Schwarzschild radius

The Schwarzschild radius is a measure of the size of a Schwarzschild black hole, that is, a black hole with spherical and static symmetry. It corresponds to the apparent radius of the event horizon, expressed in Schwarzschild coordinates.

Since the size of a black hole depends on the energy absorbed by it, the greater the mass of the black hole, the greater the Schwarzschild radius, which is given by:

rs=2GMc2{displaystyle r_{s}={2GM over c^{2}}}}

Where:

• G It's the gravitational constant,
• M is the mass of the object and
• c It's the speed of light.

This expression was found by Karl Schwarzschild in 1916 and constitutes part of an exact solution for the gravitational field formed by a star with non-rotating spherical symmetry. Schwarzschild's solution was the first exact solution found for the equations of general relativity. The Schwarzschild radius is proportional to the mass of the object. The Schwarzschild radius for the mass of the Sun is 3 km while the Schwarzschild radius for an Earth mass object is only 8.89 mm. The supermassive black hole at the galactic center has a mass of about 4 million solar masses and a radius of about 12 million kilometers (about 40 light-seconds).

History

In 1916, Karl Schwarzschild obtained the exact solution to Einstein's field equations for the gravitational field outside a non-revolving and spherically symmetrical body with mass M{displaystyle M} (see Schwarzschild metric). The solution contained terms of form 1− − rs/r{displaystyle 1-{r_{s}}/r} and 11− − rs/r{displaystyle {frac {1}{1}{1-{r_{s}}}/r}}}}which become unique r=0{displaystyle r=0} and r=rs{displaystyle r=r_{s}} respectively. The rs{displaystyle r_{s}} has become known as the radio by Schwarzschild. The physical meaning of these singularities was discussed for decades. It was discovered that the r=rs{displaystyle r=r_{s}} is a uniqueness of coordinates, which means it is an artifact of the particular coordinate system that was used; while the one located in r=0{displaystyle r=0} is a singularity of spacetime and cannot be eliminated. The Schwarzschild radio is, however, a physically relevant amount, as noted above and below.

This expression had previously been calculated, using Newtonian mechanics, as the radius of a spherically symmetric body in which the escape velocity was equal to the speed of light. It had been identified in the 18th century by John Michell and Pierre-Simon Laplace.

Parameters

The Schwarzschild radius of an object is proportional to its mass. Thus, the Sun has a Schwarzschild radius of about 3 km (1.9 mi), while Earth's is only about 9 mm (0.4 in), and the Moon's is about 0.1 mm. (0 in). The mass of the observable universe has a Schwarzschild radius of approximately 13.7 billion light-years.

Object	Masa M{textstyle M}	Radio de Schwarzschild 2GMc2{textstyle {frac {2GM}{c^{2}}}}	Real radio r{textstyle r}	Density of Schwarzschild 3c632π π G3M2{textstyle {frac {3c^{6}{32pi G^{3}M^{2}}}}} or 3c28π π Gr2{textstyle {frac {3c^{2}{8pi Gr^{2}}}
observable universe	8.8×1052 kg	1.3×1026 m (13.7 billion light years)	4,4×1026 m (46.5 billion light-years)	9,5×10-27 kg/m3.
Milky Way	1,6×1042 kg	2,4×1015 m (0.25 year-luz)	5×1020 m (52.9 thousand light years)	0.000029 kg/m3.
TON 618 (greater known black hole)	1.3×1041 kg	1.9×1014 (~1300 AU)		0.0045 kg/m3.
SMBH in NGC 4889	4,2×1040 kg	6,2×1013 m (~410 AU)		0,042 kg/m3.
SMBH in Messier 87	1.3×1040 kg	1.9×1013 m (~130 AU)		0.44 kg/m3.
SMBH in Andromeda Galaxy	3.4×1038 kg	5,0×1011 m (3.3 AU)		640 kg/m3.
Sagittarius A* (SMBH in the Milky Way)	8,2×1036 kg	1,2×1010 (0.08 AU)		1.1×106 kg/m3.
Sun	1,99×1030 kg	2,95×103 m	7,0×108 m	1,84×1019 kg/m3.
Jupiter	1,90×1027 kg	2.82 m	7,0×107 m	2,02×1025 kg/m3.
Earth	5,97×1024 kg	8,87 mm	6,37×106 m	2,04×1030 kg/m3.
Moon	7,35×1022 kg	1.09×10-4 m	1,74×106 m	1,35×1034 kg/m3.
Saturn	5,683×1026 kg	8,42×10-1 m	5,82×107 m	2,27×1026 kg/m3.
Uranus	8,681×1025 kg	1,29×10-1 m	2,54×107 m	9,68×1027 kg/m3.
Neptune	1.024×1026 kg	1.52×10-1 m	2,46×107 m	6,97×1027 kg/m3.
Mercury	3,285×1023 kg	4,87×10-4 m	2,44×106 m	6,79×1032 kg/m3.
Venus	4,867×1024 kg	7,21×10-3 m	6,05×106 m	3,10×1030 kg/m3.
Mars	6,39×1023 kg	9,47×10-4 m	3,39×106 m	1,80×1032 kg/m3.
Person	70 kg	1,04×10-25 m	~5×10-1 m	1,49×1076 kg/m3.
Planck Mass	2,18×10-8 kg	3,23×10- 35 m	(two times Planck length)	1,54×1095 kg/m3.

Classification of black holes according to the Schwarzschild radius