Why Witch Sabrina Looks Thinner While Riding On Her Broom.

[Solitary]

This might help explain it better, or not. :[youtube:1tm4qvgh]http://www.youtube.com/watch?v=rek7881OGRY[/youtube:1tm4qvgh]

As far as length contraction goes, the question is what is meant by a real physical
phenomenon.  Of course, objects are not really squeezed together in the sense that stress or strain forces could be measured.  But in three dimensions, length contraction might not even be apparent.

 Objects sometimes appear to be contracted, but only because they are rotated.  A sphere, for example, would appear to be rotated - not distorted to an ellipsoid.  Even after almost a century of STR, the subject of how moving bodies would appear if c = 100m/s remains controversial.

Solitary

[josephpalazzo]

There's something that isn't clear to me in this discussion. It's about what "actually contracted" means in this context. If there were a an absolute, fixed length dimension, then I could readily see how a length could be contracted with reference to that absolute, fixed length dimension and thus what "actually contracted" means ("actually contracted" meaning contracted with reference to a fixed, absolute length of a spatial dimension). But if there is no absolute, fixed length dimension that is a reference, then what length is being referenced to say that a contraction is actual?

I am not trying to claim that people are mistaken in saying that such a contraction is actual; what I'm wondering is if what may be going on here is that different people mean different things when they use the term "actual" in this context. I ask the above question to see if it may help ferret out any such differences in conception of what an "actual contraction of length" means, if there are any such differences.

The best way to figure out things in SR is with graphics. You can represent one observer in a t-x graph, and a second one in a t' - x' axis.

[ Image ]

In the case of length contraction, consider the ends of a meter stick that stand initially at the origin d(0,0) and at e(0,L). It is at rest in the unprimed frame of reference. And the two parallel lines in red indicate this. The dash line at 45° represents the speed of light ( t = x). The moving observer in the primed frame of reference will see a different meter stick. The moving observer will measure the meter stick along a synchronous time, which is the x'-axis, or t=vx, which will be the points d and f. But remember that df is really the proper time, a Lorentz invariant. What is the length that the moving observer measures? To find that, we must locate the x'-coordinate of the point f((0', L').

(1) t'[sup:3i08t9nq]2[/sup:3i08t9nq] – x'[sup:3i08t9nq]2[/sup:3i08t9nq]  =  t[sup:3i08t9nq]2[/sup:3i08t9nq] – x[sup:3i08t9nq]2[/sup:3i08t9nq]

Substituting,
(2) (0')[sup:3i08t9nq]2[/sup:3i08t9nq] – L'[sup:3i08t9nq]2[/sup:3i08t9nq] =  (vx)[sup:3i08t9nq]2[/sup:3i08t9nq] – x[sup:3i08t9nq]2[/sup:3i08t9nq]

(3) – L'[sup:3i08t9nq]2[/sup:3i08t9nq] =  v[sup:3i08t9nq]2[/sup:3i08t9nq]L[sup:3i08t9nq]2[/sup:3i08t9nq] – L[sup:3i08t9nq]2[/sup:3i08t9nq]
or L'[sup:3i08t9nq]2[/sup:3i08t9nq] =  – v[sup:3i08t9nq]2[/sup:3i08t9nq]L[sup:3i08t9nq]2[/sup:3i08t9nq]  +  L[sup:3i08t9nq]2[/sup:3i08t9nq]
                                                               
(4) L'[sup:3i08t9nq]2[/sup:3i08t9nq] =  (– v[sup:3i08t9nq]2[/sup:3i08t9nq]  +  1)L[sup:3i08t9nq]2[/sup:3i08t9nq]
 
(5) L'[sup:3i08t9nq]2[/sup:3i08t9nq] =  ( 1  –   v[sup:3i08t9nq]2[/sup:3i08t9nq] )L[sup:3i08t9nq]2[/sup:3i08t9nq]

Taking the square root and putting c back into the equation,
                                     
(6) moving ruler  =  stationary ruler (1  –  v[sup:3i08t9nq]2[/sup:3i08t9nq]/c[sup:3i08t9nq]2[/sup:3i08t9nq])[sup:3i08t9nq]1/2[/sup:3i08t9nq]
                                                               
We see that a moving meter will shrink.

I think I follow that, but it still isn't clear to me why this implies that the contraction is "actual" unless the stationary ruler is taken to be "the preferred" reference frame in terms of measurement of length. In a sense, it appears you want to treat the stationary ruler as though it were a fixed, absolute reference frame for length in terms of determining whether or not the moving ruler has "actually" contracted. Is there some reason to treat the stationary ruler as being preferred with respect to measuring length over the moving ruler? To put it another way, it's seems like you are saying that with respect to the stationary ruler, the moving ruler is contracted, therefore the contraction is 'actual'. But from the reference frame of the moving ruler, it is not contracted, therefore couldn't it be just as validly claimed that with respect to the reference frame of the moving ruler that there is no 'actual" contraction?

What the graphics says is:

The laws of nature should not depend on the coordinate system. So whether one chooses the coordinate x-t or the coordinate x'-t', or any other type of coordinate system for that matter, the laws of nature should be the same. So that explains equation (1) -- both see that the speed of light is constant, as required by the null result of the Michaelson-Morley experiment, and Maxwell's equations. The rest of the equations in the derivation just follow logically by the rules of math. Now, from equation (6) if you are stationary, your ruler doesn't contract. So in your frame, nothing happens. Now, you have to remember, there is also time dilation taking place. It's the combination of these two effects, that makes the speed of light constant for every observer.

Now in practice, you won't be able to measure that contraction: how do you measure the length of something which is moving away from you? However, time dilation can be measured in the half-life of cosmic muons as they come from outerspace, which differs from the half-life of muons produced in the lab. The difference is fully acounted by SR.

[entropy]

What the graphics says is:

The laws of nature should not depend on the coordinate system. So whether one chooses the coordinate x-t or the coordinate x'-t', or any other type of coordinate system for that matter, the laws of nature should be the same. So that explains equation (1) -- both see that the speed of light is constant, as required by the null result of the Michaelson-Morley experiment, and Maxwell's equations. The rest of the equations in the derivation just follow logically by the rules of math. Now, from equation (6) if you are stationary, your ruler doesn't contract. So in your frame, nothing happens. Now, you have to remember, there is also time dilation taking place. It's the combination of these two effects, that makes the speed of light constant for every observer.

Now in practice, you won't be able to measure that contraction: how do you measure the length of something which is moving away from you? However, time dilation can be measured in the half-life of cosmic muons as they come from outerspace, which differs from the half-life of muons produced in the lab. The difference is fully acounted by SR.

Okay, that helped. It seems that the issue I'm zeroing in on depends on the designation of proper time - the time of the non-moving ruler, if I understand it right. I guess what I'm getting hung up on is probably a very common stumbling block - if neither ruler is accelerating, then why is one ruler designated as being the one with proper time. But I imagine that's an elementary issue you deal with quite often with students so I won't trouble you to ask about it. Thank you for taking the time to answer my questions.

[josephpalazzo]

What the graphics says is:

The laws of nature should not depend on the coordinate system. So whether one chooses the coordinate x-t or the coordinate x'-t', or any other type of coordinate system for that matter, the laws of nature should be the same. So that explains equation (1) -- both see that the speed of light is constant, as required by the null result of the Michaelson-Morley experiment, and Maxwell's equations. The rest of the equations in the derivation just follow logically by the rules of math. Now, from equation (6) if you are stationary, your ruler doesn't contract. So in your frame, nothing happens. Now, you have to remember, there is also time dilation taking place. It's the combination of these two effects, that makes the speed of light constant for every observer.

Now in practice, you won't be able to measure that contraction: how do you measure the length of something which is moving away from you? However, time dilation can be measured in the half-life of cosmic muons as they come from outerspace, which differs from the half-life of muons produced in the lab. The difference is fully acounted by SR.

Okay, that helped. It seems that the issue I'm zeroing in on depends on the designation of proper time - the time of the non-moving ruler, if I understand it right. I guess what I'm getting hung up on is probably a very common stumbling block - if neither ruler is accelerating, then why is one ruler designated as being the one with proper time. But I imagine that's an elementary issue you deal with quite often with students so I won't trouble you to ask about it. Thank you for taking the time to answer my questions.

There is an easy trick to remember who is measuring the proper time: it's always the observer who measures two events with one clock.



In coordinate t-x, in which the ruler is at rest, the observer needs two clocks: one at d(0,0), and a second one at e(0,L). The observer in t' - x' can use one clock since the ruler is moving wrt to his frame. He can wait from one end to pass in front of him, at which time he sets t' = 0, and then when the ruler's other end passes in front of him, he measures t', and so this is the proper time in this situation.

[entropy]

Okay, that helped. It seems that the issue I'm zeroing in on depends on the designation of proper time - the time of the non-moving ruler, if I understand it right. I guess what I'm getting hung up on is probably a very common stumbling block - if neither ruler is accelerating, then why is one ruler designated as being the one with proper time. But I imagine that's an elementary issue you deal with quite often with students so I won't trouble you to ask about it. Thank you for taking the time to answer my questions.

There is an easy trick to remember who is measuring the proper time: it's always the observer who measures two events with one clock.

[ Image ]

In coordinate t-x, in which the ruler is at rest, the observer needs two clocks: one at d(0,0), and a second one at e(0,L). The observer in t' - x' can use one clock since the ruler is moving wrt to his frame. He can wait from one end to pass in front of him, at which time he sets t' = 0, and then when the ruler's other end passes in front of him, he measures t', and so this is the proper time in this situation.

Since you have been nice enough to try to explain this to me, I'm going to be bold enough to ask what I imagine are basic questions that occur to a lot of people when they encounter these issues (or, worse, don't occur to most people because the answers are so obvious that most people see the answer and don't have questions):

Hypothesize that the whole universe is made up of only two rulers. Each can send light impulses toward the other and detect a reflection and measure the time it takes the light to be emitted and the reflection detected, so the rulers are able to determine that the displacement between them is increasing at a regular rate. Wouldn't it be arbitrary as to which ruler gets designated as "at rest" - which then determines which ruler's time is considered proper time? If it is arbitrary as to which gets designated as at rest, then wouldn't it be arbitrary as to which one is considered to have actually contracted?

[Solitary]

This post is getting really tiresome, so this is the last time I'll try to explain what actually happens in the real world we live in and not the world of mathematics with regards to the special theory of relativity that makes logical sense and is easy to understand without a bunch of mathematical equation to obscure everything.

Special relativity applies the Galilean relativity principle to electromagnetic theory. Since the speed of light is determined by basic equations of that theory, if the relativity principle is to hold, you can conclude that the speed of light must be the same for observers in any inertial frame, regardless of the velocity of the light's source. This is profoundly counter-intuitive, once one explores what it means. Three of the immediate consequences of the constancy of light's velocity are the relativity of simultaneity, length contraction (apparent shortening, in the direction of motion, of rapidly moving objects), and time dilation (apparent slowing down of fast-moving clocks).

I can explain an instance of the first phenomenon here, using an example first crafted by Einstein. Suppose a fast-moving train is passing a signal light on the ground. Just as the center of the car you are in passes the lamp, it emits a flash of red light. Since you know the speed of light is a fixed constant c in inertial frames, and you are in an inertial frame, you conclude that the flash of light arrives simultaneously at the back and the front of your train car. The two arrival events, for you, are simultaneous. But now consider how an observer standing next to the signal lamp on the ground views events.

 The light travels at the same speed c toward the back of the train and toward the front. But the train is moving, fast, forward. So the light will reach the rear of your car first, and then the forward-going light rays, having to catch up to the rapidly advancing front, will arrive some time later. That is: for the observer on the ground, the two events (light arrives at back of car; light arrives at front) are definitely not simultaneous.

And if the Special Principle of relativity is to be taken seriously, we must admit that neither observer's perspective is "correct"; judgment that two events in different locations are "simultaneous" is simply relative – relative to state of motion, or reference frame. (The other two surprising effects just mentioned can be deduced from similarly simple thought-experiments.

Anyone that disagrees it's OK with me because this makes sense to me and to say solid object actually contracts to one observer and not to another and vice versa is ridiculous unless you are talking about measurement. Solitary

[entropy]

This post is getting really tiresome, so this is the last time I'll try to explain what actually happens in the real world we live in and not the world of mathematics with regards to the special theory of relativity that makes logical sense and is easy to understand without a bunch of mathematical equation to obscure everything.

Special relativity applies the Galilean relativity principle to electromagnetic theory. Since the speed of light is determined by basic equations of that theory, if the relativity principle is to hold, you can conclude that the speed of light must be the same for observers in any inertial frame, regardless of the velocity of the light's source. This is profoundly counter-intuitive, once one explores what it means. Three of the immediate consequences of the constancy of light's velocity are the relativity of simultaneity, length contraction (apparent shortening, in the direction of motion, of rapidly moving objects), and time dilation (apparent slowing down of fast-moving clocks).

I can explain an instance of the first phenomenon here, using an example first crafted by Einstein. Suppose a fast-moving train is passing a signal light on the ground. Just as the center of the car you are in passes the lamp, it emits a flash of red light. Since you know the speed of light is a fixed constant c in inertial frames, and you are in an inertial frame, you conclude that the flash of light arrives simultaneously at the back and the front of your train car. The two arrival events, for you, are simultaneous. But now consider how an observer standing next to the signal lamp on the ground views events.

 The light travels at the same speed c toward the back of the train and toward the front. But the train is moving, fast, forward. So the light will reach the rear of your car first, and then the forward-going light rays, having to catch up to the rapidly advancing front, will arrive some time later. That is: for the observer on the ground, the two events (light arrives at back of car; light arrives at front) are definitely not simultaneous.

And if the Special Principle of relativity is to be taken seriously, we must admit that neither observer's perspective is "correct"; judgment that two events in different locations are "simultaneous" is simply relative – relative to state of motion, or reference frame. (The other two surprising effects just mentioned can be deduced from similarly simple thought-experiments.

Anyone that disagrees it's OK with me because this makes sense to me and to say solid object actually contracts to one observer and not to another and vice versa is ridiculous unless you are talking about measurement. Solitary

There are one or two things you said that I might quibble with a bit, but the quibbles probably are more about how you chose to express certain notions than about the substance of what you were expressing.

I appreciate josephpalazzo taking the time to lay out the mathematics. It has helped me come to see that my questions seem to center on the issue of which object gets designated as "at rest" and therefore is said to have proper time for purposes of cranking through the equations of special relativity. I think his conversation with me has led me to realize that I don't understand on what basis the designation "at rest" is made.

In your discussion above about the two observers - one on a train and the other standing off to the side of the train - you posit the situation as the train being in motion. But doesn't that assume that the earth is at rest (and the observer standing "solidly" on the earth is also at rest)? To the observer on the train, relative to the train he is on, isn't he "at rest"? I don't get on what basis we take one observer to be at rest for purposes of the calculations. Is it a matter of determining something like momentum - e.g., that the train has momentum that the earth doesn't and so the train is said to be in motion and the earth "at rest"? If the determining factor isn't something like momentum, then it seems like designating the earth being at rest and the train in motion is just a consequence of our psychological impressions due to the relative size differences between the train and the earth.

Your explication above reminded me of the experiment Doppler did with a musician playing on note on a horn on a moving train. I'm not sure how tightly the Doppler experiment holds as an analogy, but it does seem like the interpretation of what Doppler showed in his experiment is similar to the situation with respect to special relativity and contraction. As the train approached Doppler, the pitch of the note was higher than it was for those who were riding on the train with the musician. In such a situation, does it make sense to say that the pitch was "actually" higher? Well, it was "actually" a higher pitch for those standing outside the train with the train approaching, but the pitch was not "actually" higher for those riding on the train with the musician. So in Doppler's case the "actually" is relative to the observer. What isn't evident to me yet is why the situation with the perception of length contraction isn't very much like that with Doppler's findings about what the observers on the train and off the train hear; that is, that the impression of contraction is relative to the observer and that there is no "actual" contraction taking place in an absolute sense - just like there is no absolute sense of which heard note was the "actual" note in Doppler's case.

[josephpalazzo]

Okay, that helped. It seems that the issue I'm zeroing in on depends on the designation of proper time - the time of the non-moving ruler, if I understand it right. I guess what I'm getting hung up on is probably a very common stumbling block - if neither ruler is accelerating, then why is one ruler designated as being the one with proper time. But I imagine that's an elementary issue you deal with quite often with students so I won't trouble you to ask about it. Thank you for taking the time to answer my questions.

There is an easy trick to remember who is measuring the proper time: it's always the observer who measures two events with one clock.

[ Image ]

In coordinate t-x, in which the ruler is at rest, the observer needs two clocks: one at d(0,0), and a second one at e(0,L). The observer in t' - x' can use one clock since the ruler is moving wrt to his frame. He can wait from one end to pass in front of him, at which time he sets t' = 0, and then when the ruler's other end passes in front of him, he measures t', and so this is the proper time in this situation.

Since you have been nice enough to try to explain this to me, I'm going to be bold enough to ask what I imagine are basic questions that occur to a lot of people when they encounter these issues (or, worse, don't occur to most people because the answers are so obvious that most people see the answer and don't have questions):

Hypothesize that the whole universe is made up of only two rulers. Each can send light impulses toward the other and detect a reflection and measure the time it takes the light to be emitted and the reflection detected, so the rulers are able to determine that the displacement between them is increasing at a regular rate. Wouldn't it be arbitrary as to which ruler gets designated as "at rest" - which then determines which ruler's time is considered proper time? If it is arbitrary as to which gets designated as at rest, then wouldn't it be arbitrary as to which one is considered to have actually contracted?

Which frame is at rest and which one is moving is not determined by the rulers as such but by the observers. If the two observers are at rest with each other, there is no relativistic effect according to SR. Where are your observers wrt to the two rulers? Secondly, you need to determine which "events" your observers are considering. Are they measuring a time interval? Or a distance? Measuring a time interval and measuring a distance require to perform different experiments. Hence what will be labeled as the "events" will determine who is at rest and who is moving.

Now in your scenario as it is constructed, it seems that the rulers are at rest wrt each other, and since you didn't specify any observer, there is no effect of the SR kind.

However, you say that the displacement is increasing at a regular rate. This is analogous with galaxies moving away from each other. In this case, we are talking about General Relativity, not Special Relativity. This is a different ballgame. If you want clarification on SR, you should stick to scenarios in which SR effects take place.

[Jason78]

This post is getting really tiresome

Really?

Crtl-C, Ctrl-V must really take it out of you.

[entropy]

Hypothesize that the whole universe is made up of only two rulers. Each can send light impulses toward the other and detect a reflection and measure the time it takes the light to be emitted and the reflection detected, so the rulers are able to determine that the displacement between them is increasing at a regular rate. Wouldn't it be arbitrary as to which ruler gets designated as "at rest" - which then determines which ruler's time is considered proper time? If it is arbitrary as to which gets designated as at rest, then wouldn't it be arbitrary as to which one is considered to have actually contracted?

Which frame is at rest and which one is moving is not determined by the rulers as such but by the observers. If the two observers are at rest with each other, there is no relativistic effect according to SR. Where are your observers wrt to the two rulers? Secondly, you need to determine which "events" your observers are considering. Are they measuring a time interval? Or a distance? Measuring a time interval and measuring a distance require to perform different experiments. Hence what will be labeled as the "events" will determine who is at rest and who is moving.

Now in your scenario as it is constructed, it seems that the rulers are at rest wrt each other, and since you didn't specify any observer, there is no effect of the SR kind.

However, you say that the displacement is increasing at a regular rate. This is analogous with galaxies moving away from each other. In this case, we are talking about General Relativity, not Special Relativity. This is a different ballgame. If you want clarification on SR, you should stick to scenarios in which SR effects take place.

I tried to keep the situation as simple as possible so I sort of fudged the "observer" part. I'll try to rectify that with another more carefully crafted hypothesis. I'm not sure what the ramifications are of your point about the differences in measuring time and distance, but I'll also try to see if I can deal with that in the reworked hypothesis.

Something I don't understand is why a regular rate of increasing displacement implies that the hypothetical involves the same factors as that which are causing the increasing displacement between distant galaxies to increase. If I understand it correctly, the increase in displacement between distant galaxies is caused by an expansion of spacetime between the the distant galaxies. If that is the case, I'm not sure why we would have to assume that the spacetime between the rulers in my hypothetical were in a situation where spacetime was expanding. I'll try to deal with that in the reworked hypothesis as well.

New hypothetical situation:

Imagine that there is a universe with a finite "sphere" of spacetime where there is no expansion or contraction of the spacetime "fabric" itself. The only other things (other than spacetime itself) in this universe are two rulers that have observers on them that have devices that can send out light and detect a reflection of the light and have a clock with which to measure how long it takes the light to travel from the time of emission to the time of detection of the reflected light. Each observer sends out a beam of light and measures the time it takes the beam to reflect off of the other ruler back to the detector - the time being that which each measures with their own clock. They use that time and the known speed of light to calculate the distance the light went. They do this measurement several times and each time they do it, the distance the light went is calculated to be more than the previous calculated distance. For measurements that are made at equal time intervals apart by their own clocks, the calculated distance the light travels increases the same amount.

For purposes of using the equations of special relativity to calculate contraction, how do we determine which ruler/observer is at rest? It doesn't seem like they can be at rest with respect to each other because the distance light travels between the two different ruler/observers is calculated to be increasing at a regular rate.

[josephpalazzo]

New hypothetical situation:

Imagine that there is a universe with a finite "sphere" of spacetime where there is no expansion or contraction of the spacetime "fabric" itself. The only other things (other than spacetime itself) in this universe are two rulers that have observers on them that have devices that can send out light and detect a reflection of the light and have a clock with which to measure how long it takes the light to travel from the time of emission to the time of detection of the reflected light. Each observer sends out a beam of light and measures the time it takes the beam to reflect off of the other ruler back to the detector - the time being that which each measures with their own clock. They use that time and the known speed of light to calculate the distance the light went. They do this measurement several times and each time they do it, the distance the light went is calculated to be more than the previous calculated distance. For measurements that are made at equal time intervals apart by their own clocks, the calculated distance the light travels increases the same amount.

For purposes of using the equations of special relativity to calculate contraction, how do we determine which ruler/observer is at rest? It doesn't seem like they can be at rest with respect to each other because the distance light travels between the two different ruler/observers is calculated to be increasing at a regular rate.

The only way I can explain what's happening is on a graphic (sorry if my drawings are not up to par,  :-D )



The two blue lines would be the case if you and a distant object are at rest with each other, and a signal (in yellow) is sent back and forth. You can see that you would be receiving a signal at equal interval.

The axis in red represents the object moving away from you, and a signal (in green) is sent back and forth. You can see that the time elapsed between each signal is larger and larger.

Who said that a picture is worth a thousand words?  :P

[entropy]

New hypothetical situation:

Imagine that there is a universe with a finite "sphere" of spacetime where there is no expansion or contraction of the spacetime "fabric" itself. The only other things (other than spacetime itself) in this universe are two rulers that have observers on them that have devices that can send out light and detect a reflection of the light and have a clock with which to measure how long it takes the light to travel from the time of emission to the time of detection of the reflected light. Each observer sends out a beam of light and measures the time it takes the beam to reflect off of the other ruler back to the detector - the time being that which each measures with their own clock. They use that time and the known speed of light to calculate the distance the light went. They do this measurement several times and each time they do it, the distance the light went is calculated to be more than the previous calculated distance. For measurements that are made at equal time intervals apart by their own clocks, the calculated distance the light travels increases the same amount.

For purposes of using the equations of special relativity to calculate contraction, how do we determine which ruler/observer is at rest? It doesn't seem like they can be at rest with respect to each other because the distance light travels between the two different ruler/observers is calculated to be increasing at a regular rate.

The only way I can explain what's happening is on a graphic (sorry if my drawings are not up to par,  :-D )

[ Image ]

The two blue lines would be the case if you and a distant object are at rest with each other, and a signal (in yellow) is sent back and forth. You can see that you would be receiving a signal at equal interval.

The axis in red represents the object moving away from you, and a signal (in green) is sent back and forth. You can see that the time elapsed between each signal is larger and larger.

Who said that a picture is worth a thousand words?  :P

I do think I understand the graphic but I'm not sure how it answers my question. I'm probably either asking a nonsensical question or not expressing it clearly enough. Thanks for taking the time to try to help me figure it out, though.

[josephpalazzo]

New hypothetical situation:

Imagine that there is a universe with a finite "sphere" of spacetime where there is no expansion or contraction of the spacetime "fabric" itself. The only other things (other than spacetime itself) in this universe are two rulers that have observers on them that have devices that can send out light and detect a reflection of the light and have a clock with which to measure how long it takes the light to travel from the time of emission to the time of detection of the reflected light. Each observer sends out a beam of light and measures the time it takes the beam to reflect off of the other ruler back to the detector - the time being that which each measures with their own clock. They use that time and the known speed of light to calculate the distance the light went. They do this measurement several times and each time they do it, the distance the light went is calculated to be more than the previous calculated distance. For measurements that are made at equal time intervals apart by their own clocks, the calculated distance the light travels increases the same amount.

For purposes of using the equations of special relativity to calculate contraction, how do we determine which ruler/observer is at rest? It doesn't seem like they can be at rest with respect to each other because the distance light travels between the two different ruler/observers is calculated to be increasing at a regular rate.

The only way I can explain what's happening is on a graphic (sorry if my drawings are not up to par,  :-D )

[ Image ]

The two blue lines would be the case if you and a distant object are at rest with each other, and a signal (in yellow) is sent back and forth. You can see that you would be receiving a signal at equal interval.

The axis in red represents the object moving away from you, and a signal (in green) is sent back and forth. You can see that the time elapsed between each signal is larger and larger.

Who said that a picture is worth a thousand words?  :P

I do think I understand the graphic but I'm not sure how it answers my question. I'm probably either asking a nonsensical question or not expressing it clearly enough. Thanks for taking the time to try to help x = tme figure it out, though.

Ok, I might have assumed that it was easy to read. Sorry. Perhaps this might help:


a) The graph is drawn with c = 1.
b) Therefore x =t, so a light ray would make a 45[sup:20xeksej]0[/sup:20xeksej] angle with either the t-axis, or x-axis. (Yellow and green lines)
c)The time axis is always flowing so an object at rest would be a vertical line. Two objects at rest, two vertical lines (blue lines).
d)  If you follow just the two blue line and the yellow line, this would be the case of two people at rest sending signals at the speed of light.
e) An object moving would have tilted axis ( red lines).
f) if you follow a blue line( observer at rest), the red line (observer moving) and the green lines (signal), you get the communication between the observer at rest with an observer moving away in this case.
g) Note that the inteception of the light rays with the t-axis. With two observers at rest, the time interval is constant, but between one at rest, the other moving, the time intervals get larger.

Now, in your scenario, there is no length contraction and no time dilation involved simply because you are not measuring the length of an object from one frame to the other, nor are you measuring the time on a clock from one frame to the other. What you are measuring is the velocity of the moving observer, using light signals and each observer is using his own clock in his own frame. So from the graph, if you knew the times at which the signal crosses the t-axis, and the distance between the two objects  (distance between the two parallel blue lines), you would be able to determine v, from x= vt.

Recall that I said you need to know what "events" you want to measure. This determines how you set up your graphic. In this case you are measuring a velocity. And this graph would give you the answer, provided you have a few points on that graph (at least two points with (t[sub:20xeksej]1[/sub:20xeksej],x[sub:20xeksej]1[/sub:20xeksej]), (t[sub:20xeksej]2[/sub:20xeksej],x[sub:20xeksej]2[/sub:20xeksej]))

[entropy]

The only way I can explain what's happening is on a graphic (sorry if my drawings are not up to par,  :-D )

[ Image ]

The two blue lines would be the case if you and a distant object are at rest with each other, and a signal (in yellow) is sent back and forth. You can see that you would be receiving a signal at equal interval.

The axis in red represents the object moving away from you, and a signal (in green) is sent back and forth. You can see that the time elapsed between each signal is larger and larger.

Who said that a picture is worth a thousand words?  :P

I do think I understand the graphic but I'm not sure how it answers my question. I'm probably either asking a nonsensical question or not expressing it clearly enough. Thanks for taking the time to try to help x = tme figure it out, though.

Ok, I might have assumed that it was easy to read. Sorry. Perhaps this might help:
[ Image ]

a) The graph is drawn with c = 1.
b) Therefore x =t, so a light ray would make a 45[sup:2d0y6rqg]0[/sup:2d0y6rqg] angle with either the t-axis, or x-axis. (Yellow and green lines)
c)The time axis is always flowing so an object at rest would be a vertical line. Two objects at rest, two vertical lines (blue lines).
d)  If you follow just the two blue line and the yellow line, this would be the case of two people at rest sending signals at the speed of light.
e) An object moving would have tilted axis ( red lines).
f) if you follow a blue line( observer at rest), the red line (observer moving) and the green lines (signal), you get the communication between the observer at rest with an observer moving away in this case.
g) Note that the inteception of the light rays with the t-axis. With two observers at rest, the time interval is constant, but between one at rest, the other moving, the time intervals get larger.

Now, in your scenario, there is no length contraction and no time dilation involved simply because you are not measuring the length of an object from one frame to the other, nor are you measuring the time on a clock from one frame to the other. What you are measuring is the velocity of the moving observer, using light signals and each observer is using his own clock in his own frame. So from the graph, if you knew the times at which the signal crosses the t-axis, and the distance between the two objects  (distance between the two parallel blue lines), you would be able to determine v, from x= vt.

Recall that I said you need to know what "events" you want to measure. This determines how you set up your graphic. In this case you are measuring a velocity. And this graph would give you the answer, provided you have a few points on that graph (at least two points with (t[sub:2d0y6rqg]1[/sub:2d0y6rqg],x[sub:2d0y6rqg]1[/sub:2d0y6rqg]), (t[sub:2d0y6rqg]2[/sub:2d0y6rqg],x[sub:2d0y6rqg]2[/sub:2d0y6rqg]))

I did pretty much follow the graph, but now I understand better the point you were making about what is being measured. I think the most fruitful thing for me to ask now based on this last response is to ask about the red, bolded part of your response above about the moving observer. In the hypothetical, how do you determine which observer is the moving observer?

[josephpalazzo]

I do think I understand the graphic but I'm not sure how it answers my question. I'm probably either asking a nonsensical question or not expressing it clearly enough. Thanks for taking the time to try to help x = tme figure it out, though.

Ok, I might have assumed that it was easy to read. Sorry. Perhaps this might help:
[ Image ]

a) The graph is drawn with c = 1.
b) Therefore x =t, so a light ray would make a 45[sup:36otjgkr]0[/sup:36otjgkr] angle with either the t-axis, or x-axis. (Yellow and green lines)
c)The time axis is always flowing so an object at rest would be a vertical line. Two objects at rest, two vertical lines (blue lines).
d)  If you follow just the two blue line and the yellow line, this would be the case of two people at rest sending signals at the speed of light.
e) An object moving would have tilted axis ( red lines).
f) if you follow a blue line( observer at rest), the red line (observer moving) and the green lines (signal), you get the communication between the observer at rest with an observer moving away in this case.
g) Note that the inteception of the light rays with the t-axis. With two observers at rest, the time interval is constant, but between one at rest, the other moving, the time intervals get larger.

Now, in your scenario, there is no length contraction and no time dilation involved simply because you are not measuring the length of an object from one frame to the other, nor are you measuring the time on a clock from one frame to the other. What you are measuring is the velocity of the moving observer, using light signals and each observer is using his own clock in his own frame. So from the graph, if you knew the times at which the signal crosses the t-axis, and the distance between the two objects  (distance between the two parallel blue lines), you would be able to determine v, from x= vt.

Recall that I said you need to know what "events" you want to measure. This determines how you set up your graphic. In this case you are measuring a velocity. And this graph would give you the answer, provided you have a few points on that graph (at least two points with (t[sub:36otjgkr]1[/sub:36otjgkr],x[sub:36otjgkr]1[/sub:36otjgkr]), (t[sub:36otjgkr]2[/sub:36otjgkr],x[sub:36otjgkr]2[/sub:36otjgkr]))

I did pretty much follow the graph, but now I understand better the point you were making about what is being measured. I think the most fruitful thing for me to ask now based on this last response is to ask about the red, bolded part of your response above about the moving observer. In the hypothetical, how do you determine which observer is the moving observer?

In this case, it doesn't matter. The easiest is to take that you are at rest, and your counterpart is moving. Your counterpart can do the same: he can look at his frame at rest, and yours moving. Both of you will agree to the same answer: the velocity between you and him will be v. (Both of you will draw exactly the same graph)

In the case of the twin paradox, here you are measuring two events: departure and arrival. Say the spaceship twin is going to Andromeda. She only needs one clock to measure both events, hence the proper time - this is the time measured by a moving clock. The earth twin needs two clocks: one at departure, and the other at arrival on Andromeda - both clocks are at rest wrt to the earth twin. If you compare the two times you get:

T ' [sub:36otjgkr](proper)[/sub:36otjgkr] = T (1-v[sup:36otjgkr]2[/sup:36otjgkr]/c[sup:36otjgkr]2[/sup:36otjgkr])[sup:36otjgkr]1/2[/sup:36otjgkr]

Moving clocks slow down.

At the same time, she will measure a shortened distance. From her perspective, the distance between earth and andromeda is like a moving stick.

L' [sub:36otjgkr](moving)[/sub:36otjgkr] = L(1-v[sup:36otjgkr]2[/sup:36otjgkr]/c[sup:36otjgkr]2[/sup:36otjgkr])[sup:36otjgkr]1/2[/sup:36otjgkr].

So when she calculates her velocity, V = L'/T ' , she will get the same answer as her twin on earth will have found, V = L/T.