Debate: Lorentz invariance of certain zero angles

[Pete]

1.2 Definition of rods T1 and T2

1. I can't see any useful role for $$\vec{T_1}(t,l) $$

Do you agree that it defines the inertial rod T1, which is tangent to P at t=0 in S?

2. I would expect:
$$\vec{T_2}(t,l) = \vec{P}(t) +\hat{P_t}\vec{v_p}(t)$$

The dimensions don't match in that equation.
You have displacement = displacement + displacement * velocity

Also, you're not using $$l$$ on the right hand side, and $$\hat{P_t}$$ should be a function of t.

 

[Tach]

1.2 Definition of rods T1 and T2

Do you agree that it defines the inertial rod T1, which is tangent to P at t=0 in S?

I don't understand what is its purpose, can you explain? The tangent facets rotate with the wheel, they aren't frozen in place.

The dimensions don't match in that equation.
You have displacement = displacement + displacement * velocity

I didn't think so, my understanding of your notation is that it looks like:

vector=vector+scalar_displacement*vector

so the dimensions match.

Also, you're not using $$l$$ on the right hand side, and $$\hat{P_t}$$ should be a function of t.

I don't understand the l's usefulness, so I used what I thought that was reasonable to use.

 

[Tach]

1.2 Definition of rods T1 and T2

Do you agree that it defines the inertial rod T1, which is tangent to P at t=0 in S?

Yes, it does, so what is its purpose, it doesn't have anything to do with the problem.

 

[Pete]

1.2 Definition of rods T1 and T2

I don't understand what is its purpose, can you explain?
The tangent facets rotate with the wheel, they aren't frozen in place.

T1 is an unambiguous physical tangent to point P at t=0.
I prefer it to T2, because I believe that unlike T2 it is straight in all reference frames.
I'll explore that claim further in the methodology and calculations stages.
For now, it's enough that we agree on its specification - we don't need to agree on its purpose.
It is enough that we agree that T1 is tangent to the wheel at P at t=0 in S.

I don't understand the l's usefulness, so I used what I thought that was reasonable to use.

T2 and T1 are rods, extended objects. They are made of many elements.
As defined in post 19, $$\vec{T_2}(t,l)$$ specifies the position vectors of the elements of rod T2.
$$l$$ is the parameter that distinguishes between elements.
Without $$l$$, the equation describes the position vector of a point, rather than the position vectors of every element of the rod.

I didn't think so, my understanding of your notation is that it looks like:

vector=vector+scalar_displacement*vector

so the dimensions match.

You said:

$$\vec{T_2}(t,l) = \vec{P}(t) +\hat{P_t}\vec{v_p}(t)$$

As defined in post 19:

• $$\vec{T_2}(t,l)$$ is a position vector. It has units of Length.
• $$\vec{P}(t)$$ is a position vector. It has units of Length.
• $$\hat{P_t}(t)$$ is a displacement vector (not a scalar). It has units of Length.
• $$\vec{v_P}(t)$$ is a velocity vector. It has units of Length/Time.

 

[Tach]

1.2 - Definition of rods T1 and T2

T1 is an inertial rod.
At t=0 in S, T1 is tangent to wheel element P, and moving with the same velocity as P.

T2 is a rod permanently attached to the wheel at P.
At t=0 in S, the location of T2 coincides with the location of T1 as illustrated:

T2 moves rigidly in S in such a way that it remains tangent to the wheel at P at all times.
For example, at $$t = \pi/(2\omega)$$ in S, T2 has moved around the wheel with P, while T1 has moved inertially, as illustrated:

Does that make sense? There is no coordinate dependence in that description that I can see, but here are the equations of motion in vector form anyway:
Equations of motion
$$\vec{v_P}(t)$$ is the velocity of wheel element P at time t in S.
Let $$\vec{P}(t)$$ denote the position vector of P at time t in S.
Let $$\hat{P_t}(t)$$ denote the unit displacement vector tangent to wheel element P in S at time t.

Let $$\vec{T_1}(t,l)$$ denote the position vectors of the elements of rod T1 at time t in S, where $$l$$ is distance along the rod from the element that was in contact with P at t=0.
Let $$\vec{T_2}(t,l)$$ denote the position vectors of the elements of rod T2 at time t in S, where $$l$$ is distance along the rod from P.

Thus:
$$\vec{T_1}(t,l) = \vec{P}(0) + l\hat{P_t}(0) + t\vec{v_p}(0)$$
$$\vec{T_2}(t,l) = \vec{P}(t) + l\hat{P_t}(t)$$

OK,

There is still an unresolved part, I agree with $$\vec{T_2}(t,l) = \vec{P}(t) + l\hat{P_t}(t)$$ but T1 should be : $$\vec{T_1}(t,l) = \vec{P}(0) + l\hat{P_t}(0) $$ by symmetry.

 

[Pete]

1.2 Definition of rods T1 and T2
Thanks, I agree.
$$\vec{T_1}(t,l) = \vec{P}(0) + l\hat{P_t}(0)$$

1.3 Definition of points A and B
Since we've agreed on a vector specs for the tangent rods, we don't need the points A and B, right? If things get confused later on, we could always revisit.

 

[Pete]

1.2 Definition of rods T1 and T2

Thanks, I agree.
$$\vec{T_1}(t,l) = \vec{P}(0) + l\hat{P_t}(0)$$

Sorry, I replied too hastily.

T1 is moving in S with velocity $$\vec{v_p}(0)$$.
ie at t=0, when rod T1 is tangent to wheel element P, it also has the same velocity as P.
So at t=0:
$$\vec{T_1}(0,l) = \vec{P}(0) + l\hat{P_t}(0)$$
...but at other times, we need to add a displacement due to its motion:
$$\vec{T_1}(t,l) = \vec{P}(0) + l\hat{P_t}(0) + t\vec{v_p}(0)$$

 

[Tach]

1.2 Definition of rods T1 and T2

Sorry, I replied too hastily.

T1 is moving in S with velocity $$\vec{v_p}(0)$$.
ie at t=0, when rod T1 is tangent to wheel element P, it also has the same velocity as P.
So at t=0:
$$\vec{T_1}(0,l) = \vec{P}(0) + l\hat{P_t}(0)$$
...but at other times, we need to add a displacement due to its motion:
$$\vec{T_1}(t,l) = \vec{P}(0) + l\hat{P_t}(0) + t\vec{v_p}(0)$$

Do you mean that $$\vec{T_1}(t,l)$$ keeps moving in its original plane (in the vertical direction, if we use your second drawing)? If that is the case, I fail to see what purpose is it serving.

 

[Pete]

1.2 Definition of rods T1 and T2

Do you mean that $$\vec{T_1}(t,l)$$ keeps moving in its original plane (in the vertical direction, if we use your second drawing)?

Yes.

If that is the case, I fail to see what purpose is it serving.

That's OK, just as long as we agree that at t=0, T1 is tangent to P with the same instantaneous velocity as P.

 

[Tach]

1.2 Definition of rods T1 and T2


Yes.

That's OK, just as long as we agree that at t=0, T1 is tangent to P with the same instantaneous velocity as P.

Ok, whatever

 

[Pete]

Cool. So we've agreed on the scenario, and can move on to methodology:

Stage 2 - methodology

1. We will take one post each to:
   • describe the measurements that we would like to make in each reference frame,
   • define how the measurements will be made
2. We will then discuss the suggested measurements, suggesting improvements, asking clarification, etc until we agree that the set of measurements are sufficiently well defined and mutually understood.
3. No suggested measurement may be refused, unless it can not be sufficiently well defined.
4. Disagreements about how to refer to each measurement must also be resolved at this stage.

I've half done a methodology post, but tomorrow is new kitchen day so I might not finish and post it for a day or two.

 

[Tach]

2. Methodology

Two identical lasers are attached:
- One to a square bracket mounted at the end of the spoke OP where O is the axle point and P is the point moving at speed $$\vec{v_t}=\vec{\omega} X \vec{R}$$ . This laser realizes the embodiment of the tangential speed

-The second one is attached to the rod T2, parallel to its planar surface . The second laser realizes the embodiment of the tangent microfacet

Since the observer located in O can only measure the orientation of both the velocity $$\vec{v_t}$$ and that of the rod [v]T2[/b] via remote observation (both entities are remote and they move wrt O) , the proposed method of determining if they are colinear is to observe the effect of the exact transverse Doppler effect on the frequencies arriving from each laser. Since the lasers have colinear orientation, in the frame of the axle, the observed frequencies should be identical.

 

[Pete]

2.0 Methodology
S' is a reference frame boosted relative to S by velocity v along the x-axis.

In both S and S', I propose to measure the angle between the instantaneous orientation of wheel element P and the instantaneous velocity of wheel element P, at the time when P is in contact with T1.

The instantaneous velocity of P in S is already given by $$\vec{v_p}(t)$$, and by the appropriately transformed $$\vec{v'_p}(t')$$ in S'.
If necessary, the instantaneous velocity of P can be explicitly measured by measuring the displacement over time of an inertial object that is moving with P at the instant of interest.

The instantaneous orientation of wheel element P is given by any displacement vector tangent to the wheel at P at the given instant.
Let $$\hat{D_p}(t)$$ be the unit displacement vector tangent to the wheel at P in S at time t.
Let $$\hat{D'_p}(t')$$ be the unit displacement vector tangent to the wheel at P in S' at time t'.
If necessary, whether a given displacement vector is tangent to the wheel can be tested in at least two ways:

• A displacement vector is tangent to the wheel at a given point at a given instant if it is a scalar multiple of the displacement between two points on a straight rod tangent to the wheel at that point at that instant.
• A displacement vector is tangent to the wheel at a given point at a given instant if and only if there is no other point on the wheel at that instant such that the displacement between the given point and the other point is a scalar multiple of the displacement vector.

The angle between the instantaneous orientation of P and the instantaneous velocity of P is given by the angle between the two defined vectors, that is:

• The angle between $$\vec{v_p}(t)$$ and $$\hat{D_p}(t)$$ in S.
• The angle between $$\vec{v'_p}(t')$$ and $$\hat{D'_p}(t')$$ in S'.

If necessary, this angle 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

 

[Tach]

2.0 Methodology


The angle between the instantaneous orientation of P and the instantaneous velocity of P is given by the angle between the two defined vectors, that is:

• The angle between $$\vec{v_p}(t)$$ and $$\hat{D_p}(t)$$ in S.
• The angle between $$\vec{v'_p}(t')$$ and $$\hat{D'_p}(t')$$ in S'.

Agreed, this is what I had above.

If necessary, this angle 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

While this is theoretically possible, I do not see how you plan to implement it practice, it needs a lot more in terms of actual implementation. I suggest that we use my implementation, based on the sensing the differences in the transverse Doppler effect induced in the frequency of the two lasers. Agree?

 

[Pete]

We can do both.
Which particular bit of the quoted section has the problem?

 

[Tach]

We can do both.
Which particular bit of the quoted section has the problem?

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.

 

[Pete]

Sorry for the delay. Doing home renovations.

2.0 methodology

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.

$$\vec{v_t}$$ and $$\vec{T_2}$$ are not mentioned in my proposal, so I don't understand what you don't understand.

Also, I don't think that remote observations have to be a problem.
We can use remote detectors, or use an implicit background of rods and synchronized clocks to define event coordinates.


I'm not sure whether it's best to address your methodology now, or if it will get confusing discussing both at once. Let's see...

2. Methodology

Two identical lasers are attached:
- One to a square bracket mounted at the end of the spoke OP where O is the axle point and P is the point moving at speed $$\vec{v_t}=\vec{\omega} X \vec{R}$$ . This laser realizes the embodiment of the tangential speed

If I understand this correctly, you have a laser attached to P.
Is this laser aimed in any particular direction?

$$\vec{R}$$ seems to be the time-varying position vector of P in S, which we've already labelled as $$\vec{P}(t)$$.

-The second one is attached to the rod T2, parallel to its planar surface . The second laser realizes the embodiment of the tangent microfacet

So this laser is aimed parallel to T2.
Where on T2 will this laser be attached?
I'm assuming it's not at P, otherwise it would seem redundant.

Since the observer located in O can only measure the orientation of both the velocity $$\vec{v_t}$$ and that of the rod [v]T2[/b] via remote observation (both entities are remote and they move wrt O) , the proposed method of determining if they are colinear is to observe the effect of the exact transverse Doppler effect on the frequencies arriving from each laser. Since the lasers have colinear orientation, in the frame of the axle, the observed frequencies should be identical.

I don't understand the laser on T2.
Is it aimed parallel to T2, or is it aimed so as to hit O?

If I understand correctly, we have lasers have a known proper frequency, the frequency of light from the lasers will be measured at O, and the observed doppler shift will be used to draw conclusions about the lasers' motion relative to O.

But that wouldn't give you any measure of the orientation of the tangent rod, only its velocity. Is that what you intend, or am I misunderstanding?

Also, do you intend to make measurements in any other reference frame?

 

[Tach]

Sorry for the delay. Doing home renovations.

2.0 methodology



$$\vec{v_t}$$ and $$\vec{T_2}$$ are not mentioned in my proposal, so I don't understand what you don't understand.

Look at your own definitions. $$\vec{v_t}$$ is your $$\vec{v_P}$$. $$\vec{T_2}$$ is your own $$\vec{T_2}$$.

Also, I don't think that remote observations have to be a problem.
We can use remote detectors, or use an implicit background of rods and synchronized clocks to define event coordinates.

Actually they are. I would like you to describe exactly how you plane 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.

If I understand this correctly, you have a laser attached to P.
Is this laser aimed in any particular direction?

Yes, as I explained earlier, it is aimed perpendicular to OP, i.e. in the direction of $$\vec{v_P}$$, using your notation.

$$\vec{R}$$ seems to be the time-varying position vector of P in S, which we've already labelled as $$\vec{P}(t)$$.


So this laser is aimed parallel to T2.
Where on T2 will this laser be attached?
I'm assuming it's not at P, otherwise it would seem redundant.

The second laser can be anywhere in the tangent plane, not connected in any fashion to point P.

I don't understand the laser on T2.
Is it aimed parallel to T2, or is it aimed so as to hit O?

As explained quite clearly, it is aimed "parallel to $$\vec{T_2}$$".
The two lasers are used exactly the way laser based level checkers are used in daily practice.

If I understand correctly, we have lasers have a known proper frequency, the frequency of light from the lasers will be measured at O, and the observed doppler shift will be used to draw conclusions about the lasers' motion relative to O.

But that wouldn't give you any measure of the orientation of the tangent rod, only its velocity. Is that what you intend, or am I misunderstanding?

You are understanding only partially. In the frame of the axle , the observer placed at O measures a precise TDE (Transverse Doppler Effect) frequency of $$f_0/\gamma$$ where $$f_0$$ is the native frequency of the lasers IF and ONLY IF the two lasers have directions exactly perpendicular to $$\vec{OP}$$. In other words, by detecting the expected frequency of exactly $$f_0/\gamma$$, the observer in O can determine with precision if $$\vec{v_P}(t)$$ is parallel with $$\vec{T_2}$$

Also, do you intend to make measurements in any other reference frame?

Yes, the same method is used in any other frame, except , of course that we will not see a TDE.

 

[Pete]

2.1 Tach's proposed measurements

Yes, as I explained earlier, it is aimed perpendicular to OP, i.e. in the direction of $$\vec{v_P}$$, using your notation.

The second laser can be anywhere in the tangent plane, not connected in any fashion to point P.

As explained quite clearly, it is aimed "parallel to $$\vec{T_2}$$".
The two lasers are used exactly the way laser based level checkers are used in daily practice.

So like this?

You are understanding only partially. In the frame of the axle , the observer placed at O measures a precise TDE (Transverse Doppler Effect) frequency of $$f_0/\gamma$$ where $$f_0$$ is the native frequency of the lasers IF and ONLY IF the two lasers have directions exactly perpendicular to $$\vec{OP}$$.

I don't understand how the laser frequency can be measured at O.

It seems to me that:
The lasers are not aimed toward O.
The laser beams do not meet O.
The laser frequency can't be measured at O.

 

[Pete]

Updated tracking list

1.0 Scenario (Complete)

1.1 - Coordinate dependence vs. coordinate independence (Resolved)
1.2 - Definition of rods T1 and T2 (Resolved)
1.3 - Definition of points A and B (Obsolete)​

2.0 Methodology (Active)

2.1 - Tach's proposed measurements (Active)
2.2 - Pete's proposed measurements (Pending)
2.3 - Measuring remote events using background Rods and Clocks (Pending)​

3.0 Calculations (not started)

4. Summary and reflection (not started)​