View source for Physics:Semiclassical gravity

{{Short description|Physical theory with matter as quantum fields but gravity as a classical field}}
'''Semiclassical gravity''' is the approximation to the theory of [[Physics:Quantum gravity|quantum gravity]] in which one treats [[Physics:Field|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 [[Physics:Quantum field theory in curved spacetime|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 [[Physics:Einstein tensor|Einstein tensor]] <math>G_{\mu\nu}</math>, to the expectation value of the [[Stress–energy tensor|energy–momentum tensor]] operator, <math>T_{\mu\nu}</math>, of the matter fields:

:<math> G_{\mu\nu} = \frac{ 8 \pi G }{ c^4 } \left\langle \hat T_{\mu\nu} \right\rangle_\psi </math>

where ''G'' is the [[Physics:Gravitational constant|gravitational constant]] and <math>\psi</math> indicates the quantum state of the matter fields.

==Stress–energy tensor==
There is some ambiguity in regulating the stress–energy tensor, and this depends upon the curvature. This ambiguity can be absorbed into the [[Astronomy:Cosmological constant|cosmological constant]], the [[Physics:Gravitational constant|gravitational constant]], and the [[Physics:F(R) gravity|quadratic couplings]]<ref>See Wald (1994) Chapter 4, section 6 "The Stress-Energy Tensor".</ref>
:<math>\int d^dx \,\sqrt{-g} R^2</math> and <math>\int d^dx\, \sqrt{-g} R^{\mu\nu}R_{\mu\nu}</math>. 
There's also the other quadratic term 
:<math>\int d^dx\, \sqrt{-g} R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}</math>, 
but (in 4-dimensions) this term is a linear combination of the other two terms and a surface term. See [[Physics:Gauss–Bonnet gravity|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 [[Physics:Feynman diagram|Feynman diagram]]s 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.

==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}} \left( \left| M \text{ at } A \right\rangle + \left| M \text{ at } B \right\rangle \right)</math>
where the locations ''A'' and ''B'' are spatially separated results in an expectation value of the stress–energy tensor that 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. Extensions of semi-classical gravity which incorporate decoherence have also been studied.

==Applications==
The most important applications of semiclassical gravity are to understand the [[Physics:Hawking radiation|Hawking radiation]] of [[Astronomy:Black hole|black hole]]s and the generation of random gaussian-distributed perturbations in the theory of [[Astronomy:Cosmic inflation|cosmic inflation]], which is thought to occur at the very beginning of the [[Astronomy:Big Bang|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==
*[[Physics:Quantum field theory in curved spacetime|Quantum field theory in curved spacetime]]

{{theories of gravitation}}
{{quantum gravity}}

[[Category:Quantum gravity]]

{{Sourceattribution|Semiclassical gravity}}