# How far is this definition of time correct?

1. Apr 19, 2013

### aleemudasir

How far is the definition of time that time is change in space, correct?

Last edited: Apr 20, 2013
2. Apr 20, 2013

### Staff: Mentor

I'm not sure what you're asking.

Is this related to Special Relativity and the fact that different observers may have clock running at different rates?

3. Apr 20, 2013

### aleemudasir

No, I'm asking the general statement.

4. Apr 20, 2013

### aleemudasir

And I also wish to know what will happen in this case.

5. Apr 20, 2013

### aleemudasir

When I say general definition I mean, like the definition we have for length(measure of distance), mass(amount of matter), volume(space occupied) etc.

6. Apr 20, 2013

### ghwellsjr

Time is what a clock measures. Here's how Einstein explained it in section 1 of his 1905 paper introducing Special Relativity:

It's very simple.

7. Apr 20, 2013

### WannabeNewton

George maybe you should write an FAQ or something on it. This question does seem to pop up often even though the operational definition of time is quite clearly spelled out in relativity texts and the likes.

8. Apr 20, 2013

### pervect

Staff Emeritus
There are at least two types of time - coordinate time, and proper time. And sums and differences thereof, of course.

Coordinate time tells you WHEN something happens - usually the something is abstracted as "an event".

Proper time is a duration, usually described as "wristwatch time". It's measured by some acual clock. The clock can be abstractl specified by some worldline in space-time.

The confusing part starts to happen when one tries to explain that coordinate clocks don't necessarily run at the rate as real clocks, and that this is called time dilation.

MOre confusion tends to arise when one attempts to explain that "at the same time" is an observer dependent statement, often a coordinate-dependent one.

I think a lot of the confusion involves "unlearning" some pre-relativistic ideas about time.

9. Apr 20, 2013

### ghwellsjr

Wow, I've never heard so many misstatements all in one place in a long time.

First off, Coordinate Time is an abstraction just like Coordinate Distance to permit us to define an Inertial Reference Frame (IRF) so that there are not any real coordinate clocks (or rulers) permeating all of space but if we wanted to actually put a real clock at a real coordinate location somewhere, it would keep Proper Time just like any other clock. The only difference between such a clock and some other clock that we might want to consider is that the "Coordinate Clock" must forever remain inertial which means it must remain at its coordinate location. We also set it one time to match the Coordinate Time and never set or reset it again.

But whatever scenario we want to envision in which we apply a real clock to match the Coordinate Time at some Coordinate Location, we can then apply the Lorentz Transformation process to create a new IRF moving with respect to the original one and end up with a new Coordinate Time definition in which there are no longer any clocks at any particular Coordinate Locations. It's in this sense that we never want to require that there be any real Coordinate Clocks. As Einstein said, they're imaginary.

We use the concept of an IRF then to describe and/or analyze how real clocks keep track of Proper Time but we always reserve the right to have these real clocks accelerate, in other words, they don't have to remain inertial at any particular location. Time Dilation then is the ratio of the passage of Coordinate Time to the passage of Proper Time and is a function of the instantaneous speed of the clock according to the particular IRF that we are considering.

Of course, measuring time is just like measuring distance: it is always a delta (or duration) between two points. We can always use a clock to measure a time interval and we can always use a stop watch (whose purpose is to measure a time interval) to measure what we regard as a clock time. Drawing a distinction between Proper Time and Coordinate Time with regard to one measuring duration and one measuring clock time is a red herring. Every clock has to be set at some point and then measures a delta time or duration.

And there is nothing confusing about "at the same time" if we simply point out that we mean Coordinate Time according to a chosen IRF.

It's not confusing. It's very simple. Sorry to be so blunt but I just can't let so many misstatements go unchallenged.

10. Apr 20, 2013

### A.T.

Isn't coordinate time just the proper time of some clock, that we can choose arbitrarily? In SR coordinate time is the proper time of a clock at rest in the reference frame that we have chosen. In GR we have to choose the position of the "coordinate clock" as well.

11. Apr 20, 2013

### pervect

Staff Emeritus
Not really, because coordinate clocks don't necessarily keep proper time :-(.

12. Apr 20, 2013

### pervect

Staff Emeritus
I didn't think anything I said was even mildly controversial!

I would agree that coordinate time is an abstraction. I would not, however, limit it to defining inertial reference frames. We use coordinate times every day in non-inertial reference frames - such as atomic time (also known as TAI time) on the Earth.

I'd agree that there aren't any real coordinate clocks permeating all of space, and I'd agree that if we put a real clock at a real coordinate location it would keep proper time. In fact, that's exactly what I said earlier.

This doesn't change the fact that a a real clock, placed on the Earth, will keep proper time and NOT remain in synch with the coordinate time standard. The real clock will keep proper time, and not coordinate time.

In general the rate of the real, physical clock needs to be adjusted (because it keeps proper time) when one wishes to define a coordinate time. This is routinely done with TAI time, the atomic clocks that define TAI time are rate adjusted by height above sea level before being averaged into the TAI time standard.

I thought this was well known, and totally noncontroversial, in case there's some remaining doubt, I'll post a reference from wiki:

http://en.wikipedia.org/w/index.php?title=International_Atomic_Time&oldid=550649840

This may be true in the special case you were envisioning, but it's not true in general. Or should I say In General?

Coordinate time and coordinate time standards are an important part of everyday life. Drawing the simple distinction between "time as a coordinate" and "time as in interval" is a rather elementary, but I think important, step in understanding what we mean when we say "time". We use one word ("time") to talk about a family of different concepts ("coordinate time", "proper time"). These concepts, while related, are not identical. Becoming aware of the differences is , I think, important.

And that was the purpose of my post - to point out some of these differences. When we talk about time, sometimes we are talking about "proper time", and we are not rate-adjusting the clocks. Other times, we are talking about "coordinate time", and we ARE rate adjusting them.

Last edited: Apr 20, 2013
13. Apr 20, 2013

### Fredrik

Staff Emeritus
It wasn't. My first reaction to the term "coordinate clock" was that it seems like a useless concept. I thought, why would we want a clock that measures coordinate time? But you explained that in your next post.

Pervect said that confusion arises as one tries to explain this, so he was clearly talking about how people who are new to relativity find this confusing.

14. Apr 20, 2013

### ash64449

Fredrik,i am new to relativity.

What is the difference between co-ordinate time and proper time?

15. Apr 20, 2013

### ash64449

Then which time is used to explain time dilation? Is it proper time?

16. Apr 20, 2013

### nitsuj

Agreed, didn't see anything "wrong" with what you said.

17. Apr 20, 2013

### Fredrik

Staff Emeritus
The only way to get a better understanding of reality is to find a theory that makes accurate predictions about results of experiments, and then study the theory. A common objection is that since the theory isn't reality, understanding the theory is not the same thing as understanding reality. This is true, but it doesn't change the fact that studying a theory is the only way to get a better understanding of reality. We simply don't have "direct access" to reality. We only have "indirect access" to it, via theories.

So if you want to understand what time is, the best option you have is to make sure that you understand the mathematical definition of time provided to you by our best theory of space, time and motion. At present, this is general relativity. A definition of GR includes mathematical definitions of two kinds of time, "coordinate time" and "proper time". It also includes a correspondence rule that can be stated like this: If a clock displays t at one event and t' at another, then the proper time of the part of the clock's world line from the former event to the latter event is t'-t.

(A correspondence rule is an assumption about how to interpret the mathematics of the theory as predictions about results of experiments).

The old statement "time is what a clock measures" is not a definition of time in GR, or in any other theory. It's a theory-independent statement. It's a guiding principle about what sort of thing we should be calling "time" in theories of physics. It's a way of saying that every theory of motion should use the term "time" specifically for the mathematical concept that its correspondence rules associate with clocks.

So that old statement has its place, but it's not really a definition, or an insight about what time really is. It's just a pointer that tells you what you need to look at in an actual theory to get a better understanding of what time is.

Last edited: Apr 20, 2013
18. Apr 20, 2013

### nitsuj

Agreed, I have no formal education in SR, just from reading here and there, no text book presentations.

This distinction took me quite a while to become more clear, even though it is "in your face" right from the first presentation of time dilation / differential aging.

Not surprising imo, that I never considered the proper time my watch measures is a separate concept from the age of the watch. (of course the two will always be the same for my watch, but not the always the case for spatially separated "things")

To your point of "un-learning" the idea the two are concepts are the same.

Last edited: Apr 20, 2013
19. Apr 20, 2013

### Fredrik

Staff Emeritus
A coordinate system is a function that takes points in spacetime to 4-tuples of real numbers. If p is a point in spacetime (i.e. an event), and $\phi$ is a coordinate system defined on a set that includes p, then we can write $\phi(p)=(t,x,y,z)$. The time coordinate of p in the coordinate system $\phi$ is just the number t. So coordinate time (i.e. the time coordinate) is a property of an event and a coordinate system. Change the coordinate system, and the time coordinate of p changes.

Proper time on the other hand is a property of a curve in spacetime. In SR, the definition can be stated in terms of the coordinates of some inertial coordinate system, by saying if p and q are events, then the proper time of a curve from A to B is the integral of $\sqrt{dt^2-dx^2}$ along the curve. This way of stating the definition is simple and easy to understand, but it hides the fact that the result of that integration is independent of what inertial coordinate system we're using, and it can make you think that you need to use some inertial coordinate system. This is not the case. The definition can be stated without mentioning a coordinate system at all. (It's just harder to understand that definition). So proper time is a property of a curve that doesn't involve coordinate systems in any way. Change the coordinate system, and the proper time of the curve doesn't change.

Proper time isn't defined for all curves. (There would be a problem if the thing under the square root isn't positive everywhere along the curve). But it is defined for all curves that can represent the motion of an object, like a clock.

Last edited: Apr 20, 2013
20. Apr 20, 2013

### Fredrik

Staff Emeritus
Time dilation is about how two inertial coordinate systems in SR that take the same event to 0 assign different time coordinates to events on the time axis of one of the coordinate systems. So time dilation is about coordinate time. Problems involving time dilation can however often be solved by considering proper time instead. This is e.g. the fastest way to find the correct ages of the twins in the twin paradox.

21. Apr 20, 2013

### Naty1

Fredrik:

Oh no...just when I thought I had this straight in my mind.

I would have answered: Time dilation is about proper time: Two different observers record different elapsed [proper] times on their own clocks after starting out at some location, travel different paths, then rejoin later.

What's wrong with this??

22. Apr 20, 2013

### DrGreg

I'm going to answer this in the context of inertial coordinates in Special Relativity (without gravity).

Proper time is measured by a single clock and can be used only for events that occur locally, right next to the clock.

Coordinate time allows you to measure events that are far apart in distance. To do so you can't use a single clock, you need multiple clocks, each located next to the event it is measuring. So imagine a whole network of clocks, all a fixed distance from each other, and all ticking at the same rate. The important issue is that you need to synchronise all these clocks to each other, and that's a non-trivial process. For inertial coordinates, there's an agreed way of doing this ("Einstein synchronisation"). Once you've done all that, the coordinate time at an event is determined by the nearest local clock within the network of clocks.

For time dilation, you compare the proper time on one clock that is moving relative to the network, against the coordinate time on the network of clocks.

Non-inertial coordinates

If the network of clocks is non-inertial, or if there is gravity, things get more complicated. In this case, clocks that are a fixed distance apart need not tick at the same rate ("gravitational time dilation"), so the clocks need to be adjusted to run fast or slow to remain synchronised. Also in this case there's no universally agreed method of how to synchronise your network clocks; many options are available.

23. Apr 20, 2013

### nitsuj

"different paths, then rejoin later."

Time dilation doesn't require them to rejoin. However the proper time comparison does. To "extend" your proper time to make measurements is coordinate time.

That said, I couldn't disagree that it is "about proper time".

24. Apr 20, 2013

### Fredrik

Staff Emeritus
That's the twin paradox, and as I said, the fastest way to find the correct ages of the twins is to calculate the proper time of their world lines. So there's nothing wrong with what you said.

What I mean by "time dilation" is just the fact that two inertial coordinate systems that send the same event to 0 (so that they're related by a Lorentz transformation) disagree about what time coordinates to assign to events on one of the coordinate systems' time axis. A Lorentz transformation sends (t,0) to (γt,-γvt), so the time coordinate is "dilated" by a of factor gamma. (I write the time coordinate first and the position coordinate second in these coordinate pairs).

So you can define time dilation without mentioning proper time, but I don't think there's a definition that doesn't mention inertial coordinate systems, or at least simultaneity. The problem is that we're talking about the ticking rates of two clocks that will be at the same place only at one event (assuming that they're doing inertial motion), so we will need one clock's notion of simultaneity to compare them.

Last edited: Apr 20, 2013
25. Apr 20, 2013

### Naty1

DrGreg:

Nitsuj:

I find such descriptions confusing; Are we making MEASUREMENTS? maybe 'comparisons' is better.

I don't know how to express those ideas any better, but I do know:

my proper time is your coordinate time,

your proper time is my coordinate time,

[In other words, proper time is local time, coordinate time is spacetime interval distant]

nobody can 'measure' coordinate time,

Coordinate times can only calculated from the (proper-time) readings [measurements] of clocks via time dilation relationships.

/////////////

For the OP: This might be helpful background:

http://en.wikipedia.org/wiki/Absolute_time_and_space#Impact_of_special_relativity

So in SR relative speed affects comparative elapsed times; in GR different gravity potentials [curvatures of spacetime] also affect relative elapsed times...an additional effect on time and space. Unlike Newtonian physics, there is no universal absolute time.