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Scalar–tensor theory

In theoretical physics, a scalar–tensor theory is a field 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 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. 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.

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 (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, 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) 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.

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 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 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.[1] However, no deviation from the constancy of the Newtonian gravitational constant has been measured,[2] 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.[3]

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[4] (i.e.,  ), the metric takes the following form:

 
 
 

with[5]

 
 

where   is a function depending on the coordinate gauge

 

It corresponds to the remaining diffeomorphism 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  ,[2] 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.[3] Hence, according to them, the current high value of the function   could be a simple consequence of the evolution of the universe.

The best current constraint on the post-Newtonian parameter   comes from Mercury's perihelion shift and is  .[2]

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 has ruled this out.[6][7][8][9][10]

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 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. 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 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 inflation 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,...

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.[11] 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.

Other possible scalar–tensor theories

Theories with non-minimal scalar-matter coupling

References

  1. ^ Galiautdinov, Andrei; Kopeikin, Sergei M. (2016-08-10). "Post-Newtonian celestial mechanics in scalar-tensor cosmology". Physical Review D. 94 (4): 044015. arXiv:1606.09139. Bibcode:2016PhRvD..94d4015G. doi:10.1103/PhysRevD.94.044015. S2CID 32869795.
  2. ^ a b c Uzan, Jean-Philippe (2011-12-01). "Varying Constants, Gravitation and Cosmology". Living Reviews in Relativity. 14 (1): 2. arXiv:1009.5514. Bibcode:2011LRR....14....2U. doi:10.12942/lrr-2011-2. ISSN 2367-3613. PMC 5256069. PMID 28179829.
  3. ^ a b Damour, Thibault; Nordtvedt, Kenneth (1993-04-12). "General relativity as a cosmological attractor of tensor-scalar theories". Physical Review Letters. 70 (15): 2217–2219. Bibcode:1993PhRvL..70.2217D. doi:10.1103/PhysRevLett.70.2217. PMID 10053505.
  4. ^ Damour, Thibault; Soffel, Michael; Xu, Chongming (1991-05-15). "General-relativistic celestial mechanics. I. Method and definition of reference systems". Physical Review D. 43 (10): 3273–3307. Bibcode:1991PhRvD..43.3273D. doi:10.1103/PhysRevD.43.3273. PMID 10013281.
  5. ^ Minazzoli, Olivier; Chauvineau, Bertrand (2011). "Scalar–tensor propagation of light in the inner solar system including relevant c^{-4} contributions for ranging and time transfer". Classical and Quantum Gravity. 28 (8): 085010. arXiv:1007.3942. Bibcode:2011CQGra..28h5010M. doi:10.1088/0264-9381/28/8/085010. S2CID 119118136.
  6. ^ Lombriser, Lucas; Lima, Nelson (2017). "Challenges to Self-Acceleration in Modified Gravity from Gravitational Waves and Large-Scale Structure". Physics Letters B. 765: 382–385. arXiv:1602.07670. Bibcode:2017PhLB..765..382L. doi:10.1016/j.physletb.2016.12.048. S2CID 118486016.
  7. ^ "Quest to settle riddle over Einstein's theory may soon be over". phys.org. February 10, 2017. Retrieved October 29, 2017.
  8. ^ "Theoretical battle: Dark energy vs. modified gravity". Ars Technica. February 25, 2017. Retrieved October 27, 2017.
  9. ^ Ezquiaga, Jose María; Zumalacárregui, Miguel (2017-12-18). "Dark Energy After GW170817: Dead Ends and the Road Ahead". Physical Review Letters. 119 (25): 251304. arXiv:1710.05901. Bibcode:2017PhRvL.119y1304E. doi:10.1103/PhysRevLett.119.251304. PMID 29303304. S2CID 38618360.
  10. ^ Creminelli, Paolo; Vernizzi, Filippo (2017-12-18). "Dark Energy after GW170817 and GRB170817A". Physical Review Letters. 119 (25): 251302. arXiv:1710.05877. Bibcode:2017PhRvL.119y1302C. doi:10.1103/PhysRevLett.119.251302. PMID 29303308. S2CID 206304918.
  11. ^ Damour, Thibault; Piazza, Federico; Veneziano, Gabriele (2002-08-05). "Runaway Dilaton and Equivalence Principle Violations". Physical Review Letters. 89 (8): 081601. arXiv:gr-qc/0204094. Bibcode:2002PhRvL..89h1601D. doi:10.1103/PhysRevLett.89.081601. PMID 12190455. S2CID 14136427.
  • 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, arXiv:gr-qc/0506063 v1, June 2005: Roots of scalar-tensor theories.
  • P. G. Bergmann (1968). "Comments on the scalar-tensor theory". Int. J. Theor. Phys. 1 (1): 25–36. Bibcode:1968IJTP....1...25B. doi:10.1007/BF00668828. S2CID 119985328.
  • R. V. Wagoner (1970). "Scalar-tensor theory and gravitational waves". Phys. Rev. D1 (12): 3209–3216. Bibcode:1970PhRvD...1.3209W. doi:10.1103/physrevd.1.3209.

scalar, tensor, theory, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, dec. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Scalar tensor theory news newspapers books scholar JSTOR December 2018 Learn how and when to remove this template message In theoretical physics a scalar tensor theory is a field 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 Contents 1 Tensor fields and field theory 2 Gravity as field theory 2 1 Mathematical formulation 2 1 1 The Newtonian approximation of the theory 2 1 2 The first post Newtonian approximation of the theory 3 Observational constraints on the theory 4 Higher dimensional relativity and scalar tensor theories 5 Connection to string theory 6 Other possible scalar tensor theories 6 1 Theories with non minimal scalar matter coupling 7 ReferencesTensor fields and field theory EditModern 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 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 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 Gravity as field theory EditIn 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 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 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 gt 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 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 Mathematical formulation Edit An action of such a gravitational scalar tensor theory can be written as follows S 1 c d 4 x g 1 2 m F R w F F s F 2 V F 2 m L m g m n PS displaystyle S frac 1 c int d 4 x sqrt g frac 1 2 mu times left Phi R frac omega Phi Phi partial sigma Phi 2 V Phi 2 mu mathcal L m g mu nu Psi right where g displaystyle g is the metric determinant R displaystyle R is the Ricci scalar constructed from the metric g m n displaystyle g mu nu m displaystyle mu is a coupling constant with the dimensions L 1 M 1 T 2 displaystyle L 1 M 1 T 2 V F displaystyle V Phi is the scalar field potential L m displaystyle mathcal L m is the material Lagrangian and PS displaystyle Psi represents the non gravitational fields Here the Brans Dicke parameter w displaystyle omega has been generalized to a function Although m displaystyle mu is often written as being 8 p G c 4 displaystyle 8 pi G c 4 one has to keep in mind that the fundamental constant G displaystyle G 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 F displaystyle Phi The metric and scalar field equations respectively write R m n 1 2 g m n R m F T m n 1 F m n g m n F w F F 2 m F n F 1 2 g m n a F 2 g m n V F 2 F displaystyle R mu nu frac 1 2 g mu nu R frac mu Phi T mu nu frac 1 Phi nabla mu nabla nu g mu nu Box Phi frac omega Phi Phi 2 partial mu Phi partial nu Phi frac 1 2 g mu nu partial alpha Phi 2 g mu nu frac V Phi 2 Phi and 2 w F 3 F F m F T w F F s F 2 V F 2 V F F displaystyle frac 2 omega Phi 3 Phi Box Phi frac mu Phi T frac omega Phi Phi partial sigma Phi 2 V Phi 2 frac V Phi Phi Also the theory satisfies the following conservation equation implying that test particles follow space time geodesics such as in general relativity s T m s 0 displaystyle nabla sigma T mu sigma 0 where T m s displaystyle T mu sigma is the stress energy tensor defined as T m n 2 g d g L m d g m n displaystyle T mu nu frac 2 sqrt g frac delta sqrt g mathcal L m delta g mu nu The Newtonian approximation of the theory Edit 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 g 00 1 2 U c 2 O c 3 g 0 i O c 2 g i j d i j O c 1 displaystyle g 00 1 2 frac U c 2 mathcal O c 3 g 0i mathcal O c 2 g ij delta ij mathcal O c 1 with U displaystyle U satisfying the following usual Poisson equation at the lowest order of the approximation U 8 p G e f f r O c 1 displaystyle triangle U 8 pi G mathrm eff rho mathcal O c 1 where r displaystyle rho is the density of the gravitational source and G e f f 2 w 0 4 2 w 0 3 G F 0 displaystyle G mathrm eff frac 2 omega 0 4 2 omega 0 3 frac G Phi 0 the subscript 0 displaystyle 0 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 F 0 displaystyle Phi 0 and therefore theoretically depends on time and location 1 However no deviation from the constancy of the Newtonian gravitational constant has been measured 2 implying that the scalar field background F 0 displaystyle Phi 0 is pretty stable over time Such a stability is not theoretically generally expected but can be theoretically explained by several mechanisms 3 The first post Newtonian approximation of the theory Edit 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 4 i e g i j d i j O c 3 displaystyle g ij propto delta ij mathcal O c 3 the metric takes the following form g 00 1 2 W c 2 b 2 W 2 c 4 O c 5 displaystyle g 00 1 frac 2W c 2 beta frac 2W 2 c 4 mathcal O c 5 g 0 i g 1 2 W i c 3 O c 4 displaystyle g 0i gamma 1 frac 2W i c 3 mathcal O c 4 g i j d i j 1 g 2 W c 2 O c 3 displaystyle g ij delta ij left 1 gamma frac 2W c 2 right mathcal O c 3 with 5 W 1 2 b 3 g c 2 W W 2 c 2 1 g t J 4 p G e f f S O c 3 displaystyle Box W frac 1 2 beta 3 gamma c 2 W triangle W frac 2 c 2 1 gamma partial t J 4 pi G mathrm eff Sigma mathcal O c 3 W i x i J 4 p G e f f S i O c 1 displaystyle triangle W i partial x i J 4 pi G mathrm eff Sigma i mathcal O c 1 where J displaystyle J is a function depending on the coordinate gauge J t W k W k O c 1 displaystyle J partial t W partial k W k mathcal O c 1 It corresponds to the remaining diffeomorphism degree of freedom that is not fixed by the weak isotropy condition The sources are defined as S 1 c 2 T 00 g T k k S i 1 c T 0 i displaystyle Sigma frac 1 c 2 T 00 gamma T kk qquad Sigma i frac 1 c T 0i the so called post Newtonian parameters are g w 0 1 w 0 2 displaystyle gamma frac omega 0 1 omega 0 2 b 1 w 0 2 w 0 3 2 w 0 4 2 displaystyle beta 1 frac omega 0 prime 2 omega 0 3 2 omega 0 4 2 and finally the empirical gravitational constant G e f f displaystyle G mathrm eff is given by G e f f 2 w 0 4 2 w 0 3 G displaystyle G mathrm eff frac 2 omega 0 4 2 omega 0 3 G where G displaystyle G is the true constant that appears in the coupling constant m displaystyle mu defined previously Observational constraints on the theory EditCurrent observations indicate that g 1 2 1 2 3 10 5 displaystyle gamma 1 2 1 pm 2 3 times 10 5 2 which means that w 0 gt 40000 displaystyle omega 0 gt 40000 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 w displaystyle omega toward infinity during the evolution of the universe 3 Hence according to them the current high value of the function w displaystyle omega could be a simple consequence of the evolution of the universe The best current constraint on the post Newtonian parameter b displaystyle beta comes from Mercury s perihelion shift and is b 1 lt 3 10 3 displaystyle beta 1 lt 3 times 10 3 2 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 has ruled this out 6 7 8 9 10 Higher dimensional relativity and scalar tensor theories EditAfter 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 thegauge 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 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 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 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 inflation 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 Connection to string theory EditA generic prediction of all string theory models is that the spin 2 graviton has a spin 0 partner called the dilaton 11 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 Other possible scalar tensor theories EditDegenerate Higher Order Scalar Tensor theoriesTheories with non minimal scalar matter coupling Edit Dilaton gravity Chameleon theory Pressuron theoryReferences Edit Galiautdinov Andrei Kopeikin Sergei M 2016 08 10 Post Newtonian celestial mechanics in scalar tensor cosmology Physical Review D 94 4 044015 arXiv 1606 09139 Bibcode 2016PhRvD 94d4015G doi 10 1103 PhysRevD 94 044015 S2CID 32869795 a b c Uzan Jean Philippe 2011 12 01 Varying Constants Gravitation and Cosmology Living Reviews in Relativity 14 1 2 arXiv 1009 5514 Bibcode 2011LRR 14 2U doi 10 12942 lrr 2011 2 ISSN 2367 3613 PMC 5256069 PMID 28179829 a b Damour Thibault Nordtvedt Kenneth 1993 04 12 General relativity as a cosmological attractor of tensor scalar theories Physical Review Letters 70 15 2217 2219 Bibcode 1993PhRvL 70 2217D doi 10 1103 PhysRevLett 70 2217 PMID 10053505 Damour Thibault Soffel Michael Xu Chongming 1991 05 15 General relativistic celestial mechanics I Method and definition of reference systems Physical Review D 43 10 3273 3307 Bibcode 1991PhRvD 43 3273D doi 10 1103 PhysRevD 43 3273 PMID 10013281 Minazzoli Olivier Chauvineau Bertrand 2011 Scalar tensor propagation of light in the inner solar system including relevant c 4 contributions for ranging and time transfer Classical and Quantum Gravity 28 8 085010 arXiv 1007 3942 Bibcode 2011CQGra 28h5010M doi 10 1088 0264 9381 28 8 085010 S2CID 119118136 Lombriser Lucas Lima Nelson 2017 Challenges to Self Acceleration in Modified Gravity from Gravitational Waves and Large Scale Structure Physics Letters B 765 382 385 arXiv 1602 07670 Bibcode 2017PhLB 765 382L doi 10 1016 j physletb 2016 12 048 S2CID 118486016 Quest to settle riddle over Einstein s theory may soon be over phys org February 10 2017 Retrieved October 29 2017 Theoretical battle Dark energy vs modified gravity Ars Technica February 25 2017 Retrieved October 27 2017 Ezquiaga Jose Maria Zumalacarregui Miguel 2017 12 18 Dark Energy After GW170817 Dead Ends and the Road Ahead Physical Review Letters 119 25 251304 arXiv 1710 05901 Bibcode 2017PhRvL 119y1304E doi 10 1103 PhysRevLett 119 251304 PMID 29303304 S2CID 38618360 Creminelli Paolo Vernizzi Filippo 2017 12 18 Dark Energy after GW170817 and GRB170817A Physical 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