Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a scalar field, which is required to satisfy some field equation.
Contents
- Newtonian gravity
- Nordstrms theories of gravitation
- Einsteins scalar theory
- Additional variations
- References
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 tensor field.
Newtonian gravity
The prototypical scalar theory of gravitation is Newtonian gravitation. In this theory, the gravitational interaction is completely described by the potential
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 Nordström created 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'Alembertian operator
However, several theoretical difficulties with this theory quickly arose, and Nordström dropped it.
A year later, Nordström tried again, presenting the field equation
where
Solutions of Nordström's second theory are conformally flat Lorentzian spacetimes. That is, the metric tensor can be written as
This suggestion signifies that the inertial mass should depend on the scalar field.
Nordström's second theory satisfies the weak equivalence principle. However:
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 argument that 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
Unfortunately, this theory is not diffeomorphism covariant. 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).