# Physics:Scalar theories of gravitation

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. 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 $\displaystyle{ \Phi }$, which is required to satisfy the Poisson equation (with the mass density acting as the source of the field). To wit:

$\displaystyle{ \Delta \Phi = 4 \pi G \rho }$, where

• G is the gravitational constant and
• $\displaystyle{ \rho }$ is the mass density.

This field theory formulation leads directly to the familiar law of universal gravitation, $\displaystyle{ F = m_1 m_2 G/r^2 }$.

## 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 $\displaystyle{ \square = \partial_t^2 - \nabla^2 }$. This gives the field equation

$\displaystyle{ \square \Phi = -4 \pi G \rho }$.

However, several theoretical difficulties with this theory quickly arose, and Nordström dropped it.

A year later, Nordström tried again, presenting the field equation

$\displaystyle{ \Phi \square \Phi = -4 \pi G T }$,

where $\displaystyle{ T }$ is the trace of the stress–energy tensor.

Solutions of Nordström's second theory are conformally flat Lorentzian spacetimes. That is, the metric tensor can be written as $\displaystyle{ g_{\mu\nu} = A \eta_{\mu\nu} }$, where

• ημν is the Minkowski metric, and
• $\displaystyle{ A }$ 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 precession of 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 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,

$\displaystyle{ T^{\mu\nu}_g = \frac{1}{4 \pi G} \left [\partial^\mu \phi \, \partial^\nu \phi \, - \frac{1}{2} \eta^{\mu\nu} \partial_\lambda \phi \, \partial^\lambda \phi \right] }$

represents the stress–momentum–energy of the scalar field itself. The second represents the stress-momentum-energy of any matter which may be present:

$\displaystyle{ T^{\mu\nu}_m = \rho \phi u^\mu u^\nu }$

where $\displaystyle{ u^\mu }$ is the velocity vector of an observer, or tangent vector to 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 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).

• Kaluza–Klein theory involves the use of a scalar gravitational field in addition to the electromagnetic field potential $\displaystyle{ A^\mu }$ 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.