The twin’s paradox revisited

I’m writing this article after reading a very unsatisfying discussion of the twin’s paradox. So here goes:

TLDR;

The twin’s paradox is not about relative constant speeds (special or general relativity), it’s about the effect of acceleration either in flat (special relativity) or curved (general relativity) space-time. You can travel forward in time simply by vibrating in place (Flash anyone?).

The twin paradox:

The twin paradox has one twin travel on a round trip in a rocket ship and comes back older than the twin that has stayed home. The effect is explained by the theory of relativity and has been verified using synchronized atomic clocks where one clock travels around the earth on a jet plane while the other doesn’t.

Special relativity:

According to special relativity, if someone has a different speed than yours, then you will see them age slower than you and vice-versa. Yes, it’s symmetric. Each sees the other aging slower. How strong the effect depends on the relative speeds.

This state of affairs is only paradoxical if you think of time as ‘flowing’ but if you think of all space-time as being in one piece then the interpretation is simple. While you’re looking at her ‘past’ unfolding in slow-motion, she’s looking at your ‘past’ unfolding in slow-motion.

In any case, relative constant velocity does not hold the key to the twin’s paradox.

Acceleration:

Acceleration breaks the symmetry. If one twin accelerates while the other doesn’t, then the twin who accelerates will see the other as aging slower. The other twin will agree that the accelerating twin is aging faster. Here is then the key to a change in age between the twins. Of course, as always, each twin views their own time-flow as ‘normal’.

So, if a twin accelerates long enough and then decelerates to again continuously match the speed of the other twin, then both twins will agree who aged more. The rocket twin will see the at-home twin as moving slower in time while (aging slower) and equivalently, the at-home twin will see the rocket twin as moving faster in time (aging faster). After the process, when they match speeds again, they will be traveling in time at the same rate, or in other words, they will be aging at the same rate again.

In terms of the geometry, whoever accelerates has actually been traveling in time in terms of the final situation. If all space-time is already ‘there’ then the net effect is of one twin moving to the other’s past to match their space-time coordinates. So the twin that accelerates and decelerates is a time traveler into the other twin’s past whereby both past and future are already there to ‘visit’.

Rocket man:

So what does the rocket ship have to do with the twins? Well, a good way to accelerate is to use a rocket. So one twin takes a rocket and the other stays home. Now, there’s the complication of acceleration so that the rate at which each observer sees the other’s time as changing does not remain the same. Fortunately, there’s a way to calculate what each observer sees as the other’s own time flow or the other’s proper time.

As the rocket moves away, it picks up speed. The speed differential causes the twins to occupy different inertial frames where they each see the other’s past unfolding in slow-motion. However, the one who accelerates has this added boost that breaks the symmetry. Which effect is stronger depends on the relative speeds and on the acceleration.

In the end, the rocket turns around (a complex acceleration maneuver) and moves back to the twin at home. As the speeds get closer to matching, the rate of time ‘flow’, or stretching when viewed as a dimension, starts to re-align and when the twins finally meet, they resume aging at the same rate. However, because one of the twins has accelerated, then that one has also travelled backward in time from the twin at home’s perspective.

And I think it’s gonna be a long, long time …:

Well, can we have both twins stay at the same position in space but also have one twin travel in time?

Sure. Just have one twin accelerate and decelerate in short bursts so that the twin comes back to the same spot without developing any appreciable speed. In other words, let one twin vibrate (albeit very violently and in short bursts)!

By vibrating with tremendous but very short acceleration and deceleration bursts, we get the effect of acceleration without having to travel at all. So, one twin can vibrate intensely like the flash (or quicksilver if you’re a Marvel fan) and can thus travel to someone else’s past without developing a significant mismatch in speed with the other twin.

So to vibrate is to time-travel!