I Proper (and coordinate) times re the Twin paradox

[Grimble]

May I thank everyone for being so patient and forbearing with me, it is very much appreciated.

 

[Grimble]

Neither of which makes it clear what you consider to be unique.

The way I understand English the difference between - "a unique set of coordinates" and "a set of unique coordinates" it is the placement of the word unique that is important:
When I say . "...unique set..." it is the set of coordinates that is unique; while putting 'unique' next to coordinates implies that I am referring to a set of coordinates where each coordinate is unique, whether that be unique in that set (which has to be true) or is unique amongst coordinates from any set.

 

[Grimble]

If you are using "interval" in a normal English sense then you owe it to us to define for us what that means in a scientific sense. We cannot know what specific meaning you intend by using the term.

Yes, I see this, but it isn't easy to try and do that for every word that may be used as so many everyday English terms have specific Scientific meanings.

It can be difficult to try and find the words to express what one means without misusing some term or other...

 

[jbriggs444]

The way I understand English the difference between - "a unique set of coordinates" and "a set of unique coordinates" it is the placement of the word unique that is important:
When I say . "...unique set..." it is the set of coordinates that is unique; while putting 'unique' next to coordinates implies that I am referring to a set of coordinates where each coordinate is unique, whether that be unique in that set (which has to be true) or is unique among coordinates from any set.

In context, we had a set of events, a set of coordinate systems, and a set of coordinate tuples, each of which has four "coordinates". There are a lot of ways to have meant "unique". More than the two that one word placement can distinguish between.

 

[Grimble]

Now, to go back to an earlier thought experiment

Think of proper time as something that we observe: Say we design our clock so that every time it ticks it punches a hole in a piece of paper somewhere inside; we start with a fresh piece of paper at event A and remove it at event B. How many holes are there in the piece of paper? That's a simple direct observation; all observers everywhere will agree about the answer without any rigamarole about reference frames or relative velocity or time dilation. We call the number of holes in the piece of paper "the proper time along the path from A to B", and it is a fact that has nothing to do with any other observers and their notions of time, distance and speed.

please shew me what is wrong with my logic here...

Twin B is at rest in his inertial frame of reference. After 10 seconds 10 holes will be punched in his paper.
Twin B is at rest in an inertial frame of reference and measures 10 seconds proper time.
After 10 seconds, Twin A, also at rest in her inertial frame of reference, has 10 holes punched in her paper and measures 10 seconds proper time.

Now the Twins are separating at 0.6c.

Each twin will measure the other twin who is moving at 0.6c to be time dilated and measure γt = 12.5 seconds (γ = 1.25, t = 10) to have passed (coordinate time?), for the other traveling twin's clock, yet still count only 10 holes in their paper...

 

[PeroK]

Now, to go back to an earlier thought experimentplease shew me what is wrong with my logic here...

Twin B is at rest in his inertial frame of reference. After 10 seconds 10 holes will be punched in his paper.
Twin B is at rest in an inertial frame of reference and measures 10 seconds proper time.
After 10 seconds, Twin A, also at rest in her inertial frame of reference, has 10 holes punched in her paper and measures 10 seconds proper time.

Now the Twins are separating at 0.6c.

Each twin will measure the other twin who is moving at 0.6c to be time dilated and measure γt = 12.5 seconds (γ = 1.25, t = 10) to have passed (coordinate time?), for the other traveling twin's clock, yet still count only 10 holes in their paper...

... back to square one, the basics of SR. No universal time, no universal simultaneity. Measuring the time of an event in one frame requires the time and position of the event in another frame.

 

[Dale]

How are you going to instruct them to punch holes in the paper? I.e. What is the stopping criteria. If your criteria is simply that they should each punch 10 holes then clearly they will each have 10 holes

Take a piece of paper and a ruler. Draw one line 10 cm long. From one end of that line, draw another line at an angle less than 45 deg, also 10 cm long. How long is each line?

 

[Grimble]

OK. With two twins let us specify the movement is measured along the mutual x axes, as is the convention in all such diagrams.

As for punching the holes, that was not my invention I borrowed that from Nugatory's post (#13). Let us say that each twin has a light clock with the mirror set at 0.5 light seconds from the light and that they punch a hole each time the light pulse returns.

 

[stevendaryl]

OK. With two twins let us specify the movement is measured along the mutual x axes, as is the convention in all such diagrams.

As for punching the holes, that was not my invention I borrowed that from Nugatory's post (#13). Let us say that each twin has a light clock with the mirror set at 0.5 light seconds from the light and that they punch a hole each time the light pulse returns.

Okay, I understand the scenario. You have two twins, each is punching one hole per second (according to their own lightclock). What's the question?

Let's assume that the twins start off together for their first hole punch. Then let's identify a few events:

• e_1: They each punch their first hole.
• e_{A2}: Alice (the first twin) punches her second hole.
• e_{B2}: Bob (the second twin) punches his second hole.

In a reference frame in which Alice is momentarily at rest, e_{A2} occurs before e_{B2}. In a reference frame in which Bob is momentarily at rest, e_{B2} occurs before e_{A2}. So what question are you asking?

 

[Dale]

As for punching the holes, that was not my invention I borrowed that from Nugatory's post (#13). Let us say that each twin has a light clock with the mirror set at 0.5 light seconds from the light and that they punch a hole each time the light pulse returns.

That isn't the question I asked. The question is how do they know when to stop punching holes? If you just instruct them to punch 10 holes and stop then of course both will have 10 holes.

 

[Vitro]

This is where it goes wrong:

...and measure γt = 12.5 seconds (γ = 1.25, t = 10) to have passed (coordinate time?), for the other traveling twin's clock...

That's not correct, each twin measures 12.5 seconds (coordinate time) to have passed on their own clock in order for the other twin's paper to show 10 holes (10 seconds of proper time).

The rule of thumb is: if you can measure it with a single clock then it's a proper time, if you need two (or more) clocks then it's a coordinate time. Alternatively, if you measure it at the same location it's a proper time, if you measure it at different locations it's a coordinate time.

 

[Nugatory]

The rule of thumb is: if you can measure it with a single clock then it's a proper time, if you need two (or more) clocks then it's a coordinate time. Alternatively, if you measure it at the same location it's a proper time, if you measure it at different locations it's a coordinate time.

And if either or both events don't happen at the location of the clock, it is always a coordinate time.

 

[Grimble]

That's not correct, each twin measures 12.5 seconds (coordinate time) to have passed on their own clock in order for the other twin's paper to show 10 holes (10 seconds of proper time).

I am sorry but I do not understand what you are saying here...

Surely each twin is measuring proper time on the clock they are holding in their reference frame; the coordinate time is that time, transformed by the Lorentz Transformation Equations, from proper time to coordinate time (multiplying it by gamma...)

While each twin, at rest in their inertial frame of reference will measure the 10 holes punched by their clocks in proper time?

 

[jbriggs444]

I am sorry but I do not understand what you are saying here...

Surely each twin is measuring proper time on the clock they are holding in their reference frame; the coordinate time is that time, transformed by the Lorentz Transformation Equations, from proper time to coordinate time (multiplying it by gamma...)

[emphasis mine]

The Lorentz transforms contain more than a multiplication by gamma. There is also a term for relativity of simultaneity.

 

[Grimble]

That isn't the question I asked. The question is how do they know when to stop punching holes? If you just instruct them to punch 10 holes and stop then of course both will have 10 holes.

I presume they will continue to punch holes until they are switched off.
The important point is that both clocks will punch 10 holes.
They are each at rest in an inertial frame of reference and so are keeping proper time for that clock. The clocks are identical, the laws of science are identical, the times measured will presumably be identical - what reason is there for them to be different?

It seems to me that if another clock C, were permanently mid way between A and B, then their relative velocities would be ^v/₂ and -^v/₂ with respect to the clock C. And C would measure the same time dilation for each A and B and the same length contraction for their frames(?)

 

[PeroK]

I presume they will continue to punch holes until they are switched off.
The important point is that both clocks will punch 10 holes.
They are each at rest in an inertial frame of reference and so are keeping proper time for that clock. The clocks are identical, the laws of science are identical, the times measured will presumably be identical - what reason is there for them to be different?

It seems to me that if another clock C, were permanently mid way between A and B, then their relative velocities would be ^v/₂ and -^v/₂ with respect to the clock C. And C would measure the same time dilation for each A and B and the same length contraction for their frames(?)

The basis of your argument is (assuming a third clock at C:)

In A's frame, A's clock reaching $10s$ coincides with C's clock reaching $8s$ (say).

In B's frame, B's clock reaching $10s$ coincides with C's clock reaching $8s$.

Therefore, in A's frame: A's clock reaching $10s$, B's clock reaching $10s$ and C's clock reaching $8s$ are all simultaneous. Hence, simultaneity is not relative and SR is wrong?

Although, given this, C's clock must also read $10s$ as well (just put another clock that stays half-way between A and C) and there's no time dilation either.

 

[Dale]

I presume they will continue to punch holes until they are switched off.
The important point is that both clocks will punch 10 holes.

As you have stated it they will both punch an infinite number of holes, not just 10.

It is important that you actually answer this question, not avoid it. They start punching holes when they are together, they each punch a hole when a local clock that they carry ticks 1 s, but how do they know when to stop?

 

[Ibix]

The clocks are identical, the laws of science are identical, the times measured will presumably be identical - what reason is there for them to be different?

Two rulers have their zero markings aligned but do not point in the same direction. The rulers are identical, the distances measured will presumably be identical - what reason is there for the 10cm marks to be in different places?

 

[Herman Trivilino]

OK. With two twins let us specify the movement is measured along the mutual x axes, as is the convention in all such diagrams.

Yes, but when you draw a spacetime diagram of the situation A's x-axis is not parallel to B's x-axis. The reason is because they are in relative motion. They can't each be present at the punching of the other's tenth hole if they were each present at starting of the other's clock.

 

[Ibix]

Two rulers have their zero markings aligned but do not point in the same direction. The rulers are identical, the distances measured will presumably be identical - what reason is there for the 10cm marks to be in different places?

To expand on this a bit - both rulers and clocks are devices for measuring intervals along lines in spacetime. Rulers can only measure spacelike intervals and clocks can only measure timelike intervals. But notice those likes. There is no unique direction in spacetime that is Time. There are a whole family of directions which are timelike. So, generally, a clock does not necessarily measure what I choose to call time anymore than rulers are restricted to measuring what I choose to call forwards or sideways.

Clocks always measure an interval, but this may be something I call a mix of time and distance. Just as a ruler always measures distance, but this may be something I call a mix of forwards and sideways. The only special thing about Minkowski space is that you can't map the direction you call the future onto any of the spatial directions (or vice versa) by rotation because of the way the geometry is defined.

 

[Grimble]

The basis of your argument is (assuming a third clock at C:)

In A's frame, A's clock reaching $10s$ coincides with C's clock reaching $8s$ (say).

In B's frame, B's clock reaching $10s$ coincides with C's clock reaching $8s$.

Therefore, in A's frame: A's clock reaching $10s$, B's clock reaching $10s$ and C's clock reaching $8s$ are all simultaneous. Hence, simultaneity is not relative and SR is wrong?

Although, given this, C's clock must also read $10s$ as well (just put another clock that stays half-way between A and C) and there's no time dilation either.

If Observer C measures the coordinate time for clock A to equal the coordinate time for clock B, when A and B are traveling at the same speed relative to C, then is this not measuring equal times for A's clock and B's clock. Are their Lorentz transformations not the same?

 

[Grimble]

As you have stated it they will both punch an infinite number of holes, not just 10.

It is important that you actually answer this question, not avoid it. They start punching holes when they are together, they each punch a hole when a local clock that they carry ticks 1 s, but how do they know when to stop?

I'm sorry but I do not understand why you are asking that. Clocks continue to work ad infinitum...
A clock does not have to stop to take a reading from it?

 

[stevendaryl]

I'm sorry but I do not understand why you are asking that. Clocks continue to work ad infinitum...
A clock does not have to stop to take a reading from it?

Could you state more precisely what your question is? You have two different observers, Alice and Bob, each punching a hole in a paper at the rate of once per second (according to their own clocks). What's your question about it? In Alice's rest frame, Bob is punching slower than Alice is. In Bob's rest frame, it's the other way around.

 

[Dale]

I'm sorry but I do not understand why you are asking that. Clocks continue to work ad infinitum...
A clock does not have to stop to take a reading from it?

No, but you do have to have some rule about when you are going to take the reading. That is what you need to consider. The clock ticks 1, 2, 3, ... 946737, ... What is the criteria used to determine which of those infinite numbers is the reading?

Please think this through, don't dismiss it.

 

[Grimble]

In context, we had a set of events, a set of coordinate systems, and a set of coordinate tuples, each of which has four "coordinates". There are a lot of ways to have meant "unique". More than the two that one word placement can distinguish between.

Something is unique or it is not unique. There is no half-way unique or partly unique...
A unique set of coordinates is unlike any other set of coordinates.
A set of unique coordinates is made up of a multitude of coordinates each of which is unique...
You are trying to determine what makes them unique.

 

[jbriggs444]

Something is unique or it is not unique. There is no half-way unique or partly unique...

Uniqueness is a relative property. It is a property of an item within a collection. If you do not specify the collection, you have not specified the property.

Backing up and making that relevant in context...

Suppose that you have a worldline composed of a continuous sequence of events all of which are timelike separated from one another. Suppose further that you have singled out a coordinate system covering that set of events. Then each position on the worldline corresponds to a unique event -- no other event is at that position on the worldline and no other position on the worldline is at that event. Further, each event has a unique coordinate. No other coordinate tuple denotes that event and no other event has that coordinate tuple.

But now suppose that no coordinate system has been singled out. There is still a one to correspondence between positions along the worldline and events. But there is no longer a unique correspondence between events and coordinate tuples. An event can be associated with many coordinate tuples since there are many possible coordinate systems. A coordinate tuple can be associated with many possible events since there are many possible coordinate systems. The property of uniqueness has been lost.

 

[PeroK]

If Observer C measures the coordinate time for clock A to equal the coordinate time for clock B, when A and B are traveling at the same speed relative to C, then is this not measuring equal times for A's clock and B's clock. Are their Lorentz transformations not the same?

The Lorentz Transformations are not the same, as A and B are traveling in opposite directions relative to C.

 

[Grimble]

The Lorentz Transformations are not the same, as A and B are traveling in opposite directions relative to C.

And how does that make a difference to the results?
Both A and B are moving away from C, either can be a positive or negative displacement depending on how the observer in C observes them...

 

[Grimble]

Uniqueness is a relative property. It is a property of an item within a collection. If you do not specify the collection, you have not specified the property.

Oh for goodness sake!
What I said was

...a worldline is a succession of events that have a unique set of coordinates in each frame?

.
and that means a set of coordinates that is different from any other set of coordinates in the same frame.
It does not matter how each coordinate is different from any other coordinate as I specifically referred to a unique set of coordinates.

 

[stevendaryl]

Could somebody please repeat what the question is?