Bar and ring paradox Ladder paradox

the diagram on left illustrates bar , ring in rest frame of ring @ instant centers coincide. bar lorentz-contracted , moving upward , right while ring stationary , uncontracted. diagram on right illustrates situation @ same instant, in rest frame of bar. ring lorentz-contracted , rotated respect bar, , bar uncontracted. again, ring passes on bar without touching it.

a problem similar simpler rod , grate paradox, involving inertial frames, bar , ring paradox (ferraro 2007). rod , grate paradox complicated: involves non-inertial frames of reference since @ 1 moment man walking horizontally, , moment later falling downward; , involves physical deformation of man (or segmented rod), since rod bent in 1 frame of reference , straight in another. these aspects of problem introduce complications involving stiffness of rod tends obscure real nature of paradox . bar , ring paradox free of these complications: bar, larger in length diameter of ring, moving upward , right long axis horizontal, while ring stationary , plane of ring horizontal. if motion of bar such center of bar coincides center of ring @ point in time, bar lorentz-contracted due forward component of motion, , pass through ring. paradox occurs when problem considered in rest frame of bar. ring moving downward , left, , lorentz-contracted along horizontal length, while bar not contracted @ all. how can bar pass through ring?

the resolution of paradox again lies in relativity of simultaneity (ferraro 2007). length of physical object defined distance between 2 simultaneous events occurring @ each end of body, , since simultaneity relative, length. variability in length lorentz contraction. similarly, physical angle defined angle formed 3 simultaneous events, , angle relative quantity. in above paradox, although rod , plane of ring parallel in rest frame of ring, not parallel in rest frame of rod. uncontracted rod passes through lorentz-contracted ring because plane of ring rotated relative rod amount sufficient let rod pass through.

in mathematical terms, lorentz transformation can separated product of spatial rotation , proper lorentz transformation involves no spatial rotation. mathematical resolution of bar , ring paradox based on fact product of 2 proper lorentz transformations may produce lorentz transformation not proper, rather includes spatial rotation component.