arXiv 0904.24972009-04gr-qc

宇宙学常数为正的静态天体在M/R上的界

Bounds on M/R for static objects with a positive cosmological constant

Hakan Andreasson, Christian G. Boehmer

备注：: 14 pages; Improvements and corrections, published version
发表信息：: Class.Quant.Grav.26:195007,2009
DOI：: 10.1088/0264-9381/26/19/195007

摘要：

考虑具有正宇宙学常数$Λ,$的爱因斯坦方程的球对称静态解，研究了$Λ,$对M/R界的影响，其中M是ADM质量，R是静态物体边界的面积半径.我们发现，对于满足能量条件$p+2p_{\perp}\leqρ,$的任何解，其中$p\geq 0$和$p_{\perp}$分别是径向和切向压力，$ρ\geq 0$是能量密度，并且对于其中的$0\leq ΛR^2\leq 1,$，不等式\frac{M}{R} \leq \frac 29- \frac{ΛR^2}{3} + \frac 29 \sqrt{1+3ΛR^2}成立.如果$Λ=0$它是已知的无限薄壳解决方案唯一饱和的不等式，即不等式是尖锐的，在这种情况下。情况是完全不同的，如果$Λ>0.$事实上，我们表明，无限薄壳的解决方案一般不饱和的不等式，除了在两个退化的情况$ΛR^2=0$和$ΛR^2=1$。在后一种情况下，也有一个常密度解，其中外部时空是Nariai解，它饱和了不等式，因此，饱和解是非唯一的。在这种情况下，宇宙视界和黑洞视界重合。这类似于带电的情况，其中有数值证据表明，当Reissner-Nordström解的内视界和外视界重合时，饱和解的唯一性丢失。

英文摘要：

We consider spherically symmetric static solutions of the Einstein equations with a positive cosmological constant $Λ,$ which are regular at the centre, and we investigate the influence of $Λ$ on the bound of M/R, where M is the ADM mass and R is the area radius of the boundary of the static object. We find that for any solution which satisfies the energy condition $p+2p_{\perp}\leqρ,$ where $p\geq 0$ and $p_{\perp}$ are the radial and tangential pressures respectively, and $ρ\geq 0$ is the energy density, and for which $0\leq ΛR^2\leq 1,$ the inequality \frac{M}{R}\leq\frac29-\frac{ΛR^2}{3}+\frac29 \sqrt{1+3ΛR^2}, holds. If $Λ=0$ it is known that infinitely thin shell solutions uniquely saturate the inequality, i.e. the inequality is sharp in that case. The situation is quite different if $Λ>0.$ Indeed, we show that infinitely thin shell solutions do not generally saturate the inequality except in the two degenerate situations $ΛR^2=0$ and $ΛR^2=1$. In the latter situation there is also a constant density solution, where the exterior spacetime is the Nariai solution, which saturates the inequality, hence, the saturating solution is non-unique. In this case the cosmological horizon and the black hole horizon coincide. This is analogous to the charged situation where there is numerical evidence that uniqueness of the saturating solution is lost when the inner and outer horizons of the Reissner-Nordström solution coincide.

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