I Metric for Infinite Rod in Gen. Relativity

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What is the metric for the spacetime around an infinitely thin, infinitely long, uniform rod? Could it be written in the form

ds2 = A(r)dt2 + B(r)dr2 + C(r)dh2 + r2dθ2

where h is the coordinate along the rod and r is the radial coordinate, or would it be something more complicated?
 
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Maybe I'm wrong, but my first impulse would be to model this as a special case of a Tipler cylinder, with small radius and zero angular momentum, in which case your ansatz looks pretty good. Let's wait and see what the experts say :smile:
 

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