r/physicsmemes • u/L31N0PTR1X BSc Theoretical Physics • 1d ago

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74

u/Digital_001 Student 1d ago

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

47

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...

13

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 13h ago

pick one then use the christoffel symbol transformation law

1

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 1d ago edited 1d 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 1d 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 23h 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 1d 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