Semiclassical gravity
Encyclopedia
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.
There is some ambiguity in regulating the stress-energy tensor, and this depends upon the curvature. According to Robert Wald
, this ambiguity can be absorbed into the cosmological constant
, Newton's constant, and the quadratic couplings
and 
. There's also the other quadratic term 
, but in 4D, 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.
There are cases where semiclassical gravity breaks down. For instance, 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.
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
.
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
in which one treats matter fields as being quantum and the gravitational field
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
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
Quantum field theory in curved spacetime
Quantum field theory in curved spacetime is an extension of standard, Minkowski-space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time dependent gravitational fields , or by time independent gravitational fields that contain...
. 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
Einstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
, to the expectation value of the energy-momentum tensorStress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
operator,
, of the matter fields:where G is Newton's constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
and
indicates the quantum state of the matter fields.There is some ambiguity in regulating the stress-energy tensor, and this depends upon the curvature. According to Robert Wald
Robert Wald
Robert M. Wald is a physicist who specializes in general relativity and the thermodynamics of black holes. He is well known as the author of a widely used graduate textbook, General Relativity . Wald is a professor at the Enrico Fermi Institute and the University of Chicago...
, this ambiguity can be absorbed into the cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
, Newton's constant, and the quadratic couplings
F(R) gravity
f gravity is a type of modified gravity theory first proposed in 1970by Buchdahl as a generalisation of Einstein's General Relativity. Although it is an active field of research, there are known problems with the theory...
and . There's also the other quadratic term , but in 4D, 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
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
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.
There are cases where semiclassical gravity breaks down. For instance, 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.
The most important applications of semiclassical gravity are to understand the Hawking radiation
Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...
of black holes
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
and the generation of random gaussian-distributed perturbations in the theory of cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
, which is thought to occur at the very beginnings of the big bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
.