# How does information of the parent neutron star get encoded on the event horizon of a black hole?

I was watching a video on a typical black hole formation from a neutron star and it suggested that the event horizon appears instantaneously at the surface of the star even as the stellar matter inside starts to converge into a singularity. I have also read that the information that gets encoded on an event horizon is because of the fact that all the infalling matter gets sort of smeared on the black hole's surface from the point of view of an outside observer. So what happens to the information of all the original stuff in the neutron star if the stuff is inside the event horizon to begin with and so never gets smeared over it? ( And if we say that the event horizon starts out small from the center of the neutron star then shouldn't Hawking radiation destroy or at least impede the formation of the nascent and tiny black hole? )

It is best to not think about the event horizon in this scenario, but rather the "apparent horizon" in a particular timelike slicing of spacetime designed to correspond with the time coordinate of a particular viewer. In this construction, the apparent horizon is defined as a closed surface where the time rate of change of a "sphere of outbound light"'s surface area is zero$${}^{1}$$. In common cases like Schwarzshild and Kerr spacetimes, stacks of apparent horizons correspond exactly to event horizons, but in cases of gravitational collapse, they can be quite different (and frame-dependent), though the apparent horizon will always lie inside the event horizon.

Why is this important? Because, as the collapse happens, the apparent horizon will form at the center of the collapse, and expand outward, but the last ray of light will also expand out from the outer layers of the collapsing region. As the stack of apparent horizons expands out at a superluminal rate (they don't carry information), they will eventually intersect with the last ray coming from the collapsing star, and freeze it on the surface of the now-corresponding event and apparent horizons that is now a static surface equivalent to a stack of kerr horizons.

$${}^{1}$$ An event horizon, by contrast, is defined as the boundary between "the past of everything that fell into the black hole" and "the past of everything that did not fall into the black hole", and inherently depends on the future development of the spacetime.

• Thanks. This explains what the host in the video meant by the event horizon expanding before actually coming into being. So this apparent horizon would not emit any Hawking radiation? – Midovaar Oct 3 '20 at 9:29
• @Midovaar: the question of which horizon emits the hawking radiation is a bit of an interesting one, but it's also pretty moot -- the timescale for significiant hawking radiation to be emitted from a black hole is comparable to the age of the universe, and the timescale for an object to collapse completely to a black hole, the ringdown to complete, and the apparent horizon to match the event horizon is, even if you're just giving away orders to magintude to make it longer, days at the most. – Jerry Schirmer Oct 3 '20 at 17:11

The event horizon is by definition the boundary between the black hole interior and exterior in spacetime. Any matter that starts outside the hole and ends inside it must cross the horizon at some point by definition, whether it's "founding" matter or matter that falls in later.

The horizon doesn't appear instantaneously at the star surface. It starts as a set of points that has no interior and encloses no space (possibly but not necessarily a single point), and grows outward at the speed of light.

When the event horizon appears and starts to grow, there is much less spacetime curvature locally than there would be in the vicinity of a small black hole of that size, so there's no reason to expect it to immediately evaporate by Hawking radiation.

Let me begin to state that the neutron star must have a mass of at least 1,43 times the Solar mass to form a black hole. When the neutron star starts to contract it will become a quark star, which contracts further. So the event horizon can never be formed at the surface of the neutron star.

The radius of the event horizon (Schwarzschild radius) is $$\frac{2MG}{c^2}$$, but that's not of importance here. The question about the information of the stuff inside the star was a long time called "the information paradox". In my humble opinion, this paradox hasn't been resolved yet. Süsskind says he has, but his evidence is based on string theory (ADS/CFT correspondence) which I don't believe to correspond to reality.

For an observer falling in the material doesn't get smeared out. All material is falling in according to him.