New Spacetime Solutions for Spiral Galaxies

TL;DR

This paper addresses the puzzle of flat galactic rotation curves by constructing a broad class of static, spherically symmetric spacetimes in Einstein gravity with anisotropic pressure. The solutions are labeled by the asymptotic rotation velocity parameter $\alpha$ and a spatially averaged EOS parameter $w$, with the requirement $w> -\frac{1}{3}$ ensuring flat outer rotation. It provides explicit metrics and stress-energy components, including non-ideal dust ($w=0$), non-ideal radiation ($w=\tfrac{1}{3}$), and the Einstein cluster ($w=\alpha/3$), all with positive density and anisotropic pressures. A key result is the w-dependent correction to light bending within a halo, $\delta \approx 4m_B/r_0 + f(w)\,\alpha$, which offers a concrete observational handle to constrain the EOS of the anisotropic dark matter fluid and thus discriminate among competing DM frameworks.

Abstract

We find a set of spacetime solutions whose rotation curves approach a flat profile at large radii, as observed for spiral galaxies. The associated stress-energy tensor reflects a positive mass and anisotropic pressure. The spatially averaged equation of state, which parametrizes these infinity of solutions, exhibits a lower bound: $w> -\frac{1}{3}$. As examples, we present the spacetime metrics for non-ideal `dust' and `radiation'. The Einstein clusters emerge as another special class, whose EOS parameter is predicted to be given by the limiting rotational velocity as: $w=\frac{v^2}{3c^2}$. Analyzing the gravitational deflection of a light ray penetrating a large halo, we find the correction to the Einsteinian bending angle to be $\Big(\frac{2+3w}{1+3w}\Big)π\frac{v^2}{c^2}$.