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Semiclassical gravity
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{{Short description|Physical theory with matter as quantum fields but gravity as a classical field}} '''Semiclassical gravity''' is an approximation to the theory of [[quantum gravity]] in which one treats matter and energy [[Field (physics)|fields]] as being [[quantum]] and the [[Gravitation|gravitational field]] as being classical. In semiclassical gravity, matter is represented by quantum [[matter field]]s that propagate according to the theory of [[quantum field theory in curved spacetime|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]] <math>G_{\mu\nu}</math> to the [[Expectation value (quantum mechanics)|expectation value]] of the [[Stress–energy tensor|energy–momentum tensor]] <math>\hat T_{\mu\nu}</math> (a [[quantum field theory]] operator) of the matter fields, i.e. : <math>G_{\mu\nu} = \frac{8 \pi G}{c^4} \left\langle \hat T_{\mu\nu} \right\rangle_\psi,</math> where ''G'' is the [[gravitational constant]], and <math>\psi</math> 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 [[f(R) gravity|quadratic couplings]]<ref>See Wald (1994) Chapter 4, section 6 "The Stress–Energy Tensor".</ref> : <math>\int \sqrt{-g} R^2 \, d^dx</math> and <math>\int \sqrt{-g} R^{\mu\nu} R_{\mu\nu} \, d^dx.</math> There is another quadratic term of the form : <math>\int \sqrt{-g} R^{\mu\nu\rho\sigma} R_{\mu\nu\rho\sigma} \, d^dx,</math> 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 diagram]]s 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. ==Experimental status== There are cases where semiclassical gravity breaks down. For instance,<ref>See Page and Geilker; Eppley and Hannah; Albers, Kiefer, and Reginatto.</ref> if ''M'' is a huge mass, then the superposition : <math>\frac{1}{\sqrt{2}} \big(|M \text{ at } A\rangle + |M \text{ at } B\rangle\big),</math> where the locations ''A'' and ''B'' are spatially separated, results in an expectation value of the energy–momentum tensor that is ''M''/2 at ''A'' and ''M''/2 at ''B'', but one would never observe the metric sourced by such a distribution. Instead, one would observe the [[Quantum decoherence|decoherence]] into a state with the metric sourced at ''A'' and another sourced at ''B'' with 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 the [[Hawking radiation]] of [[black hole]]s and the generation of random Gaussian-distributed perturbations in the theory of [[cosmic inflation]], which is thought to occur at the very beginning of the [[Big Bang]]. ==Notes== {{Reflist}} ==References== * Birrell, N. D. and Davies, P. C. W., ''Quantum fields in curved space'', (Cambridge University Press, Cambridge, UK, 1982). * {{cite journal | last1=Page | first1=Don N. | last2=Geilker | first2=C. D. | title=Indirect Evidence for Quantum Gravity | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=47 | issue=14 | date=1981-10-05 | issn=0031-9007 | doi=10.1103/physrevlett.47.979 | pages=979–982| bibcode=1981PhRvL..47..979P }} * {{cite journal | last1=Eppley | first1=Kenneth | last2=Hannah | first2=Eric | title=The necessity of quantizing the gravitational field | journal=Foundations of Physics | publisher=Springer Science and Business Media LLC | volume=7 | issue=1–2 | year=1977 | issn=0015-9018 | doi=10.1007/bf00715241 | pages=51–68| bibcode=1977FoPh....7...51E | s2cid=123251640 }} * {{cite journal | last1=Albers | first1=Mark | last2=Kiefer | first2=Claus | last3=Reginatto | first3=Marcel | title=Measurement analysis and quantum gravity | journal=Physical Review D | publisher=American Physical Society (APS) | volume=78 | issue=6 | date=2008-09-18 | issn=1550-7998 | doi=10.1103/physrevd.78.064051 | page=064051|arxiv=0802.1978| bibcode=2008PhRvD..78f4051A | s2cid=119232226 }} * 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]] {{theories of gravitation}} {{quantum gravity}} [[Category:Quantum gravity]]
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