User:Ems57fcva/sandbox/scalar gravity

Scalar Gravity is a term that can be applied to any model of gravitation in which the gravitational field is modelled as arising out of a single scalar value. This is in contrast to general relativity, in which gravitation is described using in a rank two tensor.

Newtonian gravity

The protoypical scalar gravitational theory is Newton's theory of gravity. In this theory, we can treat gravity as being due to a potential field Φ where the local divergence of this field is given by

$\nabla \Phi =4\pi G\rho$, where

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

This leads directly to the law of universal gravitation, $F=m_{1}m_{2}G/r^{2}$.

Nordstöm's theories of gravitation

Full article: Nordström's theory of gravitation

The first tries at describing gravity in light of relativity theory were also scalar theories. Gunnar Nordström created two such theories, and they were part of the history of general relativity. The first attempt (done in 1912) was nothing more than a replacement of the divergence operator above with the correponding d'Almebertian operator $\square =\partial _{t}^{2}-\nabla ^{2}$ to obtain

$\square \Phi =4\pi G\rho$.

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

The second attempt, presented in 1913 involved the use of the field equation

$\Phi \square \Phi =-4\pi GT_{m}$, where T_m is the trace of the stress-energy tensor.

This theory produces as solutions metrics that are conformally flat, meaning that these solutions can be written as

$g_{\mu \nu }=A\eta _{\mu \nu }$, where

• η_μν is the Minkowski metric, and
• $A$ is a scalar which is a function of position.

These metrics, while being in accord with the weak equivalence principle none the less have some serious problems with observation. Amongst them are

• No deflection of light is supported (as has been observed) and
• The anomolous perihelion precession of Mercury is predicted to be of both the wrong magnitude and direction.

Einstein's scalar theory

In 1913, Einstein (erroneously) concluded that general covariance was not viable due to the hole argument, and inspired by Nordström's work created his own scalar theory. In Einstein's 1913 scalar theory, spacetime has massless scalar field with a stress-energy of

$T_{g}^{\mu \nu }={\frac {1}{4\pi G}}\left[\partial ^{\mu }\phi \,\partial ^{\nu }\phi \,-{\frac {1}{2}}\eta ^{\mu \nu }\left(\partial _{\lambda }\phi \right)\right]$

In addition, mass-energy contributes a stress-energy of

$T_{m}^{\mu \nu }=\rho \phi u^{\mu }u^{\nu }$, where $u^{\mu }$ is the relativistic velocity vector.

Eventually, this scheme was found to lack diffeomorphism invariance (which is a self-consistency check), and Einstien dropped it in late 1914. Thus ended the use of scalar models in Einstein's search for a general theory of relativity. After this, Einstein soon realized the flaw in the hole argument, and returning permanently to general covariance. Geenral relativity was finalized using general covariance in 1915.

Additional variations

• Kaluza-Klein theory involves the use of a scalar gravitional field in addition to the electromagnetic field potential $A^{\mu }$ in an attempt to create a 5-dimensinal unification of gravity and electromagnetism.
• Brans-Dicke theory is a combination scalar-tensor theory which has not been totally ruled out by observation.

References

• Scalar Gravitation and Extra Dimensions from arXiv.org

See also

• Brans-Dicke theory
• Kaluza-Klein theory
• Relativistic Scalar Gravity: A Laboratory for Numerical Relativity from arXiv.org