Event horizon

This article is about the event horizon around a black hole. For another meaning, see Observable Universe .

Black Hole Metrics

	static ${\displaystyle (J=0)}$	rotating ${\displaystyle (J\neq 0)}$
uncharged ${\displaystyle (Q=0)}$	Schwarzschild metric	Kerr metric
loaded ${\displaystyle (Q\neq 0)}$	Reissner-Nordström metric	Kerr-Newman metric
${\displaystyle Q}$: Electric charge ; : Angular momentum ${\displaystyle \ J}$

An event horizon is the general theory of relativity , an interface in the space-time , for which it holds that events beyond this interface are generally not visible to observers who are on this side of the interface. With "events" are meant points in space-time that are determined by place and time. The event horizon forms a boundary for information and causal relationships that result from the structure of spacetime and the laws of physics, especially with regard to the speed of light . In the case of static black holes , the event horizon is a spherical surface, the radius of which is called the Schwarzschild radius .

For every mass above Planck's mass there is a Schwarzschild radius: If an object is compressed onto a spherical volume with a smaller radius than its Schwarzschild radius, it becomes a black hole. Objects with less mass have too great a spatial blur and therefore cannot be compressed sufficiently. For example, the positional uncertainty of a much lesser-mass proton is around 10 ⁻¹⁵  m, while the event horizon  would be 10 ⁻⁵⁴ m.

The shape and size of the event horizon depend only on the mass, the angular momentum and the charge of the black hole inside it. In general, the event horizon is shaped like an ellipsoid of revolution . A non-rotating, electrically uncharged black hole is spherical in shape.

introduction

Outer Schwarzschild solution (flame paraboloid)

The gravitational field of a body consists of an outer and an inner solution of the field equations , whereby the outer solution describes the gravitational field outside the body and the inner solution describes the field inside the body. In the case of a homogeneous, non-charged and non-rotating sphere, the Schwarzschild metric describes the internal and external gravitational field.

For an object that is itself larger than the Schwarzschild radius, there is no event horizon because the inner part does not belong to the outer Schwarzschild solution; the inner solution contains no singularities . Only when an object becomes smaller than its Schwarzschild radius does a singularity arise and an event horizon appears in spacetime. In the case of non-rotating and electrically uncharged black holes, the event horizon is the surface of a sphere around the central singularity. The radius of this sphere is the Schwarzschild radius.

The scalar curvature of spacetime at the event horizon of the Schwarzschild metric is zero, because the metric is a vacuum solution of Einstein's field equations, which implies that neither the scalar curvature nor the Ricci tensor can be different from zero. A measure of curvature that does not disappear at the event horizon is the Kretschmann scalar

${\displaystyle R^{\alpha \beta \gamma \delta }R_{\alpha \beta \gamma \delta }={\frac {12{r_{\mathrm {S} }}^{2}}{r^{6}}}={\frac {48G^{2}M^{2}}{c^{4}r^{6}}},}$

which takes on the value at the event horizon , where the speed of light , the gravitational constant , the mass and the Schwarzschild radius of the black hole are. ${\displaystyle {12}/{r_{\mathrm {S} }^{4}}}$ ${\displaystyle c}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle r_{\mathrm {S} }}$

In the far field , the classic law of gravitation continues to apply as an approximation. However, this approximation leads to increasingly larger deviations the closer one approaches the event horizon. In the immediate vicinity of the event horizon, the general theory of relativity must finally be used.

history

John Michell was the first to grapple with the question of how great the gravitational pull of a celestial body must be so that light can no longer escape from its surface. Using Newton's theory of gravity and corpuscle theory , he found a relationship between the radius and the mass of a celestial body in which this effect occurs. Karl Schwarzschild found this radius again in a general relativistic calculation in 1916, which is why it was named the Schwarzschild radius in his honor.

Event horizon in the Schwarzschild metric

For non-rotating black holes, the event horizon in the Schwarzschild metric is identical to the Schwarzschild radius . The Schwarzschild radius of a body of mass is given by: ${\displaystyle r_{\mathrm {S} }}$ ${\displaystyle M}$

${\displaystyle r_{\mathrm {S} }={\frac {2GM}{c^{2}}}\approx M\cdot 1{,}485\cdot 10^{-27}{\frac {\mathrm {m} }{\mathrm {kg} }}}$

Often the mass of objects in astronomy is given in solar masses , with . For the Schwarzschild radius of the sun we get: ${\displaystyle \mathrm {M_{\odot }} =1{,}988\cdot 10^{30}\,\mathrm {kg} \approx 2\cdot 10^{30}\,\mathrm {kg} }$ ${\displaystyle r_{\mathrm {S,\odot } }}$

${\displaystyle r_{\mathrm {S,\odot } }\approx 2{,}952\cdot 10^{3}\,\mathrm {m} \approx 3\cdot 10^{3}\,\mathrm {m} }$

or in general:

${\displaystyle r_{\mathrm {S} }\approx M\cdot 2{,}952\cdot 10^{3}\,{\frac {\mathrm {m} }{\mathrm {M_{\odot }} }}\;.}$

The Schwarzschild volume is accordingly

${\displaystyle V_{\mathrm {S} }={\frac {4}{3}}\pi r_{\mathrm {S} }^{3}={\frac {4}{3}}\pi {\frac {8G^{3}M^{3}}{c^{6}}},}$

with which a critical density is achieved

${\displaystyle \rho _{c}={\frac {M}{V}}={\frac {M}{{\frac {4}{3}}\pi {\frac {8G^{3}M^{3}}{c^{6}}}}}={\frac {3c^{6}}{32\pi G^{3}M^{2}}}}$

lets define. As soon as a body exceeds this density, a black hole is created. For the mass of the sun is the Schwarzschild radius , for the earth and for Mount Everest . ${\displaystyle 2952\;{\text{m}}}$ ${\displaystyle 9\;{\text{mm}}}$ ${\displaystyle 1\;{\text{nm}}}$

It should also be noted that the radius of the event horizon in general relativity does not indicate the distance from the center, but is defined via the surface of spheres. A spherical event horizon with radius has the same area as a sphere with the same radius in Euclidean space, namely . Due to the space-time curvature, the radial distances in the gravitational field are increased (that is, the distance between two spherical shells with radial coordinates defined by the spherical surface and is greater than the difference between these radii). ${\displaystyle r_{\mathrm {H} }}$ ${\displaystyle A=4\pi r_{\mathrm {H} }^{2}}$ ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$

Significance and properties of the event horizon of a black hole

Gravitational redshift

The frequency of a photon that reaches a distant observer from a gravitational field is shifted to the red (low-energy) part of the light spectrum , since the corresponding potential energy is lost to the photon . The closer the light source is to the black hole, the greater the redshift. At the event horizon the redshift becomes infinitely great.

Incident time for an outside observer

To an outside observer who watches from a safe distance as a particle hits a black hole, it appears as if it is asymptotically approaching the event horizon. This means that an outside observer never sees how it reaches the event horizon, since from his point of view it takes an infinite amount of time. This does not apply to macroscopic objects that deform space-time themselves. In particular, supernovae can be observed.

Incidence time for a freely falling observer

For an observer who is moving in free fall towards the black hole, this is of course different. This observer reaches the event horizon in finite time. The apparent contradiction to the previous result arises from the fact that both considerations are carried out in different frames of reference. An object that has reached the event horizon falls into the central singularity in finite time (viewed from the object itself).

Geometric properties

The event horizon of a black hole represents a light-like surface. Geometrically speaking, it is the amount of all radially outgoing rays of light which cannot escape the black hole and which do not fall into the black hole, i. i.e., which appear to be frozen at a constant radial coordinate. As a result, it is impossible for a body in mass to dwell on the event horizon. It must leave the event horizon in the direction of a decreasing radial coordinate.

The event horizon is not an objective limit. A freely falling observer could therefore not directly determine when he was passing the event horizon.

Angular momentum and electrical charge

Rotating black holes

For rotating black holes, the Kerr metric results in an event horizon, which, however, in contrast to the event horizon of the Schwarzschild metric, has the geometric properties of an ellipsoid of revolution. The dimensions of this ellipsoid of revolution depend on the angular momentum and the mass of the black hole.

The event horizon of a rotating black hole is through in Boyer-Lindquist coordinates ${\displaystyle r_{\mathrm {H} }}$

${\displaystyle r_{\mathrm {H} }:={\frac {GM}{c^{2}}}+{\sqrt {\left({\frac {GM}{c^{2}}}\right)^{2}-a^{2}}}}$

given with and the angular momentum . ${\displaystyle a:={\frac {J}{Mc}}}$ ${\displaystyle J}$

The solution for for a black hole with a given mass depends only on its rotation . Two special cases can be identified: ${\displaystyle r_{\mathrm {H} }}$ ${\displaystyle a}$ For , d. H. for a non-rotating black hole, is ${\displaystyle a\to 0}$

${\displaystyle r_{\mathrm {H} }=2{\frac {GM}{c^{2}}}=r_{\mathrm {S} }}$

and thus identical to the radius from the Schwarzschild metric. For , d. H. for a maximally rotating black hole, is ${\displaystyle r_{\mathrm {H} }}$
${\displaystyle a\to GM/c^{2}}$

${\displaystyle r_{\mathrm {H} }={\frac {GM}{c^{2}}}}$

and is also called the gravitational radius. In Cartesian background coordinates, on the other hand , the radius is at maximum rotation , while the physical axial gyration radius is. The poloidial radius of gyration ${\displaystyle r_{\mathrm {G} }}$ ${\displaystyle {\bar {r}}={\sqrt {2}}\ r_{\mathrm {s} }}$ ${\displaystyle {\bar {R}}_{\phi }=U_{\phi }/(2\pi )={\sqrt {|g_{\phi \phi }|}}=r_{\mathrm {s} }}$

${\displaystyle {\bar {R}}_{\theta }={\sqrt {|g_{\theta \theta }|}}={\sqrt {a^{2}\cos ^{2}\theta +r^{2}}}}$

however, is not only dependent on the radial coordinate , but also on the pole angle. The surface of the event horizon at maximum rotation is thus ${\displaystyle r}$

${\displaystyle A_{\mathrm {H} }=\int _{0}^{\pi }2\pi \ {\bar {R}}_{\phi }\ {\bar {R}}_{\theta }\,\mathrm {d} \theta =4\pi \ (r^{2}+a^{2})=8\pi \ G^{2}M^{2}/c^{4}}$

and not, as one might assume . ${\displaystyle 4\pi \ r_{\text{H}}^{2}}$

The gravitational radius is often used as a unit of length when describing the vicinity of a black hole.

Around the event horizon of the rotating black hole there is also the ergosphere , in which spacetime itself increasingly participates in the rotation of the black hole. Matter, light, magnetic fields, etc. must basically rotate with the black hole within the ergosphere. Since charges induce a strong magnetic field in the ergosphere, the observed jets and their synchrotron radiation in active galaxy nuclei can thus be explained.

Electrically charged black holes

Electrically charged, non-rotating black holes are described by the Reissner-Nordström metric , electrically charged, rotating black holes by the Kerr-Newman metric .

literature

• Ray d'Inverno: Introduction to Relativity . 2nd Edition. Wiley-VCH, Berlin 2009, ISBN 978-3-527-40912-9 , Chapters 6.7, 23.13 and 23.14.

Web links

Wiktionary: event horizon  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ^ Alan Ellis: Black Holes - Part 1 - History. ( Memento of October 6, 2017 in the Internet Archive ) In: Journal of the Astronomical Society of Edinburgh. 39 (1999), English, description of Michell's theory of the "dark stars". Retrieved February 15, 2012.
2. ↑ K. Schwarzschild: About the gravitational field of a mass point according to Einstein's theory. In: Meeting reports of the German Academy of Sciences in Berlin, class for mathematics, physics and technology. (1916) p. 189.
3. ↑ ^{a b} Florian Scheck: Theoretical Physics 3: Classical Field Theory. Springer, Berlin 2005, ISBN 3-540-23145-5 , p. 354. Online version at Google Books. Retrieved February 21, 2012.
4. ↑ Ray d'Inverno: Introduction to the theory of relativity. 2nd edition, Wiley-VCH, Berlin 2009, ISBN 978-3-527-40912-9 , p. 311.
5. ↑ ^{a b} Ray d'Inverno: Introduction to the theory of relativity. 2nd edition, Wiley-VCH, Berlin 2009, ISBN 978-3-527-40912-9 , p. 318
6. ^ Predrag Jovanović, Luka Č. Popović: X-ray Emission From Accretion Disks of AGN: Signatures of Supermassive Black Holes. Astronomical Observatory, Volgina 7, 11160 Belgrade, Serbia (PDF; 1.5 MB) p. 15. Retrieved on February 24, 2012.
7. ^ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes. P. 5 ff.
8. ^ Raine, Thomas: Black Holes: A Student Text. P. 80 ff.
9. ↑ Matt Visser: The Kerr spacetime: A brief introduction. (First published: arxiv : 0706.0622 ), p. 27, equation 118.
10. ^ Andreas Müller: Astro Lexicon G4. Entry "Gravitation radius", portal science-online of the spectrum of science publishing company mbH. Retrieved February 22, 2012.