# Twin Paradox in case of two twins that don't meet

If there are two twins. such that one of them goes on a space voyage maintaining a constant velocity, and that one never returns to earth, and the twins decide to never meet but send information about their age to each other by some light pulse or any other method.

If then one knows that he is younger, won't he know he is the one travelling thus violating the postulate of relativity that all inertial frames are equal w.r.t. all phenomena and that one can never know if he is moving with a constant velocity.

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What if without meeting they send a light pulse to each other, such that they can know each other's age

The result will still be the same - each twin judges the other twin to be ageing more slowly than themselves.

However, sending a light pulse to each other involves other factors that must be taken into account such as time of flight and unnecessarily complicates the matter.

A better way, in my opinion, to understand what's going on is as follows:

## The setup:

Position a monitoring station at a fixed distance (say, one lightyear) from the Earth along the path of the spacefaring twin and synchronize the clock on the monitoring station with the Earthbound twin's clock. The distance between the Earth and the monitoring station as well as the synchronization of the clocks is done by exchanging light signals.

In addition, position another monitoring station the same fixed distance behind the spacecraft and synchronize that clock with the spacefaring twin's clock. The distance between the spacecraft and this following monitoring station as well as the synchronization of the clocks is done by exchanging light signals.

Let's assume that the relative speed of the twins is 0.5 c and also that, at the moment the spacefaring twin passes the Earth, each twin's clock reads 0.

Now, the first monitoring station is programmed to send a message to the Earth the moment the spacecraft passes by. The message will contain the time according the monitoring station's clock and the time according the spacefaring twin's clock.

Similarly, the following monitoring station is programmed to send a message to the spacecraft the moment the Earth passes by. The message will contain the time according the following monitoring station's clock and the time according the Earthbound twin's clock.

## Results:

Since the relative speed is 0.5 c, the spacecraft passes the first monitoring station, one lightyear away from the Earth, after two Earth years have elapsed. The monitoring station sends a message to the Earthbound twin with the following:

• Station clock: 2 years
• Spacefaring clock: 1.732 years

Now, the Earthbound twin receives this message a year later but that doesn't matter. What matters is that when the spacecraft passed the monitoring station, 2 years had elapsed for the Earthbound twin and 1.732 years had elapsed for the spacefaring twin.

OK, so it seems the spacefaring twin actually is ageing more slowly. But...

The Earth passes the following monitoring station, one lightyear away from the spacecraft, after two spacecraft years have elapsed. The following monitoring station sends a message to the spacefaring twin with the following:

• Station clock: 2 years
• Earthbound clock: 1.732 years

So, according to this message, it is the Earthbound twin that actually ageing more slowly.

This seems to be a paradox but, in fact, it isn't. The paradox appears only if you assume that one or the other twin is actually, i.e., absolutely ageing slower than the other.

But, it isn't the case that one twin is actually ageing more slowly than the other. In fact, there is a frame of reference in which both twins have the same speed, in opposite directions, and thus age at exactly the same rate in that reference frame.

In other words, the question of "which twin ages more slowly?" depends on the reference frame, i.e., there is no absolute answer.

## How can this be?

You'll note that, for each monitoring station, the clocks had to be synchronized and the distance measured. What wasn't pointed out at the time is this:

According to the Earthbound twin, the first monitoring station was 1 lightyear distant and the station's clock was synchronized with hers.

Now, and this is crucial to understanding the result above, according to the spacefaring twin, the distance between the Earth and the first monitoring station is less than 1 light year and, further, the clocks aren't synchronized.

This is symmetric. The Earthbound twin sees that the distance between the spacecraft and the following monitoring station is less then 1 lightyear and the station clock and spacefaring clock are not synchronized.

This is a fundamental result from Special Relativity: clocks synchronized in one frame appear unsynchronized in a relatively moving frame; simultaneity is relative.

Once you accept this, the results above will become intuitive rather than paradoxical.

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If the twins never meet, but just continue travelling in a straight line at constant velocity then each twin will see the other as being younger. The *paradox*$^1$ only occurs if one or both of the twins is accelerated, which of course is necessary for the twins to meet again.

$^1$ it's not a paradox of course, just an unintuitive result!

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What if without meeting they send a light pulse to each other, such that they can know each other's age. – Isomorphic Nov 23 '13 at 10:25
Don't forget that there will be a time delay before they receive the other twin's message about how old he is. This delay will lead each other to assume that the other one is younger. – user1459524 Nov 23 '13 at 13:32