Benefits of time dilation / length contraction pairing?

[neopolitan]

OK, the clarified version looks clear to me.

Then shortly I will move onto the next stage. I will try to hold back (which means: write a reply, leave it for a while, then check it for typos then post)

I've started replying to other parts of your post but run out of time. Will try to address them later.

cheers,

neopolitan

 

[JesseM]

Then shortly I will move onto the next stage. I will try to hold back (which means: write a reply, leave it for a while, then check it for typos then post)

I've started replying to other parts of your post but run out of time. Will try to address them later.

OK, but in order to avoid getting into an extended discussion of the steps of your proof only to find that the ending conclusion isn't actually equivalent to the Lorentz transformation, can you state in advance the final equation(s) you intend to derive, including the physical definitions of any symbols appearing in the final equation(s) in terms of the setup you've outlined (space and time intervals involving a photon traveling towards A and B which passes them at particular points along with some Event or Events on its worldline, presumably)?

 

[neopolitan]

OK, but in order to avoid getting into an extended discussion of the steps of your proof only to find that the ending conclusion isn't actually equivalent to the Lorentz transformation, can you state in advance the final equation(s) you intend to derive, including the physical definitions of any symbols appearing in the final equation(s) in terms of the setup you've outlined (space and time intervals involving a photon traveling towards A and B which passes them at particular points along with some Event or Events on its worldline, presumably)?

Alright, I'll put some thought into that as well.

cheers,

neopolitan

 

[Rasalhague]

I haven't caught up with all the most recent posts yet, but I hope no one minds if I butt in with some general observations.

I just googled the expression "moving clocks run slow" (3190 hits), then tried "moving clocks run fast" (128 hits), several of the latter apparently referring to unorthodox theories and rejections of relativity, although not all of them. Presumably authors who present the slogan "moving clocks run slow" as a verbal equivalent to the time dilation equation are taking the word dilation to mean a bigger number when it refers to time, but a smaller number when it refers to space! On the other hand, Taylor and Wheeler in Spacetime Physics explicitly define dilation as a bigger number, and I think this is the more standard idea.

So if some (most?) of the authors who use the expression "moving clocks run slow" are, in fact, referring to the temporal analogue of length contraction, it seems strange that they avoid the obvious term "time contraction" and its formula. (Unfortunately Google results for "time contraction" are mixed up with pages about timing the contractions of labour, so I can't make a fair comparison.) The only book I've yet found to mention "time contraction" is What Does a Martian Look Like by Jack Cohen and Ian Stewart, about the possible forms that extraterrestrial life might take, rather than physics as such.

This preference for the expression "moving clocks run slow" is presumably as much a matter of convention as the preference for the expression "time dilation" and the pairing of the time dilation formula with that of length contraction. Its popularily suggests that it may be no less natural a way of conceptualising the relationship. All of this strongly inclines me to agree with Neopolitan's comments in the very first post of this thread: that the pairing of time dilation with length contraction is a source of needless confusion. Thanks to Jesse's diagram and explanation of some reasons why contraction is a more natural way to view what happens to the length of a physical object when viewed as moving as opposed to standing still, it seems to me that it would indeed be more logical to pair time contraction and length contraction (i.e. use the same terminology for time as we use for space), for the sake of comparing like with like, and of avoiding the false impression of asymmetry which the traditional pairing creates.

That said, there are genuine asymmetries between time and space, or between the ways we relate to them, which might lead us to view length contraction as more natural while either transformation (dilation or contraction) seems equally natural for time. Here is a list I came up with. It could be that some of the items are essentially the same as others, stated in different words. The fourth point about determinism is based on what Jesse said in an earlier post.

1. Persistence. Clocks and rulers are both objects with sharply defined spatial limits; they both persist in time. To make a thought experiment more symmetrical, we could imagine everlasting clocks, each static at the origin of their respective rest frames, and infinite rulers (each existing for one moment only, as defined in its own rest frame). While this may be convenient for the thought experiment, its unphysicality could point to a real difference in how we relate to time and space.

2. Degrees of freedom. Objects are free to move in either direction along a line through space, but their motion in time is unidirectional.

3. Speed. Speed is defined as length divided by time, regardless of which component we're calculating a change in.

4. Determinism. Physical laws predict events in limited space over unlimited time (with certain limits on accuracy); they have less power to predict events in limited time over unlimited space. We're used to ideas like “what goes up must come down”, but it's harder to imagine a universe where what goes up here would be a reliable guide to what goes up simultaneously somewhere else.

 

[Rasalhague]

Some thoughts on terminology and notation.

In my notes, I've taken to using the term input frame (or source frame) for the frame for which we know the coordinate values, and output frame (or target frame) for the frame for which we want to calculate the coordinate values. I've been using the terms left frame and right frame, respectively, for the frame moving left (i.e. in the negative x direction) past the other, and the frame moving right (i.e. in the positive x direction) past the other. The initials L and R stand for left and right. For example, "Clock L is at rest in frame L, the left frame, so frame L is clock L's rest frame. Clock L is moving in frame R, so frame R is a moving frame for clock L."

The terms input and output depend on the question asked, and may change their referents (change the frames they refer to) if a different question is asked. The terms left and right depend on the definition of the frames, and keep the same referents so long as the same frames are being used. The terms rest and moving are defined relative to a particular object; they may change their referents if a different object is discussed. Contraction and dilation questions can be asked of both time and space with any of these terms.

Some authors use primed and unprimed for what I'm calling input and output, but others use primed and unprimed for left and right. Others again use primed to refer to whichever frame is defined as moving in a particular example. These definitions don't necessarily coincide!

Some authors do as Wolfram Alpha does and use a subscript zero for the coordinates of the input frame if your question involves time dilation or length contraction, and for the output frame if your question involves time contraction or length dilation. Such coordinates change which frames they're referring to whenever you go from asking a dilation to a contraction question of the same coordinate, or from asking a contraction to a dilation question of the same coordinate. Wolfram Alpha calls dilated time "moving time", and contracted length "moving length".

http://www34.wolframalpha.com/input/?i=time+dilation
http://www34.wolframalpha.com/input/?i=length+contraction

To me this feels like an arbitrary switch in terminology purely for the sake of maintaining this artificial, traditional association of dilation exclusively with time, and contraction exclusively with length.

 

[JesseM]

I just googled the expression "moving clocks run slow" (3190 hits), then tried "moving clocks run fast" (128 hits), several of the latter apparently referring to unorthodox theories and rejections of relativity, although not all of them. Presumably authors who present the slogan "moving clocks run slow" as a verbal equivalent to the time dilation equation are taking the word dilation to mean a bigger number when it refers to time, but a smaller number when it refers to space! On the other hand, Taylor and Wheeler in Spacetime Physics explicitly define dilation as a bigger number, and I think this is the more standard idea.

There is no contradiction between the phrase "time dilation" and the phrase "moving clocks run slow". The "dilation" in question is not of the clock's rate of ticking, but of the period between a given pair of readings. For example, if a clock ticks forward by 10 seconds between two events on its worldline, but the time interval between these two events is 30 seconds in my frame, then that period of 10 seconds between the events as measured by the clock itself has been "dilated" by a factor of 3 from my perspective. But at the same time, if it takes 30 seconds of my time for the clock to tick forward by 10 seconds, obviously I can also say this clock is "running slow" from my perspective.

So if some (most?) of the authors who use the expression "moving clocks run slow" are, in fact, referring to the temporal analogue of length contraction

No, I'm sure that not a single one of them is referring to this, "the temporal analogue of length contraction" is a fairly arcane idea I brought up for the sake of my discussion with neopolitan that would probably never be used in practice. The idea (illustrated in the diagram I drew that neopolitan posted in post #5) is that if you have two events on a clock's worldline that are separated by a time t (say 10 seconds again) according to the clock's own readings, and then you draw surfaces of simultaneity (surfaces of constant t) in the clock's own rest frame that pass through these two events, and then consider how those surfaces would look in the frame of an observer who sees the clock in motion (where the surfaces will be 'slanted'), and work out the time between these surfaces along the vertical time axis of this observer's frame, it will be less than the time of 10 seconds, even though the time in this frame between those two events on the first clock's worldline (which is what the regular time dilation equation gives you) is greater than 10 seconds. This is analogous to length contraction where you look at two lines of constant x in an object's rest frame that represent the worldlines of the object's endpoints, then switch to a different frame where the object is moving so these same lines are slanted, and consider the distance between these slanted lines along the horizontal space axis of this frame, which is the "length" of the object in this frame.

If this is hard to follow even after looking at the diagram, it's not really worth worrying about, since like I said the "temporal analogue of length contraction" is just an artificial concept I came up with for the purposes of showing that you could imagine something analogous to length contraction, it's defined in such a weird way that it's not a concept that anyone would actually be likely to find useful for any other purpose besides illustrating that such an analogous notion is possible.

 

[Rasalhague]

Some notes I made to get my head around the symmetry which the traditional pairing of time dilation and length contraction disguises. All criticism welcome!

Assume a (-1, 1)-dimensional Minkowski spacetime described by two reference frames moving relative to one another with speed u.

Time. Let clocks be synchronised at the intersection of the origins of the two frames so that the coordinates of this coincidence are $$t_{L} = t_{R} = 0$$ and $$x_{L} = x_{R} = 0$$. The clocks last for all time. Clock L is confined in space to the location (line of collocality/syntopy) $$x_{L} = 0$$, clock R to $$x_{R} = 0$$.

Length. Let two rulers have their zero ends lined up at the intersection of the origins of the two frames so that the coordinates of this coincidence are $$t_{L} = t_{R} = 0$$ and $$x_{L} = x_{R} = 0$$. The rulers extend through all space. Ruler L exists only at the instant (line of contemporality/synchrony) $$t_{L} = 0$$, ruler R at $$t_{R} = 0$$.

We can ask contraction questions of time or of length. We can ask dilation questions of time or of length. The contraction questions we ask of time are formally the same as those we ask of length (except for the difference in coordinate). The dilation questions we ask of time are formally the same as those we ask of length (except for the difference in coordinate).


1. CONTRACTION

$$\frac{1}{\gamma} = \sqrt[]{1-\left(\frac{u}{c}\right)^{2}} = \frac{1}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}$$

1.1. Time contraction.

1.1.1. At a moment defined in frame R, the frame where clock L is moving, clock L shows this fraction of the time shown by clock R.

1.1.2. At a moment defined in frame L, the frame where clock L is still, clock R shows this fraction of the time shown by clock L.

1.2. Length contraction.

1.2.1. At a location defined in frame R, the frame where ruler L is moving, ruler L shows this fraction of the length shown by ruler R.

1.2.2. At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.


2. DILATION

$$\gamma = \frac{1}{\sqrt[]{1-\left(\frac{u}{c}\right)^{2}}} = cosh\left(artanh\left(\frac{u}{c} \right) \right)$$

2.1. Time dilation.

2.1.1. At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.

2.1.2. At a moment defined in frame L, the frame where clock L is still, clock L shows this multiple of the time shown by clock R.

2.2. Length dilation.

2.2.1. At a location defined in frame R, the frame where ruler L is moving, ruler R shows this multiple of the length shown ruler L.

2.2.2. At a location defined in frame L, the frame where ruler L is still, ruler L shows this multiple of the length shown by ruler R.

 

[JesseM]

1. CONTRACTION

$$\frac{1}{\gamma} = \sqrt[]{1-\left(\frac{u}{c}\right)^{2}} = \frac{1}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}$$

1.1. Time contraction.

1.1.1. At a moment defined in frame R, the frame where clock L is moving, clock L shows this fraction of the time shown by clock R.

But that's just what is usually called "time dilation"--if some quantity is greater in the frame of the observer who sees the instrument moving than it is when measured by the instrument itself, that's caused dilation; if some quantity is smaller in the observer's frame, that's called contraction. In this case, the quantity is the time between two events on the clock's worldline; the time between these events will be greater in the observer's frame than as measured by the clock itself, so this is time dilation. It seems unnecessarily confusing to change this convention and to say that it's "contraction" if the quantity measured by the instrument is smaller than the corresponding quantity measured in the observer's frame.

 

[Rasalhague]

But that's just what is usually called "time dilation"--if some quantity is greater in the frame of the observer who sees the instrument moving than it is when measured by the instrument itself, that's caused [called] dilation;

Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction! Wouldn't it be less confusing to call the same thing contraction for both dimensions?

if some quantity is smaller in the observer's frame, that's called contraction. In this case, the quantity is the time between two events on the clock's worldline; the time between these events will be greater in the observer's frame than as measured by the clock itself, so this is time dilation. It seems unnecessarily confusing to change this convention and to say that it's "contraction" if the quantity measured by the instrument is smaller than the corresponding quantity measured in the observer's frame.

But when we talk about rulers, the terminology is traditionally reversed. There too the distance between the corresponding pair of events is greater in the observer's frame, so why do we not call that dilation?

 

[JesseM]

Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction!

How do you figure it's the "same phenomenon?" In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.

But when we talk about rulers, the terminology is traditionally reversed. There too the distance between the corresponding pair of events is greater in the observer's frame, so why do we not call that dilation?

In the case of length you aren't measuring distance between a single pair of events, you're measuring the distance between the endpoints of the ruler at a single moment in time of whatever frame you're using. The distance between the endpoints of the ruler at a single moment of time in the observer's frame is smaller than the distance between the endpoints of the ruler at a single moment of time in the ruler's own rest frame.

If you want to talk about the distance between a single pair of events in two frames, you're right that the distance is larger in the observer's frame where they're non-simultaneous than it is in the frame where the events were simultaneous (this is what I called the 'spatial analogue for time dilation'). But this is not the same thing as measuring the length of an object in two different frames, since "length" always means the distance between the endpoints at a single moment in time.

 

[Rasalhague]

There is no contradiction between the phrase "time dilation" and the phrase "moving clocks run slow". The "dilation" in question is not of the clock's rate of ticking, but of the period between a given pair of readings.

Yes, this is the way Taylor and Wheeler define it: a dilation (lengthening) of the period. And yet, I'm not sure everyone understands it this way. We also find statements like this on Wikipedia: "Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock. This is often interpreted as time "slowing down" for the other clock, [...]" But from any perspective in which a clock is ticking slower, it will show a shorter (contracted) period as having elapsed. So presumably the writer of this article, rightly or wrongly, took dilation to refer to something other than the period. Perhaps they conceptualised it as a dilation of the units. Either way, it seems arbitrary to switch terminology when talking about space.

For example, if a clock ticks forward by 10 seconds between two events on its worldline, but the time interval between these two events is 30 seconds in my frame, then that period of 10 seconds between the events as measured by the clock itself has been "dilated" by a factor of 3 from my perspective. But at the same time, if it takes 30 seconds of my time for the clock to tick forward by 10 seconds, obviously I can also say this clock is "running slow" from my perspective.

Equally, you could say of the clock that's running slow: this clock has measured a shorter period; the period has been contracted, behold "time contraction". What else would you call it if a lengthening of the period is a dilation. Since the relationship between the events involved in this measurement is exactly analogous to the relationship between the events involved in calculating "length contraction", why not present these equations as parallel, showing the same symmetry between time and space that the full Lorentz transformation does?

If this is hard to follow even after looking at the diagram, it's not really worth worrying about, since like I said the "temporal analogue of length contraction" is just an artificial concept I came up with for the purposes of showing that you could imagine something analogous to length contraction, it's defined in such a weird way that it's not a concept that anyone would actually be likely to find useful for any other purpose besides illustrating that such an analogous notion is possible.

At this stage, it seems to me no more articial or arcane than "time dilation". It's just the reverse calculation. Wolfram Alpha calls it a transformation from "moving time" to "stationary time" ( http://www34.wolframalpha.com/input/?i=time+dilation ). What seems artificial and potentially confusing to me is their use of different definitions of "moving" and "stationary" when calculating distance as opposed to time.

 

[Rasalhague]

How do you figure it's the "same phenomenon?" In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.

Here I think is where the asymmetry creeps in. In the case of time dilation, you define the moment (the end of the period to be measured) in the output frame.

(1) At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.

In the case of length contraction, you define the location (the end of the distance to be measured) in the input frame.

(2) At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.

So you're asking different questions and getting different answers. But this is only a matter convention. We could just as well ask the latter question of time:

(3) At a moment defined in frame L, the frame where clock L is still, clock R shows this fraction of the time shown by clock L.

Pairing question (3) with question (2) shows what happens when we ask the same kind of question of time as we ask of space. We get the same kind of answer. The spatial interval is less, the temporal interval is less. This seems to me a more intuitive way to compare time and space.

In the case of length you aren't measuring distance between a single pair of events, you're measuring the distance between the endpoints of the ruler at a single moment in time of whatever frame you're using. The distance between the endpoints of the ruler at a single moment of time in the observer's frame is smaller than the distance between the endpoints of the ruler at a single moment of time in the ruler's own rest frame.

Exactly, and this is the source of the "inconsistency" (if such it be), the fact that "in the case of length (contraction)" a different operation is being carried out with respect to length from the operation being carried out in time dilation with respect to time. Or rather, the inconsistency is to present these different operations are somehow equivalent or parallel to each other.

We can look at the time calculation and the space calculation as each involving three events.

In the case of the clocks, supposing them to be arbitrarily longlasting and arbitrarily small, thus each describing a line through spacetime: (A) a coincidence at the origin (of the two clocks meeting and being set to zero), (B) the event of the clock collocal with the origin in the input frame showing a certain value, (C) the event of the clock collocal in the output frame with the origin in the output frame showing a certain value.

In the case of the rulers, supposing them to be arbitrarily long and arbitrarily shortlived, thus each describing a line through spacetime: (A) a coincidence at the origin (of the zero end of the two rulers meeting), (B) the event of the ruler contemporaneous with the origin in the input frame showing a certain value, (C) the event of the ruler contemporaneous with the origin in the output frame showing a certain value.

Which events we choose to label (B) and (C) in each case depends on what information we have and what we want to calculate. Obviously it's possible to ask whatever questions we need to get any of the available answers. As far as I can see, calling the reverse questions from the traditional ones "time contraction" and "length dilation" would be less ambiguous than, for example, Wolfram Alpha's "stationary time" and "stationary length". After all, to compare times, there have to be two notional clocks (one stationary and one moving in each frame). And when we compare lengths, we're comparing two rulers (one stationary and one moving in each frame). In the case of length, because real objects persist in time, we think naturally of an object having a certain length when stationary being contracted when seen as moving. So why not use the same convention for time, and many authors do verbally, and use the appropriate formula to match the common statement that "a moving clock ticks slow" (and therefore shows a shorter time period), just as we think of a moving object as having a shorter length?

 

[JesseM]

Yes, this is the way Taylor and Wheeler define it: a dilation (lengthening) of the period. And yet, I'm not sure everyone understands it this way. We also find statements like this on Wikipedia: "Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock. This is often interpreted as time "slowing down" for the other clock, [...]" But from any perspective in which a clock is ticking slower, it will show a shorter (contracted) period as having elapsed.

Huh? No it won't, if a clock is ticking at a slower rate it shows a longer period. For example, if a clock is slowed down by a factor of 3 in my frame, it will take a period of 30 seconds of time in my frame to tick forward by 10 seconds.

Equally, you could say of the clock that's running slow: this clock has measured a shorter period; the period has been contracted, behold "time contraction".

But then you'd be adopting the convention that dilation/contraction refers to whether a quantity measured in the instrument's own frame is greater or smaller than the corresponding quantity measured in the observer's frame. This is simply not the convention that has been adopted, dilation/contraction always refers to whether the quantity in the observer's frame is greater or smaller.

Since the relationship between the events involved in this measurement is exactly analogous to the relationship between the events involved in calculating "length contraction"

No, it isn't. Length contraction follows the normal convention that you're talking about the value in the observer's frame.

At this stage, it seems to me no more articial or arcane than "time dilation". It's just the reverse calculation.

No, it certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:

(time in observer's frame) = (time in clock's frame)/gamma

(note that this equation only works if 'time in observer's frame' refers to something other than the time between two events on the clock's worldline, such as the time in the observer's frame between planes of simultaneity from the clock's frame which I talked about earlier)

Whereas the normal time dilation equation is:

(time in observer's frame) = (time in clock's frame)*gamma

You can take the normal time dilation equation and divide both sides by gamma, but this doesn't give you the TAFLC above, instead it gives you what I called the "inverse time dilation equation":

(time in clock's frame) = (time in observer's frame)/gamma

See the difference?

What seems artificial and potentially confusing to me is their use of different definitions of "moving" and "stationary" when calculating distance as opposed to time.

It's potentially confusing if you don't make clear whether you're talking about the distance between a set pair of events in two frames or about the length of a physical object in two frames. The latter is what length contraction is dealing with, and in this case "moving" and "stationary" has an obvious meaning, it just refers to the object whose length is being measured in two frames.

 

[JesseM]

Here I think is where the asymmetry creeps in. In the case of time dilation, you define the moment (the end of the period to be measured) in the output frame.

(1) At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.

Time dilation deals with the time interval between a pair of events on the clock's worldline, not just the reading at a single moment. So you should say:

(1) For a given pair of events on the clock L's worldline, frame R measures this multiple of the time interval measured between the events by clock L.

In the case of length contraction, you define the location (the end of the distance to be measured) in the input frame.

(2) At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.

Just as you weren't comparing readings at a single time in time dilation, "length" does not refer to position coordinates at a single location, it refers to the distance between the endpoints of the thing being measured at a single moment of time in whatever frame you use. Clocks "naturally" measure the time between events that take place on their own worldline, and rulers "naturally" measure the distance between points along the ruler, like the distance between their own endpoints. So in both cases we are starting from something that is naturally measured in the instrument's own frame, and figuring out whether the corresponding quantity in the observer's frame is smaller or larger, and if it's smaller we say "contraction" and if it's larger we say "dilation".

We can look at the time calculation and the space calculation as each involving three events.

Time dilation involves only two events which occur on the clock's worldline. Length contraction can be thought of two involve three events if you like; just pick one event A on the worldline of the ruler's left end, then the distance in the ruler's rest frame between this event and the event B on the worldline of the right side that is simultaneous with event A in the ruler's rest frame can be understood as the "rest length" of the ruler, while the distance in the observer's frame between event A and the event C on the worldline of the right side that is simultaneous with A in the observer's frame can be understood as the "length" of the ruler in the observer's frame.

In the case of the clocks, supposing them to be arbitrarily longlasting and arbitrarily small, thus each describing a line through spacetime: (A) a coincidence at the origin (of the two clocks meeting and being set to zero)

The time dilation equation is not restricted to dealing with cases where one of the events is at the origin.

(B) the event of the clock collocal with the origin in the input frame showing a certain value, (C) the event of the clock collocal in the output frame with the origin in the output frame showing a certain value.

If the first event occurs at the origin, why would you be interested in an event colocal with the origin in the observer's frame? You're only interested in the difference in time coordinates between two events on the clock's worldline, so naturally since the clock is moving in the observer's frame, if the first event occurred at the origin then the second event did not.

In the case of the rulers, supposing them to be arbitrarily long and arbitrarily shortlived, thus each describing a line through spacetime: (A) a coincidence at the origin (of the zero end of the two rulers meeting), (B) the event of the ruler contemporaneous with the origin in the input frame showing a certain value, (C) the event of the ruler contemporaneous with the origin in the output frame showing a certain value.

If you are measuring the "length" of one particular ruler in its own frame and in the observer's frame, then if one event occurs on the left end of the ruler at the origin of both frames, B must be an event on the worldline of the right end of this ruler that's simultaneous with the first event in the ruler's own frame, and C must be an event on th worldline of the right end of the same ruler that's simultaneous with the first event in the observer's frame.

As far as I can see, calling the reverse questions from the traditional ones

What equations are you talking about? Can you write them out in words as I did in my previous post, and do you understand the distinction I made there between "the temporal analogue for length contraction" and the "inverse time dilation equation"?

"time contraction" and "length dilation" would be less ambiguous than, for example, Wolfram Alpha's "stationary time" and "stationary length".

"Stationary time" and "stationary length" don't refer to a comparison between two frames, they just refer to the time between events on a clock's own worldline as measured by that clock (i.e. as measured in the clock's rest frame) or the length of ruler as measured by the ruler itself (as measured in the ruler's rest frame). Since you're only talking about the value in a single frame, it would make little sense to talk about "dilation" or "contraction".

After all, to compare times, there have to be two notional clocks (one stationary and one moving in each frame).

You can just talk about the time coordinates of events in the observer's frame without worrying about how he assigns them. But if you do want to think about that, then you really need two synchronized clocks at rest in the observer's frame, since in SR a clock only assigns time coordinates to events that occur on its own worldline, and the events in question are ones that occur on the worldline of the moving clock so they'll happen at different positions in the observer's frame.

And when we compare lengths, we're comparing two rulers (one stationary and one moving in each frame).

Again, easier to just talk about coordinates in the observer's frame. But if you do want details of how the observer assigns coordinates, then when measuring length you really need to bring in clocks too, so that the observer can make sure he's measuring the position of the ends of the moving ruler relative to his own ruler at a single moment in time in his own frame.

In the case of length, because real objects persist in time, we think naturally of an object having a certain length when stationary being contracted when seen as moving. So why not use the same convention for time, and many authors do verbally, and use the appropriate formula to match the common statement that "a moving clock ticks slow" (and therefore shows a shorter time period), just as we think of a moving object as having a shorter length?

I don't understand what you mean here. Are you talking about a formula which explicitly deals with rate of ticking rather than with periods of time between a pair of events? In that case you might write:

(clock's rate of ticking relative to time coordinate in observer's frame) = (clock's rate of ticking relative to time coordinate in clock's frame) / gamma

You can see that this is physically distinct from both the "temporal analogue for length contraction" and the "inverse time dilation" equations I wrote in my previous post, in spite of the fact that they all involve dividing by gamma on the right-hand side.

 

[Rasalhague]

Huh? No it won't, if a clock is ticking at a slower rate it shows a longer period. For example, if a clock is slowed down by a factor of 3 in my frame, it will take a period of 30 seconds of time in my frame to tick forward by 10 seconds.

We seem to be talking at cross-purposes here somehow. In everyday language, if two clocks are synchronised at 00:00 and one runs slower by a factor of three, the slow clock will show 10 when the faster one shows 30. The slower clock shows the smaller number, i.e. a shorter period than the faster clock.

Likewise in your example, the clock that's slow shows 10. It's ticking at a slower rate in the sense that only ten ticks have occurred by the time a clock at rest in the other frame has ticked 30 times (with the end of the period defined in the rest frame of the clock that ticked 30 times). And 10 < 30, so the clock that shows 10 is showing a shorter period, right? Smaller number = contraction (shrinking). Bigger number = dilation (stretching). In this sense, the moving clock runs slow (as the motto goes), and shows a contracted period of time (time contraction). In the same sense, a moving ruler can be said to show a contracted interval of space (length contraction).

Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is $$\Delta t' = 1\,meter$$. Show that the time lapse observed in the laboratory frame is given by the expression \Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).

Or do you take dilation in some other sense, for example (if not an increase in the rate) a stretching of the size of each unit?

But then you'd be adopting the convention that dilation/contraction refers to whether a quantity measured in the instrument's own frame is greater or smaller than the corresponding quantity measured in the observer's frame. This is simply not the convention that has been adopted, dilation/contraction always refers to whether the quantity in the observer's frame is greater or smaller.

Perhaps the source of the contradiction here is "corresponding quantity". It's a different quantity that's treated as "corresponding" in the case of time compared to space. In the case of time dilation, the convention is that dilation refers to the greater time (bigger quantity) shown by the clock at rest in the output frame, compared to the clock at rest in the input frame, at an instant defined in the output frame (here's my clock ticking a second: which event on your clock, collocal for you with the origin, do you consider simultaneous to this event on mine). In the case of length contraction, however, the convention is that contraction refers to the shorter length (smaller quantity) shown by the ruler at rest in the output frame, compared to the ruler at rest in the input frame, at a position defined in the input frame (here's my ruler reading a meter: which event on your ruler, instantaneous for you with the origin, do I consider level with it). Only if the position in the length equation and the instant in the timeequation were both defined in the output frame, or both defined in the input frame, would we be comparing like with like. And when that's done, the asymmetry vanishes.

It isn't enough to say whether the quantity is greater or smaller in "the observer's frame" (output frame); there will be some quantity greater and smaller in each. We need to define which frame's "now" or "here" we're using, and when we do, we see that it's a different definition being used for time compared to space.

No, it isn't. Length contraction follows the normal convention that you're talking about the value in the observer's frame.

We need to know which "value in the observer's frame" we're talking about. When you say "the observer's frame", do you mean what I defined in post #355 as the output frame?

No, it certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:

(time in observer's frame) = (time in clock's frame)/gamma

Whereas the normal time dilation equation is:

(time in observer's frame) = (time in clock's frame)*gamma

You can take the normal time dilation equation and divide both sides by gamma, but this doesn't give you the TAFLC above, instead it gives you what I called the "inverse time dilation equation":

(time in clock's frame) = (time in observer's frame)/gamma

See the difference?

Not yet. Since there is no formal difference - exacly the same equation is used - and since, by the principle of relativity, there can be no asymmetry between the two inertial frames, except that they're moving in opposite directions, whatever the difference is, I'm guessing it must be a subjective difference: something about how the frames are conceived? You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify? Presumably the "observer" also has a clock to compare with the clock at rest in the other frame. In a more general sense, each clock is a kind of observer, observing/recording the passage of time, regardless of whether it's consciously observed.

The way I'm looking at it is in terms of frames identical in every way possible so as not to introduce the impression that one is favoured in some way, and thereby risk introducing some false asymmetry into the example. So I'm imagining identical clocks and rulers in each frame, and (unphysically) conceiving of the rulers as somehow only existing for one moment (simultaneous with the origin in their respective rest frames), so as to make them more exactly correspond to the clocks which are restricted in space to the location of the origin in their respective rest frames.

In #355 I tried to define a few terms that could be used to distinguish between frames that would make explicit what it was about the frames that marked them out.

input : output (synonymously: source : target)
left : right

It's potentially confusing if you don't make clear whether you're talking about the distance between a set pair of events in two frames or about the length of a physical object in two frames. The latter is what length contraction is dealing with, and in this case "moving" and "stationary" has an obvious meaning, it just refers to the object whose length is being measured in two frames.

This is a source of asymmetry: physical objects persist in time, both clocks and rulers. We intuitively think of the stationary condition of an object as more fundamental. For that reason, it seems more natural to talk of length contraction than length dilation. But that doesn't preclude asking the same question of time, as people do when they say "a moving clock runs slow".

 

[Doc Al]

We seem to be talking at cross-purposes here somehow. In everyday language, if two clocks are synchronised at 00:00 and one runs slower by a factor of three, the slow clock will show 10 when the faster one shows 30. The slower clock shows the smaller number, i.e. a shorter period than the faster clock.

Likewise in your example, the clock that's slow shows 10. It's ticking at a slower rate in the sense that only ten ticks have occurred by the time a clock at rest in the other frame has ticked 30 times (with the end of the period defined in the rest frame of the clock that ticked 30 times). And 10 < 30, so the clock that shows 10 is showing a shorter period, right? Smaller number = contraction (shrinking). Bigger number = dilation (stretching). In this sense, the moving clock runs slow (as the motto goes), and shows a contracted period of time (time contraction). In the same sense, a moving ruler can be said to show a contracted interval of space (length contraction).

In going from A to B, a moving clock measures 10 seconds. According to laboratory clocks, 30 seconds have passed. 30 > 10, thus time dilation.

A moving stick is 3 meters long in its own frame. According to laboratory measurements, it is 1 meter long. 1 < 3, thus length contraction.

What's the problem?

Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is $$\Delta t' = 1\,meter$$. Show that the time lapse observed in the laboratory frame is given by the expression \Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).

Exactly! Laboratory clocks measure a greater time interval than the moving clock, thus time dilation.

 

[Rasalhague]

In going from A to B, a moving clock measures 10 seconds. According to laboratory clocks, 30 seconds have passed. 30 > 10, thus time dilation.

A moving stick is 3 meters long in its own frame. According to laboratory measurements, it is 1 meter long. 1 < 3, thus length contraction.

What's the problem?

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples? Could we not just as well say 1 < 3, thus time contraction? That sounds simpler to me. Surely there isn't some fundamental property of time that it always has to be "greater than" ;-)

 

[JesseM]

We seem to be talking at cross-purposes here somehow. In everyday language, if two clocks are synchronised at 00:00 and one runs slower by a factor of three, the slow clock will show 10 when the faster one shows 30. The slower clock shows the smaller number, i.e. a shorter period than the faster clock.

As I keep saying, the convention is that contraction/dilation is defined in terms of the measurement in the observer's frame. Do you disagree that in this case the period in the observer's frame is 30, and that 30 is a longer period than 10?

Likewise in your example, the clock that's slow shows 10. It's ticking at a slower rate in the sense that only ten ticks have occurred by the time a clock at rest in the other frame has ticked 30 times (with the end of the period defined in the rest frame of the clock that ticked 30 times). And 10 < 30, so the clock that shows 10 is showing a shorter period, right? Smaller number = contraction (shrinking). Bigger number = dilation (stretching).

Both 10 and 30s are "numbers", so if 10 < 30 then you can also say 30 > 10, and in your own words, "Bigger number = dilation". In order to avoid confusingly referring to every difference as both a contraction and a dilation, we need to pick a convention about which frame's number to use (so that if that frame's number is smaller than the other frame's number we call it 'contraction' and if it's bigger we call it 'dilation'). The convention is to pick the measurement in the observer's frame, not the frame of the instrument which is used to define the "proper" quantity (proper time or proper length).

Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is $$\Delta t' = 1\,meter$$. Show that the time lapse observed in the laboratory frame is given by the expression \Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).

Yes, and note that they are using exactly the convention I described--since the time lapse between the two events in the observer's frame is more than the proper time measured by the moving clock, they call this time dilation.

Or do you take dilation in some other sense, for example (if not an increase in the rate) a stretching of the size of each unit?

Time dilation is not defined in terms of rates (the ratio of clock time to coordinate time), it's defined in terms of time intervals. The interval between two events on the moving clock's worldline is larger in the observer's frame than as measured by the clock, so that's why they call it "dilation".

Perhaps the source of the contradiction here is "corresponding quantity". It's a different quantity that's treated as "corresponding" in the case of time compared to space. In the case of time dilation, the convention is that dilation refers to the greater time (bigger quantity) shown by the clock at rest in the output frame, compared to the clock at rest in the input frame, at an instant defined in the output frame

As I said in the first section of my last post, time dilation does not refer to readings at a particular instant, but to intervals between a pair of events. For example, there might be a clock moving at 0.6c in my frame which reads 12 seconds at an event on its worldline that I assign a time coordinate t'=80 seconds, and then a little later the clock reads 20 seconds at an event on its worldline that I assign a time coordinate t'=90 seconds. Now take a look at the time dilation equation, which should really be written like so:

delta-t' = delta-t * gamma

If I try to plug in 80 and 12 it doesn't work, and it also doesn't work if I plug in 90 and 20. But if I plug in delta-t'=90-80=10 and delta-t=20-12=8, then with gamma=1.25 it does work.

Of course, if you make the assumption that the first event corresponds to the moving clock reading 0, and that the moving clock was synchronized so that it read 0 at t'=0 in the observer's frame, then the intervals will just be equal to the time-coordinates in each frame of the second event on the clock's worldline, so this is probably what you were doing implicitly. Still it's important to understand that the time dilation equation is fundamentally about intervals.

In the case of length contraction, however, the convention is that contraction refers to the shorter length (smaller quantity) shown by the ruler at rest in the output frame, compared to the ruler at rest in the input frame, at a position defined in the input frame (here's my ruler reading a meter: which event on your ruler, instantaneous for you with the origin, do I consider level with it).

In much the same way as time dilation doesn't deal with the times of individual events but with time-intervals between a single pair of events, length contraction doesn't deal with the positions of individual events but with the distance between the two endpoints of a ruler (though of course this is still not quite analogous to time dilation because we aren't talking about the distances between a single pair of events in both frames). In time dilation we could replace intervals with coordinates of a single event only in the very specific case where the first event was assigned a time coordinate of 0 in both frames; with length the only way to replace lengths with position coordinates of a single event is to have it so that the back end of the ruler reaches the origin of the observer's (output) frame at t'=0 in the observer's frame, and then let the event E in question be the event on the front end of the ruler that also occurs at t'=0 in the observer's frame. The position coordinate of this event E in the observer's frame will of course be equal to the length of the ruler in the observer's frame (since length involves simultaneous measurements of either end of an object in whatever frame you're using), and it works out that the position coordinate of event E in the ruler's own rest frame is also equal to its length in its own frame. The reason this works is that the Lorentz transformation tells us that x'=0, t'=0 in the observer's frame coincides with x=0, t=0 in the ruler's frame, and we set things up so that in the observer's frame the left end of the ruler would be at x'=0 (the spatial origin) at t'=0 in the observer's frame, so the left end of the ruler must also be at position x=0 at t=0 in its own frame, and since the ruler is at rest in its own frame this means the left end is at x=0 at all times in its frame, including the time of event E (which does not occur at t=0 in this frame).

With all this said, I'm confused by your above quote, especially the meaning of "at a position defined in the input frame". Position of what, exactly? I think it would be easier if we defined length contraction in terms of the distance between endpoints of the object rather than the position of some single event, but if you want to define it in terms of a single event you have to do it the way I described above, which I'm not sure you're doing. You go on to elaborate this by saying "here's my ruler reading a meter: which event on your ruler, instantaneous for you with the origin, do I consider level with it"; this is rather confusing because you haven't defined which of us is meant to be the "input frame" and which is meant to be the "output frame", but the combination of "position defined in the input frame" and "which event ... do I consider level with it" makes me think you're defining yourself as the input frame and me as the external observer in the output frame. So, in terms of my definition of length contraction in terms of a single event E, you'd be saying that E occurs at x=1 meter in the input frame...but in this case I just want to know the x' coordinate of the same event E on my own ruler, I don't understand the significance of the business about my having to worry about which event on my ruler you "consider level with it", or even what you mean by "level" in this context. (Do you just mean what reading on my ruler lines up with the reading on your ruler of x=1 meter at the moment the event E happens? Or does 'level' refer to spacetime, so you're talking about simultaneity with some distant event? When I measure the length of your ruler I certainly don't have to worry about how you define simultaneity, if that's what you're implying...)

Only if the position in the length equation and the instant in the timeequation were both defined in the output frame, or both defined in the input frame, would we be comparing like with like.

Again we are not normally referring to the position or time coordinates of a single event in these equations, but rather to the time intervals between a single pair of events in two frames, or to the distance between two endpoints of an object at a single moment (which is different from the distance between a single pair of events as in my 'spatial analogue for time dilation') in two frames. You can think of special cases where we are just referring to coordinates of a single event, but in that case contraction vs. dilation is just defined in terms of whether the coordinate of this one event is smaller or larger in the output frame. For example, in the above scenario involving an event E at the front end of a ruler whose back end was at the origin at t'=0 in the output frame, the position coordinate x' of E in the output frame would be smaller than the position coordinate x of E in the input frame. Likewise, if you set things up so the moving clock in the input frame reads t=0 at t'=0 in the output frame, and pick some later event E on the input clock's worldline, then the time coordinate t' of E in the output frame would be greater than the time coordinate t of E in the input frame.

It isn't enough to say whether the quantity is greater or smaller in "the observer's frame" (output frame); there will be some quantity greater and smaller in each.

Sure, but we know the specific quantity we're dealing with for time dilation (the time intervals between the same pair of events in each frame) and for length contraction (the lengths of the same object in each frame). In both cases the quantity is a "proper" quantity for an instrument at rest in the input frame--in the first case it's the proper time between events on the worldline of a clock at rest in the input frame, in the second case it's the proper length of a ruler at rest in the input frame.

No, it certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:

(time in observer's frame) = (time in clock's frame)/gamma

...

You can take the normal time dilation equation and divide both sides by gamma, but this doesn't give you the TAFLC above, instead it gives you what I called the "inverse time dilation equation":

(time in clock's frame) = (time in observer's frame)/gamma

See the difference?

Not yet. Since there is no formal difference - exacly the same equation is used - and since, by the principle of relativity, there can be no asymmetry between the two inertial frames, except that they're moving in opposite directions, whatever the difference is, I'm guessing it must be a subjective difference: something about how the frames are conceived?

There's only "no formal difference" only if you choose to use the same notation for quantities with a different physical interpretation. This is the same issue I criticized neopolitan for--when doing physics, you have to keep in mind the physical meaning of the symbols, just because two equations can be written the same way doesn't mean they have the same meaning! For example, if I made the weird choice to define the time interval between two events in the output frame using the notation "E", and the time interval between the same two events in the input frame (where they are colocated) using the notation "m", and the square root of the gamma factor using the notation "c", then there would be "no formal difference" between the time dilation equation and the equation E=mc^2 where the symbols are interpreted in the more conventional manner. Do you therefore conclude that the difference between the time dilation equation and the relativistic energy/mass relation is only a "subjective difference"?

You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify?

It signifies that we are talking about the time intervals in each frame between events which have been specifically selected to occur on the clock's worldline (so they are colocated in the clock's frame but not the observer's). In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame. Perhaps it would be clearer if I added even more words to my way of writing out the time dilation equation:

(time interval in observer's frame between a pair of events colocated in clock's frame) = (time in clock's frame between same pair of events)*gamma

Then of course the "inverse time dilation equation" obtained by just dividing both sides by gamma is:

(time interval in clock's frame between a pair of events colocated in clock's frame) = (time in observer's frame between same pair of events)/gamma

Whereas the point of the "temporal analogue for length contraction" is meant to keep the convention of the original time dilation equation that the frame in which the quantity we're looking at takes a "special" value stays on the right side of the equation. To make it so that this is true and that the right side is divided by gamma rather than multiplied by it, the quantity in question cannot just be the time in each frame between a pair of events. My suggestion was to consider two spacelike planes which represent surfaces of simultaneity (surfaces of constant t) in the input frame, and let the quantity be the time between these two planes in either frame (i.e. the time between the points where a line of constant x in a given frame will pierce each plane). This is analogous to length contraction where we consider two timelike paths which are lines of constant x in the input frame (these paths are just the worldlines of either end of a ruler at rest in the input frame), and define length as the distance between these two lines in either frame (i.e. the distance between the points where a line of constant t in a given frame will pierce each of these lines of constant x). So, you can write the "temporal analogue for length contraction" as:

(time interval in output frame between two spacelike surfaces that are surfaces of simultaneity in the input frame) = (time interval in the input frame between same spacelike surfaces) / gamma

Here you can see the "special" frame for this quantity is the input frame, and that we have kept it on the right side just as with the original time dilation equation.

The way I'm looking at it is in terms of frames identical in every way possible so as not to introduce the impression that one is favoured in some way, and thereby risk introducing some false asymmetry into the example.

There is no asymmetry in the laws of physics, but it's crucial to understand that all of these equations--time dilation, length contraction, and the "analogues" I defined--all assume that one of the frames is "special" in regards to the quantity that's being measured. If you don't want to make that sort of assumption, just use the full Lorentz transformation equations! For example, if I have two events and I do not assume they are colocated in either frame, then if I know the coordinate separations delta-x and delta-t between them in the input frame, the time interval in the output frame is given by:

delta-t' = gamma*(delta-t - v*delta-x/c^2)

You can see that in the special case where delta-x=0 in the input frame (i.e. they are colocated in the input frame), this reduces to the time dilation equation.

 

[JesseM]

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples?

Because he wanted to stick to the convention that dilation/contraction is consistently defined in terms of the observer's frame (the non-'special' frame as I discussed above).

 

[Doc Al]

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples? Could we not just as well say 1 < 3, thus time contraction? That sounds simpler to me.

You must be consistent, else you render the comparison meaningless. It's always lab frame ("stationary" frame) measurements compared to moving frame measurements. There's no argument here, you just need to understand how the terms "time dilation" and "length contraction" are used.

What might be throwing you off is the apparent lack of symmetry. Going back to the example of a clock (in frame S') moving from A to B while measuring 10 seconds of elapsed time. According to laboratory clocks (frame S), the time elapsed is 30 seconds. Moving clocks run slow: time dilation.

How are things viewed from frame S'? According to frame S', the clocks in frame S are moving and therefore unsynchronized. According to frame S', during the time that S' moves from A to B the clocks in frame S have only recorded an elapsed time of 10/3 seconds. As observed by S', the moving clocks in frame S run slow by that same factor of 3; thus S' measures the time interval to be 10 seconds. 10 > 10/3. Moving clocks run slow: time dilation.

The "time dilation" effect is completely symmetric. All frames observe moving clocks to run slow.

 

[Rasalhague]

Time dilation deals with the time interval between a pair of events on the clock's worldline, not just the reading at a single moment.

Yes, in this case, I should have made clear that I was still referring to the set-up described in #357, in which the clocks each have zero x coordinate indefinitely, and are synchronised (set to time = zero) at the intersection of the origins of their rest frames.

So you should say:

(1) For a given pair of events on the clock L's worldline, frame R measures this multiple of the time interval measured between the events by clock L.

Yes, that would be another way of putting it.

Just as you weren't comparing readings at a single time in time dilation, "length" does not refer to position coordinates at a single location, it refers to the distance between the endpoints of the thing being measured at a single moment of time in whatever frame you use.

Again, I should have made explicit that this was the same example I outlined in #357, and that the other end of all intervals involved here is the spacetime coincidence of the zero end of both rulers being level at time = 0.

Clocks "naturally" measure the time between events that take place on their own worldline, and rulers "naturally" measure the distance between points along the ruler, like the distance between their own endpoints. So in both cases we are starting from something that is naturally measured in the instrument's own frame, and figuring out whether the corresponding quantity in the observer's frame is smaller or larger, and if it's smaller we say "contraction" and if it's larger we say "dilation".

Okay, but this doesn't explain why the pedagogical pairing of a contraction equation for one coordinate and a dilation equation for the other.

Time dilation involves only two events which occur on the clock's worldline. Length contraction can be thought of two involve three events if you like; just pick one event A on the worldline of the ruler's left end, then the distance in the ruler's rest frame between this event and the event B on the worldline of the right side that is simultaneous with event A in the ruler's rest frame can be understood as the "rest length" of the ruler, while the distance in the observer's frame between event A and the event C on the worldline of the right side that is simultaneous with A in the observer's frame can be understood as the "length" of the ruler in the observer's frame.

Alternatively, you could conceptualise time dilation as involving three events: (1) the coincidence of the clocks being synchronised at the origin, (2) the first clock at x = 0 reading one value (its proper time), (3) the second clock at x' = 0 reading another value (its proper time) at the instant simultaneous in the second clock's rest frame with event two. This is equivalent to asking what is the time component of the separation between events one and two in some frame where they're not collocal. Or maybe that's a needless complication.

Alternatively, you could use the three events which correspond to time in the way that the three events of the length contraction relation correspond to distance.

The time dilation equation is not restricted to dealing with cases where one of the events is at the origin.

How would you define the restriction? Could we say: the equation converts the interval of a separation with no space component into the time coordinate of that same interval in an inertial frame moving at some speed relative to the frame in which the events happen at the same place?

If the first event occurs at the origin, why would you be interested in an event colocal with the origin in the observer's frame?

I guess only for the sake of comparison of the interval between it and the origin, on the one hand, with the interval between the origin and some other event simulateous with in one frame or the other - or, equivalently, to find out how its coordinates change when viewed according to a different frame.

You're only interested in the difference in time coordinates between two events on the clock's worldline, so naturally since the clock is moving in the observer's frame, if the first event occurred at the origin then the second event did not.

Okay.

If you are measuring the "length" of one particular ruler in its own frame and in the observer's frame, then if one event occurs on the left end of the ruler at the origin of both frames, B must be an event on the worldline of the right end of this ruler that's simultaneous with the first event in the ruler's own frame, and C must be an event on th worldline of the right end of the same ruler that's simultaneous with the first event in the observer's frame.

The purpose of visualising two rulers was to remind me that there is no physical difference between the frames, they're interchangeable (apart from the difference in the direction of movement) in that any conversion you can make from one to the other, you can make from the other to the one and get the same result. I made the rulers indefinitely long so as to emphasise the parallel with the clocks. Alternatively, if we wanted to think of meter rulers, we could visualise a pair of timebombs, or candle clocks, or hourglasses. I was thinking in this way to emphasise that you can chose either frame as your input frame, and either as your output frame, and can ask contraction and dilation questions of each.

What equations are you talking about? Can you write them out in words as I did in my previous post, and do you understand the distinction I made there between "the temporal analogue for length contraction" and the "inverse time dilation equation"?

I wrote them in words and symbols in #357, alongside the traditional ones. I described them there in terms of a left frame and a right frame, visualising a clock L and a ruler L at rest in the former, and a clock R and ruler R at rest in the latter. If we reworded them in terms of input (source) and output (target) frames, then the output frame, in each case, would be the rest frame of the measuring device said to "show this multiple" or "show this fraction". I'm still unclear about what distinguishes your "temporal analogue for length contraction" from the "inverse time dilation equation". To me they both sound like "time contraction". Do they match any of the situations I described in #357?

"Stationary time" and "stationary length" don't refer to a comparison between two frames, they just refer to the time between events on a clock's own worldline as measured by that clock (i.e. as measured in the clock's rest frame) or the length of ruler as measured by the ruler itself (as measured in the ruler's rest frame). Since you're only talking about the value in a single frame, it would make little sense to talk about "dilation" or "contraction".

This isn't the way Wolfram Alpha is using these terms. If there was no change of frame involved, there would be no change of coordinate, so they would just give you back whatever value you entered! Rather, Wolfram Alpha uses "stationary time" and "stationary length" to refer to what I'd call time contraction and length dilation, and what I think you'd call either the analogue or the inverse equations of the time dilation and length contraction.

You can just talk about the time coordinates of events in the observer's frame without worrying about how he assigns them. But if you do want to think about that, then you really need two synchronized clocks at rest in the observer's frame, since in SR a clock only assigns time coordinates to events that occur on its own worldline, and the events in question are ones that occur on the worldline of the moving clock so they'll happen at different positions in the observer's frame.

True, although both of those clocks at rest in one frame would keep the same time relative to each other (if I've understood this right), so this is just a matter of how physical we want to make the thought experiment, or how simple and abstract.

Again, easier to just talk about coordinates in the observer's frame. But if you do want details of how the observer assigns coordinates, then when measuring length you really need to bring in clocks too, so that the observer can make sure he's measuring the position of the ends of the moving ruler relative to his own ruler at a single moment in time in his own frame.

And by the same logic, we'd want rulers handy when talking about time to make sure the clocks are where they should be... I'm trying to go by the rule that whatever details are suppressed (or expressed) when visualising the situation with changes in time coordinates are also suppressed (or expressed) when visualising the situation with changes in space coordinates. My intention there is to identify or highlight any genuine, fundamental asymmetries between time and space, and to eliminate from the visualisation any arbitrarily asymmetrical details.

I don't understand what you mean here. Are you talking about a formula which explicitly deals with rate of ticking rather than with periods of time between a pair of events? In that case you might write:

(clock's rate of ticking relative to time coordinate in observer's frame) = (clock's rate of ticking relative to time coordinate in clock's frame) / gamma

You can see that this is physically distinct from both the "temporal analogue for length contraction" and the "inverse time dilation" equations I wrote in my previous post, in spite of the fact that they all involve dividing by gamma on the right-hand side.

No, that's not what I had in mind. What other people are thinking of when they use the expression "a moving clock runs slow" I can't say, but to me it suggests time contraction which you subdivide into "temporal analogue for length contraction" and "inverse time dilation". But I'm puzzled by the general use of the expression "a moving clock runs slow" as a verbal summary of the time dilation formula, t' = t * gamma, whose output is a bigger number. In everyday life, when we think of a clock a running slow, we'd expect it to show a smaller number than it would otherwise.

 

[Doc Al]

But I'm puzzled by the general use of the expression "a moving clock runs slow" as a verbal summary of the time dilation formula, t' = t * gamma, whose output is a bigger number.

That's not the correct formula. The "time dilation" formula is: ΔT = gamma * ΔT₀, where ΔT₀ is the time elapsed on the moving clock and ΔT is the time measured in the laboratory frame doing the observing. Of course, ΔT > ΔT₀.

In everyday life, when we think of a clock a running slow, we'd expect it to show a smaller number than it would otherwise.

And it does. You have the formula backwards.

 

[Rasalhague]

Thanks to you both for pointing me towards the more general expression of these equations with differences as opposed to coordinates. I was just used the simplification I've seen in several introductory texts of denoting one end of each of the intervals involved as the origin.

The "time dilation" effect is completely symmetric. All frames observe moving clocks to run slow.

I'm okay with that. The thing that's been troubling me is the way that, while time and space are treated symmetrically in standard presentations of the full Lorentz transformation, when it comes to these special cases "time dilation" and "length contraction", the symmetry has gone. How much is this due to convention, how much is it a fundamental difference between time and space? It isn't as if no one ever divides a time coordinate by gamma or multiplies a space coordinate by gamma, but the terms dilation and contraction seem to attach to time and length respectively, regardless of whether the calculation is making a big number small (contracting it), as here for instance http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html , or a small number big (dilating it). To me that seems inconsistent, but obviously I have a lot to learn.

As I keep saying, the convention is that contraction/dilation is defined in terms of the measurement in the observer's frame. Do you disagree that in this case the period in the observer's frame is 30, and that 30 is a longer period than 10?

Of course, you're right: 30 is bigger than 10!

Thanks for your patience in explaining. I'll get back to you when I've had a chance to mull over what your wrote some more.

 

[Rasalhague]

>You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify?

It signifies that we are talking about the time intervals in each frame between events which have been specifically selected to occur on the clock's worldline (so they are colocated in the clock's frame but not the observer's).

So the "clock's frame" is whichever frame it is in which the time interval between the events is equal to the proper time, and the "observer's frame" is the other (the frame where the time interval is bigger than the proper time because the separation also has some spatial component), regardless of which of these happens to be input or output?

In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame. Perhaps it would be clearer if I added even more words to my way of writing out the time dilation equation:

(time interval in observer's frame between a pair of events colocated in clock's frame) = (time in clock's frame between same pair of events)*gamma

Then of course the "inverse time dilation equation" obtained by just dividing both sides by gamma is:

(time interval in clock's frame between a pair of events colocated in clock's frame) = (time in observer's frame between same pair of events)/gamma

Whereas the point of the "temporal analogue for length contraction" is meant to keep the convention of the original time dilation equation that the frame in which the quantity we're looking at takes a "special" value stays on the right side of the equation. To make it so that this is true and that the right side is divided by gamma rather than multiplied by it, the quantity in question cannot just be the time in each frame between a pair of events. My suggestion was to consider two spacelike planes which represent surfaces of simultaneity (surfaces of constant t) in the input frame, and let the quantity be the time between these two planes in either frame (i.e. the time between the points where a line of constant x in a given frame will pierce each plane). This is analogous to length contraction where we consider two timelike paths which are lines of constant x in the input frame (these paths are just the worldlines of either end of a ruler at rest in the input frame), and define length as the distance between these two lines in either frame (i.e. the distance between the points where a line of constant t in a given frame will pierce each of these lines of constant x). So, you can write the "temporal analogue for length contraction" as:

(time interval in output frame between two spacelike surfaces that are surfaces of simultaneity in the input frame) = (time interval in the input frame between same spacelike surfaces) / gamma

Here you can see the "special" frame for this quantity is the input frame, and that we have kept it on the right side just as with the original time dilation equation.

So would you call the operation carried out in this example TAFLC?

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

The time interval in the input frame (Jack's rest frame) is 10 seconds. It's the time interval between two surfaces of simultaneity in the input frame, namely that through C1 = 0 = C2, and that through C1 = 10 = C2. This value is divided by gamma to give the time interval in the output frame (Jill's rest frame) between the same surfaces, namely 8 seconds.

Or would you call it "the inverse of time dilation"?

The time interval in the input frame is the time interval between a pair of events collocated in the clock's frame (i.e. the special one, the one where those events are separated by the minimum time interval). This value is divided by gamma to give the time interval in the clock's frame between a pair of events collocated in the clock's frame, namely C' passing C1 at the event of them both being set to 0, and the event of C' passing C2 when C2 = 10.

Does the fact that I can describe it in terms of both concepts indicate that they are basically the same thing after all, or have I misunderstood?

 

[JesseM]

So the "clock's frame" is whichever frame it is in which the time interval between the events is equal to the proper time, and the "observer's frame" is the other (the frame where the time interval is bigger than the proper time because the separation also has some spatial component), regardless of which of these happens to be input or output?

Yes, exactly.

So would you call the operation carried out in this example TAFLC?

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

The time interval in the input frame (Jack's rest frame) is 10 seconds. It's the time interval between two surfaces of simultaneity in the input frame, namely that through C1 = 0 = C2, and that through C1 = 10 = C2. This value is divided by gamma to give the time interval in the output frame (Jill's rest frame) between the same surfaces, namely 8 seconds.

Yes, that actually works, provided you here treat Jill as "the observer" rather than Jack. I hadn't thought of it like this, but you're right that the measurements involved in an ordinary time dilation experiment like this can be re-interpreted as a TAFLC measurement, just by switching who we call "the observer", and by switching what defines the "special" frame from the frame where the time between two events on Jill's worldline is minimized (i.e. her own frame) to the frame where two spacelike surfaces that Jill passes through are surfaces of simultaneity (i.e. Jack's frame, since we were already considering his surfaces of simultaneity when showing how he would measure the time elapsed on Jill's clock).

Or would you call it "the inverse of time dilation"?

You've made a good point in that the difference between the two is smaller than I was making it out to be, but I still think it's worth distinguishing them by specifying in words exactly what physical quantity is being measured (i.e. whether you want to say that Jack is just using two of his own surfaces of simultaneity in order to measure the time between two events on Jill's worldline, which is also what Jill's clock is measuring, or whether you want to say that both of them are explicitly trying to measure the time between two spacelike surfaces which are surfaces of simultaneity in Jack's frame), and calling the frame where this quantity takes a non-"special" value the outside observer's frame. Then the TAFLC is the equation that has the outside observer's frame as the output of the equation (the left-hand side) just like the time dilation equation (because the time between the spacelike surfaces takes a 'special' value in Jack's frame, so Jill is defined as 'the outside observer'), whereas the "inverse time dilation equation" has the outside observer's frame as the input (because the time between events on Jill's worldline takes a 'special' value in Jill's frame, so Jack is defined as 'the outside observer').

Does the fact that I can describe it in terms of both concepts indicate that they are basically the same thing after all, or have I misunderstood?

Well, you've convinced me that it's not absolutely essential to distinguish between the "inverse time dilation" equation and the "TAFLC" equation, that the type of conceptual distinction I make above is really more of an aesthetic preference; I still think it's clearer to think in these terms but if you don't want to it's kind of a matter of taste.

 

[neopolitan]

I'm still thinking about how to write the next stage in the process.

JesseM, you asked what I come up with at the end. I come up with an interval between B and an event according to B which, because B is (notionally) at the origin of the B axes corresponds with the coordinates of the event according to B.

(Even if B is not at the origin of the B axes, there will be an offset according to B at colocation with A so the coordinates I work out will be affected by the same offset (as appropriate to each frame), so there is consistency. It will just be like normal vector addition in each frame, ie

"vector 1 in A frame (offset at t_a=t'_b=0) + vector 2 in A frame (interval between colocation and event)"

will transform to

"rotated vector 1 in B frame (offset at t_a=t'_b=0) + rotated vector 2 in B frame (interval between colocation and event)" - aside)

Hopefully this satisfies.

As for leading into the next stage, I've taken on board the fact that you don't like my assumption that if there is a factor or function affecting measurements in the B frame, according to A then that same factor or function affects measurements in the A frame, according to B. Additionally, I am being more careful about extrapolations, keeping in mind that A and B can only measure times at colocation with themselves.

I take v as a given (not measured, although it could be).

I need to use another time (interval) that I have not used in the first stage and am wondering about what nomenclature will suit.

The time (interval) I need is:

(between colocation of A and B in the A frame and when the photon from the Event passes B, in the A frame)

As you pointed out, I have used t'_a to mean something else. The cause behind this is that I didn't introduce subscripts so early before, which may have led to your concern about x'_a.

My suggestion for naming this time (interval) is t'_oa, where the _o indicates a photon interception event.

I think that once we have an agreed term, we can move on (work permitting.)

cheers,

neopolitan

 

[JesseM]

I'm still thinking about how to write the next stage in the process.

JesseM, you asked what I come up with at the end. I come up with an interval between B and an event according to B which, because B is (notionally) at the origin of the B axes corresponds with the coordinates of the event according to B.

By "interval" do you mean a spatial interval or a time interval rather than a spacetime interval? Also, what do you mean by "between B and the event"? Do you mean the interval between the spacetime origin of B's frame (i.e. x=0 and t=0 in B's frame) and this other event, or do you mean the distance between B and the event at the instant the event occurs in B's frame, or something else? Also, presumably what you derive is an equation that gives you this value on the left side and something that looks like the Lorentz transformation equation on the right side, i.e. something like gamma*(x - vt) on the right, yes? If so, what is the corresponding interpretation of x and t (or whatever symbols you use on the right side) in terms of your physical setup?

As for leading into the next stage, I've taken on board the fact that you don't like my assumption that if there is a factor or function affecting measurements in the B frame, according to A then that same factor or function affects measurements in the A frame, according to B.

Such an assumption might well be justifiable in terms of the assumption that the laws of physics should work the same way in both frames (the first postulate of SR), it's just that I would need to see a little more of a detailed justification for it if you're trying to do a rigorous proof.

Additionally, I am being more careful about extrapolations, keeping in mind that A and B can only measure times at colocation with themselves.

I'm not sure this is strictly necessary in a derivation--as long as we start from the basic postulate that A and B each assume light moves at c in their own frame, we can basically take as read that whenever they deal with time coordinates of events that don't happen along their worldlines, they calculate it by noting the time they receive the light from the event and subtracting the travel time based on the distance the event happened from them (as measured by a ruler at rest relative to themselves). It's fine if you don't state this explicitly each time you talk about the time of events, and in fact it probably makes the derivation easier to follow if you just assume this is understood and don't worry about it (or just mention it once at the beginning).

The time (interval) I need is:

(between colocation of A and B in the A frame and when the photon from the Event passes B, in the A frame)

As you pointed out, I have used t'_a to mean something else.

You used it to mean the time between an event E_a on the photon's worldline which was simultaneous with A and B being colocated in the A frame, and the event of the photon passing B, in the A frame. So, this will obviously always give the same value as your definition above, even if you selected the events differently; if you only did this because you're worried about "extrapolations" then like I said my advice would just be not to worry. But if you want to introduce this interval separate from the other one that's fine too.

My suggestion for naming this time (interval) is t'_oa, where the _o indicates a photon interception event.

Don't all your time intervals include a photon interception event as one of the two events they're giving the interval between? But the terminology doesn't really matter, t'_oa is fine with me.

 

[neopolitan]

By "interval" do you mean a spatial interval or a time interval rather than a spacetime interval?

If you talk about the "Lorentz Transforms" (plural) then it is spacetime. Otherwise I would arrive at one equation each.

Also, what do you mean by "between B and the event"? Do you mean the interval between the spacetime origin of B's frame (i.e. x=0 and t=0 in B's frame) and this other event, or do you mean the distance between B and the event at the instant the event occurs in B's frame, or something else?

I mean:

(spatial and temporal separation between B and an event, in the B frame) = (a function operating on or a factor multiplied by the spatial and temporal separation between A and an event, in the A frame)

And I am not giving away the end by saying that these will end up in the form:

(spatial separation between B and an event, in the B frame) = (a factor) . ((spatial separation between A and an event, in the A frame) - (relative velocity).(temporal separation between A and an event, in the A frame))

(temporal separation between B and an event, in the B frame) = (a factor) . ((temporal separation between A and an event, in the A frame) - (relative velocity).(spatial separation between A and an event, in the A frame)/(the speed of light squared))

Also, presumably what you derive is an equation that gives you this value on the left side and something that looks like the Lorentz transformation equation on the right side, i.e. something like gamma*(x - vt) on the right, yes? If so, what is the corresponding interpretation of x and t (or whatever symbols you use on the right side) in terms of your physical setup?

Addressed above, I think.

I'm not sure this is strictly necessary in a derivation--as long as we start from the basic postulate that A and B each assume light moves at c in their own frame, we can basically take as read that whenever they deal with time coordinates of events that don't happen along their worldlines, they calculate it by noting the time they receive the light from the event and subtracting the travel time based on the distance the event happened from them (as measured by a ruler at rest relative to themselves). It's fine if you don't state this explicitly each time you talk about the time of events, and in fact it probably makes the derivation easier to follow if you just assume this is understood and don't worry about it (or just mention it once at the beginning).

Ok, happy with that.

You used it to mean the time between an event E_a on the photon's worldline which was simultaneous with A and B being colocated in the A frame, and the event of the photon passing B, in the A frame. So, this will obviously always give the same value as your definition above, even if you selected the events differently; if you only did this because you're worried about "extrapolations" then like I said my advice would just be not to worry. But if you want to introduce this interval separate from the other one that's fine too.

Prior to the summation, I introduced subscripts at a later point in my derivation process, which allowed me to use a subscripted x' differently. I don't want to introduce a x'_a which is not the same as the summation x'_a.

Don't all your time intervals include a photon interception event as one of the two events they're giving the interval between?

Yes, but not photon interception events which are separated from the observer. So to be more precise, I intend to use _o to mean a reference to a non-local photon interception event. So:

t'_oa = time (t), according to A _a, that a photon passes B ('_o) if that photon was at a distance of x_a when A and B were colocated, according to A.


_______________________________________________________________________


At the risk of repeating myself, I want to make clear that I am talking about a single event described in two frames.

I also want to make clear that A and B won't know anything about that event before a photon from the event reaches them.

I also want to make clear that, even when photon reaches them, A and B won't know more than "I received a photon".

If they are given information about when the event took place (in their own frame), they can work out where the event took place (in their own frame). But they can't work out when and where the photon was released from the mere fact that they receive a photon at a specific time and place.

So, my derivation works on the principle that if we consider an event which was simultaneous (in the A frame) with the colocation of A and B, at t=0, then we can calculate a "where" (in the A frame) for this event. We can also work out a "when" and "where" (in the A frame) for colocation of B and the photon, from the timing of colocation of A and B and the timing of the photon's colocation with A (in the A frame).

Because A and B were colocated at t=0, and at colocation t'=0, we then have sufficient information to work out the "when" and "where" in the B frame of the event that was simultaneous with colocation of A and B in the A frame, which would be the coordinates of that same event in the B frame.

If we say that the event that is simultaneous (in the A frame) with the colocation of A and B is the event which spawns the photon, this is just for the sake of convenience. Without more information, A can't really know when or where the photon was spawned.

Are you happy for me to go on to the next stage of the derivation?

cheers,

neopolitan

 

[JesseM]

If you talk about the "Lorentz Transforms" (plural) then it is spacetime. Otherwise I would arrive at one equation each.

I should have been more specific, I was wondering about the meaning of the specific variables in your final equations, like the "interval between B and an event" which you mentioned. In the Lorentz transform each variable represents a purely spatial interval or a purely temporal interval between two events (in whatever frame each variables are dealing with), so presumably the same is true for your final equations?

Also, what do you mean by "between B and the event"? Do you mean the interval between the spacetime origin of B's frame (i.e. x=0 and t=0 in B's frame) and this other event, or do you mean the distance between B and the event at the instant the event occurs in B's frame, or something else?

I mean:

(spatial and temporal separation between B and an event, in the B frame) = (a function operating on or a factor multiplied by the spatial and temporal separation between A and an event, in the A frame)

And I am not giving away the end by saying that these will end up in the form:

(spatial separation between B and an event, in the B frame) = (a factor) . ((spatial separation between A and an event, in the A frame) - (relative velocity).(temporal separation between A and an event, in the A frame))

(temporal separation between B and an event, in the B frame) = (a factor) . ((temporal separation between A and an event, in the A frame) - (relative velocity).(spatial separation between A and an event, in the A frame)/(the speed of light squared))

When you say "temporal separation between A and an event", you must mean the temporal separation between the event and some other specific event that occurs on A's worldline--can I assume this is just the event of A and B being colocated at t=0 in A's frame? And likewise for "temporal separation between B and an event"?

Also, can you tell if the derivation will be based on treating the event as one of the specific events you've already introduced, like the event of the photon crossing either A or B's worldline, or an event that's simultaneous with their being colocated in one of the frames? If so it seems to me your proof is not going to be fully general--once you've calculated the space and time intervals between this event and the event of A and B being colocated at the origin, you are of course free to move the origin if you introduce a lemma of the type I talked about in the second paragraph of post 338 and earlier in post 249, but just shifting the origin wouldn't change the fact that if you used one of these four events originally, then the line between the event and the A&B colocation event will be either purely spatial (meaning the two events are simultaneous) or purely temporal (meaning the events are colocated) in one of the two frames, so this will remain true if you shift the position of the origin. This is not to say I think it's pointless to derive a special case of the Lorentz transformation, but if that's what you're doing we should at least be clear on this.

Yes, but not photon interception events which are separated from the observer. So to be more precise, I intend to use _o to mean a reference to a non-local photon interception event. So:

t'_oa = time (t), according to A _a, that a photon passes B ('_o) if that photon was at a distance of x_a when A and B were colocated, according to A.

OK, makes sense.

I also want to make clear that, even when photon reaches them, A and B won't know more than "I received a photon".

If they are given information about when the event took place (in their own frame), they can work out where the event took place (in their own frame). But they can't work out when and where the photon was released from the mere fact that they receive a photon at a specific time and place.

So, my derivation works on the principle that if we consider an event which was simultaneous (in the A frame) with the colocation of A and B, at t=0, then we can calculate a "where" (in the A frame) for this event. We can also work out a "when" and "where" (in the A frame) for colocation of B and the photon, from the timing of colocation of A and B and the timing of the photon's colocation with A (in the A frame).

Because A and B were colocated at t=0, and at colocation t'=0, we then have sufficient information to work out the "when" and "where" in the B frame of the event that was simultaneous with colocation of A and B in the A frame, which would be the coordinates of that same event in the B frame.

If we say that the event that is simultaneous (in the A frame) with the colocation of A and B is the event which spawns the photon, this is just for the sake of convenience. Without more information, A can't really know when or where the photon was spawned.

Are you happy for me to go on to the next stage of the derivation?

This part is fine, but see my questions at the beginning.

 

[Rasalhague]

Yes, that actually works, provided you here treat Jill as "the observer" rather than Jack. I hadn't thought of it like this, but you're right that the measurements involved in an ordinary time dilation experiment like this can be re-interpreted as a TAFLC measurement, just by switching who we call "the observer", and by switching what defines the "special" frame from the frame where the time between two events on Jill's worldline is minimized (i.e. her own frame) to the frame where two spacelike surfaces that Jill passes through are surfaces of simultaneity (i.e. Jack's frame, since we were already considering his surfaces of simultaneity when showing how he would measure the time elapsed on Jill's clock).

I'm not quite sure what you mean by "switching", and I'm confused by your redefinition of "special". How can we compare TAFLC with INV(TD) if the meaning of the terms used to define INV(TD) have to be changed to define TAFLC? I thought you'd consider Jack "the observer" whether his rest frame was the input or not, and whether we're dilating the input value or contracting it, since, of the two quantities involved (input/known and output/unknown), his rest frame is the one where the variable has the greater value (is not minimised). Going back to your definition of "the observer"...

You wrote: >>In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame.

...when you say "the quantity we are dealing with", was I right to think that this quality of being minimum is the only distinguishing feature of "the quantity we are dealing with", or is there something else that marks out one quantity in this way? There are two quantities involved, one known (the time interval in the input frame), and one unknown (the time interval in the output frame). My impression was that you were defining the "quantity we are dealing with" as whichever of these quantities is minimum (i.e. a proper time in one of the frames). The difference between your TD and INV(TD) equations is that in TD the observer's frame is the output frame, while in INV(TD) the observer's frame in the input frame, so it can't be the quality of being an input or an output that defines "oberser's" value and "clock" value.

You've made a good point in that the difference between the two is smaller than I was making it out to be, but I still think it's worth distinguishing them by specifying in words exactly what physical quantity is being measured (i.e. whether you want to say that Jack is just using two of his own surfaces of simultaneity in order to measure the time between two events on Jill's worldline, which is also what Jill's clock is measuring, or whether you want to say that both of them are explicitly trying to measure the time between two spacelike surfaces which are surfaces of simultaneity in Jack's frame), and calling the frame where this quantity takes a non-"special" value the outside observer's frame. Then the TAFLC is the equation that has the outside observer's frame as the output of the equation (the left-hand side) just like the time dilation equation (because the time between the spacelike surfaces takes a 'special' value in Jack's frame, so Jill is defined as 'the outside observer'), whereas the "inverse time dilation equation" has the outside observer's frame as the input (because the time between events on Jill's worldline takes a 'special' value in Jill's frame, so Jack is defined as 'the outside observer').

What is the defining feature of "special" that applies to both TAFLC and INV(TD) if it's not the quality of being the input, or the quality of being minimum? What is the difference between "Jack is just using" and "both of them are explicitly trying to measure", given that calculation doesn't depend on who's performing it?

Well, you've convinced me that it's not absolutely essential to distinguish between the "inverse time dilation" equation and the "TAFLC" equation, that the type of conceptual distinction I make above is really more of an aesthetic preference; I still think it's clearer to think in these terms but if you don't want to it's kind of a matter of taste.

It's been a fascinating discussion. You've taught me a lot along the way, and given me a lot to think about.

 

[neopolitan]

I should have been more specific, I was wondering about the meaning of the specific variables in your final equations, like the "interval between B and an event" which you mentioned. In the Lorentz transform each variable represents a purely spatial interval or a purely temporal interval between two events (in whatever frame each variables are dealing with), so presumably the same is true for your final equations?

Yes, x'_b for instance is purely spatial in the B frame.

When you say "temporal separation between A and an event", you must mean the temporal separation between the event and some other specific event that occurs on A's worldline--can I assume this is just the event of A and B being colocated at t=0 in A's frame? And likewise for "temporal separation between B and an event"?

Temporal separation between the event and colocation of A and the photon from the event, in the A frame. (I do use t=0 such that it is simultaneous with the event in the A frame.)

Temporal separation between the event and colocation of B and the photon from the event, in the B frame. (The event that A considers to be simultaneous with t=0, is not simultaneous with t'=0 in the B frame. However, by same token, the event that is the location of the photon at t'=0 in the B frame, is not simultaneous with t=0 in the A frame.)

Also, can you tell if the derivation will be based on treating the event as one of the specific events you've already introduced, like the event of the photon crossing either A or B's worldline, or an event that's simultaneous with their being colocated in one of the frames?

The event I intend to use will be simultaneous with A and B being colocated in one frame.

What I propose is to continue with what you will tell me not a general case and I will remain aware that I have a burden of proof to show that you may be wrong about it not being a general case.

Or do you want me to provide the general case argument first? (Which is difficult, but maybe not impossible, before I have shown the derivation.)

cheers,

neopolitan

 

[Rasalhague]

You must be consistent, else you render the comparison meaningless. It's always lab frame ("stationary" frame) measurements compared to moving frame measurements. There's no argument here, you just need to understand how the terms "time dilation" and "length contraction" are used.

On consistency, see the end of this post. When you say "compared to", are you defining "lab frame" as input frame (the frame for which we know the value), and "moving frame" as output frame (the frame for which we want to calculate the interval)? If not, what are the distinguishing features of "lab frame" and "moving frame"; how are they defined?

In post #357 I described various possible questions we might ask of these formulas, calling contraction whatever operation contracted the input, and dilation whatever dilated the input. (Excuse the lack of deltas; I hope any ambiguity there is removed by the description of the set-up at the beginning of that post and by the definitions I gave in #355.) Instead of "moment defined in frame X", I could have said "surface of simultaneity in frame X". Instead of "location defined in frame X", I could have said "worldline of a mark on the ruler at rest in frame X".

Maybe it would help to match up definitions to this scenario.

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples?

Because he wanted to stick to the convention that dilation/contraction is consistently defined in terms of the observer's frame (the non-'special' frame as I discussed above).

To me, the intuitive way would be to define dilation/contraction in terms of the input frame in the sense that the value of the input variable is dilated (made bigger) by the operation, or contracted (made smaller), to give the result. But apparently this isn't the convention. When you say "in terms of the observer's frame", this suggests that the variable would take a smaller value in the observer's frame, in the case of dilation, and a dilated value in what you called the clock frame. But that's the opposite of your definition of the terms "observer's frame" and "clock frame"; you defined the clock frame as the one where the variable takes its minimum value, and the observer's frame as other other frame.

In going from A to B, a moving clock measures 10 seconds. According to laboratory clocks, 30 seconds have passed. 30 > 10, thus time dilation.

A moving stick is 3 meters long in its own frame. According to laboratory measurements, it is 1 meter long. 1 < 3, thus length contraction.

What's the problem?

The problem is that I'd have expected "dilation" to refer to the process of dilating the input to produce the output. Dilation seems to imply a starting point at which something is small, and an endpoint in the process at which it's bigger. I'm aware that this isn't the convention, but I don't understand why not. So when we're given a value of 10 seconds, and calculate a value of 30 seconds from it, it seem only natural to call this operation dilation. But when the same term, dilation, is used of the inverse operaion, as here

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

that sounds bizarre to me. Our input is a big number, our output is a smaller number, and yet we're supposed to call this operation dilation as well (or multiplication by the "time dilation factor").

In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.

* Time dilation. In the case of clocks, the time measured in the output frame between two events on the worldline of a clock at rest in the input frame is greater than the time measured by the clock between those two events.

(We take as our standard the variable of the input, in comparison to which the output variable is bigger. But when we say "a moving clock runs slow", we're conceptualising it as the inverse, taking as our standard the variable with the bigger variable: given a certain input, the output will be smaller, contracted. Hence...)

* Time contraction. The time measured in the output frame between the ends of that period (defined as surfaces of simultaneity in the input frame) is smaller than the time measured by the clock at rest in the input frame.

* Length dilation. In the case of rulers, the distance measured in the output frame between two events whose separation has no time component in the rest frame of some ruler is greater than the distance measured by that ruler itself.

* Length contraction. The distance measured in the output frame between the ends of the a ruler at rest in the input frame (defined as the worldlines of marks on the ruler, lines of collocality) is smaller than the distance measured by the ruler at rest in the input frame.

The only inconsistency, it seems to me, is in talking as if there was something inherently dilatory about time, and something inherently contractory about space, in spite of the fact that we can and do dilate or contract either, depending on the context and what we want to find out from the equations. An example of this inconsistency is the way that Wolfram Alpha is obliged to reverse its definitions of moving and stationary depending on whether you want to transform a time interval or a space interval.

 

[matheinste]

Is there perhaps some confusion here between number of ticks and duration between ticks. When we say that clocks moving relative to a given frame run slow when compared with that frame we mean that the moving clock ticks slower, that is the time between ticks is dilated ( longer or greater or bigger ). However the number of ticks will be decreased ( contracted, smaller , less in number), it records less time. I think the standard and accepted usage is that time dilates for a moving clock, that is the time between ticks is longer. I have never seen it used any other way.

Matheinste.

 

[Rasalhague]

Is there perhaps some confusion here between number of ticks and duration between ticks. When we say that clocks moving relative to a given frame run slow when compared with that frame we mean that the moving clock ticks slower, that is the time between ticks is dilated ( longer or greater or bigger ). However the number of ticks will be decreased ( contracted, smaller , less in number), it records less time. I think the standard and accepted usage is that time dilates for a moving clock, that is the time between ticks is longer. I have never seen it used any other way.

Here is an example of dilation used in the opposite sense.

Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is $$\Delta t' = 1\,meter$$. Show that the time lapse observed in the laboratory frame is given by the expression $$\Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}$$. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).

My impression so far as been that this is the standard interpretation, especially given other people's comments in this thread, although I've wondered whether some people interpreted it as you do. Many texts I've read don't make any explicit statement on how the word "dilation" is to be interpreted.

After I posted this in #365, Doc Al wrote: "Exactly! Laboratory clocks measure a greater time interval than the moving clock, thus time dilation."

JesseM wrote: "Yes, and note that they are using exactly the convention I described--since the time lapse between the two events in the observer's frame is more than the proper time measured by the moving clock, they call this time dilation."

 

[Doc Al]

On consistency, see the end of this post. When you say "compared to", are you defining "lab frame" as input frame (the frame for which we know the value), and "moving frame" as output frame (the frame for which we want to calculate the interval)?

No, just the opposite. (Assuming I understand what you mean by "input frame".)

If not, what are the distinguishing features of "lab frame" and "moving frame"; how are they defined?

In the so-called "time dilation" formula, you start with a time interval measured on a moving clock (the "input", I suppose) and use the formula to compute what the lab frame would measure for that time interval (the "output" of the formula). The "output" is always bigger than the "input".

The "lab frame" is the frame of the observer whose measurements we want to calculate; the "moving frame" is the frame in which the clock in question is at rest. For example, I observe a clock moving past me as it goes from position A to position B. My rest frame is the lab frame; my cohorts and I in our frame have measured the time interval (on our lab frame clocks, of course) for the clock to pass from A to B. Call that time interval ΔT. What time interval does the clock itself record (the "moving" frame, to us)? Call that time interval ΔT₀. The time dilation formula relates those two time intervals: ΔT = gamma*ΔT₀. This is just a precise statement of the loose phrase "moving clocks run slow".

The problem is that I'd have expected "dilation" to refer to the process of dilating the input to produce the output. Dilation seems to imply a starting point at which something is small, and an endpoint in the process at which it's bigger. I'm aware that this isn't the convention, but I don't understand why not. So when we're given a value of 10 seconds, and calculate a value of 30 seconds from it, it seem only natural to call this operation dilation.

Where do you get the idea that "dilation" means anything other than it does in normal usage? To "dilate" means to expand--get bigger.

But when the same term, dilation, is used of the inverse operaion, as here

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

that sounds bizarre to me. Our input is a big number, our output is a smaller number, and yet we're supposed to call this operation dilation as well (or multiplication by the "time dilation factor").

I'm familiar with that site. I don't see anything there that would contradict the usual usage of the term "time dilation".

Realize that the "time dilation" formula applies to time readings on a single moving clock. You cannot take a time interval measured in the moving frame using multiple clocks, blindly apply the time dilation formula, and expect it to to give the correct time interval measured in another frame*. When multiple clocks are involved you must also include the effects of clock desynchronization (the relativity of simultaneity). All of this is factored in automatically when you use the full Lorentz transformations.

*I suspect that this is at the root of your confusion.