Special relativity and the rod/slot paradox – XVIII – Lorentz Transformation of a transit scenario to the standard setup frame for a collision scenario

Let us look deeper into the rod/slot paradox of Special Relativity. We know already that one can set up very different scenarios of the rod/slot encounter in two dimensions. The easiest way to do so is to use different reference frames to define the initial conditions. We have already analyzed the perceptions of a variety of observers for two basic scenarios: a transit scenario (see the posts VIII to IX in this series) and a collision scenario (see the first six posts). We now want to directly compare data for these two scenarios in one and the same reference frame and, of course, at the same point in time, there.

In the preceding post we have already succeeded with a direct scenario comparison in a specific frame W. The origin of W approached the origin of a standard setup frame A for the collision scenario on a specific diagonal path (see the 8th post of this series). We could make the comparison after a Lorentz Transformation [LT] of the collision scenario’s data from frame A to frame W.

As expected, we saw major differences between specific quantities of the moving objects in our scenarios, namely regarding

the angle between the rod and the slot and both their inclination angles vs. the line defined by the relative motion of the frames’ origins vs. each other,
and the velocity vectors of the rod and the slot.

The differences were in part due to the impact of the Thomas-Wigner rotation.

In this post we prepare a direct comparison of the two scenarios in our standard setup frame A for the collision scenario. We perform a Lorentz Transformation [LT] of the transit scenario’s data from the diagonally moving frame W to A. If you are new to this series, a reading of previous posts may be required to understand the steps taken.

The transit scenario in the setup frame W

Let us call the slot and the rod of the collision scenario “c-slot” and “c-rod“, respectively. For the transit scenario we use the names “t-slot” and “t-rod“.

Just to remind you about the results of the preceding post:
The following schematic drawing shows on its left side a transit scenario for the “t-rod” and the “t-slot” in frame W at a time t_W = 0. On the right side of the y_s,_W-axis a collision scenario unfolds between the “c-rod” and the”c-slot“.

Note: Both the c-slot and the t-slot have the same length and velocity in our frame W. This makes our comparison especially instructive.

The x-axis x_s,W of the coordinate system for frame W was chosen such that the c-slot of the collision scenario resided on this axis. I.e., there was an alignment of the x_s,W -axis with the c-slot’s orientation.

Comparison of a transit scenario with a collision scenario in frame W

Illustration 1: Comparison of the transit scenario and the collision scenario in a common frame W at time t_W = 0. At this moment the transit scenario (still) unfolds on the left side of the y_s,W-axis, whereas the collision scenario takes place on the right side of the y_s,W-axis. The conditions of the transit scenario are equivalent to the setup in a frame Z, which we used in previous posts. The length of the t-rod is measured to be shorter than its proper length (L_tr,W < L) due to the relativistic length contraction effect.

From the perspective of W the origin of frame A approaches W‘s origin on a diagonal path with a velocity v_rel,A2W = v_rel. See preceding posts for more details.

As in previous posts, we assume that there infinitesimally thin plates outside the c-slot and the t-slot. Note: The assumed impenetrable and infinitesimally thin plates around the slots are omitted in this and further drawings for the sake of clarity. We only show the slots themselves.

Remarks on the collision scenario:
The peculiar appearance of the collision scenario in W is due to the impact of the Thomas-Wigner rotation on rod and the slot. This rotation in turn is a result for the LT from frame A to the diagonally moving frame W. We have discussed details already extensively. The length L_cs,W of the c-slot as perceived in W was found to be L_cs,W = L. The reason lies in the vertical relative motion of W‘s origin vs. the c-slot in A.

Remarks on the collision scenario: The t-rod is just passed by the vertically moving ends of the t-slot at t_W = 0 – admittedly at the last possible moment. So, no collision with surrounding plates, but just a transit of the t-rod through the t-slot. A reader has asked why the transit is set up in such an extreme way regarding the timing. This was just done to make the problem mathematically a bit more handsome. But, in your own calculations you may move the t-rod a bit to the left, without changing any of the qualitative results.

Basic quantities and relations

Let me remind you about some mathematical aspects of the properties indicated in illustration 1. We derived respective values in previous posts (see the 13th and 14th post of this series).

Both the perceived length L_cs,W of the c-slot and the set length L_ts,W of the t-slot in W are equal to the proper length L of these objects:

\[ L_{cs,W} \,=\, L_{ts,W} \,=\, L \,. \]

We have introduced this length in the very first posts already.

The relativistic factors β_x and β_y refer to the (absolute) values ν_x and v_y of the slot’s horizontal velocity and the rod’s vertical velocity, respectively, in our standard frame A. They also define the relative velocity v_rel of the A/W-frames’ origins vs. each other:

\[ |\pmb{v}_{rel,W2A}| \,=\, |\pmb{v}_{rel,A2W}| \,= \, v_{rel} \,, \]

\[ \begin{align} v_{rel}^2 \,&=\, \nu_x^2 \,+\, v_y^2\,, \\[8pt] \beta_{rel}^2 \,&=\, \beta_x^2 \,+\, \beta_y^2\,. \end{align} \]

The angle Ψ_s,rel,W between the c-slot and the line of relative motion between our frames A and W is given by

\[ \begin{align} \cos \left( \Psi_{s,rel,W} \right) \,&=\, {\gamma_x \over \gamma_{rel} } \,{\beta_x \over \beta_{rel}} \,, \\[8pt] \sin \left( \Psi_{s,rel,W} \right) \,&=\, – \, \gamma_x\,{\beta_y \over \beta_{rel}} \end{align} \]

Below we will often use the absolute value of the sine. The direction of quantities is in most cases obvious.

The common vertical velocity v_s,W of the c-slot and the t-slot in W‘s coordinate system with axes x_s,W and y_s,W has a positive value of :

\[ \begin{align} v_{s,W} \,&=\, \gamma_x \, \beta_y * c\,,\quad \beta_{s,W} = \gamma_x\, \beta_y \,, \\[8pt] v_{ts,W} \,&=\, v_{s,W} \,, \quad v_{tr,W} \,=\, v_{s,W} \,. \end{align} \]

Note that the vertical velocity vector v_s,W and its absolute value v_s,W are completely determined by the relative motion of the frame W vs. A.

Objectives of this post

We want to transform position and velocity data of the t-rod and t-slot from frame W to frame A. I present the the results below. The analysis of consequences is the topic of forthcoming posts.

The mathematics is a bit complicated and depends strongly on the choice of the t-rod/t-slot velocities. But the steps required to perform the LT follow familiar patterns which we have established already in other posts. Aside of some hints, I will not go into details. Once again, one has to take care of the correction of time points in W (and related spatial coordinates) to enforce a LT of spatially separated events to a common time t_A = 0 in A.

I have discussed the special values chosen for v_ts,W and v_tr,W in the preceding post, already:

\[ \begin{align} \pmb{v}_{ts,W} \, &=\, \pmb{v}_{s,W} \,=\, \left(\, 0, \, v_{s,W}\,\right) \,=\, \left(\, 0, \, \gamma_x \, \beta_y*c\, \right) \,, \\[8pt] \beta_{ts,W} \,&=\, v_{s,W} * |\sin \left( \Psi_{s,rel,W} \right)| \, \end{align} \]

and

\[ \pmb{v}_{tr,sW} \, =\, \left(\, v_{rel} * \cos \left(\Psi_{s,rel,W}\right), \, 0\, \right) \,=\, \left(\, v_{rel} * {\gamma_x \,\beta_x \over \gamma_{rel} \, \beta_{rel}} , \, 0\, \right) \,. \]

Within the usual limit, we could at least in principle have chosen a different value for the horizontal velocity v_tr,sW of the rod along the x_s,W-axis. I will, therefore, present results for the t-rod’s coordinates and velocity components first in form of general formulas depending on v_tr,sW (and respective factors β_tr,sW and γ_tr,sW) and only afterward as derived for the specific value of v_tr,sW given above.

Lorentz Transformation of the coordinates of the t-slot for t_A = 0

If you want to apply the LT between W and A in its common simple form, you have to perform it in a (rotated) Cartesian coordinate system with an x_rel,W -axis aligned with the line of the frames’ relative motion. The x_rel,W-axis is thereby also aligned with the vector v_rel,A2W, as perceived in W. x-coordinate-values in this coordinate system get a subscript “rel” below. W

The right end point of the t-slot poses no problem during the LT. Its t_rel, x_rel and y_rel -coordinates get a value of 0 both in W and A. However, an important requirement for a correct LT of the t-slot’s left end point is that you need to pick x– and y-coordinates at a time t_W that transforms to t_A = 0. We have gone through the necessary steps to achieve this already very often in this post series. I leave the straightforward operations to the reader. You must, of course, also perform projections of the position and velocity vectors onto the axes x_rel,W and y_rel,W to determine the required values.

I call the relevant time coordinate t_ts,l,rel,W (< 0) and the respective spatial coordinates x_ts,l,rel,W (along the x_rel,W-axis) and y_ts,l,rel,W (along the perpendicular y_rel,W -axis).

Expectations: Similar to the initial conditions set for the c-slot in frame A, we expect the length of the t-slotto become 1/γ_x * L in frame A . Its orientation there should also be along the x_A-axis.

The corrected coordinates in W, which become subject to the LT (to t_A = 0) , are:

\[ \begin{align} t_{ts,l,rel,W} \,&=\, – \, {1 \over c} \,L\, {1 \over N_{ts}} \, \beta_{rel} * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] x_{ts,l,rel,W} \,&=\, – \,L\, {1 \over N_{ts}} * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] y_{ts,l,rel,W} \,&=\, + \, L \, {1 \over N_{ts}} * \left(\, |\sin \left( \Psi_{s,rel,W} \right) | \,-\, \beta_{rel}\,\beta_{ts,W}\, \right) \end{align}\]

with

\[ N_{ts} \,=\, 1 \,-\, \beta_{ts,W} \,\beta_{rel} * |\sin \left( \Psi_{s,rel,W} \right) | \,. \]

The Lorentz Transformation (in its simple form along the line of relative motion) then gives us:

\[ \begin{align} t_{ts,l,rel,A} \,&=\, 0 \,, \\[8pt] x_{ts,l,rel,A} \,&=\, – \,{1 \over \gamma_{rel}} \,L\, {1 \over N_{ts}} * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] y_{ts,l,rel,A} \,&=\, + \, L \, {1 \over N_{ts}} * \left(\, |\sin \left( \Psi_{s,rel,W} \right) | \,-\, \beta_{rel}\,\beta_{ts,W}\, \right) \,. \end{align}\]

This in turn indeed gives us, with the uniquely defined vertical velocity v_ts,W and the values for the angle Ψ_s,rel,W, the following results

\[ L^2 _{ts,A} \,=\, {1 \over \gamma_x^2} \, L^2 \,. \]

and

\[\cos \left( \Psi_{ts,rel,A} \right) \,=\, {\beta_x \over \beta_{rel}} \,.\]

The respective angle just defines an orientation of the slot along the x_A-axis. Together with the right end of the slot coinciding with the origin of A at t_A = 0, this tells us the following:

In frame A the t-slot resides on the x_A-axis – to the left of A‘s origin.

Lorentz Transformation of the velocity components of the t-slot for t_A = 0

The velocity components v_ts,x,rel,W (along the x_rel,W-axis) and v_ts,y,rel,W (along the y_rel,W-axis) of the t-slot’s velocity are given by

\[ \begin{align} v_{ts,x,rel,W} \,&=\, \gamma_x^2 \, {\beta_y \over \beta_{rel}} * \beta_y * c \,, \\[8pt] v_{ts,y,rel,W} \,&=\, \gamma_x^2 \, {1 \over \gamma_{rel}} \, {\beta_y \over \beta_{rel}} * \beta_x * c \,. \end{align}\]

They transform to respective constant components in A:

\[ \begin{align} v_{ts,x,rel,A} \,&=\, – \, {\beta_x \over \beta_{rel} } \, \beta_x * c \,, \\[8pt] v_{ts,y,rel,A} \,&=\, + \, {\beta_y \over \beta_{rel} } \, \beta_x * c \, . \end{align}\]

The negative signs results from the difference of the component v_ts,x,rel,W and v_rel in the LT-formulas. As a consequence we find for the perceived velocity value v_ts,A of the t-slot in A and its angle Ψ_v,ts,rel,A with the x_rel,A-axis:

\[ \begin{align} |v_{ts,A}| \,&=\, \left[ v_{ts,x,rel,A}^2 \,+\, v_{ts,y,rel,A}^2 \right]^{1/2} \,=\, \nu_x \,, \\[8pt] v_{ts,A} \,&=\, -\, \nu_x\,, \\[8pt] \cos \left( \Psi_{v,ts,rel,A} \right) \,&=\, – \, {\beta_x \over \beta_{rel} } \,. \end{align} \]

Just, as we had expected:

In frame A, the t-slot moves to the left with the same velocity as the c-slot in the collision scenario.

Lorentz Transformation of the coordinates of the t-rod for t_A = 0

The rod is a bit more difficult to handle. Its length in W is contracted in comparison to its proper length:

\[ L_{tr,W} \,=\, {1 \over \gamma_{tr,sW} }* L \,. \]

With this we get after a somewhat lengthy calculation the right coordinates t_tr,l,rel,W , x_tr,l,rel,W and y_tr,l,rel,W, which we must use for the LT to A to get results for a point in time t_tr,l,rel,A = 0 :

\[ \begin{align} t_{tr,l,rel,W} \,&=\, – \, {1 \over c} \, {1 \over \gamma_{tr,sW} } \,L\, {1 \over N_{tr}} \, \beta_{rel} * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] x_{tr,l,rel,W} \,&=\, – \, {1 \over \gamma_{tr,sW} } \,L\, {1 \over N_{tr}} * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] y_{tr,l,rel,W} \,&=\, + \, {1 \over \gamma_{tr,sW} } \, L \, {1 \over N_{tr}} * |\sin \left( \Psi_{s,rel,W} \right) | \end{align}\]

with

\[ N_{tr} \,=\, 1 \,-\, \beta_{tr,sW} \,\beta_{rel} * |\cos \left( \Psi_{s,rel,W} \right) | \,. \]

The factor 1/N_tr accounts for the required shift in time to fix a LT to t_tr,l,rel,A = 0.

The formulas for the spatial coordinates have some symmetry as the left end of the rod resides on the x_rel,W -axis also at t_tr,l,rel,W .

After the LT we expect some impact of the Thomas-Wigner rotation. A lengthy calculation delivers respective data in frame A:

\[ \begin{align} t_{tr,l,rel,A} \,&=\, 0 \,, \\[8pt] x_{tr,l,rel,A} \,&=\, – \, {1 \over \gamma_{tr,sW} } \, {1 \over \gamma_{rel}} \,L\, {1 \over N_{tr}} * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] y_{tr,l,rel,A} \,&=\, + \, {1 \over \gamma_{tr,sW} } \, L \, {1 \over N_{tr}} * |\sin \left( \Psi_{s,rel,W} \right) | \,. \end{align}\]

From the sign of x_tr,l,rel,W we conclude that:

In frame A, the t-rod is located to the left of the y_A-axis.

The length L_tr,A of the t-rod and its angle Ψ_tr,rel,A with the line of the frames’ relative motion in A are given by:

\[ L^2 _{tr,A} \,=\, {1 \over \gamma_{tr,sW}^2 } \, L^2\, {1 \over N_{tr}^2} * \left[\, 1 \,-\, \beta_{rel}^2 * \cos^2 \left( \Psi_{s,rel,W} \right) \, \right] \,. \]

\[ L_{tr,A} \,=\, {1 \over \gamma_{tr,sW} } \, L\, {1 \over N_{tr}} * \left[\, 1 \,-\, \beta_{rel}^2 * \cos^2 \left( \Psi_{s,rel,W} \right) \, \right]^{1/2} \,, \]

\[ \cos \left( \Psi_{tr,rel,A} \right) \,=\, -\, {1 \over \gamma_{rel}} * { \cos \left( \Psi_{s,rel,W} \right) \over \, \left[\, 1 \,-\, \beta_{rel}^2 * \cos^2 \left( \Psi_{s,rel,W} \right) \, \right]^{1/2} }\,. \]

The angle between the t-rod and the x_A-axis

A very interesting thing is: For our specific coupling of the scenarios, the angle Ψ_tr,rel,A does not depend on the velocity of the t-rod in W. The reason lies in the horizontal alignment of the t-rod with the x_s,W-axis and resulting linear scaling of the components along the axes x_rel,W, y_rel,W.

The angle Ψ_s,rel,W is fixed (!) by the collision scenario and the relative movement of W vs. A. We use the results of previous posts (see above) for Ψ_s,rel,W to further analyze the angle Ψ_tr,rel,A:

\[ | \cos \left( \Psi_{tr,rel,A} \right) | \,=\, {\gamma_x \over \gamma_{rel}} \, {\beta_x \over \beta_{rel} } * { \left[\, 1 \,-\, \beta_{rel}^2 \, \right]^{1/2} \over \, \left[\, 1 \,-\, \beta_{rel}^2 * \cos^2 \left( \Psi_{s,rel,W} \right) \, \right]^{1/2} } \]

It follows that

\[ \begin{align} | \cos \left( \Psi_{tr,rel,A} \right) | \,&\lt\,{\beta_x \over \beta_{rel} } \,=\, | \cos \left( \Psi_{ts,rel,A} \right) | \,=\, | \cos \left( \Psi_{cs,rel,A} \right) | \\[8pt] |\Psi_{tr,rel,A}| \,&\gt\, |\Psi_{s,rel,A}| \,, \\[8pt] 0 \,\lt\ |\Psi_{tr,xA}| \,&\lt\, \pi/2 \,. \end{align} \]

I.e., the angle between the t-rod and the line of relative motion between W and A is bigger in frame A than the respective angle of the c-slot or t-slot, there. As the slots reside on the x_A-axis, the rod has an acute angle relative to the x_A-axis and thus also relative the t-slot in A. Meaning:

The left end of the t-rod resides at t_A = 0 above the x_A-axis (left to the origin), while the right end coincides with A‘s origin.

See illustration 2 in a section below. This is, of course, due to the impact of the Thomas-Wigner rotation on the t-rod.

The length of the t-rod in A for our special selection of the t-rod’s velocity

The length of the t-rod in A depends on the t-rod’s velocity in W. Let us find out what it gives for our special choice

\[ \pmb{v}_{tr,sW} \, =\, \left(\, v_{rel} * \cos \left(\Psi_{s,rel,W}\right), \, 0\, \right) \,=\, \left(\, v_{rel} * {\gamma_x \,\beta_x \over \gamma_{rel} \, \beta_{rel}} , \, 0\, \right) \,. \]

Then we have

\[ \begin{align} {1 \over \gamma_{tr,sW}^2} \,&=\, 1 \,-\, \beta_{rel}^2 \, \cos^2 \left(\Psi_{s,rel,W} \right) \,, \\[8pt] N_{tr} \rightarrow N_{tr}^s \,&=\, 1 \,-\, \beta_{rel}^2 \, \cos^2 \left(\Psi_{s,rel,W} \right) \,. \end{align}\]

It follows – a bit surprising, maybe – that we have

\[ L_{tr,A} \,=\, L \,. \]

One can show that this is a maximum value that can be reached for L_tr,A with respect to its dependence on the parameter v_tr,sW. I leave the relatively easy proof to the reader.

The angle of the t-rod with the line of relative motion in A for our special selection of the t-rod’s velocity

For

\[ N_{tr} \rightarrow N_{tr}^s \,=\, 1 \,-\, \beta_{rel}^2 \, \cos^2 \left(\Psi_{s,rel,W} \right) \,. \]

the angle Ψ_tr,rel,A is given by

\[ \begin{align} \cos \left( \Psi_{tr,rel,A} \right) \,& =\, -\, {1 \over \gamma_{rel}} * {1 \over [N_{tr}^s]^{1/2} } * \cos \left( \Psi_{s,rel,W} \right) \,, \\[8pt] \sin \left( \Psi_{tr,rel,A} \right) \,& =\, +\, {1 \over [N_{tr}^s]^{1/2} } * | \sin \left( \Psi_{s,rel,W} \right) | \,,\end{align} \]

which is equivalent to

\[ \begin{align} \cos \left( \Psi_{tr,rel,A} \right) \,& =\, -\, {1 \over \gamma_{rel}} * {\beta_x \, \gamma_x \over \beta_{rel} \, \gamma_{rel}} \, {1 \over [N_{tr}^s]^{1/2} } \,, \\[8pt] \sin \left( \Psi_{tr,rel,A} \right) \,& =\, +\, \gamma_x \, {\beta_y \over \beta_{rel}} \, {1 \over [N_{tr}^s]^{1/2} } \,. \end{align} \]

Lorentz Transformation of the velocity components of the t-rod to A (for t_A = 0)

Let us transform the velocity components of the t-rod. We first get the component v_tr,x,rel,W (along the x_rel,W-axis) and the perpendicular component v_tr,y,rel,W (along the y_rel,W-axis)

\[ \begin{align} v_{tr,x,rel,W} \,&=\, +\, v_{tr,sW} * \cos\left(\Psi_{s,rel,W} \right) \,, \\[8pt] v_{tr,y,rel,W} \,&=\, -\, v_{tr,sW} * |\sin\left(\Psi_{s,rel,W} \right)| \,. \end{align}\]

With the general form of N_tr

\[ N_{tr} \,=\, 1 \,-\, \beta_{tr,sW} \,\beta_{rel} * |\cos \left( \Psi_{s,rel,W} \right) | \,, \]

the Lorentz Transformation to frame A results in

\[ \begin{align} v_{tr,x,rel,A} \,&=\, + {1 \over N_{tr}} * \left( \, v_{tr,sW} *\cos\left(\Psi_{s,rel,W} \right) \,-\, v_{rel} \, \right) \,, \\[8pt] v_{tr,y,rel,A} \,&=\, -\, {1 \over \gamma_{rel}} \, {1 \over N_{tr}} * v_{tr,sW} * |\sin\left(\Psi_{s,rel,W} \right)| \,. \end{align}\]

\[\begin{align} \end{align}\]

Note that v_tr,x,rel,A can become negative.

Velocity components of the t-rod in A for our special case

For our special selection of v_tr,sW, we have

\[ N_{tr} \rightarrow N_{tr}^s \,=\, 1 \,-\, \beta_{rel}^2 \, \cos^2 \left(\Psi_{s,rel,W} \right) \,. \]

This results in

\[ \begin{align} v_{tr,x,rel,A} \,&=\, -\, c * \beta_{rel} \, {1 \over N_{tr}^s} \, \sin^2 \left(\Psi_{s,rel,W} \right) \,, \\[8pt] v_{tr,y,rel,A} \,&=\, -\, c * \beta_{rel} \, { 1 \over \gamma_{rel}} \, {1 \over N_{tr}^s} * \cos\left(\Psi_{s,rel,W} \right) * |\sin\left(\Psi_{s,rel,W} \right)| \,. \end{align}\]

A short calculation shows that v_tr,A = ||v_tr,A|| is given by

\[ \begin{align} v_{tr,A} \,&=\, c * \beta_{rel} * {1 \over [N_{tr}^s]^{1/2}} * |\sin\left(\Psi_{s,rel,W} \right)| \,, \\[8pt] v_{tr,A} \,&=\, c * \beta_{rel} * \gamma_x \, {\beta_y \over \beta_{rel}} \,{1 \over [N_{tr}^s]^{1/2}} \end{align} \]

This gives us

\[ \begin{align} \sin \left( \Psi_{v,tr,rel,A} \right) \,=\, {v_{tr,y,rel,A} \over v_{tr,A} } \,=\, – { 1 \over \gamma_{rel}} \, {1 \over [N_{tr}^s]^{1/2} } * \cos\left(\Psi_{s,rel,W} \right) \,, \\[8pt] \cos \left( \Psi_{v,tr,rel,A} \right) \,=\, {v_{tr,x,rel,A} \over v_{tr,A} } \,=\, – \, {1 \over [N_{tr}^s]^{1/2}} * |\sin\left(\Psi_{s,rel,W} \right)| \,, \end{align} \]

I.e., for the special case (with N_tr = N^s_tr ) we have

\[ \begin{align} \sin \left( \Psi_{v,tr,rel,A} \right) \,&= \, +\, \cos \left( \Psi_{tr,rel,A} \right) \,, \\[8pt] \cos \left( \Psi_{v,tr,rel,A} \right) \,&= \, -\, \sin \left( \Psi_{tr,rel,A} \right) \,, \end{align} \]

The velocity vector v_tr,A is in this case obviously perpendicular to the orientation of t-rod in A.

Perception of the transit scenario in frame A at t=0

Illustration 2: Schematic representation of the transit scenario in out standard frame A at t_A = 0. The drawing is based on the results of a LT of the data for the t-rod and t-slot from W to A – and a special (but very reasonable) selection of the t-rod’s velocity in W.

Conclusion

In this post we have applied the Lorentz Transformation to the setup data of a transit scenario for the rod and the slot. We have calculated both position data and components of the velocities of both the t-slot and the t-rod in our original standard frame A.

In the next post I will analyze our result in more detail, discuss consequences and perform a comparison of our two scenarios.

Special relativity and the rod/slot paradox – XIX – perception of transit and collision scenario in a common reference frame

Stay tuned …