Why can't gravitational force have repulsive counter part? The valleys in the space-time curvature signifies the existence of an attractive body which causes the curvature. So why can't we say that the hills in the fabric of space-time is a source of repulsive gravitational force? Can there be hills in the fabric at all?
 A: Repulsive Gravity would require Negative Energy and pure Negative Energy is not known to exist or at least as real particles.  Negative Energy would also have Negative inertia and so it would tend to move in the opposite direction from its momentum as well as accelerating in the opposite direction of a force being applied to it.  If there was a Planet made from Negative Energy with a mass that was exactly opposite of a Planet made from Positive Energy and with the exact same velocity then the pair of planets would both accelerate at the same rate forever without violating conservation of energy as the total energy of the system would be zero and the Planet with Negative Energy would chase the planet with Positive Energy.  This means that Negative Energy would produce perpetual motion without violating conservation of energy and momentum meaning that in order for Repulsive Gravity to be possible Perpetual Motion would also have to be possible.
A: Gravity is usually thought of as a weak boring force. But if you get spun up black holes and strong gravitational waves interacting, then you can get 'repulsion and attraction' effects happening out of purely gravitational forces.
So repulsive gravity does not exist in '0Hz' quiet spacetimes, but if you look at what happens when you shine a gravitational wave onto a rapidly spinning black hole, you will find that adjusting the frequency only slightly from the maximum interaction level allows you to push the hole away from you (when the bh absorbs a wave) or pull it closer (when the bh adds energy to the radiation beam via superradiance). 
See:
http://arxiv.org/pdf/1312.4529v2.pdf - figure 4. With a Gravitational Wave 
