The Schwarzschild solution is the first exact solution to the Einstein Field Equations of General Relativity, which makes use of Schwarzschild coordinates and Schwarzschild metric.
The Schwarzschild metric describes a static and spherically symmetric gravitation field in the empty region of spacetime near a massive spherical object. It is an exact solution to the Einstein Field Equations for non-rotating spherical objects. However, it provides a good approximation to the gravitational field of a slowly rotating bodies such as the Sun or the Earth. The solution also leads to a derivation of the Schwarzschild radius which is the size of the event horizon of a non-rotating black hole.
In this article, we are going to give a summary of the Christoffel symbols, the geodesic equations, the Riemann curvature tensor, the Ricci tensor, Ricci scalar of curvature by using the Schwarzschild metric. Since these calculations are lengthy and tedious, we will not show the details here but will only display the results.
The Schwarzschild metric is
where M is the mass of gravitational source, G is the gravitational constant, and are the Schwarzschild coordinates.
In matrix form, the Schwarzschild metric can be written as
The inverse metric then becomes
The Christoffel symbols are calculated from the formula
where is the inverse matrix of .
Next, we will list the non-zero results of our calculations.
1. The Christoffel symbols – there are nine non-vanishing components
2. The Geodesic equations
After obtaining the Christoffel symbols we can calculate the geodesic equations. The four geodesic equations allow physicists to describe how objects and photons move in Schwarzschild spacetime.
The four components of the geodesic equations are obtained by using the following basic equation
where is the proper time and is the four-velocity. In this case, .
3. The Riemann Curvature Tensor
The components of the Riemann tensor, are calculated using the definition
The nonzero components are
Note: the Riemann curvature tensor is anti-symmetric on the first and second pair of indices.
4. The Ricci tensor and Ricci scalar
The Ricci tensor can be formed by contracting the first and the third indices of the Riemann curvature tensor (which has the first index already raised).
The results of our calculation, the Ricci tensor has vanishing components. Likewise the Ricci scalar which is formed by using the inverse metric and the Ricci tensor
The Einstein tensor is found from the tensors already calculated. We obtain
This means that the Schwarzschild metric satisfied the vacuum Einstein equation.