1. 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

2. Double-null formalism : brief review

3. 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

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

5. Implementation Evolution equations - Einstein equations
- Field equation 5

6. 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

7. Example: a neutral black hole in Einstein gravity7

8. Example: a neutral black hole in Einstein gravity8

9. 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

10. Charged black holes in string-inspired models

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

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 12

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 13

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 14

15. 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

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

17. Causal structures 17

18. Causal structures 18

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

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. 20

21. 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

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

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

24. 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

25. Return of mass inflation25

26. Return of mass inflation26

27. 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

28. Thank you very much