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Comment by orlp | Hacker News Reader

This is often stated but it literally can't be true. It's true for stationary solutions to the relevant equations, but we're very much not living in a stationary world.

Here's a simple thought experiment disproving your claim. A person hovers just above the origin of a supermassive black hole. They chuck a massively charged object into the black hole. If what you said is true they should observe the charge instantly being transported to the singularity, since a black hole can't have any attributes such as where charge is distributed within the horizon.

Where it gets impossible is that someone very far away around the same supermassive black hole could observe a small charge increment. They in turn could chuck charged stuff into the black hole and now you've got faster than lightspeed communication.

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> above the origin of a supermassive black hole

I spent couple of minutes trying to understand your thought experiment and was puzzled why can't I understand it. It seems that it is probably because I don't understand what you mean by "just above the origin of supermassive black hole"?

I feel that you have something interesting to say but not clear.

Mental typo, I meant horizon, not origin.

The no-hair theorem (scientific term for what you are descriping) essentially states that black holes can be completely described by only three externally observable classical properties: mass, charge, and angular momentum. The internal distribution of matter or charge within the black hole is irrelevant from the outside once it crosses the event horizon.

The charge of an object thrown into a black hole doesn't need to "instantaneously" reach the singularity. In fact, it doesn't even "travel" in the sense that the observer would see a time delay based on where the charge is inside the horizon. The EM field generated by this charged object can still be observed outside the horizon before the object crosses the horizon. In classical GR, the exterior of a charged black hole is governed by the Reissner-Nordström metric [1], and it already includes the influence of charge. Once the object enters the horizon, the outside observer will perceive the black hole's external field to have changed. The field adjustment is instant for the outside observer because, from their point of view, the object never actually crosses the event horizon (due to infinite time dilation at the horizon from their perspective). The charge appears to have been "absorbed" by the black hole when the object is still outside the horizon.

So no-hair theorem doesn't imply that properties such as charge or angular momentum must be "smeared" instantaneously to the singularity inside the black hole. The theorem only describes how these properties manifest externally, not their behavior inside the horizon. Once a charge object passes the event horizon, the information about its charge is not causally connected to any observer inside the black hole (except in QM contexts, where the information paradox becomes relevant). However, the charges affects the external metric of the black hole immediately and completely as seen by external observers. Also it doesn't say that the charge or mass needs to be uniformly distributed or behave in any specific way inside the event horizon. It only states that from the outside, black holes look like point-like objects characterized by mass, charge, and angular momentum. The specifics of how charge is distributed inside the black hole’s event horizon aren't visible to outside observers and therefore don't affect the validity of the theorem.

[1] https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6...

Again, the no-hair theorem only talks about stationary solutions:

> The no-hair theorem (which is a hypothesis) states that all *stationary* black hole solutions of...

https://en.m.wikipedia.org/wiki/No-hair_theorem

> The field adjustment is instant for the outside observer

And as my above thought experiment shows, any instant changes like such could be used for FTL communication.

I think time would pass incredibly slowly for an observer that close to the black hole? Since the event horizon provides a singularity where time slows to zero, it makes sense that any non-zero speed along it would go to infinity for an outside observer.

I doubt you could usefully exploit this behavior, because any charges would still need to travel to and away from the black hole at no more than light speed, and because time slows down the further you get to the event horizon the shortest path "around" a black hole would probably not go through it.

What is unique about your thoughts experiment that makes it applies only to dynamical phases?

I am really not getting what you are trying to say?

The point I'm trying to make is that everyone always cites the no hair theorem but completely leaves out the "stationary solution" part, leading to misconceptions. My thought experiment highlights that one can't treat a black hole as a particle that only has mass, charge and angular momentum in a dynamic world, because such a treatment must inherently lead to instantaneous updates to large regions of spacetime in response to 'stimuli' (in the form of mass, charge or angular momentum) for super massive black holes, which is impossible.

I understand that this is valid for stationary solution which is what an astronomical black hole would be most of the time. What I don't get is why do you assume that this would violate causality. In GR the idea that changes must “instantaneously update” large regions of spacetime due to supermassive black holes ignores relativistic effects like time dilation. From the perspective of an external observer, as an object falls into a black hole, time appears to slow down near the event horizon. The infalling object’s influence on the black hole’s external field (e.g., its charge or mass) is perceived gradually by distant observers.

For supermassive black holes, this process might seem slow due to the massive scale of spacetime curvature near the event horizon, but there is no requirement for instantaneous changes. In fact, relativistic causality ensures that no information can propagate faster than light, so updates to the black hole’s charge, mass, or angular momentum are constrained by the speed at which signals (gravitational or electromagnetic) can travel.

The object will be observed asymptotically decelerating as it approaches the event horizon, never gets observed crossing by a distant observer.

IIRC the event horizon expands to encompass the object just before it gets to the original horizon, due to the Schwarzschild formula.

Also IIRC, the electric field is blurred out by the geodesics by this point, as if it were from the interior. But that's based on what I've heard, I have yet to derive useful results from the Einstein field equations, even though I think I should give it a go and I can follow them well enough to code a simple simulation…

You can’t actually observe objects going into a black hole. They just become redshifted and fall slower and slower from your perspective towards the event horizon.

The singularity is not the same thing as its event horizon.

That said, it's the no hair theorem. It could of course still be wrong...

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