Black Hole Event Horizon Hypothesis

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Wouldn't the definition of the event horizon of a black hole be the radius at which the acceleration of gravity exceeds the speed of light, instead of the radius at which the escape velocity exceeds the speed of light?


It's very clear to me that a photon emitted at a position where the acceleration of gravity is just below the speed of light, traveling exactly upwards away from the singularity of the black hole, would make it out, but extremely red-shifted. (A photon not traveling directly outwards would be pulled into the black hole at a greater radius.)

Thought experiment: a baseball-sized probe is put into a moderately rapidly descending orbit around a black hole of around 10,000 solar masses, where it's estimated that its escape velocity would exceed the speed of light at a radius of around 30,000 kilometers. The radius at which the acceleration of gravity would exceed the speed of light is of course much less. The probe emits a very tight, almost laser-like intense beam of mostly high-energy gamma rays, mixed with some lower frequencies for ease of detection, pointing straight out of the black hole.

As its orbit descends, the high-energy gamma rays become red-shifted down to lower-energy gamma rays, x-rays, ultraviolet then visual light, infrared, microwave, and the radio bands, down to the VLF band, below 30 KHz.

At 30 KHz, the time dilation would be on the order of 10^18, so one second seen by the probe equals over 30 billion years outside the black hole's gravitational well! (One problem: what began as an intense flux of gamma rays, has now become the arrival of one of the vastly lower energy photons once or less per millennium.)

The point is that the photons make it out of the black hole, albeit with extreme red-shifting, having lost almost all of their energy in the climb out. They've still gained gravitational potential energy at their lower energy level, because if they were reflected back down, they'd regain all of the energy they lost, and the probe would see the return of gamma rays of the same energy as the ones that had left.

When the probe emitted these gamma rays, the acceleration of gravity was one part in 10^18 below the speed of light.

I have what appears to be very strong logical proof of the above, but nothing else. Can anyone say why the photons so near to where the acceleration of gravity is equal to the speed of light would not make it out? And therefore, information makes it out, as well?

Note: an observer at the exact location of the photon traveling up out of the black hole would always observe it to be traveling at exactly the speed of light, but an observer some distance away might not, due to time dilation. A more practical example: what appears to be the slowing of light when passing through a gravitational field is actually time dilation within it, however slight. An observer traveling with a photon would always observe it to be traveling at exactly the speed of light.
 
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Ibix
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Wouldn't the definition of the event horizon of a black hole be the radius at which the acceleration of gravity exceeds the speed of light, instead of the radius at which the escape velocity exceeds the speed of light?
You can't meaningfully compare a speed and an acceleration, so your entire approach is mistaken. Even if that were not the case, there is no "acceleration of gravity" in general relativity. The proper acceleration of a free falling body is zero. Even if that were not the case, the definition of an event horizon is a surface separating regions that can send signals to infinity from regions that cannot. Note that escape velocity does not come in to this.

Sean Carroll's lecture notes on general relativity are a good and free to download source to learn GR if you want to do so.
 
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You can't meaningfully compare a speed and an acceleration, so your entire approach is mistaken. Even if that were not the case, there is no "acceleration of gravity" in general relativity. The proper acceleration of a free falling body is zero. Even if that were not the case, the definition of an event horizon is a surface separating regions that can send signals to infinity from regions that cannot. Note that escape velocity does not come in to this.

Sean Carroll's lecture notes on general relativity are a good and free to download source to learn GR if you want to do so.
I see you're talking general relativity, when I was talking Newtonian physics.

I meant "acceleration of gravity" in the same way as dropping a ball from a height. In freefall, the ball is accelerating, but does not feel it because gravity is pulling in the opposite direction, so the forces cancel. The ball feels having been accelerated when it hits the ground.

Using the black hole singularity as the frame of reference, a free falling body is accelerating towards it. Its velocity relative to the singularity is increasing, and the rate of increase is increasing, as well. The free falling body does not feel the acceleration, because the force of gravity is pulling in the opposite direction, and the forces cancel.

I do see a problem when the velocity of the free falling body approaches the speed of light relative to the black hole singularity, but yet it's impossible for matter to travel at or above the speed of light. I now see my error of trying to apply the Newtonian physics of gravity in a place where it clearly breaks down.

However, one question remains. How near to a black hole singularity of known mass can a photon be created, and be able to make it out of the black hole, no matter how low its energy upon its exit? This must surely be the same radius at which information can make it out, as well. How sure are we that this is the same distance as the Schwarzschild radius?

Looking at it in a different way, regarding time dilation only, how near to the black hole singularity could the highest energy gamma rays be created, and be able to escape the black hole, even at the lowest energies, say what the U.S. Navy uses for submarine communications, say 10 Hertz electromagnetic waves? In other words, at what point does time dilation reach infinity? Is this also the same as the Schwarzschild radius?

Of course, I haven't had time to read Sean Carroll's lecture notes, even just the parts about black holes. But I will do so soon.
 
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jbriggs444
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I see you're talking general relativity, when I was talking Newtonian physics.

I meant "acceleration of gravity" in the same way as dropping a ball from a height.
In Newtonian physics there is no such thing as a black hole. In Newtonian physics, just as in General Relativity, you cannot meaningfully compare a speed to an acceleration.

The "black hole singularity" is a prediction of General relativity. It is not a point in Newtonian 3-space and does not define a frame of reference.
However, one question remains. How near to a black hole singularity of known mass can a photon be created, and be able to make it out of the black hole, no matter how low its energy upon its exit? This must surely be the same radius at which information can make it out, as well. How sure are we that this is the same distance as the Schwarzschild radius?
Black holes are not something that we have explored physically. They are something that we have predicted mathematically. The predicted properties of a black hole come from the mathematics. These predictions cannot be otherwise than what the mathematics says.

The notion of "near to the singularity" suggests that you are still thinking in Newtonian terms. The path length to the singularity is not a distance. It is a duration.
 
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PeroK
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I see you're talking general relativity, when I was talking Newtonian physics.
At 30 KHz, the time dilation would be ...

There's no time dilation in Newtonian physics. Newtonian time is absolute.
 
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Ibix
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I see you're talking general relativity, when I was talking Newtonian physics
Newtonian physics does not include either time dilation or event horizons, nor an accurate model of light in gravitational fields. Newtonian "dark stars" with escape velocities exceeding ##c## were proposed centuries ago, but they are nothing like black holes. Nor does Newtonian physics allow you to compare speeds and accelerations. That's simply incoherent - it's like asking whether a volt is taller than a kilogram.
However, one question remains. How near to a black hole singularity of known mass can a photon be created, and be able to make it out of the black hole, no matter how low its energy upon its exit? This must surely be the same radius at which information can make it out, as well. How sure are we that this is the same distance as the Schwarzschild radius?
Light can escape from anywhere above the event horizon. The redshift of light increases arbitrarily high as its source approaches the horizon but is not defined for things at or below the horizon. For a source hovering at radial coordinate ##r## (not the same as distance from the centre, which is ill-defined) the frequency observed at infinity is a factor of ##\sqrt{1-GM/c^2r}## lower than that emitted.
 
  • #7
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In Newtonian physics there is no such thing as a black hole. In Newtonian physics, just as in General Relativity, you cannot meaningfully compare a speed to an acceleration.

The "black hole singularity" is a prediction of General relativity. It is not a point in Newtonian 3-space and does not define a frame of reference.

Black holes are not something that we have explored physically. They are something that we have predicted mathematically. The predicted properties of a black hole come from the mathematics. These predictions cannot be otherwise than what the mathematics says.

The notion of "near to the singularity" suggests that you are still thinking in Newtonian terms. The path length to the singularity is not a distance. It is a duration.
Of course, we haven't been to black holes physically, but we have observed them, even if only akin to telescopes that made it look as if Mars had canals. Here are two articles about supercomputer simulations we've run of them:

https://www.space.com/43151-how-particles-escape-black-holes.html

https://www.space.com/21581-black-hole-supercomputer-simulation.html
 

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