Part VI – Validation of the Theory
6 Verification that the Schwarzschild metric satisfies the Einstein field equations
We will now mathematically verify whether the Schwarzschild equation satisfies the
Einstein field equations. We first do this for the full field equations and
then for the simplified form.
6.1 Verification of the Full Field Equations
The general form of the Einstein field equations is:
\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}
\label{eq:einstein-full}
\end{equation}
Here, the left-hand side describes the geometry of spacetime, while the right-hand side
represents the content of mass and energy. The constant \( \lambda \) is the
cosmological constant, which is usually negligibly small for calculations on
astrophysical or planetary scales. Therefore, the simplified equation is generally used:
\begin{equation}
R_{\mu\nu}
- \frac{1}{2} g_{\mu\nu} R
= \frac{8\pi G}{c^{4}}\,T_{\mu\nu}
\end{equation}
In vacuum regions – that is, outside a mass – we have:
\begin{equation}
T_{\mu\nu} = 0 \quad \text{(vacuum)} \label{eq:vacuum}
\end{equation}
so that the equation reduces to:
\begin{equation}
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0 \label{eq:einstein-vacuum}
\end{equation}
The indices \( \mu \) and \( \nu \) each take four values (0 to 3), resulting in
16 coupled differential equations. The field equations are fully
dependent on the metric tensor \( g_{\mu\nu} \) and its first and second derivatives.
This is because they are constructed entirely from the Christoffel symbols and their
derivatives, and the Christoffel symbols themselves are fully determined by the metric and its
first derivatives.
Interpretation:
In vacuum regions outside a mass, the equation reduces to \eqref{eq:einstein-vacuum}.
This means that the Ricci tensor components must be zero, describing the geometry of empty space.
6.1.1 The Schwarzschild solution
Karl Schwarzschild found an exact solution to the field equations in vacuum,
assuming spherical symmetry. The metric is:
\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\varphi^2 \label{eq:schwarzschild-metric}
\end{equation}
where:
\begin{equation}
\sigma^2 = 1 - \frac{2GM}{c^2 r} \label{eq:sigma}
\end{equation}
The general form of the metric in coordinate notation is:
\begin{equation}
ds^{2}
= g_{00}\,dt^{2}
+ g_{11}\,dr^{2}
+ g_{22}\,d\theta^{2}
+ g_{33}\,d\varphi^{2}.
\end{equation}
It follows that only four of the sixteen metric components are non-zero.
Therefore, there are also only four relevant components of the Ricci tensor \(R_{\mu\nu}\),
namely:
\begin{equation}
R_{00},\quad R_{11},\quad R_{22},\quad R_{33}.
\end{equation}
Interpretation:
The Schwarzschild metric describes the curvature of spacetime around a spherically
symmetric mass.
The function \(\sigma^2\) determines the strength of gravity depending on the
distance from the mass.
6.1.2 The Ricci tensor and Christoffel symbols
The Ricci tensor is defined as:
\begin{equation}
R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu}.
\end{equation}
With the general expression:
\begin{equation}
R_{\mu\nu}
= \partial_{\rho}\Gamma^{\rho}_{\mu\nu}
- \partial_{\nu}\Gamma^{\rho}_{\mu\rho}
+ \Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\mu\nu}
- \Gamma^{\rho}_{\lambda\nu}\Gamma^{\lambda}_{\mu\rho}.
\end{equation}
This formula is composed of derivatives and products of the Christoffel symbols.
The general form of a Christoffel symbol is:
\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}
6.1.3 Simplification in vacuum
As stated earlier, in vacuum:
\begin{equation}
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0.
\end{equation}
Here \(R\) denotes the Ricci scalar and represents the total curvature of
local spacetime. The Ricci scalar is computed as the contraction of the
Ricci tensor:
\begin{equation}
R = g^{\mu\nu} R_{\mu\nu}.
\end{equation}
This means that the Ricci scalar summarizes how spacetime curves in all
directions, based on the information in the Ricci tensor.
In the case of the Einstein field equations in vacuum:
\begin{equation}
R = 0,
\end{equation}
which means that the total spacetime curvature is zero outside a massive source.
When the earlier equation is multiplied by \( g^{\mu\nu} \), we obtain:
\begin{equation}
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0,
\end{equation}
\begin{equation}
g^{\mu\nu} R_{\mu\nu}
- \frac{1}{2} g^{\mu\nu} g_{\mu\nu} R = 0.
\end{equation}
Since:
\begin{equation}
g^{\mu\nu} g_{\mu\nu} = 4,
\end{equation}
it follows that:
\begin{equation}
R - \frac{1}{2}(4)R = 0
\quad\Rightarrow\quad
R - 2R = 0
\quad\Rightarrow\quad
R = 0.
\end{equation}
This can only be true if also:
\begin{equation}
R_{\mu\nu} = 0.
\end{equation}
Thus, as a consequence of the relation between \(R\) and \(R_{\mu\nu}\), it is clear that:
\begin{equation}
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0
\quad\Rightarrow\quad
R_{\mu\nu} = 0.
\end{equation}
6.1.4 Calculations and numerical verification
Based on the general form of the Ricci tensor and the Christoffel symbols, we have
demonstrated both numerically (using a computer program) and theoretically that the
Schwarzschild metric indeed satisfies the vacuum equation:
\begin{equation}
R_{\mu\nu} = 0.
\end{equation}
The relevant expressions for the Ricci components in terms of Christoffel symbols are:
\begin{equation}
R_{00}
= \Gamma^{1}_{00,1}
+ \Gamma^{1}_{00}\Gamma^{1}_{11}
+ \Gamma^{1}_{00}\Gamma^{2}_{12}
+ \Gamma^{1}_{00}\Gamma^{3}_{13}
- \Gamma^{0}_{01}\Gamma^{1}_{00},
\end{equation}
\begin{equation}
R_{11}
= -\Gamma^{0}_{10,1}
- \Gamma^{2}_{12,1}
- \Gamma^{3}_{13,1}
+ \Gamma^{1}_{11}\Gamma^{0}_{10}
+ \Gamma^{1}_{11}\Gamma^{2}_{12}
+ \Gamma^{1}_{11}\Gamma^{3}_{13}
- \Gamma^{0}_{10}\Gamma^{0}_{01}
- \Gamma^{2}_{12}\Gamma^{2}_{21}
- \Gamma^{3}_{13}\Gamma^{3}_{31},
\end{equation}
\begin{equation}
R_{22}
= \Gamma^{1}_{22,1}
- \Gamma^{3}_{23,2}
+ \Gamma^{1}_{22}\Gamma^{0}_{10}
+ \Gamma^{1}_{22}\Gamma^{1}_{11}
+ \Gamma^{1}_{22}\Gamma^{3}_{13}
- \Gamma^{2}_{21}\Gamma^{1}_{22}
- \Gamma^{3}_{23}\Gamma^{2}_{32},
\end{equation}
\begin{equation}
R_{33}
= \Gamma^{1}_{33,1}
+ \Gamma^{2}_{33,2}
+ \Gamma^{1}_{33}\Gamma^{0}_{10}
+ \Gamma^{1}_{33}\Gamma^{1}_{11}
+ \Gamma^{1}_{33}\Gamma^{2}_{12}
- \Gamma^{3}_{31}\Gamma^{1}_{33}
- \Gamma^{3}_{32}\Gamma^{2}_{33}.
\end{equation}
These equations have been evaluated by deriving the Christoffel symbols from the
Schwarzschild metric and substituting them into the above expressions.
These symbols are summarized in Table 1 (see Appendix 1.2 ).
Note:
In the literature, the Christoffel formula is sometimes given with a minus sign (−½) or a
plus sign (+½) as the leading factor.
In our approach, a positive factor of \( \tfrac{1}{2} \) was used.
This convention proved consistent with the result that all relevant Ricci components are zero:
\begin{equation}
R_{00} = R_{11} = R_{22} = R_{33} = 0,
\end{equation}
as required by the Einstein field equations in vacuum.
Therefore, the Christoffel formula was applied in the following form:
\begin{equation}
\Gamma^{\mu}_{\nu\rho}
= \frac{1}{2} g^{\mu\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\rho}}
+ \frac{\partial g_{\rho\alpha}}{\partial x^{\nu}}
- \frac{\partial g_{\nu\rho}}{\partial x^{\alpha}}
\right).
\end{equation}
6.1.5 Conclusion
Through both analytical and numerical evaluation of the Ricci tensor components, based
on the Schwarzschild metric and the corresponding Christoffel symbols, it has been shown that this
solution indeed satisfies the Einstein field equations in vacuum. Thus, the
Schwarzschild solution is an exact and physically consistent description of the
spacetime structure outside a spherically symmetric mass.
6.2 Verification of \(R_{00}, R_{11}, R_{22}\) and \(R_{33}\) in the
Schwarzschild metric
When verifying the Einstein field equations in vacuum, the components
of the Ricci tensor — in particular \(R_{00}, R_{11}, R_{22}\) and \(R_{33}\) — must be
evaluated within the framework of the Schwarzschild solution. This is done in spherical
coordinates \((t, r, \theta, \varphi)\).
The Schwarzschild metric is:
\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\varphi^{2},
\end{equation}
with:
\begin{equation}
\sigma^{2} = 1 - \frac{R_{s}}{r},
\qquad
R_{s} = \frac{2GM}{c^{2}}.
\end{equation}
To determine the components of the Ricci tensor, we follow these steps:
- Derive the Christoffel symbols from the Schwarzschild metric in spherical coordinates;
- Substitute these symbols into the expressions for the Ricci tensor components;
- Verify that all relevant components \(R_{\mu\nu}\) are equal to zero, in accordance with the Einstein field equations in vacuum.
The Christoffel symbols used and their derivatives can be found in Appendix 1.2 )
Below are the individual verifications.
Verification of \(R_{00}\)
The component \(R_{00}\) is given by:
\begin{equation}
R_{00}
= \Gamma^{1}_{00,1}
+ \Gamma^{1}_{00}\Gamma^{1}_{11}
+ \Gamma^{1}_{00}\Gamma^{2}_{12}
+ \Gamma^{1}_{00}\Gamma^{3}_{13}
- \Gamma^{0}_{01}\Gamma^{1}_{00}
\end{equation}
After substitution of the relevant terms, we obtain:
\begin{equation}
R_{00}
= \frac{R_{s}(3R_{s} - 2r)}{2r^{4}}
+ \frac{\sigma^{2}R_{s}}{2r^2}\cdot
\frac{-R_s}{2r^2\sigma^2}
+\frac{\sigma^2R_s}{2r^2}\cdot\frac{1}{r}
+\frac{\sigma^2R_s}{2r^2}\cdot\frac{1}{r}
-\frac{R_s}{2r^2\sigma^2}\cdot\frac{\sigma^2R_s}{2r^2}
\end{equation}
Together with \(\sigma^2=1-R_s/r\), this reduces to:
\begin{equation}
R_{00}
= \frac{R_{s}(3R_{s} - 2r)}{2r^{4}}
- \frac{R_{s}^{2}}{2r^{4}}
+ \frac{2R_{s}(r - R_{s})}{2r^{4}}
= \frac{3R_{s}^{2} - 2rR_{s} - R_{s}^{2} + 2R_{s}r - 2R_{s}^{2}}{2r^{4}}
= 0.
\end{equation}
As required by the vacuum equations:
\begin{equation}
R_{00} = 0.
\end{equation}
Thus:
\begin{equation}
R_{00} = 0 \quad \text{q.e.d.}
\end{equation}
Verification of \(R_{11}\)
The component \(R_{11}\) is computed from:
\begin{equation}
R_{11}
= -\Gamma^{0}_{10,1}
- \Gamma^{2}_{12,1}
- \Gamma^{3}_{13,1}
+ \Gamma^{1}_{11}\Gamma^{0}_{10}
+ \Gamma^{1}_{11}\Gamma^{2}_{12}
+ \Gamma^{1}_{11}\Gamma^{3}_{13}
- \Gamma^{0}_{10}\Gamma^{0}_{01}
- \Gamma^{2}_{12}\Gamma^{2}_{21}
- \Gamma^{3}_{13}\Gamma^{3}_{31}.
\end{equation}
After evaluation:
\begin{equation}
R_{11} =
- \frac{R_s (R_s - 2r)}{2r^4 \sigma^4}
- \frac{-1}{r^2}
- \frac{-1}{r^2}
+ \frac{-R_s}{2r^2 \sigma^2}\cdot \frac{R_s}{2r^2 \sigma^2}
+ \frac{-R_s}{2r^2 \sigma^2}\cdot \frac{1}{r}
+ \frac{-R_s}{2r^2 \sigma^2}\cdot \frac{1}{r}
- \frac{R_s}{2r^2 \sigma^2}\cdot \frac{R_s}{2r^2 \sigma^2}
- \frac{1}{r}\cdot \frac{1}{r}
- \frac{1}{r} \cdot\frac{1}{r}
\end{equation}
\begin{equation}
R_{11}
= -\frac{R_{s}(R_{s}-2r)}{2r^{4}\sigma^{4}}
+ \frac{1}{r^{2}}
+ \frac{1}{r^{2}}
- \frac{R_{s}^{2}}{4r^{4}\sigma^{4}}
- \frac{R_{s}}{2r^{3}\sigma^{2}}
- \frac{R_{s}}{2r^{3}\sigma^{2}}
- \frac{R_{s}^{2}}{4r^{4}\sigma^{4}}
- \frac{1}{r^{2}}
- \frac{1}{r^{2}}.
\end{equation}
Further reduction yields:
\begin{equation}
R_{11}
= -\frac{R_{s}(R_{s}-2r)}{2r^{4}\sigma^{4}}
- \frac{R_{s}^{2}}{2r^{4}\sigma^{4}}
- \frac{2R_{s}r(1 - R_{s}/r)}{2r^{4}\sigma^{4}}
\end{equation}
\begin{equation}
= -\frac{R_{s}^{2} - 2rR_{s}}{2r^{4}\sigma^{4}}
- \frac{R_{s}^{2}}{2r^{4}\sigma^{4}}
- \frac{2R_{s}r - 2R_{s}^{2}}{2r^{4}\sigma^{4}}
\end{equation}
\begin{equation}
= \frac{-R_{s}^{2} + 2rR_{s} - R_{s}^{2} - 2R_{s}r + 2R_{s}^{2}}{2r^{4}\sigma^{4}}
= 0.
\end{equation}
Thus:
\begin{equation}
R_{11} = 0 \quad \text{q.e.d.}
\end{equation}
Verification of \(R_{22}\)
For the component \(R_{22}\) we have:
\begin{equation}
R_{22}
= \Gamma^{1}_{22,1}
- \Gamma^{3}_{23,2}
+ \Gamma^{1}_{22}\Gamma^{0}_{10}
+ \Gamma^{1}_{22}\Gamma^{1}_{11}
+ \Gamma^{1}_{22}\Gamma^{3}_{13}
- \Gamma^{2}_{21}\Gamma^{1}_{22}
- \Gamma^{3}_{23}\Gamma^{3}_{32}.
\end{equation}
Substitution gives:
\begin{equation}
R_{22}
= -1 + 1
- r\sigma^{2}\cdot\frac{R_{s}}{2r^{2}\sigma^{2}}
+ r\sigma^{2}\cdot
\frac{+R_{s}}{2r^{2}\sigma^{2}}
- r\sigma^{2}\cdot\frac{1}{r}
+ \frac{1}{r}\cdot r\sigma^{2}
- 0.
\end{equation}
This reduces to:
\begin{equation}
R_{22} = 0.
\end{equation}
Thus:
\begin{equation}
R_{22} = 0 \quad \text{q.e.d.}
\end{equation}
Verification of \(R_{33}\)
The component \(R_{33}\) is computed as follows:
\begin{equation}
R_{33}
= \Gamma^{1}_{33,1}
+ \Gamma^{2}_{33,2}
+ \Gamma^{1}_{33}\Gamma^{0}_{10}
+ \Gamma^{1}_{33}\Gamma^{1}_{11}
+ \Gamma^{1}_{33}\Gamma^{2}_{12}
- \Gamma^{3}_{31}\Gamma^{1}_{33}
- \Gamma^{3}_{32}\Gamma^{2}_{33}.
\end{equation}
Evaluation leads to:
\begin{equation}
R_{33}
= -1 + 1
- r\sigma^{2} \cdot \frac{R_{s}}{2r^{2}\sigma^{2}
}
+ r\sigma^{2} \cdot \frac{R_{s}}{2r^{2}\sigma^{2}}
- r\sigma^{2} \cdot \frac{1}{r}
+ \frac{1}{r} \cdot r\sigma^{2}
- 0.
\end{equation}
Thus:
\begin{equation}
R_{33} = 0 \quad \text{q.e.d.}
\end{equation}
Conclusion
All relevant components of the Ricci tensor are zero:
\begin{equation}
R_{\mu\nu} = 0 \qquad \text{for } \mu,\nu \in \{0,1,2,3\},
\end{equation}
which confirms that the Schwarzschild solution satisfies the Einstein field equations
in vacuum. This demonstrates that this solution correctly describes spacetime
outside a spherically symmetric mass in the absence of energy or matter.
This forms a fundamental confirmation of both the consistency of the
Schwarzschild solution and the correctness of general relativity in this
specific case.
6.3 Verification of the Ricci tensor components \(R_{00}, R_{11}, R_{22}, R_{33}\) in
Schwarzschild coordinates
We explicitly verify the components of the Ricci tensor for the
Schwarzschild solution in a modified coordinate system
\((t_{\infty}, x_{1}, x_{2}, x_{3})\), where the metric has the form:
\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}
The required Christoffel symbols and their derivatives are given in Appendix 1.3 ).
Below we verify that all components of the Ricci tensor are zero, as required
for a vacuum solution \((R_{\mu\nu} = 0)\).
Component \(R_{00}\)
\begin{equation}
R_{00}
= \Gamma^{1}_{00,1}
+ \Gamma^{1}_{00}\Gamma^{1}_{11}
+ \Gamma^{1}_{00}\Gamma^{2}_{12}
+ \Gamma^{1}_{00}\Gamma^{3}_{13}
- \Gamma^{0}_{01}\Gamma^{1}_{00}.
\end{equation}
Substituting the expressions for the Christoffel symbols:
\begin{equation}
R_{00}
= \frac{R_{s}^{2}}{2r^{4}}
+ \frac{R_{s}\sigma^{2}}{2} \cdot\frac{3R_{s} - 4r}{2r^{4}\sigma^{2} }
+ \frac{R_{s}\sigma^{2}}{2}\cdot \frac{1}{r^{3}}
+ \frac{R_{s}\sigma^{2}}{2}\cdot \frac{1}{r^{3}}
- \frac{R_{s}}{2r^{4}\sigma^{2}}\cdot \frac{R_s\sigma^2}{2}
\end{equation}
Further reduction yields:
\begin{equation}
R_{00}
= \frac{2R_{s}^{2}}{4r^{4}}
+ \frac{3R_{s}^{2} - 4rR_{s}}{4r^{4}}
+ \frac{4R_{s}r\sigma^{2}}{4r^{4}}
- \frac{R_{s}^{2}}{4r^{4}}
\end{equation}
\begin{equation}
= \frac{2R_{s}^{2} + 3R_{s}^{2} - 4rR_{s} - R_{s}^{2} + 4R_{s}(r - R_{s})}{4r^{4}}
\end{equation}
\begin{equation}
= \frac{4R_{s}^{2} - 4rR_{s} + 4R_{s}r - 4R_{s}^{2}}{4r^{4}}
= 0.
\end{equation}
Result:
\begin{equation}
R_{00} = 0.
\end{equation}
Thus:
\begin{equation}
R_{00} = 0 \quad \text{q.e.d.}
\end{equation}
Component \(R_{11}\)
\begin{equation}
R_{11}
= -\Gamma^{0}_{10,1}
- \Gamma^{2}_{12,1}
- \Gamma^{3}_{13,1}
+ \Gamma^{1}_{11}\Gamma^{0}_{10}
+ \Gamma^{1}_{11}\Gamma^{2}_{12}
+ \Gamma^{1}_{11}\Gamma^{3}_{13}
- \Gamma^{0}_{10}\Gamma^{0}_{01}
- \Gamma^{2}_{12}\Gamma^{2}_{21}
- \Gamma^{3}_{13}\Gamma^{3}_{31}.
\end{equation}
Substitution gives:
\begin{equation}
R_{11}
= -\frac{R_{s}(3R_{s}-4r)}{2r^{8}\sigma^{4}}
- \frac{3}{r^{6}}
- \frac{3}{r^{6}}
+ \frac{(3R_{s}-4r)}{2r^{4}\sigma^{2}}\frac{R_{s}^{2}}{r^{4}\sigma^{2}}
+ \frac{(3R_{s}-4r)}{2r^{4}\sigma^{2}}\frac{1}{r^{3}}
+ \frac{(3R_{s}-4r)}{2r^{4}\sigma^{2}}\frac{1}{r^{3}}
- \frac{R_{s}^{2}}{r^{4}\sigma^{2}}\frac{R_{s}^{2}}{r^{4}\sigma^{2}}
- \frac{1}{r^{3}}\frac{1}{r^{3}}
- \frac{1}{r^{3}}\frac{1}{r^{3}}.
\end{equation}
After simplification:
\begin{equation}
R_{11}
= -\frac{2R_{s}(3R_{s}-4r)}{4r^{8}\sigma^{4}}
+ \frac{4}{r^{6}}
+ \frac{R_{s}(3R_{s}-4r)}{4r^{8}\sigma^{4}}
+ \frac{4(3R_{s}-4r)r(1 - R_{s}/r)}{4r^{8}\sigma^{4}}
- \frac{R_{s}^{2}}{4r^{8}\sigma^{4}}.
\end{equation}
Further reduction:
\begin{equation}
R_{11}
= \frac{-6R_{s}^{2} + 8rR_{s} + 3R_{s}^{2} - 4rR_{s}
+ 12R_{s}r - 16r^{2}
- 12R_{s}^{2} + 16rR_{s} - R_{s}^{2}}{4r^{8}\sigma^{4}}
+ \frac{4}{r^{6}}.
\end{equation}
\begin{equation}
R_{11}
= \frac{-16R_{s}^{2} + 32rR_{s} - 16r^{2}}{4r^{8}\sigma^{4}}
+ \frac{4}{r^{6}}
= \frac{-16R_{s}^{2} + 32rR_{s} - 16r^{2} + 16r^{2}(1 - R_{s}/r)^{2}}{4r^{8}\sigma^{4}}
= 0.
\end{equation}
Result:
\begin{equation}
R_{11} = 0 \quad \text{q.e.d.}
\end{equation}
Calculation of \(R_{22}\)
\begin{equation}
R_{22}
= \Gamma^{1}_{22,1}
+ \Gamma^{2}_{22,2}
+ \Gamma^{1}_{22}\Gamma^{2}_{12}
+ \Gamma^{2}_{21}\Gamma^{1}_{22}
+ \Gamma^{2}_{22}\Gamma^{2}_{22}
+ \Gamma^{3}_{23}\Gamma^{3}_{23}.
\end{equation}
Substitution and simplification yield:
\begin{equation}
R_{22}
= -3 + \frac{2R_{s}}{r} + 1
- r^{3}\sigma^{2} \cdot \frac{R_{s}}{2r^{4}\sigma^{2}}
- r^{3}\sigma^{2} \cdot \frac{3R_{s}-4r}{2r^{4}\sigma^{2}}
- r^{3}\sigma^{2}\cdot\frac{1}{r^{3}}
+ \frac{1}{r^{3}} \cdot r^{3}\sigma^{2}
- 0
\end{equation}
\begin{equation}
R_{22}
= -3 + \frac{2R_{s}}{r} + 1
- \frac{R_{s}}{2r}
- \frac{3R_{s}-4r}{2r}
= 0.
\end{equation}
\begin{equation}
R_{22}
=
\frac{-4r}{2r}
+ \frac{4R_{s}}{2r}
- \frac{R_{s}}{2r}
- \frac{3R_{s}-4r}{2r}
= 0.
\end{equation}
Result:
\begin{equation}
R_{22} = 0 \quad \text{q.e.d.}
\end{equation}
Component \(R_{33}\)
\begin{equation}
R_{33}
= \Gamma^{1}_{33,1}
+ \Gamma^{2}_{33,2}
+ \Gamma^{1}_{33}\Gamma^{0}_{10}
+ \Gamma^{1}_{33}\Gamma^{1}_{11}
+ \Gamma^{1}_{33}\Gamma^{2}_{12}
- \Gamma^{3}_{31}\Gamma^{1}_{33}
- \Gamma^{3}_{32}\Gamma^{2}_{33}.
\end{equation}
Substitution gives:
\begin{equation}
R_{33}
= -3 + \frac{2R_{s}}{r} + 1
- r^{3}\sigma^{2}\cdot \frac{R_{s}}{2r^{4}\sigma^{2}}
- r^{3}\sigma^{2}\cdot \frac{3R_{s}-4r}{2r^{4}\sigma^{2}}
- r^{3}\sigma^{2}\cdot \frac{1}{r^{3}}
+ \frac{1}{r^{3}}\cdot r^{3}\sigma^{2}
- 0.
\end{equation}
Simplification leads to:
\begin{equation}
R_{33}
= -3 + \frac{2R_{s}}{r} + 1
- \frac{R_{s}}{2r}
- \frac{3R_{s}-4r}{2r}
= 0.
\end{equation}
\begin{equation}
R_{33}
=
\frac{-4r}{2r}
+ \frac{4R_{s}}{2r}
- \frac{R_{s}}{2r}
- \frac{3R_{s}-4r}{2r}
= 0.
\end{equation}
Result:
\begin{equation}
R_{33} = 0 \quad \text{q.e.d.}
\end{equation}
Conclusion
The four independent components of the Ricci tensor are all zero in
Schwarzschild coordinates, as expected for the vacuum solution of the
Einstein field equations:
\begin{equation}
R_{00} = R_{11} = R_{22} = R_{33} = 0.
\end{equation}
This confirms that the Schwarzschild metric is indeed a solution of
\begin{equation}
R_{\mu\nu} = 0
\end{equation}
outside the central mass.
This demonstrates that the Schwarzschild metric provides a correct and physically consistent
description of spacetime in vacuum around a spherically symmetric object.
6.4 Verification of the Schwarzschild solution using a
simplified form of the field equations
In this section we verify the Schwarzschild solution using a
simplified version of the Einstein field equations. This limited form is
derived from Schwarzschild’s original derivation and applies only when the
trace of the metric tensor satisfies:
\begin{equation}
\mathrm{tr}(g_{\mu\nu}) = -1.
\end{equation}
In this approach, the field equations take the form:
\begin{equation}
G_{\mu\nu}
= \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x^{\alpha}}
+ \Gamma^{\alpha}_{\mu\beta}\,\Gamma^{\beta}_{\nu\alpha}.
\end{equation}
In this expression, the Christoffel symbols are defined with a negative sign,
as Schwarzschild originally did:
\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}
Due to this negative convention, all derived expressions, including the
derivations of the Ricci tensor, differ in sign from the
standard definition.
Schwarzschild used the coordinates \((t, x, y, z)\) in his derivation. We therefore begin
with these coordinates and present the relevant components of the Ricci tensor,
as obtained from the simplified formula.
Derived components of the Ricci tensor
The following expressions hold in Schwarzschild’s notation:
1.11. For the \(R_{00}\)-component:
\begin{equation}
R_{00}
= \Gamma^{1}_{00,1}
+ \Gamma^{0}_{01}\Gamma^{1}_{00}
+ \Gamma^{1}_{00}\Gamma^{0}_{10}.
\end{equation}
1.12. For the \(R_{11}\)-component:
\begin{equation}
R_{11}
= \Gamma^{1}_{11,1}
+ \Gamma^{0}_{10}\Gamma^{0}_{01}
+ \Gamma^{1}_{11}\Gamma^{1}_{11}
+ \Gamma^{2}_{12}\Gamma^{2}_{21}
+ \Gamma^{3}_{13}\Gamma^{3}_{31}.
\end{equation}
For the \(R_{22}\)-component:
\begin{equation}
R_{22}
= \Gamma^{1}_{22,1}
+ \Gamma^{2}_{22,2}
+ \Gamma^{1}_{22}\Gamma^{2}_{12}
+ \Gamma^{2}_{21}\Gamma^{1}_{22}
+ \Gamma^{2}_{22}\Gamma^{2}_{22}
+ \Gamma^{3}_{23}\Gamma^{3}_{23}.
\end{equation}
1.13. For the \(R_{33}\)-component:
\begin{equation}
R_{33}
= \Gamma^{1}_{33,1}
+ \Gamma^{2}_{33,2}
+ \Gamma^{1}_{33}\Gamma^{3}_{13}
+ \Gamma^{2}_{33}\Gamma^{3}_{23}
+ \Gamma^{3}_{31}\Gamma^{1}_{33}
+ \Gamma^{3}_{32}\Gamma^{2}_{33}.
\end{equation}
These components can be readily evaluated once the correct Christoffel symbols
have been computed from the Schwarzschild metric. In the next section, we evaluate these
components explicitly.
6.5 t, x, y, z coordinates
We work in adapted Cartesian coordinates \((t, x, y, z)\), in which
Schwarzschild originally formulated his solution.
We explicitly verify that the components of the Ricci tensor \(R_{\mu\nu}\) are zero.
Calculation of \(R_{00}\)
\begin{equation}
R_{00}
= \Gamma^{1}_{00,1}
+ \Gamma^{0}_{01}\Gamma^{1}_{00}
+ \Gamma^{1}_{00}\Gamma^{0}_{10}.
\end{equation}
With the given values:
\begin{equation}
R_{00}
= -\frac{R_{s}^{2}}{2r^{4}}
+ \frac{R_{s}}{2r^{4}\sigma^{2}}\cdot\frac{R_{s}\sigma^{2}}{2}
+\frac{R_{s}\sigma^{2}}{2}\cdot\frac{R_{s}}{2r^{4}\sigma^{2}}
= -\frac{R_{s}^{2}}{2r^{4}} + \frac{R_{s}^{2}}{2r^{4}}
= 0.
\end{equation}
Result:
\begin{equation}
R_{00} = 0 \quad \text{q.e.d.}
\end{equation}
Calculation of \(R_{11}\)
\begin{equation}
R_{11}
= \Gamma^{1}_{11,1}
+ \Gamma^{0}_{10}\Gamma^{0}_{01}
+ \Gamma^{1}_{11}\Gamma^{1}_{11}
+ \Gamma^{2}_{12}\Gamma^{2}_{21}
+ \Gamma^{3}_{13}\Gamma^{3}_{31}.
\end{equation}
By carefully substituting and simplifying all terms, we find:
\begin{equation}
R_{11}
= -\frac{6}{r^{6}\sigma^{4}}
+ \frac{10R_{s}}{r^{7}\sigma^{4}}
- \frac{4.5R_{s}^{2}}{r^{8}\sigma^{4}}
+ \frac{R_{s}^{2}}{4r^{8}\sigma^{4}}
+ \frac{(3R_{s}-4r)^{2}}{4r^{8}\sigma^{4}}
+ \frac{1}{r^{3}}\frac{1}{r^{3}}
+ \frac{1}{r^{3}}\frac{1}{r^{3}}.
\end{equation}
\begin{equation}
R_{11}
= -\frac{6}{r^{6}\sigma^{4}}
+ \frac{10R_{s}}{r^{7}\sigma^{4}}
- \frac{4.5R_{s}^{2}}{r^{8}\sigma^{4}}
+ \frac{R_{s}^{2}}{4r^{8}\sigma^{4}}
+ \frac{9R_{s}^2+16r^2-24rR_s}{4r^{8}\sigma^{4}}
+ \frac{2}{r^{6}}
\end{equation}
After further reduction:
\begin{equation}
R_{11}
= \frac{-24r^{2} + 40rR_{s} - 18R_{s}^{2}}{4r^{8}\sigma^{4}}
+ \frac{R_{s}^{2}}{4r^{8}\sigma^{4}}
+ \frac{9R_{s}^{2} + 16r^{2} - 24rR_{s}}{4r^{8}\sigma^{4}}
+ \frac{8r^2\sigma^4}{4r^{8}\sigma^4}.
\end{equation}
\begin{equation}
R_{11}
= \frac{-8R_{s}^{2} - 8r^{2} + 16rR_{s} + 8r^{2}\sigma^{4}}{4r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
= \frac{-8R_{s}^{2} - 8r^{2} + 16rR_{s} + 8r^{2}(1 - R_{s}/r)^{2}}{4r^{8}\sigma^{4}}
= 0.
\end{equation}
Result:
\begin{equation}
R_{11} = 0 \quad \text{q.e.d.}
\end{equation}
Calculation of \(R_{22}\)
\begin{equation}
R_{22}
= \Gamma^{1}_{22,1}
+ \Gamma^{2}_{22,2}
+ \Gamma^{1}_{22}\Gamma^{2}_{12}
+ \Gamma^{2}_{21}\Gamma^{1}_{22}
+ \Gamma^{2}_{22}\Gamma^{2}_{22}
+ \Gamma^{3}_{23}\Gamma^{3}_{23}.
\end{equation}
After simplification of the trigonometric and radial terms:
\begin{equation}
R_{22}
= \frac{-2R_{s} + 3r}{r\sin^{2}\theta}
+ \frac{-1 - \cos^{2}\theta}{\sin^{4}\theta}
+ \frac{-r^{3}\sigma^{2}}{\sin^{2}\theta }\cdot\frac{1}{r^{3}}
+ \frac{1}{r^{3}}\cdot\frac{-r^{3}\sigma^{2}}{\sin^{2}\theta}
+ \frac{-\cos\theta}{\sin^{2}\theta}
\frac{-\cos\theta}{\sin^{2}\theta}
+ \frac{\cos\theta}{\sin^{2}\theta}
\frac{\cos\theta}{\sin^{2}\theta}.
\end{equation}
This reduces to:
\begin{equation}
R_{22}
= \frac{-2R_{s} + 3r}{r\sin^{2}\theta}
+ \frac{-1 - \cos^{2}\theta}{\sin^{4}\theta}
- \frac{2r^3\sigma^2}{r^3\sin^{2}\theta}
+ \frac{2\cos^{2}\theta}{\sin^{4}\theta}.
\end{equation}
\begin{equation}
R_{22}
= \frac{-2R_{s} + 3r}{r\sin^{2}\theta}
+ \frac{-1 - \cos^{2}\theta}{\sin^{4}\theta}
- \frac{2(r - R_{s})}{r\sin^{2}\theta}
+ \frac{2\cos^{2}\theta}{\sin^{4}\theta}.
\end{equation}
\begin{equation}
R_{22}
= \frac{1}{\sin^{2}\theta}
+ \frac{-1 - \cos^{2}\theta}{\sin^{4}\theta}
+ \frac{2\cos^{2}\theta}{\sin^{4}\theta}
\end{equation}
\begin{equation}
= \frac{\sin^{2}\theta}{\sin^{4}\theta}
+ \frac{-\sin^{2}\theta - \cos^{2}\theta - \cos^{2}\theta+ 2\cos^{2}\theta}{\sin^{4}\theta}
= 0.
\end{equation}
Result:
\begin{equation}
R_{22} = 0 \quad \text{q.e.d.}
\end{equation}
Calculation of \(R_{33}\)
\begin{equation}
R_{33}
= \Gamma^{1}_{33,1}
+ \Gamma^{2}_{33,2}
+ \Gamma^{1}_{33}\Gamma^{3}_{13}
+ \Gamma^{2}_{33}\Gamma^{3}_{23}
+ \Gamma^{3}_{31}\Gamma^{1}_{33}
+ \Gamma^{3}_{32}\Gamma^{2}_{33}.
\end{equation}
After combining terms with angular-dependent factors:
\begin{equation}
R_{33}
= \left(3 - \frac{2R_{s}}{r}\right)\cdot\sin^{2}\theta
+ 3\cos^{2}\theta - 1
- r^{3}\sigma^{2}\sin^{2}\theta\cdot\frac{1}{r^{3}
}
+ (-\sin^{2}\theta\cos\theta)\cdot\frac{\cos\theta}{\sin^{2}\theta}\end{equation}
\begin{equation}
- \frac{1}{r^{3}}r^{3}\sigma^{2}\cdot\sin^{2}\theta
+ \frac{\cos\theta}{\sin^{2}\theta}(-\sin^{2}\theta\cos\theta).
\end{equation}
Simplified:
\begin{equation}
R_{33}
= \left(3 - \frac{2R_{s}}{r}\right)\sin^{2}\theta
+ 3\cos^{2}\theta - 1
- 2\sigma^{2}\sin^{2}\theta
- 2\sin^{2}\theta\cos^{2}\theta\frac{\cos\theta}{\sin^2\theta}
\end{equation}
With \(\sigma^{2} = 1 - \frac{R_{s}}{r}\) we obtain:
\begin{equation}
R_{33}
= \left(3 - \frac{2R_{s}}{r}\right)\sin^{2}\theta
+ 3\cos^{2}\theta - 1
- 2\left(1 - \frac{R_{s}}{r}\right)\sin^{2}\theta
- 2\cos^{2}\theta.
\end{equation}
\begin{equation}
R_{33}
= \sin^{2}\theta
+ 3\cos^{2}\theta - 1
- 2\cos^{2}\theta
\end{equation}
\begin{equation}
R_{33}
= \sin^{2}\theta + 3\cos^{2}\theta - \sin^{2}\theta - \cos^{2}\theta - 2\cos^{2}\theta
= 0.
\end{equation}
Result:
\begin{equation}
R_{33} = 0 \quad \text{q.e.d.}
\end{equation}
Conclusion
We have shown that the components \(R_{00}, R_{11}, R_{22}\) and \(R_{33}\) of the
Ricci tensor are all zero within the Schwarzschild geometry when using
the simplified field equation with \(\mathrm{tr}(g_{\mu\nu}) = -1\).
This confirms that the Schwarzschild solution is indeed a vacuum solution of the
Einstein field equations, even under this specific derivation method.
6.6 Verification of the Ricci components in spherical coordinates
We verify whether the Schwarzschild solution in spherical coordinates satisfies the
restricted Einstein field equations, where the determinant of the metric
\(g = -1\).
The Schwarzschild metric in spherical coordinates is:
\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\varphi^{2},
\qquad
\sigma^{2} = 1 - \frac{R_{s}}{r}.
\end{equation}
We evaluate below the components of the Ricci tensor \(R_{\mu\nu}\) individually.
Component \(R_{00}\)
\begin{equation}
R_{00}
= \Gamma^{1}_{00,1}
+ \Gamma^{0}_{01}\Gamma^{1}_{00}
+ \Gamma^{1}_{00}\Gamma^{0}_{10}.
\end{equation}
After substitution and simplification:
\begin{equation}
R_{00}
= \frac{R_{s}(3R_{s} - 2r)}{2r^{4}}
+ \frac{R_{s}}{2r^{2}\sigma^{2}}
\frac{\sigma^{2}R_{s}}{2r^{2}}
+ \frac{\sigma^{2}R_{s}}{2r^{2}}
\frac{R_{s}}{2r^{2}\sigma^{2}}
\end{equation}
\begin{equation}
= \frac{R_{s}(3R_{s} - 2r)}{2r^{4}}
+ \frac{R_{s}^{2}}{2r^{4}}
\end{equation}
\begin{equation}
= \frac{R_{s}(4R_{s} - 2r)}{2r^{4}}
= \frac{2R_{s}(2R_{s} - r)}{r^{4}}.
\end{equation}
Conclusion:
\begin{equation}
R_{00} \neq 0.
\end{equation}
Component \(R_{11}\)
\begin{equation}
R_{11}
= \Gamma^{1}_{11,1}
+ \Gamma^{0}_{10}\Gamma^{0}_{01}
+ \Gamma^{1}_{11}\Gamma^{1}_{11}
+ \Gamma^{2}_{12}\Gamma^{2}_{21}
+ \Gamma^{3}_{13}\Gamma^{3}_{31}.
\end{equation}
Evaluation yields:
\begin{equation}
R_{11}=\frac{R_{s} (2r−R_{s})}{ 2r^4\sigma^4}+ \frac{R_{s}} {2r^2\sigma^2}\cdot \frac{ R_{s}}{ 2r^2\sigma^2}+\frac{−R_{s}}{ 2r^2\sigma^2}\cdot \frac{−R_{s}}{ 2r^2\sigma^2}+\frac{1}{r}\cdot \frac{ 1}{r}+\frac{1}{r}\cdot \frac{ 1}{r}
\end{equation}
\begin{equation}
R_{11}=\frac{R_{s}( 2r−R_{s})}{ 2r^4\sigma^4}+ \frac{R_{s}^2}{ 2r^4\sigma^4}+\frac{2}{r^2}
\end{equation}
\begin{equation}
R_{11}=\frac{R_{s} (2r−R_{s})}{ 2r^4\sigma^4}+ \frac{R_{s}^2}{ 2r^4\sigma^4}+\frac{4 (r^2+R_{s}^2−2r R_{s})}{ 2r^4\sigma^4}
\end{equation}
\begin{equation}
R_{11}=\frac{2r R_{s}-R_{s}^2}{2r^4\sigma^4}+\frac{ R_{s}^2}{2r^4\sigma^4}+\frac{4 (r^2+R_{s}^2−2r R_{s})}{ 2r^4\sigma^4}
\end{equation}
\begin{equation}
R_{11}=\frac{2r R_{s}-R_{s}^2}{ 2r^4\sigma^4}+\frac{ 4 (r^2+R_{s}^2−2r R_{s})}{ 2r^4\sigma^4}
\end{equation}
\begin{equation}
R_{11}=\frac{2r R_{s}-R_{s}^2+4r^2+4R_{s}^2−8r R_{s}}{ 2r^4\sigma^4}
\end{equation}
\begin{equation}
R_{11}=\frac{−6r R_{s}+2R_{s}^2+4r^2}{ 2r^4\sigma^4}=\frac{R_{s}^2+2r^2−3r R_{s}}{ r^4\sigma^4}
\end{equation}
By further simplification eliminating \( \sigma^4\):
\begin{equation}
R_{11}=\frac{R_{s}^2+2r^2−3r R_{s}}{ r^2(R_{s}^2+r^2−2r R_{s})}
\end{equation}
Conclusion:
\begin{equation}
R_{11} \neq 0.
\end{equation}
Component \(R_{22}\)
\begin{equation}
R_{22}
= \Gamma^{1}_{22,1}
+ \Gamma^{2}_{22,2}
+ \Gamma^{1}_{22}\Gamma^{2}_{12}
+ \Gamma^{2}_{21}\Gamma^{1}_{22}
+ \Gamma^{2}_{22}\Gamma^{2}_{22}
+ \Gamma^{3}_{23}\Gamma^{3}_{23}.
\end{equation}
Evaluation of these terms yields:
\begin{equation}R_{22}=1+0+(-r\sigma^2)\cdot\frac{1}{r}+\frac{1}{r}\cdot(-r\sigma^2)+0+
\frac{\cos\theta}{\sin\theta}\cdot\frac{\cos\theta}{\sin\theta}
\end{equation}
\begin{equation}
R_{22}
= 1 - 2\sigma^{2}
+ \frac{\cos^{2}\theta}{\sin^{2}\theta}
=\frac{\sin^{2}\theta}{\sin^{2}\theta}+\frac{\cos^{2}\theta}{\sin^{2}\theta}-2\sigma^2\end{equation}
\begin{equation}
R_{22}= \frac{1}{\sin^{2}\theta} - 2\sigma^{2}.
\end{equation}
Conclusion:
\begin{equation}
R_{22} \neq 0.
\end{equation}
Component \(R_{33}\)
\begin{equation}
R_{33}
= \Gamma^{1}_{33,1}
+ \Gamma^{2}_{33,2}
+ \Gamma^{1}_{33}\Gamma^{3}_{13}
+ \Gamma^{2}_{33}\Gamma^{3}_{23}
+ \Gamma^{3}_{31}\Gamma^{1}_{33}
+ \Gamma^{3}_{23}\Gamma^{2}_{33}.
\end{equation}
Evaluation yields:
\begin{equation}
R_{33}
= 1 + \cos^{2}\theta - \sin^{2}\theta
- r\sigma^{2}\sin^{2}\theta\frac{1}{r}-\cos\theta\sin\theta\cdot\frac{\cos\theta}{\sin\theta}
\end{equation}
\begin{equation}
+\frac{1}{r}\cdot(-r\sigma^2\sin\theta^2)+\frac{\cos\theta}{\sin\theta}(-\cos\theta\sin\theta)
\end{equation}
\begin{equation}
R_{33}
= 1 + \cos^{2}\theta - \sin^{2}\theta
- 2\sigma^{2}\sin^{2}\theta-2\cos\theta\sin\theta\cdot\frac{\cos\theta}{\sin\theta}
\end{equation}
\begin{equation}R_{33}
= 1 + \cos^{2}\theta - \sin^{2}\theta
- 2\sigma^{2}\sin^{2}\theta-2\cos^2\theta
\end{equation}
\begin{equation}
R_{33}=
1 - \cos^{2}\theta - \sin^{2}\theta
- 2\sigma^{2}\sin^{2}\theta
\end{equation}
\begin{equation}
R_{33}= -2\,\sigma^{2}\sin^{2}\theta.
\end{equation}
Conclusion:
\begin{equation}
R_{33} \neq 0.
\end{equation}
Interpretation of the result
We observe that all components \(R_{\mu\nu} \neq 0\) when using the restricted
Einstein field equations. For the Schwarzschild solution in vacuum, it must hold that:
\begin{equation}
R_{\mu\nu}=0
\end{equation}
Thus: the Schwarzschild formula in spherical coordinates does not satisfy this
restricted form.
This is not surprising, since the determinant of \(g\) for spherical coordinates
is not equal to \(-1\), which is a requirement for applying the restricted formula.
For the Schwarzschild metric in spherical coordinates we have:
\begin{equation}
g = -\sigma^{2} \cdot \frac{1}{\sigma^{2}} \cdot r^{2} \cdot r^{2}\sin^{2}\theta
= -r^{4}\sin^{2}\theta \neq -1.
\end{equation}
However, with respect to the full Einstein field equations, the
spherical Schwarzschild metric is fully consistent, as shown above.
Remark
The restricted formula was the result of an additional condition introduced by Einstein,
namely that the product of the elements of the trace of the metric tensor must satisfy:
\begin{equation}
g = g_{00}\cdot g_{11}\cdot g_{22}\cdot g_{33} = -1.
\end{equation}
This additional condition was introduced to simplify the calculations and reduce the
general formula.
However, the restricted formula imposes a constraint that excludes a number of possible
solutions.
Therefore, applying the general Einstein field equations is the preferred
approach. This is supported by the fact that the practical Schwarzschild metric:
\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\varphi^{2},
\end{equation}
has a determinant different from \(-1\), and therefore does not satisfy the restricted
Einstein formula, but does satisfy the general formula.
With this metric, various practical problems in general relativity can be solved, such as:
- the bending of light,
- the perihelion precession of Mercury,
- the Shapiro time delay,
- dynamics around black holes.
It has also been confirmed by numerous measurements that these calculations agree with
observations.
In short: the Schwarzschild solution demonstrates that the general formulation of general
relativity:
\begin{equation}
R_{\mu\nu} - \frac{1}{2}R\,g_{\mu\nu}
= \frac{8\pi G}{c^{4}}\,T_{\mu\nu},
\end{equation}
is to be preferred over the restricted formula.
The general Einstein field equations form the correct foundation of relativistic
gravitational theory.
The restricted formula is merely a specialization under restrictive conditions.