How do proper time and time dilation differ?

[Taylor_1989]

Homework Statement

I am currently stuck on a problem as shown below. My confusion really come from the fact that I can never seem to understand the difference between proper time and time dilation. As in some books they seem to mean the same thing and never give a clear indication

Homework Equations

time dilation
$\Delta t'=\gamma \Delta t$

where
$\Delta t'=$ the dilated time in the moving frame

$\Delta t=$ the time measure in the rest frame.

The Attempt at a Solution

What confusing me the most about this question in the change in frame. The question to me is all done in the reference frame of the earth.

so in Earth frame the velocity of the ship, one day later is $\sqrt{0.8}c^2$, but it where I go from here is the confusing part originally, i thought you then just sub this into the time dilation equation, then i realized that not correct as I am looking for proper time.

I am now thinking I need to change the reference frame using Lorentz transformation, but then that to me is incorrect as you can use the Lorentz transformation in time to derive time dilation.

 

[Apashanka]

If events are occurring at the same place (e.g fixed coordinates) with respect to a coordinate system but at different times then the time interval between the events in that frame is called proper time Δt, similarly if events are occurring at different places as well as different times with respect to a coordinate system ,then the time measured in that frame gets dilated as compared to Δt.
For exm
two frame S and S^' having relative velocity v moving along the common x axis.
If two events occur at say (x,y,z) in the time interval Δt in the S frame ,then the time interval between two events as measured by S^' frame gets dilated by the factor √(1-v²/c²)Δt.

 

[PeroK]

If, instead of a rocket, you had an accelerating particle, how would you calculate the proper time of the particle?

 

[Apashanka]

proper time of a particle will be same (e.g in the coordinte system fixed to the particle) no matter whether it is moving at a uniform speed or accelerating with respect to a certain reference frame

 

[phinds]

...My confusion really come from the fact that I can never seem to understand the difference between proper time and time dilation.

Think about it this way. Your wristwatch shows time passing at one second per second.

BUT ... according to a particle in the accelerator at CERN, your time is MASSIVELY dilated. It sees your wristwatch ticking at a very slow rate. Does your wristwatch care? Does it slow down?

According to a passing asteroid, your time is dilated a tiny bit. Does your wristwatch care?

According to your chair, your time is not dilated at all. It agrees with your wristwatch.

YOUR time in YOUR frame is proper time. It is never dilated. OTHERS see it as dilated if they are moving relative to you, just as you see their dilated because they are moving relative to you.

 

[PeroK]

@Taylor_1989 to focus on your specific problem: what is the definition of proper time? Is there, perhaps, a way of describing proper time (of a particle) that is useful in this case?

 

[Taylor_1989]

First thank you for all the response. I have currently being reading a wiki article on 'proper time', https://en.wikipedia.org/wiki/Proper_time#Example_1:_The_twin_"paradox",

and now more confused than originally, I am trying to understand some of your hints, and explanation but now with the space-time stuff for proper time, I am getting more confused.

I understand the light clock derivation of time dilation ect and the proper time measured by the stationary clock, which I have tried to relate to the above question with no luck, and I now feel I am going off on a tangent looking at the proper time stuff.

 

[Taylor_1989]

As an additional note reading the above hints, If I was on the spacecraft I understand that this would be rec and the proper time, but I have been given the time, with respect to the earth, not the space craft, so in my mind on Earth it a day may have passed but on the spacecraft is as been longer. So how can I find the time diliated on Earth with respect to the space craft, this make no sense to me, I will however keep trying.

 

[Apashanka]

First thank you for all the response. I have currently being reading a wiki article on 'proper time', https://en.wikipedia.org/wiki/Proper_time#Example_1:_The_twin_"paradox",

and now more confused than originally, I am trying to understand some of your hints, and explanation but now with the space-time stuff for proper time, I am getting more confused.

I understand the light clock derivation of time dilation ect and the proper time measured by the stationary clock, which I have tried to relate to the above question with no luck, and I now feel I am going off on a tangent looking at the proper time stuff.

If you think this way that you are at absolute reference frame and with respect to you there are two frames S and S^'
S being at rest with respect to you and S^' moving with a velocity v
Now in your watch Δt time has elapsed and at that instant you ask person (say) sitting on S and S^' frame,S says Δt time has elapsed S^' says Δt^'= √(1-v²/c²)Δt time has elapsed that means time elapsed in proper rest frame is proper time and time elapsed on moving frame is called dilated time as Δt'<Δt.

 

[PeroK]

As an additional note reading the above hints, If I was on the spacecraft I understand that this would be rec and the proper time, but I have been given the time, with respect to the earth, not the space craft, so in my mind on Earth it a day may have passed but on the spacecraft is as been longer. So how can I find the time diliated on Earth with respect to the space craft, this make no sense to me, I will however keep trying.

An important description/definition of proper time is the length of a spacetime interval (worldline).

 

[Taylor_1989]

An important description/definition of proper time is the length of a spacetime interval (worldline).

That what I was reading about in the above article, but could not understand how it relates, to the hints that where given. I must say i find it all counter intuivte and just can never seem to grasp this stuff. I have tried drawing my own graphs with the world lines but I am just plain stuck. Every article I read seem to give different interpretations, which is not helping.

My throught are at the moment, then the time dilation of the ship with respect to the Earth is one day, which would give me the following:

$$1_{day}=\frac{\Delta t}{\sqrt{1-0.8^2}},\:\rightarrow \Delta t=\frac{\sqrt{5}}{5}days$$

however reading the wiki article it defines proper time as,

$$\Delta \tau =\sqrt{\left(\Delta t\right)^2-\left(\frac{\Delta x}{c}\right)^2}$$

which dose not help as I now confused to how, this related to time dilation, and what is $\Delta x$ and $\Delta t$ ref to.

 

[PeroK]

That what I was reading about in the above article, but could not understand how it relates, to the hints that where given. I must say i find it all counter intuivte and just can never seem to grasp this stuff. I have tried drawing my own graphs with the world lines but I am just plain stuck. Every article I read seem to give different interpretations, which is not helping.

My throught are at the moment, then the time dilation of the ship with respect to the Earth is one day, which would give me the following:

$$1_{day}=\frac{\Delta t}{\sqrt{1-0.8^2}},\:\rightarrow \Delta t=\frac{\sqrt{5}}{5}days$$

however reading the wiki article it defines proper time as,

$$\Delta \tau =\sqrt{\left(\Delta t\right)^2-\left(\frac{\Delta x}{c}\right)^2}$$

which dose not help as I now confused to how, this related to time dilation, and what is $\Delta x$ and $\Delta t$ ref to.

When you are dealing with two inertial reference frames, then the time dilation between them is relatively simply defined, as is the Lorentz transformation mapping events from one frame to another. However, when you have an accelerating particle, then the same principles apply but the particle itself does not have a single IRF, but an accelerating reference frame, which equates to a continuously changing set of IRF's. In such problems, therefore, you are likely to get the measurements in the IRF (in this case the Earth frame, or in other cases the lab frame).

In any case, an accelerating particle follows a curved path. And, to measure the length of a curved path you need something from calculus called integration!

In this case, we really need to replace the $\Delta x, \Delta t$ etc. (which represent finite changes in position, time etc.) with $dt, dx$, which represent the infinitesimal changes in position, time etc. And, the time dilation becomes:

$dt = \gamma(t) d\tau$

Where $\tau$ is the proper time of the particle and $\gamma(t)$ is the particle's gamma factor, which, as I've emphasised, is a function of coordinate time from the IRF (Earth frame or lab frame).

I'm surprised if whoever is teaching you SR has not gone through this.

 

[Taylor_1989]

When you are dealing with two inertial reference frames, then the time dilation between them is relatively simply defined, as is the Lorentz transformation mapping events from one frame to another. However, when you have an accelerating particle, then the same principles apply but the particle itself does not have a single IRF, but an accelerating reference frame, which equates to a continuously changing set of IRF's. In such problems, therefore, you are likely to get the measurements in the IRF (in this case the Earth frame, or in other cases the lab frame).

In any case, an accelerating particle follows a curved path. And, to measure the length of a curved path you need something from calculus called integration!

In this case, we really need to replace the $\Delta x, \Delta t$ etc. (which represent finite changes in position, time etc.) with $dt, dx$, which represent the infinitesimal changes in position, time etc. And, the time dilation becomes:

$dt = \gamma(t) d\tau$

Where $\tau$ is the proper time of the particle and $\gamma(t)$ is the particle's gamma factor, which, as I've emphasised, is a function of coordinate time from the IRF (Earth frame or lab frame).

I'm surprised if whoever is teaching you SR has not gone through this.

Firstly thank you! I am now getting the gist of what your saying. No we have not gone through this in most undgrad course in the uk, we only get 2 to 3 weeks on SR. Which dose not give a lot of time to grasp the subject. At them moment I am trying to grasp the concepts fully.

 

[Apashanka]

indication

From the invariance of the space time interval it is given
dΓ²(time elapsed on Earth clock)=dt²(time elapsed on the rocket clock)-dx²(distance traveled by the rocket in one day as observed from Earth)
From this dt is to be calculated

 

[PeroK]

From the invariance of the space time interval it is given
dΓ²(time elapsed on Earth clock)=dt²(time elapsed on the rocket clock)-dx²(distance traveled by the rocket in one day as observed from Earth)
From this dt is to be calculated

That's not correct. That only works for a particle with constant velocity. For an accelerating particle you must integrate.

 

[Apashanka]

That's not correct. That only works for a particle with constant velocity. For an accelerating particle you must integrate.

Ya that's right
Also length contraction in dx also need to be taken (distance which will the rocket see it has gone in dt)

 

[Taylor_1989]

Ya that's right
Also length contraction in dx also need to be taken (distance which will the rocket see it has gone in dt)

how can you calculate the length contractions of acceleration? Because when I read the wiki link and it talk about the accelerating ref frame you get back the time dilation equation. So what am I integrating because I thought I understood the perivous explanation but when you talk about length contraction, I am not sure how we find the this in a acceleration.

 

[Taylor_1989]

So here is what I have

$\:\tau \:=\int _{t_1=0}^{t_{2=1}}\:\sqrt{\left(dt\right)^2-\left(\frac{dx}{c}\right)^2},\:where\:dx=traveld\:in\:earth\:frame$

by how can I have the length contraction because, I can't simply replace dx with dx' that dose not make any sense to me at all.

another issue with this is what dt, beacuse if that suppose to be the 1 day then my limits are wrong.

Dose anyone know a good book, beacuse I genearlly trying to understand the concepts but I starting to get frustrated with this, and why I can see what, what.

 

[PeroK]

So here is what I have

$\:\tau \:=\int _{t_1=0}^{t_{2=1}}\:\sqrt{\left(dt\right)^2-\left(\frac{dx}{c}\right)^2},\:where\:dx=traveld\:in\:earth\:frame$

by how can I have the length contraction because, I can't simply replace dx with dx' that dose not make any sense to me at all.

another issue with this is what dt, beacuse if that suppose to be the 1 day then my limits are wrong.

Dose anyone know a good book, beacuse I genearlly trying to understand the concepts but I starting to get frustrated with this, and why I can see what, what.

$dx$ and $dt$ are related by velocity!

I already gave you the shortcut in post #12.

Special Relativity by Helliwell would be my recommendation.

 

[Apashanka]

It would be ∫dΓ=∫dt√(1-a²t²/c²)
where a is the acceleration of the rocket .
dΓ=time measured on the rocket frame.
dt=time measures on the Earth frame.
The limit of RHS is from 0 to 1 day.

 

[Taylor_1989]

It would be ∫dΓ=∫dt√(1-a²t²/c²)
where a is the acceleration of the rocket .
dΓ=time measured on the rocket frame.
dt=time measures on the Earth frame.
The limit of RHS is from 0 to 1 day.

Thank you, I managed to clock after a very long while what was actually going, much apprecaited, and I have the same limits. Since then I have managed to order the textbook that was recommend. Hopefully with a bit of will power I will master SR

 

[Andrew Mason]

Thank you, I managed to clock after a very long while what was actually going, much apprecaited, and I have the same limits. Since then I have managed to order the textbook that was recommend. Hopefully with a bit of will power I will master SR

A "proper time" interval between two events exists only if the events are time-like. Events are time-like if there exists an inertial frame of reference in which the two events occur at the same place. If events are not time-like there is no proper time between the events. If you keep those things straight you will not get confused about proper time.

You have to be very careful because there is a lot written on relativity that is misleading or wrong. For example, there is this statement on Hyperphysics (generally a reliable source) which is somewhat misleading:

A clock in a moving frame will be seen to be running slow, or "dilated" according to the Lorentz transformation. The time will always be shortest as measured in its rest frame. The time measured in the frame in which the clock is at rest is called the "proper time".

What does "The time will always be shortest as measured in its rest frame" mean? It is a meaningless statement unless one specifies the events between which a time interval is measured. If time-like events occur at different places in the rest frame, the time interval measured between those events in the rest frame will be greater than the time interval between those same two events measured by an observer in a frame of reference in which the events occur at the same place.

Dilated means "enlarged". But the term "time dilation" is used in different ways and it is very confusing. It is often used in the sense: "time dilation: moving clocks run slow". That suggests that the time intervals measured between events in the moving frame will be smaller than we measure them to be (which, incidentally, makes it hard to understand why time in the moving frame is referred to as dilated). But the point is that this is only true for events that take place at the same location in the moving frame. For time-like events that take place at different locations in the moving frame, the clocks in the moving frame may measure a longer interval or a shorter interval than another inertial observer. The observer in the moving frame will always measure the time between such events to be longer than that measured by an observer in the frame in which they occur at the same place.

We understand proper time to be the time interval between two time-like events measured by an observer in the frame of reference in which the events occur at the same place. So "dilated time" should just refer to the time interval observed between two time-like events by any other inertial observer (ie. any observer other than the observer for whom the events occur at the same location). It is always greater than the proper time interval.

Again, you will be very confused if you read a lot of the explanations on the internet or Wikipedia. A good text is essential to help you get through this.

AM