Special relativity and the rod/slot paradox – II – setup of a collision scenario

This post series is a about a paradox based on the length contraction effect in Einstein’s Special Theory of Relativity [STR]. In the previous first post I have presented a simple version of the so called “rod/slot [RS] paradox“. In this 2nd post I will define a setup for a “collision scenario” between a rod (moving downward in vertical direction) and a plate with a slot moving horizontally at a level y = 0. The details of the setup will provide the basis for a later deep dive into the RS-paradox – guided by the Lorentz transformation and respective mathematics.

Post I: Special relativity and the rod/slot paradox – I – seeming contradictions between reference frames

Two reference frames and rod/slot movements for a collision

The graphics below shows (x,y)-coordinate systems of two reference frames, frame A and frame B, which move with a constant horizontal velocity ν_x against each other. The origins of the coordinate systems are located on the horizontal line of relative motion of the frames against each other. We will in general indicate in which frame values of certain quantities have been measured by respective subscripts A and B.

The coordinate systems are oriented such that both the x_A-axis and the x_B-axes are aligned with the constant vector ν of the frames’ relative velocity. The y-axes are perpendicular to the x-axes. I.e. the y_A-axes is parallel to the y_B-axis.

The vector of the frames’ relative velocity has a component in x-direction, only: Seen from frame A, frame B moves horizontally with a negative constant velocity –ν_x. There is no relative vertical motion between the frames A and B.

Illustration 1: Vertically moving rod and horizontally moving slot seen from frame A at time t_A = –h/v_y

Vertically moving rod in frame A

In frame A a long thin cylindrical rod is positioned such that its the rotational symmetry axis is parallel to the x_A-axis. The extensions of the rod in y-direction (as well as in z- direction) shall be infinitesimally small. We indicate quantities referring to the rod by a sub-script “r“.

The rod moves vertically downwards in frame A with a constant velocity of v_r,y,A in (negative) y-direction. We will name the absolute value of this velocity in the future v_y – always being aware of the fact that this value was defined in a special frame, namely frame A. This is important as the vertical velocity of the rod will have different values in other reference frames.

\[ {\normalsize \text{ Vertical velocity of the rod in A: }} \quad v_{r,y,A} = – v_y \,\,, \quad |v_{r,y,A}| =: v_y \, .\]

We start watching at a time t_A = t_start,A when the rod just passes a height y_A = h in frame A.

\[ t_{start, A} : \quad y_{r,A} = h \, . \]

The rod shall reach the base level y_A = 0 at a time t_A = 0. All initial conditions of our scenario are therefore given at a point in time t_A = – h/v_y in frame A:

\[ t_{start, A} = – h / v_y \, . \]

Horizontally oriented plate with slot at rest in frame B

In frame B we have a very thin impenetrable plate with a slot in it – oriented in parallel to the x_B-axis and at level y_B = y_A = 0. We indicate quantities referring to the slot by a sub-script “s“.

The slot’s left end is located at x_B = 0, i.e. it coincides with the origin of B‘s coordinate system:

\[ x_{s,B}(t_B) = 0\,, \quad \forall \, t_B \, . \]

The plate and the slot do not move in frame B. Seen from frame A, however, the plate and the slot are moving horizontally in negative x-direction with a speed of –ν_x versus frame A‘s origin.

Below the plate is nothing. So, if the rod could pass the slot’s opening under certain conditions, the rod would just move on in negative y-direction.

Timing conditions

We define the timing of the slot’s movement relative to A such that the left end of the slot (and thus also the origin of frame B) will coincide with the left end of the rod in frame A at x_A = 0 at t_A = 0. We have prepared clocks in frame B such that this event also has a time coordinate of t_B = 0 there. We indicate left ends by an additional sub-index “l “, right ends by an additional sub-index “r“:

\[ \begin{align} \pmb{\operatorname{Event \, E_C :}}\quad &t_B = t_A = 0\,, \quad x_{r,l,A} = x_{s,l,A} = 0\,, \\[10pt] &y_{r,l,A} = y_{s,l,A} = 0 \, , \quad y_{r,r,A} = y_{s,r,A} = 0 \,, \\[10pt] &\normalsize \it \text{ The origins of frames A and B coincide. } \\[8pt] &\normalsize \it \text{ Left end of slot coincides with left end of the rod. } \end{align}\]

This imposes a condition on the distance d_s,A of the left end of the slot from the origin of A as seen from an observer at t_A = t_start,A in A:

\[ \begin{align} \pmb{\operatorname{ A:\,}}\quad &d_{s,A}\left(t_{start,A}\right) \, = \, \nu_x * h / v_{y} \, \\[8pt] & d_{s,A}\left(t_A\right) \, = \, \nu_x * h / v_{y} – \nu_x \, \left(t_A – t_{start,A} \right) \, , \\[8pt] & d_{s,A}\left(t_A=0\right) \, = \, 0\,. \end{align}\]

Illustration 2: The rod, the plate with the slot and frame B at t_A = t_B = 0 as seen from frame A

Orientation of the rod and the slot

Please note that the orientation of both the rod and the slot in the illustration refer to frame A, only! The drawing does not indicate anything regarding the orientation of the rod relative to the slot in frame B. More precisely:

Frame A: Seen and measured from an observer located at a fixed point in frame A, the slot is always oriented horizontally in parallel to the rod and parallel to the x_A-axis in the collision scenario. In frame A both of the slot’s ends always have a y-coordinate y_A = 0. In frame A both ends of the rod always reside at the same y-coordinate during the fall of the rod.

These appear to be trivial statements. But we will later see that these point of our setup are really worth to be emphasized. They mark important initial conditions.

Proper lengths of rod and slot / length contraction of the slot

Extensions of the rod and the slot vertical to the x-axis are basically irrelevant in so far as they would not hinder a passage of the rod through the slot. We simply ignore them. Our thin, but lengthy rod has a proper length L. The slot in the plate also has a proper length L plus an infinitesimally small extend ε to ensure that the rod can move through.

\[ \begin{eqnarray} {\normalsize \it \text{Proper length of slot:}}& \quad &L_{slot} &= L \,\,\, (+\epsilon) \\ {\normalsize \it \text{Proper length of rod:}}& \quad &L_{rod} &= L \end{eqnarray}\]

We ignore the infinitesimal excess of the slot in x-direction in forthcoming calculations.

We know from a basic course in special relativity [SR] that the length contraction effect will give us a diminished slot length in A (compared to what observers would measure in frame B, where the slot is at rest). Remember that the proper length of an object is measured with the object being at rest in some frame. Remember also that a length measurement process is defined such that the length of an object should be determined by observers measuring the spatial coordinates of the object’s ends at the same time in a given frame. This condition is essential for the length contraction.

Due to the length contraction effect we will find that the length of the slot L_slot,A as seen in frame A at any point in time t_A is:

\[ \begin{align} {\normalsize \it \text{Length of slot in }}\, \pmb{A} \,\, : \quad &L_{slot,A} \,=\, {1 \over \gamma} * L\,, \\\mbox{with} \quad &\gamma = \gamma_x = {1 \over \sqrt{ 1 – { \nu_x² / c²}\, }} \gt 1 \end{align} \]

c is the velocity of light in vacuum. This means:

\[ \begin{align} \pmb{\operatorname{Event \, E_{Coll} :}}\quad &t_{coll} = t_B = t_A = 0\: \\[10pt] & x_{r,l,A} = x_{s,l,A} = 0\,, \quad x_{r,r,A} = L\,, \quad x_{s,r,A} = {1 \over \gamma_x } \, L \,, \\[10pt] &y_{r,l,A} = y_{s,l,A} = 0 \, , \quad y_{r,r,A} = y_{s,r,A} = 0 \,. \end{align}\]

Diagonal movement of the rod versus the slot in frame A

Seen from the slot, the rod approaches diagonally with two constant components of its velocity. We will later need to apply the Lorentz transformation to determine the orthogonal component of the relative velocity between the rod and the slot objects in frame A and frame B.

Additional Remarks regarding the scenario

No forces or accelerations are at work in our scenario.
The x-coordinate axes neither of frame A nor of frame B are oriented along the line of relative movement between the left end of the rod and the left end of the slot.
According to the Lorentz-transformation y-coordinates of events remain constant between frames moving relative to each other along the x-axis.
Seen from the slot the rod appears to approach at some diagonal line. To determine the values of the objects’ velocity components in x- and y-direction in frame B, we will have to apply the Lorentz-transformation to the defined object velocities in frame A.

A predetermined crash of the rod at the slot’s right border

From illustrations 1 and 2 you can derive a predetermined development in our scenario: In frame A the conditions are defined such that the rod has no chance to pass the slot vertically without an accident and touching the plate. In A both the right and the left end of the rod will reach the plate and the slot at the same time t_coll= t_A = 0 :

\[ \pmb{\operatorname{A} :} \: { \normalsize \it \mbox{rod reaches slot and plate : } } \, \, t_A = t_{coll} = t_{start} + h / v_{y} = 0 \,. \]

We saw that the slot’s length in A at t_A = 0 is measured to be:

\[ L_{slot, A} = 1 / \gamma * L_{slot} = 1 / \gamma * L \, \lt \, L \,. \]

Hence, in A the rod’s length L will be to big for the slot. We therefore expect the rod to crash at our plate at a an x_A-coordinate

\[ x_{crash, A} = 1 / \gamma * L \, \lt \, L, \quad \mbox{with: } \, L – {1 \over \gamma} \, L \,= \, {\gamma – 1 \over \gamma } \, L \,. \]

Note that the length contraction does not describe an apparent length. It gives us the real length of the moving slot measured in A.

The paradox seems to be that length contraction in frame B will shorten the slot by a factor \( 1 / \gamma \lt 1\). Therefore in frame B one is tempted to assume that the rod falls through the slot there. We will see in the next posts that this assumption is false.

Conclusion

We have set up a scenario with a rod moving vertically downward in a special frame A. In this frame we also considered a horizontally moving plate, with a slot, along a zero y-level. The timing was such that the rod’s left end would meet the slot’s left end when at it reaches y_A = 0. Rod and slot have the same proper length. Therefore, due to the length contraction effect, we expect a collision of the rod’s right outer part with the plate.

In the next post

Special relativity and the rod/slot paradox – III – Lorentz transformation causes inclination angles

we will apply the Lorentz transformation to the coordinates of certain key events occurring in our collision scenario. This will show that in a frame B (attached to the slot) the rod will not be oriented in parallel to the slot.

Stay tuned …