# Why isn't time contraction a thing?

I'm very naive about relativity, but if it could be possible that Alice ages less than Bob in a journey, doesn't it logically follow that from the point of view of Alice, Bob has, at some point, gone through time contraction?

Additionally, Wikipedia provides a specific example, and by applying the Doppler shift concludes :

during the trip back, both twins see their sibling's clock going 3 times faster than their own.

Isn't this the exact opposite of time dilation? If not, how different is time dilation from the observation of slowed-down clocks?

• You can talk about time contraction and length dilation, as described in this video – BioPhysicist May 26 '19 at 9:58

A proper exposition of the effects implied by special relativity concentrates on the following: when your motion does not involve any acceleration then the amount of time that elapses for you is a maximum.

So: we have Bob not accelerating. Let's say he is moving in the sense that he is moving forward in time. Alice is on a trajectory that involves an acceleration phase away from Bob, and a U-turn in any way, and then rejoining Bob. At the instant of rejoining they are once again right next to each other, so they can make a direct comparison of how much proper time has elapsed for each of them.

On the other hand, there is ambiguity when they compare their clocks while Alice is far away from Bob, because then you have to take transmission delay into account. Likewise, comparing clocks while Alice is on the move is ambiguous, because then you have to take classical Doppler effect into account.

The effect that is specific to special relativity is laid bare when you remove all other effects. You set up a scenario where at the moments of comparison there is no transmission delay, and no Doppler effect.

(So anytime you encounter an exposition of special relativity where a scenario is offered that includes transmission delay and/or classical Doppler effect you can just stop reading that exposition; it won't teach you.)

So you focus on the scenario of Bob not accelerating, just moving forward in time, and Alice accelerating away, making a U-turn, and rejoining.

For Alice, by traveling more spatial distance than Bob, less proper time elapses. That is: there is something you can actively do so that for you less proper time elapses.

On the other hand, there is nothing that Bob can actively do so that for him even more proper time elapses; he is already moving forward in time along the spatially shortest path, so he is already maxing out the amount of proper time elapsing for him.

So it's not a symmetrical situation; Bob is already maxing out, Alice can make the amount of proper time that elapses for her smaller by increasing the spatial distance she travels (in the same amount of Bob's proper time.)

• Is the difference in (or ratio of?) elapsed time uniquely determined by the acceleration a(t) of the travelers? If only acceleration is responsible, it wouldn't matter when (after 1 min or 10 years) they decide to make a U-turn, but it matters, right? – Asmani May 26 '19 at 21:08
• @Asmani A single factor determines the amount of difference in elapsed proper time: difference in length of the path travelled. Acceleration: not a factor. For the traveller who accumulates the most spatial distance there is the least amount of elapsed proper time. You can travel with one giant journey away, and one U-turn, or you can do a lot of zigzag travel. When zigzagging you accumulate a lot of acceleration, but if the spatial distance travelled is the same then the amount of elapsed proper time will be the same. – Cleonis May 27 '19 at 16:41
• I'm not sure if I understand what exactly "proper time", "length of the traveled path", and "spatial distance" mean. It's up to me to study these concepts, but a quick question, is this correct: In the Wiki example, the amount of difference in elapsed proper time is 4 years, and the difference in length of the path traveled is d=4 light years. – Asmani May 27 '19 at 22:52
• However, if the traveler twin travels the same path at a velocity very close to $c$, when he comes back he will have had aged almost 0 years while the other twin has had aged about 8 years. This means the amount of difference in elapsed proper time is about 8 years, meaning it depends on the velocity/acceleration? – Asmani May 27 '19 at 22:52

Time dilation means that one of the observers experiences a longer time period than another, as a consequence of a relativistic velocity (comparable to the speed of light c).

Time contraction would mean that traveling close to the speed "c" would make the time interval for the persion traveling fast to be smaller than for the one standing still, thus having him "travel backwards in time".

Best way to see why it is NOT a thing is to see the formula for time dilation:

$$t'=\frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$$

as you see, the equation only allows for t', which is the time that a observer that is NOT moving, to be greater than the time that the person moving experiences.