Sorry guys, Lorentz's Contraction "Paradox" again!

There is a ship moving away from earth at velocity V1=0.6c relative to earth. In V1 reference frame, the ship accelerates at acceleration a=1E6 m/s² for an elapse time of <font face=symbol>D</font>t = 1 second and reaches velocity V2 relative to earth.

Velocity u gained in V1 reference frame is approximately 1E6*1=1E6 m/s and V2:

V2 = (0.6c+1E6)/(1+0.6c*1E6/c²) = 180,638,722 m/s (based on c = 3E8 m/s)

Gamma (<font face=symbol>g</font>) factor for V1 and V2 is respectively 1/0.8 and 1/0.7984

Along the direction of motion, there are (distance are as seen by earth observer):

1) Moon, 400,000 km away
2) Sun, 150,000,000 km away
3) Alpha Centaury, 4.3 light years away

What is the change of distance for those object as seen by the ship observer when its velocity increase from V1 to V2, aka change of Lorentz's contraction? The general formula for computing this is:

<font face=symbol>D</font>L = L (1/<font face=symbol>g</font>₁ - 1/<font face=symbol>g</font>₂), where L is the object distance as listed above. Since, only L is variable, we can write:

<font face=symbol>D</font>L = L (0.8 - 0.7984) = 0.0016*L

Having this simple equation, we can now easily compute <font face=symbol>D</font>L and the respective V_M defined as
<font face=symbol>D</font>L/<font face=symbol>D</font>t, which results:

1) With respect to Moon, <font face=symbol>D</font>L=640 km and V_M = 640 km/s
2) With respect to Sun, <font face=symbol>D</font>L=240,000 km and V_M = 240,000 km/s
3) With respect to Alpha Centaury, <font face=symbol>D</font>L=6.51E10 km and V_M = 6.51E10 km/s (over 2 millions times the speed of light in vacuum, or is it?)


It should be noted that the above V_M velocities are only last for one second, that is only exist during the acceleration period. This is one of the oddest properties of this "velocity". Clearly, this "velocity" is not normal velocity. It is also useless to compare this "velocity" with, for example, the speed of light. So, when its value exceed 300,000 km/s, it doesn't mean it is faster than light...just not that kind of velocity.

It should be also clear from the above that this velocity V_M is not well defined as it could be any value; for every different L we would get different V_M. Is there any paradox? Ahhh, I think this is too much!

Paul T,
Thank you. Finally your tone has become appropriate and hence I choose to continue this dialog.

What is this?

Posted by MacM in original Lorentz Contraction Paradox (LCP) introduction post: "(for general example without computing the gamma function) would appear to have moved 10 B Ly in a matter of an hour.

Posted by MacM in LCP 5/30 @ 4:40 PM: "Therefore the spatial contraction must according to that collapse many times the speed of light as I approach v = c. As the spatial dimension contracts I am getting closer, hence the faster I fly away the faster I would appear to be getting closer.

Posted by MacM in LCP 6/1 @ 8:28 PM: "Now I already know you will scream "Time Dilation". Go ahead, at V2 time dilation is such that 1 sec is actually 7,071 seconds earth time. So while I shouldn't I'll give you time dilation and still claim that if spatial contraction in Relativity is as you claim then in one second (or 7,071 seconds) moving AWAY from earth I would see earth move CLOSER by 967,009 Lyr!!!"

AND

"Note this is Light YEARS per Second. "

I could post many more examples of my statements but I think this should suffice. Now:

Posted in "MacM's Exercise"6/7 @ 7:27 AM: " Posted by MacM to James R:“ Can you comment yet on the velocity addition situation? ”

Posted by James R in response: "I don't see how it is relevant. I will wait for Paul T to explain, if he wishes."

First I'll note that you have now tried to qualify your work by adding the "approximately" term. Because you now realize you are attempting to perform this calculation in an accelerating frame.

I'll note that you are calculating V2, where in my example I approximated an acceleration and specified V2. This is not a complaint, just a note about the changing of criteria of the example. I also note the continued value of c = 3E6 m/s. It is repeated throughout so I don't see that as a typo.

Hmmmmmm.

I find this whole thing a bit curious and amusing.

You start your own thread of the same topic and lead off with "Sorry", like you are proving something important that contridicts conclusions of my original thread which you opposed.

Even altered slightly, at your option and perogative, you ultimately come to the same conclusion. But you note that the affect is only for the duration of the acceleration. Do you not think that it would only exist during the closure (acceleration) of my example also. The ultimate distance becomes fixed once V2 in my example was achieved but during the acceleration is where the FTL observation and paradox occurs. Good job - Thanks.

But because you proved it to yourself and the rest of us after all, you now want to phoo-phoo the result and say lets ignore it, it can't be real and it only last 1 second.

Do you not realize your 2 million c velocity for one second could also be > c for just under 2 million seconds? 555 hours or 23 + days!

As you like to say Paul T, "You are a funny man".

MacM,

It was just a typo, Mr MacM. It didn't affect the whole calculation.

So Paul T
Do you feel that you have found a paradox or just a mathematical anomally?

Paul T,

I'll accept your claim but please try to remember these threads and your own error, typos and such, in the future before calling others ignorant or worse.

Have a nice day.

QQ,

He found??? Oh the burden of fleeting fame.

MacM,

This error was made by me and I admit it. Have you admitted any of your errors that I pointed out. I have seen none. Rather than admitting your errors, you went on covering them up. I remember when I was a kid, I tried one or two times to put the dead rat into plastic bag, sealed it tightly so that I could throw them into the rubbish bin...but, it still smell just the same as your errors. Should I remind you what are your errors that you still have not found as errors (at least, that what's you tried to convince us)? I don't know about you, but for me, I would respect you more if you did admit your error, corrected them and move on to the nest topic.

To agree with something some one else has found is to find that which others have found as founded.

In other words to truely learn we have to find it ( again ) for ourselves.

Quantum Quack,

No, I did not found paradox. MacM did, hehe. The velocity associated to Lorentz's Contraction was named MacM's Velocity, denoted by V_M. But, of course, MacM's Velocity is a mere product of hallucination.

Velocity (the real one) that we know goes something like this. To determine velocity, say from a moving rocket relative to a star, the observer first measure the distance to that star and after an elapse time of <font face=symbol>D</font>t, he measure again and obtain <font face=symbol>D</font>s. He then determine the relative velocity of his rocket to the star using v=<font face=symbol>D</font>s/<font face=symbol>D</font>t. Assuming that the velocity is constant. If another observer at the star measures the velocity of the rocket, she would obtain v the same (numerically) as that ontained by the rocket observer. The real velocity behaves this way.

MacM's Velocity behaves very differently. As I have shown before, it has no specific value (ir can be any value) and it last during the rocket acceleration only. MacM's Velocity is therefore not a real velocity, it cannot be compared to, say the speed of light.

MacM thought there is a paradox because he found from his exercise that the rocket accelerating away from earth could observe the earth-rocket distance to decrease instead of increase. I don't see this as paradox, as the the rocket is in the non-inertial reference frame.

So,
Paul T
Can you tell me what your thread starter was all about?
Was it a demonstration of your own approach or a demonstration of someone elses approach?

V=2000000*'c' striked me as strange

I would say, my approach and someone else problem.

It strange to you, it's unbelievable to me. The "velocity" is not of the kind that we know for the speed of light and therefore is not comparable to normal velocity. I presented the computation to show that MacM's Velocity has no specific value.

That's all I have to say.

Paul T,

Fantastic. We probably won't become best of buddies but I think we might be able to have a mutually respectful communication in the future. I see your mathematical skills exceed my own but that doesn't make me enept of the processes. Like you said to another poster. "I can bite"

Actually I have but not in a point by point case. Se my post to James R where I admitted to some errors, etc.

You got it. If you would bother to list what you think are errors I will be pleased to post a point by point reply. If they are errors I will gladly admit it. I have in the past here (SciF) in the past. However, be prepared for me to also qualify certain posts.

For example I was aware that computing the acceleration at relavistic velocities was going to be beyond me, so I qualified that post and gave a general (approximation) answer and said (1 second or 7,071 seconds). So while mathematically my number may not be correct I knew it wasn't but also commented that in any case it didn't alter the ultimate claim. But list it and I will admit it is numerically incorrect.

Also I can do the algebra that you have done. I am good at algebra. I don't do calculus but understand its principles. My problem would be not knowing if the correct algebra is being used in some cases.

PaulT
Please excuse my confusion and I know you said you wanted to say no more on the subject but from my perspective I woudl like to ask:

You state that it is a velocity that can not be compared to normal velocity?

If it can't be a velocity comparable to normal velocity what sort of velocity can it be compared to?

Obviously the figure is so extreme that it brings up the question of validity.

But are we just excersising a convenience by stating that
Could it be the theoretical speed limit may impose a restriction on having this sort of result.

I am sure that the result is well formed but unable to be explained using current thinking about the nature of light.
Am I right in feeling that your extreme figure is correctly arrived at using the matematical methods that are currently accepted as valid?

Paul T,

Maybe we did learn something here.

Just wait for my next topic. You will love it too I'm sure.


PS: I am not adding this closing statement to piss you off but as a statement of fact.

MacM's velcocity is Relativity's velocity and it MUST be just as real as Relativity's purported contraction of space. Perhaps in time you will learn why I hold some of the views I do. I think different than most of you. Part of my training I guess.

It was just not properly defined as "velocity" and therefore is not velocity at all. See below.

It's invalid not because it gives "velocity" too extreme, but because it was not correctly defined.

Strickly speaking, no. There is no problem in getting velocity (an instant velocity; don't remember if this term correct) for an accelerating object or from an accelerating reference frame. What we need is, take <font face=symbol>D</font>t as small as possible (ussually denoted as dt), so that velocity during that infinetesimal time can be considered as constant. This was not done in the first post under this thread and obviously something MacM did not consider. By not taking infinitesimal time, we are unable to determine the so-called instant velocity and also differential Lorentz's contraction (which can not be considered to exist if we use dt instead) enter into the picture.

Paul T,

Two points.

1 - Only to clarify: I did not use the delta t process but I am aware of it. Rather than have you suggest otherwise, if you are not aware, then I'll inform you that James R., and I have had that discussion on this forum before in dealing with rockets under acceleration, etc.

2 - Have you given any thought to re-doing your presentation using the delta t process?. I think it would be informative to the readers.

It does seem to me that the distance stipulated for V1 and the relavistically adjusted distance for V2 are terminal velocities. That is they are constant velocities and hence the distance change during the interval of acceleration time still leaves you with the observed paradox. Using dt can only change how one gets from point A to point B, not the final point otherwise the relavistic contraction formula would be invalid.

MacM,

No. Not necessary. If you use dt instead, you will get instant velocity starting from V1 to V2 and you won't see differential Lorentz's contraction (because of the infinitesimal dt). When you measure length/distance when the rocket at velocity V1 and then at V2 (while V1 and V2 are substantially difference), you would of course get differential length attributed to Lorentz's contraction. However, that would mean you are deducting length measured from two different inertial reference frame (you can't do this). When you use dt instead, you assume that your system is still under an inertial reference frame.

O, I just want to remind you that you were wrong about SR cannot handle acceleration; with some trick, it can. Check Baez's site about relativistic rocket, it was derived using SR and there is acceleration.

You can consider velocity at V1 and V2 constant, but they are two different constant velocities. You can't mix (length) measurement obtained for V1 and V2 reference frames and use them to compute some sort of "velocity".

Paul T,

My question is what is the ultimate difference in the end when you have two velocities, over a given time, with two distances linked by relativity to those velocities. Your process is different but the physical reality is the same.

I understand you are trying to set up a condition where you can claim ds/dt is not a velocity. I'm looking for what you want to replace it with?

Not necessary. I looked back and indeed I saw where I said something to the affect that you can't compute that in an accelerating frame. So I'll accept your statement but want to qualify that I actually meant the statement to apply to the simple application of the velocity addition formula in the accelerting frame. Before you object I ask you look at the many other comments I made on that issue. They were directed to your direct use of the VAF. I did not mean it was impossible to calculate via SR. Indeed as I have said the dt process in accelerating frames has been discussed here before.

Why not? Because it gives results that you don't want? v = ds/dt. I dare say that if you were in the captains chair you would sound the abandon ship alarm because of the velocity you saw the earth coming at you!

Paul T,

I also meant to comment on this. Rather than me put my foot in my mouth by making the statement I would like to make I'll pose it in the form of a question.

Can you explain to us how "she" would see the ds/dt measurements any differently than the rocket pilot? If not then does not the behavior mimic your "real" velocity in every respect?

Your arguements seem to require that things vanish or cease to exist during acceleration since velocity can't be observed as ds/dt. I suggest that I very much can see a Mac truck when it is accelerating my way.

MacM,

I did not suggest that you need to replace v=ds/dt, which is the correct representation for velocity, where ds and dt are infinitesimal displacement and time.

Differential Lorentz's contraction only exist if you have two velocities, V1 and V2. If you able to measure V1 and V2, your ds is no longer infinitesimal as for infinitesimal ds, you will get only one velocity, not two. In English, you should take ds (and dt) limit to zero or very very close to zero.