Rolando

In alternative (1), which is common in the literature, the frequency of the light does not change with height. In alternative (2), it does change, but in a sense opposite to that of the Fe in alernative (1). If the frequency of the Fe changes, and the light frequency changes by the same factor, there will be no observable change. If the light frequency changes by the inverse factor, the observable change will be twice as large as the observed.

The gravitational redshift is described in the literature in two ways: (1) the phenomenon is explained through the behaviour of atoms that emit radiation of lower frequency (or clocks that run more slowly) the deeper they are located in a potential well, while the frequency of the light (or photons) does not change with height. (2) the phenomenon is alternatively discussed in terms of an energy loss of a photon as it overcomes the gravitational attraction of the massive body. In this case, the frequency of the light (or photons) decreases with increasing height. Alternative (2) is rooted in Newtonian physics and it has often been repeated that it is inappropriate in the frame of GR, e.g., http://arxiv.org/pdf/physics/9907017, but it is, nevertheless, widely popularized, e.g., in http://en.wikipedia.org/wiki/Gravitational_redshift (see the colorful illustration there). It is clear to me that alternative (2) is incompatible with GR. However, alternative (1) seems to me to be incompatible with the idea behind GR as well. According to GR, time is dilated in gravitational wells, so that, e.g., the gamma rays emitted by 57Fe in the Pound-Rebka experiment at the lower altitude will have a lower frequency than those emitted at the higher altitude. In GR, this is an effect of the curvature of space-time. However, if this is so, then why is the light itself not affected by this curvature? If it was affected to the same extent as the atoms, the gravitational frequency shift could not be detected experimentally, and if it is necessary to assume that light propagates through a different space-time, this is very much in need of a logical explanation.

The fact that spacetime geometry is well behaved at the gravitational radius does not mean that there cannot be a hard surface there, which would prevent an object from falling any further. So, I think that the pathology does not reside in the coordinate systems but in the neglect of the presence of a hard surface. The Schwarzschild metric suggests such a surface to be present into future eternity. By the way, "Schwarzschild" means 'black shield'.

According to my understanding, infalling observers will not either see any further than the event horizon unitil they reach it. This does not suggest that they will be able to pass across it.

In the view of any observer who is not falling into the particular black hole in question, an infalling observer will reach the event horizon only in the infinite future. If this is true, infalling observers cannot either pass through the event horizon in their own view. They will approach the event horizon in a short time and be smashed there. Unless nature is self-contradictory, the standard teaching, which says that the event horizon will not be noticed by an infalling observer, cannot be true. (At present I am not able to put my finger on exactly where the error in the standard teaching is.)

If you wish so, you can say that I am cherry picking reality. A coordinate system that does not extend to R<r_s is not mathematically equivalent to a system that does.

In order to be sure that we see a classical black hole, it would be necessary to have free sight down to the event horizon (not across it), and I claim that this is impossible if the universe is finite in age. The objects that have been observed and called black holes could all equally well be something like gravastars, in which nothing ever can cross the event horizon. Real events must be observer independent, but I think that certain frames of reference cover all of real spacetime and nothing else, while others transcend reality – they describe imaginary regions (and imaginary events) in addition to reality. Schwarzschild coordinates (for R>r_s) belong to the first set. In my view, there can be no real but just imaginary events within the region where R<r_s, and crossing the r_s boundary is also an imaginary event.

This is off-topic, but it stroke me as even more evident than you made it appear. As seen from within a gravitational well that is close to being a black hole, the temperature of the CMBR is, of course, much higher than the 2.7 K we see. I actually thought of something like "impassable in or to the view of an observer". This is what I had put into question because it cannot happen in e.g. Schwarzschild coordinates. I think that all the matter must remain in a thin shell outside the event horizon. This would make it impossible for anything to pass through, and this cannot reasonably depend on the choice of coordinates.

Thanks for this link to a paper by Vachaspati, Stojkovic and Krauss (2007). I see that I would have them on my side in the present discussion. In their Introduction they say: “The process of black hole formation is generally discussed from the viewpoint of an infalling observer. However, in all physical settings it is the viewpoint of the asymptotic observer that is relevant.” This is exactly my point, and several of you attempt to push me into the infalling observer position – which I refuse for the reason these authors express in the second clause. Further below, they say: “A crucial aspect of our analysis is that we address the question of black hole formation and evaporation as seen by an asymptotic observer. Initially, when the domain wall is large, the spacetime is described by the Schwarzschild metric, just as for a static star. From here on, the wall and the metric are evolved forward in time, always using the Schwarzschild time coordinate.” Now I understand that the Gullstrand-Painlevé metric, as suggested by Strange, would also be fully adequate. Although my conclusions seem to be harsh, I maintain their validity for classical black holes. Observers will not agree on a duration, but they should agree on the presence/absence of an impassable boundary.

I am not sufficiently familiar with the Gullstrand-Painlevé metric, but I think that it is not useful for understanding what goes on within one or several additional gravitational wells at a distance. Trivia: I once read Painlevé's original paper on this topic - a short essay that was easy to read. Then I had a look at Gullstrand's paper - sveral pages full of equations and essentially no words. Terrible.

May be, but hardly easier to discuss, and it is also "local", in the sense of MigL. I inserted it in plain text. It may have been too long.

Thanks, MigL. Consider this figure by Brian Koberlein. ( https://lh3.googleusercontent.com/-NgZK-Ee7RAk/UJPrJIsviFI/AAAAAAAABWs/BmVe1s0Y7Hw/w480-h480/freefal.png ) The black curve is valid for an outside observer - the red one for an observer who is falling into the black hole. Now, according to my reasoning, black holes (in the strict sense of the word) do not exist. The black curve ends at the event horizon (or a little above if the universe has only existed for a finite time). Since I believe in only one reality, in that the existence/non-existence of objects is independent of the observer, the red curve has also to end at the event horizon if it is to describe reality only. The fact that the course of the line can be calculated also at shorter distances from the singularity is not a valid argument for its reality. This part of the line does not proove that motion across the event horizon is possible in reality. It presupposes it (erroneously).

I do have a basis for my choice. I am reasoning in terms of the coordinate system that is not specific to one single black hole alone. All the familiar alternatives apply only to single black holes - not to the whole set of black holes that might be there. And my reasoning is applicable whenever there is an absolute event horizon. It is not necessary to go into the details of the Kerr metric, which I judge to be appropriate. The Schwarzschild metric is just easier to discuss. By the way, can it really be proved that a metric that covers more than the space exterior to the event horizon is mathematically equivalent to the Schwarzschild metric (exterior sulution)?

Thanks for this link. I am reasoning in terms of the coordinate system that is not specific to a particular black hole.

To me, this means that there are no black holes in reality. Event horizons are the boundary of reality. They are infinitely remote from a distant observer. There are coordinate transformations by which one can get rid of the boundary (if one restricts oneself to just one black hole), but if one does that one transcends reality. The question that remains is whether physical "reality" and "existence" are relative notions. Can something in a physical sense be existing and real for an observer who is falling into a gravitational well while it is non-existing and merely imaginary for a stationary observer? This problem does not arise with gravastars.

I said: ” An object turns into a black hole only when it has contracted to its event horizon.” This was not a good formulation. I am aware that black holes are assumed to form in various ways, and this is not very relevant here. I would then say that ”An object turns into a black hole only when it has reached a state at which its extension coincides with its event horizon.” Is this acceptable? If it is untrue that it takes an eternity for light to reach us from the event horizon of a black hole, then I would like to know how to calculate how long it takes. According to my, admittedly somewhat outdated, knowledge of black holes, anything that is exactly at the event horizon would appear as frozen to an outside observer if it was observable at all, which it is not really, because of infinite time dilation and redshift. I did not want to change the subject. I had expressed myself inappropriately.

I would be grateful for more convincing comments. And let us keep the discussion to what we can observe.

I was editing my previous post, addressing this while It takes more time for light to reach us from a gravitational well, the closer it comes to a black hole in its properties. An object turns into a black hole only when it has contracted to its event horizon. At that point, it takes an eternity for light to reach us from there. If the Universe has only existed for a finite time, it cannot contain such black holes and event horizons. And, in our view, everything that is at the event horizon remains there for an eternity.

Your description of falling into a black hole agrees with how this uses to be described, except that "infinitely steep" has to be substituted for "infinitely curved". However, we on Earth are not falling into a particular black hole. We are just looking into deep gravitational wells that are claimed to be black holes. It takes more time for light to reach us from a gravitational well, the closer it comes to a black hole in its properties. An object turns into a black hole only when it has contracted to its event horizon. At that point, it takes an eternity for light to reach us from its event horizon. If the Universe has only existed for a finite time, it cannot contain such black holes and event horizons.

Those who claim to have seen blackholes are likely to express themselves sloppily. Right, Kerr blackholes are the Kerr metric is much more realistic. Kerr blackholes are similar to Schwarzschild blackholes only if looked at in a polar direction. Hovever, this is not really crucial for my argument. The presence of an event horizon alone is sufficient.

Since the problem I raised is so close at hand, I am not surprised that it has been brought up many times before, but I did not know this. I rather thought that those who claim black holes to exist in the Universe, just express themselves sloppily and do not mean classical black holes, from which no radiation can reach us, but rather any object that is close to being a black hole, be it a collapsed star or the central region of a galaxy. Even if there is just Hawking radiation, the hole is no longer black. In astronomy, such sloppiness may be tolerable, since the distinction mostly makes no difference. The outside gravitational field remains the same, being given by the mass, irrespective of its state of contraction. In response also to Mordred and imatfaal: From your reactions, I conclude that it was less appropriate to involve the formation of black holes in the statement of my problem. This complicated the reasoning unnecessarily. However, it is still completely clear to me that there can be no classical black holes anywhere in our view if the Universe has only existed for a finite time. In a system of reference in which we are at rest, it requires no less than an eternity for light to reach us from the position of an event horizon. This is immediately clear from the Schwarzschild metric and the definition of an event horizon. There can be holes in our view, but these will not be as deep as required if the Universe is finite in age.

If black holes exist empirically, they must have formed prior to the big bang (and somehow survived it). An object that collapsed later cannot yet have contracted sufficiently to have reached the event horizon, since in our view, this will require an eternity. Right or wrong?

What you say is correct but beside the point. Let me restate my question in other words: "Can there be objects that are further remote from us than the big bang is?" (Remoteness in time or in space along a geodesic.)

Evidently, you are thinking of observers in the vicinity of a black hole. This is not the topic here. The question was whether there can be black holes in a universe of finite age. This has to be considered in a frame of reference in which the universe can be said to be at rest. There is no other choice that fits each of the black holes that might be there. Having said this, I realize that in an expanding universe, comoving coordinates would be appropriate.

It makes actually no difference to the facts which coordinate system is used in for describing black holes, but Kruskal-Szekeres coordinates invite confusing the imaginary with the real. The coordinate system appropriate to my reasoning is that appropriate in flat space (where the observers are). The situation becomes different only when one discards black holes in favour of some alternative, such as gravastars.