Back
I'm sure everyone is expecting to see $g_{ab} = \eta_{ab} + h_{ab}$.
Can you tell me why this is a bad idea, unless you are working in Cartesian coordinates? And chances are none of you are working in Cartesian coordinates.
If you want to see a collection of my complaints on what's wrong with this in arbitrary background metrics, check out my generated notebook here.
But here instead I'm just going to work out a different kind of perturbation. One where the basis is perturbed, not the metric. And, like GR, hopefully we'll end up with some math that is independent of the backgroundn basis, even though that does not negate such a basis' existence.
Ok here's our background metric definition wrt a background basis: $\hat{g}_{ab} \hat{e}^a \otimes \hat{e}^b = (\hat{e}_a \cdot \hat{e}_b) \hat{e}^a \otimes \hat{e}^b $
Now we perturb that basis by a transform that is just ${\epsilon_a}^u$ away from identity, where $|{\epsilon_a}^u| << 1$:
Let $e^a = (\delta_u^a + {\epsilon_u}^a) \hat{e}^u$,
so $\hat{e}^a = (\delta_u^a - {\epsilon_u}^a) e^u$.
And $e_a = (\delta_a^u - {\epsilon_a}^u) \hat{e}_u$,
so $\hat{e}_a = (\delta_a^u + {\epsilon_a}^u) e^u$.
From there we get orthogonality up to quadratic term:
$e^a ( e_b) = (\delta_u^a + {\epsilon_u}^a) \hat{e}^u (
(\delta_b^v - {\epsilon_b}^v) \hat{e}_v
)$
$= (\delta_u^a + {\epsilon_u}^a) (\delta_b^v - {\epsilon_b}^v) \hat{e}^u (\hat{e}_v)$
$= (\delta_u^a + {\epsilon_u}^a) (\delta_b^v - {\epsilon_b}^v) \delta_v^u$
$= \delta_u^a \delta_b^v \delta_v^u - \delta_u^a {\epsilon_b}^v \delta_v^u + {\epsilon_u}^a \delta_b^v \delta_v^u - {\epsilon_u}^a {\epsilon_b}^v \delta_v^u$
$= \delta_b^a - {\epsilon_b}^a + {\epsilon_b}^a - {\epsilon_u}^a {\epsilon_b}^u$
$= \delta_b^a - {(\epsilon^2)_b}^a$
...Let $|{\epsilon_a}^u|^2 \approx 0$...
$e^a (e_b) = \delta_b^a$
Now I'm going to set a second metric to exactly the same and perturb only its basis vectors. Not sure why. Later I can change my mind and say equal in magnitude within $|{\epsilon_a}^u|^2$ error, right?
So if you want some kind of bound to $O(error)$, just use $|{\epsilon_a}^u|^2$.
$g_{ab} e^a \otimes e^b = \hat{g}_{ab} \hat{e}^a \otimes \hat{e}^b$
$g_{ab} e^a \otimes e^b = \hat{g}_{uv} (\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v) (e^a \otimes e^b)$
$g_{ab} = \hat{g}_{uv} (\delta_a^u \delta_b^v - {\epsilon_a}^u \delta_b^v - \delta_a^u {\epsilon_b}^v + {\epsilon_a}^u {\epsilon_b}^v)$
$g_{ab} = \hat{g}_{ab} - \hat{g}_{ub} {\epsilon_a}^u - \hat{g}_{av} {\epsilon_b}^v + \hat{g}_{uv} {\epsilon_a}^u {\epsilon_b}^v$
$|g_{ab}| \ge |\hat{g}_{ab}| + 2 |\hat{g}_{u(a}| |{\epsilon_{b)}}^u| + |\hat{g}_{uv}| |{\epsilon_a}^u| |{\epsilon_b}^v|$
$|g_{ab}| \ge |\hat{g}_{ab}| (1 + 2 |{\epsilon_a}^u| + |{\epsilon_a}^u|^2)$
Mind you, typical perturbations are of the form $g_{ab} = \hat{g}_{ab} + h_{ab} = \hat{g}_{uv} (\delta_a^u \delta_b^v + {\epsilon_a}^u {\epsilon_b}^v)$, so it is focusing on the quadratic term and ignoring the linear term.
Alright, now once again, but with $|\epsilon|^2 \approx 0$:
$g_{ab} = \hat{g}_{ab}
- {\epsilon_a}^u \hat{g}_{ub}
- {\epsilon_b}^u \hat{g}_{au}
$
Going the other way:
In case you need to calculate a value raised or lowered by the background metric, in terms of the perturbed metric.
$\hat{g}_{ab} \hat{e}^a \otimes \hat{e}^b = g_{ab} e^a \otimes e^b$
$\hat{g}_{ab} \hat{e}^a \otimes \hat{e}^b = g_{uv} (\delta_a^u + {\epsilon_a}^u) (\delta_b^v + {\epsilon_b}^v) \hat{e}^a \otimes \hat{e}^b$
$\hat{g}_{ab} = g_{uv} (\delta_a^u + {\epsilon_a}^u) (\delta_b^v + {\epsilon_b}^v)$
Using $|\epsilon|^2 \approx 0$
$\hat{g}_{ab} = g_{ab}
+ {\epsilon_a}^u g_{ub}
+ {\epsilon_b}^v g_{av}
$
$\hat{g}_{ab} = g_{ab}
+ {\epsilon_a}^u g_{ub}
+ {\epsilon_b}^u g_{au}
$
In terms of products:
$g_{ab} = \hat{g}_{uv} (\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v)$
$g^{ab} = \hat{g}^{uv} (\delta_u^a + {\epsilon_u}^a) (\delta_v^b + {\epsilon_v}^b)$
$\hat{g}_{ab} = g_{uv} (\delta_a^u + {\epsilon_a}^u) (\delta_b^v + {\epsilon_b}^v)$
$\hat{g}^{ab} = g^{uv} (\delta_u^a - {\epsilon_u}^a) (\delta_v^b - {\epsilon_v}^b)$
So let's make a new operator that accepts a tensor in the original basis and produces the tensor in the permuted basis:
$E({T^{a_1...a_p}}_{b_1...b_q}) =
{T^{u_1...u_p}}_{v_1...v_q}
\cdot
(\delta_{u_1}^{a_1} + {\epsilon_{u_1}}^{a_1})
\cdot
...
\cdot
(\delta_{u_p}^{a_p} + {\epsilon_{u_p}}^{a_p})
\cdot
(\delta_{b_1}^{v_1} - {\epsilon_{b_1}}^{v_1})
\cdot
...
\cdot
(\delta_{b_q}^{v_q} - {\epsilon_{b_q}}^{v_q})
$
And if you expand all those deltas, and remove the $|\epsilon|^2$'s, you get:
$E({T^{a_1...a_p}}_{b_1...b_q}) =
{T^{a_1...a_p}}_{b_1...b_q}
+ {\epsilon_u}^{a_i} {T^{a_1 ... a_{i-1} u a_{i+1} ... a_p}}_{b_1 ... b_q}
- {\epsilon_{b_i}}^u {T^{a_1 ... a_p}}_{b_1 ... b_{i-1} u b_{i+1} ... b_q}
$
Raising and lowering ${\epsilon_a}^u$:
$\epsilon_{ab}$
$= {\epsilon_a}^u g_{ub}$
$= {\epsilon_a}^u \hat{g}_{pq} (\delta_a^p - {\epsilon_a}^p) (\delta_u^q - {\epsilon_u}^q)$
$= {\epsilon_a}^u (
\hat{g}_{au}
- \hat{g}_{pu} {\epsilon_a}^p
- \hat{g}_{aq} {\epsilon_u}^q
+ \hat{g}_{pq} {\epsilon_a}^p {\epsilon_u}^q
)$
remove $|\epsilon|^2$...
$= {\epsilon_a}^u \hat{g}_{au}$
So what does this mean?
${\epsilon_a}^u$, or any product of it, or $e_b({\epsilon_a}^u)$, or any product of it, can be raised or lowered by either $g_{uv}$ or $\hat{g}_{uv}$.
Calculating the derivative, and we see already the math is getting ugly.
$e_c(g_{ab}) = e_c(\hat{g}_{uv} (\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v))$
$e_c(g_{ab}) = e_c(\hat{g}_{uv}) (\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v)
+ \hat{g}_{uv} e_c((\delta_a^u + {\epsilon_a}^u) (\delta_b^v + {\epsilon_b}^v) )
$
$e_c(g_{ab}) = e_c(\hat{g}_{ab}) - 2 (\hat{g}_{u(a|} e_c({\epsilon_{|b)}}^u)) + e_c(\hat{g}_{uv} {\epsilon_a}^u {\epsilon_b}^v)$
$e_c(g_{ab}) = e_c(\hat{g}_{ab})
- 2 e_c(\hat{g}_{u(a}) {\epsilon_{b)}}^u
- 2 \hat{g}_{u(a|} e_c({\epsilon_{|b)}}^u)
+ e_c(\hat{g}_{uv}) {\epsilon_a}^u {\epsilon_b}^v
+ \hat{g}_{uv} e_c({\epsilon_a}^u) {\epsilon_b}^v
+ \hat{g}_{uv} {\epsilon_a}^u e_c({\epsilon_b}^v)
$
$e_c(g_{ab}) = e_c(\hat{g}_{ab})
- 2 e_c(\hat{g}_{u(a}) {\epsilon_{b)}}^u
- 2 \hat{g}_{u(a|} e_c({\epsilon_{|b)}}^u )
+ e_c(\hat{g}_{uv}) {\epsilon_a}^u {\epsilon_b}^v
+ 2 \hat{g}_{uv} e_c({\epsilon_{(a}}^u) {\epsilon_{b)}}^v
$
Get rid of $|\epsilon|^2$ terms. Notice, no assumptions on $\partial(|\epsilon|)$ or $\partial(|\epsilon|^2)$:
$e_c(g_{ab}) = e_c(\hat{g}_{ab})
- 2 e_c(\hat{g}_{u(a}) {\epsilon_{b)}}^u
- 2 \hat{g}_{u(a|} e_c({\epsilon_{|b)}}^u )
+ 2 \hat{g}_{uv} e_c({\epsilon_{(a}}^u) {\epsilon_{b)}}^v
$
But should I assume $\partial(|\epsilon|^2) \approx \partial(0) = 0$? In that case so would $|\epsilon| \partial(|\epsilon|) \approx 0$ as well.
I think I have to enforce this rule, for consistency with the fact that I am already equating $g_{ab} = (\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b^v}) \hat{g}_{uv} = \hat{g}_{ab} - 2 \hat{g}_{u(a} {\epsilon_{b)}}^u$, so I am already pretending like the $|\epsilon|^2$ term within $g_{ab}$ is neglegible.
$e_c(g_{ab}) = e_c(\hat{g}_{ab})
- {\epsilon_b}^u e_c(\hat{g}_{ua})
- {\epsilon_a}^u e_c(\hat{g}_{ub})
- e_c({\epsilon_a}^u) \hat{g}_{ub}
- e_c({\epsilon_b}^u) \hat{g}_{au}
$
So if $e_a(|\epsilon|^2) = |\epsilon| e_a(|\epsilon|) \approx 0$
and $e_a = (\delta_a^u - {\epsilon_a}^u) \hat{e}_u$
then $e_a(\epsilon) = (I - \epsilon) \hat{e}_a(\epsilon) = \hat{e}_a(\epsilon) - \epsilon \hat{e}_a(\epsilon) = \hat{e}_a(\epsilon)$.
So if you are doing a partial derivative of an epsilon then you can interchange the partial operator between the background basis and the perturbed basis no problem.
In terms of $\hat{e}_a$:
$e_c(g_{ab}) = (\delta_c^v - {\epsilon_c}^v) \hat{e}_v(\hat{g}_{ab})
- {\epsilon_b}^u (\delta_c^v - {\epsilon_c}^v) \hat{e}_v(\hat{g}_{ua})
- {\epsilon_a}^u (\delta_c^v - {\epsilon_c}^v) \hat{e}_v(\hat{g}_{ub})
+ (\delta_c^w - {\epsilon_c}^w) (
- e_w({\epsilon_a}^u) \hat{g}_{ub}
- e_w({\epsilon_b}^u) \hat{g}_{au}
)
$
$e_c(g_{ab}) =
\hat{e}_c(\hat{g}_{ab})
- {\epsilon_c}^v \hat{e}_v(\hat{g}_{ab})
- {\epsilon_b}^u \hat{e}_c(\hat{g}_{ua})
- {\epsilon_a}^u \hat{e}_c(\hat{g}_{ub})
- \hat{e}_c( {\epsilon_a}^u) \hat{g}_{ub}
- \hat{e}_c( {\epsilon_b}^u) \hat{g}_{au}
$
In terms of products and transformations, re-inserting several $|\epsilon|^2 \approx 0$:
$e_c(g_{ab}) =
\hat{e}_w(\hat{g}_{uv})
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- \hat{e}_c( {\epsilon_a}^u) \hat{g}_{ub}
- \hat{e}_c( {\epsilon_b}^u) \hat{g}_{au}
$
But overall the rule for $e(E({T^{a_1...a_p}}_{b_1...b_q}))$ looks like
$E(e({T^{a_1 ... a_p}}_{b_1 ... b_1}))
+ e({\epsilon_u}^{a_i}) {T^{a_1 ... a_{i-1} u a_{i+1} ... a_p}}_{b_1 ... b_q}
- e({\epsilon_{b_i}}^u) {T^{a_1 ... a_p}}_{b_1 ... b_{i-1} u b_{i+1} ... b_q}$
Commutation of the background metric:
$[\hat{e}_a, \hat{e}_b] = 2 \hat{e}_{[a} ( \hat{e}_{b]}) = {\hat{c}_{ab}}^c \hat{e}_c$
Commutation of the perturbed metric:
$[e_a, e_b] (\phi) = [(\delta_a^u - {\epsilon_a}^u) \hat{e}_u, (\delta_b^v - {\epsilon_b}^v) \hat{e}_v] (\phi)$
$= (\delta_a^u - {\epsilon_a}^u) \hat{e}_u ((\delta_b^v - {\epsilon_b}^v) \hat{e}_v (\phi))
- (\delta_b^v - {\epsilon_b}^v) \hat{e}_v ((\delta_a^u - {\epsilon_a}^u) \hat{e}_u (\phi))
$
$= (\delta_a^u - {\epsilon_a}^u) (
\hat{e}_u (\delta_b^v - {\epsilon_b}^v) \hat{e}_v (\phi)
+ (\delta_b^v - {\epsilon_b}^v) \hat{e}_u (\hat{e}_v (\phi))
)
- (\delta_b^v - {\epsilon_b}^v) (
\hat{e}_v (\delta_a^u - {\epsilon_a}^u) \hat{e}_u (\phi)
+ (\delta_a^u - {\epsilon_a}^u) \hat{e}_v (\hat{e}_u (\phi))
)
$
$= (
\delta_a^u (
\delta_b^v \hat{e}_u (\hat{e}_v (\phi))
- {\epsilon_b}^v \hat{e}_u (\hat{e}_v (\phi))
- \hat{e}_u ({\epsilon_b}^v) \hat{e}_v (\phi)
)
- {\epsilon_a}^u (
\delta_b^v \hat{e}_u (\hat{e}_v (\phi))
- {\epsilon_b}^v \hat{e}_u (\hat{e}_v (\phi))
- \hat{e}_u ({\epsilon_b}^v) \hat{e}_v (\phi)
)
)
- (
\delta_b^v (
\delta_a^u \hat{e}_v (\hat{e}_u (\phi))
- {\epsilon_a}^u \hat{e}_v (\hat{e}_u (\phi))
- \hat{e}_v ({\epsilon_a}^u) \hat{e}_u (\phi)
)
- {\epsilon_b}^v (
\delta_a^u \hat{e}_v (\hat{e}_u (\phi))
- {\epsilon_a}^u \hat{e}_v (\hat{e}_u (\phi))
- \hat{e}_v ({\epsilon_a}^u) \hat{e}_u (\phi)
)
)
$
$=
\delta_a^u \delta_b^v \hat{e}_u (\hat{e}_v (\phi))
- \delta_a^u {\epsilon_b}^v \hat{e}_u (\hat{e}_v (\phi))
- \delta_a^u \hat{e}_u ({\epsilon_b}^v) \hat{e}_v (\phi)
- {\epsilon_a}^u \delta_b^v \hat{e}_u (\hat{e}_v (\phi))
+ {\epsilon_a}^u {\epsilon_b}^v \hat{e}_u (\hat{e}_v (\phi))
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^v) \hat{e}_v (\phi)
- \delta_b^v \delta_a^u \hat{e}_v (\hat{e}_u (\phi))
+ \delta_b^v {\epsilon_a}^u \hat{e}_v (\hat{e}_u (\phi))
+ \delta_b^v \hat{e}_v ({\epsilon_a}^u) \hat{e}_u (\phi)
+ {\epsilon_b}^v \delta_a^u \hat{e}_v (\hat{e}_u (\phi))
- {\epsilon_b}^v {\epsilon_a}^u \hat{e}_v (\hat{e}_u (\phi))
- {\epsilon_b}^v \hat{e}_v ({\epsilon_a}^u) \hat{e}_u (\phi)
$
$= (
{\hat{c}_{ab}}^c
+ {\epsilon_a}^u {\hat{c}_{bu}}^c
- {\epsilon_b}^u {\hat{c}_{au}}^c
- \hat{e}_a ({\epsilon_b}^c)
+ \hat{e}_b ({\epsilon_a}^c)
+ {\epsilon_a}^u {\epsilon_b}^v {\hat{c}_{uv}}^c
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^c)
- {\epsilon_b}^u \hat{e}_u ({\epsilon_a}^c)
) \hat{e}_c (\phi)
$
Substitute $\hat{e}_c = (\delta_c^d + {\epsilon_c}^d) e_d$:
$= (
{\hat{c}_{ab}}^c
+ {\epsilon_a}^u {\hat{c}_{bu}}^c
- {\epsilon_b}^u {\hat{c}_{au}}^c
- \hat{e}_a ({\epsilon_b}^c)
+ \hat{e}_b ({\epsilon_a}^c)
+ {\epsilon_a}^u {\epsilon_b}^v {\hat{c}_{uv}}^c
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^c)
- {\epsilon_b}^u \hat{e}_u ({\epsilon_a}^c)
) (\delta_c^d + {\epsilon_c}^d) e_d(\phi)
$
$= (
{\hat{c}_{ab}}^d
+ {\epsilon_a}^u {\hat{c}_{bu}}^d
- {\epsilon_b}^u {\hat{c}_{au}}^d
- \hat{e}_a ({\epsilon_b}^d)
+ \hat{e}_b ({\epsilon_a}^d)
+ {\epsilon_a}^u {\epsilon_b}^v {\hat{c}_{uv}}^d
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^d)
- {\epsilon_b}^u \hat{e}_u ({\epsilon_a}^d)
+ {\hat{c}_{ab}}^c {\epsilon_c}^d
+ {\epsilon_a}^u {\hat{c}_{bu}}^c {\epsilon_c}^d
- {\epsilon_b}^u {\hat{c}_{au}}^c {\epsilon_c}^d
- \hat{e}_a ({\epsilon_b}^c) {\epsilon_c}^d
+ \hat{e}_b ({\epsilon_a}^c) {\epsilon_c}^d
+ {\epsilon_a}^u {\epsilon_b}^v {\hat{c}_{uv}}^c {\epsilon_c}^d
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^c) {\epsilon_c}^d
- {\epsilon_b}^u \hat{e}_u ({\epsilon_a}^c) {\epsilon_c}^d
) e_d(\phi)
$
Now use $|{\epsilon_a}^b|^2 = 0$:
$= (
{\hat{c}_{ab}}^d
+ {\epsilon_a}^u {\hat{c}_{bu}}^d
- {\epsilon_b}^u {\hat{c}_{au}}^d
- \hat{e}_a ({\epsilon_b}^d)
+ \hat{e}_b ({\epsilon_a}^d)
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^d)
- {\epsilon_b}^u \hat{e}_u ({\epsilon_a}^d)
+ {\hat{c}_{ab}}^c {\epsilon_c}^d
- \hat{e}_a ({\epsilon_b}^c) {\epsilon_c}^d
+ \hat{e}_b ({\epsilon_a}^c) {\epsilon_c}^d
) e_d(\phi)
$
So ${c_{ab}}^c =
{\hat{c}_{ab}}^c
+ {\epsilon_a}^u {\hat{c}_{bu}}^c
- {\epsilon_b}^u {\hat{c}_{au}}^c
+ {\hat{c}_{ab}}^w {\epsilon_w}^c
- \hat{e}_a ({\epsilon_b}^c)
+ \hat{e}_b ({\epsilon_a}^c)
+ {\epsilon_a}^u \hat{e}_u ({\epsilon_b}^c)
- {\epsilon_b}^u \hat{e}_u ({\epsilon_a}^c)
- \hat{e}_a ({\epsilon_b}^w) {\epsilon_w}^c
+ \hat{e}_b ({\epsilon_a}^w) {\epsilon_w}^c
$
${c_{ab}}^c =
(\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v) (\delta_w^c + {\epsilon_w}^c) {\hat{c}_{uv}}^w
- (\delta_a^u - {\epsilon_a}^u) \hat{e}_u ({\epsilon_b}^w) (\delta_w^d + {\epsilon_w}^c)
+ (\delta_b^u - {\epsilon_b}^u) \hat{e}_u ({\epsilon_a}^w) (\delta_w^d + {\epsilon_w}^c)
$
${c_{ab}}^c =
(\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v) (\delta_w^c + {\epsilon_w}^c) {\hat{c}_{uv}}^w
- 2 (\delta_{[a}^u - {\epsilon_{[a}}^u) \hat{e}_u ({\epsilon_{b]}}^w) (\delta_w^c + {\epsilon_w}^c)
$
${c_{ab}}^c =
(\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v) (\delta_w^c + {\epsilon_w}^c) {\hat{c}_{uv}}^w
- 2 e_{[a} ({\epsilon_{b]}}^w) (\delta_w^c + {\epsilon_w}^c)
$
So this is almost turning into a rule of transform-lower-with-$(\delta_u^a - {\epsilon_u}^a)$, transform-upper-with-$(\delta_u^a + {\epsilon_u}^a)$.
This rule, combined with $|\epsilon|^2 \approx 0$, gives you a rule similar to covariant derivatives, where the result is the original tensor, then add each of its upper indexes transformed by $(+\epsilon)$, then subtract each of its lower indexes transformed by $(-\epsilon)$.
And, once again, I'm making no assumptions about $|e_c({\epsilon_a}^u)|$.
But if I did, let's assume $|{\epsilon_a}^u| |e_c({\epsilon_a}^u)| \rightarrow 0$, then we would almost fully recover the upper-with-$(+\epsilon)$, lower-with-$(-\epsilon)$ rule:
${c_{ab}}^c = (\delta_a^u - {\epsilon_a}^u) (\delta_b^v - {\epsilon_b}^v) (\delta_w^c + {\epsilon_w}^c) {\hat{c}_{uv}}^w - 2 e_{[a} ({\epsilon_{b]}}^c)$
${c_{ab}}^c =
{\hat{c}_{ab}}^c
- {\epsilon_a}^u {\hat{c}_{ub}}^c
- {\epsilon_b}^u {\hat{c}_{au}}^c
+ {\epsilon_u}^c {\hat{c}_{ab}}^u
- 2 e_{[a} ({\epsilon_{b]}}^c)
$
${c_{ab}}^c =
{\hat{c}_{ab}}^c
+ 2 {\epsilon_{[a}}^u {\hat{c}_{b]u}}^c
+ {\epsilon_u}^c {\hat{c}_{ab}}^u
- 2 e_{[a} ({\epsilon_{b]}}^c)
$
As producs:
${c_{ab}}^c =
{\hat{c}_{uv}}^w
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_w^c + {\epsilon_w}^c)
- 2 e_{[a} ({\epsilon_{b]}}^c)
$
The background commutation in terms of the perturbed commutation:
${c_{uv}}^w
\delta_a^u \delta_b^v \delta_w^c
=
{\hat{c}_{uv}}^w
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_w^c + {\epsilon_w}^c)
- 2 e_{[u} ({\epsilon_{v]}}^w)
\delta_a^u \delta_b^v \delta_w^c
$
${c_{uv}}^w
(\delta_a^u + {\epsilon_a}^u)
(\delta_b^v + {\epsilon_b}^v)
(\delta_w^c - {\epsilon_w}^c)
=
{\hat{c}_{ab}}^c
- 2 e_{[u} ({\epsilon_{v]}}^w)
(\delta_a^u + {\epsilon_a}^u)
(\delta_b^v + {\epsilon_b}^v)
(\delta_w^c - {\epsilon_w}^c)
$
${\hat{c}_{ab}}^c
=
{c_{uv}}^w
(\delta_a^u + {\epsilon_a}^u)
(\delta_b^v + {\epsilon_b}^v)
(\delta_w^c - {\epsilon_w}^c)
+ 2 e_{[a} ({\epsilon_{b]}}^c)
$
This looks very similar to our basis as a linear transform of a coordinate basis:
${c_{ab}}^c = 2 e_{[a} ({e_{b]}}^\bar{c}) {e^c}_\bar{c}$
Now for $c_{abc}$:
$c_{abc} = {c_{ab}}^e g_{ec}$
$= {\hat{c}_{ab}}^e g_{ec}
+ 2 {\epsilon_{[a}}^u {\hat{c}_{b]u}}^e g_{ec}
+ {\hat{c}_{ab}}^u {\epsilon_u}^e g_{ec}
- 2 e_{[a} ({\epsilon_{b]}}^e) g_{ec}
$
$= {\hat{c}_{ab}}^e (\hat{g}_{ec} - 2 \hat{g}_{u(e} {\epsilon_{c)}}^u)
+ 2 {\epsilon_{[a}}^u {\hat{c}_{b]u}}^e (\hat{g}_{ec} - 2 \hat{g}_{u(e} {\epsilon_{c)}}^u)
+ {\hat{c}_{ab}}^u {\epsilon_u}^e (\hat{g}_{ec} - 2 \hat{g}_{v(e} {\epsilon_{c)}}^v)
- 2 (\delta_a^u - {\epsilon_a}^u) \hat{e}_{[u} ({\epsilon_{b]}}^e) g_{ec}
$
$= \hat{c}_{abc}
- {\epsilon_a}^u \hat{c}_{ubc}
- {\epsilon_b}^u \hat{c}_{auc}
- {\epsilon_c}^u \hat{c}_{abu}
- 2 \hat{e}_{[a} ({\epsilon_{b]}}^e) g_{ec}
$
$= \hat{c}_{abc}
- {\epsilon_a}^u \hat{c}_{ubc}
- {\epsilon_b}^u \hat{c}_{auc}
- {\epsilon_c}^u \hat{c}_{abu}
- 2 \hat{e}_{[a} ({\epsilon_{b]}}^u) (
\hat{g}_{uc}
- {\epsilon_u}^v \hat{g}_{vc}
- {\epsilon_c}^v \hat{g}_{uv}
)
$
$= \hat{c}_{abc}
- {\epsilon_a}^u \hat{c}_{ubc}
- {\epsilon_b}^u \hat{c}_{auc}
- {\epsilon_c}^u \hat{c}_{abu}
- 2 \hat{e}_{[a} ({\epsilon_{b]}}^u) \hat{g}_{uc}
$
In terms of products:
$c_{abc} = \hat{c}_{uvw}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_{[a} ({\epsilon_{b]}}^u) \hat{g}_{uc}
$
And of course since is next to an $\hat{e}(|\epsilon|)$, we can interchange the metric and the basis derivative operators:
$c_{abc} =
\hat{c}_{abc}
- {\epsilon_a}^u \hat{c}_{ubc}
- {\epsilon_b}^u \hat{c}_{auc}
- {\epsilon_c}^u \hat{c}_{abu}
- 2 e_{[a} ({\epsilon_{b]}}^u) g_{uc}
$
$c_{abc} = \hat{c}_{uvw}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 e_{[a} ({\epsilon_{b]}}^u) g_{uc}
$
Levi-Civita torsion-free connection, lowered indexes:
$\Gamma_{abc} = \frac{1}{2} (
e_c(g_{ab})
+ e_b(g_{ac})
- e_a(g_{bc})
+ c_{abc}
+ c_{acb}
- c_{cba}
)$
Using $e_c(g_{ab}) =
\hat{e}_w(\hat{g}_{uv})
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_c( {\epsilon_{(a}}^u) \hat{g}_{b)u}
$
Using $c_{abc} =
\hat{c}_{uvw}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_{[a} ({\epsilon_{b]}}^u) \hat{g}_{uc}
$
$\Gamma_{abc} = \frac{1}{2} ($
$ \hat{e}_w(\hat{g}_{uv})
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_c( {\epsilon_{(a}}^u) \hat{g}_{b)u}
$
$ + \hat{e}_v(\hat{g}_{uw})
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_b( {\epsilon_{(a}}^u) \hat{g}_{c)u}
$
$ - \hat{e}_u(\hat{g}_{vw})
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
+ 2 \hat{e}_a( {\epsilon_{(b}}^u) \hat{g}_{c)u}
$
$ + \hat{c}_{uvw}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_{[a} ({\epsilon_{b]}}^u) \hat{g}_{uc}
$
$ + \hat{c}_{uwv}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- 2 \hat{e}_{[a} ({\epsilon_{c]}}^u) \hat{g}_{ub}
$
$ - \hat{c}_{wvu}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
+ 2 \hat{e}_{[c} ({\epsilon_{b]}}^u) \hat{g}_{ua}
$
$)$
$\Gamma_{abc} = \frac{1}{2} (
(
\hat{e}_w(\hat{g}_{uv})
+ \hat{e}_v(\hat{g}_{uw})
- \hat{e}_u(\hat{g}_{vw})
+ \hat{c}_{uvw}
+ \hat{c}_{uwv}
- \hat{c}_{wvu}
)
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- \hat{e}_c ({\epsilon_a}^u) \hat{g}_{bu}
- \hat{e}_c ({\epsilon_b}^u) \hat{g}_{au}
- \hat{e}_b ({\epsilon_a}^u) \hat{g}_{cu}
- \hat{e}_b ({\epsilon_c}^u) \hat{g}_{au}
+ \hat{e}_a ({\epsilon_b}^u) \hat{g}_{cu}
+ \hat{e}_a ({\epsilon_c}^u) \hat{g}_{bu}
- \hat{e}_a ({\epsilon_b}^u) \hat{g}_{uc}
+ \hat{e}_b ({\epsilon_a}^u) \hat{g}_{uc}
- \hat{e}_a ({\epsilon_c}^u) \hat{g}_{ub}
+ \hat{e}_c ({\epsilon_a}^u) \hat{g}_{ub}
+ \hat{e}_c ({\epsilon_b}^u) \hat{g}_{ua}
- \hat{e}_b ({\epsilon_c}^u) \hat{g}_{ua}
)$
$\Gamma_{abc} =
\hat{\Gamma}_{uvw}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- \hat{e}_b ({\epsilon_c}^u) \hat{g}_{au}
$
And we can take advantage of the fact that the $\hat{e}_b$ is applied to an ${\epsilon_c}^u$ to insert transforms and eliminate $|\epsilon|^2$ and $|\epsilon| e(|\epsilon|)$ to equate this to...
$\Gamma_{abc} =
\hat{\Gamma}_{uvw}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- e_b({\epsilon_c}^u) g_{au}
$
$\Gamma_{abc} =
(
\hat{\Gamma}_{uvw}
- e_v({\epsilon_w}^e) g_{eu}
)
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
$
$\Gamma_{abc} =
(
\hat{\Gamma}_{uvw}
- \hat{e}_v({\epsilon_w}^e) \hat{g}_{eu}
)
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
$
Levi-Civita torsion-free connection:
${\Gamma^a}_{bc} = g^{ae} \Gamma_{ebc}$
${\Gamma^a}_{bc} =
g^{ae} (
\hat{\Gamma}_{uvw}
(\delta_e^u - {\epsilon_e}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- e_b({\epsilon_c}^u) g_{eu}
)
$
${\Gamma^a}_{bc} =
g^{ae}
\hat{\Gamma}_{uvw}
(\delta_e^u - {\epsilon_e}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- g^{ae} e_b({\epsilon_c}^u) g_{eu}
$
${\Gamma^a}_{bc} =
\hat{g}^{rs} (\delta_r^a + {\epsilon_r}^a) (\delta_s^e + {\epsilon_s}^e)
\hat{\Gamma}_{uvw}
(\delta_e^u - {\epsilon_e}^u)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- e_b({\epsilon_c}^a)
$
${\Gamma^a}_{bc} =
(\delta_r^a + {\epsilon_r}^a)
\hat{g}^{ru}
\hat{\Gamma}_{uvw}
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- e_b({\epsilon_c}^a)
$
${\Gamma^a}_{bc} =
{\hat{\Gamma}^u}_{vw}
(\delta_u^a + {\epsilon_u}^a)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
- e_b({\epsilon_c}^a)
$
And as a sum of epsilon transforms:
${\Gamma^a}_{bc} =
{\hat{\Gamma}^a}_{bc}
+ {\epsilon_u}^a {\hat{\Gamma}^u}_{bc}
- {\epsilon_b}^u {\hat{\Gamma}^a}_{uc}
- {\epsilon_c}^u {\hat{\Gamma}^a}_{bu}
- e_b({\epsilon_c}^a)
$
${\Gamma^a}_{bc} =
(
{\hat{\Gamma}^u}_{vw}
- e_v({\epsilon_w}^u)
)
(\delta_u^a + {\epsilon_u}^a)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
$
${\Gamma^a}_{bc} =
(
{\hat{\Gamma}^u}_{vw}
- \hat{e}_v({\epsilon_w}^u)
)
(\delta_u^a + {\epsilon_u}^a)
(\delta_b^v - {\epsilon_b}^v)
(\delta_c^w - {\epsilon_c}^w)
$
So it looks like the only extra term is the $-e_b({\epsilon_c}^a)$.
This is very similar to the extra term in the difference between the Levi-Civita connection in a coordinate basis, transformed to a non-coordinate, and the Levi-Civita connection in a non-coordinate basis.
From the worksheet here: ${\hat\Gamma^a}_{bc} = {e^a}_\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} {\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a})$
Geodesic equation:
$\ddot{x}^u + {\Gamma^u}_{ab} \dot{x}^a \dot{x}^b = 0$
$\ddot{x}^u + \dot{x}^a \dot{x}^b
(
{\hat{\Gamma}^p}_{qr}
- \hat{e}_q({\epsilon_r}^p)
)
(\delta_p^u + {\epsilon_p}^u)
(\delta_a^q - {\epsilon_a}^q)
(\delta_b^r - {\epsilon_b}^r)
= 0$
$\ddot{x}^u
+ \dot{x}^a \dot{x}^b (
{\hat{\Gamma}^u}_{ab}
+ {\epsilon_v}^u {\hat{\Gamma}^v}_{ab}
- {\epsilon_a}^v {\hat{\Gamma}^u}_{vb}
- {\epsilon_b}^v {\hat{\Gamma}^u}_{av}
- \hat{e}_a({\epsilon_b}^u)
)
= 0$
$\ddot{x}^u
+ {\hat{\Gamma}^u}_{ab} \dot{x}^a \dot{x}^b
+ \dot{x}^a \dot{x}^b (
{\epsilon_v}^u {\hat{\Gamma}^v}_{ab}
- {\epsilon_a}^v {\hat{\Gamma}^u}_{vb}
- {\epsilon_b}^v {\hat{\Gamma}^u}_{av}
- \hat{e}_a({\epsilon_b}^u)
)
= 0$
Now for connection squared:
${\Gamma^a}_{be} {\Gamma^e}_{cd}
= (
{\hat{\Gamma}^p}_{qr}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_e^r - {\epsilon_e}^r)
- e_b({\epsilon_e}^a)
) (
{\hat{\Gamma}^u}_{vw}
(\delta_u^e + {\epsilon_u}^e)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
- e_c({\epsilon_d}^e)
)$
${\Gamma^a}_{be} {\Gamma^e}_{cd}
=
{\hat{\Gamma}^p}_{qu}
{\hat{\Gamma}^u}_{vw}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
- {\hat{\Gamma}^u}_{vw}
(\delta_u^e + {\epsilon_u}^e)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
e_b({\epsilon_e}^a)
- {\hat{\Gamma}^p}_{qr}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_e^r - {\epsilon_e}^r)
e_c({\epsilon_d}^e)
+ e_b({\epsilon_e}^a)
e_c({\epsilon_d}^e)
$
${\Gamma^a}_{be} {\Gamma^e}_{cd}
= {\hat{\Gamma}^p}_{qu}
{\hat{\Gamma}^u}_{vw}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
- {\hat{\Gamma}^e}_{cd} e_b({\epsilon_e}^a)
- {\hat{\Gamma}^a}_{be} e_c({\epsilon_d}^e)
+ e_b({\epsilon_e}^a) e_c({\epsilon_d}^e)
$
Now to take advantage of $|\epsilon|^2 \approx 0$
${\Gamma^a}_{be} {\Gamma^e}_{cd}
= (
{\hat{\Gamma}^p}_{qu}
{\hat{\Gamma}^u}_{vw}
- {\hat{\Gamma}^e}_{vw} e_q({\epsilon_e}^p)
- {\hat{\Gamma}^p}_{qe} e_v({\epsilon_w}^e)
+ e_q({\epsilon_e}^p) e_v({\epsilon_w}^e)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
$
${\Gamma^a}_{be} {\Gamma^e}_{cd}
= (
{\hat{\Gamma}^p}_{qu}
{\hat{\Gamma}^u}_{vw}
- {\hat{\Gamma}^e}_{vw} \hat{e}_q({\epsilon_e}^p)
- {\hat{\Gamma}^p}_{qe} \hat{e}_v({\epsilon_w}^e)
+ \hat{e}_q({\epsilon_e}^p) \hat{e}_v({\epsilon_w}^e)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
$
Antisymmetric portion:
${\Gamma^a}_{[c|e} {\Gamma^e}_{|d]b}
= {\hat{\Gamma}^p}_{qu}
{\hat{\Gamma}^u}_{vw}
(\delta_p^a + {\epsilon_p}^a)
(\delta_{[c|}^q - {\epsilon_{[c|}}^q)
(\delta_{|d]}^v - {\epsilon_{|d]}}^v)
(\delta_b^w - {\epsilon_b}^w)
- e_{[c|} ({\epsilon_e}^a) {\hat{\Gamma}^e}_{|d]b}
- {\hat{\Gamma}^a}_{[c|e} e_{|d]} ({\epsilon_b}^e)
+ e_{[c|} ({\epsilon_e}^a) e_{|d]} ({\epsilon_b}^e)
$
${\Gamma^a}_{[c|e} {\Gamma^e}_{|d]b}
= {\hat{\Gamma}^p}_{[q|u}
{\hat{\Gamma}^u}_{|v]w}
(\delta_p^a + {\epsilon_p}^a)
(\delta_c^q - {\epsilon_c}^q)
(\delta_d^v - {\epsilon_d}^v)
(\delta_b^w - {\epsilon_b}^w)
- e_{[c|} ({\epsilon_e}^a) {\hat{\Gamma}^e}_{|d]b}
- {\hat{\Gamma}^a}_{[c|e} e_{|d]} ({\epsilon_b}^e)
+ e_{[c|} ({\epsilon_e}^a) e_{|d]} ({\epsilon_b}^e)
$
Now using $|\epsilon|^2 = 0$
${\Gamma^a}_{[c|e} {\Gamma^e}_{|d]b}
= (
{\hat{\Gamma}^p}_{[v|u}
{\hat{\Gamma}^u}_{|w]q}
- e_{[v|} ({\epsilon_e}^p) {\hat{\Gamma}^e}_{|w]q}
- {\hat{\Gamma}^p}_{[v|e} e_{|w]} ({\epsilon_q}^e)
+ e_{[v|} ({\epsilon_e}^p) e_{|w]} ({\epsilon_q}^e)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
$
${\Gamma^a}_{[c|e} {\Gamma^e}_{|d]b}
= (
{\hat{\Gamma}^p}_{[v|u}
{\hat{\Gamma}^u}_{|w]q}
- \hat{e}_{[v|} ({\epsilon_e}^p) {\hat{\Gamma}^e}_{|w]q}
- {\hat{\Gamma}^p}_{[v|e} \hat{e}_{|w]} ({\epsilon_q}^e)
+ \hat{e}_{[v|} ({\epsilon_e}^p) \hat{e}_{|w]} ({\epsilon_q}^e)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
$
Ok, on to $e_d({\Gamma^a}_{bc})$:
$e_d({\Gamma^a}_{bc}) =
e_d(
{\hat{\Gamma}^a}_{bc}
+ {\epsilon_u}^a {\hat{\Gamma}^u}_{bc}
- {\epsilon_b}^u {\hat{\Gamma}^a}_{uc}
- {\epsilon_c}^u {\hat{\Gamma}^a}_{bu}
- e_b({\epsilon_c}^a)
)
$
$e_d({\Gamma^a}_{bc}) =
e_d({\hat{\Gamma}^a}_{bc})
+ e_d({\epsilon_u}^a {\hat{\Gamma}^u}_{bc})
- e_d({\epsilon_b}^u {\hat{\Gamma}^a}_{uc})
- e_d({\epsilon_c}^u {\hat{\Gamma}^a}_{bu})
- e_d(e_b({\epsilon_c}^a))
$
$e_d({\Gamma^a}_{bc}) =
e_d({\hat{\Gamma}^a}_{bc})
+ {\epsilon_u}^a e_d({\hat{\Gamma}^u}_{bc})
- {\epsilon_b}^u e_d({\hat{\Gamma}^a}_{uc})
- {\epsilon_c}^u e_d({\hat{\Gamma}^a}_{bu})
+ e_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- e_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- e_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- e_d(e_b({\epsilon_c}^a))
$
With respect to the original basis:
$e_d({\Gamma^a}_{bc}) =
\hat{e}_d({\hat{\Gamma}^a}_{bc})
+ {\epsilon_u}^a \hat{e}_d({\hat{\Gamma}^u}_{bc})
- {\epsilon_b}^u \hat{e}_d({\hat{\Gamma}^a}_{uc})
- {\epsilon_c}^u \hat{e}_d({\hat{\Gamma}^a}_{bu})
- {\epsilon_d}^u \hat{e}_u({\hat{\Gamma}^a}_{bc})
+ \hat{e}_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- \hat{e}_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- \hat{e}_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- (\delta_d^u - {\epsilon_d}^u) \hat{e}_u( (\delta_b^v - {\epsilon_b}^v) \hat{e}_v ({\epsilon_c}^a))
$
$e_d({\Gamma^a}_{bc}) =
\hat{e}_d({\hat{\Gamma}^a}_{bc})
+ {\epsilon_u}^a \hat{e}_d({\hat{\Gamma}^u}_{bc})
- {\epsilon_b}^u \hat{e}_d({\hat{\Gamma}^a}_{uc})
- {\epsilon_c}^u \hat{e}_d({\hat{\Gamma}^a}_{bu})
- {\epsilon_d}^u \hat{e}_u({\hat{\Gamma}^a}_{bc})
+ \hat{e}_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- \hat{e}_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- \hat{e}_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- \hat{e}_d( \hat{e}_b ({\epsilon_c}^a))
+ {\epsilon_d}^u \hat{e}_u( \hat{e}_b ({\epsilon_c}^a))
$
Seems if I can argue that ${\epsilon_a}^u e_v({\epsilon_u}^b) \approx 0$ then why not
${\epsilon_d}^u \hat{e}_u( \hat{e}_b ({\epsilon_c}^a)) \approx 0$?
Let's see:
$e(|\epsilon|^2) = |\epsilon| e(|\epsilon|)$
So $0 = e(0) \approx e^2(|\epsilon|^2) = 2 e(|\epsilon| e(|\epsilon|)) = e(|\epsilon|)^2 + |\epsilon| e^2(|\epsilon|)$
So $e(|\epsilon|)^2 = -|\epsilon| e^2(|\epsilon|)$
But there's an issue with this identity, which is if you relate it back to tensor expressions then where do you insert the indexes?
This was easier to do with $e(|\epsilon|^2) = |\epsilon| e(|\epsilon|) = 0$ implies ${\epsilon_a}^b e_c({\epsilon_d}^e) = 0$, because there's a zero on one side,
But with tensors on both sides, where would the indexes go?
$0 = e_c(e_d(0))$
$\approx e_c(e_d({\epsilon_b}^u {\epsilon_u}^a))$
$0 \approx e_c(
e_d({\epsilon_b}^u) {\epsilon_u}^a
+ {\epsilon_b}^u e_d({\epsilon_u}^a)
)$
$0 \approx
e_c(e_d({\epsilon_b}^u)) {\epsilon_u}^a
+ e_d({\epsilon_b}^u) e_c({\epsilon_u}^a)
+ e_c({\epsilon_b}^u) e_d({\epsilon_u}^a)
+ {\epsilon_b}^u e_c(e_d({\epsilon_u}^a))
$
Now the antisymmetric portion:
$0 \approx
e_{[c}(e_{d]}({\epsilon_b}^u)) {\epsilon_u}^a
+ e_{[d|}({\epsilon_b}^u) e_{|c]}({\epsilon_u}^a)
+ e_{[c|}({\epsilon_b}^u) e_{|d]}({\epsilon_u}^a)
+ {\epsilon_b}^u e_{[c}(e_{d]}({\epsilon_u}^a))
$
$0 \approx
\frac{1}{2} {c_{cd}}^v e_v({\epsilon_b}^u) {\epsilon_u}^a
+ e_{[d|}({\epsilon_b}^u) e_{|c]}({\epsilon_u}^a)
- e_{[d|}({\epsilon_b}^u) e_{|c]}({\epsilon_u}^a)
+ \frac{1}{2} {\epsilon_b}^u {c_{cd}}^v e_v({\epsilon_u}^a)
$
$0 \approx
{c_{cd}}^v (
e_v({\epsilon_b}^u) {\epsilon_u}^a
+ {\epsilon_b}^u e_v({\epsilon_u}^a)
)
$
And that looks like our old familiar rule of $|\epsilon| e(|\epsilon|) \approx 0$, which is based on $e(|\epsilon|^2) \approx 0$.
$0 \approx {c_{cd}}^v e_v({\epsilon_b}^u {\epsilon_u}^a)$
...which comes from $0 \approx {\epsilon_b}^u {\epsilon_u}^a$ and $0 = e_v(0)$
But notice that, when considering the antisymmetric portion of this identity, the 1st derivatives cancel.
Meaning, we have no approximation for $e_{[c}(|\epsilon|) e_{d]}(|\epsilon|)$, only an approximation for $e_{(c}(|\epsilon|) e_{d)}(|\epsilon|)$.
$e_d({\Gamma^a}_{bc}) =
\hat{e}_d({\hat{\Gamma}^a}_{bc})
+ {\epsilon_u}^a \hat{e}_d({\hat{\Gamma}^u}_{bc})
- {\epsilon_b}^u \hat{e}_d({\hat{\Gamma}^a}_{uc})
- {\epsilon_c}^u \hat{e}_d({\hat{\Gamma}^a}_{bu})
- {\epsilon_d}^u \hat{e}_u({\hat{\Gamma}^a}_{bc})
+ \hat{e}_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- \hat{e}_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- \hat{e}_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- \hat{e}_d( \hat{e}_b ({\epsilon_c}^a))
+ {\epsilon_d}^u \hat{e}_u( \hat{e}_b ({\epsilon_c}^a))
$
With the last two terms replacing $\hat{e}_a$ with $e_a$ and recombined:
$e_d({\Gamma^a}_{bc}) =
\hat{e}_d({\hat{\Gamma}^a}_{bc})
+ {\epsilon_u}^a \hat{e}_d({\hat{\Gamma}^u}_{bc})
- {\epsilon_b}^u \hat{e}_d({\hat{\Gamma}^a}_{uc})
- {\epsilon_c}^u \hat{e}_d({\hat{\Gamma}^a}_{bu})
- {\epsilon_d}^u \hat{e}_u({\hat{\Gamma}^a}_{bc})
+ \hat{e}_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- \hat{e}_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- \hat{e}_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- e_d(e_b({\epsilon_c}^a))
$
Now as a product of epsilons:
$e_d({\Gamma^a}_{bc}) =
\hat{e}_s({\hat{\Gamma}^p}_{qr})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- \hat{e}_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- \hat{e}_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- \hat{e}_d( \hat{e}_b ({\epsilon_c}^a))
+ {\epsilon_d}^u \hat{e}_u( \hat{e}_b ({\epsilon_c}^a))
$
And then maybe if you want to use $e_u$ instead of $\hat{e}_u$:
$e_d({\Gamma^a}_{bc}) =
\hat{e}_s({\hat{\Gamma}^p}_{qr})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_d({\epsilon_u}^a) {\hat{\Gamma}^u}_{bc}
- \hat{e}_d({\epsilon_b}^u) {\hat{\Gamma}^a}_{uc}
- \hat{e}_d({\epsilon_c}^u) {\hat{\Gamma}^a}_{bu}
- e_d( e_b ({\epsilon_c}^a))
$
Now the antisymmetric portion:
$e_{[c}({\Gamma^a}_{d]b}) =
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_{[c|}({\epsilon_u}^a) {\hat{\Gamma}^u}_{|d]b}
- \hat{e}_{[c|}({\epsilon_{|d]}}^u) {\hat{\Gamma}^a}_{ub}
- \hat{e}_{[c|}({\epsilon_b}^u) {\hat{\Gamma}^a}_{|d]u}
- \hat{e}_{[c}(\hat{e}_{d]} ({\epsilon_b}^a))
+ {\epsilon_{[c|}}^u \hat{e}_u( \hat{e}_{|d]} ({\epsilon_b}^a))
$
$e_{[c}({\Gamma^a}_{d]b}) =
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_{[c|}({\epsilon_u}^a) {\hat{\Gamma}^u}_{|d]b}
- \hat{e}_{[c|}({\epsilon_{|d]}}^u) {\hat{\Gamma}^a}_{ub}
- \hat{e}_{[c|}({\epsilon_b}^u) {\hat{\Gamma}^a}_{|d]u}
- \frac{1}{2} {\hat{c}_{cd}}^u \hat{e}_u ({\epsilon_b}^a))
+ {\epsilon_{[c|}}^u \hat{e}_u( \hat{e}_{|d]} ({\epsilon_b}^a))
$
In terms of the permuted basis:
$e_{[c}({\Gamma^a}_{d]b}) =
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_{[c|}({\epsilon_u}^a) {\hat{\Gamma}^u}_{|d]b}
- \hat{e}_{[c|}({\epsilon_{|d]}}^u) {\hat{\Gamma}^a}_{ub}
- \hat{e}_{[c|}({\epsilon_b}^u) {\hat{\Gamma}^a}_{|d]u}
- e_{[c}(e_{d]} ({\epsilon_b}^a))
$
$e_{[c}({\Gamma^a}_{d]b}) =
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_{[c|}({\epsilon_u}^a) {\hat{\Gamma}^u}_{|d]b}
- \hat{e}_{[c|}({\epsilon_{|d]}}^u) {\hat{\Gamma}^a}_{ub}
- \hat{e}_{[c|}({\epsilon_b}^u) {\hat{\Gamma}^a}_{|d]u}
- \frac{1}{2} {c_{cd}}^u e_u ({\epsilon_b}^a))
$
Riemann curvature tensor:
${R^a}_{bcd} =
2 e_{[c}({\Gamma^a}_{d]b})
+ 2 {\Gamma^a}_{[c|e} {\Gamma^e}_{|d]b}
- {\Gamma^a}_{ub} {c_{cd}}^u
$
${R^a}_{bcd} =
2 (
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ \hat{e}_{[c|}({\epsilon_u}^a) {\hat{\Gamma}^u}_{|d]b}
- \hat{e}_{[c|}({\epsilon_{|d]}}^u) {\hat{\Gamma}^a}_{ub}
- \hat{e}_{[c|}({\epsilon_b}^u) {\hat{\Gamma}^a}_{|d]u}
- \frac{1}{2} {c_{cd}}^u e_u ({\epsilon_b}^a))
)
+ 2 (
{\hat{\Gamma}^p}_{[v|u}
{\hat{\Gamma}^u}_{|w]q}
- \hat{e}_{[v|} ({\epsilon_e}^p) {\hat{\Gamma}^e}_{|w]q}
- {\hat{\Gamma}^p}_{[v|e} \hat{e}_{|w]} ({\epsilon_q}^e)
+ \hat{e}_{[v|} ({\epsilon_e}^p) \hat{e}_{|w]} ({\epsilon_q}^e)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^v - {\epsilon_c}^v)
(\delta_d^w - {\epsilon_d}^w)
- (
{\hat{\Gamma}^u}_{vw}
- \hat{e}_v({\epsilon_w}^u)
)
(\delta_u^a + {\epsilon_u}^a)
(\delta_e^v - {\epsilon_e}^v)
(\delta_b^w - {\epsilon_b}^w)
{c_{cd}}^e
$
${R^a}_{bcd} =
2 (
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
+ e_{[r|}({\epsilon_u}^p) {\hat{\Gamma}^u}_{|s]q}
- e_{[r|}({\epsilon_{|s]}}^u) {\hat{\Gamma}^p}_{uq}
- e_{[r|}({\epsilon_q}^u) {\hat{\Gamma}^p}_{|s]u}
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
- {c_{cd}}^u e_u({\epsilon_b}^a)
+ 2 (
{\hat{\Gamma}^p}_{[r|u}
{\hat{\Gamma}^u}_{|s]q}
- e_{[r|}({\epsilon_e}^p) {\hat{\Gamma}^e}_{|s]q}
- e_{[s|}({\epsilon_q}^e) {\hat{\Gamma}^p}_{|r]e}
+ e_{[r|}({\epsilon_e}^p) e_{|s]}({\epsilon_q}^e)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
- {\hat{\Gamma}^u}_{vw}
{c_{cd}}^e
(\delta_u^a + {\epsilon_u}^a)
(\delta_e^v - {\epsilon_e}^v)
(\delta_b^w - {\epsilon_b}^w)
+ \hat{e}_e({\epsilon_b}^a)
{c_{cd}}^e
$
${R^a}_{bcd} =
2 (
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
+ {\hat{\Gamma}^p}_{[r|u} {\hat{\Gamma}^u}_{|s]q}
+ e_{[r|}({\epsilon_u}^p) e_{|s]}({\epsilon_q}^u)
- e_{[r|}({\epsilon_{|s]}}^u) {\hat{\Gamma}^p}_{uq}
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
- {\hat{\Gamma}^p}_{fr}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^r - {\epsilon_b}^r)
(\delta_e^f - {\epsilon_e}^f)
(
{\hat{c}_{uv}}^w
- 2 e_{[u} ({\epsilon_{v]}}^w)
)
(\delta_c^u - {\epsilon_c}^u)
(\delta_d^v - {\epsilon_d}^v)
(\delta_w^e + {\epsilon_w}^e)
$
${R^a}_{bcd} =
2 (
\hat{e}_{[r|}({\hat{\Gamma}^p}_{|s]q})
+ {\hat{\Gamma}^p}_{[r|u} {\hat{\Gamma}^u}_{|s]q}
+ e_{[r|}({\epsilon_u}^p) e_{|s]}({\epsilon_q}^u)
- e_{[r|}({\epsilon_{|s]}}^u) {\hat{\Gamma}^p}_{uq}
- \frac{1}{2} {\hat{\Gamma}^p}_{wq} {\hat{c}_{rs}}^w
+ {\hat{\Gamma}^p}_{wq} e_{[r} ({\epsilon_{s]}}^w)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
$
${R^a}_{bcd} =
(
{\hat{R}^p}_{qrs}
+ 2 e_{[r|}({\epsilon_u}^p) e_{|s]}({\epsilon_q}^u)
)
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
$
${R^a}_{bcd} =
{\hat{R}^p}_{qrs}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 e_{[c|}({\epsilon_u}^a) e_{|d]}({\epsilon_b}^u)
$
TODO quadruple check
If this is correct then this is great, it means that all commutation coefficients cancel.
The only difference between the tensors is in terms of partials of the permutation tensor.
${R^a}_{bcd} =
{\hat{R}^p}_{qrs}
(\delta_p^a + {\epsilon_p}^a)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 e_{[c|}({\epsilon_u}^a) e_{|d]}({\epsilon_b}^u)
$
$R_{abcd} = g_{ae} {R^e}_{bcd}$
$=
\hat{g}_{uv}
(\delta_a^u - {\epsilon_a}^u)
(\delta_e^v - {\epsilon_e}^v)
{\hat{R}^p}_{qrs}
(\delta_p^e + {\epsilon_p}^e)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 g_{ae} e_{[c|}({\epsilon_u}^e) e_{|d]}({\epsilon_b}^u)
$
$=
(\delta_a^u - {\epsilon_a}^u)
\hat{g}_{uv}
{\hat{R}^v}_{qrs}
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 g_{ae} e_{[c|}({\epsilon_u}^e) e_{|d]}({\epsilon_b}^u)
$
$=
\hat{R}_{pqrs}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 g_{ae} e_{[c|}({\epsilon_u}^e) e_{|d]}({\epsilon_b}^u)
$
So now to prove the properties of $R_{abcd}$.
We know $R_{abcd} = R_{ab[cd]}$ because (1) $\hat{R}_{abcd} = \hat{R}_{ab[cd]}$ and (2) the remaining term $2 g_{ae} e_{[c|}({\epsilon_u}^e) e_{|d]}({\epsilon_b}^u)$ is antisymmetric on c and d.
How about proving $R_{abcd} = R_{[ab]cd}$? We already know $\hat{R}_{abcd} = \hat{R}_{[ab]cd}$. How about the other term?
Well instead of doing this, how about I just prove ${R^a}_{acd} = 0$, which means $g^{ab} R_{abcd} = 0$, which means $R_{abcd} = R_{[ab]cd}$.
${R^a}_{acd} =
{\hat{R}^p}_{qrs}
(\delta_p^a + {\epsilon_p}^a)
(\delta_a^q - {\epsilon_a}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 e_{[c|}({\epsilon_u}^a) e_{|d]}({\epsilon_a}^u)
$
$=
{\hat{R}^a}_{ars}
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 e_{[c|}({\epsilon_u}^a) e_{|d]}({\epsilon_a}^u)
$
Assuming ${\hat{R}^u}_{urs} = 0$...
$= 2 e_{[c|}({\epsilon_u}^a) e_{|d]}({\epsilon_a}^u)$
$= e_c({\epsilon_u}^a) e_d({\epsilon_a}^u) - e_d({\epsilon_u}^a) e_c({\epsilon_a}^u)$
Relabel the sum indexes on the second term and switch the multiplication order to find...
$= e_c({\epsilon_u}^a) e_d({\epsilon_a}^u)
- e_c({\epsilon_u}^a) e_d({\epsilon_a}^u)$
$= 0$.
Voila.
So ${R^a}_{acd} = 0$, so $R_{(ab)cd} = 0$, so $R_{abcd} = R_{[ab]cd}$.
How about verifying $R_{abcd} = -R_{bacd}$?
$R_{abcd} =
\hat{R}_{pqrs}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
(\delta_c^r - {\epsilon_c}^r)
(\delta_d^s - {\epsilon_d}^s)
+ 2 g_{ae} e_{[c|}({\epsilon_u}^e) e_{|d]}({\epsilon_b}^u)
$
$R_{abcd} =
E(\hat{R}_{abcd})
+ g_{ae} e_c({\epsilon_u}^e) e_d({\epsilon_b}^u)
- g_{ae} e_d({\epsilon_u}^e) e_c({\epsilon_b}^u)
$
$R_{abcd} =
E(\hat{R}_{abcd})
+ g_{ae} e_c(\epsilon_{uf} g^{ef}) e_d(\epsilon_{bv} g^{uv})
- g_{ae} e_d(\epsilon_{uf} g^{ef}) e_c(\epsilon_{bv} g^{uv})
$
$R_{abcd} =
E(\hat{R}_{abcd})
+ g_{ae} (
e_c(\epsilon_{uf}) g^{ef}
+ \epsilon_{uf} e_c(g^{ef})
) (
e_d(\epsilon_{bv}) g^{uv}
+ \epsilon_{bv} e_d(g^{uv})
)
- g_{ae} (
e_d(\epsilon_{uf}) g^{ef}
+ \epsilon_{uf} e_d(g^{ef})
) (
e_c(\epsilon_{bv}) g^{uv}
+ \epsilon_{bv} e_c(g^{uv})
)
$
$R_{abcd} =
E(\hat{R}_{abcd})
+ g_{ae} (
e_d(\epsilon_{bv}) e_c(\epsilon_{uf}) g^{uv} g^{ef}
+ \epsilon_{uf} e_d(\epsilon_{bv}) g^{uv} e_c(g^{ef})
+ \epsilon_{bv} e_c(\epsilon_{uf}) e_d(g^{uv}) g^{ef}
+ \epsilon_{bv} \epsilon_{uf} e_d(g^{uv}) e_c(g^{ef})
- e_c(\epsilon_{bv}) e_d(\epsilon_{uf}) g^{uv} g^{ef}
- \epsilon_{uf} e_c(\epsilon_{bv}) g^{uv} e_d(g^{ef})
- \epsilon_{bv} e_d(\epsilon_{uf}) e_c(g^{uv}) g^{ef}
- \epsilon_{bv} \epsilon_{uf} e_c(g^{uv}) e_d(g^{ef})
)
$
Using $|\epsilon|^2 \approx 0$ and $e(|\epsilon|^2) = 0$:
$R_{abcd} =
E(\hat{R}_{abcd})
+ g_{ae} g^{uv} g^{ef} (
e_d(\epsilon_{bv}) e_c(\epsilon_{uf})
- e_c(\epsilon_{bv}) e_d(\epsilon_{uf})
)
$
$R_{abcd} =
E(\hat{R}_{abcd})
+ 2 g^{uv} e_{[c|}(\epsilon_{ua}) e_{|d]}(\epsilon_{bv})
$
So with this we see:
$R_{bacd} =
E(\hat{R}_{bacd})
+ 2 g^{uv} e_{[c|}(\epsilon_{ub}) e_{|d]}(\epsilon_{av})
$
using $\hat{R}_{abcd} = -\hat{R}_{bacd}$ and linearity of $E(\cdot)$:
$R_{bacd} =
- E(\hat{R}_{abcd})
- 2 g^{uv} e_{[c|}(\epsilon_{au}) e_{|d]}(\epsilon_{vb})
$
I guess here I also have to invoke symmetry, $\epsilon_{ab} = \epsilon_{(ab)}$, but since it wasn't necessary to prove ${R^a}_{acd} = 0$, I wonder if it really is a necessary condition to prove $R_{abcd} = -R_{bacd}$?
$R_{bacd} = -R_{abcd}$
Ricci curvature tensor:
$R_{ab} = {R^u}_{aub}$
$R_{ab} =
{\hat{R}^v}_{pvq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
+ 2 e_{[v|}({\epsilon_u}^v) e_{|b]}({\epsilon_a}^u)
$
$R_{ab} =
\hat{R}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
+ e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
- e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
$
As a product:
$R_{ab} =
(
\hat{R}_{pq}
+ e_v({\epsilon_u}^v) e_q({\epsilon_p}^u)
- e_q({\epsilon_u}^v) e_v({\epsilon_p}^u)
)
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
$
Based on the lowered Riemann:
$R_{ab} = g^{uv} R_{uavb}$
$=
g^{uv}
\hat{R}_{pqrs}
(\delta_u^p - {\epsilon_u}^p)
(\delta_a^q - {\epsilon_a}^q)
(\delta_v^r - {\epsilon_v}^r)
(\delta_b^s - {\epsilon_b}^s)
+ 2
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$=
\hat{g}^{mn}
(\delta_m^u + {\epsilon_m}^u)
(\delta_n^v + {\epsilon_n}^v)
\hat{R}_{pqrs}
(\delta_u^p - {\epsilon_u}^p)
(\delta_a^q - {\epsilon_a}^q)
(\delta_v^r - {\epsilon_v}^r)
(\delta_b^s - {\epsilon_b}^s)
+ 2
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$=
\hat{g}^{pr}
\hat{R}_{pqrs}
(\delta_a^q - {\epsilon_a}^q)
(\delta_b^s - {\epsilon_b}^s)
+ 2
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$=
\hat{R}_{uv}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
+ 2
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
Gaussian scalar curvature:
$R = g^{ab} R_{ab}$
$R =
\hat{g}^{mn}
(\delta_m^a + {\epsilon_m}^a)
(\delta_n^b + {\epsilon_n}^b)
\hat{R}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
+ \hat{g}^{mn}
(\delta_m^a + {\epsilon_m}^a) (\delta_n^b + {\epsilon_n}^b)
(
+ e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
- e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
)$
$R =
\hat{R}
+ e_v({\epsilon_u}^v) e_a({\epsilon_b}^u) g^{ab}
- e_a({\epsilon_u}^v) e_v({\epsilon_b}^u) g^{ab}
$
Based on the lowered Riemann:
$R = g^{ab} R_{ab}$
$R =
g^{ab}
\hat{R}_{uv}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
+ 2
g^{ab}
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$R =
\hat{g}^{pq}
(\delta_p^a + {\epsilon_p}^a)
(\delta_q^b + {\epsilon_q}^b)
\hat{R}_{uv}
(\delta_a^u - {\epsilon_a}^u)
(\delta_b^v - {\epsilon_b}^v)
+ 2
g^{ab}
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$R =
\hat{g}^{uv}
\hat{R}_{uv}
+ 2
g^{ab}
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$R =
\hat{R}
+ 2
g^{ab}
g^{pq}
g^{uv}
e_{[v|}(\epsilon_{up})
e_{|b]}(\epsilon_{qa})
$
$R =
\hat{R}
+
g^{ab}
g^{pq}
g^{uv}
e_v(\epsilon_{up})
e_b(\epsilon_{qa})
-
g^{ab}
g^{pq}
g^{uv}
e_b(\epsilon_{up})
e_v(\epsilon_{qa})
$
Assuming $\epsilon_{ab}$ is symmetric:
$R =
\hat{R}
+
g^{ab}
g^{cd}
g^{ef}
e_a(\epsilon_{bc})
e_e(\epsilon_{df})
-
g^{ab}
g^{cd}
g^{ef}
e_a(\epsilon_{ce})
e_d(\epsilon_{bf})
$
For the record, I think there's only 5 possible unique permutations of performing 3 traces on $e(\epsilon)^2$ when $\epsilon_{ab}$ is symmetric (Tell me if I missed any):
$g^{ab} g^{cd} g^{ef} ( e_a(\epsilon_{bc}) e_d(\epsilon_{ef}) )$
$g^{ab} g^{cd} g^{ef} ( e_a(\epsilon_{bc}) e_e(\epsilon_{df}) )$ ... we're using +
$g^{ab} g^{cd} g^{ef} ( e_a(\epsilon_{cd}) e_b(\epsilon_{ef}) )$
$g^{ab} g^{cd} g^{ef} ( e_a(\epsilon_{ce}) e_b(\epsilon_{df}) )$
$g^{ab} g^{cd} g^{ef} ( e_a(\epsilon_{ce}) e_d(\epsilon_{bf}) )$ ... we're using -
Also, there's something interesting to the study of how many possible unique traces a (pseudo)tensor expression can have based on its (anti)symmetries.
Einstein tensor:
$G_{ab} = R_{ab} - \frac{1}{2} R g_{ab}$
$G_{ab} =
(
\hat{R}_{pq}
+ e_v({\epsilon_u}^v) e_q({\epsilon_p}^u)
- e_q({\epsilon_u}^v) e_v({\epsilon_p}^u)
- \frac{1}{2}
\hat{g}_{pq}
(
\hat{R}
+ \hat{g}^{ab}
(
e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
- e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
)
)
)
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
$
$G_{ab} =
(
\hat{G}_{pq}
+ e_v({\epsilon_u}^v) e_q({\epsilon_p}^u)
- e_q({\epsilon_u}^v) e_v({\epsilon_p}^u)
- \frac{1}{2} \hat{g}_{pq} \hat{g}^{ab}
(
e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
- e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
)
)
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
$
$G_{ab} =
\hat{G}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
+ e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
- e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
- 2 (
e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
- e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
)
$
$G_{ab} =
\hat{G}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
- e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
+ e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
$
So sure enough, the Einstein tensor is just a trace-reversal on the extra components added to the Ricci tensor (as it should).
And if we started with the lowered form then we get:
$G_{ab} =
\hat{G}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
+ 2
g^{pq}
g^{uv}
(
e_b(\epsilon_{up})
e_v(\epsilon_{qa})
-
e_v(\epsilon_{up})
e_b(\epsilon_{qa})
)
$
Now if we assume a background metric with constant coefficients then we get:
$G_{ab} =
e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
- e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
$
And if we assume a coordinate basis:
$G_{ab} =
\partial_b({\epsilon_u}^v) \partial_v({\epsilon_a}^u)
- \partial_v({\epsilon_u}^v) \partial_b({\epsilon_a}^u)
$
Now to equate with stress-energy:
Some popular stress-energies from 1973 Misner, Thorne, Wheeler "Gravitation":
$\eta \ge 0 = $ coefficient of dynamic viscosity.
$\zeta \ge 0 = $ coefficient of bulk viscosity.
$P_{ab} = g_{ab} + u_a u_b$
$\theta = \nabla_a u^a$
$\sigma_{ab} = \nabla_u u_{(a} {P^u}_{b)} - \frac{1}{3} \theta P_{ab}$
$T_{ab} = T^{hydro}_{ab} + T^{visc}_{ab} + T^{heat}_{ab} + T^{EM}_{ab}$ ...
$T^{hydro}_{ab} = \rho u_a u_b + P_{ab} P$
$T^{visc}_{ab} = - (g_{ab} + u_a u_b) \zeta \theta - 2 \eta \sigma_{ab}$
$T^{heat}_{ab} = q_a u_b + u_a q_b$
$T^{EM}_{ab} = \frac{1}{\mu_0} ({F_a}^u F_{bu} - \frac{1}{4} g_{ab} F^{uv} F_{uv})$
$G_{ab} = 8 \pi T_{ab}$
$8 \pi T_{ab} =
\hat{G}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
- e_v({\epsilon_u}^v) e_b({\epsilon_a}^u)
+ e_b({\epsilon_u}^v) e_v({\epsilon_a}^u)
$
or
$R_{ab} = 8 \pi (T_{ab} - \frac{1}{2} g_{ab} g^{uv} T_{uv})$
or
$8 \pi T_{ab} =
\hat{G}_{pq}
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
+ 2
g^{pq}
g^{uv}
(
e_b(\epsilon_{up})
e_v(\epsilon_{qa})
-
e_v(\epsilon_{up})
e_b(\epsilon_{qa})
)
$
$8 \pi T_{ab} =
(\delta_a^p - {\epsilon_a}^p)
(\delta_b^q - {\epsilon_b}^q)
(
\hat{G}_{pq}
+ 2
g^{cd}
g^{uv}
(
e_q(\epsilon_{uc})
e_v(\epsilon_{dp})
-
e_v(\epsilon_{uc})
e_q(\epsilon_{dp})
)
)
$
$8 \pi T_{pq}
(\delta_a^p + {\epsilon_a}^p)
(\delta_b^q + {\epsilon_b}^q)
- \hat{G}_{ab}
=
g^{cd}
g^{uv}
(
e_b(\epsilon_{uc})
e_v(\epsilon_{da})
-
e_v(\epsilon_{uc})
e_b(\epsilon_{da})
)
$
taking advantage of $|\epsilon|^2 = 0$...
$8 \pi T_{pq}
(\delta_a^p + {\epsilon_a}^p)
(\delta_b^q + {\epsilon_b}^q)
- \hat{G}_{ab}
=
\hat{g}^{cd}
\hat{g}^{uv}
(
\hat{e}_b(\epsilon_{uc})
\hat{e}_v(\epsilon_{da})
-
\hat{e}_v(\epsilon_{uc})
\hat{e}_b(\epsilon_{da})
)
$
Let $\hat{T}_{ab} = $ the stress-energy tensor in the background basis,
such that $\hat{T}_{ab} = E(T_{ab})$.
$8 \pi \hat{T}_{ab}
- \hat{G}_{ab}
=
\hat{g}^{cd}
\hat{g}^{uv}
(
\hat{e}_b(\epsilon_{uc})
\hat{e}_v(\epsilon_{da})
-
\hat{e}_v(\epsilon_{uc})
\hat{e}_b(\epsilon_{da})
)
$
Considering 1+1 dimensions:
$8 \pi \hat{T}_{ab} - \hat{G}_{ab} =
+ \hat{g}^{tt} \hat{g}^{tt} \hat{e}_t(\epsilon_{ta}) \hat{e}_b(\epsilon_{tt})
- \hat{g}^{tt} \hat{g}^{tt} \hat{e}_t(\epsilon_{tt}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{tt} \hat{g}^{tx} \hat{e}_t(\epsilon_{xa}) \hat{e}_b(\epsilon_{tt})
- \hat{g}^{tt} \hat{g}^{tx} \hat{e}_t(\epsilon_{tt}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{tt} \hat{g}^{tx} \hat{e}_t(\epsilon_{ta}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tt} \hat{g}^{tx} \hat{e}_t(\epsilon_{tx}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{tt} \hat{g}^{tx} \hat{e}_t(\epsilon_{ta}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tt} \hat{g}^{tx} \hat{e}_t(\epsilon_{tx}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{tt} \hat{g}^{xx} \hat{e}_t(\epsilon_{xa}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tt} \hat{g}^{xx} \hat{e}_t(\epsilon_{tx}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{tx} \hat{g}^{tx} \hat{e}_t(\epsilon_{xa}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tx} \hat{g}^{tx} \hat{e}_t(\epsilon_{tx}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{tx} \hat{g}^{tx} \hat{e}_t(\epsilon_{ta}) \hat{e}_b(\epsilon_{xx})
- \hat{g}^{tx} \hat{g}^{tx} \hat{e}_t(\epsilon_{xx}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{xx} \hat{g}^{tx} \hat{e}_t(\epsilon_{xa}) \hat{e}_b(\epsilon_{xx})
- \hat{g}^{xx} \hat{g}^{tx} \hat{e}_t(\epsilon_{xx}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{tt} \hat{g}^{tx} \hat{e}_x(\epsilon_{ta}) \hat{e}_b(\epsilon_{tt})
- \hat{g}^{tt} \hat{g}^{tx} \hat{e}_x(\epsilon_{tt}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{tt} \hat{g}^{xx} \hat{e}_x(\epsilon_{ta}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tt} \hat{g}^{xx} \hat{e}_x(\epsilon_{tx}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{tx} \hat{g}^{tx} \hat{e}_x(\epsilon_{xa}) \hat{e}_b(\epsilon_{tt})
- \hat{g}^{tx} \hat{g}^{tx} \hat{e}_x(\epsilon_{tt}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{tx} \hat{g}^{xx} \hat{e}_x(\epsilon_{xa}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tx} \hat{g}^{xx} \hat{e}_x(\epsilon_{tx}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{tx} \hat{g}^{tx} \hat{e}_x(\epsilon_{ta}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{tx} \hat{g}^{tx} \hat{e}_x(\epsilon_{tx}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{tx} \hat{g}^{xx} \hat{e}_x(\epsilon_{ta}) \hat{e}_b(\epsilon_{xx})
- \hat{g}^{tx} \hat{g}^{xx} \hat{e}_x(\epsilon_{xx}) \hat{e}_b(\epsilon_{ta})
+ \hat{g}^{xx} \hat{g}^{tx} \hat{e}_x(\epsilon_{xa}) \hat{e}_b(\epsilon_{tx})
- \hat{g}^{xx} \hat{g}^{tx} \hat{e}_x(\epsilon_{tx}) \hat{e}_b(\epsilon_{xa})
+ \hat{g}^{xx} \hat{g}^{xx} \hat{e}_x(\epsilon_{xa}) \hat{e}_b(\epsilon_{xx})
- \hat{g}^{xx} \hat{g}^{xx} \hat{e}_x(\epsilon_{xx}) \hat{e}_b(\epsilon_{xa})
$
$8 \pi \hat{T}_{tt} - \hat{G}_{tt} =
- \hat{g}^{tt} \hat{g}^{xx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{tx}) - \hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{tt}))
- \hat{g}^{tx} \hat{g}^{xx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tt}))
- \hat{g}^{xx} \hat{g}^{xx} (\hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tx}))
$
$8 \pi \hat{T}_{tx} - \hat{G}_{tx} =
\hat{g}^{tt} \hat{g}^{tx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{tx}) - \hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{tt}))
+ \hat{g}^{tx} \hat{g}^{tx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tt}))
+ \hat{g}^{tx} \hat{g}^{xx} (\hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tx}))
$
$8 \pi \hat{T}_{xt} - \hat{G}_{xt} =
\hat{g}^{tt} \hat{g}^{tx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{tx}) - \hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{tt}))
+ \hat{g}^{tx} \hat{g}^{tx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tt}))
+ \hat{g}^{tx} \hat{g}^{xx} (\hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tx}))
$
$8 \pi \hat{T}_{xx} - \hat{G}_{xx} =
- \hat{g}^{tt} \hat{g}^{tt} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{tx}) - \hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{tt}))
- \hat{g}^{tt} \hat{g}^{tx} (\hat{e}_t(\epsilon_{tt}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tt}))
- \hat{g}^{tt} \hat{g}^{xx} (\hat{e}_t(\epsilon_{tx}) \hat{e}_x(\epsilon_{xx}) - \hat{e}_t(\epsilon_{xx}) \hat{e}_x(\epsilon_{tx}))
$
So the Einstein Field Equations constraints for 1+1 dimensions appear to be symmetric, even though in general for Einstein-Cartan they do not necessarily have to be.
Considering $\hat{e}_t$ as a distinct coordinate basis:
$8 \pi T_{tt} =
\hat{G}_{pq}
(\delta_t^p - {\epsilon_t}^p)
(\delta_t^q - {\epsilon_t}^q)
- e_v({\epsilon_u}^v) e_t({\epsilon_t}^u)
+ e_t({\epsilon_u}^v) e_v({\epsilon_t}^u)
$
$8 \pi T_{tt}
-
\hat{G}_{pq}
(\delta_t^p - {\epsilon_t}^p)
(\delta_t^q - {\epsilon_t}^q)
=
- e_v({\epsilon_u}^v) e_t({\epsilon_t}^u)
+ e_t({\epsilon_u}^v) e_v({\epsilon_t}^u)
$
$8 \pi T_{ti} = $...
$8 \pi T_{ij} = $...
Side bar: In such an event that that the original basis is a coordinate system, and ${\hat{c}_{ab}}^c = 0$, how would the rest of this look?
Our relations between $g_{ab}$ and $\hat{g}_{ab}$ wouldn't change.
Our partial of our metric would be unchanged:
$e_c(g_{ab}) =
e_c(\hat{g}_{ab})
- {\epsilon_b}^u e_c(\hat{g}_{ua})
- {\epsilon_a}^u e_c(\hat{g}_{ub})
- e_c({\epsilon_a}^u) \hat{g}_{ub}
- e_c({\epsilon_b}^u) \hat{g}_{au}
$
Our commutation would be simplified a bit. $\hat{c}_{abc} = 0$ would imply:
${c_{ab}}^c = - 2 e_{[a} ({\epsilon_{b]}}^c)$
Now for de-Donder gauge:
${\Gamma^a}_{bc} g^{bc} = 0$
$
\frac{1}{2} g^{ae} g^{bc} (
e_c(g_{eb})
+ e_b(g_{ec})
- e_e(g_{bc})
+ c_{ebc}
+ c_{ecb}
- c_{cbe}
)
= 0
$
Multiply by $g_{ad}$, relabel d to a:
$
2 g^{bc} e_b(g_{ac})
- g^{bc} e_a(g_{bc})
+ 2 {c_{ab}}^b
= 0
$
In terms of the background basis:
$
\hat{g}^{bc}
(\delta_b^v + {\epsilon_b}^v)
(\delta_c^w + {\epsilon_c}^w)
(
2
(
\hat{e}_v(\hat{g}_{aw})
- \hat{e}_v( {\epsilon_a}^e) \hat{g}_{ew}
- \hat{e}_v( {\epsilon_w}^e) \hat{g}_{ea}
)
-
(
\hat{e}_a(\hat{g}_{vw})
- \hat{e}_a( {\epsilon_v}^e) \hat{g}_{ew}
- \hat{e}_a( {\epsilon_w}^e) \hat{g}_{ev}
)
)
+ 2
(
{\hat{c}_{av}}^v
- 2 e_{[a} ({\epsilon_{v]}}^v)
)
= 0
$
$
\hat{g}^{bc}
(\delta_b^v + {\epsilon_b}^v)
(\delta_c^w + {\epsilon_c}^w)
(
2 \hat{e}_v(\hat{g}_{uw})
- 2 \hat{e}_v({\epsilon_u}^e) \hat{g}_{ew}
- 2 \hat{e}_v({\epsilon_w}^e) \hat{g}_{eu}
- \hat{e}_u(\hat{g}_{vw})
+ \hat{e}_u({\epsilon_v}^e) \hat{g}_{ew}
+ \hat{e}_u({\epsilon_w}^e) \hat{g}_{ev}
)
+ 2
(
{\hat{c}_{uv}}^v
- \hat{e}_u ({\epsilon_v}^v)
+ \hat{e}_v ({\epsilon_u}^v)
)
= 0
$
$
\hat{g}^{bc}
(\delta_b^v + {\epsilon_b}^v)
(\delta_c^w + {\epsilon_c}^w)
(
2 \hat{g}^{au} \hat{e}_v(\hat{g}_{uw})
+ 2 \hat{g}^{aq} {\epsilon_q}^u \hat{e}_v(\hat{g}_{uw})
+ 2 \hat{g}^{pu} {\epsilon_p}^a \hat{e}_v(\hat{g}_{uw})
+ (
\hat{g}^{au}
+ \hat{g}^{aq} {\epsilon_q}^u
+ \hat{g}^{pu} {\epsilon_p}^a
)
(
- 2 \hat{e}_v({\epsilon_u}^e) \hat{g}_{ew}
- 2 \hat{e}_v({\epsilon_w}^e) \hat{g}_{eu}
- \hat{e}_u(\hat{g}_{vw})
+ \hat{e}_u({\epsilon_v}^e) \hat{g}_{ew}
+ \hat{e}_u({\epsilon_w}^e) \hat{g}_{ev}
)
)
+ 2
\hat{g}^{pq}
(\delta_p^a + {\epsilon_p}^a)
(\delta_q^u + {\epsilon_q}^u)
(
{\hat{c}_{uv}}^v
- \hat{e}_u ({\epsilon_v}^v)
+ \hat{e}_v ({\epsilon_u}^v)
)
= 0
$
I'm sure you're used to seeing the first two terms, which gives you $2 e_b(g_{ac}) g^{bc} = e_a(g_{bc}) g^{bc}$.
The last one is for the anholonomic basis, which nobody wants to consider, maybe because "Harmonic coordinate condition" is a coordinate condition, which implies the use of a coordinate system, which implies zero commutation.
So looking back on our section on choosing a coordinate basis, we would still be left with a commutation in the perturbed basis:
$
2 e_b(g_{ac}) g^{bc}
- e_a(g_{bc}) g^{bc}
- 4 e_{[a} ({\epsilon_{b]}}^b)
= 0
$
So we find that one new term is introduced into the typical de-Donder gauge, and that is the derivative of the trace of the perturbation epsilon.
Back