- #1
unscientific
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Homework Statement
Consider the following geodesic of a massless particle where ##\alpha## is a constant:
\dot r = \frac{\alpha}{a(t)^2}
c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)}
Homework Equations
The Attempt at a Solution
Part (a)
c \frac{dt}{d\lambda} = \frac{\alpha}{a}
a dt = \frac{\alpha}{c} d\lambda
\frac{1}{H} a = \frac{\alpha}{c} \lambda + \epsilon^{'}
a = \frac{H}{c} \left( \alpha \lambda + \epsilon \right)
Similarly,
r = \frac{c^2}{H^2} \left[ -\frac{1}{\alpha \lambda + \epsilon} + \delta \right]
Part(b)
I'm confused as to what the limits of integration are. I'm not sure if this is right:
At ##t = t_0##, ##a(t_0) = 1 = \frac{H}{c} \left(\alpha \lambda + \epsilon \right)##.
At ##r = r_e## what happens to ##\delta##?
