Infinite Rod in General Relativity

[t_r_theta_phi]

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?

 

[Markus Hanke]

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