Scalar-Tensor theory

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.

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 mechanics are derived from Hamilton's principle of least action. In this approach, the behavior of a system is not described via forces, 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 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 Einstein, 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 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 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 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 Hilbert and Einstein. 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 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 Relativity 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, Axions (from Breaking of a Symmetry, too), MACHOS,...


  • 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. of 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. of Theor. Phys. 31(1): 109, 1992: Scalar-tensor-theory with Higgs field.
  • C.H. Brans, arXiv:gr-qc/0506063 v1, June 2005: Roots of scalar-tensor theories.
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