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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?

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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.

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  • $\begingroup$ So if we are given $C$, then do we just have to show that the equation above is satisfied? $\endgroup$ Feb 22 at 23:14
  • $\begingroup$ Say, $C=\frac{R_{in}}{2}$ where $R_{in}$ is the initial radius $\endgroup$ Feb 22 at 23:15
  • $\begingroup$ the equation is satisfied for all C. C is just a part of your initial conditions. $\endgroup$ Feb 22 at 23:27
  • $\begingroup$ So if we are just given the parameteric equations and the problem is to show that the large particle's trajectory has the given parameterization, what are we supposed to do? Just take derivatives and plug into the equation I gave? $\endgroup$ Feb 22 at 23:33
  • $\begingroup$ @monoidaltransform: yeah, pretty much $\endgroup$ Feb 23 at 8:12

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