Relativity and Black Hole Question

[Nugatory]

Yeah, I appreciate DrGreg's post. But it's not a perfect analogy, is it?

Of course not. It's an analogy, not a mathematical isomorphism. But it is good enough to bring out the fundamental point about the difference between the space-time around a static spherically symmetric mass and the Schwarzschild coordinates that we sometimes use to do calculations on the trajectory of bodies moving in that spacetime.

Consider DrGreg's intrepid explorer again. There's no question about the physical reality of his path up and over the curved top of the earth. I can track his path on a globe, on a Mercator-projected flat map (poor choice, as I'd have to lift my pencil at some point), on a polar-projected flat map, by writing his $\theta$ and $\phi$ coordinates (also known as latitude and longitude) as functions of my proper time, his proper time. I could redefine latitude and longitude so that Hong Kong was at ninety degrees north latitude and the zero meridian passed through Sao Paolo and now a Mercator projection would work just fine. But no matter how I describe his motion, he doesn't know or care. It is what it is, and any difficulties I have plotting it have nothing to do with his journey and everything to do with my choice of how I attach numbers to his position at any time.

 

[Nugatory]

None of the possible coordinate systems for the black-hole spacetime have an intuitive coordinate time.

Not even Painleve-Gullstrand?

 

[PeterDonis]

the SC coordinates are geodesically incomplete at the event horizon, right?

Right.

It is not possible to follow where the geodesic goes in a finite amount of proper time.

I would say *after* a finite amount of proper time (the finite proper time it takes for the geodesic to reach the horizon).

I appreciate DrGreg's post. But it's not a perfect analogy, is it?

No. It's just a more familiar and easier to understand example of coordinates being highly distorted, so that coordinate intervals are very different from physical distances ("infinitely different" at the North Pole).

Alright, so according to the Rindler observers that stay out of 'Region A', the infalling test-mass will pass into the 'Region A' at some finite Rindler time. And after some finite proper time, the infalling test-mass will pop out of 'Region A', somewhere inside of the event horizon. But inside the event horizon, the Rindler coordinates are not defined, so we can't really follow what happens in Rindler coordinates at that stage. But, we can use Schwarzschild coordinates for the inside of the event horizon. So we can continue to map that test-mass, as it is free-falling inside the event horizon.

You can do this, but it's not quite the same as the Mercator case, because the Schwarzschild coordinate patch that covers the region inside the horizon is disconnected from the patch that covers the region outside the horizon. So the infalling object "reappears" in a *new* coordinate patch that is completely separate from the original one, unlike the Mercator case, where the object passing the North Pole reappears in the same coordinate patch it disappeared from (though in a different part of it).

There is also another weirdness that happens in the Schwarzschild interior coordinate patch (the one that covers the region inside the horizon): the coordinate time $t$ *decreases* as the proper time of the infalling object increases! As the object appears in the patch, just inside the horizon, $t$ starts at a very large positive value (close to $\infty$, since in this coordinate patch $t \rightarrow \infty$ as you approach the horizon from the inside), and it gets *smaller* as the object falls towards the singularity at $r = 0$.

I would still say to the OP that (in a certain sense), according to the Schwarzschild coordinates, the test-mass never falls through the event horizon. But now, I would also add something like "but Schwarzschild coordinates do not represent an intuitive coordinate time, so don't take it too seriously."

Wouldn't it be simpler just to leave out the first sentence? All it does is cause confusion. It's still causing you confusion; see below.

None of the possible coordinate systems for the black-hole spacetime have an intuitive coordinate time.

No, this isn't true. There is at least one whose coordinate time is reasonably "intuitive": Painleve coordinates, in which coordinate time is the same as the proper time of an infalling object (more precisely, of an object falling in "from rest at infinity", which is an idealized case but can be precisely defined mathematically).

Also, about the question of how all the matter gets inside the black hole... well I'd still say that this is only possible because the correct metric inside the matter region is a FRW (or some other) metric. And the Schwarzschild metric is only correct outside the matter region. In this way, the matter distribution can collapse to arbitrarily small size in a finite coordinate time.

And the "this is only possible" part would still be wrong. Matter can get inside a black hole even if its mass is negligible, so that the overall metric is the Schwarzschild metric to a very, very good approximation. The matter falling in doesn't have to be an isolated point particle; it could be a very, very thin spherical shell of matter, with mass much, much less than the mass of the hole. You would still analyze its infall using the Schwarzschild metric, not the FRW metric, and the analysis would still show that the shell of matter would collapse to arbitrarily small size in finite proper time, adding its own small mass to the mass of the hole.

 

[DrGreg]

Yeah, I appreciate DrGreg's post. But it's not a perfect analogy, is it?

It may be a better one than you realize. In the analogy, forget about time and consider only distances. The intention is to consider Schwarzschild time analogous Mercator map vertical distance (rather than time taken to travel a distance), and proper time of an infalling particle to be analogous to distance traveled northward on the Earth's surface. It takes an infinite amount of Mercator distance to cover a finite amount of Earth distance, so in terms of Mercator distance you could say that, according to the map, a meridian line "never" reaches the North Pole, even though we know in reality it does.

Note also that if we continue a meridian line through the North Pole, the trajectory reappears on the map with Mercator distance decreasing (just like the Schwarzschild t coordinate decreases inside the event horizon).



By the way, for what it's worth I find Kruskal coordinates the most helpful to analyse spacetime near an event horizon. The relationship between Kruskal and Schwarzschild is very similar to the relationship between Minkowski and Rindler coordinates (in that order). Rindler's book Relativity: Special, General and Cosmological uses a modified version of "Rindler coordinates" that are even more like Schwarzschild coordinates.

 

[PeterDonis]

Note also that if we continue a meridian line through the North Pole, the trajectory reappears on the map with Mercator distance decreasing (just like the Schwarzschild t coordinate decreases inside the event horizon).

Just one caveat here: the Mercator coordinate patch that the meridian line reappears in is the same one that it disappeared from (i.e., there is a single connected patch that covers both segments of the meridian line). In the Schwarzschild case, the interior and exterior patches are disconnected.

 

[BruceW]

Not even Painleve-Gullstrand?

ah, that is a pretty intuitive coordinate system (I just looked it up). With constant time, the metric is 'flat' space, so a constant-time hypersurface is just Euclidean space. And at positions very far from the black hole, the metric is the same as the special relativity metric. Also, at constant position, the time dilation is the same as for Schwarzschild coordinates. Best of all, test masses fall all the way into the centre of the black hole. I like this metric a lot. thanks.

And the "this is only possible" part would still be wrong. Matter can get inside a black hole even if its mass is negligible, so that the overall metric is the Schwarzschild metric to a very, very good approximation. The matter falling in doesn't have to be an isolated point particle; it could be a very, very thin spherical shell of matter, with mass much, much less than the mass of the hole. You would still analyze its infall using the Schwarzschild metric, not the FRW metric, and the analysis would still show that the shell of matter would collapse to arbitrarily small size in finite proper time, adding its own small mass to the mass of the hole.

I guess, it would fall through the event horizon in finite proper time (even according to the Schwarzschild metric). But there is no coordinate time associated with this process. If anything, it would be at coordinate time of plus infinity. Since we can't assign spacetime events (smoothly) to the matter which falls through the event horizon, I would not like to say Schwarzschild coordinates can give a true description of what is going on. And, it is a nice coincidence that the Schwarzschild metric is not the correct metric when the infalling matter has a significant mass. We are forced to use a different metric, for example the FRW metric. And using this metric, the matter collapses in finite coordinate time. In the Schwarzschild metric, we would have a problem explaining how the matter got to the middle of the black hole, because new infalling matter would take infinite coordinate time to get to the centre of the black hole.

 

[PeterDonis]

I guess, it would fall through the event horizon in finite proper time (even according to the Schwarzschild metric).

Yes.

But there is no coordinate time associated with this process.

No. There is no Schwarzschild coordinate time associated with it (meaning the event of the infalling object being exactly at the horizon). There is a perfectly well-defined, finite coordinate time for it in other charts: the Painleve chart, for example, but also the Eddington-Finkelstein chart and the Kruskal chart. So the correct response to your remark that there is no finite coordinate time for it in the Schwarzschild chart is "so what?"

Since we can't assign spacetime events (smoothly) to the matter which falls through the event horizon

Wrong. The events associated with the matter being at or below the horizon are perfectly well defined. You just can't assign them coordinates in the exterior Schwarzschild chart. But again, so what? There is no rule that says every event must have finite coordinates in every chart. (For example, the Rindler chart on Minkowski spacetime can't assign finite coordinates to any events outside the "wedge" bounded by the lines $t = x$ and $t = -x$, to the right of the origin. But that obviously doesn't stop those events from being well-defined.)

(Note, also, that you *can* assign finite coordinates to events inside the horizon using interior Schwarzschild chart; and you can also take limits as the horizon is approached in order to compute invariants, even though Schwarzschild coordinates are singular at the horizon. See further comments below.)

I would not like to say Schwarzschild coordinates can give a true description of what is going on.

They can if you focus on invariants instead of coordinates. You can use Schwarzschild coordinates to compute all the invariants, including the finite proper time it takes for an infalling object to reach the horizon. (You do that by taking a limit as $r \rightarrow 2M$, i.e., as the horizon is approached; the limit is perfectly well-defined even though the coordinates themselves are singular at the horizon.) And since there is also an interior Schwarzschild chart, you can compute all invariants inside the horizon using that chart; and you can also verify that all limits taken as you approach the horizon "from the inside" in that chart give the same values as limits taken as you approach the horizon "from the outside" in the exterior chart. So you are simply wrong in saying that Schwarzschild coordinates cannot give a "true description" of what is going on. It may be more cumbersome to do it than it would be in another chart, but it can be done.

And, it is a nice coincidence that the Schwarzschild metric is not the correct metric when the infalling matter has a significant mass. We are forced to use a different metric, for example the FRW metric.

Only to describe the interior of the collapsing matter that originally formed the black hole; the region of spacetime outside the collapsing matter, including both outside the horizon and inside the horizon once it forms, are described by the Schwarzschild metric.

But in any case, you are trying to extend this statement to a very different scenario, in which the black hole already exists, with vacuum outside it (and inside it, for that matter--the interior of a black hole is vacuum), but then some other piece of matter comes and falls in. That scenario is *not* described by the FRW metric; it is analyzed using the Schwarzschild metric, at least for the case where the infalling matter has a much smaller mass than the hole and is confined to a small region of space. There are other possible ways for matter to be added to the hole, of course, but the case I've just described is of great practical interest, since it describes many processes we see going on in the universe now, which involve very massive black holes swallowing objects much less massive than themselves. So you are simply wrong in thinking that matter falling into a black hole cannot be described using the Schwarzschild metric; physicists are doing exactly that.

In the Schwarzschild metric, we would have a problem explaining how the matter got to the middle of the black hole

No, we wouldn't. See above. At this point you are simply repeating the same erroneous statements without any new supporting argument.

 

[Dale]

BruceW, I would like to elaborate a little on this comment of mine:

Coordinate systems are consistent only in that they all agree on the invariants.

You have mentioned the idea of different coordinate systems being "consistent". Let's examine what you mean by this idea.

First, different coordinate systems may not cover the same regions of the manifold. You can see this easily by considering Rindler coordinates on a flat manifold. Clearly, in whatever sense we would discuss consistency, the consistency of two coordinate charts would only be relevant to the intersection between the two charts. For the Schwarzschild chart this means only the region outside (not including) the EH.

Second, coordinate charts clearly disagree on the coordinates assigned, so that cannot be the correct criteria for consistency. So what makes two charts "consistent"? Here, we are interested in physical consistency. In other words, we want to be able to use both charts to represent the exact same physics. That means that there must be a set of quantities that contain all of the physical information, and that those quantities must be the same in all "consistent" charts. Such quantities are given the name "invariant". Therefore, the thing which makes a coordinate chart consistent with another is simply that you can calculate all the same invariants in either chart (in the region of their intersection).

This is why I said above that it is the invariants which are consistent, not the charts themselves.

Once you understand that, then you can simply talk about the invariants and not worry about particular coordinate charts, except when you are going through the effort to actually compute an invariant.

 

[pervect]

Yes.
No. There is no Schwarzschild coordinate time associated with it (meaning the event of the infalling object being exactly at the horizon). There is a perfectly well-defined, finite coordinate time for it in other charts: the Painleve chart, for example, but also the Eddington-Finkelstein chart and the Kruskal chart. So the correct response to your remark that there is no finite coordinate time for it in the Schwarzschild chart is "so what?"

*humor alert*

I personally blame the chart. You'd think that people would warn poor , unsuspecting students that the Schwarzschild coordinates were singular, instead of letting them find out the hard ...

Oh, wait, we did warn them. Nevermind!

 

[PeterDonis]

*humor alert*

I personally blame the chart. You'd think that people would warn poor , unsuspecting students that the Schwarzschild coordinates were singular, instead of letting them find out the hard ...

Oh, wait, we did warn them. Nevermind!

Yep, there's that Schwarzschild chart again, lording it over all the other charts and claiming to give "real" distances and times. Haven't people caught wise to it yet?

 

[BruceW]

Only to describe the interior of the collapsing matter that originally formed the black hole; the region of spacetime outside the collapsing matter, including both outside the horizon and inside the horizon once it forms, are described by the Schwarzschild metric.

But in any case, you are trying to extend this statement to a very different scenario, in which the black hole already exists, with vacuum outside it (and inside it, for that matter--the interior of a black hole is vacuum), but then some other piece of matter comes and falls in...

let's say I'm not trying to extend the statement. The OP's question was how the black hole can form in the first place. The Schwarzschild metric (for sure) cannot describe such a process in finite coordinate time, because infalling matter only gets to the event horizon in infinite coordinate time. Or maybe you would prefer to simply say that the Schwarzschild metric is for vacuum only, so we don't even need to think about whether it could describe the formation of the black hole, since it is not the correct metric anyway. And conversely, something like the FRW metric is a correct metric, and as it happens, for the FRW metric, the infalling matter can pass through the event horizon in finite coordinate time.

This is all I was saying really. So maybe the bit you disagree with is where I was saying 'suppose we did try to use the Schwarzschild metric to describe significant infalling matter, it does not work.' If I change this to simply 'the Schwarzschild metric is not correct when there is significant infalling matter', then that would be better?

... There are other possible ways for matter to be added to the hole, of course, but the case I've just described is of great practical interest, since it describes many processes we see going on in the universe now, which involve very massive black holes swallowing objects much less massive than themselves. So you are simply wrong in thinking that matter falling into a black hole cannot be described using the Schwarzschild metric; physicists are doing exactly that.

Right. and in this case (where the mass of the infalling matter is negligible compared to the mass of the black hole), the infalling matter only reaches the event horizon (in Schwarzschild coordinates). So the matter is not really added to the hole. But that is not important anyway (to the value of the gravitational field outside the black hole), since the matter has insignificant mass compared to the mass of the black hole.

You have mentioned the idea of different coordinate systems being "consistent". Let's examine what you mean by this idea.

First, different coordinate systems may not cover the same regions of the manifold. You can see this easily by considering Rindler coordinates on a flat manifold. Clearly, in whatever sense we would discuss consistency, the consistency of two coordinate charts would only be relevant to the intersection between the two charts. For the Schwarzschild chart this means only the region outside (not including) the EH.

yeah, I think that is a key point to remember. The key thing that I wasn't thinking about earlier in the thread, is that a coordinate chart only needs to be consistent over the region of spacetime on which it is defined. So as you say, if we have two coordinate charts which are equivalent in their intersection (e.g. Schwarzschild coordinates and Gullstrand-painleve coordinates), the Gullstrand-painleve chart can describe matter falling through the event horizon, but the Schwarzschild chart cannot, since the Schwarzschild chart does not include the event horizon.

So, this automatically tells us that we can't use the Schwarzschild coordinates to tell us how that black hole came to exist. And that is fine, because the Schwarzschild coordinates are not defined over all of the spacetime described by the Gullstrand-painleve chart, so they are not required to give a consistent description of everything that can be described by the Gullstrand-painleve chart.

Also, as it happens, the Gullstrand-painleve chart does not describe the formation of the black hole, since the Gullstrand-painleve chart is also a vacuum solution. But anyway, the point is that the Schwarzschild coordinates certainly could not describe the formation of the black hole, because it does not even cover the event horizon.

edit: I keep saying 'consistent description' and 'described by', I guess what I mean by this is if the coordinate system can smoothly assign points in spacetime to geodesics which represent a certain physical process. For example, if the coordinate system can describe a certain pair of people meeting up to have coffee, then it should be able to assign points smoothly in spacetime which are geodesics corresponding to the motion of the two people as they meet each other.

 

[PeterDonis]

The OP's question was how the black hole can form in the first place. The Schwarzschild metric (for sure) cannot describe such a process in finite coordinate time, because infalling matter only gets to the event horizon in infinite coordinate time. Or maybe you would prefer to simply say that the Schwarzschild metric is for vacuum only, so we don't even need to think about whether it could describe the formation of the black hole, since it is not the correct metric anyway.

No, that's still not correct. The correct statement is that, to describe the formation of the black hole, you need a spacetime that has two regions: an FRW region containing the collapsing matter, joined to a Schwarzschild region of vacuum surrounding it. Neither one works by itself; you need both. See below.

And conversely, something like the FRW metric is a correct metric, and as it happens, for the FRW metric, the infalling matter can pass through the event horizon in finite coordinate time.

No, this is wrong. The FRW metric does not contain any event horizon. To describe the horizon and its formation, you need the Schwarzschild portion of the spacetime, even in the case of the intial collapse to form the black hole.

(It's true that Schwarzschild *coordinates* can't be used to fully describe black hole formation; but coordinates are not the same as "metric". See further comments below.)

in this case (where the mass of the infalling matter is negligible compared to the mass of the black hole), the infalling matter only reaches the event horizon (in Schwarzschild coordinates). So the matter is not really added to the hole.

Still not correct. You keep on using words like "really" when you are talking about coordinate-dependent statements. Tnat's wrong; or at least, it's an abuse of language. When you use words like "really", it should be in reference to invariants, not coordinate-dependent things. The invariant "reality" in this case is that matter reaches and passes the event horizon and enters the black hole. If you insist on talking about Schwarzschild coordinates, the only correct thing you can say is that those coordinates cannot *describe* what happens to the infalling matter once it reaches the horizon. But again, coordinates are not the same as metric; see below.

So, this automatically tells us that we can't use the Schwarzschild coordinates to tell us how that black hole came to exist.

Don't confuse coordinates with "metric". The term "Schwarzschild metric", at least as I've been using it and as you have implicitly been using it, refers to a geometric object, independent of the coordinates used to describe it. So when I said, above, that you need both the "FRW metric" and the "Schwarzschild metric" to describe the formation of a black hole, I meant that the full geometry of the spacetime in question consists of two "pieces" with different geometric shapes, the FRW shape and the Schwarzschild shape, joined together at a boundary (the surface of the collapsing matter). You need both pieces to describe the process of black hole formation; so the Schwarzschild *metric* is needed to describe how black holes come to exist.

You can't use Schwarzschild *coordinates* to fully describe the process of black hole formation, because those coordinates can't cover the event horizon. But that doesn't mean the *metric* of the vacuum portion of the spacetime isn't Schwarzschild; it just means the Schwarzschild coordinates can't fully describe the Schwarzschild metric.

Also, as it happens, the Gullstrand-painleve chart does not describe the formation of the black hole

Wrong; G-P coordinates *can* be used to fully describe the Schwarzschild metric portion of the spacetime I described above. Also, IIRC, G-P coordinates happen to match up very easily at the boundary to the "natural" coordinates to use for the portion of the spacetime occupied by the collapsing matter (I think this is more or less what Oppenheimer and Snyder did in their 1939 paper describing gravitational collapse).

 

[Dale]

yeah, I think that is a key point to remember.

I am glad that helped.

edit: I keep saying 'consistent description' and 'described by', I guess what I mean by this is if the coordinate system can smoothly assign points in spacetime to geodesics which represent a certain physical process. For example, if the coordinate system can describe a certain pair of people meeting up to have coffee, then it should be able to assign points smoothly in spacetime which are geodesics corresponding to the motion of the two people as they meet each other.

That is part of what I meant by "invariants". Whether or not two people meet is an invariant. Also, whether or not their motion is a geodesic is also invariant. So both of the things that you mention are not coordinate statements, they are invariant statements. Of course, there are many more than just those two invariants, but the invariants are the things that are "consistent descriptions" between overlapping coordinate charts.

 

[BruceW]

No, that's still not correct. The correct statement is that, to describe the formation of the black hole, you need a spacetime that has two regions: an FRW region containing the collapsing matter, joined to a Schwarzschild region of vacuum surrounding it. Neither one works by itself; you need both. See below.

oh, yes, I forgot to say we need the Schwarzschild metric outside the matter region. Although, for the general case of varying density, I guess we can just use one region. But the metric would not be nice and simple.

No, this is wrong. The FRW metric does not contain any event horizon. To describe the horizon and its formation, you need the Schwarzschild portion of the spacetime, even in the case of the intial collapse to form the black hole.

really? I thought the event horizon forms inside the matter region described by the FRW metric. And as matter falls in through the event horizon, the event horizon gets bigger, until eventually it ends up outside the matter region.

Still not correct. You keep on using words like "really" when you are talking about coordinate-dependent statements. Tnat's wrong; or at least, it's an abuse of language. When you use words like "really", it should be in reference to invariants, not coordinate-dependent things. The invariant "reality" in this case is that matter reaches and passes the event horizon and enters the black hole. If you insist on talking about Schwarzschild coordinates, the only correct thing you can say is that those coordinates cannot *describe* what happens to the infalling matter once it reaches the horizon. But again, coordinates are not the same as metric; see below.

hehe, OK I'll try to talk more about invariants, and less about coordinates in the future.

Don't confuse coordinates with "metric". The term "Schwarzschild metric", at least as I've been using it and as you have implicitly been using it, refers to a geometric object, independent of the coordinates used to describe it.

uh, well, I have just been using metric to mean the same as coordinates up until now. I guess you mean the metric in coordinate-free representation. I'm not so familiar with that. But I can intuitively understand what you mean. it's just the abstract idea of the metric that describes spacetime, when we don't care about what coordinates are being chosen.

Wrong; G-P coordinates *can* be used to fully describe the Schwarzschild metric portion of the spacetime I described above. Also, IIRC, G-P coordinates happen to match up very easily at the boundary to the "natural" coordinates to use for the portion of the spacetime occupied by the collapsing matter (I think this is more or less what Oppenheimer and Snyder did in their 1939 paper describing gravitational collapse).

yeah, sorry I meant that G-P coordinates can't describe the matter region. I should have explained myself better.

 

[Nugatory]

uh, well, I have just been using metric to mean the same as coordinates up until now. I guess you mean the metric in coordinate-free representation. I'm not so familiar with that

This is a fundamental issue - it's hard to overstate its importance.

The metric itself is a coordinate-independent object that describes the geometry of a given space. It may look very different in different coordinate systems, but it's still the same metric, just as "one kilometer north and one kilometer east" is the same vector as "negative one kilometers south and one kilometer east" or "root-two kilometers north-east". This is a basic property of all tensors, including the metric tensor.

Another example: there is only one metric for a flat two-dimensional surface. In cartesian coordinates, its components have the values: $g_{00}=g_{11}=1$ and in polar coordinates they have the values $g_{00}=1$ $g_{11}=r^2$ but they're both describing the exact same flat two dimensional space.

Likewise, there is only solution to the Einstein field equations for the vacuum outside of a static spherically symmetric mass distribution, and that is the Schwarzschild solution described by the Schwarzschild metric. We can write this metric using Schwarzschild coordinates (which, interestingly, are not the coordinates that Schwarzschild himself used in his original paper), PG coordinates, kruskal coordinates, or whatever... But they are all different ways of writing down the same solution to the same differential equation.

 

[PeterDonis]

oh, yes, I forgot to say we need the Schwarzschild metric outside the matter region. Although, for the general case of varying density, I guess we can just use one region. But the metric would not be nice and simple.

Not if there's vacuum outside the collapsing matter; then you still need two regions, a matter region and a vacuum region. You're correct that the metric in both regions would be more complicated than in the simple idealized model we've been discussing; but it would still be a *different* metric, a different geometry, in the matter region and in the vacuum region.

If there isn't vacuum outside the collapsing matter, then you're modeling a collapsing universe, not a collapse to a black hole. See further comments below.

I thought the event horizon forms inside the matter region described by the FRW metric. And as matter falls in through the event horizon, the event horizon gets bigger, until eventually it ends up outside the matter region.

This is true as long as there *is* a vacuum region outside the matter region. If there isn't, then the term "event horizon" has no meaning, because there's no null infinity in the spacetime.

Let me describe this another way, to make it clearer what I'm saying. Consider a small piece of matter at the exact center of the spherically symmetric body of matter that is collapsing. At some point on the worldline of this piece of matter, an outgoing null ray will be emitted that just happens to reach the surface of the collapsing matter at the exact instant that that surface is at $r = 2M$, i.e., at the horizon radius for the collapsing matter (as given by its total mass). That outgoing null ray then stays at $r = 2M$ forever; i.e., it marks the event horizon.

If we are looking at the spacetime as a whole, then yes, the entire path of that null ray, starting from the piece of matter at the exact center, marks the event horizon, so in that sense, yes, there is an event horizon inside the matter region. But the only reason it *is* an event horizon is that that null ray never escapes to infinity; it gets trapped at $r = 2M$, even though it has exited the collapsing matter and is in the vacuum region. But for the idea of "escape to infinity" to make sense in the first place, there has to *be* the vacuum region exterior to the collapsing matter; if there isn't, then as I said above, you're describing a collapsing universe, with finite spatial volume, so there is no "infinity" and therefore no meaningful concept of "escape to infinity". So there is no meaningful concept of "event horizon" that applies to the FRW metric by itself; the concept only applies if the FRW metric is joined to the exterior vacuum (Schwarzschild) metric.

(Also, if the collapsing FRW metric describes the entire spacetime, then there is no "center" and no "surface"--the matter occupies the entire spacetime and the spacetime is homogeneous--so the description I gave above of the null ray being emitted from the center and reaching the surface at $r = 2M$ is meaningless. That description only makes sense if the collapsing FRW metric is joined to an exterior vacuum metric.)

 

[yuiop]

If the infalling matter is negligible in mass compared to the black hole, the Schwarzschild solution is not *exactly* correct, but it's a very, very good approximation, certainly good enough that if it did in fact predict that matter could not fall through the horizon, that would be a problem. There is no problem because it does *not* predict that.

Could you elaborate on why you think there "would be a problem"?

 

[PeterDonis]

Could you elaborate on why you think there "would be a problem"?

"Would be a problem" in the sense that that model wouldn't work if it didn't match observations, so we would have to find a different model. To some extent this is circular, since we interpret observations in terms of theory; but our observations of highly compact, massive objects, such as the various stellar black hole candidates and the supermassive object at the center of our galaxy, would be, IMO, very hard to reconcile with any theory that did *not* predict that there were event horizons and matter could fall into them.

(Technically, the above is true of a classical theory, i.e., one that doesn't include quantum effects on black holes, such as Hawking radiation or the more speculative suggestions about quantum effects preventing a true event horizon from forming at all. But even in the speculative quantum models where there isn't actually an event horizon, in that any information that goes into a "black hole" region eventually comes back out, there is still a long period of time where there is a compact region containing trapped surfaces that looks like a standard classical black hole. The only difference is in the far future--"far" meaning times many, many orders of magnitude longer than the current age of the universe. So for practical purposes, such models are the same as the standard classical black hole model, since in that model, you can't locally distinguish between a true event horizon and a local trapped surface anyway; you have to know the entire future of the spacetime to do that.)