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Schwarzschild black hole

A solution of Einstein’s equations found by Karl Schwarzschild in 1916, which corresponds to a model universe that contains a single, spherically symmetric black hole.

More precisely, the Schwarzschild solution is a whole family of solutions: Schwarzschild’s formulae contain a free parameter m corresponding to the mass of the black hole. To each concrete value of m corresponds one specific solution to Einstein’s equations, a spacetime containing a spherically symmetric black hole of mass m.

The Schwarzschild solution is of practical importance as the outlying regions of the corresponding model universe describe the spacetime distortion around all kinds of objects that are spherically symmetric, or nearly so, such as the sun or the earth (cf. Birkhoff’s theorem).

Synonyms: Schwarzschild solution

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.

Schwarzschild solution

See Schwarzschild black hole


In the International System of units: the basic unit of time. Defined as a certain multiple of the oscillation period of electromagnetic radiation set free in a certain transition within the electron shell of atoms of the type Cesium-133.

second of arc

See arcminute, arcseond.

self-force, gravitational

Gravitational self-force is introduced in black hole perturbation theory in order to describe the effect of a particle’s own gravitational field on its motion.

Shapiro delay

Also: gravitational time delay. In general relativity, not only are light rays deflected, in addition gravity can lead to light taking more time in its travels through space than in classical physics. This is called Shapiro effect of Shapiro delay. It has been measured numerous times for light signals in the solar system, for instance for radar waves sent from Earth to Venus and reflected back. These radar signals took measurably longer when their path led them closely by the massive sun.

Measuring this time delay is sometimes referred to as the “fourth test of general relativity”, in addition to the three classical tests of that theory.

Synonyms: Shapiro effect


Generically, an extended body in free fall will experience deformation due to tidal effects. For instance, in a body falling towards Earth, those parts that are slightly closer to the Earth will experience a slightly stronger gravitational pull than parts which are further away. Some of the deformation will change the body’s volume. The shear is that part of the deformation which does not change the volume, only the body’s shape.

Some examples for shear can be found in the spotlight text Of singularities and breadmaking.


The international system of physical units, introduced in 1960. It is based on seven fundamental units; in the context of Einstein Online, the interesting ones are the metre as a measure of length and distance, the second as the unit of time, the kilogram as the unit of mass and the Kelvin as the unit of temperature.

By multiplication and division, the seven fundamental units can be used to construct derived units for all other physical quantities. For instance, the unit of speed is the distance unit divided by the unit of time, metre per second.

Synonyms: Système International d'Unités International System of Units


The sine, written sin(x), is a mathematical function that is perfectly regular and repetitive, with maximal and minimal values following each other in endless procession. The function is plotted here:


Sine waves are the simplest waves imaginable, with crests and valleys following each other in exactly the way described by a sine function.


Irregular boundary of spacetime in general relativity – region where spacetime simply comes to an end. Often, such boundaries are associated with spacetime curvature growing beyond all bounds and becoming infinitely large – so-called curvature singularities (notably Ricci singularities or Weyl singularities – but there are exceptions (for instance a conic singularities).

According to general relativity, a singularity exists inside every black hole, and the starting point of any universe described by a big bang model is a singularity as well. The occurrence of singularities is a failure of general relativity – and a strong indication that the theory is incomplete. Instead, one could describe the earliest universe and the interior of black holes using a theory of quantum gravity.

More information about singularities can be found in the spotlight texts Spacetime singularities and Of singularities and breadmaking.

Synonyms: spacetime singularity

singularity theorems

Theorems, proven by Roger Penrose and Stephen Hawking, that state under which circumstances singularities are inevitable in general relativity. As the theorems assume the laws of general relativity and certain general properties of matter, but nothing else, they are valid quite generally. In particular, these theorems prove that, in the frame-work of general relativity, every black hole must contain a singularity, and every expanding universe like ours must have begun in a big bang singularity.

solar mass

The sun has a mass of 1.989·1030 kilograms [see exponential notation] what is equal to 332 946 earth masses.

In astronomy, the solar mass is frequently used as a unit of mass (“Neutron stars typically have a mass of 1.4 solar masses”), sometimes written as M.

solar system

Our cosmic neighbourhood, consisting of a star – the sun -, eight planets orbiting the sun, numerous smaller bodies, dust and gas.

In the context of relativity, the solar system is interesting as a natural laboratory in which the prediction of general relativity can be tested – in particular those that differ from the predictions of classical, Newtonian gravity. Examples are the relativistic perihelion shift of planetary orbits, the deflection of light close to the sun and the Shapiro effect.

solid state

State of matter in which the atoms or molecules are bound so tightly to each other so that they form a solid, stable lump. In contrast with fluids, whose form adapts to whatever container they are placed in, solid bodies keep their form.

Compare the other states of matter: liquid, gas, plasma.

Synonyms: solid body


In the context of general relativity: a solution or, more precisely, a solution of the Einstein equations is a model universe that follows the law of gravity as prescribed by general relativity. See also exact solution.


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.

Space Telescope Science Institute

The institute operating the Hubble space telescope; located in Baltimore, USA.

Space Telescope Science Institute website


Already in special relativity, observers in motion relative to each other will not, in general, agree as to whether two events happen simultaneously, or as to how great is the distance between two objects. They do, however, agree as to what events there are, although not to when and where they happen. This observer-independent totality of all events is called spacetime. How spacetime is split into space and time can differ from observer to observer.

Every-day space has three dimensions. Adding time adds another dimension – spacetime has four dimensions, all in all.

We are used to the idea of a point in space – an object that has only one location and is completely defined once its space coordinates are given. In spacetime, a spacetime point is an object defined completely once its space coordinates and its time coordinate are given – which makes a spacetime point nothing but an elementary event.

The idea of spacetime is, in addition to its role in special relativity, a building block of general relativity. Analogous to how a plane is flat, but the surface of a sphere is curved, in general relativity, curved or distorted versions of the simple, flat spacetime of special relativity play a role. Spacetime curvature, in general relativity, is intimately connected with gravity.

For an introduction to the basics of both theories of relativity, check out the chapters Special relativity and General relativity in Elementary Einstein. Sometimes, it can be helpful to view spacetime in analogy to ordinary space – such analogies are explored in the spotlight topics Time dilation on the road (for time dilation) and Twins on the road (for the twin effect).

spacetime singularity

See entry singularity above.

special relativity

Albert Einstein’s theory of the fundamentals of space, time, and movement (but not gravity). For a brief introduction, check out the chapter Special relativity of Elementary Einstein.

Selected aspects of special relativity are described in the category Special relativity of our Spotlights on relativity.

Synonyms: special theory of relativity


The electromagnetic radiation reaching us from an astronomical object or other source is a mix of electromagnetic waves with a great variety of frequencies. The spectrum lists the composition of this mix: For every frequency, it states the amount of radiation energy contributed by waves of that particular frequency.

Synonyms: spectra


An object’s average speed is the distance it moves during a given period of time, divided by the length of the time interval. If you make the time interval infinitely small, the result is the object’s speed at one particular moment in time. The notion of speed can be applied to waves in different ways; for instance, for a simple wave, the phase speed is the speed at which any given wave crest or wave propagates through space.

See also the more general entry velocity.

speed of light

The speed at which light or, more generally, electromagnetic radiation propagates through space (especially: through empty space). Central quantity in special relativity: There, the constancy of the speed of light is a basic postulate: every observer (more precisely: every inertial observer) that measures the speed of light in vacuum obtains the same constant value, c=299,792,458 metres per second.

Another important relativistic aspect of the speed of light is that it defines an absolute upper speed limit: In special relativity, nothing can move faster than light, and information or influence at most be transmitted at light-speed. In general relativity, the same law is in force locally: No object, no matter, no information can directly overtake or catch up with light (cf. causality).

Basic information about the role of light speed in special relativity can be found in the chapter Special relativity of Elementary Einstein.


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.


Fundamental quantum property of elementary as well as of compound particles. For elementary particles, the spin determines whether the particle is a matter particle (half-integer spin such as 1/2, 3/2, 5/2 etc.) or a force particle (integer spin such as 0, 1, 2 etc.).

spin network

In loop quantum gravity (one of the candidates for a theory of quantum gravity), the underlying microscopic structure of space is a spin network – a graph consisting of lines and nodes where each line is assigned a label consisting of a half-integer number. Mathematically, that number is closely connected with the spin of elementary particles.

More information about spin networks can be found in the spotlight topic The fabric of space: spin networks.

standard acceleration

The acceleration imparted by the earth’s gravitation to a body located on the earth’s surface: If you raise such a body up a bit and let it fall, it will accelerate 9.81 metres per second squared, in other words: in every second, its speed will increase by 9.81 metres per second.

Standard acceleration, abbreviated as g, is often used as a measure for accelerations. For instance, an acceleration of 2 g corresponds to 2·9.81=19.62 m/s².

standard model of cosmology

Another name for the big bang models.

standard model of elementary particle physics

The current state of the art for describing the basic properties of matter and forces. The standard model theories are based on special relativity and quantum theory and they describe the behaviour of elementary matter particles such as electrons, neutrinos and quarks as well as their anti-particles. It also describes three quantum forces acting between them: electromagnetism, the weak nuclear force and the strong nuclear force. These forces act by the exchange of force particles. There is one elementary force for which no such quantum description exists, and which is not part of the standard model: gravity.

Synonyms: standard model of particle physics


A cosmic gas ball that is massive enough for pressure and temperatur in its core to reach values where self-sustained nuclear fusion reactions set in. The energy set free in these reactions makes stars into very bright sources of light and other forms of electromagnetic radiation.

Once the nuclear fuel is exhausted, the star becomes a white dwarf, a neutron star or a black hole.

Synonyms: stars

states (of matter)

Depending on parameters like temperature or pressure, matter can exist in different states with greatly varying physical properties. The most important states of matter are the solid, liquid, gaseous and plasma states (the latter three forms of matter are also called fluid). At very low temperatures, atoms bind together to form a solid body with a definite shape. As temperature increases, many solid bodies melt, turning into a liquid or, at even higher temperatures, a gas, an ensemble of atoms and/or molecules wildly scurrying back and forth. A further increase in temperature transforms matter into a plasma. In this state atoms disintegrate into atomic nuclei and electrons.


Roughly speaking, in relativity, a situation or a spacetime is stationary if there is no change over time.

To be more precise, one has to take into account that, in general relativity, time can be defined in many different ways, all of them equally valid for formulating the laws of physics. This leads to a modified definition: A situation or a spacetime is stationary if it is possible to define time in a way so that there is no change of its properties over time – if you follow the properties of a given region of space over time, they will not change.

Stefan-Boltzmann law

One of the laws governing the properties of the simplest form of thermal radiation – that emitted by a blackbody: The total energy emitted by such a body is proportional to the fourth power of its temperature (measured in Kelvin).

stellar black holes

Stellar black holes are black holes with between a few and a dozen solar masses that are formed when the core of a massive star collapses.

Basic information about black holes can be found in the chapter Black holes & Co. of Elementary Einstein.

straight line

In the plane, in three-dimensional everyday space or in more general flat spaces: Any line that forms the shortest connection between two given points.

In the spacetime of special relativity: The world-line of an object moving with constant speed on a path that is straight in space.

Synonyms: straight line in space straight line in spacetime

string theory

Candidate for a theory of quantum gravity; a quantum theory, where the fundamental building blocks are tiny, one-dimensional, oscillating entities, called strings.

A brief description can be found on the page String theory in the chapter  Relativity and the quantum of Elementary Einstein.

Synonyms: string, strings

strong force

One of the four fundamental forces in our universe (the others are electromagnetism, weak nuclear force and gravity). The strong force binds the quarks to form compound particles such as protons and neutrons. Indirectly, it is also responsible for holding together protons and neutrons in atomic nuclei.

Synonyms: strong nuclear force


The central (and most massive) body of our solar system; the star closest to us; a ball of gas with a radius of ca. 700000 km (for comparison the radius of the earth: 12756 km) and a mass of 1.989·1030 kilograms [see exponential notation].

In the interior of the sun, nuclear fusion processes run their course; they are responsible for the sun’s impressive brightness.


Class of models that generalize Einstein’s general theory of relativity in a way that it satisfies the requirements of supersymmetry.

Today, supergravity is of interest in the context of string theory: In the limiting case of low energies (where low energies includes everything accessible with modern particle accelerators), the physics of string theories can be described with the help of certain supergravity models.

supermassive black holes

are black holes with masses of more than a million solar masses. As far as we know, such holes can be found in the central regions of almost all galaxies. Central black holes are the energy source for radio galaxies and other active galactic nuclei.

Basic informations about black holes in general can be found in the chapter Black holes & Co. of Elementary Einstein.


Highly energetic explosion that ends the life of stars with more than about ten solar masses. In this explosion, the outer layers of the stars are ejected into space, while the core regions collapse to form a neutron star or even a black hole.

superstring theory

Synonym: supersymmetric string theory. String theory that satisfies the requirements of an abstract symmetry called superymmetry. All models of string theory that are realistic candidates for a theory of quantum gravity, are superstring theories.


Abstract symmetry that some of the models of particle physics satisfy: in such models, for every species of particles, there is a partner species with the same mass. If the particles are matter particles (fermions), then the partner particles are force particles (bosons), and vice versa.


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

surface gravity

The acceleration due to gravity which is experienced by an object resting on the surface of some solid body is called the body’s surface gravity (as most of the solid bodies in question are shaped by gravity, the value for the surface gravity tends to be the same everywhere on the body’s surface). For the Earth, the surface gravity is 9.81 metres per square second, the so-called standard acceleration.


A situation has a symmetry if certain changes make no difference. For instance: a mirror-symmetric image that you view in a mirror looks the same. A perfect sphere looks the same, even if it is rotated around an arbitrary axis through its centre point (“spherical symmetry”).

In particle physics, there are more abstract symmetries, less readily pictured, such as supersymmetry.


A particle accelerator, in which particles are accelerated with the help of electric fields, while strong magnetic fields keep them on track (the fact that ever-stronger magnetic fields are needed as acceleration proceeds is a consequence of the fact that relativistic mass increases with speed.)

synchrotron radiation

Electromagnetic radiation produced when electrically charged particles (for instance, electrons) are made to follow a curved trajectory in a particle accelerator, or when these particles undergo comparable accelerations in nature.

Due to effects that can be derived from special relativity, synchrotron radiation is densely concentrated and very intense. These properties, together with the fact that it is very easy to produce synchrotron radiation with a clearly defined frequency, make this type of radiation a valuable tool for research not only in basic physics, but also in biology and medicine.

When it was first discovered, synchrotron radiation was an (annoying!) side effect, observable at particle accelerators which were used for research into the basic properties of elementary particles. Nowadays, there are many accelerators whose main purposes is the production of this radiation!

For a list of important European Synchrotrons, have a look at > FIS-Landschaft, with BESSY II, PETRA III or the ESRF.

In the US, you can find the > National Synchrotron Light Source II at the Brookhaven National Laboratory

Syracuse University

Research university (enrolment ca. 20,000) in New York State. Research topics of the physics department include classical and quantum gravity, cosmology and gravitational waves.

Theoretical Cosmology and Elementary Particle Physics at Syracuse University
Gravitational Wave Group at Syracuse University

Système International d’Unités

See SI, International System of Units

Synonyms: International System of Units