This theory is basically that gravity is similar to other forces in that it's not solely attractive, but things of opposite "gravitational charge" (i.e. particles and antiparticles) repel.
If that holds, then an ambient gravitational field in an area would presumably be strengthened by virtual particle pairs aligning their gravitational dipole with that of the ambient field, quite neatly explaining the anomalous observations that brought about the dark matter theory in the first place.
There are more recent observations that this new theory doesn't yet explain, but it doesn't seem an insurmountable challenge.
Additionally, this theory makes one very testable hypothesis - that antimatter is oppositely-charged, gravitationally, to normal matter. If that's not the case, we should be able to determine that fairly soon (a few years, perhaps).
Why wouldn't we see this effect on earth, and have taken it into account when determining the strength of earth's own gravity?
We could be observing the impact of this without exactly realizing it. I won't speculate too much on this because I haven't read enough about it but it might make more sense in light of my next answer.
Would we see this effect with electric fields as well?
We absolutely see this effect with electric fields. In the last century we've grown to understand that in electromagnetism we don't simply deal with isolated electrons and positrons. These particles are actually surrounded with clouds of virtual electrons and photons which have an impact on the field (this is a simplified view but it's enough along the right lines to give you the right impression if you haven't studied physics). When we talk about the field around an electron we take this into account by assuming that all these contributions sum to give us the field that we actually observe. If you want to read more about this then http://en.wikipedia.org/wiki/Quantum_field_theory#Renormaliz... is a good starting point.
Could differently oriented electric and gravitational fields compete to orient a given dipole?
Yes, all relevant fields will have an impact on what's happening. I don't want to speculate too much beyond my knowledge though and when it comes to quantum gravity we're already there.
I know that this wasn't the deepest contribution but hopefully it will help shed some light to someone...
One thing I don't understand: wasn't there just discovered a ring of antimatter circling around the Earth? Wouldn't it just disperse into space if antimatter caused antigravity? Stupid question, I know, but just something that's beyond my comprehension.
No, those are charged particles interacting with the magnetic field of the earth. Gravity is such a ridiculously weak force that you can safely ignore it in situations like that.
Tiny amounts of antimatter were just discovered circling around Earth. The anti-grav effect on the antimatter particles is swamped by the effects of the Earth's gravity and magnetic field.
According to this theory, it would be the magnetic field alone--not the gravitational field--that is overwhelming the antigravity effect. After all, if these particles had an opposite "gravitational charge", then their experience of gravity wouldn't be attractive but would be equal in magnitude but opposite in orientation. Hence "anti"gravity.
The whole 'virtual photons' thing is a physical interpretation of the mathematics, but it is a mistake to conclude that this is how it 'actually' is. There is a mathematical model that works. I don't think we know how it 'actually' is. There is every reason to doubt that this sea of 'virtual photons' has any physical reality in an independently verifiable way. Occam's razor doesn't favor solutions that involve multitudes of invisible, unmeasurable, particles popping in and out of existence.
In the end, all we have are our mental models and experiments to back them up.... we'll never know what anything 'actually' is - all we can do is model things to whatever degree of accuracy we can.
A construction worker doesn't need to go around thinking about his hammer as being made up of mostly empty space, and of the electromagnetic forces that give it it's properties, it's a hammer! We all know what it "IS" right?
But it's not that. It's not even what we think it is. It's not even what the best physicists think it is.... and in the end, all we're left with are incomplete mathematical models, or models that are only accurate on a certain scale.
BTW - how do we then explain hawking radiation, if virtual particles are "just math" and not real?
Well, first and foremost: the existence of Hawking radiation hasn't been experimentally verified. There's no direct evidence (measuring radiation that could have no other source than the evaporation of the black hole) nor any indirect evidence (black holes losing mass at a rate that cannot be explained unless Hawking radiation is included). So although it seems to fit nicely into the picture, it may not exist at all.
A second type of answer is: I don't know and I admit that the physical picture of virtual particles hopping in and out of existence indeed nicely includes an explanation of Hawking radiation. However, I'm reminded of the texts that illustrate how many physical phenomena were considered to be satisfactorily 'explained' within, for instance, the aether model, even though it later turned out the model was fundamentally flawed and all of those physical interpretations were hogwash.
A third type of answer is: sometimes things simply aren't what they seem. For instance, we can describe phonons as if they're particles: they are described by the same statistics as 'real' particles are. However, they decidedly aren't particles. They aren't properties of individual particles either: they only exist in macroscopic, well-ordered, amounts of material. You could look at all individual atoms one by one and never arrive at phonons as partial explanations for their movements and properties. For that, you'd have to look at things at a different level.
That was E.E. "Doc" Smith's assumption in the Lensman series when "negaspheres" were developed (planetary scale antimatter objects), in Grey Lensman written circa 1939. They had an "opposite 'gravitational charge'", a tractor beam would repel them while a pressor beam would pull them in, etc. It would be amusing if this turned out to be true (well, ignoring the yet to be if ever invented beam technology).
If that holds, then an ambient gravitational field in an area would presumably be strengthened by virtual particle pairs aligning their gravitational dipole with that of the ambient field, quite neatly explaining the anomalous observations that brought about the dark matter theory in the first place.
There are more recent observations that this new theory doesn't yet explain, but it doesn't seem an insurmountable challenge.
Additionally, this theory makes one very testable hypothesis - that antimatter is oppositely-charged, gravitationally, to normal matter. If that's not the case, we should be able to determine that fairly soon (a few years, perhaps).