Semiclassical gravity

Semiclassical gravityis an approximation to the theory ofquantum gravityin which one treats matter and energyfieldsas being quantum and thegravitational fieldas being classical.

In semiclassical gravity, matter is represented by quantum matter fields that propagate according to the theory ofquantum fields in curved spacetime.The spacetime in which the fields propagate is classical but dynamical. The dynamics of the theory is described by thesemiclassical Einstein equations,which relate the curvature of the spacetime that is encoded by theEinstein tensor$G_{\mu \nu }$to theexpectation valueof theenergy–momentum tensor${\hat {T}}_{\mu \nu }$(aquantum field theoryoperator) of the matter fields, i.e.

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

whereGis thegravitational constant,and$\psi$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 thecosmological constant,thegravitational constant,and thequadratic couplings

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

There is another quadratic term of the form

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

but in four dimensions this term is a linear combination of the other two terms and a surface term. SeeGauss–Bonnet gravityfor 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 consideringNcopies of the quantum matter fields and taking the limit ofNgoing to infinity while keeping the productGNconstant. At a diagrammatic level, semiclassical gravity corresponds to summing allFeynman diagramsthat do not have loops of gravitons (but have an arbitrary number of matter loops). Semiclassical gravity can also be deduced from an axiomatic approach.

Experimental status

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

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

where the locationsAandBare spatially separated, results in an expectation value of the energy–momentum tensor that isM/2 atAandM/2 atB,but one would never observe the metric sourced by such a distribution. Instead, one would observe thedecoherenceinto a state with the metric sourced atAand another sourced atBwith a 50% chance each. Extensions of semiclassical gravity that incorporate decoherence have also been studied.

Applications

The most important applications of semiclassical gravity are to understand theHawking radiationofblack holesand the generation of random Gaussian-distributed perturbations in the theory ofcosmic inflation,which is thought to occur at the very beginning of theBig Bang.

References

• Birrell, N. D. and Davies, P. C. W.,Quantum fields in curved space,(Cambridge University Press, Cambridge, UK, 1982).
• Page, Don N.; Geilker, C. D. (1981-10-05). "Indirect Evidence for Quantum Gravity".Physical Review Letters.47(14). American Physical Society (APS): 979–982.Bibcode:1981PhRvL..47..979P.doi:10.1103/physrevlett.47.979.ISSN0031-9007.
• Eppley, Kenneth; Hannah, Eric (1977). "The necessity of quantizing the gravitational field".Foundations of Physics.7(1–2). Springer Science and Business Media LLC: 51–68.Bibcode:1977FoPh....7...51E.doi:10.1007/bf00715241.ISSN0015-9018.S2CID123251640.
• Albers, Mark; Kiefer, Claus; Reginatto, Marcel (2008-09-18). "Measurement analysis and quantum gravity".Physical Review D.78(6). American Physical Society (APS): 064051.arXiv:0802.1978.Bibcode:2008PhRvD..78f4051A.doi:10.1103/physrevd.78.064051.ISSN1550-7998.S2CID119232226.
• Robert M. Wald,Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.University of Chicago Press, 1994.

See also

• Quantum field theory in curved spacetime