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I haven't done the mathematics of it but is the radius of a potentially naturally occurring object such as a star held stable by quark degenerate matter or hypothesized preon degenerate matter small enough such that the object has an event horizon. I know you can just say if you had some object of this density and radius then it has an event horizon but, is it possible in nature?

Also what defines a black hole? Is it the fact it is an object with an escape velocity the speed of light or is it something that collapses to infinite density hence a star with an event horizon held together by some degenerate matter pressure?

No idea if any of this is correct. Apparently the laws of physics break down inside a black hole, so even though the pressure may be great enough just considering forces, it couldn't actually exist.

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  • $\begingroup$ Well, basically a black hole is defined by having an event horizon, so everything that has an event horizon is called a black hole. So there are no things with an event horizon that aren't black holes. As for degeneracy pressure, you might wanna look up white dwarves and neutron stars $\endgroup$ – lemdan Jul 5 '17 at 10:24
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    $\begingroup$ @Mladen Horizons are actually rather general. For example, you have cosmological horizons, Rindler horizons, etc. $\endgroup$ – Dvij Mankad Jul 5 '17 at 10:35
  • $\begingroup$ @Dvij Agreed, but as far as I'm aware it should be safe to say that an object having an event horizon is always a black hole. (I don't consider the Universe an object here.) $\endgroup$ – lemdan Jul 5 '17 at 10:40
  • $\begingroup$ @Mladen Seems like the case, but to be on the safer side, I would define a black hole as simply a singularity cloaked in at least one horizon. $\endgroup$ – Dvij Mankad Jul 5 '17 at 11:06
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If it is small enough to have an event horizon it will crush to form a singularity, at least given GR. This is because once a horizon forms, everything inside is compelled to move into the center as all time-like paths, thus all those which all particles known to exist (and all which can exist assuming strict causality) are restricted to following, end up hitting the center. So the mass of neutron star MUST crush into the center, once inside a horizon radius.

But you can have horizons without black holes, just not an object shielded by one that is not a black hole. There is an event horizon far from Earth due to the expansion of the Universe.

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The black hole has an event horizon that is an invariant. The universe has a form of event horizon called the cosmological event horizon. However, this is not quite the same. It is an apparent horizon.

The Schwarzschild metric $$ ds^2~=~\left(1~-~\frac{2GM}{rc^2}\right)dt^2~-~\left(1~-~\frac{2GM}{rc^2}\right)^{-1}dr^2~-~r^2d\Omega^2. $$ The metric term $g_{tt}~=~\left(1~-~\frac{2GM}{rc^2}\right)$ $\rightarrow~0$ as $r~\rightarrow~\frac{2GM}{c^2}$. The blow up of $g_{rr}$ can be removed, so this is not a singularity. However, it is centered around the position of a gravitating mass at $r~=~0$. The spacetime solution is a vacuum solution around this singular point, which turns out to be a whole spatial surface instead of a point!

The de Sitter spacetime in stationary coordinates is $$ ds^2~=~\left(1~-~r^2\Lambda/3\right)dt^2~-~\left(1~-~r^2\Lambda/3\right)^{-1}dr^2~-~r^2d\Omega^2. $$ This is stationary because it is fixed at a coordinate origin $r~=~0$. which is not tied to a mass. There are more technical ways of working through this. but a transformation of coordinates moves the horizon as well. This means the horizon is different than the case with a black hole.

There is a case of something with an event horizon that is a bit odd. This is the Taub-NUT spacetime. This is similar to a black hole hole, but with metric term $1~-~\frac{\mu}{t}$, where $\mu$ is the NUT parameter that is analogous to the magnetic monopole; it is a gravitational form of a magnetic monopole. Here $t$ is of course time, and what this means is the event horizon occurs at some time in the past. It is proposed as a possible universe or cosmology. For $t~<~\mu$ the spacetime is nonchronal with closed timelike curves. This is a real event horizon and not a particle or apparent horizon that transforms with coordinate transformations.

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  • $\begingroup$ Doesn't really help considering I have just finished A-level $\endgroup$ – Joshua Farrell Jul 5 '17 at 15:15

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