Event horizon

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In general relativity, the event horizon —also called the event horizon— refers to a hypersurface boundary of space-time, such that events on one side from it cannot affect an observer on the other side. Note that this relationship does not have to be symmetric or bijective, that is, if A and B are the two regions of spacetime into which the event horizon divides space, A may not be affected by events within B, but events in B are generally affected by events in A. To give a concrete example, light emitted from within the event horizon could never reach an observer located outside, but an observer inside could observe events outside. There are several types of event horizons, and they can appear in various circumstances. One of them particularly important occurs in the presence of black holes, although this is not the only type of possible event horizon, there are also Cauchy horizons, Killing horizons, particle horizons or cosmological horizons.

Event horizon of a rotating black hole

Scheme of the horizon of events and the ergonomics

The event horizon is an imaginary spherical surface that surrounds a black hole, in which the escape velocity necessary to move away from it coincides with the speed of light. Therefore, nothing inside it, including photons, can escape due to the pull of an extremely strong gravitational field.

Particles from the outside that fall into this region never come out again, since to do so they would need an escape speed greater than that of light and, so far, theory indicates that nothing can top it.

Therefore, there is no way to observe the interior of the event horizon, nor to transmit information to the exterior. This is the reason why black holes do not have visible external characteristics of any kind, which allow to determine their interior structure or their content, being impossible to establish in what state the matter is from when it exceeds the event horizon until it collapses into the center of the black hole.

If we fell into a black hole, at the moment of crossing the event horizon we would not notice any change, since it is not a material surface, but an imaginary border, far from the central zone where the mass is concentrated. The peculiar characteristic of this border is that it represents the point of no return, from which there can be no other event than falling inwards, thus giving rise to the name of this surface.

By including quantum effects in the event horizon, the emission of radiation by the black hole is possible due to the vacuum fluctuations that give rise to the so-called: “Hawking radiation”.

Horizon of an accelerated observer

A different type of horizon is the one seen by a uniformly accelerated observer. To characterize this type of horizon we need to introduce the Rindler coordinates for the Minkowski space-time. Starting from the Cartesian coordinates, the metric of said space-time:

$<math alttext="{displaystyle ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2},;;-infty <T,X,Y,Zds2=− − dT2+dX2+dAnd2+dZ2,− − ∞ ∞ .T,X,And,Z.∞ ∞ {displaystyle ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2},;;-infty θT,X,Y,Z taxinfty }<img alt="{displaystyle ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2},;;-infty <T,X,Y,Z$

Now consider the region known as Rindler's wedge, given by the set of points that verify:

$<math alttext="{displaystyle {mathcal {R}}_{Rind}={(T,X,Y,Z)in mathbb {R} ^{4}|0<X<infty;-X<TRRind={(T,X,And,Z)한 한 R4日本語0.X.∞ ∞ ,− − X.T.X!{displaystyle {mathcal {R}}_{Rind}={(T,X,Y,Z)in mathbb {R} ^{4}.<img alt="{displaystyle {mathcal {R}}_{Rind}={(T,X,Y,Z)in mathbb {R} ^{4}|0<X<infty;-X<T$

And let's define on it a change of coordinates given by:

${displaystyle t=operatorname {arctanh} (T/X),;x={sqrt {X^{2}-T^{2}}},;y=Y,;z=Z}$

Whose inverse transformation is given by:

${displaystyle T=x,sinh(t),;X=x,cosh(t),;Y=y,;Z=z}$

Using these coordinates the Rindler wedge of Minkowski space has a metric expressed in the new coordinates given by the expression:

$<math alttext="{displaystyle ds^{2}=-x^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2},;0<x<infty-infty <t,y,zds2=− − x2dt2+dx2+dand2+dz2,0.x.∞ ∞ ,− − ∞ ∞ .t,and,z.∞ ∞ {displaystyle ds^{2}=-x^{2}dt{2}+dx^{2}+dy^{2}+dz^{2},;0 taxx infty-infty ≤3⁄2}, y,z taxinfty }<img alt="{displaystyle ds^{2}=-x^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2},;0<x<infty-infty <t,y,z$

This metric has an apparent singularity ${displaystyle x=0}$ where the tensor expressed in Rindler's coordenas has a determinant that is canceled. This happens because ${displaystyle xrightarrow 0}$ the acceleration associated with the observer becomes infinite. In these coordinates the horizon of Rindler is precisely the border of Rindler's sister. It is interesting that it can be shown that this horizon is analogous in many aspects to the horizon of events of a black hole.

Horizon in the observable universe

The limit of the observable universe is a hypersurface that constitutes the barrier of what can be observed at each instant of time, beyond that there would be particles whose light has not yet had time to reach us, because the age of the universe is finite (see Big Bang). Any current or past event located beyond the event horizon is not part of the current observable universe (although it may be visible in the future when light signals from them reach our future position).

The way this horizon of the observable universe changes depending on the nature of the expansion of the universe. If the expansion has certain characteristics, that will never be observable, for example, no matter how much time elapses (this happens in a certain type of accelerated expansion, for example). The past frontier of events that can never be observed is properly an event horizon called the «particle event horizon».

The criterion for determining whether the horizon of events of the universe is different from the void is the following, defy a commune distance ${displaystyle d_{E}}$ by expression:

${displaystyle d_{E}=int _{t_{0}}^{infty }{frac {c}{a(t)}}dt.}$

In this equation, a(t) is the scale factor, c is the speed of light and t₀ is the present age of the universe. Yeah. ${displaystyle d_{E}rightarrow infty }$ , that is, the arbitrarily distant points can be observed, then the horizon of events is empty. Yeah. ${displaystyle d_{E}neq infty }$ Then there will be a horizon of events.

Examples of cosmological models without an event horizon are models of universes dominated by matter or radiation. An example of a cosmological model with an event horizon is a universe dominated by the cosmological constant, such as a De Sitter Universe.

Event horizon and topology

The study of causality in general relativity is carried out following a topological approach, so a horizon of future or past events can be characterized as the set of points of the topological closure of the dependency domain of a light hypersurface located in the "infinite" that do not belong to the past or chronological future of that domain. It should be clarified that when it is said that a hypersurface is located in the "infinite" it is meant that it is located on the points of the diagram according to Penrose that represents space-time, in signs the horizons of past events ${displaystyle H^{-}({mathcal {S}}_{L})}$ and future ${displaystyle H^{+}({mathcal {S}}_{L})}$ of a light hypersurface ${displaystyle {mathcal {S}}_{L}}$ They are given by:

${displaystyle H^{-}({mathcal {S}}_{L})={overline {D^{-}({mathcal {S}}_{L})}}-I^{+}(D^{-}({mathcal {S}}_{L}))qquad H^{+}({mathcal {S}}_{L})={overline {D^{+}({mathcal {S}}_{L})}}-I^{-}(D^{+}({mathcal {S}}_{L}))}$

Where the definition of the signs that appear is the same used in the relativity glossary.

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