- #1
June_cosmo
- 25
- 0
Homework Statement
Consider a universe described by the Friedmann-Robertson-Walker metric which describes an open, closed, or
at universe, depending on the value of k:
$$ds^2=a^2(t)[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]$$
This problem will involve only the geometry of space at some fixed time, so we can ignore thedependence of a on t, and think of it as a constant. Consider a circle described by the equations:
$$r=r_0$$
$$\theta=\pi/2$$
(a) Find the circumference S of this circle. (Hint:break the circle into infinitesimal segments of angular size dphi,
calculate the arc length of such a segment, and integrate.)
b) Find the radius Rc of this circle. Note that Rc is the length of a line which runs from the origin to the circle (r = r0), along a trajectory of theta=pi/2 and phi= const. Consider the case of open and closed universes separately, and take k= 1 or k=-1 as discussed in lecture. (Hint: Break the line into infinitesimal segments of coordinate length dr, calculate the length of such a segment, and integrate.)
Homework Equations
The Attempt at a Solution
I don't know how to derive dphi from the first equation provide?