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 4-D spherically symmetric metric. The definition of the angles and the polar radius involved comes straight from their definitions in 3-D spherical polar coordinates shown in the diagram below. 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 3-D sphere having a radius  . This surface is called a 2-sphere. 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 2-sphere.

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

 (17.3)    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 4-D.