In a certain spacetime geometry, the metric is
$$
d s^{2}=-\left(1-A r^{2}\right) d t^{2}+\left(1-A r^{2}\right) d
r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)
$$
(a) Calculate the proper distance along a radial line from the centre \(r=0\) to
a coordinate radius \(r=R\). (b) Calculate the area of the sphere of coordinate
radius \(r=R\). (c) Calculate the 3 -volume bounded inside the sphere of
coordinate radius \(r=R\). (d) Calculate the 4-volume of the four-dimensional
tube bounded by a sphere of coordinate radius \(R\) and two \(t=\) constant planes
separated by \(T\).
