Quark

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https://en.wikipedia.org/wiki/Quark

 

Quark

A proton is composed of two up quarks, one down quark, and the gluons that mediate the forces "binding" them together. The color assignment of individual quarks is arbitrary, but all three colors must be present; red, blue and green are used as an analogy to the primary colors that together produce a white color.
Composition	elementary particle
Statistics	fermionic
Generation	1st, 2nd, 3rd
Interactions	strong, weak, electromagnetic, gravitation
Symbol	q
Antiparticle	antiquark (q)
Theorized	Murray Gell-Mann (1964) George Zweig (1964)
Discovered	SLAC (c. 1968)
Types	6 (up, down, strange, charm, bottom, and top)
Electric charge	+⁠2/3⁠ e, −⁠1/3⁠ e
Color charge	yes
Spin	⁠1/2⁠ ħ
Baryon number	⁠1/3⁠

A quark (/kwɔːrk, kwɑːrk/ ^ⓘ) is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly observable matter is composed of up quarks, down quarks and electrons. Owing to a phenomenon known as color confinement, quarks are never found in isolation; they can be found only within hadrons, which include baryons (such as protons and neutrons) and mesons, or in quark–gluon plasmas. For this reason, much of what is known about quarks has been drawn from observations of hadrons.

Quarks have various intrinsic properties, including electric charge, mass, color charge, and spin. They are the only elementary particles in the Standard Model of particle physics to experience all four fundamental interactions, also known as fundamental forces (electromagnetism, gravitation, strong interaction, and weak interaction), as well as the only known particles whose electric charges are not integer multiples of the elementary charge.

There are six types, known as flavors, of quarks: up, down, charm, strange, top, and bottom. Up and down quarks have the lowest masses of all quarks. The heavier quarks rapidly change into up and down quarks through a process of particle decay: the transformation from a higher mass state to a lower mass state. Because of this, up and down quarks are generally stable and the most common in the universe, whereas strange, charm, bottom, and top quarks can only be produced in high energy collisions (such as those involving cosmic rays and in particle accelerators). For every quark flavor there is a corresponding type of antiparticle, known as an antiquark, that differs from the quark only in that some of its properties (such as the electric charge) have equal magnitude but opposite sign.

The quark model was independently proposed by physicists Murray Gell-Mann and George Zweig in 1964. Quarks were introduced as parts of an ordering scheme for hadrons, and there was little evidence for their physical existence until deep inelastic scattering experiments at the Stanford Linear Accelerator Center in 1968. Accelerator program experiments have provided evidence for all six flavors. The top quark, first observed at Fermilab in 1995, was the last to be discovered.

Classification

See also: Standard Model

Six of the particles in the Standard Model are quarks (shown in purple). Each of the first three columns forms a generation of matter.

The Standard Model is the theoretical framework describing all the known elementary particles. This model contains six flavors of quarks (q), named up (u), down (d), strange (s), charm (c), bottom (b), and top (t). Antiparticles of quarks are called antiquarks, and are denoted by a bar over the symbol for the corresponding quark, such as u for an up antiquark. As with antimatter in general, antiquarks have the same mass, mean lifetime, and spin as their respective quarks, but the electric charge and other charges have the opposite sign.

Quarks are spin-⁠1/2⁠ particles, which means they are fermions according to the spin–statistics theorem. They are subject to the Pauli exclusion principle, which states that no two identical fermions can simultaneously occupy the same quantum state. This is in contrast to bosons (particles with integer spin), of which any number can be in the same state. Unlike leptons, quarks possess color charge, which causes them to engage in the strong interaction. The resulting attraction between different quarks causes the formation of composite particles known as hadrons (see § Strong interaction and color charge below).

The quarks that determine the quantum numbers of hadrons are called valence quarks; apart from these, any hadron may contain an indefinite number of virtual "sea" quarks, antiquarks, and gluons, which do not influence its quantum numbers. There are two families of hadrons: baryons, with three valence quarks, and mesons, with a valence quark and an antiquark. The most common baryons are the proton and the neutron, the building blocks of the atomic nucleus. A great number of hadrons are known (see list of baryons and list of mesons), most of them differentiated by their quark content and the properties these constituent quarks confer. The existence of "exotic" hadrons with more valence quarks, such as tetraquarks (qqqq) and pentaquarks (qqqqq), was conjectured from the beginnings of the quark model but not discovered until the early 21st century.

Elementary fermions are grouped into three generations, each comprising two leptons and two quarks. The first generation includes up and down quarks, the second strange and charm quarks, and the third bottom and top quarks. All searches for a fourth generation of quarks and other elementary fermions have failed, and there is strong indirect evidence that no more than three generations exist. Particles in higher generations generally have greater mass and less stability, causing them to decay into lower-generation particles by means of weak interactions. Only first-generation (up and down) quarks occur commonly in nature. Heavier quarks can only be created in high-energy collisions (such as in those involving cosmic rays), and decay quickly; however, they are thought to have been present during the first fractions of a second after the Big Bang, when the universe was in an extremely hot and dense phase (the quark epoch). Studies of heavier quarks are conducted in artificially created conditions, such as in particle accelerators.

Having electric charge, mass, color charge, and flavor, quarks are the only known elementary particles that engage in all four fundamental interactions of contemporary physics: electromagnetism, gravitation, strong interaction, and weak interaction. Gravitation is too weak to be relevant to individual particle interactions except at extremes of energy (Planck energy) and distance scales (Planck distance). However, since no successful quantum theory of gravity exists, gravitation is not described by the Standard Model.

See the table of properties below for a more complete overview of the six quark flavors' properties.

History

Murray Gell-Mann (2007)

George Zweig (2015)

The quark model was independently proposed by physicists Murray Gell-Mann^[24] and George Zweig in 1964. The proposal came shortly after Gell-Mann's 1961 formulation of a particle classification system known as the Eightfold Way – or, in more technical terms, SU(3) flavor symmetry, streamlining its structure. Physicist Yuval Ne'eman had independently developed a scheme similar to the Eightfold Way in the same year. An early attempt at constituent organization was available in the Sakata model.

At the time of the quark theory's inception, the "particle zoo" included a multitude of hadrons, among other particles. Gell-Mann and Zweig posited that they were not elementary particles, but were instead composed of combinations of quarks and antiquarks. Their model involved three flavors of quarks, up, down, and strange, to which they ascribed properties such as spin and electric charge. The initial reaction of the physics community to the proposal was mixed. There was particular contention about whether the quark was a physical entity or a mere abstraction used to explain concepts that were not fully understood at the time.

In less than a year, extensions to the Gell-Mann–Zweig model were proposed. Sheldon Glashow and James Bjorken predicted the existence of a fourth flavor of quark, which they called charm. The addition was proposed because it allowed for a better description of the weak interaction (the mechanism that allows quarks to decay), equalized the number of known quarks with the number of known leptons, and implied a mass formula that correctly reproduced the masses of the known mesons.

Deep inelastic scattering experiments conducted in 1968 at the Stanford Linear Accelerator Center (SLAC) and published on October 20, 1969, showed that the proton contained much smaller, point-like objects and was therefore not an elementary particle. Physicists were reluctant to firmly identify these objects with quarks at the time, instead calling them "partons" – a term coined by Richard Feynman. The objects that were observed at SLAC would later be identified as up and down quarks as the other flavors were discovered. Nevertheless, "parton" remains in use as a collective term for the constituents of hadrons (quarks, antiquarks, and gluons). Richard Taylor, Henry Kendall and Jerome Friedman received the 1990 Nobel Prize in physics for their work at SLAC.

Photograph of the event that led to the discovery of the Σ⁺⁺
_c baryon, at the Brookhaven National Laboratory in 1974

The strange quark's existence was indirectly validated by SLAC's scattering experiments: not only was it a necessary component of Gell-Mann and Zweig's three-quark model, but it provided an explanation for the kaon (K) and pion (π) hadrons discovered in cosmic rays in 1947.

In a 1970 paper, Glashow, John Iliopoulos and Luciano Maiani presented the GIM mechanism (named from their initials) to explain the experimental non-observation of flavor-changing neutral currents. This theoretical model required the existence of the as-yet undiscovered charm quark. The number of supposed quark flavors grew to the current six in 1973, when Makoto Kobayashi and Toshihide Maskawa noted that the experimental observation of CP violation could be explained if there were another pair of quarks.

Charm quarks were produced almost simultaneously by two teams in November 1974 (see November Revolution) – one at SLAC under Burton Richter, and one at Brookhaven National Laboratory under Samuel Ting. The charm quarks were observed bound with charm antiquarks in mesons. The two parties had assigned the discovered meson two different symbols, J and ψ; thus, it became formally known as the J/ψ meson. The discovery finally convinced the physics community of the quark model's validity.

In the following years a number of suggestions appeared for extending the quark model to six quarks. Of these, the 1975 paper by Haim Harari was the first to coin the terms top and bottom for the additional quarks.

In 1977, the bottom quark was observed by a team at Fermilab led by Leon Lederman. This was a strong indicator of the top quark's existence: without the top quark, the bottom quark would have been without a partner. It was not until 1995 that the top quark was finally observed, also by the CDF and DØ teams at Fermilab. It had a mass much larger than expected, almost as large as that of a gold atom.

Etymology

For some time, Gell-Mann was undecided on an actual spelling for the term he intended to coin, until he found the word quark in James Joyce's 1939 book Finnegans Wake:

– Three quarks for Muster Mark!
Sure he hasn't got much of a bark
And sure any he has it's all beside the mark.

The word quark is an outdated English word meaning to croak and the above-quoted lines are about a bird choir mocking king Mark of Cornwall in the legend of Tristan and Iseult. Especially in the German-speaking parts of the world there is a widespread legend, however, that Joyce had taken it from the word Quark, a German word of Slavic origin which denotes a curd cheese, but is also a colloquial term for "trivial nonsense". In the legend it is said that he had heard it on a journey to Germany at a farmers' market in Freiburg. Some authors, however, defend a possible German origin of Joyce's word quark. Gell-Mann went into further detail regarding the name of the quark in his 1994 book The Quark and the Jaguar:

In 1963, when I assigned the name "quark" to the fundamental constituents of the nucleon, I had the sound first, without the spelling, which could have been "kwork". Then, in one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word "quark" in the phrase "Three quarks for Muster Mark". Since "quark" (meaning, for one thing, the cry of the gull) was clearly intended to rhyme with "Mark", as well as "bark" and other such words, I had to find an excuse to pronounce it as "kwork". But the book represents the dream of a publican named Humphrey Chimpden Earwicker. Words in the text are typically drawn from several sources at once, like the "portmanteau" words in Through the Looking-Glass. From time to time, phrases occur in the book that are partially determined by calls for drinks at the bar. I argued, therefore, that perhaps one of the multiple sources of the cry "Three quarks for Muster Mark" might be "Three quarts for Mister Mark", in which case the pronunciation "kwork" would not be totally unjustified. In any case, the number three fitted perfectly the way quarks occur in nature.

Zweig preferred the name ace for the particle he had theorized, but Gell-Mann's terminology came to prominence once the quark model had been commonly accepted.

The quark flavors were given their names for several reasons. The up and down quarks are named after the up and down components of isospin, which they carry. Strange quarks were given their name because they were discovered to be components of the strange particles discovered in cosmic rays years before the quark model was proposed; these particles were deemed "strange" because they had unusually long lifetimes. Glashow, who co-proposed the charm quark with Bjorken, is quoted as saying, "We called our construct the 'charmed quark', for we were fascinated and pleased by the symmetry it brought to the subnuclear world." The names "top" and "bottom", coined by Harari, were chosen because they are "logical partners for up and down quarks". Alternative names for top and bottom quarks are "truth" and "beauty" respectively, but these names have somewhat fallen out of use. While "truth" never did catch on, accelerator complexes devoted to massive production of bottom quarks are sometimes called "beauty factories".

Properties

Electric charge

See also: Electric charge

Quarks have fractional electric charge values – either −⁠1/3⁠ or +⁠2/3⁠ times the elementary charge (e), depending on flavor. Up, charm, and top quarks (collectively referred to as up-type quarks) have a charge of +⁠2/3⁠ e; down, strange, and bottom quarks (down-type quarks) have a charge of −⁠1/3⁠ e. Antiquarks have the opposite charge to their corresponding quarks; up-type antiquarks have charges of −⁠2/3⁠ e and down-type antiquarks have charges of +⁠1/3⁠ e. Since the electric charge of a hadron is the sum of the charges of the constituent quarks, all hadrons have integer charges: the combination of three quarks (baryons), three antiquarks (antibaryons), or a quark and an antiquark (mesons) always results in integer charges. For example, the hadron constituents of atomic nuclei, neutrons and protons, have charges of 0 e and +1 e respectively; the neutron is composed of two down quarks and one up quark, and the proton of two up quarks and one down quark.

Spin

See also: Spin (physics)

Spin is an intrinsic property of elementary particles, and its direction is an important degree of freedom. It is sometimes visualized as the rotation of an object around its own axis (hence the name "spin"), though this notion is somewhat misguided at subatomic scales because elementary particles are believed to be point-like.

Spin can be represented by a vector whose length is measured in units of the reduced Planck constant ħ (pronounced "h bar"). For quarks, a measurement of the spin vector component along any axis can only yield the values +⁠ħ/2⁠ or −⁠ħ/2⁠; for this reason quarks are classified as spin-⁠1/2⁠ particles. The component of spin along a given axis – by convention the z axis – is often denoted by an up arrow ↑ for the value +⁠1/2⁠ and down arrow ↓ for the value −⁠1/2⁠, placed after the symbol for flavor. For example, an up quark with a spin of +⁠1/2⁠ along the z axis is denoted by u↑.

Weak interaction

Main article: Weak interaction

Feynman diagram of beta decay with time flowing upwards. The CKM matrix (discussed below) encodes the probability of this and other quark decays.

A quark of one flavor can transform into a quark of another flavor only through the weak interaction, one of the four fundamental interactions in particle physics. By absorbing or emitting a W boson, any up-type quark (up, charm, and top quarks) can change into any down-type quark (down, strange, and bottom quarks) and vice versa. This flavor transformation mechanism causes the radioactive process of beta decay, in which a neutron (n) "splits" into a proton (p), an electron (e⁻
) and an electron antineutrino (ν
_e) (see picture). This occurs when one of the down quarks in the neutron (udd) decays into an up quark by emitting a virtual W⁻
boson, transforming the neutron into a proton (uud). The W⁻
boson then decays into an electron and an electron antineutrino.

n	→	p	+	e−	+	ν e	(Beta decay, hadron notation)
udd	→	uud	+	e−	+	ν e	(Beta decay, quark notation)

Both beta decay and the inverse process of inverse beta decay are routinely used in medical applications such as positron emission tomography (PET) and in experiments involving neutrino detection.

The strengths of the weak interactions between the six quarks. The "intensities" of the lines are determined by the elements of the CKM matrix.

While the process of flavor transformation is the same for all quarks, each quark has a preference to transform into the quark of its own generation. The relative tendencies of all flavor transformations are described by a mathematical table, called the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix). Enforcing unitarity, the approximate magnitudes of the entries of the CKM matrix are:

${\displaystyle \left[\begin{array}{ccc}|V_{\mathrm{u}\mathrm{d}}|& |V_{\mathrm{u}\mathrm{s}}|& |V_{\mathrm{u}\mathrm{b}}|\\ |V_{\mathrm{c}\mathrm{d}}|& |V_{\mathrm{c}\mathrm{s}}|& |V_{\mathrm{c}\mathrm{b}}|\\ |V_{\mathrm{t}\mathrm{d}}|& |V_{\mathrm{t}\mathrm{s}}|& |V_{\mathrm{t}\mathrm{b}}|\end{array}\right]\approx \left[\begin{array}{ccc}0.974& 0.225& 0.003\\ 0.225& 0.973& 0.041\\ 0.009& 0.040& 0.999\end{array}\right],}$

where V_ij represents the tendency of a quark of flavor i to change into a quark of flavor j (or vice versa).

There exists an equivalent weak interaction matrix for leptons (right side of the W boson on the above beta decay diagram), called the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix). Together, the CKM and PMNS matrices describe all flavor transformations, but the links between the two are not yet clear.

Strong interaction and color charge

See also: Color charge and Strong interaction

All types of hadrons have zero total color charge.

The pattern of strong charges for the three colors of quark, three antiquarks, and eight gluons (with two of zero charge overlapping).

According to quantum chromodynamics (QCD), quarks possess a property called color charge. There are three types of color charge, arbitrarily labeled blue, green, and red. Each of them is complemented by an anticolor – antiblue, antigreen, and antired. Every quark carries a color, while every antiquark carries an anticolor.

The system of attraction and repulsion between quarks charged with different combinations of the three colors is called strong interaction, which is mediated by force carrying particles known as gluons; this is discussed at length below. The theory that describes strong interactions is called quantum chromodynamics (QCD). A quark, which will have a single color value, can form a bound system with an antiquark carrying the corresponding anticolor. The result of two attracting quarks will be color neutrality: a quark with color charge ξ plus an antiquark with color charge −ξ will result in a color charge of 0 (or "white" color) and the formation of a meson. This is analogous to the additive color model in basic optics. Similarly, the combination of three quarks, each with different color charges, or three antiquarks, each with different anticolor charges, will result in the same "white" color charge and the formation of a baryon or antibaryon.

In modern particle physics, gauge symmetries – a kind of symmetry group – relate interactions between particles (see gauge theories). Color SU(3) (commonly abbreviated to SU(3)_c) is the gauge symmetry that relates the color charge in quarks and is the defining symmetry for quantum chromodynamics. Just as the laws of physics are independent of which directions in space are designated x, y, and z, and remain unchanged if the coordinate axes are rotated to a new orientation, the physics of quantum chromodynamics is independent of which directions in three-dimensional color space are identified as blue, red, and green. SU(3)_c color transformations correspond to "rotations" in color space (which, mathematically speaking, is a complex space). Every quark flavor f, each with subtypes f_B, f_G, f_R corresponding to the quark colors, forms a triplet: a three-component quantum field that transforms under the fundamental representation of SU(3)_c. The requirement that SU(3)_c should be local – that is, that its transformations be allowed to vary with space and time – determines the properties of the strong interaction. In particular, it implies the existence of eight gluon types to act as its force carriers.

Mass

Current quark masses for all six flavors in comparison, as balls of proportional volumes. Proton (gray) and electron (red) are shown in bottom left corner for scale.

See also: Invariant mass

Two terms are used in referring to a quark's mass: current quark mass refers to the mass of a quark by itself, while constituent quark mass refers to the current quark mass plus the mass of the gluon particle field surrounding the quark. These masses typically have very different values. Most of a hadron's mass comes from the gluons that bind the constituent quarks together, rather than from the quarks themselves. While gluons are inherently massless, they possess energy – more specifically, quantum chromodynamics binding energy (QCBE) – and it is this that contributes so greatly to the overall mass of the hadron (see mass in special relativity). For example, a proton has a mass of approximately 938 MeV/c², of which the rest mass of its three valence quarks only contributes about 9 MeV/c²; much of the remainder can be attributed to the field energy of the gluons (see chiral symmetry breaking). The Standard Model posits that elementary particles derive their masses from the Higgs mechanism, which is associated to the Higgs boson. It is hoped that further research into the reasons for the top quark's large mass of ~173 GeV/c², almost the mass of a gold atom, might reveal more about the origin of the mass of quarks and other elementary particles.

Size

In QCD, quarks are considered to be point-like entities, with zero size. As of 2014, experimental evidence indicates they are no bigger than 10⁻⁴ times the size of a proton, i.e. less than 10⁻¹⁹ metres.

Table of properties

See also: Flavour (particle physics)

The following table summarizes the key properties of the six quarks. Flavor quantum numbers (isospin (I₃), charm (C), strangeness (S, not to be confused with spin), topness (T), and bottomness (B′)) are assigned to certain quark flavors, and denote qualities of quark-based systems and hadrons. The baryon number (B) is +⁠1/3⁠ for all quarks, as baryons are made of three quarks. For antiquarks, the electric charge (Q) and all flavor quantum numbers (B, I₃, C, S, T, and B′) are of opposite sign. Mass and total angular momentum (J; equal to spin for point particles) do not change sign for the antiquarks.

Quark flavor properties

Particle		Mass* [MeV/c2]	J [ħ]	B	Q [e]	I3	C	S	T	B′	Antiparticle
Name	Symbol										Name	Symbol
First generation
up	u	2.3±0.7 ± 0.5	⁠1/2⁠	+⁠1/3⁠	+⁠2/3⁠	+⁠1/2⁠	0	0	0	0	antiup	u
down	d	4.8±0.5 ± 0.3	⁠1/2⁠	+⁠1/3⁠	−⁠1/3⁠	−⁠1/2⁠	0	0	0	0	antidown	d
Second generation
charm	c	1275±25	⁠1/2⁠	+⁠1/3⁠	+⁠2/3⁠	0	+1	0	0	0	anticharm	c
strange	s	95±5	⁠1/2⁠	+⁠1/3⁠	−⁠1/3⁠	0	0	−1	0	0	antistrange	s
Third generation
top	t	173210±510 ± 710 *	⁠1/2⁠	+⁠1/3⁠	+⁠2/3⁠	0	0	0	+1	0	antitop	t
bottom	b	4180±30	⁠1/2⁠	+⁠1/3⁠	−⁠1/3⁠	0	0	0	0	−1	antibottom	b

J: total angular momentum, B: baryon number, Q: electric charge, I₃: isospin, C: charm, S: strangeness, T: topness, B′: bottomness.
* Notation such as 173210±510 ± 710, in the case of the top quark, denotes two types of measurement uncertainty: The first uncertainty is statistical in nature, and the second is systematic.

Interacting quarks

See also: Color confinement and Gluon

As described by quantum chromodynamics, the strong interaction between quarks is mediated by gluons, massless vector gauge bosons. Each gluon carries one color charge and one anticolor charge. In the standard framework of particle interactions (part of a more general formulation known as perturbation theory), gluons are constantly exchanged between quarks through a virtual emission and absorption process. When a gluon is transferred between quarks, a color change occurs in both; for example, if a red quark emits a red–antigreen gluon, it becomes green, and if a green quark absorbs a red–antigreen gluon, it becomes red. Therefore, while each quark's color constantly changes, their strong interaction is preserved.

Since gluons carry color charge, they themselves are able to emit and absorb other gluons. This causes asymptotic freedom: as quarks come closer to each other, the chromodynamic binding force between them weakens. Conversely, as the distance between quarks increases, the binding force strengthens. The color field becomes stressed, much as an elastic band is stressed when stretched, and more gluons of appropriate color are spontaneously created to strengthen the field. Above a certain energy threshold, pairs of quarks and antiquarks are created. These pairs bind with the quarks being separated, causing new hadrons to form. This phenomenon is known as color confinement: quarks never appear in isolation. This process of hadronization occurs before quarks formed in a high energy collision are able to interact in any other way. The only exception is the top quark, which may decay before it hadronizes.

Sea quarks

Hadrons contain, along with the valence quarks (q
_v) that contribute to their quantum numbers, virtual quark–antiquark (qq) pairs known as sea quarks (q
_s). Sea quarks form when a gluon of the hadron's color field splits; this process also works in reverse in that the annihilation of two sea quarks produces a gluon. The result is a constant flux of gluon splits and creations colloquially known as "the sea". Sea quarks are much less stable than their valence counterparts, and they typically annihilate each other within the interior of the hadron. Despite this, sea quarks can hadronize into baryonic or mesonic particles under certain circumstances.

Other phases of quark matter

Main article: QCD matter

A qualitative rendering of the phase diagram of quark matter. The precise details of the diagram are the subject of ongoing research.

Under sufficiently extreme conditions, quarks may become "deconfined" out of bound states and propagate as thermalized "free" excitations in the larger medium. In the course of asymptotic freedom, the strong interaction becomes weaker at increasing temperatures. Eventually, color confinement would be effectively lost in an extremely hot plasma of freely moving quarks and gluons. This theoretical phase of matter is called quark–gluon plasma.

The exact conditions needed to give rise to this state are unknown and have been the subject of a great deal of speculation and experimentation. An estimate puts the needed temperature at (1.90±0.02)×10¹² kelvin. While a state of entirely free quarks and gluons has never been achieved (despite numerous attempts by CERN in the 1980s and 1990s), recent experiments at the Relativistic Heavy Ion Collider have yielded evidence for liquid-like quark matter exhibiting "nearly perfect" fluid motion.

The quark–gluon plasma would be characterized by a great increase in the number of heavier quark pairs in relation to the number of up and down quark pairs. It is believed that in the period prior to 10⁻⁶ seconds after the Big Bang (the quark epoch), the universe was filled with quark–gluon plasma, as the temperature was too high for hadrons to be stable.

Given sufficiently high baryon densities and relatively low temperatures – possibly comparable to those found in neutron stars – quark matter is expected to degenerate into a Fermi liquid of weakly interacting quarks. This liquid would be characterized by a condensation of colored quark Cooper pairs, thereby breaking the local SU(3)_c symmetry. Because quark Cooper pairs harbor color charge, such a phase of quark matter would be color superconductive; that is, color charge would be able to pass through it with no resistance.

Inertial frame of reference

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Inertial_frame_of_reference

In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative to the frame until acted upon by external forces. In such a frame, the laws of nature can be observed without the need to correct for acceleration.

All frames of reference with zero acceleration are in a state of constant rectilinear motion (straight-line motion) with respect to one another. In such a frame, an object with zero net force acting on it, is perceived to move with a constant velocity, or, equivalently, Newton's first law of motion holds. Such frames are known as inertial. Some physicists, like Isaac Newton, originally thought that one of these frames was absolute — the one approximated by the fixed stars. However, this is not required for the definition, and it is now known that those stars are in fact moving, relative to one another.

According to the principle of special relativity, all physical laws look the same in all inertial reference frames, and no inertial frame is privileged over another. Measurements of objects in one inertial frame can be converted to measurements in another by a simple transformation — the Galilean transformation in Newtonian physics or the Lorentz transformation (combined with a translation) in special relativity; these approximately match when the relative speed of the frames is low, but differ as it approaches the speed of light.

By contrast, a non-inertial reference frame is accelerating. In such a frame, the interactions between physical objects vary depending on the acceleration of that frame with respect to an inertial frame. Viewed from the perspective of classical mechanics and special relativity, the usual physical forces caused by the interaction of objects have to be supplemented by fictitious forces caused by inertia.Viewed from the perspective of general relativity theory, the fictitious (i.e. inertial) forces are attributed to geodesic motion in spacetime.

Due to Earth's rotation, its surface is not an inertial frame of reference. The Coriolis effect can deflect certain forms of motion as seen from Earth, and the centrifugal force will reduce the effective gravity at the equator. Nevertheless, for many applications the Earth is an adequate approximation of an inertial reference frame.

Introduction

The motion of a body can only be described relative to something else—other bodies, observers, or a set of spacetime coordinates. These are called frames of reference. According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

This simplicity manifests itself in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames has external causes. The principle of simplicity can be used within Newtonian physics as well as in special relativity:

The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:
   The laws of mechanics have the same form in all inertial frames.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

However, this definition of inertial frames is understood to apply in the Newtonian realm and ignores relativistic effects.

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving with a constant absolute velocity cannot determine this velocity by any experiment. Otherwise, the differences would set up an absolute standard reference frame. According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup. In Newtonian mechanics, inertial frames of reference are related by the Galilean group of symmetries.

Newton's inertial frame of reference

Absolute space

Main article: Absolute space and time

Newton posited an absolute space considered well-approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some "relativists", even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

The expression inertial frame of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" with a more operational definition:

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojevich:

• The existence of absolute space contradicts the internal logic of classical mechanics since, according to the Galilean principle of relativity, none of the inertial frames can be singled out.
• Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
• Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

The utility of operational definitions was carried much further in the special theory of relativity. Some historical background including Lange's definition is provided by DiSalle, who says in summary:

The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. The laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.

— Robert DiSalle Space and Time: Inertial Frames

Newtonian mechanics

Classical theories that use the Galilean transformation postulate the equivalence of all inertial reference frames. The Galilean transformation transforms coordinates from one inertial reference frame, ${\displaystyle \mathbf{s}}$, to another, ${\displaystyle {\mathbf{s}}^{\mathrm{\prime }}}$, by simple addition or subtraction of coordinates:

${\displaystyle {\mathbf{r}}^{\mathrm{\prime }}=\mathbf{r}-{\mathbf{r}}_0-\mathbf{v}t}$

${\displaystyle t^{\mathrm{\prime }}=t-t_0}$

where r₀ and t₀ represent shifts in the origin of space and time, and v is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time t₂ − t₁ between two events is the same for all reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, |r₂ − r₁|) is also the same.

Figure 1: Two frames of reference moving with relative velocity ${\displaystyle \stackrel{\overrightarrow{v}}{}}$. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid. However, the principle of special relativity generalizes the notion of an inertial frame to include all physical laws, not simply Newton's first law.

Newton viewed the first law as valid in any reference frame that is in uniform motion (neither rotating nor accelerating) relative to absolute space; as a practical matter, "absolute space" was considered to be the fixed stars In the theory of relativity the notion of absolute space or a privileged frame is abandoned, and an inertial frame in the field of classical mechanics is defined as:

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton's first law of motion), in the absence of a net force, a body at rest will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant speed. Newtonian inertial frames transform among each other according to the Galilean group of symmetries.

If this rule is interpreted as saying that straight-line motion is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then being able to determine when zero net force is applied is crucial. The problem was summarized by Einstein:

The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.

— Albert Einstein: The Meaning of Relativity, p. 58

There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so it is only needed that a body is far enough away from all sources to ensure that no force is present. A possible issue with this approach is the historically long-lived view that the distant universe might affect matters (Mach's principle). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is the possibility of missing something, or accounting inappropriately for their influence, perhaps, again, due to Mach's principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when shifting reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames and have complicated rules of transformation in general cases. Based on the universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces.

Newton enunciated a principle of relativity himself in one of his corollaries to the laws of motion:

The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.

— Isaac Newton: Principia, Corollary V, p. 88 in Andrew Motte translation

This principle differs from the special principle in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares the special principle of the invariance of the form of the description among mutually translating reference frames. The role of fictitious forces in classifying reference frames is pursued further below.

Special relativity

Main article: Special relativity

Einstein's theory of special relativity, like Newtonian mechanics, postulates the equivalence of all inertial reference frames. However, because special relativity postulates that the speed of light in free space is invariant, the transformation between inertial frames is the Lorentz transformation, not the Galilean transformation which is used in Newtonian mechanics.

The invariance of the speed of light leads to counter-intuitive phenomena, such as time dilation, length contraction, and the relativity of simultaneity. The predictions of special relativity have been extensively verified experimentally. The Lorentz transformation reduces to the Galilean transformation as the speed of light approaches infinity or as the relative velocity between frames approaches zero.

Examples

Simple example

Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame S attached to the road and moving inertial frame S′ attached to the first car.

Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 meters. The car in front is traveling at 22 meters per second and the car behind is traveling at 30 meters per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.

First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where ${\displaystyle x_1(t)}$ is the position in meters of car one after time t in seconds and ${\displaystyle x_2(t)}$ is the position of car two after time t.

${\displaystyle x_1(t)=d+v_{1}t=200+22t,x_2(t)=v_{2}t=30t.}$

Notice that these formulas predict at t = 0 s the first car is 200m down the road and the second car is right beside us, as expected. We want to find the time at which ${\displaystyle x_1=x_2}$. Therefore, we set ${\displaystyle x_1=x_2}$ and solve for ${\displaystyle t}$, that is:

${\displaystyle 200+22t=30t,}$

${\displaystyle 8t=200,}$

${\displaystyle t=25\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}.}$

Alternatively, we could choose a frame of reference S′ situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v₂ − v₁ = 8 m/s. To catch up to the first car, it will take a time of ⁠d/v₂ − v₁⁠ = ⁠200/8⁠ s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m/s.

It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. One can convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you can deduct five minutes from the time displayed on your watch to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

Additional example

Figure 2: Simple-minded frame-of-reference example

For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving to the right. However, for the person facing west, the car was moving to the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.

For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis, and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame because he is not accelerating, ignoring effects such as Earth's rotation and gravity.

Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a′ = a − A in the negative y-direction—a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a′ = a + A in the negative y-direction—a larger value than Alfred's measurement.

Non-inertial frames

Main articles: Fictitious force, Non-inertial frame, and Rotating frame of reference

Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.

General relativity

Main articles: General relativity and Introduction to general relativity

See also: Equivalence principle and Eötvös experiment

General relativity is based upon the principle of equivalence:

There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.

— Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, p. 155.

This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911. Support for this principle is found in the Eötvös experiment, which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 10¹¹. For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.

Einstein's general theory modifies the distinction between nominally "inertial" and "non-inertial" effects by replacing special relativity's "flat" Minkowski Space with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of geodesic motion, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play. Consequently, modern special relativity is now sometimes described as only a "local theory". "Local" can encompass, for example, the entire Milky Way galaxy: The astronomer Karl Schwarzschild observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the Solar System. Schwarzschild pointed out that that was invariably seen: the direction of the angular momentum of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.

Inertial frames and rotation

In an inertial frame, Newton's first law, the law of inertia, is satisfied: Any free motion has a constant magnitude and direction. Newton's second law for a particle takes the form:

${\displaystyle \mathbf{F}=m\mathbf{a},}$

with F the net force (a vector), m the mass of a particle and a the acceleration of the particle (also a vector) which would be measured by an observer at rest in the frame. The force F is the vector sum of all "real" forces on the particle, such as contact forces, electromagnetic, gravitational, and nuclear forces.

In contrast, Newton's second law in a rotating frame of reference (a non-inertial frame of reference), rotating at angular rate Ω about an axis, takes the form:

${\displaystyle {\mathbf{F}}^{\prime }=m\mathbf{a},}$

which looks the same as in an inertial frame, but now the force F′ is the resultant of not only F, but also additional terms (the paragraph following this equation presents the main points without detailed mathematics):

${\displaystyle {\mathbf{F}}^{\prime }=\mathbf{F}-2m\mathbf{\Omega }\times {\mathbf{v}}_B-m\mathbf{\Omega }\times (\mathbf{\Omega }\times {\mathbf{x}}_B)-m\frac{d\mathbf{\Omega }}{dt}\times {\mathbf{x}}_B,}$

where the angular rotation of the frame is expressed by the vector Ω pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation Ω, symbol × denotes the vector cross product, vector x_B locates the body and vector v_B is the velocity of the body according to a rotating observer (different from the velocity seen by the inertial observer).

The extra terms in the force F′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the Coriolis force, the second the centrifugal force, and the third the Euler force. These terms all have these properties: they vanish when Ω = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of Ω; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, matter. Also, fictitious forces do not drop off with distance (unlike, for example, nuclear forces or electrical forces). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis.

All observers agree on the real forces, F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

In Newton's time the fixed stars were invoked as a reference frame, supposedly at rest relative to absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton's laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force. Two experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a rotating bucket. In both cases, application of Newton's second law would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket).

As now known, the fixed stars are not fixed. Those that reside in the Milky Way turn with the galaxy, exhibiting proper motions. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to expansion of the universe, and partly due to peculiar velocities. For instance, the Andromeda Galaxy is on collision course with the Milky Way at a speed of 117 km/s. The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based on the simplicity of the laws of physics in the frame. The laws of nature take a simpler form in inertial frames of reference because in these frames one did not have to introduce inertial forces when writing down Newton's law of motion.

In practice, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center.

To illustrate further, consider the question: "Does the Universe rotate?" An answer might explain the shape of the Milky Way galaxy using the laws of physics, although other observations might be more definitive; that is, provide larger discrepancies or less measurement uncertainty, like the anisotropy of the microwave background radiation or Big Bang nucleosynthesis. The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If its apparent rate of rotation is attributed entirely to rotation in an inertial frame, a different "flatness" is predicted than if it is supposed that part of this rotation is actually due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example, dark matter is invoked to explain the galactic rotation curve. So far, observations show any rotation of the universe is very slow, no faster than once every 6×10¹³ years (10⁻¹³ rad/yr), and debate persists over whether there is any rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity.

When quantum effects are important, there are additional conceptual complications that arise in quantum reference frames.

Primed frames

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x′, y′, a′.

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r′.

From the geometry of the situation

${\displaystyle \mathbf{r}=\mathbf{R}+{\mathbf{r}}^{\prime }.}$

Taking the first and second derivatives of this with respect to time

${\displaystyle \mathbf{v}=\mathbf{V}+{\mathbf{v}}^{\prime },}$

${\displaystyle \mathbf{a}=\mathbf{A}+{\mathbf{a}}^{\prime }.}$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, Newton's second law can be written as

${\displaystyle \mathbf{F}=m\mathbf{a}=m\mathbf{A}+m{\mathbf{a}}^{\prime }.}$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried).

This arrangement leads to the equation (see Fictitious force for a derivation):

${\displaystyle \mathbf{a}={\mathbf{a}}^{\prime }+\dot{\mathit{\omega }}\times {\mathbf{r}}^{\prime }+2\mathit{\omega }\times {\mathbf{v}}^{\prime }+\mathit{\omega }\times (\mathit{\omega }\times {\mathbf{r}}^{\prime })+{\mathbf{A}}_0,}$

or, to solve for the acceleration in the accelerated frame,

${\displaystyle {\mathbf{a}}^{\prime }=\mathbf{a}-\dot{\mathit{\omega }}\times {\mathbf{r}}^{\prime }-2\mathit{\omega }\times {\mathbf{v}}^{\prime }-\mathit{\omega }\times (\mathit{\omega }\times {\mathbf{r}}^{\prime })-{\mathbf{A}}_0.}$

Multiplying through by the mass m gives

${\displaystyle {\mathbf{F}}^{\prime }={\mathbf{F}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}+{\mathbf{F}}_{\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}}^{\prime }+{\mathbf{F}}_{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}^{\prime }+{\mathbf{F}}_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}}^{\prime }-m{\mathbf{A}}_0,}$

where

${\displaystyle {\mathbf{F}}_{\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}}^{\prime }=-m\dot{\mathit{\omega }}\times {\mathbf{r}}^{\prime }}$ (Euler force),

${\displaystyle {\mathbf{F}}_{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}^{\prime }=-2m\mathit{\omega }\times {\mathbf{v}}^{\prime }}$ (Coriolis force),

${\displaystyle {\mathbf{F}}_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{l}}^{\prime }=-m\mathit{\omega }\times (\mathit{\omega }\times {\mathbf{r}}^{\prime })=m({\omega }^2{\mathbf{r}}^{\prime }-(\mathit{\omega }\cdot {\mathbf{r}}^{\prime })\mathit{\omega })}$ (centrifugal force).

Separating non-inertial from inertial reference frames

Theory

Main article: Fictitious force

See also: Non-inertial frame, Rotating spheres, and Bucket argument

Figure 2: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.

Figure 3: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

Inertial and non-inertial reference frames can be distinguished by the absence or presence of fictitious forces.

The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations…

— Sidney Borowitz and Lawrence A Bornstein in A Contemporary View of Elementary Physics, p. 138

The presence of fictitious forces indicates the physical laws are not the simplest laws available, in terms of the special principle of relativity, a frame where fictitious forces are present is not an inertial frame:

The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.

— V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129

Bodies in non-inertial reference frames are subject to so-called fictitious forces (pseudo-forces); that is, forces that result from the acceleration of the reference frame itself and not from any physical force acting on the body. Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames.

To apply the Newtonian definition of an inertial frame, the understanding of separation between "fictitious" forces and "real" forces must be made clear.

For example, consider a stationary object in an inertial frame. Being at rest, no net force is applied. But in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force. How can it be decided that the rotating frame is a non-inertial frame? There are two approaches to this resolution: one approach is to look for the origin of the fictitious forces (the Coriolis force and the centrifugal force). It will be found there are no sources for these forces, no associated force carriers, no originating bodies. A second approach is to look at a variety of frames of reference. For any inertial frame, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these frames where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating frame is not an inertial frame.

Newton examined this problem himself using rotating spheres, as shown in Figure 2 and Figure 3. He pointed out that if the spheres are not rotating, the tension in the tying string is measured as zero in every frame of reference. If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating frame), the zero tension in the string is accounted for by observing that the centripetal force is supplied by the centrifugal and Coriolis forces in combination, so no tension is needed. If the spheres really are rotating, the tension observed is exactly the centripetal force required by the circular motion. Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value. That is, the inertial frame is the one where the fictitious forces vanish.

For linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common:

If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces.

— Isaac Newton: Principia Corollary VI, p. 89, in Andrew Motte translation

This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. With this in mind, inertial frames can collectively be defined as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set.

For these ideas to apply, everything observed in the frame has to be subject to a base-line, common acceleration shared by the frame itself. That situation would apply, for example, to the elevator example, where all objects are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate.

Applications

Inertial navigation systems used a cluster of gyroscopes and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it. Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of dead reckoning that requires no external input, and therefore cannot be jammed by any external or internal signal source.

A gyrocompass, employed for navigation of seagoing vessels, finds the geometric north. It does so, not by sensing the Earth's magnetic field, but by using inertial space as its reference. The outer casing of the gyrocompass device is held in such a way that it remains aligned with the local plumb line. When the gyroscope wheel inside the gyrocompass device is spun up, the way the gyroscope wheel is suspended causes the gyroscope wheel to gradually align its spinning axis with the Earth's axis. Alignment with the Earth's axis is the only direction for which the gyroscope's spinning axis can be stationary with respect to the Earth and not be required to change direction with respect to inertial space. After being spun up, a gyrocompass can reach the direction of alignment with the Earth's axis in as little as a quarter of an hour.