Negative Gravity Fields and Time Dilation Question: Some physicists believe that it is possible to have negative mass. My questions is would this negative mass create a negative gravity field, exerting a negative force on positive masses? And if so, would the negative gravity field cause an opposite effect on time, possibly slowing it down?
My attempt: I feel like it would exert a negative force because of the Universal Law of Gravitation. So we have a negative force exerted from a negative mass ($m_1$) on a positive mass ($m_2$):
$$ F_{+mass} = G \frac{(-m_1) m_2}{r^2} \vec{r} $$
This would mean that the gravity vector field is now pointed away (I think) from the center of gravity rather than towards it, aka negative gravity.
Now I am not familiar enough with the time dilation equations to take a shot at that but any search I have done comes up empty on that matter. But intuitively, it seems to me that time dilation would also work backwards.
 A: Negative mass and by corollary negative energy have some strange consequences. If you have a set of masses in a region of space with a volume $V$ the density of energy is $\rho = \sum_im_ic^2/V$ This defines $T^{00} = \rho$ component of the stress energy tensor. The Hawking-Penrose energy conditions are that $T^{00} \ge 0$. Violations of this creates various strange configurations of closed timelike curves or time travel, wormholes, and so forth. The anti-de Sitter spacetime has this feature.
To narrow the attention to Newtonian mechanics we look at Newton's second law of motion with gravity for two masses $m_1,~m_2$ where $m_2 < 0$
$$
m_1\vec a_1 = -\frac{Gm_1m_2(\vec r_2-\vec r_1)}{|\vec r_1 - \vec r_3|^3} 
$$
$$
m_2\vec a_2 = -\frac{Gm_1m_2(\vec r_1-\vec r_2)}{|\vec r_1 - \vec r_3|^3} 
$$
Now divide through by the masses to get the acceleration vectors
$$
\vec a_1 = -\frac{Gm_2(\vec r_2-\vec r_1)}{|\vec r_1 - \vec r_3|^3} 
$$
$$
\vec a_2 = -\frac{Gm_1(\vec r_1-\vec r_2)}{|\vec r_1 - \vec r_3|^3} 
$$
If the two masses were positive the acceleration vectors would clearly be in the opposite direction. However, we see the positive mass $m_1$ is repelled by $m_2$ and that $m_2$ is attracted to $m_1$.
If we were to make $|m_1| = |m_2|$ the two would accelerate off and mutually race off to infinity, or in a relativistic setting approach the speed of light. However, if the two masses are equal in magnitude then in a sense you have nothing accelerating away; the net mass is zero. This still leaves us with a problematic issue though. Suppose you have $|m_1| = 2|m_2|$, and we partition the positive mass into two parts. We connect a very stiff rod with negligible mass to the two halves of $m_2$. One of these halves is close to the negative $m_2$ and we think of this rod being long enough so there is negligible gravitational interaction between this half of $m_1$ and $m_2$. Now finally attach a stiff rod between the close half of $m_1$ and $m_2$, and cloak them with a shroud. You now have a situation where one can see the $m_2/2$ mass connected to a black box that is mysteriously exerting a force to accelerate it away. The configuration is drawn below.

A: One has to differentiate between negative inertial mass, which is very unlikely, and a negative gravitational mass charge, which is what you are looking at. The latter has not been ruled out for antimatter. More generally, we have not confirmed, yet, that the equivalence principle holds for antimatter.This leaves room for such hypotheses and it requires precision experiments to rule them out. One such experiment is being carried out at CERN, right now. 
As far as time dilation is concerned, the situation is not so clear. If we follow the line of reasoning that time dilation is being measured by an exchange of photons, how do we translate a negative mass that is linked to the charge conjugation of antimatter to the dynamics of photon that are their own anti-particles? Based on observation ordinary matter doesn't differentiate between two kinds of photons, otherwise a gravitational potential would lead to an energy split like a magnetic field that acts on spins. If antimatter were to gravitationally blue-shift photons, i.e. lead to the opposite time dilation effect as , then how do we fit the fact that photons, which don't have antiparticles, behave like "normal matter" gravitationally? To me this is a pretty strong reason to "dislike" the idea of gravitational charge, but, of course, the experiment will decide. 
Even if we find antimatter associated with a negative gravitational charge, by the way, it's unlikely that we could confirm this time dilation effect. For that we would need a planet size chunk of antimatter... that's not cheap. 
