Scalar-tensor Theory - Gravity As Field Theory

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 is of another nature. It unifies space and time in a 4-dimensional manifold called space-time that depends upon mass itself. In General Relativity there is no gravitational force, but instead a curvature of space-time. The curvature is consequence of mass and in linear approximation it is identifiable with a force. This force is the derivative of the so called metric as potential. The metric of General Relativity possesses the characteristics of space-time and it 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 not only given through a scalar field nor through the 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.