When you say the word "antimatter", one of the first questions that the typical laymen want to know is whether antimatter gravitationally repels other matter or antimatter. They think it must be the deepest, the most puzzling, and the most open question about antimatter.While the answer is obviously "No", as I will also explain, the possibility that the answer is "Yes" must be irresistible. I don't know whether the belief that antimatter repels other stuff gravitationally is being pumped into the public conscience by some channels whose location isn't yet clear to me or whether people are just so instinctively misled by the prefix "anti-" that they believe that everything, including gravity, has to be flipped.
The word "laymen" has appeared in the text above. However, there's also a new preprint on the arXiv that actually proposes this ludicrous idea - that antimatter has repulsive gravity - to be tested at a relatively expensive experiment, the Ice Cube.
Can the new Neutrino Telescopes reveal the gravitational properties of antimatter?Even though the fonts and "PDF only" formats confirm that it is a typical crackpot paper, the author is even employed by CERN - believe me or not. This ludicrous preprint was also promoted by the physics arXiv blog.
Why the antimatter's gravity is surely attractive
Well, it's because the gravitational mass - that measures the strength of a gravitational field (with "plus" meaning "attractive") - is positive both for matter and antimatter. And because all objects, whether they are made out of something called "matter" or "antimatter" (which is just a convention!) or anything else, must behave in the gravitational fields in the same way. The latter principle is known as the equivalence principle and it is a fundamental pillar underlying Einstein's general theory of relativity.
Do we know that it is the case? You bet. Any approach - experimental or theoretical, at any level you may find appropriate - is enough to instantly settle the question.
First, consider the Earth's gravitational field. Grow an anti-apple. Is it going to accelerate towards the Earth, or away from the Earth? The former answer is obviously correct. The equivalence principle requires that in a freely falling elevator, you won't be able to figure out that the Earth is nearby: all the effects of gravity may be completely compensated by the acceleration. If an anti-apple had suddenly drifted in a different direction than the ordinary apples, the principle would be massively violated.
Do we empirically know that the equivalence principle is respected by such anti-apples? You bet. We should notice that for nearly 50 years, we've know that the protons and neutrons are not elementary particles. They're composed out of quarks - but also gluons and antiquarks. Electrons are elementary - at least so far.
The relative contribution of neutrons to the mass depends on the element and the isotope. And because neutrons contain a slightly different mass contribution from the antiquarks (antimatter!) than the protons (and obviously than the electrons which are 100% matter), where the difference is something of order 1%, it's clear that the hypothetical repulsive force acting on the antiquarks would manifest itself as different gravitational accelerations acting on different elements and isotopes.
The gravitational acceleration for different materials would have to differ by something comparable to 1 percent. However, it's experimentally known to be universal for all materials with the accuracy of 10^{-15} or so.
Clearly, the hypothesis that anti-apples are being repelled by the Earth is abruptly falsified by the evidence. We often talk about "weak signals" that are used as "statistical evidence" to support or disfavor various hypotheses. But in this case, the "signal" is 10^{13} times stronger than we needed. The hypothesis is just instantly dead. No doubts can survive. To say that it is not certain whether the anti-apples are gravitationally attracted to the Earth means to overestimate the uncertainty about this part of physics trillions of times.
In this case, we were considering the Earth - an object composed out of matter - and we probed its gravitational field by objects composed out of matter or antimatter. But you may want to create a gravitational object (source of gravity) from antimatter, too. Can you achieve some repulsion in this way?
To do so, construct an anti-world: I mean an anti-Earth. There are anti-people who live there. Some of them study anti-physics and a big part of the most important contribution to anti-physics was achieved by anti-Semites who keep themselves attractive by perspirants. ;-)
And of course, there are anti-men and anti-girls over there.
Fine. So will the anti-Earth repel the anti-apples that routinely grow on the anti-trees over there? Once again, the answer is obviously "No". The gravity will continue to be attractive. The fact that the positrons - the first known particles of antimatter, the anti-electrons - have a positive mass has been known since the very first moment when Dirac identified them with holes in the Dirac sea.
You know, the Dirac equation that controls the electron has positive-energy solutions as well as negative-energy solutions. The latter could produce anti-gravity. However, Nature loves to minimize its energy as much as it can. In particular, the vacuum is the lowest-energy state, the ground state.
To lower the energy, it's actually beneficial for Nature to occupy all the possible "boxes" with negative energies. This continuum where an electron fills each possible negative-energy state is known as the Dirac sea. The Pauli exclusion principle only allows you to place one fermion in one box (i.e. one state, as specified by a complete set of quantum numbers such as 3 components of the momentum and the binary up/down information about its spin).
The only way how you can modify the Dirac sea is to create a hole, i.e. steal one electron with negative charge and negative energy. Relatively to the vacuum where the Dirac sea is filled, it's like adding "minus one negative-energy electron". It's not hard to change the signs to see that such a hole will add a positive charge - because it is a "minus electron" - but it will also add a positive energy/mass - because the negative energy we started with was reverted as well. You know, minus times minus equals plus.
The positron has therefore a positive energy - and via E=mc2 and the equivalence principle, it has a positive gravitational mass and a positive inertial mass, too.
In this "classic" form, the Dirac sea concept only applies to fermions. However, the conclusions surely apply to bosons, too. There are many ways to see that the anti-apples fall down on - and kept bound to - the anti-Earth. One of them is the CPT-symmetry (or the approximate C- or CP-symmetries). These operations replace matter by antimatter. Because these operations are symmetries (at least the CPT-symmetry is exact), they imply that the fact that the apples are bound to the Earth has to imply that the anti-apples are bound to the anti-Earth, too.
An anti-planet that would repel its own anti-apples would be qualitatively different from our planet - so it couldn't be related by any symmetry.
Genuine negative mass
So the antimatter has the opposite charges but it has a positive energy and a positive mass (both inertial and gravitational ones). I hope that one doesn't have to spend more time on this totally trivial point which most of us understood before they were ten years old. Only people with a profound misunderstanding of basic maths - including the point that "-1 times -1 equals +1" - may have doubts.
But can you imagine a genuinely negative matter? Can it exist? Can there be a repulsive gravity between two localized objects?
The answer is almost certainly "No" in any viable world, too. Why? To get a negative energy/mass, the original starting point (the space without matter, i.e. the vacuum) has to fail to be the lowest-energy state. So it becomes unstable.
You may imagine a non-vacuum where the Dirac sea is not occupied and where you can add genuinely negative-energy electrons. However, it is not just a "label" when we say that the vacuum isn't the lowest-energy state. Such a situation has dramatic physical consequences. This kind of space will spontaneously decay, creating pairs of electrons with negative and positive energies out of nothing - because it violates no conservation laws. Such a non-vacuum would behave very differently than our peaceful vacuum.
Obviously, these things can't exist in a world where the vacuum is at least as stable as the vacuum we observe in our Universe. Let's not waste too much time with this point, either.
However, if there were tachyons, the existence of negative-energy excitations would be inevitable. After all, the tachyons' energy-momentum vector is, by definition, spacelike. And there is no "canonical" - or Lorentz-invariant - separation of the spacelike area into positive-energy and negative-energy regions. After all, whether the temporal component of a spacelike vector is positive or negative depends on the reference frame.
If tachyons existed around this vacuum, they would imply an instability, too. Since the very discovery of special relativity in 1905, it has been understood that the tachyons were a real problem and they shouldn't exist. They would allow you to travel faster than light - and, by switching from one inertial frame to another, to kill your grandfather before he had sex with your grandmother.
However, only around 2000, physicists began to look at tachyons "truly rationally" - as a source of instability of the theory. In the past, an instability (often called differently) would make you scream "disaster!" and you would stop looking at the theory. This attitude was mostly right but it wasn't quite rigorous. After the Ashoke-Sen-inspired tachyon mini-revolution in string theory a decade ago, physicists became calmer.
They know that it's an instability but an instability does not have to end by an uncontrollable catastrophe. Quite on the contrary, we must impartially and calmly asks the question where the instability leads. And in many cases, the "decay" of the Universe (or object) that is implied by the existence of the tachyons actually leads to another, stable state that no longer allows any tachyons.
For example, the open-string tachyonic states living on (unstable) D-branes were understood to be the Cassandras of the decay of this very D-brane that leads to a complete "evaporation" of the D-brane away from space. The energy difference between the two states - with and without the D-brane - could therefore be calculated, translated to a difference in the tachyon's potential energy, and verified by more detailed calculations in string theory.
Similar explanations were found for closed-string tachyons that were localized - e.g. those in the twisted sectors of orbifolds, and so on (APS which stands for Adams-Polchinski-Silverstein). The closed-string tachyons that live in the bulk (e.g. in closed-string bosonic string theory) lead to an instability that is so dramatic that its endpoint is still considered to be unbounded or catastrophic - or to say the least, if such an evolution ever becomes controllable, it's not understood what it is.
At any rate, these Universes are very different from ours. Our Universe can't admit any negative-energy or negative-mass objects.
One could also try to change the sign of the Einstein-Hilbert term in the action of general relativity and to try to make all gravitational interactions repulsive. In my opinion, this would also lead to an inconsistent theory - or, to put it mildly, a theory that can't resemble our Universe even when it comes to other issues. There are many bizarre sign flips that would be implied by the opposite sign of the Einstein-Hilbert action.
One of them would be that the gravitational waves would carry a negative energy (unlike ordinary mass whose energy's sign was kept unchanged). Note that the stress-energy tensor of the gravitational waves is derived from (and proportional to) the Einstein-Hilbert term and its sign was flipped. Again, this would lead to an instability - a spontaneous creation of matter with the negative-energy gravitational waves etc. There probably exist easier ways to show that such a theory would be sick (and can't ever emerge as a limit of string theory, either).
To summarize, all the signs are dictated by consistency. In particular, the gravitational force between any pairs of localized objects that are embedded in a nearly flat surrounding space always has to be attractive and there is no realistic doubt that this is really the case in our Universe - and all worlds that may or may not exist elsewhere in a hypothetical multiverse.
And that's the memo.
P.S.: As argued above, there are no negative-mass objects in the real world; antimatter is just a form of matter whose mass is positive. But we may still imagine a world where the instability issues are solved in some way and the negative masses exist.
In such a world, two positive-mass objects would attract - just like in our world. And two negative-mass objects would repel - producing inequivalent trajectories (hyperbolae; elliptical paths would be impossible).
But it's interesting to ask what a negative-mass object near a positive-mass object would do e.g. if the magnitudes of the masses were equal. The positive-mass object would attract everything - while the negative-mass source has to repel everything. Who would win? What would be the motion at the end?
Well, both would win. Physical laws always hold. If a situation is confusing to you, it doesn't mean that the laws of physics may be relaxed.
The resulting motion would be such that both objects would accelerate in the same direction - in the direction from the negative-mass object to the positive-mass object. Such an acceleration towards the speed of light wouldn't violate any momentum conservation law because the momentum would cancel: for positive and negative masses of the same magnitude, the momenta are opposite in direction and they cancel.
If the two objects have masses of opposite signs and different magnitudes, the motion would be a composition of a uniform motion of the center-of-mass coordinates and a hyperbolic/elliptic motion of the relative coordinates. In fact, the case of equal-magnitude masses with opposite signs is the only special case because the center-of-mass location is ill-defined: the denominator of the expression for the center-of-mass location, namely the total mass, is equal to zero!
I think that the people who have problems with these calculations must also have problems with the basics of maths - such as negative numbers. They don't really "believe" negative numbers. They either want to dismiss them regardless of any evidence, or they need to "test them". Of course, such a lack of "belief" is an even more severe barrier preventing one from understanding Nature than the lack of "belief" in complex numbers - because negative numbers are much more elementary.
People may find signs to be a little puppets that may questioned and attacked at every step.
But Nature never allows signs to be open to any question. Its fundamental equations involve real or complex numbers that can be both positive and negative - and they have to be computed with properly, according to the laws of maths. Some signs are a matter of convention - e.g. which charges are considered positive and which charges are considered negative. But some signs are physical and they don't depend on any conventions. For example, the attractive force is always qualitatively different from a repulsive force. There is no symmetry between them and Nature, unlike the humans, never confuses the two cases.
