2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}_{{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}=-\epsilon\
{}_{xyz}\otimes\mathrm{d}xyz-q_{x}\ {}_{xyz}\otimes\mathrm{d}tzy+{}\\\ p_{x}\
{}_{tzy}\otimes\mathrm{d}xyz+{}_{xx}\ {}_{tzy}\otimes\mathrm{d}tzy+\dotsb$
(62j)

where the particular signs of the components, which are not determined by
dimensional analysis, will be motivated later. Our analysis of the intrinsic
dimensions therefore restricts the stress-energy-momentum tensor to be one of
the four alternatives above, with their kinds of orientation, straight or
twisted, still undetermined. Note that if we had conflated time and length
dimensions in eqs (60) and (61) by introducing $c$ factors, then dimensional
analysis wouldn’t have led to any restrictions: all sixteen alternatives would
have been dimensionally consistent. This shows the usefulness – at times at
least – to keep dimensions well distinct.

To further restrict the possibilities let’s consider three additional and
interrelated heuristic arguments.

First, the notions of volumic energy and momentum, areic energy flux, and
stress imply some kind of integration over three-dimensional spacelike or
timelike regions. Such integration needs a 3-form and thus excludes
alternatives (62a) and (62g).

Second, the total energy measured within a _topologically_ specified three-
dimensional spatial region of an observer’s orthogonal hyperplane is
considered to be independent of the extent of that region, whether the extent
is reckoned by a physical metric or by arbitrary coordinate intervals. The
_volumic_ energy therefore _does_ depend on the extent of the region and must
change accordingly. Similar arguments hold for the surface energy flux. Only
the second tensor alternative (62d) above is consistent with these
requirements.

This extent-scaling argument also suggests that the volumic-momentum part
$p_{x}\ \mathrm{d}x\otimes\mathrm{d}xyz$ should not be interpreted as an areic
mass flux $\textsf{M}\textsf{L}^{-2}\textsf{T}^{-1}$, that is, as something
that needs to be integrated over a surface: the basis element
$\mathrm{d}x\otimes\mathrm{d}xyz$ indicates that integration should happen
over a volume instead: $\dim(p_{r})=(\textsf{M}\textsf{L}\textsf{T}^{-1})\
\textsf{L}^{-3}$. A similar situation occurs for the stress part, which should
be interpreted as an areic momentum flux:
$\dim({}_{rs})=(\textsf{M}\textsf{L}\textsf{T}^{-1})\
\textsf{L}^{-2}\textsf{T}^{-1}$. It is worth noting that scaling
reinterpretations of this kind occur, even for more components, in all other
fifteen alternatives for the stress-energy-momentum tensor. Intrinsic
dimensional analysis alone thus suggests that there is a differences between
mass flux and momentum.

Third, the value of the energy density should not change under a change in the
orientation of the spacelike coordinates. The 3-covector slot in alternative
(62d) should therefore have a twisted orientation. This means that the 3-form
$\mathrm{d}xyz$ actually has an inner orientation in the positive $t$
direction, the 3-form $\mathrm{d}tzy$ in the positive $x$ direction, and so on
[schouten1951_r1989, To visualize this cf.][and][Fig. 22.10]burke1985_r1987.

A heuristic application of intrinsic dimensional analysis combined with
integration, scaling, and orientation arguments thus tells us that the stress-
energy-momentum tensor has variance type
$\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$, that is, it’s a
covector-valued 3-covector, or a four-times-covariant tensor completely
antisymmetric in three slots. The 3-covector part has a twisted orientation.
This tensor has the dimension of an _action_ :

$\dim\bigl{(}\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}}}\bigr{)}=\Epsilon\textsf{T}\equiv\textsf{M}\textsf{L}^{2}\textsf{T}^{-1}\
.$ (63)

This result agrees with the stress-energy-momentum tensor that appears for
example in Einstein’s original work [einstein1914b, § C.9, discussion before
eq. (42a)], other early works [pauli1921_t1958, e.g.][§ 13]cartan1923[§
7]brillouin1924, and more recent works [fokker1960_t1965, § VIII.3][ch. 14
Exercise 14.18, ch. 15, §
21.3]misneretal1970_r1973hehletal1986,gotayetal1992,gronwaldetal1997,castrillonlopezetal2008,castrillonlopezetal2009[see
also][for similar conclusions in general manifolds and in Newtonian
mechanics]segevetal1999,kansoetal2007; and also in Truesdell & Toupin
[truesdelletal1960, § F.IV.288], who try to find an expression universally
valid in Newtonian, Lorentzian, and general-relativistic mechanics. The
commonly encountered versions of this tensor with only two slots are discussed
below. Note that some of these works use a once covariant and once
contravariant “V(olume)-tensor” or “tensor density”, which has variance type
$\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}^{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}{}_{{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}\,$. Such an object,
however, is geometrically equivalent to the variance type
$\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}\,$; their independent
components have the same transformation law under changes of coordinates
[schouten1951_r1989, § II.8 p. 30] (this is why I chose a calligraphic letter
to denote this tensor).

The signs of the components of $\bm{\mathcal{T}}$ depend on the signature of
the metric g. If the latter has signature $(-,+,+,+)$, then the energy
components have negative sign, as in eq. (62d). If the metric has opposite
sign, that is, signature $(+,-,-,-)$, then $\bm{\mathcal{T}}$ has opposite
sign to eq. (62d) as well, and its momentum components are negative instead.

The literature cited above arrive at this kind of stress-energy-momentum
tensor through inductive generalization, often via electromagnetic theory, of
the stress tensor of Newtonian mechanics; or from principles of virtual work;
or from variational principles with an action Lagrangean [hilbert1915,
hilbert1917, cf. also][§ 3.3]hawkingetal1973_r1994, from which it easily
follows that this tensor should have the intrinsic dimension of an action; or
from combinations of these approaches. The operational meaning of this tensor
is therefore still unclear in my opinion.

In the presence of a metric tensor we can of course obtain stress-energy-
momentum tensors of different variance types by means of inner products with
the proper volume element and its inverse, and by raising and lowering
indices. But the question of the operational meaning and primitive variance
type of this tensor are important, for example, in field theories not based on
a metric, or for the formulation of constitutive equations [truesdelletal1960,
cf.]marsdenetal1983b_r1994,gotayetal1992. Extensive investigations were made
by Gotay et al. [gotayetal1998, gotayetal2004, gotayetal2006]; and by Segev
[segev2002][see also][]segev2000b, who interprets the stress-energy-momentum
tensor as a linear map from the four-dimensional flux of a conserved quantity,
such as charge or baryonic number, to that of energy. Since such fluxes are
represented by 3-forms, he arrives at the fourth alternative (62j) above: the
3-vector part of $\bm{\mathcal{T}}\indices{{}^{{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}_{{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$ can be contracted
with a 3-form, yielding another 3-form. This interesting interpretation
doesn’t seem to work out dimensionally, however. The intrinsic dimension of
the four-dimensional charge flux, for example, is charge itself,
$\textsf{Q}\equiv\textsf{I}\textsf{T}$, and similarly for the energy flux. The
tensor $\bm{\mathcal{T}}\indices{{}^{{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}_{{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$ should then have
intrinsic dimension $\Epsilon\textsf{Q}^{-1}$ according to the results of § 6.
The dimension of charge would then have to appear in Einstein’s constant (see
§ 11), because it cannot be eliminated by using the metric tensor or the
proper volume element to obtain alternative variance types.

Some authors [fock1955_t1964, e.g.][§ 4.1]mcvittie1956_r1965[§
10.1]adleretal1965_r1975 conceive the stress-energy-momentum tensor in terms
of mass rather than energy (Fock [fock1955_t1964, § II.31] calls it the “mass
tensor”), and therefore assign to its covector-valued 3-covector form
$\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$ the dimension of
mass-time, that is, an action divided by squared velocity:

$\dim\bigl{(}\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}}}\bigr{)}=\textsf{M}\textsf{T}\equiv\Epsilon\textsf{L}^{-2}\textsf{T}^{3}\
.$ (64)

With this intrinsic dimension, however, not all components of the stress-
energy-momentum tensor have intuitive meanings and dimensions when coordinates
with dimensions $(\textsf{T},\textsf{L},\textsf{L},\textsf{L})$ are used. The
two choices (63), (64) differ by a factor $c^{2}$.

The two dimensional choices for the metric, eqs (42) and (44), and for the
stress-energy-momentum tensor, eqs (63) and (64), appear in all four
combinations in the literature. For example,
$\dim(\textbf{{{g}}})=\textsf{T}^{2}$ and
$\dim(\bm{\mathcal{T}})=\Epsilon\textsf{T}$ is used by Synge [synge1960b, §§
IV.4–5]; $\dim(\textbf{{{g}}})=\textsf{L}^{2}$ and
$\dim(\bm{\mathcal{T}})=\textsf{M}\textsf{T}$ is used by Fock [fock1955_t1964,
§§ V.54–55] and Adler et al. [adleretal1965_r1975, § 10.1];
$\dim(\textbf{{{g}}})=\textsf{T}^{2}$ and
$\dim(\bm{\mathcal{T}})=\textsf{M}\textsf{T}$ is used by McVittie
[mcvittie1956_r1965, § 4.1] and possibly Kilmister [kilmister1973, chs II–III;
he seems to shift to natural units at some point]. Most other works use
$\dim(\textbf{{{g}}})=\textsf{L}^{2}$ and
$\dim(\bm{\mathcal{T}})=\Epsilon\textsf{T}$. These combinations lead to three
possible values for Einstein’s constant, discussed in the next section.

To obtain a twice covariant tensor to be used in the Einstein equations, we
first take the inner product of the inverse proper volume element with the
3-covector (that is, antisymmetric) part of
$\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$, obtaining a co-
contravariant tensor. Then we lower the new contravariant slot by means of the
metric tensor. The combined operation yields

$\displaystyle\textbf{{{T}}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}\coloneqq\bigl{(}\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}}}\cdot\bm{\gamma}^{-1}\bigr{)}\textbf{{{g}}}\ .$ (65)

According to this definition and eqs (64), (42), (44), (50), the co-co-variant
tensor T has three possible intrinsic dimensions, depending on the choices of
dimensions of g and $\bm{\mathcal{T}}$:

$\dim(\textbf{{{T}}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}})=\dim\bigl{(}\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}}}\bigr{)}\dim(\textbf{{{g}}})\dim(\bm{\gamma})^{-1}={}\\\
\begin{cases}\textsf{M}\textsf{L}\textsf{T}^{-2}\equiv\Epsilon\textsf{L}^{-1}&\quad\text{if}\
\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}\\\\[6.0pt]
\textsf{M}\textsf{L}^{-1}\equiv\Epsilon\textsf{L}^{-3}\textsf{T}^{2}&\quad\text{if}\
\begin{cases}\dim(\textbf{{{g}}})\coloneqq\textsf{T}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}\quad\text{or}\\\
\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\textsf{M}\textsf{T}\end{cases}\\\\[12.0pt]
\textsf{M}\textsf{L}^{-3}\textsf{T}^{2}\equiv\Epsilon\textsf{L}^{-5}\textsf{T}^{4}&\quad\text{if}\
\dim(\textbf{{{g}}})\coloneqq\textsf{T}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\textsf{M}\textsf{T}\end{cases}$ (66)

All three possibilities, which differ by factors $c^{2}$, appear in the
literature: see the works cited after eq. (63) concerning the combinations of
dimensions for metric and stress-energy-momentum.

It may be useful to write the coordinate expressions of the tensors
$\textbf{{{T}}}\indices{{}_{\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}$ and
$\textbf{{{T}}}\indices{{}^{\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}$ obtained from
$\bm{\mathcal{T}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$ of eq. (62d), in the
common case with $\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}$,
$\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}$, and coordinate system
with diagonal metric
$\textbf{{{g}}}=\pm(-c^{2}\mathrm{d}t\otimes\mathrm{d}t+\mathrm{d}x\otimes\mathrm{d}x+\dotsb)$,
commonly encountered in the literature:

$\displaystyle\mbox{}\textbf{{{T}}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}$
$\displaystyle=c^{2}\epsilon\ \mathrm{d}t\otimes\mathrm{d}t-q_{x}\
\mathrm{d}t\otimes\mathrm{d}x-c^{2}p_{x}\
\mathrm{d}x\otimes\mathrm{d}t+{}_{xx}\ \mathrm{d}x\otimes\mathrm{d}x+\dotsb$
(67)
$\displaystyle\mbox{}\textbf{{{T}}}\indices{{}^{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}$
$\displaystyle=\frac{1}{c^{2}}\epsilon\
{}_{t}\otimes{}_{t}+\frac{1}{c^{2}}q_{x}\ {}_{t}\otimes{}_{x}+p_{x}\
{}_{x}\otimes{}_{t}+{}_{xx}\ {}_{x}\otimes{}_{x}+\dotsb$ (68)

where the $c$ factors can be freely interpreted as part either of the
components or of the $t$ coordinate. Remember that the stress is here
considered as compressive rather than tensile, and that the Einstein equations
require in particular that $q_{x}=c^{2}p_{x}$ and so on (heat flux carries
momentum [eckart1940c, p. 923]).

##  11 The constant in the Einstein equations

We finally arrive at the Einstein equations,

$\textbf{{{G}}}=\kappa\textbf{{{T}}}$ (69)

sometimes seen with a minus sign, depending on the signature of the metric and
of alternative definitions of the curvature tensors [misneretal1970_r1973, see
the _Table of sign conventions_ on the final pages of]. is Einstein’s
constant.

The equations above are considered in their co-co-variant form. This form is
convenient because the left side is then dimensionless (its intrinsic
dimension is 1), independently of the dimension of the metric tensor, as
explained in § 9.4, eq. (59). We therefore find that the equality

$\dim(\kappa)=\dim(\textbf{{{T}}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}})^{-1}$ (70)

must always hold, for all choices of dimensions for the metric and stress-
energy-momentum tensors. Combining this equation with the results for T, eq.
(66), we find three possible conventions:

$\dim(\kappa)={}\\\
\begin{cases}\textsf{M}^{-1}\textsf{L}^{-1}\textsf{T}^{2}\equiv{}^{-1}\textsf{L}&\quad\text{if}\
\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}\\\\[6.0pt]
\textsf{M}^{-1}\textsf{L}\equiv{}^{-1}\textsf{L}^{3}\textsf{T}^{-2}&\quad\text{if}\
\begin{cases}\dim(\textbf{{{g}}})\coloneqq\textsf{T}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}\quad\text{or}\\\
\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\textsf{M}\textsf{T}\end{cases}\\\\[12.0pt]
\textsf{M}^{-1}\textsf{L}^{3}\textsf{T}^{-2}\equiv{}^{-1}\textsf{L}^{5}\textsf{T}^{-4}&\quad\text{if}\
\dim(\textbf{{{g}}})\coloneqq\textsf{T}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\textsf{M}\textsf{T}\end{cases}$ (71)

Einstein’s constant can therefore be obtained from Newton’s gravitational
constant $\dim(G)=\textsf{M}^{-1}\textsf{L}^{3}\textsf{T}^{-2}$ (this is not
the Einstein tensor G!) and the speed of light
$\dim(c)=\textsf{L}\textsf{T}^{-1}$ only in the following three ways, with the
$8\piup$ factor coming from the Newtonian limit:

$\kappa=\begin{cases}8\piup G/c^{4}&\quad\text{if}\
\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}\\\\[6.0pt] 8\piup
G/c^{2}&\quad\text{if}\
\begin{cases}\dim(\textbf{{{g}}})\coloneqq\textsf{T}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\Epsilon\textsf{T}\quad\text{or}\\\
\dim(\textbf{{{g}}})\coloneqq\textsf{L}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\textsf{M}\textsf{T}\end{cases}\\\\[12.0pt]
8\piup G&\quad\text{if}\ \dim(\textbf{{{g}}})\coloneqq\textsf{T}^{2}\ ,\
\dim(\bm{\mathcal{T}})\coloneqq\textsf{M}\textsf{T}\end{cases}$ (72)

As we saw in the discussion of the literature cited in § 10, the first
convention is the most common. The second convention appears for example in
Fock [fock1955_t1964, § 55 eqs (55.15) and (52.06)] and Adler et al.
[adleretal1965_r1975, § 10.5 eq. (10.98)]. The third, interesting convention
would appear in McVittie if he didn’t cheat a factor $1/c^{2}$ into by writing
the Einstein equations [mcvittie1956_r1965, § 4.2 eq. (4.107)] as “
$\textbf{{{G}}}=\kappa c^{2}\textbf{{{T}}}$ ”.

##  12 Summary and conclusions

We have seen that dimensional analysis, with its familiar rules, can be
seamlessly performed in Lorentzian and general relativity and in differential
geometry if we adopt the coordinate-free approach typical of modern texts. In
this approach each tensor has an _intrinsic_ dimension, a notion introduced by
Schouten and Dorgelo. This dimension doesn’t depend on the dimensions of the
coordinates, and is determined by the physical and operational meaning of the
tensor. It is therefore generally more profitable to focus on the intrinsic
dimension of a tensor rather than on the dimensions of its components. The
dimension of each specific component is easily found by formula (12): it’s the
product of the intrinsic dimension by the dimension of the $i$th coordinate
function for each contravariant index $i$, by the inverse of the dimension of
the $j$th coordinate function for each covariant index $j$. Intrinsic
dimensional analysis seems to rest on two main conventions: the tensor product
and the action of covectors on vectors behave analogously to usual
multiplication for the purposes of dimensional analysis. Alternative,
equivalent sets of conventions could perhaps be considered.

We have also seen that intrinsic dimensional analysis can help us determine or
at least constrain the variance type of candidate tensors, as exemplified with
the stress-energy-momentum tensor in § 10. We found or re-derived some
essential results for general relativity, in particular that the Riemann
$\textbf{{{Rie}}}\indices{{}^{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern
2.0mu}\mathord{\mathchoice{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}{\mkern
2.0mu\rule[-0.3014pt]{0.75pt}{3.31529pt}\mkern 2.0mu}}}$, Ricci
$\textbf{{{Ric}}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}$, and twice covariant
Einstein $\textbf{{{G}}}\indices{{}_{\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}\mathord{\mathchoice{\hbox
to5.24997pt{\hfil\raise 0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to5.24997pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.37498pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.79163pt{\hfil\raise
0.0pt\hbox{\scalebox{0.7}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}$ curvature tensors are
dimensionless. These results could be of importance for current research
involving scales and conformal factors [roehretal2005, cadonietal2019, e.g.].
We also discussed the operational reasons behind two common choices of
dimensions for the metric and stress-energy-momentum tensors.

Since the dimensions of the components are usually different from the
intrinsic dimension and depend on the coordinates, I recommend to avoid
statements such as “the tensor $A\indices{{}_{i}^{jk}}$ has dimension X”,
which leave it unclear whether “$A\indices{{}_{i}^{jk}}$” is meant to
represent the tensor in general (as in Penrose & Rindler’s notation), or to
represent its set of components, or to represent only a specific component.

For the dimensional analysis of tensorial objects in electrodynamics, which
wasn’t discussed in this note, see for example Truesdell & Toupin
[truesdelletal1960, ch. F] and Hehl & Obukhov [hehletal2003, hehletal2004b].

Dimensional analysis remains a controversial, obscure, but fascinating subject
still today, 60 years from Truesdell & Toupin’s remark quoted in the
Introduction. For an overview of some recent and creative approaches to it,
going beyond Bridgman’s text [bridgman1922_r1963] (whose point of view is in
many respects at variance with modern developments: see the following
references), I recommend for example the works by Mari et al. [marietal2012,
frigerioetal2010], Domotor and Batitsky [domotor2017, domotoretal2016,
domotor2012], Kitano [kitano2013], the extensive analysis by Dybkaer
[dybkaer2004_r2010], the historical review by de Boer [deboer1995], and
references therein.

###### Acknowledgements.

…to (chronologically) Mariano Cadoni, Ingemar Bengtsson, Iván Davidovich,
Claudia Battistin for valuable comments on previous drafts. To the staff of
the NTNU library for their always prompt support. To Mari, Miri, Emma for
continuous encouragement and affection, and to Buster Keaton and Saitama for
filling life with awe and inspiration. To the developers and maintainers of
LaTeX, Emacs, AUCTeX, Open Science Framework, R, Python, Inkscape, Sci-Hub for
making a free and impartial scientific exchange possible.  Note: this paper
was rejected by arXiv without any kind of scientific justification. This work
was financially supported partly by the Kavli Foundation and the Centre of
Excellence scheme of the Research Council of Norway (Roudi group), partly by
the Trond Mohn Research Foundation, grant number BFS2018TMT07.

prenote(“de $X$” is listed under D, “van $X$” under V, and so on, regardless
of national conventions.)