A provocative title maybe, but only in so far as the equations: y' = y and z' = z are concerned.

In paragraph 6 Einstein demonstates that if x= ct then x' = ct' in which he is absolutely correct, but if one were to try and do the same for y' and z' the result could hardly be the same as he claims it would.

It seems to me that all Lorentz and Einstein have done is to ASSUME that as they are concerned only with movement along the x axis, the y and z axes would be unaffected.

Why should this be so???

For if it were then wouldn't time have to be diectional? So that it could be applied differently to calculate the speed of light, depending on whether the spacial element was contracted or not?

I don't believe that this point affects anything else in Special Relativity but it is a little puzzle to me.

Grimble

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Could you explain why you think it's wrong? What do you mean by time being directional? (It's commonly said to be, because the past is different from the future in ways left is not different from right, but this doesn't seem to be the kind of directionality you're worried about.)

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Don't forget that Lorentz Contraction only happens in the direction of travel - hence the 'x' term. Space is not compressed perpendicular - hence y and z being unaffected.

And there's a fairly simple thought experiment that demonstrates why there can't be length contraction in a direction perpendicular to travel. Imagine you have two meter sticks moving relative to each other, perpendicular to their length. So:

| |

At each end of both meter sticks is a piece of chalk, set to draw a line on the other meter stick as they pass. Now, if the principle of relativity holds, an observer moving along with either meter stick is perfectly justified in deciding that his meter stick is at rest, and it's the other meter stick that is moving. If there were length contraction perpendicular to the direction of the motion, the observer on the left would see the meter stick on the right as shorter, and so would expect the chalk lines from that meterstick to be drawn on his own at, say, 10 cm and 90 cm, and the chalk on his own meter stick would miss the other one entirely. So he'd expect only his own meter stick to have chalk lines on it.

However, the observer on the right would expect the opposite, that his meter stick would be the one that would end up with chalk lines on it. If we actually did this experiment, and one or the other meter sticks contracted, we'd be able to determine which was "really" moving, and that would violate the principle of relativity. For the principle of relativity to hold, there cannot be any changes in length along the directions perpendicular to the direction of motion.

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Conserve energy. Commute with the Hamiltonian.

What you are seeing is an old physics trick to keep our heads from exploding. What has been done is to assume that the motion is in the x direction for the purpose of keeping the math simple.

Yes, I did say simple.

You can do out the coordinate transforms in an arbitrary direction. As a matter of fact, I think that the wiki on SR has the equations for an arbitrary direction. It makes the math rather complicated and a real bear to try to use.

The trick is, since you are transforming coordinates anyway, if you have a linear motion in an arbitrary direction, you can just rotate the coordinate system to line the motion onto one of the axis, and simplify the equations.

The contraction and time dilation take place depending upon the scalar quantity of v, so if an observer, while considering themself stationary, were to see another travelling in any direction at v, the contraction would be observed to take place along the direction of that motion, but not perpendicularly to it, and the time dilation observed on the other's clock would take place depending upon that scalar quantity v also, both smaller by a factor of sqrt(1 - (v/c)^2) than what the other would observe in their own frame.

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"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

Hello and yes what is bothering me is that if contracted length, along the x axis; that is x' is divided by dilated time t' to give c i.e. c = x'/t' = x/t then, as y = y' for the y axis we would have y/t' and z/t' for the z axis and these obviously do not = c.
So are we saying that time is only dilated along the x axis?

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Yes, I understand that, but the speed of light must be the same in all directions, but how can it be if one direction is length contracted but time dilation applies in all directions?

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You cant just swap out the axis and get the same result. For a ray along the direction of motion, you get the result shown, but if you shoot the ray at a right angle to the direction of motion you need to use the full three dimensional treatment.

contracted in x along the direction of motion, but when you shoot the ray at a right angle, it will travel a longer distance than in the stationary case. The math works out different

But why is this assumed for I have never seen it as anything other than an assumption. What is the proof that it is only along the direction of motion?

This is after all only how the spacetime magnitudes of another moving frame of reference are distorted and why should we only be seeing the 'distortion' in only one direction?

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Well, you could look to the original papers, or any intorductory text.

The contraction is an artefact of looking at the same physical system using two different frames of reference. In particular, in looking at two frames of reference that differ by their relative motion. We can simply define these frames to be moving only along one axis. In this way, they are aligned so that the value they assign to a y or z value is always going to be the same, it is only on the x (and t) values that they will disagree on.

One can describe the two frames as being in relative motion in multiple dimensions. In that case, one has to apply the proper contractions in every dimension.

Well, that goes back to the M-M experiment, where beams of light were sent along perpendicular arms of the same length and no interference shift was found. This could not be explained in terms of an aether where light travels at c relative to the aether only until Lorentz found that if the entire apparatus contracted by sqrt(1 - (v/c)^2) in the direction of motion but not perpendicularly to that motion, where v is the speed through the aether as measured by an observer that is stationary with the aether, then the times as measured by that observer while the apparatus is also in realtive motion to the aether would match. So that is initially just how the mathematics could be made to work out. But that introduced other complications, until Einstein came along and realized that if the apparatus were to contract in the direction of motion while also time dilating by an equal amount, then the complications disappear and the idea of an aether would no longer even be necessary, because time and space are then both relative using the relative speeds between the frames of the observers themselves. I didn't go into detail for this as that would take a while but I hope this generalization helps.

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Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

It isnt an assumption. The proof is something used to terrorize graduate students, so trying to give a good explanation to someone without that kind of background is a bit difficult.

You only see distortion along the direction of movement because the effects of SR are because of the movement. As circular as that sounds, it comes from the fact that if you are moving in the x direction, then the y direction is the same for the moving and non-moving cases.

Hi Grimble,

Everything works out as it should, but maybe it is a little tricky. One can apply the maths, but I hope my example can help with the intuition a bit.

Suppose an observer is stationary in his reference frame, at x=0 y=0 z=0. At time t=0, he shines a flashlight in the direction of the y-axis. So we can ask the question, how long does the light take to reach the point x=0 y=300,000,000 z=0. Suppose there is a spaceship or mirror or some object stationary at this point. Since light travels at 300,000,000 meters per second, the answer is t=1. In one second, the light travels from x=0 y=0 z=0 to x=0 y=300,000,000 z=0.

Now suppose at t=0, another observer travels through x=0 y=0 z=0, in the direction of the x-axis. For this observer, y'=y and z'=z, but x' and t' are different than x and t. So to this observer, how long does the light take to travel to the spaceship? For this observer, the spaceship is at y'=300,000,000 z'=0, but it is moving in the x' coordinate. So to this observer, the light travels more than 300,000,000 meters (because it is traveling along a diagonal path), so it must take longer than one second. So this observer sees that the first observer has time dilation; two events that are more than one second apart for the second observer, are only one second apart for the first observer. But this does not occur because there is dilation of space in the y direction; it occurs because what is a motion only in the y dimension to the first observer, is motion in both the y dimension and the x dimension to the second observer.

I hope that is helpful with intuition. For a more mathematical treatment, it is possible to show that the Lorenz transform, which has time dilation in the direction of motion but not in the other two directions, is the only linear transform of space and time that preserves the metric

x²+y²+z²-c²t²=x'²+y'²+z'²-c²t'²

If you prefer this way of thinking, it might be helpful to prove that with this property, something that is traveling at the speed of light (in any direction) to the observer with x, y, z, and t, is also traveling at the speed of light to the observer with x', y', z', and t'.

Edit - I did not notice before, but korjik already posted a similar solution

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Principal Principle Offender

My impression is that Grimble's problem can be answered with very basic
geometry. The physics of relativity is not essential to the answer-- only
the geometry is essential. An analogy with some non-relativistic situation
might help.

-- Jeff, in Minneapolis

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Thank you Gentlemen, thank you!

You are quite right it is, after all, simple geometry.

One cannot compare y and y' directly for, as you say, if the viewpoint is moving one cannot see movement in the y' direction alone, it has to have an x' component too! Silly me

Grimble
(One cannot learn with a closed mind; draw the curtain and let the light in!)

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Patience allows one to mellow while the world turns...

One cannot learn with a closed mind; draw the curtain and let the light in!

Thank you, Grey! I don't know how I've managed to miss seeing this argument or have forgotten it if I had, but I'll remember it now.

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I am afraid that, after much contemplation, one question still remains viz. If a mirror is 1 light second above the light in the moving light clock scenario, how far is that?
I understand that the distance will not be contracted as it is wholly in the y or z axes, but the time in the 1 light second will be dilated, so how does that work out? how can the time be affected by the relative velocity but the distance not??

Grimble:co nfused:

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Patience allows one to mellow while the world turns...

One cannot learn with a closed mind; draw the curtain and let the light in!

Just to clarify, we are saying a mirror is one light second away from a source of light, and we consider motion in a direction perpendicular to the direction of the mirror? For example, the light source (observer, whatever) is moving in the x direction, but the mirror is in the y or z direction?

If that's the case, I'm not sure I entirely understand the question. One light-second is 300 million meters. This distance is the same in both frames of reference (the one where the light source is stationary, and the one where it is moving).

In the frame in which the light source and the mirror are stationary, the light travels 300 million meters, and does so in one second, then is reflected back, and travels 300 million meters in another second.

In the frame in which the light source and the mirror are moving, they are still 300 million meters apart. However, because the light takes time to travel to the mirror, it must travel diagonally to hit the mirror, because the mirror is moving. So then it still travels 300 million meters in the y (or z) direction, but also must travel some distance in the x direction. So in this frame, the light travels more than 300 million meters each way, and takes more than one second each way to do so. But in this frame, we may consider the perception of an observer moving along with the light source and mirror. This observer will perceive that the light has only traveled 300 million meters (a shorter distance). However, we will see that this observer perceives time more slowly. To us, the light travels more than 300 million meters and takes more than one second, but to the moving observer, it travels 300 million meters only, and takes one second, because less time has passed.

I don't know if that helps, because I don't know if I have understood the difficulty correctly. But please follow up if any point of confusion remains; I will explain to the best of my abilities, as I am sure many others will

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Principal Principle Offender

Yes, sorry that I was not clear about that.

If we have a light clock with a mirror 1 light second away from the light source being positioned along the y axis and an observer is travelling along the x axis at a velocity v.

What I am intersted in is, measured by the observer, how big is the light second?
It should be 300 million metres Proper time as, lying along the y axis, it is not length contracted; yet the time IS time dilated as this is non directional, so how does this affect the lightsecond as a measure of distance.

Note that I am not concerned about the view of the light travelling a diagonal path but merely the measure of 1 light second along the y axis

Grimble

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So both the observer and the mirror are traveling? Or just the observer?

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Principal Principle Offender

Grimble,

Actually, you ARE concerned about the diagonal path.

You are are concerned with what is seen by observers who are moving
relative to one another. Let's call two observers 'A' and 'B'.

Observer A sends out a flash of light. The light hits a mirror 1 light-second
(299,792,458 metres) away. The reflection is detected by observer A two
seconds after he sends it.

Observer B is moving relative to observer A at high speed, perpendicular to
the line between A and the mirror. Observer B sees the light travel a diagonal
path to the mirror and back to A, who has apparently moved a significant
distance. The length of the path B would measure with his own ruler might
be 340,000,000 metres. Observer B sees the light take 2.27 seconds to go
from observer A to the mirror and back to A. He also sees that A's clock is
apparently ticking more slowly than his own, so he knows that according to
A, only two seconds elapsed.

-- Jeff, in Minneapolis

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http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

Well...Its not like if your go off the x axis by 1 degree that those objects wouldn't be time dilated.

The basic thing to understand is that there is no "universal" x. It is just easiest to consider x as the vector of acceleration. Any change in direction of travel would change the orientation of x.

in SR the final comparison of time can only occur when the 2 clocks are close together which means the 2 objects in the end where oriented towards each other. You can do comparisons between 2 object with 2 completely different vectors but as it has been pointed out the maths is a lot more difficult and for some people on this board they then want to bring in some "universal now" that they think there is when comparing the 2 clocks but forgetting that a true comparison would require another set of accelerations to bring the 2 objects together.

Well put.

Just the observer.

No, I am not concerned about the diagonal path of the light!

I understand that, but if we draw this as the usual pythagoras triangle to shew time dilation, how big is the Light second on the Y axis???

It isn't length contracted - it is on the y axis, yet the 'light second' IS time dilated as time is non directional!

Please think about it as the measure or units on the y axis, they cannot be measured in two different sets of units at the same time.

Grimble.

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Patience allows one to mellow while the world turns...

One cannot learn with a closed mind; draw the curtain and let the light in!

Everything can be measured in two different units at the same time! However, this is not really the answer to your question. The answer is that in one frame, we are using one construction of a time coordinate and in another frame we are using another. It is in the construction of the time coordinate that any time dilation enters the picture.

In any frame, light travels a null line anyway.

But that is exactly what I am saying is NOT happening!

I am not concerned with the path the light is travelling in the observer's frame
I am concerned with only one frame of reference. the observer's frame

I am concerned with one physical distance, that between the light source and the mirror, at the time the observer passes the light source if you like, 2 points and the distance between them in a line perpendicular to the path of the observer. not the path of the light


In the moving observer's frame what is the length?

In that frame alone do we not have two measures???

1. The time-dilated light-second on the y axis.
2. The non-contracted length.

It surely can't be both?

Grimble

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Patience allows one to mellow while the world turns...

One cannot learn with a closed mind; draw the curtain and let the light in!

Well, you'd better have a reason for it.
OK, perhaps this will help clear things up: give the coordinates of the mirror in one frame, you choose which one. (You may immediately see the way out of your difficulty when you do this.)

Grimble,

Could you describe the entire setup from the beginning, please?
Are the light source and the mirror in motion relative to each other?
Are the mirror and the observer in motion relative to each other?

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves