# Negative Gravity Fields and Time Dilation [duplicate]

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.

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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.

• Thanks for the answer I will check and see if I can find more info on that experiment, is it part of ATLAS? By the way, why do we need a planet-size chunk of dark matter? Can we not measure the effects on smaller volumes of matter? – M Barbosa Jun 12 '16 at 23:11
• @MBarbosa: It's called Alpha: alpha.web.cern.ch and it is a rather complex experiment. Just making anti-hydrogen is hard, capturing it at zero momentum is even harder and then shielding it in a magnetic trap from all other forces so that one can measure the minute force of gravity on single atoms is rally tough. The reason why it's being done at CERN, right now, I believe, is because they have the ideal anti-proton/positron sources to carry this out, so it's kind of a side effect of LHC's main mission. – CuriousOne Jun 12 '16 at 23:15
• Wow those are some seriously mind blowing results. Thanks again. – M Barbosa Jun 12 '16 at 23:24
• It would violate CPT invariance where antiparticles are simply inverting C. Other arguments as to why it is unlikely, though I agree not disproven. Also notice that photon is its antiparticle, and if antimatter is antigravity then that would need some non trivial explanation. But experiment will tell. – Bob Bee Jun 13 '16 at 1:30
• @BobBee: I don't see this working, either, but since it's an untested sector of the equivalence principle, the experiment needs to be done. For what it's worth, my money is on gravity being the same for matter and antimatter. – CuriousOne Jun 13 '16 at 2:21

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.

• I don't quite see the problem in the last case. What is being proposed are gravitational charges, similar to electromagnetic charges. The latter clearly do not violate momentum conversation and neither would an analogous Newtonian gravitational force with variable charge. General relativity, which relies on the equivalence principle, would simply be invalid and so one can't argue with it if we were to find any kind of deviation from Newtonian gravity in the weak field case. – CuriousOne Jun 13 '16 at 3:04
• I understand there is a solution with negative mass and no CTC, but can't remember what other conditions. I'll look for it and post – Bob Bee Jun 13 '16 at 18:49
• Anyway, the Newtonian case has the m in ma, and m is clearly the inertial mass. If inertial mass does not change there is no issue. The idea was change only the gravitational mass, on the right side of the equations. – Bob Bee Jun 13 '16 at 18:53
• If you think about it the total mass in the grey cloak is zero, and this accelerates the outside mass. This is not like changing an electric charge, for remember there is the ma part on the left. – Lawrence B. Crowell Jun 13 '16 at 23:01