Gravity of surface

In Astronomy, the revolved of surface is the intensity of the gravitational Champ on the surface of an astrophysical object (Planet, star or other). This concept is also used, though in a slightly different way, in the physics of the black holes where it regulates the speed to which the gravitational field in the classical sense of the term diverges with the approach from surface from the black hole, i.e. of sound horizon.

Newtonian formula

In the framework of Newtonian mechanics, the gravity of surface east given by the usual formula of the gravitational field of a spherical object, namely

$\backslash \; kappa\; =\; \backslash \; frac\; \{G\; M\}\; \{R^2\}$,

where G is the Constante of gravitation, M the Masse of the object considered and R its ray, about spherical object being considered.

Case of the black holes

Within the framework of the physics of the black holes, it is possible to define an analog of the concept of gravity of surface. It is necessary however to take guard with the fact that a black hole can be almost by definition regarded as an object on the “surface” whose (i.e. on the level of sound horizon) the gravitational field is infinite. There exists however another quantity which diverges as one approaches the horizon of a black hole: it is about the Décalage towards the red of gravitational origin which the signals emitted from this zone undergo. Within this framework, one defines the revolved of surface of a black hole by the limit of the relationship between the intensity of the gravitational field in the shift towards the red caused by the black hole. One can then show that this quantity remains finished when one approaches the horizon, and that in the simplest case of a Black hole of Schwarzschild, his value is equal to that which one would naively deduce in a Newtonian treatment, i.e. it is worth again G  M  /  R 2 .

Formulate and particular cases

The exact expression of the gravity of surface is written, in geometrical Unités,

$\backslash \; kappa\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{M^2\; -\; a^2\; -\; Q^2\}\}\; \{2\; M\; \backslash \; left\; (M\; +\; \backslash \; sqrt\; \{M^2\; -\; a^2\; -\; Q^2\}\; \backslash \; right)\; -\; Q^2\}$,

where M , Q , has respectively represent the mass, the electric Charge and the kinetic Moment reduced ( L / M ) of the black hole.

In the case of a Black hole of Schwarzschild, i.e. not having neither electric charge nor kinetic moment, one obtains

$\backslash \; kappa\; =\; \backslash \; frac\; \{1\}\; \{4\; M\}$,

what gives, with the units of the international Système,

$\backslash \; kappa\_\; \{\backslash \; rm\; Sch\}\; =\; \backslash \; frac\; \{G\; M\}\; \{R^2\}$,

with

$R\; =\; \backslash \; frac\; \{2\; G\; M\}\; \{c^2\}$

corresponding to the Ray of Schwarzschild.

For a extreme Black hole, for which the $M^2\; quantity\; -\; Q^2\; -\; a^2$ is cancelled, one has

$\backslash \; kappa\_\; \{\backslash \; rm\; ext.\}\; =\; 0$.

Properties of the gravity of surface of a black hole

The principal property of the gravity of surface of a black hole is that it is strictly constant on all the surface of the black hole. This result is logical in the case of a black hole with spherical symmetry (black hole of Schwarzschild and Reissner-Nordström), but it is more surprising when the black hole is nonspherical because of his rotation (Black hole of Kerr or Kerr-Newman).

The gravity of surface can be given by calculating the Dérivée partial mass of an unspecified black hole compared to his surface has by keeping fixed its electric Charge Q and its kinetic Moment L , according to the formula

$\backslash \; kappa\; =\; \backslash \; frac\; \{G\}\; \{8\; \backslash \; pi\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; M\}\; \{\backslash \; partial\; has\}\; \backslash \; right)\; \_\; \{Q,\; L\}$.

So the Différentielle of the mass of a black hole is written in the Système of geometrical units,

$\{\backslash \; rm\; D\}\; M\; =\; \backslash \; frac\; \{\backslash \; kappa\; \backslash ;\; \{\backslash \; rm\; D\}\; has\}\; \{8\; \backslash \; pi\}\; +\dots$.

The fact that the surface of a black hole necessarily grows during time, and that the gravity of surface is telling on the horizon of a black hole is to be brought closer to the principles of the Thermodynamique which say that the temperature of one object to balance is everywhere the same one in the object and that its Entropie can only grow with time. This fact is actually not pain-killer and is at the origin of the development of a major analogy between black holes and thermodynamics: the Thermodynamic of the black holes. The demonstration of this result is relatively complex, and must with Brandon Casing, Stephen Hawking and James Bardeen, in 1973.

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