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Conformal metric:
Let $\chi = (\frac{\bar{g}}{g})^{\frac{1}{n}}$ be the conformal factor.
For $g = det(g_{ab})$ and $\bar{g}$ some arbitrary value for the conformal metric determinant... often 1 or some background metric determinant $det(\hat{g}_{ab})$.
Let:
$\bar{g}_{\bar{a}\bar{b}} = \chi \delta_\bar{a}^a \delta_\bar{b}^b g_{ab}$
$\bar{g}^{\bar{a}\bar{b}} = \frac{1}{\chi} \delta_a^\bar{a} \delta_b^\bar{b} g^{ab}$
So:
$g_{ab} = \frac{1}{\chi} \delta_a^\bar{a} \delta_b^\bar{b} \bar{g}_{\bar{a}\bar{b}}$
$g^{ab} = \chi \delta^a_\bar{a} \delta^b_\bar{b} \bar{g}^{\bar{a}\bar{b}}$
Determinant:
$det(\bar{g}_{\bar{a}{b}}) = det(\chi g_{ab})$
$= \chi^n det(g_{ab})$
$= \left( (\frac{\bar{g}}{g})^{\frac{1}{n}} \right)^n g$
$= \frac{\bar{g}}{g} \cdot g$
$= \bar{g}$
Let $\bar{e}_\bar{a} = {\bar{e}_\bar{a}}^a e_a$ be the transform from the original basis to the conformal basis.
Let $W = \sqrt{\chi} = (\frac{\bar{g}}{g})^{\frac{1}{2n}}$, this is going to show up a lot.
So $\bar{g}_{\bar{a}\bar{b}} = W^2 \delta^a_\bar{a} \delta^b_\bar{b} g_{ab}$
And $g_{ab} = \frac{1}{W^2} \delta^\bar{a}_a \delta^\bar{b}_b \bar{g}_{\bar{a}\bar{b}}$
One choice of basis transformation for reproducing our conformal metric is:
${\bar{e}_\bar{a}}^a = W \delta_\bar{a}^a$.
Such that: $\bar{e}_\bar{a} = {\bar{e}_\bar{a}}^a e_a = W \delta_\bar{a}^a e_a$.
And: $e_a = \frac{1}{W} \delta_a^\bar{a} \bar{e}_\bar{a}$.
Define an inverse tetrad $\bar{e}^\bar{a} = {\bar{e}^\bar{a}}_a e^a = \frac{1}{W} \delta^\bar{a}_a$.
Such that $\bar{e}^\bar{a} ( \bar{e}_\bar{b} )$
$= \frac{1}{W} \delta^\bar{a}_a e^a ( W \delta_\bar{b}^b e_b )$
$= \frac{W}{W} \delta^\bar{a}_a \delta_\bar{b}^b e^a ( e_b )$
$= \delta^\bar{a}_a \delta_\bar{b}^b \delta^a_b$
$= \delta^\bar{a}_\bar{b}$
So $\bar{g}_{\bar{a}{b}} = \bar{e}_\bar{a} \cdot \bar{e}_\bar{b}$
$= {\bar{e}_\bar{a}}^a e_a \cdot {\bar{e}_\bar{b}}^b e_b$
$= {\bar{e}_\bar{a}}^a {\bar{e}_\bar{b}}^b e_a \cdot e_b$
$= {\bar{e}_\bar{a}}^a {\bar{e}_\bar{b}}^b g_{ab}$
$= W^2 \delta_\bar{a}^a \delta_\bar{b}^b g_{ab}$
$= \bar{g}_{\bar{a}\bar{b}}$
structure constants, ${\bar{c}_{\bar{a}\bar{b}}}^\bar{c} \bar{e}_\bar{c}$
$\left[ \bar{e}_\bar{a}, \bar{e}_\bar{b} \right] (\phi)$
$= \bar{e}_\bar{a} (\bar{e}_\bar{b} (\phi)) - \bar{e}_\bar{b} (\bar{e}_\bar{a} (\phi))$
$= W \delta_\bar{a}^a e_a (W \delta_\bar{b}^b e_b (\phi)) - W \delta_\bar{b}^b e_b (W \delta_\bar{a}^a e_a (\phi))$
$= W \delta_\bar{a}^a \delta_\bar{b}^b (
e_a (W e_b (\phi)) - e_b (W e_a (\phi))
)$
$= W \delta_\bar{a}^a \delta_\bar{b}^b (
e_a (W) e_b (\phi)
- e_b (W) e_a (\phi)
+ W e_a (e_b (\phi))
- W e_b (e_a (\phi))
)$
$= W \delta_\bar{a}^a \delta_\bar{b}^b (
e_a (W) e_b (\phi)
- e_b (W) e_a (\phi)
+ W {c_{ab}}^c e_c (\phi)
)$
$=
W \delta_\bar{a}^a \delta_\bar{b}^b e_a (W) e_b (\phi)
- W \delta_\bar{a}^a \delta_\bar{b}^b e_b (W) e_a (\phi)
+ W^2 \delta_\bar{a}^a \delta_\bar{b}^b {c_{ab}}^c e_c (\phi)
$
$=
\frac{1}{W} \bar{e}_\bar{a} (W) \delta^\bar{c}_\bar{b} \bar{e}_\bar{c} (\phi)
- \frac{1}{W} \bar{e}_\bar{b} (W) \delta^\bar{c}_\bar{a} \bar{e}_\bar{c} (\phi)
+ W \delta_\bar{a}^a \delta_\bar{b}^b \delta_c^\bar{c} {c_{ab}}^c \bar{e}_\bar{c} (\phi)
$
$= \left(
\frac{1}{W} \bar{e}_\bar{a} (W) \delta^\bar{c}_\bar{b}
- \frac{1}{W} \bar{e}_\bar{b} (W) \delta^\bar{c}_\bar{a}
+ W \delta_\bar{a}^a \delta_\bar{b}^b \delta_c^\bar{c} {c_{ab}}^c
\right) \bar{e}_\bar{c} (\phi) $
$= {\bar{c}_{\bar{a}\bar{b}}}^\bar{c} \bar{e}_\bar{c} (\phi)$
Therefore the commutation/structure constants of the conformal basis is equal to:
${\bar{c}_{\bar{a}\bar{b}}}^\bar{c} =
\frac{1}{W} \bar{e}_\bar{a} (W) \delta^\bar{c}_\bar{b}
- \frac{1}{W} \bar{e}_\bar{b} (W) \delta^\bar{c}_\bar{a}
+ W \delta_\bar{a}^a \delta_\bar{b}^b \delta_c^\bar{c} {c_{ab}}^c
$
For some background metric commutation ${c_{ab}}^c$
$\bar{c}_{\bar{a}\bar{b}\bar{c}} = \bar{g}_{\bar{d}\bar{c}} \left(
\frac{1}{W} \bar{e}_\bar{a} (W) \delta^\bar{d}_\bar{b}
- \frac{1}{W} \bar{e}_\bar{b} (W) \delta^\bar{d}_\bar{a}
+ W \delta_\bar{a}^a \delta_\bar{b}^b \delta_d^\bar{d} {c_{ab}}^d
\right)$
$=
\frac{1}{W} \bar{e}_\bar{a} (W) \bar{g}_{\bar{b}\bar{c}}
- \frac{1}{W} \bar{e}_\bar{b} (W) \bar{g}_{\bar{a}\bar{c}}
+ W \delta_\bar{a}^a \delta_\bar{b}^b {c_{ab}}^d \delta_d^\bar{d} \delta^\bar{e}_c \delta^c_\bar{c} \bar{g}_{\bar{d}\bar{e}}
$
$=
\frac{1}{W} \bar{e}_\bar{a} (W) \bar{g}_{\bar{b}\bar{c}}
- \frac{1}{W} \bar{e}_\bar{b} (W) \bar{g}_{\bar{a}\bar{c}}
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b {c_{ab}}^d g_{cd} \delta^c_\bar{c}
$
$=
\frac{1}{W} \bar{e}_\bar{a} (W) \bar{g}_{\bar{b}\bar{c}}
- \frac{1}{W} \bar{e}_\bar{b} (W) \bar{g}_{\bar{a}\bar{c}}
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c c_{abc}
$
Metric-cancelling geodesic connection of the conformal basis:
$\bar{\Gamma}_{\bar{a}\bar{b}\bar{c}} = \frac{1}{2} (
\bar{e}_\bar{c} (\bar{g}_{\bar{a}\bar{b}})
+ \bar{e}_\bar{b} (\bar{g}_{\bar{a}\bar{c}})
- \bar{e}_\bar{a} (\bar{g}_{\bar{b}\bar{c}})
+ \bar{c}_{\bar{a}\bar{b}\bar{c}}
+ \bar{c}_{\bar{a}\bar{c}\bar{b}}
- \bar{c}_{\bar{c}\bar{b}\bar{a}}
) + \bar{K}_{\bar{a}\bar{b}\bar{c}}$
$= \frac{1}{2} \left(
W \delta_\bar{c}^c e_c (W^2 \delta_\bar{a}^a \delta_\bar{b}^b g_{ab})
+ W \delta_\bar{b}^b e_b (W^2 \delta_\bar{a}^a \delta_\bar{c}^c g_{ac})
- W \delta_\bar{a}^a e_a (W^2 \delta_\bar{b}^b \delta_\bar{c}^c g_{bc})
+ \frac{1}{W} \bar{e}_\bar{a} (W) \bar{g}_{\bar{b}\bar{c}}
- \frac{1}{W} \bar{e}_\bar{b} (W) \bar{g}_{\bar{a}\bar{c}}
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c c_{abc}
+ \frac{1}{W} \bar{e}_\bar{a} (W) \bar{g}_{\bar{c}\bar{b}}
- \frac{1}{W} \bar{e}_\bar{c} (W) \bar{g}_{\bar{a}\bar{b}}
+ W^3 \delta_\bar{a}^a \delta_\bar{c}^c \delta_\bar{b}^b c_{acb}
- \frac{1}{W} \bar{e}_\bar{c} (W) \bar{g}_{\bar{b}\bar{a}}
+ \frac{1}{W} \bar{e}_\bar{b} (W) \bar{g}_{\bar{c}\bar{a}}
- W^3 \delta_\bar{c}^c \delta_\bar{b}^b \delta_\bar{a}^a c_{cba}
\right) + \bar{K}_{\bar{a}\bar{b}\bar{c}}$
$= \frac{1}{2} \left(
W \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_c(W^2) g_{ab}
+ W \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_b(W^2) g_{ac}
- W \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_a(W^2) g_{bc}
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_c(g_{ab})
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_b(g_{ac})
- W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_a(g_{bc})
+ W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_a(W) g_{bc}
- W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_b(W) g_{ac}
+ W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_a(W) g_{cb}
- W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_c(W) g_{ab}
- W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_c(W) g_{ba}
+ W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c e_b(W) g_{ca}
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c c_{abc}
+ W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c c_{acb}
- W^3 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c c_{cba}
\right) + \bar{K}_{\bar{a}\bar{b}\bar{c}}$
$= \frac{1}{2} W \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c \left(
W e_a(W) g_{bc}
- W e_b(W) g_{ac}
+ W e_a(W) g_{cb}
- W e_c(W) g_{ab}
- W e_c(W) g_{ba}
+ W e_b(W) g_{ca}
+ 2 W e_c(W) g_{ab}
+ 2 W e_b(W) g_{ac}
- 2 W e_a(W) g_{bc}
+ W^2 \left(
e_c(g_{ab}) + e_b(g_{ac}) - e_a(g_{bc}) + c_{abc} + c_{acb} - c_{cba}
\right)
\right) + \bar{K}_{\bar{a}\bar{b}\bar{c}}$
$= W^2 \delta_\bar{a}^a \delta_\bar{b}^b \delta_\bar{c}^c \left(
e_b(W) g_{ac}
+ W \hat{\Gamma}_{abc}
\right) + \bar{K}_{\bar{a}\bar{b}\bar{c}}$
${\bar{\Gamma}^\bar{a}}_{\bar{b}\bar{c}} = \bar{g}^{\bar{a}\bar{d}} \bar{\Gamma}_{\bar{d}\bar{b}\bar{c}}$
$= \bar{g}^{\bar{a}\bar{d}}
\left(
W^2 \delta_\bar{d}^d \delta_\bar{b}^b \delta_\bar{c}^c
\left(
e_b(W) g_{dc}
+ W \hat{\Gamma}_{dbc}
\right)
+ \bar{K}_{\bar{d}\bar{b}\bar{c}}
\right)
$
$=
W^2 \bar{g}^{\bar{a}\bar{d}} \delta_\bar{d}^d \delta_\bar{b}^b \delta_\bar{c}^c e_b(W) g_{dc}
+ W^3 \bar{g}^{\bar{a}\bar{d}} \delta_\bar{d}^d \delta_\bar{b}^b \delta_\bar{c}^c \hat{\Gamma}_{dbc}
+ \bar{g}^{\bar{a}\bar{d}} \bar{K}_{\bar{d}\bar{b}\bar{c}}
$
$=
W^2 \delta^\bar{a}_a \delta^a_\bar{e} \bar{g}^{\bar{e}\bar{d}} \delta_\bar{d}^d \delta_\bar{b}^b \delta_\bar{c}^c e_b(W) g_{dc}
+ W^3 \delta^\bar{a}_a \delta^a_\bar{e} \bar{g}^{\bar{e}\bar{d}} \delta_\bar{d}^d \delta_\bar{b}^b \delta_\bar{c}^c \hat{\Gamma}_{dbc}
+ {\bar{K}^\bar{a}}_{\bar{b}\bar{c}}
$
$=
W^2 \delta^\bar{a}_a g^{ad} \delta_\bar{b}^b \delta_\bar{c}^c e_b(W) g_{dc}
+ W^3 \delta^\bar{a}_a g^{ad} \delta_\bar{b}^b \delta_\bar{c}^c \hat{\Gamma}_{dbc}
+ {\bar{K}^\bar{a}}_{\bar{b}\bar{c}}
$
$=
\delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta^a_c e_b(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c {\hat{\Gamma}^a}_{bc}
+ {\bar{K}^\bar{a}}_{\bar{b}\bar{c}}
$
$=
\delta^\bar{a}_\bar{c} \delta_\bar{b}^b e_b(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c {\hat{\Gamma}^a}_{bc}
+ {\bar{K}^\bar{a}}_{\bar{b}\bar{c}}
$
$=
\frac{1}{W} \delta^\bar{a}_\bar{c} \bar{e}_\bar{b}(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c {\hat{\Gamma}^a}_{bc}
+ {\bar{K}^\bar{a}}_{\bar{b}\bar{c}}
$
(TODO you are HERE in adding contorsion to the connection)
Also how should I define the torsion of the conformally-rescaled connection vs the torsion of the base connection?
If the two are equal then maybe the contorsion will cancel.
Otherwise I think we will keep propagating a tensor for the difference-of-contorsions between the connection of the base metric and the connection of the conformal metric.
Until I straighten this out:
Let ${\bar{\hat{\Gamma}}^\bar{a}}_{\bar{b}\bar{c}} = {\bar{\Gamma}^\bar{a}}_{\bar{b}\bar{c}} - {\bar{K}^\bar{a}}_{\bar{b}\bar{c}}$
Notice that the difference is the gradient of the square-root of the conformal factor: $e^a (\nabla_b e_c) = {\Gamma^a}_{bc}$.
If we are dealing with a coordinate basis, and the conformal metric is fixed at 1, then this is the gradient of the volume element.
Notice that, if you're a BSSN numerical relativity fan, this is a bit different from the "conformal connection" that is defined in their literature,
which is simply done by replacing the $g_{ab}$ with $\bar{g}_{\bar{a}\bar{b}}$ in the definition of the Levi-Civita torsion-free connection of the coordinate metric,
with no consideration of what the conformal transform has possibly done to the underlying basis, and therefore to the commutation.
Now for some traces of the Levi-Civita torsion-free connection of the conformal metric:
${\bar{\hat{\Gamma}}}^\bar{a} = {\bar{\hat{\Gamma}}^\bar{a}}_{\bar{b}\bar{c}} \bar{g}^{\bar{b}\bar{c}}$
$= \left(
\delta^\bar{a}_\bar{c} \delta_\bar{b}^b e_b(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c {\hat{\Gamma}^a}_{bc}
\right)
\cdot \frac{1}{W^2} \delta^\bar{b}_e \delta^\bar{c}_f g^{ef}$
$=
\frac{1}{W^2} \delta^\bar{a}_a g^{ab} e_b(W)
+ \frac{1}{W} \delta^\bar{a}_a \hat{\Gamma}^a
$
${\bar{\hat{\Gamma}}^\bar{a}}_{\bar{b}\bar{a}} =
\delta^\bar{a}_\bar{a} \delta_\bar{b}^b e_b(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{a}^c {\hat{\Gamma}^a}_{bc}
$
${\bar{\hat{\Gamma}}^\bar{a}}_{\bar{b}\bar{a}} = \delta_\bar{b}^b \left(
n e_b(W)
+ W {\hat{\Gamma}^a}_{ba}
\right)$
${\bar{\hat{\Gamma}}^\bar{a}}_{\bar{b}\bar{a}} = \delta_\bar{b}^b \left(
n e_b(W)
+ W e_b(log\sqrt{|g|})
\right)$
${\bar{\hat{\Gamma}}^\bar{a}}_{\bar{b}\bar{a}} = \delta_\bar{b}^b \left(
n e_b(W)
+ \frac{W}{2 |g|} e_b(|g|)
\right)$
Riemann curvature:
${\bar{R}^\bar{a}}_{\bar{b}\bar{c}\bar{d}} =
\bar{e}_\bar{c} ({\bar{\hat{\Gamma}}^\bar{a}}_{\bar{d}\bar{b}})
- \bar{e}_\bar{d} ({\bar{\hat{\Gamma}}^\bar{a}}_{\bar{c}\bar{b}})
+ {\bar{\hat{\Gamma}}^\bar{a}}_{\bar{c}\bar{u}} {\bar{\hat{\Gamma}}^\bar{u}}_{\bar{d}\bar{b}}
- {\bar{\hat{\Gamma}}^\bar{a}}_{\bar{d}\bar{u}} {\bar{\hat{\Gamma}}^\bar{u}}_{\bar{c}\bar{b}}
- {\bar{\hat{\Gamma}}^\bar{a}}_{\bar{u}\bar{b}} {\bar{c}_{\bar{c}\bar{d}}}^\bar{u}
$
$=
\bar{e}_\bar{c} (
\delta^\bar{a}_\bar{b} \delta_\bar{d}^d e_d(W)
+ W \delta^\bar{a}_a \delta_\bar{d}^d \delta_\bar{b}^b {\hat{\Gamma}^a}_{db}
)
- \bar{e}_\bar{d} (
\delta^\bar{a}_\bar{b} \delta_\bar{c}^c e_c(W)
+ W \delta^\bar{a}_a \delta_\bar{c}^c \delta_\bar{b}^b {\hat{\Gamma}^a}_{cb}
)
+ (
\delta^\bar{a}_\bar{u} \delta_\bar{c}^c e_c(W)
+ W \delta^\bar{a}_a \delta_\bar{c}^c \delta_\bar{u}^u {\hat{\Gamma}^a}_{cu}
) (
\delta^\bar{u}_\bar{b} \delta_\bar{d}^d e_d(W)
+ W \delta^\bar{u}_v \delta_\bar{d}^d \delta_\bar{b}^b {\hat{\Gamma}^v}_{db}
)
- (
\delta^\bar{a}_\bar{u} \delta_\bar{d}^d e_d(W)
+ W \delta^\bar{a}_a \delta_\bar{d}^d \delta_\bar{u}^u {\hat{\Gamma}^a}_{du}
) (
\delta^\bar{u}_\bar{b} \delta_\bar{c}^c e_c(W)
+ W \delta^\bar{u}_v \delta_\bar{c}^c \delta_\bar{b}^b {\hat{\Gamma}^v}_{cb}
)
- (
\delta^\bar{a}_\bar{b} \delta_\bar{u}^u e_u(W)
+ W \delta^\bar{a}_a \delta_\bar{u}^u \delta_\bar{b}^b {\hat{\Gamma}^a}_{ub}
) (
\frac{1}{W} \bar{e}_\bar{c} (W) \delta^\bar{u}_\bar{d}
- \frac{1}{W} \bar{e}_\bar{d} (W) \delta^\bar{u}_\bar{c}
+ W \delta_\bar{c}^c \delta_\bar{d}^d \delta_v^\bar{u} {c_{cd}}^v
)
$
$=
W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_c(e_d(W))
- W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_d(e_c(W))
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d e_c(W {\hat{\Gamma}^a}_{db})
- W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d e_d(W {\hat{\Gamma}^a}_{cb})
+ \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_c(W) e_d(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d e_c(W) {\hat{\Gamma}^a}_{db}
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{cb} e_d(W)
+ W^2 \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{cu} {\hat{\Gamma}^u}_{db}
- \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_c(W) e_d(W)
- W \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d e_d(W) {\hat{\Gamma}^a}_{cb}
- W \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{db} e_c(W)
- W^2 \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{du} {\hat{\Gamma}^u}_{cb}
- \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_d(W) e_c(W)
+ \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_c(W) e_d(W)
- W \delta^\bar{a}_a \delta^b_\bar{b} \delta_\bar{c}^c \delta_\bar{d}^d \delta^a_b e_u(W) {c_{cd}}^u
- W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{db} e_c(W)
+ W \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{cb} e_d(W)
- W^2 \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d {\hat{\Gamma}^a}_{ub} {c_{cd}}^u
$
$= \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d \left(
W \delta^a_b \left( e_c(e_d(W)) - e_d(e_c(W)) - e_u(W) {c_{cd}}^u \right)
+ W^2 \left(
e_c({\hat{\Gamma}^a}_{db})
- e_d({\hat{\Gamma}^a}_{cb})
+ {\hat{\Gamma}^a}_{cu} {\hat{\Gamma}^u}_{db}
- {\hat{\Gamma}^a}_{du} {\hat{\Gamma}^u}_{cb}
- {\hat{\Gamma}^a}_{ub} {c_{cd}}^u
\right)
\right)$
$= W^2 \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d \left(
e_c({\hat{\Gamma}^a}_{db})
- e_d({\hat{\Gamma}^a}_{cb})
+ {\hat{\Gamma}^a}_{cu} {\hat{\Gamma}^u}_{db}
- {\hat{\Gamma}^a}_{du} {\hat{\Gamma}^u}_{cb}
- {\hat{\Gamma}^a}_{ub} {c_{cd}}^u
\right)$
$= W^2 \delta^\bar{a}_a \delta_\bar{b}^b \delta_\bar{c}^c \delta_\bar{d}^d {R^a}_{bcd}$
That looks a lot simpler than the conformally-rescaled Riemann definition that you find in the numerical relativity textbooks.
And it is proportional to the original metric's Riemann curvature. Scalar curvature is an invariant after all.
From there, the easy conclusion:
$\bar{R}_{\bar{a}\bar{b}}$
$= {\bar{R}^\bar{c}}_{\bar{a}\bar{c}\bar{b}}$
$= W^2 \delta^a_\bar{a} \delta^b_\bar{b} R_{ab}$
$\bar{R}$
$= \bar{g}^{\bar{a}\bar{b}} \bar{R}_{\bar{a}\bar{b}}$
$= \frac{1}{W^2} g^{uv} \delta^\bar{a}_u \delta^\bar{b}_v \cdot W^2 \delta^a_\bar{a} \delta^b_\bar{b} R_{ab}$
$= g^{ab} R_{ab}$
$= R$
Yeah once again, almost as if there's something invariant about curvature.
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