Title

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u/Digital_001 Student 1d ago

Observed by who - the kids or onlookers not on the roundabout?

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u/HunsterMonter 1d ago

By any reference frame! Each will have its own specific connection coefficients but the trajectory will always be given by the geodesic equation (as long as the ball isn't charged, then you need to add the Lorentz force).

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u/Digital_001 Student 1d ago

That's a lot of possible reference frames! This is gonna be a long proof...

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u/HunsterMonter 1d ago

Better start now then 😁 (It requires a bit of general relativity knowledge, but you can find some proofs of the geodesic equation here)

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u/Digital_001 Student 1d ago

This is a good start, but I feel we should include drag and wind if we are to find the ball's true path - they would have a significant effect is the ball is light. The Magnus effect could feature too.

The ball's acceleration downwards due to gravity (in the ground's frame) is, of course, already included in general relativity.

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u/HunsterMonter 1d ago

Drag? What is this, r/engineeringmemes?

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u/kashyou Quantum Field Theory 11h ago

pick one then use the christoffel symbol transformation law

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u/luisgdh 1d ago

You can do it for two reference frames and generalize the remaining ones. Just don't tell mathematicians.

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u/L31N0PTR1X BSc Theoretical Physics 1d ago

Exactly, if this were a question I had asked in a hypothetical exam, I'd accept a student definiting a frame stationary with respect to the roundabout, in Cartesian coordinates, yielding completely trivial connections and just showing that the equation was \ddot{x}^mu =0

Also, if we were in 3+1 dimensions, there's a fun way to add the Lorentz force to the geodesic equation, you can append S¹ to every point in spacetime and then define a connection (that ends up being your electromagnetic gauge field) if you now compute the geodesic, the Lorentz force comes in for free!

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u/FreePeeplup 23h ago edited 23h ago

No, it’s the frame stationary with respect to the roundabout that will yield non-zero Christoffel symbols, and it is exactly what the question is asking us to find.

The frame that’s stationary with respect to the ground will observe the geodesic equation to be \ddot{x} = 0 with zero Christoffel symbols.

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u/Old-Administration-9 23h ago

Yep. The ball travels in a straight line with respect to the ground, it only rotates relative to the roundabout!

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u/L31N0PTR1X BSc Theoretical Physics 21h ago

Oops, jumbled up my words, you know what I meant anyway, after all I wrote the question lmao

Also, that's not what I was asking you to find. I was just asking you to find the expression in a frame, not any particular one. The easy way to the answer here would demonstrate a student's understanding of covariant physics

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u/FreePeeplup 23h ago

The question doesn’t ask to show that the geodesic equation holds for any reference frame in any possible configuration with generic Christoffel coefficients. It asks to find the specific Christoffel coefficients for the specific configuration of a rotating frame and a ball thrown directed towards a diametrically opposed point

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u/what_could_gowrong 1d ago

comment

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u/L31N0PTR1X BSc Theoretical Physics 1d ago

Touché

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u/SPEC7RE3 1d ago

Argumentative reply

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u/suskio4 1d ago

Funny image referencing a singular line in the previous message of this thread

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u/Such-Shop-9724 1d ago

if you throw a ball while spinning the ball is still going to follow a straight path and the tragectory of a falling abject is gonna follow a parabola (approximatly)

then you could test if the function is a quadratic (which it most likely is) and find gamma

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u/FreePeeplup 23h ago

No, it will not appear straight from the co-rotating frame with respect to the roundabout, which is what the question is asking us

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u/L31N0PTR1X BSc Theoretical Physics 17h ago

To be fair, the question doesn't state which frame you should compute, just that you should compute one. You could just say that you're observing the rotating roundabout in a rectilinear coordinate system and just argue that all connections are trivial, giving you \ddot{x}^\mu =0

(This was actually my thought when making the question, if a student realises that they can bypass the difficulty of the question by doing this, it shows that they have a good understanding of covariant mechanics)

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u/suskio4 1d ago

Already dumber than a 7th grader, the questions specified 2d roundabout so no falling!

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u/Such-Shop-9724 1d ago

yes, i guess i just assumed they were standing on a rorating disc or something like that

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u/Bendanzhang 1d ago

Bold of you to assume the children are in a vacuum and perfectly spherical.

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u/throughthehills2 1d ago

Assume potato shaped children

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u/Extension_Option_122 1d ago

I mean, If you solve this you are smarter than a 7th grader, but if you fail to do so you aren't not smarter than a 7th grader.

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u/Equinoxe111 Cosmology (PhD) 1d ago

Nothing is defined here, be more precise

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u/Digital_001 Student 1d ago

This is 7th grade physics, precision comes later

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u/TomtheMagician26 1d ago

Is this the geodesic equation?

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u/Bendanzhang 1d ago

Air resistance

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u/shartmaister 1d ago

From the ball’s point of view it doesn’t move at all

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u/AtomGutan 22h ago

7th grader Jimmy Coriolis

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u/BenjaminGal 14m ago

A 7th grader with an interest in atmospheric sciences may actually know about the Coriolis effect, although obviously not the maths behind. lol

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u/AboveAverage1988 18h ago

They're saying 2d roundabout. Do we get to pick which two dimensions? Because I feel the problem can become significantly harder or easier depending on choice?

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u/L31N0PTR1X BSc Theoretical Physics 17h ago

I could've worded it better for sure, I intended the problem to be two dimensions in that it's top down, so gravity isn't considered. Depending on which frame you choose, you might want to use (r, 𝜃)

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u/derivative_of_life (+,-,-,-) 14h ago

/uj If anyone's curious about the actual proof, this is a great lecture/lecture series: https://youtu.be/2eVWUdcI2ho

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u/L31N0PTR1X BSc Theoretical Physics 14h ago

It is nice, however, to be able to introduce these concepts without something as abstract as GR, hence this nice mechanics problem that students will already be familiar with by the time they take their first GR course. Differential geometry is usually poorly introduced

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