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Schwarzschild radius

A measure for the size of a spherically symmetric black hole. It is defined using the area of the black hole’s horizon: In usual high school geometry (the geometry of flat space), radius and area of a spherical surface are related as

area = 4 times pi times radius².

The Schwarzschild radius is defined indirectly by the formula

Area of the black hole’s horizon = 4 times pi times (Schwarzschild radius)².

It is directly proportional to the black hole’s mass. The Schwarzschild radius for an object the mass of the earth is 9 millimetres, for an object with the mass of the sun, 2.95 kilometres.

There is a quite general result that says: If a sphere of matter is compressed further and further, a black hole forms as soon as the sphere’s radius gets smaller than the Schwarzschild radius corresponding to the matter’s mass.

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surface

Geometric space with two dimensions. Examples include the plane or the surface of a sphere.

sphere

A spherical surface is a simple example for a curved surface. It is easily pictured as a surface embedded in three-dimensional space: a spherical surface is the set of all points at a certain fixed distance from a given point (the point being the centre of the sphere). Mathematically, a spherical surface can be described without recourse to three-dimensional space - when mathematicians talk of the geometry of such a surface, they (almost) always mean the "inner geometry": Those properties of the surface noticeable to two-dimensional beings, living and working in that surface, capable of measuring distances and angles in it. Sphere is regularly used as a synonym for spherical surface (instead of describing a solid, three-dimensional ball). And not only for the two-dimensional spherical surface described above, but also for its analogues in lower and higher dimensions. A one-sphere, for instance, is the same as a circle, a two-sphere is the spherical surface defined above, a three-sphere its three-dimensional analogue.

space

In a strict sense: Space as we know it from everyday life: the totality of all locations in which objects can sit, with three dimensions. In a more general sense used by mathematicians, all kinds of sets of points are spaces - a line for instance, which has but a single dimension, or a two-dimensional surface, but also higher-dimensional spaces. Also, in such more general spaces, geometry can be different from the standard Euclidean geometry taught in high schools - such spaces can be curved.

Schwarzschild radius

A measure for the size of a spherically symmetric black hole. It is defined using the area of the black hole's horizon: In usual high school geometry (the geometry of flat space), radius and area of a spherical surface are related as

area = 4 times pi times radius².

The Schwarzschild radius is defined indirectly by the formula

Area of the black hole's horizon = 4 times pi times (Schwarzschild radius)².

It is directly proportional to the black hole's mass. The Schwarzschild radius for an object the mass of the earth is 9 millimetres, for an object with the mass of the sun, 2.95 kilometres. There is a quite general result that says: If a sphere of matter is compressed further and further, a black hole forms as soon as the sphere's radius gets smaller than the Schwarzschild radius corresponding to the matter's mass.

matter

In general relativity: All contents of spacetime that contribute to its curvature: particles, dust, gases, fluids, electromagnetic and other fields. In particle physics: All elementary particles with half-integer spin, such as electrons and quarks, as well as their composites such as protons and neutrons, in contrast with force particles.

mass

In classical physics, mass plays a triple role. First of all, it is a measure for how easy it is to influence the motion of a body. Imagine that you're drifting in emtpy space. Drifting by are an elephant and a mouse, and you give each of them a push of equal strength. The fact that the mouse abruptly changes its path, while the elephant's course is as good as unaltered, is a sure sign that the mass (or, in the language of physics, the inertia or inertial mass) of the elephant is much greater than that of the mouse. Secondly, mass is a measure of how many atoms there are in a body, and of what type they are. All atoms of one and the same type have the same mass, and adding up all those tiny component masses, the total mass of the body results. Thirdly, in Newton's theory of gravity, mass determines how strongly a body attracts other bodies via the gravitational force, and how strongly these bodies attract it (in this sense, mass is the charge associated with the gravitational force). In special relativity, one can also define a mass that is a measure for a bodies resistance to changing its motion. However, the value of this relativistic mass depends on the relative motion of the body and the observer. The relativistic mass is the "m" in Einstein's famous E=mc² (cf. equivalence of mass and energy). The relativistic mass has a minimum for an observer that is at rest relative to the body in question. This value is the so-called rest mass of the body, and when particle physicists talk of mass, this is usually what they mean. Just as in classical physics, the rest mass is a kind of measure for how much matter the body is made up of - with one caveat: For composite bodies, the energies associated with the forces holding the body together contribute to the total mass, as well (another consequence of the equivalence of mass and energy). In general relativity, mass still plays a role as a source of gravity; however, it has been joined by physical quantities such as energy, momentum and pressure.

horizon

In general relativity: A closed surface that is the boundary of a black hole. Whatever enters through this boundary from the outside can never again leave the inside.

Synonym: event horizon.

black hole

Region in space where a sufficient amount of mass is concentrated so that it forms a gravitational prison - a region into which matter or light can enter from the outside, but from which nothing that has once entered can ever leave. Basic information on this key phenomenon of Einstein's general theory of relativity can be found in the chapter Black holes & Co. of Elementary Einstein. Selected aspects of the physics of black holes and neutron stars are described in the category Black holes & Co. of our Spotlights on relativity. In Einstein's theory, black holes are truly black, due to the fact that no radiation or light can ever escape them. Once quantum theory is taken into account, this assumption no longer holds - on the contrary, it seems as if black holes should emit so-called Hawking radiation. However, for astrophysical black holes (that typically have more or even much more than one solar mass), that radiation would be undetectable, even if we could transport today's most sensitive sensors into the immediate vicinity of the black hole.

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Schwarzschild radius

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