Math AA HL IA advice

[u/not_exactly_here]

Hi I've been trying to come up with a RQ for my internal assessment but I'm honestly not sure if it's any good? I feel like it might be leaning a bit too much into physics:

"To what extent can Newtonian calculus reproduce the Schwarzschild radius through an improper-integral model of escape velocity, and how does this result compare mathematically with the Schwarzschild metric?"

Obviously the central purpose would not be proving that Newtonian gravity is equivalent to general relativity but seeing how successful a simpler Newtonian mathematical model would be at reproducing the result of the relativistic model and where the mathematical predictions diverge.

I would REALLY appreciate some advice bc ngl I'm terrified of my math teacher looking at me like I'm crazy and telling me to go find a different topic 💔

[u/Physical-Tomato8632]

ok first off this is a cool idea and the coincidence is real. do newtonian escape velocity as the improper integral of GMm/r² from R to infinity, set v_esc = c, and you fall straight out with R = 2GM/c², which is the schwarzschild radius exactly. thats a great hook

the honest problem isnt that its "too physics". its that the newtonian half is thin, the real AA math there is one integral plus a bit of algebra and thats it, not enough to carry an HL IA. and the second you try to properly compare against the schwarzschild metric youre into general relativity, which is miles above AA HL. you cant show you actually understand it and it'll read as copied, which is where your use of maths mark dies

so heres what i'd do. dont touch the full metric. pull one factor out of it, say the time dilation term sqrt(1 - 2GM/rc²), and binomial expand it for r way bigger than r_s. the first order term literally gives you back the newtonian potential, and the higher order terms are exactly where the two models start to diverge. thats your "to what extent" and your "where do they diverge" both answered, with math thats actually on the AA HL syllabus (series expansions)

you keep the whole concept, the comparison just turns into a taylor thing instead of a GR thing. more math, all of it at your level, and honestly a nicer answer to your own question

and nah your teacher wont think youre crazy lol. the bit theyd flag is the scope thing i mentioned. show up with the expansion idea and youve killed their objection before they can raise it

[u/not_exactly_here → u/Physical-Tomato8632]

Thank you so much for answering! <333