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'''Semiclassical gravity''' is the approximation to the theory of [[quantum gravity]] in which one treats [[Field (physics)|matter fields]] as being quantum and the [[Gravitation|gravitational field]] as being classical. | |||
In semiclassical gravity, matter is represented by quantum matter fields 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 curvature of the spacetime is given by the ''semiclassical Einstein equations'', which relate the curvature of the spacetime, given by the [[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 [[Gravitational constant|Newton's 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 [[cosmological constant]], [[Newton's constant]], and the [[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 [[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 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 ''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. | |||
== | ==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 beginnings of the [[Big Bang|big bang]]. | |||
{{ | ==Notes== | ||
[[Category: | {{Reflist}} | ||
==References== | |||
* Birrell, N. D. and Davies, P. C. W., ''Quantum fields in curved space'', (Cambridge University Press, Cambridge, UK, 1982). | |||
* Don N. Page, and C. D. Geilker, "Indirect Evidence for Quantum Gravity." ''Phys. Rev. Lett.'' '''47''' (1981) 979–982. DOI:[http://dx.doi.org/10.1103/PhysRevLett.47.979 10.1103/PhysRevLett.47.979] | |||
* K. Eppley and E. Hannah, "The necessity of quantizing the gravitational field." ''Found. Phys.'' '''7''' (1977) 51–68. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1007/BF00715241 10.1007/BF00715241] | |||
* Mark Albers, Claus Kiefer, Marcel Reginatto, "Measurement Analysis and Quantum Gravity." ''Phys.Rev.D'' '''78''' 6 (2008) 064051, [http://dx.doi.org/10.1103/PhysRevD.78.064051 DOI:10.1103/PhysRevD.78.064051]. Eprint [http://arxiv.org/abs/0802.1978 arXiv:0802.1978] [gr-qc]. | |||
* Robert M. Wald, ''Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics''. University of Chicago Press, 1994. | |||
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-43587 Semiclassical gravity on arxiv.org] | |||
{{theories of gravitation}} | |||
{{quantum gravity}} | |||
[[Category:Theories of gravitation]] | |||
[[Category:Quantum field theory]] | |||
[[Category:Quantum gravity]] |
Revision as of 03:41, 13 August 2014
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 , to the expectation value of the energy–momentum tensor operator, , of the matter fields:
where G is Newton's constant and 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 cosmological constant, Newton's constant, and the quadratic couplings[1]
There's also the other quadratic term
but (in 4-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 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.
Experimental status
There are cases where semiclassical gravity breaks down. For instance,[2] if M is a huge mass, then the superposition
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.
Applications
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.
Notes
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References
- Birrell, N. D. and Davies, P. C. W., Quantum fields in curved space, (Cambridge University Press, Cambridge, UK, 1982).
- Don N. Page, and C. D. Geilker, "Indirect Evidence for Quantum Gravity." Phys. Rev. Lett. 47 (1981) 979–982. DOI:10.1103/PhysRevLett.47.979
- K. Eppley and E. Hannah, "The necessity of quantizing the gravitational field." Found. Phys. 7 (1977) 51–68. doi:10.1007/BF00715241
- Mark Albers, Claus Kiefer, Marcel Reginatto, "Measurement Analysis and Quantum Gravity." Phys.Rev.D 78 6 (2008) 064051, DOI:10.1103/PhysRevD.78.064051. Eprint arXiv:0802.1978 [gr-qc].
- Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, 1994.
- Semiclassical gravity on arxiv.org