Schwarzschild Metric
The Schwarzschild Metric is a static spherically symmetric solution of the vacuum Einstein Equations. It is one of the most important metrics in General Relativity. We will now derive this metric starting from a more general 4D spherically symmetric metric. The definition of the angles and the polar radius involved comes straight from their definitions in 3D spherical polar coordinates shown in the diagram below.
Coordinates Link: http://hyperphysics.phyastr.gsu.edu/hbase/sphc.html
We take spherical symmetry to have the following definition. A gravitational system is said to be spherically symmetric if there exists a point, taken to be the origin O, such that the system is invariant under spatial rotations about O.
Consider the surface of a 3D sphere having a radius . This surface is called a 2sphere. The distance on such a surface is described by the line element relation
(17.1) 


with
(17.2)  . 

These angle ranges cover all points on the 2sphere.
To get 4D spherically symmetric equations we augment the 2D case with a time coordinate t and a radial coordinate r such that when t and r are constants we once again get the 2sphere line element. Spherical symmetry is assured if when the angles and vary only the 2sphere distance term gets affected in the 4D line element. Our 4D metric will not depend on and . We assume that there exists a special coordinate system
such that


The coefficients A,B,C and D guarantee spherical symmetry by being independent of the angles.
(17.4)
We now start the a series of coordinate transformations to get us into the standard spherically symmetric metric worked out by Karl Schwarzschild in 1916.
Let
(17.5) 


Then
(17.6)
with
(17.7)
We now change the time coordinate such that
(17.8) 


(17.9)


Solving for and substituting into (17.3) we obtain
(17.10)
We now introduce two functions such that
(17.11) 


and
(17.12) 


Then we arrive at the final functional form for our line element:
(17.13) 


where we have now dropped the primes. This form can be proved to be the most general spherically symmetric line element in 4D.