Semiclassical gravity

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 dynamics of the theory is described by the semiclassical Einstein equations, which relate the curvature of spacetime that is encoded by the Einstein tensor ${\displaystyle G_{\mu \nu }}$ to the expectation value of the energy–momentum tensor ${\displaystyle {\hat{T}}_{\mu \nu }}$ (a quantum field theory operator) of the matter fields, i.e.

${\displaystyle G_{\mu \nu }=\frac{8\pi G}{c^4}{⟨{\hat{T}}_{\mu \nu }⟩}_{\psi },}$

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

Energy–momentum tensor

There is some ambiguity in regulating the energy–momentum tensor, and this depends upon the curvature. This ambiguity can be absorbed into the cosmological constant, the gravitational constant, and the quadratic couplings

${\displaystyle \int \sqrt{-g}{R}^2{d}^{d}x}$ and ${\displaystyle \int \sqrt{-g}{R}^{\mu \nu }{R}_{\mu \nu }{d}^{d}x.}$

There is another quadratic term of the form

${\displaystyle \int \sqrt{-g}{R}^{\mu \nu \rho \sigma }{R}_{\mu \nu \rho \sigma }{d}^{d}x,}$

but in four dimensions this term is a linear combination of the other two terms and a surface term. See Gauss–Bonnet gravity for more details.

Since the theory of quantum gravity is not yet known, it is difficult to precisely determine the regime of validity of semiclassical gravity. However, one can formally show that semiclassical gravity could be deduced from quantum gravity by considering N copies of the quantum matter fields and taking the limit of N going to infinity while keeping the product GN constant. At a diagrammatic level, semiclassical gravity corresponds to summing all Feynman diagrams that do not have loops of gravitons (but have an arbitrary number of matter loops). Semiclassical gravity can also be deduced from an axiomatic approach.