Hawking radiation for a Proca field

1. Mengjie Wang（王梦杰 ） In collaboration with Carlos Herdeiro & Marco Sampaio Hawking radiation for a Proca field Based on: PRD85(2012) 024005 王梦杰 Mengjie Wang Mengjie Wang

2. Outline 1. Introduction 2. Hawking radiation in D dimensions 3. Hawking radiation on the brane 4. Discussion & Conclusions

3. Introduction (I) What ? Hawking radiation is the most prominent quantum effect for quantum fields in a background spacetime with an event horizon. Intuitive picture × virtual pair creationof particles near the event horizon BH real particle E -E E × × Hawking radiation Methodology QFT in curved spacetime Path-integral derivation ... ... Gravitational anomaly Tunneling

4. Introduction (II) Why ? From the Brane-World scenario, black holes can be produced in colliders or in cosmic ray interactions. We can detect black hole events via Hawking radiation, and we can read the extra dimension from it. SM particles are confined on a 4-dimensional Brane. Generalization & Improving current black hole event generators. Generally speaking, the Equation of Motion in curved space-time cannot be decoupled, or variables cannot be separated. Constructing a kind of systematic numerical method to deal with the coupled Ordinary Differential Equations(ODEs), as well as Partial Differential Equations(PDEs).

5. Introduction How ? How to get the EOMs? arXiv: 0712.2703 background geometry line element Hodge decomposition theorem Suppose to be a compact Riemannian manifold, any dual vector field on can be uniquely decomposed as is a scalar field is a transverse vector How to solve the EOMs? from the second to the first define scattering matrix transmission factor matrix provide physical prescription choose scattering matrix

6. Hawking radiation in D dimensions The Lagrangian for Proca field, which describe the Z and W particles in the standard model, is is the electromagnetic field strength tensor Equations of motion for Proca field The gravitational background (m+n)-dimensional spacetime whose manifold structure is locally a warped product type with Einstein symmetric spaces spanning the m-dimensional space with metric spanning the n-dimensional Einstein space

7. Hawking radiation in D-dimensions Decomposition of the vector field in tensorial types for is the Laplacian operator in Einstein space for , which can be decomposed into a scalar and a transverse vector The above decompositons and conditions allow for an expansion of the form

8. Hawking radiation in D-dimensions Equations of motion in Schwarzschild spacetime now we specialize to Schwarzschild case, i.e. Modes with massive coupled transverse massless Modes

9. Hawking radiation in D-dimensions Boundary conditions at the horizon we can rewrite our equations as making use of Frobenius method we get recurrence relations

10. Hawking radiation in D-dimensions Asymptotic behavior at infinity To understand the asymptotic behavior of the coupled modes at infinity, we study the following asymptotic expansion The asymptotic form for

11. Hawking radiation in D-dimensions The first order equations For the numerical convenience, we rewrite the coupled equations in the first order form define a vector V coupled equations can be written as matrix form define another vector from the above asymptotic expansion, we have relation from

12. Hawking radiation in D-dimensions Definition of transmission factor We know that a general solution is parameterized by 4 independent coefficients in one of the asymptotic regions, either at the horizon or at infinity. Because of the linearity of the coupled equations, we can use a matrix to relate the coefficients at the horizon and at infinity. We denote the ingoing and outgoing wave coefficients at the horizon at infinity impose an ingoing boundary condition transmission factor=eigenvalue(T)

13. Hawking radiation in D-dimensions Physical prescription There is still some freedom in the definition of the asymptotic coefficients new reflection matrix? for a single decoupled field with definite energy, the transmission factor is the definition of flux the energy momentum tensor for complex neutral Proca field the flux at infinity the flux at the horizon

14. Hawking radiation in D-dimensions

15. Hawking radiation in D-dimension The number and energy fluxes are Results

16. Hawking radiation in D-dimension Comparison between small mass and exact zero mass

17. Hawking radiation on the Brane Specialize to charged brane Now we generalize the previous work to brane case considering the background perform the same procedure, we get the following equations of motion coupled modes with transverse mode mode

18. Hawking radiation on the Brane

19. Hawking radiation on the brane

20. Hawking radiation on the brane

21. Hawking radiation on the brane

22. Conlusion We have used a numerical strategy to solve the coupled wave equations for Proca field in D-dimensional Schwarzschild black hole. Our results show some expected features, such as the mass suppression of the Hawking fluxes as the Proca mass is increased, but also some novel features, such as the nonzero limit of the transmission factor, for vanishing spatial momentum, in n=2,3. Moreover, a precise study of the longitudinal degrees of freedom was carried out. We have shown the difference of transmission factor between in the bulk and on the brane. We found the the nonzero limit of the transmission factor is existed for all n. We have shown the charge effects on the transmission factor. We found there is contribution for nonzero limit of transmission factor from the charge. For one component of the coupled transmission factor, it is increased through the field charge vary from the negative to positive, the other component is reverse.

23. Thank You !