Gravitational coupling constant
Encyclopedia
In physics
, the gravitational coupling constant, αG, is the coupling constant
characterizing the gravitation
al attraction between two charge
d elementary particles having nonzero mass
. αG is a fundamental physical constant and a dimensionless quantity
, so that its numerical value does not vary with the choice of units of measurement
.
s that are stable and well-understood. A pair of electron
s, of proton
s, or one electron and one proton all satisfy this criterion. Assuming two electrons, the defining expression and the best current estimate of its value are:
where:
While me and ħ are known to one part in 20,000,000, mP is only known to one part in 20,000 (mainly because G is known to only one part in 10,000). Hence αG is known to only four significant digits. By contrast, the fine structure constant α can be measured directly via the Quantum Hall Effect with a precision exceeding one part per billion. Also, the meter and second
are now defined in a way such that c has an exact value by definition. Hence the precision of αG depends only on that of G, ħ, and me.
, the ratio of the rest mass of the proton
to that of the electron
. Other definitions of αG that have been proposed in the literature differ from the one above merely by a factor of μ or its square:
what the fine-structure constant
is to electromagnetism
and quantum electrodynamics
. The physics literature seldom mentions αG. This may be due to the arbitrariness of the choice among particles to use (whereas α is a function of the elementary charge
e, about which there is no debate), and the relatively low precision with which αG can be measured. Unless stated otherwise, αG is here defined in terms of a pair of electron
s.
αG has a surprisingly simple physical interpretation: it is the square of the electron mass, measured in units of Planck mass. Likewise, the fine structure constant is the square of the electron charge measured in units of Planck charge
. By virtue of this fact, αG plays a role in the Higgs mechanism
determining the masses of the elementary particle
s.
The proton
and the electron
are stable and possess a quantity of charge of the elementary charge
e. Hence the ratio α/αG measures the relative strengths of the electrostatic vs. gravitation
al attraction/repulsion between these elementary particles. Using Planck units
(so that G=c=ħ=4πε0=1), then αG = me2 and α = e2, so that α/αG = (e/me)2. Thus the ratio of the electron mass to the electron charge (both are measured in terms of Planck units
) determines the relative strengths of gravitational and electromagnetic interaction between two electrons.
α is 43 (37) orders of magnitude greater than αG calculated from two electrons (protons). Either way, the electrostatic force between two charged elementary particle
s is vastly stronger than the corresponding gravitation
al attraction between them. This is so because a charged elementary particle
has approximately one Planck charge
, but a mass many orders of magnitude smaller than the Planck unit of mass. The gravitational attraction among elementary particles, charged or not, can hence be ignored. That gravitation is relevant for macroscopic objects indicates that they are electrostatically neutral to a very high degree.
Because αG = G me2/(ħc) = (tPωC)2, where tP is the Planck time
, αG is related to ωC, the Compton angular frequency of the electron
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the gravitational coupling constant, αG, is the coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
characterizing the gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
al attraction between two charge
Charge (physics)
In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
d elementary particles having nonzero mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
. αG is a fundamental physical constant and a dimensionless quantity
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...
, so that its numerical value does not vary with the choice of units of measurement
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...
.
Definition
αG can be defined in terms of any pair of charged elementary particleElementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s that are stable and well-understood. A pair of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s, of proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s, or one electron and one proton all satisfy this criterion. Assuming two electrons, the defining expression and the best current estimate of its value are:
where:
- G is the Newtonian constant of gravitationGravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
; - me is the mass of the electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
; - c is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in a vacuum; - ħ ("h-bar") is the reduced Planck constant;
- mP is the Planck mass.
Measurement and uncertainty
There is no known way of measuring αG directly, and CODATA does not report an estimate of its value. The above estimate is calculated from the CODATA values of me and mP.While me and ħ are known to one part in 20,000,000, mP is only known to one part in 20,000 (mainly because G is known to only one part in 10,000). Hence αG is known to only four significant digits. By contrast, the fine structure constant α can be measured directly via the Quantum Hall Effect with a precision exceeding one part per billion. Also, the meter and second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
are now defined in a way such that c has an exact value by definition. Hence the precision of αG depends only on that of G, ħ, and me.
Related definitions
Let μ = mp/me = 1836.15267247(80) be the dimensionless proton-to-electron mass ratioProton-to-electron mass ratio
In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton divided by that of the electron...
, the ratio of the rest mass of the proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
to that of the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
. Other definitions of αG that have been proposed in the literature differ from the one above merely by a factor of μ or its square:
- If αG is defined using the mass of one electron, me, and one proton (mp = μme), then αG = μ1.752×10-45 = 3.217×10-42, and α/αG ≈ 1039. α/αG defined in this manner is C in Eddington (1935: 232) (except that Planck's constant appears not as the reduced Planck constant);
- (4.5) in Barrow and Tipler (1986) tacitly defines α/αG as e2/(Gmpme) ≈ 1039. Even though they do not name the α/αG defined in this manner, it nevertheless plays a role in their broad-ranging discussion of astrophysicsAstrophysicsAstrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...
, cosmologyCosmologyCosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
, quantum physics, and the anthropic principleAnthropic principleIn astrophysics and cosmology, the anthropic principle is the philosophical argument that observations of the physical Universe must be compatible with the conscious life that observes it. Some proponents of the argument reason that it explains why the Universe has the age and the fundamental...
; - N in Rees (2000) is α/αG = α/(μ21.752×10−45) = α/(5.906×10−39) ≈ 1036, where the denominator is defined using a pair of protons.
Discussion
αG is to gravitationGravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
what the fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...
is to electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
and quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
. The physics literature seldom mentions αG. This may be due to the arbitrariness of the choice among particles to use (whereas α is a function of the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
e, about which there is no debate), and the relatively low precision with which αG can be measured. Unless stated otherwise, αG is here defined in terms of a pair of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s.
αG has a surprisingly simple physical interpretation: it is the square of the electron mass, measured in units of Planck mass. Likewise, the fine structure constant is the square of the electron charge measured in units of Planck charge
Planck charge
In physics, the Planck charge , is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.The Planck charge is defined as:...
. By virtue of this fact, αG plays a role in the Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
determining the masses of the elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s.
The proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
and the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
are stable and possess a quantity of charge of the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
e. Hence the ratio α/αG measures the relative strengths of the electrostatic vs. gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
al attraction/repulsion between these elementary particles. Using Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
(so that G=c=ħ=4πε0=1), then αG = me2 and α = e2, so that α/αG = (e/me)2. Thus the ratio of the electron mass to the electron charge (both are measured in terms of Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
) determines the relative strengths of gravitational and electromagnetic interaction between two electrons.
α is 43 (37) orders of magnitude greater than αG calculated from two electrons (protons). Either way, the electrostatic force between two charged elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s is vastly stronger than the corresponding gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
al attraction between them. This is so because a charged elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
has approximately one Planck charge
Planck charge
In physics, the Planck charge , is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.The Planck charge is defined as:...
, but a mass many orders of magnitude smaller than the Planck unit of mass. The gravitational attraction among elementary particles, charged or not, can hence be ignored. That gravitation is relevant for macroscopic objects indicates that they are electrostatically neutral to a very high degree.
Because αG = G me2/(ħc) = (tPωC)2, where tP is the Planck time
Planck time
In physics, the Planck time, , is the unit of time in the system of natural units known as Planck units. It is the time required for light to travel, in a vacuum, a distance of 1 Planck length...
, αG is related to ωC, the Compton angular frequency of the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
.
See also
- Coupling constantCoupling constantIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
- Dimensionless numbers
- Fine structure constant
- Gravitational constantGravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
External links
- Hyperphysics: Gravitational coupling constant.