% \filename{gemapp.tex}
% \version{19-jul-2007< 7-mar-2002<-7-mar-2001<-17-jan-2000<-15-dec-1999<-21-nov-1997}
% Appendix for:
% "Understanding Spacetime Splittings and Their Relationships"
% by Robert T Jantzen, Paolo Carini, and Donato Bini
\Chapter{Formulas from differential geometry}
This summary of formulas from differential geometry continues the
remarks of the introduction. It is assumed that the reader has
some familiarity with many of these
concepts, at least in coordinate notation, or at the level of the
text {\it Gravitation\/} by Misner, Thorne and Wheeler \Cite{1973}.
The purpose of this
appendix is to have a reference guide
for the particular choice of conventions used here
and the specific formulas for the frame components of many geometric objects
whose coordinate frame representation is more widely known.
More details may be found by consulting Spivak \Cite{1965}, Hicks \Cite{1971},
Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick \Cite{1982},
Schouten \Cite{1954}
or other well known texts on differential geometry.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Manifold}
Let $M$ be an $n$-dimensional manifold, with local coordinates
$\{ x^\alpha \}_{\alpha=1,\ldots,n}$ when needed. These local coordinates
define a local coordinate frame $ \{ \partial/\partial x^\alpha \}$
with dual frame $ \{ d x^\alpha \} $.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Frame and dual frame}
It is often more convenient to work with a (local) noncoordinate frame
$\{ e_\alpha \} $ with dual frame $\{ \omega^\alpha \} $
(sometimes called co-frame) satisfying
the duality condition
\beq
\omega^\alpha ( e_\beta ) = \delta^\alpha{}_\beta \ .
\eeq
If one expresses these fields in local coordinates
\beq
e_\alpha = e^\beta{}_\alpha \partial/\partial x^\beta \ ,\qquad
\omega^\alpha = \omega^\alpha{}_\beta dx^\beta \ ,
\eeq
then the matrices of components $(e^\alpha{}_\beta)$ and
$(\omega^\alpha{}_\beta)$ are inverse matrices.
The differential properties of the frame are characterized by the structure
functions of the frame, which define the Lie brackets of the frame vector
fields and the exterior derivatives of the dual frame 1-forms
\beq
[e_\alpha, e_\beta ] = C^\gamma{}_{\alpha \beta} e_\gamma \ ,\qquad
d \omega^\alpha =
- \half C^\alpha{}_{\beta\gamma} \omega^\beta \otimes \omega^\gamma \ .
\eeq
A $p\choose q$-tensor field can be expressed in terms of the frame as
\beq
S =S ^{\alpha \ldots}_{\ \ \beta \ldots}
e_\alpha \otimes \cdots \otimes \omega^\beta \otimes \cdots\ ,\qquad
S ^{\alpha \ldots}_{\ \ \beta \ldots} = S(\omega^\alpha,\ldots,
e_\beta,\ldots) \ .
\eeq
It is convenient to adopt the notation
\beq
\omega^{\alpha_1\ldots \alpha_p} =
\omega^{\alpha_1}\wedge \cdots \wedge \omega^{\alpha_p}
\eeq
to more compactly represent a $p$-form, or antisymmetric $0\choose p$-tensor
\beq
S = \frac1{p!} S_{\alpha_1 \ldots \alpha_p}
\omega^{\alpha_1\ldots \alpha_p}
=S_{|\alpha_1 \ldots \alpha_p|}
\omega^{\alpha_1\ldots \alpha_p}\ ,
\eeq
where the vertical bar notation indicates a sum only over strictly increasing
sequences of indices. A similar notation $e_{\alpha_1\ldots \alpha_p}$
will be used to express $p$-vector fields, namely antisymmetric
$p\choose0$-tensor fields.
A tensor field whose components are antisymmetric in a subset of $p$
covariant indices may be interpreted as a tensor-valued $p$-form
\beq\eqalign{
S &= S^{\alpha \ldots}_{\ \ \beta \ldots \gamma_1 \ldots \gamma_p}
e_\alpha \otimes \cdots \otimes \omega^\beta \otimes \cdots
\otimes \omega^{\gamma_1} \otimes \cdots \otimes \omega^{\gamma_p} \cr
&= \frac1{p!}
S^{\alpha \ldots}_{\ \ \beta \ldots \gamma_1 \ldots \gamma_p}
e_\alpha \otimes \cdots \otimes \omega^\beta \otimes \cdots
\otimes \omega^{\gamma_1 \ldots \gamma_p} \cr
&= e_\alpha \otimes \cdots \otimes \omega^\beta \otimes \cdots
\otimes S^{\alpha \ldots}_{\ \ \beta \ldots} \ .\cr
}\eeq
where
\beq
S^{\alpha \ldots}_{\ \ \beta \ldots} = \frac1{p!}
S^{\alpha \ldots}_{\ \ \beta \ldots \gamma_1 \ldots \gamma_p}
\omega^{\gamma_1 \ldots \gamma_p}
\eeq
are the components of the tensor-valued arguments of the
tensor-valued $p$-form. In most of the literature
this collection of $p$-forms is often used
in place of the tensor-valued $p$-form it represents.
One may take the exterior derivative of
this set of $p$-forms, which in turn defines the exterior derivative
of the tensor-valued $p$-form
\beq
d S
= e_\alpha \otimes \cdots \otimes \omega^\beta \otimes \cdots
\otimes d S^{\alpha \ldots}_{\ \ \beta \ldots} \ .
\eeq
This clearly depends on the choice of frame.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Linear transformations}
A function whose value at each point is a linear transformation of the
tangent space may be identified with a $1\choose1$-tensor field in a
natural way, acting on vector fields by right contraction
\beq\eqalign{
B &= B^\alpha{}_\beta e_\alpha\otimes \omega^\beta \ ,\cr
B\rightcontract X &= B^\alpha{}_\beta X^\beta e_\beta\ . \cr
}\eeq
The matrix of components
$(B^\alpha{}_\beta)$
then acts on the $R^n$-vector of components $(X^\alpha)$ by matrix
multiplication.
Similarly right contraction between $1\choose1$-tensor fields corresponds
to matrix multiplication of their square matrices of components
\beq
[ A \rightcontract B ]^\alpha{}_\beta
= A^\alpha{}_\gamma B^\gamma{}_\beta \ .
\eeq
It is convenient to use the abbreviation
\beq
A^2 = A\rightcontract A \ .
\eeq
The identity transformation is represented by the unit tensor or
Kronecker delta tensor $\delta$
\beq
\delta = \delta^\alpha{}_\beta e_\alpha \otimes \omega^\beta\ .
\eeq
The trace of a $1\choose1$-tensor corresponds to the trace of the
linear transformation
\beq
\Tr A = A^\alpha{}_\alpha \ .
\eeq
The abbreviation $\Tr^2 A = (\Tr A)^2$ is convenient.
Any such tensor can be decomposed into its pure trace and tracefree parts
\beq\eqalign{
A &= {1\over n} [ \Tr A ] \delta + A\TF \ ,\cr
A\TF &= A - {1\over n} [ \Tr A ] \delta \ .\cr
}\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Change of frame} \label{framechange}
Given a nondegenerate $1\choose1$-tensor field $A$, i.e., such that the
determinant $\det(A^\alpha{}_\beta)$ is everywhere nonvanishing,
it has a corresponding inverse tensor field $A^{-1}$ whose component
matrix is the inverse matrix of the component matrix of $A$
\beq
A^{-1} \rightcontract A = \delta\ ,\qquad
A^{-1\,\alpha}{}_\gamma A^\gamma{}_\beta = \delta^\alpha{}_\beta \ .
\eeq
These may be used to define a change of frame by defining new frame
vector fields and dual 1-forms by
\beq
\bar e_\alpha = A^{-1} \rightcontract e_\alpha
= A^{-1\,\beta}{}_\alpha e_\beta \ ,\qquad
\bar \omega^\alpha = \omega^\alpha \rightcontract A
= A^\alpha{}_\beta \omega^\beta \ .
\eeq
Under such a change of frame,
the components of a $p\choose q$-tensor field
``transform" under the representation
$\rho^{p,q}$ of the general linear group $GL(n,R)$
\beq\eqalign{
\bar S{}^{\alpha \ldots}_{\ \ \beta \ldots} &=
A^\alpha{}_\gamma \cdots A^{-1\, \delta}{}_\beta \cdots
S{}^{\gamma \ldots}_{\ \ \delta \ldots} \cr
&\equiv [\rho^{p,q} (\bfA) S ]^{\alpha \ldots}_{\ \ \beta \ldots}\ . \cr
}\eeq
The derivative of a 1-parameter family of such transformations leads
to the associated representation $\sigma^{p,q}$ of the Lie algebra
of linear transformations $gl(n,R)$
\beq
[\sigma^{p,q}(\bfA) S]^{\alpha \ldots}_{\ \ \beta \ldots} \equiv
A^\alpha{}_\gamma S{}^{\gamma \ldots}_{\ \ \beta \ldots} + \ldots
- A^\gamma{}_\beta S{}^{\alpha \ldots}_{\ \ \gamma \ldots} - \ldots\ .
\eeq
This linear operator $\sigma^{p,q}$, which is convenient to abbreviate by just
$\sigma$ when the context is clear, appears in the component formula for any
differential operator which obeys a product rule with respect
to the tensor product. It is convenient to allow the argument of $\sigma$
either to be a matrix-valued function (depending on the choice of frame)
or the corresponding $1\choose1$-tensor, whose component matrix is then
understood to be substituted in its place.
Occasionally tensor densities and oriented tensor densities prove useful.
These transform with an additional factor which is a power (called the weight)
of the inverse of the determinant of the linear transformation matrix
\beq\eqalign{
\bar S{}^{\alpha \ldots}_{\ \ \beta \ldots} &=
(\det A )^{-W}
A^\alpha{}_\gamma \cdots A^{-1\, \delta}{}_\beta \cdots
S{}^{\gamma \ldots}_{\ \ \delta \ldots} \cr
&\equiv [\rho^{p,q}_W (\bfA) S ]^{\alpha \ldots}_{\ \ \beta \ldots}\ . \cr
}\eeq
The derivative of a 1-parameter family of such transformations leads
to the associated representation $\sigma^{p,q}_W$ of the Lie algebra
of linear transformations $gl(n,R)$
\beq
[\sigma^{p,q}_W(\bfA) S]^{\alpha \ldots}_{\ \ \beta \ldots} \equiv
A^\alpha{}_\gamma S{}^{\gamma \ldots}_{\ \ \beta \ldots} + \ldots
- A^\gamma{}_\beta S{}^{\alpha \ldots}_{\ \ \gamma \ldots} - \ldots
-W A^\gamma{}_\gamma S^{\alpha \ldots}_{\ \ \beta \ldots}
\ .
\eeq
It is difficult to find clear discussions of these objects in modern
textbooks. Oriented tensor densities instead transform with an additional
sign factor which is the sign of the determinant of the transformation matrix,
thus undergoing an additional sign change for an orientation changing
transformation with negative determinant.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Metric}
If $(g_{\alpha\beta})$ is a symmetric nondegenerate matrix-valued function
of signature
$s$, i.e., can be reduced to a diagonal matrix with $N$ negative diagonal
values and $n-N$ positive values, with $s = (n-N)-N = n - 2 N$, and if
$(g^{\alpha\beta})$ is its inverse, then
\beq
\gm = g_{\alpha\beta}\, \omega^\alpha \otimes \omega^\beta \ , \qquad
\gm^{-1} = g^{\alpha\beta} e_\alpha \otimes e_\beta
\eeq
locally defines a pseudo-Riemannian (Riemannian if $N=0$) metric
and its inverse on $M$,
satisfying
\beq
\gm^{-1} \leftcontract \gm = \delta \ , \qquad
g^{\alpha\gamma} g_{\gamma\beta} = \delta^\alpha{}_\beta \ .
\eeq
Let $g = |\det(g_{\alpha\beta})|$ be the absolute value of the determinant
of the matrix of components of the metric in this frame, the sign being
$(-1)^N$. For a Lorentz metric, one has $N = 1$ or $N= n-1$.
The metric determinant derivative satisfies
\beq
d \ln g = g^{\alpha\beta} d g_{\alpha\beta}
= \Tr [ \gm^{-1} \leftcontract d \gm ] \ .
\eeq
Given a choice of orientation on $M$, namely an everywhere nonzero
$n$-form $\Oscr$, then $\{e_\alpha\}$ is an oriented frame if
$\Oscr(e_1,\ldots,e_n) > 0 $ everywhere. In an oriented frame, the
unit volume-form associated with the metric is defined by
\beq
\eta = g^{1/2} \omega^1 \wedge \cdots \wedge \omega^n \ .
\eeq
For a Lorentz metric it is often convenient to use instead the indices
$0,\ldots, n-1$ for certain choices of frame, choosing the orientation
so that in an oriented frame of this type one has
\beq
\eta = g^{1/2} \omega^0 \wedge \cdots \wedge \omega^{n-1} \ .
\eeq
Since the metric is nondegenerate, it determines an isomorphism between
the tangent and cotangent spaces at each point which in index-notation
corresponds to ``raising" and ``lowering" indices. For a vector field $X$
and a 1-form $\theta$ one has
\beq\imeqalign{
X^\flat &= \gm \rightcontract X \ , \qquad &
X_\alpha &= g_{\alpha\beta} X^\beta \ ,\cr
\theta^\sharp &= \gm^{-1} \leftcontract \theta \ , \qquad &
\theta^\alpha &= g^{\alpha\beta} \theta_\beta \ .\cr
}\eeq
As indicated in the introduction,
the sharp and flat notation for an arbitrary tensor will be understood
to mean the tensor obtained by raising or lowering respectively all
of the indices which are not already of the appropriate type.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Unit volume $n$-form}
The Levita-Civita permutation symbols
$\epsilon_{\alpha_1 \ldots \alpha_n}$ and
$\epsilon^{\alpha_1 \ldots \alpha_n}$
are totally antisymmetric with
\beq
\epsilon_{1 \ldots n} = 1 = \epsilon_{1 \ldots n}\ ,
\eeq
so that it vanishes unless $\alpha_1 \ldots \alpha_n$ is a permutation of $1 \ldots n$, in which case its value is the sign of the permutation.
The components of the unit volume $n$-form are related to these objects by
\beq\label{eq:etaepsilon}
\eta_{\alpha_1 \ldots \alpha_n}
= g^{1/2}\epsilon_{\alpha_1 \ldots \alpha_n}\ ,
\eta^{\alpha_1 \ldots \alpha_n}
= (-1)^N g^{-1/2}\epsilon_{\alpha_1 \ldots \alpha_n}\ .
\eeq
For a Lorentz 4-dimensional spacetime, it is convenient to use the ordered index labels 0,1,2,3 so that one has the Misner, Thorne and Wheeler convention
\beq
\epsilon_{0123} = 1 = \epsilon^{0123}
\eeq
and therefore the unit-oriented 4-form has components
\beq
\eta_{0123} = g^{1/2}\ ,\
\eta^{0123} = - g^{-1/2}\ .
\eeq
Others prefer the Ellis convention using the index labels 1,2,3,4, but one finds both sign conventions
$\eta_{1234} = g^{1/2}$ and $\eta_{1234} = -g^{1/2}$ in use, making comparisons difficult.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Connection}
A general linear connection or covariant derivative $\del$
on $M$ is (locally) determined by its components in a frame.
Adopting the ``del" convention for the order of the covariant indices,
these components are defined by
\beq
\del\sub{e_\alpha} e_\beta = \Gamma^\gamma{}_{\alpha\beta} e_\gamma
\quad \leftrightarrow \quad
\del\sub{e_\alpha} \omega^\beta =
-\Gamma^\beta{}_{\alpha\gamma} \omega^\gamma \ .
\eeq
The covariant derivative $\del S$ of an arbitrary $p\choose q$-tensor $S$ is
a $p\choose q+1$-tensor with components
\beq\eqalign{
[\del S ]^{\alpha \ldots}_{\ \ \beta \ldots \gamma} &\equiv
\del_\gamma S ^{\alpha \ldots}_{\ \ \beta \ldots } \equiv
S ^{\alpha \ldots}_{\ \ \beta \ldots ; \gamma} \cr
&= S ^{\alpha \ldots}_{\ \ \beta \ldots , \gamma}
+ \Gamma^\alpha{}_{\gamma\delta} S ^{\delta \ldots}_{\ \ \beta \ldots}
+ \cdots
- \Gamma^\delta{}_{\gamma\beta} S ^{\alpha \ldots}_{\ \ \delta \ldots}
- \cdots \cr
&\equiv S ^{\alpha \ldots}_{\ \ \beta \ldots , \gamma}
+ [\sigma^{p,q}(\bfomega_\gamma) S]^{\alpha \ldots}_{\ \ \beta \ldots}
\ .\cr
}\eeq
For a $p\choose q$-tensor density of weight $W$, one need only replace
$\sigma^{p,q}(\bfomega_\gamma)$ by $\sigma^{p,q}_W(\bfomega_\gamma)$,
adding the additional term
$-W \Gamma^\delta{}_{\gamma\delta} S^{\alpha \ldots}_{\ \ \beta \ldots}$.
The $1\choose1$-tensor-valued connection 1-form for the frame $\{e_\alpha\}$
is defined by
\beq
\bfomega =
e_\alpha \otimes \bfomega^\alpha{}_\beta \otimes \omega^\beta =
\Gamma^\alpha{}_{\gamma\beta} e_\alpha \otimes
\omega^\gamma \otimes \omega^\beta \ ,
\eeq
but is often treated as a matrix-valued 1-form whose entries are
\beq
\bfomega^\alpha{}_\beta = \Gamma^\alpha{}_{\gamma\beta}
\omega^\gamma \ .
\eeq
The $\gamma$ component of this matrix of 1-forms is
understood to be the argument of the
the linear operator $\sigma$ in the definition of the covariant derivative.
The components of the torsion tensor of the connection are
\beq\label{eq:torsion}
T^\alpha{}_{\beta\gamma} = 2\Gamma^\alpha{}_{[\beta\gamma]}
- C^\alpha{}_{\beta\gamma}
\eeq
and define a vector-valued torsion 2-form
\beq
\bfTheta^\alpha = \half T^\alpha{}_{\beta\gamma} \omega^{\beta\gamma}
\ , \qquad \bfTheta = e_\alpha \otimes \bfTheta^\alpha \ .
\eeq
Of course as tensor fields $\bfTheta = T$, but the different notation helps
distinguish the tensor-valued differential form interpretation from
the tensor, and provides a notation for evaluating only the differential
form arguments of the tensor. In most of the literature $\bfTheta$ is treated
as an $R^n$-valued 2-form $(\bfTheta^\alpha)$, as are all other vector-valued
forms.
The frame independent definition of the torsion 2-form, with its
2-form arguments evaluated on a pair of vector fields $X$ and $Y$ is
\beq
\del\sub{X} Y -\del\sub{Y} X - [ X , Y ] = \bfTheta(X,Y) \ .
\eeq
Evaluation of this formula using the frame vectors themselves leads
to the above component formula.
This may also be expressed equivalently in terms of the exterior
derivative
\beq\eqalign{
& d \omega^\alpha + \bfomega^\alpha{}_\beta \wedge \omega^\beta
= \bfTheta^\alpha\ ,\cr
& d \bfvartheta + \bfomega \rightcontract \wedge \bfvartheta
= \bfTheta\ ,\cr
}\eeq
where the right contraction is between the tensor-valued indices and the
wedge is between the differential form indices, and
\beq
\bfvartheta = e_\alpha \otimes \bfvartheta^\alpha
= \delta^\alpha{}_\beta e_\alpha \otimes \omega^\beta
\eeq
is the identity tensor thought of as a vector-valued 1-form with
component 1-forms $\bfvartheta^\alpha = \omega^\alpha$ equal to the frame
1-forms.
A connection for which the torsion is zero is called
torsion-free or symmetric, since
in a coordinate frame the connection components are symmetric in the
pair of covariant indices
\beq
T^\alpha{}_{\beta\gamma}=0= C^\alpha{}_{\beta\gamma}
\quad \rightarrow \quad \Gamma^\alpha{}_{[\beta\gamma]} = 0 \ .
\eeq
For a general connection, one can always introduce a 1-parameter family
of related connections whose torsion is any real multiple of the original
torsion by using a multiple of the torsion tensor itself as a difference
tensor between the two connections.
In particular there is a symmetric connection $\SYM\del$ with
zero torsion and a transposed connection $\tilde\del$ with opposite torsion
\beq\meqalign{
\Gamma^\alpha{}_{\beta\gamma} &= \Gamma^\alpha{}_{(\beta\gamma)}
+ \Gamma^\alpha{}_{[\beta\gamma]}
&= \Gamma^\alpha{}_{(\beta\gamma)} + \half C^\alpha{}_{\beta\gamma}
+ \half T^\alpha{}_{\beta\gamma} \ ,\cr
[\SYM \Gamma]^\alpha{}_{\beta\gamma} &= \Gamma^\alpha{}_{\beta\gamma}
- \half T^\alpha{}_{\beta\gamma}
&= \Gamma^\alpha{}_{(\beta\gamma)} + \half C^\alpha{}_{\beta\gamma} \ ,\cr
\tilde \Gamma{}^\alpha{}_{\beta\gamma} &= \Gamma^\alpha{}_{\beta\gamma}
- T^\alpha{}_{\beta\gamma}
&= \Gamma^\alpha{}_{(\beta\gamma)} + \half C^\alpha{}_{\beta\gamma}
- \half T^\alpha{}_{\beta\gamma} \cr
&= \Gamma^\alpha{}_{\gamma\beta} + C^\alpha{}_{\beta\gamma} \ .\cr
}\eeq
In a coordinate frame these reduce to simple relationships
\beq
[\SYM \Gamma]^\alpha{}_{\beta\gamma} = \Gamma^\alpha{}_{(\beta\gamma)}\ ,
\qquad
\tilde \Gamma{}^\alpha{}_{\beta\gamma} = \Gamma^\alpha{}_{\gamma\beta}\ .
\eeq
Such triplets of connections play an important role in the geometry
of Lie groups.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Metric connection}
A connection is said to be metric with respect to a given metric $\gm$
if that metric is covariant constant
\beq\label{eq:dg_gamma}
\meqalign{
0&= g_{\alpha\beta;\gamma} &=
g_{\alpha\beta,\gamma} - g_{\delta\beta} \Gamma^\delta{}_{\gamma\alpha}
- g_{\alpha\delta} \Gamma^\delta{}_{\gamma\beta} \cr
& &= g_{\alpha\beta,\gamma} - \Gamma_{\alpha\gamma\beta}
- \Gamma_{\beta\gamma\alpha}
\rightarrow
g_{\alpha\beta,\gamma} = 2 \Gamma_{(\alpha|\gamma|\beta)}
\ ,\cr
}\eeq
extending the index-shifting convention to the components of the connection.
This relation may be inverted using the definition of the torsion to
express the components of the connection entirely in terms of the
torsion, the structure functions of the frame and the ordinary derivatives
of the metric components.
Using the notation
\beq
A_{\{\delta\beta\gamma\}_-} = A_{\delta\beta\gamma} -
A_{\beta\gamma\delta} + A_{\gamma\delta\beta} \ .
\eeq
and shifting indices on the structure functions
with the frame component matrix of the spacetime metric,
the connection components can be written
\beq\eqalign{
\Gamma^\alpha{}_{\beta\gamma} &=
\half g^{\alpha\delta}( g_{\{\delta\beta,\gamma\}_-}
+ C_{\{\delta\beta\gamma\}_-}
+ T_{\{\delta\beta\gamma\}_-} )\ ,\cr
&= \Gamma(\gm)^\alpha{}_{\beta\gamma} + K^\alpha{}_{\beta\gamma}
\ ,\cr
}\eeq
where
\beq\eqalign{
\Gamma(\gm)^\alpha{}_{\beta\gamma} &=
\half g^{\alpha\delta} ( g_{\{\delta\beta,\gamma\}_-}
+ C_{\{\delta\beta\gamma\}_-} ) \ ,\cr
K^\alpha{}_{\beta\gamma} &=
\half g^{\alpha\delta} T_{\{\delta\beta\gamma\}_-} \ .\cr
}\eeq
$\Gamma(\gm)^\alpha{}_{\beta\gamma}$ are the components of the unique
metric connection for which the torsion vanishes, often just
called ``the metric connection."
Its connection
components in a coordinate frame reduce to the Christoffel symbols
of the metric.
$K^\alpha{}_{\beta\gamma}$ are the components of the difference tensor
between the general metric connection and the unique one associated with
the metric. This difference tensor is
sometimes referred to as the contorsion tensor.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Curvature}
The components of the curvature tensor of a connection in a frame are
defined by
\beq
R^\alpha{}_{\beta\gamma\delta} =
\Gamma^\alpha{}_{\delta\beta,\gamma} -
\Gamma^\alpha{}_{\gamma\beta,\delta} -
C^\epsilon{}_{\gamma\delta} \Gamma^\alpha{}_{\epsilon\beta}
+ \Gamma^\alpha{}_{\gamma\epsilon} \Gamma^\epsilon{}_{\delta\beta}
- \Gamma^\alpha{}_{\delta\epsilon} \Gamma^\epsilon{}_{\gamma\beta}
\ .
\eeq
This tensor is explicitly antisymmetric in its last pair of indices and
so defines a $1\choose1$-tensor-valued 2-form $\bfOmega=R$
\beq
\bfOmega = e_\alpha \otimes \omega^\beta \otimes \bfOmega^\alpha{}_\beta =
\half R^\alpha{}_{\beta \gamma\delta}
e_\alpha \otimes \omega^\beta \otimes \omega^{\gamma\delta} \ ,
\eeq
whose tensor-valued components are often thought of as a matrix-valued
2-form whose entries are
\beq
\bfOmega^\alpha{}_\beta = \half R^\alpha{}_{\beta\gamma\delta}
\omega^{\gamma\delta} \ .
\eeq
The $1\choose1$-tensor
values of this 2-form define linear transformations of the tangent and
cotangent spaces.
The invariant definition of the curvature tensor of a connection
with the 2-form arguments evaluated on a pair of vector fields $X$ and
$Y$, and acting as a linear transformation on the vector field $Z$, is
\beq
\{ [ \del\sub{X} , \del\sub{Y} ] - \del\sub{[X,Y]} \} Z
= \bfOmega(X,Y) \rightcontract Z \ .
\eeq
When evaluated on the frame vectors themselves, one obtains the above
component formula.
This may also be represented alternatively in terms of the connection
and curvature forms
\beq\eqalign{
& d \bfomega^\alpha{}_\beta + \bfomega^\alpha{}_\gamma
\wedge \bfomega^\gamma{}_\beta = \bfOmega^\alpha{}_\beta \ ,\cr
& d \bfomega + \bfomega \rightcontract \wedge \bfomega
=\bfOmega \ \cr
}\eeq
where again the right contraction refers to the adjacent tensor-valued
indices and the wedge product to the differential form indices.
The totally covariant Riemann tensor obtained by lowering its first index can be re-expressed in terms of the covariant connection component derivatives using (\ref{eq:dg_gamma}) to re-express the metric derivatives in terms of the symmetric part of the connection. For a torsion-free metric connection one finds
\beq
R_{\alpha\beta\gamma\delta} =
\Gamma_{\alpha\delta\beta,\gamma} -
\Gamma_{\alpha\gamma\beta,\delta} -
C^\epsilon{}_{\gamma\delta} \Gamma_{\alpha\epsilon\beta}
+ \Gamma_{\epsilon\delta\alpha} \Gamma^\epsilon{}_{\gamma\beta}
- \Gamma_{\epsilon\gamma\alpha} \Gamma^\epsilon{}_{\delta\beta}
\ .
\eeq
This in turn may be used to make the second derivatives of the metric coefficients explicit in a coordinate frame ($C^\alpha{}_{\beta\gamma}=0$) leading to the Landau-Lifshitz formula (92.4) from their {\it Classical Theory of Fields}
\beq
R_{\alpha\beta\gamma\delta} =
\half( g_{\alpha\delta,\beta\gamma} + g_{\beta\gamma,\alpha\delta}
-g_{\alpha\gamma,\beta\delta} - g_{\beta\delta,\alpha\gamma} )
+ \Gamma_{\epsilon\delta\alpha} \Gamma^\epsilon{}_{\gamma\beta}
- \Gamma_{\epsilon\gamma\alpha} \Gamma^\epsilon{}_{\delta\beta}
\ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Total covariant derivative}
\label{app:tcd}
The total covariant derivative of a tensor field along a parametrized
curve generalizes the ordinary derivative of a function along such a
curve. It is called the intrinsic or
absolute derivative in the older literature
\Cite{Synge and Schild 1949, M\o ller 1952}
and occurs as a special case of an induced connection over a map in
the modern literature \Cite{Sachs and Wu 1977, Bishop and Goldberg 1968}.
The terminology ``total covariant derivative" as used in this text
may be a result of teaching too many
calculus classes while thinking about relativity.
Given a parametrized curve $c(\lambda)$, with corresponding
tangent vector $c'(\lambda)$, one can
covariantly differentiate any tensor $S(\lambda)$ defined along the curve
by correcting the ordinary derivative by the representation of the
connection 1-form matrix evaluated on the tangent vector
\beq
D S^{\alpha\ldots}_{\ \ \beta\ldots} / d \lambda =
d S^{\alpha\ldots}_{\ \ \beta\ldots} / d \lambda +
[\sigma(\bfomega(c'(\lambda)))
S(\lambda) ]^{\alpha\ldots}_{\ \ \beta\ldots} \ .
\eeq
The correction terms come from the product rule and the
total covariant derivatives of the frame vectors and dual 1-forms
along the curve
\beq
D [e_\alpha \circ c(\lambda) ] / d \lambda
= \bfomega(c'(\lambda))^\beta{}_\alpha e_\beta \ ,\qquad
D [\omega^\alpha \circ c(\lambda) ] / d \lambda
= - \bfomega(c'(\lambda))^\alpha{}_\beta \omega^\beta \ .
\eeq
For a tensor field $S$, the total covariant derivative of its restriction
to the curve $S(\lambda)= S\circ c(\lambda)$
agrees with the covariant derivative of
$S$ along the tangent vector at each point of the curve
\beq
[ \del\sub{c'(\lambda)} S ] \circ c(\lambda) = D S(\lambda) / d \lambda \ .
\eeq
It is convenient to follow the sloppy convention of not
distinguishing between the two types of derivatives and suppressing
the parametrization variable, writing simply
\beq
D S / d\lambda = \del\sub{c'} S \ .
\eeq
Given a local coordinate system, a parametrized curve is represented
by the composed functions $c^\alpha(\lambda) = x^\alpha \circ c(\lambda)$,
the derivatives of which provide the coordinate frame components of its
tangent vector
$ [c'(\lambda)]^\alpha = d c^\alpha(\lambda) / d \lambda
= c^\alpha{}'(\lambda)$, often sloppily represented by the symbols
$x^\alpha(\lambda)$ and $ d x^\alpha(\lambda) / d \lambda$ respectively.
For a vector field $X(\lambda)$ defined along the curve, the coordinate
frame expression for its total covariant derivative then takes the form
\beq
D X^\alpha(\lambda) / d \lambda
= d X^\alpha(\lambda) / d \lambda + \Gamma^\alpha{}_{\beta\gamma}
\circ c(\lambda) c^\beta{}'(\lambda) X^\gamma(\lambda) \ .
\eeq
For the tangent vector itself, this becomes
\beq
[D c'(\lambda)]^\alpha
= d^2 c^\alpha(\lambda) / d \lambda + \Gamma^\alpha{}_{\beta\gamma}
\circ c(\lambda) d c^\beta(\lambda) / d\lambda
d c^\gamma(\lambda) / d\lambda \ .
\eeq
Under a change of parametrization $\lambda=f(\bar\lambda)$ of the curve,
the total covariant derivative changes according to the familiar
chain rule
\beq
DS / d\bar\lambda = (d\lambda/d\bar\lambda)\, DS / d\lambda \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Parallel transport and geodesics}
A tensor field $S(\lambda)$ defined along a parametrized curve $c(\lambda)$
is said to be parallel transported (sometimes ``propagated")
along the curve if its total covariant
derivative is identically zero
\beq
D S(\lambda) / d\lambda = 0 \ .
\eeq
A parametrized curve whose tangent vector is parallel transported along
the curve is called a geodesic
\beq
D c'(\lambda) / d\lambda = 0 \ .
\eeq
When expressed in a coordinate system, these are second order
ordinary differential equations for the values of the
coordinate functions along the parametrized curve.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Generalized Kronecker deltas}
The following identities define the generalized Kronecker deltas
and relate them to the Levi-Civita epsilon and to the unit volume
$n$-form $\eta$ when a metric is available
\beq\eqalign{
\delta^{\alpha_1 \ldots \alpha_n}_{\ \ \beta_1 \ldots \beta_n}
&= \epsilon^{\alpha_1 \ldots \alpha_n} \epsilon_{\beta_1 \ldots \beta_n}
= (-1)^N \eta^{\alpha_1 \ldots \alpha_n} \eta_{\beta_1 \ldots \beta_n}\ ,\cr
\delta^{\alpha_1 \ldots \alpha_p}_{\ \ \beta_1 \ldots \beta_p}
& = \frac1{(n-p)!}
\delta^{\alpha_1 \ldots \alpha_p \gamma_{p+1} \ldots \gamma_n}
_{\ \ \beta_1 \ldots \beta_p \gamma_{p+1} \ldots \gamma_n} \cr
&= (-1)^N \frac1{(n-p)!}
\eta^{\alpha_1 \ldots \alpha_p \gamma_{p+1} \ldots \gamma_n}
\eta_{\beta_1 \ldots \beta_p \gamma_{p+1} \ldots \gamma_n} \cr
&= (-1)^{N +p(n-p)} \frac1{(n-p)!}
\eta^{\alpha_1 \ldots \alpha_p \gamma_{p+1} \ldots \gamma_n}
\eta_{\beta_1 \ldots \beta_p \gamma_{p+1} \ldots \gamma_n} \ .\cr
}\eeq
Alternative definitions are instead
\beq\eqalign{
\delta^{\alpha_1 \ldots \alpha_p}_{\ \ \beta_1 \ldots \beta_p}
= \left|\matrix{
\delta^{ \alpha_1}{}_{\beta_1} & \cdots & \delta^{\alpha_1}{}_{\beta_p}\cr
\vdots & & \vdots \cr
\delta^{\alpha_p}{}_{\beta_1} & \cdots & \delta^{\alpha_p}{}_{\beta_p} \cr
}\right|
& =
p! \delta^{\alpha_1}{}_{[\beta_1} \cdots
\delta^{\alpha_p}{}_{\beta_p]} \cr
&= p \delta^{\alpha_1}_{\ \ [\beta_1}
\delta^{\alpha_2 \ldots \alpha_p}_{\ \ \beta_2 \ldots \beta_p]} \cr
&= \sum^p_{i=1} (-1)^{i-1} \delta^{\alpha_i}_{\ \ \beta_1}
\delta^{\alpha_1 \ldots \hat\alpha_i \ldots \alpha_p}
_{\ \ \beta_2 \ldots\phantom{\beta_1\ldots} \beta_p} \ ,\cr
}\eeq
where here
the hatted index notation implies the removal of the index from
the indicated position.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Symmetrization/antisymmetrization}
Given an object with only covariant or contravariant indices,
or a subset of only covariant or contravariant indices from those of
a mixed object, only can always project out the purely symmetric and
purely antisymmetric parts. For example for a $0\choose p$-tensor field
one has
\beq\eqalign{
[\ALT S]_{\alpha_1 \ldots \alpha_p} &=
S_{[\alpha_1 \ldots \alpha_p]} =
\frac1{p!}
\delta^{\beta_1 \ldots \beta_p}_{\ \ \alpha_1 \ldots \alpha_p}
S_{\beta_1 \ldots \beta_p} \ ,\cr
[\SYM S]_{\alpha_1 \ldots \alpha_p} &=
S_{(\alpha_1 \ldots \alpha_p)} =
\frac1{p!}
\sum_{\sigma}
S_{\sigma(\alpha_1) \ldots \sigma(\alpha_p)} \ .\cr\message{oops??}
}\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Exterior product}
The exterior or wedge product of $p$ 1-forms
like the frame 1-forms is defined antisymmetrizing without the
factorial factor
\beq
\omega^{\alpha_1\ldots \alpha_p} =
\omega^{\alpha_1}\wedge \cdots \wedge \omega^{\alpha_p}
= p! \, \omega^{[\alpha_1}\otimes \cdots \otimes \omega^{\alpha_p]}
= \delta^{\alpha_1\ldots \alpha_p}_{\ \ \beta_1 \ldots \beta_p}
\omega^{\beta_1}\otimes \cdots \otimes \omega^{\beta_p} \ .
\eeq
With this convention the components of an antisymmetric tensor are
themselves the expansion coefficients in the basis
$\{ \omega^{|\alpha_1\ldots \alpha_p|} \}$.
This leads to a correction
factor for the exterior product of $p$-forms with $p>1$.
For a $p$-form $S$ and a $q$-form $T$, one has
\beq\eqalign{
[S\wedge T]_{\alpha_1 \ldots \alpha_{p+q}} &=
\frac{(p+q)!}{p! q!} S_{[\alpha_1\ldots\alpha_p}
T_{\alpha_{p+1}\ldots \alpha_{p+q}]} \cr
&= \delta^ {\beta_1\ldots \beta_p \gamma_1\ldots \gamma_q}
_{\ \ \alpha_1 \phantom{\beta_p}\ldots\phantom{\gamma_1\ldots}
\alpha_{p+q}}
S_{|\beta_1 \ldots \beta_p|} T_{|\gamma_1 \ldots \gamma_p|} \ .
}\eeq
One has the identity
\beq
S \wedge T = (-1)^{pq} \, T \wedge S \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Hodge star duality operation}
The Hodge star duality operation is defined for
a $p$-form $S$ to be the $(n-p)$ form $\dual S$ with components
\beq
[\dual S]_{\alpha_{p+1} \ldots \alpha_n} =
\frac1{p!} S_{\alpha_1 \ldots \alpha_p}
\eta^{\alpha_1 \ldots \alpha_p}{}_{\alpha_{p+1} \ldots \alpha_n}
\eeq
following the right dual convention of Misner, Thorne and Wheeler
in which the contraction of $S^\sharp$ with $\eta$ has $\eta$ on the right.
By introducing the $(n-p)$-forms
\beq
\eta^{\alpha_1 \ldots \alpha_p} = \dual \omega^{\alpha_1 \ldots \alpha_p}
= \frac1{(n-p)!}
\eta^{\alpha_1 \ldots \alpha_p}{}_{\alpha_{p+1} \ldots \alpha_n}
\omega^{\alpha_{p+1} \ldots \alpha_n} \ ,
\eeq
one can re-express this in the following way
\beq\eqalign{
\dual S &=
\frac1{p!} S_{\alpha_1 \ldots \alpha_p}
\eta^{\alpha_1 \ldots \alpha_p} \cr
&= \frac1{p!} S^\sharp \leftcontractp{p} \eta \ ,\cr
}\eeq
where the generalized left contraction of $p$ indices indicates
the contraction of the last $p$ contravariant indices of the left
factor with the first $p$ covariant indices of the right factor in order.
Note the extreme cases
\beq
\dual 1 = \eta\ , \qquad \dual \eta = (-1)^N \ .
\eeq
Using the above definitions and identities one establishes the identity
\beq
\dual \dual S = (-1)^{N + p(n-p)} S
\eeq
for a $p$-form $S$, where the term $N$ in the sign arises from the
signature and the second from the interchange of the group of indices.
For a 4-dimensional Lorentzian manifold and a 3-dimensional Riemannian manifold
respectively, this reduces to
\beq\meqalign{
n=4,\ & N=1 : \qquad & \dual \dual S = (-1)^{p-1} S
\qquad & {\rm(spacetime)}\cr
n=3,\ & N=0 : \qquad & \dual \dual S = S
\qquad & {\rm (space)} \ .\cr
}\eeq
This is a special case of the more general identity for a $p$-form $S$
and a $q$-form $T$ with $q\ge p$
\beq
\dual ( S \wedge \dual T) = (-1)^{N + (n-q)(q-p)} \, \frac1{p!}
S^\sharp \leftcontractp{p} T \ .
\eeq
With $p=0$ and $S=1$ this reduces to the previous identity. With
$p=q$ one instead has
\beq
\dual ( S \wedge \dual T) = (-1)^{N} \, \frac1{p!}
S^\sharp \leftcontractp{p} T \ ,
\eeq
or since $\dual \eta = (-1)^N$, removing the duality operation from each
side leads to
\beq
S \wedge \dual T = \frac1{p!}
S^\sharp \leftcontractp{p} T \, \eta = \langle S,T \rangle \, \eta\ ,
\eeq
where
\beq
\langle S , T \rangle = \frac1{p!} S^{\alpha_1 \ldots \alpha_p}
T_{\alpha_1 \ldots \alpha_p}
\eeq
defines a natural local (pointwise) inner product between $p$-forms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Complex Duality Operation}
For a 4-dimensional Lorentz spacetime, 2-forms (``bivectors") and 4-forms which are symmetric under exchange of the first and second index pairs (like the Riemann and Weyl tensors) play an important role. With $N=1$ and $p=2$, one has $\dual\dual S = -S$ for all 2-forms. The minus sign comes from the fact that raising all the indices of the unit volume 4-form in an orthonormal frame changes the sign of the ordered index component as in (\ref{eq:etaepsilon}). By using the square root of the negative metric determinant rather than of its absolute value in the definition of the volume form, the extra sign in (\ref{eq:etaepsilon}) is eliminated
\beq\label{etaepsilon}
E_{\alpha_1 \ldots \alpha_4}
= (-g)^{1/2}\epsilon_{\alpha_1 \ldots \alpha_4}
= i \eta_{\alpha_1 \ldots \alpha_n}
\ ,\
E^{\alpha_1 \ldots \alpha_4}
= (-g)^{-1/2}\epsilon^{\alpha_1 \ldots \alpha_4}
= - i \eta^{\alpha_1 \ldots \alpha_4}\ .
\eeq
Using this purely imaginary 4-form instead of $\eta$ for the duality operation on 2-form index pairs defines the ``hook" duality operation with symbol $\hook$, apparently used by Veblen and von Neumann, and later Taub.
This leads to $\hook S=i\dual S$ on 2-forms which therefore satisfy $\hook\hook S = S$, so that one can find eigentensors of the operation with eigenvalues $\pm1$, called self-dual ($+$) and anti-self-dual ($-$) respectively. Self-duality then translates back to $i\dual S = S$ or $\dual S = -i S$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Exterior derivative}
The exterior derivative is defined to obey the following identity
for a $p$-form $S$ and a $q$-form $T$
\beq
d ( S \wedge T ) = d S \wedge T + (-1)^p S \wedge d T \ ,
\eeq
and to reduce to the ordinary differential of a 0-form or function
\beq
d f = \partial_\alpha f \, \omega^\alpha \ .
\eeq
For a $p$-form
\beq
S = \frac1{p!} S_{\alpha_1 \ldots \alpha_p}
\omega^{\alpha_1 \ldots \alpha_p} \ ,
\eeq
this identity yields
\beq
d S = \frac1{p!} d S_{\alpha_1 \ldots \alpha_p} \wedge
\omega^{\alpha_1 \ldots \alpha_p}
+ \frac1{p!} S_{\alpha_1 \ldots \alpha_p}
d \omega^{\alpha_1 \ldots \alpha_p}
= \frac1{(p+1)!} [d S]_{\alpha_1 \ldots \alpha_{p+1}}
\omega^{\alpha_1 \ldots \alpha_{p+1}} \ ,
\eeq
while for the basis $p$-forms it yields
\beq
d \omega^{\alpha_1 \ldots \alpha_p} =
- \half \sum^p_{i=1} (-1)^{i-1} C^{\alpha_i}{}_{\beta\gamma}
\omega^{\alpha_1 \ldots \beta\gamma \ldots \alpha_p}\ .
\eeq
Putting these together yields the formula
\beq
[d S]_{\alpha_1 \ldots \alpha_{p+1}} =
(p+1) ( \partial_{[\alpha_1} S_{\alpha_2 \ldots \alpha_{p+1}]}
- \half p C^{\beta}{}_{[\alpha_1\alpha_2}
S_{|\beta|\alpha_3 \ldots \hat\alpha_i \ldots \alpha_{p+1}]} ) \ ,
\eeq
where the vertical bar delimiters exclude the indicated index from the
antisymmetrization.
Only the first term remains in a coordinate frame.
This result may be re-expressed in terms of a general connection as
\beq
[d S]_{\alpha_1 \ldots \alpha_{p+1}} =
(p+1) ( \del_{[\alpha_1} S_{\alpha_2 \ldots \alpha_{p+1}]}
+ \half p
T^{\gamma}{}_{[\alpha_1\alpha_2}
S_{|\gamma| \alpha_3 \ldots \hat\alpha_i \ldots \alpha_p]} ) \ .
\eeq
For a symmetric connection only the first term remains
yielding a formula which arises by
substituting a covariant derivative for the ordinary
derivative in the coordinate frame formula, i.e., the ``comma to semicolon"
rule.
The exterior derivative may be expressed in a frame-independent way
by evaluating its vector field arguments
\beq
d S (X_1,\ldots,X_{p+1})
= \sum^{p+1}_{i=1} (-1)^{i-1} X_i
S(X_1, \ldots, \hat X_j, \ldots, X_{p+1})
+ \sum_{i3$
in terms of a
vector-valued 2-form, the Cotton tensor \Cite{Cotton 1899},
which is the covariant exterior derivative
of a linear combination of the Ricci tensor and the scalar curvature
seen as a vector valued 1-form.
The twice contracted Bianchi identity of the second kind in any dimension
\beq\eqalign{
0 &= R^{\alpha\beta}{}_{[\gamma\delta;\epsilon]}
\delta^\gamma{}_\alpha \delta^\delta{}_\beta\cr
&= -2 (R^\delta{}_\epsilon -\frac12 R \delta^\delta{}_\epsilon)_{;\delta} \cr
&= - 2 G^\delta{}_{\epsilon;\delta} \ ,\cr
}\eeq
which states that the Einstein tensor is divergence-free,
immediately leads to the tracefree property of the Cotton tensor
\beq
R^\beta{}_{\alpha\beta} = -G^\beta{}_{\alpha;\beta} =0\ ,
\eeq
whose totally antisymmetric part is easily seen to be zero
using the symmetry of the Ricci and metric tensors
\beq
3 R_{[\alpha\beta\gamma]}
= R_{\alpha\beta\gamma}
+ R_{\beta\gamma\alpha}
+ R_{\gamma\alpha\beta} = 0\ .
\eeq
%Finally the divergence of the once contracted Bianchi identity of the
%second kind reduces to
%\beq\eqalign{
% 0 &=- C^{\alpha\beta}{}_{\delta\epsilon;\alpha\beta}
% + (n-3)/(n-2)\, R^\beta{}_{\delta\epsilon;\beta} \ ,\cr
% = (n-3)/(n-2)\, R^\beta{}_{\delta\epsilon;\beta} \ ,\cr
%}\eeq
%using the Ricci identities on the first term which is seen to vanish
%using the symmetry of the Ricci tensor and the symmetry of the
%triple contraction $C^{\alpha\beta\gamma\delta} R_{\alpha\beta\gamma\epsilon}$,
%showing the Cotton tensor to be divergence-free as well,
%except for $n=3$ where a direct derivation is required.
The divergence of the Cotton-tensor is also zero
\beq
R^\beta{}_{\delta\epsilon;\beta}
= 2 G^\beta{}_{[\delta;|\beta|;\epsilon]} = 0\ .
\eeq
For $n=3$ dimensions where the Weyl tensor vanishes identically,
one can take
the dual of the covariant index pair of the Cotton tensor
to obtain a two-index tensor, the Cotton-York tensor \Cite{York 1971}
\beq
C^{\alpha\beta} = R^\alpha{}_{\gamma\delta} \eta^{\beta\gamma\delta}
= ( R^\alpha{}_\gamma
- 1/4 R \delta^\alpha{}_\gamma )_{;\delta} \eta^{\beta\gamma\delta}
\ ,
\eeq
which can then be decomposed into its symmetric and antisymmetric parts, the
latter of which is the spatial dual of a vector
\beq\eqalign{
C^{\alpha\beta} &= C^{(\alpha\beta)} + C^{[\alpha\beta]}
= Y^{(\alpha\beta)} + \eta^{\alpha\beta\gamma} A_\gamma\cr
R^\alpha{}_{\beta\gamma} &= C^{\alpha\delta} \eta_{\delta\beta\gamma}
= \eta_{\beta\gamma\delta} Y^{\alpha\beta}
+ A_\delta \delta^{\delta\alpha}_{\beta\gamma} \ .\cr
}\eeq
The explicitly symmetric
York tensor and the vector (which vanishes by the twice contracted
Bianchi identity) are
\beq\eqalign{
y^{\alpha\beta} &= \eta^{\gamma\delta(\alpha} R^{\beta)}{}_{\gamma;\delta}
= -[\Scurl(Ricci)]^{\alpha\beta}
= -[\Scurl(Ricci\TF)]^{\alpha\beta}
= -[\Scurl(Einstein)]^{\alpha\beta} \ ,\cr
A_\beta &= R^\alpha{}_{\beta\alpha} = G^\alpha{}_{\beta;\alpha} = 0\ ,\cr
}\eeq
The subsequent equalities for the York tensor $y$
hold since the symmetrized covariant
exterior derivative ``$\Scurl$", see section (\ref{sec:Scurl}))
of a symmetric 2-tensor interpreted as a
vector-valued 1-form annihilates the pure trace part (and automatically
yields a tracefree result, making the York tensor tracefree
since the Ricci tensor is symmetric).
The York tensor is automatically divergence-free as a consequence of the same
property for the Cotton tensor.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Conformal Transformations in 3 Dimensions}
Introduce the notation $U_a=\del_a U$ for a scalar $U$, which satisfies
\beq
\del_a U_b = \del_a \del_b U = \del_b \del_a U = \del_b U_a
\eeq
or equivalently $\eta^{abc} \del_b U_c=\hat0$.
Then the standard conformal transformation laws for the metric and the curvature tensors are
\begin{eqnarray}
\tilde g_{ab}
&=& e^{2U} g_{ab}\ , \tilde g{}^{ab} = e^{-2U} g^{ab}\ , \tilde g{}^{1/6} = e^U g^{1/6}
\ ,\\
\tilde \eta_{abc}
&=& e^{3U} \eta_{abc}\ ,\ \tilde \eta{}^{abc} = e^{-3U} \eta{}^{abc}\ ,
\\
\tilde\del_d X^b{}_c
&=& \del_d X^b{}_c + C^b{}_{de} X^e{}_c -C^e{}_{dc} X^b{}_e\ ,
\\
C^c{}_{ab}
&=& 2 \delta^c{}_{(a} U_{b)} - g_{ab} U^c\ ,\ C^c{}_{[ab]}=\hat0\ ,
\\
\tilde R_{ac} &=& R_{ac} - \del_a U_c +U_a U_c + g_{ac} ( - \del^e U_e - U^e U_e )\ ,\
\\
\tilde R &=& e^{-2U} [ R - 4 \del^e U_e - 2 U^e U_e ],
\\
\tilde G_{ac} &=& G_{ac} - \del_a U_c +U_a U_c + g_{ac} \del^e U_e \ ,\
\\
\tilde R^a{}_c &=& e^{-2U} \tilde R_{ac}\ ,\
\tilde \del_d R^a{}_c = e^{-2U} (\del_d-2 U_d) \tilde R_{ac} \ .
\end{eqnarray}
The York tensor then satisfies
\begin{eqnarray}
g{}^{-5/6} \tilde g{}^{5/6} \tilde y^{ea}
&=& g{}^{-5/6} \tilde g{}^{5/6} \tilde \eta^{edc} \tilde \del_d \tilde R{}^a{}_c
= \eta^{edc} \tilde (\del_d - 2 U_d) \tilde R{}^a{}_c \nonumber\\
&=& y^{ea} + [\Scurl(-\del U + U\otimes U) +U\times ( \del U - R) ]^{ea} \nonumber\\
&=& y^{ea} - [\Scurl(\del U) + U\times R]^{ea} + [\Scurl(U\otimes U)+U\times\del U]^{ea}\ .
\end{eqnarray}
The first pair of terms in square brackets is identically zero for a gradient $U_a=\del_a U$ by the relevant Ricci identity,
while the second is identically zero for a gradient just by symmetry of the second derivatives alone, leading
to the conformal invariance of the Cotton-York contravariant tensor-density $Y^{ea}=g{}^{5/6} y^{ea}$ (see exercise 21.22 of Misner, Thorne and Wheeler \Cite{1973}). It must therefore be zero for any metric which is conformally flat.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{$n=3$ Structure Functions
and Orthonormal Frame Connection Components}
In any frame in $n=3$ dimensions, the structure constant functions are
antisymmetric in the lower indices, so a 2-index object can be defined
by taking the spatial dual on those indices, then decomposing the result
into its symmetric and antisymmetric parts, the latter of which can
be expressed as the dual of a 1-index object
\beq\eqalign{
C^{\alpha\delta} &= C^\alpha{}_{\beta\gamma} \eta^{\delta\beta\gamma} \cr
&= C^{(\alpha\delta)}+ C^{[\alpha\delta]} \cr
&= m^{(\alpha\delta)}+ \eta^{\alpha\delta\gamma}a_\gamma\ , \cr
C^\alpha{}_{\beta\gamma} &= \eta_{\beta\gamma\delta} m^{\beta\delta}
+ a_\delta \delta^{\delta\alpha}_{\beta\gamma}\ ,\cr
}\eeq
where
\beq
m^{\alpha\delta} = C^{(\alpha}{}_{\beta\gamma} \eta^{\delta)\beta\gamma}
\ ,\qquad a_\gamma = \half C^\alpha{}_{\gamma\alpha} \ .
\eeq
The mixed object $C^\alpha{}_\beta$ may be interpreted as a ``vector"-valued
1-form in the sense of an $R^3$-valued 1-form. This notation was introduced by
Ellis and MacCallum (1969) after the decomposition was described by
Behr (1968, 2002).
In any frame for which the metric components are constants
(an orthonormal frame in general or an invariant metric on a Lie
group, for example), the antisymmetry of the matrix-valued
connection 1-form
\beq
0= \omega_{(\alpha\beta)}
= \Gamma_{(\alpha|\gamma|\beta)} \omega
\eeq
means that the connection components are antisymmetric in their outer indices,
so one may take their spatial dual to define a matrix-valued scalar which
may then be interpreted as a ``vector"-valued 1-form
\beq
\Gamma^\beta{}_\delta
= \Gamma^{\alpha\beta\gamma}\eta_{\alpha\gamma\delta}\ ,
\eeq
which in turn can be decomposed into its symmetric and antisymmetric parts
and pure trace and tracefree parts
\beq
\Gamma^{\beta\delta}
= \Gamma^{(\beta\delta)} + \Gamma^{[\beta\delta]}\ .
\eeq
The relationship between these two ``vector"-valued 1-forms in an
orthonormal frame follows from the specialization
$\partial_\alpha g_{\beta\gamma}=0$
of the general formula for the metric connection to the following result
\beq
\Gamma^\alpha{}_{\beta\gamma}
= \half ( C^\alpha{}_{\beta\gamma} + C_\beta{}^\alpha{}_\gamma
+ C_\gamma{}^\alpha{}_\beta )
= \half C^\alpha{}_{\beta\gamma} + C_{(\beta}{}^\alpha{}_{\gamma)}
\equiv \half C^\alpha{}_{\beta\gamma} + K^\alpha{}_{\beta\gamma}
\ ,
\eeq
where
\beq
K_{\alpha\beta\gamma} = -\half \Lie_{e_\alpha} g_{\beta\gamma}\ ,
\eeq
namely
\beq\eqalign{
\Gamma^\beta{}_\delta &=
C^\beta{}_\delta - \half C^\gamma{}_\gamma \delta^\beta{}_\delta \cr
&= (m^\beta{}_\delta -\half m^\gamma{}_\gamma \delta^\beta{}_\delta)
+ \eta^\beta{}_\delta{}^\gamma a_\gamma\ . \cr
}\eeq
The double dual of the Riemann tensor yields a symmetric tensor, namely
the sign-reversed Einstein tensor
\beq\eqalign{
[\dual{} R \dual{}]^\alpha{}_\beta
&= {\textstyle\frac14}\eta^{\alpha\gamma\delta}
R_{\gamma\delta}{}^{\mu\nu} \eta_{\mu\nu\beta}
= {\textstyle\frac14}\delta^{\alpha\gamma\delta}_{\beta\mu\nu}
R_{\gamma\delta}{}^{\mu\nu}
= -G^\alpha{}_\beta \ ,\cr
R^{\alpha\beta}{}_{\gamma\delta}
&= -\eta^{\alpha\beta\mu} \eta_{\gamma\delta\nu} G^\nu{}_\mu\ .
}\eeq
For a metric with constant frame components, the Einstein tensor
then reduces to
\beq\eqalign{
{\bf G} &= (G^\alpha{}_\beta)
= (-\eta_\beta{}^{\gamma\delta} \partial_\gamma \Gamma^\alpha{}_\delta
+ \Gamma^\alpha{}_\epsilon C^\epsilon{}_\beta
- \half \eta^\alpha{}_{\mu\nu} \Gamma^\mu{}_\gamma
\Gamma^\nu{}_\delta \eta^{\gamma\delta}{}_\beta )\cr
&= 2 {\bf m}^2
- {\bf m} \Tr {\bf m}-\half(\Tr {\bf m}^2 -\half \Tr^2 {\bf m}- 6 a_f a^f)
{\bf I}
-2 {\bf a} {\bf a}^T - 2 a^f {\bf K}_f \ ,
\cr
}\eeq
using a more efficient matrix notation
${\bf m} = (m^\alpha{}_\beta)$, ${\bf a} = (a^\alpha)$, ${\bf a}^T = (a_\alpha)$,
${\bf K}_\alpha = (K_\alpha{}^\beta{}_\gamma)$, ${\bf I}=(\delta^\alpha{}_\beta)$.
Similarly
\beq\eqalign{
{\bf R} &= (R^\alpha{}_\beta)
= (-\eta_\beta{}^{\gamma\delta} \partial_\gamma \Gamma^\alpha{}_\delta
+ \Gamma^\alpha{}_\epsilon C^\epsilon{}_\beta
- \half \eta^\alpha{}_{\mu\nu} \Gamma^\mu{}_\gamma
\Gamma^\nu{}_\delta \eta^{\gamma\delta}{}_\beta )\cr
&= 2 {\bf m}^2
- {\bf m} \Tr {\bf m}-(\Tr {\bf m}^2 -\half \Tr^2 {\bf m})
{\bf I}
-2 {\bf a} {\bf a}^T - 2 a^f {\bf K}_f \ ,
\cr
\Tr {\bf R} &= -(\Tr {\bf m}^2 -\half \Tr^2 {\bf m})- 6 a_f a^f\ .
\cr
}\eeq
%TO FINISH\typeout{TO FINISH!!}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Lie derivative}
The Lie derivative along a vector field $X$ is defined so that it is the
ordinary derivative along $X$ of a function $f$ and the Lie bracket
by $X$ of another vector field $Y$,
\beq
\Lie\sub{X} f = X f \ ,\qquad
\Lie\sub{X} Y = [ X,Y ] \ ,
\eeq
and it is then extended to all other
tensor fields so that it obeys the obvious product rule for the evaluation
of a tensor field on all of its arguments (reducing it to a function).
Solving this product rule for the Lie derivative of the tensor alone yields
\beq\eqalign{
[\Lie\sub{X} S ]
(\theta\subp1, \ldots, \theta\subp{p}, & X\supp1, \ldots X\supp{q}) \cr
&= \Lie\sub{X} [ S
(\theta\subp1, \ldots, \theta\subp{p}, X\supp1, \ldots X\supp{q}) ] \cr
& \qquad
- S(\Lie\sub{X} \theta\subp1, \ldots, \theta\subp{p},
X\supp1, \ldots X\supp{q}) - \ldots \cr
& \qquad
- S(\theta\subp1, \ldots, \theta\subp{p},
\Lie\sub{X} X\supp1, \ldots X\supp{q}) - \ldots \ .\cr
}\eeq
The Lie derivative is a linear operator which also acts on differential
forms (sends $p$-forms into $p$-forms) and obeys the usual product rules
with respect to the tensor and exterior products (no extra signs).
This last equation immediately leads to a formula for the components
of the Lie derivative of a $p\choose q$-tensor field in a frame by evaluating it on
the frame vectors and 1-forms
\beq\eqalign{
\Lie\sub{X} S ^{\alpha\ldots}_{\ \ \beta\ldots}
&\equiv [ \Lie\sub{X} S ]^{\alpha\ldots}_{\ \ \beta\ldots} \cr
&= S^{\alpha\ldots}_{\ \ \beta\ldots , \gamma} X^\gamma
+ [\sigma^{p,q}( \bfL(X,e) ) S ] ^{\alpha\ldots}_{\ \ \beta\ldots} \ ,\cr
}\eeq
where the component matrix which is the argument of $\sigma^{p,q}$ is
given by
\beq\eqalign{
\bfL(X,e)^\alpha{}_\beta &= [ \Lie\sub{X} e_\beta]^\alpha =
- [ \Lie\sub{X} \omega^\alpha]_\beta \cr
&= -X^\alpha{}_{,\beta} + C^\alpha{}_{\gamma\beta} X^\gamma \cr
&= -X^\alpha{}_{;\beta}
+ \tilde\Gamma{}^\alpha{}_{\gamma\beta} X^\gamma \ , \cr
}\eeq
and $\sigma^{p,q}$ is the associated representation $\sigma^{p,q}$ of the Lie algebra
of linear transformations $gl(n,R)$ introduced in section \ref{framechange}.
The last equality holds when a linear connection $\del$ is available
and enables the Lie derivative to be expressed in terms of the covariant
derivative $\del X$ and the transposed covariant derivative $\tilde\del S$
\beq
\Lie\sub{X} S ^{\alpha\ldots}_{\ \ \beta\ldots}
= \tilde\del\sub{X} S ^{\alpha\ldots}_{\ \ \beta\ldots}
- [\sigma^{p,q}( \del X ) S ] ^{\alpha\ldots}_{\ \ \beta\ldots} \ .
\eeq
For a symmetric connection, this reduces to
\beq
\Lie\sub{X} S ^{\alpha\ldots}_{\ \ \beta\ldots}
= S ^{\alpha\ldots}_{\ \ \beta\ldots;\gamma} X^\gamma
- [\sigma^{p,q}( \del X ) S ] ^{\alpha\ldots}_{\ \ \beta\ldots} \ ,
\eeq
which is the formula which results
from the substitution of a semicolon for the comma
in the formula for the coordinate frame components of the Lie derivative.
For a differential form $S$, the Lie derivative can be expressed in terms
of the exterior derivative and the contraction operation
\beq
\Lie\sub{X} S = X \leftcontract d S + d ( X \leftcontract S ) \ .
\eeq
The geometric significance of the Lie derivative comes from its relation
to the generator of a 1-parameter family of diffeomorphisms.
If $X_\lambda$ denotes the flow of a vector field, then each tensor field $S$
may be dragged along by the flow for each value of $\lambda$ to define
a 1-parameter family of tensor fields, indicated by the abbreviation
\beq
S_\lambda = X_\lambda S \ , \qquad S_0 = S \ .
\eeq
Then the parameter derivative of this family at $\lambda=0$ defines the Lie
derivative reversed in sign
\beq
d/ d\lambda \, S_\lambda |_{\lambda=0} = - \Lie\sub{X} S\ .
\eeq
For a $p\choose q$-tensor density field of weight $W$ the formula for the components
of the Lie derivative has one extra term compared to a $p\choose q$-tensor field
\beq\eqalign{
\Lie\sub{X} S ^{\alpha\ldots}_{\ \ \beta\ldots}
&\equiv [ \Lie\sub{X} S ]^{\alpha\ldots}_{\ \ \beta\ldots}
\cr
&= S^{\alpha\ldots}_{\ \ \beta\ldots , \gamma} X^\gamma
+ [\sigma^{p,q}_W( \bfL(X,e) ) S ] ^{\alpha\ldots}_{\ \ \beta\ldots} \cr
&= S^{\alpha\ldots}_{\ \ \beta\ldots , \gamma} X^\gamma
+ [\sigma^{p,q}( \bfL(X,e) ) S ] ^{\alpha\ldots}_{\ \ \beta\ldots}
-W \bfL(X,e)^\gamma{}_\gamma S^{\alpha\ldots}_{\ \ \beta\ldots}
\ .\cr
}\eeq