Benefits of time dilation / length contraction pairing?

[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

 

[JesseM]

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.

Yes.

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".

How is that different from the above? I'm just asking for a demonstration that the specific properties of the events you chose ensure that the factor will be the same in both equations, and pointing out that since you presumably agree the factor wouldn't be the same for an arbitrary pair of events, then your demonstration will have to somehow make use of those specific properties rather than just making use of more general facts like the first postulate of SR.

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 would say any totally airtight argument for why they must be the same would constitute a "proof", so I don't really understand this distinction. Unless by "default" you just mean "the assumption that seems most plausible a priori even if we aren't sure it's actually correct", in which case I don't think that would negate the need for a proof.

Keep in mind, also, my later criticisms involving the final equations you derived, which I say have a physical meaning that's totally unlike the Lorentz transformation equations...it seems in a way a bit pointless to spend a lot of time thinking about how to justify this one step in your demonstration (which I agree happens to be true even if we haven't found a way to justify it) if the final endpoint of your demonstration is just an equation that looks superficially like the Lorentz transformation but really has almost nothing to do with it, and has no real utility outside of relating quantities in the specific physical scenario you describe where a light ray crosses path with two observers (so it cannot even really be understood as a special case of the Lorentz transformation, since the variables that appear in the equations aren't all defined in terms of the same single event or pair of events)

 

[neopolitan]

How is that different from the above?

I must be getting tired.

My second point was supposed to be that generality seems to be a stopping point, in that you think my selection of event has a significant bearing on the final equation (to the extent that it is not actually the Lorentz Transform or even equivalent to it, but something less general).

I had meant to finish the phrase "you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames" with "then my derivation would not stand". I did mean to end with the ending I had, for reasons which might become clear when, at a time when I am not so dog tired, I will try to explain that my starting scenario is so general that A should be totally interchangeable with B. Generality in, generality out.

cheers,

neopolitan

 

[JesseM]

I had meant to finish the phrase "you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames" with "then my derivation would not stand".

OK, yes, I think your derivation seems to depend on the fact that both events lie on the path of a photon. What's more, even with this restriction, your final equations don't even end up measuring "the interval between them" (i.e. the variables in your final equations don't all refer to the space and time intervals between a single pair of events).

I did mean to end with the ending I had, for reasons which might become clear when, at a time when I am not so dog tired, I will try to explain that my starting scenario is so general that A should be totally interchangeable with B. Generality in, generality out.

Showing that A could be exchanged with B might show that the constants in those equations would be the same, although it wouldn't address the issue of the final equations you derive. In any case it doesn't seem to me A can be exchanged with B here, since after all both frames agree the photon passes B before it passes A.

 

[neopolitan]

In any case it doesn't seem to me A can be exchanged with B here, since after all both frames agree the photon passes B before it passes A.

Surely you see that that depends on v and the x values being positive. If we plug in a negative v, the photon passes A before it passes B. Similarly if we start with a negative displacement for the event (while v stays positive), then the photon passes A before it passes B. To complete the picture, if we put in a negative v and a negative starting displacement, then the photon passes B first.

We're not constrained to using positive values.

Can you understand that if I say "t time units before it passes A, the photon passes B" and the scenario makes t negative, then that is the same as saying "|t| time units after it passes A, the photon passes B" where |t| is the magnitude of t.

Does this help you see that A and B are interchangeable?

If so, that will leave us with what the derived equations describe.

I do have one other question, you've not said anything about it but do you have issue with using (0,0) as an agreed location (conceptually the same as a colocation, if not an actual colocation, of A and B)? If you do have a problem with it, I will have to talk exclusively about intervals (even though I think we have agreed that coordinates are implicitly intervals anyway).

cheers,

neopolitan

 

[Rasalhague]

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 :)

That's true. Partly I wanted to take advantage of the ambiguity in words like length that could refer to a length of time or a length of space. I was wondering if we could come up with a rigorous definition of a length of travel time that would intuitively match our idea of the length of an object, but I don't know how practical this would be, or whether the risk of confusion would outweigh the benefits.

Your "rod clock" is an interesting idea, although it's taken me a while to get my tongue around "rod clock rest frame" - maybe our next example can involve a red lorry and a yellow lorry... I'd been thinking along similar (albeit more primative) lines with candle clocks, or a burning fuse, but I always seemed to come back to this apparent asymmetry, as if even an object that's both ruler and clock, using the same physical markings to measure time along its length as it does to measure distance, leads to this strange feeling that we have to treat time and space as behaving differently from each other with respect to the object, or else switch which frame we take as out starting point.

How would the situation change, I wonder, if we replace the photons in the rod clock with a massive particle? Then we could think in terms of the particle's rest frame. I was looking at the way the idea of frame dependent distance is introduced in Fishbane et al.: Physics for Scientists and Engineers with a muon formed in the Earth's atmosphere and traveling from its point of creation to the ground. It lives just long enough to reach the ground. Of all the possible values they could take in all possible reference frames, the following two variables are minimum in the muon's rest frame: the time t_{\mu} that passes between the events of the muon's birth and death, and the distance x_{\mu} that extends between the receding bit of air that marks the spot where the muon was created and the approaching ground that marks the spot where it will be annihilated. So in the ground's rest frame (where the muon is moving), both the t component and the x component of the separation of these events (muon birth and death) will be greater than they were in the muon's rest frame.

But in that example, the muon is the clock and the Earth is the ruler. Although I subscripted the space coordinate with a mu there, this seems at odds with the fact that it's the Earth that's actually doing the measuring of the muon's journey, since in the muon's rest frame, the muon makes no journey. Will this always be the case, due to the way the distortions of time and space complement each other, even when we construct an object that incorporates the roles of clock and ruler: will the part or aspect of it that's used to determine time need to be moving in the rest frame of the part or aspect of it that's used to measure distance?

 

[JesseM]

Surely you see that that depends on v and the x values being positive. If we plug in a negative v, the photon passes A before it passes B. Similarly if we start with a negative displacement for the event (while v stays positive), then the photon passes A before it passes B. To complete the picture, if we put in a negative v and a negative starting displacement, then the photon passes B first.

We're not constrained to using positive values.

Can you understand that if I say "t time units before it passes A, the photon passes B" and the scenario makes t negative, then that is the same as saying "|t| time units after it passes A, the photon passes B" where |t| is the magnitude of t.

Does this help you see that A and B are interchangeable?

I don't understand what you mean by "interchangeable" here, or how it proves the constant should be the same in your equations. It's true that whatever x_a and v we pick, then whatever is seen in A's frame vs. B's frame, we could then pick a different x_a and v so that B saw what A formerly saw and vice versa. Still, for any particular choice of x_a and v, their viewpoints are asymmetrical in that they both agree the photon passed one of them first. And your equations both assume we have picked a particular choice of x_a and v, and then assert the constants relating the times will be the same.

As an analogy, suppose that instead of the two events being the photon crossing the x=0 axis of A's frame and the event of the photon crossing the x=0 axis of B's frame, we instead chose Event #1 to be the event of the photon crossing the x=5 axis of A's frame, and Event #2 to be the event of the photon crossing the x=5 axis of B's frame. Now suppose that someone said that since the two frames are "interchangeable", we should expect the constant in the following pair of equations to be the same:

(time of Event #2 in B's frame) = (constant)*(time of Event #1 in A's frame)
(time of Event #2 in A's frame) = (constant)*(time of Event #1 in B's frame)

Without doing any new calculations to check the validity of this, are you confident enough in the meaning of "interchangeability" that you would bet a large sum of money that this guy's argument is correct or incorrect? If you feel any uncertainty here that perhaps shows why the use of "interchangeability" in your own argument is not really rigorous, and is more of what physicists would call a "handwavey" argument. You really need to provide a step-by-step proof where each statement clearly follow from the last, and you show specifically where the two postulates of relativity come into play (since I think you'd agree that if we came up with a coordinate transformation that didn't respect these postulates the constants in your equations would not be the same), and what the relevance is of the fact that the two events in question are the events of a photon's worldline intersecting each frame's x=0 axis.

I do have one other question, you've not said anything about it but do you have issue with using (0,0) as an agreed location (conceptually the same as a colocation, if not an actual colocation, of A and B)? If you do have a problem with it, I will have to talk exclusively about intervals (even though I think we have agreed that coordinates are implicitly intervals anyway).

What do you mean "an agreed location"? Are you just referring to the idea that whenever we talk about the space and time coordinates of an event, we're implicitly talking about the space and time intervals between that event at the event at (0,0)? Is so I have no problem with that, again my problem with your final equations is that all the variables don't refer to intervals between a single pair of events.

 

[neopolitan]

I started to reply to this last night, what I wrote it probably sitting on the home computer in "Preview Post" form rather than being submitted. Oh well.

I'm not going to be able to reply in depth to your first chunk because you seem to have gone off on a tangent. I've highlighted the phrases which are of concern.

I don't understand what you mean by "interchangeable" here, or how it proves the constant should be the same in your equations. It's true that whatever x_a and v we pick, then whatever is seen in A's frame vs. B's frame, we could then pick a different x_a and v so that B saw what A formerly saw and vice versa. Still, for any particular choice of x_a and v, their viewpoints are asymmetrical in that they both agree the photon passed one of them first. And your equations both assume we have picked a particular choice of x_a and v, and then assert the constants relating the times will be the same.

As an analogy, suppose that instead of the two events being the photon crossing the x=0 axis of A's frame and the event of the photon crossing the x=0 axis of B's frame, we instead chose Event #1 to be the event of the photon crossing the x=5 axis of A's frame, and Event #2 to be the event of the photon crossing the x=5 axis of B's frame. Now suppose that someone said that since the two frames are "interchangeable", we should expect the constant in the following pair of equations to be the same:

(time of Event #2 in B's frame) = (constant)*(time of Event #1 in A's frame)
(time of Event #2 in A's frame) = (constant)*(time of Event #1 in B's frame)

Without doing any new calculations to check the validity of this, are you confident enough in the meaning of "interchangeability" that you would bet a large sum of money that this guy's argument is correct or incorrect? If you feel any uncertainty here that perhaps shows why the use of "interchangeability" in your own argument is not really rigorous, and is more of what physicists would call a "handwavey" argument. You really need to provide a step-by-step proof where each statement clearly follow from the last, and you show specifically where the two postulates of relativity come into play (since I think you'd agree that if we came up with a coordinate transformation that didn't respect these postulates the constants in your equations would not be the same), and what the relevance is of the fact that the two events in question are the events of a photon's worldline intersecting each frame's x=0 axis.

Here's what you are referring back to:

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)

I never said it was a constant. You are correct in that the factor is dependent on a particular value of v, and therefore not a constant. I don't call it a "function" because within the scenario v is fixed (if we changed v, then we would be talking about a different relationship between a different pair of frames a different relative speed with respect to each other). It would be a function if it varied as a result of changing our input values from within the pair of frames to which v applies (ie x and t values).

The selection of a different x_a in the scenario does not change the factor and the factor does not vary with time.

By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values).

I'm going to redo the equations with v as a velocity, rather than a speed. This may help.

What do you mean "an agreed location"? Are you just referring to the idea that whenever we talk about the space and time coordinates of an event, we're implicitly talking about the space and time intervals between that event AND the event at (0,0)? Is so I have no problem with that, again my problem with your final equations is that all the variables don't refer to intervals between a single pair of events.

I made a correction, I think that is what you meant.

You are part of the way there. I mean what you wrote in the corrected sentence, plus the fact that both A and B see that (0,0) event as (0,0), ie x=0,t=0 is the same event as x'=0,t'=0. There's agreement between A and B as to that event.

They can agree on another event rather than (0,0) if it is convenient to them as well, it will just make the coordinate transformation a little different.

When I redo the equations with v as a velocity rather than a speed, I will do them all with only intervals involved, rather than aiming for a coordinate transformation. I will also choose an event which is not necessarily simultaneous with a colocation of A and B.

At the same time, I will attempt to maintain enough rigour to satisfy you that "some factor" is in fact the same in both frames. If I can't manage that, then I might have to try a proof by elimination, eliminating the possibility of two different factors. I've already eliminated a single factor which is equal to 1.

I'll give it some thought and get back to you.

cheers,

neopolitan

 

[JesseM]

I'm not going to be able to reply in depth to your first chunk because you seem to have gone off on a tangent. I've highlighted the phrases which are of concern.
Here's what you are referring back to:
I never said it was a constant.

All I meant was that for any given choice of x_a and v it's just a number, but I probably shouldn't have used the word "constant" since it could vary as you vary x_a and v (although in fact it turns out that it doesn't). However, I wasn't actually assuming it remained constant as you varied these, so my argument in no way depends on this assumption, nor is it a "tangent". Please just mentally replace "constant" with "factor" in the block of text you quoted and reread it.

By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values).

What "end result" are you referring to? The end result that the same factor appears in both equations? In this case you are simply asserting what you are supposed to prove, but perhaps you mean something else. In any case, my point is that you haven't made any coherent argument as to why your ill-defined notion of "interchangeability" proves that for a specific choice of x_a and v, the factor in your two equations must be the same. The actual values of the variables that appear in the equations are certainly not invariant under a switching of labels--for example, the value of (the time that the photon is colocated with B, as measured in the A frame) is not equal to (the time that the photon is colocated with A, as measured in the B frame) for most specific choices of x_a and v.

Note that since A and B's worldlines are assumed to lie along the x=0 axis of their respective frames, the event of the photon being colocated with A is equivalent to the event of the photon crossing A's x=0 axis, and likewise for B...this means your two equations can be written as:

(time of photon crossing B's x=0 axis in A's frame) = (factor)*(time of photon crossing B's x=0 axis in B's frame)

(time of photon crossing A's x=0 axis in B's frame) = (factor)*(time of photon crossing A's x=0 axis in A's frame)

So maybe now you can see why it wasn't a tangent when I asked in the previous post if you'd agree with a hypothetical guy making the argument that because of the "interchangeability" of the two frames, we could also assume the same factor will appear in the following two equations (same as one another, not same as in the previous two equations):

(time of photon crossing B's x=5 axis in A's frame) = (factor)*(time of photon crossing B's x=5 axis in B's frame)

(time of photon crossing A's x=5 axis in B's frame) = (factor)*(time of photon crossing A's x=5 axis in A's frame)

All that's changed in these two equations is that x=0 has been replaced with x=5. Please tell me whether you'd be confident about whether this guy's argument is right or wrong, without doing any SR calculations to check explicitly what factor appears in each. If you're unsure, that suggests that you don't really have good justification for being confident that "interchangeability" means the factor should be the same in your own two equations.

You are part of the way there. I mean what you wrote in the corrected sentence, plus the fact that both A and B see that (0,0) event as (0,0), ie x=0,t=0 is the same event as x'=0,t'=0. There's agreement between A and B as to that event.

They can agree on another event rather than (0,0) if it is convenient to them as well, it will just make the coordinate transformation a little different.

Agree on another event for what purposes? Do you mean some of the variables in equations in your derivation would no longer represent the interval between 0,0 and some specific event like the photon crossing B's worldline (i.e. the coordinate of the latter event), but would instead represent the interval between a different event other than 0,0 on one side of the interval, but the same event on the other side? I think I'd need to see you go through at least some of the same steps of the derivation as in post 397, but with whatever altered assumptions about "another event" you want to make, in order to follow what you're talking about here.

 

[neopolitan]

I just meant that for any given choice of x_a and v it's just a number, but I probably shouldn't have used the word "constant" since it could vary as you vary x_a and v (although in fact it turns out that it doesn't). However, I wasn't actually assuming it remained constant as you varied these, so my argument in no way depends on this assumption, nor is it a "tangent". Please just mentally replace "constant" with "factor" in the block of text you quoted and reread it.

It will vary with v, but not x_a as long as you don't do something weird.

What "end result" are you referring to? The end result that the same factor appears in both equations?

The end result is the the final pair of equations. The same factor in both equations is a middle stage in the process. But I am going to have a go at redoing it. We can continue to go over what you don't understand here though, until you are satisfied.

Note that since A and B's worldlines are assumed to lie along the x=0 axis of their respective frames, the event of the photon being colocated with A is equivalent to the event of the photon crossing A's x=0 axis, and likewise for B...this means your two equations can be written as:

(time of photon crossing B's x=0 axis in A's frame) = (factor)*(time of photon crossing B's x=0 axis in B's frame)

(time of photon crossing A's x=0 axis in B's frame) = (factor)*(time of photon crossing A's x=0 axis in A's frame)

So maybe now you can see why it wasn't a tangent when I asked in the previous post if you'd agree with a hypothetical guy making the argument that because of the "interchangeability" of the two frames, we could also assume the same factor will appear in the following two equations:

(time of photon crossing B's x=5 axis in A's frame) = (factor)*(time of photon crossing B's x=5 axis in B's frame)

(time of photon crossing A's x=5 axis in B's frame) = (factor)*(time of photon crossing A's x=5 axis in A's frame)

All that's changed in these two equations is that x=0 has been replaced with x=5. Please tell me whether you'd be confident about whether this guy's argument is right or wrong, without doing any SR calculations to check explicitly what factor appears in each. If you're unsure, that suggests that you don't really have good justification for being confident that "interchangeability" means the factor should be the same in your own two equations.

Actually no, it convinces me more that you have gone off on a tangent.

A and B are colocated at (0,0) ie (x=0,t=0) and (x'=0,t'=0) is the same event, so there is agreement about that event. There is no agreement about (5,0).

So I am pretty confident that "this guy's argument is wrong". (It might be right if the scenario is changed enough, but then it would be another scenario.)

Agree on another event for what purposes? Do you mean some of the variables in equations in your derivation would no longer represent the interval between 0,0 and some specific event like the photon crossing B's worldline (i.e. the coordinate of the latter event), but would instead represent the interval between a different event other than 0,0 on one side of the interval, but the same event on the other side? I think I'd need to see you go through at least some of the same steps of the derivation as in post 397, but with whatever altered assumptions about "another event" you want to make, in order to follow what you're talking about here.

Well, agree on another event for the purposes that your friend "this guy" could make the claim he did above about (5,0). Not that I know why they would do that.

However, as I said before: "When I redo the equations with v as a velocity rather than a speed, I will do them all with only intervals involved, rather than aiming for a coordinate transformation."

cheers,

neopolitan

 

[JesseM]

The end result is the the final pair of equations. The same factor in both equations is a middle stage in the process. But I am going to have a go at redoing it. We can continue to go over what you don't understand here though, until you are satisfied.

We seem to be going in circles here. You've been saying that the magic word "interchangeable" somehow justifies the claim that the factors in your two equations should be the same, but then you said "By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values)." So here again it sounds like you're asserting what you're supposed to be proving, that "interchangeability" somehow will lead to the same "end result" (keeping in mind that what you got for the 'end result' itself depended on the earlier step where you were supposed to show that the factors would be the same in the two equations you wrote). Can you define "interchangeability" in a way that doesn't make reference to any steps in your derivation that happened after the step where you assume the factors are in fact the same in both equations?

Actually no, it convinces me more that you have gone off on a tangent.

A and B are colocated at (0,0) ie (x=0,t=0) and (x'=0,t'=0) is the same event, so there is agreement about that event. There is no agreement about (5,0).

Huh? The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=t_a in A's frame) and the colocation of B and the photon (at x'=0, t=t'_b in B's frame). Of course the time coordinate of either event in a given frame can be understood as the time interval between that event and (0,0), but then exactly is true about the time coordinate of either event of the photon crossing the x=5 axis of one of the frames. Perhaps there is something in what you mean by "interchangeable" that equations involving the time coordinates of a pair of events in both frames can only be considered interchangeable if the two events happened on the worldlines of observers at rest in each frame who crossed paths at (0,0), but if so nothing you have written about interchangeability so far even hinted at such a requirement. And what about observers who crossed paths at (0,0) but who are not at rest? What if we considered two observers who did cross paths there, with the first observer traveling at velocity V in the A frame and the second traveling at the same velocity V in the B frame? Then if we defined our two events in terms of where the light crossed each of these observer's paths, then without doing any numerical calculations do you think "interchangeability" means the factors in the equations relating the time coordinates of these two events in each frame would be the same?

Even if your answer is once again no, I say that these questions are not "tangential" because they help pin down exactly what you do or do not mean by the slippery word "interchangeable" and how it's supposed to be relevant to showing the factors are the same in those equations--this would be unnecessary if you would be willing to actually spell this stuff out, but so far you've been exceedingly vague. Remember my other request from the "tangent" post which you just ignored:

You really need to provide a step-by-step proof where each statement clearly follow from the last, and you show specifically where the two postulates of relativity come into play (since I think you'd agree that if we came up with a coordinate transformation that didn't respect these postulates the factors in your equations would not be the same), and what the relevance is of the fact that the two events in question are the events of a photon's worldline intersecting each frame's x=0 axis.

Here's another tack: if physicists say that certain things are "interchangeability", they are usually referring to some sort of symmetry where we see that something (like a dynamical equation or the values of some variables) remains the same under a certain transformation of labels (like which charge you call positive and which you call negative, or which of two frames you label A and B, etc.) Is this the sort of thing you're referring to here? If so, what is the specific thing that remains unchanged under a switch of which frame we call A and B, and how does this relate to proving that the factors will be the same in both equations?

You are part of the way there. I mean what you wrote in the corrected sentence, plus the fact that both A and B see that (0,0) event as (0,0), ie x=0,t=0 is the same event as x'=0,t'=0. There's agreement between A and B as to that event.

They can agree on another event rather than (0,0) if it is convenient to them as well, it will just make the coordinate transformation a little different.

Agree on another event for what purposes? Do you mean some of the variables in equations in your derivation would no longer represent the interval between 0,0 and some specific event like the photon crossing B's worldline (i.e. the coordinate of the latter event), but would instead represent the interval between a different event other than 0,0 on one side of the interval, but the same event on the other side? I think I'd need to see you go through at least some of the same steps of the derivation as in post 397, but with whatever altered assumptions about "another event" you want to make, in order to follow what you're talking about here.

Well, agree on another event for the purposes that your friend "this guy" could make the claim he did above about (5,0). Not that I know why they would do that.

I have no idea what this comment has to do with the questions in my post above--what "purposes" are you talking about? It's difficult to communicate with you when you a) often seem to conceptualize things in very idiosyncratic ways, and likewise seem to have your own idiosyncratic vocabulary for describing your own concepts, and b) don't seem to be aware of the fact that these ways of thinking/talking are idiosyncratic, or just not make allowances for the fact that they are, and thus you often act as though you expect others to pick up on your meaning from a few enigmatic keywords, that somehow you think the meaning you infer from these words is the "natural" one that everyone else should infer by default too (much like with your use of the word 'interchangeable' and how you immediately dismissed my questions as a 'tangent' when I didn't read your mind and detect all the subtle nuances of what you meant by that, or like when you expected me to immediately pick up on exactly what you meant by the phrase 'number of ticks' in that other recent post...this seems to happen time and time again in my discussions with you).

 

[neopolitan]

We seem to be going in circles here. You've been saying that the magic word "interchangeable" somehow justifies the claim that the factors in your two equations should be the same, but then you said "By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values)." So here again it sounds like you're asserting what you're supposed to be proving, that "interchangeability" somehow will lead to the same "end result" (keeping in mind that what you got for the 'end result' itself depended on the earlier step where you were supposed to show that the factors would be the same in the two equations you wrote). Can you define "interchangeability" in a way that doesn't make reference to any steps in your derivation that happened after the step where you assume the factors are in fact the same in both equations?

Ok, you'll probably not like it, but I am going to have to jump past the factors stage to the end result which, for me, is:

x_b'=\gamma . (x_a - vt_a)

and

x_a=\gamma . (x_b' + vt_b')

(There is a similar pair for time, but the same argument will apply to that pair as to this pair.)

In this pair, can you see that all _b terms are also primed and all _a terms are unprimed. So we could, if we wanted to, drop the subscripts leaving us with:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')


Alternatively, we could swap the A and B terms (what we called A before is now B and what we called B before is now A). Our v was (implicitly) defined as positive in the direction that B moves away from A, and negative in the direction that A moves away from B. We keep the same priming notation, since we have shifted focus from the erstwhile A to the new A.

This gives us:

x_a=\gamma . (x_b' - vt_b')

and

x_b'=\gamma . (x_a + vt_a)

Again all our _b terms are primed and all our _a are unprimed. So we could drop our subscripts, giving:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')


A and B are interchangeable labels. The possible confusion here is that in my idiosyncratic way, I have considered A and B to be labels. You seem to want to have more concrete (and thus less general) determinations of what A and B are.

Huh? The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=t_a in A's frame) and the colocation of B and the photon (at x'=0, t=t'_b in B's frame). Of course the time coordinate of either event in a given frame can be understood as the time interval between that event and (0,0), but then exactly is true about the time coordinate of either event of the photon crossing the x=5 axis of one of the frames. Perhaps there is something in what you mean by "interchangeable" that equations involving the time coordinates of a pair of events in both frames can only be considered interchangeable if the two events happened on the worldlines of observers at rest in each frame who crossed paths at (0,0), but if so nothing you have written about interchangeability so far even hinted at such a requirement. And what about observers who crossed paths at (0,0) but who are not at rest? What if we considered two observers who did cross paths there, with the first observer traveling at velocity V in the A frame and the second traveling at the same velocity V in the B frame? Then if we defined our two events in terms of where the light crossed each of these observer's paths, then without doing any numerical calculations do you think "interchangeability" means the factors in the equations relating the time coordinates of these two events in each frame would be the same?

I've mentioned a few times that there are only three colocation events. I honestly thought that I didn't need to make explicit that the colocation of A and B was at (0,0). I do think I have done it anyway - posts https://www.physicsforums.com/showpost.php?p=2232756&postcount=397" - sadly I don't have time to go on but I think there are more references where I have stated that A and B are colocated at t=0,t'=0. In at least one of those I have even made explicit that that event is (0,0).

As for the tangent, your asking questings for clarifications doesn't constitute a tangent, but your assertion that events associated with photons crossing the x=5 axis can be profitably used in my scenario indicates that you have gone off on a tangent.

The difference is like between these:

"I don't understand what you are saying, what are you saying"

and

"You are claiming this wrong thing, you are wrong"

when I never actually claimed the wrong thing you asserted that I claimed. In the real life example where I said it seems you were going off on a tangent, it might not have been so clear that you were asserting that I claimed something that I had not claimed. That's why I said you were going off on a tangent rather than saying you were making unfounded assertions.

But yes, we are going around in circles.

cheers,

neopolitan

 

[JesseM]

Ok, you'll probably not like it, but I am going to have to jump past the factors stage to the end result which, for me, is:

x_b'=\gamma . (x_a - vt_a)

and

x_a=\gamma . (x_b' + vt_b')

(There is a similar pair for time, but the same argument will apply to that pair as to this pair.)

In this pair, can you see that all _b terms are also primed and all _a terms are unprimed. So we could, if we wanted to, drop the subscripts leaving us with:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')Alternatively, we could swap the A and B terms (what we called A before is now B and what we called B before is now A). Our v was (implicitly) defined as positive in the direction that B moves away from A, and negative in the direction that A moves away from B. We keep the same priming notation, since we have shifted focus from the erstwhile A to the new A.

This gives us:

x_a=\gamma . (x_b' - vt_b')

and

x_b'=\gamma . (x_a + vt_a)

Again all our _b terms are primed and all our _a are unprimed. So we could drop our subscripts, giving:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')A and B are interchangeable labels. The possible confusion here is that in my idiosyncratic way, I have considered A and B to be labels. You seem to want to have more concrete (and thus less general) determinations of what A and B are.

Sigh. Of course I understand that once you have the final Lorentz transformation (which is not what you actually derive, but never mind that for now), A and B are interchangeable labels, "interchangeable" in the sense that the transformation equations still work fine if you switch the subscripts for A and B and also switch which frame v is defined in terms of. How does this have anything whatsoever to do with the question of how you intend to show that the factor relating the two specific events you chose--and only those two events, not two other events like the ones I mentioned--should be the same in the two equations you wrote down, without assuming the Lorentz transformations to begin with? Do you really not understand what it means to "show" something in a derivation without engaging in circular reasoning?

I've mentioned a few times that there are only three colocation events. I honestly thought that I didn't need to make explicit that the colocation of A and B was at (0,0).

I'm not talking about how many colocation events appear in your overall derivation, I'm just talking about the two events that appear in these two equations you wrote earlier:

(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)

Can you please stop your mind from wandering all over the place and imagining I am talking about all these other aspects of your derivation, when I think I made it pretty clear I'm just talking about the specific question of how in these two equations you intend to show that the factor must be the same? If you don't in fact believe that your notion of "interchangeability" is sufficient to show this specific thing, and you haven't actually thought of a good way to show this yet, then that's fine, just say so. But if you do, then please stick to this specific topic for now.

As for the tangent, your asking questings for clarifications doesn't constitute a tangent, but your assertion that events associated with photons crossing the x=5 axis can be profitably used in my scenario indicates that you have gone off on a tangent.

I didn't say it "can be profitably used" in your scenario. I was saying that your argument about "interchangeability" is so vague that there's no apparent reason it shouldn't apply to those two events just as much as the two events you chose, and thus it "proves too much". You could respond by actually elaborating on what you meant by "interchangeability" (as I keep asking you to do but you keep not doing), giving an expanded definition which would show why interchangeability demands the factor be the same in your two equations but not in my closely analogous equations. Likewise for my question about another fairly analogous pair of events in the last post:

And what about observers who crossed paths at (0,0) but who are not at rest? What if we considered two observers who did cross paths there, with the first observer traveling at velocity V in the A frame and the second traveling at the same velocity V in the B frame? Then if we defined our two events in terms of where the light crossed each of these observer's paths, then without doing any numerical calculations do you think "interchangeability" means the factors in the equations relating the time coordinates of these two events in each frame would be the same?

Again, if your "proof by interchangeability" is supposed to make any sense, you should have some clear explanation as to why "interchangeability" does or doesn't imply that the factors in equations relating these two events would be the same, and if it doesn't what the crucial difference is that makes it imply that for your pair of events but not this pair of events.

The difference is like between these:

"I don't understand what you are saying, what are you saying"

and

"You are claiming this wrong thing, you are wrong"

when I never actually claimed the wrong thing you asserted that I claimed.

I never asserted that you claimed anything about the x=5 example, my point was that if you didn't have a clear idea of why interchangeability is supposed to imply one thing about your equations and something different about the equations I wrote, then the concept is too vague to make for a convincing demonstration that the factors should in fact be the same in your own pair of equations.

 

[neopolitan]

Going back to this point:

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.

I am going to assume that I have not convinced you that there being one single factor is the default.

That means we still have two stopping points, and I have to address both of them.

From here I suggest that I provide a general scenario, rather than one which is limited by having an event which is simultaneous with colocation of A and B in one frame, and work from there, with as much rigour as I can muster.

Included in that rigour is a requirement to prove, by elimination of other options if necessary, that a single factor applies in the intermediate step which we have been discussing.

Is that right?

cheers,

neopolitan

 

[JesseM]

I am going to assume that I have not convinced you that there being one single factor is the default.

Not sure what you mean by "default". Certainly if someone had no familiarity with SR and saw your two equations, if forced to guess, they might by default choose the same factor in each. But if this is supposed to be a derivation you need to establish that it must be the same factor.

From here I suggest that I provide a general scenario, rather than one which is limited by having an event which is simultaneous with colocation of A and B in one frame, and work from there, with as much rigour as I can muster.

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

Included in that rigour is a requirement to prove, by elimination of other options if necessary, that a single factor applies in the intermediate step which we have been discussing.

Yes, that's what must be proved.

 

[neopolitan]

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

This statement means that you clearly have not grasped something. It's good to know this now, rather than later.

The light beam is totally arbitrary, because I intended a general proof.

Take a totally arbitrary light beam, let it pass A and B (order not really important), then select an event on that light beam (I didn't make it totally arbitrary, I restricted it to being simultaneous with colocation of A and B in one frame).

You then have four events:

colocation of A and B, at (0,0) - ie x=0,t=0 and x'=0,t'=0
the event I just talked about selecting
colocation of the light beam and A
colocation of the light beam and B

With these four events (plus the laws of physics) you have enough information to derive the Lorentz Transforms (or analogues thereof).

To make my proof more general, I need to select from the events along the light beam an event which is not necessarily simultaneous with (0,0) in either frame.

You don't seem to like me using the intervals (0,0)-event in one frame and (0,0)-event in the other frame - which are the coordinates of the event in one frame and the coordinates of the event in the other frame (perhaps I haven't managed to make it abundantly clear that that is what I am doing, despite having done a bit of prepartory work on the issue of coordinates basically being intervals - latest attempt was here). Therefore, I will select from the events along another arbitrary light beam another event which is also not necessarily simultaneous with (0,0) in either frame. Then I will do the derivation based on a totally arbitrary spacetime interval.

cheers,

neopolitan

 

[JesseM]

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

This statement means that you clearly have not grasped something. It's good to know this now, rather than later.

The light beam is totally arbitrary, because I intended a general proof.

What do you think I didn't grasp? Isn't that what I just said, "a totally arbitrary light beam"?

You then have four events:

Yes, in your overall derivation you have four events. But in those two equations where you say the factor is the same, only two events appear. Can we please try to stick to the subject of those two equations and the factor in them, unless you are willing to actually acknowledge that your previous "interchangeability" argument isn't sufficient to establish that the factor should be the same?

You don't seem to like me using the intervals (0,0)-event in one frame and (0,0)-event in the other frame

Where did you like the idea that I don't like that? In post 430 I wrote:

The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=ta in A's frame) and the colocation of B and the photon (at x'=0, t=t'b in B's frame). Of course the time coordinate of either event in a given frame can be understood as the time interval between that event and (0,0), but then exactly is true about the time coordinate of either event of the photon crossing the x=5 axis of one of the frames. Perhaps there is something in what you mean by "interchangeable" that equations involving the time coordinates of a pair of events in both frames can only be considered interchangeable if the two events happened on the worldlines of observers at rest in each frame who crossed paths at (0,0), but if so nothing you have written about interchangeability so far even hinted at such a requirement.

If you're talking more generally about the final equations you derive, my objection is not that you're talking about intervals from (0,0) to some other event, it's that for different variables in those equations the "other event" is different, whereas in the Lorentz transformation equations every variable is supposed to refer to the space and time intervals between the same pair of events in different frames (or the space and time coordinates of the same single event in different frames).

 

[neopolitan]

What do you think I didn't grasp? Isn't that what I just said, "a totally arbitrary light beam"?

I'm sorry, I was taking your comment to have been meant dismissively. If that was not your intention then ok.

Look back at the whole post (I know you have read it all before responding). I said:

Take a totally arbitrary light beam, let it pass A and B (order not really important), then select an event on that light beam (I didn't make it totally arbitrary, I restricted it to being simultaneous with colocation of A and B in one frame).

You then have four events:

colocation of A and B, at (0,0) - ie x=0,t=0 and x'=0,t'=0
the event I just talked about selecting
colocation of the light beam and A
colocation of the light beam and B

With these four events (plus the laws of physics) you have enough information to derive the Lorentz Transforms (or analogues thereof).

To make my proof more general, I need to select from the events along the light beam an event which is not necessarily simultaneous with (0,0) in either frame.

I start with "the event", the other (implied) event which allows me to obtain the coordinates of this event is both frames is what I call an "agreed" event, ie the colocation of A and B at (0,0).

The other two events (colocation of A and the photon) and (colocation of B and the photon) are used for the information we can glean from them, namely the time elapsed when they occur since A and B were colocated.

I'm not probably going to get this across properly right now, but I am glad to have a little better insight into what has not been conveyed correctly.

cheers,

neopolitan

 

[JesseM]

Look back at the whole post (I know you have read it all before responding).

Ugh, please don't make comments like this, it comes across as patronizing. Especially when you have a history of imagining I didn't get some point of yours because I "didn't read the whole post" or "broke it up into pieces" when in fact the issue was your idiosyncratic communication style where you seem to have expected others to interpret some random phrase in exactly the way you were thinking of without your having to actually explain your meaning (again think of the 'number of ticks' thing I responded to at the end of post 413)

I start with "the event", the other (implied) event which allows me to obtain the coordinates of this event is both frames is what I call an "agreed" event, ie the colocation of A and B at (0,0).

The other two events (colocation of A and the photon) and (colocation of B and the photon) are used for the information we can glean from them, namely the time elapsed when they occur since A and B were colocated.

Here you just seem to be throwing out random statements about your derivation (statements which I think I already understood, unless I am misinterpreting your words in some way), without explaining their relevance to what we are actually discussing. Is this supposed to somehow address either my point that only two events appear in the pair of equations where you said the factor is the same, or my point that the variables in your final equations don't all refer to a consistent pair of events?

 

[neopolitan]

Here you just seem to be throwing out random statements about your derivation (statements which I think I already understood, unless I am misinterpreting your words in some way), without explaining their relevance to what we are actually discussing. Is this supposed to somehow address either my point that only two events appear in the pair of equations where you said the factor is the same, or my point that the variables in your final equations don't all refer to a consistent pair of events?

I took one section of your post #434 and addressed it in post #435. Here:

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

In that section, you specifically refer to two events twice (I accept that I may be reading "neither" a little too strictly).

In second part of post #437, I go over what two events I am talking about, which is (one event on the path of a totally arbitrary light beam) and (the colocation of A and B). I also point out that I use information associated with two more events which are also on the path of the totally arbitrary light beam.

All up that makes three events on the path of the totally arbitrary light beam.

Since I am not talking about two events on the path of the light beam, it is clear to me that something is amiss if you think I am talking about two events on the light beam. I could agree with one or three and I might even understand what you were talking about if you said four.

I will do my best to provide a derivation in which the consistency of the pair of events is very clear. Are we ready to proceed to that?

cheers,

neopolitan

 

[JesseM]

I took one section of your post #434 and addressed it in post #435. Here:

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

Question: what specific equations do you imagine I was referring to when I said "in your two equations" in that post? (see below if it wasn't clear)

In that section, you specifically refer to two events twice (I accept that I may be reading "neither" a little too strictly).

In second part of post #437, I go over what two events I am talking about, which is (one event on the path of a totally arbitrary light beam) and (the colocation of A and B). I also point out that I use information associated with two more events which are also on the path of the totally arbitrary light beam.

All up that makes three events on the path of the totally arbitrary light beam.

Since I am not talking about two events on the path of the light beam, it is clear to me that something is amiss if you think I am talking about two events on the light beam. I could agree with one or three and I might even understand what you were talking about if you said four.

Let me just repeat my comment from post 432 again:

I'm not talking about how many colocation events appear in your overall derivation, I'm just talking about the two events that appear in these two equations you wrote earlier:

(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)

Can you please stop your mind from wandering all over the place and imagining I am talking about all these other aspects of your derivation, when I think I made it pretty clear I'm just talking about the specific question of how in these two equations you intend to show that the factor must be the same? If you don't in fact believe that your notion of "interchangeability" is sufficient to show this specific thing, and you haven't actually thought of a good way to show this yet, then that's fine, just say so. But if you do, then please stick to this specific topic for now.

Are you claiming that more than two events appear in those two specific equations? I only see "colocation of B and photon" on both sides of the first equation, and "colocation of A and photon" on both sides of the second (and these equations are quoted from your own post 427).

Note that I had said something similar in the earlier post 430:

Huh? The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=t_a in A's frame) and the colocation of B and the photon (at x'=0, t=t'_b in B's frame).

...but you didn't seem to get it in your response which was why I elaborated in post 432. Then in post 436 I again repeated the fact that I was just talking about those two specific equations where you said the factor was supposed to be the same:

Yes, in your overall derivation you have four events. But in those two equations where you say the factor is the same, only two events appear. Can we please try to stick to the subject of those two equations and the factor in them, unless you are willing to actually acknowledge that your previous "interchangeability" argument isn't sufficient to establish that the factor should be the same?

And then in post 438 I said:

Is this supposed to somehow address either my point that only two events appear in the pair of equations where you said the factor is the same, or my point that the variables in your final equations don't all refer to a consistent pair of events?

...so I don't think the problem here is that I've failed to make clear I'm just talking about the events that appear in those specific equations.

I will do my best to provide a derivation in which the consistency of the pair of events is very clear. Are we ready to proceed to that?

Does "the consistency of the pair of events" mean the same thing as your earlier talk of "interchangeability"? If so please address my questions from post 430. If not, please tell me whether you are abandoning your earlier claims that "interchangeability" was sufficient to rigorously establish that the factor should be the same, or whether you stick by that claim.

 

[neopolitan]

Realisation dawns. I think I am talking at cross purposes.

I think that you are talking about these equations:

(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)

In terms of the scenario, this does work. If you change the scenario, it won't work.

It requires that there is a colocation of A and B at (x,t)=(0,0), (x',t')=(0,0).

In my recent emails, I wasn't talking about those pair of events or those equations, I was talking about the pair of events that the final equations refer to (the event whose coordinates we have in one frame and want to find in the other frame, and the colocation of A and B). This is entirely my fault.

I really hope this is the source of confusion, because otherwise I am completely stumped as to what the issue is.

"Interchangeability" is not clearing things up. I know what I mean, but I seem unable to convey that understanding to anyone else. Therefore it is not a useful concept. Therefore while I don't abandon the claim, I don't intend to use the concept.

cheers,

neopolitan

 

[JesseM]

In terms of the scenario, this does work. If you change the scenario, it won't work.

It requires that there is a colocation of A and B at (x,t)=(0,0), (x',t')=(0,0).

In my recent emails, I wasn't talking about those pair of events or those equations, I was talking about the pair of events that the final equations refer to (the event whose coordinates we have in one frame and want to find in the other frame, and the colocation of A and B). This is entirely my fault.

I really hope this is the source of confusion, because otherwise I am completely stumped as to what the issue is.

Yes, that's it, I think we've finally cleared up the miscommunication here (whew!)

"Interchangeability" is not clearing things up. I know what I mean, but I seem unable to convey that understanding to anyone else. Therefore it is not a useful concept. Therefore while I don't abandon the claim, I don't intend to use the concept.

OK, fair enough.

 

[neopolitan]

I've been rather busy, but trying to look at this during the quiet moments.

http://www.geocities.com/neopolitonian/checking.html"is an attempt at a proof for one of the stopping points.

cheers,

neopolitan

PS I previously only posted the proof which attempts to justify my "some factor" assumption (where "some factor" is G₁ = G₂). It was part of a longer thing which I put together in word and tried to save to pdf. The distiller I have here resulted in a 8M pdf file which I was not happy with. I'll have a go later with another distiller I have at home.

Zipped, the size of that file becomes 2M. If you can handle a file that large, it is http://www.geocities.com/neopolitonian/benefits.part1.v1.zip".

 

[neopolitan]

Referring to last post and the http://www.geocities.com/neopolitonian/checking.html" therein, I noted that a line was missing from one of the text images (I had fixed it in a later variant, but it is on a work computer). Until I can fix that, I have added a line in very ordinary hypertext script.

For JesseM, the proof which may be of more interest is at the end, following the title "Testing hypothesis that A and B measure space differently".

cheers,

neopolitan

 

[neopolitan]

I have managed to reduce the size of the pdf file, down to 500k. It is http://www.geocities.com/neopolitonian/benefits.part1.v1.pdf". Zipping it makes little difference.

cheers,

neopolitan

(The zipped file has been removed and the html file now just links to the pdf file.)