Einstein's Twin Paradox: Confusing Solution?

[phymath7]

TL;DR Summary

Einstein' twin paradox offers a solution when one of the reference frame is non-inertial.What if both are inertial?

In Einstein's twin paradox,the solution comes like this: The twin on the spaceship de-accelerates his spaceship first and then accelerates in the reverse direction.This means his reference frame is not inertial hence he doesn't measure greater time interval as the other twin does. 1.If the reference frame of the twin on the spaceship is non-inertial to the other,then isn't the reverse also true?(meaning both reference frame is non-inertial to each other) In that case how do we conclude who measures greater time interval?(seems like a new paradox to me ) 2.If I am wrong,(means only one of the reference frame is non-inertial) then is it necessary for the twin on the spaceship to return back to see who is older?What if he doesn't de-accelerate and then accelerate in the reverse diretion(rather he sends some light signal to the other twin)?This implies his reference frame isn't non-inertial .Then how do we get the solution ? 3.If it's not possible to know their time interval by sending signal and if the twin on the spaceship doesn't change speed hence velocity,then is it that we can't actually say for sure that who is older? Will that remain unknown to us forever unless they meet?

 

[Ibix]

Summary: Einstein' twin paradox offers a solution when one of the reference frame is non-inertial.What if both are inertial?

In Einstein's twin paradox,the solution comes like this: The twin on the spaceship de-accelerates his spaceship first and then accelerates in the reverse direction.This means his reference frame is not inertial hence he doesn't measure greater time interval as the other twin does.

This isn't correct. The point is that if the traveller wishes to regard himself as "at rest" then he's not using an inertial frame so cannot naively apply standard SR results that assume he is using an inertial frame. In that case, he needs to use less naive maths.

He can use an inertial frame to analyse the experiment and will conclude that he should be younger - it just cannot be a rest frame for him at all times because he is not moving inertially at all times.

1.If the reference frame of the twin on the spaceship is non-inertial to the other,then isn't the reverse also true?

No. Whether a reference frame is inertial or not is not open to interpretation. The only question is, do objects that are at rest in that frame undergo proper acceleration (the kind measurable with an accelerometer in a closed box) or not. The twin who turns round fires his rockets at some point, so undergoes proper acceleration, so his rest frame is non-inertial. The twin who does not turn around records zero proper acceleration at all times, so his rest frame is inertial.

is it necessary for the twin on the spaceship to return back to see who is older?What if he doesn't de-accelerate and then accelerate in the reverse diretion(rather he sends some light signal to the other twin)?

Sending a signal is possible, but interpreting it requires subtracting out the travel time of the signal, which depends on measuring the distance it's travelled, which means the answer depends on your choice of frame. The answer is only independent of frame choice if the twins meet again (so the signal travel time between them is negligible).

If it's not possible to know their time interval by sending signal and if the twin on the spaceship doesn't change speed hence velocity,then is it that we can't actually say for sure that who is sure? Will that remain unknown to us forever unless they meet?

If the twins don't meet again their relative ages remain frame dependent. "Which one is older" does not have a unique answer, and there can be no direct physical consequences to this, so it doesn't really matter.

 

[Nugatory]

If the reference frame of the twin on the spaceship is non-inertial to the other,then isn't the reverse also true?

A frame is not inertial relative to another frame, it’s inertial or not all by itself. If an accelerometer attached to any object moving at constant velocity in a given frame always reads zero, the frame is inertial - no need to involve any other frame, just look at the accelerometer reading.

In the twin paradox case, an accelerometer attached to the spaceship will not read zero during the turnaround. Therefore the frame in which the spaceship is at rest is not inertial. The frame in which the Earth is at rest is inertial; an accelerometer attached to the at-rest Earth always reads zero.

 

[PeroK]

@phymath7 if both twins move inertially, e.g. in opposite directions in the same orbit around a star or planet, then there will be no difference in ages when they meet after each orbit.

 

[pervect]

Summary: Einstein' twin paradox offers a solution when one of the reference frame is non-inertial.What if both are inertial?

1.If the reference frame of the twin on the spaceship is non-inertial to the other,then isn't the reverse also true?(meaning both reference frame is non-inertial to each other)

No, this is not true.

If one consider a path on plane in 2 dimensional space, in the absence of time, the shortest distance between two points is a straight line. Such a path can be distinguished from a path that is not straight by the fact that the length of the path joining two points is not the shortest possible path if the line is not straight. One can distinguish between paths that are straight, and paths that are not straight - straight paths minimize the length of a curve connecting two points, and non-straight paths do not. It is incorrect to think of a non-straight path as being straight.

In space-time, one has one or more dimensions of space, plus one dimension of time. Points, which exist in space, are instead called events in the language of space-time. A path through space-time which travels from one event to another is called a worldline. In this post we will restrict ourselves to worldlines that can be physically transveresed, this sort of traversible path is given the name of a "time-like worldline".

A time-like worldline has a property that is analogous to "length" of a line on a plane. This property is just the amount of time that an idealized small clock, often concepptualized as a "wristwatch", will measure as the clocks travel from one event on the worldine to another. We can distinguish between inertial worldlines and non-inertial wordlines in space-time just as we can distinguish between straight lines on a plane and non-straight lines. Whether or not a wordline is inertial or non-inertial is absolute and a property of the worldline itself, all observers will agree on whether a given worldline is inertial or non-inertial. Inertial worldlines have the property that they maximize the "wristwatch time", of an observer traveling along them. THis is a geometrical property of the worldline in space-time, just as length is a geometrical property of a line in space. And the equivalent of the often-heard statement that the shortest distance between two points on a plane is a straight line in special relativity is the principle of maximial aging. The principle of maximal aging states that an observer following an inertial worldline that connects two events will experience more aging than an observer following a non-inertial worldline that connects the same two events.

Note that the statements about distances in a plane would need to be slightly modified if one was considering distances not on a plane, for instance if one was doing spherical geometry rather than Euclidean geometry. Similar cautions apply to the space-time in cases that require gravity and general relativity.

 

[vanhees71]

Summary: Einstein' twin paradox offers a solution when one of the reference frame is non-inertial.What if both are inertial?

Then there is no paradox, because then the twins never meet again and cannot compare their ages/clocks.

 

[phinds]

@phymath7 if both twins move inertially, e.g. in opposite directions in the same orbit around a star or planet, then there will be no difference in ages when they meet after each orbit.

? Orbital motion is non-inertial.

 

[PeroK]

Orbital motion is non-inertial?

Inertial.

 

[phinds]

Inertial.

But if you are in orbital motion, you are accelerating. Hm ... but an accelerometer would measure zero.

OK, got myself confused for a minute there. Thanks

 

[jbriggs444]

? Orbital motion is non-inertial.

Orbital motion is inertial. But only if we account for gravity according to the rules of General Relativity. In which case space-time curvature is on the table, it is no longer the case that inertial trajectories can only meet once and there is another source of time dilation.

Orbital motion is non-inertial. But only if we account for gravity according to the rules of Newtonian Mechanics. In which case special relativity is off the table.

Be careful when trying to mix special relativity and gravity. It took Einstein a while to figure it out.

 

[Mister T]

This means his reference frame is not inertial hence he doesn't measure greater time interval as the other twin does.

There are versions of the twin paradox where each twin turns around and comes back. The twin who takes the longer path through spacetime is the one who's older.

 

[robphy]

There are versions of the twin paradox where each twin turns around and comes back. The twin who takes the longer path through spacetime is the one who's older.

This is the real essence of the clock effect (that, between two events, spacetime-lengths of worldlines between those events are path-dependent.. both could be non-inertial, or in (say) a closed universe... both could be inertial).

The twin paradox part is (in the way I like to think of it)
is mistakenly thinking that "being able to be considered at rest" is equivalent to "being inertial", then treating the non-inertial frame as is if it were inertial.