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Charged black holes in string-inspired gravity models
Based on Hansen and DY, Hansen and DY, in preparation Leung Center for Cosmology and Particle Astrophysics, National Taiwan University Dong-han Yeom
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Double-null formalism : brief review
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Double-null formalism
Double-null formalism is a numerical procedure to solve Einstein equations. Using double-null metric ansatz, we overcome the coordinate singularity problem. Solve all equations: for example, for the simplest case, 3
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Implementation Einstein tensors / energy-momentum tensors / scalar-field equation 4
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Implementation Evolution equations - Einstein equations
- Field equation 5
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Boundary condition We should give boundary conditions for all functions and their first derivatives at the initial in-going and out-going null lines. - (0,0) : From the Misner-Sharp mass function, by choosing the initial mass and radius π(π,π), we can determine πΆ π,π , π ,π (π,π), and π ,π (π,π). - In-going null surface: First, we can choose matter sector πΊ(π,π). Then, πΊ ,π (π,π) is already determined. From the Einstein equation of π ,ππ part, we can obtain πΆ ,π (π,π). Then, we can obtain πΆ π,π by integration. Second, we calculate the remained part from the equations of ππ derivatives: π ,π (π,π) from the equation for π ,ππ , πΆ ,π (π,π) from the equation for πΆ ,ππ , and πΊ ,π (π,π) from the equation for πΊ ,ππ . - Out-going null surface: we can apply the same method. 6
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Example: a neutral black hole in Einstein gravity
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Example: a neutral black hole in Einstein gravity
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Applications Double-null formalism can be extended to various ways.
Matter: neutral matter, gauge field, various potential and vacuum energy, β¦ Gravity: Einstein gravity, dilaton gravity, Brans-Dicke gravity, f(R) gravity, β¦ Dimensions: 4D, 3D, 4+nD, β¦ Symmetries: spherical, hyperbolic, planar, circular, β¦ 9
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Charged black holes in string-inspired models
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String-inspired models
We investigate the following model as a prototype of the sting-inspired models. 11
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String-inspired models
We investigate the following model as a prototype of the sting-inspired models. There are some origins from string theory: Dilaton gravity 12
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String-inspired models
We investigate the following model as a prototype of the sting-inspired models. There are some origins from string theory: Dilaton gravity Couplings to gauge sector 13
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String-inspired models
We investigate the following model as a prototype of the sting-inspired models. There are some origins from string theory: Dilaton gravity Couplings to gauge sector Brane-world 14
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String-inspired models
We investigate the following model as a prototype of the sting-inspired models. There are some origins from string theory: Dilaton gravity Couplings to gauge sector Brane-world Higher curvature corrections (e.g., f(R)) 15
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String-inspired models
We implement the following model to double-null formalism. 16
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Causal structures 17
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Causal structures 18
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Causal structures Except for the Type IIA inspired case, there is no Cauchy horizon and mass inflation inside a charged black hole. 19
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Causal structures Except for the Type IIA inspired case, there is no Cauchy horizon and mass inflation inside a charged black hole. Is this a general nature of string theory? The answer is maybe NO. 20
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Causal structures Except for the Type IIA inspired case, there is no Cauchy horizon and mass inflation inside a charged black hole. Is this a general nature of string theory? The answer is maybe NO. The key is in the dynamics of the Brans-Dicke field. 21
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Responses of the Brans-Dicke sector
Responses of the Brans-Dicke field is determined by the following equation. 22
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Responses of the Brans-Dicke sector
Responses of the Brans-Dicke field is determined by the following equation. 23
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Responses of the Brans-Dicke sector
Responses of the Brans-Dicke field is determined by the following equation. If the dilaton field is βstabilizedβ by a potential term, then the equation should have one more term. 24
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Return of mass inflation
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Return of mass inflation
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Conclusion We investigated (and are investigating) gravitational collapses of string-inspired models using double-null formalism. Especially, this is useful to investigate the fully dynamical process, e.g., internal structures of a charged black hole or a dynamical formation of the Brans-Dicke hair. As long as there is a Brans-Dicke hair, the internal structure has a space-like singularity without a Cauchy horizon. On the other hand, if the Brans-Dicke hair can be controlled by introducing a potential term, then the causal structure returns to mass inflation with a Cauchy horizion. A gravitational collapse can perturb a moduli field or a dilaton field; hence, a gravitational collapse can be a good window to see stringy effects. In addition, the observation of dilaton/moduli hair can teach us internal structure of a black hole (even though we do not fall into the black hole). 27
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Thank you very much
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