(a) It turns out that the following line element is a solution of the vacuum Einstein field equation G = 0:

Since this solution is spherically symmetric, Birkhoff’s theorem guarantees it must represent the standard Schwarzschild spacetime geometry in a coordinate system that differs from Schwarzschild’s. Show that this is so by exhibiting a coordinate transformation that converts this line element into Eq. (26.1). 

(b) Show that at large radii r ≫ M, the line element (26.16) takes the form (25.98c) discussed in Chap. 25, but with vanishing spin angular momentum J = 0.

Equations.

Transcribed Image Text:

ds² == 2 1/(27)) ² 1- M/(2F) 1+ M/(27). di² +1+ M 2F 4 [dr² +r²(do²+ sin² 0dp²)]. (26.16)