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 the Special Theory of Relativity [STR]. All of the paradoxes related to the length contraction effect are based on two apparently self-consistent descriptions of a series of events in two different reference frames moving with a constant velocity against each other. These descriptions seem to lead to a 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 a related collision scenario in detail 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 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 breakthrough in the future. Later, at university and during my PHD, I just trusted in STR-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-blowingly 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: 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 rests. 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 description of the problem for a “collision scenario” (Scenario I). I will set up a special thought experiment in which a rod approaches a plate with a slot diagonally and eventually collides with the plate. I will define two frames of reference which I am going to l use in my forthcoming lines of thought. In one of the frames a collision between the rod and the plate is clearly predicted, while in the other frame one could expect the opposite. I will name some key events and a question which we later will have to take care of. After having resolved the paradox for different perspectives and frames I will turn to a second “collision-free scenario” (Scenario II) and present a paradox-free discussion for it – including different observer perspectives.

Contents of the next three posts for the collision scenario: In the second post I will first describe why the rod in my scenario can not pass the slot without a collision. This will become obvious in one of the chosen reference frames (frame B) in which the rod will fall vertically. We will then transform coordinates of key events to our second reference frame (frame A) attached to the slot and moving horizontally. We will notice that a kind of rotation effect gets important when an observed object moves vertically in one of the reference frames and when we transform respective coordinates to another horizontally moving frame. In the third post of this series I will show that the transformation results force us to conclude that the rod will not pass through the slot in either of our reference frames A and B. In the 4th post we will confirm this by looking at a third frame co-moving with the rod. I will then summarize what kind of fishy premise caused the the paradox in the first place.

In further posts I will then discuss an opposite scenario where the rod passes through the slot – consistently described in three frames of reference. 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 nonphysical) 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 of frame A 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 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 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 were the same as in A. Observers in B registered events with coordinates [x_B, y_B, t_B]. B moved 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.

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 rod, are much clearer. Therefore, I will stick to such scenarios.

Collision scenario – focus on a frame in which a 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 and the slot. 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 and the slot are 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

Special relativity and the rod/slot paradox – II – inclination angle of the rod in the slot frame

we will describe our defined events (and their coordinates) from the perspective of observers with fixed positions in A. A proper analysis will afterward 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 ….