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 G_μν, to the expectation value of the energy-momentum tensor operator, T_μν, of the matter fields:

$G_{\mu\nu} = \frac{ 8 \pi G }{ c^4 } \left\langle \hat T_{\mu\nu} \right\rangle_\psi$

where G is Newton's constant and ψ indicates the quantum state of the matter fields. There is some ambiguity in regulating the stress-energy tensor, and this depends upon the curvature. According to Robert Wald, this ambiguity can be absorbed into the cosmological constant, Newton's constant, and the quadratic couplings $\int d^dx \,\sqrt{-g} R^2$ and $\int d^dx\, \sqrt{-g} R^{\mu\nu}R_{\mu\nu}$. There's also the other quadratic term $\int d^dx\, \sqrt{-g} R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$, but in 4D, 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 say what is 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 diagrammatic level, semiclassical gravity corresponds to summing all Feynman diagrams which do not have loops of gravitons (but have an arbitrary number of matter loops). Semiclassical gravity can also be deduced from an axiomatic approach.

There are cases where semiclassical gravity breaks down. For instance, if M is a huge mass, then the superposition

$\frac{1}{\sqrt{2}} \left( \left| M \text{ at } A \right\rangle + \left| M \text{ at } B \right\rangle \right)$

where A and B are widely separated, then the expectation value of the stress-energy tensor is M/2 at A and M/2 at B, but we would never observe the metric sourced by such a distribution. Instead, we decohere into a state with the metric sourced at A and another sourced at B with a 50% chance each.

The most important applications of semiclassical gravity are to understand the Hawking radiation of black holes and the generation of random gaussian-distributed perturbations in the theory of cosmic inflation, which is thought to occur at the very beginnings of the big bang.

References

• Birrell, N. D. and Davies, P. C. W., Quantum fields in curved space, (Cambridge University Press, Cambridge, UK, 1982).
• Semiclassical gravity on arxiv.org
• Kiefer, Claus The semiclassical approximation to quantum gravity