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Acceleration in a new Coordinate System - General Relativity
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[QUOTE="kent davidge, post: 5598954, member: 563342"] [h2]Homework Statement [/h2] Below: Jac = Jacobian matrix; ξ = d/dφ for some continuous parameter φ which labels different points on the worldline. (I'm sorry for my poor English.) Consider a new coordinate system x[SUP]µ'[/SUP] which differs from the original Cartesian coordinate system x[SUP]µ[/SUP]; the Cartesian coordinates x[SUP]µ[/SUP] can be written as a function of these new coordinates x[SUP]µ[/SUP] = x[SUP]µ[/SUP] (x[SUP]µ'[/SUP]). Show that the equation of motion can be written in these new x[SUP]µ'[/SUP] coordinates as ξ²x[SUP]µ'[/SUP] + Γ[SUP]µ'[/SUP][SUB]ν'[/SUB][SUB]λ'[/SUB]ξx[SUP]ν'[/SUP]ξx[SUP]λ'[/SUP] = 0 for some Γ[SUP]µ[/SUP]'[SUB]ν'[/SUB][SUB]λ'[/SUB] which you must compute; Γ[SUP]µ'[/SUP][SUB]ν'[/SUB][SUB]λ'[/SUB] is known as the Christoffel symbol. These extra Christoffel terms in the equation of motion can be thought of as ”fictitous forces” that arise in an accelerated reference frame. [h2]Homework Equations[/h2] x[SUP]µ'[/SUP] = Jac * x[SUP]µ[/SUP] [h2]The Attempt at a Solution[/h2] [/B] I've tried to solve this by deriving x[SUP]µ'[/SUP] with respect to φ using the chain rule, but that did not work. Unfortunately I could not put my work here, but it was just what I said: I've used the chain rule to derive x[SUP]µ'[/SUP] with respect to φ. [/QUOTE]
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