One of the topics in physics that drove me crazy more than 48 years ago was the length contraction effect in physics and related paradoxes. All of these paradoxes are based on two seemingly self-consistent descriptions of a series of events in two different reference frames moving with a constant velocity against each other – and these descriptions lead to an apparent contradiction.

On such paradox is the “rod/slot paradox” – a variant of the Rindler-Shaw paradox. Let us call it RS-paradox below. In brief it states that in one frame of reference a falling rod will fall 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 the scenario more precisely below.)

Physics students get trained to achieve consistency in the description of natural events independent of a chosen frame of reference. No wonder that unresolved paradoxes may drive them crazy. I got first confronted with the RS-paradox when I was 18 and visited a physics course in a German Gymnasium. 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 SR. 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 breakthrough in the future. Later, at university and during my PHD, I just trusted in SR-formulas and simply used them whenever necessary. No time for deeper explanations of paradoxes. However, some years ago – long after my time in physics – the problem entered my mind again. I used some relaxed time ín Norway to access it again. And this time I could resolve it in a personally satisfactory way. As the basic points of the solution are so mind-blowing simple, I would like to present them here in a post series – and maybe save some students a sleepless night.

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

Requirements are that you should be familiar with reference frames in SR and the Lorenz-transformation. 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 rests. You should 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 premise (measurement of the distance between the end points at the same local time in each of the frames).

I will start this series with a simple description of the problem. I will set up a thought experiment with a rod approaching a slot diagonally. I will define two frames of reference which I will use in my forthcoming lines of thought. I will name some key events and a question which we will have to take care of.

In the next post I will describe a scenario in which the rod will not pass the slot. To our surprise we will derive this result in a consistent way in both reference frames. This will indicate what kind of fishy premise causes the the paradox. We will also notice that a kind of rotation effect gets important when moving in two directions in SR.

In further posts I will then discuss an opposite scenario where the rod passes through the slot – consistently described in both reference frames. Before we come to this point, I just ask you to think a bit about the following question: How could we get a long diagonally moving needle with length L through a slot whose extension is too short (L_slot < L)?

A typical presentation of the paradox

My teacher described the paradox in the following way: He used a (somewhat unphysical) ski-jumper with long skis who jumped from a height h down to the flat ground. He suggested a frame of reference A fixed at the ground. Frame A was equipped with a 2-dimensional spatial coordinate system with x_A-coordinate axis stretching in parallel to the ground and a y_A-coordinate axis pointing vertically up from the ground. All clocks in the spatial grid were synchronized. Events were described via space and time coordinates [x_A, y_A, t_A].

The ski-jumper was assumed to move with two constant velocity components in frame A: v_j,x,A parallel to the ground and ν_j,y,A in vertical direction down towards the ground, i.e. ν_j,y,A < 0 (ν_j,y,A = – v_y = 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 touches ground a wide slot was assumed to exist (in the ground) with a proper length L in x-direction (measured in frame A). The slot had a fixed position in A. The velocities were 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 both directions. Anyone falling through the slot would end up in this cave.

Because of the assumption v_j,x,A = ν_x = const., our teacher could introduce a second frame B co-moving with the ski-jumper in x-direction. B has its own coordinate system with synchronized clocks everywhere. The orientation of the x- and y-axes in B shall be the same as in A. Observers in B register events with coordinates [x_B, y_B, t_B]. B moves relative to A with velocity v_B,A = v_x.

At time t_A = t_B = 0 the origins of the coordinate systems of A and B coincided, and at this point in time (t_A = 0) the ski-jumper was at a position x_A = 0 and y_A= h in A having already reached his constant velocity components. (Our teacher elegantly ignored the problem of how we had accelerated him to reach these velocities).

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

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

This is a paradox, as both descriptions appear to be right. Nevertheless, they seemingly lead to a severe contradiction. One suspects a mistake, but it is not trivial to find it.

Focus on a frame in which the rod falls vertically

Let us further reduce the problem and omit the ski-jumper. We turn the “skis” into a rod of proper length L. The ground is reduced to a infinitesimally thin, but impenetrable plate having a fixed slot with proper length L in x-direction. In a frame in which both the slot and the rod are in rest, the rod will be able to just pass through the slot vertically (we ignore an infinitesimally bigger extension L + ε of the slot in comparison to the rod in x-direction in future calculations). Below the plate there is a vacuum.

Now let us prepare a thought experiment, in which the rod, seen from the plate, moves both in horizontal and vertical direction with constant velocity components. I.e., the rod shall move in diagonal direction against the plate. The plate sees the rod’s midpoint approaching the plate along a diagonal line of sight with constant velocity.

From the perspective of an observer who sees the rod falling vertically, the plate is moving with constant velocity –ν_x in negative x -direction. As always with questions in special (and also general) relativity the reasonable choice of a reference frame, in which the description of the involved objects becomes relatively easy, is essential.

We, therefore, first pick a frame B, in which the rod falls down just vertically in negative y-direction. I.e. in frame B the rod has no horizontal movement. Seen from other frames, frame B is co-moving with the rod in x-direction. The velocity components of the rod in this frame B are: v_r,x,B = 0 in x-direction and v_r,y,B = -v_y,B = const. in y-direction.

In frame B at a point t_B = 0 the left end of the rod shall have an x-coordinate x_B = 0 and an y-coordinate y_B = h.

In B the plate with the slot moves horizontally in negative x-direction along the x-axis (y_B = 0 !) with a velocity v_p,x,B = –v_x = const. The vertical velocity of the plate in B is v_p,y,B = 0.

Illustration 1: Reference frames B and A, the rod and the plate with the slot

We define another reference frame A, in which the plate has a fixed x-position and a y-coordinate y_p,y,A = 0. The velocity components of the plate in A are v_p,x,A = 0 and v_p,y,A = 0. With respect to B, frame A moves with a relative velocity v_A,B = –v_x . Seen from A the rod and fame B move with horizontal velocity v_B,A = v_r,x,A = –v_x.

The origins of our frames B and A shall coincide at t_A = t_B = 0. At this particular time the rod’s endpoints shall pass the height h (in y-direction) in B.

The x_A-axis of frame A and the x_B-axis of B actually coincide; for reasons of visual clarity I have nevertheless drawn them at slightly different y-positions in illustration 1.

As we cannot exclude that observers in A and B may describe the movement of the rod’s points differently, we distinguish between the left end of the rod and the right end of the rod. We also distinguish respective events for these end points – as e.g. passing height h and reaching the plate. Let us define some coordinates and lengths:

In frame B the x-coordinates of the left and the right end point of the rod shall be named x_r,l,B and x_r,r,B. Respective y-coordinates are y_r,l,B and y_r,r,B . In frame A we have x_r,l,A , x_r,r,A , y_r,l,A , y_r,r,A, respectively. The coordinates of the slot are named x_s,l,A , x_s,r,A , y_s,l,A , y_s,r,A in B and x_s,l,A , x_s,r,A , y_s,l,A , y_s,r,A, in A.

At time t_A = t_B = 0 we arrange the endpoints of the rod to be in B at:

The rod will pass these points vertically with –v_y,B = const. < 0. (I.e., in the past there was some acceleration phase in B with the rod reaching this vertical velocity).

From the perspective of an observer in B, the rod will touch the plate (and its slot) at a time t_g,B = h / v_y,B .

Now, we deviate a bit from the scenario presented by my old teacher:
We arrange the position of the slot (in A) such that its left end reaches the point (x_B , y_B) = (0, 0) in B at t_B = t_g,B . I.e. in B the left end of the rod reaches the plate at (x_B= 0, y_B = 0) at a point in time when also the left end of the slot reaches (x_B= 0, y_B = 0). This point is, of course, the origin of the 2-dimensional spatial coordinate system attached to B.

How do we proceed from here? We look at events …

Instead of applying length contraction directly, we will focus on some events and transform their coordinates. I describe these events below from a B-perspective:

Event E1 (B perspective): t_B = 0: The left end of the rod passes the point (x_B , y_B) = (0, h) vertically (with velocity v_y,B = const.): [ x_r,l,B, y_r,l,B, t_B ] = [ 0, h, 0 ]

Event E2 (B perspective): t_B = 0: The right end of the rod passes the point (x_B , y_B) = (L, h) vertically (with velocity v_y,B = const.): [ x_r,r,B, y_r,r,B, t_B ] = [ L, h, 0 ]

Event E3 (B perspective): t_B = t_g,B : The left end of the rod hits the plate and passes point (x_B , y_B) = (0, 0) vertically: [ x_r,l,B, y_r,l,B, t_B ] = [ 0, 0, t_g,B ]

Event E4 (B perspective): t_B = t_g,B : The right end of the rod hits the plate and passes point (x_B , y_B) = (L, 0) vertically: [ x_r,r,B, y_r,r,B, t_B ] = [ L, 0, t_g,B ].

Event E5 (B perspective): t_B = t_g,B : The left end of the slot passes (x_B , y_B) = (0, 0) horizontally with velocity v_s,B = –v_x : [ x_s,l,B, y_s,l,B, t_B ] = [ 0, 0, t_g,B ].

To get a better understanding we want to describe the events and their spatial and time coordinates from the perspective of observers in A. We are of course also interested in the position of the right end of the slot, when the right end of the rod touches the plate according to observers in A. If we find the position of the rod at that point at a distance <= L in A (!) it will fall through the slot there. Otherwise not.

Outlook

In this post we have set up a rod/slot experiment in two different reference frames. Frame B is at rest in x-direction relative to the rod. The rod has a proper length L and falls vertically down in B with a constant velocity in y-direction. It reaches a horizontally moving plate with a slot. When the rod touches the plate, the left end of the slot coincides with the left end of the rod. Frame A, instead, is co-moving with the plate (and the slot) in negative x-direction with a constant velocity –v_x ( seen from the perspective of observers having a fixed position in B). We have defined a series of key events (and their coordinates) in B.

In the next post we will describe our defined events (and their coordinates) from the perspective of observers with fixed positions in A. A proper analysis will show that both in frame B and in frame A the rod will never fall through the slot without a major accident, namely crashing against the plate at the right end side of the slot.

We will also see that rules for the addition of velocity (based on time dilation) lead to the apprehension of observers in A that the rod is falling with its symmetry axes rotated by a fixed angle against A‘s coordinate system.

Stay tuned ….