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History

The Schwarzschild solution is named in honor of Karl Schwarzschild, who found the exact solution in 1915, only about a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild had little time to think about his solution. He died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I.

Johannes Droste in 1915 independently produced the same solution as Schwarzschild, using a simpler more direct derivation.

In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the Einstein field equations. In his 1916 paper Schwarzschild took the position that the singularity at r = r_s should be identified with the coordinate singularity at the origin present in spherical coordinates on flat space. A more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r = 0 and r = r_s. Although there was general consent that the singularity at r = 0 was 'genuine' physical singularity, the nature of the singularity at r = r_s remained unclear. In 1924 Arthur Eddington produced the first coordinate transformation (Eddington–Finkelstein coordinates) that showed that the singularity at r = r_s was a coordinate artifact, although he seems to have been unaware of the significance of this discovery. Later, in 1932, Georges Lemaître gave a different coordinate transformation (Lemaître coordinates) to the same effect and was the first to recognize that this implied that the singularity at r = r_s was not physical. In 1939 Howard Robertson showed that a free falling observer descending in the Scwharzschild metric would cross the r = r_s singularity in a finite amount of proper time even though this would take an infinite amount of time in terms of coordinate time t.

In 1950, John Synge produced a paper that showed the maximal analytic extension of the Schwarzschild metric, again showing that the singularity at r = r_s was a coordinate artifact. This result was later rediscovered by Martin Kruskal, who improved on Synge's result by providing a single set of coordinates that covered (almost) the entire spacetime. However due to the obscurity of the journals in which the papers of Lemaître and Synge were published there conclusions went unnoticed, with many of the major players in the field including Einstein believing that singularity at the Schwarzschild radius was physical.

Progress was only made in the 1960s when the more exact tools of differential geometry entered the field of general relativity allowing more exact definitions of what it means for a Lorentzian manifold to be singular. This lead to definitive identification of the r = r_s singularity in the Schwarzschild metric as an event horizon (a hypersurface in spacetime that can only be crossed in one direction).

Structure

The study of matter has –over the centuries– revealed increasingly fine structures of composite systems.

Chemical elements

Main article: chemical element

The periodic table of the chemical elements

The idea that any type of matter can be reduced be reduced to a small number of basic elements can be traced back to ancient philosophy, which posited that everything could be reduced in the basic four elements: earth, water, air and fire. An element was defined by Aristotle as one of those bodies into which other bodies can decompose, and that itself is not capable of being divided into other. In the 17th century, it was realized that the number of chemical elements was quite a bit bigger than thought by ancient philosophers.

Since then 118 distinct chemical elements, carbon, oxygen, and iron have been identified. Typically, these are organized in in the periodic table of chemical elements, which groups elements with similar chemical properties.

Atoms

Main article: Atom

The idea that matter would have a smallest possible unit, was first put forward by the ancient philosophers like Democritus, who coined the term atomos, which would become atom in modern English. However, it wasn't until the 19th century that the idea gained an experimental foundation.

The periodically recurring properties of the atoms of the chemical elements suggested that the atoms themselves had some internal structure. Ernest Rutherford concluded from an experiment scattering alpha particles of gold foil, that atoms were made of a cloud of negatively charged electrons, an a tiny positively charged nucleus. Niels Bohr further conjectured in his Bohr model that recurring chemical properties of the atoms could be explained by the electrons having discrete energy levels, which was confirmed by the further development of quantum mechanics.

Nucleons

Main article: nucleon

An atomic nucleus is a compact bundle of the two types of nucleons: Protons (red) and neutrons (blue).

With the discovery of the neutron in 1932, it became clear that the nucleus of an atom was made up of protons and neutrons (collectively known as nucleons). Particle accelerator experiments in the 1950s and 1960s showed that the proton and the neutron were just two members of a large class of particles known as hadrons. Over the years more than 100 different species of hadron have been identified, which again can be organized in groups with similar properties, which suggested that these particles where not fundamental.

The Standard Model

Main article: Standard Model

To explain the observed relations between the various hadrons Murray Gell-Mann and George Zweig independently introduced the quark model in 1964. Over next two decades this was extended into the currently best theory of matter, the Standard Model.

In this model the hadrons are composed over elementary particles called quarks, which are bound together through the strong interaction. Besides the quarks, the standard model contains a group of particles, known as leptons, which do not participate in the strong interaction. The electron it the most well-known member of this group. Together with the quarks, the leptons are sometimes referred to as the constituents of matter, as opposed to the remaining particles in the standard model, the gauge bosons, which are identified with forces.

Quarks

Main article: Quark

Quark structure of a proton: 2 up quarks and 1 down quark.

Quarks are a particles of spin-1⁄2, implying that they are fermions. They carry an electric charge of −1⁄3 e (down-type quarks) or +2⁄3 e (up-type quarks). For comparison, an electron has a charge of −1 e. They also carry colour charge, which is the equivalent of the electric charge for the strong interaction. Quarks also undergo radioactive decay, meaning that they are subject to the weak interaction. Quarks are massive particles, and therefore are also subject to gravity.

Quark properties

name	symbol	spin	electric charge (e)	mass (MeV/c2)	mass comparable to	antiparticle	antiparticle symbol
up-type quarks
up	u	1⁄2	+2⁄3	1.5 to 3.3	~ 5 electrons	antiup	u
charm	c	1⁄2	+2⁄3	1160 to 1340	~ 1 proton	anticharm	c
top	t	1⁄2	+2⁄3	169,100 to 173,300	~ 180 protons or ~ 1 tungsten atom	antitop	t
down-type quarks
down	d	1⁄2	−1⁄3	3.5 to 6.0	~ 10 electrons	antidown	d
strange	s	1⁄2	−1⁄3	70 to 130	~ 200 electrons	antistrange	s
bottom	b	1⁄2	−1⁄3	4130 to 4370	~ 5 protons	antibottom	b

Leptons

Main article: Lepton

Leptons are a particles of spin-1⁄2, meaning that they are fermions. They carry an electric charge of −1 e (charged leptons) or 0 e (neutrinos). Unlike quarks, leptons do not carry colour charge, meaning that they do not experience the strong interaction. Leptons also undergo radioactive decay, meaning that they are subject to the weak interaction. Leptons are massive particles, therefore are subject to gravity.

Lepton properties

name	symbol	spin	electric charge (e)	mass (MeV/c2)	mass comparable to	antiparticle	antiparticle symbol
charged leptons
electron	e−	1⁄2	−1	0.5110	1 electron	antielectron	e+
muon	μ−	1⁄2	−1	105.7	~ 200 electrons	antimuon	μ+
tau	τ−	1⁄2	−1	1,777	~ 2 protons	antitau	τ+
neutrinos
electron neutrino	ν e	1⁄2	0	< 0.000460	< 1⁄1000 electron	electron antineutrino	ν e
muon neutrino	ν μ	1⁄2	0	< 0.19	< 1⁄2 electron	muon antineutrino	ν μ
tau neutrino	ν τ	1⁄2	0	< 18.2	< 40 electrons	tau antineutrino	ν τ

Smaller building blocks?

The Standard Model groups matter particles into three generations, where each generation consists of two quarks and two leptons. The first generation is the up and down quarks, the electron and the electron neutrino; the second includes the charm and strange quarks, the muon and the muon neutrino; the third generation consists of the top and bottom quarks and the tau and tau neutrino. A possible explanation for this would be that quarks and leptons of higher generations are excited states of the first generations. If this turns out to be the case, it would imply that quarks and leptons are composite particles, rather than elementary particles. Searches for composite structure of quarks and leptons has, not produced any positive results.}}</ref>

Types of matter

Atoms and molecules definition

A definition of "matter" that is based upon its physical and chemical structure is: matter is made up of atoms and molecules. As an example, deoxyribonucleic acid molecules (DNA) are matter under this definition because they are made of atoms. This definition can be extended to include charged atoms and molecules, so as to include plasmas (gases of ions) and electrolytes (ionic solutions), which are not obviously included in the atoms and molecules definition. Alternatively, one can adopt the protons, neutrons and electrons definition.

Protons, neutrons and electrons definition

A definition of "matter" more fine-scale than the atoms and molecules definition is: matter is made up of what atoms and molecules are made of, meaning anything made of protons, neutrons, and electrons. This definition goes beyond atoms and molecules, however, to include substances made from these building blocks that are not simply atoms or molecules, for example white dwarf matter — typically, carbon and oxygen nuclei in a sea of degenerate electrons. At a microscopic level, the constituent "particles" of matter such as protons, neutrons and electrons obey the laws of quantum mechanics and exhibit wave–particle duality. At an even deeper level, protons and neutrons are made up of quarks and the force fields (gluons) that bind them together (see Quarks and leptons definition below).

Quarks and leptons definition

Under the "quarks and leptons" definition, the elementary and composite particles made of the quarks (in purple) and leptons (in green) would be "matter"; while the gauge bosons (in red) would not be "matter". However, interaction energy inherent to composite particles (for example, gluons involved in neutrons and protons) contribute to the mass of ordinary matter.

As may be seen from the above discussion, many early definitions of what can be called ordinary matter were based upon its structure or "building blocks". On the scale of elementary particles, a definition that follows this tradition can be stated as: ordinary matter is everything that is composed of elementary fermions, namely quarks and leptons. The connection between these formulations follows.

Leptons (the most famous being the electron), and quarks (of which baryons, such as protons and neutrons, are made) combine to form atoms, which in turn form molecules. Because atoms and molecules are said to be matter, it is natural to phrase the definition as: ordinary matter is anything that is made of the same things that atoms and molecules are made of. (However, notice that one also can make from these building blocks matter that is not atoms or molecules.) Then, because electrons are leptons, and protons and neutrons are made of quarks, this definition in turn leads to the definition of matter as being "quarks and leptons", which are the two types of elementary fermions. Carithers and Grannis state: Ordinary matter is composed entirely of first-generation particles, namely the [up] and [down] quarks, plus the electron and its neutrino. (Higher generations particles quickly decay into first-generation particles, and thus are not commonly encountered.)

This definition of ordinary matter is more subtle than it first appears. All the particles that make up ordinary matter (leptons and quarks) are elementary fermions, while all the force carriers are elementary bosons. The W and Z bosons that mediate the weak force are not made of quarks or leptons, and so are not ordinary matter, even if they have mass. In other words, mass is not something that is exclusive to ordinary matter.

The quark–lepton definition of ordinary matter, however, identifies not only the elementary building blocks of matter, but also includes composites made from the constituents (atoms and molecules, for example). Such composites contain an interaction energy that holds the constituents together, and may constitute the bulk of the mass of the composite. As an example, to a great extent, the mass of an atom is simply the sum of the masses of its constituent protons, neutrons and electrons. However, digging deeper, the protons and neutrons are made up of quarks bound together by gluon fields (see dynamics of quantum chromodynamics) and these gluons fields contribute significantly to the mass of hadrons. In other words, most of what composes the "mass" of ordinary matter is due to the binding energy of quarks within protons and neutrons. For example, the sum of the mass of the three quarks in a nucleon is approximately 12.5 MeV/c², which is low compared to the mass of a nucleon (approximately 938 MeV/c²). The bottom line is that most of the mass of everyday objects comes from the interaction energy of its elementary components.

Baryonic matter

Main article: Baryon

Baryons are strongly interacting fermions, and so are subject to Fermi-Dirac statistics. Amongst the baryons are the protons and neutrons, which occur in atomic nuclei, but many other unstable baryons exist as well. The term baryon is usually used to refer to triquarks — particles made of three quarks. "Exotic" baryons made of four quarks and one antiquark are known as the pentaquarks, but their existence is not generally accepted.

Baryonic matter is the part of the universe that is made of baryons (including all atoms). This part of the universe does not include dark energy, dark matter, black holes or various forms of degenerate matter, such as compose white dwarf stars and neutron stars. Microwave light seen by Wilkinson Microwave Anisotropy Probe (WMAP), suggests that only about 4.6% of that part of the universe within range of the best telescopes (that is, matter that may be visible because light could reach us from it), is made of baryionic matter. About 23% is dark matter, and about 72% is dark energy.

A comparison between the white dwarf IK Pegasi B (center), its A-class companion IK Pegasi A (left) and the Sun (right). This white dwarf has a surface temperature of 35,500 K.

Degenerate matter

Main article: Degenerate matter

In physics, degenerate matter refers to the ground state of a gas of fermions at a temperature near absolute zero. The Pauli exclusion principle requires that only two fermions can occupy a quantum state, one spin-up and the other spin-down. Hence, at zero temperature, the fermions fill up sufficient levels to accommodate all the available fermions, and for the case of many fermions the maximum kinetic energy called the Fermi energy and the pressure of the gas becomes very large and dependent upon the number of fermions rather than the temperature, unlike normal states of matter.

Degenerate matter is thought to occur during the evolution of heavy stars. The demonstration by Subrahmanyan Chandrasekhar that white dwarf stars have a maximum allowed mass because of the exclusion principle caused a revolution in the theory of star evolution.

Degenerate matter includes the part of the universe that is made up of neutron stars and white dwarfs.

Strange matter

Main article: Strange matter

Strange matter is a particular form of quark matter, usually thought of as a 'liquid' of up, down, and strange quarks. It is to be contrasted with nuclear matter, which is a liquid of neutrons and protons (which themselves are built out of up and down quarks), and with non-strange quark matter, which is a quark liquid containing only up and down quarks. At high enough density, strange matter is expected to be color superconducting. Strange matter is hypothesized to occur in the core of neutron stars, or, more speculatively, as isolated droplets that may vary in size from femtometers (strangelets) to kilometers (quark stars).

Two meanings of the term "strange matter"

In particle physics and astrophysics, the term is used in two ways, one broader and the other more specific.

1. The broader meaning is just quark matter that contains three flavors of quarks: up, down, and strange. In this definition, there is a critical pressure and an associated critical density, and when nuclear matter (made of protons and neutrons) is compressed beyond this density, the protons and neutrons dissociate into quarks, yielding quark matter (probably strange matter).
2. The narrower meaning is quark matter that is more stable than nuclear matter. The idea that this could happen is the "strange matter hypothesis" of Bodmer and Witten. In this definition, the critical pressure is zero: the true ground state of matter is always quark matter. The nuclei that we see in the matter around us, which are droplets of nuclear matter, are actually metastable, and given enough time (or the right external stimulus) would decay into droplets of strange matter, i.e. strangelets.

Notes

References