Physics makes it impossible even with a superhuman hero and matching fishing line composed of perfect carbon nanotube (CNT).
At geosynchronous orbit (GEO), our hero is moving at 3.07 km/s To make a perfect straight down cast our hero would have to actually drop its orbital velocity to zero horizontally plus whatever vertical component he can manage. Momentum conservation just changed our hero's orbit, he is no longer in GEO as required for an orbital fishing platform. Because even the perfect CNT line 35800 km long has a significant mass, the orbital change will be correspondingly large.
Now our hero is also smart and realizes he needs to compensate for the orbital change and casts a second ballast line in the opposite direction the same time to stay in GEO.
When the line finally hit the atmosphere, the atmosphere will be traveling at about 465 m/s (equatorial velocity) -- it just became impossible to hit any target. Wind drag will quickly accelerate the bait end of the line to 465 m/s. From this time on, the line is also pulling our hero out of GEO.
Even worse, the ballast line is running out of line at about the same time. It just became a uncontrolled space tether that will also pull him out of GEO. The ballast line will also rebound and jerk our hero out of GEO. Cutting the ballast line just before rebound eliminates this, so our hero needs very keen eyesight or impeccable timing to cut the ballast line. However, without the ballast line we also get pulled out of GEO, just as if he never cast the ballast line in the first place. Definite problem for GEO in either case.
Meanwhile, back in the upper atmosphere our bait is now falling at terminal velocity which is dead slow compared to GEO (actually slower due to the increasing tension in the line). This means the fishing line will spend a lot of time pulling him out of GEO, and our heroes orbit will be pretty far out of GEO by then. Too hard to guess what terminal velocity will be, but the terminal velocity of the CNT line itself would be very low. I would expect the descent time spent in atmosphere would be hours long, if not days long (like dropping a feather).
Somehow, defying all odds, the hook is set. Since you cannot feel the other end of the line, you know this because of your keen eyesight (0.14 seconds later due to the speed of light) and you start to reel in your catch.
So close, but you discover that pulling on the line does not result in a pull at the other end. Turns out that the speed of sound is a limiting factor, the line pull cannot propagate faster than the speed of sound -- perhaps 20km/s. So, setting the hook takes 1790 seconds or 29.83 minutes, your fish is long gone before your jerk on the line reaches the other end.
No fish for you.
Thanks to Hagen von Eitzen for the quick explanation of speed of sound dependency for setting the hook. Perhaps the following explanation will help too. If you stroke a hammer on a railroad line, the speed a sound in steel is how fast the sound propagates through the rail line. But what is sound? It is a compression wave. I.e., the speed of sound in a material is how fast a compression or tension wave propagates through the material.
To be honest, I had forgotten to reflect that the speed of sound in a solid changes with temperature and pressure, but I don't know how to compute the net effect, as I don't know how the density and modulus of elasticity of a CNT will change in the cold vacuum of space. For comparison diamond and quartz have much lower density changes due to temperature than do common metals. So I expect a similar low change in a CNT. So, I expect the overall change is small, even given the extreme cold in space. Perhaps the time required to set the hook is off by as much as a minute or two.
Also admission of dumb mistake. Originally used 42,000 km line, should have used 35,800 km line as you only have the reach the surface of the earth, not the center of the earth - I adjusted my answer.
I wanted to know just how heavy the fishing line would be. On the space elevator blog there is a space elevator spreadsheet. Appears thorough, did not try to check any of the formulas, etc.
If you want to double the end load, double the total weight of the fishing line/ space elevator. 130 GPa is very optimistic for a real world CNT (I did say perfect in my answer), a real measured CNT maxes at at 63 GPa . A safety factor of 2 is really not enough in my opinion for a space elevator (crashing to earth after a break would be very bad), but it is probably ok for a fishing line. A safety factor of 2 means that the is designed to be twice as strong everywhere as the theoretical minimum value. A safety factor of 1.0 means theoretical minimum everywhere (no safety factor)
- Strength: 130 GPa, Safety Factor: 1.0, Load: 10kg, Line mass 147 kg
- Strength: 130 GPa, Safety Factor: 1.5, Load: 10kg, Line mass 271 kg
- Strength: 130 GPa, Safety Factor: 2, Load: 10kg, Line mass 444 kg
- Strength: 130 GPa, Safety Factor: 2.5, Load: 10kg, Line mass 683 kg
- Strength: 130 GPa, Safety Factor: 3., Load: 10kg, Line mass 1010 kg
Or more realistically
- Strength: 63 GPa, Safety Factor: 1.0, Load: 10kg, Line mass 470 kg
- Strength: 63 GPa, Safety Factor: 1.5, Load: 10kg, Line mass 1084 kg
- Strength: 63 GPa, Safety Factor: 2, Load: 10kg, Line mass 2228 kg
- Strength: 63 GPa, Safety Factor: 2.5, Load: 10kg, Line mass 4300 kg
- Strength: 63 GPa, Safety Factor: 3., Load: 10kg, Line mass 7985 kg
Also in the real world you have other problems that I have neglected.
On entering the atmosphere your line would burst into flames and disappear
I've neglected to mention that your platform continues in GEO while your line is descending. You need to throw it very fast indeed to reduce this problem
Casting a fishing line works because the plumb is much heavier than the line so the line follows the plumb. We have something that cannot actually be cast
The line is an incredible safety hazard. The end diameter is so small, you would not be able to see it and would probably slice off body parts trying to attach the hook
Who fails to pack lunch for space but brings along a tiny space elevator just in case?