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The '''Callan–Giddings–Harvey–Strominger model''' or '''CGHS'''<ref>
{{cite journal
| last1 = Callan | first1 = Curtis | authorlink1 = Curtis Callan
| last2 = Giddings | first2 = Steven | authorlink2 = Steven Giddings
| last3 = Harvey | first3 = Jeffrey | authorlink3 = Jeffrey A. Harvey
| last4 = Strominger | first4 = Andrew | authorlink4 = Andrew Strominger
| year = 1992
| title = Evanescent black holes
| journal = Physical Review D
| volume = 45 | pages = 1005–1009
| doi = 10.1103/PhysRevD.45.R1005
| arxiv = hep-th/9111056
}}</ref> in short is a [[toy model]] of [[general relativity]] in 1 spatial and 1 time dimension. 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]]s exist, but not in 2+1D or 1+1D. In 2+1D, general relativity becomes a [[2+1D topological gravity|topological field theory]] with no local degrees of freedom, and all 1+1D models are locally [[Flat (geometry)|flat]]. However, a slightly more complicated generalization of general relativity which includes [[dilaton]]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.<ref>
{{cite journal
| last1 = Grumiller | first1 = Daniel | authorlink1 = Daniel Grumiller
| last2 = Kummer | first2 = Wolfgang | authorlink2 = Wolfgang Kummer
| last3 = Vassilevich | first3 = Dmitri | authorlink3 = Dmitri Vassilevich
| date = October 2002
| title = Dilaton Gravity in Two Dimensions
| journal = Physics Reports
| volume = 369 | issue = 4 | pages = 327–430
| doi = 10.1016/S0370-1573(02)00267-3
|arxiv=hep-th/0204253
}}</ref><ref>
{{cite journal
| last1 = Grumiller | first1 = Daniel | authorlink1 = Daniel Grumiller
| last2 = Meyer | first2 = Rene | authorlink2 = Rene Meyer
| year = 2006
| title = Ramifications of Lineland
| journal = Turkish Journal of Physics
| volume = 30 | issue = 5 | pages = 349–378
| url = http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=fiz&rak=0604-8
| arxiv = hep-th/0604049
}}</ref> 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 and Einstein frames|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]]s, their formation, FRW cosmological models, [[gravitational singularities]], etc. In the quantized version of such models with matter fields, [[Hawking radiation]] also shows up, just as in higher dimensional models.

== Action ==

A very specific choice of couplings and interactions leads to the CGHS model.

:<math>S = \frac{1}{2\pi} \int d^2x\, \sqrt{-g}\left\{ e^{-2\phi} \left[ R + 4\left( \nabla\phi \right)^2 + 4\lambda^2 \right] - \sum^N_{i=1} \frac{1}{2}\left( \nabla f_i \right)^2 \right\}</math>
where ''g'' is the [[metric tensor]], ''φ'' is the dilaton field, ''fi'' are the matter fields, and ''λ2'' is the [[cosmological constant]]. In particular, the cosmological constant is nonzero, and the matter fields are massless real scalars.

This specific choice is classically [[integrable]], but still not amenable to an exact quantum solution. It is also the action for [[Non-critical string theory]] and [[dimensional reduction]] 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]], and in the light-cone gauge {{nowrap|ds2 {{=}} − e2ρ du,dv}}, has the simple generic form
:<math>f_i\left( u, v \right) = A_i\left( u \right) + B_i \left( v \right)</math>,
with a factorization between left- and right-movers.

The Raychaudhuri equations are
:<math>e^{-2\phi} \left( - 2\phi_{,vv} + 4 \rho_{,v}\phi_{,v} \right) + f_{i,v}f_{i,v}/2= 0</math> and
:<math>e^{-2\phi} \left( - 2\phi_{,uu} + 4 \rho_{,u}\phi_{,u} \right) + f_{i,u}f_{i,u}/2= 0</math>.
The dilaton evolves according to
:<math>\left( e^{-2\phi} \right)_{,uv} = - \lambda^2 e^{-2\phi}e^{2\rho}</math>,
while the metric evolves according to
:<math>2\rho_{,uv} - 4\phi_{,uv} + 4\phi_{,u}\phi_{,v} + \lambda^2 e^{2\rho} = 0</math>.

The [[conformal anomaly]] due to matter induces a [[Liouville term]] in the [[effective action]].

=== Black hole ===

A vacuum black hole solution is given by
:<math>ds^2 = - \left( \frac{M}{\lambda} - \lambda^2 uv \right)^{-1} du\, dv</math>
:<math>e^{-2\phi} = \frac{M}{\lambda} - \lambda^2 uv</math>,
where ''M'' is the ADM mass.
Singularities appear at {{nowrap|uv {{=}} λ−3M}}.

The masslessness of the matter fields allow a black hole to completely evaporate away via [[Hawking radiation]]. In fact, this model was originally studied to shed light upon the [[black hole information paradox]].

== References ==
<references />

== See also ==

* [[dilaton]]
* [[general relativity]]
* [[quantum gravity]]
* [[RST model]]
* [[Jackiw–Teitelboim gravity]]
* [[Liouville gravity]]

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

[[Category:Quantum gravity]]
[[Category:General relativity]]

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