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?
3$\begingroup$ To put an experimental spin on it: no repulsive gravity has been observed and the accepted theory is modeled in such a way as not to allow for it. The theorists can explain to you how that works, but the gist is that one can only make theory for what has been observed. Having said that, there is an anti-matter gravity experiment going on at CERN at the moment: alpha.web.cern.ch. There are some people who give it a non-zero chance that anti-matter and ordinary matter may have a repelling gravitational interaction... and that would be one of the most fundamental discoveries, ever. $\endgroup$– CuriousOneDec 28, 2015 at 2:42
1$\begingroup$ You can say. Then it's a theory with no experimental or theoretical reason to be true (but no prove it is false either). That just doesn't get you anywhere. You can also say there are Fixions (xkcd.com/1621)... $\endgroup$– AganjuDec 28, 2015 at 3:22
$\begingroup$ @CuriousOne : Alan Guth introduces repulsive gravity in the inflation : slide 5 $\endgroup$– user46925Dec 28, 2015 at 4:38
$\begingroup$ @igael: That's one model under test, not observation. I don't mind models like that, I just don't want to give the wrong impression about what we know fairly well (none of which gets us beyond general relativity, unfortunately) and what we are still working on. We shall see if inflation (and what flavor of it) stands up to the tests of time. (I do like eternal inflation models quite well, by the way, so it's nothing personal.). $\endgroup$– CuriousOneDec 28, 2015 at 5:52
$\begingroup$ Related: physics.stackexchange.com/q/11542/2451 and links therein. $\endgroup$– Qmechanic ♦Dec 28, 2015 at 6:39
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.
$\begingroup$ Yes, except in the early second. The very mainstream inflation theory postulates explicitely repulsive gravity. $\endgroup$– user46925Dec 28, 2015 at 4:20
$\begingroup$ @igael Standard inflation is like dark energy: you can draw analogies with a fluid with negative pressure, but still the density is strictly positive. This violates some energy conditions but not the most important ones. $\endgroup$– user10851Dec 29, 2015 at 4:16
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