**2.2 Pete's proposed measurements**

$$\vec{T_2}(t,l)$$ was defined as the position vectors of the elements ofLook at your own definitions. $$\vec{v_t}$$ is your $$\vec{v_P}$$. $$\vec{T_2}$$ is your own $$\vec{T_2}$$.$$\vec{v_t}$$ and $$\vec{T_2}$$ are not mentioned in my proposal, so I don't understand what you don't understand.I do not understand how you propose that the remote observer (comoving with the axle) measures the co-planarity of the two remote vectors $$\vec{v_t}$$ and $$\vec{T_2}$$ in your proposal.

**T2**in

**S**.

You seem to be using $$\vec{T_2}$$ to refer to a displacement vector parallel to

**T2**in

**S**.

Can we please stick to consistent labels?

But that's a side issue. In my methodology post, I'm not proposing any measurements on

**T2**.

I propose to measure the angle between $$\vec{v_p}(t)$$ and $$\hat{D_p}(t)$$, when t=0, and between $$\vec{v_p}'(t')$$ and $$\hat{D_p}'(t')$$ at the t' of the transformed event of

**P**at t=0.

$$\hat{D_p}(t)$$ and $$\hat{D_p}'(t')$$ are unit displacement vectors tangent to the wheel at

**P**in

**S**and

**S'**at times t and t' respectively.

Actually they are. I would like you to describe exactly how you plan to use "rods and clocks" or "remote detectors", whichever you prefer in order to measure the fact that $$\vec{T_2}$$ is parallel to $$\vec{v_P}$$. Even in the axle frame , where you know that they are parallel, you are still forced to do a remote measurement.

Rods and clocks:

An inertial reference frame is a conceptual, three-dimensional latticework of measuring rods set at right angles to each other with clocks at every point that are synchronised with each other...

So, the coordinates of any event can be determined by noting which clock is at the location of the event, and the time on that clock when the event occurs.

The event coordinates can then be used to determine other measurements.

For example, the position and orientation of an object at a particular time can be measured by noting which clocks are coincident with the object when they read that particular time.

Personally, I prefer the implicit use of the rods and clocks latticework, but I did suggest a more direct (but cumbersome) measurement in my first methodology post:

The angle between $$\vec{v_p}(t)$$ and $$\hat{D_p}(t)$$ can be directly measured with a protractor as the angle between:

- The path followed by an inertial object that is moving with P at the instant of interest, and
- A rod at rest that that is coincident with a straight rod tangent to the wheel at P at the instant of interest

To explain further:

Construct a background canvas, parallel to the XY plane, at rest in

**S**.

- Attach a marker to rod
**T1**at the point that touches**P**.

The marker attached to**T1**will mark on the canvas a physical line that matches the direction of $$\vec{v_p}(0)$$. - Attach markers along the length of
**T1**that mark the canvas only at t=0

The resulting mark is a straight line that matches the direction of $$\hat{D_p}(0)$$ - The angle between the two lines is the angle between $$\vec{v_p}(0)$$ and $$\hat{D_p}(0)$$

The same process works in

**S'**:

Construct a background canvas, parallel to the XY plane, at rest in

**S'**.

Let $$t'_0$$ be the time in S' when

**P**touches

**T1**.

- Attach a marker to rod
**T1**at the point that touches**P**.

The marker attached to**T1**will mark on the canvas a physical line that matches the direction of $$\vec{v_p}'(t'_0)$$. - Attach markers along the length of
**T1**that mark the canvas only at t'=$$t'_0$$

The resulting mark is a straight line that matches the direction of $$\hat{D_p}'(t'_0)$$ - The angle between the two lines is the angle between $$\vec{v_p}(t'_0)$$ and $$\hat{D_p}(t'_0)$$