引力磁性与星系旋转曲线：一个警示性故事
Gravitomagnetism and galaxy rotation curves: a cautionary tale
AI总结：
本文证明线性化广义相对论中的引力磁效应比牛顿效应小六个数量级，无法解释星系平坦旋转曲线，并指出此前方法因误用自由空间解而导致奇异性，从而排除其作为暗物质替代解释的可能性。
摘要：
我们研究了近期的一些主张，即线性化广义相对论中的引力磁效应可以在不需要暗物质的情况下解释星系中观测到的平坦和上升的旋转曲线。如果将星系建模为引力电磁（GEM）形式体系中轴对称、稳态、旋转、非相对论且无压的恒星“尘埃”，我们证明GEM效应对恒星圆周速度$v$的影响比标准牛顿（引电）效应小$O(10^{-6})$量级。此外，我们发现引力磁效应小$O(10^{-6})$量级，不足以提供维持所假设的动力学平衡所需的垂直支撑。如果像先前考虑的那样为$v$构造单一方程，这些问题就会被掩盖。尽管如此，我们对具有Miyamoto--Nagai密度分布的星系求解了这一方程。我们证明，对于矮星系典型的质量$M$以及半长轴和半短轴$a$和$b$的取值，旋转曲线对$M$的依赖非常微弱。此外，对于轴比$a/b > 2$，旋转曲线在整个范围内都是凹的，这与任何星系中的观测都不相符。最重要的是，我们证明，若要使极向引力磁通量$ψ$提供必要的垂直支撑，它必须在原点处变为奇异。这源于无意中但被禁止地纳入了确定$ψ$的类泊松方程的自由空间解，从而排除了该方法作为解释平坦星系旋转曲线的手段。我们进一步证明，近期有意利用此类自由空间解来解决旋转曲线问题的尝试，在薄盘近似之外不产生任何确定性修正，并且在任何情况下，$ψ$的齐次贡献都被任何物理轴对称星系所提出的边值问题排除。
英文摘要：
We investigate recent claims that gravitomagnetic effects in linearised general relativity can explain flat and rising rotation curves, such as those observed in galaxies, without the need for dark matter. If one models a galaxy as an axisymmetric, stationary, rotating, non-relativistic and pressureless 'dust' of stars in the gravitoelectromagnetic (GEM) formalism, we show that GEM effects on the circular velocity $v$ of a star are $O(10^{-6})$ smaller than the standard Newtonian (gravitoelectric) effects. Moreover, we find that gravitomagnetic effects are $O(10^{-6})$ too small to provide the vertical support necessary to maintain the dynamical equilibrium assumed. These issues are obscured if one constructs a single equation for $v$, as considered previously. We nevertheless solve this equation for a galaxy having a Miyamoto--Nagai density profile. We show that for the values of the mass, $M$, and semi-major and semi-minor axes, $a$ and $b$, typical for a dwarf galaxy, the rotation curve depends only very weakly on $M$. Moreover, for aspect ratios $a/b > 2$, the rotation curves are concave over their entire range, which does not match observations in any galaxy. Most importantly, we show that for the poloidal gravitomagnetic flux $ψ$ to provide the necessary vertical support, it must become singular at the origin. This originates from the unwitting, but forbidden, inclusion of free-space solutions of the Poisson-like equation that determines $ψ$, hence ruling out the methodology as a means of explaining flat galaxy rotation curves. We further show that recent deliberate attempts to leverage such free-space solutions against the rotation curve problem yield no deterministic modification outside the thin disk approximation, and that, in any case, the homogeneous contributions to $ψ$ are ruled out by the boundary value problem posed by any physical axisymmetric galaxy.
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