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Author Topic: How should one interpret spacetime diagrams?  (Read 24107 times)

Bill S

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How should one interpret spacetime diagrams?
« on: 18/03/2015 22:03:41 »
Question about Michael Huemer’s spacetime diagrams.

In Figure 2 the Earth is considered as stationary; its world line simply follows the vertical time axis.
Space Twin is considered to be in motion relative to the Earth.  He moves to the right at 0.5c, then to the left at 0.5c.

In Figure 3 Space Twin is considered to be stationary.  His world line starts by following the vertical time axis.  The Earth is considered as being in motion relative to Space Twin, so it moves to the left at 0.5c.
The reason that Earth is considered to be moving at 0.5c is that that was the speed of the spacecraft in the first F of R.
At t = 8.66, Space Twin decides to change speed and direction.
Figure 3 then shows Space Twin as being in motion.  If we are still in Space Twin’s F of R, why are we seeing him as being anything other than stationary?
Should Fig. 3 not show Earth as moving towards the space craft?  If not - why not?

Colin2B

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Re: How should one interpret spacetime diagrams?
« Reply #1 on: 18/03/2015 22:37:23 »
Should Fig. 3 not show Earth as moving towards the space craft?  If not - why not?

The way I see it: From the point of view of inertial frames this twin as you know can can consider themselves stationary. So initially the earth appears to move away from the twin.
However, there comes a point where the twin wishes to return to earth. At this point the twin accelerates and is no longer in an inertial frame so is shown as moving. For earth to come back towards the twin would require it to do the accelerating.
From a ST diagram point of view you can only consider yourself stationary if you are in an inertial frame.

Would be interested in how others view it.

jeffreyH

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Re: How should one interpret spacetime diagrams?
« Reply #2 on: 18/03/2015 23:45:29 »
Ultimately you could plot a spacetime diagram in Planck units. The time axis in units of 1 Planck time and the space axis in units of 1 Planck length. Light will always follow a 45 degree path with respect to the origin. Any constant velocity will also describe a straight line path away from the light path. However an acceleration will follow a curved path. The curved path is non-inertial and can not be considered to be at rest in any frame.

Colin2B

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Re: How should one interpret spacetime diagrams?
« Reply #3 on: 19/03/2015 08:00:56 »
......The curved path is non-inertial and can not be considered to be at rest in any frame.

I did the diagrams for curved path and got a similar result, he spends less time at higher speeds but gets extra dilation from the greater g effect. I must look at your thread on escape velocity, similar problem.

....  If we are still in Space Twin’s F of R, why are we seeing him as being anything other than stationary?
Should Fig. 3 not show Earth as moving towards the space craft?  If not - why not?

Bill, in the example you quote I think  the straight lines are used for simplification, assume instant acceleration to new speed. The important thing is to maintain the same FofR.
While the travelling twin considers himself stationary it is relative to a particular frame he has established, when he accelerates towards earth any movement has to be considered relative to that frame. If having accelerated he then began to view earth as moving towards himself he would have changed his FofR, To retain consistency you would have to draw a new diagram in which this leg of the journey is considered as stationary, but the previous leg as moving.
In other words you have to keep one FofR throughout.

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #4 on: 26/03/2015 17:21:05 »
Yesterday my son asked: Is there any situation in which an outside observer would perceive the twins as ageing at the same rate?

My answer (off the cuff, while driving) was:  There is, but it would require that the relative speed difference between each twin and the observer be the same.  For example, if the observer placed his craft between Earth and the craft in which the twin was travelling, such that, if Earth were regarded as stationary, he was moving away from it at 0.25c, the twin’s craft would be moving away from him at 0.25c.  If we now change to the observer’s reference frame, Earth will also be moving away from him at 0.25c.  In that case, he should consider that the twins are ageing at the same rate.

Thought I'd like to run that past others.

jeffreyH

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Re: How should one interpret spacetime diagrams?
« Reply #5 on: 26/03/2015 18:02:46 »
Yesterday my son asked: Is there any situation in which an outside observer would perceive the twins as ageing at the same rate?

My answer (off the cuff, while driving) was:  There is, but it would require that the relative speed difference between each twin and the observer be the same.  For example, if the observer placed his craft between Earth and the craft in which the twin was travelling, such that, if Earth were regarded as stationary, he was moving away from it at 0.25c, the twin’s craft would be moving away from him at 0.25c.  If we now change to the observer’s reference frame, Earth will also be moving away from him at 0.25c.  In that case, he should consider that the twins are ageing at the same rate.

Thought I'd like to run that past others.

They won't be ageing at the same rate to your new observer. To maintain the constant distance the observer must be also moving at a constant velocity. So whilst the moving twin might be moving at speed s your observer is moving at 1/2s. That will add relativistic effects to the new observer's journey that merely adds more complexity to the thought experiment.

Colin2B

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Re: How should one interpret spacetime diagrams?
« Reply #6 on: 26/03/2015 18:32:07 »
Yesterday my son asked: Is there any situation in which an outside observer would perceive the twins as ageing at the same rate?

My answer (off the cuff, while driving) was:  There is, but it would require that the relative speed difference between each twin and the observer be the same.  For example, if the observer placed his craft between Earth and the craft in which the twin was travelling, such that, if Earth were regarded as stationary, he was moving away from it at 0.25c, the twin’s craft would be moving away from him at 0.25c.  If we now change to the observer’s reference frame, Earth will also be moving away from him at 0.25c.  In that case, he should consider that the twins are ageing at the same rate.

Thought I'd like to run that past others.

I think that should work. relative to the new observer both earth and the other twin are moving in opposite directions at the same speed. I don't think he even needs to be half way except it makes it easier to visualise.
As jefferyh says, the problem comes with them all coming together to one frame at some stage, choose your frame!

PmbPhy

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Re: How should one interpret spacetime diagrams?
« Reply #7 on: 26/03/2015 19:00:09 »
Yesterday my son asked: Is there any situation in which an outside observer would perceive the twins as ageing at the same rate?
You're answer should have been simply - No!  The twins don't age at the same rate as reckoned by observers at rest with respect to the stay at home twin.

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #8 on: 26/03/2015 21:58:13 »
Quote from: Jeffrey
So whilst the moving twin might be moving at speed s your observer is moving at 1/2s. That will add relativistic effects to the new observer's journey that merely adds more complexity to the thought experiment.

Cast list: Earth twin (ET), observer (O) and space twin (ST)

Imagine a spacetime diagram on which we consider ET as stationary.  O is travelling at 0.25c to the right and ST is travelling at 0,5c to the right.

If we designate ST as stationary, then O and ET are travelling to the left at 0.25c and 0.5c respectively.

If we designate O as stationary, ET is moving to the left at 0.25c and ST to the right at 0.25c.

I may have oversimplified this, but I can’t see the additional complication.

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #9 on: 26/03/2015 22:16:03 »
Quote from: Colin
As jefferyh says, the problem comes with them all coming together to one frame at some stage, choose your frame!

In this scenario they don’t come together.  We were considering the first of Heumer’s “two cases” in which there is no second meeting.

Quote from: Pete
You're answer should have been simply - No!  The twins don't age at the same rate as reckoned by observers at rest with respect to the stay at home twin.

I’ve just looked back at #4, and I can’t see where I said that the observer was at rest with respect to the stay at home twin.  Would it have been clearer if I had used “inertial frame” instead of “reference frame” when referring to the observer?

Colin2B

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Re: How should one interpret spacetime diagrams?
« Reply #10 on: 26/03/2015 22:57:23 »

In this scenario they don’t come together.  We were considering the first of Heumer’s “two cases” in which there is no second meeting.
In that case I think it works

Quote from: Bill

I’ve just looked back at #4, and I can’t see where I said that the observer was at rest with respect to the stay at home twin.  Would it have been clearer if I had used “inertial frame” instead of “reference frame” when referring to the observer?

I think Pete may have misread that.
Reference frame should be ok, and if they are all up to speed, they are inertial frames (which is what Heumer assumes for simplicity)

PmbPhy

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Re: How should one interpret spacetime diagrams?
« Reply #11 on: 26/03/2015 23:44:04 »
Quote from: Bill S
I’ve just looked back at #4, and I can’t see where I said that the observer was at rest with respect to the stay at home twin.  Would it have been clearer if I had used “inertial frame” instead of “reference frame” when referring to the observer?
Quote
Is there any situation in which an outside observer would perceive the twins as ageing at the same rate?
I assumed he had a particular frame of reference in his mind and assumed it was the stay at home twin's frame. However I see here that I was wrong. One can always choose a frame of reference in that situation where both twins move at the same speed. In that frame the twins age at the same rate during the period of time when both of them are moving at constant speed, i.e. the traveling twin is not slowing down or speeding up. So the answer really is - Yes!

Here's a simple proof. Consider two objects of identical mass at rest next to each twin. Then in all frames of reference at least one of the objects is moving. Then the total momentum of the system consisting of those two particles is non-zero. One can always choose an inertial frame in which the total momentum is zero. In that frame both particles have the same momentum in magnitude and therefore since they have the same mass they have the same speed. Therefore since each object is at rest next to each twin it follows that each twin is moving at the same speed in that frame. If they're moving at the same speed then they're aging at the same rate.
« Last Edit: 26/03/2015 23:48:37 by PmbPhy »

Colin2B

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Re: How should one interpret spacetime diagrams?
« Reply #12 on: 27/03/2015 00:07:56 »
Here's a simple proof.
Interesting proof, not sure I would have thought of using that concept.

PmbPhy

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Re: How should one interpret spacetime diagrams?
« Reply #13 on: 27/03/2015 00:26:14 »
Quote from: Colin2B
I think Pete may have misread that.
What? Me? Do you know to whom you are talking to? I'm Pete, i.e. PmbPhy! Ask JohnDuffield. He'll explain to you that I never admit that I'm wrong.

So I'm confused now. Since Colin is clearly 100% correct, what does that say about John? Lol!

JohnDuffield

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Re: How should one interpret spacetime diagrams?
« Reply #14 on: 28/03/2015 17:20:10 »
What? Me? Do you know to whom you are talking to? I'm Pete, i.e. PmbPhy! Ask JohnDuffield. He'll explain to you that I never admit that I'm wrong.

All: I think the easiest way to think about this subject is to refer to the Simple inference of time dilation due to relative velocity, which features the parallel-mirror light clock. Note that the Lorentz factor is derived from Pythagoras's theorem, that the elapsed time is the number of reflections, and that the light path length is the same for both twins. Draw their lightpaths in space, and it's all very clear. What's also clear is this: only one of them has felt the accelerations. So all participants agree that it was space twin who did the out-and-back trip. He's the one with the zigzag lightpath, and he's the one who got time-dilated.

jeffreyH

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Re: How should one interpret spacetime diagrams?
« Reply #15 on: 28/03/2015 18:10:36 »
You have had a bit of a hiatus John. Did all that googling take it out of you? BTW whose illustration have you plagiarized this time?

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #16 on: 28/03/2015 18:31:15 »
Quote from: John
So all participants agree that it was space twin who did the out-and-back trip. He's the one with the zigzag lightpath, and he's the one who got time-dilated.

You are assuming there is a return trip.  The relativistic equations have to make sense even without a return.

To assume that space twin was moving in any absolute sense because he felt acceleration is unrelativistic thinking.
If Earth and the spacecraft were moving (eg) to the left at 0.5c before take off, then after accelerating, space twin would have to be considered stationary, with Earth moving away at 0.5c as in Huemer’s Fig. 3

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #17 on: 28/03/2015 18:39:17 »
Jeffrey, as you seem to be active at the moment, may I call your attention to #5 and #8?

PhysBang

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Re: How should one interpret spacetime diagrams?
« Reply #18 on: 28/03/2015 19:28:26 »
Here's a simple proof. Consider two objects of identical mass at rest next to each twin. Then in all frames of reference at least one of the objects is moving. Then the total momentum of the system consisting of those two particles is non-zero. One can always choose an inertial frame in which the total momentum is zero. In that frame both particles have the same momentum in magnitude and therefore since they have the same mass they have the same speed. Therefore since each object is at rest next to each twin it follows that each twin is moving at the same speed in that frame. If they're moving at the same speed then they're aging at the same rate.
It is important to make one thing clear.

One can identify a reference frame in which each object has the opposite momentum; that is, the absolute value of their momentum is the same but the direction is directly opposite.

By the symmetries of special relativity, the time dilation associated with the rest frame of these objects (assuming they have the same rest mass) will be the same relative to this special frame in which they have the opposite momentum.

PhysBang

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Re: How should one interpret spacetime diagrams?
« Reply #19 on: 28/03/2015 19:32:35 »
You have had a bit of a hiatus John. Did all that googling take it out of you? BTW whose illustration have you plagiarized this time?
Mr. Duffield's presentation is almost correct, however, as usual he has left out a core feature of relativity theory to which he objects: that there is no true reference frame. The use of the Pythagorean theorem that Mr. Duffield seems to revere is only justified in its use if one assumes that one can make use of a light signal to synchronize any frame of reference relative to stationary clocks (in the form of regular physical systems that can be considered to be synchronized in some fashion). This assumption, at least in standard texts from Einstein's 1905 paper onwards, arises from the assumption that there is not one inertial reference frame in which the laws of physics are correct and the correct presentation of the laws of physics relies on identifying this frame.

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #20 on: 28/03/2015 21:21:28 »
Quote from:  PhysBang
One can identify a reference frame in which each object has the opposite momentum; that is, the absolute value of their momentum is the same but the direction is directly opposite.

By the symmetries of special relativity, the time dilation associated with the rest frame of these objects (assuming they have the same rest mass) will be the same relative to this special frame in which they have the opposite momentum.

Now I’m more confused than usual.

I thought: Momentum = mass x velocity.  If mass is the same, and they have opposite momentum, they must be moving in opposite directions, relative to each other.  Can they be in the same reference frame?

Bill S

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Re: How should one interpret spacetime diagrams?
« Reply #21 on: 28/03/2015 22:21:12 »
Returning to Huemer’s Figures 2 and 3

In the RF in which Earth is stationary, space twin is perceived as travelling at 0.5c in both the outward and return directions, there is no time dilation.

In the RF in which space twin is stationary, it is Earth that is moving, so why is time dilated (10 years become 8.66 years) in that RF?    We still seem to be saying that because space twin accelerates it is he who is “really moving”.  If Earth were really moving away from the spacecraft, wouldn’t time dilation occur in Earth’s RF?

jeffreyH

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Re: How should one interpret spacetime diagrams?
« Reply #22 on: 28/03/2015 23:28:52 »
Jeffrey, as you seem to be active at the moment, may I call your attention to #5 and #8?

Yes Bill I have noted this. I will come back to you when I have a little more time.

PmbPhy

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Re: How should one interpret spacetime diagrams?
« Reply #23 on: 29/03/2015 04:40:07 »
Quote from: Bill S
I thought: Momentum = mass x velocity.  If mass is the same, and they have opposite momentum, they must be moving in opposite directions, relative to each other.  Can they be in the same reference frame?
I'm unclear what you mean by Can they be in the same reference frame? Typically one speaks of two objects as being at rest in frame S, not simply "being" in the same reference frame. Typically I'd interpret that as the observer whose frame of reference is S observes those objects in his frame. It's measurements/observations that one speaks of when they speak of frames of reference.
« Last Edit: 29/03/2015 04:42:01 by PmbPhy »

JohnDuffield

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Re: How should one interpret spacetime diagrams?
« Reply #24 on: 29/03/2015 13:20:46 »
You are assuming there is a return trip.  The relativistic equations have to make sense even without a return.
They do. If you have two twins passing each other in space, each one sees the other one as time-dilated. But this is no more a paradox than two twins separated by distance saying each one looks smaller than the other.

To assume that space twin was moving in any absolute sense because he felt acceleration is unrelativistic thinking.
It isn't. The two twins could have started off as co-moving twins. The one who feels the acceleration is moving relative to the other.

If Earth and the spacecraft were moving (eg) to the left at 0.5c before take off, then after accelerating, space twin would have to be considered stationary, with Earth moving away at 0.5c as in Huemer’s Fig. 3
That's unrelativistic thinking.

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Re: How should one interpret spacetime diagrams?
« Reply #24 on: 29/03/2015 13:20:46 »