[Halc]

It's not a function of acceleration, so I cannot say from just that.

Then would you say that Sabine in the video is wrong?

I am actually going to go so far as to say exactly that. I caution against taking a simple comment out of context, and I'm not much on clicking videos and actually watching them (21 minutes to wade through), and it's Sabine, so I presumed the content is accurate. Well it isn't, which is a shame.

At 0:40 she complains about trying to learn relativity from pop-science sources and failing or finding them incorrect. Many are. Here she is creating her own pop-science tutorial and she does the same thing: get it wrong.

At 1:45 she gets into the length of the path between two sets of coordinates, correctly pointing out that different paths are different lengths despite the beginning and end of the paths being the same, as it is in the twins scenario.

At 6:50 she shows how the calculation of the temporal length of an arbitrary path can be done by breaking the path into pieces and integrating over the length of the path. This is what Eternal Student has done in post 16:

Do you know any math formulas to see how acceleration and time dilation are related?

the elapsed time for the travelling twin (who goes to Andromeda) is given by:

Δτ =   

[Eqn 2]

The computation above is completely scalar. Note the complete lack of acceleration reference in the formula. I see time and speed (v) and that's it. It isn't a function of acceleration, as I said above. The formula above is from special relativity, so it only applies to the special case where gravity is not involved. Hossenfelder's video is entitled "Special Relativity: This Is Why You Misunderstand It", which means the content should stay away from gravity, or the video is mistitled.

And what's with Andromeda? Sure, with enough acceleration, Bob can get there and back before he dies, but Alice (and the whole human race for that matter) isn't going to be there upon his return. Sabine should pick a closer target.

Back to the video:
At 11:00 we get into the twins thing and she correctly says that at least one of the twins needs to accelerate to turn around. That's a biased way of putting it, but true. More correctly, at least one of them needs to accelerate in order for their paths to diverge but meet up a second time. Without acceleration, any relative velocity will just have them meet once at best and forever diverge after that. But it isn't the acceleration that causes the dilation, it is the relative temporal lengths of the paths they take, as computed by the above formula.

11:25 She says acceleration is absolute. She means proper acceleration (the kind you feel with an accelerometer) is absolute. Coordinate acceleration is relative to some coordinate system and is thus not absolute. So sitting at your computer reading this, your coordinate acceleration (relative to your house maybe) is stationary, but your proper acceleration is 1g upward because that's how hard the chair under you is accelerating you.

12:52 She correctly points out that the twins scenario has nothing to do with gravity.

13:39 She correctly points out that the twins starting and ending with the same velocity is not necessary (except to explain that they're twins and presumably had reasonably identical velocity at birth. They merely have to meet twice.

15:30 We start getting into gravity, which is out of scope for a video entitled "Special relativity". She starts with pointing out that under Einstein, gravity is not a force. It is in fact spacetime which has a geometery other than flat Minkowskian spacetime. So anything not accelerating (has no force acting on it) follows a geodesic along the local spacetime.

16:55 She first says acceleration causes time dilation. This is blatantly wrong. Contradictions follow.
17:50 Things really start falling apart. The time runs slower at sea level than on a mountain due to greater acceleration at sea level. This is completely wrong. If true, clocks would run fastest at the center of Earth where acceleration would be zero, but they in fact run slowest there than anywhere else on Earth. The acceleration on the surface of Mercury is under 40% of that on Earth, but time on Mercury runs slower, directly contradicting what Sabine is saying.

19:25 She asks if her video was any better than those incomprehensible books from way back? Well it would be if she hadn't mucked it up.

Back to Special relativity, since I want to disassemble her treatment of that as well and not just her botching the gravity bit. A couple examples contradicting her assertions:

Example 1) Alice, Bob and Chuck are triplets and age 20. Alice stays home. Bob and Check set out on a trip and accelerate identically (10g say) for a month and then coast, riding side by side for a while.  After a year on his own clock, Bob accelerates towards Earth at 10g for 2 months, going back towards home at the same speed he went out. He coasts for another year and takes a month to stop. He's aged 2 years coasting and 4 months acceleration and is age 22y4m now and finds Alice at age 23y2m, or 10 months older. They wait together for Chuck to come back.
Chuck coasts twice as long and turns back. So he ages 4 years coasting and the 4 months accelerating and comes home at age 24y4m finding Alice to be 25y5.7m and Bob to be 24y7.7m.
This contradicts what Hossenfelder says since both Bob and Chuck have experienced identical accelerations, just at different times. They should be aged identically per Hossenfelder's words, but they're not. This is one trouble with doing physics in the language of laymen instead of the language of physcs. Time dilation is not a function of acceleration and there's no mathematical formula expressing it in terms of acceleration.

Example 2)
I have a pair of wheels or gears. One wheel is 1000 times the radius of the other, and they meet at one point and move at the same velocity there. I put a clock on each wheel at the point at which they meet. The wheels get turned with the small  one going around at 1000 times the RPM and hence 1000 times the centripetal acceleration. Both clocks are moving at the same speed relative to the inertial frame of the setup. The two clocks will stay in sync indefinitely despite the one acceleration being a thousand times the other. This also contradicts what Hossenfelder says in the video, but is entirely consistent with the formula that ES provided.

[alancalverd]

The two clocks will stay in sync indefinitely despite the one acceleration being a thousand times the other.

Not if they were in sync before you started the second one moving. The twin "paradox" is about two clocks initially at rest and synced with respect to one another but subsequently acquiring a relative velocity, therefore one must have accelerated.

[Halc]

The two clocks will stay in sync indefinitely despite the one acceleration being a thousand times the other.

Not if they were in sync before you started the second one moving.

Incorrect. If you read my post, they're both moving relative to Earth, and always at the exact same speed (~460 m/sec). The difference is in magnitude of acceleration, 1g (away from the axis) vs 1000g (towards the axis).

The twin "paradox" is about two clocks initially at rest and synced with respect to one another but subsequently acquiring a relative velocity, therefore one must have accelerated.

In the example above, both accelerate, but one far more than the other. The point (unlike the twin scenario) is to illustrate that differential aging is not a function of magnitude of acceleration, as the video mistakenly suggests. Neither clock will run faster than the other.

Since you're on the horn:

If they were "always" moving relative to one another, you had no means of synchronising them.

This is incorrect. Clocks in each other's presence can be objectively synchronised or compared to each other. They can either both be zeroed (such as at the departure event) or the either observer can note the time on the other clock as it passes by (such as at the return event). Physical events such as this are objective and not frame dependent. If Bob reads a certain value at some event, that value measured is the same regardless of frame choice.

[alancalverd]

Incorrect. If you read my post, they're both moving relative to Earth, and always at the exact same speed (~460 m/sec). The difference is in magnitude of acceleration, 1g (away from the axis) vs 1000g (towards the axis).

But the "polar" clock P started at rest relative to the earth's  surface and was therefore travelling 460m/s slower than the "equatorial" clock E. So if you synchronised them (to make twins) and then accelerated P to 460 m/s you will have altered its tick rate as seen from E. Alternatively you could wait until  their relative speed is zero and then synchronise them, in which case they will indeed remain in step.

[alancalverd]

either observer can note the time on the other clock as it passes by

That isn't the point. Time dilation concerns the tick rate of A as observed at B.

[MikeFontenot]

I'll just "cut to the chase" and answer the question asked in the subject line:

"What is the exact cause of the time dilation of the twin?"

It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).

The simplest case is when his change of velocity is instantaneous.  According to him, when he instantaneously changes his velocity with respect to her by delta_v, he says her age instantaneously changes by the quantity

  - L * delta_v,

where "L" is their (positive) separation then according to HER, and

  delta_v = v(inbound) - v(outbound).

Velocities are taken as positive when the twins are moving apart, so delta_v is negative in the standard twin paradox scenario.

[MikeFontenot]

I'll just "cut to the chase" and answer the question asked in the subject line:

"What is the exact cause of the time dilation of the twin?"

It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).

The simplest case is when his change of velocity is instantaneous.  According to him, when he instantaneously changes his velocity with respect to her by delta_v, he says her age instantaneously changes by the quantity

  - L * delta_v,

where "L" is their (positive) separation then according to HER, and

  delta_v = v(inbound) - v(outbound).

Velocities are taken as positive when the twins are moving apart, so delta_v is negative in the standard twin paradox scenario.

I'll give a specific example of how the above equation is used.  Let the relative speed of the twins be 0.866, using units of lightyears and years.  That gives a gamma factor of exactly 2.  So we immediately know that, when he is not changing his speed (and she never does), they each conclude that the other is ageing half as fast as they themselves are.  Suppose she says that he goes outbound for 40 years (of her time).  So she says that he is 20 years old at his turnaround.  His turnaround is an EVENT, so everyone (including him) must agree that he is 20 years old then.

Similarly, according to her, she ages 40 more years while he is returning home, and he ages by 20 years during his return (both she  and he agree about that).  So at the reunion, she is 80 years old, and he is 40 years old.  They HAVE to agree about that, because they are standing side-by-side and looking that each other at the reunion.

But while he was going at a constant speed on his outbound leg, HE says she was aging at half his rate, so HE says she only got 10 years older on his outbound leg, and likewise for his inbound leg.  So he might conclude that she should only be 20 years old when he gets home.  But she's not ... she's 80.  So where did she age the additional 60 years, according to him?  There's only one place that could have happened: she HAD to have aged (according to him) by 60 years during his instantaneous turnaround.

And that's exactly what the simple equation that I gave in my first post gives.  Their distance apart at the turnaround (according to her) is

  L  =  40 * v  =  40 * 0.866  =  34.64 ly.

In my previous post, I said

  "According to him, when he instantaneously changes is velocity with respect to her by delta_v, he says her age instantaneously changes by the quantity

  - L * delta_v,

where delta_v is his velocity change at the turnaround (and it's negative, so her age change is positive).  We know from the known results at the reunion that he must conclude that she instantaneously gets OLDER when he accelerates TOWARD her.

 Specifically,

  delta_v  =  -0.866 - (+0.866)  =  -1.732.

So he says she instantaneously gets OLDER during his instantaneous turnaround by

  - L * delta_v  =  (-34.64) * (-1.732)  =  60 years.

[Halc]

"What is the exact cause of the time dilation of the twin?"

It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).

Time dilation is an abstract coordinate effect. For inertial coordinate systems, time dilation of a clock is 'caused' by motion (speed) of that clock through the space of that coordinate system. Hence it is incorrect to restrict dilation only to change in velocity as your statement above does.
Yes, for the case of accelerated coordinate systems, dilation can be a function of the position of the dilated clock in that coordinate system, and (unlike the inertial case) the position of a reference point (an observer). Inertial dilation is not dependent on an observer, only on the choice of frame. There is also dilation (objective, so not strictly a coordinate effect) due to gravitational potential differences between two reference points.

So at the reunion, she is 80 years old, and he is 40 years old.

That, for the record, is an example of differential aging, not dilation. See the post by Janus who clearly spells out the difference. Yes, differential aging is physical. Dilation is abstract, as is your explanation. Yes, it's one way of looking at it, but an abstraction doesn't answer what the physical difference is in the two twins. She ages twice as much as him because the temporal length of his worldline is twice that of his. That's true regardless of frame, inertial, accelerating, and it also works for the gravity case. It's the physical answer of why she's older than him. The abstract part is just how each of them chooses to represent the same physical facts with different abstract numbers. Pushing a pencil around doesn't cause any physical effects. The worldline length difference does.

[alancalverd]

It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).

And apropos another thread. when did "change in velocity" cease to  mean "acceleration"?

Also important to note that "twins" implies identity, such that when the relative velocity is zero, both clocks (physical or biological) remain in sync. Thus "instantaneous synchronisation" on a fly-by is not permitted.

[MikeFontenot]

"What is the exact cause of the time dilation of the twin?"

It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).

Time dilation is an abstract coordinate effect. For inertial coordinate systems, time dilation of a clock is 'caused' by motion (speed) of that clock through the space of that coordinate system. Hence it is incorrect to restrict dilation only to change in velocity as your statement above does.

The results I gave refer strictly to the issue of how the traveling twin (he) says the home twin (she) will age during his trip.  When he is not accelerating, the answer is elementary: he says she is ageing more slowly than he is (by a factor of 2 when their relative speed is 0.866 ly/y).  When he instantaneously changes their relative velocity (from +0.866 [directed AWAY from her] to -0.866 [directed TOWARD her]), he concludes that her age instantaneously increases (by 60 years in my example, easily determined using the simple equation I gave).   The combination of those two results gives his conclusion about her age at their reunion (which must, of course, agree with her conclusion).  Anything else is just unimportant quibbling about terminology.

[MikeFontenot]

It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).

And apropos another thread. when did "change in velocity" cease to  mean "acceleration"?

I used the term "change in velocity", rather than "acceleration" because I was interested in the easier-to-analyze scenario where the acceleration is infinitely great, but lasts only for an infinitesimal time, such that the change in velocity is finite, and occurs essentially instantaneously.

The case of finite accelerations lasting for a finite time can be analyzed, but it is much more complicated, and is less enlightening.

[MikeFontenot]

In my previous posts on this thread, I gave a simple equation that computes (for a one-dimensional motion) by how much the home twin's (her) age instantaneously changes, according to the traveling twin (him), when he instantaneously changes his velocity. That equation works no matter in which direction (toward or away from her) he is moving with respect to her before his velocity change, and no matter in which of the two directions his (infinite) acceleration is pointed.  And, in contrast to the effect of an (infinite) acceleration TOWARD her, an (infinite) acceleration, in the direction AWAY from her, causes her (according to him) to instantaneously get YOUNGER, not OLDER.

That really bothers some people.  But that IS the unavoidable result.

[hamdani yusuf]

Time dilation is not a function of acceleration and there's no mathematical formula expressing it in terms of acceleration.

If acceleration has no effect on time dilation, then the observation of the accelerated twin should be equally valid as non-accelerated one.
Alternatively, there must be something else which makes their situations different, such as their relative motion against an external object, like CMB.