In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a scalar–tensor theory is a
field theory that includes both a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
and a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
field to represent a certain interaction. For example, the
Brans–Dicke theory of
gravitation
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
uses both a scalar field and a
tensor field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
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
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body r ...
as well as
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
are derived from
Hamilton
Hamilton may refer to:
* Alexander Hamilton (1755/1757–1804), first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States
* ''Hamilton'' (musical), a 2015 Broadway musical by Lin-Manuel Miranda
** ''Hamilton'' (al ...
's ''
principle of least action
Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical sy ...
''. In this approach, the behavior of a system is not described via
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s, but by functions which describe the energy of the system. Most important are the energetic quantities known as the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
function and the
Lagrangian function. Their derivatives in space are known as
Hamiltonian density and the
Lagrangian 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
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry ...
. An example of a scalar field is the temperature field. An example of a vector field is the wind velocity field. An example of a tensor field is the
stress tensor field in a stressed body, used in
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...
.
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
Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, 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
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
. The gravitational force is dependent of the distance ''r'' of the massive objects 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 Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
(GR) is of another nature. It unifies space and time in a 4-dimensional ''manifold'' called space-time. In GR there is no gravitational force, instead, the actions we ascribed to being a force are the consequence of the local curvature of space-time. That curvature is defined mathematically by the so-called
metric
Metric or metrical may refer to:
Measuring
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
...
, which is a function of the total energy, including mass, in the area. The derivative of the metric is a function that approximates the classical Newtonian force in most cases. The metric 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 given neither solely through a scalar field nor solely through a 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
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
) quantity dependent on the curvature scalar R. This Lagrangian, following Hamilton's principle, leads to the field equations of
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
and
Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
. 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 constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
of Newton is no longer a real constant but a quantity dependent of the scalar field.
Mathematical formulation
An action of such a gravitational scalar–tensor theory can be written as follows:
:
where
is the metric determinant,
is the Ricci scalar constructed from the metric
,
is a coupling constant with the dimensions
,
is the scalar-field potential,
is the material Lagrangian and
represents the non-gravitational fields. Here, the Brans–Dicke parameter
has been generalized to a function. Although
is often written as being
, one has to keep in mind that the fundamental constant
there, is not the
constant of gravitation that can be measured with, for instance,
Cavendish type experiments. Indeed, the
empirical gravitational constant is generally no longer a constant in scalar–tensor theories, but a function of the scalar field
. The metric and scalar-field equations respectively write:
:
and
:
Also, the theory satisfies the following conservation equation, implying that test-particles follow space-time
geodesics
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
such as in general relativity:
:
where
is the
stress-energy tensor defined as
:
The Newtonian approximation of the theory
Developing perturbatively the theory defined by the previous action around a Minkowskian background, and assuming non-relativistic gravitational sources, the first order gives the Newtonian approximation of the theory. In this approximation, and for a theory without potential, the metric writes
:
with
satisfying the following usual
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
at the lowest order of the approximation:
:
where
is the density of the gravitational source and
(the subscript
indicates that the corresponding value is taken at present cosmological time and location). Therefore, the
empirical gravitational constant is a function of the present value of the scalar-field background
and therefore theoretically depends on time and location. However, no deviation from the constancy of the Newtonian gravitational constant has been measured,
implying that the scalar-field background
is pretty stable over time. Such a stability is not theoretically generally expected but can be theoretically explained by several mechanisms.
The first post-Newtonian approximation of the theory
Developing the theory at the next level leads to the so-called first post-Newtonian order. For a theory without potential and in a system of coordinates respecting the weak isotropy condition (i.e.,
), the metric takes the following form:
:
:
:
with
:
:
where
is a function depending on the coordinate gauge
:
It corresponds to the remaining
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
degree of freedom that is not fixed by the weak isotropy condition. The sources are defined as
:
the so-called
post-Newtonian parameters are
:
:
and finally the
empirical gravitational constant is given by
:
where
is the (true) constant that appears in the coupling constant
defined previously.
Observational constraints on the theory
Current observations indicate that
,
which means that
. Although explaining such a value in the context of the original
Brans–Dicke theory is impossible,
Damour and
Nordtvedt found that the field equations of the general theory often lead to an evolution of the function
toward infinity during the evolution of the universe.
Hence, according to them, the current high value of the function
could be a simple consequence of the evolution of the universe.
Seven years of data from the NASA
MESSENGER
Messenger, Messengers, The Messenger or The Messengers may refer to:
People
* Courier, a person or company that delivers messages, packages, or mail
* Messenger (surname)
* Bicycle messenger, a bicyclist who transports packages through cities
* M ...
mission constraints the post-Newtonian parameter
for Mercury's perihelion shift to
.
Both constraints show that while the theory is still a potential candidate to replace general relativity, the scalar field must be very weakly coupled in order to explain current observations.
Generalized scalar-tensor theories have also been proposed as explanation for the
accelerated expansion of the universe but the measurement of the speed of gravity with the gravitational wave event
GW170817
GW170817 was a gravitational wave (GW) observed by the LIGO and Virgo detectors on 17 August 2017, originating within the shell elliptical galaxy NGC 4993, about 144 million light years away. The wave was produced by the last moments of the in ...
has ruled this out.
Higher-dimensional relativity and scalar–tensor theories
After the postulation of the
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
of Einstein and Hilbert,
Theodor Kaluza and
Oskar Klein
Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physics, theoretical physicist.
Oskar Klein is known for his work on Kaluza–Klein theory, which is partially named after him.
Biography
Klein was born ...
proposed in 1917 a generalization in a 5-dimensional manifold:
Kaluza–Klein theory
In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to ...
. This theory possesses a 5-dimensional metric (with a compactified and constant 5th metric component, dependent on the
gauge potential) and unifies
gravitation
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, 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 led 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
Robert H. Dicke of 1961, so that it is usually spoken about the ''Brans–Dicke theory''. The
Brans–Dicke theory follows the idea of modifying Hilbert-Einstein theory to be compatible with
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The ...
. 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 in 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
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
SM of
elementary particles
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a con ...
, 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
In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...
(for massless scalar fields then it is spoken of the inflaton field) after the
Big Bang
The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
as well as the
quintessence. Further, they are an option to explain dynamics usually given through the standard
cold dark matter
In cosmology and physics, cold dark matter (CDM) is a hypothetical type of dark matter. According to the current standard model of cosmology, Lambda-CDM model, approximately 27% of the universe is dark matter and 68% is dark energy, with only a sm ...
models, as well as
MOND,
Axion
An axion () is a hypothetical elementary particle originally theorized in 1978 independently by Frank Wilczek and Steven Weinberg as the Goldstone boson of Peccei–Quinn theory, which had been proposed in 1977 to solve the strong CP problem ...
s (from Breaking of a Symmetry, too),
MACHOS,...
Connection to string theory
A generic prediction of all string theory models is that the spin-2 graviton has a spin-0 partner called the
dilaton
In particle physics, the hypothetical dilaton is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compa ...
.
Hence, string theory predicts that the actual theory of gravity is a scalar–tensor theory rather than general relativity. However, the precise form of such a theory is not currently known because one does not have the mathematical tools in order to address the corresponding non-perturbative calculations. Besides, the precise effective 4-dimensional form of the theory is also confronted to the so-called
landscape issue.
See also
*
*
*
*
*
References
*P. Jordan, ''Schwerkraft und Weltall'', Vieweg (Braunschweig) 1955: Projective Relativity. First paper on JBD theories.
*C.H. Brans and R.H. Dicke, ''Phys. Rev.'' 124: 925, 1061: Brans–Dicke theory starting from Mach's principle.
*R. Wagoner, ''Phys. Rev.'' D1(812): 3209, 2004: JBD theories with more than one scalar field.
*A. Zee, ''Phys. Rev. Lett.'' 42(7): 417, 1979: Broken-Symmetric scalar-tensor theory.
*H. Dehnen and H. Frommert, ''Int. J. Theor. Phys.'' 30(7): 985, 1991: Gravitative-like and short-ranged interaction of Higgs fields within the Standard Model or elementary particles.
*H. Dehnen ''et al.'', ''Int. J. Theor. Phys.'' 31(1): 109, 1992: Scalar-tensor-theory with Higgs field.
*C.H. Brans, June 2005: Roots of scalar-tensor theories. . Discusses the history of attempts to construct gravity theories with a scalar field and the relation to the equivalence principle and Mach's principle.
*
*
{{DEFAULTSORT:Scalar-Tensor Theory
Tensors
Theories of gravity
String theory
Physical cosmology
Particle physics
Physics beyond the Standard Model
Mathematical physics