B What is the effect of the twin paradoxon?

[Ibix]

Is it not permissible, then, to conclude that two children born at the same time in different places will always be the same age at any given time, from the point of view of the frame of reference in which they rest, because their world lines do not cross?

Using that frame only, yes. Using other frames, no. This is the point of the twin paradox: the twins meet, separate, and meet again, and everyone agrees on their accumulated age difference between the meetings. In your example, almost everyone disagrees.

 

[Peter Strohmayer]

Using that frame only, yes. Using other frames, no.

Is not the age of the children (their proper time) an invariant?

 

[Ibix]

Is not the age of the children (their proper time) an invariant?

There age when? It's the "when" that is not invariant unless they meet.

 

[Sagittarius A-Star]

Is not the age of the children (their proper time) an invariant?

two children born at the same time in different places

And because they were born at different places, it is not invariant, if they were born at the "same time".

 

[Nugatory]

Is it not permissible, then, to conclude that two children born at the same time in different places will always be the same age at any given time, from the point of view of the frame of reference in which they rest, because their world lines do not cross?

It's neither more nor less permissible than concluding that one of them will always be older than the other when we calculate "ages" using coordinates in which they are both moving at a constant non-zero velocity - it all depends on our essentially arbitrary choice of coordinates.

When you inserted that additional "from the point of view" qualifier you redefined "same age" so that it is frame-dependent and no longer has any physical significance, and furthermore chose a redefinition that makes it true by definition: define "same age" to mean "has the same time coordinate" and then choose time coordinates that advance at the rate as proper time and of course they will age at the same rate.

 

[robphy]

Is not the age of the children (their proper time) an invariant?

The proper-time age of a child is the elapsed time on its wristwatch between two events A_1 and A_2 on that child's worldline (akin to an arc length along a specific curve between two points). For two specified events along that worldline, that elapsed proper-time is invariant.

To compare the ages of two children,
you have to specify events A_1 and A_2 on child-A's worldline
and events B_1 and B_2 on child-B's worldline.

For distinct spacelike-separated events A_1 and B_1 ,
not all observers will regard A_1 and B_1 as simultaneous (i.e. as "being at the same time")
.... and similarly for A_2 and B_2.

 

[PeroK]

When I first studied SR, I assumed that acceleration must be the ultimate cause of the differential ageing in twin paradox. Then I learned about Minkowski spacetime. Then someone on this forum posted the idea of replacing the turnaround with the communication of a clock reading to an inbound spacecraft. And it was clear that acceleration was only a physical constraint and a geometric red herring.

This process of continually updating one's knowledge of a subject is called learning. It's the opposite of religiously adhering to an established view in the face of evidence to the contrary.

 

[Dale]

born at the same time in different places

This is a frame variant concept.

from the point of view of the frame of reference in which they rest

In that frame it is true. Not in other frames. They do not need to use the frame in which they are at rest.

Do you accept the principle of relativity?

 

[Dale]

And it was clear that acceleration was only a physical constraint and a geometric red herring.

Yes. The acceleration breaks the symmetry, but it is not the only way to break the symmetry. Acceleration is neither necessary nor sufficient for different aging, so it cannot be the cause.

 

[Dale]

Is not the age of the children (their proper time) an invariant?

As I mentioned previously $d\tau$ is an invariant. $\tau=\int_a^b d\tau$ is invariant only if $a$ and $b$ are.

 

[Peter Strohmayer]

And because they were born at different places, it is not invariant, if they were born at the "same time".

When you inserted that additional "from the point of view" qualifier you redefined "same age" so that it is frame-dependent and no longer has any physical significance,

For distinct spacelike-separated events A1 and B1 ,
not all observers will regard A1 and B1 as simultaneous

This process of continually updating one's knowledge of a subject is called learning. It's the opposite of religiously adhering to an established view in the face of evidence to the contrary.

This is a frame variant concept.

τ=∫abdτ is invariant only if a and b are.

From the point of view of reference system B, event a is simultaneous with event a' and event b is simultaneous with event b'.

Does an invariant proper time elapse for a mass point B moving - from the point of view of reference system B - on a straight world line on the time axis from event a to event b?

Does no invariant proper time elapse for a mass point B' moving - from the point of view of reference system B - on a straight world line parallel to the time axis from event a' to event b'?

Does this have to do with the fact that the events a and a' and the events b and b' occur simultaneously from the point of view of the reference system B, but not from the point of view of other reference systems?

 

[Dale]

Asked and answered. This thread is closed.