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Physics, Astronomy & Cosmology / Re: Why does this twin paradox thought experiment fail for me?

« on: 19/01/2023 20:08:00 »

Thank you for your replies.  They seem to be unanimous that acceleration is the reason.  I tend to agree.

However, I am currently discussing the same topic on another website with many verified educated people.  They don't seem to think that acceleration is the reason. 

Here is the website in case you want to discuss this with them, https://www.physicsforums.com/forums/special-and-general-relativity.70/ , and the thread is called "Please help with another twin paradox situation (sorry)" my name is student34 there in case you want to know.

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Physics, Astronomy & Cosmology / Re: What is the exact cause of the time dilation of the twin?

« on: 13/01/2023 03:55:11 »

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. 

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Physics, Astronomy & Cosmology / Re: What is the exact cause of the time dilation of the twin?

« on: 13/01/2023 03:54:24 »

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.

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Physics, Astronomy & Cosmology / Re: Is this a paradox in general relativity?

« on: 24/05/2022 23:15:26 »

Hi.

Is your origin (0,0) at the very back of the back fin of the ship like in the video?

  Yes.  The origin is intended to be exactly where they placed it in the original video (although I only sketched it, I didn't get a ruler and compass).
   The planet based observer says the back of the fin is at x= 0 when t =0.        Spaceman says the back of the fin is at x'=0 when t' =0.   

- - - - - - - - - -
   Just to emphasize one issue,  although in my diagrams it looks like the x and x' co-ordinates of the rock collision event are both  +5,   they aren't actually exactly the same.   That's just that the diagram is only a sketch and I haven't placed all the gridlines exactly the same space apart etc.   I just want to dispel the notion that there was any reason why they had to agree on the spatial location of the event... there isn't.

    I've run the precise calculation with these figures  (they are roughly what was used in the video).
Set  the velocity of the rocket =  half the speed of light.     
Use units for measuring time and distance so that the speed of light, c = 1  in those units   (Just to be clear that's not going to be seconds and metres.  It's just conventional to set c = 1).
Set the rock collision event to co-ordinates   (x, t) = ( +5.00 , +1.00 )   as was shown in my diagram for the planetary observer.
This becomes  (x', t') =  ( +5.20  ,   -1.73) in the spacemans co-ordinate system.
So, with these figures,   the spaceman and planet based observer disagree on the both the location and time of the collision event.

Best Wishes.

Yeah this makes sense to me.  Thank a lot.