Exact Black Hole and Cosmological Solutions in a Two-Dimensional Dilaton-Spectator Theory of Gravity

Exact black hole and cosmological solutions are obtained for a special two-dimensional dilaton-spectator ($\varphi -\psi$) theory of gravity. We show how in this context any desired spacetime behaviour can be determined by an appropriate choice of a dilaton potential function $V(\varphi )$ and a coupling function'' $l(\phi)$ in the action. We illustrate several black hole solutions as examples. In particular, asymptotically flat double- and multiple- horizon black hole solutions are obtained. One solution bears an interesting resemblance to the $2D$ string-theoretic black hole and contains the same thermodynamic properties; another resembles the $4D$ Reissner-Nordstrom solution. We find two characteristic features of all the black hole solutions. First the coupling constants in $l(\phi)$ must be set equal to constants of integration (typically the mass). Second, the spectator field $\psi$ and its derivative $\psi^{'}$ both diverge at any event horizon. A test particle with spectator charge" ({\it i.e.} one coupled either to $\psi$ or ${\psi }^{{}^{\mathrm{\prime }}}$), will therefore encounter an infinite tidal force at the horizon or an ``infinite potential barrier’’ located outside the horizon respectively. We also compute the Hawking temperature and entropy for our solutions. In $2D$ $FRW$ cosmology, two non-singular solutions which resemble two exact solutions in $4D$ string-motivated cosmology are obtained. In addition, we construct a singular model which describes the $4D$ standard non-inflationary big bang cosmology ($big-bang\to radiation\to dust$). Motivated by the similaritiesbetween $2D$ and $4D$ gravitational field equations in $FRW$ cosmology, we briefly discuss a special $4D$ dilaton-spectator action constructed from the bosonic part of the low energy heterotic string action and