Relative speed and time dilation

[DaveC426913]

What you typed:

It's mostly linear algebra and a bit differential geometry

What I read:
"It's mostly ☠☠✂⚙ and a bit of ✂☠☠☠"

 

[dayalanand roy]

It's quite straightforward. The amount of time you experience is just the interval along your path through spacetime (because $\Delta s^2=c^2\Delta t^2$ in your rest frame) and, just like the distance along your path through space, two paths starting in the same place and ending in the same place need not have the same "length". Any textbook on relativity will discuss this (Taylor and Wheeler's Spacetime Physics is a popular choice), although I'm not sure if they use the term "differential aging". Google can easily find the term in the literature, but I'm not certain how widely used it is.
Their movements are not symmetrical. One clock will have to fire a rocket to turn round. The other doesn't fire a rocket.

Thanks.
But it will have to fire a rocket to move away too. So its receeding movement too is not symmetrical.
I am trying to learn the very physical basis of time and relativity( though i am a student of biology).
Kindly see the replies above. It is confusing to me for why the two movements are symmetrical and why not symmetrical?

 

[Janus]

Thanks.
But it will have to fire a rocket to move away too. So its receeding movement too is not symmetrical.
I am trying to learn the very physical basis of time and relativity( though i am a student of biology).
Kindly see the replies above. It is confusing to me for why the two movements are symmetrical and why not symmetrical?

In most twin paradox scenarios we assume that the acceleration magnitude is extremely high for a very short duration (a near instantaneous change in velocity). This means we also assume that our accelerating twin moves almost not at all during the acceleration period. In other words, after our rocket has reached its top velocity, it is still essentially right next to our Stay at home twin. But when he does the acceleration during the turn around, he is far removed from the Stay at home twin. Going back to my example with our observers A and B, we can assume that they are walking in the same direction before B changes direction, and are essentially sharing the same space. At the moment immediately after B changes direction from that that A is walking, this doesn't change, so he doesn't see A as shifting in time. But once he gets some distance between himself and A, then any change in direction he makes has a effect on A's relative position from his perspective.
Grab a couple of paper plates, put them on the floor a few feet apart. Stand on one while facing the other. Without moving from your spot, turn 180 degrees. The plate that was in front of you goes from being in front to being behind you. The plate you are standing doesn't change its position relative to you at all.

This is similar to what is happening to clocks for someone undergoing an acceleration. Nothing special happens to clocks that are co-located with you, but clock removed from you in the direction of the acceleration run fast, while clocks in the opposite direction run slow.

Now in a real situation you can't change speeds instantaneously, but do so over some time while you travel some distance. In this case, as our rocket accelerates away from the stay at home clock, it would note that not only was the stay at home clock running slow due to to the increasing relative speed but also because of the increasing distance from the rocket in the direction opposite that of the acceleration.
The rocket also sees a compound effect in the "stay at home" clock during the turn around acceleration. But all this really does is complicate the calculations to find the final results, which is why we tend to assume conditions where we can ignore these additional complications.

 

[Ilythiiri]

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html
This one was good enough for me, when i got a similar question as OP(time dilation/lightspeed but i was searching in BH context). Simple and comprehensive - prose, examples and pictures that is (:

Quotes to prove relevance:
"...Terence sits at home on Earth. Stella flies off in a spaceship at nearly the speed of light, turns around after a while, thrusters blazing, and returns. (So Terence is the terrestrial sort; Stella sets her sights on the stars.)
When our heroes meet again, what do they find? Did time slow down for Stella, making her years younger than her home-bound brother? Or can Stella declare that the Earth did the travelling, so Terence is the younger? ..."

"... Why so many different analyses? Are the relativists just trying to bamboozle their opponents, like the defence attorney who just has to stir up doubt about the plaintiff's case without giving his own theory of events? Not at all; the physical theory should and does tell a single coherent story here.
Relativity pays the price of permissiveness. It says to us, "Pick whichever frame you like to describe your results. They're all equivalent." No wonder that one analysis ends up looking like three or four. ..."

 

[dayalanand roy]

In most twin paradox scenarios we assume that the acceleration magnitude is extremely high for a very short duration (a near instantaneous change in velocity). This means we also assume that our accelerating twin moves almost not at all during the acceleration period. In other words, after our rocket has reached its top velocity, it is still essentially right next to our Stay at home twin. But when he does the acceleration during the turn around, he is far removed from the Stay at home twin. Going back to my example with our observers A and B, we can assume that they are walking in the same direction before B changes direction, and are essentially sharing the same space. At the moment immediately after B changes direction from that that A is walking, this doesn't change, so he doesn't see A as shifting in time. But once he gets some distance between himself and A, then any change in direction he makes has a effect on A's relative position from his perspective.
Grab a couple of paper plates, put them on the floor a few feet apart. Stand on one while facing the other. Without moving from your spot, turn 180 degrees. The plate that was in front of you goes from being in front to being behind you. The plate you are standing doesn't change its position relative to you at all.

This is similar to what is happening to clocks for someone undergoing an acceleration. Nothing special happens to clocks that are co-located with you, but clock removed from you in the direction of the acceleration run fast, while clocks in the opposite direction run slow.

Now in a real situation you can't change speeds instantaneously, but do so over some time while you travel some distance. In this case, as our rocket accelerates away from the stay at home clock, it would note that not only was the stay at home clock running slow due to to the increasing relative speed but also because of the increasing distance from the rocket in the direction opposite that of the acceleration.
The rocket also sees a compound effect in the "stay at home" clock during the turn around acceleration. But all this really does is complicate the calculations to find the final results, which is why we tend to assume conditions where we can ignore these additional complications.

Thanks a lot sir.
It is really educating and i am grateful that you have taken this much pain to give such a long response.
Great!