I Proper (and coordinate) times re the Twin paradox

[Battlemage!]

No, the "length" of the world-line is the proper time. There is no other concept of "length" in Minkowski space (or a general space-time). That you are drawing your Minkowski diagram on a piece of paper with a natural Euclidean metric is irrelevant. Furthermore, in GR (or even just curvilinear coordinates on Minkowski space) this completely depends on your choice of coordinates and what form the metric takes in those coordinates. Given your avatar, you must be aware of this already.

On a slightly off topic tangent, isn't mc² also the "length" of the energy/momentum equation? I.e.

$mc^2 = \sqrt {E^2 - (pc)^2}$

I wonder because I'm pretty sure mc² shares the same property of proper time, that it is universally agreed upon by all inertial observers, and time and energy tend to have a very intense romantic relationship.

 

[robphy]

On a slightly off topic tangent, isn't mc² also the "length" of the energy/momentum equation? I.e.

$mc^2 = \sqrt {E^2 - (pc)^2}$

I wonder because I'm pretty sure mc² shares the same property of proper time, that it is universally agreed upon by all inertial observers, and time and energy tend to have a very intense romantic relationship.

It is.
So, in fact, one can draw energy-momentum diagrams of collisions (analogous to spacetime-diagrams).
From my Insight, https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ , here is an elastic collision of two particles.
[edit: added italicized words to following sentence]
Note the analogy of a totally-inelastic collision to the clock-effect... although I didn't draw in the "mass-diamonds" to describe the invariant-mass of the system for the center-of-momentum frame--that is, suppose instead of the elastic collision, we had the incident particles collide totally-inelastically to form a single particle in the center-of-momentum frame. That single particle has rest-mass $\sqrt{800}\approx 28.28 > 12+8$.

 

[Grimble]

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.

Why not just count the 'ticks' of the cloak: the time read from the clock face or display?

 

[Nugatory]

Why not just count the 'ticks' of the cloak: the time read from the clock face or display?

Of course that works too.

 

[Grimble]

Perhaps, gentle folk, it would be useful to correct my 'misunderstanding' of proper time?

As I have understood it, proper time is the time measured by a clock between two events on that clock's world line.
Suppose another clock takes a different path between those same two events, then the second clock (that of the traveller for example) would read a different time at event 2. Clock 2's world line also passes through events 1 & 2. What then of clock 2's proper time? Can it also be the time read on its clock? For then we have two different proper times between the same two events and I understood that proper time was invariant...?

Is the proper time interval not the same as the Spacetime interval?

 

[Dale]

Can it also be the time read on its clock?

Yes

For then we have two different proper times between the same two events

Yes

I understood that proper time was invariant...?

Yes.

The mistake you are making is thinking that proper time is a relationship between two events. It is not. It is a property of a worldline between two events.

Consider Euclidean geometry. Draw two points on the page, and two curves joining them. The two curves have different lengths, and those lengths are invariant under rotations and translations of the page.

 

[Herman Trivilino]

Is the proper time interval not the same as the Spacetime interval?

When the intervals are timelike, which they are in this case. Invariant doesn't mean that every path between the events has the same value for the interval. It means that all observers will agree on the value of the interval for any specific path.

If one twin ages 10 years both twins will agree that he aged 10 years. If the other twin ages 20 years both twins will agree that she aged 20 years.

 

[Ibix]

Is the proper time interval not the same as the Spacetime interval?

Yes and no. It's a bit of a cheat (and is misleading you here) to talk of the interval between two events. What we actually mean when we talk of the interval between two events is, more precisely, the interval along a straight line path between two events. Just as we'd casually talk about the distance between two points, and you'd understand straight-line distance unless we specified otherwise.

Your proper time is the interval along your personal worldline - which will only be a straight line if you never accelerate. But one of the twins accelerates; the traveller moves along two straight line segments while the stay-at-home travels on one. That's why I was referencing traveling from London to Edinburgh by a straight line and via Manchester earlier. You wouldn't expect the odometers of the two cars to read the same in Edinburgh because they've followed paths of different lengths. Similarly, we don't expect the clocks of the twins to read the same because they have followed paths of different intervals.

 

[Grimble]

Hmm. OK...
So we have two events in spacetime, let us keep it simple and say they are at the same physical location - can we say that if we are not using frames of reference? -

Two events in Spacetime which have time coordinates t₁ and t₂. Without a specified frame of reference all we know is the interval t₂ - t₁.
Now the duration of that interval is fixed(?) because by adding frames of reference all we are doing is changing the framework; the coordinates to reflect the null point of a particular frame.
Now what bothers me is that we have a single defined duration of the time interval (time-like interval?) yet different proper times for the individual spacetime paths.

So, we have a fixed invariant spacetime interval (time difference - tau) between these two events; yet different fixed invariant proper times between them according to their paths (worldlines).

Now the difficulty I have with this and the analogy of your trips from London to Edinburgh is that the journey is measured in two dimensions and so it is easy to create paths of different lengths, as in your triangle.

BUT, if for time there is but a single dimension, then how can the analogy work for time? There is but a single scale of time between two events. How can there be different durations between the same two spacetime events when they are intervals in a single dimension?

 

[stevendaryl]

Now the difficulty I have with this and the analogy of your trips from London to Edinburgh is that the journey is measured in two dimensions and so it is easy to create paths of different lengths, as in your triangle.

BUT, if for time there is but a single dimension, then how can the analogy work for time? There is but a single scale of time between two events. How can there be different durations between the same two spacetime events when they are intervals in a single dimension?

The invariant interval in Special Relativity is not a measure of time, it is a measure of spacetime. So we're talking about paths in 4 dimensions, not just one.

 

[Ibix]

So we have two events in spacetime, let us keep it simple and say they are at the same physical location - can we say that if we are not using frames of reference?

No, we can't. Separating spacetime into space and time is choosing a frame of reference[1]. And you can't talk about "the same physical location" without some notion of what is space and what is time.

Two events in Spacetime which have time coordinates t1 and t2. Without a specified frame of reference all we know is the interval t2 - t1.

No. As noted above, without specifying a frame of reference you cannot talk about $t_1$ and $t_2$. You can only talk about the interval (along a specified path, which I would assume to be the straight line if not specified) between the events.

So, we have a fixed invariant spacetime interval (time difference - tau) between these two events; yet different fixed invariant proper times between them according to their paths (worldlines).

You have a fixed straight line interval between the events, but paths between points are not restricted to straight lines. It's like the "crow flies" distance between two points versus a curved path between them. When you talk about the distance between two points we assume you mean the distance "as the crow flies" unless you specify otherwise.

Now the difficulty I have with this and the analogy of your trips from London to Edinburgh is that the journey is measured in two dimensions and so it is easy to create paths of different lengths, as in your triangle.

Spacetime has four dimensions. You have far more options for paths than you do in two.

The point is that there are a large family of directions in spacetime that you can choose to call "time". Different frames are different choices of that direction. The big difference between Newtonian physics and Einsteinian physics is that there is no notion of space distinct from time. Changing frames is changing your definition of "the future" to be a direction that the original frame would have said was partially a spatial direction.

In the analogy of the London-Edinburgh trip, the twin who travels direct calls "forward" the direction to Edinburgh from London. But the twin who goes via Manchester calls "forward" the direction to Manchester from London, then changes his mind and calls "forward" the direction to Edinburgh from Manchester. Similarly, the stay-at-home twin calls "the future" (in the sense of coordinate time) the direction from departure to return. The traveling twin starts out calling "the future" the direction from departure to turn around, then changes his mind and calls "the future" the direction from turn around to return.

[1] To be precise, it is choosing a family of frames of reference that share a notion of "time" and "stationary", but whose spatial directions are rotated with respect to each other.

 

[Battlemage!]

Perhaps, gentle folk, it would be useful to correct my 'misunderstanding' of proper time?

As I have understood it, proper time is the time measured by a clock between two events on that clock's world line.
Suppose another clock takes a different path between those same two events, then the second clock (that of the traveller for example) would read a different time at event 2. Clock 2's world line also passes through events 1 & 2. What then of clock 2's proper time? Can it also be the time read on its clock? For then we have two different proper times between the same two events and I understood that proper time was invariant...?

Is the proper time interval not the same as the Spacetime interval?

Just throwing my two cents in here. Hopefully it's not horribly wrong.

This is invariant: Δs² = (cΔt)² - Δx².
Proper time is when Δx² = 0.
Everyone agrees on the value of (cΔt)² measured by the frame in which Δx² = 0.

That's how I see it anyway.

 

[Grimble]

Just throwing my two cents in here. Hopefully it's not horribly wrong.

This is invariant: Δs² = (cΔt)² - Δx².
Proper time is when Δx² = 0.
Everyone agrees on the value of (cΔt)² measured by the frame in which Δx² = 0.

That's how I see it anyway.

Yes, that is how it seems to me...

 

[Nugatory]

This is invariant: Δs2 = (cΔt)2 - Δx2.
Proper time is when Δx2 = 0.
Everyone agrees on the value of (cΔt)2 measured by the frame in which Δx2 = 0.

Yes, that is how it seems to me...

That is the formula for the interval measured along a straight line between two events $E_1$ and $E_2$ with coordinates $(t_1,x_1)$ and $t_2,x_2)$ respectively. It is indeed an invariant, the same in all frames. However, we could choose to measure the interval between the two events along some other path, and we'd get a different answer that is just as invariant - the interval is a property of the path, not the endpoints. It just so happens that there's only one straight-line path between any two given endpoints, so specifying the endpoints specifies the straight-line path so you can get the straight-line interval from the endpoints.

For an arbitrary non-straight path between two events, we have to do a line integral along the path; this integral reduces to the Δs2 = (cΔt)2 - Δx2 formula for the special case of a straight path.

 

[robphy]

Hopefully this post summarizes the various comments made by others.

First, It might be good to quote Minkowski's original definition of "proper-time".

from Minkowski's "Space and Time" ( https://en.wikisource.org/wiki/Translation:Space_and_Time )

Let us now fix our attention upon the world-line of a substantial point running through the world-point P(x, y, z, t); then as we follow the progress of the line, the quantity
${\displaystyle d\tau ={\frac {1}{c}}{\sqrt {c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}}},}$
corresponds to the time-like vector-element dx, dy, dz, dt.

The integral ${\displaystyle \tau =\int d\tau }$ of this sum, taken over the world-line from any fixed initial point $\displaystyle P_{0}$ to any variable endpoint $P$, may be called the "proper-time" of the substantial point in $P$.

The "proper-time" is a property of a timelike-curve joining two events...
think "arc-length" as in ${\displaystyle \tau =\int d\tau }$.
Physically, a wristwatch worn by the observer traveling on this timelike curve--"her worldline"--measures the proper-time of this segment of her worldline.

The "interval" is a property of [the vector joining] two events...
think "magnitude of the displacement vector" as in
${\displaystyle d\tau ={\frac {1}{c}}{\sqrt {c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}}}}$.
Geometrically, think of a straight path--a timelike "geodesic"--joining the two events.
This inertial observer's proper time between these two events agrees with the "interval of this timelike-displacement".

• In special relativity, the proper time along the geodesic path from event A to event Z is longer than all other timelike-worldlines from A to Z--this is the clock effect (sometimes called the reverse-triangle inequality for a timelike-leg and timelike-hypotenuse).

By comparison,

• in galilean relativity, the proper time from event A to event Z is independent of the timelike-worldline from A to Z---this is absolute time.
• in Euclidean geometry, the arc-length along the geodesic path from point A to point Z is shorter than all other paths from A to Z-- this is the triangle inequality.

The diagram below tries to display these features in the three cases.
The "diamonds" I use in the three cases try to suggest the notion of "perpendicularity" using the "other diagonal of the diamond".
---think "that observer's sense of space is perpendicular to that observer's sense of time"---this is a key take-away message.
(Of course, these features are preserved by their corresponding "transformations".)

 

[Dale]

So we have two events in spacetime, let us keep it simple and say they are at the same physical location - can we say that if we are not using frames of reference?

No, the "same physical location" implies a frame which identifies locations and determines if they are the same or not. However, what you can say is that the two events are timelike separated, which implies that there exists a reference frame where they have the same location.

Two events in Spacetime which have time coordinates t1 and t2. Without a specified frame of reference all we know is the interval t2 - t1.

If you have time coordinates then you already have a reference frame.

Now the duration of that interval is fixed(?)

The difference in coordinate times is not fixed. What is fixed is the spacetime interval between them (with some technical caveats).

Now what bothers me is that we have a single defined duration of the time interval (time-like interval?) yet different proper times for the individual spacetime paths.

Your terminology is all over the place, so you are likely to get responses all over the place too.

I don't understand why this bothers you. Please do the exercise I suggested earlier.

Take a blank sheet of paper, draw two points on it, then draw two different curves connecting those two points. Is it really so confusing to say that there is a single defined distance between the two points and yet different lengths for the different curves?

BUT, if for time there is but a single dimension, then how can the analogy work for time?

As @stevendaryl said time is part of spacetime. There are 4 dimensions.

 

[Grimble]

So,

The point is that there are a large family of directions in spacetime that you can choose to call "time". Different frames are different choices of that direction. The big difference between Newtonian physics and Einsteinian physics is that there is no notion of space distinct from time.

Yet in Minkowski's 'Time and Space' he declares

Let x, y, z be the rectangular coordinates of space, and t denote the time. Subjects of our perception are always places and times connected. No one has observed a place except at a particular time, or has observed a time except at a particular place. Yet I still respect the dogma that time and space have independent existences each.

 

[Dale]

Yet in Minkowski's 'Time and Space' he declares

So even Minkowski had trouble letting go. So what?

 

[Grimble]

So Minkowski was wrong?

 

[Dale]

Yes. Scientists are not infallible. Their word is not divine prophecy

 

[Battlemage!]

On the other hand, why should "I still respect the dogma" necessarily mean "I believe the dogma?"

 

[Grimble]

No, the "same physical location" implies a frame which identifies locations and determines if they are the same or not. However, what you can say is that the two events are timelike separated, which implies that there exists a reference frame where they have the same location.

OK, and a worldline is a succession of events that have a unique set of coordinates in each frame?

So, Proper time is the time measured between two points on the world line of a clock and the coordinates of those points can be different in each frame. But the time displayed by the clock is the same wherever it is viewed from, it is the measure of the time that has passed on its worldline between those two po

The interval between our two points is therefore different for each frame it is viewed from, being a function of the time displayed on the clock, and the spatial displacement in that frame.

The Spacetime interval is different because the spatial coordinate difference is excluded from the function - s² = (ct)[/SUP]2[/SUP] - x²

Therefore the Spacetime Interval is the time displayed on the clock in the frame where the clock is at rest; the clock's own frame.

 

[jbriggs444]

OK, and a worldline is a succession of events that have a unique set of coordinates in each frame?

A worldline is a continuous succession of events. Full stop. These events will generally have different coordinates in different frames.

There is no guarantee of uniqueness. The coordinate [0,0,0,0] in two different frames might happen to describe the same event on the same worldline.

So, Proper time is the time measured between two points on the world line of a clock and the coordinates of those points can be different in each frame. But the time displayed by the clock is the same wherever it is viewed from, it is the measure of the time that has passed on its worldline between those two po

The interval between our two points is therefore different for each frame it is viewed from, being a function of the time displayed on the clock, and the spatial displacement in that frame.

Here you are using "interval" to mean "difference between time coordinates for the starting and ending events on the worldline". Do not do that. Do not impose your own idiosyncratic meaning on words that already mean something else.

 

[Dale]

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

Yes, although this is not the defining feature, it is true.

Proper time is the time measured between two points on the world line of a clock and the coordinates of those points can be different in each frame. But the time displayed by the clock is the same wherever it is viewed from, it is the measure of the time that has passed on its worldline between those two po

yes

The interval between our two points is therefore different for each frame it is viewed from, being a function of the time displayed on the clock, and the spatial displacement in that frame.

Usually the unqualified word "interval" refers to the spacetime interval which is invariant. I think that you mean the coordinate time difference, which is usually not described using the word "interval" in order to avoid confusion.

The Spacetime interval is different because the spatial coordinate difference is excluded from the function - s2 = (ct)[/SUP]2[/SUP] - x2

If by excluded you mean subtracted then that is essentially true. If you mean something else then please clarify.

 

[Grimble]

A worldline is a continuous succession of events. Full stop. These events will generally have different coordinates in different frames.

There is no guarantee of uniqueness. The coordinate [0,0,0,0] in two different frames might happen to describe the same event on the same worldline.

unique set of coordinates

note: I specified a 'unique set of coordinates', not a 'set of unique coordinates'...

Here you are using "interval" to mean "difference between time coordinates for the starting and ending events on the worldline". Do not do that. Do not impose your own idiosyncratic meaning on words that already mean something else.

The interval between our two points is therefore different for each frame it is viewed from, being a function of the time displayed on the clock, and the spatial displacement in that frame.

Yes, I was using the word interval (unqualified) to mean an interval ( normal English usage). I certainly was not using it to mean

difference between time coordinates for the starting and ending events on the worldline

as I explicitly specified

time displayed on the clock, and the spatial displacement in that frame.

 

[jbriggs444]

note: I specified a 'unique set of coordinates', not a 'set of unique coordinates'...

Neither of which makes it clear what you consider to be unique. Possibly you simply meant that given an event and a coordinate system, there is a one to one mapping between coordinate tuples and events. One event per tuple and one tuple per event. If so then the word "unique" conveyed no useful meaning. That sort of uniqueness is taken for granted.

Yes, I was using the word interval (unqualified) to mean an interval ( normal English usage).

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.

Edit to add:

You responded to me stating that you used the word "interval"...

I certainly was not using it to mean
difference between time coordinates for the starting and ending events on the worldline
as I explicitly specified

and to @Dale stating that

I meant the difference between the coordinates of the two points as measured from another frame

That seems contradictory. Can you clarify?

 

[Grimble]

Usually the unqualified word "interval" refers to the spacetime interval which is invariant. I think that you mean the coordinate time difference, which is usually not described using the word "interval" in order to avoid confusion.

I meant the difference between the coordinates of the two points as measured from another frame; would it be correct to refer to that as the coordinate time difference?

If by excluded you mean subtracted then that is essentially true. If you mean something else then please clarify.

Yes, sorry, I did mean subtracted...

I must apologise for the way I refer to things with unscientific word usage, but I stopped studying physics in university in 1978, which was a while ago... and the correct usage can be a bit tricky.

 

[Herman Trivilino]

So Minkowski was wrong?

I don't see how the information you posted implies that as a conclusion. Respecting a dogma means you respect something that others believe to be true.

Time and space can still be regarded as separate entities even if the time and space coordinates are different in different frames.

 

[Herman Trivilino]

The Spacetime interval is different because the spatial coordinate difference is excluded from the function - $s^2 = (ct)^2 - x^2$.

No, the spacetime interval is the same because the spatial coordinate difference is excluded from the function - $s^2 = (ct)^2 - x^2$ whenever $x=0$. In those cases we would call both $s$ and $ct$ the proper time.

Therefore the Spacetime Interval is the time displayed on the clock in the frame where the clock is at rest; the clock's own frame.

Correct.

I must apologise for the way I refer to things with unscientific word usage, but I stopped studying physics in university in 1978, which was a while ago... and the correct usage can be a bit tricky.

If you are here to learn then there's no need to apologize for making mistakes. But realize that those errors will often get corrected by other posters, especially if they are being used to draw false conclusions, and even more especially if those false conclusions are being presented as "corrections" to valid information posted by others.

 

[Dale]

I meant the difference between the coordinates of the two points as measured from another frame; would it be correct to refer to that as the coordinate time difference?

Yes, but don't forget that when you specify a coordinate time difference in needs to be clear which coordinate system is being used.

I must apologise for the way I refer to things with unscientific word usage, but I stopped studying physics in university in 1978, which was a while ago... and the correct usage can be a bit tricky.

I understand, the concepts are so specific that even subtle terminology changes can drastically alter the intended meaning. That is why a lot of questions get seemingly contradictory answers, so it is good to check on the intended meaning