# Q: How does the Twin Paradox work?

The original question was: I have a question about the twin paradox.  Is it true that faster aging of the twin who stayed at home happens only when the other twin’s spaceship is accelerating/deceleration (btw, does it matter whether he is accelerating or decelerating?)?  Consequently, do they age at the same rate when the spaceship moves inertially?

Physicist: The very short answer is: geometry works different in spacetime than it does in just space.

The twin paradox is a result from special relativity that states that if one person, Alice, remains “stationary” and another person, Bob, takes any kind of round trip, then the stationary Alice will experience more time.  The twin paradox isn’t a paradox at all, it’s just strange and off-putting (like twins).
In relativity (that is to say: “in reality”) there’s no difference between being stationary and having smooth (non-accelerating) movement.  On the surface of it, the only difference between Alice and Bob is that, in order to return home, Bob has to accelerate (turn around) at some point.  So is acceleration the secret to the twin paradox?  Nope.

In all of the pictures that follow the “time direction” is up, and one of the (three) space directions is left/right.

Out and back: In both situations Bob (blue) experiences the same acceleration, while Alice (aqua) sits around on Earth. The only difference is that the second situation involves twice the distance, and twice the difference in experienced time. Acceleration is not what's important.

The trick is: spacetime doesn’t obey the “triangle inequality”. As a result, the bendier a path is, the shorter it is (that shouldn’t make any sense, so please read on).

The triangle inequality says that (in space) the sum of any two sides of a triangle is greater than or equal to the third side. In spacetime the inequality can be reversed. One side is often longer than the other two: a round-about route is shorter than the direct route. In this case, A+C<B.

The equation for distance that we’re used to is: $D^2=\Delta x^2+\Delta y^2+\Delta z^2$ (this is just the Pythagorean theorem).  But you find that when you start involving time and movement, this isn’t a particularly good measure of the distance between two points. Specifically, it’s different for different observers because of length contraction.
It so happens that the effects of length contraction and time dilation cancel each other perfectly, so that we can use a new (better) measure for spacetime distance, called the “Interval” or “spacetime interval” or “Lorentz interval”:

$L^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$

(as often as not the sign on the right hand side is reversed, not to worry)

The advantage to the Interval is that, no matter what, the Interval between any two points in spacetime (two locations and times) are always the same, despite relativistic weirdness.  Here’s another bonus!  The Interval of a path is the same as the amount of time experienced on that path!

No one every really feels like their own position is changing, so: $L^2=c^2\Delta t^2-0^2-0^2-0^2\Rightarrow L=c\Delta t$

Now all that’s left is to draw a picture and do a little calculating.  Here’s an example situation from Alice’s perspective, and then Bob’s (initial) perspective.  The difference between Alice and Bob’s velocity is 0.6C (60% of light speed).

The same situation from two perspectives, but since L is invarient you get the same values. Alice experiences 10 units of time while Bob experiences 8 units (since 5^2-3^2=16=4^2).

This entry was posted in -- By the Physicist, Physics, Relativity. Bookmark the permalink.

### 12 Responses to Q: How does the Twin Paradox work?

1. Rich says:

I still don’t quite follow why the situation isn’t symmetrical.

Why can we say that Alice is sitting still (moving only through time) and Bob is taking a round trip, but we can’t say that Bob is sitting still and Alice is moving? Is it because Bob changes velocity? But it’s not the acceleration that’s important, it’s just that he’s had two different velocities?

2. The Physicist says:

Regardless of your perspective, Alice is always moving in a straight line, and Bob is always moving on a crooked path. As a result of the “messed up triangle inequality” all crooked paths are shorter than straight paths (through spacetime). That’s the essential difference.
Another (clearer) way to say that is Bob has two velocities.

3. Rani Sunarso says:

Bob has two velocities, that if we say Alice is sitting still on earth, while Bob is moving away.

But from Bob’s perspective, he is sitting still (on the rocket), while Alice is moving away from him with a speed of 0.6c.

4. Anthony Rose says:

Rani, I think that if you accelerate, you can’t treat your frame of reference as inertial. We cannot treat Bob as still and Alice on Earth as accelerating and decelerating, because there is a force (the rocket jet) affecting Bob’s frame of reference. I stand to be corrected but i think that at constant velocities you can apply either frame of reference, but not when one body experiences any kind of acceleration.

5. The Physicist says:

@Anthony Rose
That sounds exactly right! I tried to cover that idea with the last picture in the post, but it’s not terribly clear.

6. Gene Kim says:

If Bob leaves Earth and travels to some point in space, he will have aged slower than Alice who stayed on Earth, but if Bob stays at that point and Alice leaves Earth and joins Bob, will they both have aged the same amount of time?

7. The Physicist says:

Yup. Weirdly, it’s impossible to objectively say which has aged more or less until they’re brought back together.

8. Chrobry says:

Are you aware of any other credible explanations for the Twin Paradox?

I don’t believe the Twin Paradox is solved by determining which twin accelerated (non-inertial frame) or which twin changed inertial reference frames. I can provide a thought experiment where the twin on Earth (who neither accelerates or changes inertial frames) ages slower than the twin in a spaceship (who does accelerate and/or change inertial frames). No tricks.

These two explanations appear to be ubiquitous. I am only looking for “mainstream worthy” credible explanations, not quack explanations such as “special relativity is not real” or some variety of Mach’s principle.

9. John Frey says:

My understanding of the twin paradox is that the uniform accelerating motion of the traveiling twin plays a big role in why the principles of special relativity and this motion cause time for the traveling twin to pass slower than before this motion but, beause this accelerating motion it not the motion of the stay-at-home twin, it causes no slowing of his time.

My questions:

1) Is the above understtanding correct?

2) Does the relativity principle of special relativity function through this uniform motion because the motion keeps slipping from one uniform moving frame to another and thus, this motion cfan be considered uniform as demanded by this principle?

3) Would the relativity prinicple of special relativity function through this accelerating motion if it were rough, erratic, and swerving from side to side?