Dark-matter-like solutions to Einstein’s unified field equations

Einstein originally proposed a nonsymmetric tensor field, with its symmetric part associated with the spacetime metric and its antisymmetric part associated with the electromagnetic field, as an approach to a unified field theory. Here we interpret it more modestly as an alternative to Einstein-Maxwell theory, approximating the coupling between the electromagnetic field and spacetime curvature in the macroscopic classical regime. Previously it was shown that the Lorentz force can be derived from this theory, albeit with deviation on the scale of a universal length constant $\ell$. Here we assume that $\ell$ is of galactic scale and show that the modified coupling of the electromagnetic field with charged particles allows a non-Maxwellian equilibrium of non-neutral plasma. The resulting electromagnetic field is “dark” in the sense that its modified Lorentz force on the plasma vanishes, yet through its modified coupling to the gravitational field it engenders a nonvanishing, effective mass density. We obtain a solution for which this mass density asymptotes approximately to that of the pseudoisothermal model of dark matter. The resulting gravitational field produces radial acceleration, in the context of a post-Minkowskian approximation, which is negligible at small radius but yields a flat rotation curve at large radius. We further exhibit a family of such solutions which, like the pseudoisothermal model, has a free parameter to set the mass scale (in this case related to the charge density) and a free parameter to set the length scale (in this case an integer multiple of $\ell$). Moreover, these solutions are members of a larger family with more general angular and radial dependence. They thus show promise as approximations of generalized pseudoisothermal models, which in turn are known to fit a wide range of mass density profiles for galaxies and clusters.

Figure 1

Coeffecients $a_l(5)$ (crosses), $-a_l(15)$ (diamonds), $a_l(25)$ (squares), and $-a_l(35)$ (circles) vs $l$. This shows that $|a_l(n)|$ is largest when $l\approx n$.

Figure 2

The potential ${\phi }_{(n)}(r,\theta =\pi /6)$ for $n=5$, $n=15$, $n=25$, and $n=35$ (as calculated numerically) plotted together with an approximation (dashed line) using only the $l=n-1$, $l=n$, and $l=n+1$ terms: $\frac{4\pi {\rho }_0}{k^2}\sum _{l=n-1}^{n+1}{a}_l(n){\mathrm{j}}_l(kr){\mathrm{P}}_l(\cos (\theta ))$. This shows that the potential is dominated by terms around $l\approx n$. This is further made clear by the global extremum apparent near $kr\approx n+2$, which approximates the position of the global extremum of ${\mathrm{j}}_n(kr)$. And, like ${\mathrm{j}}_n(kr)$, the potential is relatively negligible for $kr\lesssim n$.