Time Dilation: Orbit Earth at Light Speed, What Happens?

[Drizy]

I’m having quite a bit of trouble understanding time dilation. What will happen if you orbit the Earth close to the speed of light, 1 h passes for you and due to time dilation 2 h on earth. So what will happen when you look at Earth in that hour. Since time passes 2 times faster there will it look like everything moves 2 times faster or will everything just look normal?
Or is it something else?

Thanks

 

[Ibix]

Since time passes 2 times faster there will it look like everything moves 2 times faster or will everything just look normal?

Everything will move twice as fast as mormal.

Note that this is different from the case of inertial motion in flat spacetime, which is the case usually covered in introductory relativity sources. In that case, you would literally see someone you passed moving fast as you approached and slow as you receded. But this is due to the Doppler effect and once you correct for that you would calculate that the person was moving slow.

I would suggest that it's probably worth getting your head around the inertial case before trying to understand non-inertial motion.

 

[jartsa]

So this is the twin paradox question once again.

So the orbiting observer takes a curved path through the space-time, while a clock on Earth takes a straight path through the space-time. That's why the orbiting observer's clock falls behind the clock on earth.

 

[PeterDonis]

the orbiting observer takes a curved path through the space-time, while a clock on Earth takes a straight path through the space-time.

No. In curved spacetime, not all straight paths have maximal length, and straight paths are not always longer than curved paths.

The clock on Earth's path is curved, because it has nonzero proper acceleration. The orbiting clock's path might be straight or it might be curved, depending on the orbital speed; for the right orbital speed, the one that allows a free-fall orbit, the orbiting clock's path will be straight. The OP specified an orbital speed close to the speed of light, though, so in that case the orbiting clock's path will be curved, since it will take a very large inward proper acceleration to keep it in orbit. But even a clock in free-fall orbit, for low enough orbital altitudes, will have a shorter path through spacetime than a clock on Earth, even though a free-fall orbiting clock's path through spacetime is straight.

Even in curved spacetime, there will always be some straight path that is of maximal length, but it might take some effort to find it. In the case under discussion, consider this straight path: a clock is moving upward, radially, and passes the orbiting clock at the same instant the orbiting clock is directdly overhead of the Earth clock. (We are idealizing the Earth as non-rotating for this thought experiment; the Earth's rotation adds further complications.) The radially moving clock has just the right velocity so that it rises upward, decelerates, comes to a stop, starts falling back downward, and passes the orbiting clock again at the same instant the orbiting clock has completed exactly one orbit and is again directly overhead of the Earth clock. The radially moving clock will then have the longest path through spacetime between the two meetings.

 

[pervect]

Let's pretend the Earth didn't rotate, for simplicity. We might ask - what sort of path maximizes the proper time between a couple of events on the Earth's surface?

For short time intervals, the answer is unique. You start at the start event, throw the object upwards just hard enough so that it comes back down to the end event, and is in free fall throughlut. Writing out the proper time integral may help see why this is a longer path than just staying on the Earth's surface. In a frame dependent analysis, the velocity upwards caues velocity time dilation, which hurts, but it gets you out of the gravitational time dilation, that helps.

Feynman mentioned this , as a question he asked his students, though I don't recall where.

The solution becomes non-unique when you have a period long enough that an orbit around the Earth is possible.

This applies in general to curved spatial surfaces,as well as curves space-time. Let's consider an example of two towns on a curved surface - specificallly, they are separated by a very tall hill. What's the shortest (since this is a spatial-only problem, the metric is positive definie) path between the towns?

There is a straight-line (more formally, geodesic) path between the towns that goes over the hill. But for a tall enough hill, there's a shorter geodesic path that goes around the hill. Both paths are constructed to be geodesics, but in the case where there are multiple geodesics connecting two points, one geodesic may be shorter than the other even though they both "extremize" the geodesic.

 

[anuttarasammyak]

Because light travels seven and half turns in a second around the Earth, it takes 1/7.5 second for one turn for the clocks on the Earth. Corresponding time of the almost light speed pilot is almost zero. The Earth people interpret it by SR. The pilot who regards he is at still interprets it by GR.

 

[PeroK]

Because light travels seven and half turns in a second around the Earth, it takes 1/7.5 second for one turn for the clocks on the Earth. Corresponding time of the almost light speed pilot is almost zero. The Earth people interpret it by SR. The pilot who regards he is at still interprets it by GR.

Given that the gravitational field of the Earth is negligible in this case, the whole scenario can be analysed using SR, with one inertial and one accelerating observer.

 

[anuttarasammyak]

Yes, it can be. I said it can also be explained by the metric of rotating FOR.
$$g_{00}=1-\frac{r^2\omega^2}{c^2}$$

 

[eltodesukane]

I’m having quite a bit of trouble understanding time dilation. What will happen if you orbit the Earth close to the speed of light, 1 h passes for you and due to time dilation 2 h on earth. So what will happen when you look at Earth in that hour. Since time passes 2 times faster there will it look like everything moves 2 times faster or will everything just look normal?
Or is it something else?

Thanks

Yes, everything will moves 2 times faster
Please watch this episode for a similar situation:
Star Trek Voyager 6-12-Blink of an Eye (2000)
https://www.imdb.com/title/tt0708856/