In general relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a scalar field. Such a field may or may not be massless, and it may be taken to have minimal curvature coupling, or some other choice, such as conformal coupling.
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 g a b {\displaystyle g_{ab}} (or by defining a frame field). The curvature tensor R a
b c d {\displaystyle R^{a}{}_{bcd}} of this manifold and associated quantities such as the Einstein tensor G a b {\displaystyle G_{ab}} , 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 ψ {\displaystyle \psi } . This function is required to satisfy two following conditions:
The function must satisfy the (curved spacetime) source-free wave equation g a b ψ ; a b = 0 {\displaystyle g^{ab}\psi _{;ab}=0} , 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
G a b = κ ( ψ ; a ψ ; b − 1 2 ψ ; m ψ ; m g a b ) {\displaystyle G_{ab}=\kappa \left(\psi _{;a}\psi _{;b}-{\frac {1}{2}}\psi _{;m}\psi ^{;m}g_{ab}\right)} . Both conditions follow from varying the Lagrangian density for the scalar field, which in the case of a minimally coupled massless scalar field is
L = − g m n ψ ; m ψ ; n {\displaystyle L=-g^{mn}\,\psi _{;m}\,\psi _{;n}}
Here,
δ L δ ψ = 0 {\displaystyle {\frac {\delta L}{\delta \psi }}=0}
gives the wave equation, while
δ L δ g a b = 0 {\displaystyle {\frac {\delta L}{\delta g^{ab}}}=0}
gives the Einstein equation (in the case where the field energy of the scalar field is the only source of the gravitational field).
Physical interpretation Scalar fields are often interpreted as classical approximations, in the sense of effective field theory, to some quantum field. In general relativity, the speculative quintessence field can appear as a scalar field. For example, a flux of neutral pions can in principle be modeled as a minimally coupled massless scalar field.
Einstein tensor The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer. In the special case of a minimally coupled massless scalar field, an adapted frame
e → 0 , e → 1 , e → 2 , e → 3 {\displaystyle {\vec {e}}_{0},\;{\vec {e}}_{1},\;{\vec {e}}_{2},\;{\vec {e}}_{3}}
(the first is a timelike unit vector field, the last three are spacelike unit vector fields) can always be found in which the Einstein tensor takes the simple form
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