Relativity: Time dilation and distance calculations

[WWCY]

Homework Statement

Could someone help point out certain conceptual errors in my interpretation of the mirror clocks time dilation thought experiment? Say S' is the frame of reference traveling at speed V w.r.t to frame S, which means that events happen at the same coordinate in S'.

We know that the measurement of time between the light leaving the source and making a round trip back to the source in this frame is Δt₀= 2d/c, where d is the vertical distance between the 2 mirrors. And that the time measured in the non-proper (is this a correct way of putting this?) frame S is Δt = γΔt₀, which is in part due to the distance measured between the 2 events VΔt in this frame.

Here's where my understanding becomes messy. If i wished to calculate the distance moved by S', how do I go about doing it? Do I use VΔt? Can i use VΔt₀ as well (my current understanding is that speed V is identical in both frames after sticking values into Lorentz velocity equations)? If so won't the distance traveled be shorter in the S' frame than the S frame? And how do I interpret these facts?
Thanks for helping!

Homework Equations

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The Attempt at a Solution

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[PeroK]

Homework Statement

Could someone help point out certain conceptual errors in my interpretation of the mirror clocks time dilation thought experiment? Say S' is the frame of reference traveling at speed V w.r.t to frame S, which means that events happen at the same coordinate in S'.

We know that the measurement of time between the light leaving the source and making a round trip back to the source in this frame is Δt₀= 2d/c, where d is the vertical distance between the 2 mirrors. And that the time measured in the non-proper (is this a correct way of putting this?) frame S is Δt = γΔt₀, which is in part due to the distance measured between the 2 events VΔt in this frame.

Here's where my understanding becomes messy. If i wished to calculate the distance moved by S', how do I go about doing it? Do I use VΔt? Can i use VΔt₀ as well (my current understanding is that speed V is identical in both frames after sticking values into Lorentz velocity equations)? If so won't the distance traveled be shorter in the S' frame than the S frame? And how do I interpret these facts?
Thanks for helping!

Homework Equations

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The Attempt at a Solution

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Although S and S' agree about their relative speed, they have different results for the time between two events. In this case, the events being a tick of the light clock in S'.

Can you analyse it further?

 

[WWCY]

Although S and S' agree about their relative speed, they have different results for the time between two events. In this case, the events being a tick of the light clock in S'.

Can you analyse it further?

I'm not quite sure what you mean by analysing further, could you guide me along?Edit: Does the further analysis involve contraction?

 

[PeroK]

I'm not quite sure what you mean by analysing further, could you guide me along?Edit: Does the further analysis involve contraction?

In the S frame: a point at rest in the S' frame moves a distance ##v \Delta t## in the time between the two events.

In the S' frame: a point at rest in the S frame moves a distance ##v \Delta t_0## in the time between the two events.

Where's the contradiction?

 

[WWCY]

In the S frame: a point at rest in the S' frame moves a distance ##v \Delta t## in the time between the two events.

In the S' frame: a point at rest in the S frame moves a distance ##v \Delta t_0## in the time between the two events.

Where's the contradiction?

Is it that different frames of reference perceive different lengths/distances?

 

[PeroK]

Is it that different frames of reference perceive different lengths/distances?

That's why I said "analyse" further. This statement too vague to analyse.

Also, note that in classical physics different frames measure different distances between two events. The distance between these two events in the S' frame is 0 (and that is true in both relativitity and classical physics); and the distance between the two events in the S frame is ##v \Delta t## (and that is also true in relativity and in classical physics).

 

[WWCY]

In the S frame: a point at rest in the S' frame moves a distance ##v \Delta t## in the time between the two events.

In the S' frame: a point at rest in the S frame moves a distance ##v \Delta t_0## in the time between the two events.

Where's the contradiction?

That's why I said "analyse" further. This statement too vague to analyse.

Also, note that in classical physics different frames measure different distances between two events. The distance between these two events in the S' frame is 0 (and that is true in both relativitity and classical physics); and the distance between the two events in the S frame is ##v \Delta t## (and that is also true in relativity and in classical physics).

Apologies but I'm not sure what I'm supposed to be analysing, could you nudge me along?

 

[PeroK]

Apologies but I'm not sure what I'm supposed to be analysing, could you nudge me along?

You seem to have some vague idea from your OP that something is wrong. That somehow "lengths" and/or "distances" are not what they should be.

But, until you can define what you mean by a length or a distance and why it's not what it should be, there is nothing to say.

Can you give a precise question highlighting your concern?

 

[WWCY]

You seem to have some vague idea from your OP that something is wrong. That somehow "lengths" and/or "distances" are not what they should be.

But, until you can define what you mean by a length or a distance and why it's not what it should be, there is nothing to say.

Can you give a precise question highlighting your concern?

I'm wondering if there is something like a "proper" distance or length in the same way time would be considered proper in S' 's frame of reference. Is a measurement of length more fundamental in a particular frame of reference than others? And if i was asked to compute distance using speed and time intervals, should I be giving the distance relative to all the frames in question?

 

[PeroK]

I'm wondering if there is something like a "proper" distance or length in the same way time would be considered proper in S' 's frame of reference. Is a measurement of length more fundamental in a particular frame of reference than others? And if i was asked to compute distance using speed and time intervals, should I be giving the distance relative to all the frames in question?

There's a clue in classical physics. Objects have a length. In classical physics this is an invariant between reference frames. You may already know or guess that the length of an object is not frame-invariant in SR.

Next step: define the length of an object. (Although, this is not homework help, this is trying to teach you SR!)

Distances in classical physics are not (necessarily) invariant - it depends on what you mean by distance. So, in both classical physics and SR, you have to carefully define what distance you are talking about. A distance could be:

The distance between two fixed points; the distance an object has traveled in a time ##\Delta t##; the distance between two moving particles at time ##t##.

Anyway, this is the sort of analysis which you perhaps could have been able to do yourself. I.e. think about concepts like "distance" and analyse what is meant by them.