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.

Previous post:

**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*-axes are aligned with the

_{B}*constant*vector

*of the frames’ relative velocity. The*

**ν***y*-axes are perpendicular to the

*x*-axes. I.e. the

*y*-axes is parallel to the

_{A}*y*-axis.

_{B}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*– always being aware of the fact that this value was defined in a special frame, namely frame

_{y}**A**. This is important as the vertical velocity of the rod will have different values in other reference frames.

We start watching at a time *t _{A}* =

*t*when the rod just passes a height

_{start,A}*y*=

_{A}*h*in frame

**A**.

The rod shall reach the base level *y _{A}* = 0 at a time

*t*= 0. All initial conditions of our scenario are therefore given at a point in time

_{A}*t*= –

_{A}*h/v*in frame

_{y}**A**:

**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*= 0. We indicate quantities referring to the

_{A}*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:

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*= 0. We have prepared clocks in frame

_{A}**B**such that this event also has a time coordinate of

*t*= 0 there. We indicate left ends by an additional sub-index “

_{B}*l*“, right ends by an additional sub-index “

*r*“:

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*in

_{start,A}**A**:

**Illustration 2:** The rod, the plate with the slot and frame B at *t _{A}* =

*t*= 0 as seen from frame

_{B}**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*= 0. In frame

_{A}**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.

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*is:

_{A}*c* is the velocity of light in vacuum. This means:

**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 *:

We saw that the slot’s length in **A** at *t _{A}* = 0 is measured to be:

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

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 …