Reply to {\em Acausality of Massive Charged Spin 2 Fields}
M. Novello
S. E. Perez Bergliaffa
and R. P. Neves
We show that, contrary to the claim in hep-th/0304050, the propagation of a spin-2 field in an electromagnetic background is {\em causal}. %\noindent
gr-qc0304041.tex
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\title{Reply to {\em Acausality of Massive Charged Spin 2 Fields}}
\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 the claim in hep-th/0304050, the
propagation of a spin-2 field in an electromagnetic background is
{\em causal}.
%\noindent
\end{abstract}


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

\maketitle

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

Recently \cite{nosso} we showed that the propagation of a spin 2
field in an electromagnetic background is causal. This assertion
is against results obtained previously (see \cite{d2} and
references therein). For the proof we used
used the first-order representation for
the equations of a spin 2 field, which is based on a three-index
tensor $F_{\mu\nu\alpha}$. The first
order formalism is equivalent to the more usual
second order formalism, as can be shown from
the results in \cite{nosso}. Another ingredient for
the proof was Hadamard's method
of
discontinuities, in which
$$
[F_{\mu\nu\alpha ;\lambda}] = k_\lambda f_{\mu\nu\alpha}.
$$
on the surface of discontinuity. The authors of
\cite{com}
claim that
\begin{itemize}

\item from the equations $k_\mu f^\mu =0$ and $f_\mu f^\mu =0$,
where $f_\mu = f_{\mu\nu\alpha} \eta^{\nu\alpha}$, it cannot be
concluded that $k_\mu$ is lightlike, and

\item that the masless gauge invariance that appears in the
eikonal limit, used in \cite{nosso} to {\em gauge away
acausalities} cannot be used in that way, because the
underlying massive theory is not gauge invariant.

\end{itemize}

It was shown in \cite{com} that the first claim is correct,
but no proof or support was given for the second claim. We will give here
a proof of the causal behaviour based solely, as we shall see,
in
the symmetry
present only in the equations for
the discontinuities.

As a result of taking the discontinuity of the equations of motion
for a spin 2 field in an electromagnetic background, we got in
\cite{nosso} the following equations:
\beq
f_{\mu\nu\alpha}k^\mu =0,
\eeq
\beq
f_{\mu\nu\alpha}k^\alpha =0,
\eeq
\beq
f_{\alpha\beta\lambda}f^\lambda k^2 = 0,
\eeq
\beq
A_{\alpha\beta ,\mu}f^{\alpha\beta\mu} = 0,
\eeq
\beq
\frac 3 2 \;ie\; A^{\alpha\beta}f_{\alpha\beta\mu}-m^2f_\mu = 0,
\eeq
\begin{eqnarray}
{{f_{\alpha\beta}}^{\lambda}}{}k_\mu + {{f_{\beta\mu}}^{\lambda}}%
{}k_\alpha + {{f_{\mu\alpha}}^{\lambda}}{}k_{\beta} -\frac{1}{2}\;
\delta^{\lambda}_{\;\alpha} f_{[\mu}k_{\beta ]}
   &   \nonumber \\
 - \frac{1}{2} \;\delta^{\lambda}_{\;\mu} f_{[\beta} k_{\alpha ]}
  - \frac{1}{2} \;\delta^{\lambda}_{\;\beta} f_{[\alpha}k_{,\mu ]}  =
  0&
 \label{star22}
\end{eqnarray}

Notice that the dependence of all these equations on
the polarization $\epsilon_{\mu\nu}$ is only through the tensor
$f_{\alpha\beta\mu}$, defined by
 \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 its trace. It is patently evident that equations (1)-(6)
are {\em invariant} under the transformation
\beq
\tilde \epsilon_{\mu\nu} = \epsilon_{\mu\nu} + \Lambda k_\mu k_\nu,
\label{trafo}
\eeq
where $\Lambda $ is an arbitrary function of
the coordinates. Note that this transformation is
{\em not} a symmetry of the massless theory in the presence
of the electromagnetic field, as can be shown by explicit calculation.
It is instead a symmetry of the equations for the discontinuity of the derivative
of the field $F_{\mu\nu\alpha}$. It follows from this symmetry
that the modulus of $X_\mu \equiv \epsilon_{\mu\nu}k^\nu$
transforms as \cite{nosso}
\beq
\tilde X^2 = X^2 + \Lambda (k^2)^2( 2\epsilon + \Lambda ).
\eeq
Depending on $\Lambda$, the vector $X_\mu$
(and any other projection of the polarization tensor)
may be timelike, null, or spacelike. Only for the case
$k^2=0$ this unacceptable dependence of an observable quantity with a gauge choice
disappears . We conclude then that the propagation
of a spin-2 field in an electromagnetic background is
{\em causal}.


\begin{thebibliography}{99}

\bibitem{nosso} CAUSALITY AND CHARGED SPIN 2 FIELDS IN AN ELECTROMAGNETIC BACKGROUND,
M. Novello, S.E. Perez Bergliaffa, R.P. Neves, {\tt
hep-th/0302225}.


%\bibitem{sh} See L. P. S. Singh and C. R. Hagen, \Prd {\bf 9}, 898 (1974)
%for the bosonic case.

%\bibitem{r1} See for instance I. L. Buchbinder, D. M. Gitman, and V. D. Pershin,
%\Np {\bf B584}, 615 (2000),

%\bibitem{damour1} T. Damour and I. Kogan, Phys. Rev. D {\bf 66}, 104024 (2002),
%{\tt hep-th/0206042}.

%\bibitem{damour2} T. Damour, I. Kogan, and A. Papazoglou,
%Phys. Rev. D {\bf 66}, 104025 (2002),
%{\tt hep-th/0206044}.


%\bibitem{aragones71}  C. Aragone and S. Deser, Il Nuovo Cimento vol. 3, n. 4, 709 (1971).

%\bibitem{r2}  I. L. Buchbinder, D. M. Gitman , V. D. Pershin,
%Phys. Lett. {\bf B492}, 161 (2000).

\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),
%{\em Apparent mass of the graviton in a DeSitter
%Background}, to be published in \Cqg, {\tt gr-qc/0210058}..

%\bibitem{cou} The coupling of the spin 2 field with an EM background
%was analyzed previously by W. Tait, \Prd {\bf 5}, 3272 (1972), and
%C. R. Hagen, \Prd {\bf 6}, 984 (1972).

%\bibitem{ks} M. Kobayashi and A. Shamaly, \Prd {\bf 17}, 126 (1978), and
%\Ptp {\bf 61}, 656 (1979).

\bibitem{ds} S. Deser and A. Waldron, \Np {\bf B631}, 369 (2002).

%\bibitem{fp} M. Fierz and W. Pauli, Proc. Roy. Soc. (Lond.) A {\bf 173}, 211 (1939),
%Helvetica Physica Acta 12, 297 (1939).


%\bibitem{mio} A particular case of non-linear dynamics was studied in
%{\em Propagation of perturbations in non-linear spin-2 theories}, S. E. Perez Bergliaffa,
%contribution to M. Novello's Festschrift, {\tt gr-qc 0302031}.


%\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
%l'hydrodynamique},  J. Hadamard,  J. HadamardEd. Dunod, Paris, 1958.

\bibitem{com} ACAUSALITY OF MASSIVE CHARGED SPIN 2 FIELDS,
S. Deser, A. Waldron, {\tt hep-th/0304050}.


\end{thebibliography}




\end{document}

gr-qc0304041.extracted_bib
\begin{thebibliography}{99}

\bibitem{nosso} CAUSALITY AND CHARGED SPIN 2 FIELDS IN AN ELECTROMAGNETIC BACKGROUND,
M. Novello, S.E. Perez Bergliaffa, R.P. Neves, {\tt
hep-th/0302225}.


%\bibitem{sh} See L. P. S. Singh and C. R. Hagen, \Prd {\bf 9}, 898 (1974)
%for the bosonic case.

%\bibitem{r1} See for instance I. L. Buchbinder, D. M. Gitman, and V. D. Pershin,
%\Np {\bf B584}, 615 (2000),

%\bibitem{damour1} T. Damour and I. Kogan, Phys. Rev. D {\bf 66}, 104024 (2002),
%{\tt hep-th/0206042}.

%\bibitem{damour2} T. Damour, I. Kogan, and A. Papazoglou,
%Phys. Rev. D {\bf 66}, 104025 (2002),
%{\tt hep-th/0206044}.


%\bibitem{aragones71}  C. Aragone and S. Deser, Il Nuovo Cimento vol. 3, n. 4, 709 (1971).

%\bibitem{r2}  I. L. Buchbinder, D. M. Gitman , V. D. Pershin,
%Phys. Lett. {\bf B492}, 161 (2000).

\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),
%{\em Apparent mass of the graviton in a DeSitter
%Background}, to be published in \Cqg, {\tt gr-qc/0210058}..

%\bibitem{cou} The coupling of the spin 2 field with an EM background
%was analyzed previously by W. Tait, \Prd {\bf 5}, 3272 (1972), and
%C. R. Hagen, \Prd {\bf 6}, 984 (1972).

%\bibitem{ks} M. Kobayashi and A. Shamaly, \Prd {\bf 17}, 126 (1978), and
%\Ptp {\bf 61}, 656 (1979).

\bibitem{ds} S. Deser and A. Waldron, \Np {\bf B631}, 369 (2002).

%\bibitem{fp} M. Fierz and W. Pauli, Proc. Roy. Soc. (Lond.) A {\bf 173}, 211 (1939),
%Helvetica Physica Acta 12, 297 (1939).


%\bibitem{mio} A particular case of non-linear dynamics was studied in
%{\em Propagation of perturbations in non-linear spin-2 theories}, S. E. Perez Bergliaffa,
%contribution to M. Novello's Festschrift, {\tt gr-qc 0302031}.


%\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
%l'hydrodynamique},  J. Hadamard,  J. HadamardEd. Dunod, Paris, 1958.

\bibitem{com} ACAUSALITY OF MASSIVE CHARGED SPIN 2 FIELDS,
S. Deser, A. Waldron, {\tt hep-th/0304050}.


\end{thebibliography}