Black hole

:This article is about an object in astrophysics. For other uses, see Black hole (disambiguation).

Mathematical theory

Black holes are predictions of Albert Einstein's theory of general relativity. In particular, they occur in the Schwarzschild metric, one of the earliest and simplest solutions to Einstein's equations, found by Karl Schwarzschild in 1915. This solution describes the curvature of spacetime in the vicinity of a static and spherically symmetric object, where the metric is

Related Topics:
Albert Einstein - General relativity - Schwarzschild metric - Karl Schwarzschild - 1915 - Curvature - Spacetime - Spherically - Symmetric - Metric

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: ds^2 = - c^2 left( 1 - {2Gm over c^2 r} ight) dt^2 + left( 1 - {2Gm over c^2 r} ight)^{-1} dr^2 + r^2 dOmega^2 ,

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where dOmega^2 = d heta^2 + sin^2 heta; dphi^2 is a standard element of solid angle.

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According to Schwarzschild's solution, a gravitating object will collapse into a black hole if its radius is smaller than a characteristic distance, known as the Schwarzschild radius. Below this radius, spacetime is so strongly curved that any light ray emitted in this region, regardless of the direction in which it is emitted, will travel towards the center of the system. Because relativity forbids anything from travelling faster than light, anything below the Schwarzschild radius – including the constituent particles of the gravitating object – will collapse into the center. A gravitational singularity, a region of theoretically infinite density, forms at this point. Because not even light can escape from within the Schwarzschild radius, a classical black hole would truly appear black.

Related Topics:
Schwarzschild radius - Relativity - Faster than light - Gravitational singularity - Black

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The Schwarzschild radius is given by

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:r_s = {2,Gm over c^2}

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where G is the gravitational constant, m is the mass of the object, and c is the speed of light. For an object with the mass of the Earth, the Schwarzschild radius is a mere 9 millimeters — about the size of a marble.

Related Topics:
Gravitational constant - Mass - Speed of light - Earth - Millimeters - Marble

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The mean density inside the Schwarzschild radius decreases as the mass of the black hole increases, so while an earth-mass black hole would have a density of 2 × 1030 kg/m3, a supermassive black hole of 109 solar masses has a density of around 20 kg/m3, less than water! The mean density is given by

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: ho=rac{3,c^6}{32pi m^2G^3}

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Since the Earth has a mean radius of 6371 km, its volume would have to be reduced 4 × 1026 times to collapse into a black hole. For an object with the mass of the Sun, the Schwarzschild radius is approximately 3 km, much smaller than the Sun's current radius of about 700,000 km. It is also significantly smaller than the radius to which the Sun will ultimately shrink after exhausting its nuclear fuel, which is several thousand kilometers. More massive stars can collapse into black holes at the end of their lifetimes.

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More general black holes are also predicted by other solutions to Einstein's equations, such as the Kerr metric for a rotating black hole, which possesses a ring singularity. Then we have the Reissner-Nordström metric for charged black holes. Last the Kerr-Newman metric is for the case of a charged and rotating black hole.

Related Topics:
Kerr metric - Ring singularity - Reissner-Nordström metric - Kerr-Newman metric

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There is also the Black Hole Entropy formula:

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S = rac{Akc^3}{4hbar G}

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Where A is the area of the event horizon of the black hole, hbar is Dirac's constant (the "reduced Planck constant"), k is Boltzmann's constant, G is the gravitational constant, c is the speed of light and S is the entropy.

Related Topics:
Dirac's constant - Gravitational constant - Speed of light

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