# time dilation and inertial reference

In answering the standard 'twin paradox', the reason for the reduced age of the 'traveling' twin is given as due to the change in acceleration and direction relative to the earthbound twin. This is done to resolve the question of the 'relative motion' of the earthbound twin from the perspective of the traveling twin (since the earthbound twin is moving, relative to the traveling twin, why doesn't that twin also have a reduced age).

Isn't this bugaboo an unnecessary avoidance of inertial frames of reference?

Muons created at the earth's upper atmosphere should not survive the distance traveling at near-light speed to the earth's surface, since they should decay due to their halflife prior to reaching the surface. But they don't due to their relativistic time dilation - without any change in direction or acceleration.

• But the muon will tell you that is you and I who are aging much more slowly than we "should". So there is no twin paradox here. I agree, though, that it's often misleading to "explain" the twin paradox simply by pointing out that the acceleration induces an asymmetry. The reason the acceleration is relevant --- which should be part of the standard explanation --- is that the returning twin's frame is not the same as the outbound twin's frame. So there are not two relevant frames here; there are (at least) three. – WillO May 31 '17 at 14:11
• I was not interested in the muon's feelings in this, I only want to understand why some explanations make the change in age due to the change in direction of the traveling twin. Yes, it is a change in frame of reference, but the critical issue is the SPEED. If I ride a bicycle forward, and then suddenly change direction, that is an acceleration WITHOUT time dilation, because my speed has not reached a significant percentage of "c". – Robin Indeededo Jun 5 '17 at 21:49

In answering the standard 'twin paradox', the reason for the reduced age of the 'traveling' twin is given as due to the change in acceleration and direction relative to the earthbound twin.

This is not correct. The reason why traveling twin is younger is NOT because it accelerates. It's because it is flying fast. Nothing interesting happened to him during acceleration. Theoretically the traveling twin can change his velocity within 1 second by Earth time (and even faster by his own time), and in this case he will not age for more than 1 second during acceleration.

Still acceleration is very important for twin paradox. Because during the acceleration period weird things happen to the Earth twin as seen by traveling twin!

According to traveling twin while he is flying away from Earht the Earth one is younger. Then traveling twin stops. Within this split of second the Earth twin becomes much older! Traveling twin accelerates to go back - and Earth twin becomes even more old! From now on traveling twin moves back home with constant speed, the Earth twin is aging slower then himself, but the net effect is that when he reaches Earth, the Earth twin is older.

The Earth twin suddenly aged during acceleration not because he was moving fast/slow. This effect can't be described by usual time-dilation formula. This happens because the frame of reference is not inertial.

UPDATE

Special relativity theory describes the world using inertial frames of reference. All the famous formulae are written for inertial frames of reference.

Special relativity theory states that time is going slower on moving objects. To be precise: if in some inertial frame of reference an object has velocity $v$, than time on this object is going slower by factor $\sqrt{1-v^2/c^2}$. There is not a single word about acceleration here. If traveling twin started his journey having speed 0.8$c$ he is aging 0.6 times slower. In 1 year (according to home-sitting-twin) the traveling twin will be (1 - 0.6) = 0.4 years younger.

Now traveling twin bounces from a metal wall and is moving back home. His speed is still 0.8 $c$ and he is still aging 0.6 times slower. According to home-sitting-twin the traveling one will be back home in 1 more year and by that time he will be 0.8 years younger than himself.

Again: there is nothing about acceleration in all these calculations. And this IS NOT a twin paradox yet.

Twin paradox happens when we try to describe what was happening using the frame of reference "attached" to a traveling twin. At a first glance it looks like the situation is symmetric: the second twin is staying still all the time, the first one moves back and forth, and the final result should be that the first twin (one which was sitting all the time on the traveling Earth) should be younger. And this IS a paradox, because we can just compare both twins in the end of experiment and it can not happen that each of them is younger than the other.

The explanation of this paradox is that we mistakenly used "usual" special relativity formulae in not-inertial frame of reference. If one really wants to use the frame of reference "attached" to accelerating twin he should use not Special, but General relativity theory. This exercise is not for faint-hearted! But the final result will be consistent: the twin sitting on Earth will be older than the other one.

• change of observer's frame of reference, within your explanation, in your example is very confusing! I do not at all get why you claim that when flaying AWAY from earth, the earthbound twin is younger, util the traveling twin stops, then the earthbound twin suddenly becomes older and then state the opposite: "the earth twin suddenly aged during acceleration" My concern was that the the explanation for the change in age between the traveling and the non-traveling twin is stated elsewhere, in multiple online locations, as due to the change in direction of the traveling twin. – Robin Indeededo Jun 5 '17 at 21:43