# Can we be sure that leptons and baryons gravitationally attract each other?

Covid has renewed my interest in fundamental physics. But I notice that my knowledge has rusted a little bit over the years. So please bear with me.

Since unification of gravitation and quantum field theory is still an open topic, I became stuck with the following problem I found interesting:

According to the equivalence principle, inertial and gravitational mass are proportional. But, since this principle has evolved from macroscopic observations, I am not so sure whether it excludes negative gravitational masses (i.e. proportionality factor including a minus sign). More specifically, does any physical law (not just gravitation) exclude the possibility that gravitational mass and charge always have the same sign? To my knowledge, positivity of mass is always just silently introduced, without justification.

If there was no such constraint, electrons could be the sources of a negative gravitational potential (or a corresponding metric). However, since other electrons had the same signed mass, electrons would still attract each other. Similar would hold true for protons with a positive gravitational potential, but still attracting each other. The only difference, as far as I could see, would be a repulsive gravitational force between electrons and protons, which would be weak by a factor of the electron/proton mass ratio. Neutrons, if considered a compound of protons and electrons united by the process of electron capture would present a reduced gravitational potential just like a hydrogen atom would.

Astronomical objects then would be sources of a gravitational field that is reduced by ~1/2000th for the mass of the electrons contributing a negative fraction to the overall gravitational field. But, since the ratio of electrons to protons is always the same, the reduction would be everywhere the same, and hence, could be already absorbed into the gravitational constant / the proportionality of inertial and gravitational mass.

Last but not least, anti-matter would then repel ordinary matter due to the opposite charges of their baryons (I have seen a paper https://arxiv.org/abs/1103.4937, that derives such a repulsion from assuming CPT invariance for GRT, but I guess that is a different story). There would be probably (I haven't checked) be anti-Schwarzschild solutions of the Einstein equations, and anti-Blackholes would also repel ordinary Blackholes.

Ist there any fallacy in my thinking? Or is there any experiment that can rule out the above? I guess not, since measuring the gravitational effect on electrons fails for reasons of accuracy/weakness of gravity vs. electromagnetism. But it sure would make a difference on the theory level, wouldn't it.

Edit: I am a bit unsure about my original (speculative) conclusion, that electrons would still attract each other, in case of their mass being negative; from the point of view of newtonian mechanics we could probably argue that a mass sign would enter quadratically into the gravitational force of like charges, resulting in an overall attractive gravitation between electrons (inertial mass must always remain positive if Lorentz' force law shall not be compromised) and a repulsive gravitation between electrons and protons; but from the point of view of GRT, the weak equivalence principle would dictate that gravitation is just geodesic motion in the external field (in this case of an electron), regardless of what kind of other charged particle falls into this field; so if electrons had negative gravitational mass, their metric and the corresponding Levi-Civita connection would be somehow inverted, but it would act the same on other electrons and protons; meaning if this metric causes repulsion, it would repel protons as well as other electrons; but then again, the question remains whether the weak equivalence principle (which has been experimentally confirmed only for makroscopic matter) could also be confirmed for elementary particles...

• The gravitational interaction of antimatter with matter or antimatter has not been conclusively observed by physicists, but it is a topic of ongoing research at CERN and other places - see en.wikipedia.org/wiki/Gravitational_interaction_of_antimatter. Jan 31, 2021 at 21:18
• You would really break our understanding of gravity and make things much more complicated. Maybe Nature is like that, but there is no reason to think so (it is definitely not mainstream physics). Note that there are fundamental microscopic particle physics arguments to believe in the equivalence principle, for example Steven Weinberg argued that the equivalence principle follows from having unitary and Lorentz invariant scattering amplitudes involving the exchange of a massless spin-2 particle (the graviton): sciencedirect.com/science/article/abs/pii/0031916364903968 Jan 31, 2021 at 21:28
• Here are some questions about antimatter gravity, with useful links: physics.stackexchange.com/q/534289/123208 & physics.stackexchange.com/q/139545/123208 & physics.stackexchange.com/q/589812/123208 & physics.stackexchange.com/q/241060/123208 Feb 1, 2021 at 7:55
• @oliver You can find versions online for free, for example Section 2.7 of arxiv.org/abs/1712.10020. But it is a quite sophisticated argument, and my reason for bringing it up is not so much to say you need to follow all the details. My main point is that there are very strong theoretical (in addition to observational) reasons that the equivalence principle should apply to interactions between elementary particles; what you are asking for may sound simple but actually runs very deeply against our understanding of gravity. (...) Feb 1, 2021 at 14:34
• (...) The argument does not rely on a full quantum theory of gravity, but rather gravity as an effective field theory valid at low energies, which is a well-established idea (arxiv.org/abs/gr-qc/9512024) that is regularly used to compute observable quantities in gravitational-wave experiments (arxiv.org/abs/1601.04914). My main reason to bring this up is not necessarily that you read all these papers, but to try to impress on you that people have thought deeply about this before and what you are asking for is very hard to fit in with everything else we know about physics. Feb 1, 2021 at 14:35

Yes, we can be sure that leptons and baryons attract each other. Eotvos experiments and other tests of the equivalence principle give null results to about one part in $$10^{11}$$ these days. Under your hypothesis, different chemical substances would have different gravitational accelerations, and therefore we would get violations on the order of one part in $$10^4$$.
• In my original argument ("neutron = proton+electron+anti-neutrino") I have totally overlooked the binding energy 0.78 MeV, which is already of the order $10^{-4}$. So even if the (hypothetically anti-gravitating) electrons would partly cancel the gravitational mass of protons in the same way as the gravitational neutron mass would be reduced (internally), there would still be a remaining chemical composition effect due to the binding energy 0.78 MeV captured inside the neutron. Feb 2, 2021 at 8:36