Schwarzschild radius

The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.

The Schwarzschild radius is given as $${\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}$$ where G is the gravitational constant, M is the object mass, and c is the speed of light.^{[note 1][1][2]}

History

In 1916, Karl Schwarzschild obtained the exact solution^[3][4] to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass ${\displaystyle M}$ (see Schwarzschild metric). The solution contained terms of the form ${\displaystyle 1-{r_{\text{s}}}/r}$ and ${\displaystyle {\frac {1}{1-{r_{\text{s}}}/r}}}$, which becomes singular at ${\displaystyle r=0}$ and ${\displaystyle r=r_{\text{s}}}$ respectively. The ${\displaystyle r_{\text{s}}}$ has come to be known as the Schwarzschild radius. The physical significance of these singularities was debated for decades. It was found that the one at ${\displaystyle r=r_{\text{s}}}$ is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at ${\displaystyle r=0}$ is a spacetime singularity and cannot be removed.^[5] The Schwarzschild radius is nonetheless a physically relevant quantity, as noted above and below.

This expression had previously been calculated, using Newtonian mechanics, as the radius of a spherically symmetric body at which the escape velocity was equal to the speed of light. It had been identified in the 18th century by John Michell^[6] and Pierre-Simon Laplace.^[7]

Parameters

The Schwarzschild radius of an object is proportional to its mass. Accordingly, the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi),^[8] whereas Earth's is approximately 9 mm (0.35 in)^[8] and the Moon's is approximately 0.1 mm (0.0039 in).

Object's Schwarzschild radius

Object	Mass ${\textstyle M}$	Schwarzschild radius ${\textstyle {\frac {2GM}{c^{2}}}}$	Actual radius ${\textstyle r}$	Schwarzschild density ${\textstyle {\frac {3c^{6}}{32\pi G^{3}M^{2}}}}$ or ${\textstyle {\frac {3c^{2}}{8\pi Gr^{2}}}}$
Milky Way	1.6×1042 kg	2.4×1015 m (0.25 ly)	5×1020 m (52900 ly)	0.000029 kg/m3
SMBH in Phoenix A (one of the largest known black holes)	2×1041 kg	3×1014 m (~2000 AU)		0.0018 kg/m3
Ton 618	1.3×1041 kg	1.9×1014 m (~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[9]	1.3×1040 kg	1.9×1013 m (~130 AU)		0.44 kg/m3
SMBH in Andromeda Galaxy[10]	3.4×1038 kg	5.0×1011 m (3.3 AU)		640 kg/m3
Sagittarius A* (SMBH in Milky Way)[11]	8.26×1036 kg	1.23×1010 m (0.08 AU)		1.068×106 kg/m3
SMBH in NGC 4395[12]	7.1568×1035 kg	1.062×109 m (1.53 R⊙)		1.4230×108 kg/m3
Potential intermediate black hole in HCN-0.009-0.044[13][14]	6.3616×1034 kg	9.44×108 m (14.8 R🜨)		1.8011×1010 kg/m3
Resulting intermediate black hole from GW190521 merger[15]	2.823×1032 kg	4.189×105 m (0.066 R🜨)		9.125×1014 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
Saturn	5.683×1026 kg	8.42×10−1 m	6.03×107 m	2.27×1026 kg/m3
Neptune	1.024×1026 kg	1.52×10−1 m	2.47×107 m	6.97×1027 kg/m3
Uranus	8.681×1025 kg	1.29×10−1 m	2.56×107 m	9.68×1027 kg/m3
Earth	5.97×1024 kg	8.87×10−3 m	6.37×106 m	2.04×1030 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
Mercury	3.285×1023 kg	4.87×10−4 m	2.44×106 m	6.79×1032 kg/m3
Moon	7.35×1022 kg	1.09×10−4 m	1.74×106 m	1.35×1034 kg/m3
Human	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 (2 lP)		1.54×1095 kg/m3

Derivation

Main article: Derivation of the Schwarzschild solution

The simplest way of deriving the Schwarzschild radius comes from the equality of the modulus of a spherical solid mass' rest energy with its gravitational energy:

${\displaystyle Mc^{2}={\frac {2GM^{2}}{r}}}$

So, the Schwarzschild radius reads as

${\displaystyle r={\frac {2GM}{c^{2}}}}$

Black hole classification by Schwarzschild radius

Black hole classifications

Class	Approx. mass	Approx. radius
Supermassive black hole	105–1011 MSun	0.002–2000 AU
Intermediate-mass black hole	103 MSun	3 x 103 km ≈ RMars
Stellar black hole	10 MSun	30 km
Micro black hole	up to MMoon	up to 0.1 mm

Any object whose radius is smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body (a rotating black hole operates slightly differently). Neither light nor particles can escape through this surface from the region inside, hence the name "black hole".

Black holes can be classified based on their Schwarzschild radius, or equivalently, by their density, where density is defined as mass of a black hole divided by the volume of its Schwarzschild sphere. As the Schwarzschild radius is linearly related to mass, while the enclosed volume corresponds to the third power of the radius, small black holes are therefore much more dense than large ones. The volume enclosed in the event horizon of the most massive black holes has an average density lower than main sequence stars.

Supermassive black hole

Main article: Supermassive black hole

A supermassive black hole (SMBH) is the largest type of black hole, though there are few official criteria on how such an object is considered so, on the order of hundreds of thousands to billions of solar masses. (Supermassive black holes up to 21 billion (2.1 × 10¹⁰) M_☉ have been detected, such as NGC 4889.)^[16] Unlike stellar mass black holes, supermassive black holes have comparatively low average densities. (Note that a (non-rotating) black hole is a spherical region in space that surrounds the singularity at its center; it is not the singularity itself.) With that in mind, the average density of a supermassive black hole can be less than the density of water.

The Schwarzschild radius of a body is proportional to its mass and therefore to its volume, assuming that the body has a constant mass-density.^[17] In contrast, the physical radius of the body is proportional to the cube root of its volume. Therefore, as the body accumulates matter at a given fixed density (in this example, 997 kg/m³, the density of water), its Schwarzschild radius will increase more quickly than its physical radius. When a body of this density has grown to around 136 million solar masses (1.36 × 10⁸ M_☉), its physical radius would be overtaken by its Schwarzschild radius, and thus it would form a supermassive black hole.

It is thought that supermassive black holes like these do not form immediately from the singular collapse of a cluster of stars. Instead they may begin life as smaller, stellar-sized black holes and grow larger by the accretion of matter, or even of other black holes.^[18]

The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way is approximately 12 million kilometres.^[11] Its mass is about 4.1 million M_☉.

Stellar black hole

Main article: Stellar black hole

Stellar black holes have much greater average densities than supermassive black holes. If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 10¹⁸ kg/m³; neutron stars also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3 M_☉ and thus would be a stellar black hole.

Micro black hole

Main article: Micro black hole

A small mass has an extremely small Schwarzschild radius. A black hole of mass similar to that of Mount Everest^{[19][note 2]} would have a Schwarzschild radius much smaller than a nanometre.^{[note 3]} Its average density at that size would be so high that no known mechanism could form such extremely compact objects. Such black holes might possibly be formed in an early stage of the evolution of the universe, just after the Big Bang, when densities of matter were extremely high. Therefore, these hypothetical miniature black holes are called primordial black holes.

When moving to the Planck scale ${\displaystyle \ell _{P}={\sqrt {(G/c^{3})\,\hbar }}}$ ≈ 10⁻³⁵ m, it is convenient to write the gravitational radius in the form ${\displaystyle r_{s}=2\,(G/c^{3})Mc}$, (see also virtual black hole).^[20]

Other uses

In gravitational time dilation

Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows:^[21] $${\displaystyle {\frac {t_{r}}{t}}={\sqrt {1-{\frac {r_{\mathrm {s} }}{r}}}}}$$ where:

• t_r is the elapsed time for an observer at radial coordinate r within the gravitational field;
• t is the elapsed time for an observer distant from the massive object (and therefore outside of the gravitational field);
• r is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object);
• r_s is the Schwarzschild radius.

Compton wavelength intersection

The Schwarzschild radius (${\displaystyle 2GM/c^{2}}$) and the Compton wavelength (${\displaystyle 2\pi \hbar /Mc}$) corresponding to a given mass are similar when the mass is around one Planck mass (${\textstyle M={\sqrt {\hbar c/G}}}$), when both are of the same order as the Planck length (${\textstyle {\sqrt {\hbar G/c^{3}}}}$).

Gravitational radius and the Heisenberg Uncertainty Principle

${\displaystyle r_{s}={\frac {2GM}{c^{2}}}={\frac {2G}{c^{3}}}Mc={\frac {2G}{c^{3}}}P_{0}\Rightarrow {\frac {2G}{c^{3}}}{\frac {\hbar }{2r}}={\frac {\ell _{P}^{2}}{r}}.}$

Thus, ${\displaystyle r_{s}r\sim \ell _{P}^{2}}$ or ${\displaystyle \Delta r_{s}\Delta r\geq \ell _{P}^{2}}$, which is another form of the Heisenberg uncertainty principle on the Planck scale. (See also Virtual black hole).^[20][22]

Calculating the maximum volume and radius possible given a density before a black hole forms

The Schwarzschild radius equation can be manipulated to yield an expression that gives the largest possible radius from an input density that doesn't form a black hole. Taking the input density as ρ,

${\displaystyle r_{\text{s}}={\sqrt {\frac {3c^{2}}{8\pi G\rho }}}.}$

For example, the density of water is 1000 kg/m³. This means the largest amount of water you can have without forming a black hole would have a radius of 400 920 754 km (about 2.67 AU).

See also

• Black hole, a general survey
• Chandrasekhar limit, a second requirement for black hole formation
• John Michell

Classification of black holes by type:

• Static or Schwarzschild black hole
• Rotating or Kerr black hole
• Charged black hole or Newman black hole and Kerr–Newman black hole

A classification of black holes by mass:

• Micro black hole and extra-dimensional black hole
• Planck length
• Primordial black hole, a hypothetical leftover of the Big Bang
• Stellar black hole, which could either be a static black hole or a rotating black hole
• Supermassive black hole, which could also either be a static black hole or a rotating black hole
• Visible universe, if its density is the critical density, as a hypothetical black hole
• Virtual black hole

Notes

1. In geometrized unit systems, G and c are both taken to be unity, which reduces this equation to ${\displaystyle r_{\text{s}}=2M}$.
2. Using these values,^[19] one can calculate a mass estimate of 6.3715×10¹⁴ kg.
3. One can calculate the Schwarzschild radius: 2 × 6.6738×10⁻¹¹ m³⋅kg⁻¹⋅s⁻² × 6.3715×10¹⁴ kg / (299792458 m⋅s⁻¹)² = 9.46×10⁻¹³ m = 9.46×10⁻⁴ nm.

References

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