Was thinking about that huge black hole at the heart of M87, and that got me thinking about time dilation due to gravity and event horizons. "Cook-booking" it, the equation for escape velocity is v = Squareroot(2GM/r), where G is the gravitational constant 6.67408 x 10^-11, M the mass in kilograms of the object you're trying to escape, and r is the starting distance in meters from the center of the mass M. The escape velocity at the event horizon is the speed of light, c, so solving for r:
c = Squareroot (2GM/r); c^2=2GM/r; r = 2GM/c^2
That's straight-forward. The confusion starts with time dilation due to gravity. If you assume an observer infinitely away from the source, the time he observes happening near a mass is To = T*squareroot(1-2GM/rc^2). So, our observer, looking at a clock on the surface of the earth, would see it move 0.9999 times as fast as their own. All well and good. But applying this to black holes, well ...
We can simplify things by noticing that 2GM/c^2 is the formula for an event horizon, and substitute e.h. for that. That gives us To = T*squareroot[1-(e.v./r)]. When r = e.v., we have T*squareroot([1-1]) = T*squareroot(0) = T * 0 = 0. Time seems to stop at the event horizon for our observer. Watching an astronaut move toward the event horizon, our observer sees him red shift further and further until he vanishes. Of course, the poor soul approaching the event horizon won't notice it, but it will seem to him like the entire universe is on fast forward, blue shifting until it seems to disappear.
Now, here's where I get more confused than usual. Both observers are witnessing things concurrently, with one slowing in respect to the universe, and the universe speeding faster to the other. That's no different that an object moving near the speed of light. But here we are on the event horizon. In respect to the universe, the astronaut never crosses, and the astronaut never realizes it. So, if nothing can actually pass an event horizon, how to black holes accumulate mass?
This is my confusion. Obviously black holes accumulate mass and grow. The weird thing going through my mind is that as mass accumulates along the event horizon, the event horizon expands. The astronaut doesn't cross the event horizon; the event horizon eventually engulfs him. And that ... well, I don't know. I'm cook-booking this from a couple of canned equations; nothing more.
After that point, it involves imaginary numbers. And while that's simple enough - call the square root of -1 i, multiply it by squareroot[e.v./(r-1)], and call it a day - have no idea what that would mean. Cook-booking it only goes so far.
So, what am I missing here?
c = Squareroot (2GM/r); c^2=2GM/r; r = 2GM/c^2
That's straight-forward. The confusion starts with time dilation due to gravity. If you assume an observer infinitely away from the source, the time he observes happening near a mass is To = T*squareroot(1-2GM/rc^2). So, our observer, looking at a clock on the surface of the earth, would see it move 0.9999 times as fast as their own. All well and good. But applying this to black holes, well ...
We can simplify things by noticing that 2GM/c^2 is the formula for an event horizon, and substitute e.h. for that. That gives us To = T*squareroot[1-(e.v./r)]. When r = e.v., we have T*squareroot([1-1]) = T*squareroot(0) = T * 0 = 0. Time seems to stop at the event horizon for our observer. Watching an astronaut move toward the event horizon, our observer sees him red shift further and further until he vanishes. Of course, the poor soul approaching the event horizon won't notice it, but it will seem to him like the entire universe is on fast forward, blue shifting until it seems to disappear.
Now, here's where I get more confused than usual. Both observers are witnessing things concurrently, with one slowing in respect to the universe, and the universe speeding faster to the other. That's no different that an object moving near the speed of light. But here we are on the event horizon. In respect to the universe, the astronaut never crosses, and the astronaut never realizes it. So, if nothing can actually pass an event horizon, how to black holes accumulate mass?
This is my confusion. Obviously black holes accumulate mass and grow. The weird thing going through my mind is that as mass accumulates along the event horizon, the event horizon expands. The astronaut doesn't cross the event horizon; the event horizon eventually engulfs him. And that ... well, I don't know. I'm cook-booking this from a couple of canned equations; nothing more.
After that point, it involves imaginary numbers. And while that's simple enough - call the square root of -1 i, multiply it by squareroot[e.v./(r-1)], and call it a day - have no idea what that would mean. Cook-booking it only goes so far.
So, what am I missing here?