Scalar Field Solution - Mathematical Definition

Mathematical Definition

In general relativity, the geometric setting for physical phenomena is a Lorentzian manifold, which is physically interpreted as a curved spacetime, and which is mathematically specified by defining a metric tensor (or by defining a frame field). The curvature tensor of this manifold and associated quantities such as the Einstein tensor, are well-defined even in the absence of any physical theory, but in general relativity they acquire a physical interpretation as geometric manifestations of the gravitational field.

In addition, we must specify a scalar field by giving a function . This function is required to satisfy two following conditions:

1. The function must satisfy the (curved spacetime) source-free wave equation ,
2. The Einstein tensor must match the stress-energy tensor for the scalar field, which in the simplest case, a minimally coupled massless scalar field, can be written

.

Both conditions follow from varying the Lagrangian density for the scalar field, which in the case of a minimally coupled massless scalar field is

Here,

gives the wave equation, while

gives the Einstein equation (in the case where the field energy of the scalar field is the only source of the gravitational field).