I Einstein's definition of time

Aufbauwerk 2045

It so happens that last night I was rereading Einstein's famous 1905 paper On the Electrodynamics of Moving Bodies. I think this is one of the most fascinating scientific papers in history, but some people say it's not at all clear. In any case I love reading Einstein's papers.

Clocks obviously play a major role in this paper. I was thinking about the types of clocks that existed in 1905. As a patent clerk in Switzerland, which is famous for its clocks, he may have seen many new ideas for clocks. Perhaps he dreamed about clocks.

His clock seems to be a sort of idealized perfect clock. He of course goes into no details concerning its construction. Although he does mention it has "hands."

I would be interested in reactions to how Einstein defines time in the first section of this paper. Is it clear? Is it confusing?

First he describes clock A and clock B, and the fact that each clock can only indicate the time for events in its immediate proximity, which happen "simultaneously" with a specific position of the hands on a clock. Of course we normally define "simultaneous" to mean "at the same time" and we have not yet defined "time." So I take this to mean we perceive the hands on a clock to be at a certain position, and the event to occur, in a way the brain perceives as "simultaneous." It's a matter of perception.

Thus we have the A time and the B time. But we need to define a common time for A and B.

Then he says that in order to establish this common time for A and B, we must say by definition that the time required for a ray of light to go from A to B equals the time required for a ray of light to go from B to A. Note that this is a definition, not an inference.

Then he defines what he means by synchronized clocks. We have a clock at A, and another clock at B which is "similar in all respects" to the one at A. In accord with his earlier definition, he states that clock A and clock B are synchronized if the time for light to travel from A to B equals the time for light to travel from B to A.

The thought experiment to make this clear is that the ray leaves A, the time being recorded. Then the ray arrives at B, where it is reflected back to A. The arrival/reflection time at B is recorded. Then the arrival time back at A is recorded.

In other words, let TA be the "A" time the ray leaves A. Let TB be the "B" time the ray is reflected from B. Let T'A be the "A" time the ray arrives again at A.

Then clock A and clock B are synchronized if TB - TA = T'A - TB.

Now he says we can define the time of an event in a stationary system. He says that if a clock is stationary, and is located at the place of an event in a stationary system, then the time of the event is that given simultaneously by the clock, which is synchronized with another specified stationary clock.

This "time" is what he calls the "time of the stationary system."

He also assumes "in agreement with experience" that c = 2AB/(T'A - TA) is a universal constant, namely the speed of light in empty space.

Of course this is only the beginning of this paper.

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Aufbauwerk 2045

P.S. one of my reasons for the previous post is that I think it's important to make sure the basic ideas are clear, before venturing into questions about the actual relativity theory. I wonder if the way Einstein uses his so-called "clocks" in defining time is helpful or confusing.

Paul Colby

Gold Member
I always took this as defining time in terms of its measurement. Thus all the discussion of ideal clocks. Likewise for lengths with ideal meter sticks.

Nugatory

Mentor
wonder if the way Einstein uses his so-called "clocks" in defining time is helpful or confusing.
The 1905 paper is Einstein's first presentation of the key concepts of relativity. In the following decades he and many others refined this initial presentation and incorporated significant new insights such as Minkowski space. Thus, the 1905 paper shouldn't be taken as the authoritative last word on Einstein's thinking; and if you find any part of it more confusing than helpful, you are showing no disrespect by consulting a more modern source.

However, Einstein's treatment of clocks has been pretty consistent from the beginning. The key point is that if I remain standing next to a clock, I can use it to measure the passage of time for me and the clock and everything else in our immediate neighborhood; but to do any more than that requires additional assumptions about how other clocks might be synchronized with that clock. Exposing these additional assumptions is an essential part of any presentation of special relativity.

Ibix

It's probably worth thinking about what "immediate neighbourhood" means. If I treat my wristwatch as a master clock and I have another clock at rest 1m away from me then whatever simultaneity convention I adopt, the clock will agree with a co-located Einstein synchronised clock to within ±1m/c≈±3ns. So if I can tolerate that timing error and can fit my experiment within a 1m radius sphere I can ignore clock synchronisation. If I can tolerate ±0.01s then I can ignore clock synchronisation over a 3000km sphere.

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pervect

Staff Emeritus
It so happens that last night I was rereading Einstein's famous 1905 paper On the Electrodynamics of Moving Bodies. I think this is one of the most fascinating scientific papers in history, but some people say it's not at all clear. In any case I love reading Einstein's papers.

Clocks obviously play a major role in this paper. I was thinking about the types of clocks that existed in 1905. As a patent clerk in Switzerland, which is famous for its clocks, he may have seen many new ideas for clocks. Perhaps he dreamed about clocks.

His clock seems to be a sort of idealized perfect clock. He of course goes into no details concerning its construction. Although he does mention it has "hands."
Personally I wouldn't get too hung up on the idea that clocks need to have hands, even if Einstein mentioned clocks with hands at some point. You probably even own a digital clock, one totally lacking in hands, and the lack doesn't really matter in its ability to keep time.

I would say that the key idea is that there is such a thing as an ideal clock, and that the history of making actual implementations of clocks (which are not idea) have slowly evolved our clocks into better and better approximations of this ideal. The point is that there is an ideal clock, that measures some particular thing we call time, exists, rather than a multitude of different sorts of clocks that keep different sorts of time. Some of the earliest interest in clocks was driven by practical concerns, such as the ability to determine one's longitude via sightings of the sun. For a fascinating (though perhaps off-topic) documentary, there was a PBS TV documentary called "Longitude" on some of the history of this development. There are some interesting wrinkles here in how relativity (unthought of at that time) affects the problem of navigation, but on a practical level we can say that modern methods such as GPS, that incorporate relativity, allow us to accomplish this important task in the modern world with ease and accuracy that was not always available.

The issue of "one sort of time" comes up in discussions of relativity a frequently, when people wonder whether "light clocks", a tool commonly used to teach relativity, keep "the same sort of time" as other clocks. Light clocks are a convenient tool to teach relativity, but some people get sidetracked by the idea that they might keep "a different sort of time" than other clocks, which tends to make some discussions wander off topic. It's not logically incosistent to think that different sorts of clocks might keep different sort of times - or that every person has their own unique "time" for that matter - it's just a dead end that doesn't go anywhere useful, though it can waste a lot of words in the process of not getting anywhere.

The point that I feel is important is this. Regardless of the implementation of the clock (and specific implementations can and do have different levels of accuracy), clocks all measure the same thing. Regardless of whether you have a mechanical, chemical, electronic, nuclear, or biological clock, it keeps time. And there's only one sort of time, not a multitude of different sorts of time.

I would be interested in reactions to how Einstein defines time in the first section of this paper. Is it clear? Is it confusing?

First he describes clock A and clock B, and the fact that each clock can only indicate the time for events in its immediate proximity, which happen "simultaneously" with a specific position of the hands on a clock. Of course we normally define "simultaneous" to mean "at the same time" and we have not yet defined "time." So I take this to mean we perceive the hands on a clock to be at a certain position, and the event to occur, in a way the brain perceives as "simultaneous." It's a matter of perception.
There are some useful definitions to be made here than can clarify some of the issue. The sort of time that clocks measure has a technical name, it's called "proper time". There is another facet of time that's also important, this notion of time is called "coordinate time". Coordinate time is just a way of assingning a label to an event to tell us when it happened. For instance, "Meet me in the garden at 7:00 pm". 7:00 pm indicates a time coordinate, rather than a proper time.

It's obviously important to be able to do this, if one wants to be able to arrange a meeting in the garden.

A situation where the two differ might be useful One observer says "I was in the garden at 6:00 pm" (a time coordinate - and I stayed there until 7:00 pm (another time coordinate). In this process, my clock ticked off 36,000.000001 seconds - a proper time interval.

The 36,000.000001 seconds is a proper time interval. It's measured by a single clock, with no need to concern oneself with the issue of how to synchronize clocks, as there is only one clock needed to make the measurement.

Before relativity, when one had absolute time, one would never have a clock ticking off 36,000.000001 seconds between 6:00 pm and 7:00 pm. It would always tick off exactly 36,000 seconds, no more, no less. But in relativity, one clock might tick off 36,000.000001 seconds between 6:00 pm and 7:00 pm, and another clock, taking a different path through space and time, might tick off 35,999.999999 seconds instead. This is commonly called "the twin paradox", but there's nothing paradoxical about it. It's just an aspect of relativity that one has to come to terms with. In my opinion, disentangling the notion of proper time from the notion of coordinate time makes the whole coming-to-terms process much easier to do, and much easier to talk about.

Note that when one comes up with a way to assign time coordinates to events, the concept that an event happened "at 6:00 pm" in the example, one also has an associated notion of simultaneity. Events that happen at 6:pm in the garden are "simultaneous" or happen "at the same time" as events that happen at 6:pm at other locations - for instance, perhaps, some event happend at 6:00 pm inthe house.

The reverse is also true - coming up with a way to assign events as simultaneous also allows one to set up a coordinate system based on some "reference clock", a way to assign time coordinates to events. So once we have worked out the issue of simultaneity, we have the last piece we need to be able to systematically assign coordinates to events, to actually do physics.

One of the most important part of Einsteins' presentation is, IMO, that one needs to separate the behavior of clocks as measuring time intervals (aka proper times), from the issue of how we synchronize clocks. These are both important issues, but logically distinct.

The remainder of the paper gives a detailed accounting of how we put all the building blocks to come up with a complete theory. The building blocks as I recall them are (this may not be a complete list) the idea that clocks keep proper time, the idea that light moves at a constant speed "c", and the very important idea that there is no preferred direction in space, the concept Einstein calls "isotropy". As I recall, Einstein does not spend a lot of words on isotropy - but it's a key concpet.

Given these assumptions and the supporting logical framework, we eventually come up with a procedure that allows us to use clocks, and light, to assign coordinates to events. Einstein leads the way as the first to do this, other authors such as Bondi who use "radar methods" to carry out this task of assigning coordinates to events are also worth reading. (See for instance Bondi's book "Relativity and Common Sense", or the recent PF insight article about Bondi's k-calculus).

An important result of this analysis is how two different inertial observers, moving relative to each other, will each carry out this procedure of using clocks and light beams to assign coordinates to events. The relationship between the assigned coordinates of one observer to the coordinates of the differently-moving observer is the "Lorentz transform".

Aufbauwerk 2045

The 1905 paper is Einstein's first presentation of the key concepts of relativity. In the following decades he and many others refined this initial presentation and incorporated significant new insights such as Minkowski space. Thus, the 1905 paper shouldn't be taken as the authoritative last word on Einstein's thinking; and if you find any part of it more confusing than helpful, you are showing no disrespect by consulting a more modern source.

However, Einstein's treatment of clocks has been pretty consistent from the beginning. The key point is that if I remain standing next to a clock, I can use it to measure the passage of time for me and the clock and everything else in our immediate neighborhood; but to do any more than that requires additional assumptions about how other clocks might be synchronized with that clock. Exposing these additional assumptions is an essential part of any presentation of special relativity.
I prefer to learn from the great geniuses like Einstein first-hand. Of course he made some mistakes, but that just makes things more interesting to me.

As a matter of fact I studied special and general relativity in my physics program, using the modern texts of course. I found them all rather dry. I don't know of anyone who taught using Einstein's papers. One professor made fun of me when I stated I wished I had more time to read the original papers. "Those are antiquated" he laughed. "Do you like antiquated papers?" I refrained from saying what I was thinking, which was, "obviously, you dolt, or I would not have made my original statement." I found more value in antiquated Einstein than in anything he had to say. Einstein's papers are thrilling.

I'm very happy that now I'm free of the academic system, and I can study anything I want in just the way I want. It just so happens I prefer taking the historical approach, so I can understand how an idea developed. There are people who agree with me that this is a very useful approach. Lanczos is a good example of someone who advocated the historical approach. This is the approach I'm taking in reviewing all the physics I ever learned. Even though physics was my favorite subject by a wide margin, it is even more enjoyable to me now, because I am free to explore in my own way.

I think Einstein made a good point when he said the best job for a scientist is to be a lighthouse keeper. Being a lens grinder like Spinoza wouldn't be so bad either. Of course it's not a realistic idea for many of us. Actually in the capitalist system the best job is to not need a job. Wealthy aristocrats like de Broglie could afford to go their own way, without worrying about earning money. He was so fortunate.

Of course I do use modern texts. Some I like. For example, Born's Atomic Physics is great. I even read books published within the past five years. I'm not a fanatic.

:)

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David Lewis

I would be interested in reactions to how Einstein defines time in the first section of this paper. Is it clear? Is it confusing?
Einstein originally presented a fairly non-mathematical treatment. The theory was subsequently couched in formal geometry. But there is something to be said for comprehension before calculation.

Grimble

There are some useful definitions to be made here than can clarify some of the issue. The sort of time that clocks measure has a technical name, it's called "proper time". There is another facet of time that's also important, this notion of time is called "coordinate time". Coordinate time is just a way of assingning a label to an event to tell us when it happened. For instance, "Meet me in the garden at 7:00 pm". 7:00 pm indicates a time coordinate, rather than a proper time.
So, if "proper time" is the "sort of time" that clocks measure, then it seems you are saying that "coordinate time" is a different "sort of time"?

WIKIPEDIA(dimension) said:
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.
Dimensions referenced Cartesian coordinates are mutually perpendicular which makes their measure in one dimension independent of their measure in any other dimension.
In a similar way would it be realistic to allude to two(? if not more?) dimension in time when we refer to 'moving clocks"?

One dimension being the proper time that is measured by the local clock; I think of this as an "objective time" as it is the same, measured with that frame's rods and clocks, wherever it is observed from.
A second dimension being the time in a frame measured by a moving observer. I think of this as "subjective time" as it is the time in one frame measured against the rods and clocks from the observers frame, making the measurements subjective to that observer.

pervect

Staff Emeritus
So, if "proper time" is the "sort of time" that clocks measure, then it seems you are saying that "coordinate time" is a different "sort of time"?
Basically, yes. Proper time is independent of the observer and the observer's convention, and can be regarded as something physical. For clarity, when I say there is only one sort of time, I should have said that there is only one sort of proper time. Proper time is also the sort of time that is defined by the SI defintion of the second.

Coordinate time depends on the observer and the observer's coordinate choices. To communicate unambiguously about coordinate time, there must be agreement on what convention is being used, what coordinates, are being used. In particular , saying that two events happen "at the same time" is not meaningful in special relativity unless which specified what frame is being used. It's only meaningful when one specifies what observational frame one is using. For Einstein's discussion of the issue, see for instance http://www.bartleby.com/173/9.html, the general keywords to look for for are "The Relativity of Simultaneity" or "Einstein's Train". There's a large number of posts and discussions on this topic on PF.

Dimensions referenced Cartesian coordinates are mutually perpendicular which makes their measure in one dimension independent of their measure in any other dimension.
For the observer who uses that particular Cartesian coordinate system, yes. If we have two observers, moving relative to each other, each can construct a Cartesian coordinate system. In each coordinate system time will (by convention) be chosen to be orthogonal to space. Coordinates can be transformed from one observer's frame of reference to the other observer's frame of reference by using the Lorentz transform. Suppose we have two reference frames of reference with corresponding Cartesian coordinates, A and B. Two events that have the same time coordinate in frame A , but different spatial coordinates in A, will have different time coordinates in frame B according to the Lorentz transform.

This means that there is no shared notion of simultaneity between the two frames. Simultaneity is observer-dependent - unlike proper time, which is not observer dependent.

This doesn't mean that there are any more than four dimensions in space-time. In Newtonian physics, with its absolute time, frames of reference are generally interpreted as being spatial and having only three dimensions. For a specific cartesian coordinate system, one would have coordinates x, y, and z. A frame of reference can be regarded as a set of three basis vectors, which we'll label $\vec{x}, \vec{y}, \vec{z}$. Then the spatial position of an object can be regarded as the weighted sum of three of these three basis vectors, $x \vec{x} + y \vec{y} + z \vec{z}$

The dimensionality of this vector space is always three. One can make different choices of the basis vectors $\vec{x}, \vec{y}, \vec{z}$ corresponding to rotations of the coordinate systems, but there will always be three coordinates and three basis vectors. This is a mathematical property of vector spaces, known as the dimension of the vector space.

Time in Newtonain phyisics is regarded as absolute, and is not included as part of the reference frame.

In special relativity, one can construct a four-dimensional reference frame with vectors $\vec{x}, \vec{y}, \vec{z}, \vec{t}$ at every point. Three of the vectors represent "space", one of the vectors represents "time".

One can construct different reference frames by the usual spatial rotations, but one can also consider frames of reference that are not rotated, but moving relative to each other. This is known in the language of SR as a "Lorentz boost". The dimensionality of the vector space is equal to 4 in all cases, but the choice of basis vectors is different in different reference frames.

Dale

Mentor
There are people who agree with me that this is a very useful approach
There are also people who disagree. I am one of those.

To misquote Newton: You can see further by standing on the shoulders of a dwarf who is standing on the shoulder of a giant than you can by standing directly on the shoulder of the giant.

Dale

Mentor
So, if "proper time" is the "sort of time" that clocks measure, then it seems you are saying that "coordinate time" is a different "sort of time"?
Yes, definitely. For one thing, proper time is invariant while coordinate time is not.

bahamagreen

One observer says "I was in the garden at 6:00 pm" (a time coordinate - and I stayed there until 7:00 pm (another time coordinate). In this process, my clock ticked off 36,000.000001 seconds - a proper time interval.
One hour... 36Ks is 10hr

Mister T

Gold Member
I prefer to learn from the great geniuses like Einstein first-hand.
Some of the more modern writings are Einstein's. The 1905 paper was just his first word on the topic.

His clock seems to be a sort of idealized perfect clock. He of course goes into no details concerning its construction. Although he does mention it has "hands."
The meaning of "ideal clock" aside, what does it mean to be a real clock that's both accurate and precise? Quite simply, that collections of them agree with each other. It therefore doesn't matter if the clock has hands or a cuckoo. All that matters is that it's both accurate and precise. The more so the better.

I would be interested in reactions to how Einstein defines time in the first section of this paper. Is it clear? Is it confusing?
A good deal of your post is a discussion of how Einstein defines simultaneous. It's important that it be defined because it's a convention, not a physical quantity. Time, on the other hand, is a physical quantity. I don't think Einstein defines it as anything other than what a clock measures.

Paul Colby

Gold Member
I don't think Einstein defines it as anything other than what a clock measures.
And, this measurement is alway just an interval (time lapse between events) and not some absolute "time" frame thing. That's something theorists make up (and it's a good thing they do) to connect all the interval measurements in principle. Same comment for meter sticks.

Grimble

Proper time is independent of the observer and the observer's convention, and can be regarded as something physical.
Yes, that is how it seems to me. I think of proper time as objective, being measured with the rods and clocks of the observed frame and, therefore, being the same whatever the motion or location of the observer.
On the contrary, coordinate time, measured using the observer's own rods and clocks, is subjective and therefore unique due to the relative motion between the observer and the observed frame of reference.

pervect

Staff Emeritus
Yes, that is how it seems to me. I think of proper time as objective, being measured with the rods and clocks of the observed frame and, therefore, being the same whatever the motion or location of the observer.
I basically agree with this part of what you say, though you've expanded it slightly. I agree that proper time is objective. A possible difference in our views s the point that only a single clock is needed to measure any proper time interval. You don't say quite the same thing, as you talk about "clocks" and "rods" in the plural. I didn't discuss proper length, but I would agree that proper length is objective, and could be regarded as the non-proper length of a rod in a frame in which the rod is at rest.

On the contrary, coordinate time, measured using the observer's own rods and clocks, is subjective and therefore unique due to the relative motion between the observer and the observed frame of reference.
I don't think I agree with this characterization of coordinate time and distance, though perhaps I'm not quite following what you're trying to say here. The usual construction that describes coordinate time consists of an infinite array of clocks, one clock at every point where you might want to measure the time, in which all the clocks are at rest relative to one another in said frame and synchronized. The last point is important - we need to know how to synchronize clocks in order to measure coordinate time. This may seem to be nit-picking, but it turns out to be important because the act of synchronizing clocks is observer dependent, while the clocks themselves are not.

Comparing the non-proper length of rods is also an observer dependent operation , for similar reasons. Two rods have the same length if both ends of the rods are in the same place at the same time. But "at the same time" is only defined when one knows how to synchronize clocks.

Grimble

I agree that proper time is objective. A possible difference in our views s the point that only a single clock is needed to measure any proper time interval. You don't say quite the same thing, as you talk about "clocks" and "rods" in the plural.
Yes, I see the confusion. When I referred to rods and clocks I meant the rods (all the same length) and the clocks (all stationary and synchronised) in the frame of reference where the measurement is taken. It is but a generalisation I have seen used.

Grimble

I don't think I agree with this characterization of coordinate time and distance, though perhaps I'm not quite following what you're trying to say here.
The way I am seeing it is that there are two frames of reference involved here. (Basically it is the moving clock)
There is the 'native' frame of the clock and the frame of the observer.
Measurements can be taken with reference to either frame. Objectively with reference to the clock's native frame but subjectively for each and every observer using that observer's own frame of reference.
Those measurements taken against the native frame of the clock are objective - independent of who takes them, as long as they are taken against the clock's native frame using that clock and its associated rods.
Those measurements taken against the frame of the observer, in which the clock is moving, are naturally made against the framework of the observers frame of reference using the observers own clocks (at every point in space) and the observers rods.

In fact the difference in the measurements in these two situations is that the observers measurements include an additional vector - the relative speed of the two frames of reference.

We can see this very simply with a moving light clock.
Clock A and clock B are synchronised and are together at t=0.
Clock B is travelling at 0.6 c from clock A.
After 1 second (proper time) the light pulses in clock A and Clock B will each hit the mirror in its clock. Measured objectively in each clock's reference frame and clock B will have travelled 0.6 light seconds from clock A.

However when we look at at the time taken for the light pulse in clock B to reach its mirror, measured in clock A's frame of reference, i.e. by a moving observer, the light pulse in clock B will have travelled 1.25 light seconds from clock A's light source to point (0.75,1.25).

In the native frame of the clock the light travels 1 second directly to the mirror and reflects back to the light source.
In the observers frame the clock is moving so after 1 second in the observers frame, the light must have travelled 1 light second from the source while the clock has also travelled 0.6 light seconds from the observer reaching point (0.6,0.8). This is the Lorentz transformation in effect, where the time of 1 second in the native(objective) frame is 'dilated' by the Lorentz factor γ (1.25 @ 0.6c) to 1.25 seconds; and the length of 0.75 light seconds is contracted by γ to 0.6 light seconds.

(The measurements of the moving frame B as seen by observer A are drawn in purple as spherical coordinates as this makes more sense that trying to make the round peg of time fit in the Cartesian square hole of Space)

Dale

Mentor
There are (at least) two important differences between the proper time of an inertial clock and the coordinate time of that same clock's reference frame.

First, the coordinate time is defined over the entire spacetime while the proper time is defined only along the worldline of the clock itself. Where they are both defined they are equal, but the proper time is undefined for many events where the coordinate time is defined.

Second, the coordinate time includes a notion of simultaneity that is absent from the proper time. This notion is both conventional and frame variant.

Ibix

Here are two similar questions: how tall are you? And is your head higher than mine?

The first one is like proper time. How tall you are is a simple question you can answer easily with just a tape measure.

The second one is like coordinate time. We need to establish some sort of association between us to answer it. First we need to determine if our feet are at the same level, then make sure that we're both standing upright, or how far off upright we both are.

Defining what we mean by "feet at the same level" (sea level? ground level? something else?) is like agreeing a simultaneity convention - what it is we are calling zero. Working out the angle between us is analogous to working out our relative velocity. But once we've done both of those things we have the basis of a coordinate system - we can use the same process to compare heights with anyone. But there's a degree of arbitrariness about it. Did we pick ground level or sea level as zero? Is one righter than the other? Not really, we just needed to agree a definition.

I hope that's a helpful picture. Proper time is a duration, like a length. Coordinate time the difference in the times shown by a pre-agreed set of clocks that are notionally synchronised.

Grimble

Let us take our two inertial frames of reference, that of clock A and that of clock B. An observer local to a clock will read proper time from that clock. The clocks are identical; the conditions under which they operate are identical; the scientific laws they obey are identical; therefore the proper time measured on each clock will be of identical units.
If they were synchronised they would keep identical time.

The same would be true of any inertial clock read by a local observer at rest with that clock.

In fact, it seems to me, that any inertial clock, read by an observer at rest (and therefore in the same frame of reference) must measure proper time.

Whereas time measured for a moving clock; moving within an observer's frame of reference; must include the journey time of the clock; or the clock's frame of reference.

The proper time measured between two points on a clock's life line, must be the same as the Spacetime Interval, (both being τ).
And the Spacetime Interval added to the time journey time of the moving frame of reference between the two points (vector addition) gives the coordinate time in the observer's frame of reference.

Grimble

A possible difference in our views s the point that only a single clock is needed to measure any proper time interval. You don't say quite the same thing, as you talk about "clocks" and "rods" in the plural.
By clocks in the plural I mean synchronised clocks at rest in a frame - at every point in a frame - a generalisation I suppose... differentiating between the clocks at rest in a frame and 'moving' clocks in other frames.

First, the coordinate time is defined over the entire spacetime while the proper time is defined only along the worldline of the clock itself. Where they are both defined they are equal, but the proper time is undefined for many events where the coordinate time is defined.
The time axis of a frame of reference is effectively the time displayed on a virtual perfect clock at rest at the null point of the frame. It does not move in that frame and therefore must measure proper time. The proper time read on a clock permanently at the null point of a frame gives us the time axis of that frame. It is the same time as read on any synchronised clock at rest in that frame.
The time in a frame, measured on the time axis of that frame is an objective measure - it does not matter where an observer is as the measurements are made against the axes of the measured frame.

Second, the coordinate time includes a notion of simultaneity that is absent from the proper time. This notion is both conventional and frame variant.
Coordinate time is measured from a specific Spacetime location which makes the relative velocity of the measuring frame and the measured frame a factor.
Proper time is the absolute passage of time for a clock - the time the clock experiences.
Coordinate time is the time measured by an observer for that clock and is a subjective measurement where the location and movement of the observer are important factors in the measurement.
Every observers coordinate measurements are unique to that observers location and motion relative to what is being measured.

Dale

Mentor
The proper time read on a clock permanently at the null point of a frame gives us the time axis of that frame. It is the same time as read on any synchronised clock at rest in that frame
It is not the same for the same reason that a coordinate system is not the same as one of its axes. There exists a mapping from each event in the coordinate system to events on the axis, but it is not the identity mapping. Saying they are the same is not right.

Proper time is the absolute passage of time for a clock - the time the clock experiences.
I prefer to use the term "invariant" rather than "absolute" to avoid the negative connotation associated with the latter.

Coordinate time is the time measured by an observer for that clock and is a subjective measurement
I would similarly use the term "frame variant" instead of "subjective". The coordinate time is no more or less subjective than energy or momentum.

"Einstein's definition of time"

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We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving