Proper frame of reference and the formation of black holes

[kmarinas86]

Since in our proper frame of reference the fall of material into a black hole would appear to take for an eternity, would it not follow then that all black holes would appear as these instead?

Would in fact all black holes appear this way?

What if you have a small black hole and then have a large black hole pass by within 1.0001 of its large schwarzschild radius? Wouldn't material above 1.0001 of the schwarzchild radius of the smaller object accelerate the other way?

If we could track the positions of particles relative to our proper frame of reference could we not project that some the mass of the smaller black hole leave the system entirely as a gamma ray burst?

 

[Chris Hillman]

Hi, kmarinas86, I couldn't understand your question, but let me try to clarify one or two things:

Since in our proper frame of reference the fall of material into a black hole would appear to take for an eternity,

Careful, "frame of reference" is a term usually used only in flat spacetime in elementary str.

You are no doubt referring to the fact that when we represent an exterior region of the Schwarzschild vacuum solution in the exterior Schwarzschild chart, radially infalling null and timelike geodesics appear to bend upwards and to approach $$r=2m$$ asymptotically. This represents a failure of the coordinate chart to accurately reflect what happens at the horizon; better behaved charts show that infalling matter and radiation passes right through the horizon.

What is going on here is that due to the curvature of this spacetime, outgoing radial null geodesics are diverging, especially very near the horizon. This gives rise to the so-called "gravitational redshift/time-dilation" effect. While careless authors of popular works continue to describe this in language reminiscent of the old (incorrect) "frozen star" idea, all modern gtr textbooks (MTW, Gravitation, Freeman, 1973, or later) explain why this language is misleading.

would it not follow then that all black holes would appear as these instead?

Would in fact all black holes appear this way?

I couldn't understand the question here, but it might help to say that what a distant observer would actually see is different from what the exterior Schwarzschild chart might lead you to expect: luminous infalling matter appears to be falling until it is very near the horizon, but it doesn't really appear to "hover" there; rather, it appears to suddenly redshift into invisibility. That is, both the frequency of received radiation and the luminosity suffer exponential decay.

What if you have a small black hole and then have a large black hole pass by within 1.0001 of its large schwarzschild radius? Wouldn't material above 1.0001 of the schwarzchild radius of the smaller object accelerate the other way?

Again, I can't understand the question, but it might help to recall that the EFE is nonlinear, so an exact vacuum solution describing two black holes certainly won't be a linear superposition of two copies of a solution describing one black hole (e.g. the Schwarzschild vacuum). In particular, you shouldn't assume that "Schwarzschild radius" makes sense for two black holes in the process of merging (or suffering a very near miss); both geometric intuition (see the "pair of pants" diagram in Hawking & Ellis, Large Scale Structure of Spacetime, Cambridge University Press, 1973) and numerical simulations clearly suggest that the horizon of a black hole can become significantly distorted by the presence of a nearby compact object such as a neutron star or another black hole.

If we could track the positions of particles relative to our proper frame of reference could we not project that some the mass of the smaller black hole leave the system entirely as a gamma ray burst?

As far as I can make out, you are asking if the near miss of two black holes could produce a gamma ray burst.