Observing Black Holes in Finite Time

[kmm]

TL;DR

Are the black holes we’ve observed actually “almost" black holes, i.e. indistinguishable from the mathematically idealized form of a black hole?

My understanding from General Relativity is that if as distant observers we watch a probe or any test mass approach a black hole, time dilation goes to infinity as the probe gets closer to the event horizon. This would imply that we would never observe a black hole form, or the collision of two black holes. I understand though that if we look at the proper time of the probe, it will cross the horizon briefly, with it's clock ticking as normal in its own frame. This would mean a black hole has no trouble forming or colliding with another black hole. However, since these events would take an infinite amount of time here in our Earth frame of reference, how has the Event Horizon Telescope observed a black hole? In addition, how has LIGO observed the resulting gravitational waves of colliding black holes? In my searches, I haven't found any clear answers to these questions, but the only resolution I have come up with that reconciles these issues is that the black holes we have observed are not "true" black holes in their mathematically idealized form, but rather "almost" black holes. By "almost" black holes I mean that the distribution of mass is so close to that of a "true" black hole that from an observational standpoint, they are both indistinguishable and we may as well regard these "almost" black holes as actual black holes. Is this correct or do I appear to be making any false assumptions here? If this is correct, this would mean that no "true" black holes actually exist?

 

[Nugatory]

My understanding from General Relativity is that if as distant observers we watch a probe or any test mass approach a black hole, time dilation goes to infinity as the probe gets closer to the event horizon.

That is only exactly true if the probe has zero mass and energy so that dropping it into the black hole doesn't change the mass of the black hole. This is a really good approximation if we're dropping any normal-sized object into a stellar-mass black hole, but it's still an approximation. Although we're never able to observe it crossing the horizon, when we drop a probe of mass $m\lt\lt M$ into a black hole of mass $M$ we end up with a black hole of mass $M+m$ fairly quickly.

For the more general problem of black holes forming from collapse, google for "Oppenheimer-Snyder collapse". This is the exact solution of the Einstein field equations for a spherically symmetric shell collapsing under its own gravity to form a black hole, and is a good approximation for how astronomical black holes form from collapsing stars.

in addition, how has LIGO observed the resulting gravitational waves of colliding black holes?

The gravitational waves are emitted from outside of either black hole.

 

[PeterDonis]

how has the Event Horizon Telescope observed a black hole?

Indirectly, by observing that there is a small region of space with a very large mass in it (as shown by objects orbiting it), from which no light is coming, into which things sometimes fall but from which nothing ever comes out, and light passing close to the edge of this region of space is bent in the way we expect light passing close to a black hole to be bent.

 

[PeterDonis]

This is the exact solution of the Einstein field equations for a spherically symmetric shell collapsing under its own gravity to form a black hole

A technical point: the Oppenheimer-Snyder solution describes a continuous, spherically symmetric region of "dust" (matter with uniform density and zero pressure) collapsing under its own gravity to form a black hole. A "shell" would be a region of matter with vacuum inside it; the dust in the O-S solution occupies the entire interior region.

 

[PeterDonis]

By "almost" black holes I mean that the distribution of mass is so close to that of a "true" black hole that from an observational standpoint, they are both indistinguishable and we may as well regard these "almost" black holes as actual black holes.

There is no such thing. Black holes are not made of matter; they are made of spacetime curvature, so there is no "distribution of mass" that can be "close" to a black hole. And there is a finite gap in size between a black hole and the smallest possible object made of ordinary matter and supporting itself against its own gravity; this is due to a result called Buchdahl's Theorem, which says that an object made of ordinary matter and supporting itself against its own gravity must have a radius at least 9/8 of the Schwarzschild radius for its mass--i.e., 9/8 of the radius of a black hole with the same mass. Any object smaller than that must collapse to a black hole, so there can't be an object that is "almost" a black hole, in the sense of being just a little larger than a black hole with the same mass, that doesn't collapse quickly into a black hole.

 

[kmm]

That is only exactly true if the probe has zero mass and energy so that dropping it into the black hole doesn't change the mass of the black hole. This is a really good approximation if we're dropping any normal-sized object into a stellar-mass black hole, but it's still an approximation. Although we're never able to observe it crossing the horizon, when we drop a probe of mass $m\lt\lt M$ into a black hole of mass $M$ we end up with a black hole of mass $M+m$ fairly quickly.

I mentioned the gravitational waves observation, understanding that they are emitted outside of either black hole, since I assumed they shouldn't actually collide (in Earth's frame), since this would take infinite time. However, since the infinite time dilation only applies to an object falling into a hole with zero mass and energy, and is only approximate for normal-sized objects, does this mean that this approximation becomes less and less valid as the mass of the object increases? So in the case of a black hole colliding with another black hole, where both in general will be very massive, we can assume the collision would happen over a relatively small period of time compared to an object of ordinary size, from our frame of reference? If so, would this also be true for a large number of normal-sized objects spanning the sphere of the horizon, falling into the black hole?

For the more general problem of black holes forming from collapse, google for "Oppenheimer-Snyder collapse". This is the exact solution of the Einstein field equations for a spherically symmetric shell collapsing under its own gravity to form a black hole, and is a good approximation for how astronomical black holes form from collapsing stars.The gravitational waves are emitted from outside of either black hole.

I will definitely be looking more into this.

 

[kmm]

There is no such thing. Black holes are not made of matter; they are made of spacetime curvature, so there is no "distribution of mass" that can be "close" to a black hole. And there is a finite gap in size between a black hole and the smallest possible object made of ordinary matter and supporting itself against its own gravity; this is due to a result called Buchdahl's Theorem, which says that an object made of ordinary matter and supporting itself against its own gravity must have a radius at least 9/8 of the Schwarzschild radius for its mass--i.e., 9/8 of the radius of a black hole with the same mass. Any object smaller than that must collapse to a black hole, so there can't be an object that is "almost" a black hole, in the sense of being just a little larger than a black hole with the same mass, that doesn't collapse quickly into a black hole.

Thank you for this clarification!

 

[kmm]

Another question is, does the fact that the formation of a black hole and the collision of two black holes entail a lot of dynamical processes mean that we can't apply standard gravitational time dilation to the process? I imagine that if two black holes are colliding and creating gravitational waves, determining the actual time dilation in the general area of the collision would not be straightforward and definitely wouldn't resemble anything like a very small object approaching an isolated black hole.

 

[PeterDonis]

does the fact that the formation of a black hole and the collision of two black holes entail a lot of dynamical processes mean that we can't apply standard gravitational time dilation to the process?

Yes. Strictly speaking, the concept of gravitational time dilation isn't even well-defined in a spacetime with more than one gravitating mass in it. However, if we have two black holes that are widely separated, we can use the concept to a reasonably good approximation around each one individually; but that doesn't work well when they merge. Well after the merger, when the final hole has settled down to a stationary state, then the concept works again.

 

[kmm]

Yes. Strictly speaking, the concept of gravitational time dilation isn't even well-defined in a spacetime with more than one gravitating mass in it. However, if we have two black holes that are widely separated, we can use the concept to a reasonably good approximation around each one individually; but that doesn't work well when they merge. Well after the merger, when the final hole has settled down to a stationary state, then the concept works again.

I have spent a bit of time with Special Relativity and am just starting to learn General Relativity, so I still have a lot to learn but this thread was clarifying and made me aware of some false assumptions I was making. Thanks again!

 

[PeterDonis]

Thanks again!

You're welcome! I'm glad the discussion has been helpful.