**Does Antimatter Have Negative Mass?**

**By Kathleen A. Rosser**

[Work in progress]

April 17, 2021

In standard cosmology, generally referred to as the *ΛCDM model*, various assumptions have been made that are now widely challenged in the literature. First, it is assumed there is no significant antimatter in the universe, and therefore that the antimatter supposedly created in the big bang has inexplicably vanished. Second, it is also assumed, based on Supernova Type Ia and other astromical data, that the universal expansion rate is increasing. This phenomenon, called *cosmic acceleration,* is sometimes attributed to the presence of a pervasive exotic substance known as *dark energy*, which obeys a seemingly nonphysical equation of state imposing an anti-gravity or repulsive effect on the background matter of the universe. Alternatively, cosmic acceleration may be accounted for by adding a term proportional to the cosmological constant *Λ* in Einstein’s field equations. The cosmological constant is a purely mathematical artifice. Yet it is sometimes associated with dark energy density, which as I will show in a future paper, may not be rigorously correct in the context of the Friedmann-Robertson-Walker (FRW) metric generally used to describe the expanding universe.

A third assumption is that antiparticles interact gravitationally just as matter particles do. Thus the mass of antimatter is assumed to be positive, although all other quantum numbers are of opposite sign. The mass m of an antiparticle is also assumed to be a scalar quantity, rather than a vector or tensor.

However, there has been recent interest in the idea that antiparticles may have negative mass, and in fact, the sign of the mass has not yet been established in particle accelerator experiments. But questions arise. If antiparticles have negative mass, does that imply a repulsive gravitational force between particles and antiparticles? And do antiparticles repel or attract each other? Various combinations of properties have been investigated in the literature.

The Dirac-Milne universe, proposed recently by various researchers [REFS], is a model of the cosmos that dispenses with all of the above assumptions. First, according to this theory, the antimatter created in the big bang remains present in our universe, taking the form of a diffuse background density that avoids matter structures such as galaxies and galaxy clusters. The reason for this avoidance with become clear below. Previously, it was thought that antimatter could not permeate the universe, as it would annihilate with matter and create unobserved signatures of gamma radiation. However, the avoidance of matter by antimatter obviates this objection.

Second, the Dirac-Milne universe does not expand at an accelerating rate, but rather at a constant rate. This type of linear expansion is called the *coasting model*, and was originally investigated by Milne. Interestingly, a number of recent articles suggest that the coasting model is as well supported by astronomical data as is the standard model involving cosmic acceleration. Thus, a Milne universe is not ruled out by observation.

Third, the Dirac-Milne model postulates that antimatter has negative mass. But further, this negative mass interacts gravitationally in a novel way, as follows: Matter attracts matter, but antimatter repels everything, including itself. This behavior, however, cannot be described by a mere scalar mass with a negative sign. For, according to Newton’s law of gravity, the product of two negative masses would be positive, and thus antimatter would attract rather than repel itself. To express mathematically the novel behavior of antimatter repelling everything requires that mass be represented by a vector quantity M=Sm, where S is a gravitational charge vector and m is a positive scalar equal to the magnitude of the mass. This result will be derived below.

[To be continued …]

[Earlier work:]

Ongoing research in cosmology has produced static solutions to Einstein’s field equations which are consistent with cosmic redshift and do not require an expanding universe. The various metrics associated with these solutions differ from the ΛCDM Friedman-Lemaitre-Robertson-Walker (FLRW) metric in that the time component g^{00} of the metric is a function of r, while the radial component g^{11} is also a function of r and does not include the dynamical scale factor a(t).

It is possible for a static metric to obey a redshift-distance relation by allowing the time component g^{00} to be a function of r, as was proved in the paper entitled *A Static Cosmological Model, MOND, and the Galactic Rotation Curve* found on the home page of this website. Einstein, in his original static cosmology, set the time component equal to unity. His theory was thus unable accomodate the Hubble redshift. In response to this, he adopted the expanding universe or big bang model, based on the FLRW metric first introduced by Friedman in 1922.

Einstein’s rejection of static models has had a major influence on the development of cosmological theory. As a result of his work, few physicists have investigated static models. Notable exceptions are Fritz Zwicky, who suggested that cosmic redshift might be explained in a static universe by a phenomenon called *tired light*. A suitable mechanism for tired light, however, was never found, and Zwicky’s theory was subsequently abandoned.

A second exception is Robert H. Dicke, who in the 1950’s proposed a static model in which cosmic redshift was explained by a variable speed of light (VSL). One complication related to metric VSL theories, however, is that variations in the speed of light affect the g^{00} component of the metric, and thus the rate of time flow. Light velocity can therefore be made constant through a transformation of the time coordinate. To accommodate a varying speed of light in a nontrivial way, an extra dimension or additional free variable, such as a scalar potential, must be introduced. Perhaps this explains why Dicke abandoned his VSL model in favor of the Brans-Dicke theory, a scalar-tensor theory in which includes a scalar potential.

That static theories have been neglected seems a serious oversight, especially at a time when hundreds of theories of gravity are being investigated. The majority of current models, even while modifying or rejecting Einstein’s field equations, retain the FLRW metric as a dynamic representation of an expanding universe. Yet the FLRW metric raises question from a mathematical standpoint. One feature of the FLRW metric that may raise doubts is the fact that the co-moving coordinate r, in conjuction with an orthogonal conformal time coordinate t, requires the time axes of receeding galaxies to diverge. Indeed, beyond a certain distance, the conformal time axes of remote galaxies become superluminal or spacelike, even though the sign of g^{00} in the line element does not change. This is in contrast to the static Shwarzschild line element, in which g^{00} becomes positive and g^{11} becomes negative in the region r<2m, where timelike trajectories become spacelike.

In contrast, on the surface of the earth, the latitudinal axes, or geodesics of constant longitude, diverge at various angles from the poles. Yet here there is no contradiction. Divergence is possible because the surface of the earth is described by *Riemannian* geometry. General relativistic metrics, in contrast, are *pseudo-Riemannian*, and thus only one dimension in the line element ds^{2} can have a signature of opposite sign. If the region of opposite sign is one-dimensional, can the axes in that dimension be non-parallel? Would two dimensions not be necessary to accomodate this divergence? Interestingly, in the context of 5-dimensional variable mass theories, there are arguably two dimensions of time, t and s, where s is proper time τ.