One of the topics in physics that drove me almost crazy 48 years ago was a paradox seemingly resulting from the length contraction effect described by Einstein’s Special Theory of Relativity [STR]. Paradoxes related to the length contraction effect are based on two apparently consistent descriptions of a series of events in two different reference frames which move with a constant velocity relative to each other. These descriptions seem to lead to a contradiction.

One such paradox is the “rod/slot paradox” – a variant of the Rindler-Shaw paradox. Let us call it RS-paradox below. It says that in one frame of reference a falling rod will move through a slot in a plate (or in the ground) – and in another frame of reference it would not. In both cases due to the length contraction effect. (I will present a related scenario in more detail below.)

Physics students get trained to achieve consistent descriptions of natural events independent of a chosen frame of reference. No wonder that unresolved paradoxes resulting from transition between reference frames may drive them crazy.

I got confronted with the RS-paradox for the first time when I was 17 and visited a physics course in a German Gymnasium. Special Relativity [SR] was a topic which occupied us for a few months. Our teacher presented us the RS-paradox with the help of a “ski-jumper” (see below) falling or not falling through a slot in the ground. My teacher admitted that he could not resolve the paradox for us – and as so many teachers and text books he referred to a complicated theory of Rindler regarding a modification of stiffness and elasticity in the STR. This was, however, not very satisfactory for us students. Rightfully, we had the suspicion that a basic contradiction like the one given above should disappear already on a very basic level of argumentation.

The paradox gave me some sleepless nights in 1976, until I gave up at that time and trusted in some personal breakthrough in the future. Later, at university and during my PHD, I just applied SR-formulas whenever necessary. No time for deeper explanations of paradoxes. However, some years ago – long after my professional time in physics – the problem entered my mind again. I used some relaxed time in Norway to work a bit with it once again. And this time I could resolve it in a personally satisfactory way. As the basic points of the solution are in the end rather simple, I would like to present them here in a post series – and maybe save some physics students some of their sleepless nights.

So, this is a post series for folks interested in physics and math.

Hint: This post has been changed and rewritten on 10/24/2024. Sorry for any inconvenience.

Requirements:
Requirements are that you should be familiar with reference frames and the Lorenz-transformation in STR. You should not be afraid of some formulas and some geometry. I will use math only on the level taught at high school. You should have understood the term “proper length” for a length of an object measured in a frame in which the object is at rest. You should also have learned about the length contraction effect – i.e. the effect that the length of a moving meter stick is measured to be shorter than the length measured in a frame where the meter stick is in rest – and the related premises (measurement of the distance between the end points at the same local time in each of the frames).

Discussion of collision and collision free scenarios

I will start this series with a simple naive presentation of the paradox below. Later on, I will turn to the precise and detailed description of a “collision scenario” (Scenario I). I will set up a special thought experiment in which a rod approaches a plate with a slot with a constant relative velocity along a diagonal line. The orientation of the rod and the slot will be aligned. I will define three main reference frames to cover the scenario from different perspectives.

For the collision scenario in one of the frames a collision between the rod and the plate is clearly predicted, while in the other frame one could naively expect the opposite. I will investigate some key events and apply the Lorentz transformation to describe the events in all reference frames.

We will see that the mathematics of the transformations will provide us some valuable insights, which in the end help to grasp the cause of paradoxes. After having resolved the paradox for the collision scenario I will turn to a second “collision-free scenario” (Scenario II) and present a paradox-free discussion for it – including three different observer perspectives.

A typical naive presentation of the paradox

My teacher described the RS-paradox in the following way: He used a (somewhat nonphysical) ski-jumper with long skis who jumped from a height h down to the flat ground having a slot and a cave below it.

 

Illustration 1: Skis approaching a horizontally aligned slot in the ground.

He suggested a frame of reference B fixed at the ground. Frame B was equipped with a 2-dimensional spatial coordinate system with the x_B-coordinate axis stretching in parallel to the ground and a y_B-coordinate axis pointing vertically upwards. All clocks in the spatial grid of frame B were synchronized. Events were described via space and time coordinates [x_B, y_B, t_B].

The ski-jumper was assumed to move with two constant velocity components in frame B: v_j,x,B = const parallel to the ground and ν_j,y,B = ν_y,B = const in vertical direction down towards the ground, i.e. ν_j,y,B < 0 (ν_j,y,B = – v_y,B = const) . In contrast to a real ski-jump a constant velocity in y-direction is, of course, a simplification.

The jumper’s skis were always oriented in parallel to the ground, i.e. in x-direction. (By whatever means the ski-jumper managed to control this.) The skis had a proper length L_ski = L. I.e. this length was measured when the skis were at rest.

At the position where the ski-jumper would touch the ground ground a wide slot was assumed to exist (in the ground) with a proper length L in x-direction (measured in frame B). The slot had a fixed position in B. The velocities of the jumper were defied such that the middle of the skis touched ground in the middle of the slot (in x-direction). Below the slot some underground cave was located with big dimensions in all directions. Anyone falling through the slot would end up in this cave.

Because of the assumption v_j,x,B = ν_x = const., our teacher could introduce a second frame A co-moving with the ski-jumper in x-direction. Frame A had its own coordinate system with synchronized clocks everywhere. The orientation of the x- and y-axes in A were the same as in B. Observers in A registered events with coordinates [x_A, y_A, t_A]. A moved relative to B with velocity v_A,B = v_x. The x-axis of A overlaps with that of B. The ski-jumper thus moved in A with negative velocity v_j,y,A = -v_y,A = const. in y-direction. (v_y,A typically is not identical to ν_y,B due to the rules of velocity transformations in the STR).

At time t_A = t_B = 0 the ski-jumper was at a position x_B = 0 and y_B = h in B – having already reached his constant velocity components. (We safely ignore the problem of how we had accelerated the jumper to reach these velocities).

Description of the situation in reference frame B with the slot at rest
In our reference frame B the ski-jumper moves in diagonal direction towards the ground. Due to the length contraction effect the ski-length measured in B becomes shorter than their proper length L [L_ski,B < L], while the slot keeps up its proper length L [L_slot,B = L]. Therefore, we might think that the ski-jumper will fall into the cave. It, at least, sounds logical …

 

Illustration 2: Assumed transition of the rod as seen from frame B

Description in a reference frame A co-moving with the the ski-jumper in x-direction
Now let us look at a seemingly natural description of the same scenario by an observer in frame A.

 

Illustration 3: Assumed description of the scenario from the perspective of frame A

This observer would find a horizontal velocity of the jumper ν_j,x,A = 0. We ignore the question about the value of the constant vertical ν_y,A for the time being. In frame A the skis are in rest regarding their horizontal movement. Therefore, their length is L_ski,A = L. However, due to length contraction the length of the slot moving with velocity v_slot,A = – ν_x relative to A is found to be shorter than L: L_slot,A < L. Therefore, we come to the conclusion that the ski-jumper will not fall into the cave, but instead experience a hard landing.

 

Illustration 4: Description of a collision from the perspective of frame A.

We have a paradox, as both descriptions appear to be right. Nevertheless, they obviously lead to a severe contradiction. One suspects a mistake in the assumptions. But what is wrong might be difficult to identify.

Other variants of the paradox exist in which no vertical movement happens, but gravity is assumed to work when a fast “ski-glider” moving horizontally passes the slots left end. In my opinion scenarios without gravity, but including a vertical movement of the skis or a rod with constant velocity, are much clearer. Therefore, I will stick to such scenarios.

How do we proceed from here?

My line of work will be the following: Instead of applying the length contraction directly, I will focus on some events defined by a scenario and its initial conditions. I will then apply the Lorentz-transformation to transform the events’ coordinates between my chosen frames of reference. By doing this we will get a lot of important insights.

But a first step is a more precise definition of a scenario and its initial conditions. In the next post I will turn to the setup of a collision scenario inspired by the last two illustrations above.

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

In further forthcoming posts we will learn that the Lorentz transformation implies a rotation of some objects in question. Stay tuned ….