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% Original Author: Erik Curiel
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\fancyhead[L]{\bfseries Geometric Objects, Gravitational Energy, and
the EFE}
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\title{On Geometric Objects, the Non-Existence of a Gravitational
Stress-Energy Tensor, and the Uniqueness of the Einstein Field
Equation\thanks{This paper was revised and resubmitted (Jan.~2016)
to \emph{Studies in History and Philosophy of Modern Physics}.}}
\author{Erik Curiel\thanks{I thank Robert Geroch for many stimulating
conversations in which the seeds of several of the paper's ideas
were germinated and, in some cases, fully cultivated to fruition.
I also thank David Malament for helpful conversations on the
principle of equivalence and on gravitational energy. I am
grateful to Ted Jacobson for commenting on an earlier draft and
catching a serious error. Finally, I thank an anonymous referee
for a supererogatory report, which helped me clarify my discussion
of several issues. \textbf{Author's address}: Munich Center for
Mathematical Philosophy, Ludwigstra{\ss}e 31,
Ludwig-Maximilians-Universit\"at, 80539 M\"unchen, Deutschland;
\textbf{email}: \href{mailto:erik@strangebeautiful.com}
{\texttt{erik@strangebeautiful.com}}}}
\date{}
\renewcommand{\thefootnote}{\arabic{footnote}}
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\begin{document}
\english
\maketitle
\begin{quote}
\begin{center}
\textbf{ABSTRACT}
\end{center}
The question of the existence of gravitational stress-energy in
general relativity has exercised investigators in the field since
the inception of the theory. Folklore has it that no adequate
definition of a localized gravitational stress-energetic quantity
can be given. Most arguments to that effect invoke one version or
another of the Principle of Equivalence. I argue that not only are
such arguments of necessity vague and hand-waving but, worse, are
beside the point and do not address the heart of the issue. Based
on a novel analysis of what it may mean for one tensor to depend in
the proper way on another, which, \emph{en passant}, provides a
precise characterization of the idea of a ``geometric object'', I
prove that, under certain natural conditions, there can be no tensor
whose interpretation could be that it represents gravitational
stress-energy in general relativity. It follows that gravitational
energy, such as it is in general relativity, is necessarily
non-local. Along the way, I prove a result of some interest in own
right about the structure of the associated jet bundles of the
bundle of Lorentz metrics over spacetime. I conclude by showing
that my results also imply that, under a few natural conditions, the
Einstein field equation is the unique equation relating
gravitational phenomena to spatiotemporal structure, and discuss how
this relates to the non-localizability of gravitational
stress-energy. The main theorem proven underlying all the arguments
is considerably stronger than the standard result in the literature
used for the same purposes (Lovelock's theorem of 1972): it holds in
all dimensions (not only in four), and it does not require an
assumption about the differential order of the desired concomitant
of the metric.
\end{quote}
\skipline
\noindent \textbf{Keywords:} general relativity; gravitational energy;
stress-energy tensors; concomitants; jet bundles; principle of
equivalence; geometric objects; Einstein field equation
\tableofcontents
\skipline[1.5]
\begin{quote}
As soon as the principle of conservation of energy was grasped, the
physicist practically made it his definition of energy, so that
energy was that something which obeyed the law of conservation. He
followed the practice of the pure mathematician, defining energy by
the properties he wished it to have, instead of describing how he
measured it. This procedure has turned out to be rather unlucky in
the light of the new developments.
\begin{flushright}
Arthur Eddington \nocite{eddington-math-theor-rel} \\
\emph{The Mathematical Theory of Relativity}, p.~136
\end{flushright}
\end{quote}
\skipline
\section{Gravitational Energy in General Relativity}
\label{sec:intro}
There seems to be in general relativity no satisfactory, localized
representation of a quantity whose natural interpretation would be
``gravitational (stress-)energy''. The only physically unquestionable
expressions of energetic quantities associated solely with the
``gravitational field'' we know of in general relativity are
quantities derived by integration over non-trivial volumes in
spacetimes satisfying any of a number of special
conditions.\footnote{\citeN[pp.~271--272]{weyl-space-time-matter} and
\citeN[pp.~134--137]{eddington-math-theor-rel} were perhaps the
first to grasp this point with real clarity.
\citeN[pp.~104--105]{schrodinger-st-struc} gives a particularly
clear, concise statement of the relation between the fact that the
known energetic, gravitational quantities are non-tensorial and the
fact that integration over them can be expected to yield integral
conservation laws only under restricted conditions.} These
quantities, moreover, tend to be non-tensorial in character. In other
words, these are strictly non-local quantities, in the precise sense
that they are not represented by invariant geometric objects defined
at individual spacetime points (such as tensors or scalars).
This puzzle about the character and status of gravitational energy
emerged simultaneously with the discovery of the theory
itself.\footnote{The first pseudo-tensorial entity proposed to
represent gravitational stress-energy dates back to
\citeN{einstein-gr}, the paper in which he first proposed the final
form of the theory.} The problems raised by the seeming
non-localizability of gravitational energy had a profound, immediate
effect on subsequent research. It was, for instance, directly
responsible for Hilbert's request to Noether that she investigate
conservation laws in a quite general setting, the work that led to her
famous results relating symmetries and conservation laws
\cite{brading-energy-cons-gr}.
Almost all discussions of gravitational energy in general relativity,
however, dating back even to the earliest ones, have been plagued by
vagueness and lack of precision. The main result of this paper
addresses the issue head-on in a precise and rigorous way. Based on
an analysis of what it may mean for one tensor to depend in the proper
way on another, I prove that, under certain natural conditions, there
can be no tensor whose interpretation could be that it represents
gravitational stress-energy in general relativity. It follows that
gravitational stress-energy, such as it is in general relativity, is
necessarily non-local. Along the way, I prove a result of some
interest in own right about the structure of the associated first two
jet bundles of the bundle of Lorentz metrics over spacetime. I
conclude with a discussion of the sense in which my results also show
that the Einstein field equation is, in a natural sense, the unique
field equation in the context of a theory such as general relativity,
and discuss how this fact relates to the non-localizability of
gravitational stress-energy.
The main theorem (\ref{thm:only-einstein-tens}) proven underlying all
the arguments is considerably stronger than the standard result in the
literature used for the same purposes (the theorem of
\citeNP{lovelock-4dim-space-efe}): it holds in all dimensions, not
only in four, and it does not require an assumption about the
differential order of the desired concomitant of the metric. The
theorem has interesting consequences for a proper understanding of a
cosmological-constant in the Einstein field equation, and for
higher-dimensional Lovelock theories of gravity, which I discuss at
the end of the paper.
\section{The Principle of Equivalence: A Bad Argument}
\label{sec:princ_equiv}
\resetsec
The most popular heuristic argument used to attempt to show that
gravitational energy either does not exist at all or does exist but
cannot be localized invokes the ``principle of equivalence''.
\citeN[p.~399]{choquet83}, for example, puts the argument like this:
\begin{quote}
This `non local' character of gravitational energy is in fact
obvious from a formulation of the equivalence principle which says
that the gravitational field appears as non existent to one observer
in free fall. It is, mathematically, a consequence of the fact that
the pseudo-riemannian connexion which represents the gravitational
field can always be made to vanish along a given curve by a change
of coordinates.
\end{quote}
\citeN[pp.~469-70]{goldberg-inv-trans-cons-laws-ener-mom} makes almost exactly the same argument,
though he draws the conclusion in a slightly more explicit
fashion:\footnote{\label{fn:fungible-se-tensor}Goldberg's formulation
of the argument exhibits a feature common in the many instances of
it I have found in the literature, the conclusion that a local
gravitational energy \emph{scalar} density does not exist and not
that a gravitational stress-energy tensor does not exist. One
cannot have a scalar energy density for a physical field in general
relativity, however, without an associated stress-energy tensor.
Such a state of affairs would violate the thermodynamic principle
that all energy is equivalent in character, in the sense that any
one form can always in principle be tranformed into any other form,
since all other known forms of physical field do have a
stress-energy tensor as the fundamental representation of their
energetic content.}
\begin{quote} [I]n Minkowski space any meaningful energy density
should be zero. But a general space-time can be made to appear
Minkowskian along an arbitrary geodesic. As a result, any
nontensorial `energy density' can be made to be zero along an
arbitrary geodesic and, therefore, has no invariant meaning.
\end{quote}
\citeN[pp.~135-6]{trautman-energy-grav-cosmo} has also made
essentially the same argument. In fact, the making of this argument
seems to be something of a shared mannerism among physicists who
discuss energy in general relativity; it is difficult to find an
article on the topic in which it is not at least alluded
to.\footnote{\citeN{bondi-phys-char-grav-wvs},
\citeN{penrose-gr-enflux-elem-opt} and \citeN{geroch-energy-extrac}
are notable exceptions. I take their discussions as models of how
one should discuss energetic phenomena in the presence of
gravitational fields.}
The argument has a fundamental flaw. It assumes that, if there is
such a thing as localized gravitational energy or stress-energy, it
can depend only on ``first derivatives of the metric''---that those
first derivatives encode all information about the ``gravitational
field'' relevant to stress-energy---for it is only entities depending
only on those first derivatives that one can make vanish along curves.
But that seems wrong on the face of it. If there is such a thing as a
localized gravitational energetic quantity, then surely it depends on
the curvature of spacetime and not on the affine connection (or, more
precisely, it depends on the affine connection at least in so far as
it depends on the curvature), for any energy one can envision
transferring from the gravitational field to another type of system in
a different form in general relativity (\emph{e}.\emph{g}., as heat or
a spray of fundamental particles) must at bottom be based on geodesic
deviation,\footnote{\citeN{penrose-gr-enflux-elem-opt} and
\citeN{ashtekar-penrose-mass-pos-focus-struc-sl-inf} rely on the
same idea to very fruitful effect.} and so must be determined by the
value of the Riemann tensor at a point, not by the value of the affine
connection at a point or even along a curve. There is no solution to
the Einstein field-equation that corresponds in any natural way to the
intuitive Newtonian idea of a constant non-zero gravitational field,
\emph{i}.\emph{e}., one without geodesic deviation; that, however,
would be the only sort of field that one could envision even being
tempted to ascribe gravitational energy to in the absence of geodesic
deviation, and that attribution is problematic even in Newtonian
theory. Indeed, a spacetime has no geodesic deviation if and only if
it is everywhere locally isometric to Minkowski spacetime, which we
surely want to say has vanishing gravitational energy if any spacetime
does, if one can make such a statement precise in the first
place.\footnote{One might be tempted by the stronger claim that
Minkowski spacetime ought to be the unique spacetime with vanishing
gravitational energy. I do not think that can be right, however.
If the existence of gravitational energy is indeed intimately tied
with the presence of geodesic deviation (as argued forcefully by
\citeNP{penrose-gr-enflux-elem-opt}), then any flat spacetime, such
as that of \citeN{kasner-geom-thms-efe}, also ought to have
vanishing gravitational energy.}
An obvious criticism of my response to the standard line, related to a
popular refinement of the argument given for the non-existence or
non-locality of gravitational energetic quantities, is that it would
make gravitational stress-energy depend on second-order partial
derivatives of the field potential (the metric, so comprehended by
analogy with the potential in Newtonian theory), whereas all other
known forms of stress-energy depend only on terms quadratic in the
first partial derivatives of the field potential. To be more precise,
the argument runs like this:
\begin{quote}
One can make precise the sense in which Newtonian gravitational
theory is the ``weak-field'' limit of general relativity
\cite{malament-newt-grav-geom-spc}. In this limit, it is clear that
the metric field plays roughly the role in general relativity that
the scalar potential $\phi$ does in Newtonian theory. In Newtonian
theory, bracketing certain technical questions about boundary
conditions, there is a more or less well-defined energy density of
the gravitational field, proportional to $( \nabla \phi )^2$. One
might expect, therefore, based on some sort of continuity argument,
or just on the strength of the analogy itself, that any local
representation of gravitational energy in general relativity ought
to be a ``quadratic function of the first partials of the
metric''.\footnote{In this light, it is interesting to note that
gravitational energy pseudo-tensors do tend to be quadratic in the
first-order partials of the metric.} The stress-energy tensor of
no other field, moreover, is higher than first-order in the partials
of the field potential, so surely gravity cannot be different. No
invariant quantity at a point can be constructed using only the
first partials of the metric, however, so there can be no scalar or
tensorial representation of gravitational energy in general
relativity.
\end{quote}
(No researcher I know makes the argument exactly in this form; it is
just the clearest, most concise version I can come up with myself.)
As \citeN[p.\ 178]{pauli21} forcefully argued, however, there can be
no \emph{physical} argument against the possibility that gravitational
energy depends on second derivatives of the metric; the argument above
certainly provides none. Just because the energy of all other known
fields have the same form in no way implies that a localized
gravitational energy in general relativity, if there is such a thing,
ought to have that form as well. Gravity is too different a field
from others for such a bare assertion to carry any weight. As I
explain at the end of \S\ref{sec:conds}, moreover, a proper
understanding of tensorial concomitants reveals that an expression
linear in second partial derivatives is in the event equivalent in the
relevant sense to one quadratic in first order partials. This
illustrates how misleading the analogy with Newtonian gravity can be.
\section{Geometric Fiber Bundles, Concomitants, and Geometric
Objects}
\label{sec:concomitants}
\resetsec
\skipline[.5]
\begin{quote}
The introduction of a coordinate system to geometry is an act of
violence.
\skipline[-.5]
\begin{flushright}
Hermann Weyl \\
\emph{Philosophy of Mathematics and Natural Science}
\end{flushright}
\end{quote}
I have argued that, if there is an object that deserves to be thought
of as the representation of gravitational stress-energy of in general
relativity, then it ought to depend on the Riemann curvature tensor.
Since there is no obvious mathematical sense in which a general
mathematical structure can ``depend'' on a tensor, the first task is
to say what exactly this could mean. I will call a mathematical
structure on a manifold that depends in the appropriate fashion on
another structure on the manifold, or set of others, a
\emph{concomitant} of it (or them).
The reason I am inquiring into the possibility of a concomitant in the
first place, when the question is the possible existence of a
representation of gravitational stress-energy tensor, is a simple one.
What is wanted is an expression for gravitational energy that does not
depend for its formulation on the particulars of the spacetime, just
as the expression for the kinetic energy of a particle in classical
physics does not depend on the internal constitution of the particle
or on the particular interactions it may have with its environment,
and just as the stress-energy tensor for a Maxwell field can be
calculated in any spacetime irrespective of its
particulars.\footnote{This property of (stress-)energy for other types
of physical systems already stands in contradistinction to the
properties of all known rigorous expressions for global
gravitational energy in general relativity, \emph{e}.\emph{g}., the
ADM mass and the Bondi energy, which can be defined only in
asymptotically flat spacetimes \cite{wald-gr}, and all such
quasi-local expressions, which can be defined only in stationary or
axisymmetric ones \cite{szabados-qloc-en-mom-ang-mom-gr}.} If there
is a well-formed expression for gravitational stress-energy, then one
should be able in principle to calculate it whenever there are
gravitational phenomena, which is to say, in any spacetime
whatsoever---it should be a \emph{function} of some set of geometric
objects associated with the curvature in that spacetime, in some
appropriately generalized sense of `function'. This idea is what a
concomitant is supposed to capture.
The term `concomitant' and the general idea of the thing is due to
\citeN[p.~15]{schouten-ricci-calc}.\footnote{The specific idea of
proving the uniqueness of a tensor that ``depends'' on another
tensor, and satisfies a few collateral conditions, dates back at
least to \citeN[pp.~315--318]{weyl-space-time-matter} and
\citeN{cartan22}. In fact, Weyl proved that the only two-index
symmetric covariant tensors one can construct at a point in any
spacetime, using only algebraic combinations of the components of
the metric and its first two partial derivatives in a coordinate
system at that point, that are at most linear in the second
derivatives of the metric, are linear combinations of the Ricci
curvature tensor, the scalar curvature times the metric and the
metric itself. In particular, the only such divergence-free tensors
one can construct at a point are linear combinations of the Einstein
tensor and the metric with constant coefficients.} The definition
Schouten proposed is expressed in terms of coordinates: depending on
what sort of concomitant one deals with, the components of the
concomitant in a given coordinate system must satisfy various
conditions of covariance under certain classes of coordinate
transformations, when those transformations are also applied to the
components of the objects the concomitant is defined as a ``function''
of. His work was picked up and generalized by several other
mathematicians, such as \citeN{aczel-komit-diff-komit}, who extended
Schouten's work to treat more generalized classes of higher-order
differential concomitants.\footnote{I thank an anonymous referee for
drawing my attention to the work of Acz\'el and others who developed
Schouten's work.} \label{pg:schouten-difficult}The definitions
provided by this early work is clear, straightforward and easy to
grasp in the abstract, but becomes difficult to work with in
particular cases of interest---Schouten's covariance conditions
translate into a set of partial differential equations in a particular
coordinate system, which even in seemingly straightforward cases turn
out to be forbiddingly complicated. This makes it not only unwieldy
in practice and inelegant, but, more important, it makes it difficult
to discern what of intrinsic physical significance is encoded in the
relation of being a concomitant in particular cases of interest. It
is almost impossible to determine anything of the general properties
of a particular kind of concomitant of a particular (set of) object(s)
by looking at those equations.\footnote{\label{fn:hairy}For a good
example of just how hairy those conditions can be, see
\citeN[p.~350]{duplessis-tens-concoms-cons-laws} for a complete set written out
explicitly in the case of two covariant-index tensorial second-order
differential concomitants of a Lorentz metric.} I suspect that it
is because in particular cases the conditions are so complex,
difficult and opaque that use is very rarely made of concomitants in
arguments about spacetime structure in general relativity. This is a
shame, for the idea is, I think, potentially rich, and so calls out
for an invariant formulation.\footnote{There is a tradition, initiated
in the 1970s by \citeN{nijenhuis-natl-bunds}, that attempts a more
invariant formulation of a notion similar to Schouten's original
one, introducing the idea of ``natural bundles'' as a natural
setting for the definition and study of structures closely related
to what I call here geometrical objects. That work was elaborated
and extended by, \emph{e}.\emph{g}.,
\citeN{epstein-natl-tens-riem-mnflds} and
\citeN{epstein-thurston-trans-grps-natl-bunds}, \emph{inter alia}.
That work is essentially similar to the constructions and arguments
I give here. I did not know of it when I developed my own work. (I
thank the anonymous referee again for drawing my attention to it.)
There are two novelties I can claim for my definitions and
constructions (besides the fact that it is now all presented in a
purely invariant way, with no use of coordinates). First is my
definition of fiber bundles without reference to an associated group
of transformations, and so the consequent development of what I call
geometric bundles based on the idea of inductions. Second, the idea
of an induction allows for a simple generalization of my definition
for concomitants to more general structures than just tensorial-like
objects, \emph{e}.\emph{g}., projective structures as characterized
by an appropriate family of curves; I do not develop that
generalization here as it is not needed. Also, to the best of my
knowledge, the main result of \S\ref{sec:concomitants-metric},
theorem~\ref{thm:1-2-jet-metric}, is new, and of some interest in
its own right, besides the use I put it to in proving
theorem~\ref{thm:only-einstein-tens}. (There is some contemporary work being
done on so-called natural transformations---\emph{e}.\emph{g}.,
\citeNP{kolar-et-nat-opns-dg} and
\citeNP{fatibene-francaviglia-nat-gauge-form-cft}, dating back to
\citeNP{palais-terng-nat-bunds-fin-ord}---that bears some similarity
to all these ideas, but I do not discuss it, first because it is
formulated in category theory and so is fundamentally algebraic in
nature, whereas I aim for a formulation with clear and intuitive
geometric content, and second because my idea of an induction
differentiates my work in important ways from it.)}
I use the machinery of fiber bundles to characterize the idea of a
concomitant in invariant terms. I give a (brief) explicit formulation
of the machinery, because the one I rely on is non-standard. (We
assume from hereon that all relevant structures, mappings,
\emph{etc}., are smooth; nothing is lost by the assumption and it
simplifies exposition---all germane constructions and proofs can
easily be generalized to the case of topological spaces and continuous
structures.)
\begin{defn}
\label{defn:fiberbundle}
A \emph{fiber bundle} $\mathfrak{B}$ is an ordered triplet,
$(\mathcal{B}, \, \mathcal{M}, \, \pi)$, such that:
\begin{enumerate}
\addtolength{\itemindent}{2em}
\item[\bf{FB1}.] $\mathcal{B}$ is a differential manifold
\item[\bf{FB2}.] $\mathcal{M}$ is a differential manifold
\item[\bf{FB3}.] $\pi : \mathcal{B} \rightarrow \mathcal{M}$ is
smooth and onto
\item[\bf{FB4}.] For every $q,p \in \mathcal{M}$, $\pi^{-1} ( q
)$ is diffeomorphic to $\pi^{-1} ( p )$ (as submanifolds of
$\mathcal{B}$)
\item[\bf{FB5}.] $\mathcal{B}$ has a locally trivial product
structure, in the sense that for each $q \in \mathcal{M}$ there is
a neighborhood $U \ni q$ and a diffeomorphism $\zeta : \pi^{-1}
[U] \rightarrow U \times \pi^{-1} (q)$ such that the action of
$\pi$ commutes with the action of $\zeta$ followed by projection
on the first factor.
\end{enumerate}
\end{defn}
$\mathcal{B}$ is the \emph{bundle space}, $\mathcal{M}$ the \emph{base
space}, $\pi$ the \emph{projection} and $\pi^{-1} ( q )$ the
\emph{fiber} over $q$. By a convenient, conventional abuse of
terminology, I will sometimes call $\mathcal{B}$ itself `the fiber
bundle' (or `the bundle' for short). A \emph{cross-section} $\kappa$
is a smooth map from $\mathcal{M}$ into $\mathcal{B}$ such that $\pi
(\kappa (q)) = q$, for all $q$ in the mapping's domain.
This definition of a fiber bundle is non-standard in so far as no
group action on the fibers is fixed from the start; this implies that
no correlation between diffeomorphisms of the base space and
diffeomorphisms of the bundle space is fixed.\footnote{See,
\emph{e}.\emph{g}., \citeN{steenrod-topo-fbs} for the traditional
definition and the way that a fixed group action on the fibers
induces a correlation between diffeomorphisms on the bundle space
and those on the base space.} One must fix that explicitly. On the
view I advocate, the geometric character of the objects represented by
the bundle arises arises not from the group action directly, but only
after the explicit fixation of a correlation between diffeomorphisms
on the base space with those on the bundle space---only after, that
is, one fixes how a diffeomorphism on the base space induces one on
the bundle. For example, depending on how one decides that a
diffeomorphism on the base space ought to induce a diffeomorphism on
the bundle over it whose fibers consist of 1-dimensional vector
spaces, one will ascribe to the objects of the bundle the character
either of ordinary scalars or of $n$-forms (where $n$ is the dimension
of the base space). The idea is that the diffeomorphisms induced on
the bundle space then implicitly define the group action on the fibers
appropriate for the required sort of object.\footnote{I will not work
out here the details of how this comes about, as they are not needed
for the arguments of the paper.}
I call an appropriate mapping of diffeomorphisms on the base space to
those on the bundle space an \emph{induction}. (I give a precise
definition in a moment.) In this scheme, therefore, the induction
comes first conceptually, and the relation between diffeomorphisms on
the base space and those they induce on the bundle serves to fix the
fibers as spaces of \emph{geometric objects}, \emph{viz}., those whose
transformative properties are tied directly and intimately to those of
the ambient base space.\footnote{See
Anderson~\citeyear{anderson62,anderson67},
\citeN{friedman-fnds-st-theors} and \citeN{belot-bckgrnd-indep} for
other approaches to defining geometric or (as they refer to them)
absolute objects.} This way of thinking of fiber bundles is perhaps
not well suited to the traditional mathematical task of classifying
bundles, but it turns out to be just the thing on which to base a
perspicuous and useful definition of concomitant. Although a
diffeomorphism on a base space will naturally induce a unique one on
certain types of fiber bundles over it, such as tensor bundles, in
general it will not. There is not known, for instance, any natural
way to single out a map of diffeomorphisms of the base space into
those of a bundle over it whose fibers consist of spinorial
objects.\footnote{See, \emph{e}.\emph{g}.,
\citeN{penrose-rindler-spinors-st-1}.} Inductions neatly handle
such problematic cases.
I turn now to making this intuitive discussion more precise. A
diffeomorphism $\phi^\sharp$ of a bundle space $\mathcal{B}$ is
\emph{consistent} with $\phi$, a diffeomorphism of the base space
$\mathcal{M}$, if, for all $u \in \mathcal{B}$,
\[
\pi (\phi^\sharp (u)) = \phi (\pi (u))
\]
For a general bundle, there will be scads of diffeomorphisms
consistent with a given diffeomorphism on the base space. A way is
needed to fix a unique $\phi^\sharp$ consistent with a $\phi$ so that
a few obvious conditions are met. For example, the identity
diffeomorphism on $\mathcal{M}$ ought to pick out the identity
diffeomorphism on $\mathcal{B}$. More generally, if $\phi$ is a
diffeomorphism on $\mathcal{M}$ that is the identity on an open set $O
\subset \mathcal{M}$ and differs from the identity outside $O$, it
ought to be the case that the mapping picks out a $\phi^\sharp$ that
is the identity on $\pi^{-1} [O]$. If this holds, we say that that
$\phi^\sharp$ is \emph{strongly consistent} with $\phi$.
Let $\mathfrak{D}_\mathcal{M}$ and $\mathfrak{D}_\mathcal{B}$ be,
respectively, the groups of diffeomorphisms on $\mathcal{M}$ and
$\mathcal{B}$. Define the set
\begin{center}
$\mathfrak{D}^\sharp_\mathcal{B} = \{ \phi^\sharp \in
\mathfrak{D}_\mathcal{B} : \: \exists \phi \in
\mathfrak{D}_\mathcal{M}$ such that $\phi^\sharp$ is strongly
consistent with $\phi \}$
\end{center}
It is simple to show that $\mathfrak{D}^\sharp_\mathcal{B}$ forms a
subgroup of $\mathfrak{D}_\mathcal{B}$. This suggests
\begin{defn}
\label{defn:induction}
An \emph{induction} is an injective homomorphism $\iota :
\mathfrak{D}_\mathcal{M} \rightarrow
\mathfrak{D}^\sharp_\mathcal{B}$.
\end{defn}
$\phi$ will be said to \emph{induce} $\phi^\sharp$ (under $\iota$) if
$\iota ( \phi ) = \phi^\sharp$.\footnote{In a more thorough treatment,
one would characterize the way that the induction fixes a group
action on the fibers, but we do not need to go into that for our
purposes.}
\begin{defn}
\label{defn:geom-bundle}
A \emph{geometric fiber bundle} is an ordered quadruplet
$(\mathcal{B}, \, \mathcal{M}, \, \pi, \, \iota)$ where
\begin{enumerate}
\addtolength{\itemindent}{3em}
\item[{\bf GFB1}.] $(\mathcal{B}, \, \mathcal{M}, \, \pi)$
satisfies FB1-FB5
\item[{\bf GFB2}.] $\iota$ is an induction
\end{enumerate}
\end{defn}
Geometric fiber bundles are the appropriate spaces to serve as the
domains and ranges of concomitant mappings.
Most of the fiber bundles one works with in physics are geometric
fiber bundles. A tensor bundle $\mathcal{B}$, for example, is a fiber
bundle over a manifold $\mathcal{M}$ each of whose fibers is
diffeomorphic to the vector space of tensors of a particular index
structure over any point of the manifold; a basis for an atlas is
provided by the charts on $\mathcal{B}$ naturally induced from those
on $\mathcal{M}$ by the representation of tensors on $\mathcal{M}$ as
collections of components in $\mathcal{M}$'s coordinate systems.
There is a natural induction in this case fixed by the pull-back
action of a diffeomorphism $\phi$ of tensors on $\mathcal{M}$. Spinor
bundles provide interesting examples of physically important bundles
that have no natural, unique inductions, though there are classes of
them.
We are finally in a position to define concomitants. Let
$(\mathcal{B}_1, \, \mathcal{M}, \, \pi_1, \, \iota_1)$ and
$(\mathcal{B}_2, \, \mathcal{M}, \, \pi_2, \, \iota_2)$ be two
geometric bundles with the same base space.\footnote{One can
generalize the definition of concomitants to cover the case of
bundles over different base spaces, but we do not need this here.}
\begin{defn}
\label{def:concom}
A mapping $\chi : \mathcal{B}_1 \rightarrow \mathcal{B}_2$ is a
\emph{concomitant} if
\[
\chi (\iota_1 (\phi) (u_1)) = \iota_2 (\phi) (\chi (u_1))
\]
for all $u_1 \in \mathcal{B}_1$ and all $\phi \in
\mathfrak{D}_\mathcal{M}$.
\end{defn}
In intuitive terms, a concomitant is a mapping between bundles that
commutes with the action of the induced diffeomorphisms that lend the
objects of the bundles their respective geometrical characters,
\emph{i}.\emph{e}., the structure in virtue of which they are, in a
precise sense, \emph{geometric} objects. It is easy to see that
$\chi$ must be fiber-preserving, in the sense that it maps fibers of
$\mathcal{B}_1$ to fibers of $\mathcal{B}_2$. This captures the idea
that the dependence of the one type of object on the other is strictly
local; the respecting of the actions of diffeomorphisms captures the
idea that the mapping encodes an invariant relation. By another
convenient abuse of terminology, I will often refer to the range of
the concomitant mapping itself as `the concomitant' of the domain.
\section{Jet Bundles, Higher-Order Concomitants, and Geometric
Objects}
\label{sec:jet-bundles}
\resetsec
Just as with ordinary functions from one Euclidean space to another,
it seems plausible that the dependence encoded in a concomitant from
one geometric bundle to another may take into account not only the
value of the first geometrical structure at a point of the base space,
but also ``how that value is changing'' in a neighborhood of that
point, something like a generalized derivative of a geometrical
structure on a manifold. The following construction is meant to
capture in a precise sense the idea of a generalized derivative in
such a way so as to make it easy to generalize the idea of a
concomitant to account for it.
Fix a geometric fibre bundle
$(\mathcal{B}, \, \mathcal{M}, \, \pi, \, \iota)$, and the space of
its sections $\Gamma [\mathcal{B}]$. Two sections
$\gamma, \eta : \mathcal{M} \rightarrow \mathcal{B}$ \emph{osculate to
first-order} at $p \in \mathcal{M}$ if $T\gamma$ and $T\eta$ (the
differentials of the mappings) agree in their action on
$T_p \mathcal{M}$. (They osculate to zeroth-order at $p$ if they map
$p$ to the same point in the domain.) This defines an equivalence
relation on $\Gamma [\mathcal{B}]$. A \emph{1-jet} with source $p$
and target $\gamma(p)$, written `$j^1_p [\gamma]$', is such an
equivalence class. It is not difficult to show that the set of all
1-jets,
\[
J^1 \mathcal{B} \coloneq \bigcup_{p \in \mathcal{M}, \gamma \in \Gamma
[\mathcal{B}]} j^1_p [\gamma]
\]
naturally inherits the structure of a differentiable manifold
\cite{hirsch-diff-top}. One can naturally fibre $J^1 \mathcal{B}$
over $\mathcal{M}$. The \emph{source projection}
$\sigma^1 : J^1 \mathcal{B} \rightarrow \mathcal{M}$, defined by
\[
\sigma^1 (j^1_p [\gamma]) = p
\]
gives $J^1 \mathcal{B}$ the structure of a bundle space over the base
space $\mathcal{M}$, and in this case we write the bundle $(J^1
\mathcal{B}, \, \mathcal{M}, \, \sigma^1)$. A section $\gamma$ of
$\mathcal{B}$ naturally gives rise to a section $j^1 [\gamma]$ of $J^1
\mathcal{B}$, the \emph{first-order prolongation} of that section:
\[
j^1 [\gamma] : \mathcal{M} \rightarrow \bigcup_{p \in \mathcal{M}}
j^1_p [\gamma]
\]
such that $\sigma_1 (j^1 [\gamma] (p)) = p$. (We assume for the sake
of simplicity that global cross-sections exist; the modifications
required to treat local cross-sections are trivial.)
The points of $J^1 \mathcal{B}$ may be thought of as coordinate-free
representations of first-order Taylor expansions of sections of
$\mathcal{B}$. To see this, consider the example of the trivial
bundle $(\mathcal{B}, \, \mathbb{R}^2, \, \pi)$ where $\mathcal{B}
\coloneq \mathbb{R}^2 \times \mathbb{R}$ and $\pi$ is projection onto
the first factor. Fix global coordinates $(x^1, \, x^2, \, v^1)$ on
$\mathcal{B}$, so that the induced (global) coordinates on $J^1
\mathcal{B}$ are $(x^1, \, x^2, \, v^1, \, v^1_1, \, v^1_2)$. Then
for any 1-jet $j^1_q [\gamma]$, define the inhomogenous linear
function $\hat{\gamma} : \mathbb{R}^2 \rightarrow \mathbb{R}$ by
\[
\hat{\gamma}(p) = v^1 (\gamma(p)) + v^1_1 (j^1_q [\gamma])(p_1 - q_1)
+ v^1_2 (j^1_q [\gamma])(p_2 - q_2)
\]
where $\gamma \in j^1_q [\gamma]$, and $p, q \in \mathbb{R}^2$ with
respective components $(p_1, \, p_2)$ and $(q_1, \, q_2)$. Clearly
$\hat{\gamma}$ defines a cross-section of $J^1 \mathcal{B}$
first-order osculant to $\gamma$ at $p$ and so is a member of $j^1_q
[\gamma]$; indeed, it is the unique globally defined, linear
inhomogeneous map with this property.
A 2-jet is defined similarly, by iteration, as an equivalence class of
sections under the relation of having the same first and second
differentials (as mappings) at a point. More precisely,
$\gamma, \eta \in \Gamma [\mathcal{B}]$ \emph{osculate to second
order} at $p \in \mathcal{M}$ if they are in the same 1-jet and if
their second-order differentials equal each other,
$T(T\gamma) = T(T\eta)$. Again, this defines an equivalence relation
on $\Gamma [\mathcal{B}]$. A \emph{2-jet} with source $p$ and target
$\gamma(p)$, written `$j^1_p [\gamma]$', is such an equivalence class.
The set of all 2-jets,
\[
J^2 \mathcal{B} \coloneq \bigcup_{p \in \mathcal{M}, \gamma \in \Gamma
[\mathcal{B}]} j^2_p [\gamma]
\]
also inherits the structure of a differentiable manifold.
$J^2 \mathcal{B}$ is naturally fibered over $\mathcal{M}$ by the
source projection
$\sigma^2 : J^2 \mathcal{B} \rightarrow \mathcal{M}$, defined by
\[
\sigma^2 (j^2_p [\gamma]) = p
\]
giving $J^2 \mathcal{B}$ the structure of a bundle space over the base
space $\mathcal{M}$, $(J^2 \mathcal{B}, \, \mathcal{M}, \, \sigma^2)$.
Again, a section $\gamma$ of $\mathcal{B}$ gives rise to a section
$j^2 [\gamma]$ of $J^2 \mathcal{B}$, the \emph{second-order
prolongation} of that section:
\[
j^2 [\gamma] : \mathcal{M} \rightarrow \bigcup_{p \in \mathcal{M}}
j^2_p [\gamma]
\]
such that $\sigma_1 (j^2 [\gamma] (p)) = p$. Jet bundles of all
higher orders are defined in the same way.
There is a natural projection from $J^2 \mathcal{B}$ to $J^1
\mathcal{B}$, the \emph{truncation} $\theta^{2,1}$, characterized by
``dropping the second-order terms in the Taylor expansion''. In
general, one has the natural truncation $\theta^{n,m} : J^n
\mathcal{B} \rightarrow J^m \mathcal{B}$ for all $0 < m < n$.
For our purposes, the most important fact about these spaces is that
the jet bundles of a geometric bundle are themselves naturally
geometric bundles. Fix a geometric bundle
$(\mathcal{B}, \, \mathcal{M}, \pi, \, \iota)$ and a diffeomorphism
$\phi$ on $\mathcal{M}$. Then $\iota[\phi]$ not only defines an
action on points of $\mathcal{B}$, but, as a diffeomorphism itself on
$\mathcal{B}$, it naturally defines an action on the cross-sections of
$\mathcal{B}$ and thus on the 1-jets. by the natural pull-back of
differentials of mappings. It is easy to show that the mapping
$\iota^1$ so specified from $\mathfrak{D}_\mathcal{M}$ to
$\mathfrak{D}^\sharp_{J^1 \mathcal{B}}$ is an injective homomorphism
and thus itself an induction; therefore,
$(J^1 \mathcal{B}, \, \mathcal{M}, \sigma^1, \, \iota^1)$ is a
geometric fiber bundle. One defines inductions for higher-order jet
bundles in the same way.
We can now generalize our definition of concomitants. Let
$(\mathcal{B}_1, \, \mathcal{M}, \, \pi_1, \, \iota)$ and
$(\mathcal{B}_2, \, \mathcal{M}, \, \pi_2, \, \jmath)$ be two
geometric fiber bundles over the manifold $\mathcal{M}$.
\begin{defn}
\label{defn:nth-concoms}
An \emph{$n^{\text{th}}$-order concomitant} ($n$ a strictly positive
integer) from $\mathcal{B}_1$ to $\mathcal{B}_2$ is a smooth mapping
$\chi : J^n \mathcal{B}_1 \rightarrow \mathcal{B}_2$ such that
\begin{enumerate}
\item $( \forall u \in J^n \mathcal{B}_1 )( \forall \phi \in
\mathfrak{A}_\mathcal{M} ) \; \jmath (\phi) (\chi(u)) = \chi (
\iota^n (\phi) (u) )$
\item there is no $(n-1)^{\text{th}}$-order concomitant
$\chi' : J^{n-1} \mathcal{B}_1 \rightarrow \mathcal{B}_2$
satisfying
\[
( \forall u \in J^n \mathcal{B}_1 ) \; \chi (u) = \chi'
(\theta^{n,n-1} (u))
\]
\end{enumerate}
\end{defn}
A zeroth-order concomitant (or just `concomitant' for short, when no
confusion will arise), is defined by \ref{def:concom}.
An important property of concomitants is that, in a limited sense,
they are transitive.
\begin{prop}
\label{prop:con-composition}
If $\chi_1: J^n \mathcal{B}_1 \rightarrow \mathcal{B}_2$ is an
$n^{\text{th}}$-order concomitant and $\chi_2: \mathcal{B}_2
\rightarrow \mathcal{B}_3$ is a smooth mapping, where
$\mathcal{B}_1$, $\mathcal{B}_2$ and $\mathcal{B}_3$ are geometric
bundles over the same base space, then $\chi_2 \circ \chi_1$ is an
$n^{\text{th}}$-order concomitant if and only if $\chi_2$ is a
zeroth-order concomitant.
\end{prop}
This follows directly from the fact that inductions are injective
homomorphisms and concomitants respect the fibers.
It will be of physical interest in \S\ref{sec:conds} to consider the
way that concomitants interact with multiplication by a scalar field.
(Since we consider in this paper only concomitants of linear and
affine objects, multiplication of the object by a scalar field is
always defined.) Towards that end, let us say that a concomitant is
\emph{homogeneous of weight $w$} if for any constant scalar field
$\xi$
\[
\chi (\iota_1 (\phi) (\xi u_1)) = \xi^w \iota_2 (\phi) (\chi (u_1))
\]
\section{Concomitants of the Metric}
\label{sec:concomitants-metric}
\resetsec
As a specific example that will be of use in what follows, consider
the geometric fiber bundle
$(\mathcal{B}_{\text{\small g}}, \, \mathcal{M}, \, \pi_{\text{\small
g}}, \, \iota_{\text{\small g}})$,
with $\mathcal{M}$ a 4-dimensional, Hausdorff, paracompact, connected,
smooth manifold (\emph{i}.\emph{e}., a candidate spacetime manifold),
the fibers of $\mathcal{B}_{\text{\small g}}$ diffeomorphic to the
space of Lorentz metrics at each point of $\mathcal{M}$, all of the
same signature $(+, \, -, \, -, \, -)$, and $\iota_{\text{\small g}}$
the induction defined by the natural pull-back. Since the set of
Lorentz metrics in the tangent plane over a point of a 4-dimensional
manifold, all of the same signature, is a 10-dimensional
manifold,\footnote{In fact, it is diffeomorphic to a connected,
convex, open subset---an open cone with vertex at the origin---in
$\mathbb{R}^{10}$, and has the further structure of a Fr\'echet
manifold \cite{curiel-meas-topo-prob-cosmo}.} the bundle space
$\mathcal{B}_{\text{\small g}}$ is a 14-dimensional manifold. A
cross-section of this bundle defines a Lorentz metric field on the
manifold.
The following proposition precisely captures the statement one
sometimes hears that there is no scalar or tensorial quantity one can
construct depending only on the metric and its first-order partial
derivatives at a point of a manifold.
\begin{prop}
\label{prop:no1metric}
There is no first-order concomitant from $\mathcal{B}_{\text{\small
g}}$ to any tensor bundle over $\mathcal{M}$.
\end{prop}
To prove this, it suffices to remark that, given any spacetime
$(\mathcal{M}, \, g_{ab})$ and any two points $p,p' \in \mathcal{M}$,
there are open neighborhoods $U$ of $p$ and $U'$ of $p'$ and a
diffeomorphism $\phi: \; \mathcal{M} \rightarrow \mathcal{M}$, such
that $\phi(p) = p'$, $\phi^\sharp (g'_{ab}) = g_{ab}$ at $p$, and
$\phi^\sharp(\nabla_a g_{bc}) = \nabla_a g_{bc}$ at $p$, where
$\nabla_a$ is any derivative operator other than the Levi-Civita one
associated with $g_{ab}$, and $\phi^\sharp$ is the map naturally
induced by the pull-back action of $\phi$.
This is not to say, however, that no information of interest is
contained in $J^1 \mathcal{B}_{\text{\small g}}$. Indeed, two metrics
$g_{ab}$ and $h_{ab}$ are first-order osculant at a point if and only
if they have the same associated covariant derivative operator at that
point. To see this, first note that, if they osculate to first order
at that point, then $\hat{\nabla}_a (g_{bc} - h_{bc}) = 0$ at that
point for all derivative operators. Thus, for the derivative operator
$\nabla_a$ associated with, say, $g_{ab}$, $\nabla_a (g_{bc} - h_{bc})
= 0$, but $\nabla_a g_{bc} = 0$, so $\nabla_a h_{bc} = 0$ at that
point as well. Similarly, if the two metrics are equal and share the
same associated derivative operator $\nabla_a$ at a point, then
$\hat{\nabla}_a (g_{bc} - h_{bc}) = 0$ at that point for all
derivative operators, since their difference will be identically
annihilated by $\nabla_a$, and $g_{ab} = h_{ab}$ at the point by
assumption. Thus they are first-order osculant at that point and so
in the same 1-jet. This proves that all and only geometrically
relevant information contained in the 1-jets of Lorentz metrics on
$\mathcal{M}$ is encoded in the fiber bundle over spacetime the values
of the fibers of which are ordered pairs consisting of a metric and
the metric's associated derivative operator at a spacetime point.
The second jet bundle over $\mathcal{B}_{\text{\small g}}$ has a
similarly interesting structure. Clearly, if two metrics are in the
same 2-jet, then they have the same Riemann tensor at the point
associated with the 2-jet, since the result of doubly applying an
arbitrary derivative operator (not the Levi-Civita one associated with
the metric) to it at the point yields the same tensor. Assume now
that two metrics are in the same 1-jet and have the same Riemann
tensor at the associated spacetime point. If it follows that they are
in the same 2-jet, then essentially all and only geometrically
relevant information contained in the 2-jets of Lorentz metrics on
$\mathcal{M}$ is encoded in the fiber bundle over spacetime the points
of the fibers of which are ordered triplets consisting of a metric,
the metric's associated derivative operator and the metric's Riemann
tensor at a spacetime point. To demonstrate this, it suffices to show
that if two Levi-Civita connections agree on their respective Riemann
tensors at a point, then the two associated derivative operators are
in the same 1-jet of the bundle whose base-space is $\mathcal{M}$ and
whose fibers consist of the affine spaces of derivative operators at
the points of $\mathcal{M}$ (because they will then agree on the
result of application of themselves to their difference tensor, and
thus will be in the 2-jet of the same metric at that point).
Assume that, at a point $p$ of spacetime, $g_{ab} = \tilde{g}_{ab}$,
$\nabla_a = \tilde{\nabla}_a$ (the respective derivative operators),
and $R^a {}_{bcd} = \tilde{R}^a {}_{bcd}$ (the respective Riemann
tensors). Let $C^a {}_{bc}$ be the symmetric difference-tensor
between $\nabla_a$ and $\tilde{\nabla}_a$, which is itself 0 at $p$ by
assumption. Then by definition $\nabla_{[b} \nabla_{c]} \xi^a = R^a
{}_{bcn} \xi^n$ for any vector $\xi^a$, and so at $p$
\begin{equation*}
\begin{split}
R^c {}_{abn} \xi^n &= \nabla_{[a} \tilde{\nabla}_{b]} \xi^c \\
&= \nabla_a (\nabla_b \xi^c + C^c {}_{bn} \xi^n) -
\tilde{\nabla}_b \nabla_a\xi^c \\
&= \nabla_a \nabla_b \xi^c + \nabla_a (C^c {}_{bn} \xi^n) -
\nabla_b \nabla_a \xi^c - C^c {}_{bn} \nabla_a \xi^n + C^n {}_{ba}
\nabla_n\xi^c
\end{split}
\end{equation*}
but $\nabla_b \nabla_c \xi^a - \nabla_c \nabla_b \xi^a = 2 R^a
{}_{bcn} \xi^n$ and $C^a {}_{bc} = 0$, so expanding the only remaining
term gives
\[
\xi^n \nabla_a C^c {}_{bn} = 0
\]
for arbitrary $\xi^a$ and thus $\nabla_a C^b {}_{cd} = 0$ at $p$; by
the analogous computation, $\tilde{\nabla}_a C^b {}_{cd} = 0$ as well.
It follows immediately that $\nabla_a$ and $\tilde{\nabla}_a$ are in
the same 1-jet over $p$ of the affine bundle of derivative operators
over $\mathcal{M}$. We have proven
\begin{theorem}
\label{thm:1-2-jet-metric}
$J^1 \mathcal{B}_{\text{\small g}}$ is naturally diffeomorphic to
the geometric fiber bundle over $\mathcal{M}$ whose fibers consist
of pairs $(g_{ab}, \, \nabla_a)$, where $g_{ab}$ is the value of a
Lorentz metric field at a point of $\mathcal{M}$, and $\nabla_a$ is
the value of the covariant derivative operator associated with
$g_{ab}$ at that point, the induction being defined by the natural
pull-back. $J^2 \mathcal{B}_{\text{\small g}}$ is naturally
diffeomorphic to the geometric fiber bundle over $\mathcal{M}$ whose
fibers consist of triplets $(g_{ab}, \, \nabla_a, \, R_{abc} {}^d)$,
where $g_{ab}$ is the value of a Lorentz metric field at a point of
$\mathcal{M}$, and $\nabla_a$ and $R_{abc}{}^d$ are respectively the
covariant derivative operator and the Riemann tensor associated with
$g_{ab}$ at that point, the induction being defined by the natural
pull-back.
\end{theorem}
It follows immediately that there is a first-order concomitant from
$\mathcal{B}_{\text{\small g}}$ to the geometric bundle
$(\mathcal{B}_\nabla, \, \mathcal{M}, \, \pi_\nabla,$ $\iota_\nabla)$
of derivative operators, \emph{viz}., the mapping that takes each
Lorentz metric to its associated derivative operator. (This does not
contradict proposition~\ref{prop:no1metric}, as the bundle of
derivative operators is an affine not a tensor bundle.) Likewise,
there is a second-order concomitant from
$\mathcal{B}_{\text{\small g}}$ to the geometric bundle
$(\mathcal{B}_{\text{\small Riem}}, \, \mathcal{M}, \,
\pi_{\text{\small Riem}},$
$\iota_{\text{\small Riem}})$ of tensors with the same index structure
and symmetries as the Riemann tensor, \emph{viz}., the mapping that
takes each Lorentz metric to its associated Riemann tensor. (This is
the precise sense in which the Riemann tensor associated with a given
Lorentz metric is ``a function of the metric and its partial
derivatives up to second order''.) It is easy to see, moreover, that
both concomitants are homogeneous of degree 0.
It follows from theorem~\ref{thm:1-2-jet-metric} and
proposition~\ref{prop:con-composition} that a concomitant of the
metric will be second order if and only if it is a zeroth-order
concomitant of the Riemann tensor:
\begin{prop}
\label{prop:0-concoms-riemann}
A concomitant of the metric is second-order if and only if it can be
expressed as a sum of terms consisting of constants multiplied by
the Riemann tensor, the Ricci tensor, the Gaussian scalar curvature,
and contractions and products of these with the metric itself.
\end{prop}
\section{Conditions on a Possible Gravitational Stress-Energy Tensor}
\label{sec:conds}
\resetsec
We are almost in a position to state and prove the main result of the
paper, the nonexistence of a gravitational stress-energy tensor. In
order to formulate and prove a result having that proposition as its
natural interpretation, one must first lay down some natural
conditions on the proposed object, to show that no such object exists
satisfying the conditions. In general relativity, the stress-energy
tensor is the fundamental invariant quantity encoding all known
localized energetic properties of all known types of matter field, in
the sense that each known type of matter field has a canonical, unique
form of stress-energy tensor associated with it, and all other
invariant energetic quantities associated with the matter field are
derivable from that object. The canonical form of a stress-energy
tensor is a two-index, symmetric, covariantly divergence-free
tensor.\footnote{Thus, the Bel-Robinson tensor is ruled out from the
start, as it is a 4-index tensor. (For characterization and
discussion of the Bel-Robinson tensor and its properties, including
the way it gives rise to energy-like quantities, see
Senovilla~\citeyearNP{senovilla-superenrgy-tens,senovilla-superergy-tens-apps},
\citeNP{garecki-rmrks-bel-rob-tens} and
\citeNP{gomez-lobo-dynal-laws-supenrgy-gr}.) There are indeed
several other ``energetic quantities'' that have in general
relativity invariant representation in some form other than a
stress-energy tensor, \emph{e}.\emph{g}., the ADM mass and various
so-called quasi-local quantities
\cite{szabados-qloc-en-mom-ang-mom-gr}. Since none of those are
localized quantities, I do not consider them to be relevant to the
purposes of this paper. (One might also reasonably complain, so far
as my purposes go, that all of those quantities do not differentiate
between gravitational and non-gravitational forms of energy, but
rather represent only total, aggregate energy.) Starting with
\citeN{komar-cov-cons-laws-gr} and
\citeN{finkelstein-misner-new-cons-laws}, there is another tradition
in the context of general relativity of searching for quantities
that one might hope to be able to interpret as energetic quantities,
possibly associated in a physically relevant way with the
``gravitational field'', \emph{viz}., the search for scalar and
1-index objects satisfying various forms of ``conservation laws''.
(See as well, \emph{e}.\emph{g}., \citeNP{trautman-cons-laws-gr},
\citeNP{goldberg-inv-trans-cons-laws-ener-mom}.) As interesting as
that work is from a mathematical point of view, and as potentially
interesting as it may be from a physical point of view, I do not
consider here any of those quantities as viable candidates for
representations of a localized gravitational energetic quantity, for
several reasons. If there are localized energetic-like quantities
associated with ``the gravitational field'' in general relativity
that do not have the structure of $(0,2)$-index tensor, quantities
which are found from investigation of various possible forms of
conservation laws, then it seems to me there are two possibilities:
there is in fact a gravitational stress-energy tensor, and one can
derive those quantities from it, even though that is not how they
were discovered; or those quantities are in fact representative of
localized gravitational stress-energy, but the claim that they are
energetic in some important physical sense has to be articulated and
justified, with a particular eye to explaining how such an
energy-like quantity interacts with (or not) and is fungible with
(or not) the stress-energy content of ordinary matter. I do not
know how to do it for any of the objects associated with the search
for single-index conservation laws. Indeed, it is striking that
none of the researchers who have investigated such objects discuss
in any detail the possible physical interpretation of the
mathematical structures they were investigating, and in particular
how such quantities may relate to what we understand about ordinary
stress-energy.} Not just any such tensor will do, however, for that
gives only the baldest of formal characterizations of it. From a
physical point of view, at a minimum the object must have the physical
dimension of stress-energy for it to count as a stress-energy tensor.
That it have the dimension of stress-energy is what allows one to add
two of them together in a physically meaningful way to derive the
physical sum of total stress-energy from two different sources. In
classical mechanics, for instance, both velocity and spatial position
have the form of a three-dimensional vector, and so their formal sum
is well defined, but it makes no physical sense to add a velocity to a
position because the one has dimension \texttt{length/time} and the
other the dimension \texttt{length}. (I will give a precise
characterization of ``physical dimension'' below.)
An essential, defining characteristic of energy in classical physics
is its obeying some formulation of the First Law of Thermodynamics.
The formulation of the First Law I rely on is somewhat unorthodox:
that all forms of stress-energy are in principle ultimately
fungible---any form of energy can in principle be transformed into any
other form\footnote{\citeN[ch.~\textsc{v}, \S97]{maxwell-matt-mot}
makes this point especially clearly, including its relation to the
principle of energy conservation. See also \citeN[chs.~\textsc{i,
iii, iv, viii, xii}]{maxwell-theory-heat-1888}.}---not necessarily
that there is some absolute measure of the total energy contained in a
system or set of systems that is constant over time. In more precise
terms, this means that all forms of energy must be represented by
mathematical structures that allow one to define appropriate
operations of addition and subtraction among them, which the canonical
form of the stress-energy does allow for.\footnote{This is a
requirement even if one takes a more traditional view of the First
Law as making a statement about conservation of a magnitude
measuring a physical quantity.} I prefer this formulation of the
First Law in general relativity because there will not be in a general
cosmological context any well-defined global energetic quantity that
one can try to formulate a conservation principle for. In so far as
one wants to hold on to some principle like the classical First Law in
a relativistic context, therefore, I see no other way of doing it
besides formulating it in terms of fungibility. (If one likes, one
can take the fungibility condition as a necessary criterion for any
more traditional conservation law.) This idea is what the demand that
\emph{all} stress-energy tensors, no matter the source, have the same
physical dimension is intended to
capture.\footnote{\label{fn:einstein-seten}For what it's worth, this
conception has strong historical warrant---Einstein (implicitly)
used a very similar idea in one of his first papers laying out and
justifying the general theory \cite[p.~149]{einstein-fndn-gtr}:
\begin{quote}
It must be admitted that this introduction of the energy-tensor of
matter is not justified by the relativity postulate alone. For
this reason we have here deduced it from the requirement that the
energy of the gravitational field shall act gravitatively in the
same way as any other kind of energy.
\end{quote}
\citeN{moller-energy-mom-gr} also stresses the fact that the
formulation of integral conservation laws in general relativity
based on pseudo-tensorial quantities depends crucially on the
assumption that gravitational energy, such as it is, shares as many
properties as possible with the energy of ponderable
(\emph{i}.\emph{e}., non-gravitational) matter.}
To sum up, the stress-energy tensor encodes in general relativity all
there is to know of ponderable (\emph{i}.\emph{e}., non-gravitational)
energetic phenomena at a spacetime point:
\begin{enumerate}
\item it has 10 components representing with respect to a fixed
pseudo-orthonormal frame, say, the 6 components of the classical
stress-tensor, the 3 components of linear momentum and the scalar
energy density of the ponderable field at that point
\item that it has two covariant indices represents the fact that
it defines a linear mapping from timelike vectors at the point
(``worldline of an observer'') to covectors at that point
(``4-momentum covector of the field as measured by that observer''),
and so defines a bi-linear mapping from pairs of timelike vectors to
a scalar density at that point (``scalar energy density of the field
as measured by that observer''), because energetic phenomena,
crudely speaking, are marked by the fact that they are quadratic in
velocity and momental phenomena linear in velocity
\item that it is symmetric represents, ``in the limit of the
infinitesimal'', the classical principle of the conservation of
angular momentum; it also encodes part of the relativistic
equivalence of momentum-density flux and scalar energy density
\item that it is covariantly divergence-free represents the fact
that, ``in the limit of the infinitesimal'', the classical
principles of energy and linear momentum conservation are obeyed; it
also encodes part of the relativistic equivalence of
momentum-density flux and scalar energy density
\item the localization of ponderable stress-energy and its
invariance as a physical quantity are embodied in the fact that the
object representing it is a \emph{tensor}, a multi-linear map acting
only on the tangent and cotangent planes of the point it is
associated with\footnote{More generally, the notion of localized
quantity I use here means to be represented by a tensor-like
object (scalar, tensor, spinor, \ldots), one that has values
attributable to individual spacetime points and that in some sense
or other has properties or actions that ramify into the tangent
plane over that point in a way that can be made sense of by
restricting attention to the tangent plane.}
\item finally, the thermodynamic fungibility of energetic
phenomena is represented by the fact that the set of stress-energy
tensors forms a vector space---the sum and difference of any two is
itself a possible stress-energy tensor---all elements of which have
the same physical dimension
\end{enumerate}
Consequently, the appropriate mathematical representation of localized
gravitational stress-energy, if there is such a thing, is a two
covariant-index, symmetric, covariantly divergence-free tensor having
the physical dimension of
stress-energy.\footnote{\label{fn:pitts}\citeN{pitts-gauge-inv-local-grav-enrgs}
has proposed an infinite number of ways to define quantities that he
calls representations of localized gravitational energies (all
inequivalent). I exclude Pitts's proposal because I cannot see any
physical content to his constructed quantities. How,
\emph{e}.\emph{g}., could one use one of them to compute the energy
a gravitational-wave sensor would absorb from ambient gravitational
radiation? Precisely because his quantities depend on the frame one
fixes to make the computation, there can be no invariant, physically
well defined answer to such a question. If I stick a rod of
piezoelectric material in my cup of coffee and use it to warm the
coffee from the heat it generates by being deformed by a passing
gravitational wave, then surely the rise in temperature of the
coffee does not depend on which frame I use to perform the
calculation. How should the piezoelectric ``know'' which of Pitts's
``localized energies'' it should draw on? Since there seems to be
no way to answer such basic physical questions in an unambiguous
way, I do not see that what he has done is to characterize a
\emph{physical} quantity.} (That we demand it be covariantly
divergence-free is a delicate matter requiring special treatment,
which I give at the end of this section.)
Now, in order to make precise the idea of having the physical
dimension of stress-energy, recall that in general relativity all the
fundamental units one uses to define stress-energy, namely time,
length and mass, can themselves be defined using only the unit of
time; these are so-called geometrized units. For time, this is
trivially true: stipulate, say, that a time-unit is the time it takes
a certain kind of atom to vibrate a certain number of times under
certain conditions. A unit of length is then defined as that in which
light travels \emph{in vacuo} one time-unit. A unit of mass is
defined as that of which two, placed one length-unit apart, will
induce in each other by dint of their mutual gravitation alone an
acceleration towards each other of one length-unit per time-unit per
time-unit.\footnote{This definition may appear circular, in that it
would seem to require a unit of mass in the first place before one
could say that bodies were of the \emph{same} mass. I think the
circularity can be mitigated by using two bodies for which there are
strong prior grounds for positing that they are of equal mass,
\emph{e}.\emph{g}., two fundamental particles of the same type. It
also suffers from a fundamental lack of rigor that the definition of
length does not suffer from. In order to make the definition
rigorous, one would have to show that there exists a solution of the
Einstein field-equation (approximately) representing two particles
in otherwise empty space (as defined by the form of
$T_{ab}$)---\emph{viz}., two timelike geodesics---such that, if on a
spacelike hypersurface at which they both intersect 1 unit of length
apart (as defined on the hypersurface with respect to either) they
accelerate towards each other (as defined by relative acceleration
of the geodesics) one unit length per unit time squared, then the
product of the masses of the particles is 1. I will just assume,
for the purposes of this paper, that such solutions exist. Another
possibility for geometrizing a unit of mass would be to define one
as that of a Schwarzschild black hole with spatial radius one unit
of length, as measured with respect to a fixed radial coordinate
respecting the spherical and timelike symmetries of the spacetime.
It would be of some interest to determine the relation between these
two different ways of defining a geometrized unit of mass.} These
definitions of the units of mass and length guarantee that they scale
in precisely the same manner as the time-unit when new units of time
are chosen by multiplying the time-unit by some fixed real number
$\lambda^{-\frac{1}{2}}$. (The reason for the inverse square-root
will become clear in a moment). Thus, a duration of $t$ time-units
would become $t\lambda^{-\frac{1}{2}}$ of the new units; an interval
of $d$ units of length would likewise become $d\lambda^{-\frac{1}{2}}$
in the new units, and $m$ units of mass would become
$m\lambda^{-\frac{1}{2}}$ of the new units. This justifies treating
all three of these units as ``the same'', and so expressing
acceleration, say, in inverse time-units. To multiply the length of
all timelike vectors representing an interval of time by
$\lambda^{-\frac{1}{2}}$, however, is equivalent to multiplying the
metric by $\lambda$ (and so the inverse metric by $\lambda^{-1}$), and
indeed such a multiplication is the standard way one represents a
change of units in general relativity. This makes physical sense as
the way to capture the idea of physical dimension: all physical units,
the ones composing the dimension of any physical quantity, are
geometrized in general relativity in the most natural formulation, and
so depend only on the scale of the metric itself. By Weyl's theorem,
however, a metric times a constant represents exactly the same
physical phenomena as the original metric \cite[ch.~2,
\S1]{malament-fnds-gr-ngt}.
Now, the proper dimension of a stress-energy tensor can be determined
by the demand that the Einstein field-equation, $G_{ab} = \gamma
T_{ab}$, where $\gamma$ is Newton's gravitational constant, remain
satisfied when one rescales the metric by a constant factor. $\gamma$
has dimension $\frac{\mbox{(length)}^3} {\mbox{(mass)(time)}^2}$, and
so in geometrized units does not change under a constant rescaling of
the metric. Thus $T_{ab}$ ought to transform exactly as $G_{ab}$
under a constant rescaling of the metric. A simple calculation shows
that $G_{ab}$ $(= R_{ab} - \half R g_{ab})$ remains unchanged under
such a rescaling. Thus, a necessary condition for a tensor to
represent stress-energy is that it remain unchanged under a constant
rescaling of the metric. It follows that the concomitant at issue
must be homogeneous of weight 0 in the metric, whatever order it may
be.
We must still determine the order of the required concomitant, for it
is not \emph{a priori} obvious. In fact, the weight of a homogeneous
concomitant of the metric suffices to fix the differential order of
that concomitant.\footnote{I thank Robert Geroch for pointing this out
to me.} This can be seen as follows, as exemplified by the case of
a two covariant-index, homogeneous concomitant $S_{ab}$ of the metric.
A simple calculation based on definition~\ref{defn:nth-concoms} and on
the fact that the concomitant must be homogeneous shows that the value
at a point $p \in \mathcal{M}$ of an $n^{\text{th}}$-order concomitant
$S_{ab}$ can be written in the general form
\begin{equation}
\label{eq:Sab-form}
S_{ab} = \sum_\alpha k_\alpha \, g^{qx} \ldots g^{xr} \left(
\widetilde{\nabla}_x^{(n_1)} g_{qx} \right) \ldots \left(
\widetilde{\nabla}_x^{(n_i)} g_{xr} \right)
\end{equation}
where: $\widetilde{\nabla}_a$ is any derivative operator at $p$
\emph{other} than the one naturally associated with $g_{ab}$; `$x$' is
a dummy abstract index; `$\widetilde{\nabla}_x^{(n_i)}$' stands for
$n_i$ iterations of that derivative operator (obviously each with a
different abstract index); $\alpha$ takes its values in the set of all
permutations of all sets of positive integers $\{ n_1, \ldots, n_i \}$
that sum to $n$, so $i$ can range in value from 1 to $n$; the
exponents of the derivative operators in each summand themselves take
their values from $\alpha$, \emph{i}.\emph{e}., they are such that
$n_1 + \cdots + n_i = n$ (which makes it an $n^{\text{th}}$-order
concomitant); for each $\alpha$, $k_\alpha$ is a constant; and there
are just enough of the inverse metrics in each summand to contract all
the covariant indices but $a$ and $b$.
Now, a combinatorial calculation shows
\begin{prop}
\label{prop:nth-concom-factor}
If, for $n \geq 2$, $S_{ab}$ is an $n^{\text{th}}$-order homogeneous
concomitant of $g_{ab}$, then to rescale the metric by the constant
real number $\lambda$ multiplies $S_{ab}$ by $\lambda^{n - 2}$.
\end{prop}
In other words, the only such homogeneous $n^{\text{th}}$-order
concomitants must be of weight $\lambda - 2$.\footnote{The exponent
$(n - 2)$ in this result depends crucially on the fact that $S_{ab}$
has only two indices, both covariant. One can generalize the result
for tensor concomitants of the metric of any index structure. A
slight variation of the argument, moreover, shows that there does
not in general exist a homogeneous concomitant of a given
differential order from a tensor of a given index structure to one
of another structure---one may not be able to get the number and
type of the indices right by contraction and tensor multiplication
alone.} So if one knew that $S_{ab}$ were multiplied by, say,
$\lambda^4$ when the metric was rescaled by $\lambda$, one would know
that it had to be a sixth-order concomitant. In particular, $S_{ab}$
does not rescale when $g_{ab} \rightarrow \lambda g_{ab}$ only if it
is a second-order homogeneous concomitant of $g_{ab}$,
\emph{i}.\emph{e}., (by theorem~\ref{thm:1-2-jet-metric} and
proposition~\ref{prop:0-concoms-riemann}) a zeroth-order concomitant
of the Riemann tensor. There follows from
proposition~\ref{prop:con-composition}
\begin{lemma}
\label{lem:riem-0th-concom-0-homog}
A 2-covariant index concomitant of the Riemann tensor is homogeneous
of weight zero if and only if it is a zeroth-order concomitant.
\end{lemma}
Thus, such a tensor has the physical dimension of stress-energy if and
only if it is a zeroth-order concomitant of the Riemann tensor. It is
striking how powerful the physically motivated assumption that the
required object have the physical dimensions of stress-energy: it
guarantees that the required object will be a second-order concomitant
of the metric.
We now address the issue whether it is appropriate to demand of a
potential gravitational stress-energy tensor that it be covariantly
divergence-free. In general, I think it is not, even though that is
one of the defining characteristics of the stress-energy tensor of
ponderable matter in the ordinary formulation of general
relativity.\footnote{I thank David Malament for helping me get
straight on this point. The following argument is in part
paraphrastically based on a question he posed to me.} To see this,
let $T_{ab}$ represent the aggregate stress-energy of all ponderable
matter fields. Let $S_{ab}$ be the gravitational stress-energy
tensor, which we assume for the sake of argument to exist. Now, we
ask: can the ``gravitational field'' interact with ponderable matter
fields in such a way that stress-energy is exchanged? If it could,
then, presumably, there could be interaction states characterized (in
part) jointly by these conditions:
\begin{enumerate}
\item $\nabla^n (T_{na} + S_{na}) = 0$
\item\label{item:t-not-cons} $\nabla^n T_{na} \ne 0$
\item $\nabla^n S_{na} \ne 0$
\end{enumerate}
It is true that, as ordinarily conceived,
condition~\ref{item:t-not-cons} is incompatible with general
relativity as standardly understood and formulated. The existence of
a gravitational stress-energy tensor, however, would necessarily
entail that we modify our understanding and formulation of general
relativity. That is why this argument is only \emph{ex hypothesi},
and not meant to be one that would make sense in general relativity as
we actually know it. (One possible way to understand it,
\emph{e}.\emph{g}., would be that the ways we currently use to
calculate the stress-energy tensor of ordinary matter are mistaken,
precisely in so far as they do not take into account possible
interactions with gravitational phenomena.)
The most one can say, therefore, without wading into some murkily deep
and speculative waters about the way that a gravitational
stress-energy tensor (if there were such a thing) might enter into the
righthand side of the Einstein field-equation or that its existence
might modify the ways we calculate stress-energy for ordinary matter,
is that we expect such a thing would have vanishing covariant
divergence when the aggregate stress-energy tensor of ponderable
matter vanishes, \emph{i}.\emph{e}., that gravitational stress-energy
on its own, when not interacting with ponderable matter, would be
conserved in the sense of being covariantly divergence-free. This
weaker statement will suffice for our purposes, so we can safely avoid
those murky waters.
Finally, it seems reasonable to require one more condition: were there
a gravitational stress-energy tensor, it should not be zero in any
spacetime with non-trivial curvature, for one can always envision the
construction of a device to extract energy in the presence of
curvature by the use of tidal forces and geodesic deviation. (See,
\emph{e}.\emph{g}., \citeNP{bondi-mccrea-en-trans-ngt} and
\citeNP{bondi-phys-char-grav-wvs}.)
To sum up:
\begin{condition}
\label{cond:candidates-sab}
The only viable candidates for a gravitational stress-energy tensor
are two covariant-index, symmetric, second-order, zero-weight
homogeneous concomitants of the metric that are not zero when the
Riemann tensor is not zero and that have vanishing covariant
divergence when the stress-energy tensor of ponderable matter
vanishes.
\end{condition}
This discussion, by the way, obviates the criticism of the claim that
gravitational stress-energy ought to depend on the curvature,
\emph{viz}., that this would make gravitational stress-energy depend
on second-order partial derivatives of the field potential whereas all
other known forms of stress-energy depend only on terms quadratic in
the first partial derivatives of the field potential. It is exactly
second-order, homogeneous concomitants of the metric that possess
terms quadratic in the first partials. The rule is that the order of
a homogeneous concomitant is the sum of the exponents of the
derivative operators when the concomitant is represented in the form
of equation~\eqref{eq:Sab-form}.
\section{Gravitational Energy, Again, and the Einstein Field Equation}
\label{sec:nonexist}
\resetsec
\skipline[.5]
\begin{quote}
If we are to surround ourselves with a perceptual world at all, we
must recognize as substance that which has some element of
permanence. We may not be able to explain how the mind recognizes
as substantial the world-tensor [\emph{i}.\emph{e}., the Einstein
tensor], but we can see that it could not well recognize anything
simpler. There are no doubt minds which have not this
predisposition to regard as substantial the things which are
permanent; but we shut them up in lunatic asylums.
\begin{flushright}
Arthur Eddington \\
\emph{The Mathematical Theory of Relativity}, pp.~120--121
\end{flushright}
\end{quote}
\skipline
It follows from lemma~\ref{lem:riem-0th-concom-0-homog}, in
conjunction with condition~\ref{cond:candidates-sab}, that any
candidate gravitational stress-energy tensor must be a zeroth-order
concomitant of $\mathcal{B}_{\mbox{\small Riem}}$, the geometric
bundle of Riemann tensors over spacetime. (One can take this as a
precise statement of the fact that any gravitational stress-energy
tensor ought to ``depend on the curvature'', as I argued in
\S\ref{sec:princ_equiv}.) It follows from
proposition~\ref{prop:0-concoms-riemann} that the only possibilities
then are linear combinations of the Ricci tensor and the scalar
curvature multiplied by the metric. The only covariantly
divergence-free, linear combinations of those two quantities, however,
are constant multiples of the Einstein tensor $G_{ab}$. (To see this,
note that if there were another, say $k_1 R_{ab} + k_2 R g_{ab}$ for
constants $k_1$ and $k_2$, then
$k_1 R_{ab} + k_2 R g_{ab} - 2k_2 G_{ab}$ would also be divergence
free, but that expression is just a constant multiple of the Ricci
tensor.) The Einstein tensor, however, can still be zero even when
the Riemann tensor is not (when, \emph{e}.\emph{g}., there is only
Weyl curvature). This proves the main
result.
\begin{theorem}
\label{thm:only-einstein-tens}
The only two covariant-index, divergence-free concomitants of the
metric that are homogeneous of zero weight are constant multiples of
the Einstein tensor.
\end{theorem}
(Note the strength of the result: one need not even assume the tensor
to be symmetric; it automatically follows from the proof that all such
concomitants of the metric are symmetric.) Because the Einstein
tensor will be zero in a spacetime having a vanishing Ricci tensor but
a non-trivial Weyl tensor, there follows immediately
\begin{corollary}
\label{cor:non-exist}
There are no two covariant-index, divergence-free concomitants of
the metric that are homogeneous of weight zero that do not
identically vanish when the Riemann tensor is not zero.
\end{corollary}
The corollary does bear the required natural interpretation, for the
Einstein tensor is not an appropriate candidate for the representation
of gravitational stress-energy: it can be zero in spacetimes with
non-zero curvature; such spacetimes, however, can manifest phenomena,
\emph{e}.\emph{g}., pure gravitational radiation in the absence of
ponderable matter, that one naturally wants to say possess
gravitational energy in some (necessarily non-localized) form or
other.\footnote{As an historical aside, it is interesting to note that
early in the debate on gravitational energy in general relativity
\citeN{lorentz-zwaartekracht-3} and \citeN{levi-civita-grav-tensor}
proposed that the Einstein tensor be thought of as the gravitational
stress-energy tensor. Einstein criticized the proposal on the
grounds that this would result in attributing zero total energy to
any closed system.}
Theorem~\ref{thm:only-einstein-tens} is similar to the classic result
of \citeN{lovelock-4dim-space-efe}, but it is significantly stronger
in two important ways.\footnote{Lovelock proved the following, using
the definition of concomitant due to Schouten, and based on earlier
work by \citeN{rund-var-probs-comb-tens} and
\citeN{duplessis-tens-concoms-cons-laws}.
\begin{theorem}
\label{thm:nonexist.lovelock}
Let $(\mathcal{M},\; g_{ab})$ be a spacetime. In a coordinate
neighborhood of a point $p\in \mathcal{M}$, let
$\Theta_{\alpha \beta}$ be the components of a tensor concomitant
of
$\{g_{\lambda \mu} ; \; g_{\lambda \mu,\nu} ; \; g_{\lambda \mu ,
\nu \rho} \}$ such that
\[
\nabla^n \Theta_{nb} = 0.
\]
Then
\[
\Theta_{ab} = r G_{ab} + q g_{ab},
\]
where $G_{ab}$ is the Einstein tensor and $q$ and $r$ are
constants.
\end{theorem}
} It does not assume that the desired concomitant be
second-order; and it holds in all dimensions, not just four. Both of
those properties are grounded on the derivation of the differential
order of the desired concomitant of the metric based on analysis of
its required physical dimension, encoded in the requirement that the
concomitant of the metric be homogeneous of weight zero. The physical
interpretation of this is that the desired tensor has the physical
dimensions of stress-energy, as is the case for the Einstein tensor,
and as must be the case for any tensor that one would equate to a
material stress-energy tensor to formulate a field equation (so long
as the coupling constant is dimensionless, as is the case for Newton's
constant). This provides a physical interpretation to the conditions
of the theorem that Lovelock's theorem lacks.
The fact, moreover, that the proof relies essentially only on the
structure of the first and second jet bundles of the bundle of metrics
over a manifold, \emph{i}.\emph{e}., on the bundle of Riemann tensors
over a manifold, and how that structure places severe restrictions on
its possible concomitants, illuminates the geometrical content of the
theorem. Because Lovelock bases his theorem and its proof on
Schouten's definition of a concomitant, with the attendant complexity
and opacity of the conditions one then has to work with (as I
discussed on p.~\pageref{pg:schouten-difficult}, and in particular in
footnote~\ref{fn:hairy}), his proof consists of several pages of
Baroque and unilluminating coordinate-based, brute-force calculation,
which gives no geometrical insight into why the theorem holds.
The third difference is that the addition of constant multiples of the
metric is not allowed. I discuss the consequences of that below.
Before concluding the paper with a discussion of the bearing of the
theorem on the Einstein field equation, it behooves us to examine a
\emph{prima facie} puzzle my arguments have left us with. I argued in
\S\ref{sec:conds} that the form of the desired object, that it ought
to be a two-index tensor, followed from the idea that all forms of
stress-energy ought to be fungible, and so \emph{a fortiori} one must
be able to add in a physically significant way entities representing
the stress-energy of different kinds of systems. Now that I have
shown that there is no gravitational stress-energy tensor, one may be
tempted to conclude that gravitational energy, such as it is, is not
fungible with other forms of energy. That would be disastrous,
because, as I argued in footnote~\ref{fn:pitts}, there are
circumstances whose only reasonable interpretation is that
gravitational energy, such as it is, is in some way or other being
transformed into other, less \emph{recherch\'e} forms of energy. (For
more rigorous arguments to this effect, again see
\citeNP{bondi-mccrea-en-trans-ngt} and
\citeNP{bondi-phys-char-grav-wvs}.) I think the resolution is that,
in general relativity, there is no single framework for analyzing and
intrepreting all the phenomena one may want to characterize as
involving the coupling of physical systems based on energy transfer.
Energetic concepts that hang together in a unified framework in
classical physics come apart in general relativity. When one is
dealing with processes mediated by localizable energetic quantities,
the stress-energy tensor should do the job; otherwise, there are a
multitude of different kinds of quantities any one of which may be
physically relevant to the phenomena at issue. This should not be
surprising. We already know of cases in which concepts that formed a
unified framework in classical physics come apart in radical ways in
general relativity, such as the different ways one may characterize a
physical system as being in rotation or not
(Malament~\citeyearNP{malament-rot-nogo,malament-rel-orbit-rot}). In
any event, even in classical physics there are non-localized energetic
quantities, such as heat in thermodynamics and gravitational potential
energy in Newtonian gravitational theory, that one cannot always treat
in a unified framework with all localized forms of energy.
I conclude the paper by noting that theorem~\ref{thm:only-einstein-tens} has
another reasonable interpretation, that, in a natural sense the
Einstein field equation is the unique field equation for a theory such
as general relativity that unifies spatiotemporal structure with
gravitational phenomena by way of an appropriate relation between
spacetime curvature and the energetic content of ponderable matter.
(In particular, it follows from the result that a
cosmological-constant term in the field equation \emph{must} be
construed as forming part of the total stress-energy tensor of
spacetime.) \citeN{malament-newt-grav-geom-spc} makes precise the
sense in which geometrized Newtonian gravity is the limiting theory of
general relativity, as ``the speed of light goes to infinity''. In
geometrized Newtonian gravity, moreover, the Poisson equation is
formally almost equivalent to the Einstein field equation, and indeed
is identical with it in the vacuum case. \citeN[ch.~2,
\S7]{malament-fnds-gr-ngt} argues persuasively that, on this basis, it
is natural to adopt the Einstein field equation as the appropriate one
when one moves from the context of a Newtonian to a relativistic,
curved spacetime, in so far as any theory better in some sense than
Newtonian theory must, at an absolute minimum, have Newtonian theory
as its limit in certain weak-field regimes.
One can read theorem~\ref{thm:only-einstein-tens} as a way to
generalize this argument. We know from Newtonian gravitational theory
that the intensity of the gravitational field in a spatial region, in
so far as one can make sense of this idea, is directly proportional to
the density of mass in that region. In geometrized Newtonian gravity,
this idea is made precise in the geometrized form of the Poisson
equation, which equates a generalized mass-like quantity, which has
the form of a stress-energy tensor, to the Ricci curvature of the
ambient spacetime. In relativity, one knows that mass just is a form
of energy. In order for a relativistic theory of gravitation to have
Newtonian gravitational theory as its limiting form, therefore, one is
driven to look for the appropriately analogous equation, equating a
term representing the curvature of a Lorentz metric with a
stress-energy tensor. Once one imposes natural ancillary conditions
on the desired curvature term, such as that it must be a second-order,
homogeneous concomitant of the metric, then, by
theorem~\ref{thm:only-einstein-tens}, the Einstein field equation
falls out as the only possibility.\footnote{One may take this as a
more precise and rigorous form of the argument
\citeN[p.~149]{einstein-fndn-gtr} proposed for his introduction of
the stress-energy tensor in the first place, as I discussed in
footnote~\ref{fn:einstein-seten}.}
Theorem~\ref{thm:only-einstein-tens} implies that the addition of
constant multiples of the metric to the geometrical lefthand side of
the Einstein field equation is not allowed. I interpret that to mean
that any cosmological-constant term must be construed as part of the
total stress-energy tensor of spacetime, and so, in particular, the
cosmological constant itself must have the physical dimensions of
(mass)$^2$, so that its product with the metric will not change under
constant rescaling of the metric.
In higher dimensions, there are other tensors satisfying Lovelock's
original theorem, the so-called Lovelock tensors. (Those tensors are
not linear in the second-order partial-derivatives of the metric as
the Einstein tensor is.) Those tensors form the basis of so-called
Lovelock gravity theories in dimensions higher than four
\cite{lovelock-einst-tens-genls,padmanabhan-kothawala-lanczos-lovelock-mods},
being used to formulate field equations including concomitants of the
metric higher order than second. Because
theorem~\ref{thm:only-einstein-tens} holds in all dimensions, not just
in four, it follows that, in dimensions other than four, the Lovelock
tensors are not homogeneous of weight zero, and so do not have the
physical dimension of stress-energy. Thus, if one wants to construct
a field equation that equates a linear combination of such tensors to
the stress-energy tensor of ordinary matter, as Lovelock theories of
gravity do, then the coupling constants cannot be dimensionless like
Newton's gravitational constant; the physical dimension of each
coupling constant will be determined by the physical dimension of the
Lovelock tensor it multiplies.
The fact that the same theorem has as its natural interpretation the
uniqueness of the Einstein field equation and the non-existence of a
gravitational stress-energy tensor suggests that there may be a tight
relation between the non-localizability of gravitational stress-energy
and the form of the Einstein field equation. I have a strong
suspicion this is correct, but I have not been able to put my finger
on exactly what that relation may come to. A hint, perhaps, comes
from the pregnant remark of \citeN{choquet83} to the effect that the
principle of equivalence (on her interpretation of it) demands that
the gravitational field act as its own source, represented
mathematically by the non-linearity of the Einstein field equation.
Choquet-Bruhat's claim, if true, implies that there can be no linear
field equation for gravity satisfying the equivalence principle, which
would to my mind be a startlingly strong implication for the
equivalence principle to have. And yet my arguments here suggest that
she may, in some sense, be correct. That is a question, however, for
future work.
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