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# These twins are hurting my head!

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For anyone who is interested in exploring this type of  "popular" physics  paradox then I can recommend the following by Prof Jim Al-Khalili , it's very easy to read and thoroughly enjoyable.

Jim

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1 hour ago, saac said:

I'd be delighted to hear from anyone who can explain without reference to the curvature of space time why gravitational fields affect time

Unfortunately, this is impossible as GR is about the curvature of space-time. You might get a feel for it by considering the equivalence principle and looking at a closed box falling uniformly in a gravitational field and one undergoing a uniform acceleration. Fire a laser pulse across the box and look at the path and time off impact of the beam as observed in the box and how one out side the box would describe it. The solution is set for homework Jim! (Hint this is how the great man came up with gravity/mass bending light.)

Regards Andrew.

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1 hour ago, andrew s said:

Unfortunately, this is impossible as GR is about the curvature of space-time.

Andrew, that's why I think exploring time dilation by SR is more accessible. The problem with explaining GR as you said, is that as soon as you move beyond the qualitative approach (explored via the equivalence principle), you run into the impenetrable maths (for me anyway) of curved geometry   I may be wrong here, I think I read of it in the book I mentioned above, but didn't Einstein almost in a moment of serendipity come to see how Riemannian's equations on curved geometry, hitherto an obscure branch of maths, could be used to develop his GR?  I was lucky enough to attend an excellent in-service day course  just before Christmas. We had a presentation on GR by a lecturer from University of St Andrews;  I signed up straight away thinking here's my chance to get to grips with GR.  Oh, boy, thoroughly captivating, I sat there beguiled by the easy manner in which Einstein's great work unfolded before me.  Well it didn't last long - about 10 minutes in! As soon as the equations on curved geometry started flying around I knew I was well out of my depth    I can't complain too much, Einstein said himself that "everything should be made simple as possible but not simpler".  Anyway, thanks for the homework, I'm taking baby steps with this stuff

Jim

Edited by saac

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10 hours ago, saac said:

Andrew, that's why I think exploring time dilation by SR is more accessible. The problem with explaining GR as you said, is that as soon as you move beyond the qualitative approach (explored via the equivalence principle), you run into the impenetrable maths (for me anyway) of curved geometry   I may be wrong here, I think I read of it in the book I mentioned above, but didn't Einstein almost in a moment of serendipity come to see how Riemannian's equations on curved geometry, hitherto an obscure branch of maths, could be used to develop his GR?  I was lucky enough to attend an excellent in-service day course  just before Christmas. We had a presentation on GR by a lecturer from University of St Andrews;  I signed up straight away thinking here's my chance to get to grips with GR.  Oh, boy, thoroughly captivating, I sat there beguiled by the easy manner in which Einstein's great work unfolded before me.  Well it didn't last long - about 10 minutes in! As soon as the equations on curved geometry started flying around I knew I was well out of my depth    I can't complain too much, Einstein said himself that "everything should be made simple as possible but not simpler".  Anyway, thanks for the homework, I'm taking baby steps with this stuff

Jim

That's a great final quotation from EInstein!

Olly

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10 hours ago, saac said:

Andrew, that's why I think exploring time dilation by SR is more accessible.

Indeed it is and the effects of time dilation in SR and GR are quite separate. You don't need to invoke GR to explain the twin paradox as it can be set up as a thought experiment using identical clocks rather than people and then no deviations the assumption of SR are required (specifically keeping within inertial frames)  I will post this later as I need to draw some diagrams.

For now here is a much simpler "paradox" . Consider two twins (identical clocks in this case) moving away from each other  a speed v. Now from the point of view of an inertial observer moving with clock 1 he/she observes that clock 2 is running slow. But, an inertial observer moving with clock 2 observes that clock 1 is running slow. How can this be?

The answer is that SR requires the laws of physics to be the same in all inertial frame but not all measurements. Only special measurements call invariants need be the same.

Take kinetic energy, if say two billiard balls are moving towards each other at velocity v then with respect to the first ball it has zero kinetic energy and the ball coming towards it has Ke = 1/2 m v^2 and according to ball 2 it has zero kinetic energy and the first ball has Ke = 1/2 m v^2. The law is the same the value is different.

(For nerds - In this case it is the energy momentum 4 vector that is invariant.)

Regards Andrew

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So here is the twin paradox with 3 identical clocks. From the perspective of clock 1 clock 2 is moving in the x direction at 3/5 c while clock 3 is moving in the x direction at -3/5 c.

Looking at the left hand diagram (which is drawn in the frame of clock 1) clocks 1 & 2 are synchronised at t=0 x = 0 as they pass . Clock 2 then meets clock 3 which is synchronised with it.

Clock 3 then meets clock 1 and the times compared. I have labeled the  paths of the clocks (their world lines)  with the "Proper Time" recorded by them as they go along. The proper time of clock 1 is the time it records etc.

I have then redrawn the same events in the right hand diagram but now from the perspective of Clock 2.  You will see that the clocks record the same proper times. However, I could just be cheating so how do you get the proper times. Well it is Pythagoras with a twist. Proper time is just = ( t^2 - x^2)^1/2 (where x is in light years) So to get the 4 at the meeting of clock 2 and 3 you have (5x5 - 3x3)^/12 = 4 and to get the 10 for clock 1 in the right hand diagram you have (12.5x12.5 - 7.5x7.5) ^1/2 = 10.

So the twin paradox is resolved because the twins follow different world lines.

The invariant in this case is the proper time.

Hope this helps

Regards Andrew

Edited by andrew s
typo

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andrew s I don't know if it helps, but I'm very impressed nonetheless!

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Andrew I like your resolution of the twin paradox, I may borrow that     You do realise of course that you've gone an opened up a whole new discussion on world lines and the possibility of time travel.

Jim

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54 minutes ago, saac said:

Andrew I like your resolution of the twin paradox, I may borrow that

No problem but just to be fair it is not my solution it has been about for a long time. The trick (again not mine) is to pick 3/5 c so the sums work out simply. You should (more home work) check I did my velocity addition correctly in coming up with -15/17 c in the right hand diagram for clock 3.

57 minutes ago, saac said:

You do realise of course that you've gone an opened up a whole new discussion on world lines and the possibility of time travel.

Unfortunately, we are constrained within SR to plod on into our forward light come at a rate of one s/s in our own frame so no going back to the delights of youth!

Regards Andrew

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5 minutes ago, andrew s said:

so no going back to the delights of youth!

!  Dont want to do that, nearly didnt survive the first time round ! No, I want to see 100y hence and then 1000y

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4 hours ago, andrew s said:

No problem but just to be fair it is not my solution it has been about for a long time. The trick (again not mine) is to pick 3/5 c so the sums work out simply. You should (more home work) check I did my velocity addition correctly in coming up with -15/17 c in the right hand diagram for clock 3.

Unfortunately, we are constrained within SR to plod on into our forward light come at a rate of one s/s in our own frame so no going back to the delights of youth!

Regards Andrew

Thanks for the heads-up Andrew, I'll go through the numbers to make sure I properly understand it. So the doors to our past are firmly closed - we must abide in both the here and now of our world lines. Just as well, I think we would only cause havoc by revisiting our past, either by accident or intent.  Like Silver Astro, I'd love to see how things turn out in our future, discover what new boundaries we push back in our understanding, or park at a safe distance and watch the final dying light of our sun. I must admit, I do enjoy the time travel genre in science fiction, The Time Travellers Wife, Age of Adaline, Frequency and Predestination among my favourites.

Jim

Edited by saac

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Well after a couple of readings I think I am starting to get my head around the theory of it all-although it will probably take it while before I understand everything that you guys posted! (Not a complaint though, like I said I find it fascinating!)

@David Levi analogy comparing moving through space and time to moving in a north/West direction really helped-Cheers Dave!

Now I have a better understand, I think what was tripping me up before  was I didn't have a good enough understanding about the concept of 'space-time', I thought I did but obviously not! If anyone can throw anything at me to help better understand will be greatly recieved!

Thanks again for all the help.

Popeye.

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Popeye - there is no absolute reference for space or time. It all depends on the observer.  It is counter-intuitive, and goes against how we think it all is.  Not easy.  Not easy at all.

Doug.

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On ‎11‎/‎01‎/‎2017 at 11:31, popeye85 said:

...... the theory of time dilation and the so called 'twin paradox' and I am really struggling to get my head around it! Can someone please explain it to me in lay man's terms as it absolutely fascinates me and I'm desperate to know how it comes about.

Cheers

Popeye.

The speed of light being so fast in comparison to other objects that we encounter in everyday life leads to a great many counterintuitivisms when considering rapidly moving objects. In the case of the twins, no-one knows why they should age differently it's just that they do. It is an artefact of the way our universe is configured. Einstein predicted time dilation as a logical consequence of his SR postulates: the laws of physics are the same for all inertial observers (ie. not accelerating with respect to one another) and that the speed of light in a vacuum is constant regardless of how the source moves. To derive the same all you need are the two postulates, Pythagoras's Theorem and a carefully chosen thought experiment.

The 'Special' in SR is special as in 'special case'. The special case being no acceleration and no gravity. With this being said, it is only possible to treat the twins within the framework of SR if we don't ask about the accelerated part of the journey. This leads to a paradox (cf. Olly's post on page 1 of this thread) as to which twin should be the oldest as within the confines of SR it is impossible to differentiate between the two.

On ‎14‎/‎01‎/‎2017 at 19:31, popeye85 said:

@David Levi analogy comparing moving through space and time to moving in a north/West direction really helped-Cheers Dave!

Now I have a better understand, I think what was tripping me up before  was I didn't have a good enough understanding about the concept of 'space-time', I thought I did but obviously not! If anyone can throw anything at me to help better understand will be greatly recieved!

I agree, great analogy from David Levi.

Concentrate on the 'space-time interval' and it's invariant nature and you will soon be ready to understand SR in terms of Minkowski-space (a  geometric approach). Once you have  Minkowski-space internalized then much of GR will be accessible without the need for heavyweight differential geometry/ linear algebra.

Space-time Physics  by E.F.Taylor and the great J.A.Wheeler is always worth a look. Lots of words and lots to ponder, a thought provoking read to say the least.

On ‎12‎/‎01‎/‎2017 at 11:26, andrew s said:

So the twin paradox is resolved because the twins follow different world lines.

I'm not sure I follow. Do you contend that time dilation is a consequence of following different world lines?

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18 hours ago, Tiki said:

I'm not sure I follow. Do you contend that time dilation is a consequence of following different world lines?

Time dilation in SR is due to relative velocity and the constancy and finite seed of light as measured in any inertial frame of reference.

To have a velocity relative to something else you and the something else must follow different world lines. (All that means is you don't stay together all the time.)

The "time dilation" is the difference in the proper time their clocks show as calculated in any given inertial frame.

Sorry to be turgid but I am trying to make it precise.

Regards Andrew

Edited by andrew s
added to first sentence " and the constancy and finite seed of light as measured in any inertial frame of reference".

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This kind of makes it easy i think, well easier anyway.

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