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\Chapter{Maxwell's equations}

\Section{Introduction}

Maxwell's equations and their immediate consequences for the
electromagnetic 2-form $\four F^\flat = \half \four F_{\alpha\beta}
\omega^\alpha \wedge \omega^\beta$, or simply $\four F_{\alpha\beta} $
in index notation, can be written covariantly in many ways.
In the notation of differential forms one has
\beq\meqalign{
   d \four F^\flat &= 0 \ , 
              & \quad  \rightarrow \quad && \four F^\flat &= d \four A \ , \cr
   -\delta \four F^\flat &= 4\pi \four J^\flat \ , & \quad \rightarrow\quad && 
        -\delta \four J &= 0 \ , \cr
}\eeq
or in terms of covariant derivatives
\beq\meqalign{
   \dual\four F^{\alpha\beta}{}_{;\beta} &= 0 \ , 
       & \quad  \rightarrow \quad &&
      \four F_{\alpha\beta} &= 2 \four A_{[\beta;\alpha]} \ , \cr
   \four F^{\alpha\beta}{}_{;\beta} &= 4\pi \four J^\alpha \ , 
        & \quad  \rightarrow \quad && 
       \four J^\alpha{}_{;\alpha}  &= 0 \ , \cr
}\eeq
or in terms of ordinary derivatives and components with respect to a frame
\beq \meqalign{
   & \four F_{\alpha\beta,\gamma} + \four F_{\beta\gamma,\alpha} 
        + \four F_{\gamma\alpha,\beta} & \cr
   & \qquad - \four F_{\delta\gamma} C^\delta{}_{\alpha\beta}
            - \four F_{\delta\alpha} C^\delta{}_{\beta\gamma}
            - \four F_{\delta\beta} C^\delta{}_{\gamma\alpha} =0\ ,\cr
   &  \four g^{-1/2} ( \four g \four F^{\alpha\beta})_{,\beta}
        - \four F^{\alpha\beta} C^\gamma{}_{\beta\gamma}
        - \half C^\alpha{}_{\beta\gamma} \four F^{\beta\gamma} =
            4\pi \four J^\alpha \ ,\cr
}\eeq
and
\beq\eqalign{
 & \four F_{\alpha\beta} = 2 \four A_{[\beta,\alpha]} - \four A_\gamma
               C^\gamma{}_{\alpha\beta} \ ,\cr
  &   \four g^{-1/2} ( \four g \four J^\alpha)_{,\beta} 
        - \four J^\alpha C^\gamma{}_{\alpha\gamma} =0 \ ,\cr
}\eeq
the latter four of which reduce to the more familiar coordinate component
expressions when $C^\alpha{}_{\beta\gamma} =0$.
The equivalence of these various forms of Maxwell's equations depends
on the fact that the metric covariant derivative is symmetric, since
otherwise torsion terms lead to differences between the covariant
derivative form and the exterior derivative forms.
The existence of a vector potential $\four A$ for the electromagnetic
field is of course local, and is associated with the identity
$d^2=0$. The ``conservation law" for the current 4-vector $\four J$
instead follows from the identity $\delta^2=0$.


\vskip24pt
\message{REWRITE!}REWRITE FROM HERE!

\Section{Splitting the electromagnetic field}

\Subsection{Congruence point of view}

The decomposition of the electromagnetic 2-form into a 1-form electric part
and a 2-form magnetic part is the first step in splitting the electromagnetic
field into an electric field and a magnetic field as observered
by a test observer with 4-velocity $u$.
The electric part 
The spatial dual of the
magnetic part of the electromagnetic 2-form defines the magnetic field
1-form \Cite{Lichnerowicz 1967}.

In index-free form one has
\beq\eqalign{
   E(u)^\flat &= \four F^\flat{} \E \ , \cr
   B(u)^\flat &= \dualp{u} F(u)^\flat \ , 
       \qquad F(u)^\flat = \four F^\flat{}\M \ , \cr
}\eeq
in terms of which the 2-form has the expression
\beq \eqalign{
 \four F^\flat &= u^\flat  \wedge E(u)^\flat  
           +\dual (u^\flat  \wedge B(u)^\flat ) \ , \cr
  \dual \,\four F^\flat &= -u^\flat  \wedge B(u)^\flat  
            +\dual (u^\flat  \wedge E(u)^\flat )  \ .\cr
}\eeq 

In index form one has instead
\beq\eqalign{
    & E(u)_\alpha = -u^\beta F_{\beta\alpha}\ , \qquad
    B(u)^\alpha = u_\beta \dual F^{\beta\alpha}
   =\half \eta(u)^{\alpha\beta\gamma}F_{\beta\gamma}\ ,\cr
  & F(u)_{\alpha\beta} = P(u)^\gamma{}_\alpha P(u)^\delta{}_\beta
          \four F_{\gamma\delta} \ .\cr
}\eeq 
and
\beq \eqalign{
F_{\alpha\beta} 
          &= \delta^{\gamma\delta}_{\alpha\beta} u_\gamma E(u)_\delta
 +\half \eta^{\gamma\delta}{}_{\alpha\beta} u_\gamma B(u)_\delta \ ,\cr
\dual F_{\alpha\beta} 
           &= -\delta^{\gamma\delta}_{\alpha\beta} u_\gamma B(u)_\delta
 +\half \eta^{\gamma\delta}{}_{\alpha\beta} u_\gamma E(u)_\delta \ .\cr
}\eeq 


It is sometimes useful to introduce the complex field $Z$ which is an
eigenvector of the duality operation
\beq\eqalign{
   \four Z^\flat &= \four F^\flat +i\,\dual\,\four F^\flat\ , 
   \qquad \hook \four Z^\flat = \four Z^\flat \ , \cr
   \four Z_{\alpha\beta} &= 
      \four F_{\alpha\beta} +i\,\dual\,\four F_{\alpha\beta} \ , 
   \qquad i \eta_{\alpha\beta}{}^{\gamma\delta}   \four Z_{\gamma\delta}
         = \four Z_{\alpha\beta}  \ , \cr
}\eeq 
and its spatial projections
\beq \eqalign{
& Z^{\bot\alpha} = E(n)^\alpha -i\, B(n)^\alpha= \Zscr(n)^\alpha \ ,\cr
& Z^{\top \alpha} = E(m)^\alpha -i\, B(m)^\alpha =\Zscr(m)^\alpha \ .\cr}
\eeq 

\Subsection{Slicing and threading points of view}

The congruence point of view formulas with the replacement $u \to o$
define the slicing and threading decompositions
of the electromagnetic field for $u=n$ and $u=m$ respectively.
The most familiar way of stating this result is by
specifying the observer-adapted components of the electric and magnetic fields,
both spatial with respect to the observer.

For the slicing point of view one has
\beq\meqalign{
& E(n)_a =-\four F_{\bot a} \qquad&&
  B(n)^a =-\dual \four F^{\bot a}= -N\dual \four F^{0a}
       =\half \eta(n)^{abc} \four F_{bc}\ ,\cr
& E(n)^a=F^{\bot a}=N \four F^{0a}\ , \qquad&&
 B(n)_a = \dual \four F_{\bot a}\ ,\cr
& \Escr(n)_a=g^{1/2}E(n)_a \qquad && \Bscr(n)^a=g^{1/2}B(n)^a\ .\cr
}\eeq 
Thus the spacetime fields are
\beq \meqalign{
& E(n)=E(n)^a e_a\ , \qquad && B(n)=B(n)^a e_a\ ,\cr
& E(n)^\flat =E(n)_a \theta^a\ , \qquad && B(n)^\flat =B(n)_a \theta^a\ .\cr
}\eeq 

For the threading point of view one has
\beq \meqalign{
 & E(m)^a = \four F^{\top  a} \qquad &&
      B(m)_a =\dual \four F_{\top  a}=  M^{-1} \dual \four F_{0 a}=  
  \half \eta(m)_{abc}\four F^{bc}\ ,\cr
 & E(m)_a =-\four F_{\top  a} =-M^{-1}\four F_{0a} \qquad &&
B(m)^a =-\dual \four F^{\top  a} \ ,\cr
}\eeq 
with
\beq \meqalign{
& E(m)= E(m)^a\epsilon_a \qquad &&
      B(m)=B(m)^a\epsilon_a\ ,\cr
&  E(m)^\flat = E(m)_a\omega^a \qquad &&
     B(m)^\flat =B(m)_a\omega^a\ .\cr
}\eeq 

\Subsection{Observer Boost}

The relationship between the electric and magnetic fields measured by
the two families of observers is rather simple
\beq \eqalign{
& P(m)E(n)=\gamma_L[ E(m)-\vec V \times_m B(m)]\ ,\cr
& P(m)B(n)=\gamma_L[ B(m)+\vec V \times_m E(m)]\ .\cr}
\eeq 
The inverse of the map $P(m)$ restricted to the space orthogonal to $n$
is just the natural projection $\Pscr$,
so another way of stating this relationship is
\beq \eqalign{
& E(n)=\Pscr\gamma_L[ E(m)-\vec V \times_m B(m)]\ ,\cr
& B(n)=\Pscr\gamma_L[ B(m)+\vec V \times_m E(m)]\ .\cr}
\eeq 
The simplicity of these formulas compared to the usual special relativistic
formulas for the transformation of electric and magnetic 
fields \Cite{Anderson 1967, Misner, Thorne and Wheeler 1973}
is due to the fact that the boost $B(m,n)$ which maps $n$ onto $m$ differs by 
an additional factor of $\gamma_L$ (associated with the Lorentz contraction
of lengths) along the direction of the relative velocity compared to
the projection $P(m)$ when acting on the local rest space of $n$.
The transformation of these formulas into the more familiar ones is discussed
in the appendix.

\Subsection{Reference representation (Landau-Lifshitz-Hanni)}

\message{change LL to reference representation?}

Landau and Lifshitz \Cite{1975}
and several related discussions instead decompose the
covariant and contravariant electromagnetic field tensor using the 
%natural splitting, 
reference decomposition splitting,
modulo scale factors, leading to a hybrid
approach involving a pair of electric and magnetic fields from each of
the observers $m$ and $n$ in the case of a timelike threading and a
spacelike slicing.
Landau and Lifshitz make the following component
definitions (corrected for signature difference)
\beq \eqalign{
 & E(LL)_a = -\four F_{0a} = M E(m)_a\ ,\cr
 & H(LL)_a = \dual \four F_{0a} = M B(m)_a=\half \four g{}^{1/2}
 \epsilon_{abc} \four F^{bc}\ ,\cr
 & D(LL)^a = M \four F^{0a} = \gamma_L{}^{-1} E(n)^a\ ,\cr
 & B(LL)^a = -M \dual \four F^{0a} =\gamma_L{}^{-1} B(n)^a
 =\half \gamma^{-1/2}\epsilon^{abc}\four F_{bc}\ .\cr}
\eeq 
The reference spatial vector fields
\beq  D(LL)=D(LL)^a e_a =\gamma_L{}^{-1} E(n)\ ,\qquad
 B(LL)=B(LL)^a e_a =\gamma_L{}^{-1} B(n)
\eeq 
are proportional to the fields associated with $n$, while
the reference spatial 1-forms
\beq  E(LL)^\flat =E(LL)_a\omega^a =ME(m)^\flat\ ,\qquad
H(LL)^\flat =H(LL)_a\omega^a =MB(m)^\flat
\eeq 
are proportional to the fields associated with $m$. 
%
The corresponding
vector fields are
\beq  E(LL) =E(LL)^a\epsilon_a =ME(m)\ ,\qquad
H(LL)=H(LL)^a\epsilon_a =MB(m)\ .
\eeq 
Spatial indices are shifted with the reference spatial metric
$\gamma_{ab} \omega^a \otimes \omega^b$ and its reference spatial inverse
as discussed in section \ref{sec:rr} for the reference representation
of the threading point of view.


The interpretation      of the Landau-Lifshitz fields in terms
of the slicing or threading observers is valid only when the appropriate
causality properties of the nonlinear reference frame hold, which is often
the case in applications. However, even when neither causality property
holds, one can make this decomposition, the interpretation of which one
must then investigate.

The initial definitions of Landau and Lifshitz, before shifting indices,
lead to a decomposition of the electromagnetic 2-form into a pair of
reference spatial 1-forms $E(LL)^\flat$ and $H(LL)^\flat$
coming from the covariant
components of the electric part of the 
2-form and its dual
(equivalently from the covariant components of $Z$)
and a pair of reference
vector fields $D(LL)$ and $B(LL)$ coming from the contravariant
components of the electric part of the 2-form and its dual
(equivalently from the contravariant components of $Z$),
as expressed in the computational frame.
On the other hand the alternative
pairing $(E(LL)^\flat,B(LL))$ corresponds to the
electric and magnetic parts of the
2-form alone and $(D(LL),H(LL)^\flat)$ to its
contravariant components or to the covariant components of its dual.
     
%%%%%% THE NEXT THREE PARAGRAPHS NEED WORK.
     
The raising of indices on the 2-form and its dual
with respect to the spacetime metric
(equivalently taking $Z$ to $Z^\sharp$)
can then be interpreted as a linear map from the
first pair $(E(LL)^\flat,H(LL)^\flat)$ to the second pair
$(D(LL),B(LL))$ in the first pairing
\beq \eqalign{
& D(LL)^a= M^{-1}\gamma^{ab}
E(LL)_b - \gamma^{-1/2}\epsilon^{abc}M_b H(LL)_c \ ,\cr
& B(LL)^a= M^{-1}\gamma^{ab}
H(LL)_b + \gamma^{-1/2}\epsilon^{abc}M_b E(LL)_c \ .\cr}
\eeq 
These linear relations between the spatial fields $(E,H)$ and
$(D,B)$ represent, apart from normalization factors,
just the Lorentz
transformation from the local rest space associated with $m$
to the one associated with $n$, when both $m$ and $n$ are timelike.
     
\begincomment
On the other hand, one may re-express these relations in terms of
the second pairing which represents the raising of indices on the
2-form alone to produce its electric contravariant components and
the lowering of the contravariant indices to produce the magnetic
components     %%%%% CHECK:
\beq \eqalign{
& D(LL)^a= N^{-1}(\gamma_L{}^{-1}g^{ab}E(LL)_b
   - \eta^{abc}N_b B(LL)_c) \ ,\cr
& B(LL)^a= N^{-1}(\gamma_L{}^{-1}g^{ab} H(LL)_b
   + \eta^{abc}N_b D(LL)_c) \ .\cr}
\eeq 
\endcomment
     
Several variations of this hybrid scheme appear in the literature.
All use the same definitions of the covariant spatial components of
$E$ and $H$, but differ on the contravariant spatial components of
$D$ and $B$ by a scale factor, and use different spatial metrics.
Closely related to the Landau-Lifshitz approach is the one of
Plebanski \Cite{1960}
who, following earlier work by Skrotsky \Cite{1957},
uses instead the normally projected densities $\Escr^a$ and $\Bscr^a$
\beq \eqalign{ 
& D(P)^a = \four g{}^{1/2} \four F^{0a} = \Escr^a =\gamma^{1/2} D(LL)^a\ ,\cr
& B(P)^a = -\four g{}^{1/2} \dual \four F^{0a} 
    = \Bscr^a =\gamma^{1/2} B(LL)^a\ .\cr
}\eeq 
He then uses natural contractions of contravariant and covariant indices
in dealing with Maxwell's equations and the energy-momentum tensor, thus
simulating a Euclidean metric.
He also introduces a different threading shift convention
\beq  g(P)_a = -g(LL)_a = -M_a\ .
\eeq 

Hanni \Cite{1977}
uses the normally projected vector fields
\beq  
     D(H)^a = E(n)^a = \gamma_L D(LL)\ , \qquad 
     B(H)^a = B(n)^a = \gamma_L B(LL)^a\ ,
\eeq 
but uses the slicing spatial metric to shift indices from their natural
positions and the shift vector rather than the threading shift to
describe the mixed spacetime metric components, with the notation
$g(H)_a = -N_a$.
This corresponds to the reference representation of the slicing point
of view for the electromagnetic field.
However, the spatial curl and divergence operators
in both the Landau-Lifshitz and Hanni approaches
are rescalings of the same natural differential operators,
and natural contractions and tensor products of dual fields must in any 
case be used to obtain physically interesting quantities 
(like energy-momentum).

The relationship among the various fields in this picture is given by
%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
\begincomment
\beq \eqalign{
& D(H)^a= N^{-1}g^{ab}(
E(H)_b + g^{1/2}\epsilon_{bcd}g(H)^c B(H)^d )\ ,\cr
& B(H)^a= N^{-1}g^{ab}(
H(H)_b - g^{1/2}\epsilon_{bcd}g(H)^c D(H)^d )\ .\cr}
\eeq  \endcomment
\beq \eqalign{
& E(H)_a= N g_{ab} D(H)^b + g^{1/2}\epsilon_{abc}N^b B(H)^c \ ,\cr
& H(H)_a= N g_{ab} B(H)^b - g^{1/2}\epsilon_{abc}N^b D(H)^c \ ,\cr
}\eeq 
which represents the inverse Lorentz transformation from the local
rest space of $n$ to that of $m$, when $n$ and $m$ are both timelike.

By working with the densities and natural contractions in a coordinate
frame, Plebanski is
able to reduce the Maxwell equations to their form
in flat spacetime cartesian coordinates, appropriate for use with
asymptotically cartesian coordinate systems.
The other approaches use a curved 3-space,
but one in which the spatial metric really plays a superficial role
since natural contractions replace metric contractions.
In each case the Lorentz boost/projection which relates the pair $(E,H)$ to
$(D,B)$ allows an interpretation as the constituitive relations of
an equivalent material medium in motion \Cite{Tamm 1924, 
Sommerfeld 1964}, 
although this analogy has never really been satisfactorily discussed.
A review of this idea in connection with the Sagnac effect was given
by Post \Cite{1967}, 
while a more recent discussion appears in the text
by Van Bladel \Cite{1984}. 

\Section{Splitting the 4-current}

To split Maxwell's equations one must first split 
the spacetime current-density vector field $\four J$
(not to be confused with a vector density).
This 4-current field decomposes into the
the charge density $\rho(u)$ and the current density $J(u)$ measured
by the test observers with four-velocity
$u$ in the congruence point of view
\beq
   \four J = \rho(u) u + J(u)\ , \qquad
   \four J^\alpha = \rho(u) u^\alpha + J(u)^\alpha\ .
\eeq

The reference, slicing and threading decompositions
and the rescaled reference decomposition used by Landau and Lifshitz are
instead
\beq \eqalign{
\four J &= \four J^0 + \four J^\alpha e_\alpha \cr
       &= \rho(n) e_\bot +J(n)^a e_a \cr
       &= \rho(m) e_\top +J(m)^a \epsilon_a \cr
       &= M^{-1}[\rho(LL) e_0 +J(LL)^a e_a]\ ,\cr
}\eeq
where the charge densities are defined by
\beq \eqalign{
\rho(n) &=\four J^\bot =N \four J^0\ ,\cr
\rho(m) &=\four J^{\top } =-M^{-1} \four J_0\ ,\cr
\rho(LL)&= M \four J^0 =\gamma_L{}^{-1}\rho(n)\ ,\cr
}\eeq
while the spatial current vectors are defined by
\beq \meqalign{
J(n)^a &= g^{ab}\four J_b\ , \qquad & J(n)=J(n)^a e_a\ ,\cr
J(m)^a &= \four J^a\ , \qquad & J(m)=J(m)^a \epsilon_a\ ,\cr 
J(LL)^a&= M \four J^a\ , \qquad & J(LL)=J(LL)^a e_a=M\Pscr J(m)\ .\cr
}\eeq 
For the Landau-Lifshitz definitions it is assumed that 
both $n$ and $m$ are timelike.

Note that in the Landau-Lifshitz approach, the fields associated with the
observer $n$ are scaled by the inverse gamma factor $\gamma_L{}^{-1}$,
while those associated with $m$ are scaled by the dual lapse $M$.
Absorbing the factor of $M$ into their definitions is a matter of convenience
so that it does not appear explicitly in Maxwell's
equations. Apart from this factor, the spatial current density projects onto
the one measured by the threading observers, while the charge density is
defined so that the Lorentz contraction factor of the spatial volume is exactly
compensated for when integrating against the spatial volume three-form
$\eta(m)$ on a time slice
\beq \eqalign{
& \rho(LL)\gamma^{1/2} =\rho(n) g^{1/2}\ ,\cr
&\rho(LL)\eta(m)\,|_{slice} = \rho(n) \eta(n)\,|_{slice}\ .\cr}
\eeq 
One cannot in general integrate the charge density orthogonally to
$m$ since a vector field does not admit a family of orthogonal
hypersurfaces unless its rotation vanishes.

\Subsubsection{Charge conservation}

The vanishing of the divergence of the 4-current can be expressed
either using the formula for the splitting of the operator $\delta$ for a
1-form or for the covariant divergence of a vector field. In the former
case only the magnetic part of the resulting 0-form exists by default.
One finds
\beq
    -[\delta \four J]\M = \four J^\alpha{}_{;\alpha} =
           [\Lie(u)\sub{u} + \Theta(u)] \rho(u) +
              [ \del(u)_\alpha + a(u)_\alpha ] J(u)^\alpha = 0 \ .
\eeq

\message{major work here:??}
To express charge conservation in the remaining points of view, one must
manipulate these formulas or instead
decompose the component formula for the divergence of a vector field
\beq 
X^\alpha{}_{;\alpha} =
\four g^{-1/2} \partialslash_\alpha(\four g^{1/2} X^\alpha)=
\four g^{-1/2} \partialslash_0(\four g^{1/2} X^0)+
\four g^{-1/2} \partialslash_a(\four g^{1/2} X^a)\ ,
\eeq 
where the slashed partial notation for the computational frame derivatives
stands for
\beq \partialslash_\alpha
\equiv e_\alpha-C^\beta{}_{\alpha\beta} =
e_\alpha-\delta^a{}_\alpha C^b{}_{ab} \ .
\eeq 
One easily derives the following three versions of this formula corresponding
to the slicing, threading and Landau-Lifshitz
rescaled reference decompositions
\beq \eqalign{
\four \div \four X
&= N^{-1}[g^{-1/2} \Lie\sub{\epsilon_0}
   (g^{1/2}X^\bot) +\div\, (NP(n)X)] \cr
&= M^{-1}[\gamma^{-1/2} \Lie\sub{e_0}(\gamma^{1/2}X^{\top })
   +\div_m\, (MP(m)X) + MX^a \partial_0 M_a] \cr
&= M^{-1}[\gamma^{-1/2} \Lie\sub{e_0}(\gamma^{1/2}MX^0)
   +\div_{LL}\, (M\Pscr X)]
\ ,\cr}
\eeq 
where the symbol $\Lie\sub{\epsilon_0}=\Lie\sub{e_0}-\Lie\sub{\vec N}$ must
be interpreted as acting on a spacetime or spatial scalar density.
This distinction is irrelevant for the Lie derivative $\Lie\sub{e_0}$, which
simply produces the $e_0$ derivative of the scalar density expression, but the
shift derivative has the following form
\beq  \Lie\sub{\vec N} \Fscr = (N^a\partial_a +\partialslash_a N^a) \Fscr\ .
\eeq 
In the threading expression for the divergence, the second two terms inside
the square brackets may be re-arranged as
\beq 
(\div_m -\vec g(m)\cdot_m)\, P(m)X =M \four \div P(m)X \ .
\eeq 
which is proportional to the spacetime divergence of the spatial projection
of the vector field.
     
With the above definitions, one finds the results
\beq \eqalign{
0=\four \div \four J
&= N^{-1}[g^{-1/2} \Lie\sub{\epsilon_0}
   (g^{1/2}\rho(n)) +\div_n\, NJ(n)] \cr
&= M^{-1}[\gamma^{-1/2}\Lie\sub{e_0} (\gamma^{1/2}\rho(m))
   +M(\div_m -\vec g(m)\cdot_m) \, J(m)] \cr
&= M^{-1}[\gamma^{-1/2}\Lie\sub{e_0} (\gamma^{1/2}\rho(LL))
   +\div(\Pscr,m)\, J(LL)]
\ .\cr}
\eeq 
Since the divergence may be expressed in terms of the exterior derivative,
the slicing version of this equation corresponds exactly to the
congruence point of view associated with the normal congruence.
     
\Section{Splitting Maxwell's equations}

\Subsection{Congruence point of view}

The splitting of the exterior derivative and its adjoint divergence
operator for a 2-form together with the definitions of the electric
and magnetic fields
immediately leads to the splitting of the differential form
version of Maxwell's equations in the congruence point of view.
Equivalently one can use the covariant divergence form of Maxwell's equations
and the formulas of Eq.(\ref{eq:atdsplit}) for the
splitting of the covariant divergence of a 2-form to obtain the same
result.

The source equations are
\beq \imeqalign{
   \four F^{\top \beta}{}_{;\beta} &= 4\pi \four J^\top : \qquad &
   [\delta \four F ]\E &= \div(u) E(u) - 2 \vec \omega(u) \cdot_u B(u)
                          = 4 \pi \rho(u) \ ,\cr
   \four F^{a \beta}{}_{;\beta} &= 4\pi \four J^a : \qquad &
   [\delta \four F ]\M &= [\curl(u) + a(u)\times_u] B(u) 
                - [\Lie(u)\sub{u} + \Theta(u)] E(u) = 4 \pi J(u) \ ,\cr
}\eeq
while the source-free equations are
\beq \imeqalign{
   \dual \four F^{\top \beta}{}_{;\beta} &= 0 : \qquad &
   [\delta \dual \four F ]\E &= 
        -[ \div(u) B(u) + 2 \vec \omega(u) \cdot_u E(u) ] =0 \ ,\cr
   \dual \four F^{a \beta}{}_{;\beta} &= 0 : \qquad &
   [\delta \dual \four F ]\M &= [\curl(u) + a(u)\times_u] E(u) 
                + [\Lie(u)\sub{u} + \Theta(u)] B(u) =0\ .\cr
}\eeq
\message{check??}

In an observer-adapted frame, the Lie derivative term in Maxwell's equations
may be re-expressed using the following formula for a spatial vector field $X$
\beq
  [ \Lie(u)\sub{u} + \Theta(u) ] X^a = (Mh^{1/2})^{-1} [h^{1/2} X^a]_{,0}
         + M^{-1} C^a{}_{0b} X^b \ .
\eeq


\Subsection{Slicing and threading points of view}

%\Section{Vector potential}
%\Section{Wave equations}
%\Section{Lines of force}

\vskip 12pt
REWRITE REST:

\Newpage


\Subsubsection{Divergence equations}

The spatial divergence equations for the slicing and Landau-Lifshitz variables
result from the normal projection of
Maxwell's equations
\beq  \eqalign{
F^{0\alpha}{}_{;\alpha} =& \four g^{-1/2}\partialslash_\alpha
(\four g^{1/2} F^{0\alpha})
=N^{-1}\left[ g^{-1/2}\partialslash_a (\Escr^a) \right]
=M^{-1}\left[ \gamma^{-1/2}\partialslash_a (\gamma^{1/2}D(LL)^a) \right]\cr
&=4\pi J^0 =4\pi N^{-1} \rho =4\pi M^{-1}\rho(LL) \ .\cr}
\eeq 
The expressions in square brackets are just the slicing and Landau-Lifshitz
covariant spatial divergences
of the vector fields $E=E(n)$ and $D(LL)$, respectively
\beq  g^{-1/2} \div \Escr=
 \div E = 4\pi \rho\ ,\qquad \div(\Pscr,m) \,D(LL) =4\pi \rho(LL)\ .
\eeq 
Similarly the dual equation leads to
\beq  g^{-1/2} \div \Bscr=
\div B = 0\ ,\qquad \div(\Pscr,m)\,B(LL) =0\ .
\eeq 
On the other hand, the projection along $m$ leads to the spatial divergence
equation in the threading point of view
\beq \eqalign{
&- M^{-1}F_0{}^\alpha{}_{;\alpha} =\div_m E(m) -\vec H(m)\cdot_m B(m)
     = 4\pi \rho(m)\ ,\cr
& M^{-1}\dual F_0{}^\alpha{}_{;\alpha} =\div_m B(m) +\vec H(m)\cdot_m E(m)
     =0\ .\cr}
\eeq 
Replacing the  spatial divergence of the spatial vector fields in terms of
the spacetime divergence leads to the 
Ellis form of these equations based on the congruence point of 
view \Cite{Ellis 1973}.

\Subsubsection{Curl equations}

The fields $E(LL)$ and $H(LL)$
share a common scaling factor of the threading lapse relative
to the projections orthogonal to $m$. These enter into
the remaining equations in the Landau-Lifshitz approach which
result from the projection of Maxwell's equations
orthogonal to $m$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
\beq \eqalign{
M F^{a\alpha}{}_{;\alpha} =& M[\four g^{-1/2}\partialslash_\alpha
(\four g^{1/2} F^{a\alpha}) - \half C^a{}_{bc}F^{bc}]\cr
=&M[\four g^{-1/2}\partial_0 (\four g^{1/2} F^{a0})
+\four g^{-1/2}\partialslash_b (\four g^{1/2} F^{ab})
  -\half C^a{}_{bc}F^{bc} ]\cr
 =&-\gamma^{-1/2}\partial_0(\gamma^{1/2} D(LL)^a) +
[\gamma^{-1/2}(\epsilon^{abc}\partialslash_b H(LL)_c
-\half C^a{}_{bd}\epsilon^{bdc}H(LL)_c]
 =4\pi J(LL)^a\ .\cr}
\eeq 
where the expression in square brackets on the last line
may be shown to be equivalent to
\beq 
\gamma^{-1/2}\epsilon^{abc}(\partial_b H(LL)_c
-\half C^d{}_{bc}H(LL)_d)
=\gamma^{-1/2}\epsilon^{abc}(d H(LL))_{bc}
 \ ,
\eeq 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
which coincides with the
Landau-Lifshitz curl operator.
This leads to
\beq  
    - \gamma^{-1/2}\Lie\sub{e_0}(\gamma^{1/2} D(LL)) +
      \curl(\Pscr,m)\, H(LL) =4\pi J(LL)\ ,
\eeq 
where the Lie derivative is equivalent to the $e_0$ derivative of the
computational components.
Similarly the dual equation leads to
\beq  
    \gamma^{-1/2}\Lie\sub{e_0}(\gamma^{1/2} B(LL)) +
                \curl_{LL} E(LL) =0\ .
\eeq 
     
Expressing these same equations in terms of the threading variables leads
instead to the equations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
\beq \eqalign{
&- M^{-1}\gamma^{-1/2}\Lie\sub{e_0}(\gamma^{1/2} E(m)) +
   M^{-1}\curl_m M B(m) + (\Lie(m)\sub{e_0}\overeq M)^\sharp\times_m B(m)
   =4\pi J(m)\ ,\cr
& M^{-1}\gamma^{-1/2}\Lie\sub{e_0}(\gamma^{1/2} B(m))
   +(\Lie(m)\sub{e_0}\overeq M)^\sharp\times_m E(m)
   M^{-1}\curl_m M E(m) =0\ \cr}
\eeq 
or equivalently
\beq \eqalign{
&- M^{-1}\gamma^{-1/2}\Lie\sub{e_0}(\gamma^{1/2} E(m)) +
   (\curl_m  -\vec g(m)\times_m)\, B(m)
   =4\pi J(m)\ ,\cr
& M^{-1}\gamma^{-1/2}\Lie\sub{e_0}(\gamma^{1/2} B(m)) +
   (\curl_m  -\vec g(m)\times_m)\, E(m)
   =0\ \cr}
\eeq 
These are the Lie form of the evolution equations
For a threading spatial vector field as in these equations, the Lie
derivative coincides with the projected Lie derivative.
Replacing this projected Lie derivative by the equivalent expression in terms of
the spatial Fermi-Walker derivative leads to the 
Fermi-Walker form of these same equations given by Ellis \Cite{1973}. (?? TO DO)

The slicing versions of these equations come from the tangential projection
of Maxwell's equations.
These equations take the form
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
\beq \meqalign{
4\pi \four J =& g^{1/2} g^{ab} \four F_b{}^\alpha{}_{;\alpha}e_a &=
-N^{-1}g^{-1/2}\Lie\sub{\epsilon_0} (g^{1/2}E(n))
  +N^{-1}\curl_n (NB(n))\ ,\cr
0=& g^{1/2} g^{ab} \dual\four F(n)_b{}^\alpha{}_{;\alpha} e_a   &=
N^{-1}g^{-1/2}\Lie\sub{\epsilon_0}(g^{1/2}B(n)) +
  N^{-1}\curl_n (NE(n))\ ,\cr}
\eeq 
where the relevant Lie derivative of a spatial density $\Yscr=\Yscr^ae_a$
is defined by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
\beq  \eqalign{
&\Lie\sub{\epsilon_0} \Yscr = (\partial_0 \Yscr^a) e_a -\Lie\sub{\vec N}
\Yscr\ ,\cr
&\Lie_{\vec N} \Yscr = [\vec N,\Yscr] +(\partialslash_a  N^a)\Yscr \ .\cr}
\eeq 
??
     
The factors of the lapse which enter into
Maxwell's equations arise essentially
because of the difference in the spatial and
spacetime metric duality operation on differential forms which use the
volume 3-form and 4-form respectively.
The threading approach leads to equations first expressed by 
Benvenuti  \Cite{1960}
which are equivalent to the
congruence approach of Ellis \Cite{1973}
when the latter version of Maxwell's equations
are expressed in terms of projected Lie derivatives with respect to
a choice of parametrization for the congruence and in terms of the
spatial divergence and curl.
The slicing approach to
Maxwell's equations is equivalent to the Ellis equations for the normal 
congruence with similar qualifications.

\Section{Vector potential}

Maxwell's equations 
imply that the electromagnetic field 2-form is closed and therefore
locally representable as the exterior derivative of a potential
1-form $\four A$.
The splitting of this simple four-dimensional relation in the various
points of view is easily performed.
First the potential 1-form is split 
\beq
\four A = \phi(u) u^\flat  + A(u) \ ,
\eeq
which leads to the slicing and threading decompositions, together with
the reference and Landau-Lifshitz decompositions
\beq \eqalign{
\four A &= -\phi(n) \omega^\bot +A(n)_a \theta^a \cr
        &= -\phi(m)\theta^{\top } +A(m)_a \omega^a \cr
        &= \four A_0 \omega^0 + \four A_a \omega^a \cr
        &= -\phi(LL)\omega^0 +A(LL)_a \omega^a\ ,\cr
}\eeq
where the scalar potentials are defined by
\beq \eqalign{
\phi(n) &=    -\four A_\bot    = N \four A^0\ , \cr
\phi(m) &= -\four A_{\top } =-M^{-1} \four A_0\ , \cr
\phi(LL)&= -\four A_0= M \phi(m) \ ,\cr
}\eeq
and the spatial vector potentials by
\beq \meqalign{
A(n)_a &=    \four A_a\ , \qquad & A(n)=A(n)_a \theta^a\ , \cr
A(m)_a &= \gamma_{ab}\four A^b\ , \qquad & A(m)=A(m)_a \omega^a\ , \cr
A(LL)_a &= \four A_a\ , \qquad & A(LL)=A(LL)_a \omega^a= \Pscr A\ .\cr
}\eeq 

Then one splits the exterior derivative in the respective approaches.
In this calculation it is convenient to temporarily
assume a spatial coordinate frame
where $C^a{}_{bc}=0$ in order not to worry about the complications of
which occur in formulas expressed in frames.

For example, the electric field in the slicing point of view is
\beq  \eqalign{
E(n)_a &= -\four F_{\bot a} = -N^{-1}( \four F_{0a} - N^b \four F_{ba}) \cr
       &= -N^{-1}(\four A_{a,0} - \four A_{0,a} - N^b \four F_{ba})\ , \cr}
\eeq 
the last equation being equivalent to equation (21.103) of Misner, Thorne
and Wheeler, who use the reference decomposition of the 1-form potential
rather than the slicing decomposition, a poor choice initially followed
by Arms \Cite{1979}.
The slicing decomposition of the potential is in
fact important in the Hamiltonian formulation of the coupled 
Einstein-Maxwell equations, as discussed by Arms, Marsden, and 
Moncrief \Cite{1982}
for the more general Einstein-Yang-Mills equations.
Re-expressing $\four A_0$ in terms of the normal component
and using the Lie derivative formula in a coordinate frame
referring to components along $\omega^a$
\beq  \eqalign{
\Lie_{\vec N} A_a &= A_{a,b} N^b + A_b N^b{}_{,a}
      = A_{a,b} N^b + (A_b N^b)_{,a} - A_{b,a} N^b\cr
     & = \four F_{ba} N^b + (A_b N^b)_{,a}\ ,\cr}
\eeq 
one finds
\beq   \eqalign{
E(n)_a &= - N^{-1} [-NA_{\bot}]_{,a}
            - N^{-1} [\Lie\sub{e_0} - \Lie_{\vec N}] A_a \cr
           &= - N^{-1} [-NA_{\bot}]_{,a}     -\Lie\sub{n} A_a \ ,\cr}
\eeq 
or 
\beq  
   E(n)^\flat = 
   -N^{-1} (\grad [N\phi])^\flat   - [P(n)\Lie]\sub{n} A
\eeq 
and
\beq  
 E(n)  = -N^{-1} \grad [N\phi]  -([P(n)\Lie]\sub{n} A)^\sharp
\eeq 
for the electric 1-form field and vector field respectively.
The calculation of the magnetic field is much simpler, giving easily
\beq 
  B(n) = \curl A^\sharp\ .
\eeq 
Note that the above Lie derivative coordinate frame formula is just
a special case of the expression
\beq  
\Lie_{X}\sigma = d (X\leftcontract \sigma) + X\leftcontract d\sigma
\eeq 
for the Lie derivative of a differential form $\sigma$, where 
$X\leftcontract \sigma$ refers to
contraction of the differential form on the left by the vector field $X$,
also written $i_X \sigma$ in an alternative notation.

A similar calculation for the threading point of view leads to the
equations
\beq \eqalign{
 E(m)  &= -M^{-1} \grad_m [M\phi(m)]  -\phi(m) \Lie\sub{e_0} \overeq M
   -(\Lie(m)\sub{m} A(m))^\sharp\ ,\cr
&= (\vec g(m) -\grad_m) \phi(m)  %-\phi(m) \Lie\sub{e_0} \overeq M 
   -(\Lie(m)\sub{m} A(m))^\sharp\ ,\cr
 B(m)  &= \curl_m A(m)^\sharp  +\phi(m)\vec H(m) \ ,\cr}
\eeq 
while the Landau-Lifshitz approach is the simplest
\beq  \eqalign{
 E(LL)^\flat  &= -d_{LL}\phi(LL)  -\Lie\sub{e_0} A(LL)\ ,\cr
  B(LL) &= \curl_{LL} A(LL)\ .\cr}
\eeq 

\Section{Wave equations}

Expressing Maxwell's equations in terms of the vector potential leads
to second order equations. Using the differential form approach one
immediately finds 
\beq
   \delta d \four A = \four \Delta\dR - d \delta A = 4\pi \four J\ .
\eeq
Imposing the Lorentz gauge condition $\delta \four A=0$ leads to the
wave equation
\beq
   = \four \Delta\dR A = 4\pi \four J
\eeq
involving the deRham Laplacian.

The equations for the 2-form itself also lead to a wave equation
involving the deRham Laplacian with the exterior derivative of the
4-current 1-form as a source
\beq
    \delta \four F^\flat = -4\pi \four J^\flat \ ,\qquad
 d \four F^\flat = 0 \qquad \rightarrow 
    \Delta\dR \four F = - 4 \pi d \four J^\flat \ .
\eeq

\Section{Computational 3-space representations}

Each of the above decompositions of the electromagnetic field and
its field equations may be explicitly represented on the
appropriate computational 3-space.

?? TO DO

\Section{Lines of force}

A very useful way of representing electric and magnetic fields in
flat spacetime is in terms of their lines of force, namely the integral
curves of these vector fields on space at a given moment of time.
Similar representations hold on the various computational 3-spaces for
a curved spacetime.

?? TO DO

