Why y=y' in the Lorentz transformation?

[Johnls]

By y I mean any perpendicular axis to the axis in which there is movement.

 

[vanhees71]

By definition, a boost in $x$ direction doesn't change components of four-vectors in $y$ and $z$ direction. For an introduction to special relativity, see my SRT manuscript (unfinished, but the basic concepts on Lorentz transformations are all in):

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[Johnls]

By definition, a boost in $x$ direction doesn't change components of four-vectors in $y$ and $z$ direction. For an introduction to special relativity, see my SRT manuscript (unfinished, but the basic concepts on Lorentz transformations are all in):

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

Thanks for the response!
By definition?
I can Imagine length contraction in the y-axis without it violating any definitions.
The link doesn't work for me.

 

[vanhees71]

Well, try the link to the website. There you find a link to the pdf:

http://th.physik.uni-frankfurt.de/~hees/pf-faq/

 

[Ibix]

I can Imagine length contraction in the y-axis without it violating any definitions.

Can you really? If I have a 1m rule held perpendicular to my motion and so do you, and we are on a collision course - what happens when we collide? Whose rod is shorter? Why?

 

[PeroK]

Thanks for the response!
By definition?
I can Imagine length contraction in the y-axis without it violating any definitions.
The link doesn't work for me.

Can you come up with a simple thought experiment that shows that there cannot be length contraction in the $y$ direction when the relative motion is in the $x$ direction?

 

[SiennaTheGr8]

To Ibix's point, see here: https://books.google.com/books?id=PDA8YcvMc_QC&pg=PA65

 

[Johnls]

Can you really? If I have a 1m rule held perpendicular to my motion and so do you, and we are on a collision course - what happens when we collide? Whose rod is shorter? Why?

Can you come up with a simple thought experiment that shows that there cannot be length contraction in the $y$ direction when the relative motion is in the $x$ direction?

I was about to say:
"I can imagine that in my reference frame your ruler will be shorter and in your reference frame my ruler will be shorter."
but then I realized that if I place a chalk on to top and bottom of each ruler the argument by symmetry makes perfect sense, because there should be only one result. Is this a valid thought experiment?

 

[PeroK]

I was about to say:
"I can imagine that in my reference frame your ruler will be shorter and in your reference frame my ruler will be shorter."
but then I realized that if I place a chalk on to top and bottom of each ruler the argument by symmetry makes perfect sense, because there should be only one result. Is this a valid thought experiment?

Yes, if the two rulers (with the same rest length) move towards each other "vertically", then they must, by symmetry, be of equal height as they pass each other. Each is a valid measurement of length for the other, because the two ends meet simultaneously (in both frames).

 

[MikeLizzi]

Can you come up with a simple thought experiment that shows that there cannot be length contraction in the $y$ direction when the relative motion is in the $x$ direction?

Here’s one example of a paradox if length contraction occurred along axes other than the direction of motion:

Consider these initial conditions
1. A sphere with proper diameter 1 meter with position (x, y, z) = (-8, 0, 0) and velocity (vx, vy, vz) = (.866c, 0, 0) with respect to you the observer.
2. A vertical plate with a 1.1 meter diameter hole centered on the origin at rest with respect to you.

Using Relativistic Physics, the x-dimension if the sphere is 50% contracted but it still passes through the hole in the plate without interference.

Now switch reference frames to one in which the sphere is at rest. The vertical plate is moving at velocity (vx, vy, vz) = (-.866c, 0, 0) toward the sphere. Standard Relativistic Physics says the sphere is a back to being a real sphere and plate is now half its thickness but the hole is still a 1.1 meter circle and the sphere still passes through the hole without interference.

What if length contraction occurred in all 3 dimensions?
Consider again the initial conditions. The sphere is contracted in all dimensions and again passes through the plate without interference.
Now switch reference frames to one in which the sphere is at rest. The vertical plate is again moving at velocity (vx, vy, vz) = (-.866c, 0, 0) toward the sphere. But the plate is contracted in all dimensions requiring the hole to be contracted too. The sphere cannot make it through the hole. Paradox.

 

[Andrew Mason]

By y I mean any perpendicular axis to the axis in which there is movement.

y = y' follows from the Lorentz transformation for time, i.e. $t' = \gamma (t - vx/c^2)$. An observer in the stationary frame measures a distance y in the direction perpendicular to the direction of motion (of an observer in a rocket ship) by sending a light signal from the origin (0,0) to a mirror located at position (0,y) and measuring the time, t, that it takes to return to the point (0,0). It takes t = 2y/c. But the observer in the moving rocket frame sees that same light signal travel a longer path over a time t' = 2d/c, where d is the hypotenuse of a right angle triangle whose base is v(t'/2) and whose height is y'. This is because, in the moving frame, the light is emitted at time t'=0 at a point on the x' axis (-vt'/2,0), travels to a point (0,y') and returns to a point (vt'/2,0) at a time t'.

So let's compare y and y':

(1) $y = ct/2$
(2) $y' = \sqrt{d^2-(vt'/2)^2} = \sqrt{(ct'/2)^2 - (vt'/2)^2)} = (ct'/2)(\sqrt{1-v^2/c^2}) = (ct'/2)/\gamma$

Since x = 0, $t' = \gamma t$. So substituting this for t' in (2) results in:

(3) $y' = c(\gamma t)/2\gamma = ct/2 = y$

AM

 

[tade]

Can you come up with a simple thought experiment that shows that there cannot be length contraction in the $y$ direction when the relative motion is in the $x$ direction?

Andrew Mason's example is pretty good.

 

[tade]

Thanks for the response!
By definition?
I can Imagine length contraction in the y-axis without it violating any definitions.

Think of the Lorentz transformation with only one dimension of space, x.

Now, consider a photon moving in the x and y directions. What transformation would allow the photon to still posses a speed of c? y' = y

This is equivalent to Andrew Mason's example.

 

[tade]

Most teachers jump the gun by assuming that y' = y and use it to derive time dilation, working backwards, which ends up confusing students more ironically.

 

[PeroK]

Most teachers jump the gun by assuming that y' = y and use it to derive time dilation, which ends up confusing students more ironically.

Although, assuming the Lorentz Transformation not only jumps the gun but jumps half way to the finishing line!

 

[tade]

Although, assuming the Lorentz Transformation not only jumps the gun but jumps half way to the finishing line!

haha, but you see, teachers don't want to derive the LT the proper way, so they just skip it and use a light clock.

the LT itself is not a jump but based on Einstein's postulates.

 

[robphy]

Most teachers jump the gun by assuming that y' = y and use it to derive time dilation, working backwards, which ends up confusing students more ironically.

Well... it's not an unreasonable assumption since it is already there in the Galilean transformation and there has been no issues of length contraction in the Galilean case. It's only after deducing length contraction along the direction of relative motion in special relativity that one goes back and questions the assumption in the transverse direction.

 

[tade]

Well... it's not an unreasonable assumption since it is already there in the Galilean transformation and there has been no issues of length contraction in the Galilean case. It's only after deducing length contraction along the direction of relative motion in special relativity that one goes back and questions the assumption in the transverse direction.

hmmm, I'm not so sure. there is an issue in the case of the light clock, which is time dilation being a result of the height remaining the same.

conversely, if we assumed that there was no time dilation then there would be a height contraction of the light clock.

 

[robphy]

hmmm, I'm not so sure. there is an issue in the case of the light clock, which is time dilation being a result of the height remaining the same.

conversely, if we assumed that there was no time dilation then there would be a height contraction of the light clock.

The issue of time-dilation arises because one is invoking the principle of relativity with the light clock...
that the round-trip of the light signal in that light clock marks one tick for that clock, as it would for a clock at rest,
but that elapsed time measured by "the clock at rest" is different than that of "the clock at rest".

While one may be so unhappy with this time-dilation result that one tries to use height-contraction to avoid it,
the arguments above show that no such height-contraction exists (as in the Galilean case, which we already presumed)... since to do so violates the principle of relativity. [I guess one could have also questioned the principle of relativity... and be led to consider other implications... ultimately being tested by experiment.]

So, my point is that consideration of transverse contraction is an afterthought... since transverse-noncontraction is already practically presumed in the Galilean case.

edit: I guess one could have started from scratch altogether [not starting with Galilean physics]... then one has many more assumptions to question.

 

[tade]

The issue of time-dilation arises because one is invoking the principle of relativity with the light clock...
that the round-trip of the light signal in that light clock marks one tick for that clock, as it would for a clock at rest,
but that elapsed time measured by "the clock at rest" is different than that of "the clock at rest".

While one may be so unhappy with this time-dilation result that one tries to use height-contraction to avoid it,
the arguments above show that no such height-contraction exists (as in the Galilean case, which we already presumed).

So, my point is that consideration of transverse contraction is an afterthought... since transverse-noncontraction is already practically presumed in the Galilean case.

Although there is no height contraction in the GT, there is no length contraction either.
Ultimately, it has been shown that there should be no height contraction, though I don't agree that that is a reasonable assumption to make from the start.
The simplest assumptions would be Einstein's two postulates.

 

[strangerep]

The simplest assumptions would be Einstein's two postulates.

Mmmpff! Must... resist... temptation to talk about 1-postulate derivations...

 

[GeorgeDishman]

The sphere cannot make it through the hole. Paradox.

I've seen it as the "Rod and Tube Paradox". The Voigt Transforms suffer from this problem.

 

[tade]

I've seen it as the "Rod and Tube Paradox". The Voigt Transforms suffer from this problem.

View attachment 107973

What does it look like when you draw a Minkowski-esque spacetime diagram and then transform it?

In SR the pole-barn paradox is averted when a Minkowski spacetime transformation reveals the RoS.

The biggest issue with the Voigt transformations is that the primed and un-primed equations are not reciprocal.

 

[PeterDonis]

What does it look like when you draw a Minkowski-esque spacetime diagram and then transform it?

Like there is no length contraction perpendicular to the direction of motion.

The biggest issue with the Voigt transformations is that the primed and un-primed equations are not reciprocal.

The biggest issue with the Voigt transformations is that they give the wrong answers, whereas the Lorentz transformations give the right answers. Voigt didn't know that, of course; but we do.

 

[tade]

Like there is no length contraction perpendicular to the direction of motion.

I mean a Minkowski-esque diagram but using the Voigt transformations instead.

The biggest issue with the Voigt transformations is that they give the wrong answers, whereas the Lorentz transformations give the right answers. Voigt didn't know that, of course; but we do.

that too.

 

[PeterDonis]

I mean a Minkowski-esque diagram but using the Voigt transformations instead.

I'm not sure you could draw one, because I don't think the Voigt transformations preserve any meaningful geometric invariants. The reason the Minkowski diagram works in SR is that the Lorentz transformations preserve the appropriate geometric invariants.

 

[GeorgeDishman]

What does it look like when you draw a Minkowski-esque spacetime diagram and then transform it?

It would be harder to draw because you have to represent both the x and y axes as well as t so it would be a 3D image. For example I could modify this interactive diagram I did for the Twins Paradox to show a section along the axis of the rod and tube to show the wdiths 'into' the screen. As you move the slider, the widths would alter as well as the slope of the worldlines. One way, the tube gets wider and the rod narrower so all is well. Slide it the other way and the opposite happens so the rod becomes bigger than the tube and won't pass through.

 

[tade]

I'm not sure you could draw one, because I don't think the Voigt transformations preserve any meaningful geometric invariants. The reason the Minkowski diagram works in SR is that the Lorentz transformations preserve the appropriate geometric invariants.

If y'=y/γ , then y=γy'

Then this wouldn't happen and there wouldn't be any collision paradoxes. But if y=γy', it would violate Einstein's first postulate.