In the first posts of this series we have set up a collision scenario to study the the **rod/slot paradox** in the *Special Theory of Relativity* [**STR**]: A rod falls vertically in a reference frame **A** and eventually meets a horizontally moving plate with a slot. The slot is at rest in a frame **B**. In this 4th post look at details of the rod’s movement towards the slot in frame **B**. The already derived inclination angle of the rod against the slot and the rod’s diagonal movement ensure a collision in **B**. Despite the fact the rod and slot have the same proper length. We will also find an interesting sequence of *x _{B}*-coordinates for key points on the rod and respective events.

Previous posts:

- Special relativity and the rod/slot paradox – I – definition of reference frames
- Special relativity and the rod/slot paradox – II – inclination angle of the rod in the slot frame
- Special relativity and the rod/slot paradox – III – Lorentz transformation causes inclination angles

Just a hint to avoid misunderstandings: We could, of course, have chosen other initial conditions such that the rod would move through the slot *without* any collision. Such a “*collision-free scenario*” will be the topic of later posts in this series.

## Previous results of the Lorentz transformation for events in our “collision scenario”

For a summary of settings for our present “collision scenario”, basic coordinates, quantities and respective notations see the previous posts and illustrations 1 and 2. The schematic drawing below depicts the situation of the rod touching the slot at *t _{A}* = 0 and

*y*= 0, after a phase of the rod’s vertical downward movement in

_{A}**A**with a velocity –

*v*.

_{y}**Illustration 1:** Vertically falling rod in a horizontally co-moving frame **A** at *t _{A}* = 0 touching the line

*y*=

_{A}*0*

Initial conditions were such that the left end of the rod after having moved from a height *y _{A}* =

*h*to

*y*=

_{A}*0*also touches the

*left*end of a slot. The slot moves with horizontal velocity –

*ν*

_{x}at

*y*=

_{A}*y*=

_{B}*0*along the

*x*-axis.

_{A}A major result of applying the **Lorentz transformation** in the previous 3rd post was that the rod as seen from frame **B** (where the plate with the slot is at rest) shows an inclination angle Ψ with the x-axis. This is depicted in the next drawing at *t _{B}* = 0:

**Illustration 2:** Inclination angle of the rod with the x-axis in frame A at* t _{A}* = 0

You also see that the vertical velocity of the rod in **B** is *v _{r,y,B}* = 1/

*γ*•

*v*. The overall movement of the rod vs. the slot is a diagonal one. For the time being we use the standard factors of STR:

_{y}## A detailed view at the expected collision in frame B

The results of the 3rd post have already indicated that the rod’s right end might crash against the plate to the right of the slot. We shall look at this now in more detail. I first repeat some results for the space and time coordinates of events **E1** and **E2** at which the rod’s left and right ends pass the line *y _{B}* =

*0*(=

*y*) in

_{A}**B**.

In our shorter event notation:

The differences in space and time in **B** are given by :

Obviously, **E2** and **E1** do not happen at the same time in B. The rod’ right end passes the line *y _{B}* = 0 later than the rod’s left end – and at a bigger

*x*-coordinate than

_{B}*L*!

In contrast to this we found that the time coordinate of an event **E2R** (related to the rod’s right end) transformed to *t _{B}* =

*0*in

**B**:

The *y _{A}*-coordinate of

**E2R**transforms into an equal

*y*coordinate :

_{B }–_{ }### Interpretation: Collision !

Note the important result for the right end of the rod when it passes passing the line *y _{B}* = 0 (event

**E2**). With

*x*meaning the right end of the slot in frame

_{s,r,B}**B**:

We have already checked this result in a previous post for consistency with velocity components in **B** and the respective time difference to move down to *y _{B}* = 0 and to the right on the

*x*axis.

_{B}–So, we have a clear indication now that the rod’s right end will **not** pass the slot’s right end *x _{s,r,B}* =

*L*in frame

**B**, but instead crash against the plate! Somewhat

*later*than

*t*= 0 (see below): The rod’s right end eventually hits the plate at the time of event

_{B}**E2**.

## Frame B: A nice sequence of lengths and events

The following illustration reveals some more details of the rod’s movement ahead of the collision as registered by an observer in frame **B**. We get a nice sequence of *x _{B}*-intervals for some key points on the rod and respective events :

**Illustration 3:** Rod as seen in frame **B** close to touchdown. The rod’s symmetry axis shows an inclination angle with the slot. Velocity vectors and coordinates for the rod’s right edge crossing line *y _{B}* = 0 clearly indicate a

**collision**.

We recognize our angle Ψ again, for which we had found the following equation in the previous 3rd post:

We understand that due to this inclination and the rod’s diagonal velocity, the rod’s right end must move longer to the right than the location of the slot’s right end to arrive at a *y*-level of *y _{B}* = 0.

The diagonally oriented velocity vector of the rod *v*_{rod}_{,B} has the components:

As we have already seen in a previous post the timing is such that the rod’s right end indeed moves horizontally from *x _{B}* =

*1/γ • L*to

*x*=

_{B}*γ • L*between the events

**E2R**and

**E2**:

This is exactly what we need!

## Movement of the special point on the rod which collides with the slot’s right end

From the setup of our “*collision scenario*” we already know in which part of the rod we expect the collision with the plate to happen: In frame **A** the *leftmost* collision point on the rod has the same *x _{A}*-coordinate as the right end of the (length contracted) slot:

We apply the Lorentz transformation to get a corresponding coordinates In frame **B**:

As expected! The respective point in time, at which the collision for his special point on the rod occurs, lies between *t _{B}*|

_{E2R}and

*t*|

_{B}_{E2}:

### Coordinates of the leftmost collision point in frame B at *t*_{B} = 0

_{B}

Just for fun and as a consistency check, let us derive the *x _{B}*-coordinate of the rod’s particular point, which collides with the plate at the slots right end,

*from the given velocity components*and

*t*.

_{B}^{coll}As we have *x _{B}^{coll}* and

*t*in

_{B}^{coll}**B**, we just can get the

*x*

_{B}-coordinate of the collision point on the rod relative to the rod’s left end by calculating the distance this point must have moved to the right since

**E1**, i.e. in the time interval

*Δt*=

^{coll}*t*–

_{B}^{coll}*t*:

_{B}|_{E1}From this we can easily get the height of this point at the time of event **E1** in frame **B**:

This is consistent with the time our particular point needs to move vertically down to *y _{A}* = 0 :

So, we have an event **E0C** with coordinates for the leftmost collision point of the rod at *t _{B}* = 0:

We can back-transform the event **E0C** for the rod’s leftmost collision point at *t _{B}* = 0 from frame

**B**to frame

**A**:

We get the expected x-coordinate. But we find a backwards shifted time coordinate and thus a different y-coordinate, which in turn fits the vertical movement with velocity *-v _{y} *in

**A**for reaching the

*y*-level

*y*= 0.

_{A}## Collision as a series of events in frame B

The collision unfolds in frame B as a series of events:

- Firstly, the left end of the rod crosses the line
*y*= 0 unhindered with diagonal velocity_{A}*v*._{rod,B} - Secondly, the point with
*x*-coordinate_{B}*1/γ*(at^{2}• L*t*= 0) reaches the slot’s right end at_{B}*x*= L at time_{B}*t*=_{B}*ν*_{x}/*c*^{2}*L*. - Thirdly (and somewhat theoretically) the rod’s right end may collide with reach the plate at
*x*=_{B}*γ • L*at time*t*=_{B}*γ*•*ν*_{x}/*c*^{2}*L.*

Hmm, wait a second. We have shown that a collision will occur in frame B, too. BUT:

If a special point **on** the rod touches the plate first due to the rod’s diagonal movement, would this not have direct consequences for the rod’s outer right part? A completely stiff rod (which is not realistic) may actually turn into an anti-clockwise rotation movement around the collision point – and the rod’s right end would never reach the *y*-level *y _{B}* = 0. In contradiction to our collision setup in frame A.

Following this line of thought goes beyond the intentions of this series. We may get lost in aspects of a relativistic definition of stiffness and the respective propagation of sound waves through the solid grid of the rod at the atomic level. Let us not go there and postpone this topic to other posts.

The concerned reader may instead have a second look at the collision setup in frame **A** – and allow for an infinitesimal inclination of the rod there, too. Then the collision at the slot’s right end would also introduce an anti-clockwise rotational movement of the rod around the collision point in **A**. I hope this helps a bit.

## The length of the rod in frame A

Just for the sake of completeness let us calculate the length of the rod in frame **B**. We use Pythagoras to get:

This gives us:

So, in the slot’s frame the total length of the rod is actually shorter than L. But due to its inclination and diagonal movement it nevertheless collides with the plate.

## Conclusion

In this post we have shown the following: The Lorentz transformation for events in our rod-slot-*“collision scenario”* leads to a consistent prediction of a collision in **both** (!) our selected frames

- in frame
**A**in which the rod falls vertically. - in frame
**B**, in which the slot is at rest and which mover horizontally versus frame**A**.

If the rod is doomed to collide with the plate (around the slot) in **frame** **A**, then the Lorentz transformation inevitably leads to a collision with the plate in **frame** **B**, too. So, from all we have done so far, we may get some hope that there is no paradox for the collision scenario in the end!

What may still lead to some confusion concerning our collision scenario is the following question:

If a diagonally falling rod shows an inclination angle in a frame moving with the slot, what would an observer in a frame **C** fixed to the rod tell us about the slot’s orientation?

We get the feeling that we have not covered the situation fully, yet. Our frame **A** is only co-moving horizontally with the rod, but it is NOT completely co-moving with rod in diagonal direction towards the slot. So far, we have *not at all* regarded the Lorentz-transformation for the *relative* movement between the rod and the slot. We will turn to a description based on the relative movement between the rod and slot in the next post – and detect a further inclination angle.

Stay tuned …