# The Twin Paradox: Part III

This is the third in a series on the "Twin Paradox".

Just to remind you of the important facts:

The Earth Federation has created a series of artificial planets stretching away in space in both directions. The planets are called "P1", "P2", "P3", etc., in the positive direction (right), and "M1", "M2", "M3" and so forth in the minus direction (left). Each planet is exactly one year (or light-year) from its neighbor in each direction. The planets keep their clocks synchronized with each other.

The Federation Navy has a special fleet of red supply ships which travel through space with a constant speed v = 0.9c. There are many, many ships, all moving in a line, one after the other. These red ships keep their clocks synchronized with each other.

There is also a second fleet of green supply ships which travel through space in the OPPOSITE direction, moving with constant speed v = 0.9c. These green ships keep their clocks synchronized with each other (but not with the red ships).

Joe and Bob are twins living on Earth. On their twentieth birthday, Jan 1, 2500, Joe joins the Space Navy. He takes a shuttle into space to catch the Red Flagship, one of the Red Fleet ships. The clock on board that ship also reads Jan 1, 2500, as Joe climbs aboard.

Joe's plan is to travel to planet P27, then return home to Earth by switching to a green supply ship.

Now, on planet P27, something is happening on Jan 1, 2500. A young man who is 20 years old decides NOT to join the Space Navy, but instead, to become a Boot Salesman.

#### Joe's trip -- according to Bob and other observers on Earth

Joe and Bob say good-bye in the year 2500, as Joe prepares to leave the Earth. We could draw a space-time diagram showing Joe at the origin, since Earth is at position zero, and since we're measuring time since 2500.

The blue line shows the location of events which are simultaneous, according to Earth-based observers with Joe's departure. In other words, the blue line shows all events which take place on Jan 1, 2500, according to the Earth's time system.

(Click the image below to watch Earth-time march forward)

Here are the ships of the Red Fleet which happen to be passing Earth and its neighboring planets on Jan 1, 2500, according to the reference frame of the Earth.

Joe takes the shuttle to the Red Navy ship Red Flagship and flies to planet P27. Since he is moving at v = 0.9c, and must travel 27 light-years, he arrives 30 years later. Again, the blue line indicates all events which occur when the planetary calendars read "2530".

(Click the image below to watch Earth-time march forward)

Joe then hops onto the Green Navy ship Green Flagship and comes back to Earth. It will take another 30 years for this return trip, according to people on Earth, so he arrives in 2560.

(Click the image below to watch Earth-time march forward)

#### Joe's trip -- the first half, according to Red Navy sailors

Joe and Bob say good-bye in the year 2500, as Joe prepares to leave the Earth. The Red Navy ship Red Flagship, which picks Joe up, has a calendar which also (by coincidence) reads Jan 1, 2500. If we draw a line showing the location of all events which are simultaneous according to Red Navy observers, it looks like this:

(Click the image below to watch Red-Navy-time march forward)

In other words, events which occur anywhere along that red line will have a Red Navy ship flying past with a calendar reading "Jan 1, 2500." For example, here are the ships of the Red Fleet which happen to be passing Earth and its neighboring planets on Jan 1, 2500, according to the reference frame of the Red Navy.

When Joe jumped from the planetary frame of reference to the Red Navy frame of reference, his notion of events which are "simultaneous" changed drastically. Look what happened to the Boot Salesman on planet P27: according to the Red Ship which is passing planet P27 just as the ship's calendar reads Jan 1, 2500, the Boot Salesman is 44 years old!

As Joe and the ship travel towards planet P27, time passes on board. The rate at which time passes is slower for Joe than for Bob back on Earth. When Joe and the ship reach planet P27, only about 13 years have passed. The red line in the diagram below shows the location of events which occur when Red Navy calendars read "2513".

```

Q:  The Red Navy ship "Red Condor" flies
past the Earth just as the Red Condor's
calendar reads "2513".  If the sailors
ask people on Earth what year
it is, what will the Earth people say?

Q:  How old is Bob when the "Red Condor"
flies past the Earth?

```

#### Joe's trip -- a brief shopping excursion

Joe's ship reaches planet P27, and he hops onto a shuttle which takes him down to the planet. He goes shopping. As he purchases some boots, he asks the salesman what year it is.

"Why, it's the year 2530," says the Boot Saleman.

"Hmmm," replies Joe, "I need to move my watch ahead by 17 years."

Joe's notion of "simultaneous" changes again.

```

Q:  At this moment, according to the
Boot Saleman, back on Earth,
how old is Bob?

```

Joe goes to the spaceport to purchase a return ticket to Earth aboard the Green Navy fleet. He asks the ticket agent about the Green Navy. "Oh, it's a fine fleet," say the Ticket Agent. "One of their best ships, the Green Monkey, should be passing by the Earth right now."

#### Joe's trip -- the return segment, according to Green Navy sailors

Joe takes the shuttle from Planet P27 up into space and hops aboard one of the Green Navy ships, the Green Flagship. The Green Navy flies at the same speed, v = 0.9c , but in the opposite direction.

By a nice coincidence, the calendar aboard the Green Flagship just happens to read Jan 1, 2530, which is the same as the calendars on Planet P27.

After Joe has boarded the ship, his notion of "simultaneous" shifts once again. The green line in the diagram below shows the location of all events which occur when Green Navy ships read "Jan 1, 2530".

```

Q:  The Green Navy ship "Green Dragon" flies
past the Earth just as the Green Dragon's
calendar reads "2530".  If the sailors
ask people on Earth what year
it is, what will the Earth people say?

Q:  How old is Bob when the "Green Dragon"
flies past the Earth?

```

As he boards the ship, Joe asks the purser about the Green Navy ship which is currently passing the Earth. "Oh, that would be the Green Dragon," says the purser.

Time passes aboard the Green Flagship. In fact, about 13 years go by for Joe as he waits to get back to Earth. When he arrives, the Green Flagship's calendar reads 2543.

#### Joe's arrival back home -- according to Bob

Joe takes the shuttle from the Green Flagship back down onto Earth. As he lands, he asks a spaceport worker what year it is here on Earth.

"Why, it's 2560, of course," says the Spaceport Worker.

"Gosh," thinks Joe, "I must re-set my watch forward by 17 years again."

Now that he is back in the planetary frame of reference, his notion of "simultaneous" events shifts a final time. The blue line below shows the events that he now considers "simultaneous."

```

Q:  How old is Bob when the twins meet again?

Q:  How old is Joe when the twins meet again?

```

#### Viewing the twin paradox through diary entries

Another way to look at this story is to examine the diary of each twin. We're going to cheat a little bit, and allow each twin to write down information that isn't easily available; but we'll make sure that the information is correct.

```
The diary of Bob, the Earth Twin

Boot Seller's
Year        My age   Joe's age     age              comments

2500        20.00      20.00      20.00           Joe leaves Earth
2501        21.00      20.44      21.00
2502        22.00      20.87      22.00
2503        23.00      21.31      23.00
2504        24.00      21.74      24.00
2505        25.00      22.18      25.00
2506        26.00      22.62      26.00
2507        27.00      23.05      27.00
2508        28.00      23.49      28.00
2509        29.00      23.92      29.00
2510        30.00      24.36      30.00
2511        31.00      24.79      31.00
2512        32.00      25.23      32.00
2513        33.00      25.67      33.00
2514        34.00      26.10      34.00
2515        35.00      26.54      35.00
2516        36.00      26.97      36.00
2517        37.00      27.41      37.00
2518        38.00      27.85      38.00
2519        39.00      28.28      39.00
2520        40.00      28.72      40.00
2521        41.00      29.15      41.00
2522        42.00      29.59      42.00
2523        43.00      30.03      43.00
2524        44.00      30.46      44.00
2525        45.00      30.90      45.00
2526        46.00      31.33      46.00
2527        47.00      31.77      47.00
2528        48.00      32.20      48.00
2529        49.00      32.64      49.00

2530        50.00      33.08      50.00        Joe arrives at P27
leaves Red Fleet
shops briefly
boards Green Fleet

2531        51.00      33.51      51.00
2532        52.00      33.95      52.00
2533        53.00      34.38      53.00
2534        54.00      34.82      54.00
2535        55.00      35.26      55.00
2536        56.00      35.69      56.00
2537        57.00      36.13      57.00
2538        58.00      36.56      58.00
2539        59.00      37.00      59.00
2540        60.00      37.44      60.00
2541        61.00      37.87      61.00
2542        62.00      38.31      62.00
2543        63.00      38.74      63.00
2544        64.00      39.18      64.00
2545        65.00      39.62      65.00
2546        66.00      40.05      66.00
2547        67.00      40.49      67.00
2548        68.00      40.92      68.00
2549        69.00      41.36      69.00
2550        70.00      41.79      70.00
2551        71.00      42.23      71.00
2552        72.00      42.67      72.00
2553        73.00      43.10      73.00
2554        74.00      43.54      74.00
2555        75.00      43.97      75.00
2556        76.00      44.41      76.00
2557        77.00      44.85      77.00
2558        78.00      45.28      78.00
2559        79.00      45.72      79.00
2560        80.00      46.15      80.00         Joe returns to Earth

```
```

Q:  Do you see sudden jumps in anyone's age?

Q:  Who is ageing slowly?   By what factor?

```

Hmmmmm. That looks pretty simple: Bob and the Boot Salesman age at the ordinary rate of one year per year, while Joe ages at a continuously slower rate.

What if we look at Joe's diary? This will be somewhat more complicated ....

```
The diary of Joe, the Space-faring Twin

Boot Seller's
Year       Bob's age    my age       age              comments

2500          20.00      20.00      20.00        Joe walks to spaceport

2500.00       20.00      20.00      44.30        Joe boards Red Ship

2501.00       20.44      21.00      44.74        Joe travels on Red Ship
2502.00       20.87      22.00      45.17
2503.00       21.31      23.00      45.61
2504.00       21.74      24.00      46.04
2505.00       22.18      25.00      46.48
2506.00       22.62      26.00      46.92
2507.00       23.05      27.00      47.35
2508.00       23.49      28.00      47.79
2509.00       23.92      29.00      48.22
2510.00       24.36      30.00      48.66
2511.00       24.79      31.00      49.09
2512.00       25.23      32.00      49.53
2513.00       25.67      33.00      49.97
2513.07       25.70      33.07      50.00        Red ship reaches P27

2530.00       50.00      33.07      50.00        Joe steps off ship onto P27
and goes shopping

2530.00       74.30      33.07      50.00        Joe steps onto Green Ship

2531.00       74.73      34.07      50.44
2532.00       75.17      35.07      50.87
2533.00       75.60      36.07      51.31
2534.00       76.04      37.07      51.74
2535.00       76.48      38.07      52.18
2536.00       76.91      39.07      52.62
2537.00       77.35      40.07      53.05
2538.00       77.78      41.07      53.49
2539.00       78.22      42.07      53.92
2540.00       78.66      43.07      54.36
2541.00       79.09      44.07      54.79
2542.00       79.53      45.07      55.23
2543.00       79.96      46.07      55.67
2543.07       80.00      46.15      55.70

2560.00       80.00      46.15      80.00        Joe steps onto Earth

```

What's going on here? Look carefully at the table, and you'll notice several things:

• When Joe accelerates (changes from one inertial frame to another), the age of a distant person suddenly jumps; but the age of a local person does not.
• When Joe accelerates, he will probably have to re-set his calendar and watch. To simplify our story, I arranged for the Red Flagship to just happen to have the same calendar as the Earth, when it passed the Earth; and for the Green Flagship to just happen to have the same calendar as planet P27 when it passed P27.