- 27

- 0

^{2}α

^{3}

Hint says we must use the dV in the spherical system (dV=r

^{2}sin

^{2}θdrdθdφ) as well as the equation of the three-dimensional metric ds

^{2}= c

^{2}dt

^{2}- a

^{2}[ dr

^{2}/(1-kr

^{2}) +r

^{2}(dθ

^{2}+sin

^{2}θ dφ

^{2}) ]

For a closed universe we know k=+1 and with dt=0

My problem is, I don't understand how the 3-d metric correlates with the dV. Are they the same? Could I do V=∫ds

^{2}= ∫∫∫- a

^{2}[ dr

^{2}/(1-kr

^{2}) +r

^{2}(dθ

^{2}+sin

^{2}θ dφ

^{2}) r

^{2}sin

^{2}θdrdθdφ