[Dimensional (OP)]

Please read before commenting because I found some contradictory claims from two credible sources.

Here is a link that Fermilab put out,

.  At 12:00, it explains that the reason for the time dilation of the twin is because of the change in direction and not acceleration (although he does say that acceleration plays a small part).

The other claim is made by Sabine Hossenfelder.  She says in this video,

, at 11:00, that it is the acceleration that causes the dilation.

There must be a simple formula to calculate the time dilation by the acceleration.  This seems to be a way to know who is correct.

Any thoughts about this?

[Halc]

There are many ways to explain the difference, so there is no one correct way.
One of the best ways is simply to choose an inertial frame of reference and stick with it. Time dilation is a function of the speed of each twin in that frame. It's that simple.

it explains that the reason for the time dilation of the twin is because of the change in direction and not acceleration

This explains the asymmetry of the situation, but it is possible for both twins to change direction, and this explanation then falls apart.

The other claim is made by Sabine Hossenfelder.  She says in this video,
, at 11:00, that it is the acceleration that causes the dilation.

It is misleading to simplify it to this, totally out of context. I don't think Hossenfelder said this without qualifying context.

I can have two clocks, one accelerating at about 1/30th of a g for a year, and the other at thousands of g for a year. The clocks can be kept in sync. Or maybe the high-g one can be made to age faster or slower. Point is, it simply isn't a function of acceleration, but rather a function of speed relative to some given frame of reference. There's no way to fool that method.

[alancalverd]

A change in direction is an acceleration.

[Dimensional (OP)]

I can have two clocks, one accelerating at about 1/30th of a g for a year, and the other at thousands of g for a year. The clocks can be kept in sync. Or maybe the high-g one can be made to age faster or slower. Point is, it simply isn't a function of acceleration, but rather a function of speed relative to some given frame of reference. There's no way to fool that method.

So if the twin only accelerated instead of traveling at a velocity, there wouldn't be a difference in age?  Consider the twin accelerates away for half the distance towards the turning point, and returns by accelerating towards the twin for the second half of the distance.  Are you saying that their clocks would be equal upon return to the twin?

At 12:05 in the video, she says, "the real time dilation comes from the acceleration".

And the other video tries to explain how no acceleration is needed for time dilation. 

[Dimensional (OP)]

A change in direction is an acceleration.

Yes, I meant to say that the video from Fermilab says that no acceleration is needed.  While the other video claims that the acceleration caused the time dilation.

[Dimensional (OP)]

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?

The point of my thread is to try to figure out if there is actually a clear and definite understanding of the twin paradox. 

[Eternal Student]

Hi.

      I've watched both videos sufficiently (I would think).  Sabine's video is long and people could just jump in to it about one-third the way along and stop before she discusses general relativity (the last third).   Meanwhile, the Fermilab video probably does need to be watched almost all the way through.

    The videos aren't really contradictory,  they are just emphasizing things in slightly different ways.

The common ground is:
       They assume you are broadly familiar with the twin paradox and why it seems like a paradox (specifically they they can't both be older than the other).
       They state the paradox is resolved because there is an asymmetry between the two twins.
       They both state that Special Relativity is sufficient, you don't need General relativity.

The differences:
   Sabine's video emphasizes that one twin can measure an acceleration with an accelerometer (she uses a spring balance).   She spends some time imagining a realistic acceleration, i.e.  where one twin is smoothly accelerated.

      Don / Fermilab  video  tries to emphasize that acceleration isn't used much to determine the time dilation.   Indeed, that much is true if you understand it a certain way:  It hardly matters if the acceleration was smooth and steady or abrupt or, as Don tries to suggest if there was no acceleration at all.   Notice that Don never really gets his space traveller back home in the second half of his video, he only gets some information about time clocks back.  At best this scenario could be used to model a situation where the acceleration is abrupt, the space traveller is instantly thrown from one one rest frame into a different rest frame (i.e. the two rocket crews didn't just hold up their clocks as they passed each other, they threw the traveller out of the airlock of one ship and into the other ship, the other ship's crew presumably had a very good doctor who could put Ron back together again after their impact on the other ship at a speed close to c ).
     The key element in the Fermilab video is where Don  explains the resolution of the paradox   at  12: 00  ~  12: 30.   He states   "The moving observer existed in TWO different frames".    He does then confound the issue at 12:30 ~12:40 by explaining that the acceleration wasn't important....

Summary:
    Overall I can find more fault with the Don Lincoln's / Fermilab  video.  I can't help but mention one minor detail -  the statement "the stationary observer only existed in one frame while the moving twin existed in two different frames"   is just wrong.  They existed in all valid frames of reference,  all he means is that there was only ever one rest frame for the twin on Earth  but two different rest frames for the traveller.   People don't suddenly stop existing in any frame of reference, they just stop being being at rest in that frame.
      The more important point is clearly where he implies that the change of rest frames is important but then states that acceleration is NOT important.   You MUST read between the lines here.....   The acceleration was obviously important because otherwise the space travelling twin wouldn't have had two different rest frames.   However, it's just the way the acceleration was applied that doesn't matter and isn't used in the usual treatment of the twins paradox under Special relativity.    I get the impression that  Don Lincoln was just emphasizing that General relativity is not required.   This, I think, is where some confusion has been left with the viewer ( @Dimensional ).   Let's try and give a simple analogy:
       Alice buys some strawberry milkshake that was in a glass bottle.  She pours the milkshake into a cup and gives it to her son.   We would say that it doesn't matter how she poured the milkshake or what the original glass bottle was like.  She could have poured it slowly, quickly,  in two separate goes with a break in between... it doesn't matter.    What matters is that it's in the cup at the end.   This is what Don Lincoln was stating.   The nature of the acceleration is unimportant   (the way she poured the milkshake is unimportant)   BUT we know the travelling twin changed rest frames somehow (just as we understand that some milkshake pouring operation must have happened).  So, we end up with a mathematical formula for the time dilation in the twins paradox which does not include the variable, a (the acceleration), it only involves quantities like L  (the distance between Andromeda and Earth) and  V (the constant speed of travel).   In this sense, the time dilation is independent of the acceleration.

   I hope that helps a bit.

Best Wishes.

[Dimensional (OP)]

Hi.

      I've watched both videos sufficiently (I would think).  Sabine's video is long and people could just jump in to it about one-third the way along and stop before she discusses general relativity (the last third).   Meanwhile, the Fermilab video probably does need to be watched almost all the way through.

    The videos aren't really contradictory,  they are just emphasizing things in slightly different ways.

The common ground is:
       They assume you are broadly familiar with the twin paradox and why it seems like a paradox (specifically they they can't both be older than the other).
       They state the paradox is resolved because there is an asymmetry between the two twins.
       They both state that Special Relativity is sufficient, you don't need General relativity.

The differences:
   Sabine's video emphasizes that one twin can measure an acceleration with an accelerometer (she uses a spring balance).   She spends some time imagining a realistic acceleration, i.e.  where one twin is smoothly accelerated.

      Don / Fermilab  video  tries to emphasize that acceleration isn't used much to determine the time dilation.   Indeed, that much is true if you understand it a certain way:  It hardly matters if the acceleration was smooth and steady or abrupt or, as Don tries to suggest if there was no acceleration at all.   Notice that Don never really gets his space traveller back home in the second half of his video, he only gets some information about time clocks back.  At best this scenario could be used to model a situation where the acceleration is abrupt, the space traveller is instantly thrown from one one rest frame into a different rest frame (i.e. the two rocket crews didn't just hold up their clocks as they passed each other, they threw the traveller out of the airlock of one ship and into the other ship, the other ship's crew presumably had a very good doctor who could put Ron back together again after their impact on the other ship at a speed close to c ).
     The key element in the Fermilab video is where Don  explains the resolution of the paradox   at  12: 00  ~  12: 30.   He states   "The moving observer existed in TWO different frames".    He does then confound the issue at 12:30 ~12:40 by explaining that the acceleration wasn't important....

Summary:
    Overall I can find more fault with the Don Lincoln's / Fermilab  video.  I can't help but mention one minor detail -  the statement "the stationary observer only existed in one frame while the moving twin existed in two different frames"   is just wrong.  They existed in all valid frames of reference,  all he means is that there was only ever one rest frame for the twin on Earth  but two different rest frames for the traveller.   People don't suddenly stop existing in any frame of reference, they just stop being being at rest in that frame.
      The more important point is clearly where he implies that the change of rest frames is important but then states that acceleration is NOT important.   You MUST read between the lines here.....   The acceleration was obviously important because otherwise the space travelling twin wouldn't have had two different rest frames.   However, it's just the way the acceleration was applied that doesn't matter and isn't used in the usual treatment of the twins paradox under Special relativity.    I get the impression that  Don Lincoln was just emphasizing that General relativity is not required.   This, I think, is where some confusion has been left with the viewer ( @Dimensional ).   Let's try and give a simple analogy:
       Alice buys some strawberry milkshake that was in a glass bottle.  She pours the milkshake into a cup and gives it to her son.   We would say that it doesn't matter how she poured the milkshake or what the original glass bottle was like.  She could have poured it slowly, quickly,  in two separate goes with a break in between... it doesn't matter.    What matters is that it's in the cup at the end.   This is what Don Lincoln was stating.   The nature of the acceleration is unimportant   (the way she poured the milkshake is unimportant)   BUT we know the travelling twin changed rest frames somehow (just as we understand that some milkshake pouring operation must have happened).  So, we end up with a mathematical formula for the time dilation in the twins paradox which does not include the variable, a (the acceleration), it only involves quantities like L  (the distance between Andromeda and Earth) and  V (the constant speed of travel).   In this sense, the time dilation is independent of the acceleration.

   I hope that helps a bit.

Best Wishes.

But at 14:25 in the video, he definitely seems adamant that acceleration is not the cause of the time dilation of the classical twin paradox thought experiment.  So if acceleration is not the cause, then what is?

As you saw, he then goes on to try to "prove" in the video that acceleration is not necessary (like you say, there would have been at least some form of acceleration, but anyway), and then does not really explain what is the cause of the dilation if not acceleration. 

[alancalverd]

Yes, I meant to say that the video from Fermilab says that no acceleration is needed. 

Then it's wrong! If there is no acceleration between the twins, there is no relative speed, therefore no time discrepancy. If they were always moving relative to one another, they aren't twins because there never was any synchronicity.

[Eternal Student]

Hi.

But at 14:25 in the video, he definitely seems adamant that acceleration is not the cause....

    This might be an error.   Don Lincoln's video ends at 13:20,   there isn't anything at 14:25.

     I would be willing to explain the difference in time that each twin experienced by using a hybrid of both videos:     It is because the space traveller had two different rest frames (as per Don Lincoln's video).   However, an acceleration was necessary for that happen (as per Sabines's video).   The exact details of the acceleration aren't important only that it ultimately results in the traveller moving in the opposite direction as observed by the twin who remained on planet Earth.
    It is then a bit arbitrary to say that the acceleration did or did not cause it.   The acceleration had to be there but it only had to meet one condition (a reversal of the travelling twins velocity).  Provided that condition is met, then the acceleration is arbitrary and does not change the difference in elapsed time that the twins would experience.  Meanwhile, the distance between between Earth and Andromeda, L, along with the speed of travel of the travelling twin (V which is stated as 0.99 c in Don Lincoln's video) is very important.   Change L or V and the time difference experienced by the two twins does change.

Best Wishes.

[Eternal Student]

Hi again.

  Look, there's always a problem trying to summarise things,  inevitably you leave something out.   The difficulty here is that acceleration and speed are connected.

  The nature of the acceleration does matter a little because it could prevent the assumption of a constant speed of travel being reasonable.    The travelling twin is assumed to be moving at 0.99c almost all the time.  They change their velocity from + 0.99c to -0.99c  as far as the twin on earth is concerned and this must happen over a short amount of time,  if it happened over a long period of time then their speed wasn't 0.99c for some of the journey.

   Anyway, I could go on finding minor details and issues but it may not help.  Overall, in the simple treatment of the twins paradox, the assumption is that the acceleration is so rapid that it can be ignored, it happens in approximately 0 seconds.

Best Wishes.

[Dimensional (OP)]

Hi again.

  Look, there's always a problem trying to summarise things,  inevitably you leave something out.   The difficulty here is that acceleration and speed are connected.

  The nature of the acceleration does matter a little because it could prevent the assumption of a constant speed of travel being reasonable.    The travelling twin is assumed to be moving at 0.99c almost all the time.  They change their velocity from + 0.99c to -0.99c  as far as the twin on earth is concerned and this must happen over a short amount of time,  if it happened over a long period of time then their speed wasn't 0.99c for some of the journey.

   Anyway, I could go on finding minor details and issues but it may not help.  Overall, in the simple treatment of the twins paradox, the assumption is that the acceleration is so rapid that it can be ignored, it happens in approximately 0 seconds.

Best Wishes.

Please excuse me if I am wrong, but this would seem to suggest that the minimum that is necessary for the classical twin paradox scenario to work (outside of any wormhole or crazy geometric possibilities for the twin to decrease in age relative to the other) is that acceleration is necessary.   

But Don says at 4:25 (not 14:25, sorry) that the acceleration (in the typical scenario or the twin paradox, not the "frame changing example") is not the reason for the dilation.  While, Sabine gives an example using only acceleration.

I still can't help but see a distinct contradiction between Sabine and Don.   

[Dimensional (OP)]

 

Yes, I meant to say that the video from Fermilab says that no acceleration is needed.

Then it's wrong! If there is no acceleration between the twins, there is no relative speed, therefore no time discrepancy. If they were always moving relative to one another, they aren't twins because there never was any synchronicity.

Yes accept that it haunts me that at 4:25 (not 14:25) he also says that many physicists who don't work with relativity a lot believe that time dilation occurs during the acceleration periods (in the classical twin paradox thought experiment).

And when I think about it, in his example in which he claims there is no acceleration, there is a way to look at it where there really isn't any change in direction either.  None of the travelers change direction or accelerate, yet the result is the same as if the twin were thrown into the returning ship.

[alancalverd]

Thrown - acceleration. Change of velocity vector from one ship to another - acceleration. Acceleration is nothing more or less than a change of velocity: speed, direction or both.

If both clocks have the same velocity, there is no discrepancy between them (observed fact!).

[Dimensional (OP)]

Thrown - acceleration. Change of velocity vector from one ship to another - acceleration. Acceleration is nothing more or less than a change of velocity: speed, direction or both.

If both clocks have the same velocity, there is no discrepancy between them (observed fact!).

Then this would mean that acceleration played a role in the time dilation.  Do you know any math formulas to see how acceleration and time dilation are related?  A possible problem might be that acceleration is infinite using,   
a = (vf - vi)/t   

[alancalverd]

Time dilation requires (a) that two clocks were synchronised at some point and (b) that they are now moving relative to one another. Therefore one must have accelerated. Hence no paradox - you do something to a clock, and something changes. If you do nothing, they have no relative velocity and nothing changes.  If they were "always" moving relative to one another, you had no means of synchronising them.

I had the odd experience of hearing Hermann Bondi explaining this to Shirley Williams on a broken-down train one morning when both were famous and trains usually worked.

[Eternal Student]

Hi.

     I'm concerned you might be adding a bit more confusion @Alanacalverd but there's every chance I'm also doing the same.
    When  @alancalverd    says something like "there is no relative velocity",   he means the velocity of one thing relative to the other thing was numerically 0.     Don't assume there was no relative velocity in any other sense,  there was one, there always is a velocity of one thing relative to another thing, it just is numerically 0 sometimes.

- - - - - - - - -

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

      Yes.   It's in most textbooks.   Did you want to see it here?

Δt =   time elapsed for the twin remaining on earth  =   

[Eqn 1]

   
where  x_A = x co-ordinate position of Andromeda   and we have chosen the axis so that x=0 is where Earth is located;    v(t) = velocity of the traveller at time t  and we can assume  the position x(t) of the traveller is an invertible function of time so that we can express the time t as a function of the position,   t = t(x) because then v(t) = v(t(x)) can also be expressed as a function of x and then we can perform the integral appropriately.   (If the traveller wasn't always travelling away from earth but spent some time moving to/from it  like taking 2 steps forward with 1 step backward then we can't  invert the function x(t)  to find t(x)  but we can still perform the calculation - we would just break the integral into a sum of smaller sections where the traveller was always travelling in one direction on each section.   This is needlessly complicated, so in most simple treatments of the problem the traveller only ever moves in 1 direction along each leg of the journey and we only ever need to split it into two pieces, the outward journey and the return journey back to earth).

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

Δτ =   

[Eqn 2]

   That formula is based on using the proper time interval, dτ,  which will be the identical to the time elapsed on a clock that was moving with the traveller.

   Now, when we assume the speed v(t) was constant,  then the gamma factor γ(t) is also constant and can be taken outside the integral as a constant.    Ultimately we then have     Δτ  =   Δt / γ    as usual.

    If you want to know the difference between the elapsed times, then just subtract them.   Also, we have deliberately only looked at the outward journey.   The calculation for the return journey is much the same and indeed there is an obvious symmetry.   We would just add the times for the two sections together.  I'm sure you know that adding two things that are the same is equivalent to multiplying by 2.   So you should multiply  both  Δt    and   Δτ   by 2  if you want to consider the total elapsed time for the combined outward + return journey.

   Now the point that Don Lincoln was trying to communicate is that only  v(t) = velocity of the traveller   and  x_A = distance to Andromeda      appeared in the integrals and therefore influence the elapsed times, the acceleration does not directly appear.   However, it's a superficial comment.    v(t) is related to the acceleration, a(t) by the following:
  v(t) = v₀ +      

[Eqn 3]

    So the acceleration is there, it's just hidden in the assumptions about what v(t) is like.  (In the simplest situation the velocity is always kept constant).

   So we would just substitute  v(t) with that expression [Eqn 3] in the two formulae earlier [Eqn 1] and [Eqn 2].   It's obviously a bit messy but still useable.   

   For the purpose of getting the basic principles of  physics across, it's easier to assume a(t) is constant and especially easy if a(t) = 0 everywhere   (i.e.  to assume the traveller maintains constant speed except for a negligible and completely ignorable amount of time at Andromeda where they turn around and possibly at planet Earth if you want them to start and stop there.   So that is what is generally done.  However, if you must have a varying acceleration, a = a(t), then you can.

     Look, I don't know how to phrase this carefully:   Don Lincoln was just saying that acceleration doesn't appear in the usual formula for the twin paradox, which is true.  However, this is only because a simple pattern of movement is usually assumed, in particular the speed is constant everywhere except where the travelling twin is rapidly turned around at Andromeda, which is such a rapid turn around that it can be completely ignored.   He was also boosting or reinforcing the claim that general relativity was never required, which is also true.   
    You ( @Dimensional ) are just taking some bits of the video too literally and out of context.   Don Lincoln is not without fault here - he has promoted or up-sold his video with some short and snappy sounding phrases and it was too easy for a viewer to walk away with the wrong ideas.
    Take the phrasing at 4:25 which you point to in the broader context it was intended:   The time dilation isn't JUST happening at the time or place where the traveller experienced some acceleration (made the turn around Andromeda), it was happening somewhere or all the way through the journey.  This is why the things that are important and will affect the total time elapsed are  (i) the total distance travelled  and  (ii) the speed of travel during the whole journey.     The dilation happens everywhere along the journey.  Take a moment to let that sink in before you read the next paragraph.....
   Now, the time dilation was happening everywhere along the journey, however, there is still some discrepancy, some reason to explain why it wasn't happening exactly the same way to the twin who remained on Earth.   Recall that this is the fundamental issue in the twin paradox,  why can't we consider the situation as being perfectly symmetric (the travelling twin stayed still while the earth twin appeared to move away and come back).  Indeed, if you really do take the time and effort to study the paradox situation, then you will see that the travelling twin really does think that the clocks on planet Earth are ticking more slowly.  If nothing else happened then the travelling twin should have found that LESS time had elapsed for his Earth twin and not the MORE time which we know is the correct result.   There is a "discrepancy" of some sort.  It is the discrepancy in the elapsed times for the two twins which is explained by the acceleration that only one of them experienced around Andromeda.  When the travelling twin was abruptly shifted into a new rest frame, they "jumped tracks" or "skipped over some events for the earth twin".   Some moments of time that the twin on Earth would have experienced were skipped over.   Some events (which is a formal mathematical term, the nearest ordinary English language idea is to say those "moments of time at that place on earth") were in the travelling twins future (in the old rest frame) but they abruptly changed to being events that were in their past (in the new rest frame), they were never in their present or "now" , they were skipped over entirely.    Where you don't assume an instantaneous acceleration but just a very rapid one, then those events aren't completely skipped they are just very tightly compressed, they all occur over a very small time interval for the travelling twin.   
    I know that the idea of skipping over or missing out some events is difficult to get your head around.    LATE EDITING:  I can show you the maths but I reckon this post is long enough already.

    So I would say that Don Lincoln is mis-representing what many physicists think.  Those who have spent some time studying the problem know that "the discrepancy" in elapsed time happens because of the acceleration at Andromeda but they are well aware that it's not as if the time dilation effect was only happening during the acceleration.   It is impossible to say exactly "where" it was happening - some time dilation effect was happening everywhere along the journey but it wasn't a discrepancy in the right way (it would make the earth twin experience more time than the traveller), the major discrepancy (which does make the change in the right way) occurs at Andromeda where the traveller jumps rest frames with the result that some events the earth twin experiences are abruptly shifted from the travellers future into their past (skipped over).   That "discrepancy" is obviously due to the abrupt shift in rest frames, which is a consequence of an abrupt acceleration (no matter how hard Don Lincoln protests about it).   So, what I'm trying to say as politely as possible is that Don Lincoln has over-emphasised or over-sold something.  Many of the statements you have highlighted are just wrong IF you insist on taking them on their own, out of the general context, or as being more than just an impression or summary using English that is accessible to the general public.

Best Wishes.

[Dimensional (OP)]

Time dilation requires (a) that two clocks were synchronised at some point and (b) that they are now moving relative to one another. Therefore one must have accelerated. Hence no paradox - you do something to a clock, and something changes. If you do nothing, they have no relative velocity and nothing changes.  If they were "always" moving relative to one another, you had no means of synchronising them.

I had the odd experience of hearing Hermann Bondi explaining this to Shirley Williams on a broken-down train one morning when both were famous and trains usually worked.

Thanks, I think I get it now.

[Dimensional (OP)]

Hi.

     I'm concerned you might be adding a bit more confusion @Alanacalverd but there's every chance I'm also doing the same.
    When  @alancalverd    says something like "there is no relative velocity",   he means the velocity of one thing relative to the other thing was numerically 0.     Don't assume there was no relative velocity in any other sense,  there was one, there always is a velocity of one thing relative to another thing, it just is numerically 0 sometimes.

- - - - - - - - -

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

      Yes.   It's in most textbooks.   Did you want to see it here?

Δt =   time elapsed for the twin remaining on earth  =   

[Eqn 1]

   
where  x_A = x co-ordinate position of Andromeda   and we have chosen the axis so that x=0 is where Earth is located;    v(t) = velocity of the traveller at time t  and we can assume  the position x(t) of the traveller is an invertible function of time so that we can express the time t as a function of the position,   t = t(x) because then v(t) = v(t(x)) can also be expressed as a function of x and then we can perform the integral appropriately.   (If the traveller wasn't always travelling away from earth but spent some time moving to/from it  like taking 2 steps forward with 1 step backward then we can't  invert the function x(t)  to find t(x)  but we can still perform the calculation - we would just break the integral into a sum of smaller sections where the traveller was always travelling in one direction on each section.   This is needlessly complicated, so in most simple treatments of the problem the traveller only ever moves in 1 direction along each leg of the journey and we only ever need to split it into two pieces, the outward journey and the return journey back to earth).

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

Δτ =   

[Eqn 2]

   That formula is based on using the proper time interval, dτ,  which will be the identical to the time elapsed on a clock that was moving with the traveller.

   Now, when we assume the speed v(t) was constant,  then the gamma factor γ(t) is also constant and can be taken outside the integral as a constant.    Ultimately we then have     Δτ  =   Δt / γ    as usual.

    If you want to know the difference between the elapsed times, then just subtract them.   Also, we have deliberately only looked at the outward journey.   The calculation for the return journey is much the same and indeed there is an obvious symmetry.   We would just add the times for the two sections together.  I'm sure you know that adding two things that are the same is equivalent to multiplying by 2.   So you should multiply  both  Δt    and   Δτ   by 2  if you want to consider the total elapsed time for the combined outward + return journey.

   Now the point that Don Lincoln was trying to communicate is that only  v(t) = velocity of the traveller   and  x_A = distance to Andromeda      appeared in the integrals and therefore influence the elapsed times, the acceleration does not directly appear.   However, it's a superficial comment.    v(t) is related to the acceleration, a(t) by the following:
  v(t) = v₀ +      

[Eqn 3]

    So the acceleration is there, it's just hidden in the assumptions about what v(t) is like.  (In the simplest situation the velocity is always kept constant).

   So we would just substitute  v(t) with that expression [Eqn 3] in the two formulae earlier [Eqn 1] and [Eqn 2].   It's obviously a bit messy but still useable.   

   For the purpose of getting the basic principles of  physics across, it's easier to assume a(t) is constant and especially easy if a(t) = 0 everywhere   (i.e.  to assume the traveller maintains constant speed except for a negligible and completely ignorable amount of time at Andromeda where they turn around and possibly at planet Earth if you want them to start and stop there.   So that is what is generally done.  However, if you must have a varying acceleration, a = a(t), then you can.

     Look, I don't know how to phrase this carefully:   Don Lincoln was just saying that acceleration doesn't appear in the usual formula for the twin paradox, which is true.  However, this is only because a simple pattern of movement is usually assumed, in particular the speed is constant everywhere except where the travelling twin is rapidly turned around at Andromeda, which is such a rapid turn around that it can be completely ignored.   He was also boosting or reinforcing the claim that general relativity was never required, which is also true.   
    You ( @Dimensional ) are just taking some bits of the video too literally and out of context.   Don Lincoln is not without fault here - he has promoted or up-sold his video with some short and snappy sounding phrases and it was too easy for a viewer to walk away with the wrong ideas.
    Take the phrasing at 4:25 which you point to in the broader context it was intended:   The time dilation isn't JUST happening at the time or place where the traveller experienced some acceleration (made the turn around Andromeda), it was happening somewhere or all the way through the journey.  This is why the things that are important and will affect the total time elapsed are  (i) the total distance travelled  and  (ii) the speed of travel during the whole journey.     The dilation happens everywhere along the journey.  Take a moment to let that sink in before you read the next paragraph.....
   Now, the time dilation was happening everywhere along the journey, however, there is still some discrepancy, some reason to explain why it wasn't happening exactly the same way to the twin who remained on Earth.   Recall that this is the fundamental issue in the twin paradox,  why can't we consider the situation as being perfectly symmetric (the travelling twin stayed still while the earth twin appeared to move away and come back).  Indeed, if you really do take the time and effort to study the paradox situation, then you will see that the travelling twin really does think that the clocks on planet Earth are ticking more slowly.  If nothing else happened then the travelling twin should have found that LESS time had elapsed for his Earth twin and not the MORE time which we know is the correct result.   There is a "discrepancy" of some sort.  It is the discrepancy in the elapsed times for the two twins which is explained by the acceleration that only one of them experienced around Andromeda.  When the travelling twin was abruptly shifted into a new rest frame, they "jumped tracks" or "skipped over some events for the earth twin".   Some moments of time that the twin on Earth would have experienced were skipped over.   Some events (which is a formal mathematical term, the nearest ordinary English language idea is to say those "moments of time at that place on earth") were in the travelling twins future (in the old rest frame) but they abruptly changed to being events that were in their past (in the new rest frame), they were never in their present or "now" , they were skipped over entirely.    Where you don't assume an instantaneous acceleration but just a very rapid one, then those events aren't completely skipped they are just very tightly compressed, they all occur over a very small time interval for the travelling twin.   
    I know that the idea of skipping over or missing out some events is difficult to get your head around.    LATE EDITING:  I can show you the maths but I reckon this post is long enough already.

    So I would say that Don Lincoln is mis-representing what many physicists think.  Those who have spent some time studying the problem know that "the discrepancy" in elapsed time happens because of the acceleration at Andromeda but they are well aware that it's not as if the time dilation effect was only happening during the acceleration.   It is impossible to say exactly "where" it was happening - some time dilation effect was happening everywhere along the journey but it wasn't a discrepancy in the right way (it would make the earth twin experience more time than the traveller), the major discrepancy (which does make the change in the right way) occurs at Andromeda where the traveller jumps rest frames with the result that some events the earth twin experiences are abruptly shifted from the travellers future into their past (skipped over).   That "discrepancy" is obviously due to the abrupt shift in rest frames, which is a consequence of an abrupt acceleration (no matter how hard Don Lincoln protests about it).   So, what I'm trying to say as politely as possible is that Don Lincoln has over-emphasised or over-sold something.  Many of the statements you have highlighted are just wrong IF you insist on taking them on their own, out of the general context, or as being more than just an impression or summary using English that is accessible to the general public.

Best Wishes.

Thank you very much, I think it makes sense now. 

[alancalverd]

Don't assume there was no relative velocity in any other sense,  there was one, there always is a velocity of one thing relative to another thing, it just is numerically 0 sometimes.

And they call me pedantic!   

I didn't watch the video, but life is too short to attend lectures by physicists who ignore Newton's basic principles.