Q & A: on the track of General Relativity

Q:
I have come up with an interesting paradox and I would like to have your opinion. Travelling at relativistic speeds slows the flow of time in reference to the flow of time on Earth for example. Now imagine that we set up an HD video camera that records the daily activity on Time Square and that the video being captured by the camera is being played live in a spaceship travelling at a relativistic speed in a circle path around the camera. We know that the flow of time in Time Square goes faster that the flow of time inside the spaceship. However, the live video being sent by the camera arrives at light speed at the spaceship which is at a constant distance of the camera (travelling around it). Light speed is the same whatever the reference frame used. Now the question is, what the astronauts aboard the relativistic spaceship will see on the video? A very fast going-on video like if it were on fastfoward because the flow of time on Time Square is going faster than their's or if the video will not be altered because the information the camera sends arrives at the speed of light to them? If the video is not altered, that would mean that the astronauts are seeing into the past (because 1 spaceship second may be like 1 Earth year long) and that the transmission is somehow delayed which is impossible since it arrives at the spaceship few microseconds after it left the transmitter on Time Square in the Time Square reference frame. If the video is on fastfoward, that still doesn't make any sense because the signal is being sent into the future; few microseconds after the signal leaves the transmitter it arrives at the spaceship but it will show an image that is supposed to be long time passed for the astronauts. How do we get out of that? What did I miss or misurderstood? Thanks for your time
- Anonymous
A:
What a wonderful question!
You've put your finger on something very important. Essentially, because the transmission time in this case doesn't change, the two observers can't agree to disagree about whose clock is faster. Regardless of what one might say about who is "right", as you argue they must agree about who is fast.

The key is this. The Special Relativistic arguments only apply to a non-accelerated observer in a region where gravity is unimportant. Just to go step-by-step, let's say there was little gravity here and the satellite was kept in orbit by a string. The satellite is accelerating with respect to the earth. Therefore SR by itself doesn't describe what things look like in the satellite frame. However, we can then deduce, using your argument, that its acceleration towards the earth causes earth clocks to appear sped-up, by twice the amount that the SR effects cause them to appear slowed down (all this is just to lowest order in the effect). Then the apparent speed-up of the earth seen by the satellite will match the apparent slow-down of the satellite seen by the earth. This is the way, in some of my course notes, we derive the (lowest order) magnitude of the effects of accelerating frames, the start on the path to General Relativity.

Now when you consider gravity as well, another ingredient (the Equivalence Principle) is needed. It turns out to make the object higher-up in the gravitational field go slower. For low-orbit satellites, the first effect dominates and their clocks are slow. For high-orbit satellites (e.g. geosynchronous) the second (gravity) effect dominates and their clocks are fast. That's important for the very precise clocks used in GPS systems.

Mike W.

(published on 09/29/2010)