Benefits of time dilation / length contraction pairing?

[Rasalhague]

How is "moment" different from "surface of simultaneity"

They're synonymous as far as I know. That's how I intended them, anyway.

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.

Okay, perhaps I should have use the more general, more explicit forms of the equations with delta symbols, rather than incorporating an event at the mutual origin of the two frames into the definition (clock's being synchronised as they pass; rulers aligned as their zero ends pass).

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.

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)."

I'm a bit confused by this switching. When you use "observer's frame" and "clock frame" now are you going by your original first definition, or should I take them to have the second meaning sometimes, depending on the problem to be solved?

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? I thought this is just what you advised me the convention was with the terms unprimed and primed. 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 thought they might be handy terms to distinguish between kinds of frames when talking in the abstract about the kinds of questions that can be asked of the formulas, but I agree it would be impractical to switch back and forth in the middle of working on a complex problem. 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".)

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.

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? How would you express the problem in conventional terms? Would you just swap the names of the frames? 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.

Does your observer's frame equate with Doc Al's lab frame, and your clock frame equate with Doc Al's moving frame? 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 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).

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, seems like someone holding up a whole apple and the core of an eaten pear and saying, "Look, apples are a whole fruit, but pears are eaten." Of course, being whole is no more a defining feature of apples than being eaten is a defining feature of pears; they aren't a whole kind of fruit and an eaten kind; either can be whole or eaten. Okay, that's an absurd analogy: no one's going to think of fruit that way. 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). A different pairing (a different convention), one that compared like with like as the full Lorentz transformation does, might avert that problem and make a better mnemonic. There would be no loss in calculating convenience, since we could carry on - as now - inverting either equation as required.

What question did you give to Wolfram Alpha, exactly?

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.

 

[Rasalhague]

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".

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? 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? Would you say that your lab frame and moving frame are equivalent to Jesse's terms "observer's frame" and "clock frame" respectively? If not, how do they differ?

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]

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?

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.

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?

"Time value" is too vague a term.

Would you say that your lab frame and moving frame are equivalent to Jesse's terms "observer's frame" and "clock frame" respectively?

Absolutely. Calling the frame of the moving clock the "clock frame" makes it kind of easy to remember, doesn't it?

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?

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".

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).

 

[Rasalhague]

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.

I found input and output helpful in digging myself out of some of the confusion I was in. If they're no use to anyone else, that's fine. They may not be convenient labels to give to a pair of frames in practice if we want to calculate back and forth between them, but at least they allow us to name that concept without going round in circles. I'm not saying there aren't better labels to use to distinguish frames on the basis of some other feature.

In the example we looked at with Jill in her rocket (carrying a clock) and Jack outside the rocket, also carrying a clock, and moving relative to Jill, we could label either of these as lab frame and moving frame, and when we do so, presumably we need to specify who or what they are the lab frame / moving frame of? I get the impression that by lab frame you mean the same as rest frame, is that right? Is Jill's lab frame synonymous with Jill's rest frame, and would the latter a more precise and self-explanitory term (given that a particular example might happen to involve an actual, physical laboratory at rest in some observer's "moving frame", in which case the term lab frame would become rather confusing)?

"Time value" is too vague a term.

What would you suggest? I left it vague because I didn't know whether your definition was precisely this inexact in its requirement. By time value I meant precisely: whatever the time values involved represent. But perhaps your definition of lab frame depends on some more specific kind of time value being 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?

I suppose it will be once I've worked out what exactly it is that it's making so easy to remember ;-)

By "the frame of the moving clock" do you mean "the rest frame of a clock which is moving in the rest frame of some other specified object or person, their rest frame being labelled the observer's frame of that object or person (or synonymously their lab frame)". Or do you mean it the other way around: a frame in which the clock that it's named after is moving, i.e. not that clock's rest frame. Maybe it would help if you could apply these labels to an example such as the one with Jack and Jill that we're already familiar with.

I imagine this label could get very confusing if there are clocks explicity visualised in both frames, or if the visualisation only involves one explicit clock altogether and that clock is at rest in the lab frame, or is it always possible to chose the terms so that this doesn't happen?

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".

Ah, so here's yet another different usage of primed and unprimed frames to add to my list. So, in this system, we'd begin by making an arbitrary choice such as "let the lab frame in this example denote Jill's rest frame", and then stick to talking about "the lab frame", with this definition implicit, rather than talking variously about Jill's lab frame and Jack's lab frame?

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).

All time intervals represent the possible proper time of some notional clock though, don't they, in the sense that you could always imagine a clock following a certain trajectory whose proper time would be equal to that time interval - or is that the wrong way to look at it? Be that as it may, yes, I agree with what you say, and we have formulas to calculate from this proper time its time component in some other frame, which will be bigger than the proper time, and likewise from proper distance to the spatial component of some spacelike separation.

Why then is this operation, time dilation, conventionally paired not with its spatial equivalent, but with the inverse operation for space? Why not compare like with like?

To extend the metaphor I introduced in my previous post, it's as if someone was trying to teach people who've never seen apples and pears before about the nature of these fruit by holding up a whole apple and a mostly eaten pear and saying, "Today we're going to learn about the wholeness of apples and the eatenness of pears." Sure, this apple is whole, and this pear is eaten, but being whole or eaten isn't among the properties which distinguish apples from pairs, so as an introduction to apples and pairs it introduces an added, arbitrary complication which obscures both their similarity and the real differences between them.

I grant it's perfectly possible and conventional to avoid such a conceptualisation, but we match up time dilation with length dilation in the full Lorentz transformation, so why not match up with length contraction a time example of the sort we looked at conceptualised as time contraction, for aesthetic, pedagogic and mnemonic purposes?

As will be obvious, I'm new to this subject, and may well be missing something. I thank you all for your efforts at explaining this to me.

 

[JesseM]

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)."

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.

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?

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.

I thought this is just what you advised me the convention was with the terms unprimed and primed.

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.

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.

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?

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".)

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".

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?

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.

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.

In a problem with multiple objects you could again just denote different frames based on which object's rest frame they were.

Does your observer's frame equate with Doc Al's lab frame, and your clock frame equate with Doc Al's moving frame?

Yes.

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"?

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.

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

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.

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).

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.

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.

OK, but I don't understand what you meant when you talked about an "inconsistency" here:

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.

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.

 

[atyy]

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.

There's a cute story about this in Wilczek's Nobel lecture:
"My friend and mentor Sam Treiman liked to relate his experience of how, during World War II, the U.S. Army responded to the challenge of training a large number of radio engineers starting with very different levels of preparation, ranging down to near zero. They designed a crash course for it, which Sam took. In the training manual, the first chapter was devoted to Ohm’s three laws. Ohm’s first law is V = IR. Ohm’s second law is I = V/R. I’ll leave it to you to reconstruct Ohm’s third law.
Similarly, as a companion to Einstein’s famous equation E = mc² we have his second law, m = E/c²."
http://nobelprize.org/nobel_prizes/physics/laureates/2004/wilczek-lecture.pdf

 

[neopolitan]

https://www.physicsforums.com/showpost.php?p=2231347&postcount=381" was my last post where I asked if I should move on, I'm assuming a yes.

We've discussed a scenario in which a photon passes B then A, where A and B have a relative separation speed of v. When A and B were colocated, t=0 and t'=0, ie this event serves as the origin of the t axis for both A and B.

Once a photon reaches A, A can work out when (t'_oa B and that photon were colocated, in the A frame, and the separation between where the photon was at t=0 and where (x'_oa) B was when B and the photon were colocated, in the A frame.

The values A has are:

time of colocation of A and B, t=0
time of colocation of A and photon, t = t_a
speed of B towards where the photon originated, v
location of photon at t=0, x_a = c.t_a

Therefore, we have (all in the A frame):

(location of colocation of B and photon) = (location of photon at t=0) - (speed of B) * (time of colocation of B and photon) = (speed of light) * (time of colocation of B and photon)

x'_oa = x_a - v.t'_oa = c.t'_oa

or

c.t_a - v.t'_oa = c.t'_oa

so

t_a = t'_oa + v.t'_oa / c . . . . . . (1)

We can follow the same procedure for B to reach (all in the B frame):

(time of colocation of photon with B) = (time of colocation of A and photon) - (speed of A) * (time at which A and photon will be colocated) / (speed of light)

t'_b = t_ob - v.t_ob / c . . . . . . (2)

We've already concluded that since

t_a = time interval between Event and when a photon from the Event reaches A, in A's frame

and

t'_b = time interval between Event and when a photon from the Event reaches B, in B's frame

we have no expectation that t_a = t'_b

We can now test the hypothesis that:

(time of colocation of B and photon in the A frame) = (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (time of colocation of A and photon in the A frame)

That would mean:

t'_oa = t'_b . . . . . . (3)

and

t_ob = t_a . . . . . . (4)

Substituting (3) into (1):

t_a = t'_b + v.t'_b / c = t'_b ( 1 + v / c ) . . . . . . (5)

Substituting (4) into (2):

t'_b = t_a - v.t_a / c = t_a (1 - v / c ) . . . . . . (6)

Substituting (6) into (5):

t_a = t_a (1 - v / c ) (1 + v / c)


So (3) and (4) are not valid. This indicates that:

(time of colocation of B and photon in the A frame) does not = (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the A frame) does not = (time of colocation of A and photon in the B frame).

If we make an alternative hypothesis that:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

That would mean:

t'_oa = G.t'_b . . . . . . (7)

and

t_ob = G.t_a . . . . . . (8)

Substituting (7) into (1):

t_a = G.t'_b + v.G.t'_b / c = G.t'_b ( 1 + v / c ) . . . . . . (9)

Substituting (4) into (2):

t'_b = G.t_a - v.G.t_a / c = G.t_a (1 - v / c ) . . . . . . (10)

Substituting (10) into (9):

t_a = G.G.t_a (1 - v / c ) (1 + v / c)

so:

G² = 1/(1 - v² / c²)

so G = \gamma

Therefore:

t'_oa = \gamma . t'_b . . . . . . (11)

and

t_ob = \gamma . t_a . . . . . . (12)

Substituting (11) into (1):

t_a = \gamma . t'_b + v . \gamma . t'_b / c . . . . . . (13)

Substituting (12) into (2):

t'_b = \gamma . t_a - v. \gamma.t_a / c . . . . . . (14)

or in words:

(time of colocation of A and photon in the A frame) = gamma * ((time of colocation of B and photon in the B frame) + (speed of B in the A frame) * (time of colocation of B and photon in the B frame) / (speed of light) )

and

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) / (speed of light) )

I've put speed in bold to highlight that it is not a velocity.

Now those equations are not Lorentz transformations. I grant you that, but multiply through by c.

x_a = \gamma.(x'_b + v.t'_b)

and

x'_b = \gamma.( x_a - v . t_a)

(where the photon was when A and B were colocated, in the A frame) = gamma * ((where the photon was when A and B were colocated, in the B frame) + (speed of B in the A frame) * (time of colocation of B and photon in the B frame) )

and

(where the photon was when A and B were colocated, in the B frame) = gamma * ((where the photon was when A and B were colocated, in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) )

Making A the unprimed frame, and B the primed frame, then this latter equation (in A, the unprimed frame is at rest) is, at the very least, a spatial Lorentz Transform analogue.

Substituting x_a = c.t_a into (14) gives us:

t'_b = \gamma.t_a - v . \gamma . x_a / c^2

or, in words

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

This is not quite what we want, since the event we are talking about was back when A and B were colocated (in the A frame), but this equation does express an interval of note:

(how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame) = gamma * ((how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened), in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

or

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

which is, at the very least, a temporal Lorentz Transform analogue.

I will leave generalisation until later.

cheers,

neopolitan

 

[Rasalhague]

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.

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. ...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.

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. 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?

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). And might that not get confusing, given that it's the same physical situation we're looking at?

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?

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.

Then why do we have two different names for the same equation (dilation, contraction) that depend on which coordinate is being computed? 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.

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.

But what, in general, defines a clock as "the clock"?

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?

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.

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".

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? 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.

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.

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? 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.? Well, I suppose if the two names are settled on at the outset and stated in the definition, it might not get that bad - but you'd still be thinking implicitly in those terms, wouldn't you, and if you got confused, you might find yourself trying to work it out in such terms in order to "clarify" how the elements of the problem relate to each other, or else revert to unambiguously, explicitly relative terms like rest frame.

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.

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!

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.

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"? 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, 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. But maybe I'm exaggerating the contrivedness of that, given the genuine differences between time and space that might motivate us not to pair like with like.

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.

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. 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. Or substitute for moving and stationary whatever pair of terms is suitable (minimally ambiguous, maximally general, associated with the same mathematical operation).

But maybe there is a flaw in that plan. It's easy to talk about a moving length because we think of length as an inherent, persistent attribute of an object. Because of this, we can easily imagine a "moving length". We use length in two ways: in the abstract as a measurement, and more concretely as a persistant, physical property of a specified object. On the other hand, what would it mean to say "a moving while" or "a moving period"? We have no intuition. That creates a hurdle for anyone trying to make a neat, mnemonic pairing of the sort "a moving [INSERT SPATIAL PROPERTY HERE] does this, a moving [INSERT TEMPORAL PROPERTY HERE] does the same". Then again, at least the lack of an existing intuition makes a clear mental gap to fill with the appropriate definition.

 

[JesseM]

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 "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.

...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.

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.

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.

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.

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?

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.

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).

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.

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?

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.

Then why do we have two different names for the same equation (dilation, contraction) that depend on which coordinate is being computed?

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.

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.

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.

But what, in general, defines a clock as "the clock"?

That the pair of events you're calculating the time between both occur on this particular clock's own worldline.

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.

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?

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".

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?

Not that I know of.

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.

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".

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?

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.

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.?

"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.

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!

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.

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"?

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.

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,

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.

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.

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.

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.

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.

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.

"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".

Perhaps it would help if I say that pedagogically, the point of introducing these equations has nothing at all to do with "emphasizing the interchangeability of time and space", the point is that they are helpful when actually doing calculations about specific word-problems. Switching the terminology in the manner you suggest would make it more confusing to try to apply them to specific word-problems.

 

[neopolitan]

A comment on post https://www.physicsforums.com/showpost.php?p=2232756&postcount=397" which may go some way to explaining why I think my derivation holds generally.

The final two equations I arrive at (spatial and temporal) are:

x'_b = \gamma.( x_a - v . t_a)

and

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

In words:

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) )

and

(how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame) = gamma * ((how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened), in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

I want to highlight that in each case there is an interval, explicit or implied, between an event about which there is agreement in all frames and one other event.

We discuss the scenario in terms of there being a colocation of A and B, which is going to be agreed by both A and B - at least at the level that "A and B were colocated where they were colocated at the time at which they were colocated". Really, that is all they know.

Each of A and B are likely to have a coordinate system established before they are colocated which does not result in their colocation being at (0,0). Making their colocation (0,0) is handy, but by no means essential.

The point is that this colocation of A and B is an event which A and B have some level of agreement about. And there must be some such event - and it doesn't have to be a colocation.

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)

Hopefully this is so blatantly obvious that it doesn't really need more emphasis.

So, we have an interval between one event for which the coordinates are agreed (nominally (0,0)) and another event for which the coordinates are not agreed.

If the event for which there is agreement is the nominal (0,0), then the intervals are also coordinates.

In my scenario, the event which is agreed is colocation of A and B (0,0).

In the final equations the (possibly implied) intervals are:

(spatial interval between where A and B were colocated and where the photon was when A and B were colocated, in the B frame) = gamma * ((spatial interval between where A and B were colocated and where the photon was when A and B were colocated, in the A frame) - (speed of A in the B frame) * (time interval between colocation of A and B and colocation of A and photon in the A frame) )

and

(time interval between when A and B were colocated and when the event took place, in the B frame) = gamma * (time interval between when A and B were colocated and when the event took place, in the A frame) - (speed of A in the B frame) * (spatial interval between where A and B were colocated and where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

To make this general, A and B just need to agree on a different event. Conceptually, I know this works but while proving it mathematically won't be impossible, it might be messy.

cheers,

neopolitan

PS I am aware that to be totally consistent, I should express the spatial equation in delta format, but I think you can understand that this would be trivial mathematically.

 

[Rasalhague]

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.

And yet the way we express length is in terms of spatial coordinates, albeit not events that bear the same relationship to each other as the events in "time dilation", as normally conceived. But this led me to thinking: we now have abstract geometrical descriptions of length contraction and its temporal analogue, and a colloquial description of length contraction. To complete the picture, we'd need a colloquial way of expressing "time contraction". Of course, it may say something about the difference between space and time that we don't have such a description, or find it less intuitive, or less intuitively necessary - but still I'd like to have a try.

A physical object like a clock doesn't bear the same realtion to time, in all its particlars, as a ruler bears to space. Clock and ruler are both sharply bounded in space; both persist indefinitely in time. No wonder the symmetry between time and space is obscured if we treat them (or inadvertently let them appear by convention) as if a clock is, in all relevant respects, to time as a ruler is to space. The length of a ruler in different frames is determined by the changing relationship, in their different coordinate systems, between two worldlines (those of its ends), whereas, in the traditional conceptualisation of time dilation, we're instead talking simply about the changing relationship between one pair of points as we change the frame use to describe them. But what if we were to conceptualise this same situation in terms of the duration of a journey (as the temporal equivalent of the length of an object)?

Just as understanding of length in special relativity requires additional definitions beyond our naive intuitions about length, so too any definition of the "duration of a journey" would involve some additional convention to be defined. In fact, I've wondered at times whether the very naturalness of the idea of length is, in some sense, beguilingly natural. That is, it's all too easy as beginners to see that familiar word and think we know what it means, which can lead to paradoxes until we realize that the relativistic definition of length depends on concepts such as the relativity of simultaneity, for which we have no naive intuition. We're used to the idea of objects shrinking in everyday life, but length contraction in relativity isn't quite the same thing. Of course, the same criticism could be levelled at "duration contraction" or "travel-time contraction", which, aside from definitions, is probably every bit as ambiguous a name as the alternatives.

That said, here's my attempt at parallel geometric and colloquial definitions:

*Edit: I got a bit muddled with these next two paragraphs: see #403 for revised definitions. I'll leave these here though for the sake of continuity.

Length Contraction. The spacelike interval covered by the segment of a line of synchrony/simultaneity/now between its intersection with two worldlines in a frame where the worldlines are oblique compared to the unique frame where they're parallel to the x axis. (Colloquially: the length of an object is greatest in the unique frame where its ends are at rest. Restriction: in the frame where the ends of the object are moving, we must measure the position of both ends at the same time.)

Duration Contraction (travel-time/journey-time contraction). The timeline interval covered by the segment of a line of syntopy/collocality/here (a worldline) between two hypersurfaces of synchrony in a frame where the hypersurfaces of synchrony are oblique compared to the unique frame where they're parallel to the t axis. (Colloquially: the duration of a journey is greatest in the unique frame where its ends are at rest. Restriction: in the frame where the ends of the journey are moving, we must meaure the time of both ends in the same place.)

*

For example, last night, I worked through problem 29 in Taylor/Wheeler: Spacetime Physics, the purpose of which is to demonstrate the relativity of simultaneity, and how that relates to the problem of synchronising clocks. A pair of clocks are synchronised at the spacetime coincidence of their passing. One clock they call Big Ben, the other is being carried by a Mr Engelsberg. After some time Mr Engelsberg comes to, a third clock, called Little Ben, at rest with respect to Big Ben, and synchronises Little Ben to the time shown by the clock he's carrying. The question asks how much will Little Ben lag behind Big Ben once it's been set to the time shown by Mr Engelberg's clock as he passes.

They show multiple ways of solving the problem, one of which begins by thinking of the time shown on Mr Engelsberg's clock as he passes Little Ben (journey's end) as the interval between this event and his passing Big Ben. We calculate this interval from its time coordinate in the rest frame of Big Ben and Little Ben. In terms of "duration/travel-time contraction", this is the frame in which the journey takes place, since by definition Mr Engelsberg does no traveling in his own rest frame. The duration of Mr Engelsberg's journey between Big Ben showing a certain time and Little Ben showing some other time is biggest in the unique frame where that journey takes place. We can define this frame, as above, in a way that bears the same relation to time as the rest frame of an object bears to space.

Does the concept work? If so, are there more standard names that I could have used for any of the entities these definitionsm, and - where there are no standard terms - can we think of better, more descriptive, less ambiguous names for any of these ideas?

Perhaps it would help if I say that pedagogically, the point of introducing these equations has nothing at all to do with "emphasizing the interchangeability of time and space", the point is that they are helpful when actually doing calculations about specific word-problems. Switching the terminology in the manner you suggest would make it more confusing to try to apply them to specific word-problems.

The purpose of my attempt above to define (geometrically and colloquially) a relation that is to time what length contraction is to space is to understand the symmetry between space and time and how they relate to each other. It may be an arcane way of putting it, needlessly complicated, or unnecessary for solving word-problems, but it's often said that there's more to understanding than the ability to plug in numbers and get the right answer. Even if this duration idea turns out to be impractical or irrelevant to solving textbook excercises, without exploring the issues in these ways, I wouldn't feel confident of having really grasped what was going on, and which technique it's appropriate to apply where. Part of my motivation is that, having read some introductory texts and been confused by the accidental suggestion of asymmetry in the apparent pairing of T.D. and L.C., I suspected it would be all too easy to phrase a problem in some unconventional way that would throw me. But thanks to this discussion and your explanations, I hope I've taken a few small steps towards unravelling what confused me when I first began. Of course, I make have to retrace a few steps along the way...

 

[Rasalhague]

Or would it clash too much with the convention to call them both "length contraction"? Thus: contraction of the length (in space) of an object (when measured in any frame other than the unique frame where its ends are at rest), and contraction of the length (in time) of a journey (when measured in any frame other than the unique frame where its ends are at rest).

 

[Rasalhague]

Hmm, I got a bit muddled with my geometric definitions. For one thing, I got x and t the wrong way round. Obviously no worldline can be parallel to the x-axis! Here's a second attempt.


LENGTH CONTRACTIONS FOR SPACE AND TIME.


x_{i}' = \frac{x_{i}}{\gamma} = \frac{x_{i}}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}


1. Spacelike. Consider two events at either end of a spacelike interval. Input: the spacelike interval between two parallel lines of syntopy (worldlines) which intersect the events, that is shortest in the unique frame where these lines are parallel to the t-axis. Output: the interval between these events.

Meaning: a length of space, such as the linear extent of a physical object, is greatest in the unique frame where the locations of the object's ends are at rest.

Restriction: in a frame where the positions of the object's ends are moving, we must locate them both at the same time.


2. Timelike. Consider two events at either end of a timelike interval. Input: The timelike interval between two hypersurfaces of synchrony which intersect the events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval between these events.

Meaning: a length of time, such as the duration of a journey, is greatest in the unique frame where the locations of the journey's ends are at rest.

Restriction: in a frame where the positions of the journey's ends are moving, we must time them both at the same location.


How could this definition of the meaning of conventional spacelike length contraction be reworded in a more general way, so as to eliminate the reference to an object, and could a corresponding generalisation be made to this timelike version?

 

[JesseM]

LENGTH CONTRACTIONS FOR SPACE AND TIME.x_{i}' = \frac{x_{i}}{\gamma} = \frac{x_{i}}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}1. Spacelike. Consider two events at either end of a spacelike interval. Input: the spacelike interval between two parallel lines of syntopy (worldlines)

What do you mean "spacelike interval between two parallel lines of syntopy"? Do you mean the distance between the points that the lines intersect a surface of constant t in whatever frame you're using? And on that point, what frame are you using for the input? The frame where both "events at either end of a spacelike interval" happen simultaneously, or the frame where the "two parallel lines of syntopy" are parallel to the t axis? Are you choosing the two parallel lines so that these frames are one and the same? If not, is the idea just to pick two arbitrary (not necessarily simultaneous) events in this first frame, draw two parallel lines parallel to the t-axis which intersect the events, and then use as input the distance between the two lines in this frame where they're parallel to the t-axis?

which intersect the events, that is shortest in the unique frame where these lines are parallel to the t-axis.

What is shortest? The distance between the two lines of syntopy, or the distance between the two events? Note that if you are defining the distance between lines as "the distance between the points that the lines intersect a surface of constant t in whatever frame you're using" as I suggested above, then the distance between them is longest in the unique frame where the lines are parallel to the t-axis, not shortest. On the other hand, the distance between a single pair of events is shortest in the frame where the events are simultaneous.

Output: the interval between these events.

Interval between the events, or between the lines of syntopy? And in what frame? Since you include gamma, which is a function of v, it's presumably one moving at v relative to the first frame, but I'm not clear on what the first frame is. If the first frame was the one where the lines of syntopy are parallel to the t-axis, but the events were not simultaneous in that frame, and the input was the distance between the lines in that frame, then in order for the output to be the distance between the events themselves in another frame, the output frame would have to be the one where the events are simultaneous.

Meaning: a length of space, such as the linear extent of a physical object, is greatest in the unique frame where the locations of the object's ends are at rest.

If your output is the distance between two events on the lines of syntopy, then the only way this would be the same as the "length" of an object with those lines as its boundary worldlines in the output frame is, again, to impose the rule that the output frame must be the one where the events are simultaneous, since "length" always represents a simultaneous measurement of the positions of the boundaries of an object.

2. Timelike. Consider two events at either end of a timelike interval. Input: The timelike interval between two hypersurfaces of synchrony which intersect the events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval between these events.

Similar questions as above...are you assuming the output is the time between the events in the frame where they were co-located?

Meaning: a length of time, such as the duration of a journey, is greatest in the unique frame where the locations of the journey's ends are at rest.

I don't think that verbal summary really makes sense. What is greatest is the time between the two spacelike hypersurfaces in the frame where they are parallel to the t-axis. However, the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located, and there is no upper limit on how large the time between these events can be in other frames.

 

[JesseM]

https://www.physicsforums.com/showpost.php?p=2231347&postcount=381" was my last post where I asked if I should move on, I'm assuming a yes.

We've discussed a scenario in which a photon passes B then A, where A and B have a relative separation speed of v. When A and B were colocated, t=0 and t'=0, ie this event serves as the origin of the t axis for both A and B.

Once a photon reaches A, A can work out when (t'_oa B and that photon were colocated, in the A frame, and the separation between where the photon was at t=0 and where (x'_oa) B was when B and the photon were colocated, in the A frame.

The values A has are:

time of colocation of A and B, t=0
time of colocation of A and photon, t = t_a
speed of B towards where the photon originated, v
location of photon at t=0, x_a = c.t_a

Therefore, we have (all in the A frame):

(location of colocation of B and photon) = (location of photon at t=0) - (speed of B) * (time of colocation of B and photon) = (speed of light) * (time of colocation of B and photon)

x'_oa = x_a - v.t'_oa = c.t'_oa

Why would the location of the photon passing B be equal to (location of photon at t=0) - (speed of B) * (time of colocation of B and photon)? Seems like the leftmost side should rather be (the separation between the position of the photon at t=0 and the position of B when the photon was colocated with B), or x_a - x'_oa. For example, if we go back to the numbers from the old numerical example, at t=0 seconds the photon is at position x_a=8 light-seconds, then at t'_oa=5 s the photon passes B at position x'_oa=3 ls, with B moving at v=0.6c. So you can see here that (the separation between the position of the photon at t=0 and the position of B when the photon was colocated with B) = x_a - x'_oa = 8 - 3 = 5, and likewise (location of photon at t=0) - (speed of B) * (time of colocation of B and photon) = x_a - v*t'_oa = 8 - 0.6*5 = 8 - 3 = 5, and finally it also works that (speed of light) * (time of colocation of B and photon) = c*t'_oa = 5.

or

c.t_a - v.t'_oa = c.t'_oa

so

t_a = t'_oa + v.t'_oa / c . . . . . . (1)

OK, I see you didn't actually need to make use of the left hand side of the equation above so this is fine.

We can follow the same procedure for B to reach (all in the B frame):

(time of colocation of photon with B) = (time of colocation of A and photon) - (speed of A) * (time at which A and photon will be colocated) / (speed of light)

t'_b = t_ob - v.t_ob / c . . . . . . (2)

OK

We've already concluded that since

t_a = time interval between Event and when a photon from the Event reaches A, in A's frame

and

t'_b = time interval between Event and when a photon from the Event reaches B, in B's frame

we have no expectation that t_a = t'_b

We can now test the hypothesis that:

(time of colocation of B and photon in the A frame) = (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (time of colocation of A and photon in the A frame)

That would mean:

t'_oa = t'_b . . . . . . (3)

and

t_ob = t_a . . . . . . (4)

Substituting (3) into (1):

t_a = t'_b + v.t'_b / c = t'_b ( 1 + v / c ) . . . . . . (5)

Substituting (4) into (2):

t'_b = t_a - v.t_a / c = t_a (1 - v / c ) . . . . . . (6)

Substituting (6) into (5):

t_a = t_a (1 - v / c ) (1 + v / c)So (3) and (4) are not valid.

OK, looks like a good proof by contradiction.

This indicates that:

(time of colocation of B and photon in the A frame) does not = (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the A frame) does not = (time of colocation of A and photon in the B frame).

Well, it proves that they can't both be true. I don't think it proves that neither could be true...what if one was true and the other false? Your proof-by-contradiction above made use of the assumption that both were true.

If we make an alternative hypothesis that:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

You can make this hypothesis, but you could equally well make the hypothesis that the factors in the two equations were different. If the derivation is meant to be rigorous you need a justification for the idea that the factors must be the same in both equations.

That would mean:

t'_oa = G.t'_b . . . . . . (7)

and

t_ob = G.t_a . . . . . . (8)

Substituting (7) into (1):

t_a = G.t'_b + v.G.t'_b / c = G.t'_b ( 1 + v / c ) . . . . . . (9)

Substituting (4) into (2):

t'_b = G.t_a - v.G.t_a / c = G.t_a (1 - v / c ) . . . . . . (10)

Substituting (10) into (9):

t_a = G.G.t_a (1 - v / c ) (1 + v / c)

so:

G² = 1/(1 - v² / c²)

so G = \gamma

Therefore:

t'_oa = \gamma . t'_b . . . . . . (11)

and

t_ob = \gamma . t_a . . . . . . (12)

Substituting (11) into (1):

t_a = \gamma . t'_b + v . \gamma . t'_b / c . . . . . . (13)

Substituting (12) into (2):

t'_b = \gamma . t_a - v. \gamma.t_a / c . . . . . . (14)

or in words:

(time of colocation of A and photon in the A frame) = gamma * ((time of colocation of B and photon in the B frame) + (speed of B in the A frame) * (time of colocation of B and photon in the B frame) / (speed of light) )

and

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) / (speed of light) )

I've put speed in bold to highlight that it is not a velocity.

Now those equations are not Lorentz transformations. I grant you that, but multiply through by c.

x_a = \gamma.(x'_b + v.t'_b)

and

x'_b = \gamma.( x_a - v . t_a)

(where the photon was when A and B were colocated, in the A frame) = gamma * ((where the photon was when A and B were colocated, in the B frame) + (speed of B in the A frame) * (time of colocation of B and photon in the B frame) )

and

(where the photon was when A and B were colocated, in the B frame) = gamma * ((where the photon was when A and B were colocated, in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) )

Making A the unprimed frame, and B the primed frame, then this latter equation (in A, the unprimed frame is at rest) is, at the very least, a spatial Lorentz Transform analogue.

I don't think it's very closely analogous. Remember that the spatial Lorentz transform takes either the coordinates of a single event in one frame and finds the spatial coordinates of the same event in the other frame, or else it takes the coordinate intervals between a single pair of events in one frame and finds the spatial interval between the same pair of events in the other frame. But just looking at the first of the two equations above, if you're talking about coordinates rather than coordinate intervals, you're dealing with three separate events: the event on the photon's worldline that occurred at t=0 in the A frame, the event on the photon's worldline that occurred at t=0 in the B frame, and the event of the photon passing B. There's no way to re-interpret this so all the variables represent intervals between a single pair of events, either.

Substituting x_a = c.t_a into (14) gives us:

t'_b = \gamma.t_a - v . \gamma . x_a / c^2

or, in words

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

This is not quite what we want, since the event we are talking about was back when A and B were colocated (in the A frame), but this equation does express an interval of note:

(how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame) = gamma * ((how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened), in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

I don't understand the phrase "how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame". You seem to be mentioning three events in this phrase--the event of the photon reaching B, the event of the colocation of A and B, and "when the event happened" (which I assume means the event on the photon's worldline that occurred at t=0 in B's frame)? So how can you have an interval between three events? Likewise with "how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened)".

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

which is, at the very least, a temporal Lorentz Transform analogue.

Again, not very analogous since all the terms don't refer to the coordinates of a single event or to intervals between a single pair of events (in fact, you seem to be mixing time intervals with spatial coordinates here).

 

[neopolitan]

Would it upset things to reword your timelike like this (changes look like this):

2. Timelike. Consider two events at either end of a timelike interval. Input: The timelike interval between two hypersurfaces of synchrony which intersect the events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval between these events.

Meaning: a length of time, such as the interval between causally related events, like the ticks of a clock, is greatest in the unique frame where the causally related events are spatially colocated (or the clock is at rest).

Restriction: in a frame where causally related events are not spatially colocated (such as the ticks of a moving clock), we must consider alternative events which are simultaneous with the causally related events but at a single location or alternative events which are causally related to the first set of events such that they maintain the same temporal separation, such as the spawning and receipt of photons.

An alternative to this, given that the temporal restriction is quite complex, is to consider a specially designed "rod clock" such that the spatial and temporal intervals are intertwined.

The specifications of the rod clock are such that is has a length of L, it has two photon tubes in it with two photons and two sets of mirrors. The photons bounce between the mirrors in phase (at least while the clock is at rest) so that when one photon hits the mirror at one end of the rod, the other photon hits the mirror at the other end of the rod.

It makes sense to count ticks, doesn't it?

The number of ticks on the moving clock will be fewer than for a rest clock.

It also makes sense to measure a rod at a single moment in time, doesn't it?

The interval between where one end of a moving rod is at a single moment in a rest frame and the other end of the same moving rod at the same single moment in the same rest frame is going to be less than the interval between one end of the moving rod and the other end in the moving frame.

The thing that is all screwed up (in our basic perception of these things, but not our sophisticated SR perception), is that the moving rod is at rest in the moving frame.

That means that the moving length (in the rest frame) is less than the rest length (in the moving frame).

Going back to time to try to express it in similar terms:

The number of "moving ticks" (in the moving frame) is less than the number of "rest ticks" (in the rest frame).

We have a couple of options for consistency here: compare "in the rest frame" with "in the moving frame" in each of the pairs, ie priming the "in the moving frame" values; or consider a single rod clock which is in motion relative to a notional rest frame, ie priming the rod clock values.

Comparing these approaches, the rod clock despite having a single construction has two natures for our purposes "rod" and "clock". If we have "rod clock frame" and "observer frame", then the "rod clock frame" is the rest frame for considering the rod clock's rod-nature, but the "observer frame" is the rest frame for considering the rod clock's clock-nature.

I think it is this inconsistency that Rasalhague is getting at (and which I have touched on once or twice).

Anyways, Rasalhague might want to use the rod clock approach to frame "space-like" and "time-like" in an internally consistent way, since using the terminology "moving ends of a journey" is fraught with danger :)

cheers,

neopolitan

 

[JesseM]

Would it upset things to reword your timelike like this (changes look like this):

Meaning: a length of time, such as the interval between causally related events, like the ticks of a clock[/color], is greatest in the unique frame where the causally related events are spatially colocated (or the clock is at rest)[/color].

You should say it's smallest in that unique frame, not greatest.

The thing that is all screwed up (in our basic perception of these things, but not our sophisticated SR perception), is that the moving rod is at rest in the moving frame.

That means that the moving length (in the rest frame) is less than the rest length (in the moving frame).

It seems like this is unnecessarily confusing because of your use of the phrase "moving frame" to refer to the rod's rest frame. If you used a less ambiguous pair of names for the frames, like "the rod's rest frame" and "the observer's rest frame" (or the rest frame of whatever object you want to see the rod in motion), then this problem of terminology wouldn't arise.

Comparing these approaches, the rod clock despite having a single construction has two natures for our purposes "rod" and "clock". If we have "rod clock frame" and "observer frame", then the "rod clock frame" is the rest frame for considering the rod clock's rod-nature, but the "observer frame" is the rest frame for considering the rod clock's clock-nature.

How do you figure? We say "time dilation" because the time between two events on the worldline of one of the clocks is greater in the observer's frame than the time between the same two events in the rod/clock frame. Likewise, we say "length contraction" because the length of the rod/clock is shorter in the observer's frame than its length in the rod/clock frame. So the terminology is consistent, if that's what you're talking about (it's what Rasalhague was talking about). If you're talking about something else, can you elaborate on what you mean by the phrase "the rest frame for considering the rock clock's ___ nature"?

 

[neopolitan]

JesseM,

Just to focus in on one issue for once, does the fact that:

the selection of A and B
their separation velocity v
the direction selected as positive, and
the event under consideration

are all arbitrary, not mean that I'd have to prove a contention that one of A and B somehow have primacy? I'd think that the contention that neither have primacy could be taken as read given the selection process.

This is your comment:

"Well, it (the preceding proof) proves that they can't both be true. I don't think it proves that neither could be true...what if one was true and the other false? Your proof-by-contradiction above made use of the assumption that both were true."

You are asking me here to prove that neither A nor B have primacy. I am thinking that the default is that neither A nor B have primacy, is that an incorrect starting point somehow?

cheers,

neopolitan

 

[Rasalhague]

What do you mean "spacelike interval between two parallel lines of syntopy"? Do you mean the distance between the points that the lines intersect a surface of constant t in whatever frame you're using?

Yes. Specifically the frame you mention is a frame where the object is moving. These lines are the worldlines of the ends of the object in the object's rest frame. This interval is meant to represent its length contracted in a frame where the object is moving. Syntopy = collocation = same place = constant x (in some frame).

I should have made it explicit that two different intervals are involved in each case:

1. Spacelike. Consider two events with spacelike interval \sigma_{small}. Input: the spacelike interval \sigma_{big} between two parallel lines of syntopy (worldlines) which intersect the aforementioned events, that is shortest in the unique frame where these lines are parallel to the t-axis. Output: the interval \sigma_{small}.

2. Timelike. Consider two events with timelike interval \tau_{small}. Input: The timelike interval \tau_{big} between two hypersurfaces of synchrony which intersect the aforementioned events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval \tau_{small}.

And on that point, what frame are you using for the input? The frame where both "events at either end of a spacelike interval" happen simultaneously, or the frame where the "two parallel lines of syntopy" are parallel to the t axis? Are you choosing the two parallel lines so that these frames are one and the same? If not, is the idea just to pick two arbitrary (not necessarily simultaneous) events in this first frame, draw two parallel lines parallel to the t-axis which intersect the events, and then use as input the distance between the two lines in this frame where they're parallel to the t-axis?

They're not the same frame. The input is characterised as a spacelike interval \sigma_{big} defined in terms of two different frames. One of these frames is that in which the events with interval \sigma_{small} are intersected by the worldlines of the object's ends (the two parallel lines of syntopy). The other frame is that in which those worldlines are parallel to the t-axis. The interval of the events (the only events explicitly mentioned) is the output. Which events these are (i.e. their position in spacetime) is determined by the interval \sigma_{big} and the relative velocity between the frames, represented by the slope of the separation between the events in the rest frame of the object.

What is shortest? The distance between the two lines of syntopy, or the distance between the two events?

The distance between the two lines of syntopy, \sigma_{big}. It's the shortest possible interval between these lines in the frame where they're parallel to the t-axis, and thus orthogonal to them (and the t-axis), and thus parallel to the x-axis. The frame where the lines of syntopy (worldlines of the ends of the object) are parallel to the t-axis is the object's rest frame, so \sigma_{big} represents the length of the object in its rest frame.

Note that if you are defining the distance between lines as "the distance between the points that the lines intersect a surface of constant t in whatever frame you're using" as I suggested above, then the distance between them is longest in the unique frame where the lines are parallel to the t-axis, not shortest. On the other hand, the distance between a single pair of events is shortest in the frame where the events are simultaneous.

I'm aware of that. My definition wasn't clear enough. I should have made it explicit that these are two different intervals I'm talking about.

Output: the interval between these events.

Interval between the events, or between the lines of syntopy? And in what frame? Since you include gamma, which is a function of v, it's presumably one moving at v relative to the first frame, but I'm not clear on what the first frame is.

The interval between the events (the only events mentioned explicitly, those referred to at the beginning of the definition), labelled in the revised definition \sigma_{small}. By "interval" (without qualification) I mean "spacetime interval", which is constant in all frames.

If the first frame was the one where the lines of syntopy are parallel to the t-axis, but the events were not simultaneous in that frame, and the input was the distance between the lines in that frame, then in order for the output to be the distance between the events themselves in another frame, the output frame would have to be the one where the events are simultaneous.

If your output is the distance between two events on the lines of syntopy, then the only way this would be the same as the "length" of an object with those lines as its boundary worldlines in the output frame is, again, to impose the rule that the output frame must be the one where the events are simultaneous, since "length" always represents a simultaneous measurement of the positions of the boundaries of an object.

The events have a spacelike separation, so there will be some frame in which there're simultaneous. That they lie on the worldlines of the object's ends ensures that they will represent the length of the object as measured in the frame where they're simultaneous. Which frame it is that they're simultaneous in is determined by the relative speed, represented geometrically by the hypersurface of synchrony (simultaneity, constant t). I hope my clarifications above have answered your other questions in this section, but let me know if there's anything still unclear (or that I've just got plain wrong). Perhaps it would help to make explict all of the events involved, rather than just naming two of them as "the events".

Similar questions as above...are you assuming the output is the time between the events in the frame where they were co-located?

Yes.

I don't think that verbal summary really makes sense. What is greatest is the time between the two spacelike hypersurfaces in the frame where they are parallel to the t-axis.

I guess you mean parallel to the x-axis?

However, the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located,

Yes.

and there is no upper limit on how large the time between these events can be in other frames.

I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame of the its start and finish (point of departure and point of arrival). The faster the traveller goes from start to finish, the shorter the duration, as speed increases arbitrarily close to c.

The definition is intended to parallel the way that the upper limit of the length of an object is its linear extent, the distance it spans, from one end to the other. The faster the object travels, the shorter its length, as speed increases arbitrarily close to c.

 

[JesseM]

JesseM,

Just to focus in on one issue for once, does the fact that:

the selection of A and B
their separation velocity v
the direction selected as positive, and
the event under consideration

are all arbitrary, not mean that I'd have to prove a contention that one of A and B somehow have primacy? I'd think that the contention that neither have primacy could be taken as read given the selection process.

Not exactly sure what you mean by "primacy" here, I'm just saying you haven't proved that you haven't given any real reason to believe that the factor here:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

must be the same as the factor here:

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

And as long as we entertain the possibility that they're different, it's also possible the factor in one of the equations could be 1--is that what you meant by "primacy", in response to my comment that you hadn't disproved the possibility that "one was true and the other false"? Of course, intuitively it seems pretty unlikely that the factor would always be 1 in one of these equations but not the other regardless of your choice of v and separation of A and B at the moment the photon passes either of them. Still,"intuitively it seems pretty unlikely" is not a rigorous argument, and in any case for the sake of your derivation it's not the possibility that the factor could be 1 that you have to worry about, but rather the more general possibility that the factors could be different. I bet if we tried hard enough we could even come up with an example of a coordinate transformation where the factors would in general be different and the speed of the photon would still be c in both frames, although presumably the transformation would violate the first postulate in the sense that the coordinate transformation from A's frame to B's frame would look different from the transformation from B's frame to A's frame.

 

[neopolitan]

You should say it's smallest in that unique frame, not greatest.

Clocking ticking here on earth, causally related events are twin travels away, twin arrives back.

On the Earth clock the time elapsed is greater in the Earth frame (for this clock, all the intermediate events between the departure event and arrival home are colocated with those events). For the traveling twin (for whom all the intermediate causally related events were not colocated with the departure event and the arrival home event event), the time elapsed is less than for the Earth clock.

See what I mean?

Apart from that, you broke up my post before you read it all. Otherwise you would have seen that I addressed your second question later.

As for "rod clock's ____ nature", the rod clock is both a rod, and a clock. I described that.

So there is a spatial interval associated with the rod clock (length of rod clock = L' and a temporal interval associated with the rod clock (time between ticks = dt').

Describe those in the observer's rest frame, where an identical rod clock in the observer's rest frame has length = L and time between ticks = dt.

You'll arrive at time dilation and length contraction.

But I wasn't talking about "time between ticks", I was talking about "number of ticks" on the clocks (notionally as the ticking end of the clock travels between two locations in the observer frame). I've talked often about the fact that our normal use of clocks is to look at how many ticks have taken place, rather than how long the period between ticks is. Using number of ticks, we arrive at something other than time dilation. Call it what you will.

number of ticks on rod clock in motion is fewer than number of ticks on observer's clock

t_{rod clock} = t_{observer's clock}/gamma

length of rod clock in motion is less than it's rest length

L_{rod clock} = L_{observer's clock}/gamma

I agree that

interval between ticks on rod clock in motion is greater than interval between ticks on observer's clock (according to the observer)

cheers,

neopolitan

 

[JesseM]

Yes. Specifically the frame you mention is a frame where the object is moving. These lines are the worldlines of the ends of the object in the object's rest frame. This interval is meant to represent its length contracted in a frame where the object is moving. Syntopy = collocation = same place = constant x (in some frame).

I should have made it explicit that two different intervals are involved in each case:

1. Spacelike. Consider two events with spacelike interval \sigma_{small}. Input: the spacelike interval \sigma_{big} between two parallel lines of syntopy (worldlines) which intersect the aforementioned events, that is shortest in the unique frame where these lines are parallel to the t-axis. Output: the interval \sigma_{small}.

Still a little unclear on "shortest in the unique frame where these lines are parallel to the t-axis". Do you mean 1) that out of all frames, the interval between these parallel lines is shortest in the frame where where the lines are parallel to the t-axis? Or 2) that considering only the frame where the lines are parallel to the t-axis, you want to look at the shortest interval between a pair of events on each line? If 2), then maybe a less confusing way of stating it is that you just want to talk about the distance between two events on either line which are simultaneous in this frame where the lines are parallel to the t-axis. But yeah, if it is 2) I think I understand what you're saying overall here.

I don't think that verbal summary really makes sense. What is greatest is the time between the two spacelike hypersurfaces in the frame where they are parallel to the t-axis.

I guess you mean parallel to the x-axis?

Yeah, my mistake.

However, the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located,

Yes.

and there is no upper limit on how large the time between these events can be in other frames.

I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame of the its start and finish (point of departure and point of arrival). The faster the traveller goes from start to finish, the shorter the duration, as speed increases arbitrarily close to c.

I don't understand, didn't you just agree the time between two events (point in spacetime of departure and point in spacetime of arrival) is greater in other frames besides the traveler's frame, because "the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located"? Just as an example, suppose in the traveler's rest frame the departure point passes next to him at coordinates x=0 light years, t=0 years and the arrival point passes next to him at x=0 light years, t=10 years. So, in his rest frame the time is 10 years. Do you agree that in any other frame where the traveler is moving, the time will be greater than 10 years, not smaller? If so, what do you mean when you say "I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame"? This should be the lower limit, not the upper limit.

The definition is intended to parallel the way that the upper limit of the length of an object is its linear extent, the distance it spans, from one end to the other. The faster the object travels, the shorter its length, as speed increases arbitrarily close to c.

If you want to make a parallel with length contraction, you should use your idea above about talking about the time between two parallel spacelike surfaces (which is how I have conceptualized the 'temporal analogue of length contraction' equation), not the time between two distinct events (which is how we ordinarily conceptualize the time dilation equation). Out of all frames, the time between two events is minimized in the frame where they are colocated (so if they are both events on the worldline of a traveler, this time is minimized in the traveler's rest frame). On the other hand, out of all frames, the time between two parallel spacelike surfaces is maximized in the frame where these spacelike surfaces are surfaces of simultaneity (parallel to the x-axis). This is more like with length contraction, where the distances between two parallel timelike worldlines is maximized in the frame where these lines are parallel to the t-axis.

 

[JesseM]

Clocking ticking here on earth, causally related events are twin travels away, twin arrives back.

On the Earth clock the time elapsed is greater in the Earth frame (for this clock, all the intermediate events between the departure event and arrival home are colocated with those events). For the traveling twin (for whom all the intermediate causally related events were not colocated with the departure event and the arrival home event event), the time elapsed is less than for the Earth clock.

But in this case one of the twins is not inertial. When you rewrote Rasalhague's statement to read "Meaning: a length of time, such as the interval between causally related events, like the ticks of a clock, is greatest in the unique frame where the causally related events are spatially colocated (or the clock is at rest)", I assumed "frames" referred to inertial frames, since this is what all the discussion had been about so far.

Apart from that, you broke up my post before you read it all. Otherwise you would have seen that I addressed your second question later.

Please don't jump to uncharitable conclusions like this. In fact I did read the whole post before responding, and I saw nothing (and still see nothing) in it that allows me to understand what you meant by phrases like "the rod clock's clock nature". I'm sure it's clear to you since you wrote it, but you aren't communicating your ideas in a way that allows me (or others reading, I'd wager) to follow.

As for "rod clock's ____ nature", the rod clock is both a rod, and a clock. I described that.

I understood that part, of course.

So there is a spatial interval associated with the rod clock (length of rod clock = L' and a temporal interval associated with the rod clock (time between ticks = dt').

Describe those in the observer's rest frame, where an identical rod clock in the observer's rest frame has length = L and time between ticks = dt.

You'll arrive at time dilation and length contraction.

But I wasn't talking about "time between ticks", I was talking about "number of ticks" on the clocks (notionally as the ticking end of the clock travels between two locations in the observer frame). I've talked often about the fact that our normal use of clocks is to look at how many ticks have taken place, rather than how long the period between ticks is. Using number of ticks, we arrive at something other than time dilation. Call it what you will.

You didn't mention this distinction in your post, so why did you conclude that my failure to infer your meaning (via mind-reading?) proved I didn't read your post? In any case the time dilation equation doesn't deal with "time between ticks", it deals with the time period between some specific pair of events that occurs on the worldline of a clock (preferably events separated by a large number of ticks if we want to measure the time interval fairly accurately), and how much time the clock itself measures between these events (or the 'number of ticks' between them as measured by that clock) vs. the amount of time between the same events in the observer's frame where the clock is moving (which could be measured by 'number of ticks' on a pair of synchronized clocks at rest in the observer's frame if you like). The time between these events in the observer's frame is greater than the time between them in the clock's rest frame, no? And measuring time intervals between specific physical events of interest (like the beginning and ending of a race) is part of "our normal use of clocks", no?

 

[neopolitan]

Not exactly sure what you mean by "primacy" here, I'm just saying you haven't proved that you haven't given any real reason to believe that the factor here:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

must be the same as the factor here:

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

And as long as we entertain the possibility that they're different, it's also possible the factor in one of the equations could be 1--is that what you meant by "primacy", in response to my comment that you hadn't disproved the possibility that "one was true and the other false"? Of course, intuitively it seems pretty unlikely that the factor would always be 1 in one of these equations but not the other regardless of your choice of v and separation of A and B at the moment the photon passes either of them. Still,"intuitively it seems pretty unlikely" is not a rigorous argument, and in any case for the sake of your derivation it's not the possibility that the factor could be 1 that you have to worry about, but rather the more general possibility that the factors could be different. I bet if we tried hard enough we could even come up with an example of a coordinate transformation where the factors would in general be different and the speed of the photon would still be c in both frames, although presumably the transformation would violate the first postulate in the sense that the coordinate transformation from A's frame to B's frame would look different from the transformation from B's frame to A's frame.

By primacy I mean having privilege, which would in turn violate the first postulate.

If I explicitly state the first postulate, which I take to follow from Galilean relativity, would that satisfy you or do you demand that I do a proof by elimination for the hypothesis that one transformation has one form and the other transformation has another form?

I just can't see what the justification would be for uneven transformations given that I should be able swap B for A without affecting the result (because of the way they were selected and because everything about them was expressed in general terms).

The only addition I can see which would make sense would be to make v a velocity (with a potentially negative value in one frame and a positive value in the other frame, but the same magnitude).

cheers,

neopolitan

 

[neopolitan]

The number of ticks on the moving clock will be fewer than for a rest clock.

Reading the whole post means you don't need to read minds. (Remember to check from which post this quote was taken.)

 

[neopolitan]

I feel like we are going over old ground unprofitably, really we don't disagree on the time dilation thing other than whether there is utility in thinking about an inverse function or a temporal analogue for the spatial function. I'd prefer not to waste effort on going over it again so is it possible that we focus on the derivation (and I won't intrude on the separate strand you have going with Rasalhague).

cheers,

neopolitan

 

[JesseM]

Reading the whole post means you don't need to read minds. (Remember to check from which post this quote was taken.)

Sorry, read it all again for a third time, nothing there that would indicate that you were talking about your own weird notion that "number of ticks" contracts rather than dilates like "time between ticks". You did use the phrase "number of ticks" but did not indicate that you were using this in anything other than the normal way, where we are talking about the "number of ticks" of both the observer's time and the clock's time between a specific pair of events, where naturally the number of ticks of the observer's time is greater, so it makes sense that we use the word "dilation" to be consistent with the fact that in the case of length, the length of the object is smaller for the observer. So clearly in this (standard) sense, both "time between ticks" on the moving clock and "number of ticks" between two events on the moving clock's worldline are larger for the observer than they are for the moving clock. If you meant something else you needed to actually explain it.

 

[JesseM]

By primacy I mean having privilege, which would in turn violate the first postulate.

But why should we think that one of the frames having a different value for the constant indicates that one frame has privilege (for example, if the first equation had the constant 2 and the second equation had the constant 3, which frame would be priveleged? Your two equations each involve quantities from both frames anyway) or that the first postulate has been violated? The first postulate says the laws of physics must be the same in both frames, it doesn't say that specific physical scenarios will look the same in both frames. For example, if I had a pair of objects traveling in opposite directions with the same speed in one frame, they would not have equal speeds in a different frame, but this wouldn't violate the first postulate. The physical scenario you are considering does look different in the two frames--the photon starts at a different distant from the origin at t=0 in each frame, and A's frame B is moving towards the photon while in B's frame A is moving away from the photon--so I don't see how you can assume a priori that the first postulate says the constant in one equation must be the same as the constants in the other, although we know in retrospect that this does turn out to be true.

If I explicitly state the first postulate, which I take to follow from Galilean relativity, would that satisfy you or do you demand that I do a proof by elimination for the hypothesis that one transformation has one form and the other transformation has another form?

Yes, it needs to be proved. And those equations are not "transformation" equations, the first equation is a relationship between the time of one event in the A frame and the time of a second event in the B frame, and the second equation is the relationship between the time of the first event in the B frame and the time of the second event in the A frame. Do you agree that the constants in the equations would not necessarily be the same if we picked two totally arbitrary events, rather than the events being ones where a single photon crossed the time axis of each frame? If so maybe you can see the need for an explanation as to what specific property of the two events you chose ensures that the constants in those two equations will be the same.

 

[Rasalhague]

Still a little unclear on "shortest in the unique frame where these lines are parallel to the t-axis". Do you mean 1) that out of all frames, the interval between these parallel lines is shortest in the frame where where the lines are parallel to the t-axis? Or 2) that considering only the frame where the lines are parallel to the t-axis, you want to look at the shortest interval between a pair of events on each line? If 2), then maybe a less confusing way of stating it is that you just want to talk about the distance between two events on either line which are simultaneous in this frame where the lines are parallel to the t-axis. But yeah, if it is 2) I think I understand what you're saying overall here.

It was 2 that I meant.

I don't understand, didn't you just agree the time between two events (point in spacetime of departure and point in spacetime of arrival) is greater in other frames besides the traveler's frame, because "the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located"?

That's right, but the faster the traveller goes, the smaller the (spacetime) interval, the proper time, of the separation between the event of departure and that of arrival. So greater speed results in a smaller output. That's to say, a greater contraction of the input. The input is the time component of this separation in the rest frame of the place the traveller left and the place they're going to. A faster traveller arrives at the same point in space as a slower traveller, but not the same point in spacetime (which is to say: not the same event). So I didn't mean to give the impression that this operation consists of calculating the time component of a given separation (whose interval is the proper time between two fixed events) in some arbitrary frame other than the traveller's rest frame. Rather, I meant that a faster speed determines which separation (between which pair of events) it will be whose interval is the output. The duration of the journey through space (as I'm trying to define it) is smaller the faster the journey, even though the time component of whichever separation is selected will be greater than the interval in any frame other than the traveller's rest frame.

Just as an example, suppose in the traveler's rest frame the departure point passes next to him at coordinates x=0 light years, t=0 years and the arrival point passes next to him at x=0 light years, t=10 years. So, in his rest frame the time is 10 years. Do you agree that in any other frame where the traveler is moving, the time will be greater than 10 years, not smaller? If so, what do you mean when you say "I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame"? This should be the lower limit, not the upper limit.

The full phrase I used was "in the rest frame of its start and finish (point of departure and point of arrival)". By "its", I meant "the journey's". This it not the same frame as the traveller's rest frame; if the traveller remained at rest with respect to their destination, they'd never get there! (Unless it was a trivial journey, with the place of origin and the destination being identical, making leaving and arrival the same event, which is no journey at all, just a rod whose ends are collocated would have no length, and thus be no rod.)

I think something Neopolitan mentioned in #406 sheds light on one possible source of confusion, namely the danger of talking about the "moving ends of a journey". I can see that "start" and "finish" are potentially ambiguous as to whether what's meant is the events of departure and arrival, or the points in space where these events take place (at whatever time).

I'm trying to depict the journey itself as the entity that relates to time as a ruler relates to space. My motivation for this is that the pairing of clock and ruler is, I suspect, the source of this apparent asymmetry. Clocks and rulers are both objects with limited spatial extent and indefinite (arbitrary) temporal extent. The time dilation operation, as tratitionally conceived, is defined in terms of finding the time component for a given proper time. The length contraction operation, as traditionally conceived, could be defined in terms of finding the proper distance with a given x component (i.e. the inverse of time dilation), although it's more often expressed in terms of three explicitly defined measuring events. The ruler (or rod or rocket) in these thought experiments is a more complex entity than the clock in that it has both temporal and spatial extent. In talking about a journey, I'm trying to define some entity that would be to time what a ruler is to space. Thus the journey is a more complex entity than a single, pointlike clock (conceived of as locatable at a single point in space, and with single worldline for its trajectory). Like a ruler, a journey extends through space as well as time; as a finite ruler has a sharply defined pair of ends, a finite journey has a sharply defined beginning and end in time.

If you want to make a parallel with length contraction, you should use your idea above about talking about the time between two parallel spacelike surfaces (which is how I have conceptualized the 'temporal analogue of length contraction' equation), not the time between two distinct events (which is how we ordinarily conceptualize the time dilation equation).

This is still my intention. From what you've said, it seems that I need to come up with a clearer way of wording the definition.

Out of all frames, the time between two events is minimized in the frame where they are colocated (so if they are both events on the worldline of a traveler, this time is minimized in the traveler's rest frame). On the other hand, out of all frames, the time between two parallel spacelike surfaces is maximized in the frame where these spacelike surfaces are surfaces of simultaneity (parallel to the x-axis). This is more like with length contraction, where the distances between two parallel timelike worldlines is maximized in the frame where these lines are parallel to the t-axis.

I suppose I was trying to define the operations in terms of invariant intervals without reference to components. I'm not sure why I chose to do that (maybe just exploring the concepts, maybe looking for a definition that incorporated the idea of space and time components of a separation without naming them, to break it down into the most basic concepts), and I might have another go at that and try to do it more clearly, but reverting the language of components for now, would the following work?


1. Spacelike. Consider two events, each on a different one out of two timelike lines, these events having a separation with spacelike interval \sigma. Input: the x component of the separation of these events in a frame where the lines are parallel to the t-axis. Output: the interval \sigma.

Meaning: a length of space, such as the linear extent of a physical object, is greatest in the unique frame where the locations of the object's ends are at rest.

Restriction: in a frame where the locations of the object's ends are changing, we must locate them both at the same time.


2. Timelike. Consider two events, each on a different one out of two spacelike hypersurfaces, these events having a separation with timelike interval \tau. Input: the time component of the separation of these events in a frame where the hypersurfaces are parallel to the x-axis. Output: the interval \tau.

Meaning: a length of time, such as the duration of a journey, is greatest in the unique frame where the locations of the journey's ends are at rest.

Restriction: in a frame where the locations of the journey's ends are changing, we must time them both at the same location.

 

[neopolitan]

Yes, it needs to be proved. And those equations are not "transformation" equations, the first equation is a relationship between the time of one event in the A frame and the time of a second event in the B frame, and the second equation is the relationship between the time of the first event in the B frame and the time of the second event in the A frame. Do you agree that the constants in the equations would not necessarily be the same if we picked two totally arbitrary events, rather than the events being ones where a single photon crossed the time axis of each frame? If so maybe you can see the need for an explanation as to what specific property of the two events you chose ensures that the constants in those two equations will be the same.

It seems to me that we have two stopping points here.

First, the need to prove that there must be one single factor (as in my derivation shown earlier), rather than possibly two factors.

Second, generality, in that you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames, and that you feel that there is a "need for an explanation as to what specific property of the two events (I) chose ensures that the constants in those two equations will be the same".

Is this correct?

Do you further agree that, if I were to convince you that there being one single factor is the default, that feat would negate the need for a proof? I'm going to think about the proof angle anyway, but I am not abandoning my argument that what you are asking me to prove is actually the default.

cheers,

neopolitan