# Does an event horizon imply a singularity?

I surprisingly did not find anyone addressing this directly. The weak cosmic censorship states, very loosely that singularities imply event horizons. I am wondering whether the converse is true. I.e. is it possible for an event horizon to exist, but it harbors no singularity inside? This does not look obvious to me since the definition of a BH never mentions any singularity whatsoever.

Since there seems to be some confusion on the notion of event horizon, I define it thus,

Let $$(M,g)$$ be a spacetime that is asymptotically flat at null infinity. Define the BH region to be $$\mathcal{B} := M \setminus[M \cap J^-(\mathcal{I^+})].$$. Then the (future) event horizon is defined to be the boundary $$\mathcal{H}^+ := \dot{\mathcal{B}} = M \setminus[M \cap \dot{J}^-(\mathcal{I^+})].$$ In a sentence, the event horizon is the boundary of the BH region, which is defined to be the region of spacetime that is causally disconnected from (future) null infinity.

• The de Sitter metric has a horizon but no singularity. Jan 16 '20 at 18:14
• You need to make the distinction between an absolute event horizon (which is a global thing and is the defining feature of a black hole) and an apparent horizon. The question of whether an absolute event horizon implies a singularity is essentially what is being asked in this question: physics.stackexchange.com/questions/524742/…
– user4552
Jan 16 '20 at 18:49
• Something far enough is certainly behind an horizon but it does not mean that horizon has a singularity on the other side. I mean there are probably region out there to which we can't communicate now and forever but there the universe is supposed to be like it is here around. If all this is due to a remote singularity such as big bang is another point. But there is no singularity now and there. Jan 17 '20 at 10:31
• Ps saw that an answer and a comment already differentiate event horizons of two kinds. My comment above it is not really necessary. Jan 17 '20 at 10:37
• Now that you've clarified what kind of horizon you mean, I think this question is a duplicate of this one that I asked: physics.stackexchange.com/questions/524742/…
– user4552
Jan 17 '20 at 14:39

The crucial thing about event horizons is that they are boundaries beyond which things cannot affect an observer. While black holes have very conspicuous event horizons standard FRLW model accelerating expanding universes have cosmological event horizons. Note that there is no singularity in such models except for the big bang itself far in the past.

• The question as I read it is about black holes, meaning that the OP is asking about an absolute horizon -- one that is causally disconnected from null infinity. By that definition, this is not a counterexample.
– user4552
Jan 16 '20 at 22:52
• @BenCrowell - That is not how I interpreted the question. Jan 17 '20 at 10:07
• This is how I have interpreted the question and what I would have answered if able to be concise. Instead I have commented. Plus 1, even if OP might have asked for bh, this is a valid answer to the question, especially to the title question. Jan 17 '20 at 10:42

The weak cosmic censorship hypothesis is basically that if there are any singularities, then they are hidden from "us" by event horizons. Whether this is actually true of the universe isn't known.

Both event horizons and singularities are mathematical phenomena that arise in abstract mathematical systems such as our models of gravitation and space-time. My understanding is that most people studying this stuff still think those models are more-or-less correct in the sense that there are real phenomena in the universe that behave enough like the models of event horizons and singularities that we can safely use those words to describe them. But these are not settled questions.

Can we show abstract configurations of our models of space, time, and gravity, such that there's an even horizon and no singularity? Certainly!

• Conventional models of the formation of a black hole have the appearance of the event horizon as a separate and "earlier" event than the formation of the singularity.
• The "edge of the observable universe" is technically an event horizon, albeit of a different kind.
• Physicist over the decades have expressed all sorts of different exotic space-time configurations; by all means go looking for one that meets the specific idea of an "event horizon" and a "singularity" that you're interested in.

Are there or will there ever be closed gravitational event horizons in the universe that neither contain nor imply the future existence of singularities? It seems kinda unlikely, but I'm not taking bets.

• The "edge of the observable universe" is technically an event horizon, albeit of a different kind. The question as I read it is about black holes, meaning that the OP is asking about an absolute horizon -- one that is causally disconnected from null infinity. By that definition, this is not a counterexample.
– user4552
Jan 16 '20 at 18:57
• Conventional models of the formation of a black hole have the appearance of the event horizon as a separate and "earlier" event than the formation of the singularity. This is not right, which may be the reason you put the scare quotes on "earlier...?" Starting from any point on the horizon of a Schwarzschild black hole, you can draw a spacelike surface that intersects the singularity. Therefore we cannot say that the singularity forms after the horizon. GR doesn't have a clear enough notion of simultaneity to answer questions like that.
– user4552
Jan 16 '20 at 19:00
• Perhaps building on the comment by @BenCrowell, an event horizon is a global feature of the black hole spacetime. A spacetime either has it or it doesn't. There's this notation of it forming at a specific time is just wrong. (It is true that a spacetime with an event horizon might allow for spacelike surfaces that don't intersect the horizon, which is what I think you're getting at, but that's different than what you said.) Jan 16 '20 at 19:02
• Question : I am under the impression that while a singularity emerges from mathematics, the horizon associated with should be physical as well (of course not a wall, but dictating the physics between the two sides and the communication between events. Isn't? Jan 17 '20 at 10:35
• @safesphere: You might want to ask that in a separate question. It seems like you're confused about notions like spacelike surfaces and simultaneity in GR. "Spatially separated" is not the same as "spacelike."
– user4552
Jan 17 '20 at 14:35

It has been my understanding that from our point of view, time stops at the event horizon. If that is true for all points inside the horizon, then we would say that the collapse which caused the event horizon has stopped, and the singularity never forms.

• It has been my understanding that from our point of view, time stops at the event horizon. This is a very restrictive statement that is basically trivial. The definition of an absolute horizon is that a distant observer can't observe it. Therefore it's trivial true that we never see it. This does not mean that a singularity never forms. An observer who falls in will hit the singularity in finite time.
– user4552
Jan 16 '20 at 18:52
• @safesphere Proper time doesn't stop at the horizon. The equation of motion for a radially infalling particle is d²r/dτ²=-GM/r², that can be continued all the way down to r=0. Only t gets infinite at the horizon, but the proper time is τ Jan 17 '20 at 13:00