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\def\sesquispace{\baselineskip=18pt}    % (used for main text)
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\sesquispace                  % spacing is set here and again after abstract

%----------------------------TITLE-------------------------------------
\centerline{\bf GENERAL RELATIVITY II--PHY 786; FALL, 1994}
\smallskip
\centerline{\bf Instructor: Rafael D. Sorkin}
\smallskip
\centerline{\bf Orientations, ``Extensors'' and the General Form of Stokes'
            Theorem}

%------------------------------MAIN TEXT------------------------------------
\sesquispace                                % 3/2 spacing for main text
\bigskip\medskip

\def\Chi{{\cal \chi }}
\def\Or{{\cal O}}
\def\epsilon{\varepsilon}
\def\A{{\cal A}}  % really %gothic A
\def\curl{{\rm curl}}
\def\div{{\rm div}}

Just writing down Stokes' Theorem as a formula is easy enough; the whole
difficulty is in defining a convenient set of conventions for the
orientations, signs and normalizations involved.  The conventions made
below are chosen to agree with all the standard rules of vector analysis
when one specializes to that case (e.g. an element of the surface bounding
a compact region in flat $\Reals^3$ will be representable by the vector
$d{\vec S}  = {\vec n}dA$ where $\vec n$ is the outward unit
normal).

In the following, $\Chi$ will be a compact manifold with boundary
$\partial\Chi$, forming an embedded submanifold (a region, a surface, a
curve...) of the $n$-dimensional manifold $M$.  Lower case latin letters
$a, b, c, \cdots $ will be abstract indices, while $\alpha, \beta, \gamma,
\cdots $ will be numerical (``concrete'') indices.

The symbol `$\Chi$' should really be the ``curly Chi'' symbol we used in
class, but which I don't know how to get in TeX.  Similarly $\Or$ should be
the ideogram for orientation, and $\A$ is meant to be a gothic $A$.



\bigskip
\noindent{\bf Orientations }
\nobreak

A space $\Or$ of orientations has only two elements, but it does have some
structure.  We can multiply an orientation by $\pm 1$ and also take the
product of two orientations.  Specifically, if $\Omega'$ and $\Omega''$ are
the two elements of some orientation-space $\Or$ then we have
$\Omega' \Omega' = 1$, $\Omega' \Omega'' = -1$, $- \Omega' = \Omega''$,
etc.  Similarly, one can  form the ``tensor product'' of two
orientation-spaces, giving a third orientation-space.

For vector spaces $V \supseteq W$ we define
$$
\Or (V) \ideq \Or(V,int)
$$
as the space of {\it internal} orientations of $V$, and
$$
\Or(V/W) \ideq \Or(W,ext)
$$
as the space of {\it external} orientations of $W$ in $V$.
Notice that $\Or(V) = \Or(V/\{0\})$.


We can regard $\Or(V)$ as the
product of $\Or(V/W)$ with $\Or(W)$, and {\it we always take the factors in
that order} :
$$
            \Or(V) = \Or(V/W) \tensor \Or(W).       \eqno (1a)
$$
Similarly, for $U\supseteq V \supseteq W$ we write
$$
      \Or(U/W) = \Or(U/V) \tensor \Or(V/W),              \eqno (1b)
$$
$$
    \Or(U) = \Or(U/V) \tensor \Or(V/W) \tensor \Or(W),    \eqno (1c)
$$
etcetera.


An internal orientation for $W$ can be specified by an ordered basis
$(w_1, w_2, \cdots w_K)$ for $W$ (where $K= \dim W$), or equivalently by
an ordered basis $(\omega_1\cdots\omega_K)$ for $W^*$.  The orientations
specified thereby are {\it equal} iff det$(\omega^i \cdot w_j)$ is
positive, i.e. iff the inner product of
$\omega^1_a \wedge \cdots \wedge \omega^K_b$ with
$w_1^a \wedge \cdots \wedge w_K^b$ is positive.
An external orientation of $W$ in $V$ can be specified by an ordered basis
for the quotient-space $V/W$, or equivalently by an ordered set of $K$
linearly independent
covectors $\omega^1 \cdots \omega^K$ annihilating $W$, where
$K = {\rm codim} W := \dim V - \dim W$.
(Recall here the sign nuisance that $n^a$ and $n_a = g_{aa} n^a$ specify
{\it opposite} orientations for a spacelike hypersurface in a Lorentzian
manifold, because $n_a n^a < 0$ in this case.)

So far everything was for vector spaces.  For manifolds we just apply the
above definitions to tangent spaces to define local orientations, and we
take a global orientation to be any continuous choice of local orientations,
assuming such exists.  Notationally in our case, if $\Chi$ is (internally)
orientable then $\Or(\Chi)$ will denote the space of its internal
orientations; if it is externally orientable in $M$ then $\Or(M/\Chi)$ will
denote the space of its external orientations.  Note that an external
orientation for $\Chi$ in $M$ can be specified by giving $K$=codim($\Chi$)
independent real functions $y^1, \cdots, y^K$, because their gradients
$\partial_a y^1, \cdots, \partial_a y^K$ furnish $K$ linearly independent
covectors annihilating $T_x \Chi$.  (For example, for $K$=1 and $t$ a
time-coordinate in Minkowski space, the function $y^1 = \pm t$ gives a
future/past orientation to the t=0 hypersurface.)

Finally recall that an {\it axial} tensor of type ${\cal T}$ at $x\in M$
lives in ${\cal T}\tensor \Or(T_xM)$, i.e. it is the product of an ordinary
tensor (possibly density-weighted) with an orientation of $T_xM$.
Similarly, a $\Chi$-externally axial tensor at $x \in \Chi$ lives in
${\cal T} \tensor \Or(T_xM/T_x\Chi)$, etc.

\bigskip
\noindent{\bf Multi-indices }
\nobreak
$$
   A = a b \cdots c
$$
is a totally skew multi-index, with $|A|$ being the number of individual
indices $a, b \cdots, c$.

\noindent
Contraction of multi-indices:
$$
   v_A w^A := v_{ab\cdots c} w^{ab\cdots c} / |A|!               \eqno (2)
$$

\bigskip
\noindent{\bf The $\epsilon$-tensor }
\nobreak

$\epsilon^{abcd}$ (in 4D as an example) is an {\it axial} tensor of
density-weight +1.  Its components in {\it any} basis are
 $\epsilon^{0123}$ = 1, $\epsilon^{1023}$ = $-1$, etc.  The covariant,
weighted tensor $\epsilon_{abcd}$ is axial of weight $-1$ with identical
components; it is normalized by
$\epsilon_{abcd}\epsilon^{abcd}/4! = \epsilon_A\epsilon^A = 1$.
(Warning: when a Lorentzian metric is involved there is the sign nuisance
that
$\sqrt{-g}\epsilon_{abcd}
      = - g_{aa} g_{bb} g_{cc} g_{dd} \epsilon^{abcd} /\sqrt{-g}$.
On the other hand, you can always rely on
$\epsilon^{a \cdots b} g_{aa}\cdots g_{bb} = \epsilon_{a\cdots b} \det g$.)
Both $\epsilon^{a \cdots b}$ and $\epsilon_{a \cdots b}$ {\it exist on any
manifold, orientable or not}.

Note that $\epsilon^A_B = \epsilon^{AC} \epsilon_{BC}$ acts as the
identity (a generalized Kronecker $\delta$) in contractions with
multi-indices $A$ or $B$.

\bigskip
\noindent{\bf ``Extensors'' (or tensors of extension)}
\nobreak

The magnitude of an infinitesimal portion $d\Chi$ of a submanifold $\Chi$
of {\it codimension} $K$ can be represnted by a $K$-index skew-tensor
$d\sigma_A$ of density-weight $-1$.  We may call $d\sigma_A$ a ``tensor of
extension'' or ``extensor'' for short.  For maximum generality, $d\sigma_A$
must be regarded as {\it externally axial}, where, as just described, an
externally axial tensor at
$x\in M$ is the product of an ordinary tensor at $x$ with an external
orientation of $T_x\Chi$ in $T_xM$; in other words an externally axial
tensor of type ${\cal T}$ lives in ${\cal T}\tensor
\Or(T_xM/T_x\Chi)$.  However, if $\Chi$ carries an {\it external
orientation} $\Omega$ then $d\sigma_A$ may be construed as polar (non-axial)
via multiplication with $\Omega$.  (Thus an externally oriented $d\Chi$
may be represented by a polar extensor $d\sigma_A$.)

Consider the general case where $\Chi$ is unoriented.
If $y^1, \cdots y^n$ are $n$ independent locally defined real functions of
which the first $K$ determine $\Chi$ in the sense that (locally)
$$
   \Chi = \{ x \in M | y^1(x) = y^2(x) = \cdots = y^K(x) = 0\}
$$
then we have the formula
%
$$
d\sigma_{ab\cdots c}
%
   = {\partial_a y^1 \wedge \partial_b y^2 \wedge \cdots \wedge \partial_c y^K
               dy^{K+1}\cdots dy^n  \over
%
   |\epsilon^{p \cdots q \cdots r} \partial_p y^1 \cdots \partial_q y^K
        \cdots \partial_r y^n|}
    \tensor \Omega
$$
%
where $n$ = dim($M$) and $\Omega$ is the element of $\Or(T_xM/T_x\Chi)$
specified by the functions $y^1, \cdots y^K$.
  % that is by the $K$ covectors, $\partial_a y^1, \cdots, \partial_b y^K$.
For example, if $\Chi = \{t=0\}$ is a spacelike hyperplane in Minkowski
space and
$\Omega$ is an external orientation of $\Chi$
then in $t$-$x$-$y$-$z$-coordinates, the only nonzero component of
$\Omega d\sigma_a$ is
$ (\Omega d\sigma)_0 = \pm dx dy dz$, where
the $\pm$ is positive iff $\Omega$ agrees with the orientation specified by
the function $t$.  (It is instructive to think through in detail why this
is true and why it is meaningful.  For example, $d\sigma_a$ as such does not
have well-defined
numerical components, because a coordinate system cannot in general specify
an external orientation for any submanifold.)
Equivalently, $d\sigma_0 = + dx dy dz$ if $\Chi$ is given a future
orientation, and $d\sigma_0 = - dx dy dz$ if $\Chi$ is given a past
orientation.  In general, $d\sigma_{\alpha \beta \cdots \gamma}$ is up to
sign the magnitude of the coordinate projection of $d\Chi$ onto the $\alpha
= \beta = \gamma = 0$ coordinate-plane.

When a metric is present, we may use instead of $d\sigma_a$ the ordinary
(unweighted) tensor
$$
    d S_A = \sqrt{-g} \, d\sigma_A.                       \eqno (3)
$$
For codimension 1, $dS_a$ is up to sign $d^{n-1}V n_a$ where $d^{n-1}V
 = |d\Chi|$ and $n_a$ = unit normal to $\Chi$; or in the null case,
$dS_a = d^{n-2}V dx^a$ where $d^{n-2}V$ = sectional ``area'' and $dx^a$ is the
extent of $d\Chi$ along its null direction.

The magnitude of an infinitesimal element of extension $d\Chi$ of dimension
$K$ also admits a representation in terms of a dual ``extensor''
$d\Sigma^A$, which is a $K$-index skew tensor (of density-weight 0).
(Notice that $K$  now denotes the dimension of $\Chi$, not its codimension.)
Most generally, $d\Sigma^A$ must
be regarded as {\it internally axial}, where an internally axial tensor at
$x\in M$ is the product of an ordinary tensor at $x$ with an internal
orientation of $T_x\Chi$; in other words an internally axial
tensor of type ${\cal T}$ lives in ${\cal T}\tensor \Or(T_x\Chi)$.
However, if $\Chi$ carries an internal
orientation $\Omega$ then $d\Sigma^A$ may be construed as polar (non-axial)
via multiplication with $\Omega$.  (Thus an internally oriented $d\Chi$
may be represented by a polar extensor $d\Sigma^A$.)

In the general case of unoriented $d\Chi$, if
$x = f(y^1, \cdots , y^K)$ is a parametric representation of $\Chi$, then
$$
\eqalign{
   d\Sigma^{ab \cdots c}  &=
  {\partial x^a \over \partial y^1} \wedge
  {\partial x^b \over \partial y^2} \wedge \cdots \wedge
  {\partial x^c \over \partial y^K} dy^1 dy^2 \cdots dy^K
                                                         \tensor \Omega \cr
    &=
  {\partial x^a \over \partial y^j} \cdots
  {\partial x^c \over \partial y^l}
            \epsilon^{j \cdots l}   dy^1 dy^2 \cdots dy^K
    \tensor \Omega, }
$$
where $\Omega$ is the element of $\Or(T_x\Chi)$
specified by the functions $y^1 \cdots y^K$, that is by the $K$ vectors,
$ {\partial x^a \over \partial y^1} \cdots  {\partial x^c \over \partial y^K}$.
(For example,
if $\Chi$ is the $t$-axis in $t$-$x$-$y$-$z$-coordinates for Minkowski
space, and $\Omega$ is an internal orientation for $\Chi$, then the only
nonzero component of $\Omega d\Sigma^a$ is $(\Omega d\Sigma)^0 = \pm dt$,
where the
sign $\pm$ is positive iff $\Omega$ agrees with the orientation specified by
$\partial/\partial t$.)
In general, $d\Sigma^{\alpha \beta \cdots \gamma}$ is up to sign the
magnitude of the coordinate projection of $d\Chi$ onto the
$\alpha = \beta = \gamma = 0$ coordinate-plane.


The ``extensors'' $d\Sigma_A$ and $d\sigma^A$ are related by duality.  In
the general case of unoriented $d\Chi$ we have
$$
   d \sigma_A = \epsilon_{AB} \; d\Sigma^B,                    \eqno (4)
$$
where the internal and external orientations entering respectively into
$d\Sigma$ and $d\sigma$ are related with the overall orientation entering
into $\epsilon_{AB}$ by eq. (1a), which in this case becomes
$$
   \Or(T_xM) = \Or(T_xM/T_x\Chi) \tensor \Or(T_x\Chi).           \eqno (5)
$$

\bigskip
\noindent{\bf Divergence and Curl}
\nobreak

If $\A^{ab \cdots c}$ is a (polar or axial) skew tensor of density-weight
$+1$ then
$$
      (\div \A)^{a \cdots b} \ideq \partial_c \A^{a \cdots bc}    \eqno (6)
$$
is its {\it divergence} (with components
 $ \partial_\gamma \A^{\alpha \cdots \beta \gamma} / \partial x^{\gamma}$,
as the notation suggests).  In the multi-index notation, the divergence
appears as $\partial_b \A^{Ab}$.

If $\omega_{b \cdots c}$ is a (polar or axial) form (skew covariant tensor)
then
$$
   (\curl \, \omega)_{ab \cdots c} \ideq \partial_a \wedge \omega_{b \cdots c}
 \eqno (7)
$$
is its {\it curl} or ``exterior derivative''
(with components $\partial_\alpha \omega_{\beta \cdots \gamma}
                - \partial_\beta \omega_{\alpha \cdots \gamma} \cdots
               \pm \partial_\gamma \omega_{\beta \cdots} $).


Div and curl are dual to each other:
$$
   *_{_l} \  \div = \curl \  *_{_l}                                \eqno (8)
$$
where the `$l$' in `$*_{_l}$' stands for `left', the precise action
of $*_{_l}$  being
$$
      \A^B  \longrightarrow \epsilon_{AB} \A^B,                     \eqno(9)
$$
which has $\epsilon_{AB}$ sitting on the left of $\A$, as in (4).

\bigskip
\noindent{\bf Stokes' Theorem}
\nobreak

\noindent
First form:
$$
   \int\limits_\Chi \partial_c \A^{Ac} d\sigma_A =  \Omega
   \oint\limits_{\partial\Chi}  \A^{B} d\sigma_B
$$
In its most general guise, the theorem does not require either $\Chi$ or
$\partial\Chi$ to be oriented, which means, as we have seen, that the
``extensors'' $d\sigma_A$ and $d\sigma_B$ (with $|B| = |A| + 1$) must be
taken to be externally axial with respect to $\Chi$ and $\partial\Chi$
respectively.  For consistency, $\A^B$ must then be a
$\Chi$-externally-axial tensor as well (of density-weight 1).  That is,
$\A^B(x)$ must be the product of a tensor-density of weight +1 with an
element of $\Or(T_xM/T_x\Chi)$, in order to make the $LHS$ integrand,
$\div \A \; d\sigma$, be an ordinary scalar as required.

On the $RHS$, $\Omega \in \Or(\Chi/\partial\Chi)$ is the customary
{\it outward} orientation of $\partial\Chi$ in $\Chi$ (i.e. the one
specified for example by a function $f$ which vanishes on $\partial\Chi$
and is negative within $\Chi$).  Without the presence of $\Omega$ on the
$RHS$, the integral there would not make sense, since the integrand
involved, namely $\A^B(x) d\sigma_B(x)$,
is the product of a $\Chi$-externally axial quantity of density weight $+1$
with a $\partial\Chi$-externally axial quantity of density weight $-1$.
Such a product is not a true scalar.  Rather it lives in $\Reals \tensor
\Or( T_x\Chi / T_x\partial\Chi)$, as follows from (1) and the rules for
multiplication of orientations.  Only when multiplied
by an element $\Omega$ of $\Or( T_x\Chi / T_x\partial\Chi)$ (in our case by
the particular $\Omega$ representing outward orientation) does the product
become a valid integrand.

In the way that Stokes' Theorem is ordinarily written one does not see the
orientation $\Omega$ explicitly.  Instead there are assumptions of
orientablity and tacit rules for in effect converting one kind of axiality
to another.  Indeed, one most commonly assumes both $M$ and $\Chi$ to be
oriented, and then all reference to axial quantities can be avoided by
working  in terms of
suitable products of the given orientations with the
quantities we have been using here.  This often leads to the erroneous
impression that the theorem only works for orientable manifolds.

When stated as above, the theorem holds for {\it any} embedded
submanifold $\Chi$, even, for example, for a M\"obius strip in $\Reals^3$.
However the quantities $\A^B$ which occur in practice are much more likely
to be polar or overall-axial than $\Chi$-externally axial.  Thus
one usually must convert $\A^B$ to the required type of ``axiality'' by
using some reference orientation of $\Chi$, and this requires $\Chi$ to be
internally or externally orientable as the case may be.  Hence it is rare
that one can actually make use of Stokes' Theorem with a M\"obius strip.
On the other hand, the assumption that everything in sight is orientable,
is usually much more than one needs.

When a metric is present, it is convenient to rewrite the theorem in terms
of ordinary (unweighted) tensors:
$$
   \int_\Chi \nabla_c A^{Ac} dS_A =  \Omega
   \oint_{\partial\Chi}  A^{B} dS_B
$$

The dual form of Stokes' Theorem is
$$
   \int\limits_\Chi (\curl \, \omega)_B \; d\Sigma^B =  \Omega
   \oint\limits_{\partial\Chi}  \omega_A  d\Sigma^A,
$$
with $(\curl \, \omega)_{aA} = \partial_a \wedge \omega_A$ (or
$\nabla_a \wedge \omega_A$ when a metric is present).   Here the form
$\omega$ is $\Chi$-internally axial, while the ``extensors'' $d\Sigma^B$
and $d\Sigma^A$ are repectively $\Chi$-internally axial and
$\partial\Chi$-internally axial, as usual.  And once again, $\Omega$ is the
outward orientation of $\partial\Chi$ in $\Chi$.


The full duality relating the two forms of Stokes' Theorem is
$$
\eqalign
  {  d\sigma_A & = \epsilon_{AB} \, d\sigma^B  =: \; *_{_l} \; d\Sigma \cr
      \A^B & = \epsilon^{BC} \, \omega_C  =: \;  *_{_l} \; \omega   \cr
      *_{_l} & \;  \curl  =  \div  \; *_{_l}    \cr }
$$
On making the  substitutions given in the first two lines above, the
first-given form of Stokes's Theorem turns into its dual.  The brief
computation needed to establish this is free of signs and combinatorial
factors if one uses the identity pointed out at the end of the section on
the $\epsilon$-tensor.



\end


