相对论性黑洞吸积的多临界行为及相关分岔现象的解析研究
An Analytical Study on the Multi-critical Behaviour and Related Bifurcation Phenomena for Relativistic Black Hole Accretion
摘要：
本文应用代数多项式理论分析研究了非旋转天体物理黑洞的广义相对论流体动力学轴对称吸积的跨音速性质。对于这种吸积现象，流的守恒比能，这原来是在所研究的系统中的两个第一运动积分之一，可以表示为一个8 $^{th}$次多项式的临界点的流配置。然后，我们构造相应的Sturm链算法来计算实根的数目躺在天体物理学相关领域的$\mathbb{R}$。这使得在文献中第一次{ \it解析地}找出具有特定几何构型、时空度量和状态方程的吸积流所能具有的物理上可接受的最大解的数目，从而{ \it完全解析地}研究其多临界性质，对于临界点的位置不借助于数值格式就不能计算的吸积流。这一工作可以进一步推广到一般自治动力系统的最大平衡点数目的解析计算。我们还演示了如何从一个单临界到多临界（反之亦然）流配置的过渡可以通过鞍中心分叉现象，使用某些技术的突变理论实现。
英文摘要：
We apply the theory of algebraic polynomials to analytically study the transonic properties of general relativistic hydrodynamic axisymmetric accretion onto non-rotating astrophysical black holes. For such accretion phenomena, the conserved specific energy of the flow, which turns out to be one of the two first integrals of motion in the system studied, can be expressed as a 8$^{th}$ degree polynomial of the critical point of the flow configuration. We then construct the corresponding Sturm's chain algorithm to calculate the number of real roots lying within the astrophysically relevant domain of $\mathbb{R}$. This allows, for the first time in literature, to {\it analytically} find out the maximum number of physically acceptable solution an accretion flow with certain geometric configuration, space-time metric, and equation of state can have, and thus to investigate its multi-critical properties {\it completely analytically}, for accretion flow in which the location of the critical points can not be computed without taking recourse to the numerical scheme. This work can further be generalized to analytically calculate the maximal number of equilibrium points certain autonomous dynamical system can have in general. We also demonstrate how the transition from a mono-critical to multi-critical (or vice versa) flow configuration can be realized through the saddle-centre bifurcation phenomena using certain techniques of the catastrophe theory.
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