CGHS model
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The Callan–Giddings–Harvey–Strominger model or CGHS model in short is a
toy model In the modeling of physics, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model. * In "toy" mathematical models, ...
of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
in 1 spatial and 1 time dimension.


Overview

General relativity is a highly nonlinear model, and as such, its 3+1D version is usually too complicated to analyze in detail. In 3+1D and higher, propagating
gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s exist, but not in 2+1D or 1+1D. In 2+1D, general relativity becomes a
topological field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of math ...
with no local degrees of freedom, and all 1+1D models are locally
flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...
. However, a slightly more complicated generalization of general relativity which includes
dilaton In particle physics, the hypothetical dilaton particle is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theor ...
s will turn the 2+1D model into one admitting mixed propagating dilaton-gravity waves, as well as making the 1+1D model geometrically nontrivial locally. The 1+1D model still does not admit any propagating gravitational (or dilaton) degrees of freedom, but with the addition of matter fields, it becomes a simplified, but still nontrivial model. With other numbers of dimensions, a dilaton-gravity coupling can always be rescaled away by a conformal rescaling of the metric, converting the Jordan frame to the Einstein frame. But not in two dimensions, because the conformal weight of the dilaton is now 0. The metric in this case is more amenable to analytical solutions than the general 3+1D case. And of course, 0+1D models cannot capture any nontrivial aspect of relativity because there is no space at all. This class of models retains just enough complexity to include among its solutions
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, their formation, FRW cosmological models,
gravitational singularities A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular sp ...
, etc. In the quantized version of such models with matter fields,
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 arg ...
also shows up, just as in higher-dimensional models.


Action

A very specific choice of couplings and interactions leads to the CGHS model. :S = \frac \int d^2x\, \sqrt\left\ where ''g'' is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, \phi is the dilaton field, ''fi'' are the matter fields, and ''λ2'' is 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 equ ...
. In particular, the cosmological constant is nonzero, and the matter fields are massless real scalars. This specific choice is classically
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, but still not amenable to an exact quantum solution. It is also the action for
Non-critical string theory The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invari ...
and
dimensional reduction Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in ''D'' spacetime dimensions can be redefined in a lower number of dimensions ''d'', by taking all the fields ...
of higher-dimensional model. It also distinguishes it from Jackiw–Teitelboim gravity and Liouville gravity, which are entirely different models. The matter field only couples to the
causal structure In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian ma ...
, and in the light-cone gauge , has the simple generic form :f_i\left( u, v \right) = A_i\left( u \right) + B_i \left( v \right), with a factorization between left- and right-movers. The Raychaudhuri equations are :e^ \left( - 2\phi_ + 4 \rho_\phi_ \right) + f_f_/2= 0 and :e^ \left( - 2\phi_ + 4 \rho_\phi_ \right) + f_f_/2= 0. The dilaton evolves according to :\left( e^ \right)_ = - \lambda^2 e^e^, while the metric evolves according to :2\rho_ - 4\phi_ + 4\phi_\phi_ + \lambda^2 e^ = 0. The
conformal anomaly A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. A classically conformal theory is a theory which, when placed on a surface wi ...
due to matter induces a Liouville term in the
effective action In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective act ...
.


Black hole

A vacuum black hole solution is given by :ds^2 = - \left( \frac - \lambda^2 uv \right)^ du\, dv :e^ = \frac - \lambda^2 uv, where ''M'' is the ADM mass. Singularities appear at . The masslessness of the matter fields allow a black hole to completely evaporate away via
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 arg ...
. In fact, this model was originally studied to shed light upon the
black hole information paradox The black hole information paradox is a puzzle that appears when the predictions of quantum mechanics and general relativity are combined. The theory of general relativity predicts the existence of black holes that are regions of spacetime from wh ...
.


See also

*
dilaton In particle physics, the hypothetical dilaton particle is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theor ...
*
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
*
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 ...
*
RST model The Russo–Susskind–Thorlacius model or RST model in short is a modification of the CGHS model to take care of conformal anomalies and render it analytically soluble. In the CGHS model, if we include Faddeev–Popov ghosts to gauge-fix diffe ...
* Jackiw–Teitelboim gravity * Liouville gravity


References

Quantum gravity General relativity {{relativity-stub