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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

m2>0•

• m1<0

Newtonian mechanics F = m1m2 r2 , (1) a1 = m2 r2 > 0 respect to particle 2, (2) a2 = m1 r2 < 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

← ←

m2>0•

• m1<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

m2>0•

• m1<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

m2>0 •

·m1<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 m1 < 0 and positively massive black hole m2 > 0: γij = ψ4δij; ψ = 1 + m1 2r1 + m2 2r2 ; Kij = 0 (4) Note coordinate singularity: ψ = 0 because m1 < 0.

Zhoujian Cao Numerical study of interaction between positively and negatively mass

Involved theoretical problem Numerical method Numerical result Summary

Horizon Coordinate singularity

m2>0 •

• m1<0

Zhoujian Cao Numerical study of interaction between positively and negatively mass

Involved theoretical problem Numerical method Numerical result Summary

Horizon Coordinate singularity

m2>0 •

• m1<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.

m2>0 ·

·m1<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 → γij and Kij ˜ γij = e−4φγij, ˜ Aij = e−4φ(Kij − 1

3γijK)

˜ Γi = −˜ γij ,j Initially φ = ln ψ = ln(1 + m1

2r1 + m2 2r2 )

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 + m1

2r1 );

Step 2: Implement the standard Sommerfeld boundary condition for all variables; Step 3: φ → φ + ln(1 + m1

2r1 ).

Zhoujian Cao Numerical study of interaction between positively and negatively mass

Involved theoretical problem Numerical method Numerical result Summary

Configuration

Initially m1 = −100 locates at (R, 0, 0); m2 = 1 locates at (0, 0, 0).

m2=1 •

• m1=−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 = m1

2R t2. Comparison of

numerical result and the fitting curve of x = A + B1t + B2t2 for R = 1000. A and B1 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 B2 = m1

2R t2. Fitting numerical

results to x = A + B1t + B2t2 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