Involved theoretical problem Numerical method Numerical result Summary Numerical study of interaction between positively and negatively massive black holes Zhoujian Cao Academy of Mathematics and Systems Science, Chinese Academy of Sciences 2012-3-1 Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary m 2 > 0 • • m 1 < 0 Newtonian mechanics m 1 m 2 = , (1) F r 2 a 1 = m 2 r 2 > 0 respect to particle 2 , (2) a 2 = m 1 r 2 < 0 respect to particle 1 . (3) Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary ← ← m 2 > 0 • • m 1 < 0 Newtonian mechanics Negatively massive particle chases positively massive particle. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary m 2 > 0 • • m 1 < 0 How about general relativity? Positive mass theorem? Cosmic censorship? VS Alternative theory for gravity? Dark energy? G. Gibbons, Commun. Math. Phys. 35, 13 (1974). Robert T. Jantzen, Gen. Relativ. Gravit. 15, 115 (1983). W. B. Bonnor, Gen. Relativ. Gravit. 21, 1143 (1988). R. Gleiser and G. Dotti, Class. Quant. Grav. 23, 5063 (2006). J. Gonz´ alez and F. Guzm´ an, Phys. Rev. D 79, 121501 (2009). R. Lazkoz, J. Kroon, Proc. R. Soc. Lond. A 460, 995 (2004). J. Bicak and D. Kofron, Gen Relativ Gravit 41, 153 (2009). ...... Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary There is No an Newtonian limit: R. Lazkoz, J. Kroon, Proc. R. Soc. Lond. A 460, 995 (2004). There does be an Newtonian limit: J. Bicak and D. Kofron, Gen Relativ Gravit 41, 153 (2009). What really happens in General Relativity for the interaction between positively and negatively massive black holes? BBH with positive and negative masses Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Numerical relativity: breakthrough in BBH simulation; well developed tool for GR theoretical study. Binary Black Hole calculation Z. Cao, C. Lin, H. Yo, and J. Yu (2010) Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary NR meets BBH with positive and negative masses Difficulty: How to deal with the naked singularity? How to construct reasonable initial data? Horizon Naked singularity m 2 > 0 • · m 1 < 0 Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Extended BY initial data for BBH with positive and negative masses For initially rest negatively massive black hole m 1 < 0 and positively massive black hole m 2 > 0: γ ij = ψ 4 δ ij ; ψ = 1 + m 1 + m 2 ; K ij = 0 (4) 2 r 1 2 r 2 Note coordinate singularity: ψ = 0 because m 1 < 0. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Horizon Coordinate singularity � m 2 > 0 • m 1 < 0 Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Horizon Coordinate singularity � m 2 > 0 • m 1 < 0 Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Domain of dependence Our computational domain contains the domain of dependence. If we care about the domain of dependence only, the boundary of our computational domain is not important. � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ m 2 > 0 · · m 1 < 0 Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Our numerical code is based on Baumgarte-Shapiro-Shibata-Nakamura-Oohara-Kojima (BSSNOK) formalism. 3+1 decomposition → γ ij and K ij γ ij = e − 4 φ γ ij , ˜ A ij = e − 4 φ ( K ij − 1 ˜ 3 γ ij K ) Γ i = − ˜ ˜ γ ij , j Initially φ = ln ψ = ln(1 + m 1 2 r 1 + m 2 2 r 2 ) Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Boundary condition For numerical stability, we take following detail recipe for boundary condition. Step 1: φ → φ − ln(1 + m 1 2 r 1 ); Step 2: Implement the standard Sommerfeld boundary condition for all variables; Step 3: φ → φ + ln(1 + m 1 2 r 1 ). Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Configuration Initially m 1 = − 100 locates at ( R , 0 , 0); m 2 = 1 locates at (0 , 0 , 0). ✻ ✲ m 2 =1 • • m 1 = − 100 We use puncture position to approximate the position of the black hole. Caution: gauge effect. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary R=1000. a, largest domain. resolutions: low (1/24), medium (1/28) and high (1/32). b, low resolution. domain: (-64:64), (-128:128) and (-256:256). c, Truncation error and boundary error. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary For Newtonian mechanics x = m 1 2 R t 2 . Comparison of numerical result and the fitting curve of x = A + B 1 t + B 2 t 2 for R = 1000. A and B 1 come from gauge effect. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary For Newtonian mechanics B 2 = m 1 2 R t 2 . Fitting numerical results to x = A + B 1 t + B 2 t 2 for different R. We expect the numerical result approaches to Newtonian mechanics’ prediction in the limit. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Involved theoretical problem Numerical method Numerical result Summary Summary There is a debate about the Newtonian limit on the interaction between positively and negatively massive particles in GR. Based on BSSNOK formalism, we proposed a method to study BBH with positive and negative masses through numerical relativity techniques. Up to gauge effect, we find that the interaction between positively and negatively massive particles in GR has proper Newtonian limit. Zhoujian Cao Numerical study of interaction between positively and negatively mass
Recommend
More recommend
