- Scalar theories of gravitation
Scalar theories of gravitation are field theories of
gravitationin which the gravitational field is described using a scalar field, which is required to satisfy some field equation.
"Note:" This article focuses on relativistic classical field theories of gravitation. The best known relativistic classical field theory of gravitation,
general relativity, is a tensor theory, in which the gravitational interaction is described using a tensorfield.
The prototypical scalar theory of gravitation is Newtonian gravitation. In this theory, the gravitational interaction is completely described by the
potential, which is required to satisfy the Poisson equation(with the mass density acting as the source of the field). To wit:
* "G" is the gravitational constant and
* is the mass density.
This field theory formulation leads directly to the familiar law of universal gravitation, .
Nordström's theories of gravitation
The first attempts to present a relativistic (classical) field theory of gravitation were also scalar theories.
Gunnar Nordströmcreated two such theories.
Nordström's first idea (1912) was to simply replace the divergence operator in the field equation of Newtonian gravity with the d'Almebertian operator . This gives the field equation
However, several theoretical difficulties with this theory quickly arose, and Nordström dropped it.
A year later, Nordström tried again, presenting the field equation
where is the trace of the
Solutions of Nordström's second theory are
conformally flatLorentzian spacetimes. That is, the metric tensor can be written as , where
* ημν is the
Minkowski metric, and
* is a scalar which is a function of position. This suggestion signifies that the inertial mass should depend on the scalar field.
Nordström's second theory satisfies the weak
equivalence principle. However:
* The theory fails to predict any deflection of light passing near a massive body (contrary to observation)
* The theory predicts an anomalous
perihelion precessionof Mercury, but this disagrees in both sign and magnitude with the observed anomalous precession (the part which cannot be explained using Newtonian gravitation).Despite these disappointing results, Einstein's critiques of Nordström's second theory played an important role in his development of general relativity.
Einstein's scalar theory
In 1913, Einstein (erroneously) concluded from his
hole argumentthat general covariance was not viable. Inspired by Nordström's work, he proposed his own scalar theory. This theory employs a massless scalar field coupled to the stress-energy tensor, which is the sum of two terms. The first,:represents the stress-momentum-energy of the scalar field itself. The second represents the stress-momentum-energy of any matter which may be present::where is the velocity vector of an observer, or tangent vectorto the world line of the observer. (Einstein made no attempt, in this theory, to take account of possible gravitational effects of the field energy of the electromagnetic field.)
Unfortunately, this theory is not
diffeomorphismcovariant. This is an important consistency condition, so Einstein dropped this theory in late 1914. Associating the scalar field with the metric leads to Einstein's later conclusions that the theory of gravitation he sought could not be a scalar theory. Indeed, the theory he finally arrived at in 1915, general relativity, is a tensor theory, not a scalar theory, with a 2-tensor, the metric, as the potential. Unlike his 1913 scalar theory, it is generally covariant, and it does take into account the field energy-momentum-stress of the electromagnetic field (or any other nongravitational field).
Kaluza-Klein theoryinvolves the use of a scalar gravitational field in addition to the electromagnetic fieldpotential in an attempt to create a five-dimensional unification of gravity and electromagnetism. Its generalization with a 5th variable component of the metric that leads to a variable gravitational constant was first given by Pascual Jordan[http://www.citebase.org/cgi-bin/fulltext?format=application/pdf&identifier=oai:arXiv.org:gr-qc/0506063] .
Brans-Dicke theoryis a scalar-tensor theory, not a scalar theory, meaning that it represents the gravitational interaction using both a scalar field and a tensor field. We mention it here because one of the field equations of this theory involves only the scalar field and the trace of the stress-energy tensor, as in Nordström's theory. Moreover, the Brans-Dicke theory is equal to the independently derived theory of Jordan (hence it is often referred to as the Jordan-Brans-Dicke or JBD theory). The Brans-Dicke theory couples a scalar field with the curvature of space-time and is self-consistent and, assuming appropriate values for a tunable constant, this theory has not been ruled out by observation. The Brans-Dicke theory is generally regarded as a leading competitor of general relativity ,which is a pure tensor theory. However, the Brans-Dicke theory seems to need too high a parameter, which favours general relativity).
* Zee combined the idea of the BD theory with the Higgs-Mechanism of Symmetry Breakdown for mass generation, which led to a scalar-tensor theory with Higgs field as scalar field, in which the scalar field is massive (short-ranged). An example of this theory was proposed by H. Dehnen and H. Frommert 1991, parting from the nature of Higgs field interacting gravitational- and Yukawa (long-ranged)-like with the particles that get mass through it ("Int. J. of Theor. Phys. 29(4): 361", 1990).
* The Watt-Misner theory (1999) is a recent example of a scalar theory of gravitation. It is not intended as a viable theory of gravitation (since, as Watt and Misner point out, it is not consistent with observation), but as a toy theory which can be useful in testing numerical relativity schemes. It also has pedagogical value.
Nordström's theory of gravitation
Watt-Misner theory of gravitation
*cite web | author=Gönner, Hubert F. M. | title=On the History of Unified Field Theories | work=Living Reviews in Relativity | url=http://relativity.livingreviews.org/open?pubNo=lrr-2004-2 | accessdate=August 10 | accessyear=2005
*cite paper | author=Ravndal, Finn | title=Scalar Gravitation and Extra Dimensions| date=2004 | url=http://www.arxiv.org/abs/gr-qc/0405030
*cite paper | author=Watt, Keith, and Misner, Charles W. | title= Relativistic Scalar Gravity: A Laboratory for Numerical Relativity| date=1999 | version=October 10 | url=http://www.arxiv.org/abs/gr-qc/9910032
*P. Jordan, "Schwerkraft und Weltall", Vieweg (Braunschweig) 1955.
*H. Dehnen and H. Frommert, "Scalar Gravity and Higgs Potential"; "Int. J. of Theor. Phys. 29(4): 361", 1990.
*H. Dehnen and H. Frommert, "Higgs-Field Gravity within the Standard Model"; "Int. J. of Theor. Phys.30(7): 985", 1991.
*H. Dehnen "et al.", "Higgs-Field and a New Scalar-Tensor Theory of Gravity"; "Int. J. of Theor. Phys. 31(1): 109", 1992.
* C.H.Brans, "The roots of scalar-tensor theory: an approximate history"; arXiv:gr-qc/0506063 v1, 10 Jun 2005.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Classical theories of gravitation — The current Gold Standard Theory of Gravitation is the general theory of relativity. This is a classical, relativistic field theory (physics) of gravitation. At present, there is no quantum theory of gravitation. Contents 1 Discussion 2 See also … Wikipedia
Scalar field — In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature… … Wikipedia
Scalar-tensor theory — Scalar tensor theories are theories that include a scalar field as well as a tensor field to represent an interaction, especially the gravitational one. Tensor fields and field theory Modern physics tries to derive all physical theories from as… … Wikipedia
Gravitation — is a natural phenomenon by which objects with mass attract one another [http://math.ucr.edu/home/baez/physics/Relativity/GR/grav speed.html Does Gravity Travel at the Speed of Light?] , UCR Mathematics . 1998. Retrieved 3 July 2008] . In everyday … Wikipedia
Scalar-tensor-vector gravity — (STVG) is a theory of gravity which, like other theories including Modified Newtonian dynamics (otherwise known as MOND ), attempts to explain the rotation of galaxies by doing away with dark matter. Unlike MOND, STVG explains galactic clustering … Wikipedia
Nordström's theory of gravitation — In theoretical physics, Nordström s theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913… … Wikipedia
Mechanical explanations of gravitation — (or kinetic theories of gravitation) are attempts to explain the action of gravity by aid of basic mechanical processes, such as pressure forces caused by pushes, and without the use of any action at a distance. These theories were developed from … Wikipedia
Newton's law of universal gravitation — Classical mechanics Newton s Second Law History of classical mechanics … Wikipedia
Kretschmann scalar — In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann. DefinitionThe Kretschmann invariant is: K =… … Wikipedia
Théorie de la Gravitation d'Einstein — Relativité générale Pour les articles homonymes, voir relativité … Wikipédia en Français