Explaining C: How Space Changes with Speed

[zoobyshoe]

It's the Lorentz transformation for a time interval. It's the "interval" version of equation 1d on this page:

http://www.physics.nyu.edu/courses/V85.0020/node45.html

Tom,

That site is very interesting. I have never seen so many different manifestations of the Lorentz Transformation.

Now the one you used, you say, is the one for a time interval. The time interval you are applying it to is the interval between event one and event two, the times of emission of the rear and forward beams of light.

My first thought is to wonder if this is the right interval to be applying it to. It seems to me that the simultaneity shouldn't be obviously relative to anyone till the time of detection. I think the interval we're supposed to be applying the transformation to, is the interval between detection of the light from the two separate sources by the rail observer: the interval between his detection of the light from the rear flare and his detection of the light from the forward flare. What do you think?

 

[quantumdude]

My first thought is to wonder if this is the right interval to be applying it to. It seems to me that the simultaneity shouldn't be obviously relative to anyone till the time of detection.

No, we should be using the emissions as events, because it is the simultaneous emissions that mark the length of the rod. Whether or not the detctions are simultaneous is irrelevant to the thought experiment. Indeed, as has been noted, the pulses need not be seen by anyone at all.

I think the interval we're supposed to be applying the transformation to, is the interval between detection of the light from the two separate sources by the rail observer: the interval between his detection of the light from the rear flare and his detection of the light from the forward flare. What do you think?

As I said, the time interval between the flares being detected is not relevant. However, it just so happens that in this case (because you've got both observers equidistant from the flares) that the chronological order of emission will be the same as the chronological order of detection.

So the guy in the frame of the track sees the pulses at the same time, and the guy on the rod doesn't.

 

[zoobyshoe]

No, we should be using the emissions as events, because it is the simultaneous emissions that mark the length of the rod. Whether or not the detctions are simultaneous is irrelevant to the thought experiment. Indeed, as has been noted, the pulses need not be seen by anyone at all.

Yes, the simultaneous emissions mark the length of the rod, but the whole point of the experiment is to compare what the rail observer sees with the marks, and ask the question: will the marks be separated by a length equal to the length the rod looked to be when it was in motion?

The observer can't see the emissions at the time of emission. He is ignorant about any apparent length interval between them until their light reaches him.

As I said, the time interval between the flares being detected is not relevant.

I can't agree with this. The detection is all important. The detection tells the observer when the emission occurred. Detecting the light simultaneously when he is equidistant from the sources, assures him that the emissions were simultaneous in his frame.[/color]

In Halliday and Resnick the thought experiment goes like this:

Put flares on the ends of a rod of proper length L₀, and connect them to a switch so that an observer can ignite them. Let the rod move by at a velocity v on a track, so that the ingited flares can leave marks on the track. Now let the observer ignite the flares simultaneously, in his frame[/color] (The reason for simultaneous ignition is that it is the only way you could correctly say that the distance between the marks is equal to the length of the rod).

When you say nobody even has to be there to see them, that is only because you offer a mechanical substitute for the observer which is the same thing: an observer. There has to be an observer, human or mechanical, to detect the light simultaneously. Otherwise the conditions of the experiment are not fullfilled. The whole thing is about the difference, if any, between what happens and what an observer sees.

However, it just so happens that in this case (because you've got both observers equidistant from the flares) that the chronological order of emission will be the same as the chronological order of detection.

I can't claim to understand the formula you're using very well but it seems, from what you say, to be one you can use to transform the value of a measured time interval in one frame to what that interval will measure in another frame. If that's the case, then yes it should apply equally well to the detection interval.

However, I believe that transforming the emission interval is barking up the wrong tree. The time of emission is a time of ignorance for both observers: the rail guy and the guy on the rod. I believe the thought experiment, as presented, allow us only one objective piece of information which can be transformed from the rail frame to the rod guy: the interval of detection for the rail guy.

 

[quantumdude]

Yes, the simultaneous emissions mark the length of the rod, but the whole point of the experiment is to compare what the rail observer sees with the marks, and ask the question: will the marks be separated by a length equal to the length the rod looked to be when it was in motion?

The observer can't see the emissions at the time of emission. He is ignorant about any apparent length interval between them until their light reaches him.

Whether or not he can see the flashes is irrelevant. All he needs to know is that they are simultaneous (and he does not need to see them to know that) and how far apart they are. If he knows that they were simultaneous, then he can go up to the marks at his leisure and measure the distance between them.

Tom: As I said, the time interval between the flares being detected is not relevant.

zoobyshoe: I can't agree with this. The detection is all important. The detection tells the observer when the emission occurred. Detecting the light simultaneously when he is equidistant from the sources, assures him that the emissions were simultaneous in his frame.[/color]

I mean that the time interval between the two lights being detected is not relevant to the calculation with the Lorentz transformation. Yes, we need to know the time and location of the detection so that we can go back and determine when and where the emission occurred. But you were inquiring about what we should be applying the Lorentz transformation to, and the time interval for detection of the pulses is not it. We need to apply it to the time interval for emission to get the length of the rod in each frame.

When you say nobody even has to be there to see them, that is only because you offer a mechanical substitute for the observer which is the same thing: an observer. There has to be an observer, human or mechanical, to detect the light simultaneously. Otherwise the conditions of the experiment are not fullfilled. The whole thing is about the difference, if any, between what happens and what an observer sees.

No, the light does not have to be detected at all. The reason is that the light is not the only indicator of the ignition. For instance, you can have each flare trip a switch that stops a stopwatch at the moment of ignition.

Yes[/color], we have to take experimental data of some sort on the time and place of ignition in each frame.

No[/color], the light from the flares need not be observed by anyone or anything. The flares and the light from them are purely incidental. The exact same thought experiment could be done with lightless markers, and it wouldn't change a thing.

I can't claim to understand the formula you're using very well but it seems, from what you say, to be one you can use to transform the value of a measured time interval in one frame to what that interval will measure in another frame. If that's the case, then yes it should apply equally well to the detection interval.

The Lorentz transformation does apply to any interval, including the detection interval. But I am not bothering with Lorentz transforming the detection interval, because that interval is not related to the length of the rod in any frame. The observer could be anywhere in his frame, and the time interval between detections could be anything according to him. It makes no never mind whatsoever to the simultaneity (or lack thereof) of the events, except of course that whatever time interval he does measure between detected light pulses will be consistent with the simultaneity/nonsimultaneity of the events in his frame.

However, I believe that transforming the emission interval is barking up the wrong tree. The time of emission is a time of ignorance for both observers: the rail guy and the guy on the rod. I believe the thought experiment, as presented, allow us only one objective piece of information which can be transformed from the rail frame to the rod guy: the interval of detection for the rail guy.

No. You can use the information about the detection to calculate back to the information about the emission, and indeed you must. That's the only way we can say anything about the length of the rod according to any observer. It's not the detections that are used to measure the length of the rod, it's the emissions.

 

[zoobyshoe]

Tom,
I think I found the equations you are using in one of my books.

Are these them? :

\Delta t=\gamma(\Delta t'+v\Delta x'/c^2)

\Delta t'=\gamma(\Delta t-v\Delta x/c^2)

 

[quantumdude]

Tom,
I think I found the equations you are using in one of my books.

Are these them? :

\Delta t=\gamma(\Delta t'+v\Delta x'/c^2)

\Delta t'=\gamma(\Delta t-v\Delta x/c^2)

Those are them. I'll learn to TeX this weekend.

 

[zoobyshoe]

Here's gamma:

\gamma = \frac{1}{\sqrt{1- \beta^2}}

\beta = \frac {v}{c}

 

[zoobyshoe]

Or:

\gamma = \frac{1}{\sqrt{1 -(\frac{v}{c})^2}}



Tex tips:

gamma = \gamma

fraction = \frac followed by {stuff in numerator}{denominator} (that is: you indicate the difference with these: {}{} numerator first, then denominator

Square root = \sqrt followed by: {everything you want under the square root sign}

beta =\beta

Delta (different than \delta) = \Delta

squared = number to be squared followed by ^2

[/ tex] to end the whole line, and to begin it.<br /> <br /> Add more to your store of abilities as you go by clicking on the quote button of any post with some Tex you want to be able to use, and observe how they did it, or look through the Tex thread till you find what you need.<br /> <br /> Always good to preview your post before submitting because it&#039;s easy to miss a \ or an ending } when you&#039;re typing along. The &quot;preview post&quot; feature in any thread can be used to practise to your heart&#039;s content, I also realized. (Just abandon it when you&#039;re done, don&#039;t hit &quot;submit reply&quot;.)

 

[zoobyshoe]

In Halliday and Resnick the thought experiment goes like this:

Interestingly enough, I found these:

\Delta t=\gamma(\Delta t&#039;+v\Delta x&#039;/c^2)

\Delta t&#039;=\gamma(\Delta t-v\Delta x/c^2)

in Halliday and Resnik. I didn't even realize I had this book. I pick up physics texts all the time at the swap meet for peanuts just in case one will have a explanation of something that's easier to grasp than some other source.

Any way, what edition do you have? I can't find the flare gedanken in mine, at least not in the relativity part. They spread some relativity into other parts, though.

 

[quantumdude]

Any way, what edition do you have?

I have the 2nd, 4th, and 5th editions.

Mind you, the thought experiment in H+R may not have actually involved flares. I really don't remember, because it's been awhile. As I said, the flares are totally incidental. But the basic idea is taught there: Only simultaneous measurements of the position of the ends of the rod constitute a "length measurement".

 

[zoobyshoe]

I have the 2nd, 4th, and 5th editions.

We're clearly fated to be out of phase, Tom. I have the third edition, 1988.

Mind you, the thought experiment in H+R may not have actually involved flares. I really don't remember, because it's been awhile. As I said, the flares are totally incidental. But the basic idea is taught there: Only simultaneous measurements of the position of the ends of the rod constitute a "length measurement".

Yeah, hehehehehe. I think the flare experiment is more of a Tom Mattson original than you think. In my copy all they do is warn you to mark the positions of the head and tail of your goldfish simultaneously (in your reference frame) rather than arbitrarily. They don't propose any mechanism to do this.

What they actually say about the "reality" of length contraction is essentially what Eddington says:

"The questions, `Does the rod really shrink?' and `Do the atoms in the rod really get pushed closer together?' are not proper questions within the framework of relativity. The length of the rod is what you measure it to be and motion affects measurements."

p.962 3rd edition 1988.

They don't have any gedakens where the moving thing actually leaves physical marks on the stationary one. It is clear that you fully believe your flare and rod thing is an obvious extrapolation of what they're saying, but they, like Eddington, are carefully avoiding saying exactly what you think they're saying.

If you proposed your flare thing to Halliday and Resnik, I believe they would say what they said: "It's not a proper question in the framework of relativity."

 

[quantumdude]

Yeah, hehehehehe. I think the flare experiment is more of a Tom Mattson original than you think. In my copy all they do is warn you to mark the positions of the head and tail of your goldfish simultaneously (in your reference frame) rather than arbitrarily. They don't propose any mechanism to do this.

Well, the flares certainly provide a means to do it.

They don't have any gedakens where the moving thing actually leaves physical marks on the stationary one. It is clear that you fully believe your flare and rod thing is an obvious extrapolation of what they're saying, but they, like Eddington, are carefully avoiding saying exactly what you think they're saying.

It is an obvious extrapolation of what they say. It makes no comment whatsoever on what "really" happens to the rod. It simply provides a means of simultaneously recording the positions of the two ends of it.

If you proposed your flare thing to Halliday and Resnik, I believe they would say what they said: "It's not a proper question in the framework of relativity."

They would not say that at all, because it's a perfectly proper question in the framework of relativity.

"Proper questions" in the framework of relativity are questions that relate to intervals in spacetime. That much should be clear from even a glance at the Lorentz transformation. The measurement scheme I have used does only one thing: It records the places and times of events.

If that thought experiment is not a proper question in the framework of relativity, then there are no proper questions in the framework of relativity.

What makes you think otherwise?

 

[pmb_phy]

I'm new to this thread so please bear with me. I took a look at the first posts in this thread but there are too many for me to read (due to my back and the pain of sitting and reading). But I agree with the original post to some extent, and to the extent that I understood what he was trying to say.

He was suggesting that the reason that the speed of light is constant has something to do with space changing. That is true. The reason why c is invariant is due to a combination of Lorentz contraction and time dilation. A close and thoughful look at the transformation rules for velocity shows that.

Also there is nothing electromagnetic about Lorentz contraction. Its distances that contract and that's why the lengths of rods contract. All one need to do in order to see that is true is to consider the distance between two point particles which are separated in space. The distance between them is contracted and there is no physical connection between them.

Pete

 

[zoobyshoe]

If that thought experiment is not a proper question in the framework of relativity, then there are no proper questions in the framework of relativity.

What makes you think otherwise?

Ah! Don't ask me! Ask them. I am merely quoting their respose:"The questions`Does the rod really shrink?' and `Do the atoms in the rod really get pushed closer together?' are not proper questions in the framework of relativity"

Is your thought experiment proper in my estimation? Is it "proper" to try and pin down the most literal possible proof of length contraction? Absolutely.

My point is that "In Halliday and Resnik the thought experiment goes like this" ought to have been something more like: "Extrapolating from what Halliday and Resnik say we can use flares and a rod", and so on, and so forth. The way you phrased it gives the unequivocal impression that very set up is to be found in their book. (They do have a thing with a train and clocks, and they have a goldfish, but they don't have a rod and flares, or anything where the moving body leaves physical marks on the stationary one.)

 

[zoobyshoe]

He was suggesting that the reason that the speed of light is constant has something to do with space changing. That is true.

Hi Pete,

Yes, but he also had some point about the angle at which you encounter the light changing until at c you would theoretically be encountering it at 90º. Could you make any sense out of that?

-Zooby

 

[pmb_phy]

Hi Pete,

Yes, but he also had some point about the angle at which you encounter the light changing until at c you would theoretically be encountering it at 90º. Could you make any sense out of that?

-Zooby

No. I was totally lost on what he was talking about. Too bad we can't have a black board here huh?

Pete

 

[zoobyshoe]

No. I was totally lost on what he was talking about. Too bad we can't have a black board here huh?

I'm not sure it wouldn't just give people an even greater capacity to confuse me. :-)

 

[quantumdude]

Ah! Don't ask me! Ask them. I am merely quoting their respose:"The questions`Does the rod really shrink?' and `Do the atoms in the rod really get pushed closer together?' are not proper questions in the framework of relativity"

But I don't need to ask them. "Proper questions" in SR pertain to spacetime intervals, and that's precisely what this thought experiment is about. I could go down the hall and ask Resnick (he's at my school) to verify that, but he'd probably think I'm a simpleton.

Is your thought experiment proper in my estimation? Is it "proper" to try and pin down the most literal possible proof of length contraction? Absolutely.

It would help both this discussion and your personal understanding of relativity a great deal if you would have a good look at the Lorentz transformation. Then, you could see for yourself that questions of spacetime intervals are proper questions in SR, in the strictest sense.

My point is that "In Halliday and Resnik the thought experiment goes like this" ought to have been something more like: "Extrapolating from what Halliday and Resnik say we can use flares and a rod", and so on, and so forth. The way you phrased it gives the unequivocal impression that very set up is to be found in their book.

I double checked, and the scenario in fact does not appear exactly as I described it (more on this below). But so what? This isn't about flares or identical wires, it's about the Lorentz transformation, simultaneity, and length measurements. In other words, it's about everything that isn't being discussed, because we are so lost in these irrelevant tangents about the actual mechanisms of the thought experiments and who saw which light pulse in what order. I'm sorry I ever mentioned "flares", because it has become a focal point of the discussion.

(They do have a thing with a train and clocks, and they have a goldfish, but they don't have a rod and flares, or anything where the moving body leaves physical marks on the stationary one.)

They do in the 4th and 5th editions.

Upon double checking, I found that the length contraction thought experiments are done with stopwatches and trains, as you say. But flipping back a page to the section on the relativity of simultaneity, we find in my copies of the text the "flare" thought experiment that I tried unsuccessfully to recall in all its details. The scenario has 2 rocket ships passing each other. They are close enough so that 2 flares ignite simultaneously in one frame, but not in another, and they leave permanent marks on the ships. The thought experiment stops with the discussion of simultaneity, but since length contraction can be derived from it, I see no reason not to say what I said before: This is an obvious extrapolation of what Halliday and Resnick does say.

 

[pmb_phy]

Hi Pete,

Yes, but he also had some point about the angle at which you encounter the light changing until at c you would theoretically be encountering it at 90º. Could you make any sense out of that?

-Zooby

Perhaps the angles he was speaking of were from his vizualization of the moving body. We think in terms of math and coordinate systems. Perhaps he's thinking of what he sees as a body passes him by, like a train passing a person staning on one side of the track. Our heads are atr first looking to one direction, the direction the train is coming. Then that line of sight changes angularly. Perhaps that's the angle he was speaking of. And it is true that angles do change upon Lorentz transformation.

Pete

 

[zoobyshoe]

This is an obvious extrapolation of what Halliday and Resnick does say.

Only in the sense that it's an obvious extrapolation of any good text on relativity.

Anyway, what is the significance of a negative value for the time interval in the t' frame? What's negative time?

 

[Chronos]

Anyway, what is the significance of a negative value for the time interval in the t' frame? What's negative time?

Short answer is, the past. Negative t is a problem when you try to do the math. The solutions end up being the square root of a negative number. Those results do not have predictive value [using the term 'predictive' in the future tense].

 

[quantumdude]

Only in the sense that it's an obvious extrapolation of any good text on relativity.

Well, I'm sorry that swapping two rocket ships for a rod and rail has confused you so much, but the fact of the matter is that the thought experiment described in Halliday and Resnick can be used to teach length contraction with just a line or two of math. But if you don't understand the Lorentz transformation, then it would seem as though they were worlds apart.

Anyway, what is the significance of a negative value for the time interval in the t' frame? What's negative time?

If you set your stopwatch to t=0 when a flare ignites, then t<0 corresponds to those moments before the flare ignited. I'm sure that at some point you've heard the countdown to a rocket launch: "T-minus-10...T-minus-9...etc."

It's the same thing.

However, I wasn't talking about negative time. I was talking about a negative time interval. If t₂-t₁ is less than zero, it means that t₁ is greater than t₂. That means that t event 1 occurs later than event 2.

 

[zoobyshoe]

Short answer is, the past. Negative t is a problem when you try to do the math. The solutions end up being the square root of a negative number. Those results do not have predictive value [using the term 'predictive' in the future tense].

Thanks Chronos,

The specific situation is this, from Tom's breakdown of the flare/rod gedanken:

For observer S' (the guy standing on the rod):
?t'=t₂'-t₁'=?(?t-v?x/c²)
?t'=?(0-vL/c²)
?t'=-?vL/c²

See? ?t' is negative (not zero). This means that, according to the guy on the rod, event 1 (the rear flare igniting) occurs later than event 2 (the front flare igniting). If the two flares ignite at different times, and the observer is at their midpoint, then there is no way that he is going to receive both pulses simultaneously.

The interval in the t frame, an interval of 0, becomes a negative interval in the t' frame, which is not 0. (It has some very small value which, given a meter rod, varies according to what specific speed it has.)

How do we "locate" this negative, "past" interval of time with respect to t? Does one end of the negative interval fall on t=0, and the other at t= -.0000000001 (for instance)?

 

[quantumdude]

How do we "locate" this negative, "past" interval of time with respect to t?

The same way the folks at NASA locate "T-minus-10": with a clock.

Does one end of the negative interval fall on t=0, and the other at t= -.0000000001 (for instance)?

Yes.

 

[zoobyshoe]

The same way the folks at NASA locate "T-minus-10": with a clock.

I think you were composing your post to me at the same time I was composing mine to chronos. Your answer wasn't up yet when I submitted. I wasn't ignoring it.

So, you are saying if we get a value of, for example, t' = -.00000001 one end of the interval would be located at that position and the other at 0'?

 

[quantumdude]

So, you are saying if we get a value of, for example, t' = -.00000001 one end of the interval would be located at that position and the other at 0'?

If the event to which you ascribe t' = -.00000001 occurs .00000001 time units earlier than the event to which you ascribe 0', then the time coordintate of the first event is as you say.

 

[zoobyshoe]

OK. You want me to be more familiar with the LT. Let's practise:

\Delta t&#039; = \gamma(\Delta t - v\Delta x/c^2)

Let's plug a value of .5 c in for rod velocity.

For gamma that gives \gamma = 1.1547005

x= length of rod = 1 meter = 1

\Delta t = 0, c^2 = 9^1^0

Therefore: \Delta t&#039; = 1.1547005(-150,000/9^1^0)

so: \Delta t&#039; = 1.1547005(1.6667^-^0^6) = 1.9245^-^0^6

I am reasonably sure this is right, but it would be best if you checked it thoroughly to see if I have misconstrued anything.

 

[quantumdude]

Almost. c²=9*10¹⁶ m²/s², so your final answer should be 1.9245*10^-9 s. But I didn't intend for you to get bogged down in numerical calculations. What I think you need to do is learn what the Lorentz transformation means.

First, look at what it accepts as inputs: spacetime coordinates. So, experiments designed to test SR (aka the "proper questions in the framework of SR") are those experiments that record spacetime coordinates. This can be done with clocks, meter sticks, flares ( ), electrical switches, what have you. If it accurately captures the spacetime coordinates of events, then it is a pertinent issue in SR.

Second, think about what your numerical result means. As has been discussed, a time interval in frame S' given by \Delta t\&#039;\; = \gamma(\Delta t - v\Delta x/c^2) (look ma, I'm TeX-ing!) is taken to be the difference between times t_2-t_1. So, if \Delta t\&#039;\; = \gamma(\Delta t - v\Delta x/c^2) comes out negative, then it means that t_1 is greater than t_2. This means that Event 1 happens later than Event 2 in the frame S'.

With me so far?

 

[zoobyshoe]

Almost. c²=9*10¹⁶ m²/s², so your final answer should be 1.9245*10^-9 s.

This is because I failed to keep my units consistent, right? If I choose meters for the rod I can't use kilometers for the velocities. (DOH!) I could have stuck with the kilometers had I used .001 for x, I guess.

But I didn't intend for you to get bogged down in numerical calculations. What I think you need to do is learn what the Lorentz transformation means.

For some reason, I can't grasp any equation till I can plug values into it and work them out. Till I can do that, it's a meaningless abstraction to me.

First, look at what it accepts as inputs: spacetime coordinates.

Yep, I get this aspect.

Second, think about what your numerical result means. As has been discussed, a time interval in frame S' given by \Delta t\&#039;\; = \gamma(\Delta t - v\Delta x/c^2) (look ma, I'm TeX-ing!) is taken to be the difference between times t_2-t_1. So, if \Delta t\&#039;\; = \gamma(\Delta t - v\Delta x/c^2) comes out negative, then it means that t_1 is greater than t_2. This means that Event 1 happens later than Event 2 in the frame S'.

Glad you explained that. I was wondering how you'd determined which was first.

So the events in frame t' take place 1.9245 ^-09 seconds apart? That is: .0000000019245 of a second apart?

 

[zoobyshoe]

Ground control to Major Tom. We have lost your signal. Do you read us?