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The arxiv version of the paper is here. Since I’m quoted in the Science News piece saying something to the extent that I have my reservations but think it’s a promising direction of study, I have gotten a lot of questions about negative masses in General Relativity lately. So here a clarification.

First one has to be careful what one means with mass. There are three types of masses: inertial mass, passive gravitational mass, and active gravitational mass. In General Relativity these masses, or their generalization in terms of tensors respectively, are normally assumed to be identical.

The equality of inertial and passive gravitational mass is basically the equivalence principle. The active gravitational mass is what causes space-time to bend; the passive gravitational mass is what couples to the space-time and determines the motion of particles in that background. The active and passive gravitational masses are identical in almost all theories I know. (The SchrĂ¶dinger-Newton approach is the only exception that comes to mind). I doubt it is consistent to have them not be equal, but I am not aware of a proof for this. (I tried in the SchrĂ¶dinger-Newton case, but it’s not as trivial as it looks at first sight.)

In General Relativity one further has to distinguish between the local quantities like energy-density and pressure and so on that are functions of the coordinates, and global quantities that describe the space-time at large. The total mass or energy in some asymptotic limit are essentially integrals over the local quantities, and there are several slightly different ways to define them.

The positive mass theorem, in contrast to what its name suggests, does not state that one cannot have particles with negative masses. It states instead, roughly, that if your local matter is normal matter and obeys certain plausible assumptions, then the total energy and mass are also positive. You thus cannot have stars with negative masses, regardless of how you bend your space-time. This isn’t as trivial a statement as it sounds because the gravitational interaction contributes to the definition of these integrated quantities. In any case, the positive mass theorem holds in space that is asymptotically flat.

Now what they point out in the new paper is that for all we know we don’t live in asymptotically flat space, but we live in asymptotic de-Sitter space because observational evidence speaks for a positive cosmological constant. In this case the positive mass theorem doesn’t apply. Then they go on to construct a negative mass solution in asymptotic de Sitter space. I didn’t check the calculation in detail, part of it is numerical, but it all sounds plausible to me.

However, it is somewhat misleading to call the solution that they find a negative mass solution. The cosmological constant makes a contribution to the effective mass term in what you can plausibly interpret as the gravitational potential. Taken together both, the effective mass in the potential is positive in the region where this solution applies. The local mass (density) is also positive by assumption. (You see this most easily by looking at fig 1 in the paper.)

Selling this as a negative mass solution is like one of these ads that say you’ll save 10$ if you spend at least $100 – in the end your expenses are always positive. The negative mass in their solution corresponds to the supposed savings that you make. You never really get to see them. What really matters are the total expenses. And these are always positive. There are thus no negative mass particles in this scenario whatsoever. Further, the cosmological constant is necessary for these solutions to exist, so you cannot employ them to replace the cosmological constant.

It also must be added that showing the existence of a certain solution to Einstein’s field equations is one thing, showing that they have a reasonable chance to actually be realized in Nature is an entirely different thing. For this you have to come up with a mechanism to create them and you also have to show that they are stable. Neither point is addressed in the paper.

Advertisement break: If you want to know how one really introduces negative masses into GR, read this.

In the Science News article Andrew Grant quotes one of the authors as saying:

“Paranjape wants to look into the possibility that the very early universe contained a plasma of particles with both positive and negative mass. It would be a very strange cosmic soup, he says, because positive mass gravitationally attracts everything and negative mass repels everything.”This is wrong. Gravitation is a spin-2 interaction. It is straightforward to see that this means that like charges attract and unlike charges repel. The charge of gravity is the mass. This does not mean that negative gravitational mass repels everything. Negative gravitational mass repels positive mass but attracts negative mass. If this wasn’t so, then you’d run into the above mentioned inconsistencies. The reason this isn’t so in the case considered in the paper is that they don’t have negative masses to begin with. They have certain solutions that basically have a gravitational attraction which is smaller than expected.

In summary, I think it’s an interesting work, but so far it’s an entirely theoretical construct and its relevance for the description of cosmological dynamics is entirely unclear. There are no negative mass particles in this paper in any sensible interpretation of this term.