots,6)$ (there are no terms of the form V⁰ or V⁰¹). By writing a multipole in this way the components of a multipole are unique. The components can be extracted by acting $\mathscr{J}^{a}$ on specific test forms. The details of these test forms and how they isolate the components of $\mathscr{J}^a$ are outlined in appendix A.1. A discussion on why this means they are unique can be found in [25].

To show the components of (25) are the moments of f (equation (24)), the distribution function f is squeezed. This is the process of representing a function using a Dirac delta function and the derivative of Dirac delta functions (figure 4). The moments of f are the coefficients of this expansion. By doing this, it can be shown that the moments of the distribution function f naturally appear in the components of the Ellis representation of a multipole.

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Squeezing a distribution can be shown using the language of differential geometry, the process of which can be found in appendix A.2.

The evolution of the moments defined by (22) are governed by two conditions: the conservation of charge, described by the equation

$$\begin{equation} \partial_a \mathscr{J}^a = 0, \end{equation} \tag{ 26 }$$

and the Vlasov equation,

$$\begin{equation} \mathscr{J}^a W^b - \mathscr{J}^b W^a = 0. \end{equation} \tag{ 27 }$$

The combination of the equations (26) and (27) are called the transport equations, and are used to find the evolution of the moments in the Ellis representation. The transport equations can also be defined by their actions on test forms λ and $\alpha_{a b}$,

$$\begin{equation} \int \mathscr{J}^a \partial_a \lambda dt = 0, \quad \int \mathscr{J}^a W^b \alpha_{a b} dt = 0 \end{equation} \tag{ 28 }$$

where $\alpha_{a b}$ is antisymmetric, and both λ and $\alpha_{a b}$ have compact support. These two conditions give

$$\begin{equation} \begin{aligned} \frac{d V^{\underline{a} \underline{b}}}{dt} &= X^{\underline{a} \underline{b}} + X^{\underline{b} \underline{a}}, \quad \frac{d V^{\underline{a}}}{dt} = X^{\underline{a}}, \quad \frac{dq}{dt} = 0, \\ X^{\underline{a} \underline{b}} &= V^{\underline{a} \underline{c}} \partial_{\underline{c}} W^{\underline{b}}, \quad X^{\underline{a}} = V^{\underline{b}} \partial_{\underline{b}} W^{\underline{a}} + \frac{1}{2} V^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} W^{\underline{a}}. \end{aligned} \end{equation} \tag{ 29 }$$

Equation (18) can be found as a trivial corollary of this, so this method can also be used to find the differential equations for the moments. These differential equations are the same as directly differentiating (15).

Proof. The proof of this is in appendix A.3. □

Whilst both methods achieve the same result, there is a different philosophy to each approach. The method presented in section 3.1 gives an infinite Taylor expansion, and picks the truncation point at the end of the method. Alternatively, the Ellis representation makes the choice of truncation when first performing the multipole expansion to a specific order. This order of expansion is then kept throughout. Whilst this work only carries out this expansion to quadrupole order, it is trivial to extend this result to higher orders.

To validate the model, tests were performed in both Schwarzschild and Kruskal–Szekeres coordinates. The coordinates in Schwarzschild coordinates are denoted with lowercase letters $\xi_{(\text{Sw})}^{\mu} = (t,r,\theta,\phi,u_r,u_{\theta}, u_{\phi})$, with metric

$$\begin{equation} \begin{aligned} g_{00}^{\left(\text{Sw}\right)} &= - \left( 1 - \frac{r_s}{r} \right), \quad g_{11}^{\left(\text{Sw}\right)} = \frac{1}{1 - \frac{r_s}{r}} \\ g_{22}^{\left(\text{Sw}\right)} &= r^2, \quad g_{33}^{\left(\text{Sw}\right)} = r^2 \sin^2\left(\theta\right). \end{aligned} \end{equation} \tag{ 47 }$$

where $g_{\mu \nu}^{(\text{Sw})}$ is the metric in Schwarzschild coordinates.

Coordinates in Kruskal–Szekeres will be denoted with a capital letter, such that the coordinates in Kruskal–Szekeres are $\xi^\mu_{(\text{KS})} = (T,R,\Theta,\Phi,U_{R},U_{\Theta},U_{\Phi})$. The transformation from Schwarzschild coordinates to Kruskal–Szekeres coordinates (shown in figure 2) is given by

$$\begin{align} R &= \sqrt{\frac{r}{r_s} - 1} \,\exp\! \left( \frac{r}{2 r_s} \right) \,\cosh\! \left( \frac{t}{2 r_s} \right), \end{align} \tag{ 48 }$$

$$\begin{align} T &= \sqrt{\frac{r}{r_s} - 1} \,\exp\! \left( \frac{r}{2 r_s} \right) \,\sinh\! \left( \frac{t}{2 r_s} \right), \end{align} \tag{ 49 }$$

$$\begin{align} U^{\underline{\mu}} &= u^\nu \frac{\partial \xi^{\underline{\mu}}_{\left(\text{KS}\right)}}{\partial \xi_{\left(\text{Sw}\right)}^\nu}. \end{align} \tag{ 50 }$$

where the subscript $(\text{KS})$ indicates the coordinates in Kruskal–Szekeres. The metric in Kruskal–Szekeres coordinates is given by

$$\begin{equation} \begin{aligned} g_{00}^{\left(\text{KS}\right)} &= - \frac{4r_s^3}{r} \exp{\left(\frac{-r}{r_s} \right)}, \quad g_{11}^{\left(\text{KS}\right)} = \frac{4r_s^3}{r} \exp{\left(\frac{-r}{r_s} \right)} \\ g_{22}^{\left(\text{KS}\right)} &= r^2, \quad g_{33}^{\left(\text{KS}\right)} = r^2 \sin^2\left(\theta\right) \end{aligned} \end{equation} \tag{ 51 }$$

where r, the radial coordinate in Schwarzschild coordinates, can be found by the inverse transformation

$$\begin{equation} r = r_s \left(1+ W_0 \left(\frac{R^2 - T^2}{e} \right) \right) \end{equation} \tag{ 52 }$$

where W₀ is the principal branch of the Lambert W function.

The numerical testing performed in this article uses moments over a size that may be considered small on astrophysical scales. This is because for numerical simulations in Kruskal–Szekeres, it is impossible to run the model with large moments, or for large amounts of time. As R is updated, the particle will eventually cross the event horizon (the point $T^2 - R^2 = 1$) due to numerical errors from updating the position of the particle. This will happen with any numerical differential equation solver that overestimates the true value. This is another reason it is important to transform coordinates, so a full PIC simulation can be performed in Schwarzschild coordinates, then transformed to Kruskal–Szekeres at the end, avoiding these numerical issues. Although the variation in the metric over the domain represented by the macroparticle and its moments may be small, this does not mean the derivatives of the metric, which the moments couple to, are small.

To test the model, the motion of 200 particles that began normally distributed at $r = 30\,000$ in Schwarzschild coordinates with Schwarzschild radius $r_s = 3000$, and a time step size of $\Delta t = 0.01$ were modelled. These particles were also transformed into Kruskal–Szekeres coordinates. In both spacetimes, the particles were tracked using equation (45) and the moments taken at t = 10, and the moments were taken at t = 0 and the moments tracked using equation (46). When taking the moments, recall that f is a density, so the effect of curved spacetime adding a measure to the integrals is included in the definition of f. To convert f into a scalar field, f can be divided by $|\text{det}(g)|$, where the lack of square root is because the integral is over six dimensional phase space.

By modelling particles at a radius of $10r_s$, the accretion disc of the black hole can be studied. Once the accuracy of the moment tracking model at this distance from the black hole has been established, the accuracy of our model in the extreme environments close to the Schwarzschild radius can be examined.

The results of the tracking are shown in figure 5, with the moments used to calculate confidence ellipses in Schwarzschild spacetime. Figure 5 shows two things: firstly, whilst there is some deviation between the moment tracking and particle tracking ellipses, neither accurately reflect the underlying distribution of particles. This is because the data develops a large skew, as the faster particles orbits get thrown radially outwards. This means the particle distribution is no longer normally distributed, and as such, cannot be modelled accurately with just the first and second order moments. To improve this, higher order moments will also need to be tracked. All models correctly predict the spread in the radial coordinate, such that if the r and u_φ axes were equally scaled, all models would correctly predict a long, horizontally thin ellipse. The second feature is that whilst the data is initially uncoupled, the velocity and position very quickly develop a covariance (a V¹⁶ moment), coupling the u_φ and r motion together. In this particular case, this is an expected result, as particles with a faster magnitude of u_φ than the one required for a circular orbit are spiralling outwards to a higher radius.

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There is a noticeable displacement in the radial coordinate in the Kruskal–Szekeres coordinates tracking (figure 5(b)) from 30 000 to 30 000.08. This is because in Kruskal–Szekeres coordinates, small errors from the numerical integration will compound, and result in the ideal orbit drifting from its expected position. This compounding of errors is because Kruskal–Szekeres coordinates involves exponential functions, so small deviations can result in substantial offsets, which cannot be reduced by decreasing step size. This effect is small, and can be avoided by prescribing η beforehand, if it is known. If η is not known beforehand it is still not a major issue, as the small offset from these numerical factors is likely to be small compared to the other effects within a plasma.

To quantify the error in the model, there are six different errors that are analysed: $\epsilon_{\mathbf{{sp}}-{\mathbf{sm}}}, \epsilon_{\mathbf{{sp}}-{\mathbf{kp}}}, \epsilon_{\mathbf{{sp}}-{\mathbf{km}}}, \epsilon_{\mathbf{{kp}}-{\mathbf{km}}}, \epsilon_{\mathbf{{kp}}-{\mathbf{sp}}},$ and $\epsilon_{\mathbf{{kp}}-{\mathbf{sm}}}$. These errors are defined in table 2. The subscript sp represents that particles were tracked, then the moments taken and the end of the simulation, all in Schwarzschild coordinates. Whilst a subscript km represents moments being tracked in Kruskal–Szekeres coordinates. These subscripts are defined pictorially in figure 6. A hatted V in table 2 represents a coordinate transformation. As an example, $\epsilon_{\mathbf{{sp}}-{\mathbf{sm}}}$ represents the difference between tracking the group of particles and taking their moments at the end (the black ellipse in figure 5(b)), compared to tracking the moments using equation (18) (the blue ellipse in figure 5(b)), all in Schwarzschild coordinates. These errors assess either the error in the moment tracking model, the error in the coordinate transformations, or the combined error of both. These six errors allow both the errors in moment tracking and coordinate transformations to be quantified and measured over time.

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Table 2. The types of error the numerical testing generates. These errors show the accuracy of both the moment tracking model and the coordinate transformations by comparing the results to a particle tracking model. Figure 6 shows these errors diagrammatically.

Error and description of the error
$\epsilon_{\mathbf{{sp}}-{\mathbf{sm}}} = \sqrt{\sum_{\underline{a}} \left|V^{\underline{a}}_{\mathbf{{sp}}} - V^{\underline{a}}_{\mathbf{{sm}}} \right|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}_{\mathbf{{sp}}} - V^{\underline{a} \underline{b}}_{\mathbf{{sm}}} \right|}$The error between tracking particles then taking moments, compared to tracking moments, both in Schwarzschild coordinates.
$\epsilon_{\mathbf{{sp}}-{\mathbf{kp}}} = \sqrt{\sum_{\underline{a}} \left|V^{\underline{a}}_{\mathbf{{sp}}} - \hat{V}^{\underline{a}}_{\mathbf{{kp}}} \right|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}_{\mathbf{{sp}}} - \hat{V}^{\underline{a} \underline{b}}_{\mathbf{{kp}}} \right|}$The error between tracking particles and taking moments in Kruskal–Szekeres coordinates then transforming these into moments in Schwarzschild coordinates, compared to tracking particles and taking moments in Schwarzschild coordinates.
$\epsilon_{\mathbf{{sp}}-{\mathbf{km}}} = \sqrt{\sum_{\underline{a}} \left|V^{\underline{a}}_{\mathbf{{sp}}} - \hat{V}^{\underline{a}}_{\mathbf{{km}}} \right|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}_{\mathbf{{sp}}} - \hat{V}^{\underline{a} \underline{b}}_{\mathbf{{km}}} \right|}$The error between tracking moments in Kruskal–Szekeres coordinates then transforming these into moments in Schwarzschild coordinates, compared to tracking moments in Schwarzschild coordinates.
$\epsilon_{\mathbf{{kp}}-{\mathbf{km}}} = \sqrt{\sum_{\underline{a}} \left|V^{\underline{a}}_{\mathbf{{kp}}} - V^{\underline{a}}_{\mathbf{{km}}} \right|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}_{\mathbf{{kp}}} - V^{\underline{a} \underline{b}}_{\mathbf{{km}}} \right|}$The error between tracking particles then taking moments, compared to tracking moments, both in Kruskal–Szekeres coordinates.
$\epsilon_{\mathbf{{kp}}-{\mathbf{sp}}} = \sqrt{\sum_{\underline{a}} \left|V^{\underline{a}}_{\mathbf{{kp}}} - \hat{V}^{\underline{a}}_{\mathbf{{sp}}} \right|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}_{\mathbf{{kp}}} - \hat{V}^{\underline{a} \underline{b}}_{\mathbf{{sp}}} \right|}$The error between tracking particles and taking moments in Schwarzschild coordinates then transforming these into moments in Kruskal–Szekeres coordinates, compared to tracking particles and taking moments in Kruskal–Szekeres coordinates.
$\epsilon_{\mathbf{{kp}}-{\mathbf{sm}}} = \sqrt{\sum_{\underline{a}} \left|V^{\underline{a}}_{\mathbf{{kp}}} - \hat{V}^{\underline{a}}_{\mathbf{{sm}}} \right|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}_{\mathbf{{kp}}} - \hat{V}^{\underline{a} \underline{b}}_{\mathbf{{sm}}} \right|}$The error between tracking moments in Schwarzschild coordinates then transforming these into moments in Kruskal–Szekeres coordinates, compared to tracking moments in Kruskal–Szekeres coordinates.

The error $\epsilon_{\text{km-sm}}$ could also be calculated, to show the error in the coordinate transformations between two different of moments that were tracked. The origin of this error would not be discernible, as it would be impossible to distinguish between errors caused by moment tracking in either coordinate system, or the error from the coordinate transformation.

To examine the error, another simulation was performed, again around a black hole with Schwarzschild radius $r_s = 3000$, and an ideal circular orbit at $r = 30\,000$ in Schwarzschild coordinates. For these simulations the number of particles was decreased to 20, and the time step used was increased to $\Delta t = 0.1$, with 10⁶ total iterations. This adjustment was made to allow the simulation to run for longer, to obtain better information about the long term behaviour of the model. Note the time step size is in the respective frame, so $t = 10\,000$ in Schwarzschild coordinates corresponds to T = 1200 in Kruskal–Szekeres coordinates. Figure 7 shows the three different total errors as functions of time, where the particles are normally distributed around the ideal orbit with variance 10⁻²⁰ in all dimensions.

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The error in all methods is small, although the error from the coordinate transformations is more substantial than the error from the moment tracking. It is postulated that this increased error is because the higher order moments affect the coordinate transformations twice, once during the coordinate transformation (during equation (35)), and once in the projection (during equation (38)). This suggests that for a moment tracking code that also incorporates a coordinate transformation, a higher order of moments will be needed. The bumps and discontinuities in the results are due to particles passing the macroparticle centre. If a particle is travelling faster than the macroparticle centre, the particle tracking moments (i.e. $V_{\mathbf{{sp}}}$) will decrease, then increase once the particle passes the macroparticle centre. The moment tracking code will not see this behaviour, and will track the moments as always either decreasing or increasing, rather than the true mixture of both. These discontinuities are an intrinsic part of modelling moments. They can be avoided by sorting the particles before the modelling starts, so higher speed particles are ahead of the macroparticle centre, but this is no longer realistic.

To study the magnitude of the error, rather than just the shape, 3 main sources of error can be identified: floating point errors, numerical errors (the error arising from finite step size in numerical integration), and truncation errors (the error arising from truncating the multipole expansion at second order). These errors will dominate in different ways depending on the size of the total initial moments µ, where

$$\begin{equation} \mu = \sum_{\underline{a}} \Big| V^{\underline{a}}|_{t = 0} \Big|^2 + \sum_{\underline{a} \underline{b}} \left| V^{\underline{a} \underline{b}}|_{t = 0} \right|. \end{equation} \tag{ 53 }$$

Floating point errors can arise from a sufficiently small time step and small number of iterations in any numerical differential equation solver, but in the case of this work they also dominate if the moments are very small i.e. $\mu \approx 10^{-15}$. Numerical errors arise from the choice of integrator used, and in the case of forward Euler, are linear in $\Delta t$. The truncation errors arise from only running the moment tracking code up to quadrupole order. There is an infinite expansion of moments, which is truncated to quadrupole order in this paper. Including more moments will decrease the total error. At the quadrupole level, the truncation error is quadratic. This means total error $\epsilon(\mu) \approx \mu^2$. This is verified in figure 8. The error is linear in the low µ regime, where numerical errors dominate, and as µ increases, the total error increases at about $\epsilon(\mu) \approx \mu^{1.7}$. This is a combination of the predicted quadratic increase, and the numerical error. This quadratic behaviour suggests that if the moments are half the size, the total error will be quartered.

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In this section we present the transport equations and coordinate transformations of multipoles in the language of differential geometry and de Rham currents [24, 25]. By using this method there is a more obvious split between the $V^{\underline{a}}$ and $X^{\underline{a} \underline{b}}$ components. This split means it is simpler to isolate each term when doing complicated calculations that mix the two terms, such as during coordinate transformations. It also means the evolution of the moments can be described in a coordinate free way. This is in contrast to working directly from equation (15), which is highly dependent on the coordinate system, in particular the choice of time slicing. The Ellis method also requires a coordinate system to define the action on a test form.

On a manifold M with tangent bundle TM the coordinate time frame hypersurface E is defined such that the Z⁰ component of any vector on M is 1,

$$\begin{equation} E = \left\{Z \in TM \; | \; Z^0 = 1 \right\}. \end{equation} \tag{ 54 }$$

The bundle of p-forms is written $\Lambda^p E$ such that a specific p-form (field) is denoted as $\alpha \in \Gamma \Lambda^p E$. A vector field is denoted as $V \in \Gamma TE$.

A distributional p-form is defined by its action on a test $(7-p)$-form $\phi \in \Gamma \Lambda^{7-p} E$, this is a $(7-p)$-form with compact support. Given $\alpha \in \Gamma \Lambda^p E$ is a smooth p-form, a regular distribution α^D is constructed via

$$\begin{equation} \alpha^D \left[\phi\right] = \int_E \phi \wedge \alpha. \end{equation} \tag{ 55 }$$

The definition of the wedge product, Lie derivatives, internal contractions and exterior derivatives for an arbitrary distribution Ψ are defined as

$$\begin{equation} \begin{aligned} \left( \Psi_1 + \Psi_2 \right)\left[\phi\right] &= \Psi_1\left[\phi\right] + \Psi_2\left[\phi\right], \\ \left(\beta \wedge \Psi\right)\left[\phi\right] &= \Psi\left[\phi \wedge \beta\right], \\ i_V \Psi \left[\phi\right] = -\left(-1\right)^{\text{deg} \phi} \Psi\left[i_V \phi\right], & \ \ d \Psi \left[\phi\right] = -\left(-1\right)^{\text{deg} \phi} \Psi\left[d \phi\right], \\ L_V \Psi \left[\phi\right] &= -\Psi\left[L_V \phi\right] \end{aligned} \end{equation} \tag{ 56 }$$

where $\beta \in \Gamma \Lambda^{q} E$ and $V \in \Gamma T E$.

The order of a p-form distribution is defined as follows. If

$$\begin{align} \Psi\left[\lambda^{k+1}\phi\right] &= 0 \ \text{for all } \phi \in \Gamma \Lambda^q E \ \text{with compact support and } \nonumber\\ &\quad \lambda \in \Gamma \Lambda^0 E\ \text{such that } \eta^*\left(\lambda\right) = 0 \end{align} \tag{ 57 }$$

where $\eta^*$ is the pullback, then the order of Ψ is at most k. Note that a quadrupole (a distribution of order 2) also includes the dipole and the monopole terms. Lie derivatives and exterior derivatives both increase order by one. Internal contractions do not affect the order of a distribution.

Given $\eta: \mathbb{R} \to E$ is a closed embedding parameterised by t, the de Rham pushforward with respect to η of a p-form $\alpha \in \Gamma \Lambda^p \mathbb{R}$ is given by the distribution

$$\begin{equation} \eta_{\varsigma}\left(\alpha\right)\left[\phi\right] = \int_{\mathbb{R}} \eta^*\left(\phi\right) \wedge \alpha. \end{equation} \tag{ 58 }$$

The degree of the distribution $\eta_{\varsigma}(\alpha)$ is $6+ \text{deg}(\alpha)$. Since $\mathbb{R}$ is a curve, the degree of α is either 0 or 1, and the degree of $\eta_{\varsigma}(\alpha)$ is either 6 or 7. An internal contraction decreases the degree by one and an exterior derivative increases the degree by one. Lie derivatives do not affect the degree of a distribution.

A distribution Ψ of degree 6 is a semi-multipole of order at most l if

$$\begin{equation} \begin{aligned} \Psi \left[\lambda^l d\mu\right] &= 0 \quad \text{for all } \lambda, \mu \in \Gamma \Lambda^0 E \\ \text{such that } \eta^*\left(\lambda\right) &= \eta^*\left(\mu\right) = 0. \end{aligned} \end{equation} \tag{ 59 }$$

The integral curves of the Vlasov vector field are tangent to the de Rham pushforward, such that

$$\begin{equation} i_W \eta_{\varsigma}\left(\alpha\right) = \eta_{\varsigma} \left(i_{\frac{d}{dt}} \alpha\right). \end{equation} \tag{ 60 }$$

This work concerns the dynamics of a semi-quadrupole, which is a semi-multipole $\mathscr{J}$ of order 2 and degree 6. In coordinates, this is denoted as

$$\begin{equation} \begin{aligned} \mathscr{J} &= \tfrac{1}{2} L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(V^{\underline{a} \underline{b}}\right) - i_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(X^{\underline{a} \underline{b}} dt\right) \\ &\quad - L_{\underline{a}} \eta_{\varsigma} \left(V^{\underline{a}} \right) + i_{\underline{a}} \eta_{\varsigma} \left(X^{\underline{a}} dt\right) + \eta_{\varsigma}\left(q\right). \end{aligned} \end{equation} \tag{ 61 }$$

where $L_{\underline{a}}$ is the Lie derivative with respect to $\partial_{\underline{a}}$, and $i_{\underline{a}}$ is the internal contraction with respect to $\partial_{\underline{a}}$. Note in this representation of the semi-quadrupole, the $X^{\underline{a} \underline{b}}$ and $V^{\underline{a}}$ term are easier to distinguish by the additional internal contraction, as opposed to the Ellis representation (equation (22)), in which the separation between the terms was less clear.

The de Rham current representation of a multipole can be related to the Ellis representation through the relationship

$$\begin{equation} \mathscr{J} = \mathscr{J}^{a} i_a d^7 \xi, \end{equation} \tag{ 62 }$$

and conversely

$$\begin{equation} \mathscr{J}^a = \mathscr{J} \wedge d\xi^a. \end{equation} \tag{ 63 }$$

In $\mathscr{J}$, there is no term of the form $i_{\underline{a}} L_{\underline{b}} L_{\underline{c}} \eta_{\varsigma} (X^{\underline{a} \underline{b} \underline{c}} dt)$ even though this contains two Lie derivatives. Similarly, there is no $X^{\underline{a} \underline{b} \underline{c}}$ term in the Ellis representation. This term is included if $\mathscr{J}$ is a full quadrupole, as opposed to a semi-quadrupole. The advantage of the coordinate free approach used in this section is that it can be shown that the $X^{\underline{a} \underline{b} \underline{c}}$ term vanishes in a coordinate system adapted to η, and hence $\mathscr{J}$ is a semi-quadrupole in this adapted coordinate system. Since the definition of a semi-multipole is coordinate free, $\mathscr{J}$ is a semi-quadrupole in every coordinate system. The coordinate free semi-multipoles written in this form correspond to the semi-multipoles of [25], and the electric multipoles of [24].

By using the language of differential geometry, a more geometric approach to understanding the origin of the transport equations can be found. To find the dynamics of multipoles defined through distributions, the equivalent of equations (26) and (27) for distributions are used,

$$\begin{equation} d \mathscr{J} = 0, \quad i_W \mathscr{J} = 0, \end{equation} \tag{ 64 }$$

where $W = W^a \partial_a$. The $d\mathscr{J} = 0$ condition corresponds to the conservation of charge, and the $i_W \mathscr{J}$ condition says that the flow lines of $\mathscr{J}$ are the integral curves of the Vlasov field. These equations are the same as equations (26) and (27), i.e. it is equivalent to the Ellis representation.

Proof. Using equation (62)

$$\begin{align} d\mathscr{J} & = d\left(\mathscr{J}^a i_a d^7 \xi \right) = d\left(\mathscr{J}^a \right) \wedge i_a d^7 \xi + \mathscr{J}^a L_a d^7 \xi \nonumber \\ & = \partial_b \mathscr{J}^a d\xi^b i_a d^7 \xi + \left(0\right) = \partial_b \mathscr{J}^a \delta^b_{a} d^7 \xi \nonumber \\ & = \partial_a \mathscr{J}^a d^7 \xi. \end{align} \tag{ 65 }$$

This is zero if and only if $\partial_a \mathscr{J}^a = 0$, so $d\mathscr{J} = 0$ is equivalent to (26). For the $i_{W} \mathscr{J}$ term,

$$\begin{equation} i_W \mathscr{J} = i_W \left(\mathscr{J}^a i_a d^7 \xi \right) = \mathscr{J}^a W^b i_b i_a d^7 \xi. \end{equation} \tag{ 66 }$$

Since $i_b i_a$ is antisymmetric, we can take the symmetric part of $\mathscr{J}^a W^b$,

$$\begin{equation} i_W \mathscr{J} = \left(\mathscr{J}^a W^b + \mathscr{J}^b W^a \right) i_b i_a d^7 \xi. \end{equation} \tag{ 67 }$$

This is zero if and only if $\mathscr{J}^a W^b + \mathscr{J}^b W^a = 0$, so $i_W \mathscr{J} = 0$ is equivalent to equation (27). □

Since the conditions in the geometric approach are the same as the Ellis representation, both languages give the same differential equations for the moments.

Coordinate transformations can also be found using the language of distributions. The coordinate transformations for internal contractions are given by

$$\begin{equation} i_{\underline{a}} \alpha = A_{\underline{a}}^{\hat{b}} i_{\hat{b}} \alpha = i_{\hat{b}} \left(A_{\underline{a}}^{\hat{b}} \alpha\right) \end{equation} \tag{ 68 }$$

and for Lie derivatives the coordinate transformations are

$$\begin{equation} L_{\underline{a}} \alpha = A_{\underline{a}}^{\hat{b}} L_{\hat{b}} \alpha = L_{\hat{b}} \left(A_{\underline{a}}^{\hat{b}} \alpha\right) - \alpha \wedge L_{\hat{b}} A_{\underline{a}}^{\hat{b}}. \end{equation} \tag{ 69 }$$

Note that similarly to the transformation of $\partial_{\underline{a}}$, the indices in the transformed coordinate system run from $(0 \ldots 6)$, whilst the original indices only ran from $(1 \ldots 6)$. From these the coordinate transformations for the semi-quadrupole can be found. This is equivalent to the coordinate transformations found through the Ellis representation (equation (35)).

Proof. The proof of this is in the appendix A.6. □

As before, the transformed quadrupole is still based on the original time slicing. In this case the projections to the new time slicing are based on the internal contraction and Lie derivatives along η. The projections are given by

$$\begin{equation} i_0 = i_{\dot{\eta}} - \dot{\eta}^{\underline{a}} i_{\underline{a}}, \quad L_0 = L_{\dot{\eta}} - \dot{\eta}^{\underline{a}} L_{\underline{a}}. \end{equation} \tag{ 70 }$$

Pushing these through the distributions, (40) and (41) are found. These coordinate transformations are also the same as the ones found through the Ellis representation.

Proof. The proof of this is in appendix A.7. □

This section shows that by acting an Ellis distribution $\mathscr{J}^a$ on specific test forms, the components of a multipole can be extracted. If it is possible to do this, the components of a multipole are unique. The components are isolated using specific tests forms:

$$\begin{equation} \begin{aligned} q\left(t_0\right) &= \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^0_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi\\ V^{\underline{b}}\left(t_0\right) &= \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^0_{a} \; \psi \left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi\\ V^{\underline{b} \underline{c}}\left(t_0\right) &= \lim_{\epsilon \to 0} \; \frac{2}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^0_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \left( \xi^{\underline{c}} - \eta^{\underline{c}} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi\\ X^{\underline{b}}\left(t_0\right) + q\left(t_0\right) \dot{\eta}^{\underline{b}}\left(t_0\right) &= \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^{\underline{b}}_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right)d^7 \xi\\ X^{\underline{b} \underline{c}} \left(t_0\right) + V^{\underline{b}} \left(t_0\right) \dot{\eta}^{\underline{c}}\left(t_0\right) &= \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^{\underline{c}}_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \; \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi \end{aligned} \end{equation} \tag{ A1 }$$

where $\psi: \mathbb{R} \to \mathbb{R}$ is a test function such that $\psi_1 (0) = 1$, it is flat about zero and $\int \psi_1 (t) dt = 1$, t₀ is the point at which the moments are evaluated, and E, the coordinate time frame hypersurface, is defined by equation (54) in section 7.

Proof. Only the $V^{\underline{b} \underline{c}}$ term and $X^{\underline{b} \underline{c}} + V^{\underline{b}} \dot{\eta}^{\underline{c}}$ term will be shown as the other terms follow trivially.

Consider $\mathscr{J}^a$ acting on the $V^{\underline{b} \underline{c}}$ equation of (A1). The only non-zero derivatives are the $\xi^{\underline{b}}$ and $\xi^{\underline{c}}$ terms. There are three possibilities, each $\xi^{\underline{b}}$ or $\xi^{\underline{c}}$ can either not be differentiated, differentiated once, or differentiated twice. If it is not differentiated, then the evaluation at η gives $(\xi^{\underline{b}} - \eta^{\underline{b}})|_{\eta} = \eta^{\underline{b}} - \eta^{\underline{b}} = 0$. If it is differentiated exactly once, then $\partial_{\underline{a}} (\xi^{\underline{b}} - \eta^{\underline{b}}) = \delta^{\underline{b}}_{\underline{a}}$, a Kronecker delta. If this is differentiated twice, then the derivative of a Kronecker delta will vanish. Thus the only non zero term when acting $\mathscr{J}^a$ on the $V^{\underline{b} \underline{c}}$ equation of (A1) is the term where the number of derivatives matches the number of $\xi^{\underline{b}}$ terms. In this case this happens when there are exactly two partial derivatives. This gives

$$\begin{equation} \lim_{\epsilon \to 0} \; \frac{2}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^0_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \left( \xi^{\underline{c}} - \eta^{\underline{c}} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi = \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{\mathbb{R}} \dot{\eta}^0 V^{\underline{b} \underline{c}} \psi\left(\frac{t-t_0}{\epsilon} \right) \prod_{i = 1}^6 \psi\left(\xi^i|_{\eta} - \eta^i\right) dt. \end{equation} \tag{ A2 }$$

Noting $\xi^{a}|_{\eta} = \eta^a$, $\dot{\eta}^0 = 1$, $V^{\underline{b} \underline{c}} = V^{\underline{b} \underline{c}} (t)$ and introducing the substitution $t = t_0 + \epsilon t^{^{\prime}}$ gives

$$\begin{equation} \lim_{\epsilon \to 0} \; \frac{2}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^0_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \left( \xi^{\underline{c}} - \eta^{\underline{c}} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi = \lim_{\epsilon \to 0} \int_{\mathbb{R}} V^{\underline{b} \underline{c}}\left(t_0 + \epsilon t^{^{\prime}}\right) \psi\left(t^{^{\prime}} \right) \left(\psi\left(0\right)\right)^6 dt^{^{\prime}}. \end{equation} \tag{ A3 }$$

Integrating and taking the limit, noting $\psi(0) = 1$ gives

$$\begin{equation} \lim_{\epsilon \to 0} \; \frac{2}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^0_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \left( \xi^{\underline{c}} - \eta^{\underline{c}} \right) \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi = V^{\underline{b} \underline{c}}\left(t_0\right) \end{equation} \tag{ A4 }$$

as required. In the extension to a higher order multipole, this still works as the $\delta^0_a$ term isolates only the $V^{\underline{a} \underline{b}}$ term.

To isolate the $X^{\underline{b} \underline{c}} + V^{\underline{b}} \dot{\eta}^{\underline{c}}$ term of $\mathscr{J}^a$, consider $\mathscr{J}^a$ acting on the $X^{\underline{b} \underline{c}} + V^{\underline{b}} \dot{\eta}^{\underline{c}}$ equation of (A1),

$$\begin{equation} \begin{aligned} \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^{\underline{c}}_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \; \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi &= \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{\mathbb{R}} \dot{\eta}^{\underline{a}} V^{\underline{b}} \psi\left(\frac{t-t_0}{\epsilon} \right) \prod_{i = 1}^6 \psi\left(\xi^i|_{\eta} - \eta^i\right) dt \\ &\quad + \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \int_{\mathbb{R}} X^{\underline{a} \underline{b}} \psi\left(\frac{t-t_0}{\epsilon} \right) \prod_{i = 1}^6 \psi\left(\xi^i|_{\eta} - \eta^i\right) dt. \end{aligned} \end{equation} \tag{ A5 }$$

Repeating the previous process gives

$$\begin{equation} \lim_{\epsilon \to 0} \; \frac{1}{\epsilon} \; \int_{E} \mathscr{J}^a \; \delta^{\underline{c}}_{a} \; \psi\left(\frac{t-t_0}{\epsilon} \right) \left( \xi^{\underline{b}} - \eta^{\underline{b}} \right) \; \prod_{i = 1}^6 \psi \left( \xi^i - \eta^i \right) d^7 \xi = X^{\underline{b} \underline{c}} \left(t_0\right) + V^{\underline{b}} \left(t_0\right) \dot{\eta}^{\underline{c}}\left(t_0\right). \end{equation} \tag{ A6 }$$

$X^{\underline{a} \underline{b}}$ can be isolated by finding $V^{\underline{a}}$ using the appropriate test form. □

Since the components $V^{\underline{a}}, V^{\underline{a} \underline{b}}, X^{\underline{a}}$ and $X^{\underline{a} \underline{b}}$ can all be extracted using test forms, the components of $\mathscr{J}^a$ are unique.

This section shows that the components of the distribution $\mathscr{J}^a$ (equation (22)) are closely related to the moments of f, defined by equation (24). Consider a smooth 6-form that describes the flow of particles in a collisionless plasma,

$$\begin{equation} \theta = f \, i_W \, d^7 \xi = f \, W^a i_a \, d^7 \xi = f d \xi^{1 \ldots 6} - f \, W^{\underline{a}} dt \wedge i_{\underline{a}} d \xi^{1 \ldots 6} \end{equation} \tag{ A7 }$$

where $\wedge$ is the wedge product, i_W is an internal contraction with respect to W, $i_{\underline{a}}$ is an internal contraction with respect to $\partial_{\underline{a}}$,

$$\begin{equation} d \xi^{1 \ldots 6} = d\xi^1 \wedge d\xi^2 \wedge d\xi^3 \wedge d\xi^4 \wedge d\xi^5 \wedge d\xi^6 \end{equation} \tag{ A8 }$$

and f is a solution to the Vlasov equation. This 6-form obeys the transport equations $d\theta = 0$, $i_W \theta = 0$.

The one parameter family of smooth 6-forms θ_ɛ is defined as

$$\begin{equation}{\theta _{\varepsilon {|_{t,{\mathbf{\xi }}}}}} = {\left. {\frac{1}{{{\varepsilon ^6}}}{\mkern 1mu} f\left( {t,\frac{{{\mathbf{\xi }} - \eta }}{\varepsilon }} \right)d{\xi ^{1 \ldots 6}}} \right|_{\left( {t,{\mathbf{\xi }}} \right)}}{\textrm{ }} - {\left. {\frac{1}{{{\varepsilon ^6}}}{\mkern 1mu} \left( {f\left( {t,{\mathbf{\xi }} - \eta \varepsilon } \right){\mkern 1mu} {W^{}}} \right){\mkern 1mu} dt \wedge {i_{}}d{\xi ^{1 \ldots 6}}} \right|_{\left( {t,{\mathbf{\xi }}} \right)}} \end{equation} \tag{ A9 }$$

where ξ refers to the combination of all spatial coordinates.

By expanding θ_ɛ about ε = 0,

$$\begin{equation} \phi \wedge \theta_{\varepsilon} = \mathcal{\bar{J}}^a \phi_a + \mathcal{O}\left(\varepsilon^3\right) \end{equation} \tag{ A10 }$$

where φ is a test form (a form with compact support) $\phi_a d\xi^a$, and

$$\begin{equation}\begin{aligned} \mathcal{\bar{J}}^a &= \frac{1}{2} \int_{\mathbb{R}} \dot{\eta}^a \, \bar{V}^{\underline{b} \underline{c}} \left( \partial_{\underline{b}} \partial_{\underline{c}} \delta^{\left(6\right)} \left(\xi - \eta\right) \right) dt - \int_{\mathbb{R}} \delta^a_{\underline{b}} \, \bar{X}^{\underline{b} \underline{c}} \left( \partial_{\underline{c}} \delta^{\left(6\right)} \left(\xi-\eta\right) \right) dt - \int_{\mathbb{R}} \dot{\eta}^a \, \bar{V}^{\underline{b}} \left( \partial_{\underline{b}} \delta^{\left(6\right)}\left(\xi-\eta\right) \right) dt \\ &\quad + \int_{\mathbb{R}} \delta^a_{\underline{b}} \, \bar{X}^{\underline{b}} \, \delta^{\left(6\right)}\left(\xi-\eta\right) dt + \int_{\mathbb{R}} \dot{\eta}^a \, q \, \delta^{\left(6\right)} \left(\xi-\eta\right) dt \end{aligned} \end{equation} \tag{ A11 }$$

where

$$\begin{equation} \begin{aligned} q = \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) d\bar{\xi}^{1 \ldots 6}, \quad \bar{V}^{\underline{a}} &= \varepsilon \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \left( \bar{\xi}^{\underline{a}} - \eta^{\underline{a}} \right) d\bar{\xi}^{1 \ldots 6}, \quad \bar{V}^{\underline{a} \underline{b}} = \varepsilon^2 \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \left( \bar{\xi}^{\underline{a}} - \eta^{\underline{a}} \right) \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) d\bar{\xi}^{1 \ldots 6}, \\ \bar{X}^{\underline{a}} &= \bar{V}^{\underline{b}} \left(\partial_{{\underline{b}}} W^{\underline{a}}\right)|_{\eta} + \frac{1}{2} \bar{V}^{\underline{b} \underline{c}} \left(\partial_{{\underline{b}}} \partial_{{\underline{c}}} W^{\underline{a}}\right)|_{\eta}, \quad \bar{X}^{\underline{a} \underline{b}} = \bar{V}^{\underline{b} \underline{c}} \left(\partial_{{\underline{c}}} W^{\underline{a}}\right)|_{\eta}. \end{aligned} \end{equation} \tag{ A12 }$$

where Σ is the spatial part of E. Note $\bar{X}^{\underline{a}}$ is inhomogeneous in ɛ, since $V^{\underline{a}}$ contains an ɛ term whilst $V^{\underline{a} \underline{b}}$ contains an ɛ² term.

Proof. Begin by wedging θ_ɛ against a test form $\phi = \phi_a d\xi^a$,

$$\begin{align} \phi \wedge \theta_{\varepsilon}|_{\left(t, {\boldsymbol{\xi}}\right)} & = \left. \frac{1}{\varepsilon^6} f \left(t, \frac{{\boldsymbol{\xi}} - \eta}{\varepsilon} \right) \phi_0 dt \wedge d \xi^{1 \ldots 6} \right|_{\left(t,{\boldsymbol{\xi}}\right)} - \left. \frac{1}{\varepsilon^6} f \left(t, \frac{{\boldsymbol{\xi}} - \eta}{\varepsilon} \right) W^{\underline{a}} \phi_{\underline{b}} d\xi^{\underline{b}} \wedge dt \wedge i_{\underline{a}} d \xi^{1 \ldots 6} \right|_{\left(t,{\boldsymbol{\xi}}\right)} \end{align} \tag{ A13 }$$

$$\begin{align} & = \left. \frac{1}{\varepsilon^6} f \left(t, \frac{{\boldsymbol{\xi}} - \eta}{\varepsilon} \right) \phi_0 dt \wedge d \xi^{1 \ldots 6} \right|_{\left(t,{\boldsymbol{\xi}}\right)} + \left. \frac{1}{\varepsilon^6} f \left(t, \frac{{\boldsymbol{\xi}} - \eta}{\varepsilon} \right) W^{\underline{a}} \phi_{\underline{b}} dt \wedge d\xi^{\underline{b}} \wedge i_{\underline{a}} d \xi^{1 \ldots 6} \right|_{\left(t,{\boldsymbol{\xi}}\right)} \end{align} \tag{ A14 }$$

$$\begin{align} & = \left. \frac{1}{\varepsilon^6} f \left(t, \frac{{\boldsymbol{\xi}} - \eta}{\varepsilon} \right) \phi_0\left( {\boldsymbol{\xi}}\right) dt \wedge d \xi^{1 \ldots 6} \right|_{\left(t,{\boldsymbol{\xi}}\right)} + \left. \frac{1}{\varepsilon^6} f \left(t, \frac{{\boldsymbol{\xi}} - \eta}{\varepsilon} \right) W^{\underline{a}} \phi_{\underline{a}} dt \wedge d \xi^{1 \ldots 6} \right|_{\left(t,{\boldsymbol{\xi}}\right)}. \end{align} \tag{ A15 }$$

Making the substitution ${\boldsymbol{\bar{\xi}}} = ({\boldsymbol{\xi}}-\eta)/\varepsilon$, and making the evaluation at $(t, {\boldsymbol{\xi}})$ implicitly,

$$\begin{align} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} & = \frac{1}{\varepsilon^6} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \phi_0 \left(t, \eta + \varepsilon {\boldsymbol{\bar{\xi}}} \right) \varepsilon^6 dt \wedge d \bar{\xi}^{1 \ldots 6} + \frac{1}{\varepsilon^6} f\left(t, {\boldsymbol{\bar{\xi}}}\right) W^{\underline{a}} \left( \eta + \varepsilon {\boldsymbol{\bar{\xi}}} \right) \phi_{\underline{a}} \left(t, \eta + \varepsilon {\boldsymbol{\bar{\xi}}} \right) \varepsilon^6 dt \wedge d \bar{\xi}^{1 \ldots 6} \end{align} \tag{ A16 }$$

$$\begin{align} & = f\left(t, {\boldsymbol{\bar{\xi}}}\right) \phi_0 \left(t, \eta + \varepsilon {\boldsymbol{\bar{\xi}}} \right) dt \wedge d \bar{\xi}^{1 \ldots 6} + f\left(t, {\boldsymbol{\bar{\xi}}}\right) W^{\underline{a}} \left(t, \eta + \varepsilon {\boldsymbol{\bar{\xi}}} \right) \phi_{\underline{a}} \left(t, \eta + \varepsilon {\boldsymbol{\bar{\xi}}} \right) dt \wedge d \bar{\xi}^{1 \ldots 6}. \end{align} \tag{ A17 }$$

Taylor expanding $\phi_0, \phi_{\underline{a}}$ and $W^{\underline{a}}$ around η, noting there are only spatial derivatives as $\xi^0|_{\eta} - \eta^0 = t-t = 0$,

$$\begin{equation} \begin{aligned} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} &= f\left(t, {\boldsymbol{\bar{\xi}}}\right) \phi_0 |_{\eta} dt \wedge d \bar{\xi}^{1 \ldots 6} + f\left(t, {\boldsymbol{\bar{\xi}}}\right) W^{\underline{a}}|_{\eta} \phi_{\underline{a}}|_{\eta} dt \wedge d \bar{\xi}^{1 \ldots 6} \\ &\quad + f\left(t, {\boldsymbol{\bar{\xi}}}\right) \varepsilon \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) \partial_{{\underline{b}}} \phi_0|_{\eta}dt \wedge d \bar{\xi}^{1 \ldots 6} + f\left(t, {\boldsymbol{\bar{\xi}}}\right) \varepsilon \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) \partial_{{\underline{b}}} \left(\phi_{\underline{a}} W^{\underline{a}}\right) |_{\eta} dt \wedge d \bar{\xi}^{1 \ldots 6} \\ &\quad + \frac{1}{2} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \varepsilon^2 \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) \left( \bar{\xi}^{\underline{c}} - \eta^{\underline{c}} \right) \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_0|_{\eta}dt \wedge d \bar{\xi}^{1 \ldots 6} \\ &\quad + \frac{1}{2} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \varepsilon^2 \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) \left( \bar{\xi}^{\underline{c}} - \eta^{\underline{c}} \right) \partial_{{\underline{b}}} \partial_{{\underline{c}}} \left(\phi_{\underline{a}} W^{\underline{a}}\right)|_{\eta}dt \wedge d \bar{\xi}^{1 \ldots 6} + \mathcal{O}\left(\varepsilon^3\right). \end{aligned} \end{equation} \tag{ A18 }$$

Next, integrate this over E, splitting E into $\mathbb{R} \times \Sigma$. The terms only depending on ${\boldsymbol{\bar{\xi}}}$ can be integrated out,

$$\begin{equation}\begin{aligned} \int_{E} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} &= \int_{\mathbb{R}} \left( \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) d \bar{\xi}^{1 \ldots 6} \right) \phi_0 |_{\eta} dt + \int_{\mathbb{R}} \left( \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) d \bar{\xi}^{1 \ldots 6} \right) W^{\underline{a}}|_{\eta} \phi_{\underline{a}}|_{\eta} dt \\ &\quad + \int_{\mathbb{R}} \left( \varepsilon \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) d \bar{\xi}^{1 \ldots 6} \right) \partial_{{\underline{b}}} \phi_0|_{\eta}dt + \int_{\mathbb{R}} \left( \varepsilon \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) d \bar{\xi}^{1 \ldots 6} \right) \partial_{{\underline{b}}} \left(\phi_{\underline{a}} W^{\underline{a}}\right) |_{\eta} dt \\ &\quad + \frac{1}{2} \int_{\mathbb{R}} \left( \varepsilon^2 \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) \left( \bar{\xi}^{\underline{c}} - \eta^{\underline{c}} \right) d \bar{\xi}^{1 \ldots 6} \right) \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_0|_{\eta}dt \\ &\quad + \frac{1}{2} \int_{\mathbb{R}} \left( \varepsilon^2 \int_{\Sigma} f\left(t, {\boldsymbol{\bar{\xi}}}\right) \left( \bar{\xi}^{\underline{b}} - \eta^{\underline{b}} \right) \left( \bar{\xi}^{\underline{c}} - \eta^{\underline{c}} \right) d \bar{\xi}^{1 \ldots 6} \right) \partial_{{\underline{b}}} \partial_{{\underline{c}}} \left(\phi_{\underline{a}} W^{\underline{a}}\right)|_{\eta}dt + \mathcal{O}\left(\varepsilon^3\right). \end{aligned} \end{equation} \tag{ A19 }$$

Using the definitions of $q, \bar{V}^{\underline{a}}$, and $\bar{V}^{\underline{a} \underline{b}}$ from equation (A12),

$$\begin{equation} \begin{aligned} \int_{E} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} &= \int_{\mathbb{R}} q \phi_0 |_{\eta} dt + \int_{\mathbb{R}} q W^{\underline{a}}|_{\eta} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \partial_{{\underline{b}}} \phi_0|_{\eta}dt + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \partial_{{\underline{b}}} \left(\phi_{\underline{a}} W^{\underline{a}}\right) |_{\eta} dt \\ &\quad + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_0|_{\eta}dt + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \left(\phi_{\underline{a}} W^{\underline{a}}\right)|_{\eta}dt + \mathcal{O}\left(\varepsilon^3\right). \end{aligned} \end{equation} \tag{ A20 }$$

Next, expand the partial derivatives, and note $W^{\underline{a}}|_{\eta} = \dot{\eta}^{\underline{a}}$,

$$\begin{equation} \begin{aligned} \int_{E} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} &= \int_{\mathbb{R}} q \phi_0 |_{\eta} dt + \int_{\mathbb{R}} q \dot{\eta}^{\underline{a}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \partial_{{\underline{b}}} \phi_0|_{\eta} dt \\ &\quad + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \dot{\eta}^{\underline{a}} \partial_{{\underline{b}}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \phi_{\underline{a}}|_{\eta} \left(\partial_{{\underline{b}}} W^{\underline{a}}\right)|_{\eta} dt + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_0|_{\eta}dt \\ &\quad + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \dot{\eta}^{\underline{a}} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \left(\partial_{{\underline{c}}} W^{\underline{a}}\right)|_{\eta} \left(\partial_{{\underline{b}}} \phi_{\underline{a}}\right)|_{\eta} dt + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \phi_{\underline{a}}|_{\eta} \left(\partial_{{\underline{b}}} \partial_{{\underline{c}}} W^{\underline{a}}\right)|_{\eta} dt + \mathcal{O}\left(\varepsilon^3\right). \end{aligned} \end{equation} \tag{ A21 }$$

Inserting $\bar{X}^{\underline{a}}$ and $\bar{X}^{\underline{a} \underline{b}}$ from equation (A12) into this,

$$\begin{equation} \begin{aligned} \int_{E} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} &= \int_{\mathbb{R}} q \phi_0 |_{\eta} dt + \int_{\mathbb{R}} q \dot{\eta}^{\underline{a}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \partial_{{\underline{b}}} \phi_0|_{\eta} dt \\ &\quad + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \dot{\eta}^{\underline{a}} \partial_{{\underline{b}}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{X}^{\underline{a}} \phi_{\underline{a}}|_{\eta} dt + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_0|_{\eta}dt \\ &\quad + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \dot{\eta}^{\underline{a}} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_{\underline{a}}|_{\eta} dt + \frac{1}{2} \int_{\mathbb{R}} \bar{X}^{\underline{a} \underline{c}} \left(\partial_{{\underline{c}}} \phi_{\underline{a}}\right)|_{\eta} dt + \mathcal{O}\left(\varepsilon^3\right). \end{aligned} \end{equation} \tag{ A22 }$$

Recalling $\dot{\eta}^0 = 1$, this can be further simplified,

$$\begin{equation} \begin{aligned} \int_{E} \phi \wedge \theta_{\varepsilon}|_{t, {\boldsymbol{\xi}}} &= \int_{\mathbb{R}} q \phi_0 |_{\eta} dt + \int_{\mathbb{R}} q \dot{\eta}^{\underline{a}} \phi_{\underline{a}}|_{\eta} dt + \int_{\mathbb{R}} \bar{V}^{\underline{b}} \dot{\eta}^{a} \partial_{{\underline{b}}} \phi_a|_{\eta} dt + \int_{\mathbb{R}} \bar{X}^{\underline{a}} \phi_{\underline{a}}|_{\eta} dt \\ &\quad + \frac{1}{2} \int_{\mathbb{R}} \bar{V}^{\underline{b} \underline{c}} \dot{\eta}^{a} \partial_{{\underline{b}}} \partial_{{\underline{c}}} \phi_a|_{\eta} dt + \int_{\mathbb{R}} \bar{X}^{\underline{a} \underline{c}} \left(\partial_{{\underline{c}}} \phi_{\underline{a}}\right)|_{\eta} dt + \mathcal{O}\left(\varepsilon^3\right). \end{aligned} \end{equation} \tag{ A23 }$$

This is the same as $\mathcal{\bar{J}}^a \phi_a$, as required. □

Thus there is a close relationship between the components of a multipole and the moments of f.

In this section, the solutions to the transport equations (26) and (27) will be found. Begin by considering the conservation of charge (equation (26)), and act on a test form λ,

$$\begin{equation} \int_{E} \mathscr{J}^a \partial_a \lambda d^7 \xi = \frac{1}{2} \int_{\mathbb{R}} V^{\underline{a} \underline{b}} \dot{\eta}^c \partial_{\underline{a}} \partial_{\underline{b}} \partial_c \lambda dt + \int_{\mathbb{R}} X^{\underline{a} \underline{b}} \partial_{\underline{b}} \partial_{\underline{a}} \lambda dt + \int_{\mathbb{R}} V^{\underline{a}} \dot{\eta}^b \partial_{\underline{a}} \partial_b \lambda dt + \int_{\mathbb{R}} X^{\underline{a}} \partial_{\underline{a}} \lambda dt + \int_{\mathbb{R}} q \dot{\eta}^{a} \partial_a \lambda dt = 0. \end{equation} \tag{ A24 }$$

From here, note

$$\begin{equation} \dot{\eta}^a \frac{\partial \lambda\left(t\right)}{\partial \xi^a} = \left. \frac{d \lambda}{dt} \right|_{\eta} \end{equation} \tag{ A25 }$$

so integration by parts can be used to pass a derivative onto the $V^{\underline{a}}$ and $V^{\underline{a} \underline{b}}$ terms. This gives

$$\begin{equation} \int_{E} \mathscr{J}^a \partial_a \lambda d^7 \xi = - \frac{1}{2} \int_{\mathbb{R}} \frac{dV^{\underline{a} \underline{b}}}{dt} \partial_{\underline{a}} \partial_{\underline{b}} \lambda dt + \int_{\mathbb{R}} X^{\underline{a} \underline{b}} \partial_{\underline{b}} \partial_{\underline{a}} \lambda dt - \int_{\mathbb{R}} \frac{dV^{\underline{a}}}{dt} \partial_{\underline{a}} \lambda dt + \int_{\mathbb{R}} X^{\underline{a}} \partial_{\underline{a}} \lambda dt + \int_{\mathbb{R}} \frac{dq}{dt} \lambda dt = 0. \end{equation} \tag{ A26 }$$

Collecting terms based on derivatives of λ gives the first line of (29).

To consider the effects of the Vlasov equation (equation (27)), act on $W^b \alpha_{a b}$, and expand the partial derivatives,

$$\begin{equation} \begin{aligned} \int_E \mathscr{J}^a W^b \alpha_{a b} d^7 \xi &= \frac{1}{2} \int_{\mathbb{R}} V^{\underline{c} \underline{b}} \dot{\eta}^a \left(\alpha_{a d} \partial_{\underline{b}} \partial_{\underline{c}} W^d + \partial_{\underline{b}} \alpha_{a d} \partial_{\underline{c}} W^d + \partial_{\underline{c}} \alpha_{a d} \partial_{\underline{b}} W^d + W^d \partial_{\underline{b}} \partial_{\underline{c}} \alpha_{a d} \right) dt \\ &\quad + \int_{\mathbb{R}} X^{\underline{a} \underline{b}} \left( W^c \partial_{\underline{b}} \alpha_{\underline{a} c} + \alpha_{\underline{a} c} \partial_{\underline{b}} W^c \right) dt + \int_{\mathbb{R}} V^{\underline{b}} \dot{\eta}^{a} \left( W^c \partial_{\underline{b}} \alpha_{a c} + \alpha_{a c} \partial_{\underline{b}} W^c \right) dt \\ &\quad + \int_{\mathbb{R}} X^{\underline{a}} W^b \alpha_{\underline{a} b} dt + \int_{\mathbb{R}} q W^b \dot{\eta}^{c} \alpha_{b c} dt. \end{aligned} \end{equation} \tag{ A27 }$$

Next, look at terms of the form

$$\begin{equation} W^a \dot{\eta}^{b} \alpha_{a b}, \end{equation} \tag{ A28 }$$

recalling the implicit evaluation at η, $W|_{\eta} = \dot{\eta}^{a}$. Since $\alpha_{a b}$ is antisymmetric,

$$\begin{equation} W^a \dot{\eta}^{b} \alpha_{a b} = 0. \end{equation} \tag{ A29 }$$

This is true for the derivatives of $\alpha_{a b}$ as well, since the derivatives are also antisymmetric. Rearranging the remaining terms in derivatives of $\alpha_{a b}$ gives

$$\begin{equation}\begin{aligned} \int_E \mathscr{J}^a W^b \alpha_{a b} d^7 \xi &= \int_{\mathbb{R}} \left(V^{\underline{c} \underline{b}} \partial_{\underline{c}} W^{\underline{a}} - X^{\underline{a} \underline{b}} \right) \dot{\eta}^{d} \partial_{\underline{b}} \alpha_{d \underline{a}} dt + \int_{\mathbb{R}} \left( V^{\underline{c} \underline{b}} \partial_{\underline{c}} W^0 \right) \dot{\eta}^{d} \partial_{\underline{b}} \alpha_{d 0} dt + \int_{\mathbb{R}} X^{\underline{a} \underline{b}} \partial_{\underline{b}} W^{d} \alpha_{\underline{a} d} dt\\ &\quad + \int_{\mathbb{R}} \left( V^{\underline{b}} \partial_{\underline{b}} W^{\underline{a}} + \frac{1}{2} V^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} W^{\underline{a}} - X^{\underline{a}} \right) \dot{\eta}^d \alpha_{d \underline{a}} dt +\int_{\mathbb{R}} \left( \frac{1}{2} V^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} W^0 + V^{\underline{b}} \partial_{\underline{b}} W^0 \right) \dot{\eta}^{d} \alpha_{d 0} dt \end{aligned} \end{equation} \tag{ A30 }$$

where the minus sign in the $X^{\underline{a}}$ and $X^{\underline{a} \underline{b}}$ terms comes from flipping the $\alpha_{a \underline{b}}$ indices. Next, note that calculations are in a frame where $W^0 = 1$, so derivatives of W⁰ vanish. Two of the remaining terms give the required differential equations,

$$\begin{align} X^{\underline{a} \underline{b}} & = V^{\underline{b} \underline{c}} \partial_{\underline{c}} W^{\underline{a}}, \end{align} \tag{ A31 }$$

$$\begin{align} X^{\underline{a}} & = V^{\underline{b}} \partial_{\underline{b}} W^{\underline{a}} + \frac{1}{2} V^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} W^{\underline{a}}. \end{align} \tag{ A32 }$$

Lastly, for the remaining integral, the third integral of equation (A30), taking the antisymmetric part of $X^{\underline{a} \underline{b}} \partial_{\underline{b}} W^{\underline{d}}$ gives

$$\begin{align} X^{\underline{a} \underline{b}} \partial_{\underline{b}} W^{\underline{d}} \alpha_{\underline{a} \underline{d}} & = \left(X^{\underline{a} \underline{b}} \partial_{\underline{b}} W^{\underline{d}} - X^{\underline{d} \underline{b}} \partial_{\underline{b}} W^{\underline{a}} \right) \alpha_{\underline{a} \underline{d}} \end{align} \tag{ A33 }$$

$$\begin{align} & = \left(V^{\underline{b} \underline{c}} \partial_{\underline{b}} W^{\underline{a}} \partial_{\underline{c}} W^{\underline{d}} - V^{\underline{c} \underline{b}} \partial_{\underline{c}} W^{\underline{a}} \partial_{\underline{b}} W^{\underline{d}} \right) \alpha_{\underline{a} \underline{d}} \end{align} \tag{ A34 }$$

$$\begin{align} & = 0. \end{align} \tag{ A35 }$$

So the differential equations in the bottom line of (29) uniquely solve the system, giving the equations of motion for the moments through the Ellis representation of multipoles.

In this section the first step of the coordinate transformation for the moments is performed, to obtain non-unique components in terms of $U^a, U^{a b}, Y^{a}$ and $Y^{a b}$. Proceeding term by term using the known transformation rules: For the $V^{a b}$ term,

$$\begin{align} \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\eta}^a \partial_{\underline{b}} \partial_{\underline{c}} \phi_a dt & = \int_{{{\mathbb{R}}}} \hat{V}^{\underline{b} \underline{c}} \dot{\hat{\eta}}^a \partial_{\hat{\underline{b}}} \partial_{\hat{\underline{c}}} \hat{\phi}_a d \hat{t} \end{align} \tag{ A36 }$$

$$\begin{align} & = \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{e}}_{\underline{b}} \partial_{\hat{e}} \left( A^{\hat{f}}_{\underline{c}} \partial_{\hat{f}} \left(A^{\hat{g}}_{a} \hat{\phi}_{\hat{g}} \right) \right) \frac{d t}{d \hat{t}} d \hat{t}. \end{align} \tag{ A37 }$$

Performing these derivatives gives

$$\begin{equation} \begin{aligned} \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\eta}^a \partial_{\underline{b}} \partial_{\underline{c}} \phi_a dt &= \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{g}}_{a \underline{b} \underline{c}} \hat{\phi}_{\hat{g}} \frac{d t}{d \hat{t}} d \hat{t} + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{e}}_{\underline{b}} A^{\hat{g}}_{a \underline{c}} \partial_{\hat{e}} \left(\hat{\phi}_{\hat{g}}\right) \frac{d t}{d \hat{t}} d \hat{t} \\ &\quad + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{e}}_{\underline{c}} A^{\hat{f}}_{a \underline{b}} \partial_{\hat{e}} \left(\hat{\phi}_{\hat{f}}\right) \frac{d t}{d \hat{t}} d \hat{t} + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{e}}_{\underline{b}} A^{\hat{f}}_{\underline{c}} A^{\hat{g}}_{a} \partial_{\hat{e}} \partial_{\hat{f}} \left( \hat{\phi}_{\hat{g}}\right) \frac{d t}{d \hat{t}} d \hat{t} \\ &\quad + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{f}}_{\underline{c} \underline{b}} A^{h}_{\hat{f}} A^{\hat{g}}_{a h} \hat{\phi}_{\hat{g}} \frac{d t}{d \hat{t}} d \hat{t} + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{a}_{\hat{d}} \frac{dt}{d\hat{t}} A^{\hat{f}}_{\underline{c} \underline{b}} A^{\hat{g}}_{a} \partial_{\hat{f}} \left(\hat{\phi}_{\hat{g}}\right) \frac{d t}{d \hat{t}} d \hat{t}. \end{aligned} \end{equation} \tag{ A38 }$$

Simplifying these terms down

$$\begin{equation} \begin{aligned} \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\eta}^a \partial_{\underline{b}} \partial_{\underline{c}} \phi_a dt &= \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{{\eta}}^{a} \partial_a \left( A^{\hat{g}}_{\underline{b} \underline{c}} \right) \hat{\phi}_{\hat{g}} \frac{d t}{d \hat{t}} d \hat{t} + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} A^{\hat{e}}_{\underline{b}} \dot{{\eta}}^{a} \partial_a \left(A^{\hat{g}}_{\underline{c}} \right) \partial_{\hat{e}} \left(\hat{\phi}_{\hat{g}}\right) \frac{d t}{d \hat{t}} d \hat{t} \\ &\quad + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} A^{\hat{e}}_{\underline{c}} \dot{{\eta}}^{\hat{a}} \partial_a \left(A^{\hat{f}}_{\underline{b}}\right) \partial_{\hat{e}} \left(\hat{\phi}_{\hat{f}}\right) \frac{d t}{d \hat{t}} d \hat{t} + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{\hat{e}}_{\underline{b}} A^{\hat{f}}_{\underline{c}} \partial_{\hat{e}} \partial_{\hat{f}} \left( \hat{\phi}_{\hat{d}}\right) d \hat{t} + \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{\hat{f}}_{\underline{c} \underline{b}} \partial_{\hat{f}} \left(\hat{\phi}_{\hat{d}}\right) d \hat{t}. \end{aligned} \end{equation} \tag{ A39 }$$

For the $X^{\underline{a} \underline{b}}$ term,

$$\begin{align} \int_{{{\mathbb{R}}}} X^{\underline{a} \underline{b}} \partial_{\underline{b}} \phi_{\underline{a}} dt & = \int_{{{\mathbb{R}}}} \hat{X}^{\underline{a} \underline{b}} \partial_{\hat{\underline{b}}} \hat{\phi}_{\underline{a}} d\hat{t} \end{align} \tag{ A40 }$$

$$\begin{align} & = \int_{{{\mathbb{R}}}} X^{\underline{a} \underline{b}} A^{\hat{c}}_{\underline{b}} \partial_{\hat{c}} \left( A^{\hat{d}}_{\underline{a}} \hat{\phi}_{\hat{d}} \right) \frac{dt}{d \hat{t}} d \hat{t} \end{align} \tag{ A41 }$$

$$\begin{align} & = \int_{{{\mathbb{R}}}} X^{\underline{a} \underline{b}} A^{\hat{d}}_{\underline{a} \underline{b}} \hat{\phi}_{\hat{d}} \frac{dt}{d \hat{t}} d \hat{t} + \int_{{{\mathbb{R}}}} X^{\underline{a} \underline{b}} A^{\hat{c}}_{\underline{b}} A^{\hat{d}}_{\underline{a}} \partial_{\hat{c}} \left(\hat{\phi}_{\hat{d}}\right) \frac{dt}{d \hat{t}} d \hat{t}. \end{align} \tag{ A42 }$$

For the V^a term,

$$\begin{align} \int_{{\mathbb{R}}} {\dot{\eta}}^a V^{\underline{b}} \frac{\partial \phi_{a}}{\partial \xi^{\underline{b}}} d t & = \int_{{\mathbb{R}}} \hat{\dot{\eta}}^a \hat{V}^{\underline{b}} \frac{\partial \hat{\phi}_{a}}{\partial \hat{\xi}^{\underline{b}}} d \hat{t} \end{align} \tag{ A43 }$$

$$\begin{align} & = \int_{{\mathbb{R}}} \dot{\hat{\eta}}^{\hat{c}} A^{a}_{\hat{c}} \frac{d \hat{t}}{dt} {V}^{\underline{b}} A^{\hat{d}}_{\underline{b}} \frac{\partial}{\partial \hat{\xi}^d} \left( A^{\hat{e}}_{a} {\hat{\phi}}_{\hat{e}} \right) \frac{d \hat{t}}{dt} dt \end{align} \tag{ A44 }$$

$$\begin{align} & = \int_{{\mathbb{R}}} \dot{\eta}^{a} {V}^{\underline{b}} A^{\hat{e}}_{\underline{b} a} {\hat{\phi}}_{\hat{e}} \frac{d \hat{t}}{dt} dt + \int_{{\mathbb{R}}} \dot{\hat{\eta}}^{\hat{c}} A^{a}_{\hat{c}} {V}^{\underline{b}} A^{\hat{d}}_{\underline{b}} A^{\hat{e}}_{a} \partial_{\hat{d}} \hat{\phi}_{\hat{e}} dt \end{align} \tag{ A45 }$$

$$\begin{align} & = \int_{{\mathbb{R}}} {V}^{\underline{b}} \dot{\eta}^{a} \partial_a \left(A^{\hat{e}}_{\underline{c}}\right) {\hat{\phi}}_{\hat{e}} \frac{d \hat{t}}{dt} dt + \int_{{\mathbb{R}}} \dot{\hat{\eta}}^{\hat{c}} {V}^{\underline{b}} A^{\hat{d}}_{\underline{b}} \partial_{\hat{d}} \hat{\phi}_{\hat{c}} dt. \end{align} \tag{ A46 }$$

For the $X^{\underline{a}}$ term,

$$\begin{equation} \int_{{\mathbb{R}}} {X}^{\underline{d}} {\phi}_{\underline{d}} dt = \int_{\hat{\mathbb{R}}} \hat{X}^{\underline{d}} \hat{\phi}_{\underline{d}} d \hat{t} = \int_{\hat{\mathbb{R}}} X^{\underline{d}} A^{\hat{c}}_{\underline{d}} {\hat{\phi}}_{\hat{c}} \frac{d \hat{t}}{dt} d t. \end{equation} \tag{ A47 }$$

Summing all these terms together gives the transformed quadrupole,

$$\begin{equation} \begin{aligned} \int \mathscr{J}^a \phi_a d^7 \xi = \int \hat{\mathscr{J}}^a \hat{\phi}_a d^7 \hat{\xi} &= \frac{1}{2} \int_{{{\mathbb{R}}}} V^{\underline{b} \underline{c}} \dot{\hat{\eta}}^{\hat{d}} A^{\hat{e}}_{\underline{b}} A^{\hat{f}}_{\underline{c}} \partial_{\hat{e}} \partial_{\hat{f}} \left( \hat{\phi}_{\hat{d}}\right) d \hat{t} \\ &\quad + \int_{{{\mathbb{R}}}} \left(X^{\underline{a} \underline{b}} A^{\hat{c}}_{\underline{b}} A^{\hat{d}}_{\underline{a}} + \frac{1}{2} V^{\underline{e} \underline{f}} A^{\hat{c}}_{\underline{e}} \dot{{\eta}}^{a} \partial_a \left(A^{\hat{d}}_{\underline{f}} \right) + \frac{1}{2} V^{\underline{e} \underline{f}} A^{\hat{d}}_{\underline{f}} \dot{{\eta}}^{a} \partial_a \left(A^{\hat{c}}_{\underline{e}} \right) \right) \partial_{\hat{c}} \left(\hat{\phi}_{\hat{d}}\right) \frac{dt}{d \hat{t}} d \hat{t}\\ &\quad + \int_{{\mathbb{R}}} \dot{\hat{\eta}}^{\hat{a}} \left({V}^{\underline{b}} A^{\hat{d}}_{\underline{b}} + \frac{1}{2} V^{\underline{b} \underline{c}} A^{\hat{a}}_{\underline{c} \underline{b}} \right) \partial_{\hat{d}} \hat{\phi}_{\hat{a}} dt \\ &\quad + \int_{\hat{\mathbb{R}}} \left(X^{\underline{d}} A^{\hat{c}}_{\underline{d}} + {V}^{\underline{b}} \dot{\eta}^{a} \partial_a \left(A^{\hat{c}}_{\underline{b}}\right) + X^{\underline{a} \underline{b}} A^{\hat{c}}_{\underline{a} \underline{b}} + \frac{1}{2} V^{\underline{d} \underline{e}} \dot{{\eta}}^{a} \partial_a \left( A^{\hat{c}}_{\underline{d} \underline{e}} \right) \right) {\hat{\phi}}_{\hat{c}} \frac{d t}{d\hat{t}} d \hat{t}. \end{aligned} \end{equation} \tag{ A48 }$$

This gives the coordinate transformations for the quadrupole components,

$$\begin{equation} \begin{aligned} U^{b c} &= V^{\underline{d} \underline{e}} A^{\hat{b}}_{\underline{d}} A^{\hat{c}}_{\underline{e}}, \\ Y^{c d} &= \left(X^{\underline{a} \underline{b}} A^{\hat{c}}_{\underline{b}} A^{\hat{d}}_{\underline{a}} + \frac{1}{2} V^{\underline{e} \underline{f}} A^{\hat{c}}_{\underline{e}} \dot{{\eta}}^{a} \partial_a \left(A^{\hat{d}}_{\underline{f}} \right) \right. \left. + \frac{1}{2} V^{\underline{e} \underline{f}} A^{\hat{d}}_{\underline{f}} \dot{{\eta}}^{a} \partial_a \left(A^{\hat{c}}_{\underline{e}} \right) \right) \frac{dt}{d \hat{t}}, \\ U^{a} &= V^{\underline{b}} A^{\hat{a}}_{\underline{b}} + \frac{1}{2} V^{\underline{b} \underline{c}} A^{\hat{a}}_{\underline{b} \underline{c}}, \\ Y^{c} &= \left(X^{\underline{d}} A^{\hat{c}}_{\underline{d}} + {V}^{\underline{b}} \dot{\eta}^{a} \partial_a \left(A^{\hat{c}}_{\underline{b}}\right) + X^{\underline{a} \underline{b}} A^{\hat{c}}_{\underline{a} \underline{b}} \right. \left. + \frac{1}{2} V^{\underline{d} \underline{e}} \dot{{\eta}}^{a} \partial_a \left( A^{\hat{c}}_{\underline{d} \underline{e}} \right) \right) \frac{d t}{d\hat{t}}. \end{aligned} \end{equation} \tag{ A49 }$$

where the components of the quadrupole in the new coordinate system are no longer unique.

This section performs the projection $\partial_0 = \partial_{\dot{\eta}} - \dot{\eta}^{\underline{a}} \partial_{\underline{a}}$ into the quadrupole with non unique components defined by equation (35). By performing this projection $U^a, U^{a b}, Y^{a}$ and $Y^{a b}$ can be written in a form where the components are unique, giving the full coordinate transformation for the quadrupole.

Inserting (38) term by term into the non-unique semi-quadrupole

$$\begin{equation} \begin{aligned} \int \mathscr{J}^a \phi_a d^6 \xi = \frac{1}{2} \int_{\mathbb{R}} \dot{\eta}^a U^{b c} \partial_b \partial_c \phi_a dt + \int_{\mathbb{R}} Y^{a b} \partial_{\underline{b}} \phi_{\underline{a}} dt + \int_{\mathbb{R}} \dot{\eta}^a U^{\underline{b}} \partial_{\underline{b}} \phi_a dt + \int_{\mathbb{R}} Y^{\underline{a}} \phi_{\underline{a}} dt + \int_{\mathbb{R}} \dot{\eta}^a q \phi_a dt. \end{aligned} \end{equation} \tag{ A50 }$$

For the $U^{a b}$ term, noting the symmetry of $U^{a b}$,

$$\begin{align} \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{b c} \partial_b \partial_c \phi_{a} dt &= \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt + 2 \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{\underline{b} 0} \partial_{\underline{b}} \partial_0 \phi_{a} dt + \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{0 0} \partial_0 \partial_0 \phi_{a} dt \end{align} \tag{ A51 }$$

$$\begin{align} &= \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{\underline{b} \underline{c}} \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt + 2 \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{\underline{b} 0} \partial_{\dot{\eta}} \partial_{\underline{b}} \phi_{a} dt - 2 \int_{{\mathbb{R}}} {\dot{\eta}}^a \dot{\eta}^{\underline{d}} {U}^{\underline{b} 0} \partial_{\underline{b}} \partial_{\underline{d}} \phi_{a} dt \nonumber\\ &\quad + \int_{{\mathbb{R}}} {\dot{\eta}}^a {U}^{0 0} \partial_{\dot{\eta}} \partial_0 \phi_{a} dt - \int_{{\mathbb{R}}} {\dot{\eta}}^a \dot{\eta}^{\underline{b}} {U}^{0 0} \partial_0 \partial_{\underline{b}} \phi_{a} dt \end{align} \tag{ A52 }$$

$$\begin{align} &= \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - 2 \dot{\eta}^{\underline{c}} U^{\underline{c} 0} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt - 2 \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{\underline{b} 0} \right) \partial_{\underline{b}} \phi_{a} dt \nonumber\\ & \quad + \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \partial_0 \phi_{a} dt - \int_{{\mathbb{R}}} {\dot{\eta}}^a \dot{\eta}^{\underline{b}} {U}^{0 0} \partial_{\underline{b}} \partial_0 \phi_{a} dt \end{align} \tag{ A53 }$$

$$\begin{align} & = \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - 2 \dot{\eta}^{\underline{c}} U^{\underline{c} 0} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt - 2 \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{\underline{b} 0} \right) \partial_{\underline{b}} \phi_{a} dt + \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \partial_{\dot{\eta}} \phi_{a} dt \nonumber\\ &\quad + \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt - \int_{{\mathbb{R}}} {\dot{\eta}}^a \dot{\eta}^{\underline{b}} {U}^{0 0} \partial_{\dot{\eta}} \partial_{\underline{b}} \phi_{a} dt + \int_{{\mathbb{R}}} {\dot{\eta}}^a \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} {U}^{0 0} \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt \end{align} \tag{ A54 }$$

$$\begin{align} &= \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - 2 \dot{\eta}^{\underline{c}} U^{\underline{c} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt - 2 \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{\underline{b} 0} \right) \partial_{\underline{b}} \phi_{a} dt \nonumber\\ &\quad + \int_{{\mathbb{R}}} \frac{d^2}{dt^2} \left( {\dot{\eta}}^a {U}^{0 0} \right) \phi_{a} dt + \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt - \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a \dot{\eta}^{\underline{b}} {U}^{0 0} \right) \partial_{\underline{b}} \phi_{a} dt. \nonumber \end{align} \tag{ A55 }$$

For the $Y^{a b}$ term,

$$\begin{align} \int_{{{\mathbb{R}}}} {Y}^{a b} \partial_b \phi_{a} dt & = \int_{{{\mathbb{R}}}} {Y}^{a \underline{b}} \partial_{\underline{b}} \phi_{a} dt + \int_{{{\mathbb{R}}}} {Y}^{a 0} \partial_0 \phi_{a} dt \end{align} \tag{ A56 }$$

$$\begin{align} & = \int_{{{\mathbb{R}}}} {Y}^{a \underline{b}} \partial_{\underline{b}} \phi_{a} dt + \int_{{{\mathbb{R}}}} {Y}^{a 0} \partial_{\dot{\eta}} \phi_{a} dt - \int_{{{\mathbb{R}}}} {Y}^{a 0} \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt \end{align} \tag{ A57 }$$

$$\begin{align} & = \int_{{{\mathbb{R}}}} \left( {Y}^{a \underline{b}} - {Y}^{a 0} \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt - \int_{{{\mathbb{R}}}} \frac{d}{dt} {Y}^{a 0} \phi_{a} dt. \end{align} \tag{ A58 }$$

For the U^a term,

$$\begin{align} \int_{{{\mathbb{R}}}} \dot{\eta}^a {U}^{b} \partial_b \phi_{a} dt & = \int_{\mathbb{R}} \dot{\eta}^a U^{0} \partial_{0} \phi_{a} dt + \int_{\mathbb{R}} \dot{\eta}^a U^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt \end{align} \tag{ A59 }$$

$$\begin{align} & = \int_{\mathbb{R}} \dot{\eta}^a U^{0} \partial_{\dot{\eta}} \phi_{a} dt - \int_{\mathbb{R}} \dot{\eta}^a U^{0} \dot{\eta}^{\underline{a}} \partial_{\underline{b}} \phi_{a} dt + \int_{\mathbb{R}} \dot{\eta}^a U^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt \end{align} \tag{ A60 }$$

$$\begin{align} & = - \int_{\mathbb{R}} \frac{d}{dt} \left(\dot{\eta}^a U^{0} \right) \phi_{a} dt + \int_{\mathbb{R}} \dot{\eta}^a \left(U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt . \end{align} \tag{ A61 }$$

Summing all these terms together,

$$\begin{equation} \begin{aligned} \int \mathscr{J}^a \phi_a d^7 \xi &= \frac{1}{2} \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - 2 \dot{\eta}^{\underline{c}} U^{\underline{c} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt - \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{\underline{b} 0} \right) \partial_{\underline{b}} \phi_{a} dt + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d^2}{dt^2} \left( {\dot{\eta}}^a {U}^{0 0} \right) \phi_{a} dt \\ &\quad + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt -\frac{1}{2} \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a \dot{\eta}^{\underline{b}} {U}^{0 0} \right) \partial_{\underline{b}} \phi_{a} dt + \int_{{{\mathbb{R}}}} \left( {Y}^{a \underline{b}} - {Y}^{a 0} \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt - \int_{{{\mathbb{R}}}} \frac{d}{dt} {Y}^{a 0} \phi_{a} dt \\ &\quad - \int_{\mathbb{R}} \frac{d}{dt} \left(\dot{\eta}^a U^{0} \right) \phi_{a} dt + \int_{\mathbb{R}} \dot{\eta}^a \left(U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt + \int_{\mathbb{R}} Y^{\underline{a}} \phi_{\underline{a}} dt + \int_{\mathbb{R}} \dot{\eta}^a q \phi_a dt. \end{aligned} \end{equation} \tag{ A62 }$$

Grouping terms together,

$$\begin{equation} \begin{aligned} \int \mathscr{J}^a \phi_a d^7 \xi &= \frac{1}{2} \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - 2 \dot{\eta}^{\underline{c}} U^{\underline{c} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt \\ &\quad + \int_{{{\mathbb{R}}}} \left( Y^{a \underline{b}} - {Y}^{a 0} \dot{\eta}^{\underline{b}} - \frac{d}{dt} \left(\dot{\eta}^a U^{\underline{b} 0} \right) + \frac{1}{2} \frac{d}{dt} \left( {\dot{\eta}}^a \dot{\eta}^{\underline{b}} {U}^{0 0} \right) \right) \partial_{\underline{b}} \phi_{a} dt \\ &\quad + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d^2}{dt^2} \left( {\dot{\eta}}^a {U}^{0 0} \right) \phi_{a} dt + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt - \int_{{{\mathbb{R}}}} \frac{d}{dt} {Y}^{a 0} \phi_{a} dt + \int_{\mathbb{R}} \dot{\eta}^a \left(U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt \\ &\quad + \int_{\mathbb{R}} \left( Y^{\underline{a}} - \frac{d}{dt} \left(\dot{\eta}^a U^{0} \right) \right) \phi_{\underline{a}} dt + \int_{\mathbb{R}} \dot{\eta}^a q \phi_a dt. \end{aligned} \end{equation} \tag{ A63 }$$

Calculating some derivatives and symmetrising $U^{0 \underline{b}}$ gives

$$\begin{equation} \begin{aligned} \int {\mathscr{J}}^a_{Q} {\phi}_{a} d^7 {\xi} &= \frac{1}{2} \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - \dot{\eta}^{\underline{b}} U^{\underline{c} 0} - \dot{\eta}^{\underline{c}} U^{\underline{b} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt \\ &\quad + \int_{{{\mathbb{R}}}} \left( Y^{a \underline{b}} - {Y}^{a 0} \dot{\eta}^{\underline{b}} - \dot{\eta}^a \frac{d}{dt} \left(U^{\underline{b} 0} \right) + \frac{1}{2} {\dot{\eta}}^a \dot{\eta}^{\underline{b}} \frac{d}{dt} \left( {U}^{0 0} \right) \right) \partial_{\underline{b}} \phi_{a} dt + \int_{{{\mathbb{R}}}} \left(-U^{\underline{b} 0} \frac{d}{dt} \left( \dot{\eta}^a \right) + \frac{1}{2} {U}^{0 0} \frac{d}{dt} \left( {\dot{\eta}}^a \dot{\eta}^{\underline{b}} \right) \right) \partial_{\underline{b}} \phi_{a} dt \\ &\quad + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d^2}{dt^2} \left( {\dot{\eta}}^a {U}^{0 0} \right) \phi_{a} dt + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt - \int_{{{\mathbb{R}}}} \frac{d}{dt} {Y}^{a 0} \phi_{a} dt \\ &\quad + \int_{\mathbb{R}} \dot{\eta}^a \left(U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt + \int_{\mathbb{R}} \left( Y^{\underline{a}} - \frac{d}{dt} \left(\dot{\eta}^a U^{0} \right) \right) \phi_{\underline{a}} dt + \int_{\mathbb{R}} \dot{\eta}^a q \phi_a dt. \end{aligned} \end{equation} \tag{ A64 }$$

Noting $dU^{a b}/dt = Y^{a b} + Y^{b a}$ gives

$$\begin{equation} \begin{aligned} \int {\mathscr{J}}^a_{Q} {\phi}_{a} d^7 {\xi} &= \frac{1}{2} \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - \dot{\eta}^{\underline{b}} U^{\underline{c} 0} - \dot{\eta}^{\underline{c}} U^{\underline{b} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt \\ &\quad + \int_{{{\mathbb{R}}}} \left( Y^{a \underline{b}} - {Y}^{a 0} \dot{\eta}^{\underline{b}} - \dot{\eta}^a Y^{\underline{b} 0} - \dot{\eta}^a Y^{0 \underline{b}} + {\dot{\eta}}^a \dot{\eta}^{\underline{b}} Y^{00} \right) \partial_{\underline{b}} \phi_{a} dt + \int_{{{\mathbb{R}}}} \left(\frac{1}{2} {U}^{0 0} \frac{d}{dt} \left( {\dot{\eta}}^a \dot{\eta}^{\underline{b}} \right) -U^{\underline{b} 0} \frac{d}{dt} \left( \dot{\eta}^a \right) \right) \partial_{\underline{b}} \phi_{a} dt \\ &\quad + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d^2}{dt^2} \left( {\dot{\eta}}^a {U}^{0 0} \right) \phi_{a} dt + \frac{1}{2} \int_{{\mathbb{R}}} \frac{d}{dt} \left( {\dot{\eta}}^a {U}^{0 0} \right) \dot{\eta}^{\underline{b}} \partial_{\underline{b}} \phi_{a} dt - \int_{{{\mathbb{R}}}} \frac{d}{dt} {Y}^{a 0} \phi_{a} dt \\ &\quad + \int_{\mathbb{R}} \dot{\eta}^a \left(U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} \right) \partial_{\underline{b}} \phi_{a} dt + \int_{\mathbb{R}} \left( Y^{\underline{a}} - \frac{d}{dt} \left(\dot{\eta}^a U^{0} \right) \right) \phi_{\underline{a}} dt + \int_{\mathbb{R}} \dot{\eta}^a q \phi_a dt. \end{aligned} \end{equation} \tag{ A65 }$$

Rearranging for clarity,

$$\begin{equation} \begin{aligned} \int {\mathscr{J}}^a_{Q} {\phi}_{a} d^7 {\xi} &= \frac{1}{2} \int_{{\mathbb{R}}} {\dot{\eta}}^a \left( {U}^{\underline{b} \underline{c}} - \dot{\eta}^{\underline{b}} U^{\underline{c} 0} - \dot{\eta}^{\underline{c}} U^{\underline{b} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00} \right) \partial_{\underline{b}} \partial_{\underline{c}} \phi_{a} dt + \int_{{{\mathbb{R}}}} \left( Y^{a \underline{b}} - {Y}^{a 0} \dot{\eta}^{\underline{b}} - \dot{\eta}^a Y^{0 \underline{b}} + {\dot{\eta}}^a \dot{\eta}^{\underline{b}} Y^{00} \right) \partial_{\underline{b}} \phi_{a} dt \\ &\quad + \int_{{{\mathbb{R}}}} \left({U}^{0 0} \dot{\eta}^{\underline{b}} \frac{d}{dt} \left( {\dot{\eta}}^a \right) - U^{\underline{b} 0} \frac{d}{dt} \left( \dot{\eta}^a \right) \right) \partial_{\underline{b}} \phi_{a} dt +\int_{{\mathbb{R}}} \dot{\eta}^a \left( U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} + \frac{1}{2} \frac{d}{dt} \left(\dot{\eta}^{\underline{b}} {U}^{0 0} \right) - Y^{\underline{b} 0} \right) \partial_{\underline{b}} \phi_{a} dt \\ &\quad + \int_{{\mathbb{R}}} \left( Y^{\underline{a}} - \frac{d}{dt} \left(\dot{\eta}^a U^{0} \right) + \frac{1}{2} \frac{d^2}{dt^2} \left( {\dot{\eta}}^a {U}^{0 0} \right) - \frac{d}{dt} {Y}^{a 0} \right) \phi_{a} dt. \end{aligned} \end{equation} \tag{ A66 }$$

By noting $\dot{\eta}^0 = 1$ the $Y^{0\underline{a}}, Y^{\underline{a} 0}, Y^{00}$ and Y⁰ components vanish. Thus the projected moments are given by

$$\begin{equation} \begin{aligned} \hat{V}^{\underline{b} \underline{c}} &= {U}^{\underline{b} \underline{c}} - \dot{\eta}^{\underline{b}} U^{\underline{c} 0} - \dot{\eta}^{\underline{c}} U^{\underline{b} 0} + \dot{\eta}^{\underline{b}} \dot{\eta}^{\underline{c}} U^{00}, \\ \hat{X}^{\underline{a} \underline{b}} &= Y^{\underline{a} \underline{b}} - {Y}^{\underline{a} 0} \dot{\eta}^{\underline{b}} - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} + {\dot{\eta}}^{\underline{a}} \dot{\eta}^{\underline{b}} Y^{00} + {U}^{0 0} \dot{\eta}^{\underline{b}} \frac{d}{dt} \left( {\dot{\eta}}^{\underline{a}} \right) - U^{\underline{b} 0} \frac{d}{dt} \left( \dot{\eta}^{\underline{a}} \right), \\ \hat{V}^{\underline{b}} &= U^{\underline{b}} - U^0 \dot{\eta}^{\underline{b}} + \frac{1}{2} \frac{d}{dt} \left(\dot{\eta}^{\underline{b}} {U}^{0 0} \right) - Y^{\underline{b} 0}, \\ \hat{X}^{\underline{a}} &= Y^{\underline{a}} - \frac{d}{dt} \left(\dot{\eta}^{\underline{a}} U^{0} \right) + \frac{1}{2} \frac{d^2}{dt^2} \left( {\dot{\eta}}^{\underline{a}} {U}^{0 0} \right) - \frac{d}{dt} {Y}^{\underline{a} 0}. \end{aligned} \end{equation} \tag{ A67 }$$

Combining this with (35) give the full coordinate transformations for the quadrupole.

This section finds the first step of the coordinate transformation for the de Rham current representation of a quadrupole, obtaining non-unique components in terms of $U^a, U^{a b}, Y^a$ and Y^ab .

Consider a unique distributional quadrupole,

$$\begin{equation}{\Psi _Q} = {\textstyle{1 \over 2}}L_{}^{\left( \xi \right)}L_{}^{\left( \xi \right)}{\eta _\varsigma }\left( {{V^{}}} \right) - L_{}^{\left( \xi \right)}i_{}^{\left( \xi \right)}{\eta _\varsigma }\left( {{X^{}}dt} \right){\textrm{ }} - L_{}^{\left( \xi \right)}{\eta _\varsigma }\left( {{V^{}}} \right) + i_{}^{\left( \xi \right)}{\eta _\varsigma }\left( {{X^{}}dt} \right) + {\eta _\varsigma }\left( q \right). \end{equation} \tag{ A68 }$$

This is transformed by noting

$$\begin{equation} \frac{\partial}{\partial \xi^{\underline{a}}} = \frac{\partial \hat{\xi}^b}{\partial \xi^{\underline{a}}} \frac{\partial}{\partial \hat{\xi}^b} \end{equation} \tag{ A69 }$$

so

$$\begin{equation} i^{\left(\xi\right)}_{\underline{a}} = i^{\left(\hat{\xi}\right)}_b \frac{\partial \hat{\xi}^b}{\partial \xi^{\underline{a}}}. \end{equation} \tag{ A70 }$$

Using Cartan's identity the Lie derivative transformation can be found,

$$\begin{align} L_{\underline{a}}\left(\alpha\right) & = d i_{\underline{a}} \alpha + i_{\underline{a}} d \alpha \end{align} \tag{ A71 }$$

$$\begin{align} & = d i_{\hat{b}} \left(A^{\hat{b}}_{\underline{a}} \wedge \alpha\right) + i_{\hat{b}} \left(A^{\hat{b}}_{\underline{a}} \wedge d \alpha\right) \end{align} \tag{ A72 }$$

$$\begin{align} & = d i_{\hat{b}} \left(A^{\hat{b}}_{\underline{a}} \wedge \alpha\right) + i_{\hat{b}} d \left( A^{\hat{b}}_{\underline{a}} \wedge \alpha \right) - i_{\hat{b}} \left( dA^{\hat{b}}_{\underline{a}} \wedge \alpha \right) \end{align} \tag{ A73 }$$

$$\begin{align} & = L_{\hat{b}} \left( A^{\hat{b}}_{\underline{a}} \wedge \alpha \right) - i_{\hat{b}} \left(dA^{\hat{b}}_{\underline{a}} \wedge \alpha \right). \end{align} \tag{ A74 }$$

For the double Lie derivative term, use the standard differential geometry result

$$\begin{equation} L_U \left(fg\right) = f L_U g + g L_U f \quad \forall f,g \in \Gamma \Lambda^0 E, U \in \Gamma T E \end{equation} \tag{ A75 }$$

to find

$$\begin{align} L_{\underline{a}} L_{\underline{b}} \alpha &= L_{\underline{a}} \left( L_{\hat{d}} \left(A^{\hat{d}}_{\underline{b}} \wedge \alpha\right) - i_{\hat{d}} \left(d A^{\hat{d}}_{\underline{b}} \wedge \alpha\right)\right) \end{align} \tag{ A76 }$$

$$\begin{align} &= L_{\hat{c}} \left( A^{\hat{c}}_{\underline{a}} \wedge \left( L_{\hat{d}} \left(A^{\hat{d}}_{\underline{b}} \wedge \alpha\right) - i_{\hat{d}} \left(d A^{\hat{d}}_{\underline{b}} \wedge \alpha\right)\right) \right) - i_{\hat{c}} \left(d A^{\hat{c}}_{\underline{a}} \wedge \left( L_{\hat{d}} \left(A^{\hat{d}}_{\underline{b}} \wedge \alpha\right) - i_{\hat{d}} \left(d A^{\hat{d}}_{\underline{b}} \wedge \alpha\right)\right) \right) \end{align} \tag{ A77 }$$

$$\begin{align} &= L_{\hat{c}} L_{\hat{d}} \left( A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} \wedge \alpha \right) - L_{\hat{c}} \left( i_{\hat{d}} \left(d A^{\hat{c}}_{\underline{a}}\right) \wedge A^{\hat{d}}_{\underline{b}} \wedge \alpha \right) - L_{\hat{c}} i_{\hat{d}} \left(A^{\hat{c}}_{\underline{a}} \wedge d A^{\hat{d}}_{\underline{b}} \wedge \alpha\right) - i_{\hat{c}} L_{\hat{d}} \left(d A^{\hat{c}}_{\underline{a}} \wedge A^{\hat{d}}_{\underline{b}} \wedge \alpha \right) \nonumber\\ &\quad + i_{\hat{c}} \left( \left(d i_{\hat{d}} d A^{\hat{c}}_{\underline{a}}\right) \wedge A^{\hat{d}}_{\underline{b}} \wedge \alpha \right) - i_{\hat{c}} i_{\hat{d}} \left( d A^{\hat{c}}_{\underline{a}} \wedge dA^{\hat{d}}_{\underline{b}} \wedge \alpha \right) + i_{\hat{c}} \left( i_{\hat{d}} dA^{\hat{c}}_{\underline{a}} \wedge dA^{\hat{d}}_{\underline{b}} \wedge \alpha \right) \end{align} \tag{ A78 }$$

$$\begin{align} &= L_{\hat{c}} L_{\hat{d}} \left( A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} \wedge \alpha \right) - L_{\hat{c}} i_{\hat{d}} \left( d\left(A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} \right) \wedge \alpha \right) - L_{\hat{c}} \left( \partial_{\hat{d}} A^{\hat{c}}_{\underline{a}} \wedge A^{\hat{d}}_{\underline{b}} \wedge \alpha \right) \nonumber\\ &\quad + i_{\hat{c}} \left( d \left(\partial_{\hat{d}} A^{\hat{c}}_{\underline{a}} \wedge A^{\hat{d}}_{\underline{b}} \right) \wedge \alpha \right) - i_{\hat{c}} i_{\hat{d}} \left( d A^{\hat{c}}_{\underline{a}} \wedge dA^{\hat{d}}_{\underline{b}} \wedge \alpha \right). \nonumber \end{align} \tag{ A79 }$$

Using $\partial_{\hat{d}} A^{\hat{c}}_{\underline{a}} = A^{e}_{\hat{d}} A^{\hat{c}}_{\underline{a} \underline{b}}$, and noting the $i_{\hat{c}} i_{\hat{d}}$ term vanishes as the term inside the brackets is symmetric, gives

$$\begin{equation} {L_{}}{L_{}}\alpha = {L_{\hat c}}{L_{\hat d}}\left( {A_{}^{\hat c}A_{}^{\hat d} \wedge \alpha } \right) - {L_{\hat c}}{i_{\hat d}}\left( {d\left( {A_{}^{\hat c}A_{}^{\hat d}} \right) \wedge \alpha } \right){\textrm{ }} - {L_{\hat c}}\left( {A_{}^{\hat c} \wedge \alpha } \right) + {i_{\hat c}}\left( {dA_{}^{\hat c} \wedge \alpha } \right). \end{equation} \tag{ A80 }$$

This gives the coordinate transformation for the $V^{\underline{a} \underline{b}}$ term,

$$\begin{align} L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma}\left(V^{\underline{a} \underline{b}}\right) = L_{\hat{c}} L_{\hat{d}} \eta_{\varsigma} \left( A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} V^{\underline{a} \underline{b}} \right) - L_{\hat{c}} i_{\hat{d}} \eta_{\varsigma} \left( d\left(A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} \right) \wedge V^{\underline{a} \underline{b}} \right) - L_{\hat{c}} \eta_{\varsigma} \left( A^{\hat{c}}_{\underline{a} \underline{b}} V^{\underline{a} \underline{b}} \right) + i_{\hat{c}} \eta_{\varsigma} \left( dA^{\hat{c}}_{\underline{a} \underline{b}} \wedge V^{\underline{a} \underline{b}} \right). \end{align} \tag{ A81 }$$

For the $X^{\underline{a} \underline{b}}$ term,

$$\begin{align} i_{\underline{a}} L_{\underline{b}} \eta_{\varsigma}\left(X^{\underline{a} \underline{b}} dt\right) & = i_{\underline{a}} L_{\hat{c}}\eta_{\varsigma}\left(A^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} dt\right) - i_{\underline{a}} i_{\hat{c}} \left( \eta_{\varsigma}\left(dA^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} dt\right) \right) \end{align} \tag{ A82 }$$

$$\begin{align} & = i_{\hat{d}} A^{\hat{d}}_{\underline{a}} L_{\hat{c}} \eta_{\varsigma}\left(A^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} dt\right) \end{align} \tag{ A83 }$$

$$\begin{align} & = i_{\hat{d}} L_{\hat{c}} \eta_{\varsigma}\left( A^{\hat{d}}_{\underline{a}} A^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} dt\right) - i_{\hat{d}} \eta_{\varsigma} \left( L_{\hat{c}}\left(A^{\hat{d}}_{\underline{a}}\right) \wedge A^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} dt \right) \end{align} \tag{ A84 }$$

$$\begin{align} & = i_{\hat{d}} L_{\hat{c}} \eta_{\varsigma}\left( A^{\hat{d}}_{\underline{a}} A^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} dt\right) - i_{\hat{d}} \eta_{\varsigma}\left( A^{\hat{d}}_{\underline{a} \underline{b}} \wedge X^{\underline{a} \underline{b}} dt\right) \end{align} \tag{ A85 }$$

$$\begin{align} & = i_{\hat{d}} L_{\hat{c}} \eta_{\varsigma} \left( A^{\hat{d}}_{\underline{a}} A^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}} \frac{dt}{d\hat{t}} d\hat{t} \right) - i_{\hat{d}} \eta_{\varsigma} \left( A^{\hat{d}}_{\underline{a} \underline{b}} \wedge X^{\underline{a} \underline{b}} \frac{dt}{d\hat{t}} d\hat{t} \right) \end{align} \tag{ A86 }$$

where $dA^{\hat{c}}_{\underline{b}} \wedge X^{\underline{a} \underline{b}}dt = 0$ is used. The $V^{\underline{a}}$ and $X^{\underline{a}}$ terms are more straightforward,

$$\begin{align} L_{\underline{a}} \eta_{\varsigma} \left(V^{\underline{a}}\right) &= L_{\hat{b}} \eta_{\varsigma} \left( A^{\hat{b}}_{\underline{a}} \wedge V^{\underline{a}} \right) - i_{\hat{b}} \eta_{\varsigma} \left(dA^{\hat{b}}_{\underline{a}} \wedge V^{\underline{a}} \right) \end{align} \tag{ A87 }$$

$$\begin{align} i_{\underline{a}} \eta_{\varsigma} \left(X^{\underline{a}} dt\right) &= i_{\hat{b}} \eta_{\varsigma} \left( A^{\hat{b}}_{\underline{a}} X^{\underline{a}} \frac{dt}{d\hat{t}} d\hat{t} \right). \end{align} \tag{ A88 }$$

Summing these together (noting that the monopole term is invariant under transformation), gives

$$\begin{equation} \begin{aligned} \mathscr{J} = \hat{\mathscr{J}} &= \frac{1}{2} L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(V^{\underline{a} \underline{b}}\right) - i_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(X^{\underline{a} \underline{b}} dt\right) - L_{\underline{a}} \eta_{\varsigma} \left(V^{\underline{a}} \right) + i_{\underline{a}} \eta_{\varsigma}\left(X^{\underline{a}} dt\right) + \eta_{\varsigma}\left(q\right) \\ &= \frac{1}{2} L_{\hat{c}} L_{\hat{d}} \eta_{\varsigma} \left( A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} V^{\underline{a} \underline{b}} \right) - i_{\hat{d}} L_{\hat{c}} \eta_{\varsigma} \left( A^{\hat{d}}_{\underline{a}} A^{\hat{c}}_{\underline{b}} X^{\underline{a} \underline{b}} \frac{dt}{d\hat{t}} d\hat{t} + \frac{1}{2} d \left(A^{\hat{c}}_{\underline{a}} A^{\hat{d}}_{\underline{b}} \right) V^{\underline{a} \underline{b}} \right) - L_{\hat{c}} \eta_{\varsigma} \left(A^{\hat{c}}_{\underline{a}} V^{\underline{a}} + \frac{1}{2} A^{\hat{c}}_{\underline{a} \underline{b}} V^{\underline{a} \underline{b}} \right) \\ &\quad + i_{\hat{c}} \eta_{\varsigma} \left( A^{\hat{c}}_{\underline{a}} X^{\underline{a}} \frac{dt}{d\hat{t}} d\hat{t} + \left(d\left(A^{\hat{c}}_{\underline{a}}\right) V^{\underline{a}} + d\left(A^{\hat{c}}_{\underline{a} \underline{b}} V^{\underline{a} \underline{b}}\right) + A^{\hat{c}}_{\underline{a} \underline{b}} X^{\underline{a} \underline{b}}\right) \frac{dt}{d\hat{t}} d\hat{t} \right) + \eta_{\varsigma}\left(q\right). \end{aligned} \end{equation} \tag{ A89 }$$

By taking the external derivatives, this is equivalent to (35).

This section performs the projections $L_0 = L_{\dot{\eta}} - \dot{\eta}^{\underline{a}} L_{\underline{a}}$ and $i_0 = i_{\dot{\eta}} - \dot{\eta}^{\underline{a}} i_{\underline{a}}$ into the de Rham current representation of the quadrupole with non-unique components. By performing this projection $U^a, U^{a b}, Y^{a}$ and $Y^{a b}$ can be written in a form where the components are unique, finding the full coordinate transformation for the quadrupole.

The non-unique quadrupole is given by

$$\begin{equation} \mathscr{J} = \tfrac{1}{2} L_a L_b \eta_{\varsigma} \left(U^{a b}\right) - i_{a} L_{b} \eta_{\varsigma} \left(Y^{a b} d t\right) - L_a \eta_{\varsigma} \left(U^a\right) + i_a \eta_{\varsigma} \left(Y^a dt\right) + \eta_{\varsigma} \left(q\right). \end{equation} \tag{ A90 }$$

To find the projections, proceed term by term, beginning with the $L_a L_b$ term,

$$\begin{align} L_a L_b \eta_{\varsigma} \left(U^{a b}\right) &= L_a L_{\underline{b}} \eta_{\varsigma} \left(U^{a \underline{b}}\right) + L_a L_0 \left(U^{a 0}\right) \end{align} \tag{ A91 }$$

$$\begin{align} &= L_a L_{\underline{b}} \eta_{\varsigma} \left(U^{a \underline{b}}\right) + L_a L_{\dot{\eta}} \left(U^{a 0}\right) - L_a \dot{\eta}^{\underline{b}} L_{\underline{b}} \eta_{\varsigma}\left(U^{a 0}\right) \end{align} \tag{ A92 }$$

$$\begin{align} &= L_a L_{\underline{b}} \eta_{\varsigma} \left(U^{a \underline{b}} - \dot{\eta}^{\underline{b}} U^{a 0}\right) + L_a \eta_{\varsigma} \left(\frac{d{U}^{a 0}}{dt} \right) + L_a i_{\underline{b}} \eta_{\varsigma} \left(U^{a 0} d\dot{\eta}^{\underline{b}} \right) \end{align} \tag{ A93 }$$

$$\begin{align} &= L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(U^{\underline{a} \underline{b}} - \dot{\eta}^{\underline{b}} U^{\underline{a} 0}\right) + L_{\underline{b}} L_{\dot{\eta}} \eta_{\varsigma} \left(U^{0 \underline{b}} - \dot{\eta}^{\underline{b}} U^{0 0}\right) - L_{\underline{b}} \dot{\eta}^{\underline{a}} L_{\underline{a}} \eta_{\varsigma} \left(U^{0 \underline{b}} - \dot{\eta}^{\underline{b}} U^{0 0}\right) + L_{\underline{a}} \eta_{\varsigma} \left(\frac{d{U}^{\underline{a} 0}}{dt} \right) \nonumber\\ &\quad + L_{\dot{\eta}} \eta_{\varsigma} \left(\frac{d{U}^{0 0}}{dt} \right) - \dot{\eta}^{\underline{a}} L_{\underline{a}} \eta_{\varsigma} \left(\frac{d{U}^{0 0}}{dt} \right) + L_{\underline{a}} i_{\underline{b}} \eta_{\varsigma} \left(U^{\underline{a} 0} d\dot{\eta}^{\underline{b}} \right) + i_{\underline{b}} L_{\dot{\eta}} \eta_{\varsigma} \left(U^{0 0} d \dot{\eta}^{\underline{b}} \right) - i_{\underline{b}} \dot{\eta}^{\underline{a}} L_{\underline{a}} \eta_{\varsigma} \left(U^{0 0} d \dot{\eta}^{\underline{b}} \right) \end{align} \tag{ A94 }$$

$$\begin{align} &= L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(U^{\underline{a} \underline{b}} - \dot{\eta}^{\underline{b}} U^{\underline{a} 0} - \dot{\eta}^{\underline{a}} U^{\underline{b} 0} + \dot{\eta}^{\underline{a}} \dot{\eta}^{\underline{b}} U^{00} \right) + L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma} \left(\left(2U^{0 \underline{b}} - 2\dot{\eta}^{\underline{b}} U^{0 0}\right) d\dot{\eta}^{\underline{a}} \right) \nonumber\\ &\quad + L_{\underline{a}} \eta_{\varsigma} \left(2 \frac{d{U}^{\underline{a} 0}}{dt} - \dot{\eta}^{\underline{a}} \frac{d U^{00}}{dt} - \frac{d}{dt} \left( \dot{\eta}^{\underline{a}} U^{0 0} \right) \right) + i_{\underline{a}} \eta_{\varsigma} \left(\frac{d{U}^{0 0}}{dt} d\dot{\eta}^{\underline{a}} + \frac{d}{dt} \left(U^{0 0} d \dot{\eta}^{\underline{a}} \right) \right) + \eta_{\varsigma} \left(\frac{d^2{U}^{0 0}}{dt^2} \right) \nonumber\\ &\quad+ i_{\underline{b}} i_{\underline{a}} \eta_{\varsigma} \left(U^{0 0} d \dot{\eta}^{\underline{a}} \wedge d \dot{\eta}^{\underline{b}} \right) \nonumber \end{align} \tag{ A95 }$$

and note the last term vanishes as $\mathbb{R}$ is only 1-dimensional. For the $L_a i_b$ term,

$$\begin{align} L_b i_a \eta_{\varsigma}\left(Y^{a b} dt\right) &= L_{b} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} b} dt\right) + L_a i_{\dot{\eta}} \eta_{\varsigma}\left(Y^{0 b} dt\right) - L_{a} \dot{\eta}^{\underline{b}} i_{\underline{b}} \eta_{\varsigma} \left(Y^{0 b} dt\right) \end{align} \tag{ A96 }$$

$$\begin{align} &= L_{b} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} b} dt - \dot{\eta}^{\underline{a}} Y^{0 b} dt\right) + L_b \eta_{\varsigma}\left(Y^{0 b}\right) \end{align} \tag{ A97 }$$

$$\begin{align} &= L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} \underline{b}} dt - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} dt\right) + L_{0} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} 0} dt - \dot{\eta}^{\underline{a}} Y^{0 0} dt\right) + L_{\underline{b}} \eta_{\varsigma}\left(Y^{0 \underline{b}}\right) + L_0 \eta_{\varsigma}\left(Y^{0 0}\right) \end{align} \tag{ A98 }$$

$$\begin{align} &= L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} \underline{b}} dt - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} dt\right) + L_{\dot{\eta}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} 0} dt - \dot{\eta}^{\underline{a}} Y^{0 0} dt\right) -\dot{\eta}^{\underline{b}} L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} 0} dt - \dot{\eta}^{\underline{a}} Y^{0 0} dt\right)\nonumber \\ &\quad + L_{\underline{b}} \eta_{\varsigma}\left(Y^{0 \underline{b}}\right) + L_{\dot{\eta}} \eta_{\varsigma}\left(Y^{0 0}\right) - \dot{\eta}^{\underline{a}} L_{\underline{a}} \eta_{\varsigma}\left(Y^{0 0}\right) \end{align} \tag{ A99 }$$

$$\begin{align} & = L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} \underline{b}} dt - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} dt\right) + L_{\dot{\eta}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} 0} dt - \dot{\eta}^{\underline{a}} Y^{0 0} dt\right)\nonumber \\ &\quad - L_{\underline{b}} \dot{\eta}^{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} 0} dt - \dot{\eta}^{\underline{a}} Y^{0 0} dt\right) + L_{\underline{b}} \eta_{\varsigma}\left(Y^{0 \underline{b}}\right) + L_{\dot{\eta}} \eta_{\varsigma}\left(Y^{0 0}\right) - L_{\underline{a}} \dot{\eta}^{\underline{a}} \eta_{\varsigma}\left(Y^{0 0}\right)\nonumber \\ &\quad + i_{\underline{a}} \eta_{\varsigma}\left(Y^{00} d\dot{\eta}^{\underline{a}}\right) + i_{\underline{b}} i_{\underline{a}} \eta_{\varsigma} \left( d\dot{\eta}^{\underline{b}} \wedge\left( Y^{\underline{a} 0} - \dot{\eta} Y^{00}\right) dt \right) \end{align} \tag{ A100 }$$

$$\begin{align} &= L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} \underline{b}} dt - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} dt - \dot{\eta}^{\underline{b}} Y^{\underline{a} 0} dt + \dot{\eta}^{\underline{a}} \dot{\eta}^{\underline{b}} Y^{0 0} dt\right) + L_{\underline{b}} \eta_{\varsigma}\left(Y^{0 \underline{b}} - \dot{\eta}^{\underline{b}} Y^{00}\right)\nonumber \\ &\quad + i_{\underline{a}} \eta_{\varsigma} \left(\frac{dY^{\underline{a} 0}}{dt} dt - \frac{d}{dt} \left( \dot{\eta}^{\underline{a}} Y^{0 0} \right) dt + Y^{00} d\dot{\eta}^{\underline{a}} \right) + \eta_{\varsigma} \left(\frac{dY^{0 0}}{dt} \right). \end{align} \tag{ A101 }$$

The U^a term,

$$\begin{align} L_a \eta_{\varsigma} \left(U^{a}\right) & = L_{\underline{a}} \eta_{\varsigma} \left(U^{\underline{a}}\right) + L_{\dot{\eta}} \eta_{\varsigma} \left(U^{0}\right) - \dot{\eta}^{\underline{a}} L_{\underline{a}} \eta_{\varsigma} \left(U^0\right) \end{align} \tag{ A102 }$$

$$\begin{align} & = L_{\underline{a}} \eta_{\varsigma} \left(U^{\underline{a}} - \dot{\eta}^{\underline{a}} U^0\right) + i_{\underline{a}} \eta_{\varsigma} \left(U^0 d \dot{\eta}^{\underline{a}}\right) + \eta_{\varsigma} \left(\frac{dU^{0}}{dt} \right). \end{align} \tag{ A103 }$$

Lastly the Y^a term,

$$\begin{align} i_a \eta_{\varsigma} \left(Y^{a} dt\right) & = i_{\underline{a}} \eta_{\varsigma} \left(Y^{\underline{a}} dt\right) + i_{\dot{\eta}} \eta_{\varsigma} \left(Y^{0} dt\right) - i_{\underline{a}} \dot{\eta}^{\underline{a}} \eta_{\varsigma} \left(Y^0 dt\right) \end{align} \tag{ A104 }$$

$$\begin{align} & = i_{\underline{a}} \eta_{\varsigma} \left(Y^{\underline{a}} dt - \dot{\eta}^{\underline{a}} Y^0 dt\right) + \eta_{\varsigma} \left(Y^0\right). \end{align} \tag{ A105 }$$

Summing these together,

$$\begin{equation} \begin{aligned} \Psi_Q &= \frac{1}{2} L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(U^{\underline{a} \underline{b}} - \dot{\eta}^{\underline{b}} U^{\underline{a} 0} - \dot{\eta}^{\underline{a}} U^{\underline{b} 0} + \dot{\eta}^{\underline{a}} \dot{\eta}^{\underline{b}} U^{00} \right) \\ &\quad - L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} \underline{b}} dt - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} dt - \dot{\eta}^{\underline{b}} Y^{\underline{a} 0} dt + \dot{\eta}^{\underline{a}} \dot{\eta}^{\underline{b}} Y^{0 0} dt - \left(U^{0 \underline{b}} - \dot{\eta}^{\underline{b}} U^{0 0}\right) d\dot{\eta}^{\underline{a}} \right) \\ &\quad - L_{\underline{a}} \eta_{\varsigma} \left(U^{\underline{a}} - \dot{\eta}^{\underline{a}} U^0 + Y^{0 \underline{a}} - \dot{\eta}^{\underline{a}} Y^{00}\right) - \frac{1}{2} L_{\underline{a}} \eta_{\varsigma} \left( \dot{\eta}^{\underline{a}} \frac{d U^{00}}{dt} + \frac{d}{dt} \left( \dot{\eta}^{\underline{a}} U^{0 0} \right) - 2 \frac{d{U}^{\underline{a} 0}}{dt} \right) \\ &\quad + i_{\underline{a}} \eta_{\varsigma} \left(Y^{\underline{a}} dt - \dot{\eta}^{\underline{a}} Y^0 dt - \frac{dY^{\underline{a} 0}}{dt} dt - U^{0} d\dot{\eta}^{\underline{a}} + \frac{d}{dt} \left( \dot{\eta}^{\underline{a}} Y^{0 0} \right) dt - Y^{00} d\dot{\eta}^{\underline{a}} \right) \\ &\quad + \frac{1}{2} i_{\underline{a}} \eta_{\varsigma} \left(\frac{d{U}^{0 0}}{dt} d\dot{\eta}^{\underline{a}} + \frac{d}{dt} \left(U^{0 0} d \dot{\eta}^{\underline{b}} \right) \right) + \frac{1}{2} \eta_{\varsigma} \left(\frac{d^2{U}^{0 0}}{dt^2} \right) - \eta_{\varsigma} \left(\frac{dY^{0 0}}{dt} \right) - \eta_{\varsigma} \left(\frac{dU^{0}}{dt} \right) + \eta_{\varsigma} \left(Y^0\right) + \eta_{\varsigma}\left(q\right). \end{aligned} \end{equation} \tag{ A106 }$$

To simplify this, recall (29), giving

$$\begin{align} \Psi_Q &= \frac{1}{2} L_{\underline{a}} L_{\underline{b}} \eta_{\varsigma} \left(U^{\underline{a} \underline{b}} - \dot{\eta}^{\underline{b}} U^{\underline{a} 0} - \dot{\eta}^{\underline{a}} U^{\underline{b} 0} + \dot{\eta}^{\underline{a}} \dot{\eta}^{\underline{b}} U^{00} \right)\nonumber \\ &\quad - L_{\underline{b}} i_{\underline{a}} \eta_{\varsigma}\left(Y^{\underline{a} \underline{b}} dt - \dot{\eta}^{\underline{a}} Y^{0 \underline{b}} dt - \dot{\eta}^{\underline{b}} Y^{\underline{a} 0} dt + \dot{\eta}^{\underline{a}} \dot{\eta}^{\underline{b}} Y^{0 0} dt + \left(U^{0 \underline{b}} - \dot{\eta}^{\underline{b}} U^{0 0}\right) d\dot{\eta}^{\underline{a}} \right) - L_{\underline{a}} \eta_{\varsigma} \left(U^{\underline{a}} - \dot{\eta}^{\underline{a}} U^0 + \frac{1}{2} \frac{d}{dt} \left( \dot{\eta}^{\underline{a}} U^{0 0} \right) - Y^{\underline{a} 0} \right) \nonumber\\ &\quad + i_{\underline{a}} \eta_{\varsigma} \left(Y^{\underline{a}} dt - \dot{\eta}^{\underline{a}} Y^0 dt - U^{0} d\dot{\eta}^{\underline{a}} - \frac{dY^{\underline{a} 0}}{dt} dt + \frac{1}{2} \frac{d^2}{dt^2} \left( \dot{\eta}^{\underline{a}} U^{0 0} \right) dt \right) + \eta_{\varsigma}\left(q\right). \end{align} \tag{ A107 }$$

This is the full coordinate transformation for the quadrupole in the de Rham current representation.