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I am fully aware that there may be many misunderstandings in the following. However, in recent lectures we covered the Schwarzschild solution, and after some interesting mathematics we reached equations for $\frac{dr}{dt}$ and $\frac{dr}{d\tau}$. As far as I know, using the standard coordinates the former of these is the radial velocity of an object as seen in the frame of an observer at infinity, and the latter is the radial velocity in the frame of an object falling into the black hole. These equations could then be integrated, which in the case of $\frac{dr}{dt}$ gave the following solution:

$$\frac{ct}{Rs}\:=\:const.\:-\frac{2}{3}\left(\frac{r}{R_s}\right)^{\frac{3}{2}}-\left(\frac{2r}{R_s}\right)^{\frac{1}{2}}+\ln\left|\frac{\left(\frac{r}{R_s}\right)^{\frac{1}{2}}+1}{\left(\frac{r}{R_s}\right)^{\frac{1}{2}}-1}\right|$$

Giving a relationship between the time taken for the object to fall, as measured by an observer at infinity, and the radial coordinate of the in-falling object. Our lecturer was careful to point out that there is a logarithmic singularity in this relationship as $r \rightarrow R_ s$ - in other words, the in-falling object never reaches the Schwarzschild radius to an outside observer, as it would take an infinite amount of time to do so.

Now, while I think I understand the maths, and also understand that in the frame of the in-falling object, there is no such singularity at $R_s$ and it crosses the event horizon without noticing anything, I am struggling to get my head around how the above equation doesn't have very important consequences.

Say the observer at infinity is us on Earth. If, from Earth, it takes an infinite amount of time for objects to fall into the black hole, how does the mass of a black hole increase to such huge numbers as is found in the centers of galaxies? Surely these supermassive black holes weren't entirely formed at once from some huge star? Does the mass that it gains from in-falling matter just sort of accumulate near the event horizon, and won't this effect start to change the dynamics of the problem dramatically?

Another related problem is how do mergers work if black holes in proximity with each-other should in theory take infinitely long to merge from our perspective? Does the gravitational radiation they release somehow peter out once they get too close for time to pass at any reasonable rate for us? Or are we somehow sure that they really have merged in our frame, and somehow avoided this time singularity entirely?

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