Appendix 1 — Formulas of General Relativity

Below we provide a summary of several formulas previously derived from general relativity and the Schwarzschild solution. Subsequently, we derive all formulas that are relevant for calculations in various chapters. In this appendix we employ Einstein notation.

General Relativity — Fundamental Formulas

Einstein’s field equations: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\, T_{\mu\nu}. \]

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)

\[ ds^{2} = \left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2} - \left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2}. \]

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 therefore do not depend on \(t\) and \(\phi\), but only on \(r\) and \(\theta\).

Time Dilation for a Spherical Object (Gravitational Time Dilation)

For a stationary observer at distance \(r\) from a spherical mass: \[ d\tau = \sqrt{1 - \frac{2GM}{r c^{2}}}\; dt, \] where \(d\tau\) is the local (proper) time and \(dt\) is the coordinate time at large distance.

\[ \Delta \tau = \Delta t \,\sqrt{1 - \frac{2GM}{r c^{2}}} \]

Where:

\(\Delta \tau\): the proper time for an observer at distance \(r\),
\(\Delta t\): the time measured by a distant observer.

Trajectory of Light (null geodesics)

For light one has \(ds^{2} = 0\). This yields: \[ \left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2} = \left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2} + r^{2} d\theta^{2} + r^{2}\sin^{2}\theta\, d\phi^{2}. \]

Curvature Radius of Light Near a Mass

The deflection of a light ray near a mass is given by: \[ \delta\phi = \frac{4GM}{r c^{2}}. \]

Appendix 1.1 — Summary and Derivation of Further Relevant Formulas

In this section we derive the formulas relevant for the calculations in the chapters. This includes:

the metric tensor in different coordinate systems,
the geodesic equations,
the energy–momentum tensor in various configurations.

Coordinate Transformations

\[ dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\, dy^{r} \]

\[ ds^{2} = \eta_{mn}\, d\xi^{m} d\xi^{n} \]

\[ ds^{2} = g_{mn}(x)\, dx^{m} dx^{n} = g_{pq}(y)\, dy^{p} dy^{q} \]

\[ g_{pq}(y) = g_{mn}(x) \frac{\partial x^{m}}{\partial y^{p}} \frac{\partial x^{n}}{\partial y^{q}} \]

Transformation of Vectors and Tensors

\[ V'^{n}(y) = \frac{\partial y^{n}}{\partial x^{m}}\, V^{m}(x) \]

\[ W'_{p}(y) = \frac{\partial x^{q}}{\partial y^{p}}\, W_{q}(x) \]

\[ T_{mn}(x) = \frac{\partial V_{m}(x)}{\partial x^{n}} \]

\[ T_{mn}(y) = \frac{\partial x^{r}}{\partial y^{m}} \frac{\partial x^{s}}{\partial y^{n}} T_{rs}(x) \]

\[ T_{mn}(y) = \frac{\partial y^{m}}{\partial x^{r}} \frac{\partial y^{n}}{\partial x^{s}} T_{rs}(x) \]

\[ T_{rs}(x) = A^{x}_{r} B^{x}_{s} \]

Raising and Lowering Indices

\[ E_{\mu} = g_{\mu\nu} E^{\nu} \]

\[ E^{\mu} = g^{\mu\nu} E_{\nu} = g^{\mu\nu} g_{\nu\rho} E^{\rho} = \delta^{\mu}{}_{\rho} E^{\rho} = E^{\mu} \]

Line Element in a Small Region

Pythagoras: \[ ds^{2} = \delta_{mn} \frac{\partial x^{m}}{\partial y^{r}}dy^{r} \cdot \frac{\partial x^{n}}{\partial y^{s}} dy^{s} \]

Transformation to another frame: \[ ds^{2} = \delta_{mn} \frac{\partial x^{m}}{\partial y^{r}} \frac{\partial x^{n}}{\partial y^{s}} dy^{r} dy^{s} \]

Metric Tensor

\[ g_{mn} = \delta_{mn} \frac{\partial x^{m}}{\partial y^{r}} \frac{\partial x^{n}}{\partial y^{s}} \]

Einstein’s Field Equations

\[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\, T_{\mu\nu} \]

Geodesic Equation

\[ \frac{d^{2} x^{\lambda}}{d\tau^{2}} + \Gamma^{\lambda}{}_{\mu\nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = 0 \]

\[ \Gamma^{\lambda}{}_{\mu\nu} \equiv \frac{\partial x^{\lambda}}{\partial \xi^{\alpha}} \frac{\partial^{2} \xi^{\alpha}}{\partial x^{\mu}\partial x^{\nu}} \]

Tensor Transformations

\[ T'_{\mu\nu}(y) = \frac{\partial x^{\alpha}}{\partial y^{\mu}} \frac{\partial x^{\beta}}{\partial y^{\nu}} T_{\alpha\beta}(x) \]

\[ T'^{\mu\nu}(y) = \frac{\partial y^{\mu}}{\partial x^{\alpha}} \frac{\partial y^{\nu}}{\partial x^{\beta}} T^{\alpha\beta}(x) \]

\[ T_{\mu}{}^{\nu}(y) = \frac{\partial x^{\alpha}}{\partial y^{\mu}} \frac{\partial y^{\nu}}{\partial x^{\beta}} T_{\alpha}^{\beta}(x) \]

\[ g_{\mu\alpha} g^{\alpha\nu} = \delta_{\mu}^{\nu} \]

Contraction

\[ A^{\mu} = g^{\mu\nu} A_{\nu} \]

\[ A_{\mu} = g_{\mu\nu} A^{\nu} \]

\[ A \cdot B = g_{\mu\nu} A^{\mu} B^{\nu} \equiv A_{\nu} B^{\nu} \]

Ricci Tensor

\[ R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu} = \Gamma^{\rho}{}_{\mu\nu,\rho} - \Gamma^{\rho}{}_{\mu\rho,\nu} + \Gamma^{\rho}{}_{\lambda\rho}\Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\rho} \]

\[ G_{\mu\nu} = \Gamma^{\rho}{}_{\mu\nu,\rho} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\rho} \quad\text{(only if } g = \det(g_{\mu\nu}) = -1\text{)} \]

Christoffel Symbols

\[ \Gamma^{\rho}{}_{\mu\nu} = \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]

Ricci Scalar

\[ R=R^\mu_\mu = g^{\mu\nu} R_{\mu\nu} \]

\[ R = g^{ab} \left( \Gamma^{c}{}_{ab,c} - \Gamma^{c}{}_{ac,b} + \Gamma^{d}{}_{ab}\Gamma^{c}{}_{dc} - \Gamma^{d}{}_{ac}\Gamma^{c}{}_{db} \right) \]

\[ R = 2 g^{ab} \left( \Gamma^{c}{}_{a[b,c]} + \Gamma^{d}{}_{a[b}\Gamma^{c}{}_{c]d} \right) \]

Appendix 1.2 — Schwarzschild Metric in Polar Coordinates

The Schwarzschild metric is given by: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dr^{2}}{\sigma^{2}} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2}, \] where: \[ \sigma^{2} = 1 - \frac{R_{s}}{r}, \qquad R_{s} = \frac{2GM}{c^{2}}. \]

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 \(\theta = \frac{\pi}{2}\)

\[ g_{00} = \sigma^{2}, \qquad g^{00} = \frac{1}{\sigma^{2}}, \] \[ g_{11} = -\frac{1}{\sigma^{2}}, \qquad g^{11} = -\sigma^{2}, \] \[ g_{22} = -r^{2}, \qquad g^{22} = -\frac{1}{r^{2}}, \] \[ g_{33} = -r^{2}\sin^{2}\theta = -r^{2}, \qquad g^{33} = -\frac{1}{r^{2}\sin^{2}\theta} = -\frac{1}{r^{2}}. \]

Derivative of \(\sigma\):

\[ \frac{d\sigma}{dr} = \frac{R_{s}}{2 r^{2} \sigma}. \]

First Derivatives of the Metric

\[ \frac{\partial g_{00}}{\partial r} = \frac{R_{s}}{r^{2}}, \qquad \frac{\partial g_{11}}{\partial r} = \frac{R_{s}}{r^{2}\sigma^{4}}, \] \[ \frac{\partial g_{22}}{\partial r} = -2r, \qquad \frac{\partial g_{33}}{\partial r} = -2r\sin^{2}\theta = -2r, \] \[ \frac{\partial g_{33}}{\partial \theta} = -2 r^{2}\sin\theta\cos\theta = 0. \]

Second Derivatives of the Metric

\[ \frac{\partial^{2} g_{00}}{\partial r^{2}} = -\frac{2R_{s}}{r^{3}}, \qquad \frac{\partial^{2} g_{11}}{\partial r^{2}} = -\frac{2R_{s}}{r^{3}\sigma^{6}}, \] \[ \frac{\partial^{2} g_{22}}{\partial r^{2}} = -2, \qquad \frac{\partial^{2} g_{33}}{\partial r^{2}} = -2\sin^{2}\theta = -2, \] \[ \frac{\partial^{2} g_{33}}{\partial \theta \partial r} = -4r\sin\theta\cos\theta = 0, \] \[ \frac{\partial^{2} g_{33}}{\partial \theta^{2}} = 2r^{2}(\sin^{2}\theta - \cos^{2}\theta) = 2r^{2}. \]

Christoffel Symbols for Schwarzschild in Polar Coordinates

Non-zero components:

\[ \Gamma^{0}{}_{10} = \Gamma^{0}{}_{01} = \frac{1}{2} g^{00}\frac{\partial g_{00}}{\partial r} = \frac{R_{s}}{2 r^{2} \sigma^{2}}, \] \[ \Gamma^{1}{}_{00} = \frac{1}{2} g^{11}\left(-\frac{\partial g_{00}}{\partial r}\right) = \sigma^{2}\frac{R_{s}}{2 r^{2}}, \] \[ \Gamma^{1}{}_{11} = \frac{1}{2} g^{11}\frac{\partial g_{11}}{\partial r} = -\frac{R_{s}}{2 r^{2} \sigma^{2}}, \] \[ \Gamma^{2}{}_{21} = \Gamma^{2}{}_{12} = \frac{1}{2} g^{22}\left(\frac{\partial g_{22}}{\partial r}\right) = \frac{1}{r}, \] \[ \Gamma^{3}{}_{31} = \Gamma^{3}{}_{13} = \frac{1}{2} g^{33}\left(\frac{\partial g_{33}}{\partial r}\right) = \frac{1}{r}, \] \[ \Gamma^{1}{}_{22} = \frac{1}{2} g^{11}\left(-\frac{\partial g_{22}}{\partial r}\right) = -r\sigma^{2}, \] \[ \Gamma^{1}{}_{33} = \frac{1}{2} g^{11}\left(-\frac{\partial g_{33}}{\partial r}\right) = -r\sigma^{2}\sin^{2}\theta, \] \[ \Gamma^{3}{}_{32}, = \frac{1}{2} g^{33}\left(-\frac{\partial g_{33}}{\partial \theta}\right) = \frac{\cos\theta}{\sin\theta}, \] \[ \Gamma^{2}{}_{33} = \frac{1}{2} g^{22}\left(-\frac{\partial g_{33}}{\partial \theta}\right) = -\sin\theta\cos\theta. \]

First Derivatives of the Christoffel Symbols

\[ \frac{\partial \Gamma^{0}_{01}}{\partial r} = \frac{\partial \Gamma^{0}_{10}}{\partial r} = \frac{R_{s}(R_{s}-2r)}{2 r^{4} \sigma^{4}}, \qquad \frac{\partial \Gamma^{1}{}_{00}}{\partial r} = \frac{R_{s}(3R_{s}-2r)}{2 r^{4}}, \] \[ \frac{\partial \Gamma^{1}_{11}}{\partial r} = \frac{R_{s}(2r - R_{s})}{2 r^{4} \sigma^{4}}, \] \[ \frac{\partial \Gamma^{1}_{22}}{\partial r} = -1, \] \[ \frac{\partial \Gamma^{1}_{33}}{\partial r} = -\sin^{2}\theta, \] \[ \frac{\partial \Gamma^{2}_{12}}{\partial r} = \frac{\partial \Gamma^{2}_{21}}{\partial r} = \frac{\partial \Gamma^{3}_{13}}{\partial r} = \frac{\partial \Gamma^{3}_{31}}{\partial r} = -\frac{1}{r^{2}}, \] \[ \frac{\partial \Gamma^{2}_{33}}{\partial \theta} = -\cos^{2}\theta+\sin^{2}\theta=1, \] \[ \frac{\partial \Gamma^{3}_{23}}{\partial \theta} = \frac{\partial \Gamma^{3}_{32}}{\partial \theta} = -\frac{1}{\sin^{2}\theta} = -1, \]

First Derivative of the Christoffel Symbol (General Form)

\[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\delta}} = \frac{1}{2} \frac{\partial g^{\rho\alpha}}{\partial x^{\delta}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]\[ + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}} \right) \]

Because:

\[ \frac{\partial g^{\rho\alpha}}{\partial x^{\gamma}} = \frac{\partial \frac{1}{g_{\rho\alpha}}}{\partial x^{\gamma}} = \frac{-1}{g^2_{\rho\alpha}}\cdot\frac{\partial g_{\rho\alpha}}{\partial x^{\lambda}} = - (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \]

we get: \[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\delta}} = -\frac{1}{2} (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\delta}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]\[ + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}} \right). \]

>Appendix 1.3 — Schwarzschild Metric in x, y, z Coordinates

Coordinate Transformation

\[ x_{0} = t_{\infty}, \qquad dx_{0} = dt_{\infty} \] \[ x_{1} = \frac{r^{3}}{3}, \qquad dx_{1} = r^{2}\, dr, \qquad \frac{dr}{dx_{1}} = \frac{1}{r^{2}} \] \[ x_{2} = -\cos\theta, \qquad dx_{2} = \sin\theta\, d\theta = d\theta, \qquad \frac{d\theta}{dx_{2}} = \frac{1}{\sin\theta} \] \[ x_{3} = \phi, \qquad dx_{3} = d\phi \]

Schwarzschild Metric in xyz Coordinates

\[ ds^{2} = \sigma^{2} c^{2} dt_{\infty}^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - r^{2}\frac{dx_{2}^{2}}{\sin^{2}\theta} - r^{2}\sin^{2}\theta\, dx_{3}^{2}, \] where: \[ \sigma^{2} = 1 - \frac{R_{s}}{r}, \qquad R_{s} = \frac{2GM}{c^{2}}. \]

Assumption: Equatorial Plane \(\theta = \frac{\pi}{2}\)

\[ \sin\theta = 1 \] \[ ds^{2} = \sigma^{2} c^{2} dt_{\infty}^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - r^{2} dx_{2}^{2} - r^{2} dx_{3}^{2}. \]

Metric Components in xyz Coordinates

\[ g_{00} = \sigma^{2}, \qquad g^{00} = \frac{1}{\sigma^{2}}, \] \[ g_{11} = -\frac{1}{r^{4}\sigma^{2}}, \qquad g^{11} = -r^{4}\sigma^{2}, \] \[ g_{22} = -\frac{r^{2}}{\sin^{2}\theta}, \qquad g^{22} = -\frac{\sin^{2}\theta}{r^{2}}, \] \[ g_{33} = -r^{2}\sin^{2}\theta=-r^2, \qquad g^{33} = -\frac{1}{r^{2}\sin^{2}\theta}=\frac{-1}{r^2}. \]

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

\[ \frac{\partial g_{00}}{\partial x_{1}} = \frac{\partial g_{00}}{\partial r} \frac{dr}{dx_{1}} = 2\sigma\frac{R_s}{2r^4\sigma} = \frac{R_{s}}{r^{4}} \] \[ \frac{\partial g_{11}}{\partial x_{1}} = \frac{4r - 3R_{s}}{r^{8}\sigma^{4}} \] \[ \frac{\partial g_{22}}{\partial x_{1}} = \frac{\partial g_{22}}{\partial r} \frac{\partial r}{\partial x_{1}} = r^{-2}\left(\frac{-2r}{\sin^2\theta}\right) = \frac{-2}{r\sin^{2}\theta} =\frac{ -2}{r} \] \[ \frac{\partial g_{33}}{\partial x_{1}} = r^{-2}\left(-2r\sin^2\theta\right) = \frac{-2r\sin^{2}\theta}{r} =\frac{ -2}{r} \] \[ \frac{\partial g_{22}}{\partial x_{2}} = \frac{2r^2\cos\theta}{\sin^3\theta}.\frac{1}{\sin\theta} = \frac{2r^{2}\cos\theta}{ \sin^{4}\theta} = 0 \] \[ \frac{\partial g_{33}}{\partial x_{2}} = \frac{\partial g_{33}}{\partial \theta} \frac{\partial \theta}{\partial x_{2}} = -2r^2\sin\theta\,\cos\theta\,\frac{1}{\sin\theta} = -2r^{2}\cos\theta = 0 \]

Second Derivatives of the Metric

\[ \frac{\partial^{2} g_{00}}{\partial x_{1}^{2}} = -\frac{4R_{s}}{r^{7}} \] \[ \frac{\partial^{2} g_{11}}{\partial x_{1}^{2}} = -\frac{2(14r^{2} + 9R_{s}^{2} - 22rR_{s})}{r^{12}\sigma^{6}} \] \[ \frac{\partial^{2} g_{22}}{\partial x_{1}^{2}} = \frac{2}{r^4\sin^2\theta} = \frac{2}{r^{4}} \] \[ \frac{\partial^{2} g_{22}}{\partial x_{2}^{2}} = -2r^{2}\frac{1 + 3\cos^{2}\theta}{\sin^{6}\theta} = -2r^{2} \] \[ \frac{\partial^{2} g_{22}}{\partial x_{1}\partial x_{2}} = \frac{4\cos\theta}{r\sin^{4}\theta} = 0 \] \[ \frac{\partial^{2} g_{33}}{\partial x_{1}^{2}} = \frac2\sin^2\theta{}{r^4} = \frac{2}{r^{4}} \] \[ \frac{\partial^{2} g_{33}}{\partial x_{1}\partial x_{2}} = \frac{-4\cos\theta}{r} = 0 \] \[ \frac{\partial^{2} g_{33}}{\partial x_{2}^{2}} = 2r^2\,\sin\theta\,\frac{1}{\sin\theta} = 2r^{2} \]

Christoffel symbols in xyz coordinates

Non-zero components:

\[ \Gamma^{0}{}_{10} = \Gamma^{0}{}_{01} = \frac{R_{s}}{2 r^{4}\sigma^{2}} \] \[ \Gamma^{1}{}_{00} = \frac{R_{s}\sigma^{2}}{2} \] \[ \Gamma^{1}{}_{11} = \frac{3R_{s} - 4r}{2 r^{4}\sigma^{2}} \] \[ \Gamma^{1}{}_{22} = \frac{-r^{3}\sigma^{2}}{\sin^2\theta} \] \[ \Gamma^{1}{}_{33} = -r^{3}\sigma^{2}\sin^{2}\theta =-r^3\sigma^2 \] \[ \Gamma^{2}{}_{21} = \Gamma^{2}{}_{12} = \frac{1}{r^{3}} \] \[ \Gamma^{2}{}_{33} = -\sin^2\theta\,\cos\theta \] \[ \Gamma^2_{22} = \frac{-\cos\theta}{\sin^2\theta} =0 \] \[ \Gamma^{3}{}_{31} = \Gamma^{3}{}_{13} = \frac{1}{2} \frac{-1}{r^2\sin^2\theta} \frac{-2\sin^2\theta}{r} = \frac{1}{r^{3}} \] \[ \Gamma^{3}{}_{32} = \frac{1}{2} \frac{-1}{r^2\sin^2\theta} \left(-2r^2\cos\theta\right) = \frac{\cos\theta}{\sin^{2}\theta} =0 \]

(in the equatorial plane \(\theta = \frac{\pi}{2}\)).

First derivatives of Christoffel symbols in xyz coordinates

\[ \frac{\partial \Gamma^{0}_{10}}{\partial x_{1}} = \frac{\partial \Gamma^{0}_{01}}{\partial x_{1}} = \frac{R_{s}(3R_{s} - 4r)}{2 r^{8}\sigma^{4}} \] \[ \frac{\partial \Gamma^{1}_{00}}{\partial x_{1}} = \frac{R_{s}}{2 r^{4}} \] \[ \frac{\partial \Gamma^{1}_{11}}{\partial x_{1}} = \frac{6}{r^{6}\sigma^{4}} - \frac{10R_{s} }{r^{7}\sigma^{4}} + \frac{4.5 R_{s}^{2} }{r^{8}\sigma^{4}} \] \[ \frac{\partial \Gamma^{1}_{22}}{\partial x_{1}} = \frac{2R_s-3r}{r\sin^2\theta} = -3+\frac{2R_s}{r} \] \[ \frac{\partial \Gamma^{1}_{33}}{\partial x_{1}} = \left(-3+\frac{2R_s}{r}\right)\sin^2\theta = -3+\frac{2R_s}{r} \] \[ \frac{\partial \Gamma^{2}_{21}}{\partial x_{1}} = \frac{\partial \Gamma^{3}_{13}}{\partial x_{1}} = -\frac{3}{r^{6}} \] \[ \frac{\Gamma^2_{33}}{\partial x_{1}}= \frac{\Gamma^2_{22}}{\partial x_{1}}= \frac{\Gamma^3_{23}}{\partial x_{1}}= \frac{\Gamma^3_{32}}{\partial x_{1}} =0 \] \[ \frac{\partial \Gamma^{1}_{22}}{\partial x_{2}} = \frac{2r^3\sigma^2\cos\theta}{\sin^4\theta} =0 \] \[ \frac{\partial \Gamma^{1}_{33}}{\partial x_{2}} =-2r^3\sigma^2\cos\theta=0 \] \[ \frac{\partial \Gamma^{2}_{33}}{\partial x_{2}} =-3\cos^2\theta+1=1 \] \[ \frac{\partial \Gamma^{2}_{22}}{\partial x_{1}} = \frac{1+\cos^2\theta}{\sin^{4}\theta}=1 \] \[ \frac{\partial \Gamma^{3}_{23}}{\partial x_{2}} = \frac{\partial \Gamma^{3}_{32}}{\partial x_{2}} = -\frac{1+\cos^2\theta}{\sin^4\theta} = -1 \]

>Riemann tensor

\[ R^{i}_{jkl} = \Gamma^{i}_{jl,k} - \Gamma^{i}_{jk,l} + \Gamma^{u}_{jl}\Gamma^{i}{}_{uk} - \Gamma^{u}_{jk}\Gamma^{i}{}_{ul} \]

Ricci tensor

\[ R_{\mu\nu} = R^{\rho}_{\mu\rho\nu} = \Gamma^{\rho}_{\mu\nu,\rho} - \Gamma^{\rho}_{\mu\rho,\nu} + \Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\rho} - \Gamma^{\lambda}_{\mu\rho}\Gamma^{\rho}_{\lambda\nu} \]

Or: \[ R_{\mu\nu} = R^{\rho}_{\mu\nu\rho} = -\Gamma^{\rho}_{\mu\nu,\rho} + \Gamma^{\rho}_{\mu\rho,\nu} - \Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\rho} + \Gamma^{\lambda}_{\mu\rho}\Gamma^{\rho}_{\lambda\nu}. \]

Remark on the sign of the Christoffel symbol

From the calculations it follows that, in order for all Ricci components to vanish in vacuum, the Christoffel symbol must start with a positive +1/2: \[ \Gamma^{\rho}_{\mu\nu} = +\frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \] The sign only affects the derivative terms in the Ricci tensor, not the products of Christoffel symbols.

Schwarzschild symmetry of the Ricci tensor

\[ R_{\mu\nu} = \Gamma^0_{\mu\nu,0} - \Gamma^0_{0\mu,\nu} + \Gamma^{0}_{0\lambda}\Gamma^{\lambda}_{\mu\nu} - \Gamma^0_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu} \] \[ +\Gamma^1_{\mu\nu,1} - \Gamma^1_{1\mu,\nu} + \Gamma^{1}_{1\lambda}\Gamma^{\lambda}_{\mu\nu} - \Gamma^1_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu} \] \[ +\Gamma^2_{\mu\nu,2} - \Gamma^2_{2\mu,\nu} + \Gamma^{2}_{2\lambda}\Gamma^{\lambda}_{\mu\nu} - \Gamma^2_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu} \] \[ +\Gamma^3_{\mu\nu,3} - \Gamma^3_{3\mu,\nu} + \Gamma^{3}_{3\lambda}\Gamma^{\lambda}_{\mu\nu} - \Gamma^3_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu} \]

In compact form: \[ R_{\mu\nu} = \Gamma^{\rho}_{\mu\nu,\rho} - \Gamma^{\rho}_{\rho\mu,\nu} + \Gamma^{\rho}_{\rho\lambda}\Gamma^{\lambda}_{\nu\mu} - \Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\rho\mu} \]

Ricci tensor components for Schwarzschild

\[ R_{00} = \Gamma_{00,1}^{1} + \Gamma_{11}^{1}\Gamma_{00}^{1} + \Gamma_{21}^{2}\Gamma_{00}^{1} + \Gamma_{31}^{3}\Gamma_{00}^{1} - \Gamma_{00}^{1}\Gamma_{10}^{0} \] \[ =\frac{R_s^2}{2r^4} -\frac{1}{2}\frac{4r-3R_s}{r^4\sigma^2}\frac{1}{2}R_s\sigma^2 -\frac{1}{2}R_s\sigma^2\frac{1}{2}\frac{R_s}{r^4\sigma^2} -\frac{1}{2}\frac{R_s}{r^4\sigma^2}\frac{1}{2}R_s\sigma^2 \]

\[ R_{11} = -\Gamma_{01,1}^{0} - \Gamma_{21,1}^{2} - \Gamma_{31,1}^{3} + \Gamma_{01}^{0}\Gamma_{11}^{1} + \Gamma_{21}^{2}\Gamma_{11}^{1} + \Gamma_{31}^{3}\Gamma_{11}^{1} - \Gamma_{10}^{0}\Gamma_{10}^{0} - \Gamma_{12}^{2}\Gamma_{21}^{2} - \Gamma_{13}^{3}\Gamma_{31}^{3} \]

\[ R_{22} = \Gamma_{22,1}^{1} - \Gamma_{32,2}^{3} + \Gamma_{01}^{0}\Gamma_{22}^{1} + \Gamma_{11}^{1}\Gamma_{22}^{1} + \Gamma_{21}^{2}\Gamma_{22}^{1} + \Gamma_{31}^{3}\Gamma_{22}^{1} - \Gamma_{22}^{1}\Gamma_{12}^{2} - \Gamma_{21}^{2}\Gamma_{22}^{1} \]

\[ R_{33} = \Gamma_{33,1}^{1} + \Gamma_{01}^{0}\Gamma_{33}^{1} + \Gamma_{11}^{1}\Gamma_{33}^{1} + \Gamma_{21}^{2}\Gamma_{33}^{1} - \Gamma_{33}^{1}\Gamma_{13}^{3} \]

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:

\[ R_{00} = \Gamma_{00,1}^{1} + \Gamma_{00}^{1}\Gamma_{11}^{1} + \Gamma_{00}^{1}\Gamma_{12}^{2} + \Gamma_{00}^{1}\Gamma_{13}^{3} - \Gamma_{01}^{0}\Gamma_{00}^{1} \]

\[ R_{11} = -\Gamma_{10,1}^{0} - \Gamma_{12,1}^{2} - \Gamma_{13,1}^{3} + \Gamma_{11}^{1}\Gamma_{10}^{0} + \Gamma_{11}^{1}\Gamma_{12}^{2} + \Gamma_{11}^{1}\Gamma_{13}^{3} - \Gamma_{10}^{0}\Gamma_{01}^{0} - \Gamma_{12}^{2}\Gamma_{21}^{2} - \Gamma_{13}^{3}\Gamma_{31}^{3} \]

\[ R_{22} = \Gamma_{22,1}^{1} - \Gamma_{23,2}^{3} + \Gamma_{22}^{1}\Gamma_{10}^{0} + \Gamma_{22}^{1}\Gamma_{11}^{1} + \Gamma_{22}^{1}\Gamma_{13}^{3} + \Gamma_{22}^{2}\Gamma_{32}^{3} - \Gamma_{21}^{2}\Gamma_{22}^{1} - \Gamma_{23}^{3}\Gamma_{32}^{3} \]

\[ R_{33} = \Gamma_{33,1}^{1} + \Gamma_{33,2}^{2} + \Gamma_{33}^{1}\Gamma_{10}^{0} + \Gamma_{33}^{1}\Gamma_{11}^{1} + \Gamma_{33}^{1}\Gamma_{12}^{2} + \Gamma_{33}^{2}\Gamma_{22}^{2} - \Gamma_{31}^{3}\Gamma_{33}^{1} - \Gamma_{32}^{3}\Gamma_{33}^{2} \]

\[ R_{33} = \sin^{2}\theta\, R_{22} \]

When \(\theta \neq \frac{\pi}{2}\), additional terms occur: \[ R_{22} \to R_{22} + \Gamma_{22}^{2}\Gamma_{32}^{3}, \qquad R_{33} \to R_{33} + \Gamma_{33}^{2}\Gamma_{22}^{2}. \]