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

the schwarzschild metric
 
where M is the mass of gravitational source, G is the gravitational constant, and the schwarzschild coordinates are the Schwarzschild coordinates.

In matrix form, the Schwarzschild metric can be written as

the Schwarzschild metric matrix form

The inverse metric then becomes

the schwarzschild metric, inverse matrix form

The Christoffel symbols are calculated from the formula

the christoffel symbols

where inverse metricis the inverse matrix of metric.

Next, we will list the non-zero results of our calculations.

 

1. The Christoffel symbols – there are nine non-vanishing components

the christoffel symbols for the schwarzschild metric

 

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

the geodesic equations

where tau is the proper time and four-velocity is the four-velocity. In this case, four-velocity in the schwarzschild coordinates.

Thus,

four-velocity in the schwarzschild metric

 

3. The Riemann Curvature Tensor

The components of the Riemann tensor, the riemann tensor are calculated using the definition

the riemann tensor definition

The nonzero components are

the riemann tensor in schwarzschild metric

the riemann tensor in schwarzschild metric

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 ricci tensor

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 ricci scalar
yields zero.

The Einstein tensor the einstein tensor is found from the tensors already calculated. We obtain

vacuum einstein equation

This means that the Schwarzschild metric satisfied the vacuum Einstein equation.