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\newcommand{\Prd}{Phys. Rev D}
\newcommand{\Prl}{Phys. Rev. Lett.}
\newcommand{\Plb}{Phys. Lett. B}
\newcommand{\Cqg}{Class. Quantum Grav.}
\newcommand{\Np}{Nuc. Phys.}
\newcommand{\Ap}{Ann. Phys.}
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\title{Causality and charged spin-2 fields in an electromagnetic background}
\author{M. Novello, S. E. Perez Bergliaffa, and R. P. Neves}
\affiliation{ Centro Brasileiro de Pesquisas F\'{\i}sicas\\Rua
Dr.\ Xavier Sigaud 150, Urca 22290-180\\ Rio de Janeiro,
Brazil}
\date{\today}

\begin{abstract}
We show that, contrary to common belief,
the propagation of a spin-2 field in an electromagnetic background is
{\em causal}. The proof will be given in the Fierz formalism which, as we shall see,
is free of the ambiguity present in the more usual Einstein representation.
We shall also
review the proof in this latter representation.

%\noindent
\end{abstract}


\pacs{03.50.-z, 11.90.+t}

\maketitle

 \renewcommand{\thefootnote}{\arabic{footnote}}

\section{Introduction}

%\subsection{General comments}

The interaction of fields with spin greater than 1 (particularly $s=2$ fields)
with a fixed
gravitational or electromagnetic background has attracted
a lot of attention in the last three decades.
There are at least two reasons for this interest.
First, on the theoretical side,
interacting particles with $s>1$ present
features that are absent in Electromagnetism.
In this regard, two
items are specially important: the consistency of the
equations of motion (EOM), and the causality of the propagation.
A consistent set of EOM (and of the constraints derived from them)
has been obtained quite a long time ago for free fields (see for instance
\cite{sh}), but the consistency of the system is usually broken when interactions
are introduced. Interactions can also excite new
degrees of freedom, absent in the free-field case. This may lead
to violation of causality. In fact, it has been stated that causality can be
violated even when the higher spin fields have the correct number of degrees
of freedom, and that in order to ensure consistency, some restrictions
must be made on the kind of interaction \cite{r1}.

Second, on a more phenomenological vein,
particles
with spin 2 are known to exist as resonances, and it is desirable to have a theory
to describe their interaction with background fields. Yet another reason is furnished by
string theory, in which a tower of massive states of Kaluza-Klein type
with various spins
interact with each other.
A particular instance
of this scenario is furnished by
the so-called bigravity models \cite{damour1, damour2}, from which
a theory with a massless and a very light graviton can be obtained.
It would be very interesting then to have a consistent
theory of fields with $s>1$ (and specifically of $s=2$ fields)
interacting with given backgrounds.

A lot of work has been devoted to the case of a gravitational background. In particular,
the properties of an $s=2$ field in this kind of background were studied
for instance in
\cite{aragones71,r1,r2,d2}. Several interesting new results in this area
were obtained in \cite{nn}.
We shall analyze here instead a spin 2 field in an electromagnetic (EM) background
\cite{cou}.
It is common lore in this situation that massive spin 2 particles propagate {\em acausally}.
This result was obtained by Kobayashi and Shamaly in the late 70's
\cite{ks}, and a more recent demonstration has been given by Deser and Waldron
\cite{ds}, both proofs being based on the method of characteristics.
The main result we shall present here is that, contrary to the aforementioned claims,
a more careful application of the method of characteristics
reveals that the propagation of $s=2$ fields
in an EM background is actually {\em causal}.

Let us remind the reader that
a spin-2 field can be described in two equivalent
ways, which we shall call the Einstein representation (ER)
and the Fierz representation (FR).
The former uses a symmetric second-order tensor
$\varphi_{\mu\nu}$ to represent the field. In the FR
\cite{fp} this
role is played by a third order tensor $F_{\alpha\mu\nu},$ which is
antisymmetric in the first pair of indices, and obeys the cyclic identity and
a further condition (which will be given in Sect.\ref{fabc}) in order to represent a
single spin-2 field.
In flat spacetime
and in the absence of interactions both representations are equivalent.
Nevertheless, in the case an EM background (or in
curved spacetime) this is no longer
true. As we shall see,
in the ER there is an ambiguity which originates in the ordering
of the non-commuting covariant derivatives.
We shall show that the use of the FR
yields instead a non-ambiguous description
with the minimal coupling procedure.

Our proof of the causal propagation will be given
in the FR. For those unfamiliar with these variables, we revise in the Appendix I
the proof of our statement in the more usual Einstein variables.
We shall begin by giving in Sect.\ref{fabc}
a review of the Fierz variables to describe a spin-2
field in Minkowski spacetime (some properties of these variables are
discussed in Appendices 2 and 3). In Sect.\ref{int} it will be shown how the minimal coupling
in the FR avoids the ambiguity present in the ER. The causality of the propagation
will be discussed in Sect.\ref{caus}. We close with a discussion of the results.


\section{Spin-2 field description in the Fierz representation}
\label{fabc}

In this section we present a short review of the FR in
a Minkowskian background and in the absence of interactions
\footnote{Throughout this article we will use
the signature $(+---)$, and the notation $A_{(\alpha}B_{\beta )} = A_\alpha B_\beta
+ A_\beta B_\alpha$, $A_{[\alpha}B_{\beta ]} = A_\alpha B_\beta
- A_\beta B_\alpha$.}. Let us start by
defining a three-index tensor $F_{\alpha\beta\mu}$ which is
anti-symmetric in the first pair of indices and obeys the cyclic
identity:
\begin{equation}
F_{\alpha\mu\nu} + F_{\mu\alpha\nu} = 0,
\label{01}
\end{equation}
\begin{equation}
F_{\alpha\mu\nu} + F_{\mu\nu\alpha} + F_{\nu\alpha\mu} = 0.
\label{02}
\end{equation}
The former expression implies that the dual of $F_{\alpha\mu\nu}$ is
trace-free:
\begin{equation}
\stackrel{*}{F}{}^{\alpha\mu}{}_{\mu} = 0 ,
\label{02bis}
\end{equation}
where the asterisk represents the dual operator, defined in terms
of $\eta_{\alpha\beta\mu\nu}$ by
\[
\stackrel{*}{F}{}^{\alpha\mu}{}_{\lambda} \equiv \frac{1}{2} \,
\eta^{\alpha\mu}{}_{\nu\sigma}\,F^{\nu\sigma}{}_{\lambda}.
\]
The tensor $F_{\alpha\mu\nu}$ has 20 independent components.
The necessary and sufficient condition for $F_{\alpha\mu\nu}$ to
represent an unique spin-2 field (described by 10 components)
is \footnote{Note that this
condition is analogous to that necessary for
the existence of a potential
$A_{\mu}$ for the EM field, given by
$\stackrel{*}{A}{}^{\alpha\mu}{}_{,\alpha} = 0.$}
\begin{equation}
\stackrel{*}{F}{}^{\alpha (\mu\nu)}{}_{,\alpha} = 0,
\label{03}
\end{equation}
which can be rewritten as
\begin{eqnarray}
&& {{F_{\alpha\beta}}^{\lambda}}{}_{,\mu} + {{F_{\beta\mu}}^{\lambda}}%
{}_{,\alpha} + {{F_{\mu\alpha}}^{\lambda}}{}_{,\beta} -\frac{1}{2}
\delta^{\lambda}_{\alpha} (F_{\mu ,\beta} - F_{\beta ,\mu}) + \nonumber \\
&& - \frac{1}{2} \delta^{\lambda}_{\mu} (F_{\beta ,\alpha} - F_{\alpha
,\beta}) - \frac{1}{2} \delta^{\lambda}_{\beta} (F_{\alpha ,\mu} - F_{\mu
,\alpha}) = 0.
%\label{4002}
\end{eqnarray}
A direct consequence of the above
equation is the identity:
\begin{equation}
F^{\alpha\beta\mu}{}_{\, ,\mu} = 0 \ .  \label{z1}
\end{equation}
We will call a tensor that satisfies the conditions given in the Eqns.(\ref{01}),
(\ref{02}) and (\ref{03}) a Fierz tensor.

If $F_{\alpha\mu\nu}$ is a Fierz tensor, it represents a
unique spin-2 field. Condition (\ref{03}) yields a
connection between the ER and the FR: it implies that
there exists a symmetric second-order tensor $\varphi_{\mu\nu}$
such that
\begin{eqnarray}
2\,F_{\alpha\mu\nu} &=& \varphi_{\nu [\alpha,\mu ]} +
\left( \varphi_{,\alpha} - \varphi_{\alpha}{}^{\lambda}{}_{,\lambda}
\right)\, \eta_{\mu\nu}\nonumber \\
 &-& \left(\varphi_{,\mu} -
\varphi_{\mu}{}^{\lambda}{}_{,\lambda} \right)\, \eta_{\alpha\nu} . \label{04}
\end{eqnarray}
where $\eta_{\mu\nu}$ is the flat spacetime metric tensor, and
the factor $2$ in the l.h.s. is introduced for convenience.

Taking the trace of equation (\ref{04}) we find that
%\begin{equation}
$$
F_{\alpha} = \varphi_{,\alpha} - \varphi_{\alpha}{}^{\lambda}{}_{,\lambda},
%\label{05}
%\end{equation}
$$
where $F_{\alpha}\equiv F_{\alpha\mu\nu}\eta^{\mu\nu}$. Thus we can write
\begin{equation}
2 F_{\alpha\mu\nu} = \varphi_{\nu [\alpha,\mu ]}  +
F_{[\alpha } \,\eta_{\mu ]\nu}.  \label{06}
\end{equation}

%When a Fierz-Pauli tensor is written under the form given in equation
%(\ref{04}) or (\ref{06}) we will say that it is in the Einstein
%frame.
The following identity can proved using
the properties of the Fierz tensor:
\begin{equation}
F^{\alpha }{}_{(\mu \nu ),\alpha }\equiv -\,G^{(L)}{}_{\mu \nu } ,
\label{07}
\end{equation}
where $G^{(L)}{}_{\mu \nu }$ is the linearized Einstein tensor, defined in
terms of the symmetric tensor $\varphi _{\mu \nu }$ by
\begin{equation}
G^{(L)}{}_{\mu \nu }\equiv \Box \,\varphi _{\mu \nu }-\varphi ^{\epsilon
}{}_{(\mu ,\nu )\,,\epsilon }+\varphi _{,\mu \nu }-\eta _{\mu \nu }\,\left(
\Box \varphi -\varphi ^{\alpha \beta }{}_{,\alpha \beta }\right) . \label{08}
\end{equation}


The divergence of $F^{\alpha }{}_{(\mu \nu ),\alpha }$ yields the identity:

\begin{equation}
F^{\alpha (\mu \nu )}{}_{,\alpha \mu }\equiv 0.  \label{07bis}
\end{equation}
Indeed,
\begin{equation}
F^{\alpha \mu \nu }{}_{,\alpha \mu }+F^{\alpha \nu \mu }{}_{,\mu \alpha }=0.
\label{070}
\end{equation}
The first term vanishes identically due to the symmetry properties
of the field and the second term vanishes due to equation
(\ref{z1}). Using Eqn.(\ref{07})
the identity which states that the
linearized Einstein tensor $G^{(L)}{}_{\mu \nu }$ is
divergence-free is recovered.

We shall build now dynamical equations for the free Fierz tensor in flat spacetime.
Our considerations will be restricted here to linear dynamics \cite{mio}.
The most general
theory can be constructed from a combination of the three invariants involving
the field. These are
represented by $A$, $B$ and $W$:
$$
A \equiv F_{\alpha \mu \nu }\hspace{0.5mm}F^{\alpha \mu \nu } ,
\;\;\;\;\;\;\;\;B \equiv F_{\mu }\hspace{0.5mm}F^{\mu },
$$
$$
W \equiv F_{\alpha \beta \lambda }\stackrel{\ast }{F}{}^{\alpha \beta
\lambda }=\frac{1}{2}\,F_{\alpha \beta \lambda }\hspace{0.5mm}F^{\mu \nu
\lambda }\,\eta ^{\alpha \beta }{}_{\mu \nu } .
$$
$W$ is a topological
invariant in the linear regime, so we shall use in what follows only
the invariants $A$ and $B$.

The EOM  for the massless spin-2 field in the ER is given by

\begin{equation}
G^{(L)}{}_{\mu \nu }=0.  \label{014bis}
\end{equation}
As we have seen above, in terms of the field $F^{\lambda \mu \nu}$
this equation can be written as

\begin{equation}
F^{\lambda (\mu \nu )}{}_{,\lambda }=0.  \label{014}
\end{equation}
The corresponding action takes the form

\begin{equation}
S=\frac{1}{k}\,\int {\rm d}^{4}x\,(A-B) . \label{013}
\end{equation}
Note that the Fierz tensor has dimensionality (length)$^{-1}$,
which is compatible with the fact that Einstein constant $k$ has
dimensionality (energy)$^{-1}$ (length)$^{-1}$. From now
on we set $k=1.$ Then,
\begin{equation}
\delta S=\int 2F^{\alpha \mu \nu }{}_{,\alpha }\,\delta
\varphi _{\mu \nu}\,d^{4}x . \label{018}
\end{equation}
Using the identity
\begin{equation}
F^{\alpha}{}_{\mu \nu,\alpha }=\frac{1}{2}\,F^{\alpha}{}_{(\mu\nu),\alpha}=
-\,\frac{1}{2}\,G^{(L)}{}_{\mu \nu },  \label{01888}
\end{equation}
we obtain
\begin{equation}
\delta S=-\int G^{(L)}{}_{\mu \nu }\,\delta\varphi^{\mu\nu}\,d^{4}x,
\label{018bis}
\end{equation}
where $G^{(L)}\mbox{}_{\mu \nu }$ is given in Eqn.(\ref{08}).
Thus, the action in Eqn.(\ref{013}) when written in the ER reads
\begin{equation}
S=-\int G^{(L)}{}_{\mu \nu }\,\varphi ^{\mu \nu }\,d^{4}x .
\label{0181bis}
\end{equation}

Let us consider now the massive case.
If we include a mass for the spin 2 field in the FR, the Lagrangian
takes the form
\begin{equation}
{\cal L}=A-B-\frac{m^{2}}{2}\,\left(\varphi _{\mu \nu }\, \varphi ^{\mu
\nu}-\varphi ^{2}\right),  \label{x055}
\end{equation}
and the EOM that follow are
\begin{equation}
F^{\alpha }{}_{(\mu \nu ),\alpha }
- m^{2}\,\left( \varphi_{\mu\nu}-\varphi\,\eta _{\mu \nu }\right) =0 ,
\label{mc1}
\end{equation}
or equivalently,
\[
G^{(L)}{}_{\mu \nu } +m^{2}\,\left( \varphi _{\mu \nu }-\varphi \,\eta _{\mu\nu }\right) =0.
\]
The trace of this equation gives
\begin{equation}
F^{\alpha }{}_{,\alpha }+ \frac{3}{2}\,m^{2}\,\varphi =0,
  \label{mc12}
\end{equation}
while the divergence of Eqn.(\ref{mc1}) yields
\begin{equation}
F_{\mu }=0.  \label{mc121}
\end{equation}
This result together with the trace equation gives $\varphi =0.$

In terms of the potential, Eqn.(\ref{mc121}) is equivalent to
\begin{equation}
\varphi _{,\,\mu }-\varphi ^{\epsilon }{}_{\mu \,,\epsilon }=0.
\label{mcd121}
\end{equation}
It follows that we must
have
\[
\varphi ^{\mu \nu }{}_{,\nu }=0.
\]
Thus we have shown that the original ten degrees of freedom (DOF)
of $F_{\alpha\beta\mu}$
have been reduced to five (which is the correct number
for a massive spin-2 field) by means of the  five constraints
\beq
\varphi ^{\mu \nu }{}_{,\nu }=0,\;\;\;\;\;\;\;\;\;\varphi = 0.
\label{fc}
\eeq

\section{Interaction of the spin-2 field with an electromagnetic field}
\label{int}

As discussed for instance in \cite{ds}, the minimal coupling
prescription
$\partial_\mu \rightarrow \partial_\mu - ieA_\mu$  is ambiguous in the case of a spin 2
field interacting with an EM field. The origin of this ambiguity is rooted, as in the case
of a curved background \cite{aragones71}, in the non-commutativity of the derivative
operator, which is manifest from
\beq
\varphi_{\alpha\beta ;\mu\nu} - \varphi_{\alpha\beta ;\nu\mu}
= ieA_{\nu\mu},
\label{comm}
\eeq
where $A_{\mu\nu}$ is the EM field,
and the semicolon is the covariant derivative $\partial_\mu - ieA_\mu$.
Let us review the argument in \cite{ds}, which starts from
the free Lagrangian for a charged spin 2 field in the ER:
$$
{\cal L} = \half \varphi^{*\mu\nu} G^{(L)}_{\mu\nu}
+ \half m^2(\varphi^{*\mu\nu}\varphi_{\mu\nu}- \varphi^* \varphi).
$$
The EOM that follow from this Lagrangian
are
\beqa
\Box (\varphi_{\mu\nu} - \eta_{\mu\nu} \varphi) + \varphi_{,\mu\nu} + \eta_{\mu\nu}
\varphi^{\alpha\beta}_{\;\;,\alpha\beta} -\varphi^{\alpha }_{(\nu ,\mu )\alpha} & \nonumber\\
+ m^2 (\varphi_{\mu\nu} - \eta_{\mu\nu} \varphi) & =0.
\eeqa
It is the term before the mass term
of this equation that leads to an ambiguity when minimal coupling
is adopted. In \cite{ds}, a one-parameter family of couplings was introduced, such that
$$
\varphi^{\alpha }_{(\nu ,\mu )\alpha} \rightarrow g \;\varphi^\alpha_{(\nu ;\mu ) \alpha}
+ (1-g)\varphi^\alpha_{(\mu ;\alpha\mu)}.
$$
By studying the constraints of the
one-parameter theory,
it was shown in \cite{ds} that the only value of the gyromagnetic factor
$g$ that maintains the correct number
of DOF is $g=1/2$. The resulting EOM is
\beqa
\Box (\varphi_{\mu\nu} - \eta_{\mu\nu} \varphi) + \varphi_{;(\mu\nu )} + \eta_{\mu\nu}
\varphi^{\alpha\beta}_{\;\; ;\alpha\beta} -\varphi^{\alpha }_{(\nu ;\mu )\alpha} &
\label{eomd} \\
-\varphi^{\alpha }_{(\mu ;\nu )\alpha}
+ m^2 (\varphi_{\mu\nu} - \eta_{\mu\nu} \varphi) & =0. \nonumber
\eeqa

Let us see how the minimal coupling procedure
affects the equations for the free field in the FR, given in Sect.\ref{fabc}.
First, in the presence of an EM field Eqn.(\ref{03}) transforms to
\beq
\stackrel{*}{F}{}^{\alpha (\beta\lambda )}_{\;\;\;\;\;\;\;\;;\alpha} = -\half ie
\stackrel{*}{A}{}^{\nu (\beta}\;\varphi^{\lambda )}_{\;\;\;\nu}.
\label{star}
\eeq
From this equation, the tensor $F_{\alpha\mu\nu}$ can be written as
\begin{equation}
2 F_{\alpha\mu\nu} = \varphi_{\nu [\alpha ;\mu ]}  +
F_{[\alpha } \,\eta_{\mu ]\nu},
\label{061}
\end{equation}
with
\begin{equation}
F_{\alpha} = \varphi_{;\alpha} - \varphi_{\alpha}{}^{\lambda}{}_{;\lambda}.
\label{051}
\end{equation}
If we start with the EOM for the charged spin-2 field in the absence of interactions in the
FR (Eqn.(\ref{mc1})), and
apply the minimal coupling procedure, we get
\beq
F^{\alpha }{}_{(\mu \nu );\alpha }
- m^{2}\,\left( \varphi_{\mu\nu}-\varphi\,\eta _{\mu \nu }\right) =0.
\label{eom1}
\eeq
There is no ambiguity then in the minimal substitution. In fact, using
Eqns.(\ref{061}) and (\ref{051}) in Eqn.(\ref{eom1}),
we get the equation derived in \cite{ds}
with $g=1/2$ ({\em i.e.} Eqn.(\ref{eomd})).
In other words,
the Fierz representation automatically gives a theory
with the correct number of degrees of freedom when the minimal coupling scheme is used.

Let us now give two constraints that follow from Eqn.(\ref{eom1}).
If we take the divergence on the index $\mu$ in Eqn.(\ref{eom1}), we get
\beq
 -\frac 3 2\; ie A^{\alpha\mu}F_{\alpha\mu\nu} + \half ie A^\mu_{\;\;\nu ,\alpha}
\varphi_\mu^{\;\alpha}  + m^2 F_\nu = 0,
\label{v1}
\eeq
for a  sourceless EM field $A_{\mu\nu}$.
Notice that in this constraint only first derivatives of $\varphi_{\mu\nu}$ appear (in
$F_\mu$).
Taking the divergence of
Eqn.(\ref{v1}) we obtain
\beq
ieA^{\alpha\mu}_{\;\;\;,\beta}\;F_{\alpha\mu}^{\;\;\;\beta}
- \frac 3 2 \left( m^4 - \frac 1 2\; e^2 A^2
 \right) + \frac 3 2 \;e^2 A^{\alpha}_{\;\mu}
A^{\mu\beta}\varphi_{\alpha\beta} = 0,
\label{v2a}
\eeq
where $A^2 = A_{\alpha\beta}A^{\alpha\beta}$.
Eqns.(\ref{v1}) and (\ref{v2a}) correspond to the
free-case equations (\ref{fc}). They reduce the number of DOF to five,
and are necessary for the compatibility of the system.
Note that a
remarkable cancellation has happened: no second derivatives of $\varphi_{\mu\nu}$
are present in this second constraint.
It is precisely the absence
in the constraints of second derivatives w.r.t time that guarantees that only physical
degrees of freedom propagate. Armed with the EOM (\ref{eom1}),
we shall study in the next section the causal properties of massive spin 2
particles in an EM background.

\section{Causality in spin 2 fields interacting with an EM background}
\label{caus}

In this section it will be shown, using the FR,
that the propagation of a massive spin 2 field
in an EM background is causal. We shall recourse to
the well-known method of the characteristics,
which is in fact equivalent to the infinite-momentum limit of the
eikonal approximation \cite{hada}. To set the stage for the calculation, let us
put together the equations we shall use. They are the EOM,
\beq
F^{\alpha }{}_{(\mu \nu );\alpha }
 -m^{2}\,\left( \varphi_{\mu\nu}-\varphi\,\eta _{\mu \nu }\right) =0,
\label{eom11}
\eeq
its trace,
\begin{equation}
F^{\alpha }{}_{;\alpha }-\frac{3}{2}\,m^{2}\,\varphi =0,
\label{traco}
\end{equation}
and the two constraints
\beq
-\frac 3 2 \;ie A^{\alpha\mu}F_{\alpha\mu\nu} + \half ie A^\mu_{\;\;\nu ,\alpha}
\varphi_\mu^{\;\alpha}    + m^2 F_\nu = 0,
\label{v11}
\eeq
and
\beq
ieA^{\alpha\mu}_{\;\;\;,\beta}\;F_{\alpha\mu}^{\;\;\;\beta}
- \frac 3 2 \left( m^4 - \frac 1 2\; e^2 A^2 \right) + \frac 3 2 \;e^2 A^{\alpha}_{\;\mu}
A^{\mu\beta}\varphi_{\alpha\beta} = 0.
\label{v2}
\eeq
To these, we must add some properties of the Fierz tensor:
\beq
F_{\alpha\mu\nu}+ F_{\mu\alpha\nu} =0,
\label{antys}
\eeq
\begin{equation}
F_{\alpha\mu\nu} + F_{\mu\nu\alpha} + F_{\nu\alpha\mu} = 0,
\label{cyclic}
\end{equation}
and
\beq
\stackrel{*}{F}{}^{\alpha (\beta\lambda )}_{\;\;\;\;\;\;\;\;;\alpha} = -\half ie
\stackrel{*}{A}{}^{\nu (\beta}\;\varphi^{\lambda )}_{\;\;\;\nu}.
\label{star2}
\eeq

Let $\Sigma$ be the surface of discontinuity defined by the equation
$$\Sigma(x^{\mu}) = {\rm constant}.$$
The discontinuity of a function $J$ through $\Sigma$ will be represented by
$[J]_\Sigma$, and its definition is
$$
[J]_\Sigma \equiv \lim_{\delta\rightarrow 0^+} \left( \left. J\right|_{\Sigma +\delta}
- \left. J \right|_{\Sigma - \delta}\right) .
$$
We shall assume that $F_{\alpha\mu\nu}$ is continuous through the surface $\Sigma$
but its first derivative is not:
\beq
[F_{\alpha\mu\nu}]_\Sigma = 0, \;\;\;\;\;\;\;\;[F_{\alpha\mu\nu ;\lambda}]_\Sigma =
f_{\alpha\mu\nu}k_\lambda.
\eeq
From the discontinuity of the EOM (\ref{eom11}) we learn that
\beq
f^{\mu (\alpha \beta)}k_\mu =0.
\label{d1}
\eeq
Taking the derivative of Eqn.(\ref{antys}), the discontinuity yields
\beq
k^\alpha f_{\alpha\mu\nu}+ k^{\alpha}f_{\mu\alpha\nu} = 0.
\label{d2}
\eeq
The same procedure applied to Eqn.(\ref{cyclic}) results in
\beq
k^\alpha f_{\alpha\mu\nu} + k^\alpha f_{\nu\alpha\mu} + k^\alpha
f_{\mu\nu\alpha} =0.
\label{d3}
\eeq
This equation, together with Eqns.(\ref{d1}) and (\ref{d2}) tells us that
the contraction of $k$ with $f$ is zero on any index of $f$.
The trace equation (\ref{traco}) gives
\beq
f^\mu k_\mu =0.
\label{d4}
\eeq
Eqn.(\ref{star}) can be written as
\begin{eqnarray}
{{F_{\alpha\beta}}^{\lambda}}{}_{,\mu} + {{F_{\beta\mu}}^{\lambda}}%
{}_{,\alpha} + {{F_{\mu\alpha}}^{\lambda}}{}_{,\beta} -\frac{1}{2}\;
\delta^{\lambda}_{\;\alpha} F_{[\mu ,\beta ]} + &  \\
 - \frac{1}{2} \;\delta^{\lambda}_{\;\mu} F_{[\beta ,\alpha ]}
  - \frac{1}{2} \;\delta^{\lambda}_{\;\beta} F_{[\alpha ,\mu ]}  = & -\half ie
\stackrel{*}{A}{}^{\nu (\rho}\;\varphi^{\lambda )}_{\;\;\;\nu}.\nonumber
\label{star22}
\end{eqnarray}
Notice that the r.h.s. is continuous. Taking the discontinuity of this equation,
multiplying by $k^\mu$ and $f_\lambda$, and using Eqn.(\ref{d1}), we get that
\beq
f_{\alpha\beta\lambda}f^\lambda k^2 = 0.
\label{d5}
\eeq
We shall assume for the time being that $k^2\neq 0$.
The discontinuity of the derivatives of the constraints Eqns.({\ref{v11})
and (\ref{v2}) give
\beq
A_{\alpha\beta ,\mu}f^{\alpha\beta\mu} = 0,
\label{dv1}
\eeq
\beq
\frac 3 2 \;ie\; A^{\alpha\beta}f_{\alpha\beta\mu}-m^2f_\mu = 0.
\label{dv2}
\eeq
From Eqns.(\ref{d5}) and (\ref{dv2}) we deduce that
\beq
f_\mu f^\mu = 0.
\label{d6}
\eeq
Note that {\em all} the equations that resulted from taking the discontinuity
({\em i.e.} Eqns.(\ref{d1})-(\ref{d4}) and (\ref{d5})-(\ref{dv2}))
depend only
on $f_{\mu\nu\alpha}$ and its trace. Taking the discontinuity of
the derivative of $F_{\mu\nu\alpha}$ we get that
$$
[F_{\alpha\mu\nu ,\lambda}]_\Sigma = f_{\alpha\mu\nu}k_\lambda
$$
where
\beq
2f_{\alpha\mu\nu} = \epsilon_{\nu\alpha}k_{\mu}- \epsilon_{\nu\mu}k_{\alpha}
+ f_\alpha \eta_{\mu\nu} - f_\mu \eta_{\alpha\nu},
\label{discf}
\eeq
and
$$
f_\alpha = \epsilon k_\alpha - \epsilon _\alpha^\beta k_\beta.
$$
Consequently the equations that follow from taking
the discontinuity are invariant under the
transformation
\beq
\epsilon'_{\mu\nu} = \epsilon_{\mu\nu} + \Lambda k_\mu k_\nu,
\label{trafo}
\eeq
where $\Lambda $ is an arbitrary
function of the coordinates. This equation
implies that
\beq
\epsilon' = \epsilon + \Lambda k^2.
\label{moda}
\eeq
Notice that this
is the symmetry of the massless theory. This is not surprising though, because
when we use the method of the characteristics
the mass term does not give any contribution, since it is continuous.
In the equivalent method of the eikonal, this is analogous to
the infinite-momentum limit, which takes us
from the mass shell to the light cone, and it is the latter that
determines the causal structure.

Now, for the propagation to be {\em acausal}, $k_\mu$ must be a timelike vector,
{\em i.e.} $k^2 =1$. But we have already proved that $f^\mu k_\mu =0$ and
$f_\mu f^\mu = 0$. Consequently $k_\mu$ must be {\em either null or
spacelike}. If $k_\mu$ is null, the propagation is causal. If $k_\mu$ is
spacelike, we need to check that there are no components of $X_\mu
= \epsilon_\mu^\lambda k_\lambda$ parallel to $k_\mu$.
%From Eqn.(\ref{d6}),
%\beq
%\epsilon^2 k^2 - 2\; \epsilon\; k_\mu X^\mu + X^2 = 0.
%\label{pu}
%\eeq
From Eqn.(\ref{d4}),
\beq
X_\mu k^\mu = \epsilon k^2.
\label{kx}
\eeq
It follows that this product is not a gauge-invariant quantity, because $\epsilon$
transforms under a gauge transformation according to Eqn.(\ref{moda}). Another quantity
that is not gauge invariant is $X_\mu X^\mu$, which transforms as
\beq
X'^2= X^2 + 2\Lambda (k^2)^2 \epsilon +\Lambda ^2 (k^2)^2.
\eeq
We see that a spacelike (actually, a non-null) $k_\mu$ entails an unacceptable
dependence of observable quantities with the gauge choice. This dependence
disappears only when $k^2 = 0$.
Summing up,
the propagation of spin two fields in an EM background
is causal, with the characteristics governed by
the equation $k^2 = 0$.

\section{Conclusion}

We have given a summary of the Fierz representation for a spin 2 field, both in the
free case and for the interaction with an electromagnetic background. This representation
has some advantages over the Einstein representation. In particular,
it was shown that, while the ER of a spin-2 field in an electromagnetic background
has an inherent ambiguity related to the order of the derivatives when the
minimal coupling procedure is applied, the Fierz representation
is free from this difficulty.
Another advantage
of this representation is that it is similar to that used in Electromagnetism, and
then we can profit from work already done in this area for instance in construction
nonlinear theories for the spin 2 field
\cite{mio}. More importantly, the use of the Fierz representation
has paved the way to a clean proof of the causality in the propagation of spin 2 fields in
the presence of an electromagnetic field, thus showing that previous claims about
noncausal propagation were mistaken.

To close, we would like to point out that
in the issue of causality, the use of the Fierz representation is not mandatory. A
closer look to the relevant equations in the Einstein representation (for instance Eqns.
(64) and (66) in \cite{ds}, without choosing a timelike $k_\mu$) shows that the gauge
invariance given by Eqn.(\ref{trafo}) is present there too. However, it is important
to remark that this gauge invariance (which went unnoticed before) is clearly displayed
in the Fierz representation.

\section*{Appendix 1}

In this Appendix we shall explain why
the
argument presented in \cite{ds}, which concludes that the EOM of an
$s =2$ field in an EM background allow the propagation with a timelike $k_\mu$
({\em i.e.} acausally) is not correct.
The proof in \cite{ds} starts by the definition of the discontinuity of the spin 2 field:
\beq
[\varphi_{\alpha\beta ,\nu \mu}] = k_\mu k_\nu \epsilon_{\alpha\beta}.
\eeq
The argument is based on the discontinuity of four equations:
Eqns.(\ref{d1}) and (\ref{d4}) expressed in
the ER by means of Eqn.(\ref{discf}) (these correspond to Eqns.(61) and (62)
of \cite{ds}), and Eqns.(\ref{dv1}) and (\ref{dv2}), also expressed in the ER as before
(corresponding to Eqns.(64) and (66) of \cite{ds}).
In \cite{ds}, it is assumed that $k_\mu$ is timelike and then shown that $X_\mu$ has
a nonzero component orthogonal to $k_\mu$. This line of reasoning
has two drawbacks. First, $k_\mu$ cannot be taken as timelike, as
we have shown in the paragraph before Eqn.(\ref{kx}): the assumption of
a timelike $k_\mu$ contradicts Eqn.(\ref{d6}), which has not been taken into account
in \cite{ds}.
Second, even if it is assumed that $k_\mu$ is timelike, nothing can be
said about the product $X_\mu k^\mu$ because it is not
gauge-invariant under the transformation given in Eqn.(\ref{trafo}), as shown by
Eqns.(\ref{moda}) and (\ref{kx}).

\section*{Appendix 2}


%\subsection{Symmetries}

We shall be concerned here
with the gauge invariance of
Eqn.(\ref{014bis}) under the map
\begin{equation}
\varphi_{\mu\nu} \rightarrow \tilde{\varphi}_{\mu\nu} = \varphi_{\mu\nu} +
\Lambda_{\mu ,\nu} + \Lambda_{\nu , \mu} . \label{30011}
\end{equation}
Although the field $F_{\alpha\beta\mu}$ is invariant under this map only
if the vector $\Lambda_{\mu}$ is a
gradient, it is important to realize that
the dynamics is invariant even when $\Lambda$ is not a gradient. Indeed, we
have
\begin{equation}
\delta F_{\alpha\beta\mu} \equiv \tilde{F}_{\alpha\beta\mu} -
F_{\alpha\beta\mu} = \frac{1}{2} \, X_{\alpha\beta\mu}{}^{\,\lambda}{}_{\,
,\lambda},
\end{equation}
where
\begin{eqnarray}
X_{\alpha\beta\mu}{}^{\lambda} &\equiv&(\Lambda_{\alpha ,\beta} -
\Lambda_{\beta ,\alpha}) \delta^{\lambda}_{\mu} +
[ \Lambda^{\sigma}{}_{,\sigma} \delta^{\lambda}_{\alpha} -
\Lambda_{\alpha}{}^{,\lambda}] \eta_{\beta\mu} \nonumber \\
&-&[\Lambda^{\sigma}{}_{,\sigma} \delta^{\lambda}_{\beta} -
\Lambda_{\beta}{}^{,\lambda} ] \eta_{\alpha\mu}. \label{119}
\end{eqnarray}
Then it follows that
\begin{equation}
2\delta F_{\alpha} = X_{\alpha}{}^{\, \lambda}{}_{\, ,\lambda},
\end{equation}
with
\[
X_{\alpha}{}^{\,\lambda} \equiv X_{\alpha\beta}{}^{\,\beta\lambda}.
\]

As a consequence of this transformation, the invariants $A$ and
$B$ change in the following way:
$$
\delta A =
F^{\alpha\beta\mu}X_{\alpha\beta\mu}{}^{\,\lambda}{}_{\,
,\lambda},\;\;\;\;\;\;\;\; \delta B =
F^{\alpha}X_{\alpha}{}^{\,\lambda}{}_{,\lambda}.
$$
Note that $X_{\alpha\beta\mu}{}^{\lambda}$ is {\bf not} cyclic
in the indices $(\alpha\beta\mu )$, but the quantity
%\begin{equation}
$X_{\alpha\beta\mu}{}^{\,\lambda}{}_{,\lambda}$
%\end{equation}
has such cyclic property:
\begin{equation}
X_{\alpha\beta\mu}{}^{\,\lambda}{}_{,\lambda} +
X_{\beta\mu\alpha}{}^{\,\lambda}{}_{,\lambda} +
X_{\mu\alpha\beta}{}^{\lambda}{}_{,\lambda} =0.
\end{equation}
It is straightforward to show the associated identities:
\begin{equation}
X^{\alpha\beta\mu\lambda}{}_{,\lambda \alpha} = 0
\label{id}
\end{equation}
\begin{equation}
X^{\alpha\beta\mu\lambda}{}_{,\lambda \mu} = 0
\end{equation}
\begin{equation}
{X^{\alpha\lambda}{}_{,\alpha\lambda} = 0}.
\end{equation}
Thus,
\begin{equation}
\delta A = [\varphi^{\mu\alpha ,\beta} + F^{\alpha}\,\eta^{\mu\beta}] \
X_{\alpha\beta\mu}{}^{\lambda}{}_{\,,\lambda}  \nonumber,
\end{equation}
or, equivalently,
\begin{equation}
\delta A = \varphi^{\mu\alpha ,\beta} X_{\alpha\beta\mu}{}^{\,\lambda}{}_{\,
,\lambda} + F^{\alpha} X_{\alpha}{}^{\,\lambda}{}_{\, ,\lambda}.
\end{equation}
Then,
\begin{equation}
\delta (A - B) = \varphi_{\mu\alpha ,\beta} X^{\alpha\beta\mu\lambda}{}_{\,
,\lambda},
\end{equation}
and
\begin{equation}
\int \varphi_{\mu\alpha ,\beta} X^{\alpha\beta\mu\lambda}{}_{\, ,\lambda} = \int
div - \int \varphi_{\mu\alpha} X^{\alpha\beta\mu\lambda}{}_{ ,\lambda \beta},
\end{equation}
so that, because of (\ref{id}),
\begin{equation}
\int \delta (A - B) = 0.
\end{equation}

This shows that the transformation
\[
F_{\alpha\beta\mu}\rightarrow F_{\alpha\beta\mu} +
X_{\alpha\beta\mu}{}^{\,\lambda}{}_{\,,\lambda},
\]
for $X_{\alpha\beta\mu}{}^{\lambda}$ given in equation (\ref{119}), leaves
the dynamics invariant.

\section{Acknowledgements}

This work was partially supported by the Brazilian agency Conselho Nacional
de Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico (CNPq).


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\bibitem{sh} See L. P. S. Singh and C. R. Hagen, \Prd {\bf 9}, 898 (1974)
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\bibitem{r1} See for instance I. L. Buchbinder, D. M. Gitman, and V. D. Pershin,
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\bibitem{damour2} T. Damour, I. Kogan, and A. Papazoglou,
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\bibitem{aragones71}  C. Aragone and S. Deser, Il Nuovo Cimento vol. 3, n. 4, 709 (1971).

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\bibitem{d2} S. Deser and A. Waldron, Nucl. Phys. {\bf B631}, 369 (2002).

\bibitem{nn} M. Novello and R. P. Neves, Class. Quant. Grav. {\bf 19}, 5335 (2002),
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\bibitem{cou} The coupling of the spin 2 field with an EM background
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\bibitem{ks} M. Kobayashi and A. Shamaly, \Prd {\bf 17}, 126 (1978), and
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\bibitem{mio} A particular case of non-linear dynamics was studied in
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\bibitem{aragone2}  C. Aragone and S. Deser, Phys. Lett. 86B, 161 (1979); Il Nuovo Cimento 57B, 33 (1980).


%\bibitem{Deser}  S. Deser, General Relativity and Gravitation, 1, 9 (1970).


\bibitem{hada} {\em Le\c cons sur la propagation des ondes et les equations de
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\end{thebibliography}




\end{document}

