广义相对论中的非均匀密度星
No uniform density star in general relativity
摘要：
根据广义相对论（GR），不可能有任何能量的超光速传播。因此，连续介质中的声速，$c_s=\sqrt{dp/dρ}$，必须是亚光速。然而，如果设想一种{ \em均质}流体，则除非压力也是均质的，否则就有$c_s=\infty$。因此，人们普遍认为，首次GR内部解决方案所获得的史瓦西，涉及一个均匀的流体边界，是非物理的。然而，从来没有人证明这个精确的解决方案在现实中是如何缺乏物理现实的。此外，这个解决方案是普遍使用的近似建模的广义相对论性恒星和紧凑的对象。但在这里，我们首先表明，为了使Kretschmann标量是连续的，人们应该有$ρ=0$严格均匀的静态恒星。此外，通过调用在GR中给定一个时间标签$t$可以选择另一个时间标签$t_*=f(t)$ { \em而不失一般性}的事实，我们获得了与静态均匀球$ρ=0$相同的结果。因此，最终发现具有边界的静态均匀球只是真空的一部分，其中$c_s=0$而不是$\infty$。因此，所有广义相对论恒星都必须是非均匀的。
英文摘要：
As per general relativity (GR), there cannot be any superluminal propagation of energy. And thus, the sound speed in a continuous medium, $c_s=\sqrt{dp/dρ}$, must be subluminal. However, if one would conceive of a {\em homogeneous} fluid, one would have $c_s=\infty$ unless pressure too would be homogeneous. Thus it is universally accepted that the maiden GR interior solution obtained by Schwarzschild, involving a homogeneous fluid having a boundary, is unphysical. However no one has ever shown how this exact solution is in reality devoid of physical reality. Also, this solution is universally used for approximate modelling of general relativistic stars and compact objects. But here first we show that in order that the Kretschmann scalar is continuous, one should have $ρ=0$ for strictly homogeneous static stars. Further, by invoking the fact that in GR, given one time label $t$ one can choose another time label $t_*=f(t)$ {\em without any loss of generality}, we obtain the same result that for a static homogeneous sphere $ρ=0$. Consequently, it is eventually found that the static homogeneous sphere having a boundary is just part of the vacuum where $c_s=0$ rather than $\infty$. Therefore all general relativistic stars must be inhomogeneous.
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