Debate: Lorentz invariance of certain zero angles

[Pete]

Full topic:

The zero angle between the tangent plane of a microfacet and the tangent velocity as measured in the axle frame transforms into a zero angle through a boost along the x-axis in the ground frame​

For: Tach
Against: Pete

Rules are as agreed in the Proposal thread:
[thread=111135]Proposal: A surface moving parallel to itself in one frame is not doing so in all frames[/thread]
[thread=111143]Discussion thread[/threaD]

Tracking list

1.0 Scenario (Pending)
1.1 - Coordinate dependence vs. coordinate independence (Resolved)
1.2 - Definition of rods T1 and T2 ("Active")
1.3 - Definition of points A and B (Pending)

2.0 Methodology (not started)

3.0 Calculations (not started)

4. Summary and reflection (not started)​

 

[Tach]

A. Scenario: The zero angle between the tangent plane of any microfacet belonging to a circular wheel and the tangent velocity as measured in the axle frame transforms into a zero angle through a boost along the x-axis in the ground frame


B. Methodology and Calculations:

1. In the frame co-moving with the axle, the trajectory of a point on the circumference is:

$$x=r cos(\omega t)$$
$$y=r sin(\omega t)$$

where $$\omega$$ is the angular speed of the wheel and r is its radius.

The tangential speed of a particle $$\vec{v_p}$$ is identical to the speed of the tangent plane in any point $$\vec{v_p}=\vec{v_t}=(-r \omega sin(\omega t), r \omega cos (\omega t))$$, a well known fact. So,the angle between $$\vec{v_p}$$ and $$\vec{v_t}$$ is ZERO everywhere in the axle frame.

2. In the ground frame

The axle moves with speed $$\vec{V}=(V,0)$$ so:

$$\vec{v'_p}=\vec{v'_t}=(\frac {V-r \omega sin (\omega t)}{1-Vr \omega sin(\omega t) }, \frac{r \omega/ \gamma(V) cos(\omega t)}{1-Vr \omega sin(\omega t)})$$

so, not only that the two speeds are identical (of course they are), their angle is ALSO ZERO in the ground frame. This is true for both relativistic and "classical" case . It is also true whether the wheel motion has slippage or not.

The "no slip" situation is a particular case of the above, $$r \omega=V$$

$$\vec{v'_p}=\vec{v'_t}=(\frac{V(1- sin (\omega t))}{1-V^2 sin(\omega t) }, \frac{V/\gamma(V) cos(\omega t)}{1-V^2 sin(\omega t) })$$

Of course, the same holds, the zero angle in the axle frame transforms into a zero angle in the ground frame.

 

[Pete]

1.0 Scenario

Scenario: The zero angle between the tangent plane of any microfacet belonging to a circular wheel and the tangent velocity as measured in the axle frame transforms into a zero angle through a boost along the x-axis in the ground frame

To be clear:

It seems that the scenario you are describing is that of a wheel rolling without slipping along the ground.
In reference frame S, the axle is at rest at (x,y)=(0,0), the wheel is circular with radius r, and is spinning with angular velocity $$\omega$$.

P is a point on the circumference, located at (x,y)=(r,0) at t=0.
$$v_p$$ is the instantaneous velocity of point P at t=0 in frame S.

You mention a moving tangent plane, which is not clearly defined.
I would like add physical realisation of a specific tangent plane to the scenario:
T is a straight rod
In frame S:
At t=0, T is tangent to the wheel at point P.
T is moving inertially parallel to the y axis, with velocity $$\vec{v_t} = \vec{v_p}$$.

Diagram of frame S at t=0:

Does this match what you had in mind, and is it sufficiently well defined?

 

[Tach]

You mention a moving tangent plane, which is not clearly defined.
I would like add physical realisation of a specific tangent plane to the scenario:
T is a straight rod
In frame S:
At t=0, T is tangent to the wheel at point P.
T is moving inertially parallel to the y axis, with velocity $$\vec{v_t} = \vec{v_p}$$.

Yes, but much more general, T is moving in a direction perpendicular on the position vector connecting the center of the wheel and the point P, so $$\vec{v_t(t)}=\vec{v_p(t)}$$ for all values of time, t and for all points P on the circumference. This is general stuff, well known so I did not think I needed to get that specific but now we are as specific as we could be.

 

[Pete]

1.0 Scenario - Complete

OK, I'll edit the tracking list to mark the Scenario stage as "Complete", and the methodology stage as active, if you agree.

I'll prepare a list of the measurements I want us to consider, along with how they will be measured. You do the same (feel free to copy from your first post if you have nothing more to add.)

Don't forget to add a header to your post, like this one. This might seem anal now, but I think it will really help to keep thing on track later on.

 

[Tach]

1.0 Scenario - Complete

OK, I'll edit the tracking list to mark the Scenario stage as "Complete", and the methodology stage as active, if you agree.

I'll prepare a list of the measurements I want us to consider, along with how they will be measured. You do the same (feel free to copy from your first post if you have nothing more to add.)

Don't forget to add a header to your post, like this one. This might seem anal now, but I think it will really help to keep thing on track later on.

Cool.

 

[Pete]

1.0 Scenario

Yes, but much more general, T is moving in a direction perpendicular on the position vector connecting the center of the wheel and the point P, so $$\vec{v_t(t)}=\vec{v_p(t)}$$ for all values of time, t and for all points P on the circumference. This is general stuff, well known so I did not think I needed to get that specific but now we are as specific as we could be.

Wait. You edited this post as I was responding.

I think I know what you mean, but I don't think T can be made so general and still be well defined.
Let's keep to something specific enough that we can actually analyse it.

We've chosen a specific point on the circumference.
T is a specific object that is tangent to that point at a specific time.

It is moving inertially, so it can be easily transformed between frames, and remains straight in all frames.

Agreed?

Or, perhaps would you like to add another tangent rod?

• T1 is a straight rod, moving inertially with $$\vec{v_t} = \vec{v_p}$$ at t=0 in S, and is tangent to the wheel at P at t=0 in S.
• T2 is a straight (in S) rod, tangent and attached to the wheel at P at all times

 

[Tach]

1.0 Scenario


Wait. You edited this post as I was responding.

I think I know what you mean, but I don't think T can be made so general and still be well defined.
Let's keep to something specific enough that we can actually analyse it.

We've chosen a specific point on the circumference.
T is a specific object that is tangent to that point at a specific time.

It is moving inertially, so it can be easily transformed between frames, and remains straight in all frames.

Agreed?

Or, perhaps would you like to add another tangent rod?

• T1 is a straight rod, moving inertially with $$\vec{v_t} = \vec{v_p}$$ at t=0 in S, and is tangent to the wheel at P at t=0 in S.
• T2 is a straight (in S) rod, tangent and attached to the wheel at P at all times

I have big issues with using coordinate-dependent proofs. As such, I prefer the use of coordinate independent entities, like vectors over points. So, for me, the tangent is defined by a pair of time varying points, let's call T_1(t) and T_2(t), or, better by the time varying vector
$$\vec{v_t}=\vec{T_1T_2}$$ which is nothing but the velocity of the tangent plane expressed in the axle frame. Makes sense? Did you notice that none of my proofs contains any coordinate-dependent entities? This approach makes the proofs simpler and less susceptible to error.

 

[Pete]

1.1 Coordinate dependence

Sorry, I don't follow what you mean by "coordinate dependent".

Are you saying that the rods T1 and T2 are a coordinate dependent entities?

I also don't understand how you are defining T_1(t) and T_2(t).

 

[Tach]

1.1 - Coordinate dependence vs. coordinate independence

1.1 Coordinate dependence

Sorry, I don't follow what you mean by "coordinate dependent".

Coordinate-dependent means writing formalisms in terms of point coordinates , as in (x,y,z). By contrast, a formulation in terms of vectors is considered coordinate - independent.

Are you saying that the rods T1 and T2 are a coordinate dependent entities?

I also don't understand how you are defining T_1(t) and T_2(t).

No, what I was saying was that choosing T as a point results into a coordinate dependent formulation. Likewise , taking points T1(x1,y1,z1) and T2(x2,y2,z2) as two points defining the tangent results into a coordinate-dependent formulation. On the other hand, working with the vector $$\vec{v_t}=\vec{T_1T_2}$$ results into a coordinate-independent formalism. T1 and T2 are points, not rods. Does this make sense now?

 

[Pete]

1.1 - Coordinate dependence vs. coordinate independence

Coordinate-dependent means writing formalisms in terms of point coordinates , as in (x,y,z). By contrast, a formulation in terms of vectors is considered coordinate - independent.

I don't understand why using coordinates is a problem, as long as the coordinate system is well defined.
I note that in your opening post to this thread, you've specified the trajectory of a specific point on the circumference in terms of coordinates:

1. In the frame co-moving with the axle, the trajectory of a point on the circumference is:
$$x = r\cos(\omega t) \\
y = r\sin(\omega t)$$

This is perfectly clear, given a couple of straightforward assumptions about the origin and orientation of the coordinate system.

And I didn't notice you having a problem with coordinates in previous threads either (in [post=2847465]shape of a relativistic wheel[/post], for example), so I don't understand why it's a problem now.

I am not skilled in the formalism of using vectors, so if you insist on going down that path the pace of the discussion will become much slower.

 

[Tach]

1.1 - Coordinate dependence vs. coordinate independence

I don't understand why using coordinates is a problem, as long as the coordinate system is well defined.
I note that in your opening post to this thread, you've specified the trajectory of a specific point on the circumference in terms of coordinates:

Yes, I did just in order to be able to represent the components of the vectors $$\vec{v_p}$$, $$\vec{v_t}$$

And I didn't notice you having a problem with coordinates in previous threads either (in [post=2847465]shape of a relativistic wheel[/post], for example), so I don't understand why it's a problem now.

...because that thread was about shapes, more exactly about the implicit representation of a closed curve (the ellipse).

I am not skilled in the formalism of using vectors, so if you insist on going down that path the pace of the discussion will become much slower.

I don't insist, use what you feel more comfortable with and I will use what I think is more appropriate, including the coordinate-independent formalism. So, at times we may have to reconcile the two.

 

[Pete]

Ok, thanks for that.

Definition of T1 and T2

No, what I was saying was that choosing T as a point results into a coordinate dependent formulation. Likewise , taking points T1(x1,y1,z1) and T2(x2,y2,z2) as two points defining the tangent results into a coordinate-dependent formulation. On the other hand, working with the vector $$\vec{v_t}=\vec{T_1T_2}$$ results into a coordinate-independent formalism. T1 and T2 are points, not rods. Does this make sense now?

In post 3, I suggested adding a rod, labelled T, to the scenario.
In post 7, I suggested adding another rod, labelled T2, and relabelling T as T1:

Or, perhaps would you like to add another tangent rod?

• T1 is a straight rod, moving inertially with at t=0 in S, and is tangent to the wheel at P at t=0 in S.
• T2 is a straight (in S) rod, tangent and attached to the wheel at P at all times

• 
  
  Is there a coordinate dependence problem with the specification of those rods?
  
  In post 8, you suggested defining T1 and T2 as points:
  

  ...for me, the tangent is defined by a pair of time varying points, let's call T_1(t) and T_2(t)...

  
  But I don't understand exactly how those points are defined.

 

[Tach]

In post 8, you suggested defining T1 and T2 as points:


But I don't understand exactly how those points are defined.

Points T1 and T2 are simply two points defining the tangent in T.
since you aren't using the coordinate independent formalism, you will need two points to define the tangent. They can be T1 and T2 or T1 and T or whatever.

 

[Pete]

Definition of T1 and T2
Sorry, I still don't understand how you are defining T1 and T2 as points. Can you be more explicit?

Also,
Do you understand that when T1 and T2 were originally mentioned in post 7, they were defined as rods, not points:

Or, perhaps would you like to add another tangent rod?

• T1 is a straight rod, moving inertially with at t=0 in S, and is tangent to the wheel at P at t=0 in S.
• T2 is a straight (in S) rod, tangent and attached to the wheel at P at all times

Is there any problem with that specification?

Remember, we agreed to answer direct questions in the next post.

 

[Tach]

Definition of T1 and T2
Sorry, I still don't understand how you are defining T1 and T2 as points. Can you be more explicit?

I am defining them different from you, xcall them A and B, ok?

Also,
Do you understand that when T1 and T2 were originally mentioned in post 7, they were defined as rods, not points:

Is there any problem with that specification?

Remember, we agreed to answer direct questions in the next post.

I answered the direct question, I am not using any rods, I am defining the tangent through two points. See what happens when you insist in using coordinate-dependent representations? A lot of confusion and wasted time. Please, let's use the tangent vector $$\vec{v_t}$$ and be done with this.

 

[Pete]

Definition of rods T1 and T2

I am defining them different from you, xcall them A and B, ok?

Agree, we will call your two tangent-defining points A and B.
We will call the two tangent rods T1 and T2.

I answered the direct question

No, you did not.
The question was whether there is a problem with the specification of the rods, specifically a coordinate dependence problem.

Is there a problem with the way the rods are specified?

You seem to imply that they are defined in a coordinate dependent way, but I don't understand why.

 

[Tach]

Definition of rods T1 and T2

Agree, we will call your two tangent-defining points A and B.
We will call the two tangent rods T1 and T2.


No, you did not.
The question was whether there is a problem with the specification of the rods, specifically a coordinate dependence problem.

Is there a problem with the way the rods are specified?

You seem to imply that they are defined in a coordinate dependent way, but I don't understand why.

Can you do a drawing for each of the two rods and add their equations of motion? Thank you

 

[Pete]

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

 

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

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

2. I would expect:

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