Does spinning a massive sphere create extra gravity pull?

[Slingshot]

Does spinning a massive sphere at high speed would somehow create extra gravity pull (or some sort of phenomenon perceived as such) added to it's initial gravity force (rest state gravity/not spinning state gravity)?

I came to the assumption that accelerating the speed of an object would result of the increase of it's mass (relativistic mass). By increasing it's mass, wouldn't it be increasing it's gravitational force?

While obviously a spinning sphere is not traveling to an outward direction, it is none the less moving in a rotational manner. Shouldn't having a high angular velocity create more gravity force near the equatorial region of the sphere (since it is also matters that are traveling at a high velocity) and less to the poles or to the center within?

Yes I am aware that I might be totally off base due to probably a lot of misconception, so if you can answer and explain with clear example and analogy instead of a simply "no you don't make sense" would really help.
But thank you in advance.

 

[bcrowell]

You're sort of on the right track. The kinetic energy does add to the equivalent mass. However, you also get effects like the geodetic effect and frame dragging, which you can find out about by googling, WP, etc. The basic principle is that in GR, the source of gravity is not the scalar mass-energy but the stress-energy tensor. Therefore it's not sufficient just to use E=mc² to convert the spinning Earth's KE into an equivalent amount of mass.