Intro
You've watched Interstellar right? Go watch it right now. Anyways. Fun fact of the day: Did you know that you would have to shrink our sun to 6 km across for it to become a black hole? Okay that's fun and games but our guy Cooper who's headed to his death is crying atm.
Situation
Our man Cooper turns on his engine, while having an angle that would not be dangerous if he had a working engine but the problem is that his engine fails, and his coast trajectory will bring him close to the black hole, will he survive? He sends us a message panicking, and asking us to predict whether he'll get the succ or not, because he would like to say goodbye to his family before becoming spaghetti bolognese.
The blackhole has 2M as a radius, and our orbit is at 20M, we have an angle of 167 deg relative to the x axis. Again, our main focus is just to find out if the man will make it.
Principles
Gravitational potential:
I love this demonstration!! In a few minutes you'll have intuition for it. Consider this diagram:
What happens to ball 1? The plane is frictionless and there are no other forces other than the force of gravity. Well you can imagine that it is at height h1 and it will fall down and then make it up at height h1 again but in a different position and it will oscillate back and forth forever.
Let's now consider a another ball, ball 2, that came already with speed (kinetic energy) before it goes down, what will happen to it? Well it will go up higher than block 1, no prizes for guessing that. How about after it reaches the maximum point?
It will get yeeted back and it won't be trapped in the "well", as in it won't oscillate back and forth forever like block 1 .
I hope potential graphs make much more sense now! If they don't, you're probably the reason shampoo bottles have instructions, jk jk yk I <3 u.
But for a black whole it's difference as is everything, newtonian mechanics kind of breakdown, in fact, the general relativity version is the correct one, and newtonian version is simply an extremely good approximation for most of our cases (mercury is an exception). Back to the potential graph, we have a small caveat, if your energy is larger than a specific value, you'll get the ultimate succc.
Whether you get the succ is dependent on the extrema point of the potential (or \(E_{crit}/m\) , this criterion depends on the expression for the potential. And the expression for the potential has distance r as a variable, and is dependent on spin per mass (L/m), Size of black whole as constants. Another quantity is of course E/m of the object we're concerned with
The conserved quantity and energy:
We use also the conserved quantities and line elements to obtain expressions that we otherwise won't be able to. In anddition, we use the fact that in very short time intervals we can consider the t-shell and tau to have a lorentzian relationship (for them to have a local inertial frame) and thus we can use the Lorentz transform which helps a bunch. Think of it like velocity and position and how for small time intervals we can do this
Instead of integrating that is! We've seen from the twin paradox that we can look at infinite amount of inertial reference frames, that's sorta the same concept here
What happens if you get the succc in the black hole?
Spaghetti. Yes spaghetti. What will happen is actually quite simple, we look at the diagram, you wonder what's in it then I explain to you, pedagogy babyyyy
.png)
AS you can see (the force arrows are meant to be to scale), the ultimate succc strikes hardest at the bottom (feet), and a bit weaker at the top (head) so what happens is that he's stretched (fancy way of describing gravitational garden), however he also gets squished from the sides. The right side of the body will be pulled to the left, and the left side of the body will be pulled to the right, horizontally compressing the person. This begs the question, what if you fall sideways? Do you become thiccc and short? Like a ball? God I love physics.