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If we define our covariant derivative such that it eliminates the metric:
$0 = \nabla_c (g_{ab} e^a \otimes e^b)$
$0 = e_c (g_{ab}) e^a \otimes e^b
+ g_{ab} \nabla_c e^a \otimes e^b
+ g_{ab} e^a \otimes \nabla_c e^b
$
$0 = e_c (g_{ab}) e^a \otimes e^b
- g_{ab} {\Gamma^a}_{cu} e^u \otimes e^b
- g_{ab} e^a \otimes {\Gamma^b}_{cu} e^u
$
$e_c (g_{ab}) e^a \otimes e^b = (\Gamma_{bca} + \Gamma_{acb}) e^a \otimes e^b$
$e_c (g_{ab}) = \Gamma_{bca} + \Gamma_{acb}$
Due to this constraint, we can say that all metric-eliminating connections are symmetric possess this identity:
$\Gamma_{(a|b|c)} = \frac{1}{2} e_b (g_{ac})$
From here we can define our covariant derivative in terms of structure constants and torsion:
$e_a (g_{bc}) = \Gamma_{cab} + \Gamma_{bac}$
$e_b (g_{ac}) = \Gamma_{abc} + \Gamma_{cba}$
$e_c (g_{ab}) = \Gamma_{bca} + \Gamma_{acb}$
...subtract...
$e_c(g_{ab}) + e_b(g_{ac}) - e_a(g_{bc})
= \Gamma_{bca} + \Gamma_{acb} + \Gamma_{abc} + \Gamma_{cba} - \Gamma_{cab} - \Gamma_{bac}$
$e_c(g_{ab}) + e_b(g_{ac}) - e_a(g_{bc}) = 2 \Gamma_{abc}
- c_{acb} - T_{bac}
- c_{abc} - T_{cab}
- c_{bca} - T_{abc}$
$\Gamma_{abc} = \frac{1}{2} (
e_c(g_{ab}) + e_b(g_{ac}) - e_a(g_{bc})
+ c_{abc} + c_{acb} - c_{cba}
+ T_{cab} + T_{bac} - T_{acb}
)$
Notice that $\nabla$ can be defined to have any ${\Gamma^a}_{bc}$, however there is only one ${\hat\Gamma^a}_{bc}$ such that the torsion is zero, and that is the Levi-Civita connection.
Let $\hat{\nabla}$ be the covariant derivative associated with the zero-torsion connection $\hat{\Gamma}$.
$\hat\Gamma_{abc} = \frac{1}{2} (
e_c (g_{ab}) + e_b (g_{ac}) - e_a (g_{bc})
+ c_{abc} + c_{acb} - c_{cba}
)$
This is also known as the "Christoffel symbol of the 1st kind".
Misner Thorne Wheeler "Gravitation" list this as $\hat\Gamma_{abc} = \frac{1}{2} (
e_c (g_{ab}) + e_b (g_{ac}) - e_a (g_{bc})
+ c_{abc} + c_{acb} - c_{bca})$,
so notice the last commutation coefficient is negative'd. This coincides with its reversed definition of $\nabla_b e_c = {\Gamma^a}_{cb} e_a$, which would make sense. If you reverse b and c then the one term that is negative'd is the last commutation term.
However, in contrast, Wikipedia here holds my un-reversed definition of ${\Gamma^a}_{bc}$ but still lists the same anholonomic definition. Maybe the authors of the different sections of the page used different conventions?
Christoffel of the 2nd kind / affine connection: $ {\hat\Gamma^a}_{bc} = g^{ad} \hat\Gamma_{dbc} $
in a holonomic basis (so ${c_{ab}}^c = 0$):
$ \hat\Gamma_{abc} = \frac{1}{2} (
e_c (g_{ab}) + e_b (g_{ac}) - e_a (g_{bc})
) $
Levi-Civita connection in a non-coordinate basis:
If $e_a = {e_a}^{\tilde{a}} \partial_\tilde{a}$ and $e^a = {e^a}_{\tilde{a}} dx^\tilde{a}$ are linear combinations of a coordinate basis $\partial_\tilde{a}$ and dual $dx^\tilde{a}$ then we can calculate the following:
Using the identities that $\delta^a_b = {e^a}_\tilde{u} {e_b}^\tilde{u}$ and $\delta^\tilde{a}_\tilde{b} = {e_u}^\tilde{a} {e^u}_\tilde{b}$
Let $\tilde{g}_{\tilde{a}\tilde{b}} = \partial_\tilde{a} \cdot \partial_\tilde{b}$
Let $g_{ab} = e_a \cdot e_b = {e_a}^\tilde{a} \partial_\tilde{a} \cdot {e_b}^\tilde{b} \partial_\tilde{b} = {e_a}^\tilde{a} \cdot {e_b}^\tilde{b} \tilde{g}_{\tilde{a}\tilde{b}}$.
Let ${c_{ab}}^c e_c = [e_a, e_b]$, so (from the 'structure constants' worksheet on structure constants of a linear transform of a coordinate basis) ${c_{ab}}^c = 2 e_{[a} ({e_{b]}}^\tilde{c}) {e^c}_\tilde{c}$
$\hat\Gamma_{abc} = \frac{1}{2} (
e_c (g_{ab})
+ e_b (g_{ac})
- e_a (g_{bc})
+ c_{acb}
+ c_{abc}
- c_{cba}
)$
$= \frac{1}{2} (
e_c ({e_a}^\tilde{a} {e_b}^\tilde{b} \tilde{g}_{\tilde{a}\tilde{b}})
+ e_b ({e_a}^\tilde{a} {e_c}^\tilde{c} \tilde{g}_{\tilde{a}\tilde{c}})
- e_a ({e_b}^\tilde{b} {e_c}^\tilde{c} \tilde{g}_{\tilde{b}\tilde{c}})
+ 2 e_{[a} ({e_{c]}}^\tilde{d}) {e^d}_\tilde{d} g_{db}
+ 2 e_{[a} ({e_{b]}}^\tilde{d}) {e^d}_\tilde{d} g_{dc}
- 2 e_{[c} ({e_{b]}}^\tilde{d}) {e^d}_\tilde{d} g_{da}
)$
$= \frac{1}{2} (
e_c ({e_a}^\tilde{a}) {e_b}^\tilde{b} \tilde{g}_{\tilde{a}\tilde{b}}
+ {e_a}^\tilde{a} e_c ({e_b}^\tilde{b}) \tilde{g}_{\tilde{a}\tilde{b}}
+ {e_a}^\tilde{a} {e_b}^\tilde{b} e_c (\tilde{g}_{\tilde{a}\tilde{b}})
+ e_b ({e_a}^\tilde{a}) {e_c}^\tilde{c} \tilde{g}_{\tilde{a}\tilde{c}}
+ {e_a}^\tilde{a} e_b ({e_c}^\tilde{c}) \tilde{g}_{\tilde{a}\tilde{c}}
+ {e_a}^\tilde{a} {e_c}^\tilde{c} e_b (\tilde{g}_{\tilde{a}\tilde{c}})
- e_a ({e_b}^\tilde{b}) {e_c}^\tilde{c} \tilde{g}_{\tilde{b}\tilde{c}}
- {e_b}^\tilde{b} e_a ({e_c}^\tilde{c}) \tilde{g}_{\tilde{b}\tilde{c}}
- {e_b}^\tilde{b} {e_c}^\tilde{c} e_a (\tilde{g}_{\tilde{b}\tilde{c}})
+ e_a ({e_c}^\tilde{d}) e_{b\tilde{d}}
- e_c ({e_a}^\tilde{d}) e_{b\tilde{d}}
+ e_a ({e_b}^\tilde{d}) e_{c\tilde{d}}
- e_b ({e_a}^\tilde{d}) e_{c\tilde{d}}
- e_c ({e_b}^\tilde{d}) e_{a\tilde{d}}
+ e_b ({e_c}^\tilde{d}) e_{a\tilde{d}}
)$
$= \frac{1}{2} (
e_c ({e_a}^\tilde{a}) e_{b\tilde{a}}
+ e_c ({e_b}^\tilde{b}) e_{a\tilde{b}}
+ {e_a}^\tilde{a} {e_b}^\tilde{b} e_c (\tilde{g}_{\tilde{a}\tilde{b}})
+ e_b ({e_a}^\tilde{a}) e_{c\tilde{a}}
+ e_b ({e_c}^\tilde{c}) e_{a\tilde{c}}
+ {e_a}^\tilde{a} {e_c}^\tilde{c} e_b (\tilde{g}_{\tilde{a}\tilde{c}})
- e_a ({e_b}^\tilde{b}) e_{c\tilde{b}}
- e_a ({e_c}^\tilde{c}) e_{b\tilde{c}}
- {e_b}^\tilde{b} {e_c}^\tilde{c} e_a (\tilde{g}_{\tilde{b}\tilde{c}})
+ e_a ({e_c}^\tilde{d}) e_{b\tilde{d}}
- e_c ({e_a}^\tilde{d}) e_{b\tilde{d}}
+ e_a ({e_b}^\tilde{d}) e_{c\tilde{d}}
- e_b ({e_a}^\tilde{d}) e_{c\tilde{d}}
- e_c ({e_b}^\tilde{d}) e_{a\tilde{d}}
+ e_b ({e_c}^\tilde{d}) e_{a\tilde{d}}
)$
$= \frac{1}{2} (
{e_a}^\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} \partial_\tilde{c} (\tilde{g}_{\tilde{a}\tilde{b}})
+ {e_a}^\tilde{a} {e_c}^\tilde{c} {e_b}^\tilde{b} \partial_\tilde{b} (\tilde{g}_{\tilde{a}\tilde{c}})
- {e_b}^\tilde{b} {e_c}^\tilde{c} {e_a}^\tilde{a} \partial_\tilde{a} (\tilde{g}_{\tilde{b}\tilde{c}})
)
+ e_b ({e_c}^\tilde{c}) e_{a\tilde{c}}
$
$\hat\Gamma_{abc} = {e_a}^\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} \tilde{\Gamma}_{\tilde{a}\tilde{b}\tilde{c}} + e_{a\tilde{a}} e_b ({e_c}^\tilde{a})$
${\hat\Gamma^a}_{bc} = {e^a}_\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} {\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}} + {e^a}_\tilde{a} e_b ({e_c}^\tilde{a})$
...where ${\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}}$ is the Levi-Civita connection associated with the coordinate basis $\partial_\tilde{a}$.
From there, of course,
${\hat\Gamma^a}_{[bc]} = {e^a}_\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} {\tilde{\Gamma}^\tilde{a}}_{[\tilde{b}\tilde{c}]} + {e^a}_\tilde{a} e_{[b} ({e_{c]}}^\tilde{a})$
and a coordinate basis has no commutation, so ${\tilde{\Gamma}^\tilde{a}}_{[\tilde{b}\tilde{c}]} = 0$
so ${\hat\Gamma^a}_{[bc]} = {e^a}_\tilde{a} e_{[b} ({e_{c]}}^\tilde{a}) = \frac{1}{2} {c_{bc}}^a$
If we use un-tilde-d indexes to denote transformation by the ${e_a}^\tilde{a}$ basis from the coordinate (tilde) to non-coordinate (non-tilde) basis,
then we can represent the transformed coordinate basis Levi-Civita connections as then we can represent this as ${\tilde{\Gamma}^a}_{bc} = {e^a}_\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} {\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}}$,
and the identity becomes:
${\hat\Gamma^a}_{bc} = {\tilde{\Gamma}^a}_{bc} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a})$
Notice how this matches up with the covariant derivative definition: ${\Gamma^a}_{bc} = e^a (\nabla_b e_c)$
If we want to transforms all this to the coordinate basis:
${e_a}^\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{c} {\hat\Gamma^a}_{bc} =
{\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}}
+ {e^c}_\tilde{c} {e^b}_\tilde{b} e_b ({e_c}^\tilde{a}) $
${e_a}^\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{c} {\hat\Gamma^a}_{bc} =
{\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}}
+ {e^c}_\tilde{c} \partial_\tilde{b} ({e_c}^\tilde{a}) $
What does the non-coordinate metric-cancelling torsion-free covariant derivative of a object look like when represented with coordinate-basis indexes?
Let $\tilde\nabla$ be the coordinate-metric-cancelling torsion-free covariant-derivative (for coordinate metric $\tilde{g}_{\tilde{a}\tilde{b}}$).
Let $\hat\nabla$ be the non-coordinate-metric-cancelling torsion-free covariant-derivative (for non-coordinate metric $g_{ij}$).
From the worksheet on "covariant derivative":
$\tilde{\nabla}_\tilde{u} ( {T^\tilde{A}}_{\tilde{B}} \partial_\tilde{A} \otimes dx^\tilde{B} ) =
(
\partial_\tilde{u} ({T^\tilde{A}}_\tilde{B})
+ \underset{a_i \leftrightarrow c}{{T^\tilde{A}}_\tilde{B}} {\tilde{\Gamma}^{a_i}}_{uc}
- \underset{b_j \leftrightarrow c}{{T^A}_\tilde{B}} {\tilde{\Gamma}^c}_{u b_j}
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
So in a non-coordinate basis for a non-coordinate-metric-cancelling torsion-free covariant-derivative:
$\hat{\nabla}_u ( {T^A}_{B} e_A \otimes e^B ) =
(
e_u ({T^A}_B)
+ \underset{a_i \leftrightarrow c}{{T^A}_B} {\hat{\Gamma}^{a_i}}_{uc}
- \underset{b_j \leftrightarrow c}{{T^A}_B} {\hat{\Gamma}^c}_{u b_j}
) e_A \otimes e^B
$
... using ${\hat\Gamma^a}_{bc} = {\tilde{\Gamma}^a}_{bc} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a})$ (where tilde indexes are coordinate-basis and non-tilde are non-coordinate-basis)
$\hat{\nabla}_u ( {T^A}_{B} e_A \otimes e^B ) =
(
e_u ({T^A}_B)
+ \underset{a_i \leftrightarrow c}{{T^A}_B} (
{\tilde{\Gamma}^{a_i}}_{uc} + {e^{a_i}}_{\tilde{a}_i} e_u({e_c}^{\tilde{a}_i})
)
- \underset{b_j \leftrightarrow c}{{T^A}_B} (
{\tilde{\Gamma}^c}_{u b_j} + {e^c}_\tilde{c} e_u({e_{b_j}}^\tilde{c})
)
) e_A \otimes e^B
$
...transform all indexes outside of derivatives from non-coordinate-basis to coordinate-basis:
$=
{e_u}^\tilde{u} (
\partial_\tilde{u} ({T^A}_B)
\cdot {e_{a_i}}^{\tilde{a}_i}
\cdot ...
\cdot {e^{b_i}}_{\tilde{b}_i}
\cdot ...
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}} + \partial_\tilde{u}({e_c}^{\tilde{a}_i}) {e^{c}}_\tilde{c}
)
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j} + \partial_\tilde{u}({e_c}^\tilde{c}) {e^c}_{\tilde{b}_j}
)
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
...transform indexes within partial derivative:
$=
{e_u}^\tilde{u} (
\partial_\tilde{u} (
{T^\tilde{C}}_\tilde{D}
\cdot {e^{a_i}}_{\tilde{c}_i}
\cdot ...
\cdot {e_{b_i}}^{\tilde{c}_i}
\cdot ...
)
\cdot {e_{a_i}}^{\tilde{a}_i}
\cdot ...
\cdot {e^{b_i}}_{\tilde{b}_i}
\cdot ...
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}} + \partial_\tilde{u}({e_c}^{\tilde{a}_i}) {e^{c}}_\tilde{c}
)
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j} + \partial_\tilde{u}({e_c}^\tilde{c}) {e^c}_{\tilde{b}_j}
)
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
...distribute partial through basis transforms of indexes of T:
$=
{e_u}^\tilde{u} (
(
\partial_\tilde{u} ({T^\tilde{C}}_\tilde{D})
\cdot {e^{a_i}}_{\tilde{c}_i}
\cdot ...
\cdot {e_{b_i}}^{\tilde{c}_i}
\cdot ...
+
{T^\tilde{C}}_\tilde{D}
\cdot ...
\cdot \partial_\tilde{u} {e^{a_i}}_{\tilde{c}_i}
\cdot ...
\cdot {e_{b_i}}^{\tilde{c}_i}
\cdot ...
+
{T^\tilde{C}}_\tilde{D}
\cdot {e^{a_i}}_{\tilde{c}_i}
\cdot ...
\cdot \partial_\tilde{u} {e_{b_i}}^{\tilde{c}_i}
\cdot ...
)
\cdot {e_{a_i}}^{\tilde{a}_i}
\cdot ...
\cdot {e^{b_i}}_{\tilde{b}_i}
\cdot ...
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}} + \partial_\tilde{u}({e_c}^{\tilde{a}_i}) {e^{c}}_\tilde{c}
)
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j} + \partial_\tilde{u}({e_c}^\tilde{c}) {e^c}_{\tilde{b}_j}
)
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
$=
{e_u}^\tilde{u} (
\partial_\tilde{u} ({T^\tilde{A}}_\tilde{B})
+ \underset{ \tilde{a}_i \leftrightarrow \tilde{c} }{ {T^\tilde{A}}_\tilde{B} }
\cdot \partial_\tilde{u} ({e^{a_i}}_{\tilde{c}}) {e_{a_i}}^{\tilde{a}_i}
+ \underset{ \tilde{b}_i \leftrightarrow \tilde{c} }{ {T^\tilde{A}}_\tilde{B} }
\cdot \partial_\tilde{u} ({e_{b_i}}^{\tilde{c}}) {e^{b_i}}_{\tilde{b}_i}
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}} + \partial_\tilde{u}({e_c}^{\tilde{a}_i}) {e^{c}}_\tilde{c}
)
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}} (
{\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j} + \partial_\tilde{u}({e_c}^\tilde{c}) {e^c}_{\tilde{b}_j}
)
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
...rearrange...
$=
{e_u}^\tilde{u} (
\partial_\tilde{u} ({T^\tilde{A}}_\tilde{B})
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot {\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}}
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot {\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j}
+ \underset{ \tilde{a}_i \leftrightarrow \tilde{c} }{ {T^\tilde{A}}_\tilde{B} }
\cdot \partial_\tilde{u} ({e^{a_i}}_{\tilde{c}}) {e_{a_i}}^{\tilde{a}_i}
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot \partial_\tilde{u}({e_c}^{\tilde{a}_i}) {e^{c}}_\tilde{c}
+ \underset{ \tilde{b}_i \leftrightarrow \tilde{c} }{ {T^\tilde{A}}_\tilde{B} }
\cdot \partial_\tilde{u} ({e_{b_i}}^{\tilde{c}}) {e^{b_i}}_{\tilde{b}_i}
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot \partial_\tilde{u}({e_c}^\tilde{c}) {e^c}_{\tilde{b}_j}
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
...using e times e-inverse equals identity, therefore $\partial_\tilde{u} ({e^{a_i}}_\tilde{c}) {e_{a_i}}^{\tilde{a}_i}
= -{e^{a_i}}_\tilde{c} \partial_\tilde{u} ({e_{a_i}}^{\tilde{a}_i})
$
$=
{e_u}^\tilde{u} (
\partial_\tilde{u} ({T^\tilde{A}}_\tilde{B})
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot {\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}}
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot {\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j}
- \underset{ \tilde{a}_i \leftrightarrow \tilde{c} }{ {T^\tilde{A}}_\tilde{B} }
\cdot \partial_\tilde{u}({e_{a_i}}^{\tilde{a}_i}) {e^{a_i}}_{\tilde{c}}
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot \partial_\tilde{u}({e_c}^{\tilde{a}_i}) {e^{c}}_\tilde{c}
+ \underset{ \tilde{b}_i \leftrightarrow \tilde{c} }{ {T^\tilde{A}}_\tilde{B} }
\cdot \partial_\tilde{u}({e_{b_i}}^{\tilde{c}}) {e^{b_i}}_{\tilde{b}_i}
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot \partial_\tilde{u}({e_c}^\tilde{c}) {e^c}_{\tilde{b}_j}
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
...relabel sum indexes and cancel like terms:
$=
{e_u}^\tilde{u} (
\partial_\tilde{u} ({T^\tilde{A}}_\tilde{B})
+ \underset{\tilde{a}_i \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot {\tilde{\Gamma}^{\tilde{a}_i}}_{\tilde{u} \tilde{c}}
- \underset{\tilde{b}_j \leftrightarrow \tilde{c}}{{T^\tilde{A}}_\tilde{B}}
\cdot {\tilde{\Gamma}^\tilde{c}}_{\tilde{u} \tilde{b}_j}
) \partial_\tilde{A} \otimes dx^\tilde{B}
$
$= {e_u}^\tilde{u} \tilde{\nabla}_\tilde{u} ({T^\tilde{A}}_\tilde{B} \partial_\tilde{A} \otimes dx^\tilde{B})$
Looks like the metric-cancelling torsion-free covariant-derivative of one basis is just the transform of the metric-cancelling torsion-free covariant-derivative of another.
Hence the covariant derivative of a tensor is a tensor.
Let the contorsion tensor be the difference between an arbitrary connection and the Levi-Civita connection:
${K^a}_{bc} = {\Gamma^a}_{bc} - {\hat\Gamma^a}_{bc}$
$= \frac{1}{2} ({{T_c}^a}_b + {{T_b}^a}_c + {T^a}_{bc})$
This shows that any connection can be uniquely defined by its contorsion, which is defined by its torsion.
Now let's revisit the definition of the non-coordinate metric-cancelling torsion-free connection with regards to the coordinate:
${\hat\Gamma^a}_{bc} = {\tilde{\Gamma}^a}_{bc} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a})$
...substitute ${\hat\Gamma^a}_{bc} = {\Gamma^a}_{bc} - {K^a}_{bc}$
${\Gamma^a}_{bc} - {K^a}_{bc} = {\tilde{\Gamma}^a}_{bc} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a})$
...and solve:
${\Gamma^a}_{bc} = {\tilde{\Gamma}^a}_{bc} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a}) + {K^a}_{bc} $
So any metric-cancelling connection can be represented as the sum of the coordinate-basis torsion-free connection, plus the partial of the vielbein, plus the contorsion.
And if there is one unique Levi-Civita torsion-free metric-cancelling connection per metric, and there is one unique linear transform from coordinate to non-coordinate basis per diagonalized metric,
then there is only one unique torsion-free metric-cancelling connection per diagonalized metric (or per any non-coordinate basis?).
TODO but there isn't one unique transform for diagonalization.
Antisymmetry of contorsion tensor:
${K^a}_{[bc]} = \frac{1}{2} ({{T_{[c}}^a}_{b]} + {{T_{[b}}^a}_{c]} + {T^a}_{[bc]}) = \frac{1}{2} {T^a}_{[bc]} = \frac{1}{2} {T^a}_{bc}$
Trace of contorsion tensor:
${K^u}_{au} = \frac{1}{2} ({{T_u}^u}_a + {{T_a}^u}_u + {T^u}_{au})$
...cancelling trace of the antisymmetric components of the torsion...
${K^u}_{au} = \frac{1}{2} ({T^u}_{ua} + {T^u}_{au})$
...applying the antisymmetry of the torsion...
${K^u}_{au} = 0$
Jacobi formula applied to metric tensor (more on worksheet #13):
$e_a (g) = g g^{uv} e_a (g_{uv})$
$\frac{1}{2 g} e_a (g) = \frac{1}{2} g^{uv} e_a (g_{uv})$
$\frac{1}{2} e_a ( log |g| ) = \frac{1}{2} g^{uv} e_a (g_{uv})$
$e_a ( log \sqrt{|g|} ) = \frac{1}{2} g^{uv} e_a ( g_{uv} )$
...in a non-coordinate basis that is a linear map of a coordinate basis...
$e_a ( log \sqrt{|g|} ) = \frac{1}{2} g^{uv} e_a ( g_{uv} )$
Jacobi formula of a coordinate metric in a coordinate basis:
Remember, $g_{uv} = e_u \cdot e_v$ is a non-coordinate metric expressed in the non-coordinate basis and $\tilde{g}_{\tilde{u}\tilde{v}} = \partial_\tilde{u} \cdot \partial_\tilde{v}$ is the coordinate metric expressed in the coordinate basis.
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) = \frac{1}{2} \tilde{g}^{\tilde{u}\tilde{v}} \partial_\tilde{a} ( \tilde{g}_{\tilde{u}\tilde{v}} )$
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) = \frac{1}{2} g^{uv} {e_u}^\tilde{u} {e_v}^\tilde{v} \partial_\tilde{a} ( g_{cd} {e^c}_\tilde{u} {e^d}_\tilde{v} )$
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) = \frac{1}{2} g^{uv} {e_u}^\tilde{u} {e_v}^\tilde{v} (
\partial_\tilde{a} (g_{cd}) {e^c}_\tilde{u} {e^d}_\tilde{v}
+ g_{cd} \partial_\tilde{a} ({e^c}_\tilde{u}) {e^d}_\tilde{v}
+ g_{cd} {e^c}_\tilde{u} \partial_\tilde{a} ({e^d}_\tilde{v})
)$
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) = \frac{1}{2}
g^{uv} \partial_\tilde{a} (g_{uv})
+ {e_v}^\tilde{u} \partial_\tilde{a} ({e^v}_\tilde{u})
$
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) = \frac{1}{2}
g^{uv} \partial_\tilde{a} (g_{uv})
- {e^v}_\tilde{u} \partial_\tilde{a} ({e_v}^\tilde{u})
$
Going further:
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) =
\frac{1}{2} g^{uv} \partial_\tilde{a} (g_{uv})
+ {e_v}^\tilde{u} \partial_\tilde{a} ({e^v}_\tilde{u})
$
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) =
\frac{1}{2 g} \partial_\tilde{a} g
+ \frac{1}{e} \partial_\tilde{a} e
$
$\partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) =
\frac{1}{2} \partial_\tilde{a} (log(\sqrt{|g|}))
+ \partial_\tilde{a} (log(e))
$
transform both sides:
${e_a}^\tilde{a} \partial_\tilde{a} ( log \sqrt{|\tilde{g}|} ) =
{e_a}^\tilde{a} \partial_\tilde{a} (log(\sqrt{|g|}))
+ {e_a}^\tilde{a} \partial_\tilde{a} (log(e))
$
$e_a ( log \sqrt{|\tilde{g}|} ) =
e_a (log(\sqrt{|g|}))
+ e_a (log(e))
$
$e_a ( log \sqrt{|\tilde{g}|} ) =
{\hat{\Gamma}^b}_{ab}
+ e_a (log(e))
$
Trace of Levi-Civita connection on 1 and 3:
${\hat\Gamma^u}_{au} = \frac{1}{2} g^{uv} (e_u (g_{va}) + e_a (g_{vu}) - e_v (g_{au}) + c_{vau} + c_{vua} - c_{uav})$
${\hat\Gamma^u}_{au} = \frac{1}{2} g^{uv} e_a (g_{uv})$
${\hat\Gamma^u}_{au} = e_a ( log \sqrt{|g|} )$
Mind you, even in non-coordinate orthonormal frames where $g = $ constant, $log(\sqrt{|g|}) = $ constant, and $e(log(\sqrt{|g|})) = 0$, this identity holds.
I mean, the trace of the connection isn't necessarily the volume element. Sometimes it is just zero.
Trace of Levi-Civita connection on 1 and 2:
${\hat\Gamma^u}_{ua} = \frac{1}{2} g^{uv} (
e_a (g_{vu}) + e_u (g_{va}) - e_v (g_{ua})
+ c_{vua} + c_{vau} - c_{auv}
)$
$= \frac{1}{2} g^{uv} e_a (g_{vu}) - {c_{au}}^u$
$= e_a (log \sqrt{|g|}) + {c_{ua}}^u$
using ${\hat\Gamma^a}_{bc} = {e^a}_\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} {\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}} + {e^a}_\tilde{a} e_b ({e_c}^\tilde{a})$
and using ${\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{u}} = \frac{1}{\sqrt{|\tilde{g}|}} \partial_\tilde{a} (\sqrt{|\tilde{g}|})$
therefore
${\hat\Gamma^u}_{au} = {e_a}^\tilde{a} {\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{u}} + {e^u}_\tilde{u} e_a ({e_u}^\tilde{u})$
$\frac{1}{\sqrt{|g|}} e_a (\sqrt{|g|}) = {e_a}^\tilde{a} {\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{u}} - e_a ({e^u}_\tilde{u}) {e_u}^\tilde{u}$
$\frac{1}{\sqrt{|g|}} e_a (\sqrt{|g|}) = \frac{1}{\sqrt{|\tilde{g}|}} {e_a}^\tilde{a} \partial_\tilde{a} (\sqrt{|\tilde{g}|}) - \frac{1}{e} e_a (e)$
$\frac{1}{\sqrt{|g|}} e_a (\sqrt{|g|}) = \frac{1}{\sqrt{|\tilde{g}|}} e_a (\sqrt{|\tilde{g}|}) - \frac{1}{e} e_a (e)$
...and keep going to find...
$\frac{1}{\sqrt{|g|}} e_a(\sqrt{|g|}) = \frac{1}{e \sqrt{|g|}} e_a (e \sqrt{|g|}) - \frac{1}{e} e_a (e)$
$\frac{1}{\sqrt{|g|}} e_a(\sqrt{|g|}) = \frac{1}{e \sqrt{|g|}} e e_a (\sqrt{|g|}) + \frac{1}{e \sqrt{|g|}} \sqrt{|g|} e_a (e) - \frac{1}{e} e_a (e)$
$\frac{1}{\sqrt{|g|}} e_a(\sqrt{|g|}) = \frac{1}{\sqrt{g}} e_a (\sqrt{|g|}) + \frac{1}{e} e_a (e) - \frac{1}{e} e_a (e)$
$\frac{1}{\sqrt{|g|}} e_a(\sqrt{|g|}) = \frac{1}{\sqrt{g}} e_a (\sqrt{|g|})$
What about ${\hat\Gamma^u}_{ua} = {e_a}^\tilde{a} {\tilde{\Gamma}^\tilde{u}}_{\tilde{u}\tilde{a}} + {e^u}_\tilde{u} e_u ({e_a}^\tilde{u})$
And ${\tilde{\Gamma}^\tilde{u}}_{\tilde{u}\tilde{a}}$ is symmetric and therefore equal to ${\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{u}}$, which is equal to $\partial_\tilde{a} (log \sqrt{|\tilde{g}|})$
${\hat\Gamma^u}_{ua} = {e_a}^\tilde{a} \partial_\tilde{a} (log \sqrt{|\tilde{g}|}) + {e^u}_\tilde{u} e_u ({e_a}^\tilde{u})$
${\hat\Gamma^u}_{ua} = e_a (log \sqrt{|\tilde{g}|}) + {e^u}_\tilde{u} e_u ({e_a}^\tilde{u})$
Equate the two definitions of ${\hat\Gamma^u}_{ua}$ to find:
$e_a (log \sqrt{|g|}) + {c_{ua}}^u = e_a (log \sqrt{|\tilde{g}|}) + {e^u}_\tilde{u} e_u ({e_a}^\tilde{u})$
Now this is the connection that a covariant divergence would use:
$\hat\nabla \cdot v$
$= e^b \hat\nabla_b (v^a e_a)$
$= e_b (v^a) e^b(e_a) + e^b (\nabla_b e_a) v^a$
$= e_a (v^a) + {\hat{\Gamma}^b}_{ba} v^a$
$= e_a (v^a) + v^b {\hat{\Gamma}^a}_{ab}$
$= e_a (v^a) + v^b (e_b (log(\sqrt{|\tilde{g}|})) + e_a ( {e_b}^\tilde{b} ) {e^a}_\tilde{b} )$
$= e_a (v^a) + v^a \frac{1}{\sqrt{|\tilde{g}|}} e_a( \sqrt{|\tilde{g}|} ) + v^b e_a ( {e_b}^\tilde{b} ) {e^a}_\tilde{b}$
$= \frac{1}{\sqrt{|\tilde{g}|}} e_a (\sqrt{|\tilde{g}|} v^a) + v^b e_a ({e_b}^\tilde{b}) {e^a}_\tilde{b}$
$= \frac{1}{\sqrt{|\tilde{g}|}} \partial_\tilde{a} (\sqrt{|\tilde{g}|} {e_a}^\tilde{a} v^a)$
$= \frac{1}{\sqrt{|\tilde{g}|}} {e^b}_\tilde{a} e_b (\sqrt{|\tilde{g}|} {e_a}^\tilde{a} v^a)$
This is the Voss-Weyl identity.
Levi-Civita connection antisymmetry on 2nd and 3rd indexes:
$ {\hat\Gamma^a}_{[bc]}$
$= {\Gamma^a}_{[bc]} - {K^a}_{[bc]}$
$= \frac{1}{2} ({T^a}_{bc} + {c_{bc}}^a) - \frac{1}{2} {T^a}_{[bc]}$
$= \frac{1}{2} {c_{bc}}^a$
In a holonomic basis this gives ${\hat\Gamma^a}_{[bc]} = 0$
Notice that ${\hat\Gamma^u}_{ua} = {\hat\Gamma^u}_{au} + {c_{ua}}^u$
Therefore ${\hat\Gamma^u}_{ua} = {\hat\Gamma^u}_{au} + {c_{ua}}^u$
${\hat\Gamma^u}_{ua} = e_a ( log \sqrt{|g|} ) + {c_{ua}}^u$
Which looks just like our trace of the Levi-Civita connection above.
Trace of an arbitrary connection:
${\Gamma^u}_{au} = {\hat\Gamma^u}_{au} + {K^u}_{au} = e_a ( log \sqrt{|g|} )$
${\Gamma^u}_{ua} - {\Gamma^u}_{au} = {T^u}_{ua} + {c_{ua}}^u$
${\Gamma^u}_{ua} = e_a ( log \sqrt{|g|} ) + {c_{ua}}^u + {T^u}_{ua}$
Trace of the Levi-Civita connection of the coordinate basis (which will equal the volume gradient):
${\tilde\Gamma^\tilde{u}}_{\tilde{a}\tilde{u}} = \partial_\tilde{a} ( log \sqrt{|\tilde{g}|} )$
...transformed into non-coordinates...
${\tilde\Gamma^u}_{au} = e_a ( log \sqrt{|\tilde{g}|} )$
Difference of Levi-Civita connection of the coordinate basis vs the non-coordinate basis:
${\hat\Gamma^u}_{au} = {\tilde\Gamma^u}_{au} + {e^u}_\tilde{u} e_a({e_u}^\tilde{u})$
${\tilde\Gamma^u}_{au} = {\hat\Gamma^u}_{au} - {e^u}_\tilde{u} e_a({e_u}^\tilde{u})$
Trace of the Levi-Civita connection of the coordinate basis, in terms of the non-coordinate Levi-Civita connection:
${\tilde\Gamma^u}_{au} = e_a ( log \sqrt{|\tilde{g}|} )$
Equate:
${\hat\Gamma^u}_{au} - {e^u}_\tilde{u} e_a({e_u}^\tilde{u}) = e_a ( log \sqrt{|\tilde{g}|} )$
... subsitute $- {e^u}_\tilde{u} e_a ({e_u}^\tilde{u}) = e_a ( log \sqrt{|\tilde{g}|} ) - \frac{1}{2} g^{uv} e_a (g_{uv})$ ...
${\hat\Gamma^u}_{au} + e_a ( log \sqrt{|\tilde{g}|} ) - \frac{1}{2} g^{uv} e_a (g_{uv}) = e_a ( log \sqrt{|\tilde{g}|} )$
${\hat\Gamma^u}_{au} = \frac{1}{2} g^{uv} e_a (g_{uv})$
Levi-Civita change-of-basis:
$e_c (g_{ab}) = \frac{\partial}{\partial x^c} ( g_{ab} )$
$e_{c'} (g'_{a'b'}) = \frac{\partial}{\partial x^{c'}} ( g'_{a'b'} )$
$= \frac{\partial}{\partial x^{c'}} (
g'_{ab}
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^b}{\partial x^{b'}}
)$
$= \frac{\partial x^c}{\partial x^{c'}} \frac{\partial}{\partial x^c} (
g'_{ab}
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^b}{\partial x^{b'}}
)$
$= (\frac{\partial}{\partial x^c} (g_{ab})
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^b}{\partial x^{b'}}
+ g_{ab} (
\frac{\partial}{\partial x^c} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ \frac{\partial x^a}{\partial x^{a'}}
\frac{\partial}{\partial x^c} ( \frac{\partial x^b}{\partial x^{b'}} )
)
) \frac{\partial x^c}{\partial x^{c'}}$
$= \frac{\partial}{\partial x^c} (g_{ab})
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^c}{\partial x^{c'}}
+ g_{ab} (
\frac{\partial}{\partial x^c} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ \frac{\partial x^a}{\partial x^{a'}}
\frac{\partial}{\partial x^c} ( \frac{\partial x^b}{\partial x^{b'}} )
) \frac{\partial x^c}{\partial x^{c'}}$
$= \frac{\partial}{\partial x^c} (g_{ab})
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^c}{\partial x^{c'}}
+ g_{ab} (
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ \frac{\partial x^a}{\partial x^{a'}}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^b}{\partial x^{b'}} )
)$
change of basis of connection:
$\Gamma_{abc} = \frac{1}{2} ( e_c(g_{ab}) + e_b(g_{ac}) - e_a(g_{bc}))$
$\Gamma'_{a'b'c'} = \frac{1}{2} ( e_{c'} ( g'_{a'b'} ) + e_{b'} ( g'_{a'c'} ) - e_{a'} ( g'_{b'c'} ) )$
$= \frac{1}{2} (
\frac{\partial}{\partial x^c} (g_{ab})
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^c}{\partial x^{c'}}
+ g_{ab} (
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ \frac{\partial x^a}{\partial x^{a'}}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^b}{\partial x^{b'}} )
)
$$+ \frac{\partial}{\partial x^b} (g_{ac})
\frac{\partial x^a}{\partial x^{a'}}
\frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
+ g_{ac} (
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^c}{\partial x^{c'}}
+ \frac{\partial x^a}{\partial x^{a'}}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^c}{\partial x^{c'}} )
)
$$- \frac{\partial}{\partial x^a} (g_{cb})
\frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^a}{\partial x^{a'}}
- g_{cb} (
\frac{\partial}{\partial x^{a'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ \frac{\partial x^c}{\partial x^{c'}}
\frac{\partial}{\partial x^{a'}} ( \frac{\partial x^b}{\partial x^{b'}} )
)
)$
$= \frac{1}{2} (
\frac{\partial}{\partial x^c} (g_{ab})
+ \frac{\partial}{\partial x^b} (g_{ac})
- \frac{\partial}{\partial x^a} (g_{cb})
) \frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^a}{\partial x^{a'}}
+ \frac{1}{2} (
g_{ab}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ g_{ab}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^a}{\partial x^{a'}}
+ g_{ac}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^c}{\partial x^{c'}}
+ g_{ac}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^a}{\partial x^{a'}}
- g_{cb}
\frac{\partial}{\partial x^{a'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^b}{\partial x^{b'}}
- g_{cb}
\frac{\partial}{\partial x^{a'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^c}{\partial x^{c'}}
)$
$= \Gamma_{abc}
\frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^a}{\partial x^{a'}}
+ \frac{1}{2} (
g_{ab}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ g_{ab}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^a}{\partial x^{a'}}
+ g_{ac}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^a}{\partial x^{a'}} )
\frac{\partial x^c}{\partial x^{c'}}
+ g_{ac}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^a}{\partial x^{a'}}
- g_{cb}
\frac{\partial}{\partial x^{a'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^b}{\partial x^{b'}}
- g_{cb}
\frac{\partial}{\partial x^{a'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^c}{\partial x^{c'}}
)$
${{\Gamma'}^{a'}}_{b'c'} = g^{a'd'} {\Gamma'}_{d'b'c'}$
$= g^{a'd'} (
\Gamma_{dbc}
\frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^d}{\partial x^{d'}}
+ \frac{1}{2} (
g_{db}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^d}{\partial x^{d'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ g_{db}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^d}{\partial x^{d'}}
+ g_{dc}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^d}{\partial x^{d'}} )
\frac{\partial x^c}{\partial x^{c'}}
+ g_{dc}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^d}{\partial x^{d'}}
- g_{cb}
\frac{\partial}{\partial x^{d'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^b}{\partial x^{b'}}
- g_{cb}
\frac{\partial}{\partial x^{d'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^c}{\partial x^{c'}}
)
)$
$= \frac{\partial x^{a'}}{\partial x^a}
\frac{\partial x^{d'}}{\partial x^e}
g^{ae} (
\Gamma_{dbc}
\frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
\frac{\partial x^d}{\partial x^{d'}}
+ \frac{1}{2} (
g_{db}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^d}{\partial x^{d'}} )
\frac{\partial x^b}{\partial x^{b'}}
+ g_{db}
\frac{\partial}{\partial x^{c'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^d}{\partial x^{d'}}
+ g_{dc}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^d}{\partial x^{d'}} )
\frac{\partial x^c}{\partial x^{c'}}
+ g_{dc}
\frac{\partial}{\partial x^{b'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^d}{\partial x^{d'}}
- g_{cb}
\frac{\partial}{\partial x^{d'}} ( \frac{\partial x^c}{\partial x^{c'}} )
\frac{\partial x^b}{\partial x^{b'}}
- g_{cb}
\frac{\partial}{\partial x^{d'}} ( \frac{\partial x^b}{\partial x^{b'}} )
\frac{\partial x^c}{\partial x^{c'}}
)
)$
$= \frac{\partial x^{a'}}{\partial x^a}
{\Gamma^a}_{bc}
\frac{\partial x^c}{\partial x^{c'}}
\frac{\partial x^b}{\partial x^{b'}}
+ \frac{\partial x^{a'}}{\partial x^a}
\frac{\partial^2 x^a}{\partial x^{b'} \partial x^{c'} }
$
geodesics minimize distance along a trajectory (using a coordinate basis):
Minimize $S$ by solving for $\delta S = 0$:
Let $S = \int \sqrt{ g_{uv} \frac{dx^u}{d\tau} \frac{dx^v}{d\tau} } d\tau$
We can just as easily minimize $S \rightarrow S = \int g_{uv} \frac{dx^u}{d\tau} \frac{dx^v}{d\tau} d\tau$
Now solve for $\frac{\delta S}{\delta x^a} = 0$
$S = \int g_{uv} \dot{x}^u \dot{x}^v d\tau$
$\delta S = \int (\delta g_{uv} \dot{x}^u \dot{x}^v + 2 g_{uv} \delta \dot{x}^u \dot{x}^v ) d\tau$
Next we use integration by parts and assign the total integral to zero: $u dv = uv - v du$ for $uv = 0$
Using $u = 2 g_{uv} \dot{x}^v$, $dv = \delta \dot{x}^u$, and therefore $v = \delta x^u$, $du = 2 (\dot{g}_{uv} \dot{x}^v + g_{uv} \ddot{x}^v) d\tau$
$\delta S = \int (\delta g_{uv} \dot{x}^u \dot{x}^v - 2 \delta x^u (\dot{g}_{uv} \dot{x}^v + g_{uv} \ddot{x}^v)) d\tau$
Substitute $\delta g_{uv} = \frac{\partial g_{uv}}{\partial x^w} \cdot \delta x^w$ and $\frac{\partial g_{uv}}{\partial \tau} = \frac{\partial g_{uv}}{\partial x^w} \cdot \frac{\partial x^w}{\partial \tau}$
$\delta S = \int (\delta x^w \frac{\partial}{\partial x^w} g_{uv} \dot{x}^u \dot{x}^v - 2 \delta x^u (\frac{\partial}{\partial x^w} g_{uv} \dot{x}^w \dot{x}^v + g_{uv} \ddot{x}^v)) d\tau$
Reindex and factor out $\delta x^u$
$\delta S = \int \delta x^u (\frac{\partial}{\partial x^u} g_{ab} \dot{x}^a \dot{x}^b - 2 \frac{\partial}{\partial x^a} g_{ub} \dot{x}^a \dot{x}^b - 2 g_{uv} \ddot{x}^v) d\tau$
Solve for $\delta S = 0$
$0 = \frac{\partial}{\partial x^u} g_{ab} \dot{x}^a \dot{x}^b - 2 \frac{\partial}{\partial x^a} g_{ub} \dot{x}^a \dot{x}^b - 2 g_{uv} \ddot{x}^v$
$0 = g^{cu} ( \frac{\partial}{\partial x^u} g_{ab} \dot{x}^a \dot{x}^b - 2 \frac{\partial}{\partial x^a} g_{ub} \dot{x}^a \dot{x}^b - 2 g_{uv} \ddot{x}^v )$
$\ddot{x}^c - \frac{1}{2} g^{cu} (\frac{\partial}{\partial x^u} g_{ab} - 2 \frac{\partial}{\partial x^a} g_{ub}) \dot{x}^a \dot{x}^b = 0$
Separate the term with a factor of 2:
$\ddot{x}^c + \frac{1}{2} g^{cu} (\frac{\partial}{\partial x^b} g_{ua} + \frac{\partial}{\partial x^a} g_{ub} - \frac{\partial}{\partial x^u} g_{ab}) \dot{x}^a \dot{x}^b = 0$
$\ddot{x}^c + {\Gamma^c}_{ab} \dot{x}^a \dot{x}^b = 0$
For ${\Gamma^c}_{ab} = \frac{1}{2} g^{cu} (\frac{\partial}{\partial x^b} g_{ua} + \frac{\partial}{\partial x^a} g_{ub} - \frac{\partial}{\partial x^u} g_{ab})$
This fits with the above definition of a metric-cancelling covariant derivative that has no torsion and no commutation.
acceleration of geodesic motion:
$\hat{\nabla}_u u = 0$
$u^b \hat{\nabla}_b u = 0$
$u^b ( \partial_b ( u^a ) + {\hat\Gamma^a}_{bc} u^c ) = 0$
$u^b \partial_b ( u^a ) + {\hat\Gamma^a}_{bc} u^b u^c = 0$
$\dot{u}^a = -{\hat\Gamma^a}_{bc} u^b u^c$
...in the presence of torsion:
$\dot{u}^a = -({\Gamma^a}_{bc} - {K^a}_{bc}) u^b u^c$
$\dot{u}^a = -({\Gamma^a}_{bc} - \frac{1}{2} ({{T_c}^a}_b + {{T_b}^a}_c + {T^a}_{bc})) u^b u^c$
$\dot{u}^a = -{\Gamma^a}_{bc} u^b u^c + {{T_b}^a}_c u^b u^c$
$\dot{u}^a = -{\Gamma^a}_{bc} u^b u^c + u_b {T^b}_{dc} u^c g^{da}$
change of basis of torsion-free holonomic geodesic:
$\frac{\partial^2 x^a}{\partial \tau^2} + {\Gamma^a}_{bc} \frac{\partial x^b}{\partial \tau} \frac{\partial x^c}{\partial \tau} = 0$
$\frac{\partial}{\partial \tau} (\frac{\partial x^a}{\partial x^{a'}} \frac{\partial x^{a'}}{\partial \tau})
+ {\Gamma^a}_{bc}
\cdot \frac{\partial x^b}{\partial x^{b'}} \frac{\partial x^{b'}}{\partial \tau}
\cdot \frac{\partial x^c}{\partial x^{c'}} \frac{\partial x^{c'}}{\partial \tau}
= 0$
$\frac{\partial}{\partial \tau} \frac{\partial x^a}{\partial x^{a'}} \cdot \frac{\partial x^{a'}}{\partial \tau}
+ \frac{\partial x^a}{\partial x^{a'}} \cdot \frac{\partial}{\partial \tau} \frac{\partial x^{a'}}{\partial \tau}
+ {\Gamma^a}_{bc} \frac{\partial x^b}{\partial x^{b'}} \frac{\partial x^c}{\partial x^{c'}}
\cdot \frac{\partial x^{b'}}{\partial \tau} \frac{\partial x^{c'}}{\partial \tau}
= 0$
$\frac{\partial^2 x^{a'}}{\partial \tau^2}
+ (
\frac{\partial x^{a'}}{\partial x^a}
{\Gamma^a}_{bc} \frac{\partial x^b}{\partial x^{b'}} \frac{\partial x^c}{\partial x^{c'}}
+ \frac{\partial x^{a'}}{\partial x^a} \frac{\partial^2 x^a}{\partial x^{b'} \partial x^{c'}}
) \frac{\partial x^{b'}}{\partial \tau} \frac{\partial x^{c'}}{\partial \tau}
= 0$
This matches the change-of-basis of metric-cancelling connection, which we can now substitute:
$\frac{\partial^2 x^{a'}}{\partial \tau^2} + {\Gamma^{a'}}_{b'c'} \frac{\partial x^{b'}}{\partial \tau} \frac{\partial x^{c'}}{\partial \tau} = 0$
Holonomic Levi-Civita connection as the sum of two other connections, derived from their metrics:
$g_{ab} = g'_{ab} + g''_{ab}$
$g_{ab,c} = g'_{ab,c} + g''_{ab,c}$
Once again, (see Section 4 - metric tensor) ${r^a}_b = g'^{ac} g''_{cb}$ and $r = {r^a}_a$.
$\Gamma_{abc}$
$= \frac{1}{2} (g_{ab,c} + g_{ac,b} - g_{bc,a})$
$= \frac{1}{2} (g'_{ab,c} + g'_{ac,b} - g'_{bc,a}) + \frac{1}{2} (g''_{ab,c} + g''_{ac,b} - g''_{bc,a})$
$= \Gamma'_{abc} + \Gamma''_{abc}$
${\Gamma^a}_{bc}$
$= g^{ad} \Gamma_{dbc}$
$= (\Gamma'_{dbc} + \Gamma''_{dbc}) (g'^{ad} - (g'^{ae} g''_{ef} g'^{fd}) \cdot \frac{1}{1 + r})$
$= \Gamma'_{dbc} g'^{ad} - (\Gamma'_{dbc} g'^{ae} g''_{ef} g'^{fd}) \cdot \frac{1}{1 + r}
+ \Gamma''_{dbc} g'^{ad} - \Gamma''_{dbc} (g'^{ae} g''_{ef} g'^{fd}) \cdot \frac{1}{1 + r}$
$= {\Gamma'^a}_{bc}
+ g'^{ad} \Gamma''_{dbc}
- g'^{ae} g''_{ef} {\Gamma'^f}_{bc} \cdot \frac{1}{1 + r}
- g'^{ae} g''_{ef} g'^{fd} \Gamma''_{dbc} \cdot \frac{1}{1 + r}$
$= {\Gamma'^a}_{bc}
+ g'^{ad} g''_{de} g''^{ef} \Gamma''_{fbc}
- g'^{ae} g''_{ef} {\Gamma'^f}_{bc} \cdot \frac{1}{1 + r}
- g'^{ae} g''_{ef} g'^{fd} g''_{dg} g''^{gh} \Gamma''_{hbc} \cdot \frac{1}{1 + r}$
$= {\Gamma'^a}_{bc}
+ g'^{ad} g''_{de} {\Gamma''^e}_{bc}
- g'^{ae} g''_{ef} {\Gamma'^f}_{bc} \cdot \frac{1}{1 + r}
- g'^{ae} g''_{ef} g'^{fd} g''_{dg} {\Gamma''^g}_{bc} \cdot \frac{1}{1 + r}$
$= {\Gamma'^a}_{bc}
+ {r^a}_d {\Gamma''^d}_{bc}
- \frac{1}{1 + r} {r^a}_d {\Gamma'^d}_{bc}
- \frac{1}{1 + r} {r^a}_d {r^d}_e {\Gamma''^e}_{bc}$
Tensor density.
Assuming $\nabla$ is a metric-cancelling covariant derivative, calculate the covariant derivative of the scalar of the metric determinant:
$\nabla_a g$
$= \nabla_a \left( \tilde{\epsilon}^{u_1 ... u_n} \underset{i=1}{\overset{n}{\Pi}} g_{i u_i} \right)$
$= \tilde{\epsilon}^{u_1 ... u_n} \nabla_a \left( \underset{i=1}{\overset{n}{\Pi}} g_{i u_i} \right)$
$= \tilde{\epsilon}^{u_1 ... u_n} \underset{i=1}{\overset{n}{\Sigma}} \left( \nabla_a g_{i u_i} \cdot \underset{j=1, j \ne i}{\overset{n}{\Pi}} g_{j u_j} \right)$
Using $\nabla_a g_{uv} = 0$:
$= \tilde{\epsilon}^{u_1 ... u_n} \underset{i=1}{\overset{n}{\Sigma}} \left( 0 \cdot \underset{j=1, j \ne i}{\overset{n}{\Pi}} g_{j u_j} \right)$
$= \tilde{\epsilon}^{u_1 ... u_n} \underset{i=1}{\overset{n}{\Sigma}} 0$
$= 0$
Now we see that the covariant derivative of every scalar is not necessarily equal to the partial derivative of that scalar.
Raising it to a power:
$\nabla_a (g)^m$
$= m (g)^{m-1} \cdot \nabla_a g$
$= m (g)^{m-1} \cdot 0$
$= 0$
Equating this to the original "covariant derivative equals partial derivative plus connection" rule:
${\Gamma^u}_{au} = e_a (log(\sqrt{|g|})) = e_a( \frac{1}{2} log(|g|) ) = \frac{1}{2|g|} e_a(|g|) = \frac{1}{2g} e_a(g)$
So $2 g {\Gamma^u}_{au} = e_a(g)$
$0 = \nabla_a g = e_a ( g ) - e_a ( g ) = e_a(g) - 2 g {\Gamma^u}_{au}$
Same but for metric determinant raised to a power:
$0 = \nabla_a (g)^m$
$= e_a((g)^m) - e_a((g)^m)$
$= e_a((g)^m) - m (g)^{m-1} e_a(g)$
$= e_a((g)^m) - m (g)^{m-1} \cdot 2 g {\Gamma^u}_{au}$
$= e_a((g)^m) - 2 m (g)^m {\Gamma^u}_{au}$
Using the metric determinant square-root:
$\nabla_a (\sqrt{|g|})^m = e_a((\sqrt{|g|})^m) - m (\sqrt{|g|})^m {\Gamma^u}_{au}$
From here on we will refer to any scalar function mapping from our manifold to a real value, if it is scaled by $(\sqrt{|g|})^m$, then it will be "of weight $m$".
Combining this with the original (p, q) tensor definition of covariant derivatives:
Let $T = {T^{a_1 ... a_p}}_{b_1 ... b_q} \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}$ be a (p, q) tensor.
Let $\mathscr{T}
= (\sqrt{|g|})^m T
= {\mathscr{T}^{a_1 ... a_p}}_{b_1 ... b_q} \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
= (\sqrt{|g|})^m {T^{a_1 ... a_p}}_{b_1 ... b_q} \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
$ be a weight-m (p, q) tensor.
The covariant derivative of $\mathscr{T}$ is defined as:
$\nabla \mathscr{T} = e^u \otimes \nabla_u ((\sqrt{|g|})^m {T^{a_1 ... a_p}}_{b_1 ... b_q} \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i})$
$= e^u \otimes \nabla_u ( (\sqrt{|g|})^m ) {T^{a_1 ... a_p}}_{b_1 ... b_q} \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
+ e^u \otimes (\sqrt{|g|})^m \nabla_u ( {T^{a_1 ... a_p}}_{b_1 ... b_q} \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i} )
$
$= \left(
(e_u ( (\sqrt{|g|})^m ) - m (\sqrt{|g|})^m {\Gamma^v}_{uv} ) {T^{a_1 ... a_p}}_{b_1 ... b_q}
+ (\sqrt{|g|})^m (
e_u ( {T^{a_1 ... a_p}}_{b_1 ... b_q} )
+ {\Gamma^{a_i}}_{u v} {T^{a_1 ... a_{i-1} v a_{i+1} ... a_p}}_{b_1 ... b_q}
- {\Gamma^v}_{u b_i} {T^{a_1 ... a_p}}_{b_1 ... b_{i-1} v b_{i+1} ... b_q}
)
\right) e^u \otimes \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
$
$= \left(
e_u ( (\sqrt{|g|})^m {T^{a_1 ... a_p}}_{b_1 ... b_q} )
- m (\sqrt{|g|})^m {\Gamma^v}_{uv} {T^{a_1 ... a_p}}_{b_1 ... b_q}
+ (\sqrt{|g|})^m (
+ {\Gamma^{a_i}}_{u v} {T^{a_1 ... a_{i-1} v a_{i+1} ... a_p}}_{b_1 ... b_q}
- {\Gamma^v}_{u b_i} {T^{a_1 ... a_p}}_{b_1 ... b_{i-1} v b_{i+1} ... b_q}
)
\right) e^u \otimes \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
$
$= \left(
e_u ( {\mathscr{T}^{a_1 ... a_p}}_{b_1 ... b_q} )
+ {\Gamma^{a_i}}_{u v} {\mathscr{T}^{a_1 ... a_{i-1} v a_{i+1} ... a_p}}_{b_1 ... b_q}
- {\Gamma^v}_{u b_i} {\mathscr{T}^{a_1 ... a_p}}_{b_1 ... b_{i-1} v b_{i+1} ... b_q}
- m {\Gamma^v}_{uv} {\mathscr{T}^{a_1 ... a_p}}_{b_1 ... b_q}
\right) e^u \otimes \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
$
And now we have the up-add/down-subtract definition for the metric-cancelling covariant derivative of a tensor density of weight $m$.
Keep twisting the knobs and we get...
$= \left(
e_u( \sqrt{|g|}^m {T^{a_1 ... a_p}}_{b_1 ... b_q})
+ \sqrt{|g|}^m {\Gamma^{a_i}}_{u v} {T^{a_1 ... a_{i-1} v a_{i+1} ... a_p}}_{b_1 ... b_q}
- \sqrt{|g|}^m {\Gamma^v}_{u b_i} {T^{a_1 ... a_p}}_{b_1 ... b_{i-1} v b_{i+1} ... b_q}
- e_u(\sqrt{|g|}^m) {T^{a_1 ... a_p}}_{b_1 ... b_q}
\right) e^u \otimes \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
$
$= \sqrt{|g|}^m \left(
e_u({T^{a_1 ... a_p}}_{b_1 ... b_q})
+ {\Gamma^{a_i}}_{u v} {T^{a_1 ... a_{i-1} v a_{i+1} ... a_p}}_{b_1 ... b_q}
- {\Gamma^v}_{u b_i} {T^{a_1 ... a_p}}_{b_1 ... b_{i-1} v b_{i+1} ... b_q}
\right) e^u \otimes \underset{i=1}{\overset{p}{\otimes}} e_{a_i} \underset{i=1}{\overset{q}{\otimes}} e^{b_i}
$
$= \sqrt{|g|}^m \nabla T$
And this is where I could have ended up much earlier if I had just used the chain rule without expanding the basis vectors/one-forms of the unweighted tensor's definition.
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