Finding Null Geodesic Equations for Einstein's Metric in Curved Space

[Shadowlass]

Homework Statement

I'm given the metric for Einstein's universe,
ds² = c²dt² - dr²/(1 - kr²) - r²d(theta)² - r²sin²(theta)d(phi)²
and asked to find the null geodesic equations and show that in the plane theta=$$\pi$$/2, the curves satisfy the equation:
(dr/d(phi))² = r²(1-kr²)(mr²-1)
where m is a constant

Homework Equations

The Attempt at a Solution

I know I have to use "The Integral" (I'm not sure if that's a widely accepted name for it or if not what the real name of it is), and that for finding null geodesic equations I have to set it equal to zero, which gives
0 = g_ij$$\stackrel{.}{x}$$ⁱ$$\stackrel{.}{x}$$^j
where $$\stackrel{.}{x}$$ = dx/d(mu), or essentially dx/ds, but I'm confused about how to do this, every time I take my work into be looked at, my professor says I'm doing something else and that my work is irrelevant to the question, so I'm stumped. He's told me that I'll also need to use Christoffel symbols to get to the dr/d(phi) form in the second part of the equation, but without the first part of the problem solved I can't do that, so far as I can tell. Can anyone tell me what I'm missing here?

 

[nickjer]

Well took me awhile but I got the same answer as they got. It is a lot of messy algebra. You first need to solve for the geodesic equations using:

$$\frac{d^2 x^{\alpha}}{d\tau^2} - \Gamma^{\alpha}_{\delta \beta}\frac{dx^{\delta}}{d\tau}\frac{dx^{\beta}}{d\tau}=0$$

That will be the first step. You will then find the constants of motion in 2 of those 3 equations. The next step would be the keyword "null". You need to set $ds=0$ and solve for

$$\left(\frac{dr}{d\phi}\right)^2$$

Then you plug in your constants of motion to get the formula you wrote above.