semiclassical gravity
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Semiclassical gravity is the approximation to the theory of
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
in which one treats matter fields as being quantum and the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
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 In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field ...
G_, to the expectation value of the
energy–momentum tensor Energy–momentum may refer to: * Four-momentum * Stress–energy tensor * Energy–momentum relation {{dab ...
operator, T_, of the matter fields: : G_ = \frac \left\langle \hat T_ \right\rangle_\psi where ''G'' is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
and \psi 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 In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
, the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, and the quadratic couplings :\int d^dx \,\sqrt R^2 and \int d^dx\, \sqrt R^R_. There's also the other quadratic term :\int d^dx\, \sqrt R^R_, but (in 4-dimensions) this term is a linear combination of the other two terms and a surface term. See
Gauss–Bonnet gravity In general relativity, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet term (named after Carl Friedrich Gauss and Pierre Ossian Bonne ...
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 In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
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,See Page and Geilker; Eppley and Hannah; Albers, Kiefer, and Reginatto. if ''M'' is a huge mass, then the superposition :\frac \left( \left, M \text A \right\rangle + \left, M \text 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 Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave ...
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
Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical a ...
of
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
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 The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
.


Notes


References

* Birrell, N. D. and Davies, P. C. W., ''Quantum fields in curved space'', (Cambridge University Press, Cambridge, UK, 1982). * * * * 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 In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while givi ...
{{quantum gravity Quantum gravity