Appendix 1 — Formulas of General Relativity
Below we provide a summary of several previously derived formulas from general relativity and the Schwarzschild solution. We then derive all formulas relevant for calculations in various chapters. In this appendix, we use Einstein notation.
General Relativity — Fundamental Equations
Einstein’s field equations:
\begin{equation}
R_{\mu\nu}
- \frac{1}{2} g_{\mu\nu} R
+ \lambda g_{\mu\nu}
=
\frac{8\pi G}{c^{4}}\, T_{\mu\nu}.
\end{equation}
Where:
- \(R_{\mu\nu}\): the Ricci tensor,
- \(g_{\mu\nu}\): the metric tensor,
- \(R\): the Ricci scalar,
- \(\lambda\): the cosmological constant,
- \(T_{\mu\nu}\): the energy–momentum tensor.
Schwarzschild Metric (in spherical coordinates)
\begin{equation}
ds^{2}
=
\left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2}
-
\left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2}
-
r^{2} d\theta^{2}
-
r^{2}\sin^{2}\theta\, d\phi^{2}.
\end{equation}
Where:
- \(ds^{2}\): the spacetime interval,
- \(G\): the gravitational constant,
- \(M\): the mass of the central object,
- \(r\): the radial coordinate,
- \(\theta\) and \(\phi\): the spherical coordinates.
The metric coefficients are therefore not dependent on \(t\) and \(\phi\), but only on \(r\) and \(\theta\).
Gravitational Time Dilation (spherical mass)
For a stationary observer at distance \(r\) from a spherical mass:
\begin{equation}
\Delta \tau
=
\Delta t \,\sqrt{1 - \frac{2GM}{r c^{2}}}
\end{equation}
Where:
- \(\Delta \tau\): proper time at distance \(r\),
- \(\Delta t\): time for a distant observer.
Light Trajectories (null geodesics)
For light, \(ds^{2} = 0\). This gives:
\begin{equation}
\left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2}
=
\left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2}
+ r^{2} d\theta^{2}
+ r^{2}\sin^{2}\theta\, d\phi^{2}.
\end{equation}
Light Deflection Near a Mass
The deflection of a light ray near a mass is:
\begin{equation}
\delta\phi = \frac{4GM}{r c^{2}}.
\end{equation}
Appendix 1.1 — Summary and derivation of further relevant formulas
In this section, we derive the relevant formulas used for calculations in the chapters. This includes:
- the metric tensor in various coordinate systems,
- the geodesic equations,
- the energy–momentum tensor in different configurations.
Coordinate transformations
\begin{equation}
dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\, dy^{r}
\end{equation}
\begin{equation}
ds^{2} = \eta_{mn}\, d\xi^{m} d\xi^{n}
\end{equation}
\begin{equation}
ds^{2}
= g_{mn}(x)\, dx^{m} dx^{n}
= g_{pq}(y)\, dy^{p} dy^{q}
\end{equation}
\begin{equation}
g_{pq}(y)
=
g_{mn}(x)
\frac{\partial x^{m}}{\partial y^{p}}
\frac{\partial x^{n}}{\partial y^{q}}
\end{equation}
Transformation of vectors and tensors
\begin{equation}
V'^{n}(y)
=
\frac{\partial y^{n}}{\partial x^{m}}\, V^{m}(x)
\end{equation}
\begin{equation}
W'_{p}(y)
=
\frac{\partial x^{q}}{\partial y^{p}}\, W_{q}(x)
\end{equation}
\begin{equation}
T_{mn}(x)
=
\frac{\partial V^{m}(x)}{\partial x^{n}}
\end{equation}
\begin{equation}
T_{mn}(y)
=
\frac{\partial x^{r}}{\partial y^{m}}
\frac{\partial x^{s}}{\partial y^{n}}
T_{rs}(x)
\end{equation}
\begin{equation}
T^{mn}(y)
=
\frac{\partial y^{m}}{\partial x^{r}}
\frac{\partial y^{n}}{\partial x^{s}}
T^{rs}(x)
\end{equation}
\begin{equation}
T^{rs}(x) = A_{x}^{r} B_{x}^{s}
\end{equation}
Raising and lowering indices
\begin{equation}
E_{\mu} = g_{\mu\nu} E^{\nu}
\end{equation}
\begin{equation}
E^{\mu}
= g^{\mu\nu} E_{\nu}
= g^{\mu\nu} g_{\nu\rho} E^{\rho}
= \delta^{\mu}_{\rho} E^{\rho}
= E^{\mu}
\end{equation}
Line element in a small region
Pythagoras:
\begin{equation}
ds^{2}
=
\delta_{mn}
\frac{\partial x^{m}}{\partial y^{n}}dy^{n}
\cdot
\frac{\partial x^{n}}{\partial y^{s}}
dy^{s}
\end{equation}
Transformation to another frame:
\begin{equation}
ds^{2}
=
\delta_{mn}
\frac{\partial x^{m}}{\partial y^{r}}
\frac{\partial x^{n}}{\partial y^{s}}
dy^{r} dy^{s}
\end{equation}
Metric tensor
\begin{equation}
g_{mn}
=
\delta_{mn}
\frac{\partial x^{m}}{\partial y^{r}}
\frac{\partial x^{n}}{\partial y^{s}}
\end{equation}
Einstein’s field equations
\begin{equation}
R_{\mu\nu}
- \frac{1}{2} g_{\mu\nu} R
+ \lambda g_{\mu\nu}
=
\frac{8\pi G}{c^{4}}\, T_{\mu\nu}
\end{equation}
Geodesic equation
\begin{equation}
\frac{d^{2} x^{\lambda}}{d\tau^{2}}
+
\Gamma^{\lambda}{}_{\mu\nu}
\frac{dx^{\mu}}{d\tau}
\frac{dx^{\nu}}{d\tau}
= 0
\end{equation}
\begin{equation}
\Gamma^{\lambda}{}_{\mu\nu}
\equiv
\frac{\partial x^{\lambda}}{\partial \xi^{\alpha}}
\frac{\partial^{2} \xi^{\alpha}}{\partial x^{\mu}\partial x^{\nu}}
\end{equation}
Tensor transformations
\begin{equation}
T'_{\mu\nu}(y)
=
\frac{\partial x^{\alpha}}{\partial y^{\mu}}
\frac{\partial x^{\beta}}{\partial y^{\nu}}
T_{\alpha\beta}(x)
\end{equation}
\begin{equation}
T'^{\mu\nu}(y)
=
\frac{\partial y^{\mu}}{\partial x^{\alpha}}
\frac{\partial y^{\nu}}{\partial x^{\beta}}
T^{\alpha\beta}(x)
\end{equation}
\begin{equation}
T_{\mu}^{'\nu}(y)
=
\frac{\partial x^{\alpha}}{\partial y^{\mu}}
\frac{\partial y^{\nu}}{\partial x^{\beta}}
T_{\alpha}^{\beta}(x)
\end{equation}
\begin{equation}
g_{\mu\alpha} g^{\alpha\nu} = \delta_{\mu}^{\nu}
\end{equation}
Contraction
\begin{equation}
A^{\mu} = g^{\mu\nu} A_{\nu}
\end{equation}
\begin{equation}
A_{\mu} = g_{\mu\nu} A^{\nu}
\end{equation}
\begin{equation}
A \cdot B
=
g_{\mu\nu} A^{\mu} B^{\nu}
\equiv A_{\nu} B^{\nu}
\end{equation}
Ricci tensor
\begin{equation}
R_{\mu\nu}
=
R^{\rho}_{\mu\rho\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\mu\rho,\nu}
+
\Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\nu\mu}
-
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\rho}
\end{equation}
\begin{equation}
G_{\mu\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\rho}
\quad\text{(only if } g = \det(g_{\mu\nu}) = -1\text{)}
\end{equation}
Christoffel symbols
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
=
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Ricci scalar
\begin{equation}
R=R^\mu_\mu = g^{\mu\nu} R_{\mu\nu}
\end{equation}
\begin{equation}
R
=
g^{ab}
\left(
\Gamma^{c}{}_{ab,c}
-
\Gamma^{c}{}_{ac,b}
+
\Gamma^{d}{}_{ab}\Gamma^{c}{}_{dc}
-
\Gamma^{d}{}_{ac}\Gamma^{c}{}_{db}
\right)
\end{equation}
\begin{equation}
R
=
2 g^{ab}
\left(
\Gamma^{c}{}_{a[b,c]}
+
\Gamma^{d}{}_{a[b}\Gamma^{c}{}_{c]d}
\right)
\end{equation}
Appendix 1.2 — Schwarzschild metric in spherical coordinates
The Schwarzschild metric is given by:
\begin{equation}
ds^{2}
=
\sigma^{2} c^{2} dt^{2}
-
\frac{dr^{2}}{\sigma^{2}}
-
r^{2} d\theta^{2}
-
r^{2}\sin^{2}\theta\, d\phi^{2},
\end{equation}
where:
\begin{equation}
\sigma^{2} = 1 - \frac{R_{s}}{r},
\qquad
R_{s} = \frac{2GM}{c^{2}}.
\end{equation}
Identification of metric components:
- \(g_{00} = g_{tt}\)
- \(g_{11} = g_{rr}\)
- \(g_{22} = g_{\theta\theta}\)
- \(g_{33} = g_{\phi\phi}\)
Schwarzschild in the plane \(\theta = \frac{\pi}{2}\)
\begin{equation}
g_{00} = \sigma^{2},
\qquad
g^{00} = \frac{1}{\sigma^{2}},
\end{equation}
\begin{equation}
g_{11} = -\frac{1}{\sigma^{2}},
\qquad
g^{11} = -\sigma^{2},
\end{equation}
\begin{equation}
g_{22} = -r^{2},
\qquad
g^{22} = -\frac{1}{r^{2}},
\end{equation}
\begin{equation}
g_{33} = -r^{2}\sin^{2}\theta = -r^{2},
\qquad
g^{33} = -\frac{1}{r^{2}\sin^{2}\theta} = -\frac{1}{r^{2}}.
\end{equation}
Derivative of \(\sigma\):
\begin{equation}
\frac{d\sigma}{dr}
=
\frac{R_{s}}{2 r^{2} \sigma}.
\end{equation}
First derivatives of the metric
\begin{equation}
\frac{\partial g_{00}}{\partial r}
=
\frac{R_{s}}{r^{2}},
\qquad
\frac{\partial g_{11}}{\partial r}
=
\frac{R_{s}}{r^{2}\sigma^{4}},
\end{equation}
\begin{equation}
\frac{\partial g_{22}}{\partial r}
=
-2r,
\qquad
\frac{\partial g_{33}}{\partial r}
=
-2r\sin^{2}\theta = -2r,
\end{equation}
\begin{equation}
\frac{\partial g_{33}}{\partial \theta}
=
-2 r^{2}\sin\theta\cos\theta = 0.
\end{equation}
Second derivatives of the metric
\begin{equation}
\frac{\partial^{2} g_{00}}{\partial r^{2}}
=
-\frac{2R_{s}}{r^{3}},
\qquad
\frac{\partial^{2} g_{11}}{\partial r^{2}}
=
-\frac{2R_{s}}{r^{3}\sigma^{6}},
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial r^{2}}
=
-2,
\qquad
\frac{\partial^{2} g_{33}}{\partial r^{2}}
=
-2\sin^{2}\theta = -2,
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial \theta \partial r}
=
-4r\sin\theta\cos\theta = 0,
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial \theta^{2}}
=
2r^{2}(\sin^{2}\theta - \cos^{2}\theta)
= 2r^{2}.
\end{equation}
Christoffel symbols for Schwarzschild in polar coordinates
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
=
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Non-zero components:
\begin{equation}
\Gamma^{0}{}_{10}
=
\Gamma^{0}{}_{01}
=
\frac{1}{2} g^{00}\frac{\partial g_{00}}{\partial r}
=
\frac{R_{s}}{2 r^{2} \sigma^{2}},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{00}
=
\frac{1}{2} g^{11}\left(-\frac{\partial g_{00}}{\partial r}\right)
=
\sigma^{2}\frac{R_{s}}{2 r^{2}},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{11}
=
\frac{1}{2} g^{11}\frac{\partial g_{11}}{\partial r}
=
-\frac{R_{s}}{2 r^{2} \sigma^{2}},
\end{equation}
\begin{equation}
\Gamma^{2}{}_{21}
=
\Gamma^{2}{}_{12}
=
\frac{1}{2} g^{22}\left(\frac{\partial g_{22}}{\partial r}\right)
=
\frac{1}{r},
\end{equation}
\begin{equation}
\Gamma^{3}{}_{31}
=
\Gamma^{3}{}_{13}
=
\frac{1}{2} g^{33}\left(\frac{\partial g_{33}}{\partial r}\right)
=
\frac{1}{r},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{22}
=
\frac{1}{2} g^{11}\left(-\frac{\partial g_{22}}{\partial r}\right)
=
-r\sigma^{2},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{33}
=
\frac{1}{2} g^{11}\left(-\frac{\partial g_{33}}{\partial r}\right)
=
-r\sigma^{2}\sin^{2}\theta,
\end{equation}
\begin{equation}
\Gamma^{3}{}_{32},
=
\frac{1}{2} g^{33}\left(-\frac{\partial g_{33}}{\partial \theta}\right)
=
\frac{\cos\theta}{\sin\theta},
\end{equation}
\begin{equation}
\Gamma^{2}{}_{33}
=
\frac{1}{2} g^{22}\left(-\frac{\partial g_{33}}{\partial \theta}\right)
=
-\sin\theta\cos\theta.
\end{equation}
First derivatives of the Christoffel symbols
\begin{equation}
\frac{\partial \Gamma^{0}_{01}}{\partial r}
=
\frac{\partial \Gamma^{0}_{10}}{\partial r}
=
\frac{R_{s}(R_{s}-2r)}{2 r^{4} \sigma^{4}},
\qquad
\frac{\partial \Gamma^{1}{}_{00}}{\partial r}
=
\frac{R_{s}(3R_{s}-2r)}{2 r^{4}},
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{11}}{\partial r}
=
\frac{R_{s}(2r - R_{s})}{2 r^{4} \sigma^{4}},
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{22}}{\partial r}
=
-1,
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{33}}{\partial r}
=
-\sin^{2}\theta,
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{12}}{\partial r}
=
\frac{\partial \Gamma^{2}_{21}}{\partial r}
=
\frac{\partial \Gamma^{3}_{13}}{\partial r}
=
\frac{\partial \Gamma^{3}_{31}}{\partial r}
=
-\frac{1}{r^{2}},
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{33}}{\partial \theta}
=
-\cos^{2}\theta+\sin^{2}\theta=1,
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{3}_{23}}{\partial \theta}
=
\frac{\partial \Gamma^{3}_{32}}{\partial \theta}
=
-\frac{1}{\sin^{2}\theta} = -1,
\end{equation}
First derivative of the Christoffel symbol (general form)
\begin{equation}
\begin{aligned}
\frac{\partial \Gamma^{\rho}_{\mu\nu}}{\partial x^{\delta}}
&=
\frac{1}{2}
\frac{\partial g^{\rho\alpha}}{\partial x^{\delta}}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\\
&\quad
+
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}}
+
\frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}}
-
\frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}}
\right)
\end{aligned}
\end{equation}
Because:
\begin{equation}
\frac{\partial g^{\rho\alpha}}{\partial x^{\delta}}
=
\frac{\partial \frac{1}{g_{\rho\alpha}}}{\partial x^{\delta}}
=
\frac{-1}{g^2_{\rho\alpha}}\cdot\frac{\partial g_{\rho\alpha}}{\partial x^{\lambda}}
=
- (g^{\rho\alpha})^{2}
\frac{\partial g_{\rho\alpha}}{\partial x^{\delta}}
\end{equation}
we obtain:
\begin{equation}
\begin{aligned}
\frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\delta}}
=
-\frac{1}{2}
(g^{\rho\alpha})^{2}
\frac{\partial g_{\rho\alpha}}{\partial x^{\delta}}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\\
+
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}}
+
\frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}}
-
\frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}}
\right)
\end{aligned}
\end{equation}
Appendix 1.3 — Schwarzschild metric in x, y, z coordinates
Coordinate transformation
\begin{equation}
x_{0} = t_{\infty}, \qquad dx_{0} = dt_{\infty}
\end{equation}
\begin{equation}
x_{1} = \frac{r^{3}}{3}, \qquad dx_{1} = r^{2}\, dr, \qquad
\frac{dr}{dx_{1}} = \frac{1}{r^{2}}
\end{equation}
\begin{equation}
x_{2} = -\cos\theta, \qquad dx_{2} = \sin\theta\, d\theta = d\theta,
\qquad
\frac{d\theta}{dx_{2}} = \frac{1}{\sin\theta}
\end{equation}
\begin{equation}
x_{3} = \phi, \qquad dx_{3} = d\phi
\end{equation}
Schwarzschild metric in xyz coordinates
\begin{equation}
ds^{2}
=
\sigma^{2} c^{2} dt_{\infty}^{2}
-
\frac{dx_{1}^{2}}{r^{4}\sigma^{2}}
-
r^{2}\frac{dx_{2}^{2}}{\sin^{2}\theta}
-
r^{2}\sin^{2}\theta\, dx_{3}^{2},
\end{equation}
where:
\begin{equation}
\sigma^{2} = 1 - \frac{R_{s}}{r},
\qquad
R_{s} = \frac{2GM}{c^{2}}.
\end{equation}
Assumption: equatorial plane \(\theta = \frac{\pi}{2}\)
\begin{equation}
\sin\theta = 1
\end{equation}
\begin{equation}
ds^{2}
=
\sigma^{2} c^{2} dt_{\infty}^{2}
-
\frac{dx_{1}^{2}}{r^{4}\sigma^{2}}
-
r^{2} dx_{2}^{2}
-
r^{2} dx_{3}^{2}.
\end{equation}
Metric components in xyz coordinates
\begin{equation}
g_{00} = \sigma^{2},
\qquad
g^{00} = \frac{1}{\sigma^{2}},
\end{equation}
\begin{equation}
g_{11} = -\frac{1}{r^{4}\sigma^{2}},
\qquad
g^{11} = -r^{4}\sigma^{2},
\end{equation}
\begin{equation}
g_{22} = -\frac{r^{2}}{\sin^{2}\theta},
\qquad
g^{22} = -\frac{\sin^{2}\theta}{r^{2}},
\end{equation}
\begin{equation}
g_{33} = -r^{2}\sin^{2}\theta=-r^2,
\qquad
g^{33} = -\frac{1}{r^{2}\sin^{2}\theta}=\frac{-1}{r^2}.
\end{equation}
Dependencies:
- \(g_{\mu\nu} = g_{\mu\nu}(r,\theta)\)
- \(\displaystyle \frac{dr}{dx_{1}} = \frac{1}{r^{2}}\)
- \(\displaystyle \frac{d\sigma}{dx_{1}} = \frac{R_{s}}{2 r^{4}\sigma}\)
- \(\displaystyle \frac{d\theta}{dx_{2}} = \frac{1}{\sin\theta}\)
First derivatives of the metric
\begin{equation}
\frac{\partial g_{00}}{\partial x_{1}}
=
\frac{\partial g_{00}}{\partial r}
\frac{dr}{dx_{1}}
=
2\sigma\frac{R_s}{2r^4\sigma}
=
\frac{R_{s}}{r^{4}}
\end{equation}
\begin{equation}
\frac{\partial g_{11}}{\partial x_{1}}
=
\frac{4r - 3R_{s}}{r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
\frac{\partial g_{22}}{\partial x_{1}}
=
\frac{\partial g_{22}}{\partial r}
\frac{\partial r}{\partial x_{1}}
=
r^{-2}\left(\frac{-2r}{\sin^2\theta}\right)
=
\frac{-2}{r\sin^{2}\theta} =\frac{ -2}{r}
\end{equation}
\begin{equation}
\frac{\partial g_{33}}{\partial x_{1}}
=
r^{-2}\left(-2r\sin^2\theta\right)
=
\frac{-2r\sin^{2}\theta}{r} =\frac{ -2}{r}
\end{equation}
\begin{equation}
\frac{\partial g_{22}}{\partial x_{2}}
=
\frac{2r^2\cos\theta}{\sin^3\theta}.\frac{1}{\sin\theta}
=
\frac{2r^{2}\cos\theta}{ \sin^{4}\theta} = 0
\end{equation}
\begin{equation}
\frac{\partial g_{33}}{\partial x_{2}}
=
\frac{\partial g_{33}}{\partial \theta}
\frac{\partial \theta}{\partial x_{2}}
=
-2r^2\sin\theta\,\cos\theta\,\frac{1}{\sin\theta}
=
-2r^{2}\cos\theta = 0
\end{equation}
Second derivatives of the metric
\begin{equation}
\frac{\partial^{2} g_{00}}{\partial x_{1}^{2}}
=
-\frac{4R_{s}}{r^{7}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{11}}{\partial x_{1}^{2}}
=
-\frac{2(14r^{2} + 9R_{s}^{2} - 22rR_{s})}{r^{12}\sigma^{6}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial x_{1}^{2}}
=
\frac{2}{r^4\sin^2\theta}
=
\frac{2}{r^{4}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial x_{2}^{2}}
=
-2r^{2}\frac{1 + 3\cos^{2}\theta}{\sin^{6}\theta}
= -2r^{2}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial x_{1}\partial x_{2}}
=
\frac{4\cos\theta}{r\sin^{4}\theta} = 0
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial x_{1}^{2}}
=
\frac2\sin^2\theta{}{r^4}
=
\frac{2}{r^{4}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial x_{1}\partial x_{2}}
=
\frac{-4\cos\theta}{r} = 0
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial x_{2}^{2}}
=
2r^2\,\sin\theta\,\frac{1}{\sin\theta}
=
2r^{2}
\end{equation}
Christoffel symbols in xyz coordinates
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
=
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Non-zero components:
\begin{equation}
\Gamma^{0}{}_{10}
=
\Gamma^{0}{}_{01}
=
\frac{R_{s}}{2 r^{4}\sigma^{2}}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{00}
=
\frac{R_{s}\sigma^{2}}{2}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{11}
=
\frac{3R_{s} - 4r}{2 r^{4}\sigma^{2}}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{22}
=
\frac{-r^{3}\sigma^{2}}{\sin^2\theta}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{33}
=
-r^{3}\sigma^{2}\sin^{2}\theta
=-r^3\sigma^2
\end{equation}
\begin{equation}
\Gamma^{2}{}_{21}
=
\Gamma^{2}{}_{12}
=
\frac{1}{r^{3}}
\end{equation}
\begin{equation}
\Gamma^{2}{}_{33}
=
-\sin^2\theta\,\cos\theta
\end{equation}
\begin{equation}
\Gamma^2_{22}
=
\frac{-\cos\theta}{\sin^2\theta}
=0
\end{equation}
\begin{equation}
\Gamma^{3}{}_{31}
=
\Gamma^{3}{}_{13}
=
\frac{1}{2}
\frac{-1}{r^2\sin^2\theta}
\frac{-2\sin^2\theta}{r}
=
\frac{1}{r^{3}}
\end{equation}
\begin{equation}
\Gamma^{3}{}_{32}
=
\frac{1}{2}
\frac{-1}{r^2\sin^2\theta}
\left(-2r^2\cos\theta\right)
=
\frac{\cos\theta}{\sin^{2}\theta}
=0
\end{equation}
(in the equatorial plane \(\theta = \frac{\pi}{2}\)).
First derivatives of the Christoffel symbols in xyz coordinates
\begin{equation}
\frac{\partial \Gamma^{0}_{10}}{\partial x_{1}}
=
\frac{\partial \Gamma^{0}_{01}}{\partial x_{1}}
=
\frac{R_{s}(3R_{s} - 4r)}{2 r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{00}}{\partial x_{1}}
=
\frac{R_{s}}{2 r^{4}}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{11}}{\partial x_{1}}
=
\frac{6}{r^{6}\sigma^{4}}
-
\frac{10R_{s} }{r^{7}\sigma^{4}}
+
\frac{4.5 R_{s}^{2} }{r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{22}}{\partial x_{1}}
=
\frac{2R_s-3r}{r\sin^2\theta}
=
-3+\frac{2R_s}{r}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{33}}{\partial x_{1}}
=
\left(-3+\frac{2R_s}{r}\right)\sin^2\theta
=
-3+\frac{2R_s}{r}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{21}}{\partial x_{1}}
=
\frac{\partial \Gamma^{3}_{13}}{\partial x_{1}}
=
-\frac{3}{r^{6}}
\end{equation}
\begin{equation}
\frac{\Gamma^2_{33}}{\partial x_{1}}=
\frac{\Gamma^2_{22}}{\partial x_{1}}=
\frac{\Gamma^3_{23}}{\partial x_{1}}=
\frac{\Gamma^3_{32}}{\partial x_{1}}
=0
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{22}}{\partial x_{2}}
=
\frac{2r^3\sigma^2\cos\theta}{\sin^4\theta}
=0
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{33}}{\partial x_{2}}
=-2r^3\sigma^2\cos\theta=0
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{33}}{\partial x_{2}}
=-3\cos^2\theta+1=1
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{22}}{\partial x_{1}}
=
\frac{1+\cos^2\theta}{\sin^{4}\theta}=1
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{3}_{23}}{\partial x_{2}}
=
\frac{\partial \Gamma^{3}_{32}}{\partial x_{2}}
=
-\frac{1+\cos^2\theta}{\sin^4\theta}
=
-1
\end{equation}
Riemann tensor
\begin{equation}
R^{i}_{jkl}
=
\Gamma^{i}_{jl,k}
-
\Gamma^{i}_{jk,l}
+
\Gamma^{u}_{jl}\Gamma^{i}{}_{uk}
-
\Gamma^{u}_{jk}\Gamma^{i}{}_{ul}
\end{equation}
Ricci tensor
\begin{equation}
R_{\mu\nu}
=
R^{\rho}_{\mu\rho\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\mu\rho,\nu}
+
\Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\rho}
-
\Gamma^{\lambda}_{\mu\rho}\Gamma^{\rho}_{\lambda\nu}
\end{equation}
Or:
\begin{equation}
R_{\mu\nu}
=
R^{\rho}_{\mu\nu\rho}
=
-\Gamma^{\rho}_{\mu\nu,\rho}
+
\Gamma^{\rho}_{\mu\rho,\nu}
-
\Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\rho}
+
\Gamma^{\lambda}_{\mu\rho}\Gamma^{\rho}_{\lambda\nu}.
\end{equation}
Note on the sign of the Christoffel symbol
From the calculations it follows that, in order to obtain all Ricci components equal to zero in vacuum, the Christoffel symbol must start with a positive +1/2:
\begin{equation}
\Gamma^{\rho}_{\mu\nu}
=
+\frac{1}{2}
g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
The sign only affects the derivative terms in the Ricci tensor,
not the products of Christoffel symbols.
Schwarzschild symmetry of the Ricci tensor
\begin{equation}
R_{\mu\nu}
=
\Gamma^0_{\mu\nu,0}
-
\Gamma^0_{0\mu,\nu}
+
\Gamma^{0}_{0\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^0_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
\begin{equation}
+\Gamma^1_{\mu\nu,1}
-
\Gamma^1_{1\mu,\nu}
+
\Gamma^{1}_{1\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^1_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
\begin{equation}
+\Gamma^2_{\mu\nu,2}
-
\Gamma^2_{2\mu,\nu}
+
\Gamma^{2}_{2\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^2_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
\begin{equation}
+\Gamma^3_{\mu\nu,3}
-
\Gamma^3_{3\mu,\nu}
+
\Gamma^{3}_{3\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^3_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
In compact form:
\begin{equation}
R_{\mu\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\rho\mu,\nu}
+
\Gamma^{\rho}_{\rho\lambda}\Gamma^{\lambda}_{\nu\mu}
-
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\rho\mu}
\end{equation}
Ricci tensor components for Schwarzschild
\begin{equation}
R_{00}
=
\Gamma_{00,1}^{1}
+
\Gamma_{11}^{1}\Gamma_{00}^{1}
+
\Gamma_{21}^{2}\Gamma_{00}^{1}
+
\Gamma_{31}^{3}\Gamma_{00}^{1}
-
\Gamma_{00}^{1}\Gamma_{10}^{0}
\end{equation}
\begin{equation}
=\frac{R_s^2}{2r^4}
-\frac{1}{2}\frac{4r-3R_s}{r^4\sigma^2}\frac{1}{2}R_s\sigma^2
-\frac{1}{2}R_s\sigma^2\frac{1}{2}\frac{R_s}{r^4\sigma^2}
-\frac{1}{2}\frac{R_s}{r^4\sigma^2}\frac{1}{2}R_s\sigma^2
\end{equation}
\begin{equation}
\begin{aligned}
R_{11}
&=
-\Gamma_{01,1}^{0}
-
\Gamma_{21,1}^{2}
-
\Gamma_{31,1}^{3}
+
\Gamma_{01}^{0}\Gamma_{11}^{1}
+
\Gamma_{21}^{2}\Gamma_{11}^{1}
\\
&\quad
+
\Gamma_{31}^{3}\Gamma_{11}^{1}
-
\Gamma_{10}^{0}\Gamma_{10}^{0}
-
\Gamma_{12}^{2}\Gamma_{21}^{2}
-
\Gamma_{13}^{3}\Gamma_{31}^{3}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{22}
&=
\Gamma_{22,1}^{1}
-
\Gamma_{32,2}^{3}
+
\Gamma_{01}^{0}\Gamma_{22}^{1}
+
\Gamma_{11}^{1}\Gamma_{22}^{1}
+
\Gamma_{21}^{2}\Gamma_{22}^{1}
\\
&\quad
+
\Gamma_{31}^{3}\Gamma_{22}^{1}
-
\Gamma_{22}^{1}\Gamma_{12}^{2}
-
\Gamma_{21}^{2}\Gamma_{22}^{1}
\end{aligned}
\end{equation}
\begin{equation}
R_{33}
=
\Gamma_{33,1}^{1}
+
\Gamma_{01}^{0}\Gamma_{33}^{1}
+
\Gamma_{11}^{1}\Gamma_{33}^{1}
+
\Gamma_{21}^{2}\Gamma_{33}^{1}
-
\Gamma_{33}^{1}\Gamma_{13}^{3}
\end{equation}
Ricci tensor components for Schwarzschild (\(\theta = \frac{\pi}{2}\))
For spherical coordinates and the Schwarzschild configuration with \(𝜃=90^0\), the following elements of the Ricci tensor are relevant:
\begin{equation}
R_{00}
=
\Gamma_{00,1}^{1}
+
\Gamma_{00}^{1}\Gamma_{11}^{1}
+
\Gamma_{00}^{1}\Gamma_{12}^{2}
+
\Gamma_{00}^{1}\Gamma_{13}^{3}
-
\Gamma_{01}^{0}\Gamma_{00}^{1}
\end{equation}
\begin{equation}
\begin{aligned}
R_{11}
&=
-\Gamma_{10,1}^{0}
-
\Gamma_{12,1}^{2}
-
\Gamma_{13,1}^{3}
+
\Gamma_{11}^{1}\Gamma_{10}^{0}
+
\Gamma_{11}^{1}\Gamma_{12}^{2}
\\
&\quad
+
\Gamma_{11}^{1}\Gamma_{13}^{3}
-
\Gamma_{10}^{0}\Gamma_{01}^{0}
-
\Gamma_{12}^{2}\Gamma_{21}^{2}
-
\Gamma_{13}^{3}\Gamma_{31}^{3}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{22}
&=
\Gamma_{22,1}^{1}
-
\Gamma_{23,2}^{3}
+
\Gamma_{22}^{1}\Gamma_{10}^{0}
+
\Gamma_{22}^{1}\Gamma_{11}^{1}
+
\Gamma_{22}^{1}\Gamma_{13}^{3}
\\
&\quad
+\boxed{
\Gamma_{22}^{2}\Gamma_{32}^{3}}
-
\Gamma_{21}^{2}\Gamma_{22}^{1}
-
\Gamma_{23}^{3}\Gamma_{32}^{3}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{33}
&=
\Gamma_{33,1}^{1}
+
\Gamma_{33,2}^{2}
+
\Gamma_{33}^{1}\Gamma_{10}^{0}
+
\Gamma_{33}^{1}\Gamma_{11}^{1}
+
\Gamma_{33}^{1}\Gamma_{12}^{2}
\\
&\quad
+\boxed{
\Gamma_{33}^{2}\Gamma_{22}^{2}}
-
\Gamma_{31}^{3}\Gamma_{33}^{1}
-
\Gamma_{32}^{3}\Gamma_{33}^{2}
\end{aligned}
\end{equation}
\begin{equation}
R_{33} = \sin^{2}\theta\, R_{22}
\end{equation}
When \(\theta \neq \frac{\pi}{2}\), additional terms appear:
\begin{equation}
R_{22} \to R_{22} + \Gamma_{22}^{2}\Gamma_{32}^{3},
\qquad
R_{33} \to R_{33} + \Gamma_{33}^{2}\Gamma_{22}^{2}.
\end{equation}