$<ϕ^2>$ in the Spacetime of a Cylindrical Black Hole

The renormalised value of $<ϕ^2>$ is calculated for a massless, conformally coupled scalar field in the Hartle-Hawking vacuum state. This calculation is a first step towards the calculation of the gravitational back reaction of the field in a black cosmic string spacetime which is asymptotically anti-DeSitter and possesses a non constant dilaton field. It is found that the field is divergence free throughout the spacetime and attains its maximum value near the horizon.

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