Can they exist ?? Like Magnetic Field lines and Electric Field lines
I think you might be confused by what "exists" means in this context. http://en.wikipedia.org/wiki/Field_line" [Broken] are often physically significant, they are still just mathematical tools. In that sense, none of them "exist". But the gravitational field isn't any different, in terms of 'having' field lines, from the EM field, or any vector field, really.
Like an Electric field connects one point to the charge to the other..in that context , would a Grav. field line connect any two centre's of gravity?
Also it will always act radially inwards for a particle , so is the geometry similar to other Field lines?
Gravitational field lines "connect" massive bodies as much as electrical field lines do. Field lines are mathematical, they are not physical.
The geometry of the gravitational and EM fields are similar, but there are differences. They are both 'radial' though, in the sense that they come from the mass/charge of the particle.
Magnetic and electric fields lines are a model for describing forces which act at a distance, and since gravity acts at a distance it can be described by the same model.
The gravitational field lines from a point mass look identical to the electric field lines of a negative point charge.
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