In this section equations will be derived that should allow the future numerical assessment of the scalar field *p* and the interaction energy in a given EM field and a given *weak and slowly varying* gravitational field. To achieve this, the results of the asymptotic framework developed in some detail in Sects. 4 and 5 of Ref. [4] will be used. The only change with respect to that former work is the additional term −*G*^{μν} δ_{ν} (p) in Eq. (55), as compared with Eq. (22) in Ref. [4], and correspondingly the additional equation (57). With the additional equation (57), it is required that *δ*_{ν} (*p*) be of the same order as is *b*_{ν} (*T*_{field}) in the gravitational weak-field parameter *λ*, with *λ* = *c*^{−2} in specific *λ*-dependent units of mass and time. From Eqs. (34) and (36) in Ref. [4], for the order of *b*_{ν}:

$$\begin{array}{}{\displaystyle {b}_{\nu}({\mathit{T}}_{\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}})=\mathrm{o}\mathrm{r}\mathrm{d}({c}^{2n-5}),}\end{array}$$(59)

where

$$\begin{array}{}{\displaystyle \mathit{F}={c}^{n}\left(\stackrel{0}{\mathit{F}}+{c}^{-2}\phantom{\rule{thinmathspace}{0ex}}\stackrel{1}{\mathit{F}}+O({c}^{-4})\right)}\end{array}$$(60)

is the expansion of the EM field tensor. The gravitational field is expanded as (Eq. (28) in Ref. [4]):

$$\begin{array}{}{\displaystyle \beta :=\sqrt{{\gamma}_{00}}=1-U\phantom{\rule{thinmathspace}{0ex}}{c}^{-2}+O({c}^{-4}),}\end{array}$$(61)

where *U* is the Newtonian gravitational potential. Using this and setting

$$\begin{array}{}{\displaystyle p={c}^{q}\left(\stackrel{0}{p}+{c}^{-2}\phantom{\rule{thinmathspace}{0ex}}\stackrel{1}{p}+O({c}^{-4})\right),}\end{array}$$(62)

from (49) and (50) it is shown that *δ*_{μ} = ord(*c*^{q}), hence the requirement is satisfied iff *q* = 2*n* − 5 so that

$$\begin{array}{}{\displaystyle {p}_{,\mu}={c}^{2n-5}\left({\stackrel{0}{p}}_{,\mu}+O({c}^{-2})\right).}\end{array}$$(63)

Let us define

$$\begin{array}{}{\displaystyle \hat{\rho}:={\left({G}_{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\nu}^{\mu}\phantom{\rule{thinmathspace}{0ex}}{b}^{\nu}({\mathit{T}}_{\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}})\right)}_{;\mu},}\end{array}$$(64)

as in Eq. (23) in Ref. [4] — but now *ρ̂* ≠ *J*^{μ} _{;μ}, unlike in the latter work. With this definition, everything in Sects. 4 and 5 of Ref. [4] remains valid and we have in particular (Eq. (45) in Ref. [4]):^{5}

$$\begin{array}{}{\displaystyle \hat{\rho}={c}^{-3}{\left[\left({G}_{1}{\phantom{\rule{thinmathspace}{0ex}}}^{\mu 0}\phantom{\rule{thinmathspace}{0ex}}{T}_{1}{\phantom{\rule{thinmathspace}{0ex}}}^{jj}-{G}_{1}{\phantom{\rule{thinmathspace}{0ex}}}^{\mu i}\phantom{\rule{thinmathspace}{0ex}}{T}_{1}{\phantom{\rule{thinmathspace}{0ex}}}^{0i}\right){\mathrm{\partial}}_{T}U\right]}_{,\mu}(1}\\ {\displaystyle \phantom{\rule{1em}{0ex}}+O\left({c}^{-2}\right)),}\end{array}$$(65)

where *G*_{1} ^{μν} and *T*_{1} ^{μν} are the first approximations of *G* ^{μν} and *T*^{μν}, *i.e*.,

$$\begin{array}{}{\displaystyle {\mathit{G}}_{1}:=({\mathit{F}}_{1}{)}^{-1},\phantom{\rule{2em}{0ex}}{\mathit{F}}_{1}:={c}^{n}\stackrel{0}{\mathit{F}},}\end{array}$$(66)

and the like for *T*_{1} [4]. This can be calculated: as given by Eqs. (53) and (54) of Ref. [4],

$$\begin{array}{}{\displaystyle \hat{\rho}={c}^{-3}{\left({e}^{i}{\mathrm{\partial}}_{T}U\right)}_{,i}\left(1+O\left({c}^{-2}\right)\right),}\end{array}$$(67)

where

$$\begin{array}{}{\displaystyle {e}^{i}=\frac{1}{2\phantom{\rule{thinmathspace}{0ex}}c\phantom{\rule{thinmathspace}{0ex}}{\mu}_{0}\phantom{\rule{thinmathspace}{0ex}}\left({B}_{1}\phantom{\rule{thinmathspace}{0ex}}{E}_{1}+{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{E}_{2}+{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{E}_{3}\right)}\times}\\ {\displaystyle \left(\begin{array}{c}{c}^{2}\left({{B}_{1}}^{3}+{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{{B}_{2}}^{2}+{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{{B}_{3}}^{2}\right)+{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{{E}_{1}}^{2}-{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{{E}_{2}}^{2}-{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{{E}_{3}}^{2}\\ +2\phantom{\rule{thinmathspace}{0ex}}{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{E}_{1}\phantom{\rule{thinmathspace}{0ex}}{E}_{2}+2\phantom{\rule{thinmathspace}{0ex}}{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{E}_{1}\phantom{\rule{thinmathspace}{0ex}}{E}_{3}\\ {c}^{2}\left({{B}_{2}}^{3}+{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{{B}_{3}}^{2}+{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{{B}_{1}}^{2}\right)+{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{{E}_{2}}^{2}-{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{{E}_{3}}^{2}-{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{{E}_{1}}^{2}\\ +2\phantom{\rule{thinmathspace}{0ex}}{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{E}_{2}\phantom{\rule{thinmathspace}{0ex}}{E}_{3}+2\phantom{\rule{thinmathspace}{0ex}}{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{E}_{2}\phantom{\rule{thinmathspace}{0ex}}{E}_{1}\\ {c}^{2}\left({{B}_{3}}^{3}+{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{{B}_{1}}^{2}+{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{{B}_{2}}^{2}\right)+{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{{E}_{3}}^{2}-{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{{E}_{1}}^{2}-{B}_{3}\phantom{\rule{thinmathspace}{0ex}}{{E}_{2}}^{2}\\ +2\phantom{\rule{thinmathspace}{0ex}}{B}_{1}\phantom{\rule{thinmathspace}{0ex}}{E}_{3}\phantom{\rule{thinmathspace}{0ex}}{E}_{1}+2\phantom{\rule{thinmathspace}{0ex}}{B}_{2}\phantom{\rule{thinmathspace}{0ex}}{E}_{3}\phantom{\rule{thinmathspace}{0ex}}{E}_{2}\end{array}\right).}\end{array}$$(68)

In Eq. (68), *E*_{i} := *E*^{i} and *B*_{i} := *B*^{i} are the components of the first approximations of the electric and magnetic fields in the frame 𝓔, in coordinates of the class specified after Eq. (44). I.e., *E*^{i} and *B*^{i} are extracted (Note 2) from the first approximation *F*_{1} of the EM field tensor *F*, Eq. (66), that obeys the flat-spacetime Maxwell equations [4]. Inserting (63) and (67) into (57) using
$\begin{array}{}{\displaystyle {u}_{;\mu}^{\mu}=({u}^{\mu}\sqrt{-\gamma}{)}_{,\mu}/\sqrt{-\gamma},}\end{array}$
with
$\begin{array}{}{\displaystyle \sqrt{-\gamma}}\end{array}$
= 1 + *O*(*c*^{−2}) owing to (46) and (61), provides:

$$\begin{array}{}{\displaystyle \text{\hspace{0.17em}}{\left({G}_{1}^{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mu \nu}\phantom{\rule{thinmathspace}{0ex}}({p}_{1}{)}_{,\nu}\right)}_{,\mu}={G}_{1\phantom{\rule{1em}{0ex}},\mu}^{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mu \nu}\phantom{\rule{thinmathspace}{0ex}}({p}_{1}{)}_{,\nu}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}={c}^{-3}{\left({e}^{i}{\mathrm{\partial}}_{T}U\right)}_{,i}\left(1+O\left({c}^{-2}\right)\right),}\end{array}$$(69)

where *p*_{1} := *c*^{2n−5}
$\begin{array}{}{\displaystyle \stackrel{0}{p}}\end{array}$
is the first approximation of *p*. The first equality is due to the antisymmetry of *G*^{μν} and *G*_{1} ^{μν}. The matrix *G*′ := (*G*_{1} ^{μν}) is given explicitly by Eq. (50) in Ref. [4], which can be rewritten as

$$\begin{array}{}{\displaystyle {\mathit{G}}^{\prime}=\frac{-c}{\mathbf{E}\mathbf{.}\mathbf{B}}\phantom{\rule{thinmathspace}{0ex}}\mathit{H},\phantom{\rule{1em}{0ex}}\mathit{H}:=\left(\begin{array}{cccc}0& {B}_{1}& {B}_{2}& {B}_{3}\\ -{B}_{1}& 0& -\frac{{E}_{3}}{c}& \frac{{E}_{2}}{c}\\ -{B}_{2}& \frac{{E}_{3}}{c}& 0& -\frac{{E}_{1}}{c}\\ -{B}_{3}& -\frac{{E}_{2}}{c}& \frac{{E}_{1}}{c}& 0\end{array}\right).}\end{array}$$(70)

It is easy to check that Maxwell’s (flat-spacetime) first group, verified by *F*_{1}, implies that

$$\begin{array}{}{\displaystyle {H}_{,\mu}^{\mu \nu}=0.}\end{array}$$(71)

It follows from this and (70) that in (69):

$$\begin{array}{}{\displaystyle ({k}^{\nu}):=\left({G}_{1\phantom{\rule{1em}{0ex}},\mu}^{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mu \nu}\right)=}\\ {\displaystyle \frac{1}{{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}^{2}}\times \left(\begin{array}{c}-{B}_{i}\phantom{\rule{thinmathspace}{0ex}}c\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,i}\\ {E}_{3}\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,2}-{E}_{2}\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,3}+{B}_{1}\phantom{\rule{thinmathspace}{0ex}}c\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,0}\\ {E}_{1}\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,3}-{E}_{3}\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,1}+{B}_{2}\phantom{\rule{thinmathspace}{0ex}}c\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,0}\\ {E}_{2}\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,1}-{E}_{1}\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,2}+{B}_{3}\phantom{\rule{thinmathspace}{0ex}}c\phantom{\rule{thinmathspace}{0ex}}{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}_{,0}\end{array}\right),}\end{array}$$(72)

*i.e*.,

$$\begin{array}{}{\displaystyle {k}^{0}=\frac{-c}{{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}^{2}}\phantom{\rule{thinmathspace}{0ex}}\mathbf{B}\mathbf{.}\mathbf{(}\mathrm{\nabla}\mathbf{(}\mathbf{E}\mathbf{.}\mathbf{B}\mathbf{)}\mathbf{)},}\end{array}$$(73)

$$\begin{array}{}{\displaystyle ({k}^{i})=\frac{1}{{\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}^{2}}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{\mathrm{\partial}\left(\mathbf{E}\mathbf{.}\mathbf{B}\right)}{\mathrm{\partial}T}\mathbf{B}-\mathbf{E}\wedge (\mathrm{\nabla}(\mathbf{E}\mathbf{.}\mathbf{B}))\right).}\end{array}$$(74)

Equation (69) can be rewritten in the form

$$\begin{array}{}{\displaystyle {\mathrm{\partial}}_{T}\phantom{\rule{thinmathspace}{0ex}}{p}_{1}+{u}^{j}{\mathrm{\partial}}_{j}\phantom{\rule{thinmathspace}{0ex}}{p}_{1}=S,}\end{array}$$(75)

where

$$\begin{array}{}{\displaystyle S:=\frac{{c}^{-2}{\left({e}^{i}{\mathrm{\partial}}_{T}U\right)}_{,i}}{{k}^{0}}}\end{array}$$(76)

(no confusion can occur with the sum *S* in Subsect. 2.1), and

$$\begin{array}{}{\displaystyle {u}^{j}:=\frac{c\phantom{\rule{thinmathspace}{0ex}}{k}^{j}}{{k}^{0}}.}\end{array}$$(77)

We assume that *k*^{0} ≠ 0 in Eq. (72), *i.e*. **B**.(∇(**E.B**)) ≠ 0. Note that **E.B** ≠ 0 is required from Sect. 6. Note that here the first-approximation fields **E** and **B** are involved, and they obey the flat-spacetime Maxwell equations. Equation (75) is an advection equation with a given source *S* for the unknown field *p*_{1}. This is a hyperbolic PDE whose characteristic curves are the integral curves of the vector field **u** := (*u*^{j}). That is, on the curve 𝓒(*T*_{0}, **x**_{0}) defined by

$$\begin{array}{}{\displaystyle \frac{\text{d}\mathbf{x}}{\text{d}T}=\mathbf{u}(T,\mathbf{x}),\phantom{\rule{2em}{0ex}}\mathbf{x}({T}_{0})={\mathbf{x}}_{0},}\end{array}$$(78)

we have from (75):

$$\begin{array}{}{\displaystyle \frac{\text{d}{p}_{1}}{\text{d}T}=\frac{\mathrm{\partial}{p}_{1}}{\mathrm{\partial}T}+\frac{\mathrm{\partial}{p}_{1}}{\mathrm{\partial}{x}^{j}}\frac{\text{d}{x}^{j}}{\text{d}T}=S(T,\mathbf{x}).}\end{array}$$(79)

Note that the field **u** is *given*, Eq. (77), since it does not depend on the unknown field *p*_{1}. Therefore, the integral lines (78) are given, too, hence the characteristic curves do not cross. Thus, the solution *p*_{1} is obtained unambiguously by integrating (79):

$$\begin{array}{}{\displaystyle {p}_{1}(T,\mathbf{x}(T))-{p}_{1}({T}_{0},{\mathbf{x}}_{0})=\underset{{T}_{0}}{\overset{T}{\int}}S(t,\mathbf{x}(t))\phantom{\rule{thinmathspace}{0ex}}\text{d}t,}\end{array}$$(80)

where *T* ↦ **x**(*T*) is the solution of (78). If at time *T*_{0} the position **x**_{0} in the frame 𝓔 is far enough from material bodies, it can be assumed that *p*_{1}(*T*_{0}, **x**_{0}) = 0.

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