# Event horizons without singularities

Someone answered this question by saying that black hole entropy conditions and no-hair theorems are asymptotic in nature -- the equations give an ideal solution which is approached quickly but never actually reached from the point of view of an observer outside the event horizon.

Since then I've been wondering whether singularities are ever really created, and if not, why do we worry about naked singularities?

Quick recap: to an external observer, an object falling into a black hole experiences time dilation such that it appears to take an infinite amount of time to cross the event horizon and ends up sitting frozen at the border.

So here's my reasoning: the above should also apply during the formation of the black hole in the first place. The gravitational field approaches infinite density as the constituent matter approaches a central point, but to an outside observer, it takes an infinite amount of time for the singularity to form. In other words, it never happens.

As I understand it, naked singularities are dismissed with hand-waving, "we'll fix it when we go quantum," but I don't see that as necessary. It seems to me that singularities never actually form, although event horizons clearly do.

Does this mean that we can stop worrying? What happens in naked singularity scenarios when there is no singularity yet?

• Actually, it's believed that naked singularities cannot classically form from ordinary'' matter except in extremely exceptional circumstances. No need to invoke quantum mechanics at all. May 23, 2012 at 12:53

So here's my reasoning: the above should also apply during the formation of the black hole in the first place.

That isn't true. Black holes don't start from a point in (for example) the centre of a collapsing star and grow outwards. It's actually the opposite - the event horizon forms outside the collapsing star. That means the matter forming the black hole is already inside the event horizon and GR tells us that anything inside the event horizon falls into the singularity in a finite time. You never have to worry about whether the matter does or doesn't cross the event horizon in a finite time.

It seems odd for the event horizon to spring into existance outside the star, but it's because large black holes are easier to make than small black holes. The Schwarzschild radius depends on the mass, but if you assume a uniform star the mass depends on the star's radius cubed. This means the average density of the black hole, i.e. the mass divide by the volume within the event horizon, is lower for large black holes than for small ones.

So I think even the most sceptical would have to concede that singularities really exist. However the question of whether naked singularities exist is another and different question. If you do the maths then GR tells you that they can be created e.g. by charging a Reissner–Nordström black hole to it's extremal value. The question is whether the maths is related to reality.

• How does this picture change if one uses dynamic and isolated horizons, rather than event horizons? The latter are somewhat aesthetically unpleasing in many ways, least of which you said, that it already exists in places which do not have high matter density and is extremely teleological in nature. May 23, 2012 at 14:43
• I'm not sure I see the connection with singularities, naked or otherwise. The main problem here is the old chestnut that nothing can fall through an event horizon because it takes infinite Schwarzschild co-ordinate time to reach the horizon. May 23, 2012 at 15:12
• I almost accepted this -- "GR tells us that anything inside the event horizon falls into the singularity in a finite time" but this is from the POV of the falling object, right? From an outside-the-horizon observer isn't it also infinite? If not, why not? May 27, 2012 at 22:24
• @spraff: it's true that if you use Schwarzschild co-ordinates it takes infinite co-ordinate time for an object outside the event horizon to reach the even horizon. However if an object starts inside the event horizon, i.e. because the event horizon forms outside it, then even in Schwarzschild co-ordinates it will reach the singularity in a finite time (though the Schwarzschild co-ordinates don't make a lot of sense inside the event horizon so you need to be careful what you mean by "time"). This is the key point. The collapsing star is inside the horizon when the horizon forms. May 28, 2012 at 5:59
• @JohnRennie, I don't think your answer really answers the question. We can separate one event - "event horizon forms" and think of what happens just before it. That would be some particle finally crossing the boundary of a Schwarzschild radius sphere. So the question is, does this event happen in finite time for an outside observer or maybe from his POV the particle will approach that boundary infinitely? Jun 22, 2012 at 23:30

The currently accepted answer is entirely wrong. It says

Black holes don't start from a point in (for example) the centre of a collapsing star and grow outwards. It's actually the opposite - the event horizon forms outside the collapsing star.

The idea seems to be that the matter that initially collapses to make the black hole doesn't pass through the event horizon and so doesn't "freeze" there.

None of that is true. In an isotropic collapse, the event horizon does start from a point in the center and grow outwards (at the speed of light). In a general collapse, it starts at a one-dimensional set of spacelike separated points called the crease set. In any case, all of the collapsing matter crosses the horizon and in principle appears "frozen" near the horizon forever. (In practice it can't be seen because the redshift is far too large.)

The event horizon is the boundary of the black hole. Like any boundary, it has no gaps in it, because a gap would make the division into inside and outside meaningless. Everything that ends up inside the black hole crosses the boundary.

black hole entropy conditions and no-hair theorems are asymptotic in nature [...] Since then I've been wondering whether singularities are ever really created, and if not, why do we worry about naked singularities?

Technically, everything is asymptotic in nature. If you make waves on the surface of a perfectly still lake, the amplitude of the waves dies down over time but the lake will never be precisely still again. The settling-down time of black holes should be viewed in the same way. This has no bearing on whether the bottom of the lake exists, nor on whether the interior of a black hole exists.

Also, naked singularities are by definition singularities that aren't hidden by an event horizon, so the behavior of the horizon is even less relevant. My impression is that "naked" in the question is just an intensifier (especially given the title), so I'll ignore it.

to an outside observer, it takes an infinite amount of time for the singularity to form. In other words, it never happens.

For a classical black hole (with an event horizon), you can pick a time coordinate that respects causality (anything that happens at $$t_1$$ can only affect what happens at $$t_2$$ if $$t_1), and that covers everything that happens outside of the hole into the indefinite future, and that doesn't cover the black hole interior, or doesn't cover a part of the interior that includes the singularity. From the perspective of that definition of time, there is never a singularity. In effect, the spacetime never gets around to collapsing that last little bit.

But all that you've really done, if you do that, is fail to cover a portion of the spacetime with your coordinate system. That doesn't make it go away. Even if you aren't interested in what happens there, other people are, and the problem isn't solved for them.

Yes, the no-hair theorems are asymptotic, and yes a frozen star or red hole approach has some accuracies associated with it.

However, the issue of cosmic censorship is not well understood, and is still relevant to your situation.

In particular, if a collapsing system forms a singularity that is not naked, then the event horizon could introduce long range asymptotics that make the inside irrelevant to an external observer. But what if the singularity forms without an event horizon? Then the singularity is real.

Since then I've been wondering whether singularities are ever really created, and if not, why do we worry about naked singularities?

A naked singularity is exactly the singularity you want to worry about, because it would sit on your past light cone and causally affect you, even though the equations don't make clear predictions about what happens. So you get a full loss of determinism and causality if you have a naked singularities.

The clothed singularities, as you reference, might not be anything we need to worry at all until we get to a quantum theory or consider other ways to get out of a classical event horizon.

What happens in naked singularity scenarios when there is no singularity yet?

A naked singularity is a singularity that is not inside an event horizon. So there is no time dilation to save you from seeing it form. And so if it is naked, then there is no "not yet", it happened, and you can see things coming from the singularity all the way to your eyeball (or the theory itself is just wrong, which either way is pretty bad for the theory).

no proof of a singularity can exist as any thing that enters an event horizon can not return as there accessible future only exists in the black hole other wise they have to travel faster than the speed of light which according to Einstein is always the same.

The horizons also cannot appear, not only singularities.

Our primary result, that no event horizon forms in gravitational collapse as seen by an asymptotic observer is suggestive of the possibility of using the number of local event horizons to classify and divide Hilbert space into superselection sectors, labeled by the number of local event horizons. Our result suggests that no operator could increase the number of event horizons, but the possibility of reducing the number of pre-existing primordial event horizons is not so clear and would require that Hawking radiation not cause any primordial black hole event horizons to evaporate completely.

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