I Can't understand this argument for Lorentz transformation y'=y

[Happiness]

I don't fully understand the argument below used to derive the Lorentz transformation equation $y'=y$.

Suppose we have a rod of unit length placed stationary in frame S. According to an observer in frame S' (which is moving at a velocity v relative to frame S), this rod is moving and its length would be $\frac{1}{\gamma}$ (i.e., $\Delta l'=\frac{\Delta l}{\gamma}=\frac{1}{\gamma}$).

Following the above argument, we now place the rod stationary in frame S'. The rod must also be of unit length in frame S'; otherwise there would be an asymmetry in the frames. In this case, however, the S-observer would measure the rod's length to be $\gamma$ (i.e., $\Delta l=\gamma\Delta l'=\gamma\times1$). Now, because of the reciprocal nature of these length measurements, the first postulate (see below) requires that these measurements be identical, for otherwise the frames would not be identical physically. Hence, we must have $\gamma=\frac{1}{\gamma}$ or $\gamma=1$.

But for non-zero v, $\gamma\neq1$. Hence, I don't understand the above argument fully.

Of course, I know my argument involving $\Delta l$ is wrong. What I don't understand is why the argument works for $y$ but not for $\Delta l$.

 

[PeterDonis]

the argument below

Where is this argument from? Please give a reference.

 

[PeterDonis]

According to an observer in frame S' (which is moving at a velocity v relative to frame S), this rod is moving and its length would be $\frac{1}{\gamma}$ (i.e., $\Delta l'=\frac{\Delta l}{\gamma}=\frac{1}{\gamma}$).

Only if the rod is oriented along the direction of relative motion, which in this case, I believe (you didn't give a reference or context so I don't know for sure), is the x axis. A rod oriented along the y axis, perpendicular to the direction of motion, would have the same length in both frames.

 

[Happiness]

Where is this argument from? Please give a reference.

It's from Introduction to Special Relativity by Robert Resnick, page 58.

 

[Happiness]

Only if the rod is oriented along the direction of relative motion, which in this case, I believe (you didn't give a reference or context so I don't know for sure), is the x axis. A rod oriented along the y axis, perpendicular to the direction of motion, would have the same length in both frames.

Yes, I should have mentioned that the rod is oriented along the direction of relative motion, which is is the x axis.

 

[PeterDonis]

I should have mentioned that the rod is oriented along the direction of relative motion, which is is the x axis.

Not in the quote you gave in the OP. Read it again.

 

[PeterDonis]

Following the above argument

Which doesn't apply if the rod is oriented along the direction of motion.

 

[PeterDonis]

What I don't understand is why the argument works for yyy but not for ΔlΔl\Delta l.

Because the argument in the quote you gave only applies if the rod is oriented perpendicular to the direction of motion.

I strongly suspect that if you go back and look at the context, instead of just the limited quote you gave in the OP, you will see why. Or you could give the actual reference where you got the quote from, so that others can also see the context. Without that added information it's going to be very difficult to see where your reasoning has gone wrong.

 

[Happiness]

Or you could give the actual reference where you got the quote from, so that others can also see the context.

Special Relativity by Robert Resnick, page 58

 

[Happiness]

Not in the quote you gave in the OP. Read it again.

Yes, I did not. And I should have done so.

 

[Happiness]

These are the relevant pages.

 

[Happiness]

Which doesn't apply if the rod is oriented along the direction of motion.

I know the argument doesn't apply if the rod is oriented along the direction of motion, because I know the Lorentz transformation equations are true.

But since the text in these relevant pages is trying to derive the Lorentz transformation equations, it cannot and does not assume the Lorentz transformation equations to be true beforehand. Yet, without first assuming the Lorentz transformation equations to be true, I do not understand why the argument applies when the rod is oriented perpendicular to the direction of motion but does not apply when the rod is oriented along the direction of motion.

 

[Ibix]

You might want to consider the step leading to $y'=a_{22}y+a_{23}z$ and the similar expression for $z'$.

 

[Sorcerer]

You might want to consider the step leading to $y'=a_{22}y+a_{23}z$ and the similar expression for $z'$.

So basically in order for the x-axis to continuously coincide with the x'-axis, the coefficients in y' and z' that multiply the x and t coordinates must be zero?

Also wouldn't a23 and a32 have to be zero as well?

 

[Herman Trivilino]

Yet, without first assuming the Lorentz transformation equations to be true, I do not understand why the argument applies when the rod is oriented perpendicular to the direction of motion but does not apply when the rod is oriented along the direction of motion.

The argument about the $a_{22}$ coefficient applies because $y'=a_{22}y$.

The equation relating $x'$ to $x$ doesn't take the same form, so there is no same argument to be made.

 

[Ibix]

So basically in order for the x-axis to continuously coincide with the x'-axis, the coefficients in y' and z' that multiply the x and t coordinates must be zero?

Yes.

Also wouldn't a23 and a32 have to be zero as well?

Not for the x and x' axes to coincide. That simply requires all points (t,x,0,0) to map onto (t',x',0,0). For example, if the y' and z' axes are rotated with respect to the y and z axes the x-axis still maps on to the x' axis.

The next paragraph makes additional arguments leading to $a_{23}=a_{32}=0$.