Let $x^I$ be the Minkowski coordinates
Let $x^u$ be the curved coordinates
I'll also use $x^\bar{a}$ for flat space coordinates and $x^a$ for curved space coordintes
Let $ijklmn$ represent spatial coordinates
Let $0$ or $t$ represent the time coordiante
Let $x^\bar{0} = \phi(x^u)$, $x^\bar{i} = x^i$
The basis is defined as:
${e_u}^\bar{0} = \frac{\partial x^\bar{0}}{\partial x^u} = {x^\bar{0}}_{,u} = \phi_{,u}$
${e_u}^\bar{i} = \frac{\partial x^\bar{i}}{\partial x^u} = {x^\bar{i}}_{,u} = \delta^\bar{i}_u$
${e_u}^{\bar{v}} = \downarrow u(i) \overset{\rightarrow \bar{v}(\bar{j})}{ \left[\matrix{
\phi_{,0} & 0 \\
\phi_{,i} & \delta_i^\bar{j}
}\right] }$
The inverse basis is defined as:
${e^t}_\bar{0} = (\phi_{,t})^{-1}$
${e^t}_\bar{i} = -(\phi_{,i}) (\phi_{,t})^{-1}$
${e^u}_\bar{i} = \delta^u_i$
Metric:
$g_{uv} = {e_u}^I {e_v}^J \eta_{IJ}$
$= -{e_u}^\bar{0} {e_v}^\bar{0} + {e_u}^\bar{i} {e_v}^\bar{j} \eta_{\bar{i}\bar{j}}$
$= -\phi_{,u} \phi_{,v} + \delta^i_u \delta^j_v \eta_{ij}$
$g_{tt} = -(\phi_{,t})^2$
$g_{ti} = -\phi_{,t} \phi_{,i}$
$g_{ij} = -\phi_{,i} \phi_{,j} + \delta_{ij}$
Metric inverse:
$g^{tt} = (\phi_{,t})^{-2} (-1 + \Sigma_{k=1}^n (\phi_{,k})^2)$
$g^{it} = -(\phi_{,i}) (\phi_{,t})^{-1}$
$g^{ij} = \delta^{ij}$
Metric partial:
$g_{uv,w} = -(\phi_{,u} \phi_{,v})_{,w}$
$=-\phi_{,uw} \phi_{,v} - \phi_{,u} \phi_{,vw}$
Connection
$\Gamma_{uvw} = \frac{1}{2} (g_{uv,w} + g_{uw,v} - g_{vw,u})$
$= \frac{1}{2} (
-\phi_{,uw} \phi_{,v} - \phi_{,u} \phi_{,vw}
-\phi_{,uv} \phi_{,w} - \phi_{,u} \phi_{,wv}
+\phi_{,vu} \phi_{,w} + \phi_{,v} \phi_{,wu}
)$
$= - \phi_{,u} \phi_{,vw}$
${\Gamma^u}_{vw} = g^{ua} \Gamma_{avw}$
$ = -g^{ua} \phi_{,a} \phi_{,vw}$
${\Gamma^t}_{vw} = -g^{tt} \phi_{,t} \phi_{,vw}
- g^{ti} \phi_{,i} \phi_{,vw}$
$ = (
-(\phi_{,t})^{-2} (-1 + \Sigma_{k=1}^n (\phi_{,k})^2) \phi_{,t}
+ \phi_{,i} (\phi_{,t})^{-1} \phi_{,i}
) \phi_{,vw}$
$ = (
1 - \Sigma_{k=1}^n (\phi_{,k})^2
+ (\phi_{,i})^{2}
) (\phi_{,t})^{-1} \phi_{,vw}$
${\Gamma^i}_{vw} = g^{it} \Gamma_{tvw} + g^{ij} \Gamma_{jvw}$
$ = -g^{it} \phi_{,t} \phi_{,vw} - g^{ij} \phi_{,j} \phi_{,vw}$
$ = (\phi_{,i}) (\phi_{,t})^{-1} \phi_{,t} \phi_{,vw}
- \delta^{ij} \phi_{,j} \phi_{,vw}$
$ = \phi_{,i} \phi_{,vw} - \phi_{,i} \phi_{,vw}$
$ = 0$
So this gives the opposite of what I wanted: A controllable connection in the new dimension for vectors at rest in the subspace, and no connection in the new dimension for any vectors - especially those at rest in the new dimension.
Now let's try reversing the definition. Still sticking to holonomic spaces, but defining the basis inverse rather than the basis.
Let $x^t = \phi(x^I)$, $x^i = x^\bar{i}$
basis inverse:
${e^t}_\bar{u} = \frac{\partial x^t}{\partial x^\bar{u}} = \phi_{,\bar{u}}$
${e^i}_\bar{u} = \frac{\partial x^i}{\partial x^\bar{u}} = \delta^i_\bar{u}$
${e^u}_\bar{v} = \downarrow u(i) \overset{ \rightarrow \bar{v}(\bar{j}) }{ \left[\matrix{
(\phi_{,0})^{-1} & 0 \\
-\phi_{,\bar{i}} (\phi_{,0})^{-1} & \delta^i_\bar{j}
}\right] }$
basis:
${e_t}^0 = (\phi_{,0})^{-1}$
${e_t}^\bar{i} = 0$
${e_i}^0 = -(\phi_{,\bar{i}}) (\phi_{,0})^{-1}$
${e_i}^\bar{j} = \delta_i^\bar{j}$
Now the metric and metric inverse are switched, just like the basis and basis inverse were switched.
Metric:
$g_{tt} = -(\phi_{,0})^{-2}$
$g_{it} = -\phi_{,\bar{i}} (\phi_{,0})^{-2}$
$g_{ij} = -\phi_{,\bar{i}} \phi_{,\bar{j}} (\phi_{,0})^{-2}$
Metric inverse:
$g^{tt} = -(\phi_{,\bar{0}})^2 + \Sigma_{k=1}^n (\phi_{,\bar{k}})^2$
$g^{ti} = \phi_{,\bar{i}}$
$g^{ij} = \delta^{ij}$
Metric partials wrt flat space coordinates:
$g_{tt,\bar{u}} = 2 \phi_{,0\bar{u}} (\phi_{,0})^{-3}$
$g_{ti,\bar{u}} = - \phi_{,\bar{i}\bar{u}} (\phi_{,0})^{-2}
+ 2 \phi_{,\bar{i}} \phi_{,\bar{u}} (\phi_{,0})^{-3}$
$= (
2 \phi_{,\bar{i}} \phi_{,\bar{u}}
- \phi_{,\bar{i}\bar{u}} \phi_{,0}
) (\phi_{,0})^{-3}$
$g_{ij,\bar{u}} =
- (\phi_{,\bar{i}\bar{u}} \phi_{,\bar{j}} + \phi_{,\bar{i}} \phi_{,\bar{j}\bar{u}}) (\phi_{,0})^{-2}
+ 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{u}} (\phi_{,0})^{-3}$
$= (
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{u}}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}\bar{u}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}\bar{u}}
) (\phi_{,0})^{-3}$
Metric partials wrt curved space coordinates:
$g_{tt,t} = g_{tt,\bar{u}} {e_t}^\bar{u} = g_{tt,0} {e_t}^0 + g_{tt,\bar{i}} {e_t}^\bar{i}$
$= 2 \phi_{,00} (\phi_{,0})^{-3} \cdot (\phi_{,0})^{-1}$
$= 2 \phi_{,00} (\phi_{,0})^{-4}$
$g_{tt,i} = g_{tt,\bar{u}} {e_i}^\bar{u}$
$= g_{tt,0} {e_i}^0 + g_{tt,\bar{j}} {e_i}^{\bar{j}}$
$= 2 \phi_{,00} (\phi_{,0})^{-3}
\cdot -(\phi_{,\bar{i}}) (\phi_{,0})^{-1}
+ 2 \phi_{,0\bar{j}} (\phi_{,0})^{-3}
\cdot \delta^\bar{j}_i$
$= -2 \phi_{,\bar{i}} \phi_{,00} (\phi_{,0})^{-4}
+ 2 \phi_{,0\bar{i}} (\phi_{,0})^{-3}$
$= 2 (\phi_{,0\bar{i}} \phi_{,0} - \phi_{,\bar{i}} \phi_{,00}) (\phi_{,0})^{-4} $
$g_{ti,t} = g_{ti,\bar{u}} {e_t}^\bar{u} = g_{ti,0} {e_t}^0 + g_{ti,\bar{j}} {e_t}^\bar{j}$
$= (
2 \phi_{,\bar{i}} \phi_{,0}
- \phi_{,0\bar{i}} \phi_{,0}
) (\phi_{,0})^{-3}
\cdot (\phi_{,0})^{-1}
+ (
2 \phi_{,\bar{i}} \phi_{,\bar{j}}
- \phi_{,\bar{i}\bar{j}} \phi_{,0}
) (\phi_{,0})^{-3}
\cdot 0$
$= (
2 \phi_{,\bar{i}}
- \phi_{,0\bar{i}}
) (\phi_{,0})^{-3}$
$g_{ti,j} = g_{ti,\bar{u}} {e_j}^\bar{u} = g_{ti,0} {e_j}^0 + g_{ti,\bar{k}} {e_j}^\bar{k}$
$= (
2 \phi_{,\bar{i}} \phi_{,0}
- \phi_{,0\bar{i}} \phi_{,0}
) (\phi_{,0})^{-3}
\cdot -\phi_{,\bar{j}} (\phi_{,0})^{-1}
+ (2 \phi_{,\bar{i}} \phi_{,\bar{k}}
- \phi_{,\bar{i}\bar{k}} \phi_{,0}
) (\phi_{,0})^{-3}
\cdot \delta_j^{\bar{k}}
$
$= (
\phi_{,\bar{j}} \phi_{,0\bar{i}} \phi_{,0}
- 2 \phi_{,\bar{j}} \phi_{,\bar{i}} \phi_{,0}
) (\phi_{,0})^{-4}
+ (
2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}}
- \phi_{,\bar{i}\bar{j}} (\phi_{,0})^2
) (\phi_{,0})^{-4}
$
$= (
\phi_{,\bar{j}} \phi_{,0\bar{i}}
- \phi_{,0} \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-3}
$
$g_{ij,t} = g_{ij,\bar{u}} {e_t}^\bar{u} = g_{ij,0} {e_t}^0 + g_{ij,\bar{k}} {e_t}^\bar{k}$
$= (
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}0}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}0}
) (\phi_{,0})^{-3}
\cdot (\phi_{,0})^{-1}
+ (
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}}
) (\phi_{,0})^{-3}
\cdot 0$
$= (
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}}
) (\phi_{,0})^{-4}$
$g_{ij,k} = g_{ij,\bar{u}} {e_k}^\bar{u} = g_{ij,0} {e_k}^0 + g_{ij,\bar{l}} {e_k}^\bar{l}$
$= (
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}}
) (\phi_{,0})^{-3}
\cdot -\phi_{,\bar{k}} (\phi_{,0})^{-1}
+ (
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{l}}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}\bar{l}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}\bar{l}}
) (\phi_{,0})^{-3}
\cdot \delta^\bar{l}_k$
$= \phi_{,\bar{k}} (
- 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}}
+ \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}}
) (\phi_{,0})^{-4}
+ (
2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
- (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}}
- (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}}
) (\phi_{,0})^{-4}
$
$= (
- 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}}
+ \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,0\bar{i}}
+ 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
- (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}}
- (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}}
) (\phi_{,0})^{-4}
$
Christoffel symbols of the 1st kind:
$\Gamma_{ttt} = \frac{1}{2} g_{tt,t} $
$= \phi_{,00} (\phi_{,0})^{-4}$
$\Gamma_{tti} = \frac{1}{2} (g_{tt,i} + g_{ti,t} - g_{ti,t}) = \frac{1}{2} g_{tt,i}$
$= (\phi_{,0\bar{i}} \phi_{,0} - \phi_{,\bar{i}} \phi_{,00}) (\phi_{,0})^{-4} $
$\Gamma_{tij} = \frac{1}{2} (g_{ti,j} + g_{tj,i} - g_{ij,t})$
$= \frac{1}{2} (
(
\phi_{,\bar{j}} \phi_{,0\bar{i}}
- \phi_{,0} \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-3}
+
(
\phi_{,\bar{i}} \phi_{,0\bar{j}}
- \phi_{,0} \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-3}
-
(
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}}
) (\phi_{,0})^{-4}
)$
$= (
- \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}}
+ \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}}
- (\phi_{,0})^2 \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-4}$
$\Gamma_{itt} = \frac{1}{2} (g_{it,t} + g_{it,t} - g_{tt,i}) = g_{it,t} - \frac{1}{2} g_{tt,i}$
$= (
2 \phi_{,\bar{i}}
- \phi_{,0\bar{i}}
) (\phi_{,0})^{-3}
- (
\phi_{,0\bar{i}} \phi_{,0}
- \phi_{,\bar{i}} \phi_{,00}
) (\phi_{,0})^{-4} $
$= (
2 \phi_{,0} \phi_{,\bar{i}}
- 2 \phi_{,0} \phi_{,0\bar{i}}
+ \phi_{,\bar{i}} \phi_{,00}
) (\phi_{,0})^{-4} $
$\Gamma_{ijt} = \frac{1}{2} (g_{ij,t} + g_{it,j} - g_{jt,i})$
$= \frac{1}{2} (
(
2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,\bar{i}} \phi_{,0\bar{j}}
- \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{i}}
) (\phi_{,0})^{-4}
+
(
\phi_{,\bar{j}} \phi_{,0\bar{i}}
- \phi_{,0} \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-3}
-
(
\phi_{,\bar{i}} \phi_{,0\bar{j}}
- \phi_{,0} \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-3}
)$
$= \phi_{,\bar{i}} (
\phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,0\bar{j}}
) (\phi_{,0})^{-4}$
$\Gamma_{ijjk} = \frac{1}{2} (g_{ij,k} + g_{ik,j} - g_{jk,i})$
$= \frac{1}{2} (
(
- 2 \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}}
+ \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,0\bar{i}}
+ 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
- (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}}
- (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{i}\bar{k}}
) (\phi_{,0})^{-4}
+ (
- 2 \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,\bar{j}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
+ \phi_{,0} \phi_{,\bar{k}} \phi_{,\bar{j}} \phi_{,0\bar{i}}
+ 2 \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}}
- (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{k}\bar{j}}
- (\phi_{,0})^2 \phi_{,\bar{k}} \phi_{,\bar{i}\bar{j}}
) (\phi_{,0})^{-4}
- (
- 2 \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,\bar{i}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{i}} \phi_{,0\bar{k}}
+ \phi_{,0} \phi_{,\bar{k}} \phi_{,\bar{i}} \phi_{,0\bar{j}}
+ 2 \phi_{,0} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,0\bar{i}}
- (\phi_{,0})^2 \phi_{,\bar{j}} \phi_{,\bar{k}\bar{i}}
- (\phi_{,0})^2 \phi_{,\bar{k}} \phi_{,\bar{j}\bar{i}}
) (\phi_{,0})^{-4}
)$
$= (
- (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
- \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00}
) (\phi_{,0})^{-4}$
Christoffel symbols of the 2nd kind:
${\Gamma^i}_{tt} = g^{iu} \Gamma_{utt} = g^{it} \Gamma_{ttt} + g^{ij} \Gamma_{jtt}$
$= \phi_{,\bar{i}}
\cdot \phi_{,00} (\phi_{,0})^{-4}
+ \delta^{ij}
\cdot (
2 \phi_{,0} \phi_{,\bar{j}}
- 2 \phi_{,0} \phi_{,0\bar{j}}
+ \phi_{,\bar{j}} \phi_{,00}
) (\phi_{,0})^{-4}$
$= (
2 \phi_{,0} \phi_{,\bar{i}}
- 2 \phi_{,0} \phi_{,0\bar{i}}
+ 2 \phi_{,\bar{i}} \phi_{,00}
) (\phi_{,0})^{-4}$
${\Gamma^i}_{tj} = g^{iu} \Gamma_{utj} = g^{it} \Gamma_{ttj} + g^{ik} \Gamma_{ktj}$
$= \phi_{,\bar{i}}
(\phi_{,0\bar{j}} \phi_{,0} - \phi_{,\bar{j}} \phi_{,00}) (\phi_{,0})^{-4}
+
\delta^{ik}
\phi_{,\bar{k}} (
\phi_{,\bar{j}} \phi_{,00}
- \phi_{,0} \phi_{,0\bar{j}}
) (\phi_{,0})^{-4} $
$= 0$
${\Gamma^i}_{jk} = g^{iu} \Gamma_{ujk} = g^{it} \Gamma_{tjk} + g^{il} \Gamma_{ljk}$
$ = \phi_{,\bar{i}} (
- \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00}
+ \phi_{,0} \phi_{,\bar{j}} \phi_{,0\bar{k}}
+ \phi_{,0} \phi_{,\bar{k}} \phi_{,0\bar{j}}
- (\phi_{,0})^2 \phi_{,\bar{j}\bar{k}}
) (\phi_{,0})^{-4}
+ \delta^{il} (
- (\phi_{,0})^2 \phi_{,\bar{l}} \phi_{,\bar{j}\bar{k}}
+ \phi_{,0} \phi_{,\bar{l}} \phi_{,\bar{k}} \phi_{,0\bar{j}}
+ \phi_{,0} \phi_{,\bar{l}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
- \phi_{,\bar{l}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00}
) (\phi_{,0})^{-4}
$
$ = 2 (
- (\phi_{,0})^2 \phi_{,\bar{i}} \phi_{,\bar{j}\bar{k}}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,0\bar{k}}
+ \phi_{,0} \phi_{,\bar{i}} \phi_{,\bar{k}} \phi_{,0\bar{j}}
- \phi_{,\bar{i}} \phi_{,\bar{j}} \phi_{,\bar{k}} \phi_{,00}
) (\phi_{,0})^{-4}
$
TODO check the other geodesic components
Now let $\phi_{,0} = 1$
${\Gamma^i}_{tt} = 2 \phi_{,\bar{i}}$
So let $\phi(t,x^\bar{i}) = \bar{t} - \frac{1}{2} \psi(x^\bar{i})$
Then $\phi_{,0} = 1$
And $\phi_{,\bar{i}} = -\frac{1}{2} \psi_{,\bar{i}}$
Therefore ${\Gamma^i}_{tt} = -\psi_{,\bar{i}}$