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Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. You are welcome to comment upon or question anything said on my website (relativity-4-engineers), in the eBook or in the snippets I post here.

Comments/questions of a general nature should preferably be posted to the FAQ section of this Blog (http://cr4.globalspec.com/blogentry/316/Relativity-Cosmology-FAQ).

A complete index to the Relativity and Cosmology Blog can be viewed here: http://cr4.globalspec.com/blog/browse/22/Relativity-and-Cosmology"

Regards, Jorrie

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Paradoxes of Relativity Part 3 - Ladder Paradox (i)

Posted December 28, 2006 9:24 AM by Jorrie

In this variant of the "long ladder in the short garage paradox", we will get "out of the box" a little and use an arrow and an open-ended box to illustrate, as shown in figure 1. The conventional discussion will first be given and then followed by a more 'intuitively accessible' one.

Figure 1

If the arrow is moving lengthwise at a significant portion of the speed of light, then in the rest frame of the box, the arrow is length contracted so that it could, at least for a moment, fit completely into the box, before beginning to exit on the far side.

Figure 2

Light moves at very, very close to 1 nanosecond (ns) per ft, so even without relativity, you can check that the front of the arrow will take 12.5ns to travel the length of the 10ft box. Two clocks, one at each end of the box and synchronized in the reference frame of the box measure this time interval.

However, we are entitled to view the arrow as the rest frame and let the box move towards the arrow at the same speed. Now, the box is length contracted and the arrow cannot, even for a moment, fit into the box.

Figure 3

Again, without using relativity, one can check that the left end of the box, moving at 0.8c, takes 18.75ns to travel the 15ft length of the "stationary" arrow. Two clocks synchronized in the reference frame of the arrow, one at the head and one at the tail, measure this time interval. Note the peculiar order of events - it appears as if events 2 and 3 are chronologically swapped around, compared to what it was in the frame of the box.

A natural question at this point is: "are these differences in the timing of events caused by relativistic time dilation?" The answer is, quite surprisingly, no, not quite! This mini-series will show how the "paradox" it is properly explained by a combination of time dilation and the way clocks are synchronized in inertial frames (which is essentially the same thing as relative simultaneity).

As usual, you can read more on these concepts on the website Relativity 4 Engineers.

Jorrie

PS: "Relativistic length contraction" is a very ugly concept, is it not? We will use a slightly "out of the box" model that does not require contraction, so watch this space!

J

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#1

Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

12/29/2006 12:08 PM

If all this is based on a fixed position in space (Which I can not even begin to define) how do we know that the fixed postioin is not itself moving relative to the presumed moving object? Is it a matter of acceleration that determines this supposed phenomenon? And which object is accellerating? Me thinks me thinking may not right be!

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#2
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

12/29/2006 1:19 PM

Keywalker, you wrote: "... how do we know that the fixed position is not itself moving relative to the presumed moving object? Is it a matter of acceleration that determines this supposed phenomenon?"

This is the cause of many headaches in relativity! Only relative motion is accounted for and it does not matter which frame you choose as your reference. This reference frame can be moving (uniformly) at any velocity relative to the fixed stars, or the universe at large, or the CMB...

Observation has confirmed that only the relative motion shows up in measurements - if there is an absolute rest frame, nobody have yet detected it. Einstein believed that it is impossible to detect it and reportedly said something like: "If nature conspires against us in this way, we may just as well forget about the aether."

Jorrie

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#3
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

12/30/2006 5:18 PM

If it is impossible to detect the absolute at rest position then how do we know which object is actually moving? Is it because it was set into motion last?

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#4
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

12/31/2006 1:45 AM

Keywalker, you wrote: "... then how do we know which object is actually moving?"

In relativity, it does not matter. As far as we can tell, everything is moving relative to something. We may choose any one of the "moving" things as our reference and measure everything relative to it.

The beauty of Einstein's relativity is that it gives consistent results when we do just that - we do not have to worry about an absolute reference, which would have complicated things considerably. As an example, take Doppler radars in space - they would be very difficult to interpret if our measurements depended on an absolute frame that we cannot detect!

You can read some more on relativistic Doppler effects in a free download from this web page, pdf page 19, document page 44.

Jorrie

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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

01/03/2007 10:53 AM

The mathematics is hard for me to understand as I don't have a strong background in this area. This much I admit. But, even given that I am ignorant of many things it is still hard for me to believe that something can be shorter or longer simply by the mere fact that we have or have not chosen it as a frame of referrence or a rapidly travelling object relative to the frame of referrence. It seems that our perspective must be that which is changing, not the object. Wouldn't you agree?

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#6
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

01/03/2007 1:06 PM

Hi Keywalker, when you wrote: "It seems that our perspective must be that which is changing, not the object. Wouldn't you agree?", you were perfectly right!

If you follow the "paradox" mini-series to the end, you will see that clearly. It is simply the way we are observing or measuring things that is playing all sorts of tricks on us.

The idea of Einstein's relativity was, IMO, to put those "tricks" inside a solid theoretical framework so that we would no longer be surprised by it...

Math helps, but is not absolutely essential. One must however maintain a fairly open mind and not get too dogmatic about things. What is observed and measured in a repeatable, or verifiable, fashion is king!

Jorrie

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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

01/03/2007 1:38 PM

I agree: "What is observed and measured in a repeatable, or verifiable, fashion is king!", given that the same parameters are present in all cases of observation and measurement or there are no other parameters to consider. I am also very much against dogma and vow to remain open minded. Thank you for your replies to my queries.

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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/23/2008 6:57 PM

I can't wait for the explanation without length contraction. I think both length contraction and mass increase are phantoms of equations used to explain wave properties. For example take the doppler effect on a water wave and indeed you get wavelength contraction but it's not real. Just as a siren approaching you doesn't really increase in pitch at all, it just sounds that way to you. Your time observation has been affected by the relative speed between frames. Same with mass increase. By moving protons close to the speed of light you're not increasing their mass, you're shortening their wavelength and increasing their energy by increasing their frequency. Hence, infinite frequency requires infinite energy so you can't reach the speed of light. No mass particles are suddenly gluing themselves onto the proton. If there were, the proton would become even more massive as we input energy into slowing them down. We have galaxies moving at close to light speed and we don't observe mass increase or length contraction in them and we'll never see it happening because it doesn't really exist except as a mathematical trick.

The real effect in the ladder paradox is only time dilation but not as described in relativity. A moving object will slow down the observer's clock much like the doppler effect affects the pitch we hear. Hence assuming length contraction exists and that the two garage doors close simultaneously are false assumptions. The speed of the ladder affects the time we see the last door close just as in the doppler effect. This means there are no separate realities for each reference frame. The ladder does fit into the time warped garage and the time warped doors do close simultaneously time warped. I mean what's the mystery? But let's get rid of the artificial constructs of length contraction and mass increase and de-mystify relativity so all can understand.

Oh and while we're at it, let's get rid of the myth that moving clocks slow down. A moving clock will slow your observation of it. This is a subtle but important difference which can be borne out experimentally by quantum entangling the outputs of two clocks. Once you remove the problems of observation limited by the speed of light, there'll be a firmer understanding of the underlying reality.

Ralf Chlipalski

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#9
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/25/2008 12:15 PM

That was my first post. I've been trying for 4 days to derive a formula to relate the time warp of simultaneity. That is relating the speed of the ladder (moving frame) to how it delays the closing of the second door with respect to the closing of the first door separated by a garage length G. Does this formula already exist because I just don't seem to be intelligent enough to figure it out. I'm sure if I did figure it out, I would then undertand Lorentz transforms and minkowski diagrams.

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#10
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/27/2008 1:47 AM

Hi ralfcis, you did notice that the "solution without length contraction" was given in part iii: "http://cr4.globalspec.com/blogentry/854/Paradoxes-of-Relativity-Part-3c-Ladder-Paradox-iii"?

Jorrie

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#11
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/27/2008 2:17 PM

Thanks, saves me from checking for it everyday.

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#12

Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

04/05/2010 9:37 AM

With all do respect to those who are proficient with Minkowski Diagrams, I don't like to use them. My reason is that a Minkowski Diagram is a tool, a complex tool. A and there is a tendency to spend a lot of time arguing about how to use the tool and very little time on the real physics.

So, I wrote a 3d simulation program to verify what was going on. Now I spend all my time doing computer programming instead of the real physics. But at least you don't have to.

I've written a tutorial to guide a new user in using the simulation program to see what is going on. I call it the "Pole in the Barn Paradox". My tutorial is at:

http://mysite.verizon.net/mikelizzi/Tutorials/TutorialPoleInBarn.html

Even if you don't want to (or can't) run the simulation, you can still get a lot out of the tutorial since it has lots of pictures from the simulation.

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#13

Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

04/26/2010 5:13 PM

This concept of varying simultanaity of the front and rear of the ladder and garage does help explain this. I know this problem as the letter and letterbox paradox where the letter may or may not be larger than the letterbox. But now imagine that the letter is trying to get through the letterbox and is constrained to move perpendicularly to the letterbox opening. I can see that the two observers may have differing opinions of whether the letter (=ladder) is larger or smaller than the box (=garage) but does it get through the letterbox? (or does the ladder slip into the garage). Either it does or it does not. Both observers must see the same outcome??

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#14

Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/10/2015 11:55 AM

I haven't yet seen your explanation here but I assume it uses length contraction from one perspective and the explanation using relativity of simultaneity to match that perspective's fit or no fit from the contrary perspective. However, although this method allows processing to correlate fit or no fit from both perspectives, it does not match "by how much" it fits or no fits. For the length contraction perspective, the max length is the max length of the ladder or barn but from the relativity of simultaneity perspective of the length contraction perspective, the max "length" (simultaneous separation of one end of the ladder to the other) can approach infinity as c is approached. Hence I don't see a quantitative proof that there is no paradox. There's a huge difference between 10 or 15 ft going toward 0 ft and an object length approaching infinity. Is there a proof where the numbers match from either perspective?

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#15
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/11/2015 12:41 AM

"For the length contraction perspective, the max length is the max length of the ladder or barn but from the relativity of simultaneity perspective of the length contraction perspective, the max "length" (simultaneous separation of one end of the ladder to the other) can approach infinity as c is approached. Hence I don't see a quantitative proof that there is no paradox. There's a huge difference between 10 or 15 ft going toward 0 ft and an object length approaching infinity."

I do not see any infinities creeping in here. From either perspective the "moving" object is contracting, not expanding, so where does "object length approaching infinity" come from?

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#16
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/11/2015 5:21 AM

From the barn's perspective the ladder is length contracting and fits but from the ladder's perspective the barn is contracting and it can't fit. So the barn wonders how the ladder's perspective is possible. So it calls in the effects of relativity of simultaneity where both ends of the ladder are "simultaneously" inside the barn while in actual fact the front of the ladder is way outside the barn. The faster the relative velocity, the more time between the ladder's ends and the closer to infinity the front of the ladder is actually outside the barn while its rear is inside while simultaneity says both ends are inside the barn simultaneously.

Oh I may have it wrong. Is it possible the faster the ladder, the faster its rear end is through the barn and the shorter it'll look inside the barn no matter how far outside the barn the front is?

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#17
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Re: Paradoxes of Relativity Part 3 - Ladder Paradox (i)

08/11/2015 7:23 AM

"Oh I may have it wrong. Is it possible the faster the ladder, the faster its rear end is through the barn and the shorter it'll look inside the barn no matter how far outside the barn the front is?"

Sure, you had it wrong, but you are getting closer now. Just one thing, as far as the barn is concerned, the whole ladder was inside the barn at one moment in time (i.e. simultaneously) - no ends sticking out. That's the barn observer's perspective according to his own definition of what is simultaneous. Obviously others, e.g. the ladder observer, will disagree, observing that the barn is too short for the ladder.

Remember, we have started with an arrow that is too long for the box when stationary relative to each other, so there is no complementary aspect to this 'paradox'. It is just the way different observers will measure the situation when relative motion is involved. Again, it is about relative, not absolute motion!

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