Schwarschild radius and paramaterizing path

Question

Consider the metric $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.$$

Suppose a particle very large starts at the initial radius $R$ and then radially infalls in a Schwarzchild manifold

My text then states:

It can be shown that the parameter defining the particle's trajectory (assuming $m$ is large enough) is expressed as $r(\lambda)=C(1+cos\lambda)$ and $\tau(\lambda)=C(\frac{R}{m})^{\frac{1}{2}}(\lambda + sin\lambda)$ where $C$ is a constant and $\tau$ is the proper time along the geodesic.

My questions are: How in the world does one show the equations above? How does one find $C$? and how does one show that the statement above is true?

If an object radially infalls in the vicinity of a spherically symmetric object, then we can take $\dot{\theta}=\dot{\phi}=0$ Thus, assuming that the mass is large, we can get the equation:

$$g_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=-1$$ which implies $$g_{tt}\left(\frac{dt}{d\tau}\right)^2+g_{rr}\left(\frac{dr}{d\tau}\right)^2=-1.$$

However, I am unable to do anything beyond the above and plugging in the components of the metric tensor.

Please can someone help?

Answer 1

Start with your equation, which reduces to:

$$-1 = -\left(1 - \frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^{2} + \frac{1}{1-\frac{2M}{r}}\left(\frac{dr}{d\tau}\right)^{2}$$

Now, you can leverage the fact that $\partial_t$ is a Killing vector to show that $E = \left(1-\frac{2M}{r}\right)\frac{d t}{d \tau}$ is a constant of the motion (easiest way to prove this: use the fact that the arc length of a geodesic is an extremum of the motion, and the fact that the arc length of a path is an integral of the line element to treat the geodesic as the same sort of maximization problem that the Lagrangian is)

This makes the above reducible to:

$$\left(\frac{dr}{d\tau}\right)^2 = (E^{2}-1)\left(1-\frac{2M}{r}\right)$$

It is a bit of algebra, and you'll need to work out the relationship between $E$, $R$, and $C$, but you can work out that the parametric equations in your expression is a solution to this equation, knowing that $\frac{dr}{d\tau} = \frac{dr}{d\lambda}/\frac{d\tau}{d\lambda}$ by the chain rule.

As far as deriving the parametric equations, I have generally only seen "one makes an inspired guess as to the form of $r(\lambda)$, uses that to replace the values for $r$ and $\frac{dr}{d \lambda}$ in the above equation, and then solves the remaining differential equation in $\tau$ and $\lambda$ for $\tau$", so most textbooks skip that and just say "this equation has this parametric solution."

As for the constants, you need to set up initial conditions like "at $\tau =0$, ${\dot r} = 0$ and $r = R$, and you're going to need constants to enforce that.