B Why the stay-at-home twin is not considered to be accelerating?

[A.T.]

Here the "this" that is a convention is what Andrew Kirk described as "the inertial frame changing". And, as my quote just above says, that is not what causes the twins' clocks to have different readings at the end; the cause of that is the different lengths of the paths they take through spacetime.

I'm not sure if there is an ultimate answer to those "what is the best explanation/reason" questions.

To me both, "changing frames" and "paths through spacetime" are abstractions, which rely on conventions and geometrical interpretations. But the difference/asymmetry in elapsed proper-times between two meetings, doesn't rely on such. So there must be something else that differentiates the twins, that also doesn't rely on such. And that is the difference in proper acceleration.

This should be part on any explanation. Then you use the tools like spacetime intervals, to compute how much proper time will elapse for either of them, etc.

 

[PeterDonis]

I'm not sure if there is an ultimate answer to those "what is the best explanation/reason" questions.

Probably not, but there is a key distinction to be made between actual measurable quantities and abstractions. See below.

To me both, "changing frames" and "paths through spacetime" are abstractions

"Changing frames" is, since "frames" are (at least in any sense in which a single observer can be said to "change" frames). But "paths through spacetime" are not: the length of each twin's path through spacetime is directly measured by the clock carried by that twin.

the difference/asymmetry in elapsed proper-times between two meetings, doesn't rely on such

The difference in elapsed proper times is the difference in path lengths through spacetime.

there must be something else that differentiates the twins, that also doesn't rely on such. And that is the difference in proper acceleration

I agree with @Dale's position on this in post #54.

 

[A.T.]

The difference in elapsed proper times is the difference in path lengths through spacetime.

- The difference in elapsed proper times is what their clocks measure.
- The difference in their proper accelerations is what their accelerometers measure.

These are direct measurements, that do not rely on the notion of spacetime.

I agree with @Dale's position on this in post #54.

Me to.

 

[Dale]

I am sure feeling the love here!

 

[Dale]

I'm not sure if there is an ultimate answer to those "what is the best explanation/reason" questions.

I think that is probably true. There are a lot of different ways that students learn and sometimes one way or another really “clicks” for a particular student. For me it was the geometric explanation, so I have a particular affinity to that one and tend to push it more than others.

 

[Dale]

The twin paradox is explained by the fact that the moving twin changes their frame of reference. That is the essential kernel of the solution - everything else is second order.

I wouldn’t say this either. You can use a single non-inertial frame to represent the traveling twin without any change of reference frame. In fact, that is a more direct (though less familiar) approach to the problem. Also, it is possible to do everything in a geometrical approach without even introducing reference frames at all.

 

[Sagittarius A-Star]

There is an equation for how an accelerating observer measures non-local clocks:
$$ T = \frac{t}{\sqrt{1-\frac{2ah}{c^2}}}$$

Yes. Via the general 1st order approximation formula ...
$$(1-x)^n \to 1-nx$$
... this equation can also be approximately written as:
$$T/t \approx 1 + \frac {ah} {c^2} = 1 + \frac {\Delta\phi} {c^2}$$
This formula can be easily derived in SR. The rocket turns by 180 degrees, so that the front end is directed towards the "stay at home" twin. Then the rocket engine is switched on, and the rocket is uniformly accelerating. At the front of the rocket is a lamp, which sends a short light pulse. At the rear end of the rocket with length Δh is a sensor, which receives this light pulse. I will show, that it is received blue-shifted.

First, I define a “co-moving” inertial reference frame.

The accelerated rocket shall have in this frame the velocity Zero at the point in time, when the light-pulse is sent. The light needs approximately
Δt ≈ Δh / c until it reaches the sensor. After that time, the sensor has approximately the velocity

v = a * Δt ≈ a * Δh / c.

The sensor moves with that velocity into the light, that was sent out, when the lamp had the velocity Zero in the defined inertial frame. For small “v”, the formula for the classical Doppler effect can be used:
$$f(received) / f(sent) \approx 1 + v/c = 1 + \frac {a * \Delta h} {c^2} = 1 + \frac {\Delta\phi} {c^2}$$
In the accelerated rest frame of the sensor, it is not a Doppler effect, but time-dilation between different pseudo-gravitational potentials Φ of lamp and sensor.

 

[PeterDonis]

The difference in their proper accelerations is what their accelerometers measure.

Yes, agreed, the proper accelerations are directly measurable.

The reason I prefer focusing on the difference in path lengths (the geometric explanation) is that it is the only explanation that generalizes to all cases. In flat spacetime you can have scenarios where both twins have nonzero proper acceleration but their elapsed times are not the same. In curved spacetime you can even have scenarios where both twins have zero proper acceleration but their elapsed times are not the same. There is no general rule involving proper accelerations that will always work. Looking at the spacetime geometry is the only technique that will always work.

 

[A.T.]

Yes, agreed, the proper accelerations are directly measurable.

The reason I prefer focusing on the difference in path lengths (the geometric explanation) is that it is the only explanation that generalizes to all cases.

Well, as soon you point out the directly measurable different proper accelerations (break in symetry), people tend to get the wrong idea, that proper acceleration directly affects the clock rate (in contradiction to the clock hypothesis). And that's where you need geometry, to explain how accelerations can affect the total proper time, without directly affecting the clock rate. Like in the analogy I used in post #53.

 

[Herman Trivilino]

For example, suppose both A and B are standing side by side. Their relative speed is zero. A starts moving and his speed increases from zero to 100 m/s in 1 second. His acceleration is 100 m /s 2. Relativity says that both are moving in relation to each other. So B's speed also increases from zero to 100 m /s. Thus, B should also be accelerating at 100 m /s2. So, why can't B too said to be accelerating.

Another thought I had on this issue. Suppose you have a third spaceship C moving away from B with an acceleration of 200 m/s². Do you think that now, all of a sudden, the acceleration of B is 200 m/s² instead of 100 m/s².

Note that this difficulty does not arise with speed. If A and B are moving relative to each other with a speed of 100 m/s, and B and C are moving relative to each with a speed of 200 m/s, there are no contradictions.

 

[Nugatory]

If, as in the twin paradox, the acceleration causes a change in speed then this does does affect the clock rate.

A change in speed does not affect the clock rate. If it did we could identify a true rest frame of the universe - it would be the only frame in which a clock at rest in that frame is unaffected.
Or consider that right now you are moving at almost $c$ using a frame in which a charged particle in a linear accelerator is at rest; you are moving at a few kilometers per second using a frame in which someone on the opposite side of the Earth is at rest; and you aren’t moving much at all in a frame in which your computer screen is at rest while you’re reading this. Which is the speed that is affecting your clock rate?

Velocity-based time dilation, the stuff they talk about in introductory presentations where they say that a moving clock ticks slow, is a consequence of the relativity of simultaneity - and has next to nothing to do with why that the traveller ages less than the stay-at-home twin. One way of seeing this is that the stay-at-home twin’s clock is dilated relative to the traveller’s clock at every moment of the journey - yet the traveller ages less.

 

[PAllen]

Velocity-based time dilation, the stuff they talk about in introductory presentations where they say that a moving clock ticks slow, is a consequence of the relativity of simultaneity - and has next to nothing to do with why that the traveller ages less than the stay-at-home twin. One way of seeing this is that the stay-at-home twin’s clock is dilated relative to the traveller’s clock at every moment of the journey - yet the traveller ages less.

I have to disagree with this, as a general statement. Consider the traveling twin to start out with some outgoing velocity, and uniformly accelerate toward the stay at home twin until they reunite. Then by any reasonable coordinate choice for the traveler, the home twins clock runs uniformly faster between meetups. There is no period of time dilation at all.

 

[Nugatory]

I have to disagree with this, as a general statement.

As a general statement, yes, you're right. I was speaking in terms of the most common intro presentation of the twin paradox, the one in which the traveller heads out at a constant velocity relative to stay-at-home, turns around quickly, and then returns at a constant velocity. In this case, the time dilation formula says what I said... and indeed that naive application of the time dilation formula is what makes this a "paradox".

Another subtlety that I glossed over: I said "every moment", but if we assume an instantaneous turnaround we can't apply the time dilation formula at every moment - it's not defined at the moment of the instantaneous turnaround. We dig our way out of that rathole by assuming a very large but not infinite acceleration applied for a very small but not instantaneous turnaround so that we always have an usable MCIF in which the traveller is momentarily at rest.

 

[dayalanand roy]

Another thought I had on this issue. Suppose you have a third spaceship C moving away from B with an acceleration of 200 m/s². Do you think that now, all of a sudden, the acceleration of B is 200 m/s² instead of 100 m/s².

Note that this difficulty does not arise with speed. If A and B are moving relative to each other with a speed of 100 m/s, and B and C are moving relative to each with a speed of 200 m/s, there are no contradictions.

Thanks. Nice thought.

 

[dayalanand roy]

Because B's accelerometers read zero at all times.

Imagine being in a train traveling next to another train at the same speed. You can see the other train's passengers through your window. One train puts on its brakes. According to the passengers on both trains, the previously stationary passengers in the other train start to move. This is coordinate acceleration, where the velocity of something relative to you changes. However, only one set of passengers will feel a jolt and be pushed back in their chairs - that's proper acceleration.

In the twin paradox only one of the twins feels proper acceleration. It's the proper acceleration that's important because coordinate acceleration is an effect of your choice of what "at rest" means whereas proper acceleration is actually felt.

Thanks. Nice explanation of coordinate and proper acceleration. Now I am getting the point. The problem is, in most of the books dealing with SR or GR, that I have read, and as far as I remember, only the term acceleration is used, not proper acceleration. More so, doesn't it mean that, "motion is relative" this statement is not fully applicable in case of motion experiencing proper acceleration?

 

[dayalanand roy]

An accelerometer attached to B reads 0. Only A is accelerating according to attached accelerometers.

Because accelerometers show that it is not relative.

Thanks. Now getting the point.

 

[dayalanand roy]

The key point to take away from the twin paradox is that the time difference is due to the moving twin's change of frame.

The acceleration is essentially irrelevant except for the fact that the moving twin must decelerate and accelerate so that they can change their frame.

You can easily set up a twin paradox without any accelerations. I use the values from above.

Set up a stationary observer on earth, another at the turning poiint and another at twice the distance of turning point. All three synchronise their clocks to read 0.

The moving twin_1 accelerates before the experiment starts in such a way that she is traveling at V when she passes earth, and she passes when the Earth clock reads 0. She looks at the Earth clock and sets her clock to 0.

A similarly moving twin_2 passes the point twice the turning point distance away, traveling at V towards earth.

The two moving travellers meet at the turning point. The stationary person's clock reads 5 but twin_1's clock reads only 4.5.

twin_2 sets her clock to be the same at twin_1, or 4.5. twin_2 continues to Earth and arrives at Earth when Earth says it is 10. However, twin_2's clock reads 9, made up of the 4.5 which she set, plus the 4.5 for her journey time, totalling 9.

So, whereas the Earth bound twin says 10 years have elapsed since twin_1 left, the total travel time measured by the two moving twins is only 9 years.

So, we have the same time difference but there have been no accelerations.

Acceleration has nothing to do with the twin paradox other than it is necessary for the moving twin to accelerate and decelerate to change her frame. The twin paradox is about changing frames.

Thanks. A new thought for me.

 

[PAllen]

Thanks. Nice explanation of coordinate and proper acceleration. Now I am getting the point. The problem is, in most of the books dealing with SR or GR, that I have read, and as far as I remember, only the term acceleration is used, not proper acceleration. More so, doesn't it mean that, "motion is relative" this statement is not fully applicable in case of motion experiencing proper acceleration?

Actually, Einstein, probably the biggest proponent of ‘motion is relative’, clearly and mathematically specified that if motion involving proper acceleration is compared to motion without, you need to include time dilation due to potential difference in the calculation. This means that the traveling twin must have a period where the stay at home twin’s clock runs faster due to being higher in a potential well.

 

[PAllen]

All this talk of changing frames leads me to the following analogy. In Euclidean plane geometry we have a simple law, the triangle inequality, closely related to the more general law that between any two points, a path which never changes direction is always shorter than any path with direction changes. These are considered geometric properties of a plane, not related to any coordinates we may place on the plane, nor which of any techniques may be used to measure lengths. The fundamental fact is that the minimizing distance is along a path that doesn’t change direction.

Making a correspondence to special relativity, change of direction corresponds to proper acceration, the latter being exactly change of normalized tangent vector, which can only be change of direction in spacetime. A frame in SR may be said to correspond to using a ruler in a particular orientation in Euclidean geometry.

The claims that acceleration does’t matter, only change of frames matters is like saying the triangle inequality isn’t at all about change in direction (acceleration), instead it is about change of rulers. The triplet form of twin scenario is like claiming that change of direction is irrelevant because instead of changing the direction of one ruler, we can use two rulers in different orientations.

Note that by analogy, the clock hypothesis corresponds to the notion that when measuring the length of a general curve by running a string along it, then measuring the string along a flat ruler, we don’t need to introduce any adjustments for where we had to bend the string. This has nothing to do with the prior, IMO, silly statements about change of rulers and change of reference frames.

 

[dayalanand roy]

If, as in the twin paradox, the acceleration causes a change in speed then this does does affect the clock rate.

I think therefore your statement is both incorrect and very misleading.

99% of the explanations of the twin paradox say "It is caused by the fact that the traveling twin accelerates" but they never give a calculation showing how it comes about, nor provide an equation relating the age difference to the rate and duration of the acceleration experienced. I suggest the statement is therefore as meaningless as saying "It is caused by the fact that the traveling twin is wearing a bikini".

The twin paradox is explained by the fact that the moving twin changes their frame of reference. That is the essential kernel of the solution - everything else is second order. It is a simple application of the Lorentz transformation equations to get the resultant time difference.

Secondly, no-one citing acceleration as the cause ever shows how acceleration can account for the fact that both twins see each each other age more slowly than themselves during the entire time. This can only be resolved by invoking a frame change. It has nothing to do with acceleration.

Perhaps we could ask the original poster dayalanand roy , who has obviously worked on the subject, whether he found the change of frame explained things to him.

Thanks for your participation and making the debate so illuminating. But, as I told in the very beginning of the post, I am not a physicist. I am a biologist. Relativity and twin paradox is quite naive to me. And my main aim is to understand what is time itself. I am working on this very problem. The current dispute that whether it is just frame change, or acceleration that accounts for the twins gaining different ages can be better solved by you learned physicists itself, and I hope you will do it. I am just here to learn from you people.
Thanks again and regards.

 

[dayalanand roy]

Actually, Einstein, probably the biggest proponent of ‘motion is relative’, clearly and mathematically specified that if motion involving proper acceleration is compared to motion without, you need to include time dilation due to potential difference in the calculation. This means that the traveling twin must have a period where the stay at home twin’s clock runs faster due to being higher in a potential well.

Thanks. But here, the potential difference and potential well are a bit new terms for me. Is the potential difference term used here conveys the same meaning as in case of any electric current?

 

[jbriggs444]

Thanks. But here, the potential difference and potential well are a bit new terms for me. Is the potential difference term used here conveys the same meaning as in case of any electric current?

Similar. They both involve the notion of a mathematical object known as a "scalar field". Which pretty just means associating a number with every point in a space. They both have units of energy per unit charge. It is just that in one case the "charge" is the electrical charge on a test particle and in the other case the "charge" is the mass of the test particle.

But different. The "potential field" associated with a stationary electrical charge is a more or less real thing that is present (in some form) regardless of one's choice of coordinates. The "potential field" associated with an acceleration in special relativity is more of a mathematical fiction -- it reflects a choice of coordinates more than an underlying physical reality.

 

[A.T.]

Is the potential difference term used here conveys the same meaning as in case of any electric current?

More like gravitational potential. When a rocket is accelerating you effectively have a gravitational field in the frame of the rocket. Clocks at different potentials in that field run at different rates, even when they are not moving in that frame.

 

[Janus]

. More so, doesn't it mean that, "motion is relative" this statement is not fully applicable in case of motion experiencing proper acceleration?

No. While you can tell who accelerated ( changed velocities), you can't tell what velocity they began with or ended with, only that they differ by a certain amount. So for example, with two starting ships side by side, and one accelerates by 100 km/sec. There is no way to distinguish between both rockets starting "at rest" and one accelerating to 100 km/sec, and both rockets already moving at 100 km/sec and one accelerating to a stop while the other continues on.

 

[dayalanand roy]

No. While you can tell who accelerated ( changed velocities), you can't tell what velocity they began with or ended with, only that they differ by a certain amount. So for example, with two starting ships side by side, and one accelerates by 100 km/sec. There is no way to distinguish between both rockets starting "at rest" and one accelerating to 100 km/sec, and both rockets already moving at 100 km/sec and one accelerating to a stop while the other continues on.

Thanks. Now I am getting the point.

 

[dayalanand roy]

More like gravitational potential. When a rocket is accelerating you effectively have a gravitational field in the frame of the rocket. Clocks at different potentials in that field run at different rates, even when they are not moving in that frame.

Yes. Thanks for a simpler explanation.