Understanding the Twin Paradox: Time and Length Contractions Explained

[yuiop]

as i said before:
time elapsed for Stella on outward journey + time elapsed for alf on inward journey=total time elapsed for moving twin=acceleration is not necessary to explain the paradox.

All that is necessary is the path through spacetime. The twin that has taken the shortest path through spacetime when they meet again will age the most.

 

[granpa]

All that is necessary is the path through spacetime. The twin that has taken the shortest path through spacetime when they meet again will age the most.

shortest path through spacetime=shortest path through time?

if all objects always move through spacetime at c then all paths of all objects through spacetime are the same length.

 

[yuiop]

shortest path through spacetime=shortest path through time?

if all objects always move through spacetime at c then all paths of all objects through spacetime are the same length.

shortest path through spacetime = longest proper time

this is sort of opposite to the normal intuition of the shortest distance between two points in 3-space being a straight line.

the 4-velocity may be interpreted as c for all objects but they can take different paths between events and the proper time recorded by the object taking the longest path will be the shortest time interval.

 

[yuiop]

THe attached image shows the paths taken through spacetime according to Terra, Stella and Alf. The red path taken by Terra is always the shortest according to any inertial observer so all observers agree that Terra ages the most.

 

[granpa]

ok. but what about the usual version of the twin paradox. don't both twins consider the other to be the one that is moving? what do their paths through spacetime look like?

just to be clear, path through spacetime is not the same as 'interval', right?

 

[matheinste]

Hello granpa.

Quote!

----ok. but what about the usual version of the twin paradox. don't both twins consider the other to be the one that is moving?-----

Yes but one is undergoes acceleration, the traveling one

Matheinste.

 

[granpa]

ok. i didnt understand the image the first time but i get it now. so all you are saying is that the one that travels in a straight line without accelerating always travels the shortest route and is always least time dilated. i agree.

but it is a fact that no individual undergoes acceleration during this thought experiment. every individual is already time dilated at the start and remains at that time dilation throughout. the full solution can be calculated without ever referring to acceleration at all. all you need to know is the velocities of the rockets and the distances involved.

when people refer to path through spacetime i always think of Minkowski space. that's quite different.

 

[yuiop]

ok. but what about the usual version of the twin paradox. don't both twins consider the other to be the one that is moving? what do their paths through spacetime look like?

just to be clear, path through spacetime is not the same as 'interval', right?

In the usual version of the twin's paradox, where one twin goes on a long fast journey and returns to the the static twin, their paths will look like the red and green paths in the left most diagram of the picture I posted showing Terra's point of view. It is not possible to depict the point of view according to the traveling twin in a single space time drawing because of the change of direction and it that change of direction that breaks the symmetry and the paradox. The traveling might consider himself to stationary for part of the journey but after he changed direction he would have felt acceleration and would have to conclude that he can not be stationary for all of the "journey".

The symmetry is also broken if each sends signals at yearly intervals on their respective birthdays. When the traveling twin changes direction he sees an immediate increase in the frequency of birthday signals coming from his stationary sibling ,while the stationary twin only sees an increase in birthday frequency of his traveling sibling later on in the experiment. When they count up the number of birthday siganls they each received during the entire experiment it will agree with their difference in ages. As soon as one twin changes direction the symmetry is broken. Before the change in direction, no one can say with certainty which twin is ageing faster. I hope that sort of makes sense :P

 

[yuiop]

ok. i didnt understand the image the first time but i get it now. so all you are saying is that the one that travels in a straight line without accelerating always travels the shortest route and is always least time dilated. i agree.

but it is a fact that no individual undergoes acceleration during this thought experiment. every individual is already time dilated at the start and remains at that time dilation throughout. the full solution can be calculated without ever referring to acceleration at all. all you need to know is the velocities of the rockets and the distances involved.

when people refer to path through spacetime i always think of Minkowski space. that's quite different.

Although I did not put labels on the axes, the paths I drew were plotted on a graph with distance on the hororizontal aaxis and time on the vertical axis which is exactly how Minkowski spacetime diagrams are plotted. There is no difference.

 

[granpa]

i meant that the interval is calculated differently. as i am sure you know.

 

[yuiop]

... but it is a fact that no individual undergoes acceleration during this thought experiment. every individual is already time dilated at the start and remains at that time dilation throughout. the full solution can be calculated without ever referring to acceleration at all. all you need to know is the velocities of the rockets and the distances involved.

In your slightly modified version with 3 observers and no acceleration it is impossible to prove whether Terra, Stella or Alf aged the least.

 

[granpa]

In your slightly modified version with 3 observers and no acceleration it is impossible to prove whether Terra, Stella or Alf aged the least.

you may be right, but it is a fact that it produces exactly the same result as the usual version. the total transit time of stella + alf = the total transit time of the moving twin.

 

[Garth]

you may be right, but it is a fact that it produces exactly the same result as the usual version. the total transit time of stella + alf = the total transit time of the moving twin.

So there is no paradox.

Two observers meet at one event and meet again at a later event.

The non-inertial observer experiences the smaller time elapse.

As I pointed out, Stella+Alf is a non-inertial frame of reference.

Garth

 

[granpa]

i know there is no paradox and the supposed paradox can be explained quite easily without reference to acceleration. all you need to know is the velocity of the rockets and the distances involved.