I Proper (and coordinate) times re the Twin paradox

[tophatphysicist]

Your suspicions notwithstanding, why should the universe care whether I do my calculations in my instantaneous rest frame or yours?

Nature wouldn't care about that, because our spaces are both special. I'm just saying that nature has selected out special frames for which the laws of physics work in a natural and special way. The laws of physics work in spaces like the ones below. Notice I'm talking about the spaces for which the photon four-fiber bisects the world line of an observer and his 3-D space.

 

[jbriggs444]

Nature wouldn't care about that

Right. Nature does not care about simultaneity. Simultaneity is a human invention.

 

[Dale]

You seem to be using the idea of causality in a different way, perhaps as a description of the related geometric patterns of bundles of 4-D fibers along the 4th dimension. You have already indicated that the universe is a 4-D object with everything already there. So, if the "future" part of the 4-D structure is already there, how do you now cause it?

Umm, I am using causality in the usual manner. I don't know why you think the dimensionality of the universe is relevant to causality.

My favorite example of causality in physics is Jefimenko's equations.

I'm not so sure about that. I have a hunch that the universe cares about those special sets of coordinates ...

Can you provide any evidence that the universe cares about any set of coordinates? I mean, something more peer-reviewed than your hunch.

From what I can tell nature doesn't care about any coordinates, inertial or not. Only we do.

 

[Dale]

I agree 4D universe 'doesn't care' about simultaneity. But we, observers do care

Yes, I agree.

 

[Herman Trivilino]

Correct, the dividing plane has zero tickness. But we can take a slice with slight non-zero thickness. That difference is the issue discussed.

Sorry, but I just don't see where the issue I'm thinking of was discussed.

If the slice has zero thickness it's two-dimensional. It it has some nonzero thickness, however slight, it's three-dimensional. I know a lot of things were said about 3D, and how it compares to 2D, but I don't recall seeing this discussed.

 

[Ebeb]

Sorry, but I just don't see where the issue I'm thinking of was discussed.

If the slice has zero thickness it's two-dimensional. It it has some nonzero thickness, however slight, it's three-dimensional. I know a lot of things were said about 3D, and how it compares to 2D, but I don't recall seeing this discussed.

Most part of page 10, 11, and 12 of this thread deal with the issue.

 

[Herman Trivilino]

Most part of page 10, 11, and 12 of this thread deal with the issue.

Your response is vague. Do you understand that a 2 dimensional slice has zero thickness and that if you give it a width, however small, it has 3 dimensions?

 

[Ebeb]

I thought I started to understand what you guys have in mind, but I see I'm still nowhere...
Maybe we differ in what 4D means. Let's start with this:

Yes. And physical objects are 4D.

There is no such thing. Clocks are 4D objects.

There is no such thing. Measuring sticks are 4D objects.

Agreed, and this automatically makes 4D universe (Block Universe) also a 4D physical object.

 

[Ebeb]

Your response is vague. Do you understand that a 2 dimensional slice has zero thickness and that if you give it a width, however small, it has 3 dimensions?

Really unbelievable you ask me this kind of question.
But we wouldn't call it a 2D slice. It's a 2D plane.
A slice has thickness, hence 3D.

 

[jbriggs444]

Agreed, and this automatically makes 4D universe (Block Universe) also a 4D physical object.

The point of that discussion was that if it's physical, it's four dimensional. And if it's three dimensional, it's not physical.

The converse does not apply. Just because it's four dimensional, that does not mean it must be physical. The definition of "physical" that we are using is "measurable". You have to be able to run an experiment. One cannot measure the future (except by waiting -- and then it's not the future any more). That means that the block universe does not qualify as "physical". It is, as @Dale refers to it in #256, an "interpretation".

Edit: Calling the block universe an "object" is also a bit of a stretch. And measuring the past is even more difficult than measuring the future -- we can't go back and take a new snapshot of the grassy knoll..

 

[Ebeb]

The point of that discussion was that if it's physical, it's four dimensional. And if it's three dimensional, it's not physical.

The converse does not apply. Just because it's four dimensional, that does not mean it must be physical. The definition of "physical" that we are using is "measurable". You have to be able to run an experiment. One cannot measure the future (except by waiting -- and then it's not the future any more). That means that the block universe does not qualify as "physical". It is, as @Dale refers to it in #256, an "interpretation".

Two observers at same location but driving different direction measure a different physical clock display event of the 4D clock located at a distance.
That's measuring a 4D object, be it a 4D clock, or the 4D universe/block universe. What else would above measuring be?

Edit: Calling the block universe an "object" is also a bit of a stretch.

What then is an object if it's not 3D, nor 4D?

 

[jbriggs444]

Two observers at same location but driving different direction measure a different physical clock display event of the 4D clock located at a distance.

You seem to be putting a lot of weight on word choice and not much effort at clarity of expression.

Two observers at the same location will be momentarily seeing the same thing through their telescopes: an image of the remote clock from some time in the past. The two observers can each adopt a frame of reference in which they are at rest and calculate a "current" clock reading on that remote clock. The two calculations will produce two different results. Those two calculated results will match the physical clock readings on that remote clock at two different events.

 

[jbriggs444]

you have to consider 'present world' non-existent

One can accept that there are objects in the neighborhood of an arbitrarily chosen surface of simultaneity. One can accept that those objects "physically exist". That does not require one to classify the surface or even its neighborhood as "physically existent". Mathematically existent is good enough.

 

[Herman Trivilino]

Really unbelievable you ask me this kind of question.
But we wouldn't call it a 2D slice. It's a 2D plane.
A slice has thickness, hence 3D.

That is not my understanding of the standard way the language is used. I could be wrong, but if that's the case it would seem there's a lot of confusion.

I thought I started to understand what you guys have in mind, but I see I'm still nowhere...
Maybe we differ in what 4D means.

Many people have trouble with even fewer dimensions. I know I do.

If you slice a potato you create two pieces. The potato is 3D and the slice is 2D. Thus the number of dimensions is always reduced by $1$ when we slice, as I understand the way the language is usually used to describe the math. Thus a slice of 4D spacetime is a 3D "hypersurface", and I guess a slice of a 2D surface would be a 1D line.

It could be that we don't all share the same common use of the language here, but as far as I know my understanding of it is consistent with the general understanding.

 

[Vitro]

That is not my understanding of the standard way the language is used. I could be wrong, but if that's the case it would seem there's a lot of confusion.
Many people have trouble with even fewer dimensions. I know I do.

If you slice a potato you create two pieces. The potato is 3D and the slice is 2D. Thus the number of dimensions is always reduced by $1$ when we slice, as I understand the way the language is usually used to describe the math. Thus a slice of 4D spacetime is a 3D "hypersurface", and I guess a slice of a 2D surface would be a 1D line.

It could be that we don't all share the same common use of the language here, but as far as I know my understanding of it is consistent with the general understanding.

It seems some participants here use the meaning of slice as in a slice of bread, where the bread is 3D and the slice is also 3D since it has a thickness.

 

[Herman Trivilino]

It seems some participants here use the meaning of slice as in a slice of bread, where the bread is 3D and the slice is also 3D since it has a thickness.

I was speaking of the general community of professional physicists and mathematicians. I would certainly agree that when I ask for a slice of bread I'm going to get a three-dimensional object. But that is not the way the word is used in the literature.

But arguing about the meaning of a word is not the the issue. The issue is a working grasp of the mathematics and how it applies to the physics. If you want to talk to physicists and mathematicians about this stuff they are going to use the technical meanings of the words, not the everyday meanings. That is a major cause of confusion for learners.

 

[Dale]

Let's please drop the philosophical discussion at this point and stick to experimentally measurable things.

 

[PeterDonis]

Let's please drop the philosophical discussion at this point and stick to experimentally measurable things.

To help keep this thread on topic, I have moved discussion arising from my Insights article on the block universe to a separate thread:

https://www.physicsforums.com/threads/block-universe-discussion.920295/

Please direct all comments on the article and topics arising from it to that thread.

 

[Grimble]

@Ebeb, that was a long post. I will try to keep this response brief.

It seems that we have reached agreement about the distinction between a thin section of an object (e.g. a slice of bread) and an infinitesimal dividing plane through an object (e.g. the place where we intend to cut the bread).
Physically, a car hitting a tree is a process with four-dimensional extent. It extends right, left, up, down, forward into the tree and back into the car. It does not just crumple a point on the bumper. It crumples a three dimensional region. It does not just crush the bark on the tree at one point. It makes a three dimensional scar. It does not occur at an instant. It takes place over a [short] time interval.

For most purposes, we do not care about the complete details of the collision. For purposes of our models, it is enough that we know the mass, momentum and energy of the car and of the tree. For the police report, we only need to know that it occurred on Fourth and Elm at 2:00 pm. [we assume that it took place at ground level]. The "event" associated with the collision is the exact four dimensional location which is only approximately where and when the diffuse process took place. The process is physical. The "event" is a labelled feature in our model.

But I agree that one could reasonably say that a collision "event" is physically a four-dimensional process.

I have the feeling that much of this discussion involves the mixing up of an 'event' which has a specific definition in Relativity/spacetime, and the noun 'event' in the English language, which denotes a specific iteration of a process.

 

[Grimble]

That sounds likely to me also. Or at least a big part of it. The other part is just the math vs physical bit.So mathematically an event is 0 dimensional. Just a point at an instant. Nothing physical meets that criteria, but we can use the math as a simplification when the actual 4D interaction is very small compared to our scale of interest.

But is that not the point? That a spacetime event is, in essence a way to refer to a 4D location? The word 'event' has connotations of an action or a process but that is really not the right way to envisage what we mean in spacetime, where an event is a point in space at a point in time and as such cannot be 'doing' anything.
It seems to me, in view of the preceding discussions that an event could be likened to a slice through the 4 dimensions which as has been pointed out is not an object of any number of dimensions.
A spacetime event can have no content; it may be described by a verb, for example, when we say the event where a photon of light was emitted by a light source, but that is using the process of light emission to describe/define the point in time (and space) that is the event referred to.

Or am I misunderstanding again?

 

[Dale]

But is that not the point? That a spacetime event is, in essence a way to refer to a 4D location?

Yes. That is correct

 

[sweet springs]

Hi. Meanings of word event in physics and in daily life are not different, I think. Information of when and where via a certain coordinate is indispensable.
E.g.
Birth of J.C. 4 B.C. @Bethlehem
Rise of French revolution 14 July 1789 @Paris

My birth was an Event no matter how any people say the birthplace is xxx and birthday is yyy differently with their coordinates. Yours also.

 

[Grimble]

Hi. Meanings of word event in physics and in daily life are not different, I think. Information of when and where via a certain coordinate is indispensable.
E.g.
Birth of J.C. 4 B.C. @Bethlehem
Rise of French revolution 14 July 1789 @Paris

My birth was an Event no matter how any people say the birthplace is xxx and birthday is yyy differently with their coordinates. Yours also.

Yes, thank you, I can see what you mean, it depends on the time scale that is used, in geological terms the last ice age was a specific event that was just a point on an appropriate time scale.
Yet it seems to me that there is an implication in physics that an event happens at an indivisible point in time and that being a point in time it must measure a specific instant in a process, rather than denoting a process...
(Or am I just being picky?)

 

[sweet springs]

Hi. Both OK. Accumulation of points denote process. My birth and my death are two point events. Between them my life so to say continuous sequence of events are expressed as line, my world line. Best.

 

[Dale]

Yes, thank you, I can see what you mean, it depends on the time scale that is used, in geological terms the last ice age was a specific event that was just a point on an appropriate time scale.
Yet it seems to me that there is an implication in physics that an event happens at an indivisible point in time and that being a point in time it must measure a specific instant in a process, rather than denoting a process...
(Or am I just being picky?)

Mathematically a point is an indivisible 0D geometric primitive. Physically it is usually understood to be an approximation for something that is just far smaller than the scale of interest.

 

[Grimble]

I have been rereading and working through this thread for some time now and I think I understand what you were trying to instil in me.

Any clock measures proper time. It doesn't matter if they are at rest or moving, if they are inertial or non inertial, in curved spacetime or flat. They always measure proper time along their worldline.

Proper time is what a clock measures and reads/displays.

The important thing about proper time is its invariance. Everyone agrees the length of your path.[worldline?] The important thing about coordinate time is its arbitrariness. We could have chosen a different way to define our zero point, and we could have chosen a different direction to move in.

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.

In any reference frame. It is invariant.
The worldline is the geometric figure itself, irrespective of the coordinate system that you might use for describing it. One worldline will have different coordinates in different coordinate systems but it is the same worldline.

If we assign coordinates using a frame in which the object is at rest we'll label the points that the worldline passes through (t,0,0,0) and if we use a frame in which the object is moving these points might be labeled (t',vt',0,0), but they're the same points and the same worldline either way. When we change frames we're changing the axes of the coordinate system we're using to assign coordinates to points, but this doesn't change the points themselves.

So the invariant proper time is measured on the same clock at the same location – in a frame in which the object is at rest, while in a frame where the object is moving we measure (calculate?) coordinate time?

If the two events occur in the same place (in some frame) then the time that elapses in that frame is a proper time. In the rest frames of observers who move relative to those events the time that elapses will not be a proper time, for them those two events occur in different places. They will therefore need two clocks, one located (local!) at each of the two events. And the time that elapses in those frames will always be larger than the proper time.

If we were to mark the point of each completed tick between two events, then regardless of which frame we use, resting of moving, there will still be the same number of ticks between events? That it is the length of the ticks that will vary not the quantity? A longer coordinate time, but the same proper time?

Suppose we have individualized time and space in any manner; then a substantial point as a world-line corresponds to a line parallel to the t-axis; a uniformly moving substantial point corresponds to a world-line inclined to the t-axis;

The above quotes lead me to conclude that proper time is measured vertically parallel to the t-axis.
So Proper Time may be likened to the spacetime Interval between two events on a resting clock's worldline – a straight vertical line parallel to the t-axis, measured at the same location, on the same clock; the spacetime interval being s²=(ct)²-x² where x=0.
The coordinate time occurs between two events on the t' axis of a moving clock, - being at different locations, measured on different clocks – x≠0. The spacetime interval s²=(ct')²-x² being invariant, must be the same value.
The proper time between two events is also invariant, it is the distance traveled between the events in the coordinate time, that makes the difference.
Compare these two measurements: while the proper time τ=√(ct)²-x²the spacetime interval s²=(ct)²-x²
They are in effect the same thing; only, because the Spacetime Interval could be space-like,
it is S² that is used rather than S.

 

[Nugatory]

The above quotes lead me to conclude that proper time is measured vertically parallel to the t-axis.

No, a thousand times no!

The proper time between two events is measured along the wordline of a clock that is present at both events. If the clock does not experience any acceleration this worldline will be a straight line. (If the clock is accelerated its worldline will not be straight, and the proper time along the non-straight worldline will be less than the proper time along the straight worldline of an unaccelerated clock). There is no reason why that straight line has to be vertical and parallel to the t axis.

 

[stevendaryl]

Proper time is what a clock measures and reads/displays.

Correct.

The above quotes lead me to conclude that proper time is measured vertically parallel to the t-axis.

Not correct. I really do think that it helps to think of the analogy with Euclidean geometry. You are on a one-way road that twists and turns. There is an intrinsic, coordinate-independent way to talk about your location on the road: You say "I am at a point 12.6 kilometers from exit number 12". There is also a coordinate-dependent way to talk about your location: I am at 36.2134 degrees north latitude, 12.0987 degrees west longitude.

To figure out the first, all you need is a car with an odometer. To figure out the second, you need a whole system of conventions for latitude and longitude, and a lot of information, some of which is completely arbitrary, such as the choice of Greenwich, England as the 0 of longitude.

Proper time is like distance along a road. It has nothing to do with coordinates. Although you can measure proper time using coordinates in the same way that you can measure distance along a road using coordinates. If you know that your longitude has changed by $\Delta \phi$ and you know that your latitude has changed by $\Delta \theta$, then your distance along the road has changed by approximately:

$\Delta s = \sqrt{g_{\theta \theta} (\Delta \theta)^2 + g_{\phi \phi} (\Delta \phi)^2}$

where $g_{\theta \theta} = R^2 (\frac{\pi}{180})^2$ and $g_{\phi \phi} = R^2 cos^2(\frac{\pi \theta}{180})$ are the metric components for the coordinate system of latitude and longitude ($R$ is the radius of the Earth).

 

[Grimble]

No, a thousand times no!

OK, thank you. What I understand from this:

[...] 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 this:

The proper time between two events is measured along the wordline of a clock that is present at both events.

is that if the two locations are the same and it is measured on the same clock, then this is represented by a vertical line where x is constant...
If the x location changes, whether it does so at a constant speed or accelerating then the line will be angled or curved, but will, necessarily have a varying x coordinate.
If, as in the case of the traveling twin, he moves away and then back to the original location, he will still have changed his x coordinate.
He will effectively have to a different location and then returned - so two coordinate times, not proper times.
I just don't know which of your explanations to believe...

 

[Grimble]

Sorry, the highlighted word was missing in the original.

He will effectively have moved to a different location and then returned - so two coordinate times, not proper times.

Consider a clock moving at v.
After time t it will have moved from event E1 to event E2 a distance x=vt
The time measured by the observer relative to whom the clock is moving from E1 to E2 would be ct - the diagonal (rotated) line.
In the clock's frame E1 and E2 would be measured at the same location by the same clock giving the proper time τ.

OK this is measured in a different frame but it is invariant so would be the same in any frame in which it is moving.

We have then a right angled triangle where the hypotenuse = ct and the catheti =vt and τ. So:
(cτ)² = (ct)²-(vt)²
cτ = √((ct)² - (vt)²)
τ = t √((1 - v²)/c²)
τ = t/γ

The proper time is of course the time displayed by the clock at E2, which is less (by the factor γ) than the coordinate time ct.

The invariant proper time is the same in every frame - a measurement between events on a line parallel to the ct axis (usually vertical)

This is what the posts I quoted tell me.

So where have I misunderstood?