Spherically Symmetric Line Element

Spherically symmetric means an invariance under any arbitrary rotation of axes at a particular point, called the center of symmetry. Using θ and ϕ (polar coordinates) and choosing the center of symmetry at origin, we have the general form of the line element with spherical symmetry.

For the surfaces r = constant and t = constant, the line elements reduces to form two spheres on which a typical point is labeled by coordinate θ and ϕ and line element takes the form

This spherical symmetric line element is invariant when θ and ϕ are varied. The center of symmetry is the point O, which is given by r = 0.

Now we introduce new coordinates by the transformations:

where the function K will be chosen later.

From the above transformation equations, we have

Then the line element becomes

Now we choose K such that coefficient of dr′dt′ is zero.

Thus, we have

Hence we get general line element on

where σ, ω, and μ are functions of r′ and t′.

Now we take another transformation,

where q is so chosen that the coefficient of dR dT in ds2 is zero.

Thus, finally, we get the line element

where 𝜈 and μ are functions of R and T.

Note 6.1

Here R coordinate has specific significance:

The area of the surface of the sphere, R = constant is given by A = 4πR2.

Three volume of the sphere with radius R

e.g., for eμ = (1 - ar2)2, one can get

For the four-dimensional tube that is bounded by the sphere with radius R and two planes, t = constant, separated by a time T, the four volume is

e.g., for eμ = e𝜈 = (1 - ar2)2, one can get

Schwarzschild Solution or Exterior Solution

The exact solution of the Einstein field equation in empty space was obtained by Schwarzschild in 1916, which describes the geometry of spacetime outside a spherically symmetric distribution of matter.

Consider the metric of the empty spacetime outside of a spherically symmetric distribution of matter of mass M as