Consider a hypothetical situation: In which a time traveler travels in a spaceplane at close to speed of light circling the earth and I get a live telecast of inside of that plane to my television. How time dilation will effect him and what will I observe sitting next to the tv seeing him live. Considering time runs slower at speed of light, will he be looking like a slow motion video? (Or let he also watching live streaming of us at earth, what it will look to him) P.S. I'm not sure if it even makes some sense but if it does kindly explain in logical details. Sorry for the the bad english, I'm not a native speaker.
Since the traveller is circling Earth, he/she is actually accelerated. So his/her frame of reference is not inertial (if we take that of the beholder to be inertial). The two points of views are then not equivalent, unlike those in the twins paradox.
If I'm not mistaken, one may show that the beholder will "see" the traveller be slower, and the traveller will "see" the beholder be faster. I put double quotes because this is not mere appearance: time does pass slower for the traveller than for the beholder. Given they remain close to each other, they can easily meet (the traveller just has to land) to compare their watches.
If the traveller were moving in a straight line at constant speed (simple Lorentz transform, both persons are in an inertial frame), both would see the other one be slower. This is only an apparent paradox, because in such a situation they won't meet again. If the traveller were then to come back to Earth, during his/her U-turn he/she would see the beholder become very fast, so that when they meet again there is indeed asymmetry between them, and the traveller is younger (this is called Langevin's twins paradox).
I am adding this answer to respond to your last followup comment on L.Levrel's answer.
Point One: Alice and Bob have to agree on how many times Alice's clock ticks per rotation, and they have to agree on how many times Bob's clock ticks per rotation. Therefore, if Alice says Bob's clock is slow, Bob must say Alice's clock is fast, and vice versa.
Point Two: Alice is an inertial observer. Bob is in motion with respect to Alice, so Alice must say that Bob's clock is slow. Between this and Point One, we can conclude that Bob must conclude Alice's clock runs fast, i.e. he sees the video she transmits as speeded up.
Point Three: At any given instant, Alice is in motion with respect to Bob, causing Bob to say her clock runs slow. You seem to be worried that this contradicts Point Two. But it's also true that any given instant, Bob is accelerating toward Alice, which causes him to see her clock jumping ahead in that instant, and this more than counteracts the effect of the time dilation we've already acknowledged.
To see the effect of Bob's acceleration (which is indeed directed toward Alice, since he's orbiting her), draw a spacetime diagram in which Alice's worldline is vertical, and Bob's, initially tilted toward Alice (so that he's moving toward her), tilts a bit more in the same direction. Then his line of simultaneity also tilts, so as to hit Alice's worldline at a later time on Alice's clock. That is, Bob's acceleration causes him to see Alice's clock move forward.
So: Bob's velocity with respect to Alice (which of course he thinks of Alice's velocity with respect to Bob) makes him say that her clock is running slow, but at the same time his acceleration makes him say that her clock is running fast. If you carefully calculate the two effects (over a given infinitesimal time increment) and add them together, you should get exactly the speed-up factor promised in Point Two. I confess that I have not done this calculation, but it should be easy and I am confident of how it would come out.
I dont think it's a problem, that the satellite is accelerating. Locally the time will nevertheless run slower.
If the camera is directed onto a watch, you would see it ticking more slowly.
If the webcam is directed on the outside, you will see the normal speed of happenings, e.g. it would show you the same number of rounds as you have seen by looking at the vehicle. Only it would have to be a very fast camera, otherwise you would see less frames per second.
And it probably would have to be a magical webcam, since the usual connection to a WLAN would probably not work. But I'm not sure about the engineering details. (Well, I'm sure it would break from the acceleration, so it has to be magical anyways)
Well, and besides, you would see the world distorted, everything shifted to the forward direction, and brighter there and less bright backwards. Just like the pictures you find on the internet, e.g. here