How to derive the equation of motion of a test particle free falling into a Schwarzschild blackhole from distance $R$?

Question

This answer gives the equation for $t(r)$ without the derivation.

I have found the derivation for particles free falling from infinite distance using the Schwarzchild metric and Lagrangian, but I don't know how to apply this to the scenario that a particle free falls from a fixed distance $R$ at rest or with an intial velocity.

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Below is the results I have got so far.

For simplicity, take $c=G=r_s=1$.

For radial free-fall, we have $d\Omega=0$, so the metric is

$$ ds^2=-(1-\frac1r)dt^2+(1-\frac1r)^{-1}dr^2=-d\tau^2 $$

Let $\dot{x}$ represent $\frac{dx}{d\tau}$, the Lagrangian is $$ L=\dot{s}^2=-(1-\frac1r)\dot{t}^2+(1-\frac1r)^{-1}\dot{r}^2=-1 $$ As $\frac{d}{d\tau}(\frac{\partial L}{\partial \dot{t}})=\frac{\partial L}{\partial t}$, we have conservation of energy $(1-\frac1r)\dot{t}=E$.

In resources I found online, $E$ is set as $1$ for the ease of derivation. Then we have

$$ \dot{r}^2=\frac1r~\Rightarrow~\tau=\frac23(r_0^{3/2}-r^{3/2}) $$

$$ \Rightarrow~d\tau=-r^{1/2}dr $$ $$ \Rightarrow~dt=(1-\frac1r)^{-1}d\tau=-r^{1/2}(1-\frac1r)^{-1}dr $$

Integrate both sides to obtain $t=F(r_0)-F(r)$, where $$ F(r)=\frac23r^{3/2}+2r^{1/2}+ln\frac{r^{1/2}-1}{r^{1/2}+1}. $$

The solution has nothing to do with the initial position $R$. I guess $E=1$ implies the particle is at rest at inifinity? What should I set $E$ to be if the particle is at rest at distance $R$?

Answer 1

Using your notation $$ (1 - r^{-1}) \dot{t} = E\ , $$ where $E$ is the energy at infinity. But what you want is an initial condition of $E_{\rm shell}=1$, where this represents an object at rest according to some stationary "shell" observer at $r=R$.

The translation between the two comes from the Schwarzschild metric, where we know that since $$ d\tau_{\rm shell} = dt(1-R^{-1})^{1/2}\ , $$ and thus in the frame of the shell observer $$E_{\rm shell} = \dot{t}_{\rm shell} = \dot{t}(1 - R^{-1})^{1/2}\ . $$

Substituting for $\dot{t}$ in the initial energy conservation equation, we get $$(1 - R^{-1}) (1-R^{-1})^{-1/2} E_{\rm shell} = E$$ and thus if $E_{\rm shell}=1$ at $R$, then $$E = (1 - R^{-1})^{1/2}\ . $$ This can be put back into the first equation to get an expression for $\dot{t}(r)$. $$\dot{t} = (1 - R^{-1})^{1/2} (1 - r^{-1})^{-1}\ . $$