Indeed, this is how
&
could tell they are living on a curved surface.
In Physics & Mathematics, this way of moving a vector along a curved surface is known as 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍 𝒕𝒓𝒂𝒏𝒔𝒑𝒐𝒓𝒕.
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Suppose&Start at point A.
walks A→C pointing North the whole time
walks A→B→C pointing North the whole time
When they both arrive at the North pole they will be pointing different directions.pic.twitter.com/UIO7iIDTTU
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A nice demonstration of parallel transport. Note
: The animation keeps the vector facing East (as opposed to North, in my previous example)
source:https://www.naturelovesmath.com/en/mathematical-physics/general-relativity-loud-and-clear/ …pic.twitter.com/ZIsGMQEUo8Show this thread
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There’s a tool called holonomy (I think?), and it’s a measure of the difference between
and’s directions
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Nice to consider also besides the Riemann tensor is the "Geodesic Deviation Equation" describing the tendency of objects to approach or recede from one another while moving under the influence of a spatially varying gravitational field.
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"Geodesic deviation eqn" not to be confused with the "geodesic eqn" ;-) Note: Also nice to consider "Schild's ladder": https://en.m.wikipedia.org/wiki/Schild%27s_ladder … (Apologies for using Wikipedia, but during the last years the quality of Wikipedia content has been improved, see also: MTW).
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This strikes me like it could be an illustration of why momentum is not conserved if space-time is curved (i.e. there is a gravitational field/acceletation in that region in space-time)
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parallel translation is path dependent
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Both will be pointing to the South Pole! :P
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