# Using position of free particle to measure time

• I
• Kashmir
In summary, the conversation discusses the concept of inertial frames and how they can be defined as Cartesian reference frames where Newton's first law holds. The author also mentions that an observer in an inertial frame can construct a clock using the position of one free particle, which changes at a constant rate in time. However, this does not guarantee that all other free particles will have a double derivative of zero, and thus, more than one free particle is needed to establish an inertial frame. The author also suggests that any changing entity can be used as a clock, including the position of a moving object. Finally, the conversation ends with a discussion on how to establish if a frame is inertial, which requires finding a parameter 'a' that
Kashmir
Hartle, Gravity

"An observer in an inertial frame can discover a parameter ##t##with
respect to which the positions of all free particles are changing at constant rates.
This is time"

Then goes on to say
"Indeed, inertial frames
could be defined as Cartesian reference frames for which Newton’s first law holds
in the form ##\mathbf{\ddot r}=0##
Using the laws of mechanics, an observer in an inertial frame can construct a
clock that measures the time ##t##. For instance, the position of one free particle could
be used to measure ##t##, since its position changes at a constant rate in ##t##"

Using the position of one free particle to measure time, how does that guarantee that for other free particles this definition of time leads to ##\mathbf{\ddot r}=0##

Kashmir said:
Using the position of one free particle to measure time, how does that guarantee that for other free particles this definition of time leads to ##\mathbf{\ddot r}=0##
It doesn’t. We still have to observe the other particles to verify that ##\mathbf{\ddot r}=0## holds for them as well; if we find exceptions then we know that we don’t have an inertial frame.

Kashmir and vanhees71
You need three particles moving in uniform motion in non-coplanar directions to establish an inertial frame of reference.

PeroK, Kashmir and phinds
Kashmir said:
Using the position of one free particle to measure time, how does that guarantee that for other free particles this definition of time leads to ##\mathbf{\ddot r}=0##
It does not. An explicit counter example is a rotating reference frame where the one free particle is moving along the axis of rotation. As @vanhees71 you need multiple particles to accomplish this.

Kashmir and vanhees71
@vanhees71 then what does the author mean by constructing a clock using a free particle ? Can you please explain it? Thank you

Nugatory said:
It doesn’t. We still have to observe the other particles to verify that ##\mathbf{\ddot r}=0## holds for them as well; if we find exceptions then we know that we don’t have an inertial frame.
What does the author mean then by constructing a clock using a free particle? Can you please explain it. Thank you :)

Kashmir said:
Using the position of one free particle to measure time, how does that guarantee that for other free particles this definition of time leads to ##\mathbf{\ddot r}=0##
You are looking at it backwards. You have to establish that you have an inertial frame (which requires more than one free particle) first. Then, once you have done that, you can pick anyone of the free particles and use its position to define time.

vanhees71 and PeroK
Kashmir said:
What does the author mean then by constructing a clock using a free particle? Can you please explain it. Thank you :)
Anything that changes can be used as a clock: in the past we’ve measured time with sand moving through a sandglass, the length of a burning candle, the shadow of the sun, nowadays we sometimes use the rate of decay of radioactive materials…. Why not use the position of a moving object as a clock? Come to think of it…. Is that not what’re doing with an ordinary dial clock or stopwatch? The hands are moving, we tell the time by where they’ve moved to.

vanhees71 and PeroK
PeterDonis said:
You are looking at it backwards. You have to establish that you have an inertial frame (which requires more than one free particle) first. Then, once you have done that, you can pick anyone of the free particles and use its position to define time.
To establish you've an inertial frame you need that for free particles there is no acceleration i.e ##\ddot r=0##. so you already have defined time to prove the frame is inertial. Isn't your logic circular?

What’s wrong with circular logic? It is certainly self consistent.

Nugatory said:
Anything that changes can be used as a clock: in the past we’ve measured time with sand moving through a sandglass, the length of a burning candle, the shadow of the sun, nowadays we sometimes use the rate of decay of radioactive materials…. Why not use the position of a moving object as a clock? Come to think of it…. Is that not what’re doing with an ordinary dial clock or stopwatch? The hands are moving, we tell the time by where they’ve moved to.
Author says "An observer in an inertial frame can discover a parameter ##t## with respect to which the positions of all free particles are changing at constant rates.
This is time"To establish I've a inertial frame I've to find that free particles move in straight lines and the double derivative with respect to a parameter, call it ##a##, is zero for all the particles.
So I've to search for a parameter ##a## that gives ##d^2 r/da =0## for all the free particles.

The author chooses ##a## as related to the distance of one particular free particle and says this can be used to measure time.
But how does one know that we choose the correct paramete?

We were guaranteed that there exists a parameter such that ##d^2 r/da =0## but it didn't tell me that the position of one free particle is a correct parameter.

Kashmir said:
To establish you've an inertial frame you need that for free particles there is no acceleration i.e ##\ddot r=0##. so you already have defined time to prove the frame is inertial. Isn't your logic circular?
This physical determination of an inertial reference frame goes back to Lange (1885). The idea is that you can establish an inertial frame by taking one free particle, moving in a straight line relative to the frame to define the time measure via measurements of the distance traveled by this particle. In addition you then need to establish that with this measure of time three free particles moving in non-coplanar directions in fact move with constant velocity to that reference frame.

For a translation of this paper, see

dextercioby, PeroK, Kashmir and 1 other person
Kashmir said:
But how does one know that we choose the correct parameter?
what would be an “incorrect” parameter? And is it possible for the position of a free particle to be one?

vanhees71 said:
This physical determination of an inertial reference frame goes back to Lange (1885). The idea is that you can establish an inertial frame by taking one free particle, moving in a straight line relative to the frame to define the time measure via measurements of the distance traveled by this particle. In addition you then need to establish that with this measure of time three free particles moving in non-coplanar directions in fact move with constant velocity to that reference frame.

For a translation of this paper, see

Thank you. This is what I was asking for.
I don't have access to the article but I think your comment did resolve much of my doubt .

But is it sufficient that the measure of time you define, related to distance of a free particle, which leads to zero accelerations for three particles moving along non coplanar paths is sufficient enough that all other free particles will have zero accelerations as well if we use this time?

Nugatory said:
what would be an “incorrect” parameter? And is it possible for the position of a free particle to be one?
An incorrect one would be that the accelerations are non zero for free particles.

I'm not sure about the second part.

Dale said:
What’s wrong with circular logic? It is certainly self consistent.
Please see post #11 above. I've written more clearly about my doubt. Thank you :)

Kashmir said:
But how does one know that we choose the correct paramete?
Usually you just plug the parameter back into the definition and see if it works. If you find that for some free particle ##d^2 r/da^2 \ne 0## then you know the choice is incorrect. Otherwise it is correct.

This is very circular, but it is also correct.

Kashmir said:
An incorrect one would be that the accelerations are non zero for free particles.
Then we know we have made a poor choose of parameter, so we try choosing another one that works better.

Start with Einstein's "Time is what a clock measures"; anything that parameterizes the passage of time can be used as a clock. We generally try to choose our clocks so that ##\mathbf{\ddot r}=0## holds for free particles.

Kashmir said:
But is it sufficient that the measure of time you define, related to distance of a free particle, which leads to zero accelerations for three particles moving along non coplanar paths is sufficient enough that all other free particles will have zero accelerations as well if we use this time?
@vanhees71 . Could you comment on it?

## 1. What is the concept of using the position of a free particle to measure time?

The concept is based on the idea that the position of a free particle can be used as a reliable and accurate way to measure time. This is because the position of a free particle is constantly changing and can be easily tracked and measured.

## 2. How does using the position of a free particle to measure time work?

Using the position of a free particle to measure time involves tracking the movement of the particle over a specific distance and using this information to calculate the time it takes for the particle to travel that distance. This can be done using mathematical equations and precise measurements.

## 3. What are the advantages of using the position of a free particle to measure time?

One advantage is that it does not require any external factors such as clocks or other timekeeping devices. It also allows for more accurate measurements as it eliminates potential errors from these external factors. Additionally, this method can be used in various environments and does not rely on a specific set of conditions.

## 4. Are there any limitations to using the position of a free particle to measure time?

One limitation is that it requires precise measurements and calculations, which can be challenging and time-consuming. Additionally, this method may not be suitable for measuring very short time intervals or in situations where the particle's movement is affected by external forces.

## 5. How is the position of a free particle used to measure time in practical applications?

This concept is used in various fields such as physics, engineering, and astronomy. For example, in physics experiments, the position of a free particle can be used to determine the time it takes for a reaction to occur. In engineering, it can be used to measure the speed of a moving object. In astronomy, it can be used to track the movement of celestial bodies and determine their orbital periods.

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