# How did Einstein Define Time

#### ghwellsjr

Gold Member
Yes, I see your confusion but it is easily cleared up once you realize that a clock is keeping track of the time between two events.

To help you understand, let's think first about stopwatches. They display an elapsed time between two events, the first event occurs when you start the stopwatch and the second event is when you stop it. Or you could just let the stopwatch keep running without stopping it and then the second event is whenever you look at it.

A stopwatch has a reset button that lets you set the initial time to zero but a clock has several buttons or knobs that let you set the display to any time you want. This would be the first event. The second event would be whenever you look at it.

So the first event with coordinate time is when you synchronize the clock. The second event is when you look at it.

With proper time, the first event is when you set it to the local coordinate clock or when you reset two clocks or set two clocks to the same time and then take one of them on a journey so that its time goes out of sync and later return it to the second clock to observe that it has elapsed less time, for example.

So Einstein defined coordinate time when describing his convention of synchronizing all the clocks that are used to define a Reference Frame. This is what he talked about in the first two or three sections of his 1905 paper (see link in post #2). He defined proper time when he described what would happen to a moving clock in relation to a coordinate clock. This is what he talked about at the end of section 4 of his paper. Now he may not have used the same terminology that is in use today, but that is what he meant. You can see another change in terminology at the end of section 3 where he uses β for what we now call γ. We now use β for a speed as a fraction of the speed of light. So we shouldn't get overly concerned about the specific terms that were used over a hundred years ago compared to the terms that are in use today.

Does that clear things up for you?

#### Salman2

Does that clear things up for you?

1. But just so I am sure I understand you....we can conclude then that there are two completely different and non-contradictory "time" concepts for Einstein, (1) coordinate time that we use t to symbolize within (x,y,z,t), and (2) proper time that uses tau as symbol. This then would be the final answer to my OP question....how did Einstein define time ? Makes sense to me, but please let me know if I error.

2. Also, in your stopwatch example. Suppose we have a runner moving between points A and B. Would it be true that Einstein would say there are two different measurements of "time" possible for this circumstance. So, the first, would be coordinate time, and would be measured by an observer using a stationary stopwatch (say at a position midway between A and B) to determine the time for the runner to move from A to B. The second would be proper time measured if the runner carried the stopwatch and was both runner and observer. So, in theory, Einstein would say the two different time measurements should not be the same for the same race (if we could design a watch sensitive enough to measure the difference at such slow speeds)...would this be correct understanding of Einstein and his use of the concepts of coordinate time and proper time ?

#### ghwellsjr

Gold Member

1. But just so I am sure I understand you....we can conclude then that there are two completely different and non-contradictory "time" concepts for Einstein, (1) coordinate time that we use t to symbolize within (x,y,z,t), and (2) proper time that uses tau as symbol. This then would be the final answer to my OP question....how did Einstein define time ? Makes sense to me, but please let me know if I error.
Einstein is credited with having said, "Time is what a clock measures". This would be the proper time. Then he uses the proper time on a non-accelerating clock to synchronize the proper times on remote clocks at rest with the first one as part of his definition of a Frame of Reference. It's because all these clocks are not allowed to accelerate that allows the proper times on all these clocks to be used as coordinate time for the FoR.
2. Also, in your stopwatch example. Suppose we have a runner moving between points A and B. Would it be true that Einstein would say there are two different measurements of "time" possible for this circumstance. So, the first, would be coordinate time, and would be measured by an observer using a stationary stopwatch (say at a position midway between A and B) to determine the time for the runner to move from A to B. The second would be proper time measured if the runner carried the stopwatch and was both runner and observer. So, in theory, Einstein would say the two different time measurements should not be the same for the same race (if we could design a watch sensitive enough to measure the difference at such slow speeds)...would this be correct understanding of Einstein and his use of the concepts of coordinate time and proper time ?
Not just two, but an infinite number. Time is relative, not absolute or narrowed down to just two types.

The proper time is what any clock measures, whether it is accelerated or not. But if it's never allowed to accelerate, then we can use it as coordinate time. But remember, another clock moving with respect to the first one will be running at a different rate and if it is not allowed to accelerate (once it gets to its constant speed), it can be used as coordinate time in a different coordinate system (Frame of Reference).

Let me take the opportunity of your question about a runner to have a little fun with this. I'm going to assume that these people don't know anything about relativity. Here we go:

Let's say that a bunch of runners start their own stopwatches when the gun fires at the start of a race at point A and then they each punch the lap button on their stopwatches when they each arrive at point B. The lap button displays the elapsed time but allows the stopwatch to continue to run and you can see this accumulating time when you punch another button. So they all record their individual elapsed times and punch the other button to allow their stopwatches to continue to display the time since the start of the race. They do this because they want to see if all the runners started their stopwatches at the same time.

And what do they find? First, they realize that all the stopwatches are now ticking at the same rate. In other words, whatever the differences are between the stopwatches remains the same, so they believe the clocks are accurate.

But the second thing they find is that the accumulating time on the first place winner's stopwatch is earlier than all the others implying that he may have punched his start button later than the others in an attempt to give himself a better time. And the same with all the others in the order in which they arrived at the end of the race. The slowest runner's stopwatch is displaying the latest accumulating time. So this puts in to serious doubt the actual race statistics.

So these people decide they will get some unbiased measurements of the race times for each runner. They get a whole bunch of people, each with their own stopwatches and they place them all along the race route. They are each instructed to start the stopwatch when the gun fires at the start of the race and to record the time displayed on the stopwatch when they see each runner reach the end of the race. What will they find?

Again, each one will measure a different time for each runner. The one closest to the end of the race will have the shortest times and the one closest to the start of the race will have the longest times. Why is this? It's because the one at the end of the race is starting his stopwatch later than the one at the start because he has to wait for the image of the gun firing to travel to him and he doesn't have to wait any time for the image of each runner to reach him because he is right there at the end of the race. Not only does the one at the start of the race not have to wait any time to start his stopwatch but he has to wait for the image of each runner to get back to him, making his measured time even longer than the one at the end. And all the timers located along the track will have proportionately different times between these two extremes. So there are so many different measurements of the time it takes for each runner to get from point A to point B. How are they going to make sense of it all?

Well, some smart guy among them realizes that if they ran the race in the other direction, and assuming that the runners ran at the same speed as before, they should get the same elapsed time for each runner as before. When they do this, they discover that the person who is measuring the racers at the midpoint is the only one who gets the same time for each runner in both cases. So they decide from here on out to make a new rule when timing races: they will put their official timekeeper at the midpoint of each race.

That was fun. But keep in mind that they have just defined the meaning of time on remote clocks at rest with one another, the same way that Einstein did. They still realize that the accelerated clocks (the ones the runners carry) will have a different concept of time.

#### Salman2

Einstein is credited with having said, "Time is what a clock measures". This would be the proper time.
Excellent ! I was always under the impression this was the coordinate time that Einstein was talking about in this famous quote.

So, what you say below seems to me to be the correct answer to my OP question about how Einstein defined "time"....I never made all these mental connections between proper time and coordinate time....your explanation is incredibly clear....thank you ! This thread is closed for me.

ghwellsjr said:
"Time is what a clock measures". This would be the proper time....

Then he [Einstein] uses the proper time on a non-accelerating clock to synchronize the proper times on remote clocks at rest with the first one as part of his definition of a Frame of Reference (FoR). It's because all these clocks are not allowed to accelerate that allows the proper times on all these clocks to be used as coordinate time for the Frame of Reference....

The proper time is what any clock measures, whether it is accelerated or not. But if it's never allowed to accelerate, then we can use it as coordinate time. But remember, another clock moving with respect to the first one will be running at a different rate and if it is not allowed to accelerate (once it gets to its constant speed), it can be used as coordinate time in a different coordinate system (Frame of Reference).

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