Extremely Basic Question Regarding Time Dilation

[Capngarrett]

Hi all

I am new to relativity, and through the books that I have chosen to learn the subject (that cater to my level) I seemed to be following with ease. However, I was posed with a simple question that I could not answer, and whatever it is, the thing that I am overlooking still eludes me.

Consider two observers moving relative to each other on an inertial frame of reference. Gravitational and other external effects aside, each observer considers himself to be at rest on his own rigid body and believes the other to be in motion. I have been told numerous times that each person witnesses the other experiencing a slower rate of time than their own.

Now, imagine two people at one point on the circumference of a large circle. Person A remains at rest while person B accelerates along the circumference until he is at the furthest point from A, and then decelerates on the way back to meet him. The Twins Paradox tells us that person B's clock would show an earlier time than A's when compared. This would mean that B would have (if he could) witnessed A's clock ticking at a faster rate than his own.

My question to you is, which example is correct? and (pre-emptively) if both are true, then what is the difference?

 

[zhermes]

Hi Capngarrett, welcome to PF!

The resolution is the same as the twin-paradox itself. Person B has aged less than Person A. One reference frame (person B) is not an inertial reference frame, it has been accelerating (as you point out). Accelerating reference frames behave differently, and require a 'general' relativistic treatment.

Hope that helps

 

[Capngarrett]

Of course! How did I miss it? I was practically answering my own question but didn't make the connection.

Thanks for the speedy resolution, I expected no less

 

[Janus]

Of course! How did I miss it? I was practically answering my own question but didn't make the connection.

Thanks for the speedy resolution, I expected no less

The same answer applies even in the following scenario:

Assume that B is already traveling along the circumference of the circle when he passes A and they sync their clocks together. When B meets up with A again, they both agree that B's clock is behind A's, even though B never changes his speed. This is because B undergoes acceleration just by the fact that he is traveling in a circle.

 

[Capngarrett]

Assume that B is already traveling along the circumference of the circle when he passes A and they sync their clocks together. When B meets up with A again, they both agree that B's clock is behind A's, even though B never changes his speed. This is because B undergoes acceleration just by the fact that he is traveling in a circle.

You're a mindreader - I was already bending the scenario in my mind to find out where the line was drawn.

 

[ghwellsjr]

Hi all

I am new to relativity, and through the books that I have chosen to learn the subject (that cater to my level) I seemed to be following with ease. However, I was posed with a simple question that I could not answer, and whatever it is, the thing that I am overlooking still eludes me.

Consider two observers moving relative to each other on an inertial frame of reference. Gravitational and other external effects aside, each observer considers himself to be at rest on his own rigid body and believes the other to be in motion. I have been told numerous times that each person witnesses the other experiencing a slower rate of time than their own.

Now, imagine two people at one point on the circumference of a large circle. Person A remains at rest while person B accelerates along the circumference until he is at the furthest point from A, and then decelerates on the way back to meet him. The Twins Paradox tells us that person B's clock would show an earlier time than A's when compared. This would mean that B would have (if he could) witnessed A's clock ticking at a faster rate than his own.

My question to you is, which example is correct? and (pre-emptively) if both are true, then what is the difference?

You might find it interesting that your scenario is exactly the one Einstein presented in his original introduction of the Twin Paradox to the world (although he didn't call it that). You can find it near the end of section 4 of his 1905 paper.

I don't think anyone addressed your issue about B witnessing A's clock ticking at a faster rate than his own. This is an ambiguous issue because you might be asking what B actually sees A's clock doing. Afterall, isn't that what witnessing means? Well if that is what you are wondering and if you also meant that with regard to your first example of two people traveling with no acceleration in relative motion, then in both examples, they can each see the other ones clock running both faster and slower than their own. It all depends on their relative positions and directions.

In your first example, if the two people are coming towards each other, they will see the other ones clock running faster than their own. After they pass each other, they will see the other ones clock running slower than their own.

In your second example, they will also see the other ones clock running faster during part of the trip and running slower during the rest of the trip. But they don't see the same thing. B will see A's clock running slow during exactly half his trip and running fast during the other half. A will see B's clock running slow during more than half the time and fast during less than half the time. The net result is that when they get together, A has seen B's clock accumulate less time than his own and B has seen A's clock accumulate more time than his own.

It's only when we assign a particular inertial reference frame in which to describe the motions of both people, whether traveling in a straight line, or at rest, or traveling in a circle, that we can determine the rates at which the clocks tick independent of the relative position between the two people.

So in either example, if we select an inertial reference frame such that one person remains at rest, then only the other ones clock is running slower than the coordinate time of the reference frame. We can do this for either person in your first example (this is usually why people say the Twin Paradox is a paradox) but we can only do this for one person in your second example.

However, it doesn't matter which frame you use to describe or analyze any scenario, it doesn't have any bearing on what each observer witnesses (sees) of the other ones clock.

 

[Capngarrett]

Oh, that is interesting. I am reading through his paper but I've only just finished the Special Relativity sections. Perhaps I should've read it through before posting this topic as it may have provided the answer, but impatience got the better of me - it was driving me insane!

In your first example, if the two people are coming towards each other, they will see the other ones clock running faster than their own. After they pass each other, they will see the other ones clock running slower than their own.

In your second example, they will also see the other ones clock running faster during part of the trip and running slower during the rest of the trip. But they don't see the same thing. B will see A's clock running slow during exactly half his trip and running fast during the other half. A will see B's clock running slow during more than half the time and fast during less than half the time. The net result is that when they get together, A has seen B's clock accumulate less time than his own and B has seen A's clock accumulate more time than his own.

It's only when we assign a particular inertial reference frame in which to describe the motions of both people, whether traveling in a straight line, or at rest, or traveling in a circle, that we can determine the rates at which the clocks tick independent of the relative position between the two people.

So in either example, if we select an inertial reference frame such that one person remains at rest, then only the other ones clock is running slower than the coordinate time of the reference frame. We can do this for either person in your first example (this is usually why people say the Twin Paradox is a paradox) but we can only do this for one person in your second example.

However, it doesn't matter which frame you use to describe or analyze any scenario, it doesn't have any bearing on what each observer witnesses (sees) of the other ones clock.

Ah, okay. So the further away they move from each other, the bigger the observed time drift between the compared clocks (which normalises if they return to meet given that acceleration/deceleration is neglible) - kind of like a Doppler effect of time.

The fact that an observed clock seems to tick faster as it approaches is the key piece of information that I was missing in my mental jigsaw. Thanks for the input, I think I've got this one pinned down but I'm sure I'll be back on this forum soon enough with another problem.