I have encountered a difficulty which arises from my attempt to combine non collinear Lorentz transformations in analyzing the following problem:(adsbygoogle = window.adsbygoogle || []).push({});

A thin rod is cut from a metal plate leaving a slot of exactly the same size. The two are separated and set in motion thus: the rod lies along x and moves in the x direction with velocity Vx . The plate is oriented along x but at position -y, is set in motion along y direction with velocity Vy. The center of the slot and the center of the rod will coincide at time tc. Will the rod pass through the slot? Consider the point of view of an observer on the ground G, and one on the rod R, and one on the plate P.

The observers P and G both agree the rod should pass as it is Lorentz contracted so is smaller than the slot but the observer R will say the slot is contracted so how can the rod pass?

To resolve the apparent paradox I attempted to relate the space time coordinates of events in each frame as follows:

Coords of an event in G = (t, X, Y) = { Xi } .

Coords of same event in R = {Xir} = L(Vx). {Xi} ie a Lorentz boost in the direction of x by the rod’s speed. In this frame the rod is stationary and the plate moves with velocity components –Vx and Vy’= Vy . √( 1 – (Vx/c)^{2}), so the correct relative velocity of the plate and rod are

Vr = Vx^{2}+ Vy^{2}– (Vx^{2 })( Vy^{2})/c^{2 }and the angle this makes to x axis is tan Θr = Vy’/Vx

Now comes the problem!To get the coordinates of the same event in frame P , so all the motion is carried by the rod, I could either perform the transformation from the rod’s frame by the relative velocity vector Vr relating the rod and the plate so

Coords of event in P = L (Vr) . {Xir} = L(Vr). L(Vx). {Xi}

Alternativley I could go from frame G by transforming along y by L(Vy) to give

Coords of event in P = L(Vy) . {Xi}

However these two methods give different results! Despite both methods leaving the plate stationary and the rod moving- so apparently the same physical situation.

i.e. L(Vr). L(Vx) is not equal to L(Vy). The second method gives the same magnitude to the relative velocity between rod and plate but a different angle of this velocity to the x axis (call it Θp) such that

tan Θp = tan Θr /√( 1- (Vr/c)^{2})

So which of these two methods gives the correct relation of the coordinates of an event in frame P to that in Frame R and to G?

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# Problem analyzing rod-slot 2-D Lorentz contraction paradox

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