Periodic summation

2013-09-24T14:19:08Z

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In [[general relativity]], an '''event horizon''' is a boundary in [[spacetime]] beyond which events cannot affect an outside observer. In layman's terms, it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case of an event horizon is that surrounding a [[black hole]]. Light emitted from beyond the event horizon can never reach the outside observer. Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon, with its image becoming more and more [[redshift]]ed as time elapses. The traveling object, however, experiences no strange effects and does, in fact, pass through the horizon in a finite amount of [[proper time]].

More specific types of horizon include the related but distinct [[Absolute horizon|absolute]] and [[apparent horizon]]s found around a black hole. Still other distinct notions include the [[Cauchy horizon|Cauchy]] and [[Killing horizon]]; the [[photon sphere]]s and [[ergosphere]]s of the [[Kerr solution]]; [[particle horizon|particle]] and [[cosmological horizon]]s relevant to [[cosmology]]; and [[isolated horizon|isolated]] and [[dynamical horizon]]s important in current black hole research.
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== Existence and evolution of the particle horizon ==

A nice and simple example of event horizon emerges from some cases of the [[FLRW]] cosmological model. In that context, the universe can be approximated as composed by non-interacting constituents, each one being a perfect fluid with density <math>\rho_i</math>, partial pressure <math>p_i</math> and state equation <math>p_i=\omega_i \rho_i</math>, such that they add up to the total density <math>\rho</math> and total pressure <math>p</math>.<ref>{{cite journal
|last=Berta Margalef-Bentabol
|coauthors=Juan Margalef-Bentabol; Jordi Cepa
|title=Evolution of the cosmological horizons in a concordance universe
|journal=Journal of Cosmology and Astroparticle Physics
|date=21 December 2012
|volume=2012
|issue=12
|doi=10.1088/1475-7516/2012/12/035
|url=http://iopscience.iop.org/1475-7516/2012/12/035
|arxiv=1302.1609
|pages=035}}</ref> Let us now define the following functions:

* Hubble function <math>H=\frac{\dot a}{a}</math>

* The critical density <math>\rho_c=\frac{3}{8\pi}H^2</math>

* The i-th dimensionless energy density <math>\Omega_i=\frac{\rho_i}{\rho_c}</math>

* The dimensionless energy density <math>\Omega=\frac{\rho}{\rho_c}=\sum \Omega_i</math>

* The redshift <math>z</math> given by the formula <math>1+z=\frac{a_0}{a(t)}</math>

Any function with a zero subscript denote the function evaluated at the present time <math>t_0</math> (or equivalently <math>z=0</math>). The last term can be taken to be <math>1</math> including the curvature state equation.<ref name="Evolution2">{{cite journal
|last=Berta Margalef-Bentabol
|coauthors=Juan Margalef-Bentabol; Jordi Cepa
|title=Evolution of the cosmological horizons in a universe with countably infinitely many state equations
|journal=Journal of Cosmology and Astroparticle Physics
|date=8 February 2013
|volume=2013
|series=015
|issue=2
|doi=10.1088/1475-7516/2013/02/015
|url=http://iopscience.iop.org/1475-7516/2013/02/015
|arxiv=1302.2186
|pages=015}}</ref> It can be proved that the Hubble function is given by

:<math> H(z)=H_0\sqrt{\sum \Omega_{i0}(1+z)^{n_i}}</math>

where <math>n_i=3(1+\omega_i)</math>. Notice that the addition ranges over all possible partial constituents and in particular there can be countably infinitely many. With this notation we have:<ref name="Evolution2" />

:<math> \text{The event horizon } H_e \text{ exists if and only if } m<2</math>

where <math>m</math> is the lowest <math>n_i</math> (possibly <math>-\infty</math>). The evolution of the event horizon for an expanding universe (<math>\dot{a}>0</math>) is:<ref name="Evolution2" />

:<math> \frac{dH_e}{dt}=H_e(z)H(z)-c</math>

where <math>c</math> is the speed of light and can be taken to be <math>1</math> (natural units). Notice that the derivative is made with respect to the FLRW-time <math>t</math>, while the functions are evaluated at the redshift <math>z</math> which are related as stated before. We have an analogous but slightly different result for [[Particle horizon]].

==Event horizon of a black hole==
{{Main|Black hole}}
{| class="wikitable" style="float:right; margin:1ex; width:400px;"
|- width
| [[File:BH-no-escape-1.svg]] Far away from the black hole a particle can move in any direction. It is only restricted by the speed of light.
|-
| [[File:BH-no-escape-2.svg]] Closer to the black hole spacetime starts to deform. In some convenient coordinate systems, there are more paths going towards the black hole than paths moving away.{{#tag:ref|The set of possible paths, or more accurately the future [[light cone]] containing all possible [[world line]]s (in this diagram represented by the yellow/blue grid), is tilted in this way in [[Eddington–Finkelstein coordinates]] (the diagram is a "cartoon" version of an Eddington–Finkelstein coordinate diagram), but in other coordinates the light cones are not tilted in this way, for example in [[Schwarzschild coordinates]] they simply narrow without tilting as one approaches the event horizon, and in [[Kruskal–Szekeres coordinates]] the light cones don't change shape or orientation at all.<ref>{{harvnb|Thorne|Misner|Wheeler|1973|p=848}}</ref>|group="Note"}}
|-
| [[File:BH-no-escape-3.svg]] Inside of the event horizon all paths bring the particle closer to the center of the black hole. It is no longer possible for the particle to escape.
|}

One of the best-known examples of an event horizon derives from general relativity's description of a [[black hole]], a celestial object so massive that no nearby matter or radiation can escape its [[gravitational field]]. Often, this is described as the boundary within which the black hole's [[escape velocity]] is greater than the [[speed of light]]. However, a more accurate description is that within this horizon, all [[lightlike]] paths (paths that light could take) and hence all paths in the forward [[light cone]]s of particles within the horizon, are warped so as to fall farther into the hole. Once a particle is inside the horizon, moving into the hole is as inevitable as moving forward in time, and can actually be thought of as equivalent to doing so, depending on the spacetime coordinate system used.<ref>{{Cite book| title=The large scale structure of space-time | author=S. W. Hawking and G. F. R. Ellis | publisher=Cambridge University Press | year=1975}}{{Page needed|date=September 2010}}</ref><ref>{{Cite book| author=Thorne, Kip S.; Misner, Charles; Wheeler, John | title=Gravitation | publisher=W. H. Freeman and Company | year=1973}}{{Page needed|date=September 2010}}</ref><ref>{{Cite book| author=Wald, Robert M. | title=General Relativity | location=Chicago | publisher=University of Chicago Press | year = 1984}}{{Page needed|date=September 2010}}</ref><ref>{{Cite book| title=Cosmological Physics | author=J. A. Peacock | publisher=Cambridge University Press | year=1999}}{{Page needed|date=September 2010}}</ref><ref>[http://astroreview.com/issue/2012/article/black-hole-horizons-and-how-they-begin Dieter Brill, “Black Hole Horizons and How They Begin”, Astronomical Review (2012); Online Article, cited Sept.2012.]</ref>

The surface at the [[Schwarzschild radius]] acts as an event horizon in a non-rotating body that fits inside this radius (although a [[rotating black hole]] operates slightly differently). The Schwarzschild radius of an object is proportional to its mass. Theoretically, any amount of matter will become a black hole if compressed into a space that fits within its corresponding Schwarzschild radius. For the mass of the [[Sun]] this radius is approximately 3 kilometers and for the [[Earth]] it is about 9 millimeters. In practice, however, neither the Earth nor the Sun has the necessary mass and therefore the necessary gravitational force, to overcome [[Electron degeneracy pressure|electron]] and [[neutron degeneracy pressure]]. The minimal mass required for a star to be able to collapse beyond these pressures is the [[Tolman-Oppenheimer-Volkoff limit]], which is approximately three solar masses.

Black hole event horizons are especially noteworthy for three reasons. First, there may be examples near enough (on an astronomical scale) to study. Second, black holes tend to pull in matter from their environment, which provides examples where matter about to pass through an event horizon is expected to be observable. Third, the description of black holes given by general relativity is known to be an approximation and it is expected that [[quantum gravity]] effects become significant in the vicinity of the event horizon. This allows observations of matter in the vicinity of a black hole's event horizon to be used to indirectly study [[general relativity]] and proposed extensions to it.
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==Particle horizon of the observable universe==
{{Main|Cosmological horizon}}

The [[particle horizon]] of the [[observable universe]] is the boundary that represents the maximum distance at which events can ''currently'' be observed. For events beyond that distance, light has not had time to reach our location, even if it were emitted at the time the universe began. How the particle horizon changes with time depends on the nature of the [[expansion of the universe]]. If the expansion has certain characteristics, there are parts of the universe that will never be observable, no matter how long the observer waits for light from those regions to arrive. The boundary past which events cannot ever be observed is an event horizon, and it represents the maximum extent of the particle horizon.

The criterion for determining whether a particle horizon for the universe exists is as follows. Define a [[comoving distance]] <math>d_p</math> by

: <center><math>d_p=\int_{0}^{t_0} \frac{c}{a(t)}dt\ .</math></center>

In this equation, ''a'' is the [[scale factor (universe)|scale factor]], ''c'' is the [[speed of light]], and ''t0'' is the age of the universe. If <math>d_p \rightarrow \infty</math> (i.e. points arbitrarily as far away as can be observed), then no event horizon exists. If <math>d_p \neq \infty</math>, a horizon is present.

Examples of cosmological models without an event horizon are universes dominated by [[matter]] or by [[light|radiation]]. An example of a cosmological model with an event horizon is a universe dominated by the [[cosmological constant]] (a [[de Sitter universe]]).

==Apparent horizon of an accelerated particle==
{{See also|Hyperbolic motion (relativity)}}
[[Image:Event-horizon-particle.svg|right|thumb|Space-time diagram showing a uniformly accelerated particle, '''P''', and an event '''E''' that is outside the particle's apparent horizon. The event's forward [[light cone]] never intersects the particle's [[world line]].]]

If a particle is moving at a constant velocity in a non-expanding universe free of gravitational fields, any event that occurs in that universe will eventually be observable by the particle, because the forward [[light cone]]s from these events intersect the particle's [[world line]]. On the other hand, if the particle is accelerating, in some situations light cones from some events never intersect the particle's world line. Under these conditions, an [[apparent horizon]] is present in the particle's (accelerating) reference frame, representing a boundary beyond which events are unobservable.

For example, this occurs with a uniformly accelerated particle. A space-time diagram of this situation is shown in the figure to the right. As the particle accelerates, it approaches, but never reaches, the speed of light with respect to its original reference frame. On the space-time diagram, its path is a [[hyperbola]], which [[asymptote|asymptotically approaches]] a 45 degree line (the path of a light ray). An event whose light cone's edge is this asymptote or is farther away than this asymptote can never be observed by the accelerating particle. In the particle's reference frame, there appears to be a boundary behind it from which no signals can escape (an apparent horizon).

While approximations of this type of situation can occur in the real world {{Citation needed|date=October 2008}} (in [[particle accelerator]]s, for example), a true event horizon is never present, as the particle must be accelerated indefinitely (requiring arbitrarily large amounts of energy and an arbitrarily large apparatus).

==Interacting with an event horizon==
A misconception concerning event horizons, especially [[black hole]] event horizons, is that they represent an immutable surface that destroys objects that approach them. In practice, all event horizons appear to be some distance away from any observer, and objects sent towards an event horizon never appear to cross it from the sending observer's point of view (as the horizon-crossing event's [[light cone]] never intersects the observer's [[world line]]). Attempting to make an object near the horizon remain stationary with respect to an observer requires applying a force whose magnitude increases unbounded (becoming infinite) the closer it gets.

For the case of a horizon perceived by a uniformly accelerating observer in empty space, the horizon seems to remain a fixed distance from the observer no matter how its surroundings move. Varying the observer's acceleration may cause the horizon to appear to move over time, or may prevent an event horizon from existing, depending on the acceleration function chosen. The observer never touches the horizon and never passes a location where it appeared to be.

For the case of a horizon perceived by an occupant of a [[de Sitter Universe]], the horizon always appears to be a fixed distance away for a [[inertial frame of reference|non-accelerating]] observer. It is never contacted, even by an accelerating observer.

For the case of the horizon around a black hole, observers stationary with respect to a distant object will all agree on where the horizon is. While this seems to allow an observer lowered towards the hole on a rope (or rod) to contact the horizon, in practice this cannot be done. The proper distance to the horizon is finite,<ref>Charles Misner, Kip Thorne, and John Wheeler (1973). ''Gravitation'', p. 824</ref> so the length of rope needed would be finite as well, but if the rope were lowered slowly (so that each point on the rope was approximately at rest in [[Schwarzschild coordinates]]), the [[proper acceleration]] (G-force) experienced by points on the rope closer and closer to the horizon would approach infinity, so the rope would be torn apart. If the rope is lowered quickly (perhaps even in freefall), then indeed the observer at the bottom of the rope can touch and even cross the event horizon. But once this happens it is impossible to pull the bottom of rope back out of the event horizon, since if the rope is pulled taut, the forces along the rope increase without bound as they approach the event horizon and at some point the rope must break. Furthermore, the break must occur not at the event horizon, but at a point where the second observer can observe it.

An observer crossing a black hole event horizon can calculate the moment they have crossed it, but will not actually see or feel anything special happen at that moment. In terms of visual appearance, observers who fall into the hole perceive the black region constituting the horizon as lying at some apparent distance below them, and never experience crossing this visual horizon.<ref>http://jila.colorado.edu/~ajsh/insidebh/schw.html</ref> Other objects that had entered the horizon along the same radial path but at an earlier time would appear below the observer but still above the visual position of the horizon, and if they had fallen in recently enough the observer could exchange messages with them before either one was destroyed by the [[gravitational singularity]].<ref>http://casa.colorado.edu/~ajsh/singularity.html</ref> Increasing [[tidal force]]s (and eventual impact with the hole's singularity) are the only locally noticeable effects.

==Beyond general relativity==
The description of event horizons given by [[general relativity]] is thought to be incomplete. When the conditions under which event horizons occur are modeled using a more comprehensive picture of the way the universe works, that includes both relativity and [[quantum mechanics]], event horizons are expected to have properties that are different from those predicted using general relativity alone.

At present, it is expected that the primary impact of quantum effects is for event horizons to possess a [[temperature]] and so emit radiation. For [[black hole]]s, this manifests as [[Hawking radiation]], and the larger question of how the black hole possesses a temperature is part of the topic of [[black hole thermodynamics]]. For accelerating particles, this manifests as the [[Unruh effect]], which causes space around the particle to appear to be filled with matter and radiation.

A complete description of event horizons is expected to, at minimum, require a theory of [[quantum gravity]]. One such candidate theory is [[M-theory]]. Another such candidate theory is [[Loop Quantum Gravity]].

==See also==

*[[Acoustic horizon]]
*[[Cosmic censorship]]
*[[Dynamical horizon]]
*[[Event Horizon Telescope]]
*[[Hawking radiation]]
*[[Rindler coordinates]]

==Notes==
{{Reflist|group=Note}}

==References==
{{reflist|2}}

==Further reading==
*[[The Universe in a Nutshell]] by [[Stephen Hawking]]
*{{Cite book| title=[[Black Holes and Time Warps]] | author=Kip Thorne | publisher=W. W. Norton | year=1994}}
*[http://www.livingreviews.org/lrr-2004-10 Abhay Ashtekar and Badri Krishnan, “Isolated and Dynamical Horizons and Their Applications”, Living Rev. Relativity, 7, (2004), 10; Online Article, cited Feb.2009.]

{{Relativity}}

{{DEFAULTSORT:Event Horizon}}
[[Category:Astrophysics]]
[[Category:General relativity]]
[[Category:Black holes]]

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Periodic summation