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. Outline. 1. Introduction. 2. Hawking radiation in D dimensions. 3. Hawking radiation on the brane. 4.
Mengjie Wang（王梦杰 ）
In collaboration with Carlos Herdeiro & Marco Sampaio
Based on: PRD85(2012) 024005
Hawking radiation in D dimensions
Hawking radiation on the brane
Discussion & Conclusions
Hawking radiation is the most prominent quantum effect for quantum fields in a
background spacetime with an event horizon.
virtual pair creationof particles near the event horizon
QFT in curved spacetime
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
How to get the EOMs?
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
provide physical prescription
choose scattering matrix
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
with Einstein symmetric spaces
spanning the m-dimensional space with metric
spanning the n-dimensional Einstein space
Decomposition of the vector field in tensorial types
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
Equations of motion in Schwarzschild spacetime
now we specialize to Schwarzschild case, i.e.
Boundary conditions at the horizon
we can rewrite our equations as
making use of Frobenius method
we get recurrence relations
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
The first order equations
For the numerical convenience, we rewrite the coupled equations in the first order
define a vector V
coupled equations can be written as matrix form
define another vector
from the above asymptotic expansion, we have relation
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
impose an ingoing boundary condition
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
The number and energy fluxes are
Comparison between small mass and exact zero mass
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
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.