Einstein’s General Theory of Relativity

Derivations, Applications and Reflections – by Albert Prins

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 solution satisfies the Einstein field equations. We do this first for the full field equations and then for the simplified form.

6.1 Verification Against the Full Field Equations

The general form of the Einstein field equations is: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\,T_{\mu\nu}. \]

Here, the left-hand side describes the geometry of space‑time, 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 astrophysical or planetary calculations. Therefore, one typically works with the simplified equation: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8\pi G}{c^{4}}\,T_{\mu\nu}. \]

In vacuum regions – that is, outside a mass – we have: \[ T_{\mu\nu} = 0, \] so the equation reduces to: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0. \]

The indices \( \mu \) and \( \nu \) each take four values (0 to 3), resulting in 16 coupled differential equations. The field equations depend entirely on the metric tensor \( g_{\mu\nu} \) and its first and second derivatives. This is because they are constructed solely from the Christoffel symbols and their derivatives, and the Christoffel symbols themselves are fully determined by the metric and its first derivatives.

6.1.1 The Schwarzschild Solution

Karl Schwarzschild found an exact solution of the field equations in vacuum, assuming spherical symmetry. The metric is: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dr^{2}}{\sigma^{2}} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\varphi^{2}, \] where: \[ \sigma^{2} = 1 - \frac{2GM}{c^{2}r}. \]

\[ \sigma = \sqrt{1 - \frac{2GM}{c^{2}r}}. \]

The general form of the metric in coordinate notation is: \[ ds^{2} = g_{00}\,dt^{2} + g_{11}\,dr^{2} + g_{22}\,d\theta^{2} + g_{33}\,d\varphi^{2}. \]

This shows that only four of the sixteen metric components are non‑zero. Therefore, only four components of the Ricci tensor \(R_{\mu\nu}\) are relevant: \[ R_{00},\quad R_{11},\quad R_{22},\quad R_{33}. \]

6.1.2 The Ricci Tensor and Christoffel Symbols

The Ricci tensor is defined as: \[ R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu}. \]

With the general expression: \[ R_{\mu\nu} = \partial_{\rho}\Gamma^{\rho}_{\mu\nu} - \partial_{\nu}\Gamma^{\rho}_{\mu\rho} + \Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\mu\nu} - \Gamma^{\rho}_{\lambda\nu}\Gamma^{\lambda}_{\mu\rho}. \]

This formula is composed of derivatives and products of the Christoffel symbols. The general form of a Christoffel symbol is: \[ \Gamma^{\rho}_{\mu\nu} = \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right). \]

6.1.3 Simplification in Vacuum

As stated earlier, in vacuum: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0. \]

Here, \(R\) is the Ricci scalar and represents the total curvature of local space‑time. The Ricci scalar is computed as the contraction of the Ricci tensor: \[ R = g^{\mu\nu} R_{\mu\nu}. \]

This means that the Ricci scalar summarizes how space‑time curves in all directions, based on the information in the Ricci tensor. In the case of the Einstein field equations in vacuum: \[ R = 0, \] which means that the total space‑time curvature is zero outside a massive source.

When the earlier formula is multiplied by \( g^{\mu\nu} \), we obtain: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0, \] \[ g^{\mu\nu} R_{\mu\nu} - \frac{1}{2} g^{\mu\nu} g_{\mu\nu} R = 0. \]

Because: \[ g^{\mu\nu} g_{\mu\nu} = 4, \] it follows that: \[ R - \frac{1}{2}(4)R = 0 \quad\Rightarrow\quad R - 2R = 0 \quad\Rightarrow\quad R = 0. \]

This can only be true if also: \[ R_{\mu\nu} = 0. \]

Thus, due to the relationship between \(R\) and \(R_{\mu\nu}\), it is clear that: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0 \quad\Rightarrow\quad R_{\mu\nu} = 0. \]

6.1.4 Calculations and Numerical Verification

Based on the general form of the Ricci tensor and the Christoffel symbols, we have shown both numerically (using a computer program) and theoretically that the Schwarzschild metric indeed satisfies the vacuum equation: \[ R_{\mu\nu} = 0. \]

The relevant expressions for the Ricci components in terms of Christoffel symbols are:

\[ R_{00} = \Gamma^{0}_{00,1} + \Gamma^{0}_{01}\Gamma^{1}_{11} + \Gamma^{0}_{01}\Gamma^{2}_{22} + \Gamma^{0}_{01}\Gamma^{3}_{33} - \Gamma^{0}_{10}\Gamma^{0}_{01}, \] \[ R_{11} = -\Gamma^{1}_{01,01} - \Gamma^{1}_{12,12} - \Gamma^{1}_{13,13} + \Gamma^{1}_{11}\Gamma^{0}_{00} + \Gamma^{1}_{11}\Gamma^{2}_{22} + \Gamma^{1}_{11}\Gamma^{3}_{33} - \Gamma^{0}_{10}\Gamma^{0}_{10} - \Gamma^{2}_{22}\Gamma^{2}_{12} - \Gamma^{3}_{33}\Gamma^{3}_{13}, \] \[ R_{22} = \Gamma^{2}_{22,11} - \Gamma^{2}_{23,23} + \Gamma^{2}_{21}\Gamma^{0}_{00} + \Gamma^{2}_{21}\Gamma^{1}_{11} + \Gamma^{2}_{21}\Gamma^{3}_{33} - \Gamma^{2}_{12}\Gamma^{2}_{21} - \Gamma^{3}_{33}\Gamma^{3}_{23}, \] \[ R_{33} = \Gamma^{3}_{33,11} + \Gamma^{3}_{33,22} + \Gamma^{3}_{31}\Gamma^{0}_{00} + \Gamma^{3}_{31}\Gamma^{1}_{11} + \Gamma^{3}_{31}\Gamma^{2}_{22} - \Gamma^{3}_{13}\Gamma^{3}_{31} - \Gamma^{3}_{23}\Gamma^{3}_{32}. \]

These equations were evaluated by deriving the Christoffel symbols from the Schwarzschild metric and substituting them into the expressions above. 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) in front of the leading factor. In our approach, we used a positive factor of \( \tfrac{1}{2} \). This convention proved consistent with the result that all relevant Ricci components are zero: \[ R_{00} = R_{11} = R_{22} = R_{33} = 0, \] as required by the Einstein field equations in vacuum.

Therefore, the Christoffel formula was applied in the following form: \[ \Gamma^{\mu}_{\nu\rho} = \frac{1}{2} g^{\mu\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\rho}} + \frac{\partial g_{\rho\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\nu\rho}}{\partial x^{\alpha}} \right). \]

6.1.5 Conclusion

Through both analytical and numerical evaluation of the Ricci tensor components, based on the Schwarzschild metric and the associated 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 space‑time 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 in the context of the Schwarzschild solution. This is done in spherical coordinates \((t, r, \theta, \varphi)\).

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

To determine the components of the Ricci tensor, we follow these steps:

  1. Derive the Christoffel symbols from the Schwarzschild metric in spherical coordinates;
  2. Substitute these symbols into the expressions for the Ricci tensor components;
  3. 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 follow the individual checks.

Verification of \(R_{00}\)

The component \(R_{00}\) is given by: \[ R_{00} = \Gamma^{0}_{00,1} + \Gamma^{0}_{01}\Gamma^{1}_{11} + \Gamma^{0}_{01}\Gamma^{2}_{22} + \Gamma^{0}_{01}\Gamma^{3}_{33} - \Gamma^{0}_{10}\Gamma^{0}_{01}. \]

After substituting the corresponding terms, we obtain: \[ R_{00} = \frac{R_{s}(3R_{s} - 2r)}{2r^{4}} + \sigma^{2}\frac{R_{s}^{2}}{r^{2}} - \frac{R_{s}^{2}}{r^{2}\sigma^{2}} + \sigma^{2}\frac{R_{s}^{2}}{r^{2}}\frac{1}{r} + \sigma^{2}\frac{R_{s}^{2}}{r^{2}}\frac{1}{r} - \frac{R_{s}^{2}}{r^{2}\sigma^{2}} \sigma^{2}\frac{R_{s}^{2}}{r^{2}}. \]

This reduces to: \[ R_{00} = \frac{R_{s}(3R_{s} - 2r)}{2r^{4}} - \frac{R_{s}^{2}}{2r^{4}} + \frac{2R_{s}(r - R_{s})}{2r^{4}} = \frac{3R_{s}^{2} - 2rR_{s} - R_{s}^{2} + 2R_{s}r - 2R_{s}^{2}}{2r^{4}} = 0. \]

As required by the vacuum equations: \[ R_{00} = 0. \]

Thus: \[ R_{00} = 0 \quad \text{q.e.d.} \]

Verification of \(R_{11}\)

The component \(R_{11}\) is computed from: \[ R_{11} = -\Gamma^{1}_{01,01} - \Gamma^{1}_{12,12} - \Gamma^{1}_{13,13} + \Gamma^{1}_{11}\Gamma^{0}_{00} + \Gamma^{1}_{11}\Gamma^{2}_{22} + \Gamma^{1}_{11}\Gamma^{3}_{33} - \Gamma^{0}_{10}\Gamma^{0}_{10} - \Gamma^{2}_{22}\Gamma^{2}_{12} - \Gamma^{3}_{33}\Gamma^{3}_{13}. \]

After expansion: \[ R_{11} = -\frac{R_{s}(R_{s}-2r)}{2r^{4}\sigma^{4}} + \frac{1}{r^{2}} + \frac{1}{r^{2}} - \frac{R_{s}^{2}}{4r^{4}\sigma^{4}} - \frac{R_{s}^{2}}{r^{3}\sigma^{2}} - \frac{R_{s}^{2}}{r^{3}\sigma^{2}} - \frac{R_{s}^{2}}{4r^{4}\sigma^{4}} - \frac{1}{r^{2}} - \frac{1}{r^{2}}. \]

Further reduction yields: \[ R_{11} = -\frac{R_{s}(R_{s}-2r)}{2r^{4}\sigma^{4}} - \frac{R_{s}^{2}}{2r^{4}\sigma^{4}} - \frac{2R_{s}r(1 - R_{s}/r)}{2r^{4}\sigma^{4}} \] \[ = -\frac{R_{s}^{2} - 2rR_{s}}{2r^{4}\sigma^{4}} - \frac{R_{s}^{2}}{2r^{4}\sigma^{4}} - \frac{2R_{s}r - 2R_{s}^{2}}{2r^{4}\sigma^{4}} \] \[ = \frac{-R_{s}^{2} + 2rR_{s} - R_{s}^{2} - 2R_{s}r + 2R_{s}^{2}}{2r^{4}\sigma^{4}} = 0. \]

Thus: \[ R_{11} = 0 \quad \text{q.e.d.} \]

Verification of \(R_{22}\)

For the component \(R_{22}\), we have: \[ R_{22} = \Gamma^{2}_{22,11} - \Gamma^{2}_{23,23} + \Gamma^{2}_{21}\Gamma^{0}_{00} + \Gamma^{2}_{21}\Gamma^{1}_{11} + \Gamma^{2}_{21}\Gamma^{3}_{33} - \Gamma^{2}_{12}\Gamma^{2}_{21} - \Gamma^{3}_{33}\Gamma^{3}_{23}. \]

Substitution gives: \[ R_{22} = -1 + 1 - r\sigma^{2}\frac{R_{s}^{2}}{r^{2}\sigma^{2}} + r\sigma^{2} + \frac{R_{s}^{2}}{r^{2}\sigma^{2}} - r\sigma^{2}\frac{1}{r} + r\sigma^{2}\frac{1}{r} - 0. \]

This reduces to: \[ R_{22} = 0. \]

Thus: \[ R_{22} = 0 \quad \text{q.e.d.} \]

Verification of \(R_{33}\)

The component \(R_{33}\) is computed as: \[ R_{33} = \Gamma^{3}_{33,11} + \Gamma^{3}_{33,22} + \Gamma^{3}_{31}\Gamma^{0}_{00} + \Gamma^{3}_{31}\Gamma^{1}_{11} + \Gamma^{3}_{31}\Gamma^{2}_{22} - \Gamma^{3}_{13}\Gamma^{3}_{31} - \Gamma^{3}_{23}\Gamma^{3}_{32}. \]

Expansion yields: \[ R_{33} = -1 + 1 - r\sigma^{2}\frac{R_{s}^{2}}{r^{2}\sigma^{2}} + r\sigma^{2}\frac{R_{s}^{2}}{r^{2}\sigma^{2}} - r\sigma^{2}\frac{1}{r} + r\sigma^{2}\frac{1}{r} - 0. \]

Thus: \[ R_{33} = 0 \quad \text{q.e.d.} \]

Conclusion

All relevant components of the Ricci tensor vanish: \[ R_{\mu\nu} = 0 \qquad \text{for } \mu,\nu \in \{0,1,2,3\}, \] confirming that the Schwarzschild solution satisfies the Einstein field equations in vacuum. This demonstrates that the Schwarzschild metric correctly describes the spacetime outside a spherically symmetric mass in the absence of matter or energy.

6.4 Verification of the Schwarzschild Solution Using a Simplified Form of the Field Equations

In this chapter, we verify the Schwarzschild solution using a simplified version of the Einstein field equations. This restricted form originates from Schwarzschild’s original derivation and applies only when the trace of the metric tensor satisfies: \[ \mathrm{tr}(g_{\mu\nu}) = -1. \]

Under this condition, the field equations take the form: \[ G_{\mu\nu} = \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x^{\alpha}} + \Gamma^{\alpha}_{\mu\beta}\,\Gamma^{\beta}_{\nu\alpha}. \]

In this expression, the Christoffel symbols are defined with a negative sign, exactly as Schwarzschild used: \[ \Gamma^{\rho}_{\mu\nu} = -\frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right). \]

Because of this negative convention, all derived expressions—including the Ricci tensor components—will differ in sign from the standard modern definition.

Schwarzschild performed his derivation in the coordinates \((t, x, y, z)\). We therefore begin with these coordinates and list the relevant Ricci tensor components as they follow from the restricted formula.

Derivative Components of the Ricci Tensor

The following expressions apply in Schwarzschild’s notation:

These components can be evaluated directly 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 (modified polar) Coordinates

We work in modified Cartesian–polar coordinates \((t, x, y, z)\), the same coordinates Schwarzschild originally used in his derivation. In this section, we explicitly verify that the Ricci tensor components \(R_{\mu\nu}\) vanish.

Calculation of \(R_{00}\)

\[ R_{00} = \Gamma^{0}_{00,11} + \Gamma^{0}_{10}\Gamma^{0}_{01} + \Gamma^{0}_{01}\Gamma^{1}_{00}. \]

With the given values:

\[ R_{00} = -\frac{R_{s}^{2}}{2r^{4}} + \frac{R_{s}^{2}}{r^{4}\sigma^{2}}\frac{R_{s}}{\sigma^{2}} + \frac{R_{s}}{\sigma^{2}}\frac{R_{s}^{2}}{r^{4}\sigma^{2}} = -\frac{R_{s}^{2}}{2r^{4}} + \frac{R_{s}^{2}}{2r^{4}} = 0. \]

Result: \[ R_{00} = 0 \quad \text{q.e.d.} \]

Calculation of \(R_{11}\)

\[ R_{11} = \Gamma^{1}_{11,11} + \Gamma^{1}_{00}\Gamma^{0}_{10} + \Gamma^{1}_{11}\Gamma^{1}_{11} + \Gamma^{1}_{22}\Gamma^{2}_{12} + \Gamma^{1}_{33}\Gamma^{3}_{13}. \]

By carefully substituting and simplifying all terms, we find: \[ R_{11} = -\frac{6r^{6}}{\sigma^{4}} + \frac{10R_{s}r^{7}}{\sigma^{4}} + \frac{-4.5R_{s}^{2}r^{8}}{\sigma^{4}} + \frac{R_{s}^{2}}{4r^{8}\sigma^{4}} + \frac{(3R_{s}-4r)^{2}}{4r^{8}\sigma^{4}} + \frac{1}{r^{3}}\frac{1}{r^{3}} + \frac{1}{r^{3}}\frac{1}{r^{3}}. \]

After further reduction: \[ R_{11} = \frac{-24r^{2} + 40rR_{s} - 18R_{s}^{2}}{4r^{8}\sigma^{4}} + \frac{R_{s}^{2}}{4r^{8}\sigma^{4}} + \frac{9R_{s}^{2} + 16r^{2} - 24rR_{s}}{4r^{8}\sigma^{4}} + \frac{2r^{6}}{4r^{8}\sigma^{4}}. \]

\[ R_{11} = \frac{-8R_{s}^{2} - 8r^{2} + 16rR_{s}}{4r^{8}\sigma^{4}} + \frac{2r^{6}}{4r^{8}\sigma^{4}} \] \[ = \frac{-8R_{s}^{2} - 8r^{2} + 16rR_{s} + 8r^{2}\sigma^{4}}{4r^{8}\sigma^{4}} \] \[ = \frac{-8R_{s}^{2} - 8r^{2} + 16rR_{s} + 8r^{2}(1 - R_{s}/r)^{2}}{4r^{8}\sigma^{4}} = 0. \]

Result: \[ R_{11} = 0 \quad \text{q.e.d.} \]

Calculation of \(R_{22}\)

\[ R_{22} = \Gamma^{2}_{22,11} + \Gamma^{2}_{22,22} + \Gamma^{2}_{21}\Gamma^{1}_{22} + \Gamma^{2}_{12}\Gamma^{2}_{21} + \Gamma^{2}_{22}\Gamma^{2}_{22} + \Gamma^{2}_{33}\Gamma^{3}_{23}. \]

After simplifying the trigonometric and radial terms: \[ R_{22} = \frac{-2R_{s} + 3r}{r\sin^{2}\theta} + \frac{-1 - \cos^{2}\theta}{\sin^{4}\theta} - \frac{2(r - R_{s})}{r\sin^{2}\theta} + \frac{2\cos^{2}\theta}{\sin^{4}\theta}. \]

\[ R_{22} = \frac{1}{\sin^{2}\theta} + \frac{-1 - \cos^{2}\theta}{\sin^{4}\theta} + \frac{2\cos^{2}\theta}{\sin^{4}\theta} = \frac{\sin^{2}\theta}{\sin^{4}\theta} + \frac{-\sin^{2}\theta - \cos^{2}\theta + 2\cos^{2}\theta}{\sin^{4}\theta} = 0. \]

Result: \[ R_{22} = 0 \quad \text{q.e.d.} \]

Calculation of \(R_{33}\)

\[ R_{33} = \Gamma^{3}_{33,11} + \Gamma^{3}_{33,22} + \Gamma^{3}_{31}\Gamma^{1}_{33} + \Gamma^{3}_{32}\Gamma^{2}_{33} + \Gamma^{3}_{13}\Gamma^{3}_{31} + \Gamma^{3}_{23}\Gamma^{3}_{32}. \]

After combining the angle-dependent terms: \[ R_{33} = \left(3 - \frac{2R_{s}}{r}\right)\sin^{2}\theta + 3\cos^{2}\theta - 1 - 2\sigma^{2}\sin^{2}\theta - 2\sin^{2}\theta\cos^{2}\theta. \]

Using \(\sigma^{2} = 1 - \frac{R_{s}}{r}\): \[ R_{33} = \sin^{2}\theta + 3\cos^{2}\theta - 1 - 2\cos^{2}\theta = \sin^{2}\theta + 3\cos^{2}\theta - \sin^{2}\theta - \cos^{2}\theta - 2\cos^{2}\theta = 0. \]

Result: \[ R_{33} = 0 \quad \text{q.e.d.} \]

Conclusion

We have shown that the components \(R_{00}, R_{11}, R_{22}, R_{33}\) of the Ricci tensor all vanish 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 now verify that the Schwarzschild solution in spherical coordinates satisfies the restricted Einstein field equations, in which the determinant of the metric obeys \(g = -1\).

The Schwarzschild metric in spherical coordinates is: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dr^{2}}{\sigma^{2}} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\varphi^{2}, \qquad \sigma^{2} = 1 - \frac{R_{s}}{r}. \]

We evaluate the components of the Ricci tensor \(R_{\mu\nu}\) separately below.

Component \(R_{00}\)

\[ R_{00} = \Gamma^{0}_{00,11} + \Gamma^{0}_{10}\Gamma^{0}_{01} + \Gamma^{0}_{01}\Gamma^{1}_{00}. \]

After substitution and simplification: \[ R_{00} = -\frac{R_{s}(3R_{s} - 2r)}{2r^{4}} + \frac{R_{s}^{2}}{r^{2}\sigma^{2}} \frac{\sigma^{2}R_{s}^{2}}{r^{2}} + \frac{\sigma^{2}R_{s}^{2}}{r^{2}} \frac{R_{s}^{2}}{r^{2}\sigma^{2}} \] \[ = -\frac{R_{s}(3R_{s} - 2r)}{2r^{4}} + \frac{R_{s}^{2}}{2r^{4}} \] \[ = -\frac{R_{s}(2R_{s} - 2r)}{2r^{4}} = -\frac{R_{s}(R_{s} - r)}{r^{4}}. \]

Conclusion:
\[ R_{00} \neq 0. \]

Component \(R_{11}\)

\[ R_{11} = \Gamma^{1}_{11,11} + \Gamma^{1}_{00}\Gamma^{0}_{10} + \Gamma^{1}_{11}\Gamma^{1}_{11} + \Gamma^{1}_{22}\Gamma^{2}_{12} + \Gamma^{1}_{33}\Gamma^{3}_{13}. \]

Carrying out the computation yields: \[ R_{11} = -\frac{R_{s}}{2r} - \frac{R_{s}}{2r^{4}\sigma^{4}} + \frac{R_{s}^{2}}{2r^{4}\sigma^{4}} + 2r^{2} \] \[ = -\frac{2rR_{s}}{2r^{4}\sigma^{4}} + \frac{R_{s}^{2}}{2r^{4}\sigma^{4}} + 2r^{2}. \]

Further simplification: \[ R_{11} = \frac{3R_{s}^{2} + 2r^{2} - 5rR_{s}}{r^{4}\sigma^{4}} \neq 0. \]

Conclusion:
\[ R_{11} \neq 0. \]

Component \(R_{22}\)

\[ R_{22} = \Gamma^{2}_{22,11} + \Gamma^{2}_{22,22} + \Gamma^{2}_{21}\Gamma^{1}_{22} + \Gamma^{2}_{12}\Gamma^{2}_{21} + \Gamma^{2}_{22}\Gamma^{2}_{22} + \Gamma^{2}_{33}\Gamma^{3}_{23}. \]

Evaluation of these terms gives: \[ R_{22} = 1 - 2\sigma^{2} + \frac{\cos^{2}\theta}{\sin^{2}\theta} = \frac{1}{\sin^{2}\theta} - 2\sigma^{2}. \]

Conclusion:
\[ R_{22} \neq 0. \]

Component \(R_{33}\)

\[ R_{33} = \Gamma^{3}_{33,11} + \Gamma^{3}_{33,22} + \Gamma^{3}_{31}\Gamma^{1}_{33} + \Gamma^{3}_{32}\Gamma^{2}_{33} + \Gamma^{3}_{13}\Gamma^{3}_{31} + \Gamma^{3}_{23}\Gamma^{3}_{32}. \]

The computation yields: \[ R_{33} = 1 + \cos^{2}\theta - \sin^{2}\theta - 2\sigma^{2}\sin^{2}\theta - 2\cos^{2}\theta. \]

\[ R_{33} = 1 - \cos^{2}\theta - \sin^{2}\theta - 2\sigma^{2}\sin^{2}\theta = -2\sigma^{2}\sin^{2}\theta. \]

Conclusion:
\[ R_{33} \neq 0. \]

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

Conclusion:
\[ R_{33} \neq 0. \]

General conclusion

We see that all components \(R_{\mu\nu} \neq 0\) when using the restricted Einstein field equations. Thus, the Schwarzschild formula in spherical/polar coordinates does not satisfy this restricted formula.

This is not surprising, since the determinant of \(g\) for the spherical coordinates is not equal to \(-1\), which is required in order to use the restricted formula. For the Schwarzschild metric in spherical coordinates we have: \[ g = -\sigma^{2} \cdot \frac{1}{\sigma^{2}} \cdot r^{2} \cdot r^{2}\sin^{2}\theta = -r^{4}\sin^{2}\theta \neq -1. \]

However, with respect to the full Einstein field equations, the spherical/polar Schwarzschild metric is fully consistent, as was shown above.

Remark

The restricted formula was the result of an additional condition introduced by Einstein, namely that the product of the diagonal elements of the metric tensor must satisfy: \[ g = g_{00} g_{11} g_{22} g_{33} = -1. \] This extra condition was introduced to simplify the calculations and to reduce the general formula. However, the restricted formula is a constraint that excludes a number of possible solutions.

Therefore, applying the general Einstein field equations is the better approach. This is supported by the fact that the practical Schwarzschild metric: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dr^{2}}{\sigma^{2}} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\varphi^{2}, \] has a determinant that is not equal to \(-1\), and therefore does not satisfy the restricted Einstein formula, but it does satisfy the general formula.

With this metric, many practical problems in general relativity can be solved, such as:

Moreover, various measurements have now confirmed that the calculations agree with observations.

In short, the Schwarzschild solution shows that the general formula of general relativity: \[ R_{\mu\nu} - \frac{1}{2}R\,g_{\mu\nu} = \frac{8\pi G}{c^{4}}\,T_{\mu\nu}, \] is to be preferred over the restricted formula.

The general Einstein field equations form the correct foundation of the relativistic theory of gravitation. The restricted formula is merely a specialization under restrictive conditions.