A moving observer perceives stationary objects as contracted in the direction of motion. So presumably this means that an observer orbiting at relativistic velocities around a black hole would perceive the circumference of his orbit as less than 2 pi times the radius of his orbit. Is this effect equivalent to the warping of space caused by the gravity of the black hole?
As he orbits faster and faster, the circumference of his orbit should appear more and more contracted. If he’s able to orbit so fast that the apparent circumference is zero, is that equivalent to the event horizon?
Consider two observers orbiting in opposite directions around the same black hole. As they pass each other, each perceives that the other is experiencing time dilation. However, when they are on opposite sides of the black hole, they’re moving parallel to each other, which suggests that their clocks are running in sync. Over multiple orbits, how do the observers’ clocks vary with respect to each other?
With your last bit, the observers are moving parallel in opposite directions, wouldn’t that mean the time dilation is increased rather than their clocks running in sync.
Edit: Never mind, the orbits are opposite so they are travelling in the same direction on opposite sides of the orbit, understood.
Photons can orbit a black hole at a distance of 1.5 times the radius of the event horizon.
They’d have to be dilated by the same amount relative to a distant observer, so they’d have to stay in sync with each other from one orbit to the next.
In normal euclidean space, a disc spinning at relativistic rates will have an apparent circumference decrease of 1/gamma, where gamma is the typical [1-v^2/c^2]^-1/2 factor. You can imagine it like spinning a disc of dough, and having the center bulge upward, to keep the diameter the same but the apparent circumference smaller. Instead of bulging up into the z direction, however, a relativistic spinning disc bulges up into the time direction.
Around a black hole things are a lot trickier, because the circumference you measure for the event horizon will depend not only on your speed but the black hole’s mass and your distance from the horizon. If the black hole is rotating it’s even more complicated. But as was mentioned, for a non-rotating black hole, there is a radius at which you can orbit at the speed of light, which is actually outside the event horizon (1.5 times it, to be precise). Below that line nothing can orbit so it has to either escape or fall in.
This is just a variation of the twin paradox. One guy goes off on a relativistic rocket and comes back younger than his earthbound twin. The “paradox” lies in the fact that you could follow the rocket man, and from his point of view the Earth flies away relativistically and comes back, so the earthbound twin should be younger, right? However, the two situations aren’t symmetrical, because the rocket man changes direction, and therefore changes inertial frames. So the rocket twin will initially see the Earth’s clocks slow down, but when he changes direction and heads home, he’ll see the Earth’s clocks suddenly speed up to make up for lost time, and he’ll find that much more time has passed on Earth than for him.
So for our two orbiters, as they move away from each other they see each other’s clocks slow down, but as they reach the opposite ends of the black hole and start heading back toward each other, they see each other’s clocks speed up, and when they pass each other again their clocks will be back in sync.