Special relativity and the rod/slot paradox – V – perspective of the rod and aspects of the relative velocity between the rod and the slot

In this post series on the rod/slot paradox in the Special Theory of Relativity we have shown that a collision setup in a special reference frame A leads to a consistent collision description also in an inertial frame B co-moving with the slot. Despite the expected length contraction which affects the rod in frame B! For details see the previous posts. In this new post we start with a description of our collision scenario from the perspective of the rod. We introduce a related new reference frame C, which we atach to the rod and which will be found to be an inertial system, too.

As a first step we have to re-evaluate the relative motions, the orientations and the relative velocities of the slot versus the rod and vice versa. We will find that the Lorentz Transformations [LTs] between the different reference frames forces us to introduce further inclination angles to fully cover the scenario. In particular the angle between the relative velocity vector and the x-axes of the frames attached to the slot (frame B) and to the rod (frame C) will be found to be different.

What is the difference to previous considerations?

Einstein’s Special Theory Relativity discusses the descriptions of phenomena by observers in different inertial frames moving with constant velocities versus one another. In our study case we describe the movement of a rod and a slot in two dimensions as seen from three different frames of reference. Our special setup in a frame A (neither attached to the rod or the slot) ensured a collision between the rod and an impenetrable plate which surrounds the slot. For details and naming conventions see the previous posts and the schematic figure below.

Illustration 1: Setup of the collision scenario as seen in frame A. The relative diagonal motion of the rod versus the slot ensures that the left ends of the rod and slot meet at collision time t_A = t_B = 0. Length contraction affecting the slot causes a collision of the rod with an assumed plate around the slot.

The rod moves vertically downward along the y_A-axis in frame A. The slot moves horizontally along the x_A-axis vs. the origin of frame A. The dotted velocity vectors are drawn only to illustrate the relative motion of the rod vs. the slot at the chosen particular point in time .

Frame B is attached to the slot and obviously co-moving with the slot. However, frame A is not fully co-moving with the rod. Therefore, we have not presented and discussed a description from the rod’s perspective, yet!

This is kind of a deficit in our argumentation, because the typical presentations of the rod/slot paradox refer to the rod’s perspective in contrast to the slot’s perspective. What really counts is the relative motion of the rod versus the slot and vice versa – and we should use it directly for a LT between respective frames.

Another point is that, up to now, the Lorentz transformation could easily be applied, as we only had to refer to the relative motion of the frames A and B along their x-axes. Another simplifying aspect was that the slot had an orientation aligned with these axes. But to our surprise we found that from the perspective of an observer in frame B the diagonally moving rod showed an inclination angle with the x_B-axis. This gave us a first hint that diagonal movements and the Lorentz transformation may lead to different inclination angles between elongated objects in different frames of reference.

Using the Lorentz transformation along the connection line between the rod and the slot is certainly a more complicated endeavor. Note that the orientation of this connection line changes permanently in frame A. There are two points we have to care about:

As the slot moves diagonally relative to the rod, we cannot exclude an inclination angle of the slot against the rod from the perspective of an observer attached to the rod.
We even have to ask whether observers attached to the slot and the rod measure the same inclination angle of the relative velocity vector against their own x-axes. Despite the fact that these angles appear to be the same in frame A.

Point 1 is founded in our experience with the rod as seen in frame B. The reason why we have to be suspicious about the 2nd point is that velocity components of the moving objects transform differently between reference frames due to time dilation!

But an observer fixed to the rod should at least give us a consistent description of the collision, too. So, we will perform the probably complicated Lorentz transformation between the slot and the rod also as a kind of consistency check for our previous considerations. And we will be rewarded by some new insights.

Lorentz transformation from frame A to a frame “C” fixed to the rod

Let us start by introducing a new reference frame C, which fully co-moves with the rod. As the rod moves with constant velocity downwards in A, frame C is an inertial system: We can apply a LT to get transform coordinates for events measured in A to coordinates measured in C.

rod, slot, frames B and C as seen from frame A

In one of the last posts we performed a Lorentz transformation between frames A and B. Now, you can do the same for frames A and C. For the general formulas of the Lorentz transformation see post III. But watch out! In the present case we have a relative velocity between C and A along the y-axis! So, to apply our old formulas correctly, you have to reinterpret and adjust the indices there.

The good thing about frame C is that it just moves vertically downwards relative to the origin of frame A. So we have a simple relative velocity vector v_rel,CA between these frames. In frame A it is given by:

\[ \pmb{v}_{rel, \,CA} = (\,0, -v_y) \,. \]

This relative velocity is central for a Lorentz transformation of event coordinates and velocities from A to C. The velocity v_slot,C of the slot vs. the origin of A has, of course, only one non-zero component, namely –ν_x . This velocity is perpendicular to the relative velocity v_rel,CA in frame A and transforms to have two components in C:

\[ \pmb{v}_{slot,C} = \left( -{1 \over \gamma_y} \, \nu_x, \,\, v_y \right) \,. \]

In C the slot moves diagonally upwards.

For our present Lorentz transformation between the frames A and C (moving against each other in y-direction!) we also have new β and γ factors:

\[ \beta_y^2 = \left({v_y \over c}\right)^2\,, \quad \gamma_y = \left[ \, 1 – \beta_y^2 \, \right]^{-1/2} \,. \]

You can use this information now to apply the (adjusted) Lorentz Transformation to the space and time coordinates of special events in frame A. I leave this to the reader. Note that for our specific collision scenario the x_A-coordinates of events in A remain identical with respective x_C-coordinates as vectors in x-direction are perpendicular to the relative velocity between C and A.

Predictions from observes in frame C: We come to the conclusion that also from the perspective of C we get a collision with the slot being oriented in parallel to the rod there (i.e. in C) and the slot’s length being too small for the rod’s length:

\[ L_{s,C} = {1 \over \gamma_x} \, L \, \lt \, L \,. \]

This is something a direct Lorentz Transformations between frame B and frame C should reproduce. To get control over such a transformation we have to find the relative velocity vector between the slot and the rod, the absolute value of the vector’s length (norm) and its orientation versus the x-axes, both in frame B and frame C.

Relative velocity between the rod and the slot

Let us look again at the initial conditions of our collision scenario as set up in our special frame A. You may think that a standard observer in A can determine the relative velocity of the left end of the rod vs. the left end of the slot in a frame-independent way. But this is wrong. The reason lies in the occurrence of a difference between the time dilation effect between frames B and A compared to the time dilation measured between C and A.

Relative velocity of the rod vs. the slot as perceived by the slot

While the horizontal velocity of the rod in the slot’s frame B is given by the relative frame velocity ν_x, the rod’s vertical velocity in frame B is reduced due to time dilation

\[ v_{r,x,B} = \nu_x\,, \quad v_{r,y,B} = – {1 \over \gamma_x}\, v_y \,. \]

Thus, we get a constant velocity vector v_rod,B for the rod’s movement vs. the slot in B:

\[ \pmb{\operatorname{B}} : \pmb{v}_{rod,B} \, = \, \left( \, \nu_x, \,\, -{1 \over \gamma_x} \, v_y \, \right) \,. \]

This and other aspects of the Lorentz transformation between A and B gave us the following perception of the relative motion of the rod towards the slot as seen in B :

rod's relative velocity and angles as seen by the slot

Illustration 2: Relative velocity of the rod as seen by the slot (frame B)

In the drawing I have marked an inclination angle Φ_rel,B of the velocity vector v_rod,B with the x_B-axis and thus also with the slot:

\[ \begin{align} &\pmb{\operatorname{B}},\,\, \mbox{inclination of rod’s velocity vector vs. x-axis: } \\[8pt] &\quad \tan (\Phi_{rel,B}) = – {1 \over \gamma_x} {v_y \over \nu_x} \, . \end{align}\]

This angle is negative due to the rod’s diagonal movement downwards. An interesting quantity is the length (or Euclidean norm) of the vector v_rod,B :

\[ \pmb{\operatorname{B}} : \quad v_{rod,B} = ||\pmb{v}_{rod,B}|| \, = \, \left[ \, {1 \over \gamma_x^2} \, v_y^2 \, + \, \nu_x^2 \, \right]^{1/2} \,. \]

The velocity v_rod,B is identical to the relative velocity v_rel,CB of frame C vs. frame B:

\[ \pmb{v}_{rel, \,CB} = \pmb{v}_{rod,B} \,. \]

As this velocity vector is constant, frame C is an inertial system from the perspective of B.

The next drawing shows more or less the same as illustration 2, but more compact and with focus on the relative movement of frame C vs. frame B. Note that it is only a schematic drawing; scales and velocity components were chosen relatively freely:

rel. velocity of rod and frame C vs. lot

Illustration 3: Relative velocity of rod and frame C vs. the slot – as seen by an observer in frame B attached to the slot

I have indicated that the coordinate axes of frame C may appear rotated by some angle(s) in frame B. That we find an alignment of the perceived x_C-axis with the rod’s orientation is, by now, just a plausible assumption. Note that the inclination of the vector for the relative velocity of C vs. B is determined by respective velocity components measured in B. We have to pick them from the Lorentz Transformation that couples setup data in frame A to coordinates determined by an observer with synchronized clocks in frame B.

Note that we cannot get rid of frame A, logically. The reason is that frame A was used to define the setup. But, all inertial frames should lead to equivalent descriptions of events. And as we have found a relative velocity of frame C vs. frame B, we have everything required to perform an LT from B to C. And, of course, as good physicists we hope that it reproduces the predictions of the LT between A and C.

Relative velocity of the slot vs. the rod as perceived by the rod

Let us now see what our LT from A to C predicts for the measurement of the relative velocity v_slot,C of the slot vs. the rod in frame C. Illustration 4 shows what we might expect:

Illustration 4: Relative velocity of the slot and frame B vs. the origin of frame C and thus also vs. the rod.

To be careful and also to cover more general cases than ours, I have indicated that we might find an inclination angle Ψ_S between the slot and the rod. Although we, for our special collision setup, expect this angle turns out to be Ψ_S = 0 in the end.

Are there any expectations regarding the value of the relative velocity v_rel,BC = v_slot,C? As the symmetry of the LT between two inertial systems (with their x-coordinate axes aligned with the relative velocity vector) depends on the absolute value of the velocity, only, we expect that at least this value is is identical in B and C, i.e. ||v_rel,BC|| = ||v_rel,CB|| .

Actually, we have determined the slot’s velocity in C, already above and get :

\[ \begin{align} &\pmb{v}_{rel,BC} \,=\, \pmb{v}_{slot,C} = \left( -{1 \over \gamma_y} \, \nu_x, \,\, v_y \right) \quad \Rightarrow \\[10pt] &\pmb{\operatorname{C}} : \quad v_{rel,BC} = ||\pmb{v}_{rel,BC}|| \, = \, \left[ {1 \over \gamma_y^2} \, \nu_x^2 + v_y^2 \, \right]^{1/2} \, . \end{align} \]

That v_rel,BC is identical to v_rel,CB can be seen by some simple algebra:

\[ \begin{align} & \left[ \, {1 \over \gamma_y^2} \, \nu_x^2 \,+\, v_y^2 \, \right] \,=\, (1 – {v_y^2 \over c^2 } ) \, \nu_x^2 + v_y^2 = \nu_x^2 + v_y^2 – {1\over c^2} \nu_x^2 \, v_y \\[10pt] &\left[ \, {1 \over \gamma_x^2} \, v_y^2 \, + \, \nu_x^2 \, \right] \, = \, (1 – {\nu_x^2 \over c^2} ) \, v_y^2 + \nu_x^2 \, = \, \nu_x^2 + v_y^2 – {1\over c^2} \nu_x^2 \, v_y \, . \end{align} \]

So, indeed,

\[ \begin{align} & \left[ \, {1 \over \gamma_y^2} \, \nu_x^2 \,+\, v_y^2 \, \right] \,=\, \left[ \, {1 \over \gamma_x^2} \, v_y^2 \, + \, \nu_x^2 \, \right] \quad \Rightarrow \\[10pt] & v_{rel,BC} = v_{rel,CB} \,. \end{align} \]

The reader can easily derive another interesting key equation for the gamma-factor related to this relative velocity between the rod and the slot:

\[ \gamma_{rel,BC} = \left[1 – {v_{rel,BC} \over c^2} \right]^{-1/2} \,=\, \gamma_ {rel,CB} = \left[1 – {v_{rel,CB} \over c^2} \right]^{-1/2} \]

with

\[ \gamma_{rel,BC} = \gamma_ {rel,CB} =: \gamma_x * \gamma_y \,. \]

Angles of the relative velocity between the rod and the slot vs. the x-axes of frames B and C

The components of the velocities transformed from frame A to B and to C, determine the inclination angle of the relative velocity vectors in each of the frames. And, in contrast to the absolute value of the relative velocity these angles are not identical:

\[ \begin{align} &\pmb{\operatorname{B}},\,\, \mbox{inclination of slot’s velocity vector vs.} \, x_B\mbox{ -axis: } \\[8pt] &\quad \tan (\Phi_{rel,B}) = – {1 \over \gamma_x} \, {v_y \over \nu_x} \, ,. \end{align}\]

whereas

\[ \begin{align} &\pmb{\operatorname{C}},\,\, \mbox{inclination of slot’s velocity vector vs.} \, x_C\mbox{ -axis: } \\[8pt] &\quad \tan (\Phi_{rel,C}) = – \,\gamma_y \, {v_y \over \nu_x} \, . \end{align}\]

This gives us

\[ { \tan (\Phi_{rel,C}) \over \tan (\Phi_{rel,B}) } = \gamma_x * \gamma_y = \gamma_{rel,CB} \,. \]

Regarding cos- and sin-values we find:

\[ \begin{eqnarray*} & \pmb{\operatorname{B : }} \quad &\cos (\Phi_{rel,B}) = {\nu_x \over v_{rel,CB}}, &\quad \sin (\Phi_{rel,B}) = {1 \over \gamma_x} {v_y \over v_{rel,CB}} \,, \\[8pt] & \pmb{\operatorname{C : }} \quad &\cos (\Phi_{rel,C}) = {1 \over \gamma_y} {\nu_x \over v_{rel,BC}}, &\quad \sin (\Phi_{rel,C}) = {v_y \over v_{rel,BC}} \end{eqnarray*} \]

These are all useful relations which we will apply during a LT from frame B to C in the next post.

Conclusion

Warned by the result of previous posts that diagonal movements of elongated objects may lead to different inclination angles in different inertial frames, we applied a Lorentz Transformation between frame A and a new frame C attached to the rod. From the results we expect that we get a collision in frame C, too.

If all our previous considerations were correct, we must, therefore, reproduce a consistent description in frame C also by applying a Lorentz Transformation between frame B and frame C.

So, far we have understood that the absolute value of the relative velocity between the slot and the rod is the same in B and C. However, the orientation of the relative velocity vectors as determined for B and C show different inclination angles versus the respective x-axes. The reason lies in different transformations of velocity components as measured in A to frames B and C.

By having determined the angles and relation for the gamma-factor related to the relative velocity between rod and slot, we have laid the base for further work. In the next post we will apply the Lorentz Transformation between the frames B and C.