JesseM said:OK, the clarified version looks clear to me.
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 said: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.
JesseM said: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)?
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.Rasalhague said: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.
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.Rasalhague said: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
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 said: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.
JesseM said: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;
JesseM said: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.
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.Rasalhague said:Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction!
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.Rasalhague said: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 said: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.
JesseM said: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.
JesseM said: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.
JesseM said: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.
JesseM said: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.
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.Rasalhague said: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.
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.Rasalhague said: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".
No, it isn't. Length contraction follows the normal convention that you're talking about the value in the observer's frame.Rasalhague said: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 certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:Rasalhague said:At this stage, it seems to me no more articial or arcane than "time dilation". It's just the reverse calculation.
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.Rasalhague said: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.
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:Rasalhague said: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.
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".Rasalhague said: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.
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.Rasalhague said:We can look at the time calculation and the space calculation as each involving three events.
The time dilation equation is not restricted to dealing with cases where one of the events is at the origin.Rasalhague said: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)
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.Rasalhague said:(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 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.Rasalhague said: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.
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"?Rasalhague said:As far as I can see, calling the reverse questions from the traditional ones
"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".Rasalhague said:"time contraction" and "length dilation" would be less ambiguous than, for example, Wolfram Alpha's "stationary time" and "stationary length".
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.Rasalhague said:After all, to compare times, there have to be two notional clocks (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.Rasalhague said:And when we compare lengths, we're comparing two rulers (one stationary and one moving in each frame).
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:Rasalhague said: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 said: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.
JesseM said: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.
JesseM said: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?
JesseM said: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?
JesseM said: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.
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.Rasalhague said: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).
Exactly! Laboratory clocks measure a greater time interval than the moving clock, thus time dilation.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).
Doc Al said: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?
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?Rasalhague said: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.
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).Rasalhague said: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).
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.Rasalhague said: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).
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".Rasalhague said: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?
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:Rasalhague said: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
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).Rasalhague said: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).
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.Rasalhague said: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.
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.Rasalhague said: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.
JesseM said: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?
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"?Rasalhague said: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?
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:Rasalhague said:You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify?
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:Rasalhague said: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.
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).Rasalhague said:Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples?
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.Rasalhague said: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.
JesseM said: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.
JesseM said: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.
JesseM said: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.
JesseM said: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".
JesseM said: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.
JesseM said:The time dilation equation is not restricted to dealing with cases where one of the events is at the origin.
JesseM said: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?
JesseM said: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.
JesseM said: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.
JesseM said: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"?
JesseM said:"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".
JesseM said: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.
JesseM said: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.
JesseM said: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.
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.Rasalhague said: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.
And it does. You have the formula backwards.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 said:The "time dilation" effect is completely symmetric. All frames observe moving clocks to run slow.
JesseM said: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?
JesseM said:>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).
JesseM said: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.
Yes, exactly.Rasalhague said: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, 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).Rasalhague said: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.
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').Rasalhague said:Or would you call it "the inverse of time dilation"?
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.Rasalhague said: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?
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?neopolitan said: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.
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.neopolitan said: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.
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).neopolitan said:Additionally, I am being more careful about extrapolations, keeping in mind that A and B can only measure times at colocation with themselves.
You used it to mean the time between an event Ea 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.neopolitan said: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.
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 said:My suggestion for naming this time (interval) is t'oa, where the o indicates a photon interception event.
JesseM said:By "interval" do you mean a spatial interval or a time interval rather than a spacetime interval?
JesseM said: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?
JesseM said: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?
JesseM said: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).
JesseM said:You used it to mean the time between an event Ea 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.
JesseM said:Don't all your time intervals include a photon interception event as one of the two events they're giving the interval between?
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?neopolitan said:If you talk about the "Lorentz Transforms" (plural) then it is spacetime. Otherwise I would arrive at one equation each.
JesseM said: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?
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"?neopolitan said: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))
OK, makes sense.neopolitan said: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 xa when A and B were colocated, according to A.
This part is fine, but see my questions at the beginning.neopolitan said: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?
JesseM said: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).
JesseM said: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').
JesseM said: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.
JesseM said: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?
JesseM said: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"?
JesseM said: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?
Doc Al said: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.
JesseM said: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).Rasalhague said:Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples?
Doc Al said: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?
JesseM said: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.
matheinste said: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.
No, just the opposite. (Assuming I understand what you mean by "input frame".)Rasalhague said: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)?
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".If not, what are the distinguishing features of "lab frame" and "moving frame"; how are they defined?
Where do you get the idea that "dilation" means anything other than it does in normal usage? To "dilate" means to expand--get bigger.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.
I'm familiar with that site. I don't see anything there that would contradict the usual usage of the term "time 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").
Doc Al said:No, just the opposite. (Assuming I understand what you mean by "input frame".)
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.
Doc Al said: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".
Doc Al said:Where do you get the idea that "dilation" means anything other than it does in normal usage? To "dilate" means to expand--get bigger.
Doc Al said: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.
Obviously, when you are dealing with an equation the "input" and "output" are entirely arbitrary. You can take ΔT = gamma*ΔT0 and rewrite it as ΔT0 = ΔT/gamma. If you choose to call ΔT your "input", then of course your "output" (ΔT0) will be smaller. So what?Rasalhague said:By input I mean the value you start with (what's given, known), and by output the value you want to calculate (what's unknown). But if your lab frame is what I was calling input frame (the frame in which the known value is a coordinate), and your moving frame what I was calling output frame, what happens when we have a question of the inverse type and want to calculate a smaller output, as in Michael Fowler's example? Well, we've seen what happens (output < input), but what would you call this operation, how would you labels the frames and values involved in that operation?
Again, the moving frame is just the one where the variable being measured takes the "special" value--in the case of time dilation and length contraction, it's the frame where the time interval and length in that frame are equal to the proper time and rest length. The lab/observer's frame is the one where they don't, because the object in question (a clock or a ruler) is moving. The equation is usually written under the assumption that we know the proper time/rest length (so that's the input) and want to find the time interval/length in the lab frame, but of course with any equation you can rearrange it to solve for whatever variable you don't know.Rasalhague said: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?
But that just isn't the convention. The convention is that it's based on whether the value is bigger or smaller in the non-special frame.Rasalhague said: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.
How is "moment" different from "surface of simultaneity"? Again, time dilation isn't defined in terms of readings at one particular moment/surface of simultaneity, it's defined by the interval of time between two specified events.Rasalhague said:(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".
...and length contraction isn't defined by a single worldline, it's defined by the distance between two worldlines.Rasalhague said:Instead of "location defined in frame X", I could have said "worldline of a mark on the ruler at rest in frame X".
But that would mean that in the exact same physical scenario you could call it either contraction of dilation depending on the whims of what variable your teacher gave you first. In any case, the convention is also that the time dilation and length contraction equations are written in a form where the observer's frame is the output of the equation, although of course you can rearrange to solve for the proper time/proper length if you wish.Rasalhague said: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.
Dilation means an increase, so why do you say it suggests the variable would take a smaller value in the observer's frame? I said several times that when I said it was defined "in terms of the observer's frame", I meant that you used the word "dilation" if the value was bigger in the observer's frame, and "contraction" if the value was smaller in the observer's frame. If you think it's better to describe this as defining it "in terms of the moving frame", I would find that very confusing, but go ahead and do so as long as we're clear on the previous sentence.Rasalhague said: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 again, the convention is that the moving frame is the "input", and in fact most problems will give you this first. But as I said I think it would be confusing to have the phrase dilation and contraction depend on the whims of which value a textbook or teacher provided you with first.Rasalhague said: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 understand the concern, but see above for why I think this alternate convention would be confusing. In any case, the issue of the convention is already decided for us.Rasalhague said: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").
But there is something inherently dilatory about the proper time between two events and something inherently contractory about the length of an object in its rest frame--do you agree? On the other hand, there is also something inherently dilatory about the proper distance between two spacelike-separated events, where "proper distance" refers to the distance between the events in the inertial frame where they're simultaneous (the distance in other frames will always be greater).Rasalhague said: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.
What question did you give to Wolfram Alpha, exactly?Rasalhague said: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.
Doc Al said:Obviously, when you are dealing with an equation the "input" and "output" are entirely arbitrary. You can take ΔT = gamma*ΔT0 and rewrite it as ΔT0 = ΔT/gamma. If you choose to call ΔT your "input", then of course your "output" (ΔT0) will be smaller. So what?
Doc Al said:As I said many times now, the reason it's called time dilation is that a lab frame observes a larger time interval than recorded by a moving clock. That's what "moving clocks run slow" means.
Doc Al said:Note that even though you can reverse the equation to solve for either quantity given the other, that doesn't change the meaning of the quantities. ΔT0 is always the time interval recorded by the moving clock, thus ΔT > ΔT0.
The terms "lab frame" and "moving frame" are relative terms, of course, since each observer views him or herself as stationary in his or her own frame. From your frame as an observer, if you see a clock moving with respect to you, then from your frame it is a moving clock. Simple as that! And from your frame you can observe the "time dilation" effect expressed as "moving clocks run slow".Rasalhague said:What makes a lab frame a lab frame, or a moving clock a moving clock, given that each of these frames is static from its own perspective, and moving from the perspective of the other frame? In other words, what are the defining features of a lab frame and a moving frame?
JesseM said:How is "moment" different from "surface of simultaneity"
JesseM said:Again, time dilation isn't defined in terms of readings at one particular moment/surface of simultaneity, it's defined by the interval of time between two specified events.
...and length contraction isn't defined by a single worldline, it's defined by the distance between two worldlines.
JesseM said:But that would mean that in the exact same physical scenario you could call it either contraction of dilation depending on the whims of what variable your teacher gave you first. In any case, the convention is also that the time dilation and length contraction equations are written in a form where the observer's frame is the output of the equation, although of course you can rearrange to solve for the proper time/proper length if you wish.
Dilation means an increase, so why do you say it suggests the variable would take a smaller value in the observer's frame? I said several times that when I said it was defined "in terms of the observer's frame", I meant that you used the word "dilation" if the value was bigger in the observer's frame, and "contraction" if the value was smaller in the observer's frame. If you think it's better to describe this as defining it "in terms of the moving frame", I would find that very confusing, but go ahead and do so as long as we're clear on the previous sentence.
JesseM said:And again, the convention is that the moving frame is the "input", and in fact most problems will give you this first. But as I said I think it would be confusing to have the phrase dilation and contraction depend on the whims of which value a textbook or teacher provided you with first.
JesseM said:I understand the concern, but see above for why I think this alternate convention would be confusing. In any case, the issue of the convention is already decided for us.
But there is something inherently dilatory about the proper time between two events and something inherently contractory about the length of an object in its rest frame--do you agree? On the other hand, there is also something inherently dilatory about the proper distance between two spacelike-separated events, where "proper distance" refers to the distance between the events in the inertial frame where they're simultaneous (the distance in other frames will always be greater).
JesseM said:What question did you give to Wolfram Alpha, exactly?
Doc Al said:The terms "lab frame" and "moving frame" are relative terms, of course, since each observer views him or herself as stationary in his or her own frame. From your frame as an observer, if you see a clock moving with respect to you, then from your frame it is a moving clock. Simple as that! And from your frame you can observe the "time dilation" effect expressed as "moving clocks run slow".
Of course, observers in that other frame moving along with that clock can just as well observe a clock at rest in your frame. And to them your clock is a moving clock so "time dilation" applies; to them, your clock "runs slow".
Forget about the terms "input" and "output"--they just add to the confusion. "Lab frame" and "moving frame" are relationship terms. If you are doing the measuring, then your frame is the lab frame; if a frame is moving with respect to you, then that frame is the "moving" frame.Rasalhague said:So are you saying that your lab frame and moving frame are completely arbitrary terms, and that we're free to choose which frame to call lab, and which moving, according to taste or convenience: either can be the input or the output frame?
"Time value" is too vague a term.Or is the rule that we have to choose which frame is lab and which moving in such a way that the time value being transformed will always be bigger in the frame labelled lab?
Absolutely. Calling the frame of the moving clock the "clock frame" makes it kind of easy to remember, doesn't it?Would you say that your lab frame and moving frame are equivalent to Jesse's terms "observer's frame" and "clock frame" respectively?
Generally, one calls the "moving" frame the primed frame (S') and the "lab" frame the unprimed frame (S). But the main thing is that each frame is moving with respect to the other. And you can, using the Lorentz transformations, transform measurements made in one frame to measurements made in the other. No need to "switch labels".Suppose you were working on a complicated problem which involved taking inputs first from one frame, then from the other - would you switch the labels of the frames if need be to avoid having to reverse the convention that the time value in any calculation is bigger in the lab frame? Wouldn't that be potentially confusing? Or have I got this all wrong?
Doc Al said:Forget about the terms "input" and "output"--they just add to the confusion. "Lab frame" and "moving frame" are relationship terms. If you are doing the measuring, then your frame is the lab frame; if a frame is moving with respect to you, then that frame is the "moving" frame.
Doc Al said:"Time value" is too vague a term.
Doc Al said:Absolutely. Calling the frame of the moving clock the "clock frame" makes it kind of easy to remember, doesn't it?
Doc Al said:Generally, one calls the "moving" frame the primed frame (S') and the "lab" frame the unprimed frame (S). But the main thing is that each frame is moving with respect to the other. And you can, using the Lorentz transformations, transform measurements made in one frame to measurements made in the other. No need to "switch labels".
Doc Al said:Careful with vague terms like "time value". There's no rule that all time intervals measured in the lab frame must be greater than the time interval in the moving frame. The rule is moving clocks run slow. A time interval recorded by a single moving clock will be smaller than the time interval measured in any other frame. It is certainly possible to choose events such that the time interval between them is greater in the moving frame--but such an interval does not correspond to an interval measured on a single clock (as JesseM might say, it does not represent a proper time).
I wasn't redefining "special", I was just saying that which frame is treated as the "special" one depends on conceptually what it is you say that you want to know the value of in both frames. If you want to know the value in both frames of the time interval between two events on Jill's worldline, in this case Jill's frame is the special one. If you want to know the time in both frames between two spacelike surfaces that happen to be surfaces of simultaneity in Jack's frame, then it's Jack's frame that's special. However, the idea of wanting to know "the time between two spacelike surfaces" is sort of a contrived idea that doesn't really ever come up in normal problems, it's much more natural to want to know the time between two particular events, like the events of Jill passing the two different clocks on that webpage. In any case the only two equations of this type this that have standard names are the time dilation equation and the length contraction equation, the other "equivalent" equations I introduced have no standard name so no one will understand what you mean if you try to refer to them by name.Rasalhague said:When I asked what does it signify to call one frame the "observer's frame" and the other the "clock frame", you said:
"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."
But in #375, in response to my description of Michael Fowler's example in terms of your definition of the "temporal analogue for length contraction", you redefined special:
"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)."
It's not whimsical that you could be given either as input, it's whimsical that you would change the name of the equation you use to get the answer based on which you happen to know first. Communication is much less confusing if you adopt a naming convention that allows you to call the equation by the same name all the time. As an analogy, we call E=mc^2 the "mass-energy equivalence equation", it would be confusing if there were two separate names for it depending on whether you knew the mass and wanted to find the energy or if you knew the energy and wanted to find the mass.Rasalhague said:Yes the same physical scenario could be described with value A as input and value B as output, or vice versa, but what's so whimsical about that?
No, the usual convention is that unprimed is the frame where the time or length is proper time or rest length, primed is the frame where the clock or object is in motion.Rasalhague said:I thought this is just what you advised me the convention was with the terms unprimed and primed.
But "input" and "output" don't refer to anything physical, they just refer to which quantity you happen to have been given first. If the goal is communication, don't you want words for equations and the symbols that appear in them to refer to physical details?Rasalhague said:Input and output are more explicit names than unprimed and primed, given the various different uses that prime symbols are put to in this context by different textbooks.
I don't think it makes sense to talk about an interval having a rest frame since it isn't an object that persists over time, but when talking about an interval between a pair of events perhaps you could say something like "the co-location frame" (though this isn't a standard term). Normally in these types of problems the two events in question are events on the worldline of some object like a clock or Jill's ship, so you can tailor the description to the problem and say things like "the clock rest frame" or "Jill's rest frame".Rasalhague said:If we want to use fixed labels for frames that don't vary depending on the question, we need use some other names, like left and right, or something that expressed the idea of - would it be correct to say - "the rest frame of the (spacetime) interval"? (I.e. what you were describing in your first definition of "special frame".)
No, with any equation of two variables you're free to take either variable as input and use it to find the value of the other variable as output.Rasalhague said:So what do you make of Doc Al's example in #385, equivalent to Michael Fowler's with Jack and Jill, where the moving frame is the output frame? Is that unconventional?
In a problem with multiple objects you could again just denote different frames based on which object's rest frame they were.Rasalhague said:What if you were making a series of calculations of various qualities back and forth between too frames; would you switch labels every time you needed to divide by gamma in moving from a frame that you'd previously multiplied by gamma in order to find a time value for? That sounds even more complicated to me than continually switching which frame we call the primed or output frame.
Yes.Rasalhague said:Does your observer's frame equate with Doc Al's lab frame, and your clock frame equate with Doc Al's moving frame?
No, because again, the name you use for the equation has nothing to do with which variable you put in as input. Doc Al was just calculating the time between a single pair of events in two different frames, so that means he was using the time dilation equation.Rasalhague said:And if so, is that your observer's frame and clock frame as originally defined, or as redefined in the example that involved dividing my gamma which we described in terms of "temporal analogue for length contraction"?
I don't think there is any such convention, I've never seen anyone say they are "representative of time and space" or anything along those lines. They are just representative of what they actually give you, namely the time between two events in different frames and the length of an object in two different frames. Both of these are quantities that actually come up regularly in ordinary SR problems, whereas there are fewer situations where you'd want to know the distance between two spacelike-separated events in two frames, and I can't think of any non-contrived situations where you'd be directly interested in finding out the time between two parallel spacelike surfaces in two frames.Rasalhague said:Yes and, to complete the picture, something inherently contractory about the time period (whatever we call it) which bears the same relation to time as the length of an object does to space. The convention of matching up time dilation with length contraction, as somehow representative of time and space respectively
Yes, I do agree with you there--when presenting the two equations it'd be a good idea to point out that one shouldn't jump to the sort of conclusion you describe that they reflect some basic difference between time and space, since this is probably not an uncommon misunderstanding.Rasalhague said:But because relativity is so counterintuitive when we first meet it, we don't know what the distinguishing properties of time and space might be. We don't have everyday experience of passing macroscopic objects at "relativistic" speed. So when we meet this combination of equations and their associated names, it's easy get confused or jump to the (mistaken) conclusion that the pairing directly embodies some fundamental difference or asymmetry between how time and space behave in special relativity, when really it's a matter of convention (albeit there might be reasons motivating that convention).
OK, but I don't understand what you meant when you talked about an "inconsistency" here:Rasalhague said:I didn't ask a specific question. I just typed "time dilation" then toggled between "moving time" and "stationary time" in the "calculate" menu directly under the input field. Likewise with "length contraction" ("moving length", "stationary length"). It takes as its default input 1 second, in the case of time, and 1 meter, in the case of length.
What do you mean "reverse its definitions"? If the time dilation equation is understood to give you the time in two frames between events that occur on the worldline of a clock, and the length contraction equation is understood to give you the length of a ruler in two frames, then in both cases Wolfram Alpha uses "stationary" to refer to the frame in which the clock/ruler is at rest and "moving" to refer to the frame where the clock/ruler is in motion.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.
JesseM said:Communication is much less confusing if you adopt a naming convention that allows you to call the equation by the same name all the time. As an analogy, we call E=mc^2 the "mass-energy equivalence equation", it would be confusing if there were two separate names for it depending on whether you knew the mass and wanted to find the energy or if you knew the energy and wanted to find the mass.
JesseM said:I wasn't redefining "special", I was just saying that which frame is treated as the "special" one depends on conceptually what it is you say that you want to know the value of in both frames. If you want to know the value in both frames of the time interval between two events on Jill's worldline, in this case Jill's frame is the special one. If you want to know the time in both frames between two spacelike surfaces that happen to be surfaces of simultaneity in Jack's frame, then it's Jack's frame that's special. However, the idea of wanting to know "the time between two spacelike surfaces" is sort of a contrived idea that doesn't really ever come up in normal problems, it's much more natural to want to know the time between two particular events, like the events of Jill passing the two different clocks on that webpage. In any case the only two equations of this type this that have standard names are the time dilation equation and the length contraction equation, the other "equivalent" equations I introduced have no standard name so no one will understand what you mean if you try to refer to them by name.
JesseM said:It's not whimsical that you could be given either as input, it's whimsical that you would change the name of the equation you use to get the answer based on which you happen to know first. Communication is much less confusing if you adopt a naming convention that allows you to call the equation by the same name all the time. As an analogy, we call E=mc^2 the "mass-energy equivalence equation", it would be confusing if there were two separate names for it depending on whether you knew the mass and wanted to find the energy or if you knew the energy and wanted to find the mass.
JesseM said:No, the usual convention is that unprimed is the frame where the time or length is proper time or rest length, primed is the frame where the clock or object is in motion.
JesseM said:But "input" and "output" don't refer to anything physical, they just refer to which quantity you happen to have been given first. If the goal is communication, don't you want words for equations and the symbols that appear in them to refer to physical details?
JesseM said:I don't think it makes sense to talk about an interval having a rest frame since it isn't an object that persists over time, but when talking about an interval between a pair of events perhaps you could say something like "the co-location frame" (though this isn't a standard term). Normally in these types of problems the two events in question are events on the worldline of some object like a clock or Jill's ship, so you can tailor the description to the problem and say things like "the clock rest frame" or "Jill's rest frame".
JesseM said:No, with any equation of two variables you're free to take either variable as input and use it to find the value of the other variable as output.
In a problem with multiple objects you could again just denote different frames based on which object's rest frame they were.
JesseM said:No [in answer to "[...] is that your observer's frame and clock frame as originally defined, or as redefined in the example that involved dividing my gamma which we described in terms of "temporal analogue for length contraction"?], because again, the name you use for the equation has nothing to do with which variable you put in as input. Doc Al was just calculating the time between a single pair of events in two different frames, so that means he was using the time dilation equation.
JesseM said:I don't think there is any such convention, I've never seen anyone say they are "representative of time and space" or anything along those lines. They are just representative of what they actually give you, namely the time between two events in different frames and the length of an object in two different frames. Both of these are quantities that actually come up regularly in ordinary SR problems, whereas there are fewer situations where you'd want to know the distance between two spacelike-separated events in two frames, and I can't think of any non-contrived situations where you'd be directly interested in finding out the time between two parallel spacelike surfaces in two frames.
JesseM said:OK, but I don't understand what you meant when you talked about an "inconsistency" here:
What do you mean "reverse its definitions"? If the time dilation equation is understood to give you the time in two frames between events that occur on the worldline of a clock, and the length contraction equation is understood to give you the length of a ruler in two frames, then in both cases Wolfram Alpha uses "stationary" to refer to the frame in which the clock/ruler is at rest and "moving" to refer to the frame where the clock/ruler is in motion.
Yeah, but "time dilation inverse" just means rearranging terms in the time dilation equation without changing their physical meaning, so I wouldn't really consider it a physically distinct equation, "inverse" is just an adjective to indicate that the equation has been rearranged in this way.Rasalhague said:You want to know the interval between two events on Jill's worldline, given that you already know its time component in a frame where Jill is not at rest. The time component is minimised in Jill's rest frame. Jill's rest frame is special. Time dilation.
You want to know the time component of the spacetime interval between two events on Jill's worldline, given that you already know the interval. The time component is minimised in Jill's rest frame. Jill's rest frame is special. Time dilation inverse.
Yeah, but as you pointed out the difference is kind of cosmetic, any given problem involving clocks can be thought of in either way. And in practice no one ever thinks in terms of the time between spacelike surfaces as opposed to the time between events, so if you find it confusing to talk about "the temporal analogue of length contraction" it might be better to just forget the whole thing and assume by default that in any problem involving clocks the thing you're interested in is the time between events on some clock's worldline.Rasalhague said:...Unless you chose to conceptualise the same relation in a subjectively different way, as "the temporal analogue of length contraction" (time/period/while contraction), in which case Jack's frame is special.
Not "a space coordinate". A length. Remember that you can also talk about the distance between two spacelike-separated events, in which case the value is minimum in the frame where they're simultaneous.Rasalhague said:I guess what's changed is that, in the original concept, special frame (=moving frame =clock frame?) was defined as: the frame where the value is minimum if we're talking about a time coordinate; the frame where the value is maximum if we're talking about a space coordinate.
The point about "specialness" was just to explain the conceptual similarity between time dilation and length contraction, since in time dilation the time between events is greater (dilated) in the non-special frame, and in length contraction the length is smaller (contracted) in the non-special frame. "Special" is not some officially defined term with a rigorous meaning, it's just my way of saying why I think the terminology is consistent. It's true that a calculation involving clocks can in principle be conceptualized in terms of the time between two spacelike surfaces in both frames rather than the time between two events in both frames, but this is a very contrived-seeming way of conceptualizing it that no one ever does in practice. Likewise you can conceptualize the ordinary length contraction equation in terms of the distance between two events on the front and back worldline of an object that are simultaneous in the frame where the object is moving, making this frame the "special" one, but in practice most people conceptualize it just as the length of the object in both frames since that's an a more natural way to think about it. It makes sense that the terminology would naturally reflect the way that's most natural to conceptualize what it is that's being calculated, doesn't it? Perhaps you're confusion is that you're trying to understand a mere naming convention for two equations as having some super-rigorous justification, when it's really just an aesthetic choice that arises from how the equations are normally used and conceptualized in practice, you could call them the "time whatsit equation" and the "length whosit equation" if you preferred.Rasalhague said:In the revised, or expanded, definition, we add the proviso that the space definition will apply to a time calculation if-and-only-if we make an arbitrary decision to view that calculation in this certain subjective way (which doesn't affect the calculation itself). That seems like a sort of redefnition to me. Or have I misunderstood?
Keep in mind there are no official "labels" for frames, just ones that I've introduced for the purpose of explaining the thought processes behind the names of the equations. But if we stick to the convention of special/non-special or moving frame/observer's frame, why do you think we'd have to switch labels? If the clock and ruler are at rest in the same frame, then their rest frame would be the special or moving frame according to the labels I've introduced, and the frame where they were moving would be the non-special or observer's frame.Rasalhague said:Do you switch labels for frames if you have a situation involving a clock and ruler at rest in the same frame if, having first performed a calculation of the sort we call time dilation (according to the conventional definition of time dilation), you then want to perform a calculation of the sort we call length contraction (according to the conventional definition of length contraction).
Again I don't see why there'd be a need to switch labels. Just as the frame where the time between events on Jill's worldline is minimized would be Jill's rest frame, so the frame where the length of Jill's rocket is maximized would also be Jill's rest frame. In each case I would therefore call Jill's frame the special/moving frame.Rasalhague said:E.g. suppose you follow the convention you outlined for a time calculation (time dilation) and label Jill's rest frame the clock/moving frame, and Jack's rest frame the observer/lab frame. Having got the answer to that, suppose you then want to know how long Jill's rocket is in Jill's rest frame from Jack's measurement of it, or you want to know how long Jack will measure Jill's rocket to be, given that at has such-and-such a length in Jill's rest frame. Does the convention dictate that you reverse the labels you gave to the frames when you were performing a time calculation simply because now Jill's rest frame is special, because special means "where the coordinate is maximum" in the case of space?
Not coordinates. What's being computed is either a time interval or a length. And conceptually, the difference in terms is just based on whether the thing being a computed increases in the observer's "non-special" frame or whether it decreases in this frame.Rasalhague said:Then why do we have two different names for the same equation (dilation, contraction) that depend on which coordinate is being computed?
Is there anything in this analogy corresponding to the notion of their being one particular frame where the quantity in question takes a special value? If not I don't see the relevance.Rasalhague said:It's as if we had a convention when using Cartesian coordinates of only calling multiplication of the y component of a vector "multiplication", and insisting on calling the same operation "inverse division" when we perform it on an x component.
That the pair of events you're calculating the time between both occur on this particular clock's own worldline.Rasalhague said:But what, in general, defines a clock as "the clock"?
As I said, if you're trying to make it so the conventions about terminology can be justified in a super-rigorous way then you're going up a blind alley. But I don't see it as particularly confusing; it doesn't matter if multiple clocks may be present if you're calculating the time between events on the worldline of one specific clock, and likewise it doesn't matter if multiple rulers or other physical objects are present if you're calculating the length of a specific one. If you don't like that, what about the alternate special/non-special terminology I introduced for the two frames? If you're calculating the time between two events, isn't it unambiguous which frame is the "special" one where the time between them is minimized?Rasalhague said:The whole point of them is that they don't refer to anything physical. I chose them to explicitly communicate that. Lab and clock and rocket and observer are all physical objects. I appreciate that they may be used by convention to refer to a frame defined by the role it plays in the calculation, or by a free choice of how to visualise the situation, but until I've understood, absorbed and become familiar with that convention, they seem cumbersome in that they inevitably conjure up images of labs and rockets and clocks and observers which may either not correspond directly to the way a particular problem is worded, or - even worse - actually clash with the way the problem is worded, e.g. if the problem happened to involve an actual physical lab in the frame you've defined as the moving frame, or two physical labs, or a whole bunch of explicit, physical clocks, in which case we need a rule to define which clock is going to be called "the clock" and which lab "the lab" and which observer "the observer". That's why I'm looking for some general, abstract terms that would express what we're doing mathematically in any such problem, to see which kinds of operation are really the same mathematically, underneath all the varied trappings of clocks and labs and rockets and identical twins that change from problem to problem, so that I'll have a general terminology with which to view any problem of this kind, no matter what the parochial details are.
JesseM said:I don't think it makes sense to talk about an interval having a rest frame since it isn't an object that persists over time, but when talking about an interval between a pair of events perhaps you could say something like "the co-location frame" (though this isn't a standard term). Normally in these types of problems the two events in question are events on the worldline of some object like a clock or Jill's ship, so you can tailor the description to the problem and say things like "the clock rest frame" or "Jill's rest frame".
Not that I know of.Rasalhague said:I'd certainly like to have so way to to express the concept: collocation frame and contemporary frame, or something like that. Is there just no standard term at all?
But again, although this may be a confusion sometimes experienced by students when they see the two equations, I don't think any textbook author intends any sort of implication that the equations prove that "time and space seem to be behaving differently".Rasalhague said:I think we're touching on something really interesting here which is related to a genuine difference between time and space, or our ways of relating to them, namely that the term "rest frame" doesn't bear the same relation to time as it does to space. With time dilation and length contraction, in both cases, we're talking about physical object with well defined spatial limits, and in both cases we're talking about a physical object that persists indefinitely in time. So when we define dilation or contraction in terms of an object's rest frame, we naturally find that time and space seem to be behaving differently.
It depends on the context. If only one observer or lab is mentioned in the problem, in that case it's fine to just say "the lab frame" or "the observer's frame" and the meaning will be clear.Rasalhague said:So when we use labels like "lab/observer's" or "moving/clock's/special", these being as Doc Al said relative terms, we always have to explicitly state "Jill's lab frame" = "Jill's rest frame" - is that right?
"Clock frame" and "lab frame" and "observer's frame" are not official terms that you have to use in any situation, and they make very little sense the way you use them above (I don't know what a 'clock's clock frame' even means, for example). If there are multiple clocks, then presumably they are given different names, and you can just talk about "[NAME X]'s rest frame" or even just "[NAME X]'s frame". All that's important is that you talk in a way that it's clear from the context of the problem what frame you're referring to.Rasalhague said:Does this not become confusing if the problem is such that we're forced to talk about "this clock's clock frame" and "that clock's lab frame", or "that observer's observer's frame", "this observer's clock's clock frame", etc.?
Again, conceptually it makes sense to me to use this terminology, since in the case of time intervals between events on the worldline of an object (which is usually a clock though it doesn't have to be) the time interval will be larger in the non-special frame where the object is moving, and in the case of the length of some object the length will be smaller in the non-special frame where the object is moving. But it is just a naming convention, it just has to do with how people tend to conceptualize the equations and is not meant to have an ultra-rigorous justification.Rasalhague said:It does depend on "which variable" in a different sense though: namely it depends on whether it's a temporal or spatial variable, hence "calculating the time [...] so that means he was using the time dilation equation". If he'd been calculating the same relation with respect to length, I suppose he'd have used the "length contraction" equation, which is the same equation!
Yes, and naming conventions are based on how things are actually conceptualized in practice, and I imagine there are very few people who regularly conceptualize time dilation problems in terms of the time between two spacelike surfaces.Rasalhague said:But didn't you agree in #375 that any perfectly regular situation that involves what you call "inverse time dilation" can also be conceptualised as "the temporal analogue of length contraction" and that "the type of conceptual distinction" between the two "is really more of an aesthetic preference [...] kind of a matter of taste"?
More specifically, conceptualizing what the equation is telling you as "the time between two spacelike surfaces" which just happen to pass through whatever events are mentioned in the problem, rather than the time between the two events themselves, seems contrived to me.Rasalhague said:So it's not that there are a set of contrived situations that are the only application of this, and no non-contrived situations. Rather it's idea of conceptualising the relation as time contraction that seems contrived to you,
But neither equation "transforms coordinates" of single events, not unless you introduce the artificial condition that the event in question is either on one system's time axis, or that it's on one system's space axis. The time dilation equation is ordinarily understood to transform time intervals between pairs of events, and the length contraction is ordinarily understood to transform lengths (i.e. the spatial distances between two worldlines in each frame). If you want to make them more analogous you can either conceptualize the time equation in a weirder way where it's transforming the time between two spacelike surfaces in each frame, or conceptualize the spatial equation in a weirder way where it's transforming the distance between two specific events on the two worldlines in the frame where those worldlines have a constant position. But in practice people don't normally conceptualize them in either of these last two ways.Rasalhague said:and what seems contrived to me is the idea of conceptualising the same equation for transforming coordinates as two different operations with different names that depend only on which coordinate is being transformed.
What do you mean by "pairing"? It's not like they are introduced as analogous to one another, it's just that in combination they are two useful equations that make it possible to deal with many types of relativity problems that would otherwise require you to use the full Lorentz transformation equations. Personally I like to mention them along with a third equation that deals with the relativity of simultaneity, saying that if two clocks are a distance d apart and synchronized in their own rest frame, then in a frame where they are moving at speed v along the axis between them, they will be out-of-sync by vd/c^2.Rasalhague said:I'm not saying that it's inconsistent within those (conventional) terms. It's internally consistent. It's the correct terminology, given those definitions. I'm just saying that the pairing of those defenitions seems arbitrary.
"Moving" and "stationary" are not official terms for the two frames dealt with by the time dilation and length contraction equations, they are just understood in terms of the context of a particular problem. If you just have one observer who is trying to calculate the length of a ruler moving relative to him and the time interval between two readings on a clock moving relative to him, it would be confusing to switch whether the observer or the ruler/clock is called "stationary" or "moving".Rasalhague said:If we wanted to emphasise the interchangeability of time and space, we could define which clock is moving so that clicking on "moving" coordinate resulted in multiplication by gamma, in each case, and selecting "stationary" coordinate resulted in division by gamma, in each case.
neopolitan as a very vague analogy said:Say A and B bump into each other in the street, literally, in their cars.
The coordinate system they use could be such that this collision is (0,0,0,0), but in reality we know it is not likely to be.
Much more likely it will be something like (t=number of days, hours and minutes since a notional event, h=ground level, N/S=degrees from the equator, E/W=degrees from the Greenwich Meridian) or (t=number of days, hours and minutes since a notional event, h=ground level, x=distance along a road from a specific junction, y=distance from the edge of the road on one side)
