A&A 626, A5 (2019) Astronomy https://doi.org/10.1051/0004-6361/201935317 & c ESO 2019 Astrophysics Can a negative-mass cosmology explain dark matter and dark energy?? H. Socas-Navarro1,2 1 Instituto de Astrofísica de Canarias, Avda Vía Láctea s/n, La Laguna 38205, Tenerife, Spain 2 Departamento de Astrofísica, Universidad de La Laguna, La Laguna 38205, Tenerife, Spain e-mail: [email protected] Received 20 February 2019 / Accepted 17 April 2019 ABSTRACT A recent study by Farnes (2018, A&A, 620, A92) proposed an alternative cosmological model in which both dark matter and dark energy are replaced with a single fluid of negative mass. This paper presents a critical review of that model. A number of problems and discrepancies with observations are identified. For instance, the predicted shape and density of galactic dark matter halos are incorrect. Also, halos would need to be less massive than the baryonic component, otherwise they would become gravitationally unstable. Perhaps the most challenging problem in this theory is the presence of a large-scale version of the “runaway effect”, which would result in all galaxies moving in random directions at nearly the speed of light. Other more general issues regarding negative mass in general relativity are discussed, such as the possibility of time-travel paradoxes. Key words. cosmology: theory – dark matter – galaxies: kinematics and dynamics – large-scale structure of Universe 1. Introduction crucial difference is that, in the case of electromagnetism, two In a recent paper, Farnes(2018; hereafter F18) put forward an equal (opposite) charges repel (attract) each other, whereas in a audacious proposal to explain dark matter and dark energy in dual-sign gravitation the opposite process would take place. The a unified scenario with a single fluid of negative-mass particles. difference may appear subtle but it is of paramount importance. This fluid would be continuously created everywhere to maintain Gravity is able to attract many particles of the same sign together its density at a constant level in spite of cosmological expansion. and make them coalesce into arbitrarily large structures. To support his hypothesis, F18 presented a number of simple In general relativity, the solution to the simple scenario where numerical simulations showing the formation of halos around we have a point-like mass in an otherwise empty Minkowskian galaxies and indications of filament and void structures in a cos- space-time is the Schwarzschild(1916) metric. This solution has mological volume. In the spirit of healthy scientific debate, this a singularity at the mass position but the singularity is hidden paper presents the results of a critical look at the F18 model and from external observers by an event horizon that surrounds it. some objections derived from the analysis of similar simulations. No information may propagate from the singularity to the out- The idea of matter with negative mass is not new in physics. side world. The equivalent of the Schwarzschild solution for a In fact, some speculative work dates back to the nineteenth cen- negative mass does not represent a symmetric situation. It also tury (see e.g., the introduction of Bonnor 1989). After the devel- has a singularity at the mass position but now there is no hori- opment of general relativity, when physicists were struggling zon. The singularity is naked, in violation of the weak cosmic with the conflict between the crunching action of a universal censorship conjecture (Penrose 1969). While the cosmic censor- gravity and the prevailing notion at the time of a static universe, ship remains an unproven conjecture, it is widely recognized by negative mass was seen by some authors as a promising solution the community as a sensible hypothesis and solutions with naked to one of the biggest problems in cosmology. A repulsive form of singularities are generally regarded as unsatisfactory. Of course, gravity emanating from negative masses could provide the nec- point-like masses do not exist in reality but it is often the case that essary force to counterbalance the universal collapse (Einstein one may mathematically construct a continuous mass distribu- 1918). In reality, this statement is deceivingly simple and many tion by integration of infinitesimal points. This raises yet another important complications lurk behind it. To begin with, one has problem. If we consider a finite-size negative mass object, how to define exactly what is meant by negative mass and this is not are its pieces held together? In principle, each portion of it would straightforward. In general relativity, the source of gravitation is be subject to a repulsive force from the rest of the object, with the energy–stress tensor, a complex set of quantities that involves the force increasing to infinity for decreasing distances. Particles the density and flux of mass, energy, and stress. Remarkably, dif- with electric charge are held together by quantum physics and ferent observers in different reference frames might even mea- the existence of an elementary charge. However, there is no such sure opposite signs of some of these quantities. thing for gravity, at least in the context of general relativity. All of It is tempting to draw a parallelism between a dual-sign grav- these are conceptual problems which, while not necessarily ruling itation (with positive and negative masses) and the dual-charge out the existence of negative mass in the Universe, should at least nature of electromagnetism. However, this would be a very naïve serve as a warning sign that it is a complex issue and certainly not conception, as the two situations are completely different. The a symmetric counterpart of ordinary matter. The concept of mass may have four different meanings ? The simulation video is available at https://www.aanda.org in physics: the ability of matter to produce gravity (“active” Article published by EDP Sciences A5, page 1 of7 A&A 626, A5 (2019) gravitational mass), the response of matter to gravity (“pas- cosmological structures, etc.). While we have rather robust con- sive” gravitational mass), the inertia of matter (its resistance straints on the mass of dark matter concentrations by means of to accelerate when subject to a force), or its energy equiva- gravitational lensing observations, we are not able in most sit- lence. Classically, all of these definitions of mass refer to the uations to determine the sign of the lensing mass (Takahashi & same quantity and are positive by definition. General relativity Asada 2013). If the F18 ideas were proven correct, the implica- requires by construction (by virtue of the equivalence principle) tions for all of physics would be formidable. A close scrutiny of that passive gravitational and inertial mass must be the same. The this model is therefore clearly warranted. theory does not explicitly require that they be positive, although This paper presents a critical analysis of the F18 results, iden- some restrictions exist, as discussed in Sect.4 below. tifying a number of problems and incompatibilities with exist- Various ideas were proposed and discussed in the decades ing observations. Section2 has an analytical derivation of the following the development of general relativity regarding the expected properties of galactic halos in the F18 scenario. This interpretation and implications of negative sources of gravitation will be helpful to understand and validate some of the results in in that theory. Critically important was the work of Bondi(1957). Sect.3, which presents simulations and the discrepancies with First Luttinger(1951) and then Bondi(1957) (using Newtonian observations. Section4 emphasizes other conceptual di fficulties and relativistic frameworks, respectively), studied the concept facing the model that are not directly testable against observations. of what is known as the “runaway motion”: two particles with Finally, the overall conclusions are summarized in Sect.5. opposite mass would start accelerating in one direction. In the Newtonian framework, this happens because the negative mass is attracted towards the positive one (the force is in the oppo- 2. Galactic halos site direction but the negative inertial mass makes it accelerate One of the remarkable results of F18 was the formation of neg- towards the positive mass), whereas the positive mass is repelled ative mass halos around galaxies with a density structure strik- and accelerates away from the negative. The same effect occurs ingly similar to a Navarro–Frenk–White (NFW, Navarro et al. in general relativity. 1996) dark matter profile. This paper presents new simulations Following these developments, interest in the physics of neg- very similar in nature to those of F18 but the resulting halo den- ative masses declined, at least as a practical idea (exploratory sities exhibit what appears to be an exponential stratification theoretical works were still being pursued), for two main rea- instead. In order to develop a better understanding of the simula- sons. First, the runaway effect was considered so “prepos- tion results, or even to serve as a sanity check, it is usually help- terous” that it was viewed by many authors as unphysical. ful to conduct an analytical study of the system. In this section Second, the discovery of the Hubble–Lemaître law (Lemaître we seek the functional form of the density profile of a spherical 1927; Hubble 1929) and the later confirmation of the Big Bang negative mass halo surrounding a central core of positive masses. model with the detection of the cosmic microwave background Throughout this section we assume spherical symmetry, non- (Penzias & Wilson 1965) removed what had been the original interacting particles (except for gravitation), and a steady-state motivation for a repulsive gravitational force, that is, a means to equilibrium in which the system has reached a stable macro- sustain a steady-state universe (e.g., Bondi & Gold 1948). scopic configuration.