Understanding Relativity Through Spacetime Diagrams

[Involute]

A spaceship takes off from Earth. An astronomer on Earth can look though a window in the ship's hull and view a clock on the inside. He notes (I assume) it's indicating that time is flowing more slowly inside the ship than on Earth. The astronaut in the ship, however, also has a telescope and can view the clock inside the astronomer's observatory. Since (from his perspective) the spaceship is stationary and the Earth is moving away from it, he notes that the time on the terrestrial clock is flowing more slowly than it is on his ship. This seems like a paradox, though, since the astronomer should age faster than the astronaut. What am I missing? Thanks.

 

[PeterDonis]

This seems like a paradox, though, since the astronomer should age faster than the astronaut.

Not as long as the astronaut keeps on flying away; as long as he does so, there is no way to tell who is "aging faster", because they don't have any common standard of time to use since their clocks aren't synchronized. In other words, as long as their relative motion remains the same, there is no fact of the matter about who is "aging faster".

If the astronaut turns around and comes back to Earth and meets up again with the astronomer, then he will find that yes, the astronomer has aged more. But that's a different situation.

 

[Involute]

> then he will find that yes, the astronomer has aged more. But that's a different situation.

Why? How?

 

[PAllen]

> then he will find that yes, the astronomer has aged more. But that's a different situation.

Why? How?

Because the one who changes direction feels acceleration and sees something very different from the one who doesn't turn around. The one who turns around immediately starts seeing the other twin's clock going fast. The one who doesn't turn around continues to see a slow clock for the distant twin for long after the turnaround. The up shot is that each one's accumulated telescopic observations are consistent with the final result that the twin who turns around ages less.

 

[ghwellsjr]

A spaceship takes off from Earth. An astronomer on Earth can look though a window in the ship's hull and view a clock on the inside. He notes (I assume) it's indicating that time is flowing more slowly inside the ship than on Earth. The astronaut in the ship, however, also has a telescope and can view the clock inside the astronomer's observatory. Since (from his perspective) the spaceship is stationary and the Earth is moving away from it, he notes that the time on the terrestrial clock is flowing more slowly than it is on his ship. This seems like a paradox, though, since the astronomer should age faster than the astronaut. What am I missing? Thanks.

Even if it weren't for relativity, both would expect to see each others clock progressing more slowly than their own due to the Doppler effect. However, what's significant, is that they both see exactly the same amount of Doppler shift in the scenario you described. We call this Relativistic Doppler. If it were the ordinary kind of Doppler, like what happens with sound waves, then we wouldn't expect them to age differently and we could establish an absolute rest frame for the medium that is carrying the waves. But with light, we can't establish an absolute rest frame and so we can't establish which one is aging faster than the other in your scenario. It is legitimate to apply any rest frame and we can assign the slowing of time arbitrarily to either or both compared to the time of the rest frame.

But if we arrange for the two observers to meet up again, then we can determine unambiguously the age difference between them. The previous posters both mentioned one obvious way for this to happen by having the astronaut turn around and come back to Earth. But turning around is not the only way. Instead, we could have the astronomer get in his own spaceship and take off at a higher speed and catch up to the astronaut.

I have drawn some spacetime diagrams to illustrate this addition to your scenario. The progress of the astronaut is depicted by the thick red line and I'm having him take off at 0.6c with respect to the Earth. The astronomer, depicted in blue, stays on Earth for 5 years before he takes off at 0.8c. The dots indicate one-year increments of time according to each observer's clock. The thin lines show you the progress of the image of each observer's clock on its way to the other observer. In all cases, these lines are drawn at 45-degree angles which is a speed of c.

Here's a diagram for the Earth's rest frame:

Note after two years that the blue astronomer sees the red astronaut's clock at one year and similarly after two years of the red astronaut's time, he sees the blue astronomer's clock at one year. After four years for each of them, they see the other ones clock at two years.

But after the astronomer takes off at five years, he starts seeing the astronaut's clock running faster than his own. Between his sixth and eighth years, he sees the astronaut's clock progress through three years, 50% faster than his own clock.

Meanwhile the astronaut continues to see the astronomer's clock progressing at half the rate of his own until he reaches ten years. Then he sees the astronomer get in his own spaceship and take off toward him and from that point on, he sees the astronomer's clock running 50% faster than his own.

When they finally meet, the astronaut has aged 16 years while the astronomer has aged 14 years.

The spacing of the dots is called Time Dilation and is a function of the speed of the clock according to the rest frame. The faster the clock travels, the slower its time progresses. At 0.6c the Time Dilation factor (also called gamma) is 1.25 and you can see that the red dots are spaced 1.25 times the Coordinate Time. After the astronomer takes off at 0.8c, the blue dots are spaced 1.667 times the Coordinate Time.

But now let's see what happens when we use the Lorentz Transformation process to see this same scenario in the rest frame of the astronaut:

Now there is no Time Dilation for the red astronaut but the blue astronomer has two different Time Dilations depending on his two different speeds. At the beginning, his Time Dilation is exactly what the astronaut's Time Dilation was in the first diagram, 1.25. Then it switches to a smaller factor about 1.07. But note how this diagram depicts the same Doppler ratios that the first diagram depicted. This will continue to be true for the remaining spacetime diagrams.

Here's a diagram in which the astronomer is at rest while he is in the spaceship:

But we don't have to use frames in which the observers are at rest. Here's one in which both observers are moving in opposite directions at the same speed prior to the astronomer getting into his spaceship:

Note how both observers have the same Time Dilation at the beginning.

Finally, a diagram showing both observers traveling away from each other at the same speed while the astronomer is in his spaceship:

We could also make more diagrams for frames moving at any arbitrary speed with respect to the Earth rest frame and they would all show the light signals traveling at c along 45-degree angles and they would all show the same Doppler effects (what the observers actually see) and the same aging for each observer even though they assign different Time Dilations to each observer.