Charged black holes in string-inspired gravity models Based on Hansen and DY, 1406.0976 Hansen and DY, in preparation Leung Center for Cosmology and Particle Astrophysics, National Taiwan University Dong-han Yeom 2015. 6. 9.

Double-null formalism : brief review

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

Implementation Einstein tensors / energy-momentum tensors / scalar-field equation 4

Implementation Evolution equations - Einstein equations - Field equation 5

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 𝒓(𝟎,𝟎), we can determine 𝜶 𝟎,𝟎 , 𝒓 ,𝒖 (𝒖,𝟎), and 𝒓 ,𝒗 (𝟎,𝒗). - In-going null surface: First, we can choose matter sector 𝑺(𝒖,𝟎). Then, 𝑺 ,𝒖 (𝒖,𝟎) is already determined. From the Einstein equation of 𝒓 ,𝒖𝒖 part, we can obtain 𝜶 ,𝒖 (𝒖,𝟎). Then, we can obtain 𝜶 𝒖,𝟎 by integration. Second, we calculate the remained part from the equations of 𝒖𝒗 derivatives: 𝒓 ,𝒗 (𝒖,𝟎) from the equation for 𝒓 ,𝒖𝒗 , 𝜶 ,𝒗 (𝒖,𝟎) from the equation for 𝜶 ,𝒖𝒗 , and 𝑺 ,𝒗 (𝒖,𝟎) from the equation for 𝑺 ,𝒖𝒗 . - Out-going null surface: we can apply the same method. 6

Example: a neutral black hole in Einstein gravity 7

Example: a neutral black hole in Einstein gravity 8

Applications Double-null formalism can be extended to various ways. Matter: neutral matter, gauge field, various potential and vacuum energy, … Gravity: Einstein gravity, dilaton gravity, Brans-Dicke gravity, f(R) gravity, … Dimensions: 4D, 3D, 4+nD, … Symmetries: spherical, hyperbolic, planar, circular, … 9

Charged black holes in string-inspired models

String-inspired models We investigate the following model as a prototype of the sting-inspired models. 11

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

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

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

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

String-inspired models We implement the following model to double-null formalism. 16

Causal structures 17

Causal structures 18

Causal structures Except for the Type IIA inspired case, there is no Cauchy horizon and mass inflation inside a charged black hole. 19

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

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

Responses of the Brans-Dicke sector Responses of the Brans-Dicke field is determined by the following equation. 22

Responses of the Brans-Dicke sector Responses of the Brans-Dicke field is determined by the following equation. 23

Responses of the Brans-Dicke sector Responses of the Brans-Dicke field is determined by the following equation. If the dilaton field is ‘stabilized’ by a potential term, then the equation should have one more term. 24

Return of mass inflation 25

Return of mass inflation 26

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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