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According to Resnick’s Introduction to Special Relativity,Lorentz transformation in general can be given as:
$x'=a_{11}x+a_{12}y + a_{13}z+a_{14}t$

$y'=a_{21}x+a_{22}y + a_{23}z+a_{24}t$

$z'=a_{31}x+a_{32}y + a_{33}z+a_{34}t$

$t'=a_{41}x+a_{42}y + a_{43}z+a_{44}t$

Firstly, I am unable to make a symmetry argument to prove that $y=y'$ and $z=z'$. I could prove $x'$ should be independent of $y$ and $z$.

Next, I disagree with the mathematical derivation of this in Resnick.

If $S$ and $S'$ are two frames with $S'$ moving with velocity $\vec v$ relative to $S$, then without loss of generality we can orient $\vec v$ to be along $X$ axis (for $S$) and $X'$ axis (for $S'$).

Then to prove $y=y'$ and $z=z'$ he uses an argument that $X'Y'$ plane and $XY$ plane should be the same. But just because their velocity vectors are along the X axis, their XY planes need not be the same right. . . the Y' and Z' axis could lie anywhere in the YZ plane as shown in fig. So how can we prove $y=y'$ and $z=z'$?

enter image description here

Question:
What symmetry argument can be used to prove that $y=y'$

is it true that we can derive y=y' and x=x' without assuming any two axes to be aligned?

I know we can align one more axis (say z along z') and prove the invariance... But isn't it wrong otherwise?

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  • $\begingroup$ Length contraction only happens along the axis parallel to the direction of relative travel, if you rotate your axis you'll find that $y$ and/or $z$ will be changed under the Lorentz transformations. $\endgroup$
    – Charlie
    Commented May 18, 2021 at 12:36
  • $\begingroup$ No... I meant to keep the motion along the same direction (X axis) and then rotate the coordinate system about X axis... ie: Y and Y' (Z and Z') need not be in same direction. Direction of velocity is not changed $\endgroup$ Commented May 18, 2021 at 12:56
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    $\begingroup$ You can, of course, do the rotation that you suggested. It's really not clear what you're asking. The point in these textbook demonstrations is typically to demonstrate the thing that's new in SR not the thing that's the same. $\endgroup$
    – Brick
    Commented May 18, 2021 at 13:21
  • $\begingroup$ Related question: A problem in deriving Lorentz transformation from homogeneity and isotropy of spacetime and the principle of relativity. In fact, it's essentially a duplicate, though it isn't phrased the same way... $\endgroup$
    – Philip
    Commented May 18, 2021 at 18:43
  • $\begingroup$ @Philip The Related question mentioned does address my query and I am satisfied with the answer given there. But my question : how to make a symmetry argument to prove $y'=y$ still persist. $\endgroup$ Commented May 19, 2021 at 0:29

3 Answers 3

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This is mostly an arbitrary choice. The $y'$ and $z'$ axes are chosen to be aligned with the $y$ and $z$ axes for convenience. They don't need to be: the transformation you've sketched (with the $y'$ and $z'$ related to the original $y$ and $z$ by a fixed rotation about the $x$/$x'$ axis) is perfectly allowed.

As a consequence of this, it is impossible to, as you say, "derive $y=y'$ and $z=z'$". It is a possibility that can be true, but it doesn't need to be true.

That said, you do need to prove that the choice of $y=y'$ and $z=z'$ is a choice you actually can make, and this is not entirely trivial. The argumentation in Resnick is (presumably) an attempt to justify that this is choice is possible.

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Choosing a different $y'$ and $z'$ (in the y-z plane) means that you are rotating your body in space. This presumably is a transformation you're already well familiar with. So, per Emilio Pisanty's answer, there's no reason to clutter things up by including it.

Fly eastward in your airplane, with the wings parallel to the ground. Now tilt your airplane to the left and continue flying eastward. If you use $x$ to denote the direction of your travel, $y$ for the direction from your feet to your head and $z$ for the direction from your left arm to your right arm, you've just rotated your $y$ and $z$ axes. You can call the new axes $y'$ and $z'$, and they are linear combinations of $y$ and $z$. That's a perfectly good example of a Lorentz transform.

If we define $x$ to be the direction of your travel, then any Lorentz transformation is a composition of a) a Lorentz transformation where $y'=y,z'=z$ and b) a Lorentz transformation that describes a rotation in space, such as what you just did with your airplane. To understand all Lorentz transformations, you need to understand the transformations of type a) and those of type b). You don't need relativity to understand the transformations of type b), so the relativity textbooks concentrate on type a).

(There is nothing here that Emilio hasn't already said, but I thought the additional explicitness might help.)

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You can take components of your y' and z' and then use the argument that the author uses. Obviously they have no component along x axis and hence there is no change in them.

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