How Does the Twin Paradox Challenge Our Understanding of Time Dilation?

[jtbell]

it seems the clock which is ahead is ahead in time in contradiction with your explanation.

Draw a diagram, with A at the left side and A2 at the right. If B moves to the right in A's and A2's frame, then A and A2 move to the left in B's frame, with A "ahead" of A2.

A2's clock reads a later time than A's clock does, in B's frame, so A2's clock is "ahead."

 

[ratn_kumbh]

Time dilation is caused due to the relative motion(between two inertial frames) isn't it?.In this case One of the frame is accelerating.So the equation for time dilation is not apllicale to both frames.or if there is any equation of time dilation between one inertial frame and one accelerating frame,it is applicable to both in the same manner.
So when traveller turns around and accelerate,whatever equation applicable for traveller,is true for the men at rest also.

Finally both will give the same result of same ages for both twins.Am i correct?

Observation for such case should be done from an inertial frame. Since B returns (thus accelerats & it is only felt by B). its frame reference can't be selected. So u can choose A's reference Or any other inertial frame. U'll find that twin who returns, is younger.

 

[newTonn]

Observation for such case should be done from an inertial frame. Since B returns (thus accelerats & it is only felt by B). its frame reference can't be selected. So u can choose A's reference Or any other inertial frame. U'll find that twin who returns, is younger.

Time dilation equation is applicable from one frame to another frame which is moving with a uniform velocity with respect to the first frame.So it is not possible from any frame (at rest or uniform motion),to apply time dilation equation for calculation of time in an accelerating frame(here B).
So somebody has to find an equation for that.The equation contains v(not a),which is the relative velocity(not acceleration)

 

[Janus]

Time dilation equation is applicable from one frame to another frame which is moving with a uniform velocity with respect to the first frame.So it is not possible from any frame (at rest or uniform motion),to apply time dilation equation for calculation of time in an accelerating frame(here B).
So somebody has to find an equation for that.The equation contains v(not a),which is the relative velocity(not acceleration)

They already exist:
t = \frac{c}{a} sinh \left( \frac{aT}{c} \right)

T = \frac{c}{a} sinh^{-1} \left( \frac{at}{c} \right)

They are derived from the standard Relativity equations.

 

[newTonn]

They already exist:
t = \frac{c}{a} sinh \left( \frac{aT}{c} \right)

T = \frac{c}{a} sinh^{-1} \left( \frac{at}{c} \right)

They are derived from the standard Relativity equations.

Could you please give me an explanation of notations used and if possible,a link to derivations.
And please explain me how somebody knows,who is accelerating?

 

[JesseM]

Could you please give me an explanation of notations used and if possible,a link to derivations.

The notation is explained on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html page, not sure where you'd find a derivation.

And please explain me how somebody knows,who is accelerating?

In flat spacetime (no sources of gravity), the accelerating individual will feel G-forces in the direction of acceleration, the inertial individual will feel weightless.