Special relativity and the rod/slot paradox – VI – consistency and the Lorentz Transformation between the slot and the rod

We are on a good way to resolve one of the length contraction paradoxes in Special Relativity – the rod-slot paradox – with rather simple considerations. We have already found a consistent description of a collision scenario by observers in three different inertial frames. We have used Lorentz Transformations [LTs] from a setup frame A to a frame B, moving with the slot, and another LT to a frame C, moving with the slot. We now prove consistency by a direct transformation of rod and slot coordinates measured in frame B to the diagonally approaching frame C attached to the rod.

Major Hurdles and Objectives

The Lorentz transformations from A to B and from A to C were rather easy to handle as we dealt with

movements of the target frames B and C along the x_A-axis and the y_A-axis of frame A, respectively,
and an orientation of both the elongated rod and the slot parallel to the x_A-axis.

The situation between frames attached to the rod and to the slot is more complicated as we must deal with diagonal movements of the rod vs. the slot and vice versa. I have used the plural “movements” intentionally, as these movements look a bit different in B and C.

The following drawing shows the situation we already have derived for frame B:

Rod’s movement as seen in frame B

rel. velocity of rod and frame C vs. lot

Illustration 1: The rod’s diagonal movement as seen from the slot and frame B

In the previous post V we have already found that observers co-moving with the rod may have the following perception of the slot’s movement.

Slot’s movement as seen in frame C

Illustration 2: The slot’s movement as seen from the rod and frame C (according to a LT from frame A to C; see post V)

The angle Ψ_S is marked with a question mark to indicate that this is a point which we have to verify via the LT. But we have a prediction for it.

Objective: Reproduce some previous predictions by a LT from frame B to frame C

The x_C-axis of the standard coordinate system for a description of spatial coordinates of objects in frame C is aligned with the symmetry axis of the rod. For our special collision scenario (see post II) previous calculations (see post V) predicted that the slot should appear aligned with the rod’s orientation in frame C and the standard x_C-axis there:

\[ \Psi_s = 0 \,. \]

We also expect that the slot’s length L_slot,C in C fulfills:

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

In addition we know by our scenario setup and the LT between A and C:

\[ L_{rod,C} = \, L \,. \]

All of these equations should be reproduced by a direct LT from B to C – guided by the relative velocity between the rod and the slot. This is the main objective of the following steps.

When you think about it in the light of what we already know this is not a trivial endeavor. Instead it is a major consistency check for the equivalence of the collision prediction between different inertial frames.

Hurdle: The relative velocity vector has different inclination angles with the presently chosen x-axes in our frames B and C

The 5th post of this series provided us with some formulas regarding

the spatial components of the relative velocity vector v_rel,_CB between the rod and the slot in frame B,
the spatial components of the relative velocity vector v_rel,_BC between the slot and the rod in frame C,
respective inclination angles of v_rel,_CB and v_rel,_BC vs. the x-axes of B and C, respectively
and the gamma-factor related to the absolute value of the relative velocity.

Due to a different time dilation between A and B in comparison to the time dilation between A and C, we got different measurement values

for the inclination angle between the vector of relative velocity v_rel,CB with the slot and the x_B-axis in frame B
and for inclination angle between the vector of relative velocity v_rel,_BC with the rod and the x_C-axis in frame C.

This did not surprise us too much as we have shown already (see posts III and IV) that the rod appears rotated vs. the x_B-axis in the slot’s frame. We take it therefore as given that the inclination angles of the relative velocity vector with the x-axes appear different in B and C, i.e. |Φ_rel,B| ≠ |Φ_rel,C|:

\[ \Phi_{rel,C} \,\ne \Phi_{rel,B}. \]

This will make our life a bit harder. A first thing we have to think about is choosing suiting spatial coordinate systems for B and C to master our task.

Suitable spatial coordinate systems for the frames attached to the rod and the slot

We use the abbreviation “ECS” for a Cartesian coordinate system (in our flat Euclidean space of Special Relativity). To be able to use formally simple form of the Lorentz Transformation leading from the slot’s frame B to the rod-attached frame C, we have to consider spatial ECS for B and C, whose x-axes are aligned with the relative velocity vector in the respective frame. We call such an ECS with such an orientation in each of the frames a “rotated ECS” below, because it is rotated vs. the standard ECS oriented like A.

Furthermore, we have to evaluate the components of the spatial vectors to event locations in each of the rotated coordinate systems with respect to the rotated x– and y-axes there. I have illustrated this in the following drawing:.

Projections of rod and slot extensions onto axis of rotated ECS in B

Illustration 3: Rotated ECS in frame B with x-axis aligned to the relative velocity vector v_rel,_CB – and projections of rod and slot to the rotated coordinate axes

Our approach for the LT from B to C therefore consists of the following steps:

Determine the components of the relative velocity vector of the rod in the slot’s frame B and its respective angle Φ_rel,B with the x_B-axis.
Create an ECS with an x-axis (x_B,rel – axis) aligned with the relative velocity vector.
Perform a projection of the spatial vectors describing the position of rod and slot-related events to get components parallel and perpendicular to the the relative velocity vector. This corresponds to determining the spatial event coordinates in the rotated ECS.
Apply the standard Lorentz Transformation to the event-coordinates in the rotated ECS (with a gamma-factor derived from the absolute value of the relative velocity).
Re-transform the resulting coordinates parallel and perpendicular to the relative velocity vector in frame C to coordinates of the standard ECS in C, whose x-coordinate axis is aligned with the rod.

In a later post we will perform an analogous reverse transformation from C to B.

Calculation steps in frame B

According to our previous post V, the components , the length and the gamma-factor for the relative velocity vector between frames C and B (i.e. between the rod and the slot) are given by:

\[ \begin{align} \pmb{\operatorname{B}} : & \pmb{v}_{rel,CB} = \pmb{v}_{rod,B} \, = \, \left( \, \nu_x, \,\, -{1 \over \gamma_x} \, v_y \, \right) \,, \\[8pt] & v_{rel} = | \pmb{v}_{rel,CB} | = | \pmb{v}_{rel,BC} | = \left[ \, {1 \over \gamma_x^2} \, v_y^2 \, + \, \nu_x^2 \, \right]^{1/2} \,, \\[8pt] & \quad \quad = \left[\, {1 \over \gamma_y^2}\, \nu_x^2 \,+\, v_y^2 \, \right]^{1/2} \,, \\[8pt] & \gamma_{rel} = \left[1 – {v_{rel}^2 \over c^2} \right]^{-1/2} \, = \, \gamma_x * \gamma_y \,\,. \end{align} \]

Here we have used the velocity values ν_x and v_y of our collision setup (see post II). Remember that we have already shown that the absolute value of v_rel is, of course, the same in both frames B and C, and that the respective gamma-factor has the given simple form (see post V).

The values of the angles Φ_rel,B and Ψ_r are given by (see posts IV, V):

\[\begin{align} \cos \left(|\Phi_{rel,B}| \right) = {\nu_x \over v_{rel} }, &\quad \sin \left(|\Phi_ {rel,B}|\right) = {1 \over \gamma_x} \,{v_y\over v_{rel} } \, , \\[10pt] \cos \left(\Psi_r \right) = {1 \over \gamma_x} \, {L \over L_{rod,B}}, &\quad \sin \left(\Psi_r \right) = {\nu_x\, v_y \over c^2} \, {L \over L_{rod,B}} \,. \end{align}\]

I have used the absolute values of the angle Φ_rel,B, because we had defined it to be negative in post V. For the rest of our calculations below, we focus on absolute angle values. L_rod,B is the total length of the rod as seen in frame B (see post IV for details). Looking at the diagrams above we see that the projections of the rod and the slot parallel and perpendicular to the relative velocity vector are given by

\[\begin{eqnarray*} &\mbox{rod : }&\quad x_{r,v,B} = L_{rod,B} * \cos \left(|\Phi_{rel,B}| + \Psi_r\right) , &\quad y_{r,v,B} = L_{rod,B} * \sin \left(|\Phi_{rel,B}| + \Psi_r\right) \,, \\[10pt] &\mbox{slot : }&\quad x_{s,v,B} = L * \cos \left(|\Phi_{rel,B}|\right) , &\quad y_{s,v,B} = L * \sin \left(|\Phi_{rel,B}|\right) \, . \end{eqnarray*}\]

At t_B = t_C = 0 the left ends of the rod and the slot coincide with the origin of the coordinates systems of B (see posts I, II). With the help of some elementary trigonometry

\[ \begin{align} \cos \left(|\Phi_{rel,B}| + \Psi_r\right) &= \cos \left(|\Phi_{rel,B}| \right) * \cos \left(\Psi_r\right) \,-\, \sin \left(|\Phi_{rel,B}| \right) * \sin \left(\Psi_r\right) \,, \\[10pt] \sin \left(|\Phi_{rel,B}| + \Psi_r\right) &= \sin \left(|\Phi_{rel,B}| \right) * \cos \left(\Psi_r\right) \,+\, \cos \left(|\Phi_{rel,B}| \right) * \sin \left(\Psi_r\right) \,, \end{align} \]

we get for the projected components of the spatial vector to the right end of the rod at t_B = 0:

Rod’s vector components parallel and perpendicular to relative velocity vector

\[ \begin{align} x_{r,v,B} \,&=\, \left[ \, { 1\over \gamma_x} \,{\nu_x \over v_{rel}} \,-\, {1\over \gamma_x} \,{v_y \,\beta_x\, \beta_y \over v_{rel}} \,\right] * L \\[8pt] &= {1 \over \gamma_x} \, {\nu_x \over v_{rel}} \left[ 1 – \beta_y^2 \right] \,=\, {1\over \gamma_x}\,{1\over \gamma_y^2} \, {\nu_x \over v_{rel}} * L \,. \end{align} \]

\[ \begin{align} y_{r,v,B} \,&=\, \left[ \, { 1\over \gamma_x} \,{v_y \over v_{rel}} \, { 1\over \gamma_x} \, + \, { \nu_x \over v_{rel} } \,\beta_x\, \beta_y \,\right] * L \\[8pt] &= {v_y \over v_{rel}} \, \left[ \, {1 \over \gamma_x^2} + \beta_x^2 \, \right] * L \,=\, {v_y \over v_{rel}} * L \,. \end{align} \]

For the slot’s components we get:

Slot’s vector components parallel and perpendicular to relative velocity vector

\[ x_{s,v,B} \,=\, {\nu_x \over v_{rel}} * L, \quad y_{s,v,B} \,=\, {1 \over \gamma_x} \, {v_y \over v_{rel}} * L \,. \]

Event selection for the transformation of the rod’s coordinates

We now have to consider which events we select for the Lorentz transformation to frame C. The event coordinates for the rod’s left and right ends, [x_r,l,v,B, y_r,l,v,B, t_r,l,v,B] and [x_r,r,v,B, y_r,r,v,B, t_r,r,v,B], respectively, should chosen such that they get transformed to simultaneous time coordinates t_r,r,v,C = t_r,l,v,C .

Why is this? Because the observer in C naturally watches or measures different spatial data for the rod (or the slot) there at the same point in time. And because our reference data in post V were based on simultaneity of measurements in C. For reasons of simplicity we choose our time coordinates to be t_C = t_B = 0 .

The coordinates of some distinct event at a distance x_v,B along the rotated axis x_B,rel occurring at time t_v,B will be transformed to respective coordinates in frame C as

\[ \begin{align} x_{v,C} &= \gamma_x\, \gamma_y * \left( \, x_{v,B} \,-\, v_{rel} * t_{v,B} \right) \,, \\[8pt] t_{v,C} &= \gamma_x\, \gamma_y * \left(t_{v,B} \,-\, { v_{rel} \over c^2} * x_{v,B} \right) \quad ( = 0 ) \,. \end{align} \]

So, if we want to achieve t_v,C= 0, we must request :

\[ t_{v,B} = {v_{rel} \over c^2}\, x_{v,B} \,. \]

Let us mark the x_B-position of the rod’s right end at such shifted time (t_v,B = t_r,v,B) by a hat:

\[ t_{r,v,B} \,=\, {v_{rel} \over c^2} * \hat{x}_{r,v,B} \, . \]

But, on the other side, any point of the rod (!) having a coordinate x_r,v,B at t_B = 0 will at a later time t_r,v,B have an x-coordinate (in B) given by

\[ \hat{x}_{r,v,B} = x_{r,v,B} \,+\, v_{rel} * t_{r,v,B} . \]

We must, of course, use this specific coordinate for a suitable event to fulfill the condition for its transformed time coordinate to be t_r,v,C = 0. Filling in t_r,v,B gives :

\[ \begin{align} \quad x_{r,v,B} &= \hat{x}_{r,v,B} * \left( 1 – {v_{rel}^2 \over c^2} \right) \quad \Rightarrow \\[8pt] \hat{x}_{r,v,B} &= \gamma_{rel}^2 \, x_{r,v,B} \, =\, \gamma_x\, \gamma_y * x_{r,v,B} \,. \end{align} \]

We use again a short notation for events: E^v ∼ [x_v,B, y_v,B, t_v,B]_B – this time referring to coordinates in the rotated ECs with the x_B,rel and the y_B,rel axes. I have indicated this by a sub-script “v”.

Let us call the x-, y-, t-coordinates of events regarding the left and the right ends of the rod in the rotated coordinate systems of frames B and C

x_r,l,v,B , y_r,l,v,B , t_r,l,v,B and x_r,r,v,B , y_r,r,v,B , t_r,r,v,B
and
x_r,l,v,C , y_r,l,v,C , t_r,l,v,C and x_r,r,v,C , y_r,r,v,C , t_r,r,v,C,

respectively. We want to transform from the rotated ECS in B to a rotated ECS in C.

Our considerations above mean that we should select the following events for the left end of the rod (E^v_r,l) and the right end of the rod (E^v_r,r) in frame B to get respective spatial coordinates in frame C at t_C = 0:

\[ \begin{align} \mbox{Rod’s left end :} \quad &\pmb{\operatorname{E}}_{r,l}^v \sim \left [\, x_{r,l,v,B},\,\, y_{r,l,v,B},\,\, t_{r,l,v,B} \,\right ]_B \\[8pt] & = \left [\, 0,\,\, 0,\,\, 0\,\right ]_B , \\[10pt] \mbox{Rod’s right end :} \quad &\pmb{\operatorname{E}}_{r,r}^v \sim \left [\, x_{r,r,v,B},\,\, y_{r,r,v,B},\,\, t_{r,r,v,B} \,\right ]_B \\[8pt] &= \left [\, \hat{x}_{r,v,B},\,\,\, y_{r,v,B}, \,\,\, v_{rel}/c^2 * \hat{x}_{r,v,B} \,\right ]_B \,. \end{align} \]

Event selection for the transformation of the slot’s coordinates

For the slot things are a bit simpler because the slot does not move in frame B. The relevant events are:

\[ \begin{align} \mbox{Slot’s left end :} \quad &\pmb{\operatorname{E}}_{s,l}^v \sim \left [\, x_{s,l,v,B},\,\, y_{s,l,v,B},\,\, t_{s,l,v,B} \,\right ]_B \\[8pt] & = \left [\, 0,\,\, 0,\,\, 0\,\right ]_B , \\[10pt] \mbox{Slot’s right end :} \quad &\pmb{\operatorname{E}}_{s,r}^v \sim \left [\, x_{s,r,v,B},\,\, y_{s,r,v,B},\,\, t_{s,r,v,B} \,\right ]_B \\[8pt] &= \left [\, x_{s,v,B},\,\,\, y_{s,v,B}, \,\,\, v_{rel}/c^2 * x_{s,v,B} \,\right ]_B \,. \end{align} \]

Transformation of rod-coordinates from frame B to frame C

Before we perform the Lorentz Transformation from B to C, let me remind you that we found the following value for the angle Φ_rel,B between the standard x_C-axis and the vector of the relative velocity in C in post V:

\[ \cos (\Phi_{rel,C}) = {1 \over \gamma_y} {\nu_x \over v_{rel}} \,. \]

See the following drawing:

Illustration 4: Rotated ECS in frame C with x-axis aligned to the relative velocity vector v_rel,_BC – and projections of rod and slot to the rotated coordinate axes

We get the following transformed coordinates parallel and perpendicular to the relative velocity vector in frame C:

\[ \begin{align} x_{r,r,v,C} \,&=\, \gamma_x * \gamma_y * \left[ \, x_{r,r,v,B} \,-\, v_{rel} \, {v_{rel} \over c^2} * x_{r,r,v,B} \,\right] \\[8pt] &= \, \gamma_x * \gamma_y * \left[ \, \gamma_x^2 * \gamma_y^2 * x_{r,v,B} \,- \, v_{rel} \, {v_{rel} \over c^2} * \gamma_x^2 * \gamma_y^2 * x_{r,v,B} \,\right] \\[8pt] &= \gamma_x^3* \gamma_y^3 * \left( 1 \,-\, {v_{rel}^2 \over c^2} \right) * x_{r,v,B} \,=\, \gamma_x * \gamma_y * x_{r,v,B}\\[8pt] &= \, {\gamma_x * \gamma_y \over \gamma_x * \gamma_y^2} \, { \nu_x \over v_{rel} } * L \,=\, {1 \over \gamma_y}\, {\nu_x \over v_{rel}} * L \,, \\[10pt] y_{r,r,v,C} \,&=\, y_{r,v,B} \,= \, {v_y \over v_{rel}} * L \,, \\[10pt] t_{r,r,v,C} \,&=\, \gamma_x * \gamma_y * \left[ {v_{rel} \over c^2} * x_{r,r,v,B} \,-\, {v_{rel} \over c^2} * x_{r,r,v,B} \right] = 0 \, . \end{align} \]

Note again that these coordinates are defined in the rotated coordinate system in C with an x-axis (= x_C,rel -axis) parallel to the relative velocity vector, there. These are not the coordinates given in the standard coordinate system with an x_C-axis parallel to the rod.

From the results above we can derive two interesting relations – one for length of the rod in C (here via the components in the rotated coordinate system)

\[ \begin{align} L_{rod,C} \,&=\, \left[ \, {1 \over \gamma_y^2} \, {\nu_x^2 \over v_{rel}^2} * L^2 \,+\, {v_y^2 \over v_{rel}^2} * L^2 \,\right]^{1/2} \\[8pt] & =\, \left[ \, {1 \over v_{rel}^2} \, \left( {1 \over \gamma_y^2}\, \nu_x^2 \,+\, v_y^2 \,\right) \right]^{1/2} * L \,= \, L \, \end{align} \]

and one for the cosine of the angle Θ_r,rel,C between the component x_r,l,v,C parallel to the relative velocity vector and the position vector to the endpoint of the rod in C:

\[ \begin{align} \cos \left( |\Theta_{r,rel,C} | \right) \,&=\, { x_{r,r,v,C} \over L} \,=\, {1 \over \gamma_y}\, {\nu_x \over v_{rel}} \\[8pt] &= \, \cos\left( |\Phi_{rel,C} | \right)\,. \end{align} \]

This means that the angle Ψ_rod,C between the rod and the standard x_C-axis indeed becomes 0:

\[ \Psi_{rod, C} \,=\, 0 \,. \]

Our complicated transformation from frame B to C via the relative velocity between rod and slot confirms that the rod (in a frame C co-moving with the the rod) indeed is oriented along the standard x_C-axis there and has length L_rod,C = L !

This may seem trivial, but it is not. Actually, we have in parts shown that Lorentz Transformations

between a frame A and a frame B,
between a frame A and a frame C

are consistent to a much more complex Lorentz transformation

between frame B and frame C approaching each other diagonally.

Transformation of slot-coordinates from frame B to frame C

Let us now look at the transformation of the slot’s coordinates. I leave it to the reader to go through the full LT for the component parallel to the relative velocity vector. We just the usual length contraction formula as we know that we transform to simultaneous time coordinates in C – and therefore transform results of length measurements between inertial systems.

\[ \begin{align} x_{s,r,v,C} \, &=\, {1 \over \gamma_{rel} } \, x_{s,r,v,B} \,=\, {1 \over \gamma_x * \gamma_y} \, {\nu_x \over v_{rel} } * L \, , \\[10pt] y_{s,r,v,C} \, &=\, y_{s,r,v,B} \,=\, {1 \over \gamma_x } \, {v_y \over v_{rel}} * L \,. \end{align} \]

Thus, we get the following value for the length L_slot,C of the slot in frame C (by using the components of a respective position vector in the rotated ECS in C (with an x-axis parallel to the relative velocity vector):

\[ \begin{align} L_{slot,C} \,&=\, \left[\, {1 \over \gamma_x^2 * \gamma_y^2} \, {\nu_x^2 \over v_{rel}^2} * L^2 \,+\, {1 \over \gamma_x^2 }\, {v_y^2 \over v_{rel}^2 * L^2 } \,\right]^{1/2} \\[8pt] &= \, \left[\, {1 \over \gamma_x^2} \, {1 \over v_{rel}^2} \,\left( {1 \over \gamma_y^2}\, \nu_x^2 \,+\, v_y^2 \right) * L^2 \,\right]^{1/2} \quad \Rightarrow \\[10pt] L_{slot,C} \,&=\, {1 \over \gamma_x} * L \,. \end{align} \]

What about the orientation? What is the angle Θ_s,rel,C between the slot and the x_rel,C -axis of the rotated coordinates system in C?

\[ \begin{align} \cos \left( | \Theta_{s,rel,C} | \right) \, &= \, { x_{s,r,v,C} \over L_{slot,C} } \,=\, {1\over \gamma_x*\gamma_y} {\nu_x \over v_rel} * L * ( \gamma_x * {1 \over L} ) \\[8pt] &= \, {1 \over \gamma_y} \, {\nu_x \over v_{rel} } \,= \, \cos\left( |\Phi_{rel,C} | \right)\,. \end{align} \]

It follows

\[ \Psi_s = 0 \,.\]

We obviously get an alignment of the slot with the standard x_C-axis in frame C! And, as we had hoped, the length of the slot is reduced due to length contradiction as previously predicted by a LT from frame A to frame C.

This means also that a transformation from frame B to frame C consistently predicts a collision between the rod and the slot’s surrounding plate (exactly at the expected leftmost collision point on the rod). See the following drawing:

Illustration 5: Collision as seen by the rod – predicted by a LT from the slot’s frame to the rod’s frame

Conclusion

All in all we have shown that Lorentz Transformations between 3 inertial systems consistently reproduce a collision between a rod and a slot (with the same proper length) by a relative movement in two dimensions – if the setup is explicitly done such that a collision occurs in an easily analyzable frame A :

Frame A: Setup for a collision between a rod moving vertically along the y_A-axis and a slot moving horizontally along the x_A-axis.
LT from frame A to frame B (co-moving with the slot): A collision is predicted due to a diagonal movement of the length contracted rod, which shows an inclination angle Ψ_r with the x_B-axis. The collision point on the rod with the plate surrounding the slot is defined by a proportional length contraction of the interval between the left end and the collision point in frame A.
LT from frame A to frame C (co-moving with the rod): A collision is predicted very similar to the one described in frame A, but shifted vertically into the rod’s coordinate system.
LT between frame B (co-moving with the slot) and frame C (co-moving with the rod): A collision is consistently predicted in C when we perform a LT from B to C along the vector of relative velocity.

The only missing thing now for complete consistency is a transformation from frame C to frame B. But in a way this is just a reversal of what we have done in his post. I will show the basic steps in the next post:

Special relativity and the rod/slot paradox – VII – no paradox for a collision scenario and consistency between all relevant LTs

But, we now can be very confident that there is no paradox at all for the case of an explicit collision scenario. The reader, which has followed my line of thought so far, has, of course, already identified the problematic point already, which lead to the wrong claim of a paradox in the first place.