物质向黑洞的塌缩问题：对Birkhoff定理的理解和史瓦西度规的奇异性问题 张双南 高能物理研究所 Liu & Zhang, “Exact solutions for shells collapsing towards a pre-existing black hole”, 2009, Physics Letters B, Volume 679, Issue 2, p. 88-94

Question: Metric in Zone I and III? Schwarzschild metric Reasons for the answer: Birkhoff’s Theorem: the vacuum metric is Schwarzschild for spherically symmetric mass. Gauss’s Theorem: only need to calculate mass or charge inside the given radius Birkhoff’s Theorem can be understood in terms of Gauss’s Theorem. III II Zone I m 2009/9/26

The Real Birkhoff’s Theorem: Metric outside a spherically symmetric body is Schwarzschild 2009/9/26

Common understanding of Birkhoff’s Theorem x Commonly phrased as: metric anywhere is determined by only the total inside mass-energy with spherically symmetric distribution. M This is true in Newtonian gravity, due to the 1/r2 law, but also true in GR only if nothing is outside (the original Birkhoff’s Theorem). 2009/9/26

(1) BH can be formed due to gravitational collapse, for co-moving observer; (2) “BH” is “frozen star” for external observer 2009/9/26

In “George's secret key to the universe”: Publication Date: October 23, 2007 对外部观测者宇航员永远不能进入黑洞 2009/9/26

OS39 solution for self-collapsing dust ball Interior metric is not Schwarzschild ht(R>R0)=1, ht(R<R0)<1 ht also depends on mass outside: a real surprise! but nobody seems to care since it is not in a vacuum. 2009/9/26

A possible consequence of OS39 solution: lensing calculation R0 For lensing calculation, it has been assumed commonly that ht(R>R0)=1 and ht(R<R0)=1 Therefore such approximation under-estimates the light-bending angle, as well as the delay time. Gravitating mass is over-estimated from lensing data. R Also causes additional Shapiro delay 2009/9/26

One dust shell falling to a BH Zone III ht(III)=1 and ht(II)<1 Continuity ht(I)<1 Therefore the metric in Zone I is not Schwarzschild metric of mass m Clearly against common (mis)-understanding. ht(I) depends upon the mass distribution (and location) of the shell, thus varies when the shell falls. Zone II a Zone I a’ BH m 2009/9/26

Numerical example of h(t;I) As the shell falls in, the clock becomes slower and slower gravity becomes stronger and stronger 2009/9/26

More on Birkhoff’s Theorem’s Applicability Zone III Yes Birkhoff Theorem Zone II a’ Zone I No Birkhoff Theorem BH m 2009/9/26

In-falling process of a dust shell Co-moving observer External stationary observer Test particle Conclusions of the external observer: (1) the shell can pass through the event horizon within finite time, except its outer surface ; (2) the shell can never arrive at the singularity. 2009/9/26

Double in-falling shells: mimicking the real astrophysical setting Any matter between shell 1 and the observer a1 a1’ III I V m IV II a2’ a2 m1 m2 2009/9/26

Double shell in-falling process Co-moving observer External stationary observer Conclusion of the external observer: shell 1 passed through the event horizon completely, but still can never arrive at the singularity. The mathematical singularity at r=0 has no physical meaning? 2009/9/26

Detailed in-falling process of shell 1: influence of shell 2 The existence of an external shell does influence the motion of the internal shell This is very different from the common (mis-)understanding of Birkhoff’s Theorem. With shell 2 No shell 2 2009/9/26

我对几种“黑洞”的定义 数学黑洞：点质量（可以带电和角动量）的广义相对论场方程的解 所有质量都在奇点处 物理黑洞：所有的引力质量都在事件视界以内 但是不一定在奇点处 天体物理黑洞：有明确形成机制的物理黑洞 引力塌缩形成的黑洞 我们的研究表明：天体物理黑洞不是数学黑洞 其质量在黑洞视界内部是有分布的，但是在奇点处没有任何质量，彻底避免了黑洞的奇点问题。 在超大质量黑洞内部生活并不可怕。 2009/9/26

Astrophysical black hole My definition: physical BH with astrophysically viable formation mechanism For macroscopic BHs: gravitational collapse of matter towards a singularity seems to be the inevitable fate of a large mass assembly Revent horizon =3 km for the Sun 2009/9/26

总结：几个新的广义相对论效应和理解 球壳内部以及球壳以内的“真空区”度规不是常见的史瓦西形式。 定性影响引力透镜的计算。 随着球壳下落，球壳内“真空区” 的时钟越来越慢，这一变化可以产生物理效果，比如光线穿越暗物质团的时间 广义Shapiro Time Delay。 单球壳：外部观测者看到球壳外边界趋向了视界，内边界在有限时间内穿过了视界。 物质不会在黑洞外部堆积，黑洞可以形成和增长。 双球壳：内球壳在外部观测者的有限时间内穿过了视界 物理宇宙中不存在“冻结星” 。 在外部观测者的有限时间内，物质不可能到达黑洞的“数学‘奇点 物理宇宙中没有黑洞的奇点问题（!）。 2009/9/26

论文中引用（批评）的著名专著、教科书和科普书 C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (1973) S.W. Hawking, G.F.R. Ellis, The large scale structure of space-time (1973) S. Weinberg, Gravitation And Cosmology: Principles and Applications of the General Theory of Relativity (1977) S.L. Shapiro, S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars (1983) V.P. Frolov, I.D. Novikov, Black Hole Physics (1998) B.F. Schutz, A first course in general relativity (1990) D. Raine, E. Thomas, Black Holes - An Introduction (2005) J.P. Luminet, Black Holes (1992) K.S. Thorne, Black Holes & Time Warps - Einstein's Outrageous Legacy (1994) M.C. Begelman, M.J. Rees, Gravity's fatal attraction - black holes in the universe (1998) 2009/9/26

论文中致谢的学者 Sumin Tang（唐素敏）, Richard Lieu, Kinwah Wu（胡建华）, Kazuo Makishima, Neil Gehrels, Masaruare Shibata, Ramesh Narayan, Zheng Zhao（赵铮）, Zonghong Zhu（朱宗宏）,Chongming Xu（须重明）, Remo Ruffini, Roy Kerr and Hernando Quevedo Cubillos. 2007年黄山致密天体物理会议（袁业飞、赖东） 2009年中意相对论天体物理会议（方励之、Ruffini） Liu & Zhang, “Exact solutions for shells collapsing towards a pre-existing black hole”, 2009, Physics Letters B, Volume 679, Issue 2, p. 88-94 2009/9/26