Scalar-tensor theory
Encyclopedia
In theoretical physics
, a scalar-tensor theory is a theory that includes both a scalar field
and a tensor
field to represent a certain interaction. For example, the Brans-Dicke theory
of gravitation
uses both a scalar field and a tensor field
to mediate the gravitational interaction.
are derived from Hamilton
's principle of least action
. In this approach, the behavior of a system is not described via force
s, but by functions which describe the energy of the system. Most important are the energetic quantities known as the Hamilton function (or Hamiltonian
) and the Lagrange function (or Lagrangian
). Their derivatives in space are known as Hamiltonian or Hamilton density and Lagrangian or Lagrange density. Going to these quantities leads to the field theories.
Modern physics uses field
theories to explain reality. These fields can be scalar, vectorial or tensorial. For them, there is:
Scalars are numbers, quantities of the form f(x), like the temperature. Vectors are more general and show a direction. In them, every component of the direction is a scalar.
Tensors (degree 2) are a wider generalization, the most well known example of which are matrices (that can give equation systems). Higher order tensors are found for example in the deformation theory and in General Relativity.
, gravitation was given in the same way, as consequence of a gravitational force (vectorial), given through a scalar potential field, dependent of the mass of the particles. Thus, Newtonian
gravity is called a scalar theory. The gravitational force is dependent of the distance r of the massive objetcts to each other (more exactly, their centre of mass). Mass is a parameter and space and time are unchangeable.
of Einstein and Hilbert, Theodor Kaluza
and Oskar Klein
proposed in 1917 a generalization in a 5-dimensional manifold: Kaluza-Klein theory. This theory possesses a 5-dimensional metric (with a compactified and constant 5th metric component, dependent on the gauge potential) and unifies gravitation
and electromagnetism
, i.e. there is a geometrization of electrodynamics.
This theory was modified in 1955 by P. Jordan
in his Projective Relativity theory, in which, following group-theoretical reasonings, Jordan took a functional 5th metric component that lead to a variable gravitational constant G. In his original work, he introduced coupling parameters of the scalar field, to change energy conservation as well, according to the ideas of Dirac
.
Following the Conform Equivalence theory, multidimensional theories of gravity are conform equivalent to theories of usual General Relativity in 4 dimensions with an additional scalar field. One case of this is given by Jordan's theory, which, without breaking energy conservation (as it should be valid, following from microwave background radiation being of a black body), is equivalent to the theory of C. Brans
and R, Dicke of 1961, so that it is usually spoken about the Jordan-Brans-Dicke theory. The Brans-Dicke theory
follows the idea of modifying Hilbert-Einstein theory to be compatible with Mach's Principle
. For this, Newton's gravitational constant had to be variable, dependent of the mass distribution in the universe, as a function of a scalar variable, coupled as a field in the Lagrangian. It uses a scalar field of infinite length scale (i.e. long-ranged), so, in the language of Yukawa's theory of nuclear physics, this scalar field is a massless field. This theory becomes Einsteinian for high values for the parameter of the scalar field.
In 1979, R. Wagoner proposed a generalization of scalar-tensor theories using more than one scalar field coupled to the scalar curvature.
JBD theories although not changing the geodesic equation for test particles, change the motion of composite bodies to a more complex one. The coupling of a universal scalar field directly to the gravitational field gives rise to potentially observable effects for the motion of matter configurations to which gravitational energy contributes significantly. This is known as the “Dicke-Nordtvedt” effect, which leads to possible violations of the Strong as well as the Weak Equivalence Principle for extended masses.
JBD-type theories with short-ranged scalar fields use, according to Yukawa's theory, massive scalar fields. The first of this theories was proposed by A. Zee 1979. He proposed a Broken-Symmetric Theory of Gravitation, combining the idea of Brans and Dicke with the one of Symmetry Breakdown, which is essential within the Standard Model
SM of elementary particles, where the so called Symmetry Breakdown leads to mass generation (as a consequence of particles interacting with the Higgs field). Zee proposed the Higgs field of SM as scalar field and so the Higgs field to generate the gravitational constant.
The interaction of the Higgs field with the particles that achieve mass through it is short-ranged (i.e. of Yukawa-type) and gravitational-like (one can get a Poisson equation from it), even within SM, so that Zee's idea was taken 1992 for a scalar-tensor theory with Higgs field as scalar field with Higgs mechanism. There, the massive scalar field couples to the masses, which are at the same time the source of the scalar Higgs field, which generates the mass of the elementary particles through Symmetry Breakdown). For vanishing scalar field, this theories usually go through to standard General Relativity and because of the nature of the massive field, it is possible for such theories that the parameter of the scalar field (the coupling constant) does not have to be as high as in standard JBD theories. Though, it is not clear yet which of these models explains better the phenomenology found in nature nor if such scalar fields are really given or necessary in nature. Nevertheless, JBD theories are used to explain inflation
(for massless scalar fields then it is spoken of the inflaton field) after the Big Bang
as well as the quintessence
. Further, they are an option to explain dynamics usually given through the standard Cold Dark Matter
models, as well as MOND
, Axion
s (from Breaking of a Symmetry, too), MACHOS
,...
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a scalar-tensor theory is a theory that includes both a scalar field
Scalar field theory
In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
and a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
field to represent a certain interaction. For example, the Brans-Dicke theory
Brans-Dicke theory
In theoretical physics, the Brans–Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a well-known competitor of Einstein's more popular theory of general relativity...
of 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...
uses both a scalar field and a tensor field
Tensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
to mediate the gravitational interaction.
Tensor fields and field theory
Modern physics tries to derive all physical theories from as few principles as possible. In this way, Newtonian mechanics as well as quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
are derived from Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
's principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
. In this approach, the behavior of a system is not described via force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s, but by functions which describe the energy of the system. Most important are the energetic quantities known as the Hamilton function (or Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
) and the Lagrange function (or Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
). Their derivatives in space are known as Hamiltonian or Hamilton density and Lagrangian or Lagrange density. Going to these quantities leads to the field theories.
Modern physics uses field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
theories to explain reality. These fields can be scalar, vectorial or tensorial. For them, there is:
- Scalars are tensors of rank zero.
- Vectors are tensors of rank one.
- Matrices are tensors of rank two.
Scalars are numbers, quantities of the form f(x), like the temperature. Vectors are more general and show a direction. In them, every component of the direction is a scalar.
Tensors (degree 2) are a wider generalization, the most well known example of which are matrices (that can give equation systems). Higher order tensors are found for example in the deformation theory and in General Relativity.
Gravity as field theory
In physics, forces (as vectorial quantities) are given as the derivative (gradient) of scalar quantities named potentials. In classical physics before EinsteinAlbert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, gravitation was given in the same way, as consequence of a gravitational force (vectorial), given through a scalar potential field, dependent of the mass of the particles. Thus, Newtonian
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
gravity is called a scalar theory. The gravitational force is dependent of the distance r of the massive objetcts to each other (more exactly, their centre of mass). Mass is a parameter and space and time are unchangeable.
- Einstein's theory of gravity, the General RelativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
is of another nature. It unifies space and time in a 4-dimensional manifold called space-time that depends upon mass itself. In General Relativity there is no gravitational force, but instead a curvature of space-time. The curvature is consequence of mass and in linear approximation it is identifiable with a force. This force is the derivative of the so called metricMetric (mathematics)In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
as potential. The metric of General Relativity possesses the characteristics of space-time and it is a tensorial quantity of degree 2 (it can be given as a 4x4 matrix, an object carrying 2 indices).
- Another possibility to explain gravitation in this context is by using both tensor (of degree n>1) and scalar fields, i.e. so that gravitation is not only given through a scalar field nor through the metric. These are scalar-tensor theories of gravitation.
- The field theoretical start of General Relativity is given through the Lagrange density. It is a scalar and gauge invariant (look at gauge theories) quantity dependent on the curvature scalar R. This Lagrangian, following Hamilton's principle, leads to the field equations of HilbertDavid HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and EinsteinAlbert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
. If in the Lagrangian the curvature (or a quantity related to it) is multiplied with a square scalar field, field theories of scalar-tensor theories of gravitation are obtained. In them, the 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...
of Newton is no longer a real constant but a quantity dependent of the scalar field.
Higher-dimensional relativity and scalar-tensor theories
After the postulation of the General RelativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
of Einstein and Hilbert, Theodor Kaluza
Theodor Kaluza
Theodor Franz Eduard Kaluza was a German mathematician and physicist known for the Kaluza-Klein theory involving field equations in five-dimensional space...
and Oskar Klein
Oskar Klein
Oskar Benjamin Klein was a Swedish theoretical physicist.Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Dr. Gottlieb Klein from Homonna in Hungary and Antonie Levy...
proposed in 1917 a generalization in a 5-dimensional manifold: Kaluza-Klein theory. This theory possesses a 5-dimensional metric (with a compactified and constant 5th metric component, dependent on the gauge potential) and unifies 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...
and electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, i.e. there is a geometrization of electrodynamics.
This theory was modified in 1955 by P. Jordan
Jordan
Jordan , officially the Hashemite Kingdom of Jordan , Al-Mamlaka al-Urduniyya al-Hashemiyya) is a kingdom on the East Bank of the River Jordan. The country borders Saudi Arabia to the east and south-east, Iraq to the north-east, Syria to the north and the West Bank and Israel to the west, sharing...
in his Projective Relativity theory, in which, following group-theoretical reasonings, Jordan took a functional 5th metric component that lead to a variable gravitational constant G. In his original work, he introduced coupling parameters of the scalar field, to change energy conservation as well, according to the ideas of Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
.
Following the Conform Equivalence theory, multidimensional theories of gravity are conform equivalent to theories of usual General Relativity in 4 dimensions with an additional scalar field. One case of this is given by Jordan's theory, which, without breaking energy conservation (as it should be valid, following from microwave background radiation being of a black body), is equivalent to the theory of C. Brans
Carl H. Brans
Carl Henry Brans is an American mathematical physicist best known for his research into the theoretical underpinnings of gravitation elucidated in his most widely publicized work, the Brans–Dicke theory....
and R, Dicke of 1961, so that it is usually spoken about the Jordan-Brans-Dicke theory. The Brans-Dicke theory
Brans-Dicke theory
In theoretical physics, the Brans–Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a well-known competitor of Einstein's more popular theory of general relativity...
follows the idea of modifying Hilbert-Einstein theory to be compatible with Mach's Principle
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
. For this, Newton's gravitational constant had to be variable, dependent of the mass distribution in the universe, as a function of a scalar variable, coupled as a field in the Lagrangian. It uses a scalar field of infinite length scale (i.e. long-ranged), so, in the language of Yukawa's theory of nuclear physics, this scalar field is a massless field. This theory becomes Einsteinian for high values for the parameter of the scalar field.
In 1979, R. Wagoner proposed a generalization of scalar-tensor theories using more than one scalar field coupled to the scalar curvature.
JBD theories although not changing the geodesic equation for test particles, change the motion of composite bodies to a more complex one. The coupling of a universal scalar field directly to the gravitational field gives rise to potentially observable effects for the motion of matter configurations to which gravitational energy contributes significantly. This is known as the “Dicke-Nordtvedt” effect, which leads to possible violations of the Strong as well as the Weak Equivalence Principle for extended masses.
JBD-type theories with short-ranged scalar fields use, according to Yukawa's theory, massive scalar fields. The first of this theories was proposed by A. Zee 1979. He proposed a Broken-Symmetric Theory of Gravitation, combining the idea of Brans and Dicke with the one of Symmetry Breakdown, which is essential within the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
SM of elementary particles, where the so called Symmetry Breakdown leads to mass generation (as a consequence of particles interacting with the Higgs field). Zee proposed the Higgs field of SM as scalar field and so the Higgs field to generate the gravitational constant.
The interaction of the Higgs field with the particles that achieve mass through it is short-ranged (i.e. of Yukawa-type) and gravitational-like (one can get a Poisson equation from it), even within SM, so that Zee's idea was taken 1992 for a scalar-tensor theory with Higgs field as scalar field with Higgs mechanism. There, the massive scalar field couples to the masses, which are at the same time the source of the scalar Higgs field, which generates the mass of the elementary particles through Symmetry Breakdown). For vanishing scalar field, this theories usually go through to standard General Relativity and because of the nature of the massive field, it is possible for such theories that the parameter of the scalar field (the coupling constant) does not have to be as high as in standard JBD theories. Though, it is not clear yet which of these models explains better the phenomenology found in nature nor if such scalar fields are really given or necessary in nature. Nevertheless, JBD theories are used to explain inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
(for massless scalar fields then it is spoken of the inflaton field) after the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
as well as the quintessence
Quintessence (physics)
In physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...
. Further, they are an option to explain dynamics usually given through the standard Cold Dark Matter
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
models, as well as MOND
Mond
Mond may refer to:* MOND - Modified Newtonian Dynamics. A proposed adjustment to the classical inverse-square law of gravity.* Der Mond, an opera in one act* Mond Nickel Company, a defunct mining company...
, Axion
Axion
The axion is a hypothetical elementary particle postulated by the Peccei-Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics...
s (from Breaking of a Symmetry, too), MACHOS
Machos
Machos is a community in western Elis in Greece. The population is around 200. Its dialing code is 26230...
,...