# I Einstein's definition of time

#### Dale

Mentor
Several speculative posts and responses have been removed. The participants are reminded that all posts must be consistent with the professional literature.

#### Grimble

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.
Thank you, Dale; I am not quite sure just what you are saying here - is it that, being synchronised, each clock at rest in a frame, will read the same time? That is they will, each and every one, be mapped to the same point on the time axis?

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.
So: Proper time is the invariant passage of time for a clock - the time the clock experiences; the time a clock displays.

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.
Coordinate time is a frame variant measurement of time measured by an observer for a clock moving in the observer's frame.

As for being subjective, are not (potential?) energy and momentum subjective? their values are surely dependent on the bodies location/movement relative to the observer and are only valid for that observer?

#### Dale

Mentor
Thank you, Dale; I am not quite sure just what you are saying here - is it that, being synchronised, each clock at rest in a frame, will read the same time? That is they will, each and every one, be mapped to the same point on the time axis?
Yes, that mapping is what defines synchronization. It is an additional structure beyond the proper time of a clock at rest at the origin.

As for being subjective, are not (potential?) energy and momentum subjective?
Again, I would use the term "frame variant". To me "subjective" carries the connotation of being an opinion, so I would use "subjective" for things like whether a painting is pretty or ugly, or whether a meal was satisfying or not. The momentum depends on the reference frame, but given a reference frame it is not a matter of opinion.

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#### Mister T

Gold Member
So: Proper time is the invariant passage of time for a clock - the time the clock experiences; the time a clock displays.
In Einstein's native German, proper time is eigenzeit, from eigen meaning one's own and zeit meaning time. Proper time is the time that elapses on a clock that you carry with you everywhere you go. It is a relativistic invariant in that all observers, regardless of their motion relative to that clock, will agree on its value.

When we try to understand why someone else's proper time is different from our own we need to consider coordinate time.

Note that time dilation involves a comparison of a proper time to a coordinate time. We say that the coordinate time is dilated relative to the proper time. The classic example of this is muon decay.

Also note that in the twin paradox we are comparing one person's proper time with another person's proper time. It is not the same thing as time dilation, and not understanding that can be a stumbling block to understanding the twin paradox. Many if not most published explanations of the twin paradox fail to mention this, and in my opinion that can cause confusion for the reader who is trying to understand it.

#### Grimble

Yes, that mapping is what defines synchronization. It is an additional structure beyond the proper time of a clock at rest at the origin.[\quote]
Ok. but is it true then to say that the time lines of all those synchronized clocks lie along the time axis?

Again, I would use the term "frame variant". To me "subjective" carries the connotation of being an opinion, so I would use "subjective" for things like whether a painting is pretty or ugly, or whether a meal was satisfying or not. The momentum depends on the reference frame, but given a reference frame it is not a matter of opinion.
Yes, I see those implications of the term subjective and they are certainly not what is meant; I agree that 'frame variant' is preferable.

When we try to understand why someone else's proper time is different from our own we need to consider coordinate time.
Surely the difference is very simple and straightforward...
Coordinate time is the proper time passing on a clock, between two events, as would be measured by a local observer at rest with the clock; plus the duration of time taken for the clock to move the physical distance between those two events as measured by an observer relative to whom the clock is moving.

In the clock's frame the clock is at rest and therefore measures proper time.
In the observer's frame the clock is moving and the travel time for the clock is added (by vector addition) to create coordinate time.
(In the observer's frame it is coordinate time as the clock is travelling between spatial coordinates)

For the proper time, the clock is at rest in the clock's frame; measured in an observer's frame where the clock is moving, this is the 'spacetime or invariant' interval.

#### Mister T

Gold Member
Surely the difference is very simple and straightforward...
No. If you look at the Lorentz transformation equations and try to find an expression for the difference between the coordinate time and the proper time you will find the task to be neither simple nor straight forward. As far as I can tell the best you can do is an infinite series.

#### arydberg

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.
I think you are right on. The real problem is that we do not know how to think about time.

#### Dale

Mentor
I think you are right on. The real problem is that we do not know how to think about time.
We know how to think about time quite well. We have thought about it so well that we have devised machines to measure it with errors on the order of 10^-16 on a routine basis.

#### Prathyush

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.
I have read the paper long ago, and I found his treatment of the subject to be perfectly clear. One can define what a clock is constructively in any manner and they would be equivalent. A clock is anything exhibits periodic phenomenon. Clocks measure proper time is the definition I usually use for practical considerations. However in the logical treatment of things one starting with with defining clocks as things that exhibit periodic phenomenon. And using postulates that speed of light in invariant in all reference frames and physics is the same all inertial reference one can infer that clocks indeed do measure proper time.

#### JulianM

I can agree that this provides a means of determining if the clocks are synchronized, however it says nothing about what an observer, as defined by Einstein, observes even within a simple stationary system.

Taking the example of 2 clocks separated by a distance of one light hour. If clock A reads 5.00 pm then a light ray will reach clock B at 6.00 pm and arrive back at A at 7.00 pm.
Putting these values into the equation we get 6.00 pm - 5.00 pm = 7.00 pm - 6.00 pm.
We agree at this point and the clocks are synchronized to the same time.

The problem comes when we have an observer who is reading time from these clocks when the observation involves the one way transmission of light. When the observer at A reads his time as 6.00 pm he reads the time at B as 5.00 pm.

If we now move clock B to a different place in the same stationary space it is still synchronized, but the observer now reads a different time. The implication of this is that in making any observation of time then the observer must know the distance to the other clock and must calculate backwards in order to know what the time is in that part of, even, stationary space.

Where this becomes important is in Einstein's next step in choosing the position of his observers. His choice of placing an observer in the middle of a train is a special and unique choice, and is the only position that supports his train thought experiment.

If the observer on the train was not uniquely equidistant from the flashes of light, let's say he is one carriage back, and arrives at the same position as the observer on the platform just as the flashes arrive at the observer on the platform then he will see exactly what the platform observer sees, and will conclude the simultaneous flashes were indeed simultaneous. The stationary and moving observers now agree and Einstein would be unable to continue with his theory because lack of simultaneity due to speed or movement is no longer a factor.

Stepping back to Simultaneous Time. Look around you, wherever you are, and recognize that all the objects around you are at different distances away from you. You are therefore not observing them at the same time. Take a look at the leaves on a tree. You are not seeing them in simultanoeus time. If you truly want to perform any measurements on the leaves using light then you must first know the distance each one is from you.

Relating this to molecules moving in a container then their distance from you is constantly changing and therefore their relative time is constantly changing. The same would be true of the leaves on the tree blowing in the wind.

Observing simulataneity is therefore dependent on the position of the observer. When Einstein uses rods and trains as examples he neglects the fact that every observer comes to a different conclusion. An observer in the middle of the train sees something different from the observer in the carriage behind, even though they are moving at the same speed in the same frame of reference. When Einstein talks about moving rods he simplifies the situation by considering that all parts of the train or the rod are simultaneous in time. Yes they are, but no two observers, stationary or moving, would observe that.

The fact is that everything we observe - stars, molecules, trains, etc are observed using light or other electoromagnetic wave sensors. We therefore are not measuring what is actually happening unless we take into account the distance of the object from us, and the inherent difference in the observed time, and then back calculate.

Einstein has simplified the situation which leads to his conclusions on length shortening etc. The "shortening" of a moving rod is easily shown to be an illusion when we take in to account the difference in time of each end when observed by a stationary observer.

In conclusion - simultaneity of time is merely an expression of Newtonian "absolute" time and not helpful when considering more than one observer or the movement of an observer.

#### Ibix

Where this becomes important is in Einstein's next step in choosing the position of his observers. His choice of placing an observer in the middle of a train is a special and unique choice, and is the only position that supports his train thought experiment.
Not really. It's just that this case is symmetric in one of the reference frames so is easy to analyse. You can use any other place on the train and come to the same conclusion, but the situation is more complex to describe and analyse.

#### Dale

Mentor
The implication of this is that in making any observation of time then the observer must know the distance to the other clock and must calculate backwards in order to know what the time is in that part of, even, stationary space.
Yes, this is correct

His choice of placing an observer in the middle of a train is a special and unique choice, and is the only position that supports his train thought experiment
This is not correct. This unique choice simplifies the analysis, but is not necessary. The rest of your post, particularly the conclusion, does not follow.

#### Nugatory

Mentor
The "shortening" of a moving rod is easily shown to be an illusion when we take in to account the difference in time of each end when observed by a stationary observer.
Length contraction and time dilation are what's left over after you have properly corrected for light travel time, so they cannot be just an illusion of observation. Bell's Spaceship Paradox, in which length contraction generates stresses that cause a string to break, is a good example of the way in which the contraction of a moving rod is not an illusion (it is frame-dependent, in the sense that other frames will have different explanations of the breaking string, but it is not an illusion).

It is reasonable to consider relativity of simultaneity as the "real" explanation for observations of both length contraction and time dilation, but that doesn't make either phenomenon an illusion.

#### russ_watters

Mentor
Einstein has simplified the situation which leads to his conclusions on length shortening etc. The "shortening" of a moving rod is easily shown to be an illusion when we take in to account the difference in time of each end when observed by a stationary observer.
The idea that the length of an object is observer dependent causes a lot of people a lot of discomfort. It's easier to accept time dilation because you can fly a clock around the world, sit them next to each other and see that they show different elapsed time. But you can't do that with length contraction, can you?

Actually, yes you can: If you travel to Alpha Centurai at just under the speed of light, it might seem to you to take just 1 year. Since you can't exceed the speed of light (and indeed didn't measure yourself to be exceeding it), that means you must accept that you only traveled a touch under 1 light-year during your trip, not the 4.5 light years your friends on Earth saw.

The difference between length contraction and time dilation that people tend to miss is that it isn't elapsed time that dilates, it is the rate of time passage that dilates. That goes away when objects are brought together, just like length contraction does. So in both cases, only by measuring the accumulated/elapsed time/distance do you see the restults.

Time dilation and length contraction really are opposite sides of the same coin and each exactly as "real" as the other.

#### JulianM

Not really. It's just that this case is symmetric in one of the reference frames so is easy to analyse. You can use any other place on the train and come to the same conclusion, but the situation is more complex to describe and analyse.
In fact I presented another position on the train where the same conclusion is not reached. A passenger sitting behind the center who arrives at the position of the stationary observer at the same time as the flashes does see them as simultaneous. Therefore you cannot "use any or all other places".

#### Ibix

In fact I presented another position on the train where the same conclusion is not reached. A passenger sitting behind the center who arrives at the position of the stationary observer at the same time as the flashes does see them as simultaneous. Therefore you cannot "use any or all other places".
Your analysis is incorrect. He receives the flashes simultaneously, but he's not equidistant from the ends of the train so he does not conclude that they were emitted simultaneously. Whereas the platform observer is equidistant from the emission points and receives the flashes simultaneously so concludes that they were emitted simultaneously.

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#### Aufbauwerk 2045

I changed my mind about reading the old stuff. It's old. Give it to me short and sweet in language I can understand. This is the 21st century and I don't have all day. Thank you.

#### Ibix

I changed my mind about reading the old stuff. It's old. Give it to me short and sweet in language I can understand. This is the 21st century and I don't have all day. Thank you.
Taylor and Wheeler, Spacetime Physics. Some chapters are online for free if you want to try before you buy.

#### nitsuj

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.
I agree with the interpretation, its exactly as he said. However I'd argue his use of clocks is significant. The ruler and photon became a perfect measure of time. "Ticks" of said clock being ab in length. Photon moves 299.7k.km in one second..the dang thing is great for measurements of Spacetime.

#### Mister T

Gold Member
I wonder if the way Einstein uses his so-called "clocks" in defining time is helpful or confusing.
I agree with the interpretation, its exactly as he said. However I'd argue his use of clocks is significant. The ruler and photon became a perfect measure of time. "Ticks" of said clock being ab in length. Photon moves 299.7k.km in one second..the dang thing is great for measurements of Spacetime.
I don't think it's true that Einstein used any so-called "clocks" to define time. Time is simply the thing we measure with clocks. Metrologists tell us how to measure time, but there is no way that I know of to define time.

And I don't think it's true that photons are used to measure time. First of all there is no such thing as a perfect measure of anything. (The most precise measurements of distance, as far as I know, use digital interferometry; that technique makes use of waves of light as far as I know and has nothing to do with photons.) Photons are a part of quantum theory; it's true that quantum mechanics has provided us with very precise ways to measure things, and metrologists use those measurements to create better standards. They are even going so far as to set exact values for fundamental constants to establish standards rather than relying on artifacts to do so. They did that with the speed of light and they will likely do that with Avogadro's Number and other fundamental constants next year. But using that technique to establish standards does not mean we are making exact measurements. For example, when making measurements of the distance a light beam travels from $a$ to $b$ and the time it takes to make that trip, we are not measuring the speed of light but are instead calibrating the devices we're using to measure length.

#### nitsuj

I don't think it's true that Einstein used any so-called "clocks" to define time. Time is simply the thing we measure with clocks. Metrologists tell us how to measure time, but there is no way that I know of to define time.

And I don't think it's true that photons are used to measure time. First of all there is no such thing as a perfect measure of anything. (The most precise measurements of distance, as far as I know, use digital interferometry; that technique makes use of waves of light as far as I know and has nothing to do with photons.) Photons are a part of quantum theory; it's true that quantum mechanics has provided us with very precise ways to measure things, and metrologists use those measurements to create better standards. They are even going so far as to set exact values for fundamental constants to establish standards rather than relying on artifacts to do so. They did that with the speed of light and they will likely do that with Avogadro's Number and other fundamental constants next year. But using that technique to establish standards does not mean we are making exact measurements. For example, when making measurements of the distance a light beam travels from $a$ to $b$ and the time it takes to make that trip, we are not measuring the speed of light but are instead calibrating the devices we're using to measure length.
I meant to say "However I'd argue his use of clocks is NOT significant." I may have not made my point clear...I find he essentially said "I'll use this well known and easily measured length, to equate to the time it takes light to make a round trip. I define that half the distance traveled is half the time of the round trip." He referred to this as defining a "common time". Imo that is making a "clock" out of a "ruler and photon". a perfect and idealized clock.

For the underlined part of your post...YES. Same idea, leaning on the constant for making measurements of spacetime. in the case Einstein made the easily measured length IS known and the time (one half the full trip) is defined.

#### Mister T

Gold Member
He referred to this as defining a "common time". Imo that is making a "clock" out of a "ruler and photon". a perfect and idealized clock.
I think that's just a simultaneity convention. In that context "common time" means "common clock-reading". You need two clock-readings to measure the passage of time. In our everyday language we use the word "time" to refer to both $t$ and $\Delta t$. When we speak of defining time we are referring to the establishment of a standard for measuring $\Delta t$, but when we synchronize clocks we are referring to a way of establishing a common clock-reading $t$, and that does not require a standard. The former is a matter of metrology requiring the most precise devices we can invent. The latter is instead a mere convention (unless the clocks are co-located in which case it's trivial).

#### nitsuj

I think that's just a simultaneity convention. In that context "common time" means "common clock-reading". You need two clock-readings to measure the passage of time. In our everyday language we use the word "time" to refer to both $t$ and $\Delta t$. When we speak of defining time we are referring to the establishment of a standard for measuring $\Delta t$, but when we synchronize clocks we are referring to a way of establishing a common clock-reading $t$, and that does not require a standard. The former is a matter of metrology requiring the most precise devices we can invent. The latter is instead a mere convention (unless the clocks are co-located in which case it's trivial).

I agree, it is a simultaneity convention.

What do you mean you need two clocks to measure the passage of time? The second clock is to be sync'd to the first, due to the "perfect timing" (invariant speed) of this idealized light clock it can be done across a distance. One clock is used to do this, with a round trip for light. Do whatever with the results...in this case synchronize clocks..

#### Mister T

Gold Member
What do you mean you need two clocks to measure the passage of time?
You need two clock-readings.

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