Hollygrove said:
I'm fairly new to the concepts provided by special and general relativity, and was wondering if someone could provide an answer to a thought exercise I came across regarding time dilation on an object going near light speed.
Say a spaceship were connected by a live video feed to a monitor on Earth, and the feed was being monitored at both ends.
The spaceship then accelerates to near light speed away from Earth, which in my understanding would slow the passage of time on board the ship in relation to time passing on Earth.
Actually, a better, more precise, way of saying this is that it would slow the passage of time on board the moving ship according to the Inertial Reference Frame (IRF) in which the Earth is at rest. Saying it the way you said it is what led you to the false conclusion that the passage of time on Earth will be faster than time passing on the moving ship.
This slowing of time for an object moving according to an IRF is what is called Time Dilation. So in the IRF in which the Earth is at rest, it does not experience Time Dilation, only the ship does because only the ship is moving (after it leaves Earth).
Hollygrove said:
So my question is what would the video show?
It doesn't show Time Dilation. Rather it shows the
Relativistic Doppler Effect. Let me give you an example. Let's suppose that the ship leaves the Earth at 60% of the speed of light. The
Time Dilation factor at that speed is 1.25 which means that the clock on the ship will take longer to tick out each second than the Coordinate Time of Earth's rest IRF. But the Relativistic Doppler factor at that speed is 2 which means it takes twice as long for the Earth to see a particular amount of time passing on the ship through the video feed.
Here's a spacetime diagram for Earth's IRF showing the Earth as the thick blue line and the ship as the thick red line. The dots mark off one-second intervals of time for each of them. The thin lines represent one-second intervals of the video feed, red for the video sent from the ship and blue for the video sent from the Earth:
As you can see, the blue dots are aligned with the Coordinate Time of the diagram because the Earth is not moving in this IRF but the red dots are spaced 1.25 times the Coordinate Time.
Hollygrove said:
Would time actually appear to be moving slower from a person on Earth's view of the video feed, would it appear to be moving faster from someone onboard the ship? Or have I misunderstood the concept?
Yes, you have misunderstood the concept. As I stated earlier, you have incorrectly concluded that if Earth sees time going slower on the ship then the ship must see time going faster on Earth.
But if you look at the diagram, you can also see that it takes twice as long, two seconds according to the dots, for each of them to receive each second of the video feed from the other one. So even though only the ship is Time Dilated in this IRF, they both see the same Relativistic Doppler Effect of the other one. In other words, they cannot see the Time Dilation, they can only see the Relativistic Doppler Effect.
To emphasize this point, I'm going to use the
Lorentz Transformation process on the coordinates of each of the events (dots) in the above diagram to see what things look like in the IRF in which the ship is at rest after the Earth and ship separate:
As you can see, the Relativistic Doppler Effect remains the same as in the first diagram, that is, they each see the others time taking twice as long as their own even though it is the Earth that is now Time Dilated according to this IRF.
We could also transform to other IRF's and they would all show the same Relativistic Doppler Effect even though the Time Dilation of each object can be different. Just remember that Time Dilation is dependent on the speed of an object according to an IRF so since speeds are different in different IRF's, so are Time Dilations but they all show the same observations of each observer.
Does this all make perfect sense to you now? Any questions?
