# Formation of the event horizon seems impossible with singularity inside seems impossible [duplicate]

According to what I was reading, the formation of an event horizon appears if the mass becomes compact enough. Unfortunately, due to time dilation the time for mass to fall into the event horizon becomes infinite, so that nothing can cross the horizon within in the life time of the universe. Another thing is, that energy loss because of hypothetical hawking radiation would take place in finite time. Consequently, there must be something wrong with these models. Which means, there is neither a singularity nor a real event horizon existing out there.

• "so that nothing can cross the horizon within in the life time of the universe" - that fact that nothing can be observed to cross the horizon by an exterior observer doesn't mean that nothing in fact crosses the horizon. Apr 1, 2015 at 11:32
• No, according to the equation no mass can pass the horizon.
– user65208
Apr 1, 2015 at 11:33
• possible duplicate of Can black holes form in a finite amount of time? Apr 1, 2015 at 11:42
• Seems so, then it really shouldn't be possible. I was wondering, but why is wikipedia not writing about the fact that black holes cannot exist. Just stars which approach their properties?
– user65208
Apr 1, 2015 at 11:56

"Unfortunately, due to time dilation the time for mass to fall into the event horizon becomes infinite, so that nothing can cross the horizon within in the life time of the universe."

This can not be understood as any objective statement about coordinate-independent physical realities, since it's only true in certain coordinates like Schwarzschild coordinates. In general relativity it's the geometry of spacetime, defined by the proper time along arbitrary timelike curves and the proper distance along arbitrary spacelike curves, that's fundamental and coordinate-invariant. To elaborate on this a little, if you have two coordinate systems on the same patch of spacetime with a coordinate transformation between them, then if you describe a curve (a one-dimensional physical path through spacetime, which in the case of timelike curves could be some slower-than-light particle's world line) in terms of one set of coordinates, you can use the coordinate transformation to describe the same curve in the other set of coordinates. Then you can express the metric in terms of each coordinate system, and use it to calculate the proper time or proper distance along the curve (assuming it's timelike or spacelike rather than lightlike), and both should give the same answer, which is why proper time/proper distance is considered more "physical" and objective than any coordinate-dependent measurement.

When people talk about things taking an infinite time to reach the horizon, they aren't talking about the falling particle's own proper time, but rather about the coordinate time in some coordinate system like Schwarzschild coordinates. But even in these coordinates, you can show that the particle's own proper time approaches some finite value T in the limit as coordinate time goes to infinity. It's possible to construct other coordinate systems where the same particle crosses the horizon in finite coordinate time, and these coordinates will agree that the particle's own proper time reaches T at the moment of the crossing. One example is ingoing Eddington-Finkelstein coordinates, graphed below in a diagram from this page (the event horizon is the vertical line which has the label r=2m at the bottom):

Another example is Kruskal-Szekeres coordinates; shown below is a diagram from the textbook Gravitation by Misner/Thorne/Wheeler, showing the surface of a collapsing star (the boundary of the gray region) graphed in both Schwarzschild coordinates (which has the weird property that inside the horizon, the collapsing surface is actually moving backwards in time relative to the time coordinate) and Kruskal-Szekers coordinates (where the surface crosses the horizon, the dotted line marked r=2M, at a finite coordinate time and then continues moving forward in coordinate time until reaching the singularity, whose coordinate position is marked by the black sawtooth hyperbola):

Note that the Kruskal-Szekeres diagram also shows what curves of constant Schwarzschild position coordinate (the hyperbolas labeled r=2.5M, r=3M etc.) and constant Schwarzschild time coordinate (the straight lines labeled t=35.1M, t=44.3M, t=45.4M etc.) look like when transformed into Kruskal-Szekeres coordinates.

Another point is that this kind of issue can also crop up in the flat spacetime of special relativity--for example, in the non-inertial coordinate system known as Rindler coordinates in which a group of observers with constant proper acceleration are represented as being at rest, there is a boundary called the Rindler horizon which the accelerating observers can never see beyond, and any particle moving towards it will take an infinite coordinate time to reach it in Rindler coordinates. But in an inertial coordinate system, the same particle crosses the Rindler horizon in finite coordinate time. Drawn in the coordinates of an inertial frame, the Rindler horizon is a lightlike boundary, which would be the edge of some future light cone; and the accelerating observers, who in Rindler coordinates have a fixed coordinate position, would in the inertial frame have worldlines that look like hyperbolas which get closer and closer to the Rindler horizon but never quite reach it. The diagram below shows how things look in the inertial frame, with the Rindler horizon represented as the diagonal dotted line and the accelerating observers (or equivalently, the lines of constant Rindler position coordinate) represented as black curves, and gray :

Note that these curves of constant Rindler position and time coordinate, when transformed into inertial coordinates, look just the same as the curves of constant Schwarzschild position and time coordinate when transformed into Kruskal-Szekeres coordinates. So conceptually it may be a useful analogy to think "Kruskal-Szekeres coordinates are to Schwarzschild coordinates as Inertial coordinates are to Rindler coordinates".

• You say the particle/surface "crosses the horizon in finite coordinate time" in Kruskal-Szekeres coordinates, but that's not what I'm seeing in the figure from the book. The horizon I'm seeing it cross is a solid line labeled $r = 2M, t = \infty$ (not the "dotted line" you mentioned, which is at $t = -\infty$). After it crosses, you say it "continues moving forward in coordinate time until reaching the singularity", but I'm seeing coordinate time ($t$) decrease just as it does in Schwarzschild coordinates. Apr 15 at 4:15
• @GumbyTheGreen - I'm defining the time coordinate as the vertical "v" coordinate. I guess this is just a matter of terminology, but see the part of my answer where I said the diagram shows "curves of ... constant Schwarzschild time coordinate (the straight lines labeled t=35.1M, t=44.3M, t=45.4M etc.)", i.e. it's showing what happens when you take a line that has constant t-coordinates in Schwarzschild coordinates and map it into the Kruskal-Szekeres diagram. Also note that whatever you choose to call the v-coordinate, any line parallel to the v-axis is "timelike" in the physical sense. Apr 15 at 21:34

No, relativity is about reference frames. The person falling into a black hole will pass through the event horizon, but due to time dilation, no one else can observe it.

• If the formation of the black hole takes infinite time, how the person will ever fall into the hole? Even if you swap the time frame and even if this is theoretically possible it would also mean, that the person will see the universe to end before crossing the horizon AND that the black hole with decay because of hawking radiation before the person will cross the border.
– user65208
Apr 1, 2015 at 11:44
• @dgrat The problem is that you are confusing reference frames. The falling person would experience no time dilation. Apr 1, 2015 at 11:59
• Ah I got it, but the universe around the scope of the black hole would end before the persons crosses the border anyway?
– user65208
Apr 1, 2015 at 12:02
• @dgrat exactly, only to the observers would (s)he never cross Apr 1, 2015 at 12:13
• @dgrat all the effects of relativity are on the "other guy" Apr 1, 2015 at 12:26

I can only offer an opinion rather than a peer-reviewed authoritative response. But I hope it's of some use.

We can read various descriptions about falling into a black hole, such as this on Baez. There's also Andrew Hamilton's website. I'm afraid to say I think they're wrong. Take a look at The Formation and Growth of Black Holes on mathspages, and you can read this:

"Historically the two most common conceptual models for general relativity have been the "geometric interpretation" (as originally conceived by Einstein) and the "field interpretation" (patterned after the quantum field theories of the other fundamental interactions). These two views are operationally equivalent outside event horizons, but they tend to lead to different conceptions of the limit of gravitational collapse. According to the field interpretation, a clock runs increasingly slowly as it approaches the event horizon (due to the strength of the field), and the natural "limit" of this process is that the clock asymptotically approaches "full stop" (i.e., running at a rate of zero). It continues to exist for the rest of time, but it's "frozen" due to the strength of the gravitational field. In contrast, according to the geometric interpretation, all clocks run at the same rate, measuring out real distances along worldlines in curved spacetime. This leads us to think that, rather than slowing down as it approaches the event horizon, the clock is following a shorter and shorter path to the future time coordinates. In fact, the path gets shorter at such a rate that it actually reaches the future infinity of Schwarzschild coordinate time in finite proper time..."

This ought to say depth of potential rather than strength of field, and the author originally referred to Wheeler and Weinberg until I queried it with Weinberg, but no matter. The important point is that there are two interpretations of GR. One leads to a black hole akin to Oppenheimer's original frozen star. The other leads to the black hole as its usually described, with a point singularity in the middle. But see where the author talks about the infalling clock reaching future infinity in finite proper time? I think that can't be right because it hasn't happened yet, and because it goes to the end of time and back. Then like Susskind's elephant, it's is in two places at once. You can kind of see this on this picture cropped from figure 32.1 of Misner/Thorne/Wheeler. Note that the plot is truncated vertically, and that the infalling body is at two locations horizontally.

Because of the above, and because of issues to do with the speed of light, I think the original frozen-star interpretation is the one that's correct. Nothing passes through the event horizon. But I don't think that this means a black hole can't form. Think in terms of a hailstone. Imagine you're a water molecule. You alight upon the surface. You can't pass through this surface. But you are surrounded by other water molecules, and buried. So whilst you can't pass through the surface, the surface passes through you. I think there's something similar going on with the event horizon, which expands outwards from the centre of a collapsing star. Black holes exist. There is something very small and very massive at the centre of our galaxy. But I would venture to suggest that point singularities do not.