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Newtonian mechanics seems to allow for both positive and negative gravitational mass as long as the inertial mass is always positive. The situation is analogous to electrostatics but with the opposite sign. Two positive masses or two negative masses are attracted to each other whereas one positive and one negative mass repel each other.
General relativity says gravitational and inertial mass are the same thing through the equivalence principle. This has been confirmed experimentally to a very high degree of accuracy, though not for very small masses and only for normal matter.

Antimatter is known to have positive inertial mass from observing the trajectories of particles in electric or magnetic fields. Presumably it is also known that the $m$ in the famous $E=mc^2$ is positive. The gravitational mass of elementary particles is currently too small to measure, but is it possible that antimatter could have negative gravitational mass - or is this absolutely precluded in general relativity?

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    $\begingroup$ As you yourself explained, “General relativity says gravitational and inertial mass are the same thing through the equivalence principle”. So why in the next paragraph do you ask “is it possible that antimatter could have negative gravitational mass - or is this absolutely precluded in general relativity?” $\endgroup$ – G. Smith Oct 27 at 4:38
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    $\begingroup$ More on mass of antimatter. $\endgroup$ – Qmechanic Oct 27 at 4:46
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    $\begingroup$ I think this answer of mine to a possible duplicate is relevant ,has links to CERN experiments physics.stackexchange.com/questions/352563/… $\endgroup$ – anna v Oct 27 at 4:58
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    $\begingroup$ I think this exact question has been studied, though not extensively, and is known as Dirac-Milne cosmology. I believe it has had some success in describing the universe without requiring Dark Matter or Dark Energy or even Inflation, but is also fraught with many difficulties including a violation of the Equivalence principle when negative mass is included. See here for a good review, and also here. $\endgroup$ – Philip Oct 27 at 5:22
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    $\begingroup$ @G. Smith I'm wondering, for example, if you could put together a version of general relativity with inertial mass equal to the absolute value of gravitational mass. $\endgroup$ – Roger Wood Oct 27 at 5:33
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A long comment:

AEGIS is a collaboration of physicists from all over Europe. In the first phase of the experiment, the AEGIS team is using antiprotons from the Antiproton Decelerator to make a beam of antihydrogen atoms. They then pass the antihydrogen beam through an instrument called a Moire deflectometer coupled to a position-sensitive detector to measure the strength of the gravitational interaction between matter and antimatter to a precision of 1%.

A system of gratings in the deflectometer splits the antihydrogen beam into parallel rays, forming a periodic pattern. From this pattern, the physicists can measure how much the antihydrogen beam drops during its horizontal flight. Combining this shift with the time each atom takes to fly and fall, the AEGIS team can then determine the strength of the gravitational force between the Earth and the antihydrogen atoms.

Also new experiments are in the process. In total there are three experiments at CERN to measure the effect of the earth's gravitational field on antimatter. Patience.

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    $\begingroup$ great to hear these measurements are being seriously pursued. I'm surprised. it sounds like most physicists are convinced the particles will fall, not rise :-( $\endgroup$ – Roger Wood Oct 27 at 7:55
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    $\begingroup$ @RogerWood There is a german saying "Probieren geht über studieren", roughly "trying is better than studying". First of all this confirms the sign, but also the strength. If you allow for negative gravitational mass, it might also be that it has less or more, but still positive. Or negative but not with the same magnitude. Even if everyone agrees it SHOULD not be the case that both are positive and equal, testing this rules out any theories where this would not be the case. $\endgroup$ – kutschkem Oct 27 at 16:51
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    $\begingroup$ @RogerWood Falling at a rate of roughly 9.81 m/s per second would indeed be the most expected outcome; since no other experiment so far has contradicted the equivalence principle, it would be silly to expect a contradiction here. If I put a vial of antimatter in a rocket, and fire up the rocket, I would expect the antimatter to be pushed towards the floor of the rocket, not the ceiling, simply because it has inertial mass. Gravity, as far as we can tell, does the exact same thing. It would be exciting if it went the other way (hence, the experiments), but not expected. $\endgroup$ – Arthur Oct 28 at 23:15
  • $\begingroup$ You've encouraged patience here as well :-) What is the Current Status of Measurement of the Gravitational Mass of Antimatter? $\endgroup$ – uhoh Oct 29 at 2:20
  • $\begingroup$ I'll accept this particular answer, but I've received many really excellent informative andwers and comments from a lot of individuals. I like to think I'm a tad wiser now :-) FYI I've posted another question physics.stackexchange.com/questions/590262/… It's a rather different but I wanted to see if I could construct something that behaved effectively like a negative gravitational mass. $\endgroup$ – Roger Wood Oct 29 at 4:42
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I suppose nothing is impossible, including the sign of gravity for ordinary matter reversing direction tomorrow, but it seems really extraordinarily unlikely that antimatter falls up.

It is absolutely precluded by general relativity (more specifically the equivalence principle) that different particles gravitate in different ways. For antihydrogen to fall up would require either that general relativity is wrong or that there is a new, hitherto undetected long-range force countering gravity. And GR would have to be hugely wrong, and the new long-range force would have to be quite strong, to make antihydrogen actually fall up rather than merely falling down at a slightly different acceleration. It's just not plausible that we would have failed to see a fundamental effect of that size for all this time.

Another reason to doubt that antimatter has weird properties is that "antiness" isn't actually an attribute of particles in quantum field theory. Protons and antiprotons are antiparticles of each other; neither one is the antiparticle. Antiprotons get the "anti" prefix merely because they're less common. Some particles (photons, for example) are antiparticles of themselves. We know that photons fall down (from the bending of starlight by the sun, for example). If antiparticles antigravitate then photons would have to fall up also, which isn't even self consistent. I wrote about this in more detail in another answer from which I copied some of the above text.

Other answers mention the AEGIS and ALPHA experiments, but note that AEGIS is looking for deviations in the acceleration "to a precision of 1%", and ALPHA seems to be going for similar precision. They aren't expecting antihydrogen to fall up; testing for that would require only a precision of... 200%, I suppose. No one, to my knowledge, expects antihydrogen to fall up.

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    $\begingroup$ I gather that a 'white hole' is a valid solution to Einstein's field equations. I see these described as black holes running backwards in time. This means that ordinary matter in their vicinity will be repelled. If this is correct, it doesn't seem like general relativity precludes negative gravitational mass. $\endgroup$ – Roger Wood Oct 28 at 3:49
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    $\begingroup$ @RogerWood White holes have positive gravitational and inertial mass and are gravitationally attractive. You can write down GR metrics with negative mass (like the Schwarzschild metric with $m<0$) but they have negative gravitational and inertial mass. What's absolutely impossible is gravitational mass ≠ inertial mass, and antiparticles are known to have positive inertial mass. $\endgroup$ – benrg Oct 28 at 4:43
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    $\begingroup$ If you play an attractor backwards in time, it's still an attractor. Think about making a video of throwing a ball, then playing the video backwards. The ball doesn't follow an "inverted parabola" path. @RogerWood $\endgroup$ – Dawood ibn Kareem Oct 28 at 5:37
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    $\begingroup$ I do not think the equivalence principle is useful as part of this answer. Recall the equivalence principle is grounded in observation, so can be stated "for everything we have ever seen gravitational and interrail mass are the same". Therefore in response to the question "could there be something with different inertial and gravitational masses", the principle is useless. It states only that if we ever did see such a thing GR would need some changes. $\endgroup$ – Dast Oct 28 at 10:45
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    $\begingroup$ @Dast The question asked "is this absolutely precluded in general relativity?" and the answer to that is yes. I didn't mean to imply that GR has to be right, only that if the equivalence principle is wrong then GR is wrong. $\endgroup$ – benrg Oct 30 at 15:36
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You are asking whether antiparticles have different gravitational mass than particles.

There is a very good example of why the answer to your question is no, and that is light itself.

Whether you treat light classically or quantum mechanically, in both cases you will see that light or their quanta, photons do bend spacetime, they have their own gravitational effects and they do bend spacetime.

Given that photons have energy and momentum, it would surprise me if they do not induce curvature. I also note that the expansion of the "radiation-dominated" early universe was caused by what is generally described as a photon gas and not as a classical electromagnetic field. So the idea that photons bend spacetime is part of mainstream cosmology, such as the standard Lambda-CDM model.

Do photons bend spacetime or not?

Since photons have stress-energy, they do bend spacetime. And since they are their own antiparticles, the answer to your question is that particles and antiparticles have the same gravitational mass.

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  • $\begingroup$ Nice deductive reasoning. $\endgroup$ – HolgerFiedler Oct 28 at 16:55
  • $\begingroup$ You can take this further: a electron-positron pair will annihilate to photons with energy equivalent to their mass. If antimatter had negative gravitational mass, this would mean the net gravitational mass jumping between 0 and 2 electron masses when electron-positron annihilation or pair production occurs. You could actually produce work from a gravity field by alternately annihilating and producing electron-positron pairs if that were true. $\endgroup$ – Christopher James Huff Oct 29 at 6:16
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Mass enters gravitational physics in two ways: as a way to talk about the source of gravity (the stress energy tensor), called active gravitational mass, and as a way to talk about the response to gravity, called passive gravitational mass. It is the second one that has to equal inertial mass in a geometric theory in which freefall motion follows a geodesic. The passive gravitational mass is not really gravitational at all, rather it quantifies how much of some other, non-gravitational, force would be required to counteract the mutual gravitational acceleration of two nearby objects. It is just another name for inertial mass.

Coming back to active gravitational mass, and generalizing the question, you are asking whether the energy part of the stress-energy tensor can be of negative sign. One good reason to think that it can not is because then the equations would be unstable, and one would expect to see physical results of such instability, such as weird explosions and implosions in empty space or something like that.

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