Time Dilation Formula: Clarifying Confusion

[Grimble]

Gentlemen, it appears to me that the difficulty here is that you are both trying to agree that you have a common understanding while holding opposing concepts of what time dilation means.

If the time between 'ticks' increases, the clock runs slow.
If the number of 'ticks' increases, then, surely, the clock runs fast.

What seems to me to be the fundamental problem here is that we are overlooking the fact that if the transformed 'ticks' are longer, or shorter, then the units are different; and if the units are different then the time dilation formula isn't comparing 'like with like', i.e. it is not so much an equation as a conversion.

What we are ascertaining is how many co-ordinate time units are equal to one Proper time unit, as measured by a local observer in either Inertial Frame of Reference (IFoR).

Einstein shewed that the space-time co-ordinates from one IFoR could be converted and transferred to another IFoR by application of the Lorentz Transformation Equations whilst still complying with his two Postulates.

In such a transformation the two IFoRs would, locally, have common time and space dimensions; which I will refer to as Inertial units (as they are a special case of Proper time).
But each IFoR's observation of the other would be in transformed, or Co-ordinate units, giving rise to the Time Dilation and Length Contraction phenomena.

It is obvious from the above that, Time Dilation and Length Contraction, will be observed in another IFoR but cannot be experienced; (This thread is limited to Special Relativity so Gravitational Time Dilation is not addressed) so how can we talk of a traveller, in an IFoR, experiencing dilated time or contracted distances? For he has to experience Inertial time and distance, it is only an observer that will see the transformed units.

Everything becomes very clear and straightforward if we consider diagrams of Minkowski Spacetime.
An important factor here is that relative velocity between two IFoRs is shewn by rotation between the frames of reference.

So, taking this in the simplest case we have the following diagrams:

http://img193.imageshack.us/img193/5910/fig1fig2.jpg

Shewing the effect of rotation on the ct and the x axes where perpendicular projections from the primed axes onto the unprimed axes depict time dilation and length contraction.

Combining these into a single diagram demonstrates the rotation between two IFoRs. In the following diagram one can see the rotated frame of reference, in red and how it relates to the observer's frame of reference.

http://img16.imageshack.us/img16/5218/figure3g.jpg

An important point to note here is that the IFoR of the moving body has the same origin as that of the observer. So the moving body is progressing at a constant velocity within its own frame of reference.
If this were not so, the origin of the moving IFoR would have to be progressing along the x-axis or else we would have two bodies moving at a constant relative velocity, whilst remaining at the same location.

 

[Rasalhague]

Gentlemen, it appears to me that the difficulty here is that you are both trying to agree that you have a common understanding while holding opposing concepts of what time dilation means.

Maybe so... We have a common understanding of the phenomenon, I think (albeit only a basic one in my case), but I'm unsure of how exactly this word "dilation" is to be understood with respect to the various aspects of the phenomenon.

If the time between 'ticks' increases, the clock runs slow.
If the number of 'ticks' increases, then, surely, the clock runs fast.

Indeed. I was trying to work out whether it was the running slow (counting off a smaller total of ticks) or the running fast (counting off a larger number of ticks) that people call "dilation". The textbooks I looked at generally seemed to refer to the larger number as the "dilated time", but Wikipedia referred to the clock that ticked fewer times as being "time-dilated". That seemed to me to indicate two different usages of the word, but I could be mistaken. Others apparently see no contradiction in talking about a time-dilated clock that displays non-dilated time, and a non-time-dilated clock that displays dilated time. I suppose it's no less logical than talking about a contracted ruler that shows a longer distance (a bigger number of units) next to an uncontracted ruler that shows a shorter distance (a smaller number of units). Is this how the words dilation and contraction are generally understood?

What we are ascertaining is how many co-ordinate time units are equal to one Proper time unit, as measured by a local observer in either Inertial Frame of Reference (IFoR).

Proper time and coordinate time are nicely unambiguous (as long as we're clear about which events we mean).

 

[matheinste]

The "stationary" observer will observe the "moving" clock to have a longer period between ticks than he observes on his own clock. There is no disagreement between texts on this point, only misinterpretations by the readers. There are however many ways of describing the effect, some of which decribe the effect more clearly than others.

Matheinste.

 

[Rasalhague]

The "stationary" observer will observe the "moving" clock to have a longer period between ticks than he observes on his own clock. There is no disagreement between texts on this point, only misinterpretations by the readers. There are however many ways of describing the effect, some of which decribe the effect more clearly than others.

So maybe I was wrong to think that there's anything contradictory about characterising the moving clock as the "time-dilated clock" which displays "non-dilated time", and the stationary clock the "non-time dilated clock" which displays "dilated time". Is that how you look at it? Because that's the only way, as far as I can see, to reconcile sources like Lerner, Petkov and Schröder (who contrast dilated time with proper time) with sources like the Wikipedia entry (which characterise the clock itself as being "time-dilated").

Add Freund to the "dilated time"-constrasts-with-"proper time" list:
http://books.google.co.uk/books?id=...nepage&q="proper time" "dilated time"&f=false

Googling for "dilated time" and "dilated clock" produces mainly forum discussions, and there are a few crank sources to weed out (I get that impression that "dilated clock" is a more informal term), but here's a reasonable-looking site that uses both terms:

http://www.geocities.com/syzygywjp/RelativeI.html

It's talking about general relativity, but still relevant, I think. It says, "A body orbting at this range would experience a time dilation of about 1.07 longer than non dilated time." And "According to calculations based on relativistic motion near a black hole, the time dilation nearest the black hole would amount to 1.414 times slower than a non time dilated clock."

So for this writer, dilated time is "longer" than non-dilated time. (But would you say they meant by this that "dilated time" has more or less ticks?) And a dilated clock, as for Wikipedia, is slow and therefore makes fewer ticks than a non-time-dilated clock.

 

[Rasalhague]

One of the sites that I cited:

"But that hour and a half elapsed on the pilot's clock, in the pilot's frame. The Modesto and Fresno clocks tick off dilated time in that frame, each minute dilated to fill two minutes of the pilot's time. So one and a half hours on the pilot's clock corresponds to forty-five minutes elapsed on the Modesto and Fresno clocks, which means that in the pilot's frame the Modesto clock appears to be running forty-five minutes "fast" relative to the Fresno clock. [...] Half an hour later, by John's watch, the train arrives in Modesto, where the station clock, having ticked off fifteen dilated minutes, shows one o'clock, in perfect agreement with Jane's calculation."

http://bado-shanai.net/Map of Physics/moptempoff.htm

Surely this use of the term "dilated time" is the exact opposite of the definition given by Freund, Lerner, Petkov and Schröder, for whom it would be the pilot's clock that showed the dilated time, wouldn't it? For this writer (Dennis Anthony), dilated time is the smaller value, as recorded/measured/displayed (ticked off) by the clock that's running slow. For him, "dilated time" is a reduced number of expanded units, rather than an expanded number of reduced units.

How do you all feel about that? If you see no contradiction, could you explain to me how they amount to the same thing? If you do see a contradiction, could you tell me which interpretation matches your own, or the one that you're most familiar with, and which you feel is the standard way of understanding the term.

 

[Rasalhague]

Here are a couple more sources, Adams and Weinert, agreeing with Freund, Lerner, Petkov, Schröder.

http://books.google.co.uk/books?id=1RV0AysEN4oC&pg=PA152&lpg=PA152#v=onepage&q=&f=false

http://books.google.co.uk/books?id=8eN9zoprUT4C&pg=PA175#v=onepage&q=&f=false

For all these authors, dilated time is the expanded number of reduced units, derived from a proper time interval by the formula

$$\Delta t = \gamma * \Delta \tau.$$

I notice that Weinert, rather than talking about a "dilated clock", writes: "The spacio-temporal stages of space-time, which show the history of geodesics, can be measured by clocks, which are either attached to the trajectory (recording the proper time of the system at successive spacio-temporal stages) or from external clocks (recording a dilated time for the system undergoing linear translation)."

Here's a site which confirms what JesseM and Matheinste have been saying that there's no contradiction between, for example, the Wikipedia article and these textbooks. It agrees with all of the above that dilated time is an expanded number of reduced units:

http://www.relativitycalculator.com/stationary_moving_clocks.shtml

And also agrees with the Wikipedia characterisation of a clock which measures non-dilated time as being a "time-dilated clock":

"an observer at either of the two clocks will be stationary relative to the other clock and therefore it will reciprocally be the other clock which will be time dilated to the ( relatively stationary ) observer."

 

[matheinste]

So maybe I was wrong to think that there's anything contradictory about characterising the moving clock as the "time-dilated clock" which displays "non-dilated time", and the stationary clock the "non-time dilated clock" which displays "dilated time". Is that how you look at it? Because that's the only way, as far as I can see, to reconcile sources like Lerner, Petkov and Schröder (who contrast dilated time with proper time) with sources like the Wikipedia entry (which characterise the clock itself as being "time-dilated").

Add Freund to the "dilated time"-constrasts-with-"proper time" list:
http://books.google.co.uk/books?id=...nepage&q="proper time" "dilated time"&f=false

Googling for "dilated time" and "dilated clock" produces mainly forum discussions, and there are a few crank sources to weed out (I get that impression that "dilated clock" is a more informal term), but here's a reasonable-looking site that uses both terms:

http://www.geocities.com/syzygywjp/RelativeI.html

It's talking about general relativity, but still relevant, I think. It says, "A body orbting at this range would experience a time dilation of about 1.07 longer than non dilated time." And "According to calculations based on relativistic motion near a black hole, the time dilation nearest the black hole would amount to 1.414 times slower than a non time dilated clock."

So for this writer, dilated time is "longer" than non-dilated time. (But would you say they meant by this that "dilated time" has more or less ticks?) And a dilated clock, as for Wikipedia, is slow and therefore makes fewer ticks than a non-time-dilated clock.

With regards to the first point the wording seems over complicated and still confuses me. Clocks just show time

Regarding the extract from Freund I am not completely familiar with four vectors in that context, but earlier in the book his meaning of time dilation will be no different from others.

With regard to GR, of which I know little, time dilation is the same effect through a different mechanism whereby clocks at different gravitational potentials click at different rates. The use of the words longer in the quote is ambiguous.

Perhaps I can give examples, in my view, of faulty and correct reasoning with regard to the often used example of the muon's lifetime as an aid to illustrating time dialtion. These two methods lead to exactly the opposite outcome.

Let the lab frame be regarded as the stationary frame and the muon's frame the moving frame with repect to it. We can use the values of 2 microseconds 60 microseconds as being the figures used for the decay times of the muon measured by clocks in the muon and lab frame respectivley. Both explanations are non rigorous.

WRONG reasoning:- The muon's lifetime of 2 micoseconds its own frame is extended to 60 microseconds in the lab frame. This is an example of time dilation this shows that the number of seconds which the muon lives is dilated, made bigger, to 60 microseconds.

Now bear in mind the definition which says that a moving clock viewed from a stationary frame runs slow, and reason as follows.

CORRECT reasoning:- The muon has a lifetime as measured in its own frame, the time measured by a clock carried with it, its proper time, of 2 microseconds. This is an invariant and is the same for everyone, it cannot be changed. In the lab frame this is measured as 60 microseconds. This is an example of time dilation and shows that 2 microseconds in the muon's frame takes 60 microseconds to pass in the lab frame. So the time it takes 2 microseconds to pass in the moving frame is dilated to 60 microseconds as viewed from the stationary frame. That is, the preiod is extended.

Remember that although the lab frame measures 60 microseconds, the lab observers still agree that the muon's clock reads a proper, invariant time of 2 microseconds.

All other things having been said, look at it from this point of view. Time dilation is so fundamental to the theory that taking opposite stances at to what it means would lead to serious differences and contradictions between authors at later stages in these texts.

Matheinste.

 

[Rasalhague]

With regards to the first point the wording seems over complicated and still confuses me. Clocks just show time

Anyone who takes "dilated time" to mean an expanded total (of reduced units), and who also talks about moving clocks as being "time-dilated clocks", would presumably be thinking in these terms. I agree it sounds confusing; so confusing that I assumed at first that the author(s) of the Wikipedia article couldn't possibly have been in agreement with Adams, Freund, Lerner, Petkov and Schröder. But JesseM saw no contradiction, and the Relativity Calculator site did indeed combine these viewpoints. On the other hand, Dennis Anthony takes "dilated time" in the opposite sense, to refer to the shorter interval made up of expanded units. So presumably, for him, time dilation refers to a dilation of unit size. But the textbooks almost all seem to agree with Taylor & Wheeler's view of dilation as referring to a dilation of the total when measured in one frame as opposed to another, Lawden being a possible exception.

Regarding the extract from Freund I am not completely familiar with four vectors in that context, but earlier in the book his meaning of time dilation will be no different from others.

My point is just that Freund, like Adams, Lerner, Petkov and Schröder, takes "dilated time" to mean an expanded total (of reduced units). So for all of these authors, dilation seems to refer to the quantity of units, the total, rather than--as I thought you originally suggested--the size of individual units. If these authors had taken dilation to refer to the size of units, then surely they'd have used the label "dilated time" for the interval made up of a reduced quantity of these dilated units, wouldn't they?

WRONG reasoning:- The muon's lifetime of 2 micoseconds its own frame is extended to 60 microseconds in the lab frame. This is an example of time dilation this shows that the number of seconds which the muon lives is dilated, made bigger, to 60 microseconds.

Now bear in mind the definition which says that a moving clock viewed from a stationary frame runs slow, and reason as follows.

CORRECT reasoning:- The muon has a lifetime as measured in its own frame, the time measured by a clock carried with it, its proper time, of 2 microseconds. This is an invariant and is the same for everyone, it cannot be changed. In the lab frame this is measured as 60 microseconds. This is an example of time dilation and shows that 2 microseconds in the muon's frame takes 60 microseconds to pass in the lab frame. So the time it takes 2 microseconds to pass in the moving frame is dilated to 60 microseconds as viewed from the stationary frame. That is, the preiod is extended.

In both of these two ways of wording it, dilation refers to the process by which a bigger total, a bigger number of units, is derived from a smaller one. If we can say that "the time [...] is dilated" from 2 to 60 units, then it's the total that's being dilated, not the size of each unit relative to some other standard. The difference I see is that your first description is just shorter and less precise.

Remember that although the lab frame measures 60 microseconds, the lab observers still agree that the muon's clock reads a proper, invariant time of 2 microseconds.

Yes, what's changing is not the proper time itself, which is invariant, meaning the same in all inertial frames. Rather we're using the time dilation formula or, more generally, the Lorentz transformation, to transform the value of the proper time into the value of the coordinate time in some other frame. We're just changing which reference frame we're measuring coordinate time with respect to. We begin with an input of 2 * 10^-6 seconds, which is the time interval between two events on the muon's worldline, and hence the proper time between these events. (This is coordinate time measured in the muon's rest frame, this being the unique intertial frame where the proper time between these events coincides with the coordinate time.) Then we use the time dilation formula to dilate this value, i.e. make the number bigger, the resulting bigger number being the coordinate time between these events in a frame with respect to which they're not located in the same places as each other (at the same spatial coordinates). Is that a fair way of putting it?

All other things having been said, look at it from this point of view. Time dilation is so fundamental to the theory that taking opposite stances at to what it means would lead to serious differences and contradictions between authors at later stages in these texts.

It would certainly be helpful to get the terminology straight and for everyone to agree on what they meant by dilation or by "dilated time". This discussion shows how important it is for authors to be explicit about what they mean by dilation (what is getting bigger) to avoid misunderstandings, although in practice, if each writer is clear, precise, explicit and self-consistent, at least we stand a chance of understanding them, even if they differ from other authors in the labelling of some concepts. And it's quite possible for an author to describe concepts in later in other terms that don't depend on how exactly they understood the word dilation.

It'd be interesting to know where the term was first applied to this concept in relativity and whether the person who coined it was clear about which sense they had in mind.

 

[matheinste]

Originally Posted by matheinste
WRONG reasoning:- The muon's lifetime of 2 micoseconds its own frame is extended to 60 microseconds in the lab frame. This is an example of time dilation this shows that the number of seconds which the muon lives is dilated, made bigger, to 60 microseconds.

Now bear in mind the definition which says that a moving clock viewed from a stationary frame runs slow, and reason as follows.

CORRECT reasoning:- The muon has a lifetime as measured in its own frame, the time measured by a clock carried with it, its proper time, of 2 microseconds. This is an invariant and is the same for everyone, it cannot be changed. In the lab frame this is measured as 60 microseconds. This is an example of time dilation and shows that 2 microseconds in the muon's frame takes 60 microseconds to pass in the lab frame. So the time it takes 2 microseconds to pass in the moving frame is dilated to 60 microseconds as viewed from the stationary frame. That is, the preiod is extended.


Your reply.

In both of these two ways of wording it, dilation refers to the process by which a bigger total, a bigger number of units, is derived from a smaller one. If we can say that "the time [...] is dilated" from 2 to 60 units, then it's the total that's being dilated, not the size of each unit relative to some other standard. The difference I see is that your first description is just shorter and less precise.-------

The two descriptions are the opposite of each other so only one can be correct. In the first, incorrect example, the NUMBER of seconds enlarged. In the second, correct example, it is the PERIOD of each second that is enlarged.

I really cannot say any more to convince you than I already have so I only hope others can do so. I will keep in touch with the thread and if at a later date I can think of something else to add then I will do so.

Matheinste.

 

[Rasalhague]

The two descriptions are the opposite of each other so only one can be correct. In the first, incorrect example, the NUMBER of seconds enlarged. In the second, correct example, it is the PERIOD of each second that is enlarged.

I really cannot say any more to convince you than I already have so I only hope others can do so. I will keep in touch with the thread and if at a later date I can think of something else to add then I will do so.

Your first version: "the number of seconds which the muon lives is dilated, made bigger, to 60 microseconds"

Your second version: "the time it takes 2 microseconds to pass in the moving frame is dilated to 60 microseconds as viewed from the stationary frame. That is, the period is extended."

In both instances, you begin with a small number (a short period) and convert it into a bigger number (longer period) and call this process dilation. What else is the period if not some number of seconds? What else is the number of seconds if not how we represent that period? What do we gain by introducing a distinction between "period (as represented by a number)" and "number (that represents the period)" or lose by ignoring it? Both of these descriptions seem to me to match the way Freund, Lerner and the rest label the output of this conversion "dilated time". We convert a short interval of time to a longer one and call what we've done "dilation"; it's the same process whether the interval in question is one second or any number of seconds. Luckily, either way results in people taking time dilation to refer to the same operation.

On the other hand, we could informally visualise the two "times" as physical objects, like rubber tape measures, and identify one with the interval between events on the muon's worldine, as measured with respect to the muon's rest frame (i.e. the proper time between these events), and dilate that "time" by stretching this mental image of a rubber tape measure. We could say its seconds are bigger (the period of each of its seconds is enlarged, the muon's clock is ticking slow, it's been time-dilated) compared to our unstretched, undilated tape measure by which we represent the same period in the rest frame of the laboratory. This seems to be the conception behind some of the language used by Wikipedia and the Hyperphysics site. This is what I thought you had in mind when you said dilation referred to the expansion of each second (hence less seconds needed to cover a given period). Then, as Wikipedia says of clocks, we could say that this muon has been time-dilated. And if we thought in those terms, it might not seem unnatural to call the shorter period the "dilated time" (since it's represented by the mental image of an unstretched tape), as Dennis Anthony does. But this last step, at least, reverses the usual naming convention, as represented by Adams, Freund, Lerner, Schröder, Petkov and Taylor & Wheeler. So there we would have a real contrast that could lead to contradictory use of terminology.

As Grimble put it,

"If the time between 'ticks' increases, the clock runs slow.
If the number of 'ticks' increases, then, surely, the clock runs fast."

If we think of one of the first as dilation, we have one operation. If we think of the second as dilation, we have the inverse of that operation. So this is a difference with consequences.

 

[matheinste]

As Grimble put it,

"If the time between 'ticks' increases, the clock runs slow.
If the number of 'ticks' increases, then, surely, the clock runs fast."

If we think of one of the first as dilation, we have one operation. If we think of the second as dilation, we have the inverse of that operation. So this is a difference with consequences.

I will not repeat the more accurate desriptions but, putting it loosely, the interpretation that all authors agree upon, is, moving clocks run slow. Time between ticks increases. The number of elapsed seconds is not increased but the duration of seconds is increased. You cannot correctly interpret it both ways. There are not many more ways of putting it. That is time dilation. Any other interpretation is incorrect.

Matheinste.

 

[Rasalhague]

I will not repeat the more accurate desriptions but, putting it loosely, the interpretation that all authors agree upon, is, moving clocks run slow. Time between ticks increases. The number of elapsed seconds is not increased but the duration of seconds is increased. You cannot correctly interpret it both ways. There are not many more ways of putting it. That is time dilation. Any other interpretation is incorrect.

And how is duration expressed? By some number of elapsed seconds! We have a formula which we call the time dilation formula. In the form most often presented, it takes as its input a number of seconds. Its output is also a number of seconds. The output is bigger than the input. In that sense, the number has increased. Our calculation has increased it. If the input is the "time between ticks" (i.e. one elapsed second), we could say, as you do, that the time between ticks has increased. If the input is any other number of elapsed seconds, then that number of elapsed seconds (that duration) has increased.

I don't understand what distinction you're making between a number of seconds and a duration, particularly as either interpretation gives the same result.

$$\Delta t' = \Delta t \: cosh \theta = \gamma \: \Delta t$$

The "time lapse", as Taylor and Wheeler call it (i.e. a number of elapsed seconds), expressed by the output $$\Delta t'$$ is "more than" our input $$\Delta t$$ (also a number of elapsed seconds). Taylor and Wheeler say, "Such lengthening is called time dilation." That's pretty explicit. What has been lengthened? A time lapse, an interval of time, a duration, a number of seconds (all the same sort of thing, as far as I can see). Either way, almost all of the authors we've looked at call the outut of such a calculation "dilated time", in keeping with Taylor and Wheeler's definition.

On the other hand, as I've illustrated, there are ways of interpreting the expression "time dilation" that could lead someone to call the output of the inverse calculation "dilated time", and such interpretations do need to be distinguished from Taylor & Wheeler's. At least one of the authors I cited did see it that way, and perhaps Lawden would agree, as he presents the inverse formula under the name "time dilation". Others, such as Wikipedia, called the clock that runs slow a "time-dilated clock", without making clear which value they'd call "dilated time". At least one source spoke explicitly of dilated time as the value shown by a clock that isn't time dilated, and non-dilated time as the value shown by a clock that is time dilated, which--while not necessarily a contradiction, as JesseM points out--does strike me as a potentially confusing way to label things.

 

[Austin0]

=Rasalhague;2369858]Your first version: "the number of seconds which the muon lives is dilated, made bigger, to 60 microseconds"

Your second version: "the time it takes 2 microseconds to pass in the moving frame is dilated to 60 microseconds as viewed from the stationary frame. That is, the period is extended."

It seems to me that, in both these cases, a consistent interpretation would be that what is dilated is the duration of interval of the moving frames clock.
This is of course relative to the labs clock.
SO #1 the proper muon "second" is dilated from 1 lab second ==>30 lab seconds.
The expanded lab lifetime is a result of this but is not itself the object of the term dilation.

#2 Actually same as #1
Semantics is a mindfield :-)

 

[Rasalhague]

They seem to use "dilated" in the normal way in the second-to-last paragraph of the overview section of the time dilation article, where they write: "Thus, in special relativity, the time dilation effect is reciprocal: as observed from the point of view of either of two clocks which are in motion with respect to each other, it will be the other clock that is time dilated." From either clock's "point of view" (rest frame), when they it is "the other clock that is time dilated", presumably they mean that it takes longer to tick forward by a given amount (its seconds are longer).

Maybe it's not such a good idea to talk, as the Wikipedia article and some other online sources do, of clocks themselves as being time-dilated in special relativity when we really mean only that some interval of time is dilated when transformed from one frame to another. After all, the clocks in these thought experiments are physical objects that exist in all frames. That way we avoid having to contort our minds into remembering that a "time-dilated clock" shows "non-dilated time" and vice-versa. I suppose it's more natural though when the clock is a muon, say, and the only time interval it displays is its own lifespan.

 

[Rasalhague]

It seems to me that, in both these cases, a consistent interpretation would be that what is dilated is the duration of interval of the moving frames clock.
This is of course relative to the labs clock.
SO #1 the proper muon "second" is dilated from 1 lab second ==>30 lab seconds.
The expanded lab lifetime is a result of this but is not itself the object of the term dilation.

#2 Actually same as #1
Semantics is a mindfield :-)

The way I looked at it was that a second is just a special case of a duration, and for that reason#2 is the same as #1.

 

[matheinste]

The way I looked at it was that a second is just a special case of a duration, and for that reason#2 is the same as #1.

It can be semantically confusing and so we need some agreed definitions for our purposes. Let us define the length of time passed as a number of seconds. Let us define the basic period of time as one second, that is, the duration of time between two ticks of a clock. Let us also allow ourselves access to a master clock which ticks along unchagingly for all observers. This means it has a constant basic period for all observers. Of course such a clock does not exist in nature. Now for any clock the length of time passed and the basic period for the same amount of time on the master clock are reciprocal, if one increases the other decreases. However, and this is a possible area of confusion, there is a case where they can have the same value of 1. If we have a length of time passed of one second it is equal to the basic period of one second. So for a value of one second we have length of time passed equals one basic period of time. But if you dilate the basic period of time (relative to the master clock) to more than one second in length, you decrease the number of ticks, length of time passed, number of seconds, (relative to the master clock), to less than one.

Also bear in mind that a moving (in fact any) clock records its own proper time, number of ticks, and this is always less than the number of ticks recording the difference between the necesary two staionary clock readings, coordinate time, which are required to record the time in the stationary frame whcih it is moving relative to. And what do less ticks for the same time imply? A longer (dilated) basic period.

So dilation refers to the basic period being lenghened, made large, dilated, relative to another clock, and the length of time passed made a smaller number of the baisc periods, seconds, relative to the same other clock.

Matheinste.

 

[Rasalhague]

It can be semantically confusing and so we need some agreed definitions for our purposes. Let us define the length of time passed as a number of seconds. Let us define the basic period of time as one second, that is, the duration of time between two ticks of a clock. Let us also allow ourselves access to a master clock which ticks along unchagingly for all observers. This means it has a constant basic period for all observers. Of course such a clock does not exist in nature.

Shouldn't be a problem. For our purposes, we can arbitrarily define some clock as our master clock. It may not be visible to all observers on all occasions, but if they can't see it, they can calculate what it would say, so we can call it a master clock for the sake of definitions.

Now for any clock the length of time passed and the basic period for the same amount of time on the master clock are reciprocal, if one increases the other decreases.

Right, so if there are two interpretations of what dilation refers to--(1) the total number of units, (2) the size of each unit--the difference between them is a matter of reciprocity, and the dilation factors in the equations expressing each definition should be reciprocal.

$$(1) \; \Delta t'_{1} = \gamma \Delta t_{1}$$
$$(2) \; \Delta t'_{2} = \gamma^{-1} \Delta t_{2}$$

Here I use prime symbols simply to indicate output, what we might call the result of time dilation, "dilated time", depending on how we understand the word. In each case, we select some clock as our arbitrary standard, "master clock" if you like. And this is indeed what we find, with most authors calling (1) time dilation (and its output "dilated time"), but with a few authors, Lawden and Anthony, thinking of (2) as time dilation (and its output "dilated time"). A significant difference in terminology!

In (1), the formula converts a given time interval recorded by our arbitrarily selected "master clock" (the invariant interval between two events on our master clock's worldline) into the corresponding coordinate time in some frame where the master clock is moving. That's to say: the result is the invariant proper time interval between another pair of events, E_1 and E_2, the first of which is simultaneous in the frame where our master clock is moving with the beginning of our input interval, while E_2 is simultaneous in that same frame with the end of our input interval. We dilate the total number of seconds (and contract the size of our seconds) relative to our arbitrarily chosen standard.

Equation (2) converts the proper time between one pair of events on our master clock's worldline into the proper time between another pair of events, E_1 and E_2, on the worldline of a clock at rest in a frame where our master clock is moving, E_1 being simultaneous in the master clock's rest frame with the beginning of our input interval, and E_2 being simultaneous in the master clock's rest frame with the end of our input interval. Equivalently, it converts a coordinate time between E_1 and E_2 into the invariant proper time interval between them. We dilate the size of our seconds (and contract the total number of them) relative to our arbitrarily chosen standard.

However, and this is a possible area of confusion, there is a case where they can have the same value of 1. If we have a length of time passed of one second it is equal to the basic period of one second. So for a value of one second we have length of time passed equals one basic period of time. But if you dilate the basic period of time (relative to the master clock) to more than one second in length, you decrease the number of ticks, length of time passed, number of seconds, (relative to the master clock), to less than one.

But if you think of dilation as referring to the total number of seconds and dilate one second, the result is $$\gamma$$ seconds. If you think of dilation as referring to unit size and dilate that, thus reducing the total, the result is $$\gamma^{-1}$$ seconds. Not the same thing at all. This applies no matter whether the length of time of the input is one unit or any multiple of one unit. Nor does the effect depend on what units we use. Only when our input is zero does the effect vanish (unless there's some spatial component too, in which case we need the full Lorentz transformation).

Also bear in mind that a moving (in fact any) clock records its own proper time, number of ticks, and this is always less than the number of ticks recording the difference between the necesary two staionary clock readings, coordinate time, which are required to record the time in the stationary frame whcih it is moving relative to. And what do less ticks for the same time imply? A longer (dilated) basic period.

Since there is no absolute master clock in nature, we have to specify what arbitrary standard we're using to define "less than" or "more than". See above. If we've converted a smaller total of ticks into a larger total of ticks, we've dilated our number of ticks (and contracted the size of our seconds). If we've converted a larger number of ticks into a smaller number of ticks, we've dilated the size of our seconds (and contracted the total of them).

So dilation refers to the basic period being lenghened, made large, dilated, relative to another clock, and the length of time passed made a smaller number of the baisc periods, seconds, relative to the same other clock.

Not if, like Adams, Freund, Lerner, Petkov, Schröder and Taylor & Wheeler, we call the following operation time dilation and refer to $$\Delta t'_{1}$$ as "dilated time":

$$(1) \; \Delta t'_{1} = \gamma \Delta t_{1}$$

But yes if, like Anthony and Lawden, we call the following operation time dilation and refer to $$\Delta t'_{2}$$ as "dilated time":

$$(2) \; \Delta t'_{2} = \gamma^{-1} \Delta t_{2}$$

 

[matheinste]

But if you think of dilation as referring to the total number of seconds and dilate one second, the result is $$\gamma$$ seconds. If you think of dilation as referring to unit size and dilate that, thus reducing the total, the result is $$\gamma^{-1}$$ seconds. Not the same thing at all. This applies no matter whether the length of time of the input is one unit or any multiple of one unit. Nor does the effect depend on what units we use. Only when our input is zero does the effect vanish (unless there's some spatial component too, in which case we need the full Lorentz transformation).



Since there is no absolute master clock in nature, we have to specify what arbitrary standard we're using to define "less than" or "more than". See above. If we've converted a smaller total of ticks into a larger total of ticks, we've dilated our number of ticks (and contracted the size of our seconds). If we've converted a larger number of ticks into a smaller number of ticks, we've dilated the size of our seconds (and contracted the total of them).



Not if, like Adams, Freund, Lerner, Petkov, Schröder and Taylor & Wheeler, we call the following operation time dilation and refer to $$\Delta t'_{1}$$ as "dilated time":

$$(1) \; \Delta t'_{1} = \gamma \Delta t_{1}$$

But yes if, like Anthony and Lawden, we call the following operation time dilation and refer to $$\Delta t'_{2}$$ as "dilated time":

$$(2) \; \Delta t'_{2} = \gamma^{-1} \Delta t_{2}$$

Just a very quick reply. I will try to check the quouted references if I have them, but it seems to me its fairly likely to be the case of some authors using different conventions as to whether they designate staionary frame as the as primed or unprimed. All authors will use the term dilation to mean the same but may have different conventions for illustrating it.

I'll get back later.

Matheinste

 

[matheinste]

Hello Rassalhague,

A semi apology. I have only checked Petkov because of limited time. This seems to be what is going on in that text, but I also would like someone to take a closer look at Petkov as I find him hard to follow.

My definition has always been the generally accepted version, moving clocks run slow. In other words the period of a moving clock appears dilated. Petkov, and perhaps others say that a moving observer sees the stationary observers measurement of length of time passed (coordinate time) to be longer, a greater number of seconds. So, yes, there is a differing usage of the term. BUT as everything is reciprocal, the physical effect described is exactly the same as I hope the following shows.

My assumed normal wording :-A staionary observer sees a moving clock running slow compared to his own. We call this time dilation.

Petkov’s, and perhaps others translates to:- A stationary observer will see his own clock running fast as compared to a moving clock. We can also call this time dilation.

Matheinste.

 

[Rasalhague]

My assumed normal wording :-A staionary observer sees a moving clock running slow compared to his own. We call this time dilation.

I see what you mean. This popular, informal expression does suggest that it's unit size that's dilated. Likewise when Lawden presents equation (2) as "time dilation", this implies that he thinks of time dilation as a dilation of units. Anthony explicitly calls the output of equation (2) dilated time, so he clearly takes it to be the units that are dilated. But then Adams, Freund, Lerner, Petkov, Schröder, Taylor & Wheeler, Tipler & Mosca all take it in the opposite sense: as being a total that gets bigger. For them it's the output of equation (1) that's "dilated time". Also, in my experience, it's most often equation (1) that's called "the time dilation equation" which gives us a dilated total rather than the reduced total that would result from a dilation of units.

Given this confusion, it might be helpful for anyone writing an introductory text on relativity to be explicit about what exactly is being dilated in their view--what they think of as getting bigger: the size of each unit in a "standard" interval, or the total number of "standard" units (the standard being arbitrary, of course)--and probably worth a footnote explaining that there is this difference of usage among writers on the subject. I wonder where the term originated and what the person who first used it meant.

Just a very quick reply. I will try to check the quouted references if I have them, but it seems to me its fairly likely to be the case of some authors using different conventions as to whether they designate staionary frame as the as primed or unprimed. All authors will use the term dilation to mean the same but may have different conventions for illustrating it.

As I said, I used primed t just to denote output, unprimed t to denote input. Authors do differ in how they use the prime symbol, which is why I made sure to state which convention I used on this occasion. My intention was to present the equations in a way that would make the only difference between them depend on how the word dilation is understood and not on their choice of symbols. In that way, we can see what effect that particular difference in viewpoint would have.

 

[matheinste]

Hello again.

As both usages give the same physical effect, and a careful reading will always conclude this, it may not be so important after all. If someone says time dilation we know what the physical effect will be irrespective of how it is explained. It always boils down to the fact that, for inertial observers, proper time is less than or equal to coordinate time.

Matheinste.

Matheinste.

 

[Rasalhague]

Hello Rassalhague,

A semi apology.

Half thanks ;-)

My definition has always been the generally accepted version, moving clocks run slow.

You're right, it's is a frequently used, informal, verbal definition. That said, I also think there's a strong tendency to define it the other way around when it comes to writing an equation. The majority of writers call $$\Delta t' = \gamma \: \Delta t$$ the time dilation equation, making $$\Delta t' = \gamma^{-1} \: \Delta t$$ the inverse of time dilation.

In other words the period of a moving clock appears dilated. Petkov, and perhaps others say that a moving observer sees the stationary observers measurement of length of time passed (coordinate time) to be longer, a greater number of seconds. So, yes, there is a differing usage of the term. BUT as everything is reciprocal, the physical effect described is exactly the same as I hope the following shows.

My assumed normal wording :-A staionary observer sees a moving clock running slow compared to his own. We call this time dilation.

Petkov’s, and perhaps others translates to:- A stationary observer will see his own clock running fast as compared to a moving clock. We can also call this time dilation.

I think that's a good, clear verbal summary of the difference. It all depends on which value we take as the standard. No more mystery to it than that. Since our choice is arbitrary, we can't appeal to a universal standard, but having stated our choice of a standard (implicit in expressions like "running fast/slow compared to..."), surely the word dilation must be defined in some unique, consistent way relative to that standard.

Suppose one doctor reported "pupil dilation" because a patient's pupils were dilated in comparison to normal pupil size, while another doctor reported "pupil dilation" because normal sized pupils were dilated in comparison to a patient's pupils! In this analogy "normal sized pupils" represents our arbitrarily chosen standard. Or suppose I demand twice as much money as some stated standard of comparison, and someone gives me half as much. Will I be impressed by the argument that what I wanted was "more than" what I got, and so effectively this was a still a "money dilation" situation because the relations "twice as much" and "half as much" are reciprocal?

 

[Rasalhague]

As both usages give the same physical effect, and a careful reading will always conclude this, it may not be so important after all. If someone says time dilation we know what the physical effect will be irrespective of how it is explained. It always boils down to the fact that, for inertial observers, proper time is less than or equal to coordinate time.

If we want a neutral term, maybe we could innovate and say "time distortion", the result of which could be bigger or smaller, and so it wouldn't matter how we verbally defined bigger or smaller. But "time dilation" is an operation which claims to make something bigger. If we can't make up our minds which of two reciprocal quantities is being made bigger than our input (our standard), then two people could be desribing opposite things as "dilated time", or what one calls "dilated time" the other might be thinking of as "reduced/contracted/shrunk time". So it's not entirely without consequence.

Obviously, in practice, people get by somehow though, and if all the aspects of a particular problem are carefully defined, and we've correctly understood the question and how to use the Lorentz transformation, then the meaning of "dilation" is not going to affect the result. But I think this discussion has been helpful in showing a source of potential confusion and suggesting issues to be mindful of when teaching these ideas.

 

[Rasalhague]

As both usages give the same physical effect, and a careful reading will always conclude this, it may not be so important after all.

The danger is that unless we settle on one definition, we'd have two distinct functions each called "time dilation". They may describe the same physical effect, but they don't give the same result because they don't describe the same relationship: rather one is the inverse of the other. What one dilates, the other contracts. Obviously this is an important difference.

 

[matheinste]

The danger is that unless we settle on one definition, we'd have two distinct functions each called "time dilation". They may describe the same physical effect, but they don't give the same result because they don't describe the same relationship: rather one is the inverse of the other. What one dilates, the other contracts. Obviously this is an important difference.

Dilation, no matter who uses it, always means making larger. Time dilation in all the references we have quoted always means the same thing. The resting observer sees the other clock running slow. Some refernces translate this to "the moving observer sees his proper time projected onto the resting frame coordinates as being increased, running fast, more seconds passed", it still means the same, it just expresses it differently. Moving clocks run slow, very loosely, says all that needs to be said.

Matheinste.

 

[Rasalhague]

Dilation, no matter who uses it, always means making larger. Time dilation in all the references we have quoted always means the same thing.

For Adams, Freund, Lerner, Petkov, Schröder, Weinert, Taylor/Wheeler and Tipler/Mosca, it means dilating the total number of units. For you, Lawden and Anthony it means dilating the size of each unit and thus contracting the total number of units. It's as if doctors had one term "pupil dilation", but some doctors used it to mean that a patient's pupils are bigger than normal, while others used it to mean that normal is bigger than a patient's pupils (i.e. the opposite of how their colleagues understand the term), claiming that it makes no difference because in either case dilation refers to something being larger!

The resting observer sees the other clock running slow. Some refernces translate this to "the moving observer sees his proper time projected onto the resting frame coordinates as being increased, running fast, more seconds passed", it still means the same, it just expresses it differently.

Since "observer" and "clock" are each resting in some frame and each moving in another (observer being a colloquial short-hand in this context for "intertial reference frame", and any inertial reference frame being populated by its own notional clocks), we need to pick some standard to say what is being compared to what. If we have a particular example, the standard is chosen for us by the details of the example, by which time interval we're given and which we need to calculate. In the most general case, what is there to break the symmetry and escape reasoning round in circles? The only thing I can think of here is that we have our input, the data we know, as our standard, then the formula gives us some output expressed in terms of that standard. This notion of input and output introduces a natural way of ordering the pair of time intervals, and this allows us to talk meaningfully about which is to be made bigger by which operation. Since we're calling this action time dilation, something relating to time must have been made bigger by it. If $$\Delta t' = \gamma \; \Delta t$$ is time dilation, then a total is dilated, as we're told by Adams, Freund, Lerner, Petkov, Schröder, Weinert, Taylor/Wheeler and Tipler/Mosca. But if, like Lawden, we use a similar expression--"a moving clock [...] will appear [...] to have its rate reduced--to describe the inverse formula, and call this time dilation, then presumably it's the size of each unit that's been dilated, because it isn't the total: that's got smaller.

Moving clocks run slow, very loosely, says all that needs to be said.

But every clock at rest in some inertial frame is moving in another, the physical situation being perfectly symmetrical. So if dilation is to have any meaning at all, there must be some convention as to what it refers to. Otherwise, why not call it time distortion and save awkward questions. Sure, we could dodge the question by switching our definition of what dilation refers to whenever we want to change from using one of these reciprocal expressions to the other, so as to disguise the fact that they're reciprocal and not identical, but that's hardly a recipe for clarity.

 

[matheinste]

Its not that complicated.

ANY inertial observer will reckon that ANY clock moving inertially with respect to him is running slow. This may be expressed in other ways but, however it is expresed, it means this and the fact that it does mean this can be inferred from the given scenario. That's all there is to it.

So with regards to our overall discussion, yes, different authors do seem to illustrate time dilation diferently, but however they do it, they are describing the same effect. I can explain in more detail but for me it is long and winding road to express the two views completely unambiguously because although they are in a way reciprocal the use of that word in this context can cause more confusion.

Matheinste.

 

[Rasalhague]

Its not that complicated.

ANY inertial observer will reckon that ANY clock moving inertially with respect to him is running slow. This may be expressed in other ways but, however it is expresed, it means this and the fact that it does mean this can be inferred from the given scenario. That's all there is to it.

Sure, it's not complicated when we leave the dilation out of it. But wasn't this a discussion about the complications relating to people's varying uses of that word?

So with regards to our overall discussion, yes, different authors do seem to illustrate time dilation diferently, but however they do it, they are describing the same effect. I can explain in more detail but for me it is long and winding road to express the two views completely unambiguously because although they are in a way reciprocal the use of that word in this context can cause more confusion.

There are two functions, each the inverse of the other, and some people call one of them time dilation, and some call the other time dilation. What one function dilates, the other contracts, so they don't both dilate time in the same sense. Take my "pupil dilation" analogy. You could argue that each set of doctors is "describing the same effect", but they're using the word dilation in opposite ways to describe that effect; one doctor's dilation is another's contraction: not a healthy situation!

 

[Grimble]

Its not that complicated.

ANY inertial observer will reckon that ANY clock moving inertially with respect to him is running slow. This may be expressed in other ways but, however it is expresed, it means this and the fact that it does mean this can be inferred from the given scenario. That's all there is to it.

Matheinste.

It is all very well to make a statement like that, but if the number of units can in one view increase and in the other decrease, in one sense the moving clock reads more time has passed and in the other that less time has passed, for do we not reckon time by the number of units of time passing rather than by the size of them?

So does the clock slow because each tick takes longer?

Another interesting fact is that however one measures it the total duration of whatever we are measuring is the same, moving or not.
The number of seconds multiplied by the length of one second gives the same total whether it is proper time or co-ordinate time. The difference is that the unit of measurement changes: take the muon experiment referred to earlier where we have 2.2 microseconds proper time and approximately 65 microseconds co-ordinate time and the conversion is made applying the Lorentz factor which was 29.4

So in which way is it slowing?

Grimble.

 

[matheinste]

Sure, it's not complicated when we leave the dilation out of it. But wasn't this a discussion about the complications relating to people's varying uses of that word?



There are two functions, each the inverse of the other, and some people call one of them time dilation, and some call the other time dilation. What one function dilates, the other contracts, so they don't both dilate time in the same sense. Take my "pupil dilation" analogy. You could argue that each set of doctors is "describing the same effect", but they're using the word dilation in opposite ways to describe that effect; one doctor's dilation is another's contraction: not a healthy situation!

Its no complicated even with time dilation.

Longer period and more ticks are reciprocal. But they are obtained by using reciprocal scenarios. They "cancel out" to produce the same effect. There is only ONE effect. We are using TWO, opposite ways of describing the same effect.

In one scenario the "STATIONARY" observer is saying "according to MY reckoning the other guy's clock is showing LESS ticks than mine and so his basic period is LONGER". In the other scenario the "MOVING" observer is saying " the other guy is saying that according to HIS reckoning my clock is showing less ticks than his and so according to HIS reckoning his clock must be showing MORE ticks than mine and so according to HIS reckoning his clock's basic period is SHORTER" . They are both correct. They are both describing the same effect. Compared to stationary observer the moving observer's clock is running slow.

Notice that in the second scenario the moving guy is saying, not what he himself sees, but what the stationary observer sees and making a correct inference from this. This is because, in fact, the stationary observer does see his own clock as running faster than the other.

Ita bit convoluted but that seems to be how it works out.

I much prefer the much simpler description based on simulatnaous clock readings.

Matheinste

 

[Rasalhague]

In one scenario the "STATIONARY" observer is saying "according to MY reckoning the other guy's clock is showing LESS ticks than mine and so his basic period is LONGER". In the other scenario the "MOVING" observer is saying " the other guy is saying that according to HIS reckoning my clock is showing less ticks than his and so according to HIS reckoning his clock must be showing MORE ticks than mine and so according to HIS reckoning his clock's basic period is SHORTER" . They are both correct. They are both describing the same effect. Compared to stationary observer the moving observer's clock is running slow.

It's like saying "more than" means the same as "less than" because if x < y then y > x. It's true that these particular inequalities say the same thing--and that we could get by perfectly well with one or the other sign on all occasions--but that doesn't mean that we can disregard the difference between "more than" and "less than" because they're "opposite ways of describing the same effect". No matter how we look at it, x < y is the opposite of x > y.

Here what you've done is to apply the two mutually inconsistent definitions of time dilation to the same situation, and to make this work, you've had to switch your standard of comparison when you start talking from the other perspective. But if we apply a consistent standard of comparison, the only way to make it work is to also use a single, consistent defnition of time dilation. Much less fraught!

When I compared x < y and x > y, all I changed was the inequality. I did this to demonstrate the effect of changing the inequality. If I'd changed the inequality and simultaneously switched variables, then all I'd have done would have been to disguise the genuine difference between "more than" and "less than". That's why, in post #52, I referred each definition to the same arbitrarily chosen standard.

 

[Rasalhague]

It's as if someone were to argue that inflation and deflation both "describe the same effect" since the inflation of one currency is equivalent to the deflation of another. But see what we did there? We can only argue that they're equivalent by changing which currency we refer to whenever we switch from calling the phenomenon inflation to deflation or vice-versa. If we consistently refer to one currency, and consistently define inflation as rising prices, there's no way we can claim that inflation means the same as deflation (with respect to that same currency).

 

[matheinste]

It's as if someone were to argue that inflation and deflation both "describe the same effect" since the inflation of one currency is equivalent to the deflation of another. But see what we did there? We can only argue that they're equivalent by changing which currency we refer to whenever we switch from calling the phenomenon inflation to deflation or vice-versa. If we consistently refer to one currency, and consistently define inflation as rising prices, there's no way we can claim that inflation means the same as deflation (with respect to that same currency).

No it is not. You miss the point. Both scenarios, illustrations, explanations do describe the same effect, but they describe the same phenomena whereby "moving" clocks appear to be running slow compared to a "stationary" observers clock as observed by the "stationary" observer. One says it explicitly that way, the other says "moving" observers reckon "stationary" observers will see the "stationary" clock running fast compared to their, the "moving" observers, own clock. The first uses extended basic time periods the other uses the extended passage of time or more ticks as part of their explnation. The outcome is the same and the rest of SR which is greatly dependent on the phenomena is, obviously, unaffected by which is used otherwise there would be great disagreement at the later stages of teaching the theory.

Its a shame to end our discusiion without agreement and it is probably as frustrating for you as it is for me. Among the many possible problems the two most likely ones are that I lack the verbal skills to get my point across or I am incorrect in my interpretations. In either case, mea culpa.

Its been an interesting exercise and I have learned from it.

Matheinste.

 

[Rasalhague]

No it is not. You miss the point. Both scenarios, illustrations, explanations do describe the same effect, but they describe the same phenomena whereby "moving" clocks appear to be running slow compared to a "stationary" observers clock as observed by the "stationary" observer. One says it explicitly that way, the other says "moving" observers reckon "stationary" observers will see the "stationary" clock running fast compared to their, the "moving" observers, own clock.

My point is that in order to make the two conflicting definitions correctly describe the same situation, you've had to switch your standard of comparison. In my analogy, we define inflation to mean that one can buy less given a fixed total of money (prices get bigger). Of course, this also means that a fixed total of the affected currency can be bought for less of another currency (price gets smaller). But then we'd have redefined our original currency as a comodity, and taken a different currency as our standard against which to compare it. To see what difference the different definitions make, we need to apply them to the same standard.

So we might say that the inflation of one currency is equiavalent to the deflation of another, but that doesn't make inflation the same thing as deflation. Yes, your two verbal summaries of time "distortion" describe the same situtation. Whether both of these verbal summaries are characterised as time dilation or time contraction depends on whether we're referring to units or total. But as soon as we specify what kind of value is already known and what kind of value, relative to that, we want to calculate from it (a bigger one or a smaller one), then the difference in terminology becomes apparent, because then the definition shared by Adams, Freund, Lerner, Petkov, Schröder, Taylor & Wheeler, Tipler & Mosca leads to a dilated value being called "dilated time" (the dilation being explicit, a dilation of the quantity given), while the definition shared by Lawden and Anthony leads to a contracted value being called "dilated time" (the dilation being implict, a dilation of units).

Luckily, so long as the relevant details are known, and the Lorentz transformation and how to apply it are understood, there shouldn't be any disagreement over results. So in that sense it's not catastrophic if people take dilation to refer to different aspects of the same situation. The main problem that I see with such conflicting definitions is that it can be distracting for people trying to learn the subject.

Its been an interesting exercise and I have learned from it.

Me too! Thanks for your patience.

 

[Grimble]

Thank you Gentlemen for an illuminating discussion, but what are your answers to the original subject of this thread; "Which is the correct formula for time Dilation?"

Not if, like Adams, Freund, Lerner, Petkov, Schröder and Taylor & Wheeler, we call the following operation time dilation and refer to $$\Delta t'_{1}$$ as "dilated time":

$$(1) \; \Delta t'_{1} = \gamma \Delta t_{1}$$

But yes if, like Anthony and Lawden, we call the following operation time dilation and refer to $$\Delta t'_{2}$$ as "dilated time":

$$(2) \; \Delta t'_{2} = \gamma^{-1} \Delta t_{2}$$

This seems to be the nearest you have come to answering, stating that there are differing opinions; but as those opinions are in direct conflict where does this leave one?

And as Einstein derives his formula, http://www.bartleby.com/173/12.html" , in this passage:

“Let us now consider a seconds-clock which is permanently situated at the origin (x' = 0) of K'. t' = 0 and t' = 1 are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:
t = 0
and

which is $$t = \frac {t^'}{\sqrt{1 - \frac{v^2}{c^2}}}$$ with ${t^'}$ set to 1

Which gives us $t = \gamma {t^'}$ as Einstein's formula

Or option (2) in the quote above...

So would I be right in deducing that (2) is, in fact, the correct formula??

Grimble