Last month an international team of researchers reported in Physical Review Letters that it had engineered Bose–Einstein condensates to behave as if they have negative mass. Multiple media outlets jumped on the paper, excitedly explaining to readers that the ultracold gas would move toward you if you tried to push it away. The result is no less strange for physicists: The free expansion of the condensate spontaneously, asymmetrically, and self-consistently arrests.
After a day or two, a new wave of popular-press coverage emerged, this time debunking the idea that the condensate truly has negative mass. Indeed, the paper created quite a controversy. It is interesting that the notion of negative mass can engender strong feelings of mistrust.
The idea that matter could come toward you when you push it away or that it could rise freely against the pull of the Earth certainly sounds outrageous. But there are reasons embedded in physics’ most important equations why we should take the idea of negative mass seriously. Within Einstein’s general theory of relativity, we can analyze the possibility of negative-mass matter, which would act as a source of repulsive gravitational fields. Various investigations, including my own work, suggest that as strange as they sound, such objects might exist in our universe.
Although the idea of antigravity has fascinated humanity since the beginning of recorded history, scientists never dug into the details of negative-mass matter until Joaquin Luttinger submitted an entry to the 1951 Gravity Research Foundation essay competition. He won fourth prize for his analysis of the classical mechanical equations of motion for objects with negative mass. Luttinger observed that negative-mass objects would gravitationally repel all objects of any mass (positive or negative), while positive-mass objects would gravitationally attract all objects of any mass. That gives rise to the amusing scenario that a –5 kg object would repel a +5 kg object, while the +5 kg object would attract the –5 kg object. As a result, the two of them will move off in the same direction along the line joining the two, with the negative-mass object following the positive-mass object indefinitely. Negative mass acts even more strangely if you try to manipulate it in vessels made of positive-mass matter. If negative-mass matter impinges on a wall, it moves into the wall with ever more strength until it finally breaks through.
In 1957 Hermann Bondi performed an analysis more detailed than Luttinger’s. In the context of Newtonian physics, Bondi identified the possibility of three kinds of mass: inertial, passive gravitational, and active gravitational. Inertial mass is defined by Newton’s second law, but only through the action of a nongravitational force. The buzz-inducing Bose–Einstein condensates behave as if they have negative inertial mass. Passive gravitational mass is the constant of proportionality between the force produced on an object and an external gravitational field. The gravitational field produced by an object, via Newton’s law of universal gravitation, is proportional to its active gravitational mass. That last variety of mass is what would potentially give rise to gravitational fields that repel rather than attract.
Bondi then considered the dynamics of two gravitating bodies—the two-body problem—within Einstein’s general theory of relativity. The gravitation of each body affects the movement of the other and the ambient spacetime. Bondi succeeded in demonstrating that there exists a solution to the full, coupled equations of general relativity with matter, which corresponds to the spacetime seen by a uniformly accelerating observer, with two regions in the interior where the active gravitational mass in one region is positive while in the other region it is negative. Furthermore, the energy density in the region of positive active gravitational mass is also positive, while it is negative in the region of active negative gravitational mass. The energy density gives rise to the passive gravitational mass and hence, via the equivalence principle, the inertial mass.
That brings us to the question of active gravitational mass and whether it can be negative, allowing for an antigravity effect. In his paper Bondi observed that the Schwarzschild solution (known colloquially as the black hole solution) contains an integration constant M, which can be interpreted as the active gravitational mass of the solution. This M can be positive or negative; the metric satisfies the vacuum Einstein equations for both signs of the mass.
Why does a negative-mass solution exist? It seems to be a spurious solution. Indeed, no initial distribution of positive-mass matter will ever evolve to even a local region of spacetime with a negative Schwarzschild mass. Although both the negative-mass and positive-mass solutions are similar, the negative-mass solution has a naked singularity—there is no event horizon to cloak the singularity from the view of an external observer. Naked singularities do not bother us in other contexts. In electrodynamics, for example, we blithely entertain the possibility of point charges because we can smooth out the singularity by a nonsingular charge density.
Smoothing out the singularity of the negative-mass Schwarzschild solution is impossible in a spacetime that is asymptotically flat. But fortunately, our universe is not asymptotically flat. An asymptotic de Sitter spacetime admits an exact solution corresponding to a point mass in the presence of a background cosmological constant. The mass can be positive or negative. There is observational evidence for accelerated expansion of the universe, and the cause of the acceleration is deemed to be due to a mysterious dark energy, which is the vacuum energy density of a heretofore unidentified field.
In 2013 my student Jonathan Belletête and I won fourth prize, just as Luttinger had, in the Gravity Research Foundation essay competition. We demonstrated how to deform the Schwarzschild–de Sitter metric so that the singularity could be smoothed out. In a follow-up paper, student Saoussen Mbarek and I showed that it was possible to find solutions of the Einstein equations with the matter corresponding to a perfect fluid. The matter that gives rise to negative mass is contained in a spherical bubble. Outside the bubble, the spacetime corresponds exactly to negative-mass Schwarzschild–de Sitter spacetime. Test bodies outside the bubble will be repelled.
Critics suggest that the negative active gravitational mass we have found, just like the negative inertial mass found in the Bose–Einstein condensate, is somehow not the real thing. They suggest that it is just relative negative mass or effective negative mass. That criticism misses the point. With such a point of view, one would be of the opinion that there are no protons or neutrons, just quarks and gluons, or that in condensed matter there are no anyons or Cooper pairs or fractional charge, just electrons and atomic nuclei. In principle, effective or emergent degrees of freedom can be completely equivalent to the original degrees of freedom.
The concept of negative mass explains the behavior of certain systems in a much more efficient and compact manner than continuing to try to describe them in terms of positive-mass particles. In the cosmological context, it creates a repulsive gravitational field as can be ascertained by studying the behavior of test bodies outside the negative-mass bubble. The negative-mass bubbles as we have described could have existed in the early universe, with important potential consequences for cosmology. They could even exist today.
Another frequent concern expressed over the existence of negative mass is that it would cause an untenable instability of the universe. Stephen Hawking once told me that if negative mass existed, “the universe would be unstable and we would not be here to this day.” But negative mass exists only in an expanding universe, and because of energy conservation it can only be produced in positive–negative mass pairs. If there is a backreaction of the production of these pairs on the background cosmological energy, the production of negative mass should drive that energy density to zero, thus terminating the possibility of its production and quenching any instability. This mechanism could offer a means of resolving the long-standing problem of why the cosmological constant is so small. In the realm of speculation, the possibility of creating negative-mass bubbles in the laboratory could have incredible applications for energy production, warp-drive transportation, and armaments.
Rather than dismissing the idea of negative mass, researchers should try to use it to their advantage. Avenues for further study include looking for dynamical models of matter that would give rise to stable negative-mass configurations and exploring the consequences of negative mass in the inflationary phase of the early universe. A plasma of positive- and negative-mass particles during the inflationary epoch would have an important influence on the propagation of gravitational waves, an effect that might be observable in the cosmic microwave background.
Manu Paranjape is a theoretical physicist at the University of Montreal who studies fractional charge, the Skyrme model, conformal gravity, noncommutative geometry, and quantum spin tunneling.
