The condition for flat space in two dimensions is
∂
∂
x
1
(
g
12
g
11
(
g
11
g
22
−
g
12
2
)
∂
g
11
∂
x
2
−
1
(
g
11
g
22
−
g
12
2
)
∂
g
22
δ
x
1
)
{\displaystyle {\frac {\partial }{\partial x_{1}}}\left({\frac {g_{12}}{g_{11}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}}{\frac {\partial g_{11}}{\partial x_{2}}}-{\frac {1}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{22}}{\delta x_{1}}}\right)}
+
∂
∂
x
2
(
2
(
g
11
g
22
−
g
12
2
)
∂
g
12
∂
x
1
−
1
(
g
11
g
22
−
g
12
2
)
∂
g
11
∂
x
2
{\displaystyle +{\frac {\partial }{\partial x_{2}}}\left({\frac {2}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{12}}{\partial x_{1}}}-{\frac {1}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{11}}{\partial x_{2}}}\right.}
−
g
12
g
11
(
g
12
g
22
−
g
12
2
)
∂
g
11
∂
x
1
)
=
0
{\displaystyle \left.-{\frac {g_{12}}{g_{11}{\sqrt {\left(g_{12}g_{22}-{g_{12}}^{2}\right)}}}}{\frac {\partial g_{11}}{\partial x_{1}}}\right)=0}
.
Let
g
{\displaystyle g}
be the determinant of four rows and columns formed with the elements
g
μ
ν
{\displaystyle g_{\mu \nu }}
.
Let
g
μ
ν
{\displaystyle g^{\mu \nu }}
be the minor of
g
μ
ν
{\displaystyle g_{\mu \nu }}
, divided by
g
{\displaystyle g}
.
Let the "3-index symbol" {
μ
ν
{\displaystyle \mu \nu }
,
λ
{\displaystyle \lambda }
} denote
1
2
g
λ
a
(
∂
g
μ
a
∂
x
ν
+
∂
g
ν
a
∂
x
μ
−
∂
g
μ
ν
∂
x
a
)
{\displaystyle {\tfrac {1}{2}}g^{\lambda a}\left({\frac {\partial g_{\mu a}}{\partial x_{\nu }}}+{\frac {\partial g_{\nu a}}{\partial x_{\mu }}}-{\frac {\partial g_{\mu \nu }}{\partial x_{a}}}\right)}
summed for values of
a
{\displaystyle a}
from 1 to 4. There will be 40 different 3-index symbols.
Then the Riemann-Christoffel tensor is
B
μ
ν
σ
ρ
=
{
μ
σ
,
ϵ
}
{
ϵ
ν
,
ρ
}
−
{
μ
ν
,
ϵ
}
{
ϵ
σ
,
ρ
}
+
∂
∂
x
ν
{
μ
σ
,
ρ
}
−
∂
∂
x
σ
{
μ
ν
,
ρ
}
{\displaystyle B_{\mu \nu \sigma }^{\rho }=\{\mu \sigma ,\epsilon \}\{\epsilon \nu ,\rho \}-\{\mu \nu ,\epsilon \}\{\epsilon \sigma ,\rho \}+{\frac {\partial }{\partial x_{\nu }}}\{\mu \sigma ,\rho \}-{\frac {\partial }{\partial x_{\sigma }}}\{\mu \nu ,\rho \}}
, the terms containing
ϵ
{\displaystyle \epsilon }
being summed for values of
ϵ
{\displaystyle \epsilon }
from 1 to 4.
The "contracted" Riemann-Christoffel tensor
G
μ
ν
{\displaystyle G_{\mu \nu }}
can be reduced to
G
μ
ν
=
−
∂
∂
x
a
{
μ
ν
,
a
}
+
{
μ
a
,
β
}
{
ν
β
,
a
}
{\displaystyle G_{\mu \nu }=-{\frac {\partial }{\partial x_{a}}}\{\mu \nu ,a\}+\{\mu a,\beta \}\{\nu \beta ,a\}}
+
∂
2
∂
x
μ
∂
x
ν
log
−
g
−
{
μ
ν
,
a
}
∂
∂
x
a
log
−
g
{\displaystyle +{\frac {\partial ^{2}}{\partial x_{\mu }\partial x_{\nu }}}\log {\sqrt {-g}}-\{\mu \nu ,a\}{\frac {\partial }{\partial x_{a}}}\log {\sqrt {-g}}}
,
where in accordance with a general convention in this subject, each term containing a suffix twice over (
a
{\displaystyle a}
and
β
{\displaystyle \beta }
) must be summed for the values 1, 2, 3, 4 of that suffix.
The curvature
G
=
g
μ
ν
G
μ
ν
{\displaystyle G=g^{\mu \nu }G_{\mu \nu }}
, summed in accordance with the foregoing convention.
The electric potential due to a charge
e
{\displaystyle e}
is
ϕ
=
e
[
r
(
1
−
v
r
/
C
)
]
{\displaystyle \phi ={\frac {e}{\left[r\left(1-v_{r}/C\right)\right]}}}
,