Singularities in Graviton-Dilaton System: Their Implications on the PPN Parameters and the Cosmological Constant

Alternatives to Einstein's theory of general relativity can be distinguished by measuring the parametrised post Newtonian parameters. Two such parameters $\beta$ and $\gamma$, equal to one in Einstein theory, can be obtained from static spherically symmetric solutions. For the graviton-dilaton system, as in Brans-Dicke or low energy string theory, we find that if $\gamma \ne 1$ for a charge neutral point star, then there exist naked singularities. Thus, if $\gamma$ is measured to be different from one, then it cannot be explained by these theories, without implying naked singularities. We also couple a cosmological constant $\Lambda$ to the graviton-dilaton system, a la string theory. We find that static spherically symmetric solutions in low energy string theory, which describe the gravitational field of a point star in the real universe atleast upto a distance $r_* \simeq {\cal O} ({\rm pc})$, always lead to curvature singularities. These singularities are stable and much worse than the naked ones. Requiring their absence upto a distance $r_*$ implies a bound $| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$ in natural units. If $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$, and if $r_*$ extends all the way upto the edge of the universe ($10^{28} {\rm cm}$) then $| \Lambda | < 10^{- 122}$ in natural units.