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\rightline{SU--GP--01/06--01}
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%: Title
\sesquispace
\centerline {\bf An example relevant to the Kretschmann-Einstein debate}
\bigskip
%: Authors
\singlespace % (spacing for addresses etc.)
% \smallskip\centerline {by}
%:: FIRST AUTHOR
\centerline {\it Rafael D. Sorkin}
\medskip
\smallskip
%:: FIRST ADDRESS
\centerline {\it Department of Physics,
Syracuse University,
Syracuse, NY 13244-1130, U.S.A.}
\smallskip
%:: FURTHER ADDRESS
\centerline { \it and }
\smallskip
\centerline
{\it Queen Mary College, University of London, Mile End Road, London E1 4NS}
\smallskip
%:: EMAIL ADDRESS
\centerline {\it \qquad\qquad internet address: sorkin@physics.syr.edu}
\AbstractBegins
%
We cast the flat space theory of a scalar field in generally covariant
form by introducing an auxiliary field $\lambda$. The resulting theory
is couched in terms of an action integral $S$, and all the fields (the
scalar, the spacetime metric, and $\lambda$) are dynamical in the sense
of being varied freely in $S$. Conservation of energy-momentum emerges
as a formal consequence of diffeomorphism invariance, in close analogy
with the situation in ordinary general relativity.
%
\AbstractEnds
%: Set spacing for body of paper
\sesquispace
\bigskip\medskip
%: main text
Is it possible to lend the theory of a (classical) scalar field
propagating in Minkowski spacetime a diffeomorphism-invariant
formulation a la Kretschmann, and if so is this formulation in some
sense artificial? If we define a theory by its field equations then the
answer to the first part of our question clearly is ``yes''.
Let us take a real, free scalar field for illustration. Its
equation of motion in flat spacetime is simply
$$
\dal \phi = 0 \eqno(1)
$$
where $\dal=\eta^{ab}\grad_a\grad_b$ is the wave operator
or ``d'Alembertian''
and
$\eta_{ab}$ is the Minkowski metric. To reformulate this equation
covariantly, we need only replace $\eta_{ab}$ by a general Lorentzian
metric $g_{ab}$ and adjoin to (1) a condition stating that
$g_{ab}$ is flat:
$$
R_{abcd} = 0 \ , \eqno(2)
$$
$R_{abcd}$ being the Riemann tensor of $g_{ab}$.
These days, however, one might want to require that a theory be more
than just a set of field equations. One might want it to admit a
variational formulation, and one might then expect that a ``genuine''
diffeomorphism invariance would have to entail, in the manner of
E. Noether, identities leading to conservation of energy-momentum.
The purpose of this note is merely to point out that
there exists a ``Kretschmannian'' formulation of flat space scalar field
theory enjoying all of these attributes.
To obtain (1) and (2) from a variational principle, it seems
simplest just to introduce an auxiliary tensor field $\lambda^{abcd}$
with the same symmetries as the Riemann tensor $R_{abcd}$, and then to
write the action integral $S$ in a such a way that $\lambda^{abcd}$
plays the role of a Lagrange multiplier:\footnote{*}
%
{There is no loss of generality in restricting $\lambda^{abcd}$ to the
same symmetry type as $R_{abcd}$ because any piece of
$\lambda^{abcd}$ with a different symmetry (belonging to a distinct
Young tableau) would automatically drop out of the contraction
$\lambda^{abcd} R_{abcd}$. On the other hand, taking $\lambda^{abcd}$
to be a tensor is perhaps less natural than taking it to be a
density of weight 1, but in the presence of a nondegenerate metric, the
distinction is only one of style.}
$$
S = - \int dV \; \half \; g^{ab} \grad_a\phi \grad_b\phi \;
+ \int dV (1/4) \lambda^{abcd} R_{abcd}
\eqno(3)
$$
In this formulation $\phi$, $g$ and $\lambda$ are all treated as
dynamical variables, and accordingly are to be varied freely in $S$.
The only background structure is therefore the spacetime manifold
itself, exactly as in general relativity.
(To specify the theory completely, one should add topological and
boundary conditions, requiring, say, that the manifold be $\Reals^4$ and
that the metric be geodesically complete.)
Let us see what field equations result from our choice of $S$. The
variation of $\lambda^{abcd}$ immediately yields spacetime flatness in
the form of equation (2), which of course is what it was designed
to do. The variation of $\phi$ goes exactly as in Minkowski space, and
yields the same equation of motion (1), except that $\eta^{ab}$
gets replaced by $g^{ab}$ in the definition of the operator $\dal$.
However, this operator immediately reduces back to the flat space
d'Alembertian (with respect to $\eta_{ab}=g_{ab}$) once one takes into
account the result (2) of the $\lambda$ variation.
The only real novelty arises when one varies the spacetime metric $g$ to
obtain the ``Einstein equation'' of this theory. In discussing this
variation, I will refer to the first term in (3) as the ``matter
term'' $S_m$ and the second as the ``gravity term'' $S_g$. By
definition, variation of the metric in $S_m$ yields the stress-energy
tensor of the scalar field:
$$
\delta S_{m} = \int dV \, \half \, T^{ab} \, \delta g_{ab}
\eqno(4)
$$
where $T^{ab} = \grad^a\phi \grad^b\phi - \half (\grad\phi)^2 g^{ab}$.
Variation of $g_{ab}$ in the ``gravity term'' then has an interesting
consequence. In ordinary general relativity it would have produced the
Einstein tensor $G^{ab}$, but here we get instead a double divergence of
the Lagrange multiplier field $\lambda^{abcd}$ (see the Appendix).
Combining, then,
$\delta{S_m}$ with $\delta{S_g}$, we obtain our third and last
field equation:
$$
\grad_m \grad_n \lambda^{ambn} = T^{ab} \ , \eqno(5)
$$
which says that $T^{ab}$ acts as a ``source'' for $\lambda^{abcd}$.
Requiring that $\delta{S}=0$ has thus reproduced precisely the structure
of the earlier formulation (1) (2),
except that an extra equation (5) has
appeared, corresponding to the presence of the extra variable
$\lambda^{abcd}$. This extra equation, however is not entirely
trivial. Rather it yields very directly the conservation of $T^{ab}$,
much as the Maxwell equations yield conservation of charge, or (in even
closer analogy) as the (usual) Einstein equation yields
$T^{ab}$-conservation via the contracted Bianchi identities. In the
present case, one has instead of the latter, the identity
(for vanishing Riemann curvature)
$$
\grad_b \, (\grad_m\grad_n\lambda^{ambn}) \ideq 0 \ ,
\eqno(6)
$$
which
follows straightforwardly from the symmetry of $\lambda^{abcd}$.
In light of the field equations (5) and (2),
this identity immediately
implies the conservation law,
$$
\grad_b T^{ab} = 0 \ , \eqno(7)
$$
which we therefore obtain ``twice over'' as a consequence of
diffeomorphism invariance, exactly as in ordinary general relativity.
This is the end of the story, except for one thing. One might worry
that, besides yielding (7), our extra equation (5) could
also place artificial restrictions on $\phi$, beyond those implied by
the Klein-Gordon equation (1). But it is easy to see that this is
not the case, because (at least in flat spacetime) every conserved
symmetric tensor can be expressed as the double divergence of a tensor
with the symmetry type of the Riemann tensor.
(A proof is sketched in the the Appendix.)
Notice, however, that $\lambda^{abcd}$ is not uniquely determined
thereby, and our theory has in this sense a further ``gauge
invariance'', probably traceable to the (uncontracted) Bianchi
identities for $R_{abcd}$.
From all this, one can conclude, I think, that the Kretschmann version
of flat spacetime scalar field theory exhibited above not only yields
the desired equations of motion, but also enjoys all the formal features we
normally associate with generally covariant theories.\footnote{*}
%
{As suggested to me by John Earman, a further instructive exercise
along these lines might be to carry out a full ``Dirac constraint
analysis'' of (3).}
%
In this way, far from being ``artificial'', it affords (just as general
relativity does) a particularly simple derivation of the stress-energy
tensor $T^{ab}$ and a simple proof of its conservation.
Our considerations here have concerned only the classical theory, but it
seems unlikely that a quantal treatment would change anything essential
in our conclusions.
Adding interaction terms like $\phi^4$ or using different matter fields
clearly would not change anything either.
In a philosophical debate, examples can be helpful, but in writing this
note, I have not attempted to draw any definite
conclusions from the example presented above. In particular I have not
claimed that the ``Kretschmannian'' theory formulated herein is
necessarily ``physically equivalent'' to the corresponding special
relativistic theory, or that its existence necessarily proves that
``general covariance is empty''. My primary purpose has been rather to
help sharpen such questions by further delineating a simple example some
of whose relevant features seem not to be widely known.\footnote{**}
%
{Many of these same features show up also in certain of
the so called ``topological quantum field theories''. To the
extent that people studying such theories typically take for
granted that they incorporate general covariance in the physical
sense, one may say that, in certain quarters, the debate is effectively
over and Kretschmann has won!}
Having done so, however, I would like, in closing, to offer an opinion
which I believe to be consistent with the facts presented herein. I
believe that general covariance (or perhaps better, {\it
background independence}, since it seems to be true that one can
trivially express any theory without exception in generally covariant
{\it language}) is indeed empty if taken by itself, but that it does
have meaning if taken in conjunction with a ``specification of
substance'' (and possibly also with other requirements, like the
existence of a local Lagrangian). In the case at hand, for example, we
had to add a new ``substance'', namely the field $\lambda^{abcd}$, in
order to be able to write our action $S$. Had we stuck with just the
fields $\phi$ and $g_{ab}$, it is hard to see that we would have been
able to promote the latter from the status of background structure to
that of dynamical variable, as general covariance demands.
Finally, I would like to thank John Earman for raising some of the
questions addressed herein and then encouraging me to write up the
answers. I would also like to thank Karel Kucha{\v{r}}, John Norton,
Don Marolf and Abhay Ashtekar for discussions concerning the
implications of some of these results.
\noindent
{\it Added Note:} In working through the example presented herein, I
have realized that there may be something special about the case of a
Minkowskian background, because the demand for a flat metric is easily
expressed in local form through the equation $R^{abcd}=0$. It is not at
all obvious that something similar could be done, even for so simple a
geometry as a dust-filled Friedmann universe, let alone for a still less
symmetrical background metric. In this sense, it seems an open question
how viable the Kretschmann view would have been had we been comparing
general relativity with a generically curved background spacetime,
rather than with the flat spacetime metric that was historically given
to us.
%: Acknowledgments including grant citation
\bigskip\noindent
This research was partly supported by NSF grant PHY-0098488, by
a grant from the Office of Research and Computing of Syracuse
University, and by an EPSRC Senior Fellowship at Queen Mary
College. I would also like to express my gratitude to
Goodenough College for providing a fine working and living
environment during my stay in London where this paper was
completed.
\section{APPENDIX:\ Further details of some of the derivations}
In this appendix, I will sketch the derivation of two mathematical facts
used above, the first concerning the variation of the ``gravity term'' in
the action-integral
(3) and the second being the existence of a potential for any
conserved symmetric tensor in flat space.
\subsection {Variation of $\lambda^{abcd}R_{abcd}$}
For variations of the metric tensor about an originally flat metric
(one with vanishing Riemann curvature) one has in general
$$
\delta R_{abcd}
= \half
(- \ptl_{ac}\dg_{bd} - \ptl_{bd}\dg_{ac}
+ \ptl_{ad}\dg_{bc} + \ptl_{bc}\dg_{ad})
\ ,
\eqno(8)
$$
$\partial_a$ being the derivative operator of the flat, unvaried metric and
$\partial_{ab}$ being shorthand for $\partial_a\partial_b$.
This equation (cf. \S 92 of [1])
follows straightforwardly from standard formulas expressing
$\delta R^a_{bcd}$ in terms of $\delta\Gamma^a_{bc}$ ,
and
$\delta\Gamma^a_{bc}$ in terms of $\dg_{ab}$. Within an overall factor,
its right hand side is the only expression with the correct
symmetries that one can construct from second derivatives of the metric
variation $\dg_{ab}$.
Plugging (8) into the variation of $\lambda^{abcd}R_{abcd}$,
integrating by parts twice,
and combining with (4)
results in equation (5) of the main text.
\subsection{Existence of a potential for $T^{ab}$}
Every conserved $T^{ab}$ in Minkowski spacetime can be written
as the double divergence of some ``potential'' $H^{abcd}$ with the
symmetries of the Riemann tensor. One can
demonstrate this,
proceeding along
the lines of [2]. Beginning with the conservation equation
$$
\ptl_b T^{ab} = 0
$$
one obtains
from the Poincar{\'e} lemma
a potential $W^{abc}$, skew in its last two indices, such
that
$$ T^{ab} = \ptl_c W^{abc}
$$
The symmetry of $T^{ab}$ then implies that
$W^{abc}-W^{bac}$ is
divergence free (in the index $c$) so that, once again, a potential
exists:
$$
W^{abc}-W^{bac} = \ptl_d E^{abcd}
$$
where $E^{abcd}$ can be arranged to be skew in both the index pair $ab$
and the index pair $cd$. Solving this equation for $W^{abc}$ yields
$$
T^{ab} = \ptl_c W^{abc} = \ptl_{mn} \half (E^{ambn} + E^{bnam})
$$
Denoting the argument of $\ptl_{mn}$
in this equation
by $H^{ambn}$ and replacing it
by the combination
$$
1/3 \; (2H^{ambn} - H^{mban} -H^{bamn})
$$
yields an equivalent expression
enjoying the full Riemann tensor symmetry.
Denoting this new expression by the same letter $H$, one obtains finally,
$$
T^{ab} = \ptl_{mn} H^{ambn} \ .
$$
\ReferencesBegin
\ref
[1]
L.D.~Landau and E.M.~Lifshitz, {\it The Classical Theory of Fields},
second edition
(Add\-ison-Wesley 1962)
(translated by Morton Hamermesh)
\ref
[2]
R.D.~Sorkin,
``On Stress-Energy Tensors'',
{\it Gen. Rel. Grav.} {\bf 8:} 437-449 (1977)
\end
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