Semiclassical gravity

Semiclassical gravity is the approximation to the theory of quantum gravity in which one treats matter fields as being quantum and the gravitational field as being classical.

In semiclassical gravity, matter is represented by quantum matter fields that propagate according to the theory of quantum fields in curved spacetime. The spacetime in which the fields propagate is classical but dynamical. The curvature of the spacetime is given by the semiclassical Einstein equations, which relate the curvature of the spacetime, given by the Einstein tensor ${\displaystyle G_{\mu \nu }}$, to the expectation value of the energy–momentum tensor operator, ${\displaystyle T_{\mu \nu }}$, of the matter fields:

${\displaystyle G_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left\langle {\hat {T}}_{\mu \nu }\right\rangle _{\psi }}$

where G is the gravitational constant and ${\displaystyle \psi }$ indicates the quantum state of the matter fields.