Stone's derivation of Thomas rotation

[yuiop]

Therefore, perpendicular components of velocity don't contribute to length contraction.

I think you meant to say something like spatial components that are perpendicular to the velocity are not subject to length contraction. Perpendicular components of the velocity contribute to the magnitude and direction of the total velocity vector and therefore contribute to the magnitude and orientation of the length contraction.

 

[starthaus]

Are you sure about this?

The first part:

2\frac{1-\gamma}{\gamma}+(\frac{1-\gamma}{\gamma})^2=\frac{1-\gamma^2}{\gamma^2}

seems OK, but your algebra seems to go wrong after that.

Sorry, "my" algebra is correct.

I get:

\frac{1-\gamma^2}{\gamma^2} = \frac{1-(1-\beta^2)}{(1-\beta^2)} = \frac{\beta^2}{(1-\beta^2)} = \frac{\beta^2}{\gamma^2}

Maybe that is why it does not seem obvious to Mathematica?

If I am right, this means the expression x*Sqrt[1 + 2 bx^2 b^(-2) ((1 - g)/g) + bx^2 b^(-2) ((1 - g)/g)^2]
should simplify to x*Sqrt[1+bx^2/g^2].

Err , you got it all wrong , try again with the correct expression for \gamma as a function of \beta.

 

[yuiop]

I get the same as starthaus. I've been using the definition gamma = 1/Sqrt[1-beta^2], rather than gamma = Sqrt[1-beta^2].

Oops, your right. I fell into the trap of using gamma = Sqrt[1-beta^2]. Using the correct definition of gamma, the expression does simplify to -b^2 as you both got.

 

[starthaus]

I get the same as starthaus. I've been using the definition gamma = 1/Sqrt[1-beta^2], rather than gamma = Sqrt[1-beta^2].

Correct.

 

[starthaus]

I think you meant to say something like spatial components that are perpendicular to the velocity are not subject to length contraction.

I just proved that.

Perpendicular components of the velocity contribute to the magnitude and direction of the total velocity vector

Yes

and therefore contribute to the magnitude and orientation of the length contraction.

You are contradicting yourself, you got it right in the first sentence, now you are getting it wrong.

 

[Rasalhague]

In[20]:= bx = .9; by = 0.3; bz = .2; x = 1; b = Sqrt[bx^2 + by^2 + bz^2]; q = 
 x ({1, 0, 0} + bx*(Sqrt[1 - b^2] - 1)*b^(-2) {bx, by, bz}); Norm[q]

Out:= 0.43589

In[21]:= g = 1/Sqrt[
   1 - b^2]; p = {-bx^2*g*x + (1 + bx^2 (g - 1)*b^(-2))*x, -by*g*bx*
    x + x*bx*by (g - 1)*b^(-2), -bz*g*bx*x + bx*bz (g - 1)*x*b^(-2)}; Norm[p]

Out[21]:= 0.43589

In[22]:= x*Sqrt[1 - bx^2]

Out[22]:= 0.43589

In[23]:= x*Sqrt[1 + 2 bx^2 b^(-2) ((1 - g)/g) + bx^2 b^(-2) ((1 - g)/g)^2]

Out[23]:= 0.43589

But

In[24]:= x*Sqrt[1 + bx^2/g^2]

Out[24]:= 1.02401

 

[Rasalhague]

You must have b_x^2+b_y^2+b_z^2=1, you don't have that in the above.

Surely the magnitude of beta has to be less than 1 = c, doesn't it? In this case, it's 0.969536.

 

[starthaus]

Surely the magnitude of beta has to be less than 1 = c, doesn't it? In this case, it's 0.969536.

Yes, my mistake. You still have something wrong in the Mathematica expression.

 

[Rasalhague]

You still have something wrong in the Mathematica expression.

What?

 

[starthaus]

What?

Your formula is fine, you are simply stopping before getting rid of \beta_y and \beta_z. If you read the attachment, you will see how I did that.

 

[yuiop]

I just proved that.

I must have missed your proof. In all sincerity I would like to see it.

You are contradicting yourself, you got it right in the first sentence, now you are getting it wrong.

I think it is a question of semantics. I am saying that perpendicalur components of the velocity indirectly affect the length contraction. For example an object with Vx=0.6, Vy=0, Vz=0 will be length contracted by a factor of 0.8 in the x direction. An object with Vx=0.6, Vy=0.6, Vz=0 will be length contracted by a factor of about 0.529 and the orientation of the length contraction will no longer be parallel to the x axis. I suppose technically you could say there is no such thing as the "perpedicular components of the velocity" if we mean components perpendicular to the total resultant velocity vector, because by definition these components are always zero.

 

[starthaus]

I must have missed your proof. In all sincerity I would like to see it.

See my blog https://www.physicsforums.com/blog.php?b=1959 .

I think it is a question of semantics. I am saying that perpendicalur components of the velocity indirectly affect the length contraction.

This is incorrect since the proof shows clearly:

L'=L\sqrt{1-\beta_x^2}

No dependency whatsoever of \beta_y or \beta_z

This is not self-evident, it required some heavy lifting to prove.

For example an object with Vx=0.6, Vy=0, Vz=0 will be length contracted by a factor of 0.6 in the x direction.

This is not correct, the contraction is 0.8.

An object with Vx=0.6, Vy=0.6, Vz=0 will be length contracted by a factor of about 0.529 and the orientation of the length contraction will no longer be parallel to the x axis.

This is false as well, the correct contraction is 0.8. See the correct formula I gave above.

 

[Rasalhague]

The correct multiplication factor is :

\frac{1-\gamma}{\gamma}

Is this what you mean: s = x ({1, 0, 0} + g^(-1)*(1 - g) {bx, by, bz}) ? It doesn't work. Norm is generally different from that of the other expressions, including x*Sqrt[1 - bx^2], and the norm of s changes when I change the values of by or bz.

As you saw, Norm[q] gave the same answers as all of these other expressions, including x*Sqrt[1 - bx^2], and changing by or bz had no effect on it.

 

[starthaus]

Is this what you mean: s = x ({1, 0, 0} + g^(-1)*(1 - g) {bx, by, bz}) ?

No, this is not what I mean. You can find the correct expression in the blog attachment, I inserted a step between expression (2) and (3) specifically for your benefit. It isn't bx*by+by*bz+bz*bx, as you input into Mathematica, it is : bx^2+bx(by+bz)

It doesn't work. Norm is generally different from that of the other expressions, and the norm of s changes when I change the values of by or bz.

As you saw, Norm[q] gave the same answers as all of these other expressions, and changing by or bz had no effect on it.

Try working on the symbolic expressions, your approach using Mathematica is wrong. I am tired of figuring out what goes wrong in your derivation.

 

[Rasalhague]

This is false as well, the correct contraction is 0.8. See the correct formula I gave above.

With velocity all in x direction, magnitude 0.6, I get a contractrion of 0.8. Wolfram Alpha agrees. This with my supposedly wrong method which so far has always agreed exactly with your formula x*Sqrt[1 - bx^2]!

 

[starthaus]

With velocity all in x direction, magnitude 0.6, I get a contractrion of 0.8. Wolfram Alpha agrees. This with my supposedly wrong method which so far has always agreed exactly with your formula x*Sqrt[1 - bx^2]!

Work on the symbolic formulas. You have all the information in the attachment. I don't trust your verification via Mathematica because it suffers from errors, if you input the incorrect thing, don't be surprised to get the wrong output.

 

[Rasalhague]

Work on the symbolic formulas. You have all the information in the attachment. I don't trust your verification via Mathematica because it suffers from errors, if you input the incorrect thing, don't be surprised to get the wrong output.

Your claim that it's wrong would be more convincing if you could come up with an example where Norm[q] gives a different answer from Norm[p] and x*Sqrt[1 - bx^2]. With their current definitions, so far, they've all agreed exactly. While that doesn't prove they're identical, it does make it unlikely that they're not. All of them are independent of by and bz. It seems that like Mathematica, we humans are also struggling to establish the identity of these expressions symbolically.

 

[yuiop]

This is incorrect since the proof shows clearly:

L'=L\sqrt{1-\beta_x^2}

No dependency whatsoever of \beta_y or \beta_z

This is not self-evident, it required some heavy lifting to prove.

This is only true if \beta_y=0 and \beta_z=0. This is easy to demonstrate. If \beta_x=0 and \beta_y=0.8 then the length contraction is 0.6 and dependent on \beta_y and is orientated in the direction of the y axis.

This is not correct, the contraction is 0.8.

That was a typo that did not really affect the argument. I started with vx=0.8 and length contraction =0.6 and later changed vx to 0.6 because the resulant velocity of vx=0.8, vy=0.8 was greater than 1.0 so I changed the velocities to 0.6 and overlooked changing the contraction to 0.8. This does not change the fact that the all the velocity components contribute to the direction and magnitude of the total velocity and therefore they all contribute to the magnitude and direction of the length contraction.

 

[Rasalhague]

starthaus's example only deals with the contraction of a space vector having no y or z component. Could that be the source of the confusion? If instead of a rod aligned along the x axis, imagined as having no thickness, we had an object with spatial extent in all directions, then (if I've understood this) velocity components along the y and z axes would affect the extent of the object along those axes. I think that's the situation you're describing, kev, isn't it?

 

[yuiop]

This is incorrect since the proof shows clearly:

L'=L\sqrt{1-\beta_x^2}

No dependency whatsoever of \beta_y or \beta_z

This is not self-evident, it required some heavy lifting to prove.

I think in the end you will find the general solution for the total length contraction is given by:

L' \ = \ L\sqrt{1-(\beta_x^2+\beta_y^2+\beta_z^2)}\ =\ L\sqrt{1-\beta^2}

 

[yuiop]

starthaus's example only deals with the contraction of a space vector having no y or z component. Could that be the source of the confusion? If instead of a rod aligned along the x axis, imagined as having no thickness, we had an object with spatial extent in all directions, then (if I've understood this) velocity components along the y and z axes would affect the extent of the object along those axes. I think that's the situation you're describing, kev, isn't it?

Yes, I was talking about the more general solution, which is what I thought you guys were looking for.

If you wish to break the total length contraction down into its components then you get:

L'_x \ =\ L_x \sqrt{1-\beta^2_x}

L'_y \ =\ L_y \sqrt{1-\beta^2_y}

L'_z \ =\ L_z \sqrt{1-\beta^2_z}

and the total length contraction is:

L' \ = \sqrt{L'_x^2 + L'_y^2 + L'_z^2} = \sqrt{(L_x^2 + L_y^2 + L_z^2)(1-\beta_x^2-\beta_y^2-\beta_z^2)} =\ L \sqrt{1-\beta^2}

so yes, maybe we are at cross purposes and maybe my fault for not reading all the thread.

As for the proof that spatial components orthogonal to the motion are not length contracted, Starthaus starts with the Lorentz transformations which explicitly assume that in the first place. There are other possible formulations of the transformations that allow length contraction in the transverse plane that are consistent with a constant speed of light and MM experiment etc. and they had to be ruled out using the logic of rulers and markers that you gave or considering rings moving past each other. If the radius of a moving orthogonal ring changed, then you could have have ring A passing inside ring B in one frame and ring B passing inside ring A in another frame which is physically impossible.

 

[starthaus]

This is only true if \beta_y=0 and \beta_z=0.

This is false, you have not read the proof, it assumes non null v_y and non-null v_z

 

[yuiop]

This is false, you have not read the proof, it assumes non null v_y and non-null v_z

I think where we differ is that I am defining

L' \ = \sqrt{L'_x^2 + L'_y^2 + L'_z^2}

while you are defining L' as L'_x and the confusion comes about because you have not made your definition of L' clear.

 

[starthaus]

Yes, I was talking about the more general solution, which is what I thought you guys were looking for.

If you wish to break the total length contraction down into its components then you get:

L'_x \ =\ L_x \sqrt{1-\beta^2_x}

L'_y \ =\ L_y \sqrt{1-\beta^2_y}

L'_z \ =\ L_z \sqrt{1-\beta^2_z}

and the total length contraction is:

L' \ = \sqrt{L'_x^2 + L'_y^2 + L'_z^2} = \sqrt{(L_x^2 + L_y^2 + L_z^2)(1-\beta_x^2-\beta_y^2-\beta_z^2)} =\ L \sqrt{1-\beta^2}

so yes, maybe we are at cross purposes and maybe my fault for not reading all the thread.

Err, the correct math would say that your final formula is incorrect. The error is just glaring.

 

[Rasalhague]

I think in the end you will find the general solution for the total length contraction is given by:

L' \ = \ L\sqrt{1-(\beta_x^2+\beta_y^2+\beta_z^2)}\ =\ L\sqrt{1-\beta^2}

Let \textbf{r}=\left ( x,y,z \right ), where y and z are not necessarily equal to zero. Then I get

\left \| \textbf{r}' \right \|=\sqrt{\textbf{r} \cdot \textbf{r}-(\textbf{r}\cdot\pmb{\beta})^2}

=\sqrt{\left \| \textbf{r} \right \|^2-x^2\beta_x^2-y^2 \beta_y^2-z^2\beta_z^2}

This agrees with Norm[p] and Norm[q], as previously defined, but gives a different answers to Norm[r]*Sqrt[1 - bx^2 - by^2 - bz^2] = Norm[r]*Sqrt[1 - b^2] in the special case where y = z = 0, and, in this case, kev's formula Norm[r]*Sqrt[1 - b^2] does depend on by and bz, which I think we agree would lead to contradictions.

 

[starthaus]

In[20]:= bx = .9; by = 0.3; bz = .2; x = 1; b = Sqrt[bx^2 + by^2 + bz^2]; q = 
 x ({1, 0, 0} + bx*(Sqrt[1 - b^2] - 1)*b^(-2) {bx, by, bz}); Norm[q]

Out:= 0.43589

In[21]:= g = 1/Sqrt[
   1 - b^2]; p = {-bx^2*g*x + (1 + bx^2 (g - 1)*b^(-2))*x, -by*g*bx*
    x + x*bx*by (g - 1)*b^(-2), -bz*g*bx*x + bx*bz (g - 1)*x*b^(-2)}; Norm[p]

Out[21]:= 0.43589

In[22]:= x*Sqrt[1 - bx^2]

Out[22]:= 0.43589

In[23]:= x*Sqrt[1 + 2 bx^2 b^(-2) ((1 - g)/g) + bx^2 b^(-2) ((1 - g)/g)^2]

Out[23]:= 0.43589

But

In[24]:= x*Sqrt[1 + bx^2/g^2]

Out[24]:= 1.02401

I re-read this post, it makes no sense. Where do you think I use anything remotely similar to:

x*Sqrt[1 + bx^2/g^2]

?

The correct formula is :

x*Sqrt[1-bx^2]

No wonder you don't get the outputs to agree.

 

[starthaus]

Let \textbf{r}=\left ( x,y,z \right ), where y and z are not necessarily equal to zero. Then I get

\left \| \textbf{r}' \right \|=\sqrt{\textbf{r} \cdot \textbf{r}-(\textbf{r}\cdot\pmb{\beta})^2}

=\sqrt{\left \| \textbf{r} \right \|^2-x^2\beta_x^2-y^2 \beta_y^2-z^2\beta_z^2}

This agrees with Norm[p] and Norm[q], as previously defined, but gives a different answers to Norm[r]*Sqrt[1 - bx^2 - by^2 - bz^2] = Norm[r]*Sqrt[1 - b^2] in the special case where y = z = 0, and, in this case, kev's formula Norm[r]*Sqrt[1 - b^2] does depend on by and bz, which I think we agree would lead to contradictions.

Your formula is correct while kev's is obviously marred by an elementary algebraic mistake.

Now, having said that, there is absolutely no reason for attempting to add up the contracted dimensions the way kev did. So he compunded the algebraic mistake with a physics mistake. The length contraction of a rod with arbitrary orientation needs to be derived from basic principles. This is not what kev did.

 

[starthaus]

As for the proof that spatial components orthogonal to the motion are not length contracted, Starthaus starts with the Lorentz transformations which explicitly assume that in the first place.

This is also false. The general Lorentz transforms do not "assume" any such thing.

 

[Rasalhague]

I re-read this post, it makes no sense. Where do you think I use anything remotely similar to:

x*Sqrt[1 + bx^2/g^2]

?

That was kev's erroneous formula from #29.

 

[starthaus]

That was kev's erroneous formula from #29.

Arrgh, I see. So he made both of us waste time. We are good.