CGHS model

The Callan–Giddings–Harvey–Strominger model or CGHS model^[1] in short is a toy model of general relativity 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

. 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

also shows up, just as in higher-dimensional models.

Action

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

${\displaystyle S={\frac {1}{2\pi }}\int d^{2}x\,{\sqrt {-g}}\left\{e^{-2\phi }\left[R+4\left(\nabla \phi \right)^{2}+4\lambda ^{2}\right]-\sum _{i=1}^{N}{\frac {1}{2}}\left(\nabla f_{i}\right)^{2}\right\}}$

where g is the metric tensor, ${\displaystyle \phi }$ is the dilaton field, f_i 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

, which are entirely different models.

The matter field only couples to the causal structure, and in the light-cone gauge ds² = − e^2ρ du,dv, has the simple generic form

${\displaystyle f_{i}\left(u,v\right)=A_{i}\left(u\right)+B_{i}\left(v\right)}$,

with a factorization between left- and right-movers.

The Raychaudhuri equations are

${\displaystyle e^{-2\phi }\left(-2\phi _{,vv}+4\rho _{,v}\phi _{,v}\right)+f_{i,v}f_{i,v}/2=0}$ and

${\displaystyle e^{-2\phi }\left(-2\phi _{,uu}+4\rho _{,u}\phi _{,u}\right)+f_{i,u}f_{i,u}/2=0}$.

The dilaton evolves according to

${\displaystyle \left(e^{-2\phi }\right)_{,uv}=-\lambda ^{2}e^{-2\phi }e^{2\rho }}$,

while the metric evolves according to

${\displaystyle 2\rho _{,uv}-4\phi _{,uv}+4\phi _{,u}\phi _{,v}+\lambda ^{2}e^{2\rho }=0}$.

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

Black hole

A vacuum black hole solution is given by

${\displaystyle ds^{2}=-\left({\frac {M}{\lambda }}-\lambda ^{2}uv\right)^{-1}du\,dv}$

${\displaystyle e^{-2\phi }={\frac {M}{\lambda }}-\lambda ^{2}uv}$,

where M is the ADM mass. Singularities appear at 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.

See also

• dilaton
• general relativity
• quantum gravity
• RST model
• Jackiw–Teitelboim gravity
• Liouville gravity

References